Strong Coupling Gauge Theories in LHC Era
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Strong Coupling Gauge Theories in LHC Era Proceedings of the Workshop in Honor of Toshihide Maskawa’s 70th Birthday and 35th Anniversary of Dynamical Symmetry Breaking in SCGT Nagoya University, Japan
8 – 11 December 2009
editors
H f*ckaya • M Harada M Tanabashi • K Yamawaki Nagoya University, Japan
World Scientific NEW JERSEY
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LONDON
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SINGAPORE
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BEIJING
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SHANGHAI
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HONG KONG
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TA I P E I
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CHENNAI
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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
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STRONG COUPLING GAUGE THEORIES IN LHC ERA Proceedings of the Workshop in Honor of Toshihide Maskawa’s 70th Birthday and 35th Anniversary of Dynamical Symmetry Breaking in SCGT Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN-13 978-981-4329-51-4 ISBN-10 981-4329-51-7
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PREFACE
Spontaneous symmetry breaking has been a central paradigm of particle physics. It was actually born as dynamical symmetry breaking in analogy with the BCS theory of superconductor in the original work by Y. Nambu about half a century ago. The dynamical symmetry breaking was realized in QCD, a prototype of strong coupling gauge theory (SCGT). It was founded by T. Maskawa and H. Nakajima back in 1974 that in order for the gauge theories to develop spontaneous chiral symmetry breaking the gauge coupling must be strong, larger than a certain nonzero critical coupling, in the ladder Schwinger-Dyson equation with the coupling being nonrunning. This was the beginning of the SCGT with conformal/scale-invariant dynamics. In 1986 it was founded by K. Yamawaki, M. Bando and K. Matumoto that this conformal/scale-invariant dynamics has a large anomalous dimension of unity at criticality, so as to resolve the FCNC problem in the technicolor (TC) models. Subsequently, the first Nagoya SCGT workshop was held in 1988, largely focusing on such a dynamical symmetry breaking with conformality/criticality and/or large anomalous dimension. Along this line new types of dynamical symmetry breaking such as the top quark condensate were proposed soon after the workshop. Since then the Nagoya SCGT workshop series were further held in 1990, 1996, 2002, 2006 and this time, the sixth in 2009. Actually, over last 30 years theorists have been desperately seeking physics beyond the standard model. Now it is the era when the LHC will make a decisive test of various theoretical proposals. Certainly the new SCGT is one of such attractive possibilities. Apart from the LHC experiments, there has been remarkable progress of lattice gauge theories in recent years, so that the conformal dynamics so far difficult to be analyzed in reliable ways could be solved by computers to give definite predictions at the level to be compared with the LHC experiments. We then organized the sixth Nagoya SCGT workshop “Strong Coupling Gauge Theories in LHC Era (SCGT 09) in December 8-11, 2009. The SCGT 09 was the first SCGT workshop after Y. Nambu, M. Kobayashi and T. Maskawa shared the 2008 Nobel Prize in physics. It should be mentioned that Nambu is the initiator of the dynamical symmetry breaking, and actually was a repeated participant of the Nagoya SCGT workshops, while Maskawa and Kobayashi are both alumni of Nagoya University (both undergraduate and graduate). Moreover, Maskawa was to
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be the Director of the new institute, Kobayashi-Maskawa Institute (KMI), Nagoya University, starting April 2010. The SCGT 09 was then held in honor of Maskawa’s 70th birthday (February 7, 2010) and 35th anniversary of the Maskawa-Nakajima paper initiating the dynamical symmetry breaking in conformal/scale-invariant gauge dynamics of SCGT. During four days of sessions, we had 92 participants (28 from abroad), with 44 talks in the main program and 15 talks for the poster sessions. The topics were broad, not just conformal gauge dynamics but various aspects of strong coupling gauge theories including QCD in varieties of approaches covering lattice studies, holography, effective theories, and so on. Also several experimental talks were presented. We do believe that these presentations included in the Proceedings are very useful for the future developments in particle physics in the revolutionary era of LHC. The workshop was financially supported by the Nagoya University Global COE Program “Quest for Fundamental Principles in the Universe: from Particles to the Solar System and the Cosmos” from the Ministry of Education, Culture, Sports, Science and Technology, Japan, and also by Daiko Foundation and by Nagoya University Foundation. We would like to express sincere thanks to these organizations for generous supports. We would also like to thank all the members of the International Advisory Committee, the Organizing Committee, the Local Organizing Committee, and the SCGT 09 Secretariat and the GCOE support stuff office, and the graduate students at Nagoya University for their devoted contributions to make the workshop successful. Editors Hidenori f*ckaya Masayasu Harada Masaharu Tanabashi Koichi Yamawaki
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OPENING ADDRESS
Good morning, ladies and gentlemen. On behalf of Nagoya University, I would like to express hearty welcome to all of you attending the 2009 Nagoya Global COE workshop on “Strong Coupling Gauge Theories in LHC Era”, SCGT 09. I heard that this workshop is made in honor of Professor Toshihide Maskawa’s 70th birthday and also for celebration of the 35th anniversary of his paper which is cornerstone of the subject of this workshop, namely the symmetry breaking in the strong coupling gauge theories. I understand that the subject is to reveal the Origin of Mass, the most urgent problem in particle physics today and the main research target of the largest accelerator experiments, LHC, just getting in operation. I would like to comment that we, at Nagoya University, are proud of our alumni, Professor Maskawa and Professor Kobayashi, who shared 2008 Nobel Prize in physics for the symmetry breaking of CP symmetry. We are also proud of Professor Maskawa’s work on another type of symmetry breaking which has grown up into the present active field, the subject of this workshop, where Nagoya group has made major contributions. This is the sixth of the Nagoya SCGT workshops following the meetings in 1988, 1990, 1996, 2002 and 2006, ranging over two decades. The Nagoya SCGT workshops appear to be prestigious in the sense that every meeting had attendance of at least one of the Nobel laureates; Professor Maskawa (1988, 2009), Professor Kobayashi (1990), Professor Nambu (1988, 1990, 1996, 2002) and Professor ’t Hooft (2006). Here I should mention that this building, here you are, was constructed to celebrate Professor Noyori’s 2001 Nobel Prize in chemistry. We are also proud of our alumnus Professor Osamu Shimomura who won the 2008 Nobel Prize in chemistry. It is my pleasure to announce to you that, next April, Professor Maskawa will come back to Nagoya University as Director of a newly created institute, KobayashiMaskawa Institute for the Origin of Particles and the Universe. The new building of the institute will be ready in 2011 spring. We are hoping that particle physics at Nagoya will be boosted even more. I hope that the discussions at this workshop will contribute to the next Nobel Prize(s) in the near future. Thank you for your attention and please enjoy year stay at Nagoya and Nagoya University. Michinari Hamaguchi, D. Med President of Nagoya University
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WORKSHOP ORGANIZATION
International Advisory Committee: T.W. Appelquist (Yale) S.J. Brodsky (SLAC) C.T. Hill (FNAL) J.B. Kogut (DOE/Maryland) V.A. Miransky (W. Ontario) M. Rho (Saclay) R. Sundrum (Johns Hopkins) V.I. Zakharov (MPI/ITEP)
W.A. Bardeen (FNAL) R.S. Chivukula (MSU) M. Kobayashi (JSPS) T. Maskawa (KSU/Nagoya) Y. Nambu (Chicago) E.H. Simmons (MSU) G. ’t Hooft (Utrecht)
Organizing Committee: Chairperson Co-chairperson
K. Yamawaki (Nagoya) M. Harada (Nagoya) M. Tanabashi (Nagoya)
K-I. Aoki (Kanazawa) M. Hayakawa (Nagoya) K. Kanaya (Tsukuba) R. Kitano (Tohoku) T. Kugo (Kyoto) T. Sakai (Nagoya)
H. f*ckaya (Nagoya) Y. Hosotani (Osaka) Y. Kikukawa (Tokyo-Komaba) K.-I. Kondo (Chiba) T. Onogi (Osaka) K. Tobe (Nagoya)
Sponsored by Nagoya University Global COE Program “Quest for Fundamental Principles in the Universe: from Particles to the Solar System and the Cosmos.”
Financial support also from Daiko Foundation and Nagoya University Foundation.
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CONTENTS Preface
v
Opening Address
vii
Workshop Organization
viii
AdS/QCD, Light-Front Holography, and the Nonperturbative Running Coupling Stanley J. Brodsky, Guy de T´eramond and Alexandre Deur New Results on Non-Abelian Vortices — Further Insights into Monopole, Vortex and Confinement K. Konishi Study on Exotic Hadrons at B-Factories Toru Iijima Cold Compressed Baryonic Matter with Hidden Local Symmetry and Holography Mannque Rho
1
17 32
39
Aspects of Baryons in Holographic QCD T. Sakai
54
Nuclear Force from String Theory K. Hashimoto
67
Integrating Out Holographic QCD Back to Hidden Local Symmetry Masayasu Harada, Shinya Matsuzaki and Koichi Yamawaki
70
Holographic Heavy Quarks and the Giant Polyakov Loop Gianluca Grignani, Joanna Karczmarek and Gordon W. sem*noff
78
Effect of Vector–Axial-Vector Mixing to Dilepton Spectrum in Hot and/or Dense Matter Masayasu Harada and Chihiro Sasaki
86
Infrared Behavior of Ghost and Gluon Propagators Compatible with Color Confinement in Yang-Mills Theory with the Gribov Horizon Kei-Ichi Kondo
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Chiral Symmetry Breaking on the Lattice Hidenori f*ckaya [for JLQCD and TWQCD Collaborations]
99
Gauge-Higgs Unification: Stable Higgs Bosons as Cold Dark Matter Yutaka Hosotani
106
The Limits of Custodial Symmetry R. Sekhar Chivukula, Roshan Foadi, Elizabeth H. Simmons and Stefano Di Chiara
120
Higgs Searches at the Tevatron Kazuhiro Yamamoto [for the CDF and DØ Collaborations]
135
The Top Triangle Moose R. S. Chivukula, N. D. Christensen, B. Coleppa and E. H. Simmons
143
Conformal Phase Transition in QCD Like Theories and Beyond V. A. Miransky
150
Gauge-Higgs Unification at LHC Nobuhito Maru and Nobuchika Okada
159
WL WL Scattering in Higgsless Models: Identifying Better Effective Theories Alexander S. Belyaev, R. Sekhar Chivukula, Neil D. Christensen, Hong-Jian He, Masafumi Kurachi, Elizabeth H. Simmons and Masaharu Tanabashi
166
Holographic Estimate of Muon g − 2 Deog Ki Hong
173
Gauge-Higgs Dark Matter T. Yamash*ta
180
Topological and Curvature Effects in a Multi-Fermion Interaction Model T. Inagaki and M. Hayashi
184
A Model of Soft Mass Generation J. Hoˇsek
191
TeV Physics and Conformality Thomas Appelquist
198
Conformal Higgs, or Techni-Dilaton — Composite Higgs Near Conformality Koichi Yamawaki Phase Diagram of Strongly Interacting Theories Francesco Sannino
213 230
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Resizing Conformal Windows O. Antipin and K. Tuominen
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Nearly Conformal Gauge Theories on the Lattice Zolt´ an Fodor, Kieran Holland, Julius Kuti, D´ aniel N´ ogr´ adi and Chris Schroeder
254
Going Beyond QCD in Lattice Gauge Theory G. T. Fleming
267
Phases of QCD from Small to Large Nf : (Some) Lattice Results A. Deuzeman, E. Pallante and M. P. Lombardo
275
Lattice Gauge Theory and (Quasi)-Conformal Technicolor D. K. Sinclair and J. B. Kogut
283
Study of the Running Coupling Constant in 10-Flavor QCD with the Schr¨ odinger Functional Method N. Yamada, M. Hayakawa, K.-I. Ishikawa, Y. Osaki, S. Takeda and S. Uno
290
Study of the Running Coupling in Twisted Polyakov Scheme T. Aoyama, H. Ikeda, E. Itou, M. Kurachi, C.-J. D. Lin, H. Matsufuru, H. Ohki, T. Onogi, E. Shintani and T. Yamazaki
297
Running Coupling in Strong Gauge Theories via the Lattice Zolt´ an Fodor, Kieran Holland, Julius Kuti, D´ aniel N´ ogr´ adi and Chris Schroeder
304
Higgsinoless Supersymmetry and Hidden Gravity Michael L. Graesser, Ryuichiro Kitano and Masafumi Kurachi
311
The Latest Status of LHC and the EWSB Physics S. Asai
324
Continuum Superpartners from Supersymmetric Unparticles Hsin-Chia Cheng
338
Review of Minimal Flavor Constraints for Technicolor Hidenori S. f*ckano and Francesco Sannino
350
Standard Model and High Energy Lorentz Violation Damiano Anselmi
357
Dynamical Electroweak Symmetry Breaking and Fourth Family Michio Hashimoto
364
Holmorphic Supersymmetric Nambu–Jona-Lasino Model and Dynamical Electroweak Symmetry Breaking Dong-Won Jung, Otto C. W. Kong and Jae Sik Lee
371
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Ratchet Model of Baryogenesis Tatsu Takeuchi, Azusa Minamizaki and Akio Sugamoto
378
Classical Solutions of Field Equations in Einstein Gauss-Bonnet Gravity P. Suranyi, C. Vaz and L. C. R. Wijewardhana
385
Black Holes Constitute All Dark Matter Paul H. Frampton
390
Electroweak Precision Test and Z → bb in the Three Site Higgsless Model Tomohiro Abe
395
Chiral Symmetry and BRST Symmetry Breaking, Quaternion Reality and the Lattice Simulation Sadataka Furui
398
Holographic Techni-Dilaton, or Conformal Higgs Kazumoto Haba, Shinya Matsuzaki and Koichi Yamawaki
401
Phase Structure of Topologically Massive Gauge Theory with Fermion Yuichi Hoshino
404
New Regularization in Extra Dimensional Model and Renormalization Group Flow of the Cosmological Constant Shoichi Ichinose
407
Spectral Analysis of Dense Two-Color QCD T. Kanazawa, T. Wettig and N. Yamamoto
412
NJL Model with Dimensional Regularization at Finite Temperature T. Fujihara, T. Inagaki, D. Kimura, H. Kohyama and A. Kvinikhidze
415
A New Method of Evaluating the Dynamical Chiral Symmetry Breaking Scale and the Chiral Restoration Temperature in General Gauge Theories by Using the Non-Perturbative Renormalization Group Analyses with General 4-Fermi Effective Interaction Space Ken-Ichi Aoki, Daisuke Sato and Kazuhiro Miyash*ta
418
The Effective Chiral Lagrangian with Vector Mesons and Hadronic τ Decays D. Kimura, Kang Young Lee, T. Morozumi and K. Nakagawa
421
Spontaneous SUSY Breaking with Anomalous U (1) Symmetry in Metastable Vacua and Moduli Stabilization Hiroyuki Nishino
424
A New Description of the Lattice Yang-Mills Theory and Non-Abelian Magnetic Monopole Dominance in the String Tension Akihiro Shibata
427
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Thermodynamics with Unbroken Center Symmetry in Two-Flavor QCD S. Takemoto, M. Harada and C. Sasaki Masses of Vector Bosons in Two-Color QCD Based on the Hidden Local Symmetry T. Yamaoka, M. Harada and C. Nonaka
430
433
Walking Dynamics from String Duals Maurizio Piai
436
The Quark Mass Dependence of the Nucleon Mass in AdS/QCD Hyo Chul Ahn
436
Structure of Thermal Quasi-Fermion in QED/QCD from the Dyson-Schwinger Equation Hisao Nakkagawa
437
Critical Behaviors of Sigma-Mode and Pion in Holographic Superconductors Cheonsoo Park
437
List of Participants
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AdS/QCD, Light-Front Holography, and the Nonperturbative Running Coupling Stanley J. Brodsky SLAC National Accelerator Laboratory, Stanford University Stanford, California 94309, USA E-mail: [emailprotected] Guy de T´ eramond Universidad de Costa Rica San Jos´ e, Costa Rica E-mail: [emailprotected] Alexandre Deur Thomas Jefferson National Accelerator Facility Newport News, VA 23606, USA E-mail: [emailprotected] The combination of Anti-de Sitter space (AdS) methods with light-front (LF) holography provides a remarkably accurate first approximation for the spectra and wavefunctions of meson and baryon light-quark bound states. The resulting bound-state Hamiltonian equation of motion in QCD leads to relativistic light-front wave equations in terms of an invariant impact variable ζ which measures the separation of the quark and gluonic constituents within the hadron at equal light-front time. These equations of motion in physical space-time are equivalent to the equations of motion which describe the propagation of spin-J modes in anti–de Sitter (AdS) space. The eigenvalues give the hadronic spectrum, and the eigenmodes represent the probability distributions of the hadronic constituents at a given scale. A positive-sign confining dilaton background modifying AdS space gives a very good account of meson and baryon spectroscopy and form factors. The light-front holographic mapping of this model also leads to a non-perturbative (Q2 ) which agrees with the effective charge defined by the Bjorken effective coupling αAdS s sum rule and lattice simulations. It displays a transition from perturbative to nonperturbative conformal regimes at a momentum scale ∼ 1 GeV. The resulting β-function appears to capture the essential characteristics of the full β-function of QCD, thus giving further support to the application of the gauge/gravity duality to the confining dynamics of strongly coupled QCD. Keywords: AdS/CFT correspondence, AdS/QCD, light-front QCD, light-front quantization, nonperturbative QCD coupling.
1. Introduction The AdS/CFT correspondence1 between a gravity or string theory on a higher dimensional Anti–de Sitter (AdS) space-time with conformal gauge field theories
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(CFT) in physical space-time has brought a new set of tools for studying the dynamics of strongly coupled quantum field theories, and it has led to new analytical insights into the confining dynamics of QCD. The AdS/CFT duality provides a gravity description in a (d + 1)-dimensional AdS space-time in terms of a flat ddimensional conformally-invariant quantum field theory defined at the AdS asymptotic boundary.2 Thus, in principle, one can compute physical observables in a strongly coupled gauge theory in terms of a classical gravity theory. Since the quantum field theory dual to AdS5 space in the original correspondence1 is conformal, the strong coupling of the dual gauge theory is constant, and its β-function is zero. Thus one must consider a deformed AdS space in order to simulate color confinement and (Q2 ) for the gauge theory side of the correspondence. have a running coupling αAdS s As we shall review here, a positive-sign confining dilaton background modifying AdS space gives a very good account of meson and baryon spectroscopy and their elastic form factors. The light-front holographic mapping of this model also leads (Q2 ) which agrees with the effective to a non-perturbative effective coupling αAdS s charge defined by the Bjorken sum rule and lattice simulations.3 In the standard applications of AdS/CFT methods, one begins with Maldacena’s duality between the conformal supersymmetric SO(4, 2) gauge theory and a semiclassical supergravity string theory defined in a 10 dimension AdS5 × S 5 spacetime.1 The essential mathematical tool underlying Maldacena’s observation is the fact that the effects of scale transformations in a conformal theory can be mapped to the z dependence of amplitudes in AdS5 space. QCD is not conformal but there is in fact much empirical evidence from lattice, Dyson Schwinger theory and effective charges that the QCD β function vanishes in the infrared.4 The QCD infrared fixed point arises since the propagators of the confined quarks and gluons in the loop integrals contributing to the β function have a maximal wavelength.5 The decoupling of quantum loops in the infrared is analogous to QED where vacuum polarization corrections to the photon propagator decouple at Q2 → 0. We thus begin with a conformal approximation to QCD to model an effective dual gravity description in AdS space. One uses the five-dimensional AdS5 geometrical representation of the conformal group to represent scale transformations within the conformal window. Confinement can be introduced with a sharp cut-off in the infrared region of AdS space, as in the “hard-wall” model,6 or, more successfully, using a dilaton background in the fifth dimension to produce a smooth cutoff at large distances as in the “soft-wall” model.7 The soft-wall AdS/CFT model with a positivesign dilaton-modified AdS space leads to the potential U (z) = κ4 z 2 +2κ2 (L+S−1),8 in the fifth dimension coordinate z. We assume a dilaton profile exp(+κ2 z 2 ),8–11 with opposite sign to that of Ref. 7. The resulting spectrum reproduces linear Regge trajectories, where M2 (S, L, n) is proportional to the internal spin, orbital angular momentum L and the principal quantum number n. The modified metric induced by the dilaton can be interpreted in AdS space as a gravitational potential for an object of mass m in the fifth dimension: V (z) = 2 2 √ mc2 g00 = mc2 R e±κ z /2 /z. In the case of the negative solution, the potential
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decreases monotonically, and thus an object in AdS will fall to infinitely large values of z. For the positive solution, the potential is non-monotonic and has an absolute minimum at z0 = 1/κ. Furthermore, for large values of z the gravitational potential increases exponentially, confining any object to distances z ∼ 1/κ.8 We thus will choose the positive sign dilaton solution. This additional warp factor leads to a well-defined scale-dependent effective coupling. Introducing a positive-sign dilaton background is also relevant for describing chiral symmetry breaking,10 since the expectation value of the scalar field associated with the quark mass and condensate does not blow-up in the far infrared region of AdS, in contrast with the original model.7 Glazek and Schaden12 have shown that a harmonic oscillator confining potential naturally arises as an effective potential between heavy quark states when one stochastically eliminates higher gluonic Fock states. Also, Hoyer13 has argued that the Coulomb and a linear potentials are uniquely allowed in the Dirac equation at the classical level. The linear potential becomes a harmonic oscillator potential in the corresponding Klein-Gordon equation. Light-front (LF) quantization is the ideal framework for describing the structure of hadrons in terms of their quark and gluon degrees of freedom. The light-front wavefunctions (LFWFs) of bound states in QCD are relativistic generalizations of the Schr¨ odinger wavefunctions, but they are determined at fixed light-front time τ = x+ = x0 + x3 , the time marked by the front of a light wave,14 rather than at fixed ordinary time t. They play the same role in hadron physics that Schr¨ odinger 15 wavefunctions play in atomic physics. The simple structure of the LF vacuum provides an unambiguous definition of the partonic content of a hadron in QCD. Light-front holography8,16–20 connects the equations of motion in AdS space and the Hamiltonian formulation of QCD in physical space-time quantized on the light front at fixed LF time. This correspondence provides a direct connection between the hadronic amplitudes Φ(z) in AdS space with LF wavefunctions φ(ζ) describing the quark and gluon constituent structure of hadrons in physical space-time. In
the case of a meson, ζ = x(1 − x)b2⊥ is a Lorentz invariant coordinate which measures the distance between the quark and antiquark; it is analogous to the radial coordinate r in the Schr¨ odinger equation. In effect ζ represents the off-lightfront energy shell or invariant mass dependence of the bound state; it allows the separation of the dynamics of quark and gluon binding from the kinematics of constituent spin and internal orbital angular momentum.19 Light-front holography thus provides a connection between the description of hadronic modes in AdS space and the Hamiltonian formulation of QCD in physical space-time quantized on the light-front at fixed LF time τ. The mapping between the LF invariant variable ζ and the fifth-dimension AdS coordinate z was originally obtained by matching the expression for electromagnetic current matrix elements in AdS space21 with the corresponding expression for the current matrix element, using LF theory in physical space time.16 It has also been shown that one obtains the identical holographic mapping using the matrix
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elements of the energy-momentum tensor,18,22 thus verifying the consistency of the holographic mapping from AdS to physical observables defined on the light front. The resulting equation for the mesonic q q¯ bound states at fixed light-front time has the form of a single-variable relativistic Lorentz invariant Schr¨ odinger equation 1 − 4L2 d2 + U (ζ) φ(ζ) = M2 φ(ζ), (1) − 2 − dζ 4ζ 2
where the confining potential is U (ζ) = κ4 ζ 2 + 2κ2 (L + S − 1) in the soft-wall model. Its eigenvalues determine the hadronic spectra and its eigenfunctions are related to the light-front wavefunctions of hadrons for general spin and orbital angular momentum. This LF wave equation serves as a semiclassical first approximation to QCD, and it is equivalent to the equations of motion which describe the propagation of spin-J modes in AdS space. The resulting light-front wavefunctions provide a fundamental description of the structure and internal dynamics of hadronic states in terms of their constituent quark and gluons. There is only one parameter, the mass scale κ ∼ 1/2 GeV, which enters the confinement potential. In the case of mesons S = 0, 1 is the combined spin of the q and q¯, L is their relative orbital angular momentum as determined by the hadronic light-front wavefunctions. The concept of a running coupling αs (Q2 ) in QCD is usually restricted to the perturbative domain. However, as in QED, it is useful to define the coupling as an analytic function valid over the full space-like and time-like domains. The study of the non-Abelian QCD coupling at small momentum transfer is a complex problem because of gluonic self-coupling and color confinement. We will show that the lightfront holographic mapping of classical gravity in AdS space, modified by a positive(Q2 ) sign dilaton background, leads to a non-perturbative effective coupling αAdS s which is in agreement with hadron physics data extracted from different observables, as well as with the predictions of models with built-in confinement and lattice simulations. 2. The Hadron Spectrum and Form Factors in Light-Front AdS/QCD The meson spectrum predicted by Eq. (1) has a string-theory Regge form M2 = 4κ2 (n + L + S/2); i.e., the square of the eigenmasses are linear in both L and n, where n counts the number of nodes of the wavefunction in the radial variable ζ. This is illustrated for the pseudoscalar and vector meson spectra in Fig. 1, where the data are from Ref. 23. The pion (S = 0, n = 0, L = 0) is massless for zero quark mass, consistent with the chiral invariance of massless QCD. Thus one can compute the hadron spectrum by simply adding 4κ2 for a unit change in the radial quantum number, 4κ2 for a change in one unit in the orbital quantum number L and 2κ2 for a change of one unit of spin S. Remarkably, the same rule holds for three-quark baryons as we shall show below. In the light-front formalism, one sets boundary conditions at fixed τ and then evolves the system using the light-front (LF) Hamiltonian P − = P 0 − P 3 = id/dτ .
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The invariant Hamiltonian HLF = P + P − − P⊥2 then has eigenvalues M2 where M is the physical mass. Its eigenfunctions are the light-front eigenstates whose Fock state projections define the frame-independent light-front wavefunctions. The eigensolutions of Eq. (1) provide the light-front wavefunctions of the valence Fock state of the hadrons ψ(x, b⊥ ) as illustrated for the pion in Fig. 2 for the soft (a) and hard wall (b) models. The resulting distribution amplitude has a broad form φπ (x) ∼ x(1 − x) which is compatible with moments determined from lattice gauge theory. One can then immediately compute observables such as hadronic form factors (overlaps of LFWFs), structure functions (squares of LFWFs), as well as the generalized parton distributions and distribution amplitudes which underly hard exclusive reactions. For example, hadronic form factors can be predicted from the overlap of LFWFs in the Drell-Yan West formula. The prediction for the spacelike pion form factor is shown in Fig. 2(c). The pion form factor and the vector meson poles residing in the dressed √ current in the soft wall model require choosing a value of κ smaller by a factor of 1/ 2 than the canonical value of κ which determines the mass scale of the hadronic spectra. This shift is apparently due to the fact that the transverse current in e+ e− → q q¯ creates a quark pair with Lz = ±1 instead of the Lz = 0 q q¯ composition of the vector mesons in the spectrum. Individual hadrons in AdS/QCD are identified by matching the power behavior of the hadronic amplitude at the AdS boundary at small z to the twist τ of its interpolating operator at short distances x2 → 0, as required by the AdS/CFT dictionary. The twist also equals the dimension of fields appearing in chiral supermultiplets;24 thus the twist of a hadron equals the number of constituents plus the relative orbital angular momentum. One then can apply light-front holography to relate the amplitude eigensolutions in the fifth dimension coordinate z to the LF wavefunctions in the physical space-time variable ζ.
Fig. 1. Parent and daughter Regge trajectories for (a) the π-meson family with κ = 0.6 GeV; and (b) the I = 1 ρ-meson and I = 0 ω-meson families with κ = 0.54 GeV.
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Equation (1) was derived by taking the LF bound-state Hamiltonian equation of motion as the starting point.19 The term L2 /ζ 2 in the LF equation of motion (1) is derived from the reduction of the LF kinetic energy when one transforms to twodimensional cylindrical coordinates (ζ, ϕ), in analogy to the ( + 1)/r2 Casimir term in Schr¨ odinger theory. One thus establishes the interpretation of L in the AdS equations of motion. The interaction terms build confinement corresponding to the dilaton modification of AdS space.19 The duality between these two methods provides a direct connection between the description of hadronic modes in AdS space and the Hamiltonian formulation of QCD in physical space-time quantized on the light-front at fixed LF time τ. The identification of orbital angular momentum of the constituents is a key element in the description of the internal structure of hadrons using holographic principles. In our approach quark and gluon degrees of freedom are explicitly introduced in the gauge/gravity correspondence,25 in contrast with the usual AdS/QCD framework26,27 where axial and vector currents become the primary entities as in effective chiral theory. Unlike the top-down string theory approach, one is not limited to hadrons of maximum spin J ≤ 2, and one can study baryons with finite color NC = 3. Higher spin modes follow from shifting dimensions in the AdS wave equations. In the soft-wall model the usual Regge behavior is found M2 ∼ n + L, predicting the same multiplicity of states for mesons and baryons as observed experimentally.28 It is possible to extend the model to hadrons with heavy quark constituents by introducing nonzero quark masses and short-range Coulomb corrections. For other recent calculations of the hadronic spectrum based on AdS/QCD, see Refs. 29–40. Other recent computations of the pion form factor are given in Refs. 41, 42. For baryons, the light-front wave equation is a linear equation determined by the LF transformation properties of spin 1/2 states. A linear confining potential
Fig. 2. Pion LF wavefunction ψπ (x, b⊥ ) for the AdS/QCD (a) hard-wall (ΛQCD = 0.32 GeV) and (b) soft-wall (κ = 0.375 GeV) models. (c) Space-like scaling behavior for Q2 Fπ (Q2 ). The continuous line is the prediction of the soft-wall model for κ = 0.375 GeV. The dashed line is the prediction of the hard-wall model for ΛQCD = 0.22 GeV. The triangles are the data compilation of Baldini.
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U (ζ) ∼ κ2 ζ in the LF Dirac equation leads to linear Regge trajectories.43 For fermionic modes the light-front matrix Hamiltonian eigenvalue equation DLF |ψ = 2 M|ψ, HLF = DLF , in a 2 × 2 spinor component representation is equivalent to the system of coupled linear equations −
ν + 12 d ψ− − ψ− − κ2 ζψ− = Mψ+ , dζ ζ ν + 12 d ψ+ − ψ+ − κ2 ζψ+ = Mψ− , dζ ζ
(2)
with eigenfunctions 1
2 2
Lνn (κ2 ζ 2 ),
3
2 2
2 2 Lν+1 n (κ ζ ),
ψ+ (ζ) ∼ z 2 +ν e−κ
ψ− (ζ) ∼ z 2 +ν e−κ
ζ /2 ζ /2
(3)
and eigenvalues M2 = 4κ2 (n + ν + 1). The baryon interpolating operator O3+L = ψD{1 . . . Dq ψDq+1 . . . Dm } ψ, L = m i=1 i , is a twist 3, dimension 9/2 + L with scaling behavior given by its twistdimension 3 + L. We thus require ν = L + 1 to match the short distance scaling behavior. Higher spin modes are obtained by shifting dimensions for the fields. Thus, as in the meson sector, the increase in the mass squared for higher baryonic state is ∆n = 4κ2 , ∆L = 4κ2 and ∆S = 2κ2 , relative to the lowest ground state, the proton. Since our starting point to find the bound state equation of motion for baryons is the light-front, we fix the overall energy scale identical for mesons and baryons by imposing chiral symmetry to the pion20 in the LF Hamiltonian equations. By contrast, if we start with a five-dimensional action for a scalar field in presence of a positive sign dilaton, the pion is automatically massless.
Fig. 3.
56 Regge trajectories for the N and ∆ baryon families for κ = 0.5 GeV.
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The predictions for the 56-plet of light baryons under the SU (6) flavor group are shown in Fig. 3. As for the predictions for mesons in Fig. 1, only confirmed PDG23 states are shown. The Roper state N (1440) and the N (1710) are well accounted for in this model as the first and second radial states. Likewise the ∆(1660) corresponds to the first radial state of the ∆ family. The model is successful in explaining the important parity degeneracy observed in the light baryon spectrum, such as the L = 2, N (1680) − N (1720) degenerate pair and the L = 2, ∆(1905), ∆(1910), ∆(1920), ∆(1950) states which are degenerate within error bars. Parity degeneracy of baryons is also a property of the hard wall model, but radial states are not well described in this model.44 As an example of the scaling behavior of a twist τ = 3 hadron, we compute the spin non-flip nucleon form factor in the soft wall model.43 The proton and neutron Dirac form factors are given by F1p (Q2 )
F1n (Q2 )
1 =− 3
=
dζ J(Q, ζ) |ψ+ (ζ)|2 ,
(4)
dζ J(Q, ζ) |ψ+ (ζ)|2 − |ψ− (ζ)|2 ,
(5)
where F1p (0) = 1, F1n (0) = 0. The non-normalizable mode J(Q, z) is the solution of the AdS wave equation for the external electromagnetic current in presence of a dilaton background exp(±κ2 z 2 ).17,45 Plus and minus components of the twist 3 nucleon LFWF are √ 2 2 ψ+ (ζ) = 2κ2 ζ 3/2 e−κ ζ /2 ,
Ψ− (ζ) = κ3 ζ 5/2 e−κ
2 2
ζ /2
.
(6)
The results for Q4 F1p (Q2 ) and Q4 F1n (Q2 ) and are shown in Fig. 4.46
Fig. 4.
Predictions for Q4 F1p (Q2 ) and Q4 F1n (Q2 ) in the soft wall model for κ = 0.424 GeV.
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3. Nonperturbative Running Coupling from Light-Front Holography The definition of the running coupling in perturbative quantum field theory is scheme-dependent. As discussed by Grunberg,47 an effective coupling or charge can be defined directly from physical observables. Effective charges defined from different observables can be related to each other in the leading-twist domain using commensurate scale relations (CSR).48 The potential between infinitely heavy quarks can be defined analytically in momentum transfer space as the product of the running coupling times the Born gluon propagator: V (q) = −4πCF αV (q)/q 2 . This effective charge defines a renormalization scheme – the αV scheme of Appelquist, Dine, and Muzinich.49 In fact, the holographic coupling αAdS (Q2 ) can be considered s to be the nonperturbative extension of the αV effective charge defined in Ref. 49. We can also make use of the g1 scheme, where the strong coupling αg1 (Q2 ) is determined from the Bjorken sum rule.50 The coupling αg1 (Q2 ) has the advantage that it is the best-measured effective charge, and it can be used to extrapolate the definition of the effective coupling to large distances.51 Since αg1 has been measured at intermediate energies, it is particularly useful for studying the transition from small to large distances. We will show3 how the LF holographic mapping of effective classical gravity in AdS space, modified by a positive-sign dilaton background, can be used to identify an analytically simple color-confining non-perturbative effective coupling αAdS (Q2 ) s 2 2 as a function of the space-like momentum transfer Q = −q . This coupling incorporates confinement and agrees well with effective charge observables and lattice simulations. It also exhibits an infrared fixed point at small Q2 and asymptotic freedom at large Q2 . However, the fall-off of αAdS (Q2 ) at large Q2 is exponens AdS 2 −Q2 /κ2 tial: αs (Q ) ∼ e , rather than the perturbative QCD (pQCD) logarithmic fall-off. We also show in Ref. 3 that a phenomenological extended coupling can be defined which implements the pQCD behavior. As will be discussed below, the β-function derived from light-front holography becomes significantly negative in the non-perturbative regime Q2 ∼ κ2 , where it reaches a minimum, signaling the transition region from the infrared (IR) conformal region, characterized by hadronic degrees of freedom, to a pQCD conformal ultraviolet (UV) regime where the relevant degrees of freedom are the quark and gluon constituents. The β-function is always negative: it vanishes at large Q2 consistent with asymptotic freedom, and it vanishes at small Q2 consistent with an infrared fixed point.5,52 Let us consider a five-dimensional gauge field G propagating in AdS5 space in presence of a dilaton background ϕ(z) which introduces the energy scale κ in the five-dimensional action. At quadratic order in the field strength the action is 1 1 √ d5 x g eϕ(z) 2 G2 , S=− (7) 4 g5
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√ where the metric determinant of AdS5 is g = (R/z)5 , ϕ = κ2 z 2 and the square of the coupling g5 has dimensions of length. We can identify the prefactor g5−2 (z) = eϕ(z) g5−2 ,
(8)
in the AdS action (7) as the effective coupling of the theory at the length scale z. The coupling g5 (z) then incorporates the non-conformal dynamics of confinement. The five-dimensional coupling g5 (z) is mapped, modulo a constant, into the Yang-Mills (YM) coupling gY M of the confining theory in physical space-time using light-front holography. One identifies z with the invariant impact separation variable ζ which appears in the LF Hamiltonian: g5 (z) → gY M (ζ). Thus (ζ) = gY2 M (ζ)/4π ∝ e−κ αAdS s
2 2
ζ
.
(9)
In contrast with the 3-dimensional radial coordinates of the non-relativistic Schr¨ odinger theory, the natural light-front variables are the two-dimensional cylindrical coordinates (ζ, φ) and the light-cone fraction x. The physical coupling measured at the scale Q is the two-dimensional Fourier transform of the LF transverse (ζ) (9). Integration over the azimuthal angle φ gives the Bessel transcoupling αAdS s form ∞ 2 (Q ) ∼ ζdζ J0 (ζQ) αAdS (ζ), (10) αAdS s s 0
+
in the q = 0 light-front frame where Q2 = −q 2 = −q2⊥ > 0 is the square of the space-like four-momentum transferred to the hadronic bound state. Using this ansatz we then have from Eq. (10) 2
αAdS (Q2 ) = αAdS (0) e−Q s s
/4κ2
.
(11)
In contrast, the negative dilaton solution ϕ = −κ2 z 2 leads to an integral which (Q2 ) with the physical QCD running coupling diverges at large ζ. We identify αAdS s in its nonperturbative domain. The flow equation (8) from the scale dependent measure for the gauge fields can be understood as a consequence of field-strength renormalization. In physical QCD we can rescale the non-Abelian gluon field Aµ → λAµ and field strength Gµν → λGµν in the QCD Lagrangian density LQCD by a compensating rescaling of the coupling strength g → λ−1 g. The renormalization of the coupling 1/2 gphys = Z3 g0 , where g0 is the bare coupling in the Lagrangian in the UV-regulated theory, is thus equivalent to the renormalization of the vector potential and field −1/2 µ −1/2 µν A0 , Gµν G0 with a rescaled Lagrangian density strength: Aµren = Z3 ren = Z3 −1 ren 0 −2 LQCD = Z3 LQCD = (gphys /g0 ) L0 . In lattice gauge theory, the lattice spacing a serves as the UV regulator, and the renormalized QCD coupling is determined from the normalization of the gluon field strength as it appears in the gluon propagator. The inverse of the lattice size L sets the mass scale of the resulting running coupling. As is the case in lattice gauge theory, color confinement in AdS/QCD reflects nonperturbative dynamics at large distances. The QCD couplings defined from lattice gauge theory and the soft wall holographic model are thus similar in concept,
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and both schemes are expected to have similar properties in the nonperturbative domain, up to a rescaling of their respective momentum scales. 4. Comparison of the Holographic Coupling with Other Effective Charges The effective coupling αAdS (Q2 ) (solid line) is compared in Fig. 5 with experimental and lattice data. For this comparison to be meaningful, we have to impose the same AdS normalization on the AdS 2coupling
as the g1 coupling. This defines αs normalized AdS to the g1 scheme: αg1 Q = 0 = π. Details on the comparison with other effective charges are given in Ref. 53. The couplings in Fig. 5 (a) agree well in the strong coupling regime up to Q ∼ 1 GeV. The value κ = 0.54 GeV is determined from the vector meson Regge trajectory.8 The lattice results shown in Fig. 5 from Ref. 54 have been scaled to match the perturbative UV domain. The effective charge αg1 has been determined in Ref. 53 from several experiments. Fig. 5 also displays other couplings from different observables as well as αg1 which is computed from the Bjorken sum rule50 over a large range of momentum transfer (cyan band). At Q2 = 0 one has the constraint on the slope of αg1 from the Gerasimov-Drell-Hearn (GDH) sum rule55 which is also shown in the figure. The results show no sign of a phase transition, cusp, or other nonanalytical behavior, a fact which allows us to extend the functional dependence of the coupling to large distances. As discussed below, the smooth behavior of the AdS strong coupling also allows us to extrapolate its form to the perturbative domain. The hadronic model obtained from the dilaton-modified AdS space provides a semi-classical first approximation to QCD. Color confinement is introduced by the harmonic oscillator potential, but effects from gluon creation and absorption are not included in this effective theory. The nonperturbative confining effects vanish
Fig. 5. (a) Effective coupling from LF holography for κ = 0.54 GeV compared with effective QCD couplings extracted from different observables and lattice results. (b) Prediction for the β function compared to lattice simulations, JLab and CCFR results for the Bjorken sum rule effective charge.
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exponentially at large momentum transfer (Eq. (11)), and thus the logarithmic fall-off from pQCD quantum loops will dominate in this regime. given by Eq. (11) is obtained from a color-confining The running coupling αAdS s potential. Since the strong coupling is an analytical function of the momentum by matchtransfer at all scales, we can extend the range of applicability of αAdS s ing to a perturbative coupling at the transition scale, Q ∼ 1 GeV, where pQCD contributions become important. In order to have a fully analytical model, we write 2 AdS 2 2 f it 2 2 αAdS Modif ied,g1 (Q ) = αg1 (Q )g+ (Q ) + αg1 (Q )g− (Q ),
(12)
2 2 2 where g± (Q2 ) = 1/(1 + e±(Q −Q0 )/τ ) are smeared step functions which match the two regimes. The parameter τ represents the width of the transition region. Here is given by Eq. (11) with the normalization αAdS αAdS g1 g1 (0) = π – the plain black line in Fig. 5 – and αfg1it in Eq. (12) is the analytical fit to the measured coupling αg1 .53 The couplings are chosen to have the same normalization at Q2 = 0. The smoothly extrapolated result (dot-dashed line) for αs is also shown on Fig. 5. We use the parameters Q20 = 0.8 GeV2 and τ 2 = 0.3 GeV2 . The β-function for the nonperturbative effective coupling obtained from the LF holographic mapping in a positive dilaton modified AdS background is
β AdS (Q2 ) =
d πQ2 −Q2 /(4κ2 ) αAdS (Q2 ) = e . 2 d log Q 4κ2
(13)
The solid line in Fig. 5 (b) corresponds to the light-front holographic result Eq. (13). Near Q0 2κ 1 GeV, we can interpret the results as a transition from the nonperturbative IR domain to the quark and gluon degrees of freedom in the perturbative UV regime. The transition momentum scale Q0 is compatible with the momentum transfer for the onset of scaling behavior in exclusive reactions where quark counting rules are observed.56 For example, in deuteron photo-disintegration the onset of scaling corresponds to momentum transfer of 1.0 GeV to the nucleon involved.57 Dimensional counting is built into the AdS/QCD soft and hard wall models since the AdS amplitudes Φ(z) are governed by their twist scaling behavior z τ at short distances, z → 0.6 Also shown on Fig. 5 (b) are the β-functions obtained from phenomenology and lattice calculations. For clarity, we present only the LF holographic predictions, the lattice results from,54 and the experimental data supplemented by the relevant sum rules. The width of the aqua band is computed from the uncertainty of αg1 in the perturbative regime. The dot-dashed curve corresponds to the extrapolated approximation given by Eq. (12). Only the point-to-point uncorrelated uncertainties of the JLab data are used to estimate the uncertainties, since a systematic shift cancels in the derivative. Nevertheless, the uncertainties are still large. The β-function extracted from LF holography, as well as the forms obtained from the works of Cornwall,52 Bloch, Fisher et al.,58 Burkert and Ioffe59 and Furui and Nakajima,54 are seen to have a similar shape and magnitude.
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Judging from these results, we infer that the actual β-function of QCD will extrapolate between the non-perturbative results for Q < 1 GeV and the pQCD results for Q > 1 GeV. We also observe that the general conditions β(Q → 0) = β(Q → ∞) = 0,
dβ dQ
β(Q) < 0, for Q > 0, dβ dQ Q=Q0 = 0,
< 0, for Q < Q0 ,
dβ dQ
> 0, for Q > Q0
(14) (15) (16) (17)
are satisfied by our model β-function obtained from LF holography. Eq. (14) expresses the fact that QCD approaches a conformal theory in both the far ultraviolet and deep infrared regions. In the semiclassical approximation to QCD without particle creation or absorption, the β-function is zero and the approximate theory is scale invariant in the limit of massless quarks.60 When quantum corrections are included, the conformal behavior is preserved at very large Q because of asymptotic freedom and near Q → 0 because the theory develops a fixed point. An infrared fixed point is in fact a natural consequence of color confinement:52 since the propagators of the colored fields have a maximum wavelength, all loop integrals in the computation of the gluon self-energy decouple at Q2 → 0.5 Condition (15) for Q2 large, expresses the basic anti-screening behavior of QCD where the strong coupling vanishes. The β-function in QCD is essentially negative, thus the coupling increases monotonically from the UV to the IR where it reaches its maximum value: it has a finite value for a theory with a mass gap. Equation (16) defines the transition region at Q0 where the beta function has a minimum. Since there is only one hadronicpartonic transition, the minimum is an absolute minimum; thus the additional conditions expressed in Eq. (17) follow immediately from Eqs. (14–16). The conditions given by Eqs. (14–17) describe the essential behavior of the full β-function for an effective QCD coupling whose scheme/definition is similar to that of the V -scheme. 5. Conclusions As we have shown, the combination of Anti-de Sitter space (AdS) methods with light-front (LF) holography provides a remarkably accurate first approximation for the spectra and wavefunctions of meson and baryon light-quark bound states. We obtain a connection between a semiclassical first approximation to QCD, quantized on the light-front, with hadronic modes propagating on a fixed AdS background. The resulting bound-state Hamiltonian equation of motion in QCD leads to relativistic light-front wave equations in the invariant impact variable ζ which measures the separation of the quark and gluonic constituents within the hadron at equal light-front time. This corresponds to the effective single-variable relativistic Schr¨ odinger-like equation in the AdS fifth dimension coordinate z, Eq. (1). The eigenvalues give the hadronic spectrum, and the eigenmodes represent the probability distributions of the hadronic constituents at a given scale. As we have shown,
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the light-front holographic mapping of effective classical gravity in AdS space, modified by a positive-sign dilaton background, provides a very good description of the spectrum and form factors of light mesons and baryons. There are many phenomenological applications where detailed knowledge of the QCD coupling and the renormalized gluon propagator at relatively soft momentum transfer are essential. This includes the rescattering (final-state and initial-state interactions) which create the leading-twist Sivers single-spin correlations in semiinclusive deep inelastic scattering,61,62 the Boer-Mulders functions which lead to anomalous cos 2φ contributions to the lepton pair angular distribution in the unpolarized Drell-Yan reaction,63 and the Sommerfeld-Sakharov-Schwinger correction to heavy quark production at threshold.64 The confining AdS/QCD coupling from light-front holography can lead to a quantitative understanding of this factorizationbreaking physics.65 We have also shown that the light-front holographic mapping of effective classical gravity in AdS space, modified by the same positive-sign dilaton background pre(Q) and its β-function. dicts the form of a non-perturbative effective coupling αAdS s The AdS/QCD running coupling is in very good agreement with the effective coupling αg1 extracted from the Bjorken sum rule. Surprisingly, the Furui and Nakajima lattice results54 also agree better overall with the g1 scheme rather than the V scheme. Our analysis indicates that light-front holography captures the essential dynamics of confinement. The holographic β-function displays a transition from nonperturbative to perturbative regimes at a momentum scale Q ∼ 1 GeV. It appears to captures the essential characteristics of the full β-function of QCD, thus giving further support to the application of the gauge/gravity duality to the confining dynamics of strongly coupled QCD. Acknowledgments Presented by SJB at SCGT09, 2009 International Workshop on Strong Coupling Gauge Theories in the LHC Era, Nagoya, December 8-11, 2009. We thank Volker Burkert, John Cornwall, Sadataka Furui, Philipp H¨ agler, Wolfgang Korsch, G. Peter Lepage, Takemichi Okui, Joannis Papavassiliou and Anatoly Radyushkin for helpful comments. This research was supported by the Department of Energy contracts DE–AC02–76SF00515 and DE-AC05-84ER40150. SLAC-PUB-13998. JLAB-PHY10-1130. References 1. J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200]. 2. S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428, 105 (1998) [arXiv:hep-th/9802109]; E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998) [arXiv:hepth/9802150]. 3. S. J. Brodsky, G. F. de Teramond and A Deur, arXiv:1002.3948 [hep-ph]. 4. A. Deur, V. Burkert, J. P. Chen and W. Korsch, Phys. Lett. B 665, 349 (2008) [arXiv:0803.4119 [hep-ph]].
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5. S. J. Brodsky and R. Shrock, Phys. Lett. B 666, 95 (2008) [arXiv:0806.1535 [hep-th]]. 6. J. Polchinski and M. J. Strassler, Phys. Rev. Lett. 88, 031601 (2002) [arXiv:hepth/0109174]. 7. A. Karch, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. D 74, 015005 (2006) [arXiv:hep-ph/0602229]. 8. G. F. de Teramond and S. J. Brodsky, arXiv:0909.3900 [hep-ph]. 9. O. Andreev and V. I. Zakharov, Phys. Rev. D 74, 025023 (2006) [arXiv:hepph/0604204]. 10. F. Zuo, arXiv:0909.4240 [hep-ph]. 11. S. S. Afonin, arXiv:1001.3105 [hep-ph]. 12. S. D. Glazek and M. Schaden, Phys. Lett. B 198, 42 (1987). 13. P. Hoyer, arXiv:0909.3045 [hep-ph]. 14. P. A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949). 15. S. J. Brodsky, H. C. Pauli and S. S. Pinsky, Phys. Rept. 301, 299 (1998) [arXiv:hepph/9705477]. 16. S. J. Brodsky and G. F. de Teramond, Phys. Rev. Lett. 96, 201601 (2006) [arXiv:hepph/0602252]; 17. S. J. Brodsky and G. F. de Teramond, Phys. Rev. D 77, 056007 (2008) [arXiv:0707.3859 [hep-ph]]. 18. S. J. Brodsky and G. F. de Teramond, Phys. Rev. D 78, 025032 (2008) [arXiv:0804.0452 [hep-ph]]. 19. G. F. de Teramond and S. J. Brodsky, Phys. Rev. Lett. 102, 081601 (2009) [arXiv:0809.4899 [hep-ph]]. 20. G. F. de Teramond and S. J. Brodsky, arXiv:1001.5193 [hep-ph]. 21. J. Polchinski and M. J. Strassler, JHEP 0305, 012 (2003) [arXiv:hep-th/0209211]. 22. Z. Abidin and C. E. Carlson, Phys. Rev. D 77, 095007 (2008) [arXiv:0801.3839 [hepph]]. 23. C. Amsler et al. (Particle Data Group), Phys. Lett. B 667, 1, (2008). 24. N. J. Craig and D. Green, JHEP 0909, 113 (2009) [arXiv:0905.4088 [hep-ph]]. 25. S. J. Brodsky and G. F. de Teramond, Phys. Lett. B 582, 211 (2004) [arXiv:hepth/0310227]. 26. J. Erlich, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 95, 261602 (2005) [arXiv:hep-ph/0501128]. 27. L. Da Rold and A. Pomarol, Nucl. Phys. B 721, 79 (2005) [arXiv:hep-ph/0501218]. 28. E. Klempt and A. Zaitsev, Phys. Rept. 454, 1 (2007) [arXiv:0708.4016 [hep-ph]]. 29. H. Boschi-Filho, N. R. F. Braga and H. L. Carrion, Phys. Rev. D 73, 047901 (2006) [arXiv:hep-th/0507063]. 30. N. Evans and A. Tedder, Phys. Lett. B 642, 546 (2006) [arXiv:hep-ph/0609112]. 31. D. K. Hong, T. Inami and H. U. Yee, Phys. Lett. B 646, 165 (2007) [arXiv:hepph/0609270]. 32. P. Colangelo, F. De Fazio, F. Jugeau and S. Nicotri, Phys. Lett. B 652, 73 (2007) [arXiv:hep-ph/0703316]. 33. H. Forkel, Phys. Rev. D 78, 025001 (2008) [arXiv:0711.1179 [hep-ph]]. 34. A. Vega and I. Schmidt, Phys. Rev. D 78 (2008) 017703 [arXiv:0806.2267 [hep-ph]]. 35. K. Nawa, H. Suganuma and T. Kojo, Mod. Phys. Lett. A 23, 2364 (2008) [arXiv:0806.3040 [hep-th]]. 36. W. de Paula, T. Frederico, H. Forkel and M. Beyer, Phys. Rev. D 79, 075019 (2009) [arXiv:0806.3830 [hep-ph]]. 37. P. Colangelo, F. De Fazio, F. Giannuzzi, F. Jugeau and S. Nicotri, Phys. Rev. D 78 (2008) 055009 [arXiv:0807.1054 [hep-ph]].
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38. H. Forkel and E. Klempt, Phys. Lett. B 679, 77 (2009) [arXiv:0810.2959 [hep-ph]]. 39. H. C. Ahn, D. K. Hong, C. Park and S. Siwach, Phys. Rev. D 80, 054001 (2009) [arXiv:0904.3731 [hep-ph]]. 40. Y. Q. Sui, Y. L. Wu, Z. F. Xie and Y. B. Yang, arXiv:0909.3887 [hep-ph]. 41. H. J. Kwee and R. F. Lebed, JHEP 0801, 027 (2008) [arXiv:0708.4054 [hep-ph]]. 42. H. R. Grigoryan and A. V. Radyushkin, Phys. Rev. D 76 (2007) 115007 [arXiv:0709.0500 [hep-ph]]; Phys. Rev. D 78 (2008) 115008 [arXiv:0808.1243 [hepph]]. 43. S. J. Brodsky and G. F. de Teramond, arXiv:0802.0514 [hep-ph]. 44. G. F. de Teramond and S. J. Brodsky, Phys. Rev. Lett. 94, 201601 (2005) [arXiv:hepth/0501022]. 45. H. R. Grigoryan and A. V. Radyushkin, Phys. Rev. D 76 (2007) 095007 [arXiv:0706.1543 [hep-ph]]. 46. The data compilation is from M. Diehl, Nucl. Phys. Proc. Suppl. 161, 49 (2006) [arXiv:hep-ph/0510221]. 47. G. Grunberg, Phys. Lett. B 95, 70 (1980); Phys. Rev. D 29, 2315 (1984); Phys. Rev. D 40, 680 (1989). 48. S. J. Brodsky and H. J. Lu, Phys. Rev. D 51, 3652 (1995) [arXiv:hep-ph/9405218]; S. J. Brodsky, G. T. Gabadadze, A. L. Kataev and H. J. Lu, Phys. Lett. B 372, 133 (1996) [arXiv:hep-ph/9512367]. 49. T. Appelquist, M. Dine and I. J. Muzinich, Phys. Lett. B 69, 231 (1977). 50. J. D. Bjorken, Phys. Rev. 148, 1467 (1966). 51. A. Deur, arXiv:0907.3385 [nucl-ex]. 52. J. M. Cornwall, Phys. Rev. D 26, 1453 (1982). 53. A. Deur, V. Burkert, J. P. Chen and W. Korsch, Phys. Lett. B 650, 244 (2007) [arXiv:hep-ph/0509113]. 54. S. Furui and H. Nakajima, Phys. Rev. D 70, 094504 (2004); S. Furui, arXiv:0908.2768 [hep-lat]. 55. S. D. Drell and A. C. Hearn, Phys. Rev. Lett. 16, 908 (1966); S. B. Gerasimov, Sov. J. Nucl. Phys. 2 (1966) 430 [Yad. Fiz. 2 (1966) 598]. 56. S. J. Brodsky and G. R. Farrar, Phys. Rev. Lett. 31, 1153 (1973). 57. H. Gao and L. Zhu, AIP Conf. Proc. 747, 179 (2005) [arXiv:nucl-ex/0411014]. 58. J. C. R. Bloch, Phys. Rev. D 66, 034032 (2002) [arXiv:hep-ph/0202073]; P. Maris and P. C. Tandy, Phys. Rev. C 60, 055214 (1999) [arXiv:nucl-th/9905056]; C. S. Fischer and R. Alkofer, Phys. Lett. B 536, 177 (2002) [arXiv:hep-ph/0202202]; C. S. Fischer, R. Alkofer and H. Reinhardt, Phys. Rev. D 65, 094008 (2002) [arXiv:hepph/0202195]; R. Alkofer, C. S. Fischer and L. von Smekal, Acta Phys. Slov. 52, 191 (2002) [arXiv:hep-ph/0205125]; M. S. Bhagwat, M. A. Pichowsky, C. D. Roberts and P. C. Tandy, Phys. Rev. C 68, 015203 (2003) [arXiv:nucl-th/0304003]. 59. V. D. Burkert and B. L. Ioffe, Phys. Lett. B 296, 223 (1992); J. Exp. Theor. Phys. 78, 619 (1994) [Zh. Eksp. Teor. Fiz. 105, 1153 (1994)]. 60. G. Parisi, Phys. Lett. B 39, 643 (1972). 61. S. J. Brodsky, D. S. Hwang and I. Schmidt, Phys. Lett. B 530, 99 (2002) [arXiv:hepph/0201296]. 62. J. C. Collins, Phys. Lett. B 536, 43 (2002) [arXiv:hep-ph/0204004]. 63. D. Boer, S. J. Brodsky and D. S. Hwang, Phys. Rev. D 67, 054003 (2003) [arXiv:hepph/0211110]. 64. S. J. Brodsky, A. H. Hoang, J. H. Kuhn and T. Teubner, Phys. Lett. B 359, 355 (1995) [arXiv:hep-ph/9508274]. 65. J. Collins and J. W. Qiu, Phys. Rev. D 75, 114014 (2007) [arXiv:0705.2141 [hep-ph]].
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New Results on Non-Abelian Vortices — Further Insights into Monopole, Vortex and Confinement K. Konishi Department of Physics, “E. Fermi”, University of Pisa, and INFN, Sez. di Pisa Largo Pontecorvo, 3, 56127, Pisa, Italy E-mail: [emailprotected] http://www.df.unipi.it/∼konishi/ We discuss some of the latest results concerning the non-Abelian vortices. The first concerns the construction of non-Abelian BPS vortices based on general gauge groups of the form G = G ×U (1). In particular detailed results about the vortex moduli space have been obtained for G = SO(N ) or U Sp(2N ). The second result is about the “fractional vortices”, i.e., vortices of the minimum winding but having substructures in the tension (or flux) density in the transverse plane. Thirdly, we discuss briefly the monopole-vortex complex. Keywords: Monopoles, vortex, non-Abelian gauge theories, duality.
1. Introduction The last few years have witnessed a remarkable progress in our understanding of the non-Abelian vortices and their relation to monopoles, both of which are thirty-year old problems in theoretical physics, and which can bear important implications to some deep issues such as quark confinement. The plan of this talk is: (i) a very brief review of non-Abelian monopoles; (ii) a brief review of ’03-’07 results on nonAbelian vortices; (iii) a new result on non-Abelian vortices based on general gauge groups; (iv) the fractional vortices; and (v) a brief discussion on the monopole-vortex complex and non-Abelian duality. It has become customary to think of quark confinement as a dual superconductor, in which (chromo-) electric fields are confined in a medium in which a magnetic charge is condensed. The original suggestion by ’t Hooft and Mandelstam is essentially Abelian: the effective low-energy degrees of freedom are magnetic monopoles arising from the Yang-Mills gauge fields. It is however possible that the dual superconductor relevant to quark confinement is of a non-Abelian kind, in which case we must better understand the quantum mechanical properties of these degrees of freedom. We would like to understand how the ’t Hooft-Polyakov monopoles1 (arising from a gauge symmetry breaking, G → H) and Abrikosov-Nielsen-Olesen vortices2 (of a broken gauge theory H → ) are generalized in situations where the relevant gauge group H is non-Abelian.
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The key developments which allowed us a qualitatively better understanding of these solitons are the following. First, the Seiberg-Witten solutions of N = 2 supersymmetric gauge theories3,4 revealed the quantum-mechanical behavior of the magnetic monopoles in an unprecedented fashion. In the presence of matter fields (quarks and squarks) these theories have, typically, vacua with non-Abelian dual gauge symmetry in the infrared.5,6 Thus in these systems non-Abelian monopoles do exist and play a central role in confinement and dynamical symmetry breaking. Second, the discovery of non-Abelian vortex solutions,7,8 i.e., soliton vortices with continuous, non-Abelian moduli, has triggered an intense research activity on the classical and quantum properties of these solitons, leading to a rich variety of new interesting results.9–11 2. Monopoles When the gauge symmetry is spontaneously broken φ1 =0
G −→ H
(1)
where H is some non-Abelian subgroup of G, the system possesses a set of regular magnetic monopole solutions in the semi-classical approximation. They are natural generalizations of the Abelian ’t Hooft-Polyakov monopoles,1 found originally in the G = SO(3) theory broken to H = U (1) by a Higgs mechanism. The gauge field looks asymptotically as rk (2) Fij = ijk Bk = ijk 3 (β · H), r in an appropriate gauge, where H are the diagonal generators of H in the Cartan subalgebra. A straightforward generalization of the Dirac’s quantization condition leads to12 2β · α ∈ Z
(3)
where α are the root vectors of H. In the simplest such case, G = SU (3), H = SU (2) × U (1)/2 ∼ U (2), a straightforward idea that the degenerate monopole solutions to be a doublet of the unbroken SU (2) leads however to the well-known difficulties.14,15 On the other hand, the quantization condition Eq. (3) implies that the monopoles should transform, if any, under the dual of U (2): the individual solu˜ generated by the tions are labelled by β which live in the weight vector space of H, dual roots, α . (4) α∗ = α·α ˜ are relatively non-local, the sought-for As transformation groups of fields H and H transformations of monopoles must look as non-local field transformations from the point of view of the original theory.16
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But the most significant fact is that fully quantum mechanical monopoles appears in the low-energy dual description of a wide class of N = 2 supersymmetric QCD.5,6 There must be ways to understand the physics of non-Abelian monopoles starting from a more familiar, semiclassical soliton picture. 3. Vortices Attempts to understand the semi-classical origin of the non-Abelian monopoles appearing in the so-called r-vacua of the N = 2 supersymmetric SU (N ) gauge theory, has eventually led to the discovery of the non-Abelian vortices.7,8 They are natural generalizations of the Abrikosov-Nielsen-Olesen (ANO) vortex. Unlike the ANO vortex, however, the non-Abelian vortices carry continuous zeromodes, i.e., they have a nontrivial moduli. The simplest model in which these vortices appear is an SU (N ) × U (1) gauge theory with Nf = N flavors of squarks in the fundamental representation. The secret of the non-Abelian vortices lies in the so-called color-flavor locked phase, in which the squark fields (written as N × N color-flavor mixed matrix) takes the VEV of the form, q(x) = v
N ×N
.
(5)
The SU (N ) gauge symmetry is completely broken, but the color-flavor mixed diagonal symmetry remains unbroken. The vortex configuration in this vacuum involves scalar fields of the form, iφ e f (ρ) 0 0 0 0 g(ρ) 0 0 q(x) = v (6) . 0 0 .. 0 0
0 g(ρ)
where ρ, φ, z (the static vortex does not depend on z) are the cylindrical coordinates. The gauge fields take appropriate form, in order to ensure that the kinetic term tends to zero asymptotically, Dq(x) → 0. In Eq. (6) the first flavor of the squark winds, but the full solution Ai , q(x) can be rotated in the color flavor SU (N ) transformations, Ai , → U (Ai + i∂i )U † ,
q(x) → U q(x)U † ,
leaving the tension invariant. In other words, individual vortices break the exact SU (N )C+F symmetry of the system, developing therefore non-Abelian orientational zeromodes. Its nature is seen from the fact that the vortex Eq. (6) breaks the global symmetry as SU (N ) → SU (N − 1) × U (1)/N −1 ;
(7)
the vortex moduli is given by CP N −1 ∼
SU (N ) . SU (N − 1) × U (1)/N −1
(8)
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They are Nambu-Goldstone modes, which however can propagate only inside the vortex: far from it they are massive. The quantum properties of the non-Abelian orientational modes (the effective CP N −1 sigma model), the study of non-Abelian vortices of higher winding numbers, the generalization to the cases of larger number of flavors and the study of the resulting, much richer vortex moduli spaces, the question of vortex stability in the presence of small non-BPS corrections, extension to more general class of gauge theories, etc. have been the subjects of an intense research activity recently. 4. Non-Abelian vortices with general gauge groups One of the new results by us17 is the construction of non-Abelian vortex solutions based on a general gauge group G × U (1), where G = SU (N ), SO(N ), U Sp(2N ), etc. As in models based on SU (N ) gauge groups studied extensively in the last few years, we work with simple models which have the structure of the bosonic sector of N = 2 supersymmetric models. The model contains a FI (Fayet-Iliopoulos)-like term in the U (1) sector, allowing the system to develop stable vortices. A crucial aspect is that we work in a fully Higgsed vacuum, but with an unbroken color-flavor diagonal symmetry. We take as our model system 1 1 † L = Trc − 2 Fµν F µν − 2 Fˆµν Fˆ µν + Dµ H (Dµ H) 2e 2g 2 g 2 e2 2 |X a ta | , − X 0 t0 − 2ξt0 − 4 4 with the field strength, gauge fields and covariant derivative denoted as 0 0 Fµν = Fµν t ,
Aµ = Aaµ ta ,
0 Fµν = ∂µ A0ν − ∂ν A0µ ,
Fˆµν = ∂µ Aν − ∂ν Aµ + i [Aµ , Aν ] ,
Dµ = ∂µ + iA0µ t0 + iAaµ ta ,
A0µ is the gauge field associated with U (1) and Aaµ are the gauge fields of G . The matter scalar fields are written as an N × NF complex color (vertical) – flavor † (horizontal) mixed matrix H. be expanded as = X 0 t0 + X a ta +
It can
X0 = HH α α 0 † 0 a † a a X t , X = 2 Trc HH t , X = 2 Trc HH t . t and t stand for the U (1) and G generators, respectively, and finally, tα ∈ g⊥ , where g⊥ is the orthogonal complement of the Lie algebra g in su(N ). The traces with subscript c are over the color indices. e and g are the U (1) and G coupling constants, respectively, while ξ is a real constant. We choose the maximally “color-flavor-locked” vacuum of the system, v H = √ 1N , N
v2 . ξ= √ 2N
(9)
We have taken NF = N which is the minimal number of flavors allowing for such a vacuum. Note that, unlike the U (N ) model studied extensively in the last several years, the vacuum is not unique in these cases (i.e., with a general gauge group), even with such a minimum choice of the flavor multiplicity. This difference may be
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traced to the fact that groups such as SO(N ) × U (1) and U Sp(N ) × U (1) form strictly smaller subgroups of U (N ). The existence of a continuous vacuum degeneracy implies the emergence of vortices of semi-local type; this aspect will be crucial in the discussion of the fractional vortices in the second part of this talk. However, for now, we stick to the particular vacuum Eq. (9) and consider vortices and their moduli in this theory. The standard Bogomol’nyi completion reads 2 1 2 e2 0 0 1 g 2 a a X t − 2ξt0 + 2 Fˆ12 − X t T = d2 x Trc 2 F12 − e 2 g 2 2 ¯ − 2ξF12 t0 ≥ −ξ d2 x F 0 , +4 DH (10) 12 ¯≡ where D
D1 +iD2 , 2
z = x1 + ix2 . In the BPS limit one has √ 1 2 0 ν=− √ , T = 2 2N πξν = 2πv ν , d2 x F12 2π 2N
(11)
where ν is the U (1) winding number of the vortex. This leads immediately to the vortex BPS equations ¯ + iAH ¯ = ∂H ¯ =0, DH
0 F12 = e2 Trc HH † t0 − ξ ,
a F12
(12)
= g Trc HH † ta . 2
(13)
The matter BPS equation (12) can be solved by the Ansatz
¯ A¯ = −iS −1 (z, z¯)∂S(z, z¯) ,
H = S −1 (z, z¯)H0 (z) ,
(14) C
where S belongs to the complexification of the gauge group, S ∈ C∗ × G . H0 (z), holomorphic in z, is the moduli matrix, which contains all moduli parameters of the vortices. A gauge invariant object can be constructed from S as Ω = SS † . This can be conveniently split into the U (1) part and the G part, so that S = s S and † analogously Ω = ω Ω , ω = |s|2 , Ω = S S . The tension (11) can be rewritten as 1 ν= (15) T = 2πv 2 ν = 2v 2 d2 x ∂ ∂¯ log ω , d2 x ∂ ∂¯ log ω , π and ν determines the asymptotic behavior of the Abelian field as 2ν
ω = ss† ∼ |z|
,
for |z| → ∞ .
The minimal vortex solutions can then be written down by making use of the holomorphic invariants for the gauge group G made of H, which we denote as i i IG (H). If the U (1) charge of the i-th invariant is ni , IG (H) satisfies −1 i i i IG s−1 S H0 = s−ni IG (H) = IG (H0 (z)) ,
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i while the boundary condition is IG (H)
i = Ivev eiνni θ , where ν ni is the |z|→∞
i number of the zeros of IG . This leads then to the following asymptotic behavior |z|→∞
i ni i i IG IG (H) −→ Ivev z νni . (H0 ) = s i It shows that IG (H0 (z)), being holomorphic in z, are actually polynomials. Therefore ν ni must be positive integers for all i:
ν ni ∈ Z+
→
ν=
k , n0
k ∈ Z+ ,
i i
= 0 . The U (1) gauge transformation e2πi/n0 leaves IG with n0 ≡ gcd ni |Ivev (H) invariant and thus the true gauge group is G = [U (1) × G ] /Zn0 , where Zn0 is the center of the group G . The minimal winding in U (1) found here, 1 n0 , corresponds to the minimal element of π1 (G) = Z: it represents a minimal loop in our group manifold G. As a result we find the following non-trivial constraints for H0 kni kni −1 i i n0 . IG + O z n0 (H0 ) = Ivev z 4.1. GNO quantization for non-Abelian vortices Certain special solutions of a given theory can be found readily, as follows. It turns out that each such solution is characterized by a weight vector of the dual group, and is parametrized by a set of integers νa (a = 1, · · · , rank(G )) C
H0 (z) = z ν1N +νa Ha ∈ U (1)C × G ,
(16)
where ν = k/n0 is the U (1) winding number and Ha are the generators of the Cartan subalgebra of g . H0 must be holomorphic in z and single-valued, which gives the constraints for a set of integers νa (ν1N + νa Ha )ll ∈ Z≥0
∀l .
Suppose that we now consider scalar fields in an r-representation of G . The constraint is equivalent to ν + νa µ(i) a ∈ Z≥0
∀i ,
(17)
(i)
where µ(i) = µa (i = 1, 2, · · · , dim(r)) are the weight vectors for the rrepresentation of G . Subtracting pairs of adjacent weight vectors, one arrives at the quantization condition ν · α ∈Z, for every root vector α of G .
(18)
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Some pairs of dual groups.
G
˜ G
SU (N ) U (N )
SU (N )/ZN U (N )
SO(2M )
SO(2M )
U Sp(2M ) SO(2M + 1)
SO(2M + 1) U Sp(2M )
Now Eq. (18) is formally identical to the well-known GNO monopole quantization condition,12 as well as to the na¨ive vortex flux quantization rule.13 There is however a crucial difference here, from these earlier results. Because of an exact flavor (color-flavor diagonal GC+F ) symmetry, broken by individual vortex solutions, our vortices possess continuous orientational moduli. These zero modes are normalizable, unlike those encountered in the earlier attempts to define quantum “non-Abelian monopoles”. These non-Abelian modes of our vortices—they are a kind of Nambu-Goldstone modes—can fluctuate and propagate along the vortex length. In systems with a hierarchical symmetry breaking, G0 → G = G × U (1) → 1, where our G = G × U (1) model might emerge as the low-energy approximation, these orientational zero modes get absorbed by massive monopoles at the vortex extremities. This process endows the monopoles with fully quantum-mechanical non-Abelian (GNO-dual) charges, as has been suggested by the author and others in several occasions,16 but we shall not dwell on this subject further here. The solution of the quantization condition (18) is that µ ˜ ≡ ν /2 , is any of the weight vectors of the dual group of G . The dual group, denoted as ˜ , is defined by the dual root vectors12 α G ∗ = α /( α·α ) . Examples of dual pairs ˜ of groups G , G , are shown in Table 1. Note that (17) is actually stronger than (18), the l.h.s. must be a nonnegative integer. This positive quantization condition allows for a few weight vectors only. For concreteness, let us consider scalar fields in the fundamental representation, and choose a basis where the Cartan generators of G = SO(2M ), SO(2M + 1), U Sp(2M ) are given by 1 1 (19) Ha = diag 0, · · · , 0, , 0, · · · , 0, − , 0, · · · , 0 , 2 2 a−1
M−1
with a = 1, · · · , M . In this basis, special solutions H0 have the form for G = SO(2M ) and U Sp(2M ) + + − − (˜ µ ,··· ,˜ µM ) (20) H0 1 = diag z k1 , · · · , z kM , z k1 , · · · , z kM ,
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while for SO(2M + 1) (µ ˜1 ,··· ,˜ µM )
H0
+ + − − = diag z k1 , · · · , z kM , z k1 , · · · , z kM , z k ,
(21)
where ka± = ν ± µ ˜a . For example, in the cases of G = SO(4), U Sp(4) with a ν = 1/2 vortex, there are four special solutions with µ ˜ = ( 12 , 12 ), ( 12 , − 21 ), (− 12 , 12 ), (− 12 , − 12 ) ( 1 , 12 )
H0 2
( 1 ,− 1 ) H0 2 2 (− 1 , 1 ) H0 2 2 (− 1 ,− 1 ) H0 2 2
1
= diag(z, z, 1, 1) = z 2 14 +1·H1 +1·H2 ,
(22)
= diag(z, 1, 1, z) = z
1 2 14 +1·H1 −1·H2
,
(23)
= diag(1, z, z, 1) = z
1 2 14 −1·H1 +1·H2
,
(24)
= diag(1, 1, z, z) = z
1 2 14 −1·H1 −1·H2
.
(25)
These four vectors are the same as the weight vectors of two Weyl spinor represen˜ = SO(4) for G = SO(4), and the same as those of the Dirac tations 2 ⊕ 2 of G ˜ = Spin(5) for G = U Sp(4). spinor representation 4 of G The weight vectors corresponding to the k = 1 vortex in various gauge groups are shown in Fig. 1. In all cases the results found are consistent with the GNO duality. 1 2
− 21
1 2
− 12
SO(2)
USp(2)
( 12 , 12 )
−1
1
SO(3)
(1, 1)
( 12 , 12 )
(1, 0) (0, 0)
SO(4)
USp(4) SO(5)
( 21 , 12 , 12 )
SO(6) Fig. 1.
( 12 , 21 , 12 )
USp(6) The special points for the k = 1 vortex.
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5. Fractional vortices and lumps Another exciting recent result concerns the fractional vortex and lumps.18 We have pointed out above that in a general class of gauge theories the vacuum is not unique, even if the Fayet-Iliopoulos term is present and even if the number of the flavors is the minimum possible for a “color-flavor” locked vacuum to exist. In other words, there is a nontrivial vacuum degeneracy, or the vacuum moduli M. In the first part of the talk, we were mainly interested in the vortex moduli V, in a particular, maximally color-flavor locked vacuum. Here we are going to consider all possible vortices—the vortex moduli V—on all possible points of the vacuum moduli M at the same time. There are in fact two crucial ingredients for the fractional vortex: the vacuum degeneracy and the BPS saturated nature of the vortices. The first point was emphasized just above: the situation is schematically illustrated in Fig. 2. Even if we restrict ourselves to the minimally winding vortex solutions only, the vortices represent non-trivial fiber bundles over the vacuum moduli M. The BPS-saturated nature of the vortices, on the other hand, implies that the vortex equations become first-order equations. The matter equations of motion are solved by the moduli-matrix Ansatz. The other equations–the gauge field equations– reduce, in the strong coupling limit or, anyway, sufficiently far from the vortex center, to the vacuum equations for the scalar fields. In other words, the vortex solutions approximate the sigma model lumps.
Fig. 2.
Vacuum moduli M, fiber F over it, and possible singularities.
5.1. Structures of the vacuum moduli A vortex solution is defined on each point of M, in the sense that the scalar configuration along a sufficiently large circle (S 1 ) surrounding it traces a non-trivial closed orbit in the fiber F (hence a point in M). The existence of a vortex solution
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requires that π1 (F, f ) = 1 ,
(26)
where f is a point in M , the minima of the potential. The field configuration on a disk D2 encircled by S 1 traces M, apart from points at finite radius where it goes off M (hence from M). Thus it represents an element of π2 (M, p), where p is the gauge orbit containing f , or p = π(f ): π is the projection of the fiber onto a point of the base M. The exact sequence of hom*otopy groups for the fiber bundle reads · · · → π2 (M, f ) → π2 (M, p) → π1 (F, f ) → π1 (M, f ) → π1 (M, p) → · · · where π2 (M/F, f ) ∼ π2 (M, p). See Fig. 3.
Fig. 3.
Given the points f, p and the space M, the vortex solution is still not unique. Any exact symmetry of the system broken by an individual vortex solution gives rise to vortex zero modes (moduli), V. Our main interest here however is the vortex moduli which arises from the non-trivial vacuum moduli M itself. Due to the BPS nature of our vortices, the gauge field equation
I = gI2 q † T I q − ξ I , F12
reduces, in the strong-coupling limit (or in any case, sufficiently far from the vortex center), to the vacuum equation defining M . This means that a vortex configuration can be approximately seen as a non-linear σ-model (NLσM) lump with target space M, as was already anticipated. Various distinct maps S 2 → M of the same hom*otopy class correspond to physically inequivalent solutions; each of these corresponds to a vortex with the equal tension I I >0, d2 x F12 Tmin = −ξ by their BPS nature. They represent non-trivial vortex moduli. The semi-local vortices of the so-called extended-Abelian Higgs (EAH) model arise precisely this way. In an Abelian Higgs model with N flavors of (scalar) electrons, M = S 2N −1 , F = S 1 , M = S 2N −1 /S = CP N −1 , and the exact hom*otopy
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sequence tells us that π2 (CP N −1 ) and π1 (S 1 ) are isomorphic: each (i.e. minimum) vortex solution corresponds to a minimal σ-model lump solution. In most cases discussed in our paper,18 however, the base space M will be various kinds of singular manifolds: a manifold with singularities, unlike in the EAH model. The nature of the singularities depends on the system and on the particular point(s) of M. Our BPS degenerate vortices represent (generalized) fiber bundles with the singular manifolds M as the base space. 5.2. Two mechanisms for fractional vortex–lump There are two distinct mechanisms leading to the appearance of a fractional vortex. The first is related to the presence of orbifold singularities in M. For example, let us consider a 2 point p0 such as the one appearing in a simple U (1) model with two scalars, one of which has charge 2. At this singularity, both π2 (M, p) and π1 (F, f ) make a discontinuous change. The minimum element of π1 (F0 , f0 ) is half of that of π1 (F, f ) defined off the singularity, and similarly for π2 (M, p0 ) with respect to π2 (M, p), p = p0 . Even though the exact hom*otopy sequence continues to hold on and off the orbifold point, the vortex defined near such a point will look like a doubly-wound vortex, with two centers (if the vortex moduli parameters are chosen appropriately). Analogous multi-peak vortex solution occurs near a N orbifold point of M. Another cause for the appearance of fractional peaks is simple and very general: a deformed sigma model geometry. This phenomenon can be best seen by considering our system in the strong coupling limit. Even if the base point p is a perfectly generic, regular point of M, not close to any singularity, the field configurations in the transverse plane (S 2 ) trace the whole vacuum moduli space M. The energy distribution reflects the nontrivial structure of M as the volume of the target space is mapped into the transverse plane, C ∂2K I ¯ †J¯ ¯ ∂φ = 2 ∂φ ∂∂K . E=2 I †J¯ C ∂φ ∂φ C The field configuration may hit for instance one of the singularities (conic or not), or simply the regions of large scalar curvature. Such phenomena thus occur very generally if the underlying sigma model has a deformed geometry a . At such points the energy density will show a peak, not necessarily at the vortex center. Even at finite coupling, the vortex tension density will exhibit a similar substructure. 5.3. Some models A simple model showing the fractional vortex is an extended Abelian Higgs model, with two scalar fields A and B with charges 2 and 1, respectively. Depending on the point of M (which is CP 1 ) the minimum vortex shows doubly-peaked substructure a Basically
the same phenomenon was found also by Collie and Tong.19
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clearly, see Fig. 4. The fractional vortex structure in this model nicely illustrates the first mechanism discussed above: the point B = 0 is a Z2 orbifold point, since there the only nonvanishing field, A, having charge 2, must wind only half of the U (1) to be single-valued.
Fig. 4. The energy (the left-most and the 2nd left panels) and the magnetic flux (the 2nd right panels) density, together with the boundary values (A, B) (the right-most panel) for the minimal vortex.
Another interesting model is a U1 (1) × U2 (1) gauge theory with three flavors of scalar electrons H = (A, B, C) with charges Q1 = (2, 1, 1) for U (1)1 and Q2 = (0, 1, −1) for U (1)2 . Even though the model has the same CP 1 as the vacuum moduli M as the first model, the vortex properties are quite different. This model turns out to provide a good example of fractional vortex of the second type (deformed sigma-model geometry). Fractional vortex occurs also in non-Abelian gauge theories, such as the one with gauge group G = SO(N ) × U (1). An illustrative example of fractional vortex in an SO(6) × U (1) model is shown in Fig. 6. 6. Why non-Abelian vortices imply non-Abelian monopoles A more recent work of our research group concerns the monopole-vortex complex solitons occurring in systems with hierarchical gauge symmetry breaking, φ1 =0
G −→ H
φ2 =0
−→
,
|φ1 | |φ2 | .
(27)
The hom*otopy-group sequence · · · → π2 (G) → π2 (G/H) → π1 (H) → π1 (G) → · · · .
(28)
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Fig. 5. (1) F12
(1)
The energy density (left-most) and the magnetic flux density F12 (2nd from the left),
(2nd from the right) and the boundary condition (right-most).
Fig. 6. The energy density of three fractional vortices (lumps) in the U (1)√ × SO(6) √ √ √ model in the strong coupling approximation. The positions are z1 = − 2 + i 2, z2 = − 2 − i 2, z3 = 2. The two figures correspond to two different choices of certain size moduli parameters.
tells us that the properties of the regular monopoles arising from the breaking G → H are related to the vortices of the low-energy system. In particular, the fact that π2 (G) = for any group G, implies that π2 (G/H) ∼ π1 (H)/π1 (G) .
(29)
For instance, in the case of the symmetry breaking, SU (N +1) → SU (N )×U (1)/N the first set of checks (on the Abelian and non-Abelian magnetic flux matching) have been done20 soon after the discovery of the non-Abelian vortex in the U (N ) theory. We wish to study more carefully the monopole-vortex configurations, taking into account a small non-BPS correction term.
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For instance one might study the model based on hierarchically broken gauge symmetry, SU (3) → SU (2) × U (1) → , with the Hamiltonian, 1 1 1 1 H = d3 x (F 3 )2 + 2 (Fij0 )2 + 2 |Di φa |2 + 2 |Di φ0 |2 + |Di q|2 4g 2 ij 4g0 g g0 √ √ + g02 |µφ8 + 2Q† t8 Q|2 + g 2 |µφ3 + 2Q† t3 Q|2 + 2 Q† λ† λQ , (30)
describing the system after the first breaking. Such a low-energy theory is of the type studied in our original work on non-Abelian vortex,8 except for small terms involving φ3 (x) and φ8 (x) (which were set to their constant in that paper). In fact, the model is the same as the one studied by Auzzi et. al. recently.21 The system has unbroken, exact color-flavor diagonal SU (2)C+F symmetry. Neglecting the fields which get mass of the order of the higher symmetry breaking scale, and the fields which go to zero exponentially outside the monopole size, one makes an Ansatz (in the monopole and vortex singular gauge): √ 1 1 A3φ = − f3 (ρ, z), A8φ = − 3 f8 (ρ, z), Aφ = t3 A3φ (ρ, z) + t8 A8φ (ρ, z); ρ ρ
v0 0 φ(r) = 0 v 0 + λ(ρ, z), 0 0 −2v q(x) =
λ(ρ, z) = t3 λ3 (ρ, z) + t8 λ8 (ρ, z) .
w1 (ρ, z) 0 0 w2 (ρ, z)
,
with appropriate boundary conditions. The equations for the profile functions f3 , f8 , w1 w2 , λ3 , λ8 may be studied numerically. Some qualitative features can be read off from the structure of these equations. (i) The Dirac string of the monopole is hidden deep in the vortex core; the zero of the squark field at the vortex core makes the singularity harmless. (ii) The whole monopole-vortex complex breaks SU (2)C+F : the orientational zeromodes develops which live on SU (2)/U (1) ∼ CP 1 . (iii) The degeneracy between the monopole of the broken “U spin” and the monopole of the broken “V -spin”, which are na¨ively related by the unbroken SU (2) of the high-mass-scale breaking SU (3) → SU (2) × U (1), is explicitly broken in the vacuum with small squark VEV. (iv) Nevertheless, there is a new, exact continuous degeneracy among the monopole-vortex complex configurations, related by the color-flavor symmetry (CP 1 moduli). It is possible that such an exact non-Abelian symmetry possessed by the monopole is at the origin of the non-Abelian dual gauge symmetry which emerges at low energies of the softly broken N = 2 supersymmetric QCD.6
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Acknowledgments The main new results reported here are the fruit of a collaboration with M. Eto, T. Fujimori, S. B. Gudnason, T. Nagashima, M. Nitta, K. Ohashi, and W. Vinci. The last part on unpublished work on the monopole-vortex is based on a collaboration with S. B. Gudnason, D. Dorigoni, A. Michelini and M. Cipriani. I thank them all. References 1. G. ’t Hooft, Nucl. Phys. B 79, 817 (1974), A.M. Polyakov, JETP Lett. 20, 194 (1974). 2. A. Abrikosov, Sov. Phys. JETP 32, 1442 (1957); H. Nielsen, P. Olesen, Nucl. Phys. B 61, 45 (1973). 3. N. Seiberg, E. Witten, Nucl. Phys. B 426, 19 (1994); Erratum ibid. B 430, 485 (1994). 4. N. Seiberg, E. Witten, Nucl. Phys. B 431, 484 (1994). 5. P. C. Argyres, M. R. Plesser, N. Seiberg, Nucl. Phys. B 471, 159 (1996); P.C. Argyres, M.R. Plesser, A.D. Shapere, Nucl. Phys. B 483, 172 (1997); K. Hori, H. Ooguri, Y. Oz, Adv. Theor. Math. Phys. 1, 1 (1998). 6. G. Carlino, K. Konishi, H. Murayama, JHEP 0002, 004 (2000); Nucl. Phys. B 590, 37 (2000). 7. A. Hanany and D. Tong, JHEP 0307, 037 (2003) [arXiv:hep-th/0306150]. 8. R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung, Nucl. Phys. B 673, 187 (2003) [arXiv:hep-th/0307287]. 9. D. Tong, “TASI lectures on solitons,” arXiv:hep-th/0509216. 10. M. Shifman and A. Yung, Rev. Mod. Phys. 79, 1139 (2007), [arXiv:hep-th/0703267]. 11. M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, J. Phys. A 39, R315 (2006), [arXiv:hep-th/0602170]. 12. P. Goddard, J. Nuyts and D. Olive, Nucl. Phys. B 125 (1977) 1; F. A. Bais, Phys. Rev. D 18 (1978) 1206; B. J. Schroers and F. A. Bais, Nucl. Phys. B 512 (1998) 250, hep-th/9708004; E. J. Weinberg, Nucl. Phys. B 167 (1980) 500. 13. K. Konishi and L. Spanu, Int. J. Mod. Phys. A18 (2003) 249, arXiv: hep-th/0106175. 14. A. Abouelsaood, Nucl. Phys. B 226, 309 (1983); P. Nelson, A. Manohar, Phys. Rev. Lett. 50, 943 (1983); A. Balachandran, G. Marmo, M. Mukunda, J. Nilsson, E. Sudarshan, F. Zaccaria, Phys. Rev. Lett. 50, 1553 (1983); P. Nelson, S. Coleman, Nucl. Phys. B 227, 1 (1984) 15. N. Dorey, C. Fraser, T.J. Hollowood, M.A.C. Kneipp, “NonAbelian duality in N=4 supersymmetric gauge theories,” [arXiv: hep-th/9512116]; Phys.Lett. B 383, 422 (1996) 16. M. Eto, L. Ferretti, K. Konishi, G. Marmorini, M. Nitta, K. Ohashi, W. Vinci, N. Yokoi, Nucl.Phys. B780, 161-187 (2007), [arXiv: hep-th/0611313]. 17. M. Eto, T. Fujimori, S. B. Gudnason, K. Konishi, T. Nagashima, M. Nitta, K. Ohashi, W. Vinci, JHEP 0906:004 (2009), arXiv:0903.4471 [hep-th]; M. Eto, T. Fujimori, S. B. Gudnason, K. Konishi, M. Nitta, K. Ohashi, W. Vinci, Phys. Lett. B669: 98-101 (2008), arXiv:0802.1020 [hep-th]. 18. M. Eto, T. Fujimori, S. B. Gudnason, K. Konishi, T. Nagashima, M. Nitta, K. Ohashi, W. Vinci, Phys. Rev. D80:045018,2009, arXiv:0905.3540 [hep-th]. 19. B. Collie, D. Tong, JHEP 0908:006,2009, e-Print: arXiv:0905.2267 [hep-th]. 20. R. Auzzi, S. Bolognesi, J. Evslin and K. Konishi, Nucl. Phys. B 686, 119 (2004), [arXiv:hep-th/0312233]. 21. R. Auzzi, M. Eto, W. Vinci, JHEP 0711:090,2007, arXiv:0709.1910 [hep-th].
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Study on Exotic Hadrons at B-Factories Toru Iijima Department of Physics, Nagoya University, Furo-cho, Chikusa-ku Nagoya 464-8602, Japan
The high luminosity e+e- collision at the B-factory experiments (Belle/BaBar) have revealed rich spectra of hadron resonances in the charmonium region. Many of the newly found states do not fit to the unfilled level of the conventional cc spectrum. Some of them are thought to be exotic states, having sub-structure more complex than the quark anti-quark mesons. In fact, Belle has found some states with non-zero electric charge, that require a minimum quark content of ccud . In this paper, we review the present status of studies on such exotic hadrons at the B-factory experiments.
1
Introduction 0B
Thanks to the great success of the B-factory experiments at KEK (KEKB/Belle) and SLAC (PEP II/BaBar), our knowledge on CP violation phenomena has been greatly improved in the past years. The success resulted in the Nobel Prize of Physics in 2008 awarded to Professors Kobayashi and Maskawa. Another success of the B-factory experiments, which is an unexpected bonus, is the discoveries of many new hadronic states, as shown in Figure 1. Of particular interest are a number of charmonium-like meson states, collectively known as “ X Y Z ” mesons. Their measured mass does not fit to the mass spectrum predicted by QCD-motivated conventional quark model calculations. Although they lie above the open charm threshold at 3.73 GeV, they decay preferably to charmonium and one or two pions rather than two charm mesons ( DD ). Because of their peculiar natures, some of them are thought to be “exotics”, and have been extensively discussed in literatures [1]. Quantum Chromodynamics (QCD), the fundamental theory for strong interaction, does not exclude the possible existence of hadrons with a substructure that is more complex than the ordinal three-quark baryons and quark-antiquark mesons. In fact, such quark configurations were considered already in the original Gell-Mann’s paper for the quark model [2]. The possible exotic state includes; • Tetraquark: diquark and anti-diquark state, usually denoted as [Qq ][Qq′] , where Q is the heavy quark. •
Molecule: bound state of two mesons, [Qq ][Qq′] . • Hybrid: bound state of a quark-antiquark pair and a number of gluons. Although the interpretation is uncertain at this stage, the increasing number of XYZ meson states reported by B-factories as well as CLEO and CDF experiments, acquire a lot of interests in the community, together with the pentaquark state Θ+ ( uudd s ) reported first by the LEPS experiment at the Spring-8 [3]. Studies of such exotic hadrons
1
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have now evolved to make an interdisciplinary research area between particle and nuclear physics. In this paper, we review the present status of studies on such exotics hadrons at the Bfactory experiments. Results are shown mainly using the Belle data.
Figure 1: History for discoveries of new hadron resonances at the Belle experiment.
2
Study on exotic hadrons at B-Factories 1B
Such many discoveries of exotic hadrons have been brought by the high luminosity accelerators and the excellent detectors, which are capable to reconstruct all of the interesting reactions exclusively in clean environment of e+e− collisions. In the case of the KEK B-factory, KEKB is an asymmetric energy collider of 8GeV electron and 3.5GeV positron beams, running on the Υ(4S) resonance (center-of-mass energy is 10.58GeV). The peak luminosity has reached 2.1 × 1034cm-2s-1, the world highest luminosity and two times more than the design value. The integrated luminosity by December 2009 is about 950fb-1, among which 710fb-1 is taken on the Υ(4S). This data set corresponds to production of about 800M BB pairs and 960M charm and anti-charm meson pairs. The Belle detector surrounds the interaction point with a large solid angle (about 90% of 4π), and records the production of qq ( q = u , d , s , c, b ) with minimum bias trigger condition. The detector has also excellent particle identification system, which allows one to distinguish the final state particles, e. µ. π. K, p and γ. There are several ways to produce charmonium states at the B-factories (Figure 2); • B decays: the dominant B meson decay occurs through b→cW transition, followed by W → cs . Charmonium is produced when c and c quarks are arranged close enough in the phase space. The s quark is combined with the spectator u or d quark to make the associated K meson. The possible J PC quantum number of the produced cc system is 0-+, 1--or 1++. The well-known X(3872) and Z(4430) were found in this process. • ISR process: in e+e- collision, the initial state e+ and e− occasionally radiate a high energy γ ray, and the e+ and e − subsequently annihilate with a correspondingly
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reduced energy. This produces a charmonium state with J PC=1--. The charmoniumlike states, Y(4260), Y(4360) and Y(4660) were observed by this method. • Double charm production: the Belle collaboration discovered that when a J/ψ is produced in the process, e+e− →J/ψ +anything, the accompanying system contains another cc system with high probability. This process offers a way to search for states with PC = 0+− , 0++ in the missing mass, therefore, independent of the decay mode. • Two-photon collision: charmonium-like state, in J PC = 0- +, 0++, 2++ can be produced also via collisions of virtual photons, each produced by e+ and e − beams. The Z(3940) state was found in this process. More aggressively, one can study exotic hadrons by changing the accelerator energy, as discussed later in Section 4, which describes search for the bottom counterpart of XYZ.
Figure 2: Processes to produce charmonium-like states at B-factories; a) B decays, b) ISR process, c) double charm production and d) two-photon collision.
Status of XYZ
3 2B
Table 1 summarizes the candidate XYZ mesons found by now. Among the 15 claimed states, X(3872) was first discovered by Belle, and has been studied most extensively. There are three candidates of charged states, Z(4430), Z1(4051), Z2(4248). There are also candidates for strangeness and bottom counterparts, Ys(2175) and Yb(10890). In the following, we present status of X(3872) and Z(4430), Z1(4051), Z2(4248).
3.1. X(3872) 6B
The X(3872) was first observed in 2003 by Belle as a narrow peak near 3872 MeV in the π+π− J/ψ invariant mass distribution in B − →K− π+π-J/ψ decays [4]. It was subsequently observed also by CDF [5], D0 [6] and BaBar[7]. The world average values for its mass and width are M = 3872±0.8 MeV and Γ = 3.0 +2.1/-1.7 MeV [8]. Both Belle [9] and BaBar [10] have reported evidence for the radiative transition X(3872) →γ J/ψ, which
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Table 1: Summary of the X Y Z states which have been claimed by now. indicates that the X(3872) has C = +. This implies that the dipion in the X(3872) →π+π− J/ψ decay has C = − , suggesting it originates from a ρ. In fact, the dipion invariant mass distribution measured by CDF is consistent with originating from ρ →π+π − [11]. The decay of a charmonium state to ρ J/ψ would violate isospin, and the evidence for this process provides a strong argument against a charmonium explanation for this state. Angular correlations among the final state particles from X(3872) →π+π− J/ψ rule out all JPC assignments for X(3872) other than J PC = 1++ and 2− + [12]. The possibility to interpret X(3872) as a tetraquark state was proposed by Maiani et al. [13]. This scenario predicts nearly degenerate states including charged states which have not observed yet. It also predicts slight mass difference between two states produced by charged and neutral B decays. The neutral B decay, B0 →X(3872) K0 has been observed by Belle, and the mass difference is found to be ; δMX = M(X from B+) – M(X from B0) = 0.18 ±0.89 ±0.26 MeV, which is consistent with zero, as shown in Figure 3 [14]. This measurement disfavors the tetraquark interpretation. The molecule explanation seems to be the most likely interpretation of the X(3872), due to its proximity to the D0D*0 threshold at 3871.8 ±0.3 MeV. The decay
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Figure 3: Measured J/ψπ+π- invariant mass distribution from charged (left) and neutral (right) B decays.
X (3872) → D 0 D 0π 0 has been seen by Belle [15] and the decay X (3872) → D 0 D 0γ by BaBar [16]. These decay implies that the X(3872) decays predominantly via D 0 D*0 .
3.2. Z+(4430), Z1+(4051), Z2+(4248) 7B
The Belle collaboration has claimed three charmonium-like states with an electric charge, which cannot be conventional cc states. They require the minimum configuration of ccud . They represent clear evidence for some sort of multiquark state, either a molecule or tetraquark. The Z+(4430) was first observed in 2008 in the π+ψ’ invariant mass distribution from the B→K π+ψ’decay [17] in the 6.57M BB sample. More recently, Belle performed a +17 complete Dalitz analysis, which confirms the Z+(4430) with M = 4443+−15 MeV and 12 −13 +57 MeV [18]. The Belle observation was not confirmed by the BaBar analysis Γ = 109 +−86 43 −52
using 4.55M BB sample [19], although two π+ψ’ invariant mass distribution was consistent within statistical errors. The Belle collaboration has also observed two resonance structures in the π+χc1 mass 20 distribution from the the B → Kπ+χc1 decay with M 1 = 4051 ± 14 +−41 MeV, +21 +47 +44 +180 +54 +316 Γ1 = 82 −17 −22 MeV, and M 2 = 4248−29 −35 MeV, Γ 2 = 177 −39 −61 MeV [20].
4
XYZ-like states in the s- and b-quark sectors 3B
Having observed so many interesting charmonium-like states, it would be natural to ask whether or not there are counterpart states in the s- and b-quark sectors. Recent results show that such candidates exist. In a BaBar study of the ISR process e+e− →γ ISR π+π-φ, where the π+π- comes from f0(980) →π+π-, a bump is seen at 2175MeV in the π+π− φ mass distribution [21]. This bump has been confirmed in an ISR measurement by Belle [22] and seen in J/ψ → η f0φ decays by BES [23]. Although a conventional ss assignment cannot be ruled out, this state has properties similar to what one would expect for a s-quark counterpart of the Y(4260).
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Figure 4: a) The π+ψ’ invariant mass distribution from the B→K π+ψ’decay showing the Z(4430) peak (left). b) The π+χc1 mass distribution from the the B → Kπ+χc1 decay showing the Z1(4051) and Z2(4248) peaks.
The Belle collaboration has reported measurements of the energy dependence of the e+e− →π+π− Υ (nS) (n=1,2 and 3) cross section around ECM = 10.9 GeV, and found peaks in all three channels at 10.899 GeV [24]. The mass and width values of the peak are quite distinct from those of the nearby Υ(5S) bottomonium ( bb ) state, and the cross section values are more than two-orders-of-magnitude above expectations for a conventional ( bb ) state, for a conventional bb system. One interpretation for this peak is that it is a b-quark sector equivalent of the 1− − Y states seen in the c-quark sector [25].
5
Summary 4B
The B factory experiments at KEK (KEKB/Belle) and SLAC (PEP II/BaBar) have observed many new hadrons, which cannot be explained by conventional qq meson pictures. The Belle collaboration has observed states with non-zero electric charge, which require the minimum of contents of four quarks ccud .There is some evidence for similar structures in the s- and b-quark sectors. More studies on both experimental and theoretical sides are necessary to interpret these states. Both B factories have large data samples at hand, and will continue providing interesting results in coming years. In future, super B-factories will provide opportunities to search for much more states, and study detailed properties (spin, parity, decay modes etc.) of the observed states.
Acknowledgments 5B
This work is partially supported by a Grant-in-Aid for Scientific Research on Priority Areas “Elucidation of New Hadrons with a Variety of Flavors” from the ministry of Education, Culture, Sports, Science and Technology of Japan.
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Figure 5: The f0(980) φ mass distribution for e+e- →γ ISR π+π-φ (left), and the energy dependence of e+e− →π+π− Υ (nS) (n=1,2 and 3) cross section around ECM = 10.9 GeV.
References 1. Recent review can be found in, for example, Stephan Godfrey and Stephan L. Olsen, Ann.Rev.Nucl.Part.Sci. 58, 51 (2008) , available also as arXiv:0801.3867. 2. M. Gell-Mann, Phys. Lett. 8, 214 (1964). 3. T. Nakano et al. (LEPS Collaboration), Phys. Rev. Lett. 91, 012002 (2003); T. Nakano et al. (LEPS Collaboration), Phys. Rev. C 79, 025210 (2009). 4. S. K. Choi et al. (Belle Collaboration), Phys. Rev. Lett. 91, 262001 (2003). 5. D. E. Acosta et al. (CDF II Collaboration), Phys. Rev. Lett. 93, 072001 (2004). 6. V. M. Abazov et al. (D0 Collaboration), Phys. Rev. Lett. 93, 162002 (2004). 7. B. Aubert et al. (BaBar Collaboration), Phys. Rev. D 71, 071103 (2005). 8. C. Amsler et al. (Particle Data Group), Phys. Lett. B 667, 1 (2008). 9. K. Abe et al. (Belle Collaboration), arXiv: hep-ex/0505037. 10. B. Aubert et al. (BaBar Collaboration), Phys. Rev. D 74, 071101 (2006). 11. A. Abulencia et al. (CDF Collaboration), Phys. Rev. Lett. 96, 231801 (2006). 12. A. Abulencia et al. (CDF Collaboration), Phys. Rev. Lett. 98, 132002 (2007). 13. L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, Phys. Rev. D 71, 014028 (2005). 14. I. Adachi et al. (Belle Collaboration), arXiv: hep-ex/0809.1224. 15. G. Gokhroo et al. (Belle Collaboration), Phys. Rev. Lett. 97, 162002 (2006). 16. B. Aubert et al. (BaBar Collaboration), Phys. Rev. D 77, 011102 (2008). 17. S. K. Choi et al. (Belle Collaboration), Phys. Rev. Lett. 100, 142001 (2008). 18. R. Mizuk et al. (Belle Collaboration), Phys. Rev. D 80, 031104 (2009). 19. B. Aubert et al. (BaBar Collaboration), Phys. Rev. D 79, 112001 (2009). 20. R. Mizuk et al. (Belle Collaboration), Phys. Rev. D 78, 072004 (2008). 21. B. Aubert et al. (BaBar Collaboration), Phys. Rev. D 74, 091193 (2006). 22. I. Adachi et al. (Belle Collaboration), arXiv: hep-ex/0808.0008. 23. M. Ablikim et al. (BES Collaboration), Phys. Rev. Lett. 100, 102003 (2008). 24. I. Adachi et al. (Belle Collaboration), arXiv: hep-ex/0808.2445. 25. W. S. Hou, Phys. Rev. D 74, 017504 (2006).
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Cold Compressed Baryonic Matter with Hidden Local Symmetry and Holography Mannque Rho Institut de Physique Th´ eorique, CEA Saclay, 91191 Gif-sur-Yvette C´ edex, France and Department of Physics, Hanyang University, 133-791 Seoul, Korea E-mail: [emailprotected] http://ipht.cea.fr, //hadron.hanyang.ac.kr I describe a novel phase structure of cold dense baryonic matter predicted in a hidden local symmetry approach anchored on gauge theory and in a holographic dual approach based on the Sakai-Sugimoto model of string theory. This new phase is populated with baryons with half-instanton quantum number in the gravity sector which is dual to halfskyrmion in gauge sector in which chiral symmetry is restored while light-quark hadrons are in the color-confined phase. It is suggested that such a phase that aries at a density above that of normal nuclear matter and below or at the chiral restoration point can have a drastic influence on the properties of hadrons at high density, in particular on shortdistance interactions between nucleons, e.g., multi-body forces at short distance and hadrons – in particular kaons – propagating in a dense medium. Potentially important consequences on the structure of compact stars will be predicted.
1. The Issue Baryonic matter near the nuclear matter density n0 ≈ 0.16 fm−3 is very well understood, thanks to many years of nuclear experimental and theoretical efforts, but there is a dearth of experimental information beyond n0 . Unlike in high temperature where lattice QCD calculations come in tandem with experiments performed at relativistic heavy-ion colliders, there are no reliable theoretical tools available for dense matter for which lattice gauge calculation suffers from the notorious sign problem. Thus beyond n0 , one knows very little of what’s going on. At asymptotically high density, perturbative QCD with weak coupling allows a clear-cut prediction, but the density required there is so high that it is hardly likely to be relevant to the physics of dense matter in terrestrial laboratories or in compact stars. A large number of model calculations are nonetheless available in the literature with a plethora of predictions for nucleon stars, quark stars etc., but the problem with these predictions is that constrained or aided neither by experiments nor by theory, it is difficult to assess the reliability of the calculations with wildly varying results at densities going beyond that of the nuclear matter. The prospect for the future, however, is pretty good, in particular experimentally. Indeed the forthcoming accelerators devoted to the physics of dense matter,
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such as FAIR/GSI, will help build realistic phenomenological models just like the ones in standard nuclear physics where the wealth of accurate data played a pivotal role in arriving at realistic nuclear structure models with no guidance from the fundamental theory of strong interactions QCD. It is now established, thanks to the recent development in QCD-based effective field theories of nuclei and nuclear matter, that what the nuclear physicists have been doing was on the right track all along. While awaiting the advent of experimental inputs for going toward dense matter, the challenge for the theorists is: Make predictions, relying solely on theoretical schemes that are as faithful as possible to QCD proper, for the poorly understood regime between the normal nuclear matter density n0 and the chiral transition density nχ – at which the restoration of chiral symmetry is expected to take place. In this talk, I would like to describe one such (theoretical) effort being carried out in the context of the World Class University (WCU) Program at Hanyang University in Seoul, Korea, that is anchored on the exploitation of the notion of hidden local symmetry in strong interactions and its holographic generalization following from gauge/string duality. What I will describe is just the beginning. As such, what’s predicted at present cannot be immediately confronted with nature. But it will be easily subjected to confirmation or falsification when the data become available. This talk is not strictly in the main line of this conference where the principal focus is on strong coupling phenomena in the LHC era. My talk concerns rather physics under extreme conditions at a much lower energy scale which will be probed at other laboratories. The basic concept involved, however, such as hidden local symmetry and gauge/string duality, turns out to be closely related to the main theme of this conference. 2. Hidden Local Symmetry (HLS) 2.1. HLS Lagrangian Hidden local symmetry (HLS) in the strong interactionsa is a flavor gauge symmetry, the effective degrees of which consist of Goldstone bosons (or more realistically pseudo-Goldstone bosons in Nature), i.e., the pions (π = 12 τ · π ) and the vector µ + 12 ωµ ), which elevates the energy scale fields Vµ valued in U (2) (i.e., Vµ = 12 τ · ρ of the chiral symmetric interactions from the current algebra scale with soft pions to the vector-meson scale mV ∼ 800 MeV. There are two ways to view how hidden gauge symmetry in the strong interactions arises. One is to view the local symmetry as an “emergent” symmetry that exploits the redundancy of local gauge symmetry and the other is to view it as descending (via Kaluza-Klein reduction) from higher dimensions. In Nature, in addition to the pions and the vector mesons Vµ , there are other light mesonic excitations such as scalar mesons and axial-vector mesons. As for the baryons – that are indispensable for dense matter, they will be generated as a Here I will be focusing mainly on the two-flavor (u,d) case in the chiral limit, i.e., with zero quark mass, but will introduce the strangeness (s) quantum number as needed.
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solitons from the HLS Lagrangian, instead of positing them as is done in standard treatments. This approach is most natural from the point of view of large Nc QCD. Now limited to the minimum number of degrees of freedom, the gauge-invariant Lagrangian takes the simple formb Fπ2 γ 1 Tr {|Dµ ξL |2 + |Dµ ξR |2 + |Dµ U |2 } − Tr [Vµν V µν ] + · · · (1) 2 2 2 π σ π σ † ξR , ξL = e−i Fπ ei Fσ , ξR = ei Fπ ei Fσ , Dµ ξ is the covariant where U = e2iπ/Fπ = ξL derivative, and γ = 1/a − 1 with a ≡ Fσ2 /Fπ2 . In this form, this is an elegant, though highly nonlinear, Lagrangian. However being an effective Lagrangian defined with a cutoff, higher derivative terms at loop orders indicated by the ellipsis in (1) must figure at the quantum level. In matter-free space, namely, at zero temperature (T = 0) and zero density (n = 0), the HLS Lagrangian is gauge-equivalent to the standard and well-tested chiral Lagrangian with the pions alone, and as such leads to the same chiral perturbative procedure for pion interactions at very low energy as the standard (pion-only) chiral Lagrangian.1 At tree order, the hidden local symmetry is of no special power or advantage; gauge fixed to unitary gauge, it will simply reproduce the current algebra results of the standard chiral Lagrangian. The local symmetry, however, picks up its power at higher loop orders since it then renders – in principle – feasible a systematic power counting including the vector mesons, a task that is highly cumbersome if not impossible in a gauge-fixed theory.2 What is perhaps not properly appreciated among nuclear theorists investigating hadronic matter under extreme conditions (at high T and/or high n) is that hidden local symmetry can incorporate with ease certain properties of hadronic matter that involve highly correlated many-body interactions that are difficult to access by models that do not possess flavor local symmetry. The purpose of this talk is to discuss what transpires when HLS in its simplest form and in certain approximations is applied to cold dense matter beyond the nuclear matter density. I will present certain intriguing predictions that differ from standard phenomenological many-body models found in the literature. It has been shown at one-loop order in matter-free space that the HLS theory matched to QCD (in terms of correlators) flows to a unique fixed point called “vector manifestation” when temperature or density approaches the critical point where chiral restoration takes place1c . In the presence of dense matter, the parameters of the Lagrangian governed by renormalization group equations flow as density increases.d When matched to the QCD vector and axial vector correlators via the L=
b In
what follows, the parametric pion decay constant will be denoted Fπ while the physical pion decay constant will be written as fπ . c It is easy to see that this statement should hold to higher loop orders although there is no mathematically rigorous proof. In practice, higher orders are not very useful at present in view of the large number of unknown parameters that arise at higher orders. This means that the results I will show cannot be taken to be quantitatively accurate, even if qualitatively sound. d Unless specifically mentioned otherwise, similar arguments apply to temperature. In contrast to temperature, to take into account the density, fermion degrees of freedom, here baryons, need be introduced. This problem will be addressed below.
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in-medium OPE, that is, with the quark condensate ¯ q q and the gluon condensate G2 endowed with background density dependence, the parameters of the HLS Lagrangian (1) tend, as density approaches the chiral transition density nχ , to g ∝ ¯ q q → 0, a ≡ Fσ /Fπ → 1
(2)
where the asterisk stands for density dependence. In contrast, there turns out to be nothing special with the flow of the decay constants Fσ,π for varying densities although the physical decay constant fπ , which receives radiative corrections, should go to zero as ¯ q q goes to zero. The consequence of the VM point (2) is that the parametric mass mV vanishes mV ≈ a Fπ g ∝ g → 0.
(3)
Equations (2) and (3) are the essential ingredients of the hidden local symmetry theory with the VM – denoted “HLS/VM” for short – that I will resort to in what follows. Now the important practical question is: Where and how are these fixed point values manifested? In other words, how can one “see” the presence of these fixed points that are the potential signals for the chiral symmetry structure of the hadronic vacuum? This represents one key question nowadays asked by nuclear physicists. To answer this question, one should recognize the caveats in applying this Lagrangian to Nature. First of all, the Lagrangian itself is a highly truncated one, with the minimum number of relevant degrees of freedom, so cannot be expected to work accurately for certain processes that require other degrees of freedom. Secondly, to confront experimental observables, dense loop corrections must be calculated in conformity with chiral perturbation theory, what is most important, with the parameters of the Lagrangian running with density and satisfying thermodynamic and symmetry constraints. This means that the signals of HLS/VM cannot in general be singled out in a simple manner. They are likely to be compounded with density dependence brought in by dense loop corrections. As in the case of hightemperature processes in relativistic heavy-ion collisions, e.g., dilepton production, a careful weeding-out of “trivial effects” of many-body nature will be required before one can see the desired effects. This issue is discussed in detail in .3 2.2. Gauge-fixed Lagrangian I will take the Lagrangian (1) in unitary gauge. It has the form Lhls =
fπ2 f2 Tr(∂µ U † ∂ µ U ) − π aTr[ µ + rµ + i(g/2)(τ · ρ µ + ωµ )]2 4 4 − 41 ρµν · ρ µν − 14 ωµν ω µν + · · · ,
(4)
with U = exp(iτ · π /fπ ) ≡ ξ 2 , µ = ξ † ∂µ ξ, and rµ = ξ∂µ ξ † , ρ µν = ∂µ ρν − ∂ν ρ µ + g ρµ × ρν and ωµν = ∂µ ων − ∂ν ωµ . Because of the presence of the isoscalar vector
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meson ω, there is an additional Lagrangian which is parity-odd LhW Z = + 23 gωµ B µ
(5)
with the baryon current 1 µναβ ε Tr(U † ∂ν U U † ∂α U U † ∂β U ). (6) 24π 2 This is a part of the gauged Wess-Zumino term that survives for Nf < 3 while the topological 5D Wess-Zumino term is absent in the two-flavor case I am discussing. In what follows, it will be called “hWZ term” – “h” standing for hom*ogenous. In general the hWZ Lagrangian has four (hom*ogeneous) terms with arbitrary coefficients but if one imposes vector dominance in the isoscalar channel and consider the ρ meson to be so massive that one can substitute ρaµ = iTr [τ a (lµ + rµ )], then it reduces to one term of the form (5).5e Bµ =
2.3. Describing dense matter The Lagrangian L = Lhls + LhW Z contains no explicit baryon degrees of freedom. However there is a topological baryon current Bµ in (5), so one can think of the ω field as a chemical potential conjugate to the baryon charge. I will parallel this structure below to a holographic dual form in 5D where a similar structure arises. One might attempt to describe dense baryonic matter with (4) and (5) in the mean field. However this approach simply does not work in the absence of explicit baryon degrees of freedom. One natural way is to generate skyrmions as solitons from the Lagrangian. In principle, there can be soliton solutions with baryon number going from 1 to infinity, the latter corresponding to nuclear matter. Indeed, there have been attempts to build finite nuclei with mass number up to, say, ∼ 27 which would correspond to a skyrmion with baryon charge B = 27.6 The resulting theory with skyrmions coupled to fluctuating vector and pion fields, properly quantized, could be expressed in terms of local baryon fields coupled to the mesons satisfying the same symmetry structure as the starting theory. This is in the same spirit as the baryon chiral Lagrangian where local baryon fields are coupled to pions in a chirally invariant away, which is the basis of chiral perturbation theory for pion-nucleon interactions. Given such a baryon chiral Lagrangian, then one can do a mean field calculation for many-body systems that would correspond to what is called Walecka mean field theory in nuclear physics. This strategy, backed by experimental data, can be found to be successful up to the density for which data are available. But how to apply it to a matter beyond the nuclear matter density is e There are two caveats in this. One is that vector dominance is most likely violated with a approaching 1 in hot and dense matter.4 The second is that the “large ρ mass limit” is at odds with the VM where the vector meson mass goes to zero. These two caveats can be avoided if one works with all the hWZ terms which make the calculation considerably more involved. The merit of the form (5) is that it is consistent with the Chern-Simons term in 5D that I will consider in the holographic approach discussed below.
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not known since there have been no experiments that probed the given high density regime. This is because at higher densities, higher dimension operators must enter, which correspond, in the localized form, to rendering the constants of the mean field Lagrangian density-dependent and the density dependence is not known by theory alone. What is in principle more powerful and closer to QCD proper is to treat manybody systems in terms of the solitonic solution of the Lagrangian. The soliton of winding number A is the solution for an A-body baryonic system. The system with A = ∞ will then be a nuclear matter. Nobody has been able to perform an analytic or continuum calculation of such multi-body systems but there have been a variety of calculations putting the skyrmions on a crystal lattice. At high density, one expects the system to be in a crystal and so this is a natural approach to the problem although at lower density nuclear matter is known to be in a liquid than in a crystal. In fact, it has been established that nearly independently of detailed structure of the Lagrangian, there is a phase change from a matter of skyrmions at low density to a matter consisting of half-skyrmions at higher density,8 and this phase change takes place almost independently of whether the Lagrangian contains local gauge fields or not. The resulting half-skyrmion phase possesses certain scale invariance. What could be different is the critical density at which this takes place, which depends on the parameters of the Lagrangian as well as the degrees of freedom involved. The crystal structure of dense skyrmion matter with a variety of different ansatze for the skyrmoion configuration is reviewed recently by Park and Vento.7 For latter purpose, I will restrict myself to the approach9 based on the Atiyah-Manton ansatz,10 since it is closely connected to the instanton picture that arises in 5D Yang-Mills Lagrangian of holographic dual QCD. The Atiyah-Manton ansatz for the chiral field U is given by the holonomy in (Euclidean) time direction ∞ A4 (x, t)dr C † (7) U (x) = CS P exp − −∞
where A4 is the (Euclidean) time component of an instanton field of charge B – which is A for the A-baryon system, P is the path ordering, S is a constant matrix to make U approach 1 at infinity and C denotes an overall SU (2) rotation. The hom*otopy assures that the static soliton configuration carries the same baryon number as the total charge of the instanton. Note that here YM gauge field Aµ is a fictitious field, not present in the theory (1). In holographic QCD described with a 5D YM field, A4 is the gauge field present in the 5th direction in the theory. In,9 the ansatz (7) is employed in putting A instantons in an FCC crystal to compute the ground state energies as a function of the crystal lattice size Lf . By shrinking the size L, the multi-skyrmion system is by fiat compressed. In Nature, f In
FCC, the baryon number density is given by n = 1/2L3 .
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gravity does the job of squeezing. In Fig. 1 is shown the local baryon number density profile before and after the splitting. These results are obtained with the Skyrme Lagrangian with massive pions but the generic features are the same with the HLS Lagrangian (1). What is important with this phase change is that the half-skyrmion q q = 0 phase has ¯ q q ∝ TrU = 0 but fπ = 0 whereas the skyrmion phase has ¯ and fπ = 0, which is symptomatic of chiral symmetry in the Nambu-Goldstone mode. What is novel here is that the chiral symmetry is “restored” whereas there is propagating pion with non-vanishing decay constant. This means that quarks are still confined in that phase although chiral symmetry is “restored.” This feature is quite robust. This result implies that in the half-skyrmion phase, the nucleon mass going roughly like fπ will not drop appreciably as does the vector meson mass. This will have an important consequence on nuclear physics at high density. I should mention that a similar half-skyrmion phase structure is found in condensed matter physics (in 3D).11 There the half-skyrmions get deconfined, that is, they are unbound. In the strong-interaction case at hand, one cannot say whether the half-skyrmions are bound or not g .
Fig. 1. Local baryon number densities at L = 3.5 and L = 2.0. For L = 2.0 the system is a half-skyrmion (or nearly half-skyrmion with mass) in a CC crystal configuration.
2.4. Meson fluctuations and dilatons Given the background with skyrmions and half-skyrmions, the interesting question is how mesons behave in dense medium. This question has been addressed with the Lagrangian given by Eqs. (4) and (5). What is found there is illuminating: It g In
holographic QCD, it appears that the half-instantons are bound although maximally separated by Coulomb repulsion.12
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indicates that without additional degree(s) of freedom, the VM property of HLS fails to manifest with the unitary gauge Lagrangian in medium.13 It is found that a dilaton field is rquired.14 What happens to meson properties in dense medium is crucially controlled by the hWZ term (5). In mean field in medium, it contributes LhW Z = + 23 gω0 (x)n(x)
(8)
where n(x) is the nuclear matter distribution. Now the ω meson gives rise to a Coulomb potential, so the hWZ term leads to a repulsive interaction. What is most important is that this repulsive interaction turns out to dominate over other terms, and the repulsion diverges as density increases. In order to prevent the energy per baryon coming from the hWZ term from diverging, m∗ω has to increase sufficiently fast. And since m∗ω ∼ fπ∗ g in this model, for a fixed g, fπ∗ must therefore increase. This is at variance with Nature: QCD predicts that it should decrease and go to zero – in the chiral limit – at the chiral transition. The problem here is that the VM property – that g → 0 as density approaches the critical – is missing in this gaugefixed mean field approach. Stated concisely, the presence of the ω field prevents both the pion decay constant fπ and the ω mass from decreasing as demanded by the HLS/VM. The quark condensate ¯ q q does ultimately vanish at a critical density but with an increasing fπ . The ω field thus brings havoc in the theory. This defect can be remedied by the dilaton field associated with the trace anomaly of QCD14h . In,14 two-component scalar fields denoted as “soft” χS and “hard” χH satisfying the QCD trace anomaly were introduced. Roughly speaking the soft component represents the scalar that is locked to the quark condensate, hence directly connected to chiral symmetry, while the hard component represents the glue-ball degrees of freedom tied to the scale-invariance breaking associated with the asymptotic running of the color coupling constant gc . Given that χH involves high-energy excitations at a scale much greater than that of χS , it can be integrated out for our purpose. Thus retaining only the χS field, one can modify Eqs. (1) and (5) so that the trace anomaly is suitably implemented. How this can be done in consistency with the QCD trace anomaly is described in.16
h There
is no elegant and simple way known of introducing scalar fields in the HLS theory. What I describe here may not be the most efficient way for HLS per se but it is one simple and elegant way for an effective field theory. The role of the dilaton field χS in what is referred to as Brown-Rho scaling15 has been incorrectly understood by numerous authors who have argued against it on theoretical grounds. Although it is not in the main line of discussions of this talk, it is worth mentioning here that the dilaton introduced therewith resolves one embarrassing feature of the Skyrme model for the nucleon, that is that, with the parameters of the Lagrangian given by the meson sector, the soliton mass comes out much too high, say, ∼ 1.5 GeV. When the dilaton is put with a relatively low mass, say, < ∼ 1 Gev, then the soliton mass drops to near the physical nucleon mass. The scalar degree of freedom carries a strong attraction that cannot be ignored in the single baryon as well as baryonic matter structure.
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With the dilaton included consistently with the scale anomaly, the effective HLS Lagrangian in unitary gauge is L = Lhlsχ + +LhW Z
(9)
where LHLSχ =
fπ2 f2 (χ/fχ )2 Tr(∂µ U † ∂ µ U ) − π a(χ/fχ )2 Tr[ µ + rµ + i(g/2)(τ · ρµ + ωµ )]2 4 4 − 14 ρ µν · ρ µν − 14 ωµν ω µν + 12 ∂µ χ∂ µ χ + V (χ) (10)
LhW Z = 32 g(χ/fχ )n ωµ B µ
(11)
with χ ≡ χS where fχ is the χ decay constant and V (χ) = Bχ4 ln fχ eχ1/4 is the potential that gives the energy-momentum tensor whose trace gives the soft component of the trace anomaly. The exponent n which cannot be fixed by first principles can be arbitrary with n > 1 without violating scale symmetry.16 In the analysis of,14 n = 3 was taken for simplicity, but n = 2 gives an equally satisfying result at least qualitatively. The lesson from this model is that the dilaton field plays a crucial role in making HLS theory compatible with nature. It makes both the ω meson mass and the pion decay constant decrease as density increases in consistency with HLS/VM. An additional important consequence of the dilaton is that it reveals a novel phase with half-skyrmions as the relevant degrees of freedom where the quark condensate vanishes, ¯ q q = 0, while fπ = 0. This phase appears at a density n1/2 above the nuclear matter density n0 but below the chiral transition density nχ . This phase is characterized by the restored chiral symmetry but with quarks confined. At what density this phase appears depends on the parameters of the Lagrangian that defines the background but not on the effective mass of the dilaton. 3. Holographic Dense Matter Before going to describing the predictions of the new structure uncovered in HLS/VM, let me describe the approach taken from the gravity dual of dense matter in a holographic QCD model. There is a huge literature on the approach to dense and hot matter from the AdS/CFT angle which I cannot possibly review here. Let me focus uniquely on a particular holographic QCD model that is closely related to what is described above in HLS/VM, i.e., the Sakai-Sugimoto model.17 This is the only model that I know of that has the chiral symmetry of QCD in the chiral limit. It describes well – in the large Nc and large ’t Hooft constant λ limit where the duality should be reliable and in the probe approximation with Nf /nc 1 – both meson properties17 and baryon, specially nucleon, properties18 which are known to be well described in the quenched approximation in lattice calculations. What is particularly relevant to this discussion is that the model gives a fairly accurate description of nucleon static properties and provides the first “derivation” of the vector dominance of the nucleon form factor as emphasized in.19 The latter resolves
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a long-standing puzzle why Sakurai’s vector dominance is strongly violated in the nucleon form factors while it works very well in the meson form factorsi . 3.1. Sakai-Sugimoto model The Sakai-Sugimoto action17 valid in the large Nc and λ limit and in the probe (or quenched) approximation can be reduced to the form S = SY M + SCS where SY M = −
dx4 dw
1 2e2 (w)
trFmn F mn + · · · ,
(12)
(13)
with (a, b) = 0, 1, 2, 3, w is the 5D YM action coming from DBI action where the contraction is with respect to the flat metric dxµ dxµ + dw2 . The position-dependent electric coupling e(w) is given in terms of Nc , λ and the Kaluza-Klein mass MKK (see Hong et al in18 ) and Nc SCS = ω5 (A) (14) 24π 2 4+1 with dω5 (A) = trF 3 is the Chern-Simons action that encodes anomalies. There are two important features in this hQCD model. One is that an action of the same form arises as an “emergent” gauge action from low-energy chiral Lagrangians with the 5th dimension “deconstructed”.2 The other is that when viewed in 4D by KaluzaKlein reduction, it contains an infinite tower of both vector and axial-vector mesons. Now with the low-energy strong interaction dynamics given in the form (12), baryons arise as instantons from SY M whose size shrinks to zero as ∼ λ−1/2 as λ → ∞.18 The shrinking to zero size of the instanton is prevented by the Chern-Simons term which takes the form in the presence of the instanton Nc Aˆ0 tr(F ∧ F ) (15) SCS = 8π 2 where Aˆ is the U (1) gauge field (analogous to the abelian gauge field, ω, in HLS). The Chern-Simons term provides a Coulomb repulsion that stabilizes the instanton just as the ω field does to the skyrmion in HLS. When applied to dense matter, the instanton density distribution tr(F ∧ F ) in (15) is nothing but the baryon number density distribution and hence Aˆ0 can be interpreted as a variable conjugate to density, i.e., the baryon chemical potential µB . One can have a simple but instructive idea of what this theory contains for dense matter by noting the close parallel between (4) in 4D and (13) in 5D and between
iI
should point to the remarkable observation by Hashimoto et al. in18 that the form factor of the nucleon vector-dominated by the infinite tower of vector mesons can be re-expressed by a dipole form as found empirically!
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(5) in 4D and (15) in 5D. The instanton in (13) stabilized by the Aˆ0 field in the CS term (15) is an analog of the skyrmion in (4) stabilized by the ω field in (5). This suggests that if the action (12) is treated in mean field for the point-like instantons as baryons balanced by the Coulomb force given by the CS term, there will be the same defect found with the HLS theory without the dilaton degree of freedom that simulates the HLS/VM property. One expects that the pion decay constant in this holographic theory will increase instead of decrease with increasing density. Indeed this was found in a recent analysis of the action (12) in cold dense matter.20 3.2. Half-instanton/dyonic phase There is at present no quantitative work in this holographic QCD model of the sort described above for the crystal structure in HLS/VM. But one can reveal a phase that is a gravity dual to the half-skyrmion phase. Given that the action (12) can support instanton solutions for many-baryon systems with the instanton number A, the pertinent question one can ask vis-` a-vis with dense matter is what happens to the instanton at high density. This question was recently answered with the discovery that there can arise a half-instanton matter that is in a dyonic salt form in which chiral symmetry is restored while the baryons are color-singlet hadrons, very much like in the half-skyrmion matter. In fact, the two half-instantons fractionized from one instanton are found to be bound in a bcc configuration, separated by Coulomb repulsion. There is no quantitative description of the dyonic salt configuration at the moment but one can make a plausible qualitative argument using geometry, which, I believe, is robust.12 The basic idea is that the flavor instanton in the action (12) consists of a superposition of constituents that are BPS dyons in the leading Nc λ order. In close analogy to colorons21,22 with the colored instanton splitting into constituent dyons as a mechanism for color deconfinement where the Polyakov line plays the central role, here it is also a nontrivial holonomy that works to split the flavor instanton into BPS dyons of opposite charges e = g = ±1/2. In,12 geometric reasoning is used for the splitting phenomenon and a dynamical mechanism with topological repulsion balancing the Coulomb forces mediated by the charges of the constituents is invoked for the half-and-half non-BPS structure. It is estimated that the splitting occurs at a density comparable to what was observed in the skyrmion-half-skyrmion transition7j . Also the binding energy between the two dyons of opposite charges comes out roughly about 180 MeV, similar to the binding of half-skyrmions in the gauge model.7 The arguments leading to the instanton-dyon phase transition is admittedly heuristic and needs to be supported, in the absence of analytical method, by nuj In
the skyrmion case, the transition density n1/2 was found to be sensitively dependent on the parameters of the Lagrangian. For the parameters fixed in the meson sector, it comes out to be somewhere between ∼ 1.3 times and ∼ 3 times the nuclear matter density. This range is roughly what is found in this hQCD model.
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merical work. But there is no reason to think it is not qualitatively correct. It is easy to understand that with the action (12), were it not for the warping in the conformal direction w in the charge e(w), the constituents of the instanton would be BPS dyons, and hence there would be no reason why the instanton should split into two equal 1/2 instantons. It is the interaction encoded in the curved geometry that turns the BPS dyons with the instanton charge partitioned in v and 1 − v with v arbitrary into the half-and-half configuration. In this connection, what is highly relevant is the observation made by Hashimoto et al that at short distances, the curvature effect can be ignored and many-body forces between instantons get suppressed.23 In the standard nuclear physics picture, short-range repulsion in three-body forces (ignoring four-body forces) is understood to be responsible for pushing phase transitions (e.g., kaon condensation, chiral restoration etc.) in nuclear matter to much higher densities. If holographic duality were to hold in this phenomenon, this feature of many-body forces in the gravity description would imply in the gauge sector that the many-body repulsion problematic in standard nuclear theory is simply absent at high density. This would have an extremely important implication on the EOS of compact stars. In HLS, many-body forces are also suppressed by the vector manifestation of the vanishing hidden gauge coupling, thus making a valuable prediction that is inaccessible to QCD proper. The two predictions are qualitatively the same although they seem to involve different mechanisms, RG flow matched to QCD for the gauge sector and the infinite tower encoded in the instanton structure in the gravity sector. It would be interesting to “see” the connection if there is any.
4. Signals of Half-Skyrmion/Dyonic Phase The search for direct signals of the HLS/VM has been mainly focused on dilepton production in relativistic heavy ion collisions and in EM interactions. The former is for high temperature matter and the latter precision measurement in finite nuclei at nuclear matter density and at zero temperature. The expectation was that via vector dominance, dileptons will be profusely produced from the ρ vector meson in medium with its mass shifted downwards due to the HLS/VM. The searches so far made, however, came out more or less negative, that is, the mass shift associated with the HLS/VM has not been observed. It should, however, be pointed out that this by no means implies that the HLS/VM prediction is absent in nature. In fact, one of the most remarkable – and unexpected – predictions of HLS/VM, which is found in none of the treatments by the theorists in the field, is that vector dominance is maximally violated as a → 1 and g → 0, and the dileptons, ironically, tend to largely decouple from the ρ vector meson, so the observed dileptons carry practically no information on chiral symmetry properties. In fact, what’s observed in the ρ spectral function are predominantly mundane nuclear effects due to many-body interactions and do not exhibit clear-cut signals for the chiral structure of the vacuum change that one wants to isolate. The same suppression mechanism takes place both in T and in density. This could indicate that singling out of a “smoking-gun” signal for
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the manifestation of chiral symmetry in dense (and hot) medium at the forthcoming laboratory FAIR/GSI will not be an easy task. Let me turn to a case which appears to be more promising, i.e., the property of kaons in cold dense matter. Particularly interesting is the behavior of negatively charged kaons (i.e., anti-kaons) fluctuating in the medium described as a dense solitonic background. This concerns two very important issues in physics of dense hadronic matter. One is the possibility that K − can trigger strongly correlated mechanisms to compress hadronic matter in finite nuclei to high density as proposed by Akaishi et al24k . The other issue is the role of kaon condensation in dense compact star matter which has ramifications on the minimum mass of black holes in the Universe and cosmological natural selection.26 It would be of the greatest theoretical interest to address this problem by means of the instanton matter given in hQCD, which has the potential to also account for shorter-distance degrees of freedom via an infinite tower of vector and axial vector mesons. However, no numerical works in this framework are presently available. I shall therefore take the simple Skyrme model with the vector field Vµ integrated out, implemented with three important ingredients, viz, the Wess-Zumino term, the soft dilaton field χ that accounts for the scale symmetry tied to spontaneously broken chiral symmetry as explained above and the kaon mass term. Though perhaps oversimplified, I expect it to capture the generic qualitative feature of the novel phenomenon. The objective is to see how kaons behave on top of the background provided by the skyrmion matter as density is raised. I will assume that the back-reaction of the kaon fluctuation on the background is ignorable. The basic idea is to treat the kaon as “heavy,” not light as in chiral perturbation theory, and wrap it with the skyrmion in the way that Callan and Klebanov did for hyperons.27 The appropriate ansatz is l U (x, t) = UK (x, t)U0 (x) UK (x, t), (16) where UK and U0 are, respectively, the fluctuating kaon (doublet) field and the SU (2) soliton field. The results of this model recently worked out in29 are summarized in Fig. 2. The mass of the dilaton that figures importantly in this approach is presently unknown. What’s taken here are two values, one corresponding to the lowest observed scalar mass ∼ 600 MeV and the other a value that is used in mean-field theory calculations of nuclear matter using an effective Lagrangian that implements Landau Fermi liquid theory incorporating HLS/VM. Noteworthy in these results is that up to
kI
must mention that there is a considerable controversy on this matter, pros and cons to the Akaishi-Yamazaki idea. For the most recent summary of both theoretical and experimental status, see.25 Ultimately experiments will be the jury, and the verdict will have to await the results of those experiments that are being – and will be – performed. l This ansatz is more convenient for treating kaon fluctuations than the one taken by Callan and Klebanov where the kaon field is sandwiched by the square-root of the soliton field. The two ansatze are found to give an equally good description of the hyperons.28
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m∗K ± vs. n/n0 in dense skyrmion matter that contains three phases as indicated, with √ the middle being the half-skyrmion phase. The parameters are fixed at fπ = 93 MeV, 2efπ = mρ = 780 MeV and dilaton mass mχ = 600 MeV (left panel) and 720 MeV (right panel).
Fig. 2.
the threshold of the half-skyrmion phase, the antikaon mass behaves more or less according to chiral perturbation theory;25 however once the half-skyrmion phase sets in, the kaon mass drops more steeply, going to zero at about the same density as the chiral transition density nχ ∼ (2.3 − 4)n0 . This strong attraction that results from the combination of the topological WZ term and the dilaton field is most probably inaccessible in standard chiral perturbation theory, and hence missing from the chiral perturbation treatments in the literature. This additional attraction could help lead toward the Akaishi-Yamazaki scenario of deeply bound kaonic nuclei. Implementing the mechanism found here in finite nuclei is not an easy task that needs to be worked out. Another physically interesting question is what does the half-skyrmion matter do to compact stars? What we have done above is to subject the kaon to the skyrmion background given in the large Nc limit. In the large Nc limit, there is no distinction between symmetric matter and asymmetric matter. Compact stars have neutron excess which typically engenders repulsion at densities above n0 , and hence the asymmetry effect needs to be taken into account. This effect will arise when the system is collective-quantized, which gives rise to the leading 1/Nc correction to the energy of the bound system. Most of this correction could be translated into a correction to the effective mass of the kaon, so the rapid dropping of the kaon will have an important impact on the physics of compact stars.
Acknowledgments I am grateful for the invitation by Koichi Yamawaki to give this talk. I would also like to acknowledge delightful collaborations with Hyun Kyu Lee, Byung-Yoon Park, Sang-Jin Sin and Ismail Zahed. This work was partly supported by the WCU project of Korean Ministry of Education, Science and Technology (R33-2008-000-10087-0).
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References 1. M. Harada and K. Yamawaki, Phys. Rept. 381, 1 (2003). 2. H. Arkani-Hamed, H. Georgi and M.D. Schwartz, Ann. Phys. 305, 96 (2003). 3. G. E. Brown et al, Prog. Theor. Phys. 121, 1209 (2009); M. Rho, “Dileptons get nearly ’blind’ to mass-scaling effects in hot and/or dense matter,” arXiv:0912.3116 [nucl-th]. . 4. M. Harada and C. Sasaki, Nucl. Phys. A 736, 300 (2004). 5. U.-G. Meissner, Phys. Rept. 161, 213 (1988). 6. R.A. Battye, N.S. Manton and P.M. Sutcliffe, “Skyrmions and nuclei,” in The Multifaceted Skyrmion (World Scientific 2010), ed. G.E. Brown and M. Rho. 7. B.-Y. Park and V. Vento, “Skyrmion approach to finite density and temperature,” in The Multifaceted Skyrmion (World Scientific 2010), ed. G.E. Brown and M. Rho. 8. A. S. Goldhaber and N. S. Manton, Phys. Lett. B 198, 231 (1987); A. D. Jackson and J. J. M. Verbaarschot, Nucl. Phys. A 484, 419 (1988); M. Kugler and S. Shtrikman, Phys. Rev. D 40, 3421 (1989). 9. B. Y. Park, D. P. Min, M. Rho and V. Vento, Nucl. Phys. A 707, 381 (2002). 10. M. F. Atiyah and N. S. Manton, Phys. Lett. B 222, 438 (1989). 11. See, e.g., K. Moon, “Skyrmions and merons in bailayer quantum Hall systems” and T. Senthil et al, “Deconfined quantum critical points” in The Multifaceted Skyrmion (World Scientific 2010), ed. G.E. Brown and M. Rho. 12. M. Rho, S. J. Sin and I. Zahed, “Dense QCD: A holographic dyonic salt,” arXiv:0910.3774 [hep-th]. 13. B. Y. Park, M. Rho and V. Vento, Nucl. Phys. A 736, 129 (2004). 14. B. Y. Park, M. Rho and V. Vento, Nucl. Phys. A 807, 28 (2008). 15. G. E. Brown and M. Rho, Phys. Rev. Lett. 66, 2720 (1991). 16. H. K. Lee and M. Rho, Nucl. Phys. A 829, 76 (2009). 17. T. Sakai and S. Sugimoto, Prog. Theor. Phys. 113, 843 (2005); Prog. Theor. Phys. 114, 1083 (2005). 18. D. K. Hong, M. Rho, H. U. Yee and P. Yi, Phys. Rev. D 76, 061901 (2007). H. Hata, T. Sakai, S. Sugimoto and S. Yamato, Prog. Theor. Phys. 117, 1157 (2007); K. Hashimoto, T. Sakai and S. Sugimoto, Prog. Theor. Phys. 120, 1093 (2008); K. Y. Kim and I. Zahed, JHEP 0809, 007 (2008). 19. D. K. Hong, M. Rho, H. U. Yee and P. Yi, Phys. Rev. D 77, 014030 (2008). 20. K. Y. Kim, S. J. Sin and I. Zahed, JHEP 0801, 002 (2008). 21. T. C. Kraan and P. van Baal, Nucl. Phys. B 533, 627 (1998); Phys. Lett. B 435, 389 (1998). 22. D. Diakonov, “Topology and confinement,” arXiv:0906.2456 [hep-ph]. 23. K. Hashimoto, N. Iizuka and T. Nakatsukasa, “N-body nuclear forces at short distances in holographic QCD,” arXiv:0911.1035 [hep-th]. 24. Y. Akaishi, A. Dote and T. Yamazaki, Phys. Lett. B 613, 140 (2005). 25. A. Gal, “Overview of strangeness nuclear physics,” arXiv:0904.4009 [nucl-th]; W. Weise, “Antikaon interactions with nucleons and nuclei,” arXiv:1001.1300 [nuclth]. 26. G. E. Brown, C. H. Lee and M. Rho, Phys. Rev. Lett. 101, 091101 (2008). 27. C.G. Callan and I. Klebanov, Nucl. Phys. B262, 365 (1985). 28. E. M. Nyman and D. O. Riska, Rept. Prog. Phys. 53, 1137 (1990). 29. B. Y. Park, J. I. Kim and M. Rho, “Kaons in dense half-skyrmion matter,” arXiv:0912.3213 [hep-ph].
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Aspects of Baryons in Holographic QCD T. Sakai Department of Physics, Nagoya University Nagoya 464-8602, Japan E-mail: [emailprotected] We discuss some recent results on a holographic description of QCD on the basis of a D4-D8-D8-brane configuration in type IIA superstring theory. A special attention is paid to studies of the baryon dynamics in that model. Using the fact that baryons in large Nc QCD should be regarded as a soliton, it is seen that this model reproduces the experimental data quite nicely. Keywords: Gauge/string duality, holographic QCD.
1. Introduction It has been now realized that the gauge/string correspondence provides us with a powerful machinery for analyzing the strong coupling dynamics of gauge theory. Initiated by the pioneering work by Maldacena (for a review, see Ref. 1), there have been lots of attempts to apply this idea to quantum chromodynamics(QCD). In Ref. 2, we proposed a model on the basis of a D4-D8-D8-brane configuration in type IIA superstring theory. This model defines a string dual of large Nc QCD with Nf massless flavor quarks. We reach the conclusion that the low energy dynamics of the hadron physics is described by a five-dimensional Yang-Mills-Chern-Simons(YMCS) gauge theory with the gauge group U (Nf ): S = SYM + SCS , (1) 1 2 2 h(z)Fµν + k(z)Fµz SYM = −κ d4 xdz tr , (2) 2 Nc Nc i 1 U(Nf ) SCS = ω (A) = tr AF 2 − A3 F − A5 , (3) 24π 2 M 4 ×R 5 24π 2 M 4 ×R 2 10 with λNc 2 , λ = gYM Nc , h(z) = (1 + z 2 )−1/3 , k(z) = 1 + z 2 216π 3 From Kaluza-Klein(KK) reduction of the five-dimensional gauge potential along the z direction, this model yields not only the massless pion associated with spontaneous chiral symmetry breaking, but also an infinite tower of massive vector mesons such as ρ, a1 , · · · . It is found that this model reproduces many quantitative aspects of the κ=
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hadron physics by tuning two “free parameters” λ and MKK . The latter is the only dimensionful parameter of our model, and we mostly work in units of MKK = 1. Moreover, it is worth emphasizing that the YM part of the action (1) is equal to a five-dimensional gauge theory that are constructed from hidden local symmetry (for a review, see Ref. 3) with an infinite number of hidden local gauge group incorporated, each of which corresponds to a massive vector meson.4 Contrary to the holographic description of the mesons as KK modes, baryons in our model can be described as a soliton.5 This idea has been well-known in large Nc QCD,6 and in fact, was first applied to the Skyrmion for making a quantitative analysis of baryons.7 Compared to the studies that have been done so far, Ref. 5 has an advantage in that our model allows us to construct a soliton solution that takes into account the effects from an infinite number of massive mesons as well as the massless pion. In fact, we can find out a systematic framework for describing negative parity baryons. The main purpose of this contribution is to present recent developments in the holographic description of baryons that are obtained in Refs. 5, 8, 9. In the next section, we give a brief review of a holographic description of large Nc QCD on the basis of an intersecting D-brane configuration in type IIA superstring theory. A focus is given on how to obtain the currents associated with the chiral flavor symmetry U (Nf )L × U (Nf )R . In section 3, we solve a soliton solution corresponding to a baryon. It is argued that a YM instanton of a fixed size plays a central role. Using these results, we work out the baryon form factors and then compare these results with experiments. We then sketch a computation of the short distance behavior of nuclear force using our model. We conclude this paper with summary in section 4. 2. D-branes and holographic QCD We start with the following brane configuration in type IIA superstring2 0 1 2 3 (4) 5 6 7 8 9 Nc D4 ◦ ◦ ◦ ◦ ◦ Nf D8-D8 ◦ ◦ ◦ ◦ ◦◦◦◦◦
(4)
Here x4 is compactified as x4 ∼ x4 + 2π/MKK . Nc D4-branes are wrapped around S 1 with the antiperiodic boundary condition imposed on the world volume fermions. These define four-dimensional SU (Nc ) pure YM theory with the UV cut-off given by MKK , with the bare coupling constant 2 Nc . The weak coupling description of the YM theory is at MKK equal to λ = gYM valid for λ 1. Nf D8-D8-branes are responsible for the chiral flavor symmetry U (Nf )L × U (Nf )R : it is found that open strings that extend from D4-branes to D8and D8-branes lead to the chiral fermions qL and qR , respectively. We thus obtain QCD with massless flavor quarks. When λ 1, the weak coupling description of QCD is not valid any more. For the purpose of understanding the strong coupling phenomena, the gauge/string duality is useful. The point here is that D-branes allow two different description of
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D8 qL qR
D8
x4
D4 gluon
D8 x 5,..., x 9
Fig. 1.
1
u
D4-D8-D8-brane configuration
a common physics. One is the open string picture, where D-branes can be identified with a hypersurface on which open strings can end. The other is to regard D-branes as a source of gravity so that they create a curved background as a solution of type IIA supergravity(SUGRA). The SUGRA solution of the D4-branes is obtained by Witten.10 This is reliable when Nc → ∞ with λ 1, which ensures that the SUGRA approximation of the D4-brane solution is reliable. Now add to this background the D8-D8-brane pairs. It is in general highly difficult to find a SUGRA solution of intersecting D-branes. To evade this problem, we work in the probe approximation Nf Nc , for which the added D8-D8 pairs can be treated as a probe with their backreaction into the D4-background neglected. From the gauge theory point of view, this is identical to neglecting the quark loop effects. As shown in Ref. 2, the probe D8-D8 pairs must be interpolated with each other smoothly because of the curved D4 background. We emphasize that this is a manifestation of spontaneous chiral symmetry breaking U (Nf )L × U (Nf )R → U (Nf )V . It can be seen that the embedding of the smoothly connected D8-branes into the Witten’s D4 geometry is specified by y = 0, such that the two-dimensional cigar-like geometry appearing at Fig. 1 is mapped to a two-dimensional plane spanned by y, z. Then, the world volume coordinate of the D8-branes is given by xµ , z and S 4 . By taking into account only the KK zero mode along the S 4 , we end up with the five-dimensional YM-CS theory given in (1). Let us first discuss how the meson degrees of freedom arise from the fivedimensional gauge potential. To this end, we expand it in terms of a complete set of the wavefunctions of z: Aµ (x, z) = Az (x, z) =
Bµ(n) (x) ψn (z) ,
n≥1
n≥1
ϕ(n) (x)φn (z) + ϕ(0) (x)φ0 (z) ,
(5)
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where − k 1/3 ∂z (k ∂z ψn ) = λn ψn , κ
dzk −1/3 ψn ψm = δnm , φn ∝ ∂z ψn .
(6)
(n)
Then, the four-dimensional gauge field Bµ (x) can be identified with the vector mesons, and ϕ(0) (x) as the massless pion. The nonzero mode of ϕ(n) (x) are eaten (n) up by Bµ (x) to gain a nonvanishing mass. It is important that we can assign P and C quantum numbers of the mesons uniquely, because these symmetries originate from a discrete symmetry of the D4-D8-D8-brane system. For details, see Ref. 11. The mass spectrum of the mesons is summarized as (2n−1)
ϕ(0) vµn = Bµ J P C 0−+ mass2
(2n)
anµ = Bµ
1−−
1++
2 λ2n−1 MKK
2 λ2n MKK
Next, we discuss spontaneous breaking of chiral flavor symmetry. Note that the field profile in (5) is left unchanged under the constant gauge transformation acting at z = ±∞: g± = g(z = ±∞) ,
(7)
because the wavefunctions ψ and ϕ are normalizable so that they dump rapidly enough as z grows. We interpret g± as the elements of chiral flavor symmetry group U (Nf )L,R . This flavor symmetry can be gauged by introducing the corresponding gauge potential AµL,R , which should be identified with the gauge potential A(x, z) at z = ±∞: Aµ (x, z → ±∞) = AµL,R (x) .
(8)
This can be incorporated by using the nonnormalizable zero mode ψ0 (z). Using the standard AdS/CFT dictionary, we find that the chiral flavor current can be read off of the linear coupling to AµL,R (x)as8 cl cl JLµ = −κ k(z) Fµz , JRµ = +κ k(z) Fµz . (9) z=+∞
z=−∞
Here the superscript cl refers to the fact that the field strength F is evaluated on-shell. For more details of a study of the meson physics, see also Ref. 12. 3. Baryons in holographic QCD
As argued before, baryons in large Nc QCD may be regarded as a soliton. In fact, Adkins, Nappi and Witten found that the semiclassical quantization of the Skyrmion leads to quantitative results in baryon properties that are consistent with experiments.7 Here the baryon number is identified with the topological number associated
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with the mapping S 3 → SU (2). Now the question is what happens when we incorporate the effects of heavy mesons into the baryon soliton. There have been some attempts toward it so far (see e.g. Ref. 13). In Ref. 5, we derived a baryon solution of the five-dimensional YM-CS theory (1). The point is that as in the case of the Skyrmion, the baryon number is equal to a topological number. To see this, recall that the baryon number is defined by the baryon number current as 1 µ µ µ JL + JR = −κ dz k(z)Fclµz . = (10) JB Nc Here the quantities without hat denote the U (1) part of U (Nf ) in (1), and the factor 1/Nc is due to the fact that a single baryon composed of Nc quarks carries a unit baryon number. For simplicity, we focus on the case Nf = 2. Using the equations of motion of (1), the baryon number can be rewritten as 1 0 QB = d3 x JB = (11) d3 xdz tr(FMN FP Q ) . MN P Q 64π 2
This is exactly the instanton number. It is also interesting that the instanton number can be shown to equal the winding number of the Skyrmion.5 In the context of holographic QCD, the baryon solutions can be obtained by finding a topological soliton with a nontrivial instanton number. However, since the theory (1) is defined on a curved background with the CS five-form added, it is difficult to find an analytic expression of the solitons. Instead, we work in the large λ limit to make the problem tractable. That is, define the new variables x
M = λ+1/2 xM ,
x
0 = x0 ,
A 0 (t, x
) = A0 (t, x) ,
A M (t, x
) = λ−1/2 AM (t, x) ,
) = λ−1 FMN (t, x) , F MN (t, x
F 0M (t, x
) = λ−1/2 F0M (t, x) ,
(12)
and regard the quantities with tildes as being O(λ0 ). Hereafter, we omit the tilde for simplicity. We then find that for λ 1, the equations of motion for the SU (2) part reduce to 1 MN P Q FMN FP Q + O(λ−1 ) = 0 , 64π 2 a + O(λ−1 ) = 0 .
DM F0M +
(13)
DN FMN
(14)
Also, the equations of motion for the U (1) part become 1 1 MN P Q tr(FMN FP Q ) + FMN FP Q + O(λ−1 ) = 0 , ∂M F0M + 64π 2 a 2 −1 ∂N FMN + O(λ ) = 0 .
(15) (16)
A baryon solution with a unit baryon number reads for the SU (2) sector A0 = 0 ,
AM = ABPST (xM ) , M
(17)
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where M, N = 1, 2, 3, z, and ABPST (xM ) refers to the BPST one-instanton solution M 4 in a flat R . For the U (1) part, 27π ρ4 M = 0 , A0 = 2 1 − 2 (18) , A ξ (ρ + ξ 2 )2
2 + (z − Z)2 , with X,
Z being the center of the instanton, and where ξ = ( x − X) ρ the instanton size. The parameters ρ and Z are not genuine moduli but they must be fixed by minimizing the energy of the soliton solution, which is given by 2 ρ 36 π 1 Z2 −2 + + ) , (19) + O(λ M (ρ, Z) =M0 1 + λ−1 6 5 ρ2 3 with M0 = 8π 2 κ. In particular, this implies that the baryon solution is given by a YM instanton with a fixed size of O(λ1/2 ) in the original coordinate. This gives a natural explanation of the observation by Atiyah and Manton in Ref. 14. They discuss that the Skyrmion can be well approximated with a YM instanton that is path-ordered integrated along an artificial fifth direction. In Ref. 14, the physical meaning of the extra direction was unclear, while in our model it is nothing but the holographic direction in the gauge/string duality. Now that we find the soliton solution, we next formulate a Lagrangian that characterizes the dynamics of the collective coordinates. For this purpose, we promote the instanton moduli to time-dependent variables. The kinetic term is given by a one-dimensional nonlinear σ model whose target space is equal to the the moduli space of the SU (2) one-instanton solution. On top of this, we have the potential term of ρ and Z, which is given by the soliton mass given above. In summary, the Lagrangian reads
where
L = LX + LZ + Ly + O(λ−1 ) , mX ˙ 2 LX = −M0 + X , 2 2 mZ ˙ 2 mZ ω Z Z2 , Z − LZ = 2 2 my ωρ2 2 Q my 2 my ωρ2 2 Q my 2 y˙ I − ρ − 2 = ρ − 2 , ρ˙ + ρ2 a˙ 2I − Ly = 2 2 ρ 2 2 ρ
2 = mX = mZ = my /2 = 8π 2 κλ−1 , ωZ
2 , 3
ωρ2 =
1 , 6
Q=
Nc2 . 5mX
(20)
(21)
(22)
4 Here yI = ρ aI , (I = 1, 2, 3, 4) with I=1 aI aI = 1. aI are an element of SU (2) that specifies the global gauge transformation of the BPST instanton. Quantization of this system is straightforward with the energy eigenstate given by |l, nρ , nz .
(23)
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This belongs to (l + 1, l + 1) of SU (2)I × SU (2)J = SO(4) with l = 1, 3, 5, · · · and nρ , nz = 0, 1, 2, · · · . Here SU (2)I,J is the isospin and spin group, respectively. nρ and nz denote the quantum excitation number associated with ρ and Z, respectively. The energy eigenvalue is
2 2(nρ + nz ) + 2 (l + 1)2 √ + Nc2 + 6 15 6 1 5 (l + 1)2 2(nρ + nz ) + 2 2 √
M0 + Nc + . + 15 4 6 Nc 6
Ml,nρ ,nz =M0 +
(24)
A comment is in order. In comparison with the analysis in the Skyrme model, we notice two new quantum numbers nρ and nz . This enables us to describe higher excited baryons including negative parity baryons. In fact, it is found that the eigenstate |l, nρ , nz carries the parity P = (−1)nz . Note that the collective motion due to Z arises only after incorporating an infinite number of heavy mesons into the effective Lagrangian of the pion. Now we propose the following dictionary that relates our prediction with experimental data: (nρ , nz )
(0, 0) (1, 0) (0, 1) (1, 1) (2, 0)/(0, 2) (2, 1)/(0, 3) (1, 2)/(3, 0)
N (l = 1) 940+ 1440+ 1535− 1655− 1710+ , ? + ∆ (l = 3) 1232+ 1600+ 1700− 1940− ∗ 1920 , ?
2090− ∗, ?
2100+ ∗, ?
?, ?
?, ?
Here the experimental values are taken from PDG.15 ∗ indicates a poor evidence of the existence of a meson. Using this rule, we compare mass differences of the two results:
m∆ − mN =
mN (1440) − mN =
Ml=3,nρ =nz =0 − Ml=1,nρ =nz =0 569MeV (our model)
1232 − 940 = 292MeV (experiments) Ml=1,nρ =0,nz =1 − Ml=1,nρ =nz =0 774MeV (our model)
1440 − 940 = 500MeV (experiments)
Here we have used MKK 949MeV, which is due to fitting our prediction of the ρ mass with experiments. One of the interesting features of our result is that both N (1440) and N ∗ (1530) are predicted to have the same mass. This is hard to explain in the phenomenological quark model: it predicts mN (1440) − mN 2(mN ∗ (1530) − mN ). A resolution of this puzzle is made by a recent lattice simulation of QCD.16 Our prediction is consistent with this result. Moreover, since our mass formula depends on nρ and nz only through nρ + nz , the baryon states with a common value of nρ + nz are all predicted to be degenerate.
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3.1. Baryon form factors In this subsection, we compute the baryon form factors using the formalism given above. See Ref. 8 for details. For closely related works, see also Refs. 17–21. µC (x) for Consider the matrix element of the vector and axial-vector current JV,A a spin-1/2 baryon state B: p, B, s p , B, s |JVµC (0)| −3
= i(2π)
τC 1 (C) 2 µ (C) 2 µν u( p ,s ) σ kν F2 (k ) u( p, s) γ F1 (k ) − 2 2mB
p , B, s |JAµC (0)| p, B, s = (2π)−3 u( p , s )
τC 1 (C) (C) iγ5 γ µ gA (k 2 ) + k µ γ5 gP (k 2 ) u( p, s) 2 2mB
with k = p − p . Here, the baryon state |B is given by the mass eigenstates found before. The current JV,A can be obtained from the formula (9). Hence the left hand side of the two equations is easy to compute in our model. Actually, we find F1 ( k 2 ) = Nc F1 ( k 2 ) =
gvn ψ2n−1 (Z) ,
k 2 + λ2n−1
n≥1
gvn ψ2n−1 (Z) ,
k 2 + λ2n−1
n≥1
g g n ψ I=0 v 2n−1 (Z) −1 F2 ( k 2 ) = Nc ,
k 2 + λ2n−1 2 n≥1
gI=1 gvn ψ2n−1 (Z) F2 ( k 2 ) = ,
k 2 + λ2n−1 2 n≥1
Nc gan ∂Z ψ2n (Z) gA ( k 2 ) = ,
k 2 + λ2n 4M0 n≥1
gP (k 2 ) =
8π 2 κ 2 gan ∂Z ψ2n (Z) ρ , gA ( k 2 ) =
k 2 + λ2n 3 n≥1
(26)
2mB gA (k 2 ) , k2
gP (k 2 ) =
2mB gA (k 2 ) , k2
(27)
with gI=0 =
mB , M0
gI=1 =
32π 2 κmB 2 ρ , 3
ρ2 = nρ |ρ2 |nρ , ψ2n−1 (Z)= nz |ψ2n−1 (Z)|nz , ∂Z ψ2n (Z)= nz |∂Z ψ2n (Z)|nz .
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It is interesting to note that these results exhibit the vector meson dominance. In other words, they can be reproduced by the effective action of the form 0 a v n µτ na µτ n n B + gv BB vµ Biγ B Lint = gv BB vµ Biγ 2 2 n≥1 µν τ 0 µν τ a 1 n n na na + B + hvn BB ∂µ vν − ∂ν vµ Bσ B , vν − ∂ν vµ Bσ hvn BB ∂µ 4mB 2 2 n≥1 a τ0 µτ Laint = B anµ Biγ5 γ µ B + gan BB ana Biγ γ gan BB 5 µ 2 2 n≥1 τ0 τa + 2i gπBB π Bγ5 B + gπBB π a Bγ5 B . 2 2
The whole tower of the cubic coupling constants can be fixed uniquely in holographic QCD. By noting that the electromagnetic current equals µ = JVa=3,µ + Jem
1 µ J , Nc
(28)
the electromagnetic form factors of the proton and neutron of Sachs type is found to take the form GpE ( k 2 ) =
gvn ψ2n−1 (Z) 2 p 2 2 GM (k ) = GnM ( k 2 ) = ,
k 2 + λ2n−1 gp gn n≥1
GnE ( k 2 ) = 0 , (29)
with the g-factors given by gp,n =
1 (gI=0 ± gI=1 ) . 2
(30)
We now compare these results with experiments. First, it is known experimentally that the electric form factor of the proton is well approximated with the dipole behavior −2
k 2 p 2
1+ 2 , (31) GE (k ) Λ exp
with Λ2 = 0.758 GeV2 from experiments. We can verify that GpE ( k 2 ) obtained above matches this dipole profile well. Next, the result GnE ( k 2 ) = 0 is also in good agreement with experiments. Note that the vanishing form factor is due to cancellation between the isotriplet and isosinglet form factors. This is reminiscent of the old work by Nambu, which predicted the ω meson, the lightest isosinglet vector meson, from the requirement of GnE ( k 2 ) = 0. In our model, the cancellation of the isotriplet and isosinglet form factors occurs for an infinite number of isotriplet and isosinglet vector mesons.
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In summary, we list our prediction of the physical quantities of baryons our model Skyrmion experiment 2 1/2 r I=0 0.742 fm 0.59 fm 0.806 fm 2 1/2 r M, I=0 0.742 fm 0.92 fm 0.814 fm 2 r (0.742 fm)2 ∞ (0.875 fm)2 2 E,p r 0 −∞ −0.116 fm2 2 E,n r (0.742 fm)2 ∞ (0.855 fm)2 2 M,p r (0.742 fm)2 ∞ (0.873 fm)2 2 M,n 1/2 r A 0.537 fm − 0.674 fm µp 2.18 1.87 2.79 −1.34 −1.31 −1.91 µ n µp 1.63 1.43 1.46 µn gA gπN N gρN N
0.734 7.46 5.80
0.61 8.9 −
1.27 13.2 4.2 ∼ 6.5
Here we used as an input MKK = 949 MeV , κ = 0.00745, which follows from a fitting with the experimental value of the ρ mass and the pion decay constant. In most cases, our predictions are improved compared with those from the Skyrme model analyses. 3.2. Nuclear force from string theory We end this section with a sketch of Ref. 9. The purpose of this paper is to compute the short distance potential of two nuclei. A direct evaluation of it by means of QCD is one of the outstanding issues in the hadron physics. That was first done in Ref. 22 using a lattice simulation. Here we see that holographic QCD is powerful enough to enable us to obtain an exact expression of it. For related works, see also Refs. 23, 24. The idea is to construct a two-baryon solution and then compute the energy of this system. As in a single baryon case, the exact solution of the two-baryon system is hard to find. We thus work in the λ 1 case, where the solution can be approximated with a two-instanton solution in a flat R4 . Collective motion is described by a Lagrangian of the 16 variables, which are time-dependent moduli of the two-instanton solution. The kinetic term is given by a one-dimensional nonlinear ω-model on the two-instanton moduli space. The potential takes the form: (32) U = 2M0 + USU(2) + UU(1) , 2 Nc z Nc 2 2 USU(2) = ), d3 xdz tr − Fij2 + z 2 Fiz d3 xdz tr(z 2 FMN = 216π 3 6 6 · 216π 3 (33) Nc 2 . (34) d3 xdz F0M UU(1) = 512π 3
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z
λ0
{
}λ
-1/2
U(1) flux
xi
U(1) flux
r< O (λ0) Fig. 2.
Sketch of two-baryon system in the original coordinate
Although USU(2) can be computed analytically, UU(1) cannot. Assuming R, the relative distance of the baryons in a flat R4 , to be O(λ−1/2 ) < R < O(λ0 ), the total Hamiltonian reduces to 1 H = H(1)ins + H(2)ins + 2 U (1) + O(R−3 ) . R Here H(i)ins , (i = 1, 2) is the Hamiltonian of a single baryon we have worked with so far. U (1) measures the potential energy of two baryons. In order to evaluate it, consider the matrix element 1 V = (I1 , J1 ); (I2 , J2 ) 2 U (1) (I1 , J1 ); (I2 , J2 ) . R Here Ii and Ji denotes the of the isospin and spin of the ith baryon. third component (I1 , J1 ); (I2 , J2 ) and (I , J ); (I , J ) are an eigenfunction of H(1)ins + H(2)ins , 1 1 2 2 being given by the tensor product of two wavefunctions. It can be shown that V = VC (| r|) + S12 VT (| r |) , 27 32 Nc 1 + (I1 · I2 )(J1 · J2 ) , VC (| r|) = π 2 5 λ | r|2 8π Nc 1 VT (| r|) = (I1 · I2 ) , 5 λ | r|2
(35)
where S12 is the tensor operator defined by S12 = 3( σ1 · r )( σ2 · r )/ r 2 − σ1 · σ2 .
(36)
r is the relative position of the two baryons in R3 . We interpret this as a short distance behavior of nuclear force between two nuclei. One of the most interesting features of this result is that it behaves as 1/| r|2 . This is identical to the typical
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behavior of Coulomb potential in four spatial dimensions. In fact, the U (1) gauge potential in (1) accounts for this Coulomb behavior. 4. Summary In this paper, we gave an overview of recent developments in the holographic description of QCD on the basis of the D4-D8-D8-brane configuration. A particular emphasis is put on baryon properties. By a semiclassical analysis of the baryon solution, we worked out the physical quantities of baryons. It was shown that this model reproduces many of the experimental results quantitatively. Furthermore, it is worth stressing that our model unites many insightful ideas that have been studies so far independently, such as hidden local symmetry, baryons as instantons, the role of the isosinglet vector mesons in the short distance repulsive core of nuclear force, etc. These successes may be regarded as a strong evidence that the gauge/string duality is a powerful tool to analyze the strong coupling dynamics of nonabelian gauge theories. A thorough investigation in this line would be worthwhile. Acknowledgments We would like to thank S. Sugimoto, H. Hata, S. Yamato and K. Hashimoto for fruitful collaborations. References 1. O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, “Large N field theories, string theory and gravity,” Phys. Rept. 323 (2000), 183, hep-th/9905111. 2. T. Sakai and S. Sugimoto, “Low energy hadron physics in holographic QCD,” Prog. Theor. Phys. 113 (2005), 843, hep-th/0412141. 3. M. Bando, T. Kugo and K. Yamawaki, “Nonlinear Realization and Hidden Local Symmetries,” Phys. Rept. 164, 217 (1988). 4. D. T. Son and M. A. Stephanov, “QCD and dimensional deconstruction,” Phys. Rev. D 69, 065020 (2004) [arXiv:hep-ph/0304182]. 5. H. Hata, T. Sakai, S. Sugimoto and S. Yamato, “Baryons from instantons in holographic QCD,” Prog. Theor. Phys. 117, 1157 (2007) [arXiv:hep-th/0701280]. 6. E. Witten, “Current Algebra, Baryons, And Quark Confinement,” Nucl. Phys. B 223, 433 (1983). 7. G. S. Adkins, C. R. Nappi and E. Witten, “Static Properties Of Nucleons In The Skyrme Model,” Nucl. Phys. B 228, 552 (1983). 8. K. Hashimoto, T. Sakai and S. Sugimoto, “Holographic Baryons : Static Properties and Form Factors from Gauge/String Duality,” Prog. Theor. Phys. 120, 1093 (2008) [arXiv:0806.3122 [hep-th]]. 9. K. Hashimoto, T. Sakai and S. Sugimoto, “Nuclear Force from String Theory,” Prog. Theor. Phys. 122, 427 (2009) [arXiv:0901.4449 [hep-th]]. 10. E. Witten, “Anti-de Sitter space, thermal phase transition, and confinement in gauge theories,” Adv. Theor. Math. Phys. 2 (1998), 505; hep-th/9803131. 11. T. Imoto, T. Sakai and S. Sugimoto, “Mesons as Open Strings in a Holographic Dual of QCD,” arXiv:1005.0655 [hep-th].
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12. T. Sakai and S. Sugimoto, “More on a holographic dual of QCD,” Prog. Theor. Phys. 114 (2005), 1083, hep-th/0507073. 13. U. G. Meissner, “Low-Energy Hadron Physics From Effective Chiral Lagrangians With Vector Mesons,” Phys. Rept. 161, 213 (1988). 14. M. F. Atiyah and N. S. Manton, “SKYRMIONS FROM INSTANTONS,” Phys. Lett. B 222 (1989), 438. 15. W. M. Yao et al. [Particle Data Group], “Review of particle physics,” J. of Phys. G 33 (2006), 1. 16. K. Sasaki, S. Sasaki and T. Hatsuda, “Spectral analysis of excited nucleons in lattice QCD with maximum entropy method,” Phys. Lett. B 623, 208 (2005) [arXiv:heplat/0504020]. 17. D. K. Hong, M. Rho, H. U. Yee and P. Yi, “Chiral dynamics of baryons from string theory,” Phys. Rev. D 76, 061901 (2007) [arXiv:hep-th/0701276]. 18. D. K. Hong, M. Rho, H. U. Yee and P. Yi, “Dynamics of Baryons from String Theory and Vector Dominance,” JHEP 0709, 063 (2007) [arXiv:0705.2632 [hep-th]]. 19. D. K. Hong, M. Rho, H. U. Yee and P. Yi, “Nucleon Form Factors and Hidden Symmetry in Holographic QCD,” Phys. Rev. D 77, 014030 (2008) [arXiv:0710.4615 [hep-ph]]. 20. J. Park and P. Yi, “A Holographic QCD and Excited Baryons from String Theory,” JHEP 0806, 011 (2008) [arXiv:0804.2926 [hep-th]]. 21. H. Hata, M. Murata and S. Yamato, “Chiral currents and static properties of nucleons in holographic QCD,” Phys. Rev. D 78, 086006 (2008) [arXiv:0803.0180 [hep-th]]. 22. N. Ishii, S. Aoki and T. Hatsuda, “The nuclear force from lattice QCD,” Phys. Rev. Lett. 99 (2007), 022001, nucl-th/0611096; “Nuclear Force from Monte Carlo Simulations of Lattice Quantum Chromodynamics,” arXiv:0805.2462. H. Nemura, N. Ishii, S. Aoki and T. Hatsuda, “Hyperon-nucleon force from lattice QCD,” arXiv:0806.1094. S. Aoki, J. Balog, T. Hatsuda, N. Ishii, K. Murano, H. Nemura and P. Weisz, “Energy dependence of nucleon-nucleon potentials,” arXiv:0812.0673. 23. K. Y. Kim and I. Zahed, “Nucleon-Nucleon Potential from Holography,” JHEP 0903, 131 (2009) [arXiv:0901.0012 [hep-th]]. 24. Y. Kim, S. Lee and P. Yi, “Holographic Deuteron and Nucleon-Nucleon Potential,” JHEP 0904, 086 (2009) [arXiv:0902.4048 [hep-th]].
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Nuclear Force from String Theory K. Hashimoto Nishina Center, RIKEN 2-1 Hirosawa, Wako, Saitama 351-0198, Japan E-mail: [emailprotected] We compute nuclear forces at short distance, by applying gauge/gravity correspondence to large Nc QCD. This talk is based on work in collaboration with T. Sakai and S. Sugimoto. Keywords: Holographic QCD, nuclear force, AdS/CFT correspondence.
1. From String Theory to Nuclear Physics The application of AdS/CFT correspondence to QCD has been quite successful, at least for delivering qualitative understanding of strong coupling physics expected in QCD. However, it is important to notice that most of the researches of this application have been on hadron physics, not on nuclear physics. In fact, there is a big gap between particle physics community and nuclear physics community, and this gap exists because of the strong coupling of QCD, we can say. Once we can solve QCD at strong coupling, then it should reveal various fundamental inputs in nuclear physics. The AdS/CFT correspondence, although it can be applied only for large Nc limit of QCD-like theories, may bring some hope for this quite important direction of merger of two physics communities. Light nuclei are studied by a few body quantum mechanics under an assumption that nucleons are point particles, with given nuclear forces. Once QCD is solved at low energy, i.e. at strong coupling, to give the force among baryons, then we can use it for the basis of the nuclear physics. Since holographic QCD, the application of the AdS/CFT correspondence to QCD, can solve a certain limit of QCD-like theories, it is very natural to try to understand the nuclear forces from this duality point of view. In this talk, I describe such computations of the nuclear force in holographic QCD (based on the work1,2 which have been done in collaboration with T. Sakai and S. Sugimoto).a On the other hand, heavy nuclei are very difficult to treat, as a complicated quantum many body problem. One approach which I propose along the line of the AdS/CFT correspondence is to use this large number for the number of the a See
also related works.3,4
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nucleons as a large N limit of holography. Using the AdS/CFT applied to heavy nuclei, we can get a gravitational dual description of the nuclei naturally. This I call “holographic nuclei”,5,6 and the details will be delivered elsewhere. It has been known for many decades that nucleons always experience a repulsive force at short distances. This is known as a “repulsive core” of the nucleon. It is one of the keystones if any approximation of strongly coupled QCD may exhibit this repulsive core or not. In this talk, I show that the holographic QCD can reproduce beautifully the repulsive core of the nucleon. 2. Derivation of Nuclear Force at Short Range The idea of how to derive the nuclear forces in holographic QCD is quite simple. We rely on the following two key ideas. • By the AdS/CFT, strong coupling QCD is rephrased by a higher dimensional U (Nf ) non-Abelian gauge theory, where Nf is the number of flavors in the large Nc QCD. • Mesons are the gauge fields of this theory, while baryons are solitons of it. There are various holographic QCD models, but here we use the so-called D4D8 model (Sakai-Sugimoto model).7,8 Then the gauge theory is effectively a 4+1dimensional Yang-Mills-Chern-Simons U (Nf ) theory in a curved spacetime. Solitons are Yang-Mills instantons in this theory (instantons are localized in 4-dimensional space, while we have 5 dimensional spacetime as a starting point, so in effect the instanton is a particle-like excitation, identified as a baryon). Fortunately, we know generic 2-instanton solution in flat space, so we can use the generic solution to explore the nuclear forces. Quantization of a single instanton soliton in this curved geometry was done by Hata et al.,9 and this is used to specify the baryon states. Here we show only the results of our computation, for simplicity. V = VC (r) + (3ˆr · σ1 ˆr · σ2 − σ1 · σ2 ) VT (r). The central and the tensor forces are written as 27 8 1 1 ~ ~ VC (r) = π + σ1 · σ 2 I 1 · I 2 , 2 2 5 gQCD MKK r2 8π ~ ~ 1 1 VT (r) = I1 · I2 2 . 5 gQCD MKK r2
(1)
(2) (3)
As one can see in the central force, whatever the spin (σi /2) or the isospin I~i is, it is repulsive. So we found a repulsive potential for any nucleons. Note that our nuclear force has only two parameters, gQCD and MKK . This is a peculiar feature of this holographic QCD model. These two numbers are usually used to fit the ρ meson mass and the pion decay constant, then all the other observables are predictions of the model, and this nuclear force is one of the predictions.
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3. Conclusion and Discussions We computed2 nuclear forces at short distances, by solving a large Nc QCD-like theory using the AdS/CFT correspondence. This would serve as a first analytic result reproducing the repulsive core of nucleons, via a solution of the strongly coupled QCD-like theory. In the other work1 we computed static properties of baryons, including mesonbaryon couplings. This in fact gives a long-range nuclear force via the meson exchange picture. Along the direction of this research, it would be quite natural to extend the computation to a system with a larger number of nucleons. Indeed, the 3-body nuclear force is one of key issues in nuclear physics: it is known that it exists from few-body calculations of energy levels of light nuclei, while the intrinsic nature of the force is to be revealed. I have tackled this problem with the AdS/CFT correspondence with my collaborators and found that the 3-body forces are suppressed.10 It would be very interesting to apply the AdS/CFT correspondence to various mysteries in nuclear physics. References 1. K. Hashimoto, T. Sakai and S. Sugimoto, Prog. Theor. Phys. 120 (2008) 1093 [arXiv:0806.3122 [hep-th]]. 2. K. Hashimoto, T. Sakai and S. Sugimoto, Prog. Theor. Phys. 122 (2009) 427 [arXiv:0901.4449 [hep-th]]. 3. D. K. Hong, M. Rho, H. U. Yee and P. Yi, JHEP 0709 (2007) 063 [arXiv:0705.2632 [hep-th]]. 4. Y. Kim, S. Lee and P. Yi, JHEP 0904 (2009) 086 [arXiv:0902.4048 [hep-th]]. 5. K. Hashimoto, Prog. Theor. Phys. 121 (2009) 241 [arXiv:0809.3141 [hep-th]]. 6. K. Hashimoto, JHEP 0912 (2009) 065 [arXiv:0910.2303 [hep-th]]. 7. T. Sakai and S. Sugimoto, Prog. Theor. Phys. 113 (2005) 843 [arXiv:hep-th/0412141]. 8. T. Sakai and S. Sugimoto, Prog. Theor. Phys. 114 (2005) 1083 [arXiv:hep-th/0507073]. 9. H. Hata, T. Sakai, S. Sugimoto and S. Yamato, Prog. Theor. Phys. 117 (2007) 1157 [arXiv:hep-th/0701280]. 10. K. Hashimoto, N. Iizuka and T. Nakatsukasa, arXiv:0911.1035 [hep-th].
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Integrating Out Holographic QCD Back to Hidden Local Symmetry∗ Masayasu Haradaa,† , Shinya Matsuzakib,§ and Koichi Yamawakia,‡ a Department
of Physics, Nagoya University Nagoya, 464-8602, Japan † E-mail: [emailprotected] ‡ E-mail: [emailprotected] b Department
of Physics, Pusan National University Busan 609-735, Korea § E-mail: [emailprotected]
We develop a previously proposed gauge-invariant method to integrate out infinite towers of vector and axialvector mesons arising as Kaluza-Klein (KK) modes in a class of holographic models of QCD (HQCD). We demonstrate that HQCD can be reduced to the chiral perturbation theory (ChPT) with the hidden local symmetry (HLS) (so-called HLS-ChPT) having only the lowest KK mode identified as the HLS gauge boson, and the Nambu-Goldstone bosons. The O(p4 ) terms in the HLS-ChPT are completely determined by integrating out infinite towers of vector/axialvector mesons in HQCD: Effects of higher KK modes are fully included in the coefficients. As an example, we apply our method to the Sakai-Sugimoto model.
1. Introduction Holography, based on gauge/gravity duality,1 has been of late fashion to reveal a part of features in strongly coupled gauge theories involving the application to QCD (so-called holographic QCD (HQCD)). There are two types of holographic approaches: One is called “top-down” approach starting with a stringy setting; the other is called “bottom-up” approach beginning with a five-dimensional gauge theory defined on an AdS (anti-de Sitter space) background. It is a key point to notice that in whichever approach one eventually employs a five-dimensional gauge model with a characteristic induced-metric and some boundary conditions on a brane configuration. In the low-energy region, any model of HQCD is reduced to a certain effective hadron model in four-dimensions. Such effective models include vector and axialvector mesons as infinite towers of Kaluza-Klein (KK) modes together with the Nambu-Goldstone bosons (NGBs) associated with the spontaneous chiral symmetry breaking. Green functions in QCD are evaluated straightforwardly from the effective model following the holographic dictionary. Full sets of infinite towers ∗ Talk
given by S. M.
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of exchanges of KK modes (vector and axialvector mesons) contribute to Green functions involving current correlators and form factors and could mimic ultraviolet behaviors in QCD, although such a hadronic description would not be reliable above a certain high-energy scale. This implies that appropriate/gauge-invariant holographic results require calculations including full set of KK towers, which, however, would not be practical because of forms written in terms of an infinite sum. It was pointed out2 that the infinite tower of KK modes is interpreted as a set of gauge bosons of the hidden local symmetries (HLSs).3,4 This implies an interesting possibility that, in the low-energy region, any holographic models can be reduced to the simplest HLS model, provided that the infinite tower of KK modes is integrated out keeping only the lowest one identified with the ρ meson and its flavor partners. Effects from the higher KK modes would then be fully incorporated into higher derivative terms (O(p4 ) terms) in the HLS effective field theory as an extension of the conventional chiral perturbation theory (ChPT),5 so-called the HLS-ChPT4,6 which is manifestly gauge-invariant formulation and makes it possible to calculate any Green functions order by order in derivative expansion. Once holographic models are expressed in terms of the HLS-ChPT, one can even calculate meson-loop corrections of subleading order in 1/Nc expansion. This would give a new insight into the HQCD which as it stands is valid only in the large Nc limit. Indeed, in the previous work,7 we proposed a consistent method of integrating out the infinite towers of vector and axialvector mesons in the Sakai-Sugimoto (SS) model 8,9 into the HLS-ChPT, with the O(p4 ) terms explicitly given by the integrated higher modes effects. Then we were able to do the first calculations of 1/Nc corrections to the SS model. In this talk, we report the results of our work12 developing in detail the integrating-out method proposed in Ref. 7. First of all, we work in a class of holographic models to introduce our integrating-out method. Next, as an example, we apply our procedure to the Sakai-Sugimoto (SS) model8,9 to give the HLS model with a full set of O(p4 ) terms determined. We then calculate the pion electromagnetic form factor to demonstrate how powerful our formulation is even before including the 1/Nc effects through loop. As we will see explicitly, the momentumdependence of the form factor is evaluated including full set of contributions from KK modes without performing infinite sums. Our method can straightforwardly be applied to other types of holographic models such as those given in Refs. 10 and 11. More details are presented in Ref. 12.
2. A gauge-invariant way to integrate out HQCD In this section, starting with a class of HQCD models including the SS model,8,9 we introduce a way to obtain a low-energy effective model in four-dimension described only by the lightest vector meson identified as ρ meson based on the HLS together with the NGBs. Suppose that the fifth direction, spanned by the coordinate z,
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extends from minus infinity to plus infinity (−∞ < z < ∞) a . We employ a fivedimensional gauge theory which has a vectorial U (N ) gauge symmetry defined on a certain background associated with the gauge/gravity duality. As far as gaugeinvariant sector such as the Dirac-Born-Infeld part of the SS model8,9 is concerned, the five-dimensional action in large Nc limit can be written as b ! Z 1 2 S5 = Nc d4 xdz − K1 (z)tr[Fµν F µν ] + K2 (z)MKK tr[Fµz F µz ] , (1) 2 where K1,2 (z) denote a set of metric-functions of z constrained by the gauge/gravity duality. MKK is a typical mass scale of KK modes of the gauge field AM with M = (µ, z). The boundary condition of AM is chosen as AM (xµ , z = ±∞) = 0. A transformation which does not change this boundary condition satisfies ∂M g(xµ , z)|z=±∞ = 0, where g(xµ , z) is the transformation matrix of the gauge symmetry. This implies an emergence of global chiral U (N )L × U (N )R symmetry in four-dimension characterized by the transformation matrices gR,L = g(z = ±∞). Following Refs. 7, 8, and 9, we work in Az = 0 gauge. There still exists a four-dimensional gauge symmetry under which Aµ (xµ , z) transforms as Aµ → h · Aµ · h† − i∂µ h · h† with h = h(xµ ). This gauge symmetry is identified7–9 with the HLS.3,4 In the Az ≡ 0 gauge, the NGB fields π(xµ ) disappear from the chiral field U = e2iπ/Fπ since U → 1. They are instead included7 in Aµ (xµ , z) at the boundary † † as Aµ |z=±∞ = αR,L = iξR,L ∂µ ξR,L , where ξL,R form the chiral field U as U = ξL ·ξR . µ † 3,4 R,L R,L † Since ξL,R → h · ξL,R · gL,R , αR,L transform as α → h · α · h − i∂ h · h† . µ µ µ µ (n)
We introduce infinite towers of massive KK modes for vector (Vµ (xµ )) and (n) (n) axialvector (Aµ (xµ )) meson fields, where we treat Vµ as the HLS gauge fields (n) (n) (n) transforming under the HLS as Vµ → h · Vµ · h† − i∂µ h · h† , while Aµ as matter (n) (n) fields transforming as Aµ → h·Aµ ·h† . The five-dimensional gauge field Aµ (xµ , z) is now expanded as c µ R L µ L Aµ (xµ , z) = αR µ (x )φ (z) + αµ (x )φ (z) +
∞ X (n) (n) Aµ (xµ )ψ2n (z) − Vµ (xµ )ψ2n−1 (z) .
n=1
(2)
The functions {ψ2n−1 (z)} and {ψ2n (z)} are the eigenfunctions satisfying the eigenvalue equation obtained from the action (1): −K1−1 (z)∂z (K2 (z)∂z ψn (z)) = λn ψn (z), where λn denotes the nth eigenvalue. On the other hand, the gauge-invariance requires the functions φR,L (z) to be different from the eigenfunctions: From the trans(n) (n) formation properties for Aµ (xµ , z), αR,L we see that the functions µ , Aµ , and Vµ a In
an application to another type of HQCD,10 the z coordinate is defined on a finite interval, which is different from the z coordinate used here. They are related by an appropriate coordinate transformation as done in Refs. 8, 9. b Models of HQCD having the left- and right-bulk fields such as F , F 10,11 can be described by L R the same action as in Eq.(1) with a suitable z-coordinate transformation prescribed. c In Eq.(2) we put a relative minus sign in front of the HLS gauge fields V (n) for a convention. µ
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φR,L (z), {ψ2n−1 (z)}, and {ψ2n (z)} are constrained as φR (z) + φL (z) −
∞ X
ψ2n−1 (z) = 1 .
(3)
n=1
Using this, we may rewrite Eq.(2) to obtain Aµ (xµ , z) = αµ|| (xµ )+αµ⊥ (xµ )(φR (z)− P∞ P∞ (n) µ (n) µ µ φL (z)) + n=1 Aµ (x )ψ2n (z) + n=1 αµ|| (x ) − Vµ (x ) ψ2n−1 (z), where αR ±αL
αµ||,⊥ = µ 2 µ transform under the HLS as αµ|| → h · αµ|| · h† − i∂µ h · h† and αµ⊥ → h · αµ⊥ · h† , respectively. Note that αµ⊥ includes the NGB fields as αµ⊥ = F1π ∂µ π + · · · . The corresponding wave function (φR − φL ) should therefore be the eigenfunction for the zero mode, ψ0 : φR (z) − φL (z) = ψ0 (z). From this and Eq.(3) we see that the wave functions φR and φL are not the eigenfunctions but are P given as φR,L (z) = 21 [1 + ∞ n=1 ψ2n−1 (z) ± ψ0 (z)]. By substituting Eq.(2) with Eq.(3) into the action (1), the five-dimensional theory is now described by the NGB fields along with the infinite towers of the vector and axialvector meson fields in four dimensions. We first naively try to truncate towers of the vector and axialvector meson fields (n) (n) simply eliminating Vµ and Aµ for n > N . Then we find Z ∞ X truncation 4 2 S5 3 dzd x K2 (z) λ2n−1 ψ2n−1 (z)tr[αµ|| (xµ )]2 , (4) n=N +1
which explicitly breaks the chiral symmetry as well as the HLS, because αµ|| → h · αµ|| · h† − i∂µ h · h† . Naive truncation of tower of vector meson fields thus forces us to encounter the explicit violation of the chiral symmetry. Now we shall propose a method to truncate towers of vector and axialvector meson fields in a gauge-invariant manner. Consider a low-energy effective theory below the mass of n = N + 1 level. Such an effective theory can be obtained by integrating out mesons with n ≥ N + 1 via the equations of motion. Neglecting (k) (k) terms including the derivatives acting on the heavy fields Vµ and Aµ with k > N , (k) (k) the equations of motion for them take the following forms: Vµ = αµ|| , Aµ = 0 (k = N + 1, N + 2, · · · , ∞). Putting these solutions into the action, we have, instead of Eq.(4), Z N X 2 λ2n−1 ψ2n−1 S5integrate out 3 dzd4 x K2 (z) (z)tr[αµ|| (xµ ) − Vµ(n) (xµ )]2 , (5) n=1
which is certainly gauge-invariant. Note also that the gauge-invariance now requires PN not the constraint in Eq.(3) but φR (z) + φL (z) − n=1 ψ2n−1 (z) = 1. Let us now consider a low-energy effective model obtained by integrating out all the higher vector and axialvector meson fields except the lowest vector meson (1) field Vµ ≡ Vµ . Following the gauge-invariant way proposed above, the expansion of Aµ (xµ , z) is expressed as Aµ (xµ , z) = αµ⊥ (xµ )ψ0 (z) + (ˆ αµ|| (xµ ) + Vµ (xµ )) + α ˆ µ|| (xµ )ψ1 (z) ,
(6)
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where α ˆµ|| = −Vµ + αµ|| . One can further introduce the external gauge fields by gauging the global chiral U (N )L ×U (N )R symmetry. (For details, see Ref. 12.) Then we obtain the low-energy effective model including only the lightest HLS field Vµ and the NGB fields π described by the HLS-ChPT with O(p4 ) terms:4,6 The O(p4 ) terms include the effects from infinite towers of higher KK modes and are completely determined as explicitly shown in Ref. 12; sum rules such as those introduced in Ref. 9 are also fully built in the HLS-ChPT Lagrangian. Our formulation is thus more practical and useful. Finally, we once again emphasize that our methodology presented here is applicable to any models of HQCD. 3. Application to Sakai-Sugimoto Model In this section, we apply our methodology to the Sakai-Sugimoto (SS) model8,9 ¯ based on D8/D8/D4 brane configuration. The five-dimensional gauge-invariant portion (so-called the Dirac-Born-Infeld (DBI) part) of the low-energy effective action in the SS model is given by8,9 ! Z 1 −1/3 µν 2 µz DBI 4 (z)tr[Fµν F ] + K(z)MKK tr[Fµz F ] , (7) SSS = Nc G d xdz − K 2 where K(z) = 1 + z 2 is the induced metric of the five-dimensional space-time; the 2 overall coupling G is the rescaled ’t Hooft coupling expressed as G = Nc gYM /(108π 3 ) with gYM being the gauge coupling of the U (Nc ) gauge symmetry on the Nc D4-branes;8,9 the mass scale MKK is related to the scale of the compactification of the Nc D4-branes onto the S 1 . Comparing Eq.(7) with Eq.(1), we read off K1 (z) = GK −1/3 (z), K2 (z) = GK(z), so that we find the equation of motion, −K 1/3 (z)∂z (K(z)∂z ψn ) = λn ψn with the eigenvalues λn and the eigenfunctions ψn of the KK modes of the five-dimensional gauge field Aµ (xµ , z). Application to the Chern-Simons term is straightforward that is explicitly demonstrated in Ref. 12. As emphasized in the end of the previous section, without introducing any sum rules, we are able to calculate amplitudes straightforwardly from the effective Lagrangian which includes contributions from full set of higher KK modes. To see it more explicitly, as an example, we shall study the pion electromag± ± netic (EM) form factor FVπ at tree-level of the present model. FVπ is readily constructed from the Lagrangian written in terms of the HLS-ChPT presented ± g (Q2 )gρππ (Q2 ) , where Q2 = −p2 denotes in Ref. 12: FVπ (Q2 )|HLS = gγππ (Q2 ) + ρ m2 +Q 2 ρ 2 Q2 a momentum-squared in space-like region and12 gγππ (Q2 ) = 1 − a2 + ag4 z6 m 2, ρ 2 2 2 2 m Q g z Q ρ 1 2 2 2 4 gρ (Q ) = g 1 + g z3 m2 , gρππ (Q ) = 2 ag 1 + 2 m2 . The applicable moρ
ρ
mentum range should be restricted to 0 ≤ Q2 {m2ρ0 , m2ρ00 , · · · } since we have integrated out higher KK modes keeping only the ρ meson. Note that our form fac± ± tor FVπ (Q2 )|HLS automatically ensures the EM gauge-invariance, FVπ (0)|HLS = 1:
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One can easily show12 that if towers of vector and axialvector mesons had naively ± been truncated as in Eq.(4) one would have FVπ (0)|truncation 6= 1, leading to a violation of the EM gauge-invariance. (It turns out12 that higher KK modes actually play the crucial role to maintain the EM gauge-invariance.) We further rewrite ± FVπ (Q2 )|HLS as d 1 Q2 gρ gρππ π± 2 FV (Q )|HLS = 1 − a ˜ + z˜ 2 + 2 , (8) 2 mρ mρ + Q 2 2 2 2 z4 ) and z˜ = 41 a g 2 z6 + (g 2 z3 )(g 2 z4 ) 12 where a ˜ = a 1 − g 2z4 − g 2 z3 + (g z3 )(g 2 2g g
which are expressed in terms of the five-dimensional theory as e12 a ˜ = ρm2ρππ = ρ hψ1 ihψ1 (1−ψ02 )i hψ1 ihψ1 (1−ψ02 )i π π 2 , z˜ = 8 λ1 − h1 − ψ0 i . These a ˜ and z˜ are calcu4 λ1 hψ12 i hψ12 i lated independently of any inputs to be a ˜ ' 2.62 and z˜ ' 0.08, where we have used λ1 ' 0.669. Using the expression (8) and the values of a ˜ and z˜, we evaluate the momentum± dependence of FVπ which was actually not possible in the original SS model9 because of the form written in terms of the infinite summation. In Fig. 1 we ± show the predicted curve of FVπ with respect to Q2 together with the experimental data from Ref. 13. The χ2 -fit results in good agreement with the data (χ2 /d.o.f = 147/53 ' 2.77). Comparison with the result derived from the lowest vector meson dominance (LVMD) hypothesis with a ˜ = 2 and z˜ = 0 is shown by a dashed curve in the left panel of Fig. 1. The right panel of Fig. 1 shows a comparison with the result obtained by fitting the parameters (˜ a, z˜) to the experimental data. It is interesting to note that the best-fit values of a ˜ and z˜ are quite close to those in the predicted curve. This fact reflects that the predicted curve fits well with the experimental data. 4. Summary In this talk, we developed a methodology to integrate out arbitrary parts of infinite towers of vector and axialvector mesons arising as KK modes in a class of HQCD models. It was shown that our method is gauge-invariant in contrast to a naive truncation [See Eq.(4)]. It was demonstrated that any models of HQCD in the lowenergy region can be described by the HLS-ChPT. We applied our method to the SS model and evaluated the momentum-dependence of the pion EM form factor as an example, which demonstrated power of our formulation. The predicted form factor was shown to be fitted well with the experimental data in the low-energy (space-like momentum) region. This was difficult in the original SS model due to the forms of the form factors written in terms of infinite sum of vector meson exchanges. More on phenomenological applications of our formulation to the SS model is presented in Ref. 12. d Here e hAi
gρ ≡ gρ (Q2 = −m2ρ ) and gρππ ≡ gρππ (Q2 = −m2ρ ). R ≡ dzK −1/3 (z)A(z).
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Fig. 1. The predicted curve of the pion EM form factor FVπ with respect to space-like momentum-squared Q2 (denoted by solid line) fitted with the experimental data from Ref. 13 with χ2 /d.o.f = 147/53 ' 2.77. In the left panel, the dashed curve corresponds to the form factor in the LVMD hypothesis of the HLS model with a ˜ = 2 and z˜ = 0 taken. (χ2 /d.o.f = 226/53 ' 4.26). The dashing curve in the right panel corresponds to the form factor fitted with the experimental data, yielding the best fit values of a ˜ and z˜, a ˜|best = 2.44, z˜|best = 0.08 (χ2 /d.o.f = 81/51 ' 1.56).
Acknowledgments This work was supported in part by the JSPS Grant-in-Aid for Scientific Research; (B) 18340059 (K.Y.), (C) 20540262 (M.H.), Innovative Areas #2104 “Quest on New Hadrons with Variety of Flavors” (M.H.), the Global COE Program “Quest for Fundamental Principles in the Universe” (K.Y. and M.H.), and the Daiko Foundation (K.Y.). S.M. is supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-341-C00008).
References 1. J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998); Int. J. Theor. Phys. 38, 1113 (1999); S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428, 105 (1998); E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998); O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, Phys. Rept. 323, 183 (2000). 2. C. T. Hill, S. Pokorski and J. Wang, Phys. Rev. D 64, 105005 (2001) [arXiv:hepth/0104035]. 3. M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett. 54, 1215 (1985); M. Bando, T. Kugo and K. Yamawaki, Nucl. Phys. B 259, 493 (1985); M. Bando, T. Fujiwara, and K. Yamawaki, Prog. Theor. Phys. 79 (1988) 1140; M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988). 4. M. Harada and K. Yamawaki, Phys. Rept. 381, 1 (2003). 5. J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984); Nucl. Phys. B 250, 465 (1985). 6. M. Tanabashi, arXiv:hep-ph/9306237; Phys. Lett. B 316, 534 (1993). 7. M. Harada, S. Matsuzaki and K. Yamawaki, Phys. Rev. D 74, 076004 (2006) [arXiv:hep-ph/0603248]. 8. T. Sakai and S. Sugimoto, Prog. Theor. Phys. 113, 843 (2005) [arXiv:hep-th/0412141]. 9. T. Sakai and S. Sugimoto, Prog. Theor. Phys. 114, 1083 (2006) [arXiv:hepth/0507073]. 10. L. Da Rold and A. Pomarol, Nucl. Phys. B 721, 79 (2005) [arXiv:hep-ph/0501218].
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11. J. Erlich, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 95, 261602 (2005) [arXiv:hep-ph/0501128]. 12. M. Harada, S. Matsuzaki, and K. Yamawaki, in preparation. 13. S. R. Amendolia et al. [NA7 Collaboration], Nucl. Phys. B 277, 168 (1986); J. Volmer et al. [The Jefferson Lab F(pi) Collaboration], Phys. Rev. Lett. 86, 1713 (2001); V. Tadevosyan et al. [Jefferson Lab F(pi) Collaboration], Phys. Rev. C 75, 055205 (2007) [arXiv:nucl-ex/0607007]; T. Horn et al. [Jefferson Lab F(pi)-2 Collaboration], Phys. Rev. Lett. 97, 192001 (2006) [arXiv:nucl-ex/0607005].
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Holographic Heavy Quarks and the Giant Polyakov Loop Gianluca Grignani Dipartimento di Fisica, Universit` a di Perugia and I.N.F.N. Sezione di Perugia Via Pascoli, 06123 Perugia, Italy Joanna Karczmarek and Gordon W. sem*noff Department of Physics and Astronomy, University of British Columbia 6224 Agricultural Road, Vancouver, British Columbia, Canada V6T 1Z1 We consider the Polyakov loop in finite temperature planar N = 4 supersymmetric Yang-Mills theory defined on a spatial S 3 and in representations where the number of k remains finite in the large N limit. boxes in the Young Tableau k scales so that N We review the argument that, in the deconfined phase of the gauge theory, and for symmetric representations with row Young tableau, there is a quantum phase transition in the expectation value of the Polyakov loop operator which occurs as the size of the representation is increased. Keywords: Holography, confinement, character.
1. Introduction AdS/CFT holography identifies the deconfinement phase transition of planar N = 4 Yang-Mills theory with the collapse of hot Anti-de Sitter space to an anti-de SitterSchwarzchild black hole.1 The planar limit of Yang-Mills theory is achieved by taking the large N limit while holding the ’t Hooft coupling λ = gY2 M N fixed. The finite temperature gauge theory is defined on S 3 × S 1 where space is the 3-sphere and S 1 is periodic Euclidean time. Since N = 4 is an adjoint gauge theory, it has a center symmetry whose spontaneous breaking is associated with de-confinement. 6,7 As we shall review in the following, a deconfining phase transition can be identified by analyzing the gauge theory in the perturbative regime.2,3 In this Article, we shall review some of our recent work on the properties of the deconfined phase of the gauge theory.4 Before that, in the next Section, we shall discuss the motivation. 2. Giant Wilson loops In the duality between large N gauge field theory and string theory, the expectation value of the Wilson loop normally corresponds to a disc amplitude for an open fundamental string. This has been made precise for the Maldacena-Wilson loop10
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of the N = 4 theory,
D E H µ I I ˙ )|) WM [C] = TrPe C dτ (iAµ (x)x˙ (τ )+φ (x)θ |x(τ
(1)
where xµ (τ ) parametrizes a closed curve C. φI (x), I = 1, ..., 6 are the scalar quark fields of N = 4 super Yang-Mills theory and θ I is a unit 6-vector. When λ is large, this loop is computed by finding the minimum area of a fundamental string worldsheet which has boundary located on the contour C placed at the boundary of AdS 5 and at a point θI on S 5 . If the contour C in (1) links periodic Euclidean time in the finite temperature Yang-Mills theory, WM [C] carries center charge and it can be nonzero only when the center symmetry is spontaneously broken. Then, the boundary of the disc must links periodic Euclidean time in AdS background. Whether such a disc exists depends on whether the time circle is contractible. It is not contractible in the hot AdS background, and it is contractable on the black hole background. This was pointed out by Witten as further evidence for the identification of the black hole transition with deconfinement.1 It is known that, in the zero temperature Yang-Mills theory defined on a spatial R3 , an interesting phenomenon occurs for loops in large representations. When the k is finite in the large number of boxes k in the Young tableau is large so that N N limit, the dual fundamental string worldsheet is replaced by a probe D-brane with k units of world-volume electric flux,11 studying 21 -BPS loops, where some results are known for all values of the coupling constant.12 For the anti-symmetric representation, the dual is a D5-brane whose world volume is a direct product of AdS2 ⊂ AdS5 and S 4 ⊂ S 5 . For a symmetric representation, it is a D3-brane with world volume AdS2 × S 2 ⊂ AdS5 . The interesting question of whether these D-branes exist in the finite temperature geometry, particularly where they would be dual to a gauge theory loop linking periodic Euclidean time was studied by Hartnoll and Kumar5 who searched for solutions of the appropriate Born-Infeld actions on the AdS black hole background. For the D5-brane wrapped on S 4 ⊂ S 5 which corresponds to a totally antisymmetk ric representation on the gauge theory side, there seem to be solutions for any N with the usual cutoff at k = N dictated by the maximum size of an antisymmetric representation in gauge theory side and a maximum radius for embedding S 4 in S 5 supergravity. However, in the case of the D3-brane, which should correspond to a totally symmetric representation, Hartnoll and Kumar could not find any solutions at all. This fundamental difference between the two cases is what motivated our work on the gauge theory which is summarized in Ref.4 and which we are reviewing here. 3. Effective field theory The confinement problem in N = 4 super Yang-Mills theory on spatial S 3 can be studies at weak coupling,2,3 Due to the curvature of S 3 , the vector, scalar and spinor fields are gapped. The temporal component of the gauge field A0 has a zero
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mode on the S 3 . Then, in the regime where the temperature is much less than the gap, T = β1 0, around z = zc [for convenience, we assume that z > zc (z < zc ) in the nonsymmetric (symmetric) phase].a The conformal phase transition (CPhT), whose conception was introduced in Ref. 3, is a very different continuous phase transition. It is defined as a phase transition in which an order parameter X is given by Eq. (1) where f (z) has such an essential singularity at z = zc that while lim f (z) = 0
z→zc
(2)
as z goes to zc from the side of the non-symmetric phase, lim f (z) 6= 0 as z → zc from the side of the symmetric phase (where X ≡ 0). Notice that since the relation (2) ensures that the order parameter X → 0 as z → zc , the phase transition is continuous. a Strictly speaking, Landau considered the mean-field phase transition. By the Landau phase transition, we understand a more general class, when fields may have anomalous dimensions. 2
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There are the following basic differences between the Landau phase transition (LPhT) and the CPhT one:3 (1) In the case of the LPhT, masses of light excitations are continuous functions of the parameters z around the critical point z = zc (though they are nonanalytic at z = zc ). In the case of the CPhT, the situation is different: there is an abrupt change of the spectrum of light excitations, as the critical point z = zc is crossed. This implies that the effective actions describing low energy dynamics in the phases with z < zc and z > zc are different in a system with CPhT. (2) Unlike the LPhT, the parameter z governing the CPhT is connected with a marginal operator [in the LPhT phase transition, such a parameter is connected with a relevant operator; it is usually a mass term]. (3) The fact that the parameter z is connected with a marginal operator in the CPhT implies that in the continuum limit, when z → zc + 0, the conformal symmetry is broken by a marginal operator in nonsymmetric phase, i.e., there is a conformal anomaly. (4) Unlike the LPhT, in the case of CPhT, the structures of renormalizations (i.e., the renormalization group at high momenta) are different in symmetric phase and nonsymmetric one. In relativistic field theory, the CPhT is realized in the two dimensional GrossNeveu (GN) model4 at the critical coupling constant gc = 0, reduced (or defect) QED,5,6 and quenched QED.7–10 It was suggested that the chiral phase transition with respect to the number of fermion flavors Nf in QCD is a CPhT one.3,11 In condensed matter physics, a CPhT like phase transition is realized in the BerezinskiiKosterlitz-Thouless (BKT) model12 and, possibly, graphene.13 Recently, the interest to the dynamics with the CPht phase transition has essentially increased. It is in particular connected with a progress in numerical lattice studies of gauge theories with a varied number of fermion flavors (for a recent review, see Ref. 14), the revival of the interest to the electroweak symmetry breaking based on the walking technicolor like dynamics15,16 (for a recent review, see Ref.17 ), and intensive studies of graphene, a single atomic layer of graphite (for a review, see Ref. 18). 2. Dynamics in the conformal window in QCD-like theories 2.1. General description In this section, we will consider the problem of the existence of a nontrivial conformal dynamics in 3+1 dimensional non-supersymmetric vector like gauge theories, with a relatively large number of fermion flavors Nf . We will discuss their phase diagram in the (α(0) , Nf ) plane, where α(0) is the bare coupling constant. We also discuss a clear signature for the conformal window in lattice computer simulations of these theories suggested quite time ago in Ref. 19.
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The roots of this problem go back to a work of Banks and Zaks20 who were first to discuss the consequences of the existence of an infrared-stable fixed point α = α ∗ for Nf > Nf∗ in vector-like gauge theories.21 The value Nf∗ depends on the gauge group: in the case of SU(3) gauge group, Nf∗ = 8 in the two-loop approximation. In Nineties, a new insight in this problem3,11 was, on the one hand, connected with using the results of the analysis of the Schwinger-Dyson (SD) equations describing chiral symmetry breaking in quenched QED7–10 and, on the other hand, with the discovery of the conformal window in N = 1 supersymmetric QCD.22 In particular, Appelquist, Terning, and Wijewardhana11 suggested that, in the case of the gauge group SU(Nc ), the critical value Nfcr ' 4Nc separates a phase with no confinement and chiral symmetry breaking (Nf > Nfcr ) and a phase with confinement and with chiral symmetry breaking (Nf < Nfcr ). The basic point for this suggestion was the observation that at Nf > Nfcr the value of the infrared c π , presumably needed to fixed point α∗ is smaller than a critical value αcr ' N2N 2 c −1 3 7–10 generate the chiral condensate. The authors of Ref. 11 considered only the case when the running coupling constant α(µ) is less than the fixed point α∗ . In this case the dynamics is asymptotically c free (at short distances) both at Nf < Nfcr and Nfcr < Nf < Nf∗∗ ≡ 11N 2 . Yamawaki 3 (0) and the author analyzed the dynamics in the whole (α , Nf ) plane and suggested the (α(0) , Nf )-phase diagram of the SU(Nc ) theory, where α(0) is the bare coupling constant (see Fig 1 below).b In particular, it was pointed out that one can get an interesting non-asymptotically free dynamics when the bare coupling constant α(0) is larger than α∗ , though not very large. The dynamics with α(0) > α∗ admits a continuum limit and is interesting in itself. Also, its better understanding can be important for establishing the conformal window in lattice computer simulations of the SU(Nc ) theory with such large values of Nf . In order to illustrate this, let us consider the following example. For Nc = 3 and Nf = 16, the value of the infrared fixed point α∗ calculated in the two-loop approximation is small: α∗ '0.04. To reach the asymptotically free phase, one needs to take the bare coupling α(0) less than this value of α∗ . However, because of large finite size effects, the lattice computer simulations of the SU(3) theory with such a small α(0) would be unreliable. Therefore, in this case, it is necessary to consider the dynamics with α(µ) > α∗ . In Ref. 19, this author suggested a clear signature of the existence of the infrared fixed point α∗ , which in particular can be useful for lattice computer simulations. The signature is based on two characteristic features of the the spectrum of low energy excitations in the presence of a bare fermion mass in the conformal window: a) a strong (and simple) dependence of the masses of all the colorless bound states (including glueballs) on the bare fermion mass, and b) unlike QCD with a small Nf (Nf = 2 or 3), glueballs are lighter than bound states composed of fermions, if the value of the infrared fixed point is not too large. b This
phase diagram is different from the original Banks-Zaks diagram.20
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2.2. Phase diagram The phase diagram in the (α(0) , Nf )-plane in the SU (Nc ) gauge theory is shown in Fig. 1. The left-hand portion of the curve in this figure coincides with the line of the infrared-stable fixed points α∗ (Nf ):21 b α(0) = α∗ = − , c
(3)
where 1 (11Nc − 2Nf ), 6π N2 − 1 1 (34Nc2 − 10Nc Nf − 3 c Nf ). c= 2 24π Nc
b=
(4) (5)
It separates two symmetric phases, S1 and S2 , with α(0) < α∗ and α(0) > α∗, c π respectively. Its lower end is Nf = Nfcr (with Nfcr ' 4Nc if αcr ' N2N ): at 2 c −1 3 ∗ cr Nf < Nf < Nf the infrared fixed point is washed out by generating a dynamical fermion mass (here Nf∗ is the value of Nf at which the coefficient c in Eq. (5) becomes positive and the fixed point disappears). The horizontal, Nf = Nfcr , line describes a phase transition between the symmetric phase S1 and the phase with confinement and chiral symmetry breaking. As it was suggested in Ref.11, based on a similarity of this phase transition with that in quenched QED4 ,8,9 there is the following scaling law for m2dyn : C , (6) m2dyn ∼ Λ2cr exp − q ∗ α (Nf ) − 1 αcr
where the constant C is of order one and Λcr is a scale at which the running coupling is of order αcr . It is a CPhT phase transition with an essential singularity at Nf = Nfcr . At last, the right-hand portion of the curve on the diagram occurs because at large enough values of the bare coupling, spontaneous chiral symmetry breaking takes place for any number Nf of fermion flavors. This portion describes a phase transition called a bulk phase transition in the literature, and it is presumably a first order phase transition. c The vertical line ends above Nf =0 since in pure gluodynamics there is apparently no phase transition between weak-coupling and strong-coupling phases. 2.3. Signature for the conformal window Up to now we have considered the case of a chiral invariant action. But how will the dynamics change if a bare fermion mass term is added in the action? This question c The fact that spontaneous chiral symmetry breaking takes place for any number of fermion flavors, if α(0) is large enough, is valid at least for lattice theories with Kogut-Susskind fermions. Notice however that since the bulk phase transition is a lattice artifact, the form of this portion of the curve can depend on the type of fermions used in simulations.
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**
Nf
S2
cr
S1
Nf
A *
Nf
A
2
g(0) cr
g(0) Figure 3
Fig. 1. The phase diagram in an SU(Nc ) gauge model. The coupling constant g (0) = and S and A denote symmetric and asymmetric phases, respectively.
√ 4πα(0)
is in particular relevant for lattice computer simulations: for studying a chiral phase transition on a finite lattice, it is necessary to introduce a bare fermion mass. As was pointed out in Ref.,19 adding even an arbitrary small bare fermion mass results in a dramatic changing the dynamics both in the S1 and S2 phases. Recall that in the case of confinement SU(Nc ) theories, with a small, Nf < Nfcr , number of fermion flavors, the role of a bare fermion mass m(0) is minor if m(0) Nfcr the effective coupling is relatively weak,
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it is impossible to form bound states from massless fermions and gluons (recall that the S1 and S2 phases are chiral invariant). Therefore the absence of a mass for fermions and gluons is a key point for not creating bound states in those phases. The situation changes dramatically if a bare fermion mass is introduced: indeed, even weak gauge, Coulomb-like, interactions can easily produce bound states composed of massive constituents, as it happens, for example, in QED, where electron-positron (positronium) bound states are present. To be concrete, let us consider the case when all fermions have the same bare mass m(0) . It leads to a mass function m(q 2 ) ≡ B(q 2 )/A(q 2 ) in the fermion propagator G(q) = (ˆ q A(q 2 ) − B(q 2 ))−1 . The current fermion mass m is given by the relation m(q 2 )|q2 =m2 = m.
(7)
For the clearest exposition, let us consider a particular theory with a finite cutoff Λ and the bare coupling constant α(0) = α(q)|q=Λ being not far away from the fixed point α∗ . Then, the mass function is changing in the “walking” regime16 with α(q 2 ) ' α∗ . It is γm M (8) m(q 2 ) ' m(0) q ¯ γm = 3 − d ¯ with where γm is the anomalous dimension of the operator ψψ: ψψ dψψ ¯ being the dynamical dimension of this operator. In the walking regime, γ m ' ∗ 1 − (1 − ααcr )1/2 (see Refs. 9,16). Eqs.(7) and (8) imply that m'Λ
m(0) Λ
1+γ1 m
.
(9)
Recall that the anomalous dimension γm ≥ 0, and γm . 2 in the “walking” regime. There are two main consequences of the presence of the bare mass: (a) bound states, composed of fermions, occur in the spectrum of the theory. The mass of a n-body bound state is M (n) ' nm. Therefore they satisfy the scaling 1+γ1 m . (10) M (n) ' nm ∼ n m(0) (b) At low momenta, q < m, fermions and their bound states decouple. There is a pure SU(Nc ) Yang-Mills theory with confinement. Its spectrum contains glueballs. To estimate glueball masses, notice that at momenta q < m, the running of the coupling is defined by the parameter ¯b of the Yang-Mills theory, ¯b = 11 Nc . 6π
(11)
Therefore the glueball masses Mgl are of order 1 ΛY M ' m exp(− ¯ ∗ ). bα
(12)
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For Nc = 3, we find from Eqs.(4), (5), and (11) that exp(− ¯bα1∗ ) is 6 × 10−7 , 2 × 10−2 , 10−1 , and 3 × 10−1 for Nf =16, 15, 14, and 13, respectively. Therefore at Nf =16, 15 and 14, the glueball masses are essentially lighter than the masses of the bound states composed of fermions. The situation is similar to that in confinement QCD with heavy (nonrelativistic) quarks, m >> ΛQCD . However, there is now a new important point. In the conformal window, any value of m(0) (and therefore m) is “heavy”: the fermion mass m sets a new scale in the theory, and the confinement scale ΛY M (12) is less, and rather often much less, than this scale m. One could say that the latter plays a role of a dynamical ultraviolet cutoff for the pure YM theory. This leads to a spectacular “experimental” signature of the conformal window in lattice computer simulations: the masses of all colorless bound states, including 1 glueballs, decrease as (m(0) ) 1+γm with the bare fermion mass m(0) for all values of m(0) less than cutoff Λ. Moreover, one should expect that glueball masses are lighter than the masses of the bound states composed of quarks. Few comments are in order: (1) The phases S1 and S2 have essentially the same long distance dynamics. They are distinguished only by their dynamics at short distances: while the dynamics of the phase S1 is asymptotically free, that of the phase S2 is not. Also, while around the infrared fixed point α∗ the sign of the beta function is negative in S1 , it is positive in S2 .3 When all fermions are massive (with the current mass m), the continuum limit Λ → ∞ of the S2 -theory is a non-asymptotically free confinement theory. Its spectrum includes colorless bound states composed of fermions and gluons. For q < m the running coupling α(q) is the same as in pure SU(Nc ) Yang-Mills theory, and for all q m α(q) is very close to α∗ (“walking”, actually, “standing” dynamics). For those values Nf for which α∗ is small (as Nf =16, 15 and 14 at Nc =3), glueballs are much lighter than the bound states composed of fermions. Notice that unlike the case with m = 0, corresponding to the unparticle dynamics,23 there exists a conventional S-matrix in this theory. (2) In order to get the clearest exposition, we assumed such estimates as Nfcr ' q ∗ 4Nc for Nfcr and γm = 1 − 1 − ααcr for the anomalous dimension γm . While the
latter should be reasonable for α∗ < αcr (and especially for α∗ 4, keeping the NGB Compton wavelength well inside the lattice. For Nf = 6, there are also results for mf = 0.025, 0.03, but
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2
2
1.5 Mm / 2 m
1.5
2
Rm
Nf=6 / Nf=2
1 0.5
1 0
0.005
0.01! m
0.015
0 0
0.02
2 /2mF 2 2 m ]6f /[Mm 2 Fig. 6. From.29 Rm ≡ [Mm ]2f , versus m ≡ (m(2f ) + m(6f ))/2, showing Rm ≡ [M /2mFm/2mF ] N m/[M m ≡ m /2mFm ]2f enhancement of hψψi/F 3 at Nfm = 6 relative to6f f = 2. The open symbol at m = 0.005 denotes the(m(2f presence )of + possible systematic m(6f ))/2 errors. "ψψ#/F 3
FIG. 1:
at
FIG. 2: The slope lattice units, as a is a joint fit to M
at Nf = 2. The resulting chiral extrapolation of these quantities is shown in Fig. 5, and shows good agreement, so that the lattice cutoffs are well matched between Nf = 2 and Nf = 6. Determination of thelittle presence condensate enhancement is done and Fm change in ortheabsence rangeof m f = 0.01 − 0.02, inthrough comparison of the quantity hψψi/F 3 between the Nf = 6 and Nf = 2 dicating along with the Nf M dependence in Eq. 3 that NLO 2 theories, by way of the equivalent ratio m /(2mFm ). We can directly construct a “ratio of ratios” χPT should provide a reliable fit. Our results for "ψψ#m 2 /2mF M m m show the expectedRdominance ofNfthe =6 large linear contact , (17) m ≡ 2 [Mm /2mFthe m ]Nf use =2 term (which does not preclude of χPT). We there-
mf = 0.01 − 0.0
A value of Rmout > 1 then of the condensate fore carry the implies Nf =enhancement 2 extrapolation to m as=Nf0increases. using The result is shown in Fig. 6, and indicates that Rm > 1.5 in the chiral limit, barring ∼ the combined NLO chiral expansions of Eqs. (1) – (3). The a downturn in Rm - an unlikely outcome, as the curvature of the NLO logarithm in Figs. 4, leads to extrapois five-parameter naturally upwards infit, the shown chiral expansion of R2m -itself. The magnitude of Rm 2 is significant and larger than expected; an MS perturbation theory estimate of the , lated values F = 0.0209(41) and "ψψ#/F = 0.99(17) enhancement from Nf = 2 to 6 by integrating the anomalous dimension of the mass giving the NLO χPT result operator γm leads to an expected increase on the order of 5-10%. Some care must be taken in comparing this value to our lattice result, since the condensate hψψi and by "ψψ# extension hψψi/F 3 depends on the renormalization scheme chosen. The conversion Nf = 2 : = 47.1(17.6). (4) MS factor Z between the lattice-cutoff scheme F 3 with domain wall fermions and MS
Fm
, versus , indicating enhancement of Nf = 6 relative to Nf = 2.
The other fit parameters are αM = 0.31(62), αF = 0.64(47) and αC = 83(29). The values of the fit parameters excluding αC indicate that we are near the limit of applicability of χPT. We note also that χ2 /d.o.f. = 6.50 with an uncorrelated fit and 4 degrees of freedom. We next compare our Nf = 2 results for "ψψ#/F 3 and Mρ /F to the QCD quantities "qq#/fπ3 and mρ /fπ , a reasonable comparison since the light-quark masses mq
0.05 0.04 0.03 0.02
N N
0.01 0 0
FIG. 3: The Gold as a function of 2 fit to Mm , Fm a 0.01 − 0.02, cons 0.04 0.001
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is known from Ref.30 From that reference, for this simulation the required factor to convert Rm is Z6MS /Z2MS = 1.449(29)/1.227(11) = 1.18(3). This increases the perturbative estimate of expected enhancement to the order of 20 − 30%, so the observed Rm > ∼ 1.5 is still significantly larger than anticipated. A direct computation of the S-parameter is also important to the study of general Yang-Mills theories with a focus on technicolor, and is well within the reach of existing lattice techniques. Some results have been reported,31,32 and more work is underway by the LSD collaboration. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.
T. van Ritbergen, J. A. M. Vermaseren and S. A. Larin, Phys. Lett. B400, 379 (1997). W. E. Caswell, Phys. Rev. Lett. 33, p. 244 (1974). T. Banks and A. Zaks, Nucl. Phys. B196, p. 189 (1982). M. A. Luty, JHEP 04, p. 050 (2009). B. Holdom, Phys. Lett. B150, p. 301 (1985). K. Yamawaki, M. Bando and K.-i. Matumoto, Phys. Rev. Lett. 56, p. 1335 (1986). T. W. Appelquist, D. Karabali and L. C. R. Wijewardhana, Phys. Rev. Lett. 57, p. 957 (1986). M. L¨ uscher, R. Narayanan, P. Weisz and U. Wolff, Nucl. Phys. B384, 168 (1992). S. Sint, Nucl. Phys. B421, 135 (1994). A. Bode, P. Weisz and U. Wolff, Nucl. Phys. B576, 517 (2000), Erratumibid.B608:481,2001. T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. Lett. 100, p. 171607 (2008). T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. D79, p. 076010 (2009). E. Bilgici et al. (2009). E. Bilgici et al., Phys. Rev. D80, p. 034507 (2009). Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, Phys. Lett. B681, 353 (2009). M. L¨ uscher, P. Weisz and U. Wolff, Nucl. Phys. B359, 221 (1991). S. Caracciolo, R. G. Edwards, S. J. Ferreira, A. Pelissetto and A. D. Sokal, Phys. Rev. Lett. 74, 2969 (1995). A. Bode et al., Phys. Lett. B515, 49 (2001). R. Sommer (2006). M. L¨ uscher, R. Sommer, P. Weisz and U. Wolff, Nucl. Phys. B413, 481 (1994). U. M. Heller, Nucl. Phys. B504, 435 (1997). F. Bursa, L. Del Debbio, L. Keegan, C. Pica and T. Pickup, Phys. Rev. D81, p. 014505 (2010). X.-Y. Jin and R. D. Mawhinney, PoS LAT2009, p. 049 (2009). L. Del Debbio, B. Lucini, A. Patella, C. Pica and A. Rago, Phys. Rev. D80, p. 074507 (2009). D. B. Leinweber, A. W. Thomas, K. Tsushima and S. V. Wright, Phys. Rev. D64, p. 094502 (2001). R. Sommer, Nucl. Phys. B411, 839 (1994). J. Gasser and H. Leutwyler, Phys. Lett. B184, p. 83 (1987). J. Bijnens and J. Lu, JHEP 11, p. 116 (2009). T. Appelquist, A. Avakian, R. Babich, R. C. Brower, M. Cheng, M. A. Clark, S. D. Cohen, G. T. Fleming, J. Kiskis, E. T. Neil, J. C. Osborn, C. Rebbi, D. Schaich and P. Vranas, Phys. Rev. Lett. 104, p. 071601 (2010).
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30. S. Aoki, T. Izubuchi, Y. Kuramashi and Y. Taniguchi, Phys. Rev. D67, p. 094502 (2003). 31. E. Shintani et al., Phys. Rev. Lett. 101, p. 242001 (2008). 32. P. A. Boyle, L. Del Debbio, J. Wennekers and J. M. Zanotti, Phys. Rev. D81, p. 014504 (2010).
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Conformal Higgs, or Techni-Dilaton — Composite Higgs Near Conformality Koichi Yamawaki∗ Department of Physics, Nagoya University Nagoya, Japan E-mail: [emailprotected] In contrast to the folklore that Technicolor (TC) is a “Higgsless theory”, we shall discuss existence of a composite Higgs boson, Techni-Dilaton (TD), a pseudo-Nambu-Goldstone boson of the scale invariance in the Scale-invariant/Walking/Conformal TC (SWC TC) which generates a large anomalous dimension γm 1 in a wide region from the dynamical mass m = O (TeV) of the techni-fermion all the way up to the intrinsic scale ΛTC of the SWC TC (analogue of ΛQCD ), where ΛTC is taken typically as the scale of the Extended TC scale ΛETC : ΛTC ΛETC ∼ 103 TeV ( m). All the techni-hadrons have mass on the same order O(m), which in SWC TC is extremely smaller than the intrinsic scale ΛTC ΛETC , in sharp contrast to QCD where both are of the same order. The mass of TD arises from the non-perturbative scale anomaly associated with the techni-fermion mass generation and is typically 500-600 GeV, even smaller than other techni-hadrons of the same order of O(m), in another contrast to QCD which is believed to have no scalar q¯q bound state lighter than other hadrons. We discuss the TD mass in various methods, Gauged NJL model via ladder Schwinger-Dyson (SD) equation, straightforward calculations in the ladder SD/ Bethe-Salpeter equation, and the holographic approach including techni-gluon condensate. The TD may be discovered in LHC. Keywords: Walking technicolor, scale invariance, conformal symmetry, techni-dilaton, fixed point, composite Higgs, large anomalous dimension, holographic gauge theory.
1. Introduction Toshihide Maskawa is famous for 2008 Nobel prize-winning paper with Makoto Kobayashi on CP violation but did also fundamental contributions particularly to the SCGT, the topics of this workshop: Back in 1974 he found with Hideo Nakajima1 that spontaneous chiral symmetry breaking (SχSB) solution does exists for and only for the strong gauge coupling, with the critical coupling of order 1, based on the ladder Schwinger-Dyson (SD) equation with non-running (scale-invariant) coupling, namely the walking gauge dynamics what is called today. This turned out to be the origin of SCGT activities toward understanding the Origin of Mass. The present workshop SCGT 09 was held in honor of his 70th birthday on February 7, 2010 ∗ Present
Address: Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI), Nagoya University.
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and the 35th anniversary of his crucial contributions to SCGT. I will later explain impact of Maskawa-Nakajima solution on the conformal gauge dynamics. The Origin of Mass is the most urgent issue of the particle physics today and is to be resolved at the LHC experiments. In the standard model (SM), all masses are attributed to a single parameter of the vacuum expectation value (VEV), H of the hypothetical elementary particle, the Higgs boson. The VEV simply picks up the mass scale of the input parameter M0 which is tuned to be tachyonic (M02 < 0) in such a way as to tune H 246 GeV (“naturalness problem”). As such SM does not explain the Origin of Mass. Technicolor (TC)2 is an attractive idea to account for the Origin of Mass without introducing ad hoc Higgs boson and tachyonic mass parameter: The mass arises dynamically from the condensate of the techni-fermion and the anti technifermion pair T¯T which is triggered by the attractive gauge forces between the pair analogously to the quark-antiquark condensate ¯ q q in QCD. For the TC with SU (NTC ) gauge symmetry and Nf flavors (Nf /2weak doublets) of technifermions, the techni-pion decay constant Fπ = H/ Nf /2 corresponds to the pion decay constant fπ 93 MeV in QCD, and hence the TC may be a scale-up of QCD by the factor Fπ /fπ 2650/ Nf /2. Then the mass scale of the condensate Λχ = (−T¯T /NTC )1/3 as the Origin of Mass may be estimated as √ 1/3 1/2 −¯ q q Nc /NTC Fπ / NTC √ Λχ · 450 GeV · , (1) Nc Nf /2 fπ / Nc where we have used a typical value (−¯ q q)1/3 250 MeV (Nc = 3). The dynamically generated mass scale of the condensate Λχ , or the dynamical mass of the techni-fermion, m (∼ Λχ ∼ Fπ ), in fact picks up the intrinsic mass scale ΛTC of the theory (analogue of ΛQCD in QCD) already generated by the scale anomaly through quantum effects (“dimensional transmutation”) in the gauge theory which is scale-invariant at classical level (for massless flavors): α(µ) α(Λ0 ) dα dα = Λ0 · exp − , (2) ΛTC = µ · exp − β(α) β(α) where the running of the coupling constant α(µ), with non-vanishing beta function β(α) ≡ µ dα(µ) dµ = 0, is a manifestation of the scale anomaly and Λ0 is a fundamental scale like Planck scale. Note that ΛTC is independent of the renormalization point µ, dΛdµTC = 0, and can largely be separated from Λ0 through logarithmic running (“naturalness”). Thus the Origin of Mass is eventually the quantum effect in this picture: In the simple scale-up of QCD we would have Naturalness (QCD scale up) :
m ∼ Λχ ∼ ΛTC Λ0 .
(3)
The original version of TC, just a simple scale-up of QCD, however, is plagued by the notorious problems: Excessive flavor-changing neutral currents (FCNCs), and excessive oblique corrections of O(1) to the Peskin-Takeuchi S parameter3 compared with the typical experimental bound about 0.1.
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The FCNC problem was resolved long time ago by the TC based on the near conformal gauge dynamics with γm 1,4,5 initially dubbed “scale-invariant TC” and then “walking TC”, with almost non-running (conformal) gauge coupling, based on the pioneering work by Maskawa and Nakajima1 who discovered nonzero critical coupling, αcr (= 0), for the SχSB to occur. We may call it “Scaleinvarinat/Walking/Conformal TC” (SWC TC) (For reviews see Ref.6 ). In addition to solving the FCNC problem, the theory made a definite prediction of “Techni-dilaton (TD)”,4 a pseudo Nambu-Goldstone (NG) boson of the spontaneous breaking of the (approximate) scale invariance of the theory. This will be the main topics of this talk in the light of modern version of SWC TC. The modern version7–9 of SWC TC is based on the Caswell-Banks-Zaks (CBZ) infrared (IR) fixed point 10 , α∗ = α∗ (Nf , NTC ), which appears at two-loop beta function for the number of massless flavors Nf (< 11NTC /2) larger than a certain number Nf∗ ( NTC ). See Fig. 1 and later discussions. Due to the IR fixed point the
«
¬
« «
¼
Á
ÁÁ
ÁÁÁ
Fig. 1.
The beta function and α(µ) for SWC-TC.
coupling is almost non-running (“walking”) all the way up to the intrinsic scale ΛTC which is generated by the the scale anomaly associated with the (two-loop) running of the coupling analogously to QCD scale-up in Eq. (2). For µ > ΛTC (Region I of Fig. 1) the coupling no longer walks and runs similarly to that of QCD. When we set α∗ slightly larger than αcr , we have a condensate or the dynamical mass of the techni-fermion m (∼ Λχ ), much smaller than the intrinsic scale of the theory m ΛTC . The CBZ-IR fixed point α∗ actually disappears (then becoming wouldbe IR fixed point) at the scale µ < ∼ m where the techni-fermions have acquired the mass m and get decoupled from the beta function for µ < m (Region III in Fig. 1). Nevertheless, the coupling is still walking due to the remnant of the CBZ-IR fixed point conformality in a wide region m < µ < ΛTC (Region II in Fig. 1). Thus the symmetry responsible for the natural hierarchy m ∼ Λχ ΛTC is the (approximate) conformal symmetry, while the naturalness for the hierarchy ΛTC Λ0 is the same as that of QCD scale-up in Eq.(3): Naturalness (SWC TC) :
m ∼ Λχ ΛTC
( Λ0 ) .
(4)
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The theory acts like the SWC-TC4,5 : It develops a large anomalous dimension γm 1 for the almost non-running coupling in the Region II .8,9 Here ΛTC plays a role of cutoff Λ identified with the ETC scale: ΛTC = Λ = ΛETC . Moreover, there also exists a possibility11,12 that the S parameter can be reduced in the case of SWC-TC. In this talk I will argue13 a that in contrast to the simple QCD scale-up which is widely believed to have no composite Higgs particle (“higgsless”), a salient feature of SWC TC is the conformality which manifests itself by the appearance of a composite Higgs boson (“conformal Higgs”) as the Techni-dilaton (TD)4 with mass relatively lighter than other techni-hadrons: MTD < Mρ , Ma1 · · · = O(Λχ ) ΛTC = ΛETC , where Mρ , Ma1 · · · denote the mass of techni-ρ, techni-a1 , etc. This is contrasted to the QCD dynamics where there are no scalar bound states lighter than others. Note that there is no idealized limit where the TD becomes exactly massless to be a true NG boson, in sharp contrast to the chiral symmetry breaking. Scale symmetry is always broken explicitly as well as spontaneously b . For the phenomenological purpose, I will argue through several different calculations13,15,16 that the techni-dilaton mass in the typical SWC TC models will be in the range (see the footnote below Eq.(30), however): mTD = 500 − 600 GeV,
(5)
which is definitely larger than the SM Higgs bound but still within the discovery region of the LHC experiments. 2. Scale-Invariant/Walking/Conformal Technicolor Let us briefly review the SWC TC. The FCNC problem is related with the mass generation of quarks/leptons mass. In order to communicate the techni-fermion condensate to the quarks/leptons masses mq/l , we would need interactions between the quarks/leptons and the technifermions which are typically introduced through Extended TC (ETC)17 c with much T¯T ΛETC , where T¯T ΛETC is the condenhigher scale ΛETC ( Λχ ): mq/l ∼ Λ−1 2 ETC sate measured at the scale of ΛETC . (We here do not refer to the origin of the mass scale ΛETC which should also be of dynamical origin such as the tumbling. ) Since the newly introduced ETC interactions characterized by the same scale ΛETC should induce extra FCNC’s, we should impose a constraint ΛETC > 106 GeV in a Preliminary
discussions on the revival of the techni-dilaton4 were given in several talks14 . straightforward calculations near the conformal edge indicated16 that there is no isolated massless spectrum: MTD /Fπ , MTD /Mρ , · · · → const. = 0 even in the limit of α∗ → αcr (Nf → Nfcrit ) where Fπ /ΛTC , MTD /ΛTC , Mρ /ΛTC , · · · → 0. In the case of holographic TD,13 this fact is realized in a different manner: Although there apparently exists an isolated massless spectrum, MTD /Fπ → 0 while Mρ /Fπ , Ma1 /Fπ → const. = 0, the decay constant of the TD diverges FTD /Fπ → ∞ in that limit and hence it gets decoupled. See later discussions. c The same can be done in a composite model where quarks/leptons and techni-fermions are composites on the same footing.18 b The
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order to avoid the excessive FCNC’s (typically involving s quark). If we assume a simple QCD scale up, T¯T ΛETC T¯T Λχ = −NTC · Λ3χ , we would have mq/l ∼
Λ3χ · NTC < 0.1 MeV · NTC Λ2ETC
Nc /NTC Nd
3/2
,
(6)
which implies that the typical mass (s-quark mass) would be roughly 10−3 smaller than the reality. We would desperately need 103 times enhancement. This was actually realized dynamically by the TC based on the near conformal gauge dynamics,4,5 based on the Maskawa-Nakajima solution1 of the (scaleinvariant) ladder Schwinger-Dyson (SD) equation for fermion full propagator SF (p) p − B(p2 ) with non-running (conformal, an ideal parameterized as iSF−1 (p) = A(p2 )/ limit of the “walking”) gauge coupling, α(Q) ≡ α = constant, with Q2 ≡ −p2 > 0. (See Fig. 2) Maskawa and Nakajima discovered that the SχSB can only take place for strong coupling α > -1 αcr = O(1), non-zero critical coui SF ( p ) p = pling.d The critical value reads:20 p q C2 (F )αcr = π/3, or full fermion propagator 2 αcr = (π/3) · 2NTC /(NTC − 1) (7)
Fig. 2. Graphical expression of the SD equation in the ladder approximation.
in the SU (NTC ) gauge theory, where C2 (F ) is the quadratic Casimir of the techni-fermion representation of the TC. The asymptotic form of the Maskawa-Nakajima SχSB solution of the fermion mass function Σ(Q) = B(p2 )/A(p2 ) in Landau gauge (A(p2 ) ≡ 1) reads,1,20 Σ(Q) ∼ 1/Q (Q Λχ ) .
(8)
We then proposed a “Scale-invariant TC” 4 , based on the observation that Eq.(8) implies a special value of the anomalous dimension γm = −Λ
∂ ln Zm = 1, ∂Λ
(9)
to be compared with the operator product expansion (OPE), Σ(Q) ∼ 1/Q2 · −1 · (Q/Λχ )γm . Accordingly, we had an enhanced condensate T¯T ΛETC = Zm 2 ¯ T T Λχ −NTC (ΛETC Λχ ), with the (inverse) mass renormalization constant being −1 = (ΛETC /Λχ )γm ΛETC /Λχ 103 , which in fact yields the desired enhanceZm ment. We actually obtained a different formula than Eq.(6):4 mq/l ∼ d Earlier
Λ2χ · NTC . ΛETC
(10)
works19 in the ladder SD equation with non-running coupling all confused explicit breaking solution with the SSB solution and thus implied αcr = 0
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The model4 was formulated in terms of the Renormalization Group Equation (RGE) a la Miransky 21 for the Maskawa-Nakajima solution Σ(m) = m which takes the form20,21 Λχ ∼ m ∼ 4Λ exp −π/ α/αcr − 1 , (11) where Λ is the cutoff of the SD equation. This has an essential singularity often called “Miransky scaling” and implies the non-perturbative beta function having a multiple zero e : β(α)NP = Λ
2 ∂α(Λ) =− ∂Λ 3C2 (F )
α −1 αcr
3/2
,
(12)
with the critical coupling αcr identified with a nontrivial ultraviolet (UV) stable fixed point α = α(Λ) → αcr as Λ/m → ∞. Subsequently, similar enhancement effects of the condensate were also studied5 within the same framework of the ladder SD equation, without use of the RGE concepts of anomalous dimension and fixed point, rather emphasizing the asymptotic freedom of the TC theories with slowly-running (walking) coupling which was implemented into the ladder SD equation (“improved ladder SD equation”). Today the Scale-invariant/Walking/Conformal TC (SWC TC) is simply characterized by near conformal property with γm 1 (For a review see Ref.6 ). Such a theory should have an almost non-running and strong gauge coupling (larger than a certain non-zero critical coupling for SχSB) to be realized either at UV fixed point or IR fixed point, or both (“fusion” of the IR and UV fixed points), as was characterized by “Conformal Phase Transition (CPT)”.9 The essential feature of the above is precisely what happens in the modern version 7–9 of the SWC TC based on the CBZ IR fixed point10 of the large Nf QCD, the QCD-like theory with many flavors Nf ( NTC ) of massless techni-fermions, f see Fig. 1. The two-loop beta function is given by d α(µ) = −bα2 (µ) − cα3 (µ), where b = (11NTC − 2Nf ) /(6π), c = β(α) = µ dµ
2 2 − 10Nf NTC − 3Nf (NTC − 1)/NTC /(24π 2 ) . When b > 0 and c < 0, i.e., 34NTC ∗ Nf∗ < Nf < 11 2 NTC (Nf 8.05 for NTC = 3), there exists an IR fixed point (CBZ IR fixed point) at α = α∗ , β(α∗ ) = 0, where α∗ = α∗ (NTC , Nf ) = −b/c.
(13)
Note that α∗ = α∗ (Nf , NTC ) → 0 as Nf → 11NTC /2 (b → 0) and hence there exists a certain range Nfcr < Nf < 11NTC /2 (“Conformal Window”) satisfying α∗ < αcr , where the gauge coupling α(µ) (< α∗ ) gets so weak that attractive forces are no longer strong enough to trigger the SχSB as was demonstrated by Maskawae Simple
zero of the beta function, β(α) ∼ (α − αcr )1 , never reproduces the essential singularity scaling, as is evident from Eq.(2). f For SWC TC based on higher representation/other gauge groups see, e.g., Ref.22
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Nakajima.1 Nfcr such that α∗ (NTC , Nfcr ) = αcr may be evaluated by using the value of αcr from the ladder SD equation Eq.(7): 8 Nfcr 4NTC (= 12 for NTC = 3) g . Here we are interested in the SχSB phase slightly off the conformal window, 0 < α∗ − αcr 1 (Nf Nfcr ). We may use the same equation as the ladder SD equation with α(µ) const. = α∗ , yielding the same form as Eq.(11) :8 m ∼ 4ΛTC exp −π/ α∗ /αcr − 1 ΛTC (α∗ αcr ) , (14) where the cutoff Λ was identified with ΛTC (= ΛETC ). We also have the same result as Eqs. (8),(9): Σ(Q) ∼ 1/Q ,
γm 1 .
(15)
Hence it acts like SWC TC. Incidentally, Eq.(14) implies a multiple zero at α∗ = αcr in a non-perturbative beta function for α∗ = α∗ (Λ) similar to Eq.(12), which would suggest “running” of the IR fixed point α∗ with its UV fixed point α∗ = αcr in the limit ΛTC /m → ∞. The actual running of the coupling largely based on two-loop perturbation is already depicted in Fig. 1. The critical coupling αcr can be regarded as the UV fixed point viewed from the IR part of the Region II (m < µ < µcr , with µcr such that α(µcr ) = αcr ), while it is regarded as the IR fixed point from the UV part of the Region II (ΛT C > µ > µcr ), with the Region II regarded as the fusion of the IR and UV fixed points in the idealized limit of non-running (perturbative) coupling in Region II (or ΛTC /m → ∞). Although the perturbative (two-loop) beta function has a simple zero, which never corresponds to the essential singularity scaling as we noted before, the coupling near αcr should be sensitive to the non-perturbative effects in such as way that the beta function looks like the multiple zero as in Eq. (12) from both sides, corresponding to the essential singularity scaling as in Eq. (11). This should be tested by the fully non-perturbative studies like lattice simulations. A possible phase diagram (Fig 3 of Ref.9 ) of the large Nf QCD on the lattice is also waiting for the test by simulations. 3. Conformal Phase Transition9,14 Such an essential singularity scaling law like Eq.(11),(14), or equivalently the multiple zero of the non-perturbative beta function, characterizes an unusual phase transition, what we called “Conformal Phase Transition (CPT)”, where the GinzburgLandau effective theory breaks down:9 Although it is a second order (continuous) phase transition where the order parameter m (α∗ > αcr ) is continuously changed to m = 0 in the symmetric phase (conformal window, α∗ < αcr ), the spectra do not, i.e., while there exist light composite particles whose mass vanishes at the critical g The value should not be taken seriously, since α = α ∗ cr is of O(1) and the perturbative estimate of α∗ is not so reliable there, although the chiral symmetry restoration in large Nf QCD has been supported by many other arguments, most notably the lattice QCD simulations,23,24 which however suggest diverse results as to Nfcr ; See e.g.,25 for recent results.
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point when approached from the side of the SSB phase, no isolated light particles do not exist in the conformal window, recently dubbed “unparticle”.26 This reflects the feature of the conformal symmetry in the conformal window. In fact explicit computations show no light (composite) spectra in the conformal window, in sharp contrast to the SχSB phase where light composite spectra do exist with mass of order O(m) which vanishes as we approach the conformal window Nf Nfcr .8,9,16 The essence of CPT was illustrated 9 by a simpler model, 2-dimensional GrossNeveu Model. This is the D → 2 limit of the D-dimensional Gross-Neveu model (2 < D < 4) which has the beta function and the anomalous dimension:27,28 β(g) = −2g(g − g∗ ),
γm = 2g ,
(16)
where g = g∗ (≡ D/2 − 1) = gcr and g = 0 are respectively the UV and IR fixed points of the dimensionless four-fermion coupling, g, properly normalized (as g∗ = 1 for the D = 4 NJL model). There exist light composites π, σ near the UV fixed point (phase boundary) g g∗ in both sides of symmetric (0 < g < g∗ ) and SSB (g > g∗ ) phases as in the NJL model. Now we consider D → 2 ( g∗ → 0) where we have a well-known effective potential: V (σ, π) ∼ (1/g−1)ρ2+ρ2 ln(ρ2 /Λ2 ), or ∂ 2 V /∂ρ2 |ρ=0 = −∞, where ρ2 = π 2 +σ 2 . This implies breakdown of the Ginzburg-Landau theory which distinguishes the SSB (< 0) and symmetric (> 0) phases by the signature of the finite ∂ 2 V /∂ρ2 at the critical point g = 0. Eq. (16 ) now reads: β(g) = −2g 2 , γm |g=0 = 0
(D = 2) ,
(17)
namely a fusion of the UV and IR fixed points at g = 0 as a result of multiple zero (not a simple zero) at g = 0. Now the symmetric phase is squeezed out to the region g < 0 (conformal phase) which corresponds to a repulsive four-fermion interaction and no composite states exist, while in the SSB phase (g > 0) there exists a composite state σ of mass Mσ = 2m where the dynamical mass of the fermion is given by m2 ∼ Λ2 exp(−1/g) → 0 (g → +0), which shows an essential singularity scaling, in accord with the beta function with multiple zero, β(g) = Λ∂g/∂Λ = −2g 2 . Note the would-be composite mass in the symmetric phase |M |2 ∼ Λ2 exp(−1/g) → ∞ (g → −0). Now look at the SWC TC as modeled by the large Nf QCD: When the walking coupling α(Q) α∗ is close to the critical coupling, α∗ αcr , we should include 2 ¯ ¯ 5 ψ 2 ], which becomes relthe induced four-fermion interaction, (G/2)[ ψψ + ψiγ evant operator due to the anomalous dimension γm = 1, and the system becomes “gauged Nambu-Jona-Lasinio” model29 whose solution in the full parameter space was obtained in Ref.30 Thus we may regard the SWC TC as the gauged Nambu-Jona-Lasinio (NJL) 30 model. It was found2 that SχSB solution exists for the parameter space g > g(+) = (1 + 1 − α∗ /αcr ) /4 (α∗ < αcr ) as well as the region α∗ > αcr , where the dimensionless four-fermion coupling g ≡ GΛ2 (NTC /4π 2 ) is normalized as g = 1 for α∗ = 0 (pure NJL model without gauge interaction). Based on the solution (including the
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running coupling case), the RGE flow in (α, g) space was found to be along the line of α = α∗ (α does not run), h on which the four-fermion coupling g runs, with the beta function and anomalous dimension given by 28,32,33 β(g) = −2(g − g(+) )(g − g(−) ), γm = 2g + α∗ /(2αcr ) (18) where g = g(±) ≡ (1 ± 1 − α∗ /αcr )2 /4 are regarded as the UV/IR fixed points (fixed ∗ ≤ αcr . The above anomalous dimension takes the values: γm = lines) for α31 1 + 1 − α∗ /αcr at the UV fixed line while γm = 1 − 1 − α∗ /αcr at the IR fixed line. Light composite spectra only exist near the UV fixed line (phase boundary) g g(+) in both SSB (g > g(+) ) and symmetric (g > g(+) ) phases as in NJL model. Thus it follows that as α∗ → αcr Eq. (18) takes the form β(g) = −2(g − g∗ )2 , γm |g=g∗ = 1 ,
(α∗ = αcr ) ,
(19)
with g(±) → 1/4 ≡ g∗ , and hence we again got a multiple zero and fusion of UV and IR fixed lines 28,32,33 which corresponds to the essential singularity scaling;30 m2 ∼ Λ2 exp(−1/(g − g∗ )). A similar observation was also made recently.34 In passing, it should be stressed that the anomalous dimension never changes discontinuously across the phase boundary as is seen from Eq.(16) and Eq.(18).27,28,32 The scale anomaly in this case is given by:9 2 β(g) G ¯ 2 ¯ · ψψ + ψiγ5 ψ g 2 −m4 · (4Nf NTC /π 4 ) = O(Λ4χ ) ,
∂ µ Dµ = θµµ = 4θ00 =
(20)
where the second line was from the explicit computation35 of the vacuum energy θ00 in the limit Λ/m → ∞ (g → g∗ ) at α ≡ αcr (The result coincides with the one for α → αcr with g ≡ 0, see Eq.(24).36 ). Again there is a composite state this time having mass15 √ Mσ → 2m , (21) as g → g∗ + 0, while there are no composites |M |2 ∼ Λ2 exp(−1/(g − g∗ )) → ∞ for g → g∗ − 0. Eq.(21) is compared with Mσ = 2m in the pure NJL case with α ≡ 0. This slightly lighter scalar may be identified with the techni-dilaton in the SWC TC. I will come back to this later. The absence of the composites in the symmetric phase g < g∗ may be understood as in the 2-dimensional Gross-Neveu model for g < 0, namely the repulsive fourfermion interactions: From the analysis of the RG flow, it was argued32 that the IR fixed line g = g(−) is due to the induced four-fermion interaction by the walking TC dynamics itself, while deviation from that line, g − g(−) , is due to the additional four-fermion interactions, repulsive (g < g(−) ) and attractive (g > g(−) ), from UV h The beta function in Eq.(12) may be regarded as an artificial one keeping g ≡ const. which is not along the renormalized trajectory in the extended parameter space (α, g).
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dynamics other than the TC (i.e., ETC). It is clear that no light composites exist for repulsive four-fermion interaction g < g(−) , which becomes g < g∗ at α∗ = αcr . 4. S Parameter Constraint Now we come to the next problem of TC, so-called S, T, U parameters3 measuring possible new physics in terms of the deviation of the LEP precision experiments from the SM. In particular, S parameter excludes the TC as a simple scale-up of QCD which yields S = (Nf /2) · Sˆ with SˆQCD = 0.32 ± 0.04. For a typical ETC model with one-family TC, Nf = 8,2 we would get S = O(1) which is much larger than the experiments S < 0.1. This is the reason why many people believe that the TC is dead. However, since the simple scale-up of QCD was already ruled out by the FCNC as was discussed before, the real problem is whether or not the walking/conformal TC which solved the FCNC problem is also consistent with the S parameter constraint above. There have been many arguments11,12 that the S parameter value could be reduced in the walking/conformal TC than in the simple scale-up of QCD. Recently such a reduction has also been argued37,38 in a version of the holographic QCD39 deformed to the walking/conformal TC by tuning a parameter to simulate the large anomalous dimension γm 1. Here we present the most straightforward computation of the S parameter for the large Nf QCD, based on the SD equation and (inhom*ogeneous) Bethe-Salpeter (BS) equation in the ladder approximation.12 The S parameter S = (Nf /2)Sˆ is defined by the slope of the the current correlators ΠJJ (Q2 ) at Q2 = 0:
d Sˆ = −4π 2 ΠV V (Q2 ) − ΠAA (Q2 )
, (22) dQ Q2 =0 where δ ab qµ qν /q 2 − gµν ΠJJ (q 2 ) = F .T . i0|T Jµa (x)Jνb (0)|0, (Jµa (x) = Vµa (x), Aaµ (x)) , with Fπ2 = ΠV V (0) − ΠAA (0) . The current correlators are ob(J)
tained by closing the fermion legs of the BS amplitudes χµ (p; q) ∼ F.T . ¯ 0|T ψ(r/2) ψ(−r/2) Jµ (x) |0, which is determined by the ladder BS equation (Fig.3). Solving the BS equation with the fermion propagator given as the solution p +q2
χ (p ; q ) p −q2
Fig. 3.
p +q2
p +q2 q
q
p −q2
+
k +q2
χ (k ; q ) p −q2
q
k −q2
Graphical expression of the BS equation in the ladder approximation.
of the ladder SD equation, we can evaluate the ΠV V (Q2 ) − ΠAA (Q2 ) numerically. ˆ From this result we may read its slope at Q2 = 0 to get S. ˆ The results show definitely smaller values of S than that in the ordinary QCD and moreover there is a tendency of the Sˆ getting reduced when approaching the
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conformal window α∗ αcr (Nf Nfcr ). However, due to technical limitation of the present computation getting very close to the conformal window, the reduction does not seem to be so dramatic as the walking TC being enough to be consistent with the experimental constraints. It is highly desirable to extend the computation further close to the conformal window. Another approach to this problem is the deformation of the holographic QCD by the anomalous dimension. The reduction of S parameter in the SWC TC has been argued in a version of the hard-wall type bottom up holographic QCD39 deformed to the SWCTC by tuning a parameter to simulate the large anomalous dimension γm 1.37 We examined38 such a possibility paying attention to the renormalization point dependence of the condensate. We explicitly calculated the S parameter in entire parameter space of the holographic SWC TC. We here take a set of Fπ /Mρ and γm . We find that S > 0 and it monotonically decreases to zero in accord with the previous results37 . However, our result turned out fairly independent of the value of the anomalous dimension γm , yielding no particular suppression solely by tuning the anomalous dimension large, Sˆ ∼ B(Fπ /Mρ )2 → 0 as Fπ /Mρ → 0, with B 27(32) for γm 1(0), in sharp contrast to the previous claim37 . Although B contains full contributions from the infinite tower of the vector/axial-vector KaluzaKlein modes (gauge bosons of hidden local symmetries)40 of the 5-dimensional gauge bosons, the resultant value of B turned out close to B 4πa 8π of the single ρ meson dominance, where a 2 is the parameter of the hidden local symmetry only for the ρ meson.40 This implies that as far as the pure TC dynamics (without ETC dynamics, etc.) is concerned, an obvious way to dynamically reduce S parameter is to tune Fπ /Mρ very small, namely techni-ρ mass very large to several TeV region. ( See, however, footnote below Eq.(30).) We would need more dynamical information other than the holographic recipe, since the parameter corresponding to Fπ /Mρ as well as the scale parameter is a pure input in all the holographic models, whether bottom up or top down approach, in contrast to the underlying gauge theory which has only a single parameter, a scale parameter like ΛQCD . Curiously enough, when we calculate Fπ /Mρ from the SD and the hom*ogeneous BS equations16 and S from the SD and the inhom*ogeneous BS equation12 both in the straightforward calculation in the ladder approximation, a set of the calculated values of (Fπ /Mρ , S) lies on the line of the holographic result.38 5. Techni-Dilaton Now we come to the discussions of Techni-dilaton (TD). Existence of two largely separated scales, Λχ ∼ m and ΛTC such that Λχ ΛTC , is the most important feature of SWC-TC, in sharp contrast to the ordinary QCD with small number of flavors (in the chiral limit) where all the mass parameters like dynamical mass of quarks are of order of the single scale parameter of the theory ΛQCD , m ∼ Λχ ∼ ΛQCD . See Fig. 1. The intrinsic scale ΛTC is related with the scale anomaly
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corresponding to the perturbative running effects of the coupling, with the ordinary two-loop beta function β(α) in the Region I, in the same sense as in QCD. ∂ µ Dµ = θµµ =
β(α) αG2µν = O(Λ4TC ), 4α2
(23)
which implies that all the techni-glue balls have mass of O(ΛTC ). On the other hand, the scale Λχ is related with totally different scale anomaly due to the dynamical generation of m (∼ Λχ ) which does exist even in the idealized case with non-running coupling α(µ) ≡ α(> αcr ) such as the Maskawa-Nakajima solution,1 as was discussed some time ago.36 Such an idealized case well simulates the dynamics of Region II of Fig. 1 ,8,9 with anomalous dimension γm 1 and m ΛTC in the numerical calculations,16 with the perturbative coupling constant in Region II being almost constant slightly larger than αcr , αcr < α(µ)(< α∗ ), for a wide infrared region. The coupling α ≡ α∗ in the “idealized Region II” actually runs non-perturbatively according to the essential-singularity scaling (Miransky scaling21 ) of mass generation, Eq.(11), with the non-perturbative beta function βNP (α), Eq.(12), having a multiple zero at α = αcr . Then the non-perturbative scale anomaly reads9,i ∂ µ Dµ NP = θµµ NP =
βNP (α) 4Nf NTC αG2µν NP = −m4 · = −O(Λ4χ ), 4α2 π4
(24)
where · · · NP is the quantity with the perturbative contributions subtracted:36 · · · NP ≡ · · · − · · · Perturbative . Eq.(24) coincides with Eq.(20) and ∂ µ Dµ NP /Λ4TC vanishes with ∂ µ Dµ NP /m4 → const. = 0, when we approach the conformal window from the broken phase α∗ αcr (m/ΛTC → 0). All the techni-fermion bound states have mass of order of m,41 while there are no light bound states in the symmetric phase (conformal window) α∗ < αcr , a characteristic feature of the conformal phase transition.9 The TD is associated with the latter scale anomaly and should have mass on order of m( ΛTC ). 5.1. Calculation from Gauged NJL Model in the Ladder SD Equation15 More concretely, the mass of TD or scalar bound state in the SWC-TC was estimated in various methods: The first method 15 was based on the the ladder SD equation for the gauged NJL model which well simulates8,9 the conformal phase transition in the large Nf QCD. The result was already given by Eq.(21): √ MTD 2m . (25) i In
terms of the gauged NJL model mentioned in Section 3 this is the expression of the scale anomaly for α → αcr with g = const. = g∗ , in contrast to Eq.(20) for g → g∗ with α = const. = α∗ = αcr . Both yield the same vacuum energy and hence the same scale anomaly.
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5.2. Straightforward Calculation from Ladder SD and BS Equations Also a straightforward calculation16 of mass of TD, the scalar bound state was made in the vicinity of the CBZ-IR fixed point in the large Nf QCD, based on the coupled use of the ladder SD equation and (hom*ogeneous) BS equation lacking the first term in Fig. 3: All the bound states masses are M = O(m) and M/ΛTC → 0, when approaching the conformal window α∗ → αcr (Nf → Nfcr ) such that m/ΛTC → 0. Near the conformal window (Nf Nfcr ) the calculated values are Mρ /Fπ 11, Ma1 /Fπ 12 (near degenerate !). On the other hand, the scalar mass sharply drops near the the conformal window, MTD /Fπ 4, or √ MTD 1.5m 2m (< Mρ , Ma1 ) . (26) Note that in this calculation MTD /Fπ → const. = 0 and hence there is no isolated massless scalar bound states even in the limit Nf → Nfcr . The result is consistent with Eq.(25) and is contrasted to the ordinary QCD where the scalar mass is larger than those of the vector mesons (“higgsless”) within the same framework of ladder SD/BS equation approach. The result would imply mTD 500 GeV
(27)
in the case of the one-family TC model with Fπ 125 GeV. 5.3. Holographic Techni-Dilaton13 Recently, we have calculated13 mass of TD in an extension of the previous paper38 on the hard-wall-type bottom-up holographic SWC-TC by including effects of (techni-) gluon condensation parameterized as 1/4 1 2 4 αG /F µν π (28) Γ ≡ 1 π 2 4 π αGµν /fπ QCD
through the bulk flavor/chiral-singlet scalar field ΦX , in addition to the conventional bulk scalar field Φ dual to the chiral condensate. The five-dimensional action is given by zm 1
√ 2 1 d4 x d z −g 2 ecg5 ΦX (z) − Tr LMN LMN + RMN RMN S5 = g5 4
1 (29) +Tr DM Φ† DM Φ − m2Φ Φ† Φ + ∂M ΦX ∂ M ΦX , 2 radius L of AdS where the anti-de-Sitter space (AdS5 ) with the curvature 5 is 2 described by the metric ds2 = gMN dxM dxN = (L/z) ηµν dxµ dxν − dz 2 with ηµν = diag[1, −1, −1, −1], g = det[gMN ] = −(L/z)10; g5 denotes the gauge coupling in five-dimension and c is the dimensionless coupling constant, and LM (RM ) =
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226 a LaM (RM )T a with the generators of SU (Nf ) are normalized by Tr[T a T b ] = δ ab ; L(R)MN = ∂M L(R)N − ∂N L(R)M − i[L(R)M , L(R)N ]. The covariant derivative acting on Φ is defined as DM Φ = ∂M Φ + iLM Φ − iΦRM . The TD, a flavor-singlet scalar bound state of techni-fermion and anti-technifermion, will be identified with the lowest KK mode coming from the bulk scalar field Φ, not ΦX . Thanks to the additional explicit bulk scalar field ΦX , we naturally improve the matching with the OPE of the underlying theory (QCD and SWCTC) for current correlators so as to reproduce gluonic 1/Q4 term, which is clearly distinguished from the same 1/Q4 terms from chiral condensate in the case of SWCTC with γm 1. Our model with γm = 0 and Nf = 3 well reproduces the real-life QCD. It is rather straightforward37,39 to compute masses of the techni-ρ meson (Mρ ), the techni-a1 meson (Ma1 ), while for that of the TD, flavor-singlet scalar meson (MTD ), we would need additional IR potential with quartic coupling λ to stabilize the SχSB vacuum.42 Such an IR potential might be regarded as generated by techni-fermion loop effects and we naturally expect λ ∼ NTC /(4π)2 . The S parameter was also calculated through the current correlators by the standard way. We found general tendency of the dependence of the meson masses relative to Fπ , (Mρ /Fπ , Ma1 /Fπ , MTD /Fπ ) on γm , S, and Γ. We find a characteristic feature of the techni-dilaton mass related to the conformality of SWC-TC: For fixed S and γm , absolute values of (Mρ /Fπ ) and (Ma1 /Fπ ) are not sensitive to Γ, although they get degenerate for large Γ. On the contrary, (MTD /Fπ ) substantially decreases as Γ increases. Actually, in the formal limit Γ → ∞ we would have (MTD /Fπ ) → 0 (This is contrast to the straightforward computation through ladder SD and BS equations mentioned before16 ). For fixed S and Γ, again (Mρ /Fπ ) and (Ma1 /Fπ ) are not sensitive to γm , while (MTD /Fπ ) substantially decreases as γm increases. Particularly for the case of γm = 1, we study the dependence of the S parameter on (Mρ /Fπ ) for typical values of Γ. It is shown that the techni-gluon contribution reduces the value of S about 10% in the region of Sˆ 0.1, although the general tendency is similar to the previous paper38 without techni-gluon condensation: Sˆ decreases to zero monotonically with respect to (Fπ /Mρ ). This implies (Mρ /Fπ ) necessarily increases when Sˆ is required to be smaller. To be more concrete, we consider a couple of typical models of SWC-TC with γm 1 and NTC = 2, 3, 4 based on the CBZ-IRFP in the large Nf QCD. Using the non-perturbative conformal anomaly Eq.(24) together with the non-perturbative beta function Eq.(12) and Eq.(11), we find a concrete relation between Γ and (ΛETC /Fπ ): In the case of NTC = 3 (Nf = 4NTC ) and S 0.1, we have Γ 7 for (ΛETC /Fπ ) = 104 –105 (required by the FCNC constraint). Thanks to the large anomalous dimension γm 1 and large techni-gluon condensation Γ 7, we obtain a relatively light techni-dilaton with mass
MTD 600 GeV
(30)
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compared with Mρ Ma1 3.8 TeV (almost degenerate). Eq. (30) is consistent with the perturbative unitarity of WL WL scattering even for large Mρ , Ma1 . Note that largeness of Mρ and Ma1 is essentially determined by the requirement of S = 0.1 fairly independently of techni-gluon condensation. j The essential reason for the large Γ is due to the existence of the wide conformal region Fπ (∼ m) < µ < ΛETC with ΛETC /Fπ = 104 –105 , which yields the smallness of the beta function (see Eq.(12) and Eq. (11) ) and hence amplifies the technigluon condensation compared with the ordinary QCD with Γ = 1. Actually, in the idealized (phenomenologically uninteresting) limit ΛETC /Fπ → ∞, we would have Γ → ∞, which in turn would imply MTD /Fπ → 0 as mentioned above. k To conclude, various methods predicted the mass of the techni-dilaton (“conformal Higgs”) in the range of MTD 500 − 600 GeV, which is within reach of LHC discovery. Since the SWC TC models are strong coupling theories and the ladder approximation/holographic calculations would be no more than a qualitative hint, more reliable calculations are certainly needed, including the lattice simulations, before drawing a definite conclusion about the physics predictions. Besides the phase diagram including the TC-induced/ETC-driven four-fermion couplings on the lattice, more reliable calculations such as the spectra as well as anomalous dimensions, non-perturbative beta functions, S parameter, etc. are highly desired. Acknowledgments We thank K. Haba, M. Harada, M. Hashimoto, M. Kurachi, and S. Matsuzaki for collaborations on the relevant topics and useful discussions. This work was supported by the JSPS Grant-in-Aid for Scientific Research(S) # 22224003 and by the Nagoya University Foundation. References 1. T. Maskawa and H. Nakajima, Prog. Theor. Phys. 52 (1974), 1326; 54 (1975), 860. 2. S. Weinberg, Phys. Rev. D 13 (1976), 974; D 19 (1979), 1277; L. Susskind, Phys. Rev. D 20 (1979), 2619: See for a review of earlier literature, E. Farhi and L. Susskind, Phys. Rep. 74 (1981), 277. j The calculated S parameter here was from the TC dynamics alone and could be drastically changed by incorporating contributions from the generation mechanism of the mass of SM fermions such as the strong ETC dynamics. For instance, the fermion delocalization43 in the Higgsless models as a possible analogue of certain ETC effects in fact can cancel large positive S arising from the 5-dimensional gauge sector which corresponds to the pure TC dynamics. If it is the case in the explicit ETC model, then the overall mass scale of techni-hadrons including TD would be much lower than the above estimate down to, say, 300 GeV. k This does not mean existence of true (exactly massless) NG boson of the broken scale symmetry, since such a would-be NG boson gets decoupled in our case: The decay constant of techni-dilaton FTD would diverge, FTD /Fπ → ∞, in that limit through the PCDC relation 2 = −4θ µ /M 2 4 2 4 2 FTD µ TD ∼ m /MTD ∼ Fπ /MTD . The scale symmetry is broken explicitly as well as spontaneously.
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3. M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65 (1990), 964; Phys. Rev. D 46 (1992), 381; B. Holdom and J. Terning, Phys. Lett. B 247 (1990), 88; M. Golden and L. Randall, Nucl. Phys. B 361 (1991), 3. 4. K. Yamawaki, M. Bando and K. Matumoto, Phys. Rev. Lett. 56 (1986), 1335; M. Bando, K. Matumoto and K. Yamawaki, Phys. Lett. B 178 (1986), 308. See also M. Bando, T. Morozumi, H. So and K. Yamawaki, Phys. Rev. Lett. 59 (1987), 389 . The solution of the FCNC problem by the large anomalous dimension was first considered by B. Holdom, based on a pure assumption of the existence of UV fixed point without explicit dynamics and hence without definite prediction of the value of the anomalous dimension. See B. Holdom, Phys. Rev. D 24 (1981), 1441. 5. T. Akiba and T. Yanagida, Phys. Lett. B 169 (1986), 432; T. W. Appelquist, D. Karabali and L. C. R. Wijewardhana, Phys. Rev. Lett. 57 (1986), 957; T. Appelquist and L. C. R. Wijewardhana, Phys. Rev. D 36 (1987), 568. For an earlier work on this line based on pure numerical analysis see B. Holdom, Phys. Lett. B 150 (1985), 301. 6. C. T. Hill and E. H. Simmons, Phys. Rept. 381 (2003), 235 [Erratum-ibid. 390 (2004), 553]; K. Yamawaki, Lecture at 14th Symposium on Theoretical Physics, Cheju, Korea, July 1995, arXiv:hep-ph/9603293. 7. K. D. Lane and M. V. Ramana, Phys. Rev. D 44, 2678 (1991). 8. T. Appelquist, J. Terning and L. C. Wijewardhana, Phys. Rev. Lett. 77 (1996), 1214; T. Appelquist, A. Ratnaweera, J. Terning and L. C. Wijewardhana, Phys. Rev. D 58 (1998), 105017. 9. V. A. Miransky and K. Yamawaki, Phys. Rev. D 55 (1997), 5051 [Errata; 56 (1997), 3768]. 10. W. E. Caswell, Phys. Rev. Lett. 33, 244 (1974); T. Banks and A. Zaks, Nucl. Phys. B 196 (1982), 189. 11. R. Sundrum and S. D. H. Hsu, Nucl. Phys. B 391 (1993), 127; T. Appelquist and G. Triantaphyllou, Phys. Lett. B 278 (1992), 345; T. Appelquist and F. Sannino, Phys. Rev. D 59 (1999), 067702. 12. M. Harada, M. Kurachi and K. Yamawaki, Prog. Theor. Phys. 115 (2006), 765 ; M. Kurachi and R. Shrock, Phys. Rev. D 74, 056003 (2006); M. Kurachi, R. Shrock and K. Yamawaki, ibid D 76, 035003 (2007). 13. K. Haba, S. Matsuzaki and K. Yamawaki, arXiv:1003.2841 [hep-ph]; arXiv:1006.2526 [hep-ph]. 14. K. Yamawaki, Prog. Theor. Phys. Suppl. 180, 1 (2010) [arXiv:0907.5277 [hep-ph]]; See also Prog. Theor. Phys. Suppl. 167, 127 (2007). 15. S. Shuto, M. Tanabashi and K. Yamawaki, in Proc. 1989 Workshop on Dynamical Symmetry Breaking, Dec. 21-23, 1989, Nagoya, eds. T. Muta and K. Yamawaki (Nagoya Univ., Nagoya, 1990) 115-123; M. S. Carena and C. E. M. Wagner, Phys. Lett. B 285, 277 (1992); M. Hashimoto, Phys. Lett. B 441, 389 (1998). 16. M. Harada, M. Kurachi and K. Yamawaki, Phys. Rev. D 68 (2003), 076001; M. Kurachi and R. Shrock, JHEP 0612, 034 (2006). 17. S. Dimopoulos and L. Susskind, Nucl. Phys. B 155 (1979), 237; E. Eichten and K. D. Lane, Phys. Lett. B 90 (1980), 125. 18. K. Yamawaki and T. Yokota, Phys. Lett. B 113 (1982), 293; Nucl. Phys. B 223 (1983), 144. 19. K. Johnson, M. Baker and R. Willey, Phys. Rev. 136 (1964), B1111; M. Baker and K. Johnson, Phys. Rev. D3 (1971), 2516; R. Jackiw and K. Johnson Phys. Rev. D8 (1973), 2386; J.M. Cornwall and R.E. Norton, ibid, 3338. 20. R. f*ckuda and T. Kugo, Nucl. Phys. B 117 (1976), 250. 21. V. A. Miransky, Nuovo Cim. A 90 (1985), 149.
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22. F. Sannino and K. Tuominen, Phys. Rev. D 71, 051901 (2005); D. K. Hong, S. D. H. Hsu and F. Sannino, Phys. Lett. B 597, 89 (2004); For a review F. Sannino, arXiv:0804.0182 [hep-ph]. 23. Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai and T. Yoshie, Nucl. Phys. (Proc. Suppl.) 53 (1997), 449; Prog. Theor. Phys. Suppl. No. 131 (1998), 415. 24. T. Appelquist, G. T. Fleming and E. T. Neil, Phys.Rev.Lett.100:171607,2008, Erratum-ibid.102:149902,2009; Phys. Rev. D 79, 076010 (2009); A. Deuzeman, M. P. Lombardo and E. Pallante, Phys. Lett. B 670, 41 (2008); Talks presented at SCGT 09, http://www.eken.phys.nagoya-u.ac.jp/scgt09/ . 25. Talks at SCGT 09: http://www.eken.phys.nagoya-u.ac.jp/scgt09/ . 26. H. Georgi, Phys. Rev. Lett. 98, 221601 (2007). 27. Y. Kikukawa and K. Yamawaki, Phys. Lett. B 234, 497 (1990). 28. K. i. Kondo, M. Tanabashi and K. Yamawaki, Prog. Theor. Phys. 89, 1249 (1993). 29. W. A. Bardeen, C. N. Leung and S. T. Love, Phys. Rev. Lett. 56, 1230 (1986). 30. K. i. Kondo, H. Mino and K. Yamawaki, Phys. Rev. D 39, 2430 (1989); K. Yamawaki, in Proc. Johns Hopkins Workshop on Current Problems in Particle Theory 12, Baltimore, June 8-10, 1988, edited by G. Domokos and S. Kovesi-Domokos (World Scientific Pub. Co., Singapore 1988); T. Appelquist, M. Soldate, T. Takeuchi, and L. C. R. Wijewardhana, ibid. 31. V. A. Miransky and K. Yamawaki, Mod. Phys. Lett. A 4 (1989), 129. 32. K. i. Kondo, S. Shuto and K. Yamawaki, Mod. Phys. Lett. A 6, 3385 (1991). 33. K. I. Aoki, K. Morikawa, J. I. Sumi, H. Terao and M. Tomoyose, Prog. Theor. Phys. 102, 1151 (1999). 34. D. B. Kaplan, J. W. Lee, D. T. Son and M. A. Stephanov, Phys. Rev. D 80, 125005 (2009). 35. T. Nonoyama, T. B. Suzuki and K. Yamawaki, Prog. Theor. Phys. 81, 1238 (1989). 36. V. A. Miransky and V. P. Gusynin, Prog. Theor. Phys. 81 (1989) 426. 37. D. K. Hong and H. U. Yee, Phys. Rev. D 74 (2006), 015011; M. Piai, arXiv:hepph/0608241; K. Agashe, C. Csaki, C. Grojean and M. Reece, JHEP 0712, 003 (2007). 38. K. Haba, S. Matsuzaki and K. Yamawaki, Prog. Theor. Phys. 120, 691 (2008). 39. J. Erlich, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 95 (2005), 261602; L. Da Rold and A. Pomarol, Nucl. Phys. B 721, 79 (2005). 40. M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett. 54 (1985), 1215; M. Bando, T. Kugo and K. Yamawaki, Nucl. Phys. B 259 (1985), 493; Phys. Rept. 164 (1988), 217; M. Harada and K. Yamawaki, Phys. Rept. 381 (2003), 1. 41. R. S. Chivukula, Phys. Rev. D 55 (1997), 5238. 42. L. Da Rold and A. Pomarol, JHEP 0601, 157 (2006). 43. G. Cacciapaglia, C. Csaki, C. Grojean and J. Terning, Phys. Rev. D 71, 035015 (2005); R. Foadi, S. Gopalakrishna and C. Schmidt, Phys. Lett. B 606, 157 (2005). R. S. Chivukula, E. H. Simmons, H. J. He, M. Kurachi and M. Tanabashi, Phys. Rev. D 72, 015008 (2005)
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Phase Diagram of Strongly Interacting Theories Francesco Sannino CP3 -Origins, University of Southern Denmark, Odense, 5230 M, Denmark E-mail: [emailprotected], [emailprotected] www.cp3-origins.dk We summarize the phase diagrams of SU, SO and Sp gauge theories as function of the number of flavors, colors, and matter representation as well as the ones of phenomenologically relevant chiral gauge theories such as the Bars-Yankielowicz and the generalized Georgi-Glashow models. We finally report on the intriguing possibility of the existence of gauge-duals for nonsupersymmetric gauge theories and the impact on their conformal window.
1. Phases of Gauge Theories Models of dynamical breaking of the electroweak symmetry are theoretically appealing and constitute one of the best motivated natural extensions of the standard model (SM). We have proposed several models1–8 possessing interesting dynamics relevant for collider phenomenology9–12 and cosmology.7,13–31 The structure of one of these models, known as Minimal Walking Technicolor, has led to the construction of a new supersymmetric extension of the SM featuring the maximal amount of supersymmetry in four dimension with a clear connection to string theory, i.e. Minimal Super Conformal Technicolor.32 These models are also being investigated via first principle lattice simulations33–48 a . An up-to-date review is Ref. 59 while an excellent review updated till 2003 is Ref. 60. These are also among the most challenging models to work with since they require deep knowledge of gauge dynamics in a regime where perturbation theory fails. In particular, it is of utmost importance to gain information on the nonperturbative dynamics of non-abelian four dimensional gauge theories. The phase diagram of SU (N ) gauge theories as functions of number of flavors, colors and matter representation has been investigated in.1,61–64 The analytical tools which will be used here for such an exploration are: i) The conjectured physical all orders beta function for nonsupersymmetric gauge theories with fermionic matter in arbitrary representations of the gauge group;63 ii) The truncated Schwinger-Dyson equation (SD)65–67 (referred also as the ladder approximation in the literature); The Appelquist-Cohen-Schmaltz (ACS) conjecture68 which makes use of the counting of the thermal degrees of freedom at high and low temperature. These are the methods which we have used in our investigations. However several very interesting and competing analytic approaches69–80 have been proa Earlier
interesting models49–51 have contributed triggering the lattice investigations for the conformal window with theories featuring fermions in the fundamental representation52–58
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posed in the literature. What is interesting is that despite the very different starting point the various methods agree qualitatively on the main features of the various conformal windows presented here. 1.1. Physical all orders Beta Function - Conjecture Recently we have conjectured an all orders beta function which allows for a bound of the conformal window63 of SU (N ) gauge theories for any matter representation. The predictions of the conformal window coming from the above beta function are nontrivially supported by all the recent lattice results.33,34,81–85 In86 we further assumed the form of the beta function to hold for SO(N ) and Sp(2N ) gauge groups and further extended in59 to chiral gauge theories. Consider a generic gauge group with Nf (ri ) Dirac flavors belonging to the representation ri , i = 1, . . . , k of the gauge group. The conjectured beta function reads: Pk 2 g 3 β0 − 32 i=1 T (ri ) Nf (ri ) γi (g ) β(g) = − , (1) 0 2β g2 (4π)2 1 − 8π 1 + β00 2 C2 (G) with
β0 =
k k X 4X 11 C2 (G) − T (ri )Nf (ri ) and β00 = C2 (G) − T (ri )Nf (ri ) . 3 3 i=1 i=1
(2)
The generators Tra , a = 1 . . . N2 − 1 of the gauge group in the representation r are normalized according to Tr Tra Trb = T (r)δ ab while the quadratic Casimir C2 (r) is given by Tra Tra = C2 (r)I. The trace normalization factor T (r) and the quadratic Casimir are connected via C2 (r)d(r) = T (r)d(G) where d(r) is the dimension of the representation r. The adjoint representation is denoted by G. The beta function is given in terms of the anomalous dimension of the fermion mass γ = −d ln m/d ln µ where m is the renormalized mass, similar to the supersymmetric case.87–89 The loss of asymptotic freedom is determined by the change of sign in the first coefficient β0 of the beta function. This occurs when k X 4 T (ri )Nf (ri ) = C2 (G) , 11 i=1
Loss of AF.
(3)
At the zero of the beta function we have
k X 2 T (ri )Nf (ri ) (2 + γi ) = C2 (G) , 11 i=1
(4)
Hence, specifying the value of the anomalous dimensions at the IRFP yields the last constraint needed to construct the conformal window. Having reached the zero of the beta function the theory is conformal in the infrared. For a theory to be conformal the dimension of the non-trivial spinless operators must be larger than one in order not to contain
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negative norm states.90–92 Since the dimension of the chiral condensate is 3 − γi we see that γi = 2, for all representations ri , yields the maximum possible bound k X 8 T (ri )Nf (ri ) = C2 (G) , 11 i=1
γi = 2 .
(5)
In the case of a single representation this constraint yields Nf (r)BF ≥
11 C2 (G) , 8 T (r)
γ =2.
(6)
The actual size of the conformal window can be smaller than the one determined by the bound above, Eq. (3) and (5). It may happen, in fact, that chiral symmetry breaking is triggered for a value of the anomalous dimension less than two. If this occurs the conformal window shrinks. Within the ladder approximation65,66 one finds that chiral symmetry breaking occurs when the anomalous dimension is close to one. Picking γi = 1 we find: k X 6 T (ri )Nf (ri ) = C2 (G) , 11 i=1
γ =1.
(7)
In the case of a single representation this constraint yields Nf (r)BF ≥
11 C2 (G) , 6 T (r)
γ =1.
(8)
When considering two distinct representations the conformal window becomes a three dimensional volume, i.e. the conformal house.62 Of course, we recover the results by Banks and Zaks93 valid in the perturbative regime of the conformal window. We note that the presence of a physical IRFP requires the vanishing of the beta function for a certain value of the coupling. The opposite however is not necessarily true; the vanishing of the beta function is not a sufficient condition to determine if the theory has a fixed point unless the beta function is physical. By physical we mean that the beta function allows to determine simultaneously other scheme-independent quantities at the fixed point such as the anomalous dimension of the mass of the fermions. This is exactly what our beta function does. In fact, in the case of a single representation, one finds that at the zero of the beta function one has: 11C2 (G) − 4T (r)Nf . (9) γ= 2T (r)Nf 1.2. Schwinger-Dyson in the Rainbow Approximation For nonsupersymmetric theories another way to get quantitative estimates is to use the rainbow approximation to the Schwinger-Dyson equation.94,95 After a series of approximations (see59 for a review) one deduces for an SU (N ) gauge theory with Nf Dirac fermions transforming according to the representation r the critical number of flavors above which chiral symmetry maybe unbroken: NfSD =
17C2 (G) + 66C2 (r) C2 (G) . 10C2 (G) + 30C2 (r) T (r)
(10)
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Comparing with the previous result obtained using the all orders beta function we see that it is the coefficient of C2 (G)/T (r) which is different. We note that in96 it has been advocated a coefficient similar to the one of the all-orders beta function. 1.3. The SU , SO and Sp phase diagrams We consider here gauge theories with fermions in any representation of the SU (N ) gauge group1,61–63,97 using the various analytic methods described above. Here we plot in Fig. 1 the conformal windows for various representations predicted with the physical all orders beta function and the SD approaches.
Fig. 1. Phase diagram for nonsupersymmetric theories with fermions in the: i) fundamental representation (black), ii) two-index antisymmetric representation (blue), iii) two-index symmetric representation (red), iv) adjoint representation (green) as a function of the number of flavors and the number of colors. The shaded areas depict the corresponding conformal windows. Above the upper solid curve the theories are no longer asymptotically free. In between the upper and the lower solid curves the theories are expected to develop an infrared fixed point according to the all orders beta function. The area between the upper solid curve and the dashed curve corresponds to the conformal window obtained in the ladder approximation.
The ladder result provides a size of the window, for every fermion representation, smaller than the maximum bound found earlier. This is a consequence of the value of the anomalous dimension at the lower bound of the window. The unitarity constraint corresponds to γ = 2 while the ladder result is closer to γ ∼ 1. Indeed if we pick γ = 1 our conformal window approaches the ladder result. Incidentally, a value of γ larger than one, still allowed by unitarity, is a welcomed feature when using this window to construct walking technicolor theories. It may allow for the physical value of the mass of the top while avoiding a large violation of flavor changing neutral currents98 which were investigated in99 in the case of the ladder approximation for minimal walking models.
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1.3.1. The Sp(2N ) phase diagram Sp(2N ) is the subgroup of SU (2N ) which leaves the tensor J c1 c2 = (1N ×N ⊗ iσ2 )c1 c2 invariant. Irreducible tensors of Sp(2N ) must be traceless with respect to J c1 c2 . Here we consider Sp(2N ) gauge theories with fermions transforming according to a given irreducible representation. Since π 4 [Sp(2N )] = Z2 there is a Witten topological anomaly100 whenever the sum of the Dynkin indices of the various matter fields is odd. The adjoint of Sp(2N ) is the two-index symmetric tensor. In Figure 2 we summarize the relevant zero temperature and matter density phase diagram as function of the number of colors and Weyl flavors (NW f ) for Sp(2N ) gauge theories. For the vector representation NW f = 2Nf while for the two-index theories NW f = Nf . The shape of the various conformal windows are very similar to the ones for SU (N ) gauge theories1,61,63 with the difference that in this case the two-index symmetric representation is the adjoint representation and hence there is one less conformal window. 35
A.F.
B.F. & γ= 2
SD
30
NWf
25 20
SP(2N)
15 10 5 0 2
3
4 N
5
6
Fig. 2. Phase Diagram, from top to bottom, for Sp(2N ) Gauge Theories with NW f = 2Nf Weyl fermions in the vector representation (light blue), NW f = Nf in the two-index antisymmetric representation (light red) and finally in the two-index symmetric (adjoint) (light green). The arrows indicate that the conformal windows can be smaller and the associated solid curves correspond to the all orders beta function prediction for the maximum extension of the conformal windows.
1.3.2. The SO(N ) phase diagram We shall consider SO(N ) theories (for N > 5) since they do not suffer of a Witten anomaly100 and, besides, for N < 7 can always be reduced to either an SU or an Sp theory. In Figure 3 we summarize the relevant zero temperature and matter density phase diagram as function of the number of colors and Weyl flavors (Nf ) for SO(N ) gauge theories. The shape of the various conformal windows are very similar to the ones for SU (N ) and
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SD
B.F. & γ= 2
Nf
15
10
SO(N)
5
0 5
6
7
8
9
10
N
Fig. 3. Phase diagram of SO(N ) gauge theories with Nf Weyl fermions in the vector representation, in the two-index antisymmetric (adjoint) and finally in the two-index symmetric representation. The arrows indicate that the conformal windows can be smaller and the associated solid curves correspond to the all orders beta function prediction for the maximum extension of the conformal windows.
Sp(2N ) gauge with the difference that in this case the two-index antisymmetric representation is the adjoint representation. We have analyzed only the theories with N ≥ 6 since the remaining smaller N theories can be deduced from Sp and SU using the fact that SO(6) ∼ SU (4), SO(5) ∼ Sp(4), SO(4) ∼ SU (2) × SU (2), SO(3) ∼ SU (2), and SO(2) ∼ U (1). The phenomenological relevance of orthogonal gauge groups for models of dynamical electroweak symmetry breaking has been shown in.7
1.4. Phases of Chiral Gauge Theories Chiral gauge theories, in which at least part of the matter field content is in complex representations of the gauge group, play an important role in efforts to extend the SM. These include grand unified theories, dynamical breaking of symmetries, and theories of quark and lepton substructure. Chiral theories received much attention in the 1980’s.101,102 Here we confront the results obtained in Ref.103,104 using the thermal degree of count freedom with the generalization of the all orders beta function useful to constrain chiral gauge theories appeared in.59 The two important class of theories we are going to investigate are the Bars-Yankielowicz (BY)105 model involving fermions in the two-index symmetric tensor representation, and the other is a generalized Georgi-Glashow (GGG) model involving fermions in the two-index antisymmetric tensor representation. In each case, in addition to fermions in complex representations, a set of p anti fundamental-fundamental pairs are included and the allowed phases are considered as a function of p. An independent relevant study of the phase diagrams of chiral gauge theories appeared in.75 Here the authors also compare their results with the ones presented below.
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1.4.1. All-orders beta function for Chiral Gauge Theories A generic chiral gauge theory has always a set of matter fields for which one cannot provide a mass term, but it can also contain vector-like matter. We hence suggest the following minimal modification of the all orders beta function63 for any nonsupersymmetric chiral gauge theory: Pk g 3 β0 − 32 i=1 T (ri )p(ri )γi (g 2 ) , βχ (g) = − (11) (4π)2 1 − g22 C (G) 1 + 2βχ0 2 8π β0
where pi is the number of vector like pairs of fermions in the representation ri for which an anomalous dimension of the mass γi can be defined. β0 is the standard one loop coefficient of the beta function while βχ0 expression is readily obtained by imposing that when expanding βχ one recovers the two-loop coefficient correctly and its explicit expression is not relevant here. According to the new beta function gauge theories without vector-like matter but featuring several copies of purely chiral matter will be conformal when the number of copies is such that the first coefficient of the beta function vanishes identically. Using topological excitations an analysis of this case was performed in.74 1.4.2. The Bars Yankielowicz (BY) Model This model is based on the single gauge group SU (N ≥ 3) and includes fermions trans{ab} forming as a symmetric tensor representation, S = ψL , a, b = 1, · · · , N ; N + 4 + p c , where i = 1, · · · , N + 4 + p; and conjugate fundamental representations: F¯a,i = ψa,iL a,i a,i p fundamental representations, F = ψL , i = 1, · · · , p. The p = 0 theory is the basic chiral theory, free of gauge anomalies by virtue of cancellation between the antisymmetric tensor and the N + 4 conjugate fundamentals. The additional p pairs of fundamentals and conjugate fundamentals, in a real representation of the gauge group, lead to no gauge anomalies. The global symmetry group is Gf = SU (N + 4 + p) × SU (p) × U1 (1) × U2 (1) .
(12)
Two U (1)’s are the linear combination of the original U (1)’s generated by S → eiθS S , F¯ → eiθF¯ F¯ and F → eiθF F that are left invariant by instantons, namely that for which P j NRj T (Rj )QRj = 0, where QRj is the U (1) charge of Rj and NRj denotes the number of copies of Rj . Thus the fermionic content of the theory is where the first SU (N ) is the gauge group, indicated by the square brackets. Table 1.
The Bars Yankielowicz (BY) model.
Fields [SU (N )] SU (N + 4 + p) SU (p) S F¯ F
¯
1 ¯ 1
1 1
U1 (1)
U2 (1)
N +4 2p −(N + 2) −p N + 2 −(N − p)
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From the numerator of the chiral beta function and the knowledge of the one-loop coefficient of the BY perturbative beta function the predicted conformal window is: 3
3 (3N − 2) ≤ p ≤ (3N − 2) , 2 + γ∗ 2
(13)
with γ ∗ the largest possible value of the anomalous dimension of the mass. The maximum value of the number of p flavors is obtained by setting γ ∗ = 2: 3 3 (3N − 2) ≤ p ≤ (3N − 2) , γ∗ = 2 , (14) 4 2 while for γ ∗ = 1 one gets: 3 (3N − 2) ≤ p ≤ (3N − 2) , γ∗ = 1 . (15) 2 The chiral beta function predictions for the conformal window are compared with the thermal degree of freedom investigation as shown in the left panel of Fig. 4. In order to derive ϒ=1
ϒ=1
20
20
ACS
ϒ=2
ϒ=2 15
p
p
15
10
10
BY 5 2
4
GGG
5
ACS
6 N
8
10
2
4
6 N
8
10
Fig. 4. Left panel: Phase diagram of the BY generalized model. The upper solid (blue) line corresponds to the loss of asymptotic freedom; the dashed (blue) curve corresponds to the chiral beta function prediction for the breaking/restoring of chiral symmetry. The dashed black line corresponds to the ACS bound stating that the conformal region should start above this line. We have augmented the ACS method with the Appelquist-DuanSannino104 extra requirement that the phase with the lowest number of massless degrees of freedom wins among brk+sym all the possible phases in the infrared a chiral gauge theory can have. We hence used fIR and fU V to determine this curve. According to the all orders beta function (B.F.) the conformal window cannot extend below the solid (blue) line, as indicated by the arrows. This line corresponds to the anomalous dimension of the mass reaching the maximum value of 2. Right panel: The same plot for the GGG model.
a prediction from the ACS method we augmented it with the Appelquist-Duan-Sannino104 extra requirement that the phase with the lowest number of massless degrees of freedom wins among all the possible phases in the infrared a chiral gauge theory can have. The thermal critical number is: i p 1h pTherm = −16 + 3N + 208 − 196N + 69N 2 . (16) 4 1.4.3. The Generalized Georgi-Glashow (GGG) Model This model is similar to the BY model just considered. It is an SU (N ≥ 5) gauge theory, but with fermions in the anti-symmetric, rather than symmetric, tensor representation. The
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complete fermion content is A = ψL , a, b = 1, · · · , N ; an additional N − 4 + p fermions c in the conjugate fundamental representations: F¯a,i = ψa,iL , i = 1, · · · , N − 4 + p; and p a,i a,i fermions in the fundamental representations, F = ψL , i = 1, · · · , p. The global symmetry is Gf = SU (N − 4 + p) × SU (p) × U1 (1) × U2 (1) .
(17)
where the two U (1)’s are anomaly free. With respect to this symmetry, the fermion content is Table 2.
The Generalized Georgi-Glashow (GGG) model.
Fields [SU (N )] SU (N − 4 + p) SU (p) A F¯ F
¯
1 ¯
1 1
1
U1 (1)
U2 (1)
N −4 2p −(N − 2) −p N − 2 −(N − p)
Following the analysis for the BY model the chiral beta function predictions for the conformal window are compared with the thermal degree of freedom investigation and the result is shown in the right panel of Fig. 4. 1.5. Conformal Chiral Dynamics Our starting point is a nonsupersymmetric non-abelian gauge theory with sufficient massless fermionic matter to develop a nontrivial IRFP. The cartoon of the running of the coupling constant is represented in Fig. 5. In the plot ΛU is the dynamical scale below which the IRFP is essentially reached. It can be defined as the scale for which α is 2/3 of the fixed point value in a given renormalization scheme. If the theory possesses an IRFP the chiral condensate must vanish at large distances. Here we want to study the behavior of the condensate when a flavor singlet mass term is added to the underlying Lagrangian e + h.c. with m the fermion mass and ψ f as well as ψec left transforming ∆L = −m ψψ c f two component spinors, c and f represent color and flavor indices. The omitted color and � �*
�U
Energy
Fig. 5. Running of the coupling constant in an asymptotically free gauge theory developing an infrared fixed point for a value α = α∗ .
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flavor indices, in the Lagrangian term, are contracted. We consider the case of fermionic matter in the fundamental representation of the SU (N ) gauge group. The effect of such a term is to break the conformal symmetry together with some of the global symmetries of 0 f0 the underlying gauge theory. The composite operator Oψψ = ψef ψf has mass dimene f sion dψψ e = 3 − γ with γ the anomalous dimension of the mass term. At the fixed point γ is a positive number smaller than two.90 We assume m ΛU . Dimensional analysis demands ∆L → −m ΛγU Tr[Oψψ e ] + h.c. . The mass term is a relevant perturbation around the IRFP driving the theory away from the fixed point. It will induce a nonzero f0 vacuum expectation value for Oψψ e itself proportional to δf . It is convenient to define Tr[Oψψ e ] = Nf O with O a flavor singlet operator. The relevant low energy Lagrangian term is then −m ΛγU Nf O + h.c.. To determine the vacuum expectation value of O we follow.106,107 The induced physical mass gap is a natural infrared cutoff. We, hence, identify ΛIR with the physical value of the condensate. We find: hψecf ψfc i ∝ −mΛ2U ,
0 8. Asymptotic freedom of the newly found theory is dictated by the coefficient of the one-loop beta function : β0 =
11 2 (2Nf − 5N ) − Nf . 3 3
(25)
To this order in perturbation theory the gauge singlet states do not affect the magnetic quark sector and we can hence determine the number of flavors obtained by requiring the dual theory to be asymptotic free. i.e.: Nf ≥
11 N 4
Dual Asymptotic Freedom .
(26)
Quite remarkably this value coincides with the one predicted by means of the all orders conjectured beta function for the lowest bound of the conformal window, in the electric variables, when taking the anomalous dimension of the mass to be γ = 2. We recall that for any number of colors N the all orders beta function requires the critical number of flavors to be larger than: NfBF |γ=2 =
11 N. 4
(27)
For N=3 the two expressions yield 8.25 b . We consider this a nontrivial and interesting result lending further support to the all orders beta function conjecture and simultaneously suggesting that this theory might, indeed, be the QCD magnetic dual. The actual size of the conformal window matching this possible dual corresponds to setting γ = 2. We note that although for Nf = 9 and N = 3 the magnetic gauge group is SU (3) the theory is not trivially QCD given that it features new massless fermions and their interactions with massless mesonic type fields. Recent suggestions to analyze the conformal window of nonsupersymmetric gauge theories based on different model assumptions74 are in qualitative agreement with the precise results of the all orders beta function conjecture. It is worth noting that the combination 2Nf − 5N appears in the computation of the mass gap for gauge fluctuations presented in.74,114 It would be interesting to explore a possible link between these different approaches in the future. We have also find solutions for which the lower bound of the conformal window is saturated for γ = 1. The predictions from the gauge duals are, however, entirely and surprisingly consistent with the maximum extension of the conformal window obtained using the all orders beta function.63 Our main conclusion is that the ’t Hooft anomaly conditions alone do not exclude the possibility that the maximum extension of the QCD conformal window is the one obtained for a large anomalous dimension of the quark mass. By computing the same gauge singlet correlators in QCD and its suggested dual, one can directly validate or confute this proposal via lattice simulations. b Actually
given that X must be at least 2 we must have Nf ≥ 8.5 rather than 8.25
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1.7. Conclusions We investigated the conformal windows of chiral and non-chiral nonsupersymmetric gauge theories with fermions in any representation of the underlying gauge group using four independent analytic methods. For vector-like gauge theories one observes a universal value, i.e. independent of the representation, of the ratio of the area of the maximum extension of the conformal window, predicted using the all orders beta function, to the asymptotically free one, as defined in.62 It is easy to check from the results presented that this ratio is not only independent on the representation but also on the particular gauge group chosen. The four methods we used to unveil the conformal windows are the all orders beta function (BF), the SD truncated equation, the thermal degrees of freedom method and possible gauge - duality. They have vastly different starting points and there was no, a priori, reason to agree with each other, even at the qualitative level. Several questions remain open such as what happens on the right hand side of the infrared fixed point as we increase further the coupling. Does a generic strongly coupled theory develop a new UV fixed point as we increase the coupling beyond the first IR value115 ? If this were the case our beta function would still be a valid description of the running of the coupling of the constant in the region between the trivial UV fixed point and the neighborhood of the first IR fixed point. One might also consider extending our beta function to take into account of this possibility as done in.76 It is also possible that no non-trivial UV fixed point forms at higher values of the coupling constant for any value of the number of flavors within the conformal window. Gauge-duals seem to be in agreement with the simplest form of the beta function. The extension of the all orders beta function to take into account fermion masses has appeared in.116 Our analysis substantially increases the number of asymptotically free gauge theories which can be used to construct SM extensions making use of (near) conformal dynamics. Current Lattice simulations can test our predictions and lend further support or even disprove the emergence of a universal picture possibly relating the phase diagrams of gauge theories of fundamental interactions. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
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Resizing Conformal Windows O. Antipin Department of Physics, P.O.Box 35 FI-00014 University of Jyv¨ askyl¨ a, Finland and Helsinki Institute of Physics, P.O. Box 64 FI-00014, University of Helsinki, Finland E-mail: [emailprotected] K. Tuominen CP3-Origins, Campusvej 55 DK-5230, Odense N, Denmark and Helsinki Institute of Physics, P.O. Box 64 FI-00014 University of Helsinki, Finland E-mail: [emailprotected] Different mechanisms for the loss of conformality and analytic ans¨ atze for the betafunction of a generic gauge theory are reviewed and the implications on the conformal windows considered. Keywords: Conformal window, gauge theories.
1. Introduction Recent progress on the phase diagrams of strongly coupled gauge theory as a function of number of colors and flavors as well as matter representations1,2 shows that infrared conformal theories with simple particle content exist. The key feature in this analysis is the use of higher representations with respect to the new strong dynamics. Recently these models have been investigated also on the lattice.3,5,7–9 These theories are applied in beyond the Standard Model physics as phenomenologically viable candidates of walking Technicolor theories.10,11 If the β-function for a generic SU(N) gauge theory with matter was known beyond perturbation theory, the determination of the conformal window would be a simple matter. For N = 1 super-QCD (SQCD) the exact β-function is known,12 and for 3Nc /2 < Nf < 3Nc SQCD has an infrared stable fixed point.13,14 In15 a β-function ansatz for non-supersymmetric gauge theories was introduced. The parallel between SQCD and QCD was pursued further in16 where a proposal for the electromagnetic dual of QCD was constructed. In17 it was argued that there are three generic mechanisms how conformality is lost in quantum theories. Namely,
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1) the fixed point can go to zero coupling, 2) it can go to infinite coupling or 3) the Infrared fixed point (IRFP) may annihilate with an ultraviolet fixed point (UVFP) whence they both disappear into the complex plane. Mechanism 3) seems not to be allowed either with the β-function of SQCD or with the β-function ansatz of 15 for non-supersymmetric SU(Nc ) gauge theory. On the other hand, it was shown in17 that mechanism 3) is realized in a wide class of non-supersymmetric theories. Quantitative indications for this phenomena in QCD with many flavors were first presented in.18 In this talk we will discuss how the ideas of 15 and17 could be combined.19 We propose a concrete implementation of the mechanism 3) in an all order β-function for a generic Yang–Mills (YM) theory and show how this determines conformal windows; see also.19 2. An all order β-function ansatz We begin by a brief review of the basic results of 15 to define some notations. The rescaled ’t Hooft coupling is defined as a ≡ g 2 Nc /(4π)2 . In the perturbative βfunction for a generic non-supersymmetric YM theory with Nf Dirac fermions in a given representation R of the gauge group, the two lowest order coefficients are β 0 (x) = 11
C2 (G) − 4T (R)x Nc
(1)
β 1 (x) = 34
C22 (G) C2 (G) C2 (R) − 20 T (R)x − 12 T (R)x . 2 Nc Nc Nc
(2)
These two coefficients are universal, and the two-loop β-function is independent of the renormalization scheme, β(a) = − 23 β 0 (x)a2 − 23 β 1 (x)a3 . The conventions for the group theory factors for the representations we will consider can be found explicitly in.15 The proposal of 15 was to write all-order β-function as 2 β 0 (X) − X γ(a) , β(a) = − a2 3 1 − 2aC2 (G) 1 + 6β 0 (X) Nc
(3)
β 0 (X)
with β 0 (X) = 11 C2N(G) −2X, β 0 (X) = C2 (G)/Nc −X/2 and γ denotes the anomalous c dimension of the quark mass operator. For the group SU(Nc ), C2 (G) = Nc which simplifies the above equations. Perturbative two-loop β-function is obtained upon expansion to O(a3 ). Since only the two-loop β-function has universal coefficients, one assumes that there is a scheme in which the proposed β-function is complete. The ansatz in (3) determines the conformal window by assuming the value of γ at the lower boundary of the conformal window; by the unitarity bound γ ≤ 2. As discussed in the introduction, we will next modify Eq. (3) to be able to include conformality loss via mechanism 3).
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3. Fixed point merger in an all order β-function ansatz The simplest way to allow for the additional non-trivial UVFP in β-function of Eq. (3), and thus anticipate for the ‘fixed point merger’, is to include a term ∼ γ 2 in the numerator of the β-function suggesting a modified form 2 β (X) − X γ(a) + r(X)γ 2 (a) , β(a) = − a2 · 0 6β (X) 3 1 − 2aCN2c(G) 1 + β 0(X)
(4)
β 0 (X)
where β 0 (X) and are as defined earlier and we introduced unknown function r(X). We will determine this function first it in the limit Nc → ∞; possible O(1/Nc ) corrections will be discussed briefly in a later subsection. 3.1. Large Nc limit In the large Nc limit we will apply the holographic expectation17,20 that operator dimensions of the quark mass operators at the two fixed points satisfy ∆+ + ∆− = d = 4, which translates for the anomalous dimensions into equation γ1 + γ2 = 2.
(5)
Note, that this implies that the fixed point merger, a point where conformality is lost, will occur at γ1 = γ2 = 1. Setting the numerator of the β-function (4) to zero and solving the resulting quadratic equation, with the constraint Eq. (5) for the two roots, we find that r(X) = X/2. The numerator of Eq. (4) hence becomes X γ(γ − 2). (6) 2 Note the following features: First, the modified β-function, in the large Nc limit, predicts the lower end of the conformal window. For a given theory, we have to solve for X by setting the numerator of modified beta-function, Eq. (6), equal to zero under the condition γ = 1 on the basis of the constraint (5) from holography. Therefore, by construction, the maximum value of the anomalous dimension at the IRFP is γ ∗max = 1. This explicitly shows that the β-function of Eq. (4) is different from the one proposed in15 since scheme independent quantities, like the location of the conformal window and the values of the anomalous dimension, are different. Second, the upper end of the conformal window corresponds to γ = 0 and β0 = 0. Notice also that the second value of the anomalous dimension satisfying β0 = 0 in Eq. (6) is γ = 2 which coincides with the maximum value generally allowed in conformal field theory. We can then predict the size of the conformal window with the new β-function ansatz. Setting the expression in Eq. (6) to zero and using γ = 1 leads to the prediction that, in the large Nc limit, the lower end of the conformal window is at β 0 (X) +
Xmin =
22C2 (G) 22 . = 5Nc 5
(7)
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where we used G2 (G) = Nc . For the SU(Nc ) gauge theory with the fundamental quarks this predicts that Nfcr /Nc ≡ xmin = 22/5 in the large Nc limit. Note that for fundamental fermions this implies also that we have Nf → ∞, i.e. the Veneziano limit. The extrapolation of this result for the SU(3) theory with fundamental matter is shown in the left panel of Fig. 2. There, above x = 11/2 = 5.5 asymptotic freedom is lost (corresponding to the upper end of the conformal window), while below x = 22/5 = 4.4 the two zeros for γ become complex. Hence, for the SU(3), our extrapolation from large Nc result gives the critical number of flavors Nfcr = 13.2 below which the conformality is lost. This disagrees with the lattice result that the lower end of the conformal window occurs for 8 ≤ Nf ≤ 129 while agrees with e.g.21 Clearly more refined lattice studies are needed. As another example we consider calculating the lower end of the conformal window in terms of critical number of Dirac fermions Nfcr from (7) for fermions in the two-index symmetric (2S) representation of the SU(Nc ) gauge group. The results are presented in Fig.1 together with results corresponding to the β-function ansatz of 15 and with recent results22 obtained using deformation theory to establish the role of topological excitations in generating a mass gap in a gauge theory. Also the result from the ladder approximation is shown.
Nf
5
New Β
4
ladder D.T.
3
RSΓ1 ASF
2 2
RSΓ2
4
6
8
10
Nc
Fig. 1. Size of the conformal window determined using different methods discussed in the text with Nf fermions in 2-index symmetric representation of SU(Nc ). The topmost thick solid curve denotes the upper boundary of the conformal window, i.e. loss of asymptotic freedom. For the lower boundary the lower solid curve is the prediction from our β-function proposal, Eq.(7), the long dashed curve is the prediction from22 and the curve with shorter dashes correspond to the prediction from15 for γ = 2 (the dotted curve shows the corresponding result for γ = 1). The thin line between the solid and long-dashed curves corresponds to the ladder approximation.
3.2. Away from the large Nc limit However, in principle there could be important O(1/Nc ) corrections to (5) which, in turn, would modify the r(X) coefficient of the γ 2 term in (4). To get an idea of how such corrections would affect the results, we estimate these effects by means of
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a phenomenological modifying parameter as γ1 + γ2 = 2 + . Repeating then the exercise of Sec. 3.1, we arrive at the following coefficient of the γ 2 term X , 2+ and, thus, zero of the numerator of the new beta-function will occur at: γ − 1) = 0. β 0 (X) + X γ( 2+
(8)
r(X) =
X
X
5.4
5.4
5.2
5.2
5.0
5.0
4.8
4.8
4.6
4.6
4.4
4.4
4.2
4.2
0.0
(9)
0.5
1.0
1.5
2.0
Γ
0.0
Ε0.5
Ε0
Ε0.5
0.5
1.0
1.5
2.0
Γ
Fig. 2. Left: The line X(γ) along which β(a) = 0. Above X = 11/2 = 5.5 asymptotic freedom is lost while below X = 22/5 = 4.4 no real solutions exist. Right: The effects of 1/Nc corrections. Solid curve is the same as in the left panel. Dashed curve corresponds to = −0.5 and dotted to = 0.5
In the right panel of Fig. 2 we plot the solution X(γ) of this equation for the SU(3) fundamental with the illustrative values = ±0.5 and = 0. The latter coincides with the curve in the left panel of Fig. 2. The upper end of the conformal window at X=11/2=5.5 corresponding to γ1 = 0 remains unchanged, but the size of the conformal window increases (decreases) with positive (negative) . For positive the anomalous dimension at the UVFP will exceed the unitarity limit γ ≤ 2 before the IRFP merges with the free UVFP, i.e. before the other solution reaches γ = 0 value. In other words, as we decrease γ at the IRFP (by increasing with fixed X or vice versa) there is a point where UVFP disappears due to violation of the unitarity bound. Thus, past this point our picture becomes qualitatively similar to the one predicted by Ryttov-Sannino β-function. The value of γ at the IRFP below which the UVFP ceases to exist is given by γ = . Above, we have treated as a phenomenological parameter whose value is expected to be O(1). Considering positive and solving Eq. (9) at the point where the UV and IR fixed points merge (i.e. where γ1 = γ2 = 1 + /2) we obtain Xmin =
22 22C2 (G) = , Nc (5 + /2) 5 + /2
(10)
where the second equality again applies for SU(Nc ) gauge theory. For illustration we set = 1 which leads to Xmin = 4. Let us apply this result first for SU(3) with fundamental fermions. Using xmin = Xmin = 4 this translates to Nfmin = 12. Then let us consider various values of Nc
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Nf,min Fund. 8 12 16 20 24 40 4Nc
Nf,min 2AS 12 8 6.67 6 5 4
Nf,min 2S 2 2.4 2.67 2.86 3 3.33 4
Nf,min A. 2 2 2 2 2 2 2
and also higher representations in addition to the fundamental one. We collect the results into the Table 1. Interestingly, our results are consistent with present lattice results. Remarkably, our results match exactly on the corresponding predictions from deformation theory both for fundamental23 and higher representations.22 Of course any relation between our β-function ansatz and some underlying microscopic dynamics is pure speculation; nevertheless this coincidence, although unexpected, is temptingly systematic. Our results also agree with the ones in.24 4. Conclusions In this paper we exploited recently proposed mechanism of IR and UV fixed point annihilation as a key difference between supersymmetric and non-supersymmetric YM gauge theories. We incorporated this mechanism of fixed point merging into a proposal for a nonperturbative β-function whose functional form is similar to the Ryttov-Sannino β-function. We also compared our results with results from Ryttov-Sannino β-function15 as well as with results of.22 As a speculative note, the agreements and disagreements between these three approaches suggest that there might be important differences in how conformality is lost in supersymmetric versus non-supersymmetric gauge theories. Combining the insights from the analytic approaches with the numerical studies, performed recently and currently in progress, is expected to lead to significant improvement in our understanding of non-perturbative dynamics of (quasi)conformal gauge theories. References 1. F. Sannino and K. Tuominen, Phys. Rev. D 71, 051901 (2005) [arXiv:hep-ph/0405209]. 2. D. D. Dietrich, F. Sannino and K. Tuominen, Phys. Rev. D 72, 055001 (2005) [arXiv:hep-ph/0505059]; D. D. Dietrich, F. Sannino and K. Tuominen, Phys. Rev. D 73, 037701 (2006) [arXiv:hep-ph/0510217]. 3. A. J. Hietanen, J. Rantaharju, K. Rummukainen and K. Tuominen, JHEP 0905, 025 (2009) [arXiv:0812.1467 [hep-lat]]. 4. A. J. Hietanen, K. Rummukainen and K. Tuominen, Phys. Rev. D 80, 094504 (2009) [arXiv:0904.0864 [hep-lat]].
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5. S. Catterall and F. Sannino, Phys. Rev. D 76, 034504 (2007) [arXiv:0705.1664 [heplat]]; L. Del Debbio, A. Patella and C. Pica, arXiv:0805.2058 [hep-lat]; 6. S. Catterall, J. Giedt, F. Sannino and J. Schneible, JHEP 0811, 009 (2008) [arXiv:0807.0792 [hep-lat]]. 7. Y. Shamir, B. Svetit*ky and T. DeGrand, Phys. Rev. D 78, 031502 (2008) [arXiv:0803.1707 [hep-lat]]. 8. T. DeGrand, Y. Shamir and B. Svetit*ky, Phys. Rev. D 79, 034501 (2009) [arXiv:0812.1427 [hep-lat]]. 9. T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. Lett. 100 (2008) 171607 [Erratum-ibid. 102 (2009) 149902] [arXiv:0712.0609 [hep-ph]]; T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. D 79, 076010 (2009) [arXiv:0901.3766 [hep-ph]]. 10. O. Antipin, M. Heikinheimo and K. Tuominen, JHEP 0910, 018 (2009) [arXiv:0905.0622 [hep-ph]]. M. T. Frandsen, I. Masina and F. Sannino, arXiv:0905.1331 [hep-ph]; R. Foadi, M. T. Frandsen and F. Sannino, Phys. Rev. D 77, 097702 (2008) [arXiv:0712.1948 [hep-ph]]; O. Antipin and K. Tuominen, Phys. Rev. D 79, 075011 (2009) [arXiv:0901.4243 [hep-ph]]. 11. A. Belyaev, R. Foadi, M. T. Frandsen, M. Jarvinen, F. Sannino and A. Pukhov, Phys. Rev. D 79, 035006 (2009) [arXiv:0809.0793 [hep-ph]]. 12. V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 229 (1983) 381; M. A. Shifman and A. I. Vainshtein, Nucl. Phys. B 277 (1986) 456 [Sov. Phys. JETP 64 (1986 ZETFA,91,723-744.1986) 428]. 13. N. Seiberg, Nucl. Phys. B 435 (1995) 129 [arXiv:hep-th/9411149]. 14. K. A. Intriligator and N. Seiberg, Nucl. Phys. Proc. Suppl. 45BC, 1 (1996) [arXiv:hepth/9509066]. 15. T. A. Ryttov and F. Sannino, Phys. Rev. D 78 (2008) 065001 [arXiv:0711.3745 [hepth]]. 16. F. Sannino, Phys. Rev. D 80, 065011 (2009) [arXiv:0907.1364 [hep-th]]. 17. D. B. Kaplan, J. W. Lee, D. T. Son and M. A. Stephanov, Phys. Rev. D 80, 125005 (2009) [arXiv:0905.4752 [hep-th]]. 18. H. Gies and J. Jaeckel, Eur. Phys. J. C 46, 433 (2006) [arXiv:hep-ph/0507171]. 19. O. Antipin and K. Tuominen, arXiv:0909.4879 [hep-ph]. 20. I. R. Klebanov and E. Witten, Nucl. Phys. B 556, 89 (1999) [arXiv:hep-th/9905104]. 21. Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, arXiv:0911.2463 [hep-lat]. 22. E. Poppitz and M. Unsal, JHEP 0909, 050 (2009) [arXiv:0906.5156 [hep-th]]. 23. E. Poppitz and M. Unsal, arXiv:0910.1245 [hep-th]. 24. A. Armoni, Nucl. Phys. B 826, 328 (2010) [arXiv:0907.4091 [hep-ph]].
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Nearly Conformal Gauge Theories on the Lattice Zolt´ an Fodor Department of Physics, University of Wuppertal, Gauss Strasse 20, D-42119, Germany and Institute for Theoretical Physics, Eotvos University Budapest, Pazmany P. 1, H-1117, Hungary E-mail: [emailprotected] Kieran Holland Department of Physics, University of the Pacific 3601 Pacific Ave, Stockton CA 95211, USA E-mail: [emailprotected] Julius Kuti∗ Department of Physics 0319, University of California, San Diego 9500 Gilman Drive, La Jolla CA 92093, USA E-mail: [emailprotected] D´ aniel N´ ogr´ adi Institute for Theoretical Physics, Eotvos University Budapest, Pazmany P. 1, H-1117, Hungary E-mail: [emailprotected] Chris Schroeder Department of Physics, University of Wuppertal, Gauss Strasse 20, D-42119, Germany E-mail: [emailprotected] We present selected new results on chiral symmetry breaking in nearly conformal gauge theories with fermions in the fundamental representation of the SU(3) color gauge group. We found chiral symmetry breaking (χSB) for all flavors between Nf = 4 and Nf = 12 with most of the results discussed here for Nf = 4, 8, 12 as we approach the conformal window. To identify χSB we apply several methods which include, within the framework of chiral perturbation theory, the analysis of the Goldstone spectrum in the p-regime and the spectrum of the fermion Dirac operator with eigenvalue distributions of random matrix theory in the -regime. Chiral condensate enhancement is observed with increasing Nf when the electroweak symmetry breaking scale F is held fixed in technicolor language. Important finite-volume consistency checks from the theoretical understanding of the SU (Nf ) rotator spectrum of the δ-regime are discussed. We also consider these gauge theories at Nf = 16 inside the conformal window. Our work on the running coupling is presented separately.1 Keywords: Beyond Standard Model, lattice, near-conformal, running coupling. ∗ Speaker
at the conference.
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1. Introduction It is an intriguing possibility that new physics beyond the Standard Model might take the form of some new strongly-interacting gauge theory building on the original technicolor idea.2–4 This approach has lately been revived by new explorations of the multi-dimensional theory space of nearly conformal gauge theories.5–8 Model building of a strongly interacting electroweak sector requires the knowledge of the phase diagram of nearly conformal gauge theories as the number of colors Nc , number of fermion flavors Nf , and the fermion representation R of the technicolor group are varied in theory space. For fixed Nc and R the theory is in the chirally broken phase for low Nf , and asymptotic freedom is maintained with a negative β function. On the other hand, if Nf is large enough, the β function is positive for all couplings, and the theory is trivial. There is some range of Nf for which the β function might have a non-trivial zero, an infrared fixed point, where the theory is in fact conformal.11,12 This method has been refined by estimating the critical value of Nf , above which spontaneous chiral symmetry breaking no longer occurs.13–15 Interesting models require the theory to be very close to, but below, the conformal window, with a running coupling which is almost constant over a large energy range .16–21 The nonperturbative knowledge of the critical Nfcrit separating the two phases is essential and this has generated much interest and many new lattice studies.22–55 To provide theoretical framework for the analysis of simulation results, we review first a series of tests expected to hold in the setting of χPT in finite volume and in the infinite volume limit. 2. Chiral symmetry breaking below the conformal window We will identify in lattice simulations the chirally broken phases with Nf = 4, 8, 12 flavors of staggered fermions in the fundamental SU(3) color representation using finite volume analysis. We deploy staggered fermions with exponential (stout) smearing56 in the lattice action to reduce well-known cutoff effects with taste breaking in the Goldstone spectrum.57 The presence of taste breaking requires careful analysis of staggered χPT following the important work of Lee, Sharpe, Aubin and Bernard.58–60 2.1. Finite volume analysis in the p-regime Three different regimes can be selected in simulations to identify the chirally broken phase from finite volume spectra and correlators. For a lattice size L3s × Lt in euclidean space and in the limit Lt Ls , the conditions Fπ Ls > 1 and Mπ Ls > 1 select the the p-regime, in analogy with low momentum counting.61,62 For arbitrary Nf , in the continuum and in infinite volume, the one-loop chiral corrections to Mπ and Fπ of the degenerate Goldstone pions are given by Λ3 M2 2 2 ln , (1) Mπ = M 1 − 2 8π Nf F 2 M
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Λ4 Nf M 2 ln Fπ = F 1 + , 16π 2 F 2 M
(2)
where M 2 = 2B · mq and F, B, Λ3 , Λ4 are four fundamental parameters of the chiral Lagrangian, and the small quark mass mq explicitly breaks the symmetry.63 The chiral parameters F, B appear in the leading part of the chiral Lagrangian, while Λ3 , Λ4 enter in next order. There is the well-known GMOR relation Σcond = BF 2 in the mq → 0 limit for the chiral condensate per unit flavor.64 It is important to note that the one-loop correction to the pion coupling constant Fπ is enhanced by a factor Nf2 compared to Mπ2 . The chiral expansion for large Nf will break down for Fπ much faster for a given Mπ /Fπ ratio. The NNLO terms have been recently calculated65 showing potentially dangerous Nf2 corrections to Eqs. (1,2). The finite volume corrections to Mπ and Fπ are given in the p-regime by M2 1 · ge1 (λ, η) , (3) Mπ (Ls , η) = Mπ 1 + 2Nf 16π 2 F 2 Nf M 2 Fπ (Ls , η) = Fπ 1 − · g e (λ, η) , 1 2 16π 2 F 2
(4)
where ge1 (λ, η) describes the finite volume corrections with λ = M · Ls and aspect ratio η = Lt /Ls . The form of ge1 (λ, η) is a complicated infinite sum which contains Bessel functions and requires numerical evaluation.62 Eqs. (1-4) provide the foundation of the p-regime fits in simulations. 1 Lt 1 Ls
4 F 2 Ls3
! chiral p-regime
!
3 2F 2 Ls3
rotator pion energy gap 1 F 2 Ls3
M = 2Bmq
1 Ls
Fig. 1. Schematic plot of the regions in which the three low energy chiral expansions are valid. The vertical axis shows the finite temperature scale (euclidean time in the path integral) which probes the rotator dynamics of the δ-regime and the -regime. The first two low lying rotator levels are also shown on the vertical axis for the simple case of Nf = 2. The fourfold degenerate lowest rotator excitation at mq = 0 will split into an isotriplet state (lowest energy level), which evolves into the p-regime pion as mq increases, and into an isosinglet state representing a multi-pion state in the p-regime. Higher rotator excitations have similar interpretations.
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2.2. δ-regime and -regime At fixed Ls and in cylindrical geometry Lt /Ls 1, a crossover occurs from the pregime to the δ-regime when mq → 0, as shown in Fig. 1. The dynamics is dominated by the rotator states of the chiral condensate in this limit66 which is characterized by the conditions F Ls > 1 and M Ls 1. The densely spaced rotator spectrum scales with gaps of the order ∼ 1/F 2 L3s , and at mq = 0 the chiral symmetry is apparently restored. However, the rotator spectrum, even at mq = 0 in the finite volume, will signal that the infinite system is in the chirally broken phase for the particular parameter set of the Lagrangian. This is often misunderstood in the interpretation of lattice simulations. Measuring finite energy levels with pion quantum numbers at fixed Ls in the mq → 0 limit is not a signal for chiral symmetry restoration of the infinite system.40 0.16
0.12 (a M)2
πi5
Nf = 4 4 Stout β = 3.80 163 x 32
π
0.08
0.04
0 0
0.01
0.02
0.03 a mq
0.04
0.05
Fig. 2. The crossover from the p-regime to the δ-regime is shown for the π and πi5 states at Nf = 4.
If Lt ∼ Ls under the conditions F Ls > 1 and M Ls 1, the system will be driven into the -regime which can be viewed as the high temperature limit of the δ-regime quantum rotator. Although the δ-regime and -regime have an overlapping region, there is an important difference in their dynamics. In the δ-regime of the quantum rotator, the mode of the pion field U (x) with zero spatial momentum dominates with time-dependent quantum dynamics. The -regime is dominated by the four-dimensional zero momentum mode of the chiral Lagrangian. We report simulation results of all three regimes in the chirally broken phase of the technicolor models we investigate. The analysis of the three regimes complement each other and provide cross-checks for the correct identification of the phases. First, we will probe Eqs. (1-4) in the p-regime, and follow with the study of Dirac spectra and RMT eigenvalue distributions in the -regime. The spectrum in the δ-regime is used as a signal to monitor p-regime spectra as mq decreases. Fig. 2 is an illustrative
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example of this crossover in our simulations. It is important to note that the energy levels in the chiral limit do not always match the rotator spectrum at the small F ·Ls values of the simulations. This squeezing of insufficiently large enough F · Ls for undistorted, finite volume chiral behavior in the p-regime, -regime, and δ-regime is a serious limitation of all recent simulations.
3. Goldstone spectrum and χSB at Nf = 12 We find the chiral symmetry breaking pattern for the controversial Nf = 12 case similar to the Nf = 8, 9 cases. The Goldstone spectrum remains separated from the technicolor scale of the ρ-meson. The true Goldstone pion and two additional split pseudo-Goldstone states are shown again in Fig. 3 with different slopes as a · mq increases. The trends and the underlying explanation are similar to the Nf = 8, 9 cases. The chiral fit to Mπ2 /mq shown at the top right side of Fig. 3 is based on Eq. (1) only since the Fπ data points are outside the convergence range of the chiral expansion. At β = 2.2 the fitted value of B is a · B = 2.7(2) in lattice units with a·F = 0.0120(1) and a·Λ3 = 0.50(3) also fitted. The fitted ρ-mass in the chiral limit is a · Mρ = 0.115(15) from a · mq = 0.025 − 0.045 with Mρ /F = 10(1). The fitted value of B/F = 223(17) is not very reliable but consistent with the enhancement of the chiral condensate found at Nf = 8, 9 without including renormalization scale effects. Again, at fixed lattice spacing, the small chiral condensate hψψi summed over all flavors is dominated by the linear term in mq from UV contributions. The linear fit gives hψψi = 0.0033(13) in the chiral limit which came out unexpectedly close the GMOR relation of hψψi = 12F 2 B with 12F 2 B = 0.0046(4) fitted. Issues and concerns in the systematics are similar to the Nf = 8, 9 cases but finite volume limitations and the convergence of the chiral expansion are more problematic. Currently we are investigating the important Nf = 12 model on larger lattices to probe the possible influence of unwanted squeezing effects on the spectra. This should also clarify the mass splitting pattern of the ρ and A1 states we are seeing in the chiral limit as Nf is varied. Our findings at Nf = 12 are not consistent with some earlier work31,32 where the runing coupling was studied. Lessons from the Dirac spectra and RMT to complement p-regime tests are discussed in the next section.
4. Epsilon regime, Dirac spectrum and RMT If the bare parameters of a gauge theory are tuned to the -regime in the chirally broken phase, the low-lying Dirac spectrum follows the predictions of random matrix theory. The corresponding random matrix model is only sensitive to the pattern of chiral symmetry breaking, the topological charge and the rescaled fermion mass once the eigenvalues are also rescaled by the same factor Σcond V . This idea has been confirmed in various settings both in quenched and fully dynamical simulations. The same method is applied here to nearly conformal gauge models.
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ρ
Nf = 12 2 Stout β = 2.00 243 x 32
0.3
Nf = 12 2 Stout β = 2.20 mq < 0.033: 324 mq > 0.033: 243x32
πi5 5
0.25
π
2
(a M)
5.5
πij
a Mπ2 / mq
0.4 0.35
0.2 0.15 0.1
4.5
4
0.05 0
3.5
0.4
0.02 0.03 a mq
Nf = 12 2 Stout β = 2.20 mq < 0.033: 324 mq > 0.033: 243x32
0.35 0.3 0.25 (a M)2
0.01
0.04
πij 0.06 π
0.2
0.02 0.03 a mq
0.04
Nf = 12 2 Stout β = 2.20 mq < 0.033: 324 mq > 0.033: 243x32
0.08
ρ
πi5
0.01
a Fπ
0.15
0.04
0.1 0.05
0.02
0 0
0.4
0.01
0.02 0.03 a mq
0.04
3.2
Nf = 12 2 Stout β = 2.40 243 x 32
0.3
2.8 2.4
(a M)2
0.25 ρ πij πi5 π
0.2 0.15
a3 < ψ ψ >
0.35
2 1.6 1.2
0.1
0.8
0.05
0.4
0.01
0.02 0.03 a mq
0.04
Nf = 12 2 Stout 203x32 - 324 β=1.0 β=1.4 β=1.8 β=2.0 β=2.2 β=2.4
0 0
0.01
0.02 0.03 a mq
0.04
0.01
0.02 a mq
0.03
0.04
0.05
Fig. 3. The pseudo-Goldstone spectrum and chiral fits are shown for Nf = 12 simulations with lattice size 243 ×32 and 324 . The left column shows the pseudo-Goldstone spectrum with decreasing taste breaking as the gauge coupling is varied from β = 2.0 to β = 2.4. Although the bottom figure on the left at β = 2.4 illustrates the continued restoration of taste symmetry, the volume is too small for the Goldstone spectrum. The middle value at β = 2.2 was chosen in the top right figure with fitting range a · mq = 0.015 − 0.035 of the NLO chiral fit to Mπ2 /mq which approaches 2B in the chiral limit. The middle figure on the right shows the Fπ data with no NLO fit far away from the chiral limit. The bottom right figure, with its additional features discussed in the text, is the linear fit to the chiral condensate with fitting range a·mq = 0.02−0.04. The physical fit parameters B, F, Λ3 are discussed in the text. Two stout steps were used in all Nf = 12 simulations.
The connection between the eigenvalues λ of the Dirac operator and chiral symmetry breaking is given in the Banks-Casher relation,70 Σcond = −hΨΨi = lim lim lim
λ→0 m→0 V →∞
πρ(λ) , V
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integrated distribution
a mq = 0.001
0.02 λn
1
Nf = 4 β = 3.60
0.025
244
0.015
0.01
Nf = 4 β = 3.60 a mq = 0.001 244
0.8 0.6 0.4 0.2
1st quartet 2nd quartet 0
0.005 0
0.07
2
4
6
8
10 n
12
14
16
18
integrated distribution
a mq = 0.001 244
0.05
0.004
0.008 0.012 0.016 quartet average, a units
0.02
1
Nf = 4 β = 3.80
0.06
λn
20
0.04 0.03
Nf = 4 β = 3.80 a mq = 0.001 244
0.8 0.6 0.4 0.2
1st quartet 2nd quartet
0.02
0 0.01 0
0.18
2
4
6
8
10 n
12
14
16
18
λn
0.12
integrated distribution
0.14
0.01
0.02 0.03 0.04 quartet average, a units
0.05
1
Nf = 4 β = 4.00 a mq = 0.001 2044 24
0.16
20
0.1 0.08
Nf = 4 β = 4.00 a mq = 0.001 204
0.8 0.6 0.4 0.2
1st quartet 2nd quartet
0.06 0
0.04 0
2
4
6
8
10 n
12
14
16
18
20
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 quartet average, a units
Fig. 4. From simulations at Nf = 4 the first column shows the approach to quartet degeneracy of the spectrum as β increases. The second column shows the integrated distribution of the two lowest quartets averaged. The solid line compares this procedure to RMT with Nf = 4.
where Σcond designates the quark condensate normalized to a single flavor. To generate a non-zero density ρ(0), the smallest eigenvalues must become densely packed as the volume increases, with an eigenvalue spacing ∆λ ≈ 1/ρ(0) = π/(Σcond V ). This allows a crude estimate of the quark condensate Σcond . One can do better by exploring the -regime: If chiral symmetry is spontaneously broken, tune the volume and quark mass such that F1π L M1π , so that the Goldstone pion is much lighter than the physical value, and finite volume effects are dominant as we discussed in Section 2. The chiral Lagrangian is dominated by the zero-momentum mode from the mass term and all kinetic terms are suppressed. In this limit, the distributions of the lowest eigenvalues are identical to those of random matrix theory, a theory of large matrices obeying certain symmetries.71–73 To connect with RMT,
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the eigenvalues and quark mass are rescaled as z = λΣcond V and µ = mq Σcond V , and the eigenvalue distributions also depend on the topological charge ν and the number of quark flavors Nf . RMT is a very useful tool to calculate analytically all of the eigenvalue distributions.74 The eigenvalue distributions in various topological sectors are measured via lattice simulations, and via comparison with RMT, the value of the condensate Σcond can be extracted. After we generate large thermalized ensembles, we calculate the lowest twenty eigenvalues of the Dirac operator using the PRIMME package.75 In the continuum limit, the staggered eigenvalues form degenerate quartets, with restored taste symmetry. The first column of Fig. 4 shows the change in the eigenvalue structure for Nf = 4 as the coupling constant is varied. At β = 3.6 grouping into quartets is not seen, the Goldstone pions are somewhat still split, and staggered perturbation theory is just beginning to kick in. At β = 3.8 doublet pairing appears and at β = 4.0 the quartets are nearly degenerate. The Dirac spectrum is collapsed as required by the Banks-Casher relation. In the second column we show the integrated distributions of the two lowest eigenvalue quartet averages, Z λ pk (λ0 )dλ0 , k = 1, 2 (5) 0
which is only justified close to quartet degeneracy. All low eigenvalues are selected with zero topology. To compare with RMT, we vary µ = mq Σcond V until we satisfy hz1 iRMT hλ1 isim = , m µ
(6)
where hλ1 isim is the lowest quartet average from simulations and the RMT average hziRMT depends implicitly on µ and Nf . With this optimal value of µ, we can predict the shapes of pk (λ) and their integrated distributions, and compare to the simulations. The agreement with the two lowest integrated RMT eigenvalue shapes is excellent for the larger β values. The main qualitative features of the RMT spectrum are very similar in our Nf = 8 simulations as shown in Fig. 5. One marked quantitative difference is a noticeable slowdown in response to change in the coupling constant. As β grows the recovery of the quartet degeneracy is considerably delayed in comparison with the onset of p-regime Goldstone dynamics. Overall, for the Nf = 4, 8 models we find consistency between the p-regime analysis and the RMT tests. Earlier, using Asqtad fermions at a particular β value, we found agreement with RMT even at Nf = 12 which indicated a chirally broken phase.23 Strong taste breaking with Asqtad fermions leaves the quartet averaging in question and the bulk pronounced crossover of the Asqtad action as β grows is also an issue. Currently we are investigating the RMT picture for Nf = 9, 10, 11, 12 with our much improved action with stout smearing. This action shows no artifact transitions and handles taste breaking much more effectively. Firm conclusions on the Nf = 12 model to support our findings of χSB in the p-regime will require continued investigations.
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integrated distribution
1 Nf = 8 β = 1.40 a mq = 0.001 244
0.8 0.6 0.4 0.2
1st quartet 2nd quartet 0 0
0.01
0.02 0.03 0.04 quartet average, a units
0.05
0.06
Fig. 5. The solid lines compare the integrated distribution of the two lowest quartet averages to RMT predictions with Nf = 8.
5. Inside the conformal window A distinguished feature of the Nf = 16 conformal model is how the renormalized coupling g 2 (L) runs with L, the linear size of the spatial volume in a Hamiltonian or Transfer Matrix description. On very small scales the running coupling g 2 (L) grows with L as in any other asymptotically free theory. However, g 2 (L) will not grow large, and in the L → ∞ limit it will converge to the fixed point g ∗2 which is rather weak,76 within the reach of perturbation theory. There is non-trivial, small-volume dynamics which is illustrated first in the pure gauge sector. At small g 2 , without fermions, the zero-momentum components of the gauge field are known to dominate the dynamics.77–79 With SU (3) gauge group, there are twenty-seven degenerate vacuum states, separated by energy barriers which are generated by the integrated effects of the non-zero momentum components of the gauge field in the Born-Oppenheimer approximation. The lowest-energy excitations of the gauge field Hamiltonian scale as ∼ g 2/3 (L)/L evolving into glueball states and becoming independent of the volume as the coupling constant grows with L. Non-trivial dynamics evolves through three stages as L grows. In the first regime, in very small boxes, tunneling is suppressed between vacua which remain isolated. In the second regime, for larger L, tunneling sets in and electric flux states will not be exponentially suppressed. Both regimes represent small worlds with zeromomentum spectra separated from higher momentum modes of the theory with energies on the scale of 2π/L. At large enough L the gauge dynamics overcomes the energy barrier, and wave functions spread over the vacuum valley. This third regime is the crossover to confinement where the electric fluxes collapse into thin string states wrapping around the box.
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It is likely that a conformal theory with a weak coupling fixed point at Nf = 16 will have only the first two regimes which are common with QCD. Now the calculations have to include fermion loops.80,81 The vacuum structure in small enough volumes, for which the wave functional is sufficiently localized around the vacuum configuration, remains calculable by adding in one-loop order the quantum effects of the fermion field fluctuations. The spatially constant abelian gauge fields parametrizing the vacuum valley are given by Ai (x) = T a Cia /L where Ta are the (N-1) generators for the Cartan subalgebra of SU (N ). For SU (3), T1 = λ3 /2 and T2 = λ8 /2. With Nf flavors of massless fermion fields the effective potential of the constant mode is given by X X (i) (j) (i) k Veff (Cb ) = V (Cb [µb − µb ]) − Nf V (Cb µb + πk), (7) i>j
i
with k = 0 for periodic, or k = (1, 1, 1), for antiperiodic boundary conditions on the fermion fields. The function V (C) is the one-loop effective potential for Nf = 0 and the weight vectors µ(i) are determined by the eigenvalues of the abelian genera√ tors. For SU(3) µ(1) = (1, 1, −2)/ 12 and µ(2) = 12 (1, −1, 0). The correct quantum vacuum is found at the minimum of this effective potential which is dramatically changed by the fermion loop contributions. The Polyakov loop observables remain center elements at the new vacuum configurations with complex values; for SU (N ) 1 X 1 (n) tr exp(iCjb Tb ) = exp(iµb Cjb ) = exp(2πilj /N ). (8) Pj = N N n (n)
k This implies µb Cb = 2πl/N (mod 2π), independent of n, and Veff = −Nf N V (2πl/N +πk). In the case of antiperiodic boundary conditions, k = (1, 1, 1), this is minimal only when l = 0 (mod 2π). The quantum vacuum in this case is the naive one, A = 0 (Pj = 1). In the case of periodic boundary conditions, k = 0, the vacua have l 6= 0, so that Pj correspond to non-trivial center elements. For
0.4
Imaginary
0.2 0
0.6
Nf = 16 3 Stout β = 30.0 m = 0.005 3 12 x 36 pbc
0.4 x 0.2 Imaginary
0.6
y -0.2 -0.4
Nf = 16 3 Stout β = 18.0 m = 0.001 3 12 x 36 apbc
x
-0.2 z
-0.6 -0.8 -0.6 -0.4 -0.2 0 Real
y
z
-0.4
0.2
0.4
0.6
-0.6 -0.8 -0.6 -0.4 -0.2 0 Real
0.2
0.4
0.6
Fig. 6. The time evolution of complex Polyakov loop distributions are shown from our Nf = 16 simulations with 123 × 36 lattice volume. Tree-level Symanzik-improved gauge action is used in the simulations and staggered fermions with three stout steps and very small fermion masses.
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SU(3), there are now 8 degenerate vacua characterized by eight different Polyakov loops, Pj = exp(±2πi/3). Since they are related by coordinate reflections, in a small volume parity (P) and charge conjugation (C) are spontaneously broken, although CP is still a good symmetry.80 Our simulations of the Nf = 16 model below the conformal fixed point g ∗2 confirm the theoretical vacuum structure. Fig. 6 shows the time evolution of Polyakov loop distributions monitored along the three separate spatial directions. On the left side, with periodic spatial boundary conditions, the time evolution is shown starting from randomized gauge configuration with the Polyakov loop at the origin. The system evolves into one of the eight degenerate vacua selected by the positive imaginary part of the complex Polyakov loop along the x and y direction and negative imaginary part along the z direction. On the right, with antiperiodic spatial boundary conditions, the vacuum is unique and trivial with real Polyakov loop in all three directions. The time evolution is particularly interesting in the z direction with a swing first from the randomized gauge configuration to a complex metastable minimum first, and eventually tunneling back to the trivial vacuum and staying there, as expected. The measured fermion-antifermion spectra and the spectrum of the Dirac operator further confirm this vacuum structure. Acknowledgments Julius Kuti is greatful to the organizers for a most enjoyable meeting. We thank Sandor Katz and Kalman Szabo for the Wuppertal RHMC code. For some calculations, we used the publicly available MILC code. We performed simulations on the Wuppertal GPU cluster, Fermilab clusters under the auspices of USQCD and SciDAC, and the Ranger cluster of the Teragrid organization. This research is supported by the NSF under grant 0704171, by the DOE under grants DOE-FG03-97ER40546, DOE-FG-02-97ER25308, by the DFG under grant FO 502/1 and by SFB-TR/55. References 1. K. Holland, Talk in this Proceedings; Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, PoS LATTICE2009, 058 (2009). 2. S. Weinberg, Phys. Rev. D 19, 1277 (1979). 3. L. Susskind, Phys. Rev. D 20, 2619 (1979). 4. E. Farhi and L. Susskind, Phys. Rept. 74, 277 (1981). 5. F. Sannino, arXiv:0902.3494 [hep-ph]. 6. T. A. Ryttov and F. Sannino, Phys. Rev. D 76, 105004 (2007). 7. D. D. Dietrich and F. Sannino, Phys. Rev. D 75, 085018 (2007). 8. D. K. Hong et al., Phys. Lett. B 597, 89 (2004). 9. H. Georgi, Phys. Rev. Lett. 98, 221601 (2007). 10. M. A. Luty and T. Okui, JHEP 0609, 070 (2006). 11. W. E. Caswell, Phys. Rev. Lett. 33, 244 (1974). 12. T. Banks and A. Zaks, Nucl. Phys. B 196, 189 (1982). 13. T. Appelquist et al., Phys. Rev. Lett. 61, 1553 (1988). 14. A. G. Cohen and H. Georgi, Nucl. Phys. B 314, 7 (1989).
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15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62.
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Going Beyond QCD in Lattice Gauge Theory G. T. Fleming Sloane Physics Laboratory, Yale University New Haven, CT 06520-8120, USA E-mail: [emailprotected] Strongly coupled gauge theories (SCGT’s) have been studied theoretically for many decades using numerous techniques. The obvious motivation for these efforts stemmed from a desire to understand the source of the strong nuclear force: Quantum Chromodynamics (QCD). Guided by experimental results, theorists generally consider QCD to be a well-understood SCGT. Unfortunately, it is not clear how to extend the lessons learned from QCD to other SCGT’s. Particularly urgent motivators for new studies of other SCGT’s are the ongoing searches for physics beyond the standard model (BSM) at the Large Hadron Collider (LHC) and the Tevatron. Lattice gauge theory (LGT) is a technique for systematically-improvable calculations in many SCGT’s. It has become the standard for non-perturbative calculations in QCD and it is widely believed that it may be useful for study of other SCGT’s in the realm of BSM physics. We will discuss the prospects and potential pitfalls for these LGT studies, focusing primarily on the flavor dependence of SU(3) gauge theory. Keywords: Beyond the standard model, dynamical electroweak symmetry breaking, lattice gauge theory, technicolor.
1. Introduction The Tevatron is now scheduled to run through the end of 2011 with the expectation of providing three-sigma evidence against the existence of a standard model Higgs boson if, in fact, such a boson does not exist in nature. Given the current commissioning status, it is possible that the Large Hadron Collider (LHC) may be able to meaningfully contribute to the Higgs search in the same time frame. If the Higgs boson is ruled out, the case for new strong interactions to be discovered at higher energies in LHC detectors will be very much enhanced. If a strongly coupled gauge theory (SCGT) is responsible for the dynamical breaking of electroweak symmetry (DEWSB) in Nature, then the Tevatron and LHC may not observe a Standard Model Higgs boson. Such experimental nonobservation of the Higgs boson will accelerate already growing interest in DEWSB models. It is fortuitous that calculational capabilities of lattice gauge theory (LGT) are now able to make a significant impact just when new strong physics has the best chance of being discovered at the TeV scale. As a result, many lattice groups have now started calculations in theories which describe possible new TeV-scale strong interactions. See2,3 for reviews and additional references therein.
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95% CL Limit/SM
Tevatron Run II Preliminary, = 5.9 fb
LEP Exclusion
10
1
Expected Observed ±1σ Expected ±2σ Expected
-1
Tevatron Exclusion
SM=1 Tevatron Exclusion
July 19, 2010
100 110 120 130 140 150 160 170 180 190 200 2 mH(GeV/c ) Fig. 1. Observed and expected (median, for the background-only hypothesis) 95% C.L. upper limits on the ratios to the SM cross section, as functions of the Higgs boson mass for the combined CDF and D0 analyses. The limits are expressed as a multiple of the SM prediction for test masses (every 5 GeV/c2 ) for which both experiments have performed dedicated searches in different channels. The points are joined by straight lines for better readability. The bands indicate the 68% and 95% probability regions where the limits can fluctuate, in the absence of signal. The limits displayed in this figure are obtained with the Bayesian calculation.1
2. Conformal Window With a small number of massless fermions, a vector-like gauge field theory such as QCD exhibits confinement and dynamical chiral symmetry breaking. If the number of massless fermions in the fundamental representation, Nf , is larger than some easily computed number Nfaf = (11/2)Nc, then asymptotic freedom is lost and the theory is not even well defined because the renormalized coupling diverges at short distances. But if Nf is just below value Nfaf , the theory is conformal in the infrared, governed by a weak infrared fixed point (IRFP) which appears already in the twoloop beta function.4,5 There is no confinement, and chiral symmetry is unbroken. This IRFP persists down to some critical value Nfc , where the coupling grows sufficiently strong at long distances that a transition to the confined, chirally broken phase takes place. The range Nfc < Nf < Nfaf is the “conformal window”, where the theory is in the “non-Abelian Coulomb phase”. The initial indications, based on the non-perturbative LGT calculations of the running coupling in the Schr¨ odinger functional scheme6–8 as shown in Fig. 2, lead us to the tentative conclusion that 8 < Nfc < 12. However, the most recent LGT studies14–17,19–34 for SU(3) gauge theory with Nf flavors in the fundamental representation use a wide array of calculational tech-
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Fig. 2. Non-perturbative LGT calculations of the SU(3) Yang-Mills running coupling in the Schr¨ odinger functional scheme for Nf =0,2,4,8,10,12 flavors in the fundamental representation.9–18 The preliminary Nf =10 points were presented by N. Yamada at the second Workshop on Lattice Gauge Theory for LHC Physics, 6–7 Nov. 2009, Boston, MA. The horizontal bar at the bottom of the figure indicates the range of scales covered by a lattice calculation on a 323 spatial volume.
niques to address the question of whether Nf =12 is inside or outside the conformal window and the situation is very unclear. In particular, LGT methods commonly used to study QCD seem poorly suited for the study of theories with IRFP’s. Thus, it will likely be some time before these methods can definitively confirm (or rule out) the existence of an IRFP for Nf = 12 massless flavors. Just below this critical number of flavors, it is plausible that Yang-Mills exhibits behavior at intermediate scales that is more like the physics of the non-Abelian Coulomb phase than of QCD. Of course, at very long distances, the theory must eventually confine otherwise it would be inside the conformal window. Such behavior is called “walking” because the renormalized coupling is expected to run much slower or “walk” over intermediate distance scales when compared to QCD.35–39 This “walking” phenomenon can play an important phenomenological role in a technicolor theory of electroweak symmetry breaking. If there are Nf /2 electroweak doublets, then F = FEW / Nf /2 250 GeV/ Nf /2. Flavor-changing neutral currents (FCNC’s), which are present when the technicolor theory is extended to provide for the generation of quark masses, can be too large unless the associated scale ΛET C is high enough. But then the first- and second-generation quark masses are typically much too small. They are proportional to the quantity ψψ/Λ2ET C , where ψ is a technifermion field and ψψ is the bilinear fermion condensate defined
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0.6
, M ρ , MN r-1 0
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Nf=2 MN,m Nf=6 MN,m Nf=2 Mρ,m Nf=6 Mρ,m Nf=2 r-1 0,m Nf=6 r-1 0,m
0.3 0.2 0.1 0 0
0.01
m
0.02
0.03
Fig. 3. Chiral extrapolations of the nucleon mass MN , the vector meson mass Mρ and the Sommer scale r0,m , in lattice units. All three quantities show good agreement between Nf = 2 and Nf = 6 in the chiral limit, suggesting that the lattice spacings in the two calculations are matched within 10% accuracy.
(cut off) at ΛET C . Walking can lift the quark masses by enhancing the condensate ψψ significantly above its value (O(4πF 3 )) in a QCD-like theory while keeping Λ2ET C large enough to suppress FCNC’s. One main difficulty of studying a walking theory using LGT methods is the emergence of two dynamically-generated, and presumably widely-separated, scales: approximate conformal invariance at intermediate distances and confinement at long distances. It has been quite challenging to produce lattices large enough to simultaneously exhibit both phenomena without significant distortions due to finite volume or finite lattice spacing effects. An examination in Fig. 2 of the range of scales covered by a typical LGT calculation on a 323 spatial volume indicates the challenge of simultaneously observing both weakly-coupled behavior on short distance scales and strongly-coupled behavior at long distance scales in a single ensemble as the running coupling slows with the increase in Nf . Thus, it is prudent to perform studies away from the conformal window and closer to QCD, where finite volume effects are less severe. 3. Lattice Strong Dynamics Recently, the Lattice Strong Dynamics (LSD) collaboration has formed to compute the relevant observables in SU(3) Yang-Mills with Nf = 2, 6, 10 light flavors and
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compare these results with Nf =2+1 calculations of the RBC collaboration.40–42 In Fig. 3 we show the chiral extrapolation (at fixed lattice spacing) of three physical quantities whose energy scales are characteristically associated with confinement: the nucleon mass, the vector meson mass, and the Sommer parameter. Our inverse lattice spacings for the Nf = 2 and 6 flavor calculations are matched at a scale of a−1 ≈ 5Mρ , about twice as large as the a−1 used by the RBC collaboration. We intentionally chose the finer lattice spacing in order to study the sensitivity of the chiral condensate to the short distance cutoff. The focus of our first analysis43 has been to calculate the dimensionless ratio of the chiral condensate to the pseudoscalar decay constant ψψ/F 3 by looking at three different dimensionless ratios CF
RXY,m
CM
FM
Mπ3 Mπ2 ψψ = = = 3 Fπ 2mFπ (2m)3 ψψ
as m → 0
(1)
where the labels “X” and “Y” should be replaced with appropriate combinations of “C”, “F” and “M” to reflect the pair of observables used to construct the ratio. The three ratios are equivalent in the chiral limit by the Gell-Mann-Oakes-Renner (GMOR) relation44 2mψψ = Mπ2 Fπ2
as m → 0 .
(2)
The ratio of ratios for two SU(Nc ) theories with different number of flavors, e.g. 2 2 RF M,m ≡ [Mπ /2mFπ ]Nf /[Mπ /2mFπ ]Nf =2 , can be used to compare the relative amount of condensate enhancement provided we have computed all ratios using the same UV cutoff (lattice spacing) in physical units. In terms of the geometric mean of the quark masses, an expansion of the ratio of ratios around the chiral limit produces a simple expression to next-to-leading order (NLO) RXY,m =
R(Nf ) [1 + m (αXY 10 + α11 log m)] , R(2)
m =
m(Nf ) m(2)
(3)
with the remarkable feature that the coefficient of the log, α11 , is identical for all three ratios. The ratios and their chiral extrapolation is shown in Fig. 4. Since this enhancement arises dominantly from physics at the UV cutoff, where even the Nf = 6 coupling is relatively weak, we compare it with an estimate based on the perturbative computation of the anomalous dimension γ of ψψ,45 finding a perturbative enhancement in the range 5 − 10% in the MS scheme. Converting to the lattice regularization scheme could increase the enhancement to 20 − 30%, quite a bit smaller than the factor of two computed non-perturbatively. The much larger value suggests that γ(µ) for Nf = 6 is considerably underestimated by perturbation theory over scales ranging from µ = Mρ – 5Mρ .
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CF CM FM
2
1.5
1
0.5 0
0.01
~ m
0.02
0.03
Fig. 4. The ratio of ratios RXY,m implies a surprisingly large amount of enhancement in the chiral limit for ψψ/Fπ3 between Nf =2 and Nf =6 when a−1 ≈ 5Mρ .
4. Conclusions The theoretical exploration of SCGT’s for the last several decades has been largely dominated by the study of QCD, for the obvious reason that a large number of experimental observations are available to guide our understanding. Recently, LGT has advanced to become the method of choice for calculating most non-perturbative quantities in QCD. The use of SCGT’s in building models of BSM physics has not been nearly as successful, presumably due to the lack of experimental observations. LGT offers the possibility of first-principles calculations of SCGT’s for BSM physics. But, current phenomenology favors SCGT’s that exhibit “walking”, i.e. nearly conformal behavior over a range of intermediate scales. LGT, as currently optimized for the study of QCD, may not yet have developed the optimal set of tools for studying walking SCGT’s. We emphasized the conservative approach of the LSD collaboration: to slowly approach the walking regime while varying the number of flavors looking for trends which indicate the onset of anticipated walking behavior while keeping an eye on studying the reliability of the methods at hand. Current indications are that the methods are still reliable up to Nf =6. Acknowledgments We would like to thank the current members of the Lattice Strong Dynamics (LSD) collaboration without whom this work would not have been possible: T. Appelquist,
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R. Babich, R. Brower, M. Buchoff, M. Cheng, M. Clark, S. Cohen, J. Kiskis, M. Lin, E. Neil, J. Osborn, C. Rebbi, D. Schaich, G. Voronov, P. Vranas, J. Wasem. This material is based upon work supported by the National Science Foundation under Grant No. PHY-0801068. References 1. The Tevatron New-Phenomena and Higgs Working Group (2010), arXiv:1007.4587 [hep-ex]. 2. G. T. Fleming, PoS LATTICE2008, p. 021 (2008). 3. E. Pallante, PoS LAT2009, p. 015 (2009). 4. W. E. Caswell, Phys. Rev. Lett. 33, p. 244 (1974). 5. T. Banks and A. Zaks, Nucl. Phys. B196, p. 189 (1982). 6. M. L¨ uscher, R. Narayanan, P. Weisz and U. Wolff, Nucl. Phys. B384, 168 (1992). 7. S. Sint, Nucl. Phys. B421, 135 (1994). 8. U. M. Heller, Nucl. Phys. B504, 435 (1997). 9. M. L¨ uscher, R. Sommer, P. Weisz and U. Wolff, Nucl. Phys. B413, 481 (1994). 10. U. M. Heller, Nucl. Phys. Proc. Suppl. 63, 248 (1998). 11. S. Capitani, M. L¨ uscher, R. Sommer and H. Wittig, Nucl. Phys. B544, 669 (1999). 12. J. Heitger, H. Simma, R. Sommer and U. Wolff, Nucl. Phys. Proc. Suppl. 106, 859 (2002). 13. M. Della Morte et al., Nucl. Phys. B713, 378 (2005). 14. T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. Lett. 100, p. 171607 (2008). 15. T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. D79, p. 076010 (2009). 16. N. Yamada et al., PoS LAT2009, p. 066 (2009). 17. N. Yamada et al. (2010), arXiv:1003.3288 [hep-lat] and these proceedings. 18. F. Tekin, R. Sommer and U. Wolff (2010), arXiv:1006.0672 [hep-lat]. 19. A. Deuzeman, M. P. Lombardo and E. Pallante, Phys. Lett. B670, 41 (2008). 20. Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, PoS LATTICE2008, p. 066 (2008). 21. A. Deuzeman, M. P. Lombardo and E. Pallante, PoS LATTICE2008, p. 060 (2008). 22. A. Deuzeman, E. Pallante and M. P. Lombardo, PoS LATTICE2008, p. 056 (2008). 23. X.-Y. Jin and R. D. Mawhinney, PoS LATTICE2008, p. 059 (2008). 24. E. T. Neil, T. Appelquist and G. T. Fleming, PoS LATTICE2008, p. 057 (2008). 25. A. Deuzeman, M. P. Lombardo and E. Pallante (2009), arXiv:0904.4662 [hep-ph]. 26. A. Hasenfratz, Phys. Rev. D80, p. 034505 (2009). 27. Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, Phys. Lett. B681, 353 (2009). 28. K.-i. Nagai, G. Carrillo-Ruiz, G. Koleva and R. Lewis, Phys. Rev. D80, p. 074508 (2009). 29. X.-Y. Jin and R. D. Mawhinney, PoS LAT2009, p. 049 (2009). 30. A. Hasenfratz, PoS LAT2009, p. 052 (2009). 31. A. Deuzeman, E. Pallante and M. P. Lombardo, PoS LAT2009, p. 044 (2009). 32. Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, PoS LAT2009, p. 055 (2009). 33. Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, PoS LAT2009, p. 058 (2009). 34. A. Hasenfratz, Phys. Rev. D82, p. 014506 (2010). 35. B. Holdom, Phys. Rev. D24, p. 1441 (1981). 36. B. Holdom, Phys. Lett. B150, p. 301 (1985).
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37. 38. 39. 40. 41. 42. 43. 44. 45.
K. Yamawaki, M. Bando and K.-i. Matumoto, Phys. Rev. Lett. 56, p. 1335 (1986). T. Appelquist and L. C. R. Wijewardhana, Phys. Rev. D35, p. 774 (1987). T. Appelquist and L. C. R. Wijewardhana, Phys. Rev. D36, p. 568 (1987). D. J. Antonio et al., Phys. Rev. D75, p. 114501 (2007). C. Allton et al., Phys. Rev. D76, p. 014504 (2007). C. Allton et al., Phys. Rev. D78, p. 114509 (2008). T. Appelquist et al., Phys. Rev. Lett. 104, p. 071601 (2010). M. Gell-Mann, R. J. Oakes and B. Renner, Phys. Rev. 175, 2195 (1968). J. A. M. Vermaseren, S. A. Larin and T. van Ritbergen, Phys. Lett. B405, 327 (1997).
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Phases of QCD from Small to Large Nf : (Some) Lattice Results A. Deuzeman and E. Pallante Centre for Theoretical Physics, University of Groningen 9747 AG, Netherlands E-mail: [emailprotected], [emailprotected] M. P. Lombardo INFN-Laboratori Nazionali di Frascati I-00044, Frascati (RM), Italy E-mail: [emailprotected] We analyze the phases of strong interactions in the space of the flavor number Nf , bare coupling gL , and temperature T by lattice MonteCarlo simulations for two and three unrooted staggered flavors, corresponding to eight and twelve continuum flavors, respectively. We observe a Coulomb-like phase at intermediate lattice couplings which we interpret as the avatar of a continuum conformal theory for Nf = 12. We comment on the possible occurrence of an UVFP associated with the bulk phase transition between the strong coupling lattice phase and the Coulomb-like phase.
1. Introduction The subject of this investigation is the phase diagram of continuum QCD,1 in the chiral limit, and as a function of the number of flavors, that we sketch in Fig 1. The motivations for this study has been beautifully introduced in early talks in this meeting, and we do not need to reiterate them here. We just emphasize that the chiral dynamics which is at the very hearth of the QCD phase diagram is eminently non-perturbative. Lattice simulations of QCD are then the method of choice, and indeed the properties of gauge theories with a potential conformal or near-conformal dynamics have been the subject of several lattice studies.2–14 Once QCD is discretized on a lattice, another direction automatically appears on the phase diagram, the lattice bare coupling. Lattice simulations cannot study directly the continuum limit: they work at a non-zero bare coupling, or, equivalently, at non-zero lattice spacing. Understanding the phase structure in the lattice parameters space is a mandatory step towards continuum physics. Let us then consider the lattice phase diagram for Nf massless flavors proposed by V. Miransky and K. Yamawaki,15 which we sketch in Fig. 2. The continuum physics - Fig.1 – appears at gL = 0, the leftmost axes. Let us mention right at the onset of the talk that we could as well discuss this phase diagram as a function of one
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Fig. 1.
The phase diagram of QCD as a function of the number of flavor.
component staggered flavors: nothing would change, since the continuum number of flavors does not play any role in this analysis (that would contrast, for instance, with a study of the thermal equation of state where the continuum number of flavor enters as a parameter). The line stemming from Nf = 16.5 , where asymptotic freedom is lost, is the line of Banks-Zaks fixed points:16 its precise location and shape depends on the choice of the lattice regularization. Since the first two coefficients of the beta function are universal, the shape of the line close to gL = 0 depends very little on the regularization, and different choices of regularization are likely to deform that line only smoothly. The line of the phase diagram which separates the Coulomb phase from the strong coupling phase, instead, depends non trivially on the lattice action: for instance, positivity-violating operators which are often used by improvement schemes, and which are irrelevant in the continuum, might well originate a rich and peculiar phase structure at strong coupling. There is a robust feature, though: chiral symmetry must be broken in the strong coupling limit regardless the number of flavors. Hence, if a continuum symmetric phase exists, a bulk phase transition separating such a symmetric phase from a broken phase should exist as well.16 Strictly speaking, the reverse is not true: in principle, one could conceive a so called re-entrant behavior, and we will comment on this possibility later on. The goal of our investigation4,5 is to map out quantitatively the phase diagram of Fig. 2. We will first describe our results from Nf = 12. We will base our analysis on the chiral condensate and the spectrum properties, and we will show how the mass spectrum, besides being a probe of the symmetry of the system, can be used to build lines of constant physics and to infer the sign of the beta function, much in the same spirit as in early work for Nf = 16.17 Next, we will turn to our Nf = 8 results. It will be useful to consider one further parameter, the temperature. A bona fide transition to Quark Gluon Plasma persisting till the continuum limit will be
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Fig. 2. The phase diagram in the space of the bare lattice coupling gL and the number of continuum flavor Nf .15 The continuum physics – Fig.1– is the gL = 0 axes. See text for discussions.
clearly exposed there. This will confirm the hadronic nature of the theory with eight continuum flavors, and allow a natural interpretation of the conformal phase as the zero temperature limit of the Quark Gluon Plasma phase. 2. Zero temperature: Nf = 12 The results for the chiral condensate show a clear crossover, or transition, at zero temperature, and call for a careful statistic and systematic analysis in order to distinguish between a real transition from a broken to a symmetric phase from a rapid crossover. To interpret quantitatively the results and establish the pattern of chiral symmetry, it might be useful to consider lattice QED as a paradigm for the conformal phase of QCD. Let us consider Fig. 3: in either cases a symmetric theory in the continuum has a bulk phase transition to a broken phase. In QCD the conformal symmetry, in the proximity of the bulk transition, is perturbatively broken by Coulomb forces: ¯ = amd , d = 3−γ for the chiral condensate. this suggests the functional form ψψ 1+γ Note that d was actually measured in QED.18 It was found d 0.95, increasing towards d = 1., as expected, by weakening the coupling towards the free limit. ¯ 0 , either with ¯ = amd + ψψ Motivated by this discussion, we performed fits to ψψ ¯ all parameters open, or constraining d = 1 or ψψ0 = 0 To make a long story short, data favor chiral symmetry restoration. For instance, at β = 3.9 and on a 244 lattice a fit to Y = bX d gives a (χ2 /ndf ) = 0.4, b = 2.697(2) d = 0.9642(3), which, taking into account d = 3−γ 1+γ gives γ 1.05 > 1. A linear fit with the same
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Fig. 3. A scenario for conformal QCD on the lattice vis-a-vis the known structure of lattice QED: in either cases a symmetric theory in the continuum has a bulk phase transition to a broken phase. We will note several similarities between the two bulk transitions. At a variance with QED, of course, QCD in the conformal phase still enjoys asymptotic freedom.
number of free parameters Y = bX + K has instead a (χ2 /ndf ) = 3.9, so chiral symmetry breaking seems excluded within 2 σ. Once more the interested reader is referred to our work5 for all the details. To avoid confusion, here we just note that we do not know yet how the anomalous dimension γ we have measured on the lattice at intermediate coupling would evolve towards the continuum limit: the anomalous dimension in the conformal phase of continuum QCD at the moment is only estimated by approximate analytic studies.1 Obviously one could consider more complicated fits motivated by different considerations: certainly there are many good reasons to go beyond the leading scaling form! The overall outcome is that chiral symmetry restoration remains favored. Most physically motivated corrections would enhance the bending of the chiral condensate at lower quark masses, so the practical effect of subleading corrections, if detectable at this level of accuracy, is to push chiral symmetry restoration towards stronger couplings. Spectrum dynamics carries important information on the symmetry of a phase. For instance, in the conformal phase mπ mσ 2mq , and this limit has been searched for on the lattice.11 It is interesting, by carrying over our analogy with QED, to see how this limit can be approached. Consider the ratio (Fig.4) mπ /mσ as a function of the bare mass in the symmetric and the broken phase, for a fixed value of the lattice coupling.19 At m = 0 the ratio is fixed by symmetry to be zero in the strong coupling and one in the weak coupling side. As mentioned above, close to the transition conformality is broken and the dynamics is dominated by weak Coulomb forces, so one observes deviations from the conformal value, mπ /mσ = 1. By further decreasing the coupling, towards the continuum, conformal limit, indeed the ratio approaches one. One further comment concerns the heavy quark limit, where the ratio is one as well. Note that in the same limit the chiral condensate becomes a decreasing function of the quark mass: the only way to have degeneracy between chiral partners is a zero chiral condensate, which in the heavy quark limit
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Fig. 4. A sketchy view of the behavior of the π to σ mass ratio as a function of the bare mass. Each line corresponds to one lattice coupling, and the coupling decreases from bottom to top. In the chiral limit the ratio is constrained by chiral symmetry: zero in the symmetric phase (bottom), one in the broken phase. In the conformal phase the ratio approaches one for all masses, which is the same as the heavy quark limit, and the residual Coulomb forces of the symmetric phase are responsible for a mass ratio which is below one. Note that the behavior of the chiral condensate can be used to discriminate between the heavy quark limit and the conformal limit.
¯ >∝ 1/m. So the trend of the chiral condensate as a function is achieved since < ψψ of the mass can be used to discriminate between a trivial heavy quark limit and physical conformal limit when the ratio is close to one. In Fig. 5(left) we show the ratios of measured pseudoscalar and vector masses from our simulations, for a fixed coupling and as a function of the bare quark mass. We have superimposed the ratios of the best fits to the raw mass data, confirming the good quality of the interpolations. The analogy with the predicted trend in the symmetric phase, Fig.4, including the ratio at fixed mass approaching one when decreasing the coupling, is very well exposed by our data. Combining this observation with the decreasing of the chiral condensate with the bare mass, we conclude again in favor of chiral symmetry in this phase, with the slow approach to one a precursor effect of the conformal phase (it would be desirable to have a direct measurement of the σ mass, which turns out to be noisy, but anyway very close to the ρ mass). Further, we used the spectrum results to determine the lines of constant physics in the two dimensional parameter space gL and am, the bare quark mass of degenerate fermions: we found that the lattice spacing increases for weaker couplings, signature of the Coulomb phase(see Fig.5,right). The same trend is observed in Nf = 16 QCD,17 and indeed our strategy has been inspired by this study. Let us look again at Fig.2: our Coulomb phase looks pretty much like the one of that figure. So it seems that an emerging conformal phase in the continuum limit is a physical scenario consistent with our data. Can we challenge Fig.2 - and hence Ref.3? Of course we can - but to do so we should imagine a rather unusual scenario in which chiral
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Fig. 5. The measured π to ρ mass ratio as a function of the bare mass and decreasing coupling 2 = 3.5 to 4. The superimposed lines are ratios of the best fits as a function gL , bottom to top 6/gL of the bare quark mass (left); Pseudoscalar mass along lines of constant physics. The physical pion mass is identical along lines of constant physics, such that a decreasing pion mass in lattice units for increasing gL , implies an increase of the lattice spacing for weaker couplings, signature of the Coulomb phase. Labels give the value of the ratio mπ /mρ along the isolines (right).
Fig. 6.
The thermal behavior of QCD in the hadron (left) and conformal phase (right).
symmetry is spontaneously broken again by decreasing the coupling. A somewhat analogous behavior has been observed in condensed matter - for instance one can have a reversible liquid-to-crystal transition, with passage from a low-temperature liquid to a high-temperature crystal. Such re-entrant behavior usually emerges as a result of a complicated interplay among competing mechanisms. Indeed we have scanned the coupling space from the values discussed here towards the continuum, finding no trace of any further chiral restoring phase transition. 3. Nf = 8, and Quark Gluon Plasma We now extend further our phase space and include the temperature axes. We note that in ordinary QCD, as well as in the strong coupling phase of conformal QCD, there will be a high temperature chiral restoration transition at g = gT .
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Fig. 7. The phase diagram in the T, Nf plane.20 The circles indicate earlier lattice results, which fit very nicely the RG prediction. Our results for Nf = 8 indicate the existence of a thermal transition (marked as a square), whose temperature in physical units has yet to be estimated.
In the conformal phases limNt →∞ gT2 = gc2 , with gc the bulk phase transition discussed above, while in ordinary QCD limNt →∞ gT2 = 0 as 1/Nt = Tphys a(g 2 ). In a three dimensional picture a sheet of phase transitions in the Nt , g, Nf plane continues smoothly till gL = 0 in ordinary QCD, while it is bounded at g = gc in the conformal phase. We can see the conformal phase as the T=0 limit of the ordinary, albeit cold, Quark Gluon Plasma phase, either on the lattice and in the continuum. We have indeed checked that for Nf = 8 the critical temperature computed on lattice of various sizes Tc = a(βc1)Nt satisfies asymptotic scaling within 2 % The results can be placed on the phase diagram in the T, Nf plane,20 see Fig.7: results for a small number of flavors are available in the literature, and our results for Nf = 8 nicely extend these studies. Remarkably, the lattice results fare quite well on the near-linear behavior found in RG studies. The Nf = 8 critical temperature in physical units remains to be computed, and checked against the Braun-Gies20 prediction.
4. Summary and discussion The most plausible scenario accommodating our data is a continuum conformal phase for and SU(3) Yang–Mills gauge theory with Nf = 12, in agreement with calculations of the renormalized coupling in this model.6,7 At the intermediate couplings which we have studied the physics is that of a symmetric phase of three unrooted staggered flavors, where conformal symmetry is broken by weak Coulombic forces. Upon further decreasing the coupling Coulombic forces progressively weaken, flavor symmetry will be restored and the physics of the continuum, conformal window should emerge. We underscore that this happens without any change of chiral symmetry. In this scenario, the Coulomb phase at intermediate coupling is just an avatar of the continuum conformal phase. Clearly, it would be extremely desirable to go beyond the avatar and expose directly the conformal physics of the continuum limit.
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One final comment concerns the search for an UVFP in a strongly interacting four dimensional gauge theory. This search has a long history, which was also recorded by past editions of the Nagoya SCGT Meetings. Most attempts in the past focused on the strong coupling transition of QED, which, as we have discussed above, has a natural analogy with the bulk transition in the conformal window. Indeed, since the very first lattice studies of this model an UVFP was searched along the strong coupling line.21,22 Clearly if such theory existed in large Nf QCD, the non–abelian nature of the gauge fields should play a major role, given that the QED transition appears to be trivial. More recently, Kaplan, Lee, Son and Stephanov have further elaborated on this fascinating possibility,23 providing more motivation for such a search. It is interesting to note that even if the bulk phase transition for Nf > Nfc appeared to be of first order, its endpoint at Nf = Nfc , where ”conformality is lost”23 should indeed be of a second order, following general statistical mechanics arguments, and might be a natural candidate for the existence of a new theory. Acknowledgments MPL wishes to thank the organisers of this enjoyable and interesting workshop for their most kind hospitality, and the participants for many lively and stimulating discussions. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
F. Sannino, arXiv:0911.0931 [hep-ph]. G. T. Fleming, PoS LATTICE2008 (2008) 021. E. Pallante, arXiv:0912.5188 [hep-lat]. A. Deuzeman, M. P. Lombardo and E. Pallante, Phys. Lett. B 670, 41. A. Deuzeman, M. P. Lombardo and E. Pallante arXiv:0904.4662 [hep-ph]. T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. Lett. 100, 171607 . T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. D 79, 076010 . Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, arXiv:0911.2463 [hep-lat]. S. Catterall and F. Sannino, Phys. Rev. D 76,p. 34504 (2007). S. Catterall, J. Giedt, F. Sannino and J. Schneible, J. High Energy Phys. 2008,p. 9. L. Del Debbio, B. Lucini, A. Patella, C. Pica and A. Rago, Phys. Rev. D 80, 074507. T. DeGrand and A. Hasenfratz, Phys. Rev. D 80, 034506. Y. Shamir, B. Svetit*ky and T. DeGrand, Phys. Rev. D 78, 031502 J. B. Kogut and D. K. Sinclair, arXiv:1002.2988 [hep-lat]. V. A. Miransky and K. Yamawaki, Phys. Rev. D 55, p. 5051. T. Banks and A. Zaks, Nuc. Phys. B 196, 189. P. Damgaard, U. Heller, A. Krasnitz and P. Olesen, Phys. Lett. B 400, 169. A. Kocic, S. Hands, J. B. Kogut and E. Dagotto, Nucl. Phys. B347, 217 . A. Kocic, J. B. Kogut and M.P. Lombardo, Nucl. Phys. B398, 376. J. Braun and H. Gies, J. High Energy Phys. 2006, p. 024 . J. B. Kogut, J. Polonyi, H. W. Wyld and D. K. Sinclair, Phys. Rev. Lett. 54, 1475. J. B. Kogut and D. K. Sinclair, Nucl. Phys. B295, 465 . D. B. Kaplan, J.-W. Lee, D. T. Son and M. A. Stephanov, Phys. Rev. D 80, 125005.
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Lattice Gauge Theory and (Quasi)-Conformal Technicolor D. K. Sinclair HEP Division, Argonne National Laboratory, 9700 South Cass Avenue Argonne, Illinois 60439, USA E-mail: [emailprotected] J. B. Kogut Department of Energy, Division of High Energy Physics Washington, DC 20585, USA and Department of Physics – TQHN, University of Maryland, 82 Regents Drive College Park, Maryland 20742, USA E-mail: [emailprotected] QCD with 2 flavours of massless colour-sextet quarks is studied as a theory which might exhibit a range of scales over which the running coupling constant evolves very slowly (walks). We simulate lattice QCD with 2 flavours of sextet staggered quarks to determine whether walks, or if it has an infrared fixed point, making it a conformal field theory. Our initial simulations are performed at finite temperatures T = 1/Nt a (Nt = 4 and Nt = 6), which allows us to identify the scales of confinement and chiral-symmetry breaking from the deconfinement and chiral-symmetry restoring transitions. Unlike QCD with fundamental quarks, these two transitions appear to be well-separated. The change in coupling constants at these transitions between the two different temporal extents Nt , is consistent with these being finite temperature transitions for an asymptotically free theory, which favours walking behaviour. In the deconfined phase, the Wilson Line shows a 3-state signal. Between the confinement and chiral transitions, there is an additional transition where the states with Wilson Lines oriented in the directions of the complex cube roots of unity disorder into a state with a negative Wilson Line. Keywords: Lattice gauge theory, Walking Technicolor.
1. Introduction Technicolor theories are QCD-like gauge theories with massless fermions, whose pion-like excitations play the role of the Higgs field in giving masses to the W and Z.1,2 We search for Yang-Mills gauge theories whose fermion content is such that the running coupling constant evolves very slowly – walks. Such theories can avoid the phenomenological problems which plague other (extended-)Technicolor theories.3–6 While many studies have used fermions in the fundamental representation, with large numbers of flavours,7–22 we are concentrating on higher representations of the colour group (in particular the symmetric tensor), where conformality/walking can be achieved at much lower Nf . There have been some studies with SU (2) colour
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with two adjoint (symmetric tensor) fermions.23–30 We are considering QCD (SU (3) colour) with colour-sextet (symmetric tensor) quarks. The 2-loop β-function for QCD with Nf massless flavours of colour-sextet 28 3 ≤ Nf < 3 10 , either this theory will have an infraredquarks, suggests that for 1 125 stable fixed point, or a chiral condensate will form and this fixed point will be avoided. In the first case the theory will be conformal; in the second case it will walk. For Nf = 3 conformal behaviour is expected. Nf = 2 could, a priori, exhibit either behaviour. Because the quadratic Casimir operator for sextet quarks is 2 21 times that for fundamental quarks, it is easier for them to form a chiral condensate. Lattice QCD gives us a direct method to determine which option the Nf = 2 theory chooses. We are studying the Nf = 2 theory using staggered fermions. We are currently performing simulations at finite temperature (T ). Finite temperature enables us to study the scales of confinement and chiral symmetry breaking, and yields information on the running of the coupling constant. Simulations using Wilson fermions by DeGrand, Shamir and Svetit*ky suggest that this theory is conformal.31,32 Our simulations suggest that it walks. The scales of confinement and chiral symmetry breaking appear to be very different. (This also contrasts with what was reported by DeGrand, Shamir and Svetit*ky for simulations using Wilson quarks.33 ) Hence the phenomenology is expected to be different from that of QCD with fundamental quarks and Nf in the walking window, where these two scales appear to be the same. Preliminary studies of this theory using domainwall quarks have been reported in.34 In the deconfined phase we observe states where the phase of the Wilson Line is ±2π/3, and at weaker couplings, π in addition to the expected states with positive Wilson Lines. These have since been predicted and observed by Machtey and Svetit*ky using Wilson quarks.35 2. Simulations and Results We use the standard Wilson (triplet) plaquette action for the gauge fields and an unimproved staggered-fermion action for the quarks. The only new feature is that the quark fields are six-vectors in colour space and the gauge fields on the links of the quark action are in the sextet representation of colour. The RHMC algorithm is used to tune the number of flavours, Nf , to 2. We run on 83 ×4, 123 ×4 and 123 ×6 lattices at quark masses m = 0.005, m = 0.01 and m = 0.02 in lattice units to allow extrapolation to the chiral (m = 0) limit. β = 6/g 2 is varied over a range of values large enough to include the deconfinement and chiral transitions. Run lengths of 10,000–200,000 trajectories per (m, β) are used. More details of the results presented here are given in ref.36 Since the results from the 2 Nt = 4 lattices are consistent, we present only results from our 123 ×4 simulations. Figure 1 shows the colour-triplet Wilson Line(Polyakov ¯ Loop) and the chiral condensate(ψψ) as functions of β = 6/g 2 , for each of the 3
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Fig. 1.
¯ as functions of β on a 123 × 4 lattice. Wilson line and ψψ
quark masses on a 123 × 4 lattice. The deconfinement transition is marked by an abrupt increase in the value of the Wilson Line. Chiral symmetry restoration occurs where the chiral condensate vanishes in the chiral limit. In contrast to what was found by DeGrand, Shamir and Svetit*ky, we find well separated deconfinement and chiral-symmetry restoration transitions. The deconfinement transition occurs at β = βd where βd (m = 0.005) = 5.405(5), βd (m = 0.01) = 5.4115(5) and βd (m = 0.02) = 5.420(5). The chiral transition, estimated from the peaks in the chiral susceptibility curves, occurs at βχ = 6.3(1). Figure 1 only accounts for the state with a real positive Wilson Line in the deconfined regime. However, from the deconfinement transition up to β ≈ 5.9 there exist long-lived states with the Wilson Line oriented in the directions of the other 2 cube roots of unity. However, these states are metastable, eventually decaying into the state with a positive Wilson Line. Above β ≈ 5.9, these complex Wilson Line states disorder into a state with a negative Wilson Line. Let us now consider our 123 × 6 simulations. Again, the deconfinement and chiral-symmetry restoring transitions are well-separated. Above the deconfinement
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Fig. 2. Wilson Line and chiral condensate for the state with a real positive Wilson Line as functions of β for each of the 3 masses on a 123 × 6 lattice.
transition, we again find a clear 3-state signal. This time, however, all 3 states appear equally stable. The system tunnels between these 3 states for the duration of the run until we are so far above the transition that the relaxation time for tunneling exceeds the lengths of our runs. We therefore artificially bin our ‘data’ according to the phase of the Wilson Line, into bins (−π, −π/3), (−π/3, π/3), (π/3, π). ¯ for the central Figure 2 shows the Wilson Lines and chiral condensates ψψ ‘positive’ Wilson Line bin. Figure 3 shows the Wilson Lines and chiral condensates for the first and last ‘complex’ and ‘negative’ Wilson Line bins. The deconfinement transitions occur at βd (m = 0.005) = 5.545(5), βd (m = 0.01) = 5.550(5) and βd (m = 0.02) = 5.560(5). Chiral-symmetry restoration occurs at βχ = 6.6(1). As for Nt = 4, there is further transition between the deconfinement and chiral transitions where the states with complex Wilson Lines disorder to produce a state with a negative Wilson Line. This transition occurs at β ≈ 6.4 for m = 0.01 and β ≈ 6.5 for m = 0.02. This transition can be seen in fig. 3.
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Fig. 3. Magnitude of the Wilson Line and chiral condensate for the state with a complex or negative Wilson Line as functions of β for each of the 3 masses on a 123 × 6 lattice.
3. Discussion and conclusions We are studying the thermodynamics of Lattice QCD with 2 flavours of staggered colour-sextet quarks. We find well separated deconfinement and chiral-symmetry restoration transitions. This contrasts with the case of fundamental quarks, where these 2 transitions are coincident, but is similar to the case of adjoint quarks where again these 2 transitions are separate.37,38 We denote the value of β = 6/g 2 at the deconfinement transition by βd and that at the chiral transition by βχ . In the chiral limit βd ≈ 5.40 and βχ = 6.3(1) at Nt = 4. At Nt = 6 these values become βd ≈ 5.54 and βχ = 6.6(1). The increase in the βs for both transitions from Nt = 4 to Nt = 6 is consistent with their being finite temperature transitions for an asymptotically free theory (rather than bulk transitions). If there is an IR fixed point, we have yet to observe it. Our results suggest a Walking rather than a conformal behaviour.
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Why is this phase diagram so different from that for Wilson quarks (DeGrand, Shamir and Svetit*ky)? Is it because there is an infrared fixed point, and we are on the strong-coupling side of it? Are our quark masses too large to see the chiral limit? Is it because the flavour breaking of staggered quarks does not allow a true chiral limit at fixed lattice spacing? For the deconfined phase there is a 3-state signal, the remnant of now-broken Z3 symmetry. For Nt = 4 the states with complex Polyakov Loops appear metastable. For Nt = 6 all 3 states appear stable. Breaking of Z3 symmetry is seen in the magnitudes of the Polyakov Loops for the real versus complex states. Between the deconfinement and chiral transitions, we find a third transition where the Wilson Lines in the directions of the 2 non-trivial roots of unity change to real negative Wilson Lines. This transition occurs for β ≈ 5.9 (Nt = 4) and β ≈ 6.4–6.5 (Nt = 6). The existence of these extra states with Polyakov Loops which are not real and positive has been predicted and observed by Machtey and Svetit*ky. Drawing conclusions from Nt = 4 and Nt = 6 is dangerous. We have recently started simulations with Nt = 8. We should also use smaller quark masses. At Nt = 6, we need a second spatial lattice size. To understand this theory more fully, we need to study its zero temperature behaviour, measuring its spectrum, string tension, potential, fπ .... Measurement of the running of the coupling constant for weak coupling is needed. We have recently started simulations with Nf = 3, which is expected to be conformal, to determine if it is qualitatively different from Nf = 2. Acknowledgments DKS is supported in part by the U.S. Department of Energy, Division of High Energy Physics, Contract DE-AC02-06CH11357, and in part by the Argonne/University of Chicago Joint Theory Institute. JBK is supported in part by NSF grant NSF PHY03-04252. These simulations were performed on the Cray XT4, Franklin at NERSC under an ERCAP allocation, and on the Cray XT5, Kraken at NICS under an LRAC/TRAC allocation. DKS thanks J. Kuti, D. Nogradi and F. Sannino for helpful discussions. References 1. 2. 3. 4. 5. 6.
S. Weinberg, Phys. Rev. D 19, 1277 (1979). L. Susskind, Phys. Rev. D 20, 2619 (1979). B. Holdom, Phys. Rev. D 24, 1441 (1981). K. Yamawaki, M. Bando and K. i. Matumoto, Phys. Rev. Lett. 56, 1335 (1986). T. Akiba and T. Yanagida, Phys. Lett. B 169, 432 (1986). T. W. Appelquist, D. Karabali and L. C. R. Wijewardhana, Phys. Rev. Lett. 57, 957 (1986). 7. J. B. Kogut, J. Polonyi, H. W. Wyld and D. K. Sinclair, Phys. Rev. Lett. 54, 1475 (1985). 8. M. f*ckugita, S. Ohta and A. Ukawa, Phys. Rev. Lett. 60, 178 (1988).
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9. S. Ohta and S. Kim, Phys. Rev. D 44, 504 (1991). 10. S. y. Kim and S. Ohta, Phys. Rev. D 46, 3607 (1992). 11. F. R. Brown, H. Chen, N. H. Christ, Z. Dong, R. D. Mawhinney, W. Schaffer and A. Vaccarino, Phys. Rev. D 46, 5655 (1992) [arXiv:hep-lat/9206001]. 12. Y. Iwasaki, K. Kanaya, S. Sakai and T. Yoshie, Phys. Rev. Lett. 69, 21 (1992). 13. Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai and T. Yoshie, Phys. Rev. D 69, 014507 (2004) [arXiv:hep-lat/0309159]. 14. A. Deuzeman, M. P. Lombardo and E. Pallante, Phys. Lett. B 670, 41 (2008) [arXiv:0804.2905 [hep-lat]]. 15. A. Deuzeman, M. P. Lombardo and E. Pallante, arXiv:0904.4662 [hep-ph]. 16. T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. D 79, 076010 (2009) [arXiv:0901.3766 [hep-ph]]. 17. T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. Lett. 100, 171607 (2008) [Erratum-ibid. 102, 149902 (2009)] [arXiv:0712.0609 [hep-ph]]. 18. X. Y. Jin and R. D. Mawhinney, PoS LATTICE2008, 059 (2008) [arXiv:0812.0413 [hep-lat]]. 19. X. Y. Jin and R. D. Mawhinney, PoS LAT2009, 049 (2009) [arXiv:0910.3216 [heplat]]. 20. Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, arXiv:0907.4562 [hep-lat]. 21. Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, arXiv:0911.2463 [hep-lat]. 22. N. Yamada, M. Hayakawa, K. I. Ishikawa, Y. Osaki, S. Takeda and S. Uno, arXiv:0910.4218 [hep-lat]. 23. S. Catterall and F. Sannino, Phys. Rev. D 76, 034504 (2007) [arXiv:0705.1664 [heplat]]. 24. S. Catterall, J. Giedt, F. Sannino and J. Schneible, JHEP 0811, 009 (2008) [arXiv:0807.0792 [hep-lat]]. 25. S. Catterall, J. Giedt, F. Sannino and J. Schneible, arXiv:0910.4387 [hep-lat]. 26. L. Del Debbio, A. Patella and C. Pica, arXiv:0805.2058 [hep-lat]. 27. L. Del Debbio, B. Lucini, A. Patella, C. Pica and A. Rago, arXiv:0907.3896 [hep-lat]. 28. F. Bursa, L. Del Debbio, L. Keegan, C. Pica and T. Pickup, arXiv:0910.4535 [hep-ph]. 29. A. J. Hietanen, J. Rantaharju, K. Rummukainen and K. Tuominen, JHEP 0905, 025 (2009) [arXiv:0812.1467 [hep-lat]]. 30. A. J. Hietanen, K. Rummukainen and K. Tuominen, arXiv:0904.0864 [hep-lat]. 31. Y. Shamir, B. Svetit*ky and T. DeGrand, Phys. Rev. D 78, 031502 (2008) [arXiv:0803.1707 [hep-lat]]. 32. T. DeGrand, Phys. Rev. D 80, 114507 (2009) [arXiv:0910.3072 [hep-lat]]. 33. T. DeGrand, Y. Shamir and B. Svetit*ky, Phys. Rev. D 79, 034501 (2009) [arXiv:0812.1427 [hep-lat]]. 34. Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, arXiv:0809.4888 [hep-lat]. 35. O. Machtey and B. Svetit*ky, Phys. Rev. D 81, 014501 (2010) [arXiv:0911.0886 [heplat]]. 36. J. B. Kogut and D. K. Sinclair, arXiv:1002.2988 [hep-lat]. 37. F. Karsch and M. Lutgemeier, Nucl. Phys. B 550, 449 (1999) [arXiv:hep-lat/9812023]. 38. J. Engels, S. Holtmann and T. Schulze, Nucl. Phys. B 724, 357 (2005) [arXiv:heplat/0505008].
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Study of the Running Coupling Constant in 10-Flavor QCD with the Schr¨ odinger Functional Method N. Yamadaa,b , M. Hayakawac , K.-I. Ishikawad , Y. Osakid , S. Takedae and S. Unoc a KEK Theory Center, Institute of Particle and Nuclear Studies High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan b School of High Energy Accelerator Science The Graduate University for Advanced Studies (Sokendai), Tsukuba 305-0801, Japan c Department of Physics, Nagoya University, Nagoya 464-8602, Japan d Department of Physics, Hiroshima University, Higashi-Hiroshima 739-8526, Japan e School of Mathematics and Physics, College of Science and Engineering Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa 920-1192, Japan
The electroweak gauge symmetry is allowed to be spontaneously broken by the strongly interacting vector-like gauge dynamics. When the gauge coupling of a theory runs slowly in a wide range of energy scale, the theory is a candidate for walking technicolor. This may open up the possibility that the origin of all masses may be traced back to the gauge theory. We use the Schr¨ odinger functional method to see whether the gauge coupling of 10-flavor QCD “walks” or not. Preliminary result is reported. Keywords: Lattice gauge theory, LHC.
1. Introduction The main goal of Large Hadron Collider (LHC) is to confirm the Higgs mechanism and to find particle contents and the physics law above the electroweak scale. So far many new physics models beyond the standard model have been proposed. Among them, Technicolor (TC)1 is one of the most attractive candidates2 as it does not require elementary scalar particles which cause, so-called, the fine-tuning problem. This model is basically a QCD-like, strongly interacting vector-like gauge theory. Therefore, lattice gauge theory provides the best way to study this class of model,3 and the predictions can be as precise as those for QCD, in principle. The simple, QCD-like TC model, i.e. an SU(3) gauge theory with two or three flavors of techniquarks, has been already ruled out by, for instance, the S-parameter4 and the FCNC constraints. However, it has been argued that, if the gauge coupling runs very slowly (“walks”) in a wide range of energy scale before spontaneous chiral symmetry breaking occurs, at least, the FCNC problem may disappear.5 Such TC models are called walking technicolor (WTC) and several explicit candidates are discussed in semi-quantitative manner in Ref.6 Since the dynamics in WTC might be completely different from that in QCD and hence the use of the naive scaling in Nc or Nf to estimate various quantities may not work, the S-parameter must
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be evaluated from the first principles.7 Although really important quantity is the ¯ operator, looking for theories showing the walking anomalous dimension of ψψ behavior is a good starting point. Recently many groups started quantitative studies using lattice technique to answer the question what gauge theory shows walking behavior. In Ref.,8 the running couplings of 8- and 12-flavor QCD are studied on the lattice using the Schr¨ odinger functional (SF) scheme.9 Their conclusion is that while 8-flavor QCD does not show walking behavior 12-flavor QCD reaches an infrared 2 ∼ 5. In spite of the scheme-dependence of running and its fixed point (IRFP) at gIR value of IRFP, the speculation inferred from Schwinger-Dyson equation10 suggests 2 ∼ 5 is not large enough to trigger spontaneous chiral symmetry breaking. that gIR Although 12-flavor QCD is still an attractive candidate and is open to debate,11 we explore other Nf . In the following, we report the preliminary results on the running coupling in 10-flavor QCD. Since the conference, statistics and analysis method are changed. The following analysis is based on the increased statistics and a slightly different analysis method.
2. Perturbative analysis Before going into the simulation details, let us discuss some results from perturbative analysis. In this work, we adopt the β function defined by β(g 2 (L)) = L
∂ g 2 (L) = b1 g 4 (L) + b2 g 6 (L) + b3 g 8 (L) + b4 g 10 (L) + · · · , (1) ∂L
where L denotes a length scale. The first two coefficients are scheme-independent, and given by b1 =
2 2 N 11 − f , (4π)2 3
b2 =
2 (4π)4
102 −
38 Nf . 3
(2)
Other higher order coefficients are scheme-dependent and are known only in the limited schemes and orders. In this section, we analyze the perturbative running in the four different schemes/approximations: i) two-loop (universal), ii) three-loop in odinger the MS scheme, iii) four-loop in the MS scheme, iv) three-loop in the Schr¨ functional scheme. The perturbative coefficients relevant to the following analysis are 2857 5033 2 325 2 , (3) = + − N N bMS f 3 (4π)6 2 18 54 f bMS = 4
2 29243.0 − 6946.30 Nf + 405.089 Nf2 + 1.49931 Nf3 , 8 (4π)
(4)
2
MS bSF 3 = b3 +
b1 (cθ3 − cθ2 ) b2 cθ2 − , 2π 8π 2
(5)
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Nf 2-loop universal 3-loop SF 3-loop MS 4-loop MS
4
6
8
43.36
23.75 159.92
15.52 18.40 19.47
10 27.74 9.45 9.60 10.24
12 9.47 5.18 5.46 5.91
14 3.49 2.43 2.70 2.81
16 0.52 0.47 0.50 0.50
Tab. 1 The IRFP from perturbative analysis.
where the coefficients cθ2 and cθ3 depend on the spatial boundary condition of the SF used in calculations, i.e θ. Those for θ = π/5 and cθ2 for θ = 0 are known as θ=π/5
c2
θ=π/5 c3 cθ=0 2
= 1.25563 + 0.039863 × Nf , =
θ=π/5 2 (c2 )
+ 1.197(10) + 0.140(6) × Nf − 0.0330(2) ×
= 1.25563 + 0.022504 × Nf ,
(6) Nf2 ,
(7) (8)
is not. Therefore, the case iv) is studied with θ=π/5. Notice that in our but cθ=0 3 numerical simulation θ=0 and thus, rigorously speaking, the example iv) is not applied to our numerical result. The perturbative infrared fixed point (IRFP) are numerically solved and summarized in Tab. 1. As seen from Tab. 1, with Nf ≥ 8 the fixed point value is, to some extent, stable against the change of schemes/approximations. It is interesting that the IRFP of 12-flavor QCD in the SF scheme is consistent with the non-perturbative calculation by Ref.8 Now looking at the perturbative IRFP at Nf = 10, it appears to be stable at g 2 ∼ 10. Furthermore, according to the analysis based on SchwingerDyson equation, there is an argument that chiral symmetry breaking occurs at around g 2 ∼ 4π 2 /(3 C2 (R)) = π 2 .10 In summary, the perturbative analysis suggests that 10-flavor QCD is the most attractive candidate for WTC. 3. Simulation parameters and setup We employ the Schr¨ odinger functional method9 to calculate the running coupling constant. Unimproved Wilson fermion action and the standard plaquette gauge action without any boundary counter terms are used. The parameter of the spatial boundary condition for fermions, θ, is set to zero. The bare gauge coupling β = 6/g02 is explored in the range of 4.4–24.0. In this analysis, we report the results obtained from (L/a)4 = 64 , 84 and 124 lattices. The calculation on 164 lattice is in progress. The numerical simulation is carried out on several architectures including GPGPU and PC cluster. The standard HMC algorithm is used with some improvements in the solver part like the mixed precision algorithm. So far, we have accumulated 5,000 to 200,000 trajectories depending on (β, L/a). Since the Wilson type fermion explicitly violates chiral symmetry, the critical value of κ has to be tuned to the massless limit. We performed this tuning for every pair of (β, L/a). At around β ∼ 4.4, for massless fermions we encounter a (probably first order) phase transition independently of L/a, where the plaquette
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0.3
2
g0 /gSF
0.2
2
0.6
2
g0 /gSF
2
0.8
0.4
L/a=6 8 12 16
0.2 0
0.2
0.4
L/a=6 8 12 16
0.1
0.8
0.6
g0
2
1
1.2
1.4
1.25
1.3
g0
2
1.35
2 (L) at L/a = 6, 8, 12, 16. The right panel is just an enlargement Fig. 1. g02 -dependence of g02 /gSF of the left.
value suddenly jumps to a smaller value. Since this bulk phase transition is inferred to be lattice artifact, whenever this happens we discard the configurations. Thus the position of the critical β (∼ 4.4) sets the lower limit on β at which simulations make sense. 4. Preliminary results Figure 1 shows the Schr¨ odinger functional coupling calculated on the lattices, where 2 2 g0 /gSF is plotted as a function of the bare coupling. The solid curve is the result (and statistical error) of the fit to g02 2
lat (g 2 , L/a) gSF 0
=
1 − aL/a,1 g04 , N 1 + p1,L/a × g02 + n=2 aL/a,n × g02 n
where p1,L/a is the L/a-dependent coefficient lation, to be 0.4477107831 p1,L/a = 0.4624813408 0.4756888260
(9)
and is found, by perturbative calcufor L/a = 6 for L/a = 8 . for L/a = 12
(10)
We optimize the degree of polynomial N in the denominator of (9) by monitoring χ2 /dof, and take N = 5 for L/a = 6 and N = 4 for L/a = 8, 12. Since we do not implement any O(a) improvements, large scaling violation is expected to exist. One promising prescription to improve discretization errors has been proposed in Ref.12 Let us parameterize the lattice artifact in the step scaling by Σ0 (u, s, L/a) − σ(u, s) 1 + δ (1) (s, L/a) u = δ (2) (s, L/a)u2 + · · · , δ(u, s, L/a) = σ(u, s) 2
(11)
2 2 lat (L), σ(u, s) = gSF (sL) and Σ0 (u, s, L/a) is gSF (g02 , sL/a) at g02 where u = gSF 2 lat satisfying gSF (g02 , L/a) = u.
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δ(u, s, L/a)
0.08 0.06 0.04 0.02 0 0
1
u
2
3
Fig. 2. δ as a function of u. δ(2) (u, s, L/a) is determined as a coefficient of the quadratic term in u. (solid and dashed curves are obtained from different fit ranges.)
The coefficient δ (1) (s, L/a) in eq. (11) is given by
δ (1) (s, L/a) = p1,sL/a − b1 ln(sL/a) − p1,L/a − b1 ln(L/a) .
(12)
Dividing the lattice data Σ0 (u, s, L/a) by the factor (1 + δ (1) (s, L/a)) improves the O(u) discretization error and hence δ(u, s, L/a) starts from O(u2 ) as already indicated in eq. (11). σ(u, s) has perturbative expansion, σ(u, s) = u + s0 u2 + s1 u3 + s2 u4 + s3 u5 + · · · , s1 = ln(s) b1 2 ln(s) + b2 , s0 = b1 ln(s),
5 3 2 s2 = ln(s) b1 ln (s) + b1 b2 ln(s) + b3 , 2
13 2 3 2 4 3 2 s3 = ln(s) b1 ln (s) + b1 b2 ln (s) + ln(s) 3b1 b3 + b2 + b4 , 3 2
(13) (14) (15) (16)
where bi ’s are the coefficients of the β-function introduced in sec. 2. Since we know the first two coefficients b1 and b2 and hence σ(u, s) can be numerically determined to O(u3 ), with such σ the O(u2 ) term in δ(u, s, L/a) is attributed to discretization error. The coefficient of u2 term, δ (2) (s, L/a), is then obtained by fitting δ(u, s, L/a) to a quadratic function of u. This fit must be done in the small-coupling region where the perturbative series is reliable. Since at this moment we have only a limited number of data points in such a region, the fit range is forced to extend to u ∼ 2.5. Figure 2 shows the u dependence of δ(u, s, L/a) and the fit results. The obtained coefficients are 0.0062(9) for s = 4/3, L/a = 6 δ (2) (s, L/a) = 0.0138(26) for s = 2, L/a = 6 , (17) 0.0070(24) for s = 3/2, L/a = 8 0.0053(8) for s = 4/3, L/a = 6 (18) δ (2) (s, L/a) = 0.0122(21) for s = 2, L/a = 6 . 0.0062(19) for s = 3/2, L/a = 8
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-0.01
-0.01
lat
-0.02 -0.03 -0.04
-0.02
0.2
0.4
1/u
0.6
0.8
2-loop univ. 3-loop SF
0.02
B (u, s, L/a)
lat
s=3/2, L/a=8
0.03 2-loop univ. 3-loop SF
B (u, s, L/a)
0.01
s=2, L/a=6
0.01 2-loop univ. 3-loop SF
lat
B (u, s, L/a)
0.02
0.01 0 -0.01 -0.02
-0.05 0
0.2
0.4
1/u
0.6
0.8
,
-0.03 0
0.2
0.4
1/u
0.6
0.8
Fig. 3. Discrete beta functions for (s, L/a)=(4/3, 6), (2, 6) and (3/2, 8) from left to right. Two colored bands are the results with statistical error, obtained by the two-loop improvement with eq. (17) (red) and eq. (18) (blue), respectively. The black solid (dashed) curve is the corresponding discrete beta function obtained by integrating perturbative two-loop universal (three-loop SF scheme) beta function.
In the fit, two different fit ranges, u ∈ [0, 2.02] and [0, 2.50], are applied to examine the fit range dependence of the result. Eqs. (17) and (18) correspond to the former and the latter fit range, respectively. Using δ (2) (s, L/a) thus extracted, we define the improved lattice data by Σimp (u, s, L/a) =
Σ0 (u, s, L/a) . 1 + δ (1) (s, L/a) u + δ (2) (s, L/a) u2
To see the running in detail, we introduce the discrete beta function,13 1 1 − . B lat (u, s, L/a) = imp Σ (u, s, L/a) u
(19)
(20)
Figure 3 shows the 1/u dependence of B lat (u, s, L/a). As usual β function, large negative value of B lat means rapid increase with length scale of the coupling, and B lat flipping the sign indicate the existence of IRFP. As seen from the figure, in either pairs of (s, L/a) the discrete beta function approaches to zero from below when 1/u decreases from 0.8 to 0.2, and this happens independently of the choice of δ (2) . This means that at around u=1.25 the running starts to slow down, i.e. “walk”. In order to confirm that this behavior remains even in the continuum limit, simulations on a larger lattice is necessary. 5. Summary The running coupling constant of 10-flavor QCD is studied. The perturbative analysis suggests that this theory is extremely interesting. The preliminary result obtained without continuum limit seems to indicate the walking behavior. In order to draw definite conclusions, we clearly need larger lattices to take the continuum limit. Such calculations are in progress. A part of numerical simulations is performed on Hitachi SR11000 and the IBM System Blue Gene Solution at High Energy Accelerator Research Organization (KEK) under a support of its Large Scale Simulation Program (No. 09-05), on GCOE (Quest for Fundamental Principles in the Universe) cluster system at Nagoya University and on the INSAM (Institute for Numerical Simulations and Applied
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Mathematics) GPU cluster at Hiroshima University. This work is supported in part by the Grant-in-Aid for Scientific Research of the Japanese Ministry of Education, Culture, Sports, Science and Technology (Nos. 20105001, 20105002, 20105005, 21684013, 20540261 and 20740139), and by US DOE grant #DE-FG02-92ER40699. References 1. S. Weinberg, Phys. Rev. D 13, 974 (1976); L. Susskind, Phys. Rev. D 20, 2619 (1979); 2. For a recent review, see, for example, C. T. Hill and E. H. Simmons, Phys. Rept. 381, 235 (2003) [Erratum-ibid. 390, 553 (2004)]; F. Sannino, arXiv:0804.0182 [hep-ph]. 3. For recent review, see, for example, G. T. Fleming, PoS LATTICE2008, 021 (2008) [arXiv:0812.2035 [hep-lat]]; E. Pallante, in these proceedings. 4. M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964 (1990); Phys. Rev. D 46, 381 (1992). 5. B. Holdom, Phys. Rev. D 24, 1441 (1981); K. Yamawaki, M. Bando and K. i. Matumoto, Phys. Rev. Lett. 56, 1335 (1986); T. W. Appelquist, D. Karabali and L. C. R. Wijewardhana, Phys. Rev. Lett. 57, 957 (1986); T. Akiba and T. Yanagida, Phys. Lett. B 169, 432 (1986); M. Bando, T. Morozumi, H. So and K. Yamawaki, Phys. Rev. Lett. 59, 389 (1987). 6. D. D. Dietrich and F. Sannino, Phys. Rev. D 75, 085018 (2007) [arXiv:hepph/0611341]. 7. E. Shintani et al. [JLQCD Collaboration], Phys. Rev. Lett. 101, 242001 (2008); P. A. Boyle et al. [RBC and UKQCD collaborations], arXiv:0909.4931 [hep-lat]. 8. T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. Lett. 100, 171607 (2008); Phys. Rev. D 79, 076010 (2009). 9. M. Luscher, R. Narayanan, P. Weisz and U. Wolff, Nucl. Phys. B 384 (1992) 168; M. Luscher, R. Sommer, P. Weisz and U. Wolff, Nucl. Phys. B 413, 481 (1994); S. Sint and R. Sommer, Nucl. Phys. B 465, 71 (1996); M. Luscher and P. Weisz, Nucl. Phys. B 479, 429 (1996); A. Bode, P. Weisz and U. Wolff [ALPHA collaboration], Nucl. Phys. B 576, 517 (2000) [Erratum-ibid. B 600, 453 (2001 ERRAT,B608,481.2001)]. 10. T. Appelquist, K. D. Lane and U. Mahanta, Phys. Rev. Lett. 61, 1553 (1988); A. G. Cohen and H. Georgi, Nucl. Phys. B 314, 7 (1989). 11. For the works suggesting the other possibility, see, for example, A. Hasenfratz, Phys. Rev. D 80, 034505 (2009) [arXiv:0907.0919 [hep-lat]]; X. Y. Jin and R. D. Mawhinney, arXiv:0910.3216 [hep-lat]. 12. S. Aoki et al. [PACS-CS Collaboration], JHEP 0910, 053 (2009) [arXiv:0906.3906 [hep-lat]]. 13. Y. Shamir, B. Svetit*ky and T. DeGrand, Phys. Rev. D 78, 031502 (2008) [arXiv:0803.1707 [hep-lat]].
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Study of the Running Coupling in Twisted Polyakov Scheme T. Aoyama1 , H. Ikeda2 , E. Itou3 , M. Kurachi4 , C.-J. D. Lin5 , H. Matsufuru6 , H. Ohki7 , T. Onogi8 , E. Shintani8, ∗ and T. Yamazaki9 1 Graduate
School of Science, Nagoya University, Aichi 464-8602, Japan 2 School of High Energy Accelerator Science The Graduate University for Advanced Studies (Sokendai), Ibaraki 305-8081, Japan 3 Academic Support Center, Kogakuin University, Nakanomachi Hachioji, 192-0015, Japan 4 Department of Physics, Tohoku University, Sendai, 980-8578, Japan 5 National Chiao-Tung University, and National Center for Theoretical Sciences Hsinchu 300, Taiwan 6 KEK Computing Center, High Energy Accelerator Research Organization (KEK) Ibaraki 305-8081, Japan 7 Department of Physics, Kyoto University, Kyoto 606-8501, Japan 8 Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan 9 Center for Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan We present a non-perturbative study of the running coupling constant in the Twisted Polyakov Loop (TPL) scheme. We investigate how the systematic and statistical errors can be controlled via a feasibility study in SU(3) pure Yang-Mills theory. We show that our method reproduces the perturbative determination of the running coupling in the UV and gives consistent results with the theoretical prediction in the IR. We also present our preliminary results for Nf = 12 QCD.
1. Introduction The search for a non-trivial infrared fixed point (IRFP) in gauge theories is an interesting topic. In addition to its intrinsic field-theoretic interests, it is relevant to the phenomenology of dynamical electroweak symmetry breaking models such as the technicolor (TC) models [1]. The existence of the (approximate) IRFP provides a slowly running, or “walking,” coupling, which is required to suppress the flavorchanging neutral current (FCNC) by a large anomalous dimension of the bilinear techni-fermion operator [2, 3]. The walking behavior of the coupling might also help to reduce the technicolor contribution to the S parameter. Non-trivial IRFP is also interesting to understand phase structure, e.g. whether the chiral broken phase and confinement phase are different or not. SU(N) gauge theory with Nf massless fermions (also known as large flavor QCD) has been extensively investigated as a possible candidate for the theory with walking dynamics [4]. An analysis using the two-loop beta function and Schwinger-Dyson ∗ Speaker,
E-mail: [emailprotected]
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equation suggests the existence of the IRFP for Nfcr < Nf ≤ 16 with Nfcr 12 in the case of Nc = 3. However to search for the conformal window the full nonperturbative analysis is necessary. Recently non-perturbative analysis of the running coupling constant using lattice gauge theory with the Schr¨ odinger functional (SF) scheme has been done by Appelquist et al. [5]. While they discovered IRFP-like behavior in the case of Nf = 12 QCD, there is no such evidence in Nf = 8 QCD. (Two groups have also studied the phase structure of the Nf = 12 theory [6, 7].) In this work we focus on the confirmation of IRFP in Nf = 12 QCD with different scheme from previous IRFP studies. We here employ the Twisted Polyakov Loop (TPL) scheme [8, 9]. This scheme is not only automatically evaded from O(a/L) discretization errors, but also predicts the behavior of running coupling toward infrared limit. In addition compared with Wilson loop scheme [10] TPL scheme done not need relatively large lattice volume and link smearing. Therefore we can precisely search non-trivial IRFP with the present machine resources. We first give a brief review of TPL scheme in section 2, and in section 3 we present a validity study of this scheme by calculating in SU(3) pure Yang-Mills theory. Our preliminary results for Nf = 12 SU(3) gauge theory is reported in section 4. 2. Twisted Polyakov Loop scheme In this section, we present the definition of TPL scheme in SU(3) gauge theory. Here we consider twisted boundary condition for the link variables in x and y directions on the lattice: Uµ (x + νˆL/a) = Ων Uµ (x)Ω†ν .
(ν = 1, 2)
(1)
Ων are the twist matrices which satisfy Ω1 Ω2 = ei2π/3 Ω2 Ω1 ,
Ωµ Ω†µ = 1,
(Ωµ )3 = 1,
Tr[Ωµ ] = 0.
(2)
In order to keep the gauge invariance and translation invariance, the Polyakov loops in the twisted directions are defined as (3) P1 (y, z, t) = Tr U1 (x = j, y, z, t)Ω1 ei2πy/3L , j
for x direction, and for y direction we obtain by 1 → 2 and x ↔ y. In the above definition we add Ων for the invariance under gauge transformation, and phase ei2πy/3L for the translational invariance. The renormalized coupling is then given by † 1 y,z P1 (y, z, L/2a)P1 (0, 0, 0) . (4) gT2 P = k x,y P3 (x, y, L/2a)P3 (0, 0, 0)† At the tree level, the ratio of twisted and untwisted Polyakov-loop correlator is proportional to the bare coupling. The coefficient k is obtained by analytical calculation
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of the one-gluon-exchange diagram: 1 (−1)n = 0.03184 · · · . k= 24π 2 n2 + (1/3)2
(5)
in the case of SU(3) gauge group. The twist matrices are chosen in the same form as those in Ref.[11], −i2π/3 0 0 010 e (6) Ω1 = 0 0 1 , Ω2 = 0 ei2π/3 0 . 0 0 1 100
In the infrared limit (L → ∞), twisted and untwisted Polyakov-loop correlators are not different since the correlation between Polyakov loops vanishes. Therefore the ratio of those is unit, namely gT2 P consists of 1/k ∼ 32. This fact is also applicable to check our simulation. The naive twisted boundary condition for lattice fermions is inconsistent when we change the order of translations, i.e. ψ(x + νˆL/a + ρˆL/a) = Ωρ Ων ψ(x) = Ων Ωρ ψ(x). To avoid this problem, we introduce ”smell” degree of freedom [12], which is a copy of color degree of freedom. We identify the fermion field as a Nc ×Ns matrix (ψαa (x)). Then we impose the twisted boundary condition for fermion fields to be † b ψαa (x + νˆL/a) = eiπ/3 Ωab ν ψβ (Ων )βα
(7)
for ν = 1, 2 directions. Here, the smell index can be interpreted as a “flavor” index, then the number of flavors should be a multiple of Ns (= Nc = 3). We use staggered fermion in our simulation. Since this describes four tastes of fermions, we can perform simulations in Nf = 12 QCD, which is same number of flavor as that in Ref. [5], with twisted boundary condition. 3. Quenched QCD case Before carrying out the simulation for Nf = 12, we first measure the TPL running coupling in quenched QCD to confirm that the scheme provides the expected behavior of running coupling constant. The gauge configurations are generated by the pseudo-heatbath algorithm and the overrelaxation algorithm mixed in the ratio 1:5. To reduce large statistical fluctuation of the TPL coupling, we measure Polyakov loops at every Monte Calro sweep and perform a jackknife analysis with large bin size, typically of O(103 ). The simulations are carried out with several lattice sizes (L/a = 4, 6, 8, 10, 12, 14, 16) in the range 6.2 ≤ β ≤ 16. We generate 200,000–400,000 configurations for each data point. Figure 1 shows the β dependence of the renormalized coupling in our scheme at various lattice sizes. We fit the data at each fixed lattice size by an interpolating function: n Ai 2 (β) = , (8) gTP (β − B)i i=1
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where Ai and B are the fit parameters, 4 ≤ B ≤ 5, and n = 3, 4 are employed. We can see that the interpolating function in Eq.(8) well describes all lattice data as shown in Figure 1. In the step scaling we use L/a = 4, 6, 8, 10 for starting points. The step-scaling parameter is s = 1.5, and we estimate the coupling constant for L/a = 9, 15 from interpolations. 8 7 L/a=4 L/a=6 L/a=8 L/a=10 L/a=12 L/a=14 L/a=16
6 5 4 3 2 1
Fig. 1.
2
gTP(L/a,β) 6
7
β
8
9
10
TPL renormalized coupling in the each β and L/a in quenched QCD.
We take the continuum limit using a linear function in (a/L)2 in each step of step scaling procedure. TPL scheme involves tiny O((a/L)2 ) discretization effect, and no O(a/L) error below gT2 P 5, as shown in Figure 2. Figure 3 shows the gT2 P normalized by 1/k in Eq.(5) in the infrared limit. As expected gT2 P goes to 1/k in the L → ∞. gT2 P in quenched QCD as a function of energy scale is shown in Figure 4 together with one- and two-loop perturbative results. The step-scaling function starts from g 2 (L0 /L) = 0.65. The running coupling constant obtained in TPL scheme is consistent with perturbation theory above L0 /L = 0.01. On the other hand, in L0 /L < 0.01, the running is slower than one-loop and two-loop perturbation because of higher order effects. From this quenched test, we can see that TPL scheme provides reasonable behavior for running coupling constant. 4. Nf = 12 case In this section, we present a preliminary result in Nf = 12. The simulation parameters are 4.0 ≤ β ≤ 25.0 with lattice sizes L/a = 4, 6, 8, 10, 12, 16. Figure 5 shows the β dependence of gT2 P normalized by 1/k for each lattice size. Note that in low-β region the growth of gT2 P toward coarse lattice saturates, especially for small lattice.
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2 Fig. 2. The continuum extrapolation of gT P with s = 1.5. The fit function is a linear function of (a/L)2 . The statistical error bars are of the same size of the symbols.
1.2
R(β)
1 0.8
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2
3
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β
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2 k as a function of β, which corresponds to the inverse lattice spacing, in L/a = 2, 4, 6. gT P
It might be due to the lattice artifacts e.g. the effect of taste breaking for staggered fermions, but it is necessary to do further detailed analysis. To perform the global fit, we consider the following interpolation function kgT2 P (β, a/L) = f (β, a/L): f (β, a/L) =
2 3 6 6 6 +c1 (L/a) +c2 (L/a) , (9) β + c0 (L/a) β + c0 (L/a) β + c0 (L/a)
with ci (L/a) = xi + yi ln(L/a), i = 1, 2. xi , yi denote the fit parameters. The solid lines in Figure 5 denote the fit function with fit range 6 ≤ β ≤ 20. In the step-
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Fig. 4.
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The running coupling constants in TPL scheme, one-loop and two-loop.
L=4 L=6 L=8 L=10 L=12 L=16
R(β)
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Fig. 5.
β
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The ratio of Polyakov loop in each β and L/a.
scaling function, we use three points, L = 4, 6, 8 and sL = 8, 12, 16. Figure 6 shows the continuum extrapolation in each steps with s = 2. In Nf = 12 case the O(a/L) error is also absent. Accumulating more statistics and data points of low-β region, we will check the consistency with perturbation theory in high-β and study the running behavior in strong coupling region. These works are under way.
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2.5 s 16 s 20 s 25 s 27 s
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References 1. S. Weinberg, Phys. Rev. D 13, 974 (1976); Phys. Rev. D 19, 1277 (1979); L. Susskind, Phys. Rev. D 20, 2619 (1979). 2. B. Holdom, Phys. Rev. D 24, 1441 (1981). 3. K. Yamawaki, M. Bando and K. i. Matumoto, Phys. Rev. Lett. 56, 1335 (1986). 4. T. Banks and A. Zaks, Nucl. Phys. B 196 (1982) 189; T. Appelquist, J. Terning and L. C. R. Wijewardhana, Phys. Rev. Lett. 77, 1214 (1996) [arXiv:hep-ph/9602385]. 5. T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. Lett. 100 (2008) 171607, [Erratum-ibid. 102 (2009) 149902], Phys. Rev. D 79 (2009) 076010 6. A. Deuzeman, M. P. Lombardo and E. Pallante, arXiv:0904.4662 [hep-ph]. 7. Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. Schroeder, arXiv:0809.4890 [hep-lat]. 8. G. M. de Divitiis, R. Frezzotti, M. Guagnelli and R. Petronzio, Nucl. Phys. B 422 (1994) 382 9. G. M. de Divitiis, R. Frezzotti, M. Guagnelli and R. Petronzio, Nucl. Phys. B 433 (1995) 390 10. E. Bilgici, et al., Phys. Rev. D 80 (2009) 034507 11. H. D. Trottier, N. H. Shakespeare, G. P. Lepage and P. B. Mackenzie, Phys. Rev. D 65 (2002) 094502 12. G. Parisi,Published in Cargese Summer Inst. 1983:0531 13. G. M. de Divitiis et al., (Alpha Collaboration), Nucl. Phys. B 437 (1995) 447
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Running Coupling in Strong Gauge Theories via the Lattice Zolt´ an Fodor Department of Physics, University of Wuppertal, Gauss Strasse 20, D-42119, Germany and Institute for Theoretical Physics, Eotvos University Budapest, Pazmany P. 1, H-1117, Hungary E-mail: [emailprotected] Kieran Holland Department of Physics, University of the Pacific 3601 Pacific Ave, Stockton CA 95211, USA E-mail: [emailprotected] Julius Kuti Department of Physics 0319, University of California, San Diego 9500 Gilman Drive, La Jolla CA 92093, USA E-mail: [emailprotected] D´ aniel N´ ogr´ adi Institute for Theoretical Physics, Eotvos University Budapest, Pazmany P. 1, H-1117, Hungary E-mail: [emailprotected] Chris Schroeder Department of Physics, University of Wuppertal, Gauss Strasse 20, D-42119, Germany E-mail: [emailprotected] New strongly-interacting gauge theories are possible Beyond Standard Model candidates. It is essential to determine if a given non-abelian gauge theory is QCD-like or conformal. One way is to calculate the running of the renormalized gauge coupling. We describe a method to do this using Wilson loop ratios measured in lattice simulations. We demonstrate this in SU (3) pure gauge theory and show initial results for dynamical fermions in the fundamental representation. Keywords: Beyond Standard Model, lattice, near-conformal, running coupling.
1. Running coupling There is great interest in strongly-coupled gauge theories as new physics candidates,1–21 one example being technicolor. As a starting point, it is crucial to distinguish conformal gauge theories from those with a mass gap like QCD. There are many recent lattice studies of various theories, attempting to answer this question.22–54 One signal of conformal behavior is if the running gauge coupling flows to
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an infrared fixed point. In a QCD-like theory, no such fixed point exists. For a given theory, this should be consistent with other signals, such as the mass spectrum, the Dirac operator eigenvalues, the existence of finite-temperature transitions, or the renormalization group flow in bare parameter space. We define the renormalized coupling using Wilson loops W (R, T, L), R and T being the space- and time-like loop extents, and L the finite volume extent. One definition uses the quark-antiquark potential and its derivative dV g 2 (R) ∂ lnW (R, T, L)), F = = CF , (1) T →∞ ∂T dR 4πR2 where CF = 4/3, and L is large enough that finite-volume effects are absent. The infinite-volume coupling g 2 (R) runs with the quark separation R. In lattice simulations, the separation of scales L, T, R and lattice spacing a is challenging. An alternate definition is55 R2 ∂ 2 lnW (R, T, L) |T =R , g 2 (R, L) = − (2) k ∂R∂T for convenience restricted to square Wilson loops. The numerical factor k depends on R/L and is calculated perturbatively, such that the leading term in g 2 (R, L) is the tree-level gauge coupling. In the infinite-volume limit, the coupling g 2 (R) again runs with the loop size. Alternatively, one can hold R/L fixed in which case g 2 (L) runs with the finite volume. This may be advantageous in simulations. The fixed value of R/L defines a renormalization scheme. The role of lattice simulations is to measure expectation values ... non-perturbatively, with derivatives replaced by finite differences, giving 1 g 2 (R, L) = (R + 1/2)2 χ(R + 1/2, L) , k W (R + 1, T + 1, L)W (R, T, L) χ(R + 1/2, L) = − ln , (3) | W (R + 1, T, L)W (R, T + 1, L) T =R V (R) = lim (−
χ being the Creutz ratio56 and the renormalization scheme is r = (R+ 1/2)/L fixed. The continuum limit corresponds to L → ∞, where the physical length scale Lphys is held fixed while the lattice spacing a → 0. One starts the RG flow from some reference physical point Lphys,0 which is implicitly set by the initial choice e.g. g 2 (r, Lphys,0 ) = 0.8. In a QCD-like theory, g 2 increases with increasing Lphys , flowing in the infrared direction. However in a conformal theory, g 2 flows towards some non-trivial infrared fixed point g ∗2 as Lphys increases. One way to take the continuum limit of the RG flow is via step-scaling. The bare lattice gauge coupling is defined in the usual way β = 6/g02 in the Wilson lattice gauge action. On a sequence of lattice sizes L1 , L2 , ..., Ln , the bare coupling is tuned on each lattice so that identical values g 2 (r, Li , βi ) = g 2 (r, Lphys ) are measured in simulations. Next a new set of simulations is performed, on a sequence of lattice sizes 2L1 , 2L2 , ..., 2Ln , using the corresponding tuned couplings β1 , β2 , ..., βn . From the simulations, one measures g 2 (r, 2Li , βi ), which vary with the bare coupling i.e. the lattice spacing. These data are extrapolated to the continuum as a function of 1/L2i .
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1.8
linear fit data
0.6
linear fit data extrapolation: 1.636(23) 1.75
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g (2L)
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(R+1/2) χ
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lattice at β = 6.99. We interpolate the data linearly Fig. 1. (a) The rescaled Creutz ratio on a to r = (R + 1/2)/L = 0.25, giving a chi squared per degree of freedom 2.0/4. (b) The measured coupling g 2 (2Li , βi ) for 2Li = 20, 24, 28 and 32, where βi is tuned such that g 2 (Li , βi ) = 1.44. A linear continuum extrapolation gives g 2 (2Lphys ) = 1.636(23), with χ2 /dof = 0.57/2.
This gives one blocking step g 2 (r, Lphys ) → g 2 (r, 2Lphys ) in the continuum RG flow. The whole procedure is then iterated. The chain of measurements gives the flow g 2 (r, Lphys ) → g 2 (r, 2Lphys ) → g 2 (r, 4Lphys ) → g 2 (r, 8Lphys ) → ..., as far as is feasible. One is free to choose a different blocking factor, say Lphys → (3/2)Lphys, in which case more blocking steps are required to cover the same energy range. 2. SU (3) pure gauge theory As a first test, we study SU (3) pure gauge theory in four dimensions, calculating the renormalized coupling using square Wilson loops. We simulate using the standard Wilson lattice gauge action, with a mixture of five over-relaxation updates for every heatbath update. Our renormalization scheme is r = 0.25, for brevity we omit the label r. In this R/L range, the Creutz ratio can be accurately measured, and the factor k converges quickly to its continuum-limit value. In the pure gauge theory test, we use the finite-L values of k, as this may remove some of the cutoff dependence of the renormalized coupling. For the RG flow we choose the blocking step L → 2L. We simulate on small lattices of size L = 10, 12, 14, 16, 18, 20 and 22, and the corresponding doubled lattices 2L = 20, 24, 28, 32, 36, 40 and 44. Wilson loop measurements are separated by 1000 sweeps, which we find is sufficient to generate statistically independent configurations. To tune βi for each small lattice size Li , we typically run separate simulations at 5 – 10 different β values in the relevant range of renormalized coupling g 2 . Each of these runs contains 300 – 500 measurements i.e. up to 5 × 105 sweeps. Simulating on the doubled lattices at the tuned βi values, we generate between 200 and 1000 measurements each, typically more than 500. The signal of the Creutz ratio disappears into the noise as the size of the Wilson loop increases. One way to suppress the noise in measurements is to gauge fix the configurations to Coulomb gauge, and replace the thin-link Wilson loop with the correlator of the products of the time-like gauge
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(a)
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ln(L/L0)
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(b)
Fig. 2. (a) The continuum extrapolations of four discrete RG steps. (b) The RG flow g 2 (Lphys ), combining analytic lattice perturbation theory and the simulation results. The running starts at g 2 (Lphys,0 ) = 0.825. There is excellent agreement with continuum 2-loop running, at the strongest coupling, the simulation results begin to break away from perturbation theory.
links.57,58 Note that gauge fixing is not implemented in the actual Monte Carlo updating algorithm. An alternative method to suppress noise is to smear the gauge links and measure the fat-link Wilson loop operator. In the pure gauge theory test, we use the gauge-fixing method, in the dynamical fermion simulations we describe later, we use the smearing method. These improvement methods do not correspond to calculating the original thin-link Wilson loop operator. We show in Fig. 1(a) a typical result for the rescaled Creutz ratio (R + 1/2)2 χ. The doubled lattice is 284 and the bare coupling β = 6.99 is tuned from simulations on 144 volumes. Errorbars are calculated using the jackknife method. We interpolate the data linearly to the point r = (R + 1/2)/L = 0.25, obtaining χ2 /dof = 2.0/4. The data at different R are highly correlated, being measured on the same gauge configurations. For error estimation, we bin the gauge configurations, and analyze and interpolate separately each bin. An example of the step-scaling method is shown in Fig. 1(b). The bare couplings are tuned such that g 2 (Li , βi ) = 1.44 for Li = 10, 12, 14 and 16, the figure shows the data g 2 (2Li , βi ) for 2Li = 20, 24, 28 and 32. The leading lattice artifacts in the Creutz ratio are expected to be of order O(a2 ), corresponding to O(1/L2 ), since the physical lattice size is La. Extrapolating linearly in 1/(2L)2 gives g 2 (2Lphys ) = 1.636(23) and χ2 /dof = 0.57/2. For a systematic error, we omit one data point at a time and repeat the extrapolation. Combined in quadrature with the statistical error, our continuum result is g 2 (2Lphys ) = 1.636(25). We iterate the procedure, giving four discrete RG steps as shown in Fig. 2(a). At stronger coupling, we need lattices up to 2Li = 44 for the continuum extrapolation. The use of the finite-L values of k does not appear to reduce the cutoff effects. The continuum RG flow is shown in Fig. 2(b). At weak coupling, we use analytic/numeric lattice perturbation theory to calculate the Wilson loop ratios in finite volumes.59,60 The Wilson loops are calculated to 1-loop in the bare coupling in finite volume. The series is reexpanded in the boosted coupling constant at the relevant scale of the
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0.2 0.5
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Fig. 3. (a) The renormalized coupling g 2 (L, β) at fixed bare coupling for Nf = 16 fundamental flavors. For g 2 (L, β) > 0.5 the coupling decreases with increasing L, for g 2 (L, β) < 0.5 the coupling is independent of L within the errors. This is consistent with the existence of an infrared fixed point. (b) The quark-antiquark potential V (R) for Nf = 12 fundamental flavors measured in a 323 × 64 lattice at β = 2.2 and m = 0.015. The data indicate QCD-like behavior.
Creutz ratio. Step-scaling of the finite volume ratios can be used in exactly the same way as for the simulations results, to determine the RG flow in the continuum. The analytic RG flow starts at the reference point g 2 (Lphys,0 ) = 0.825. At weak coupling there is complete agreement with 2-loop perturbation theory. We connect lattice perturbation theory to the simulation results by matching the flows at g 2 (Lphys ) = 1.44, where the simulation RG flow begins. There is continued agreement with 2loop perturbation theory at even stronger coupling, only at the strongest coupling do we see deviation from the perturbative flow. 3. Fundamental fermions We have begun studies of SU (3) gauge theory with fermions in the fundamental representation. For Nf = 16 flavors, 2-loop perturbation theory predicts the theory is conformal with an infrared fixed point g ∗2 ≈ 0.5. At such weak coupling, this perturbative result is quite trustworthy. Because of the computational expense of step-scaling with dynamical fermions, we have not yet performed a continuum extrapolation. Hence the running coupling is contaminated with cutoff effects. If the linear lattice size L is large enough, the trend from the volume dependence of g 2 (L, β) should indicate the location of the fixed point. For g 2 (L, β) > g ∗2 we expect decrease in the running coupling as L grows, although the cutoff of the flow cannot be removed above the fixed point. Below the fixed point with g 2 (L, β) < g ∗2 we expect the running coupling to grow as L increases and the continuum limit of the flow could be determined. The first results are shown in Fig. 3(a). We use stoutsmeared61 staggered fermions62,63 and the RHMC algorithm, simulating at quark mass mq = 0.01, with some runs at mq = 0.001 to test that the mass dependence is negligible. For the Wilson loop ratios, we smear the gauge fields and measure the fat-link Wilson operator. Our experience in the pure gauge theory test is cutoff effects are not reduced using finite-L values of k, hence we use the infinite-volume k
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value to convert the Wilson loop ratios to a renormalized coupling. The results are consistent with the above picture. For g 2 (L, β) > 0.5, the cutoff-dependent renormalized coupling decreases with L. For g 2 (L, β) < 0.5, the renormalized coupling is L-independent within errors. The theory appears conformal but precise determination of the infrared fixed point requires further study. We also show results for Nf = 12 flavors in the fundamental representation. Perturbatively this theory is predicted to be conformal, at 2-loop g ∗2 ≈ 9, with significant corrections coming from 3-loop for various schemes, for example g ∗2 ≈ 4.3 in the Wilson-loop scheme. This theory is controversial, some recent lattice studies find conformal behavior, others indicate it is QCD-like. In Fig. 3(b) we show the quark-antiquark potential V (R) in a 323 × 64 lattice, using the same improved action as for Nf = 16. The data suggest the theory is in fact QCD-like, V (R) does not display the purely 1/R dependence of a conformal theory. This is consistent with our other findings for Nf = 12, that the pion mass spectrum and eigenvalues of the Dirac operator indicate spontaneously broken chiral symmetry. The running coupling study of fundamental fermions is continuing. Acknowledgments We wish to thank Urs Heller for the use of his code to calculate Wilson loops in lattice perturbation theory, and Paul Mackenzie for related discussions. We are grateful to Sandor Katz and Kalman Szabo for helping us in using the Wuppertal RHMC code. In some calculations we use the publicly available MILC code, and the simulations were performed on computing clusters at Fermilab, under the auspices of USQCD and SciDAC, and on the Wuppertal GPU cluster. This research is supported by the NSF under grant 0704171, by the DOE under grants DOE-FG0397ER40546, DOE-FG-02-97ER25308, by the DFG under grant FO 502/1 and by SFB-TR/55. References 1. S. Weinberg, Phys. Rev. D 19, 1277 (1979). 2. L. Susskind, Phys. Rev. D 20, 2619 (1979). 3. E. Farhi and L. Susskind, Phys. Rept. 74, 277 (1981). 4. F. Sannino and K. Tuominen, Phys. Rev. D 71, 051901 (2005) 5. D. D. Dietrich, F. Sannino and K. Tuominen, Phys. Rev. D 72, 055001 (2005) 6. D. D. Dietrich and F. Sannino, Phys. Rev. D 75, 085018 (2007). 7. T. A. Ryttov and F. Sannino, Phys. Rev. D 76, 105004 (2007). 8. D. K. Hong, S. D. H. Hsu and F. Sannino, Phys. Lett. B 597, 89 (2004). 9. H. Georgi, Phys. Rev. Lett. 98, 221601 (2007). 10. M. A. Luty and T. Okui, JHEP 0609, 070 (2006). 11. W. E. Caswell, Phys. Rev. Lett. 33, 244 (1974). 12. T. Banks and A. Zaks, Nucl. Phys. B 196, 189 (1982). 13. T. Appelquist, K. D. Lane and U. Mahanta, Phys. Rev. Lett. 61, 1553 (1988). 14. A. G. Cohen and H. Georgi, Nucl. Phys. B 314, 7 (1989). 15. T. Appelquist et al., Phys. Rev. Lett. 77, 1214 (1996).
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16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63.
B. Holdom, Phys. Rev. D 24, 1441 (1981). K. Yamawaki, M. Bando and K. i. Matumoto, Phys. Rev. Lett. 56, 1335 (1986). T. W. Appelquist et al., Phys. Rev. Lett. 57, 957 (1986). V. A. Miransky and K. Yamawaki, Phys. Rev. D 55, 5051 (1997). E. Eichten and K. D. Lane, Phys. Lett. B 90, 125 (1980). T. Appelquist, M. Piai and R. Shrock, Phys. Rev. D 69, 015002 (2004) T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. Lett. 100, 171607 (2008) T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. D 79, 076010 (2009) T. Appelquist et al., arXiv:0910.2224 [hep-ph]. S. Catterall and F. Sannino, Phys. Rev. D 76, 034504 (2007) S. Catterall, J. Giedt, F. Sannino and J. Schneible, JHEP 0811, 009 (2008) S. Catterall, J. Giedt, F. Sannino and J. Schneible, arXiv:0910.4387 [hep-lat]. Z. Fodor et al., PoS LATTICE2008, 058 (2008) Z. Fodor et al., PoS LATTICE2008, 066 (2008) Z. Fodor et al., JHEP 0908, 084 (2009) Z. Fodor et al., Phys. Lett. B 681, 353 (2009) Z. Fodor et al., JHEP 0911, 103 (2009) Z. Fodor et al., PoS LATTICE2009, 055 (2009) Z. Fodor et al., PoS LATTICE2009, 058 (2009) A. J. Hietanen et al., JHEP 0905, 025 (2009) A. J. Hietanen, K. Rummukainen and K. Tuominen, Phys. Rev. D 80, 094504 (2009) A. Deuzeman, M. P. Lombardo and E. Pallante, Phys. Lett. B 670, 41 (2008) A. Deuzeman, M. P. Lombardo and E. Pallante, arXiv:0904.4662 [hep-ph]. A. Deuzeman, M. P. Lombardo and E. Pallante, arXiv:0911.2207 [hep-lat]. L. Del Debbio, A. Patella and C. Pica, arXiv:0805.2058 [hep-lat]. L. Del Debbio, B. Lucini, A. Patella, C. Pica and A. Rago, arXiv:0907.3896 [hep-lat]. F. Bursa et al., Phys. Rev. D 81, 014505 (2010) Y. Shamir, B. Svetit*ky and T. DeGrand, Phys. Rev. D 78, 031502 (2008) T. DeGrand, Y. Shamir and B. Svetit*ky, Phys. Rev. D 79, 034501 (2009) O. Machtey and B. Svetit*ky, Phys. Rev. D 81, 014501 (2010) T. DeGrand and A. Hasenfratz, Phys. Rev. D 80, 034506 (2009) T. DeGrand, Phys. Rev. D 80, 114507 (2009) A. Hasenfratz, Phys. Rev. D 80, 034505 (2009) X. Y. Jin and R. D. Mawhinney, PoS LATTICE2008, 059 (2008) X. Y. Jin and R. D. Mawhinney, PoS LAT2009, 049 (2009) D. K. Sinclair and J. B. Kogut, arXiv:0909.2019 [hep-lat]. J. B. Kogut and D. K. Sinclair, arXiv:1002.2988 [hep-lat]. E. Bilgici et al., Phys. Rev. D 80, 034507 (2009) E. Bilgici et al., arXiv:0910.4196 [hep-lat]. M. Campostrini, P. Rossi and E. Vicari, Phys. Lett. B 349, 499 (1995) M. Creutz, Phys. Rev. D 21, 2308 (1980). A. Bazavov et al., arXiv:0903.3598 [hep-lat]. C. W. Bernard et al., Phys. Rev. D 62, 034503 (2000) U. M. Heller and F. Karsch, Nucl. Phys. B 251, 254 (1985). G. P. Lepage and P. B. Mackenzie, Phys. Rev. D 48, 2250 (1993) C. Morningstar and M. J. Peardon, Phys. Rev. D 69, 054501 (2004) Y. Aoki, Z. Fodor, S. D. Katz and K. K. Szabo, JHEP 0601, 089 (2006) Y. Aoki, Z. Fodor, S. D. Katz and K. K. Szabo, Phys. Lett. B 643, 46 (2006)
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Higgsinoless Supersymmetry and Hidden Gravity∗ Michael L. Graesser Theoretical Division T-2, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Ryuichiro Kitano and Masafumi Kurachi Department of Physics, Tohoku University, Sendai 980-8578, Japan We present a simple formulation of non-linear supersymmetry where superfields and partnerless fields can coexist. Using this formalism, we propose a supersymmetric Standard Model without the Higgsino as an effective model for the TeV-scale supersymmetry breaking scenario. We also consider an application of the Hidden Local Symmetry in nonlinear supersymmetry, where we can naturally incorporate a spin-two resonance into the theory in a manifestly supersymmetric way. Possible signatures at the LHC experiments are discussed.
1. Introduction Technicolor is an attractive idea in which electroweak symmetry is dynamically broken by a strong dynamics operating around the TeV energy scale.2,3 The big hierarchy, mW MPl , is elegantly explained by the very same reason as ΛQCD MPl . However, it is well-known that there are two phenomenological difficulties in this idea. One is that the electroweak precision measurements seem to prefer scenarios with a weakly coupled light Higgs boson.4 Another is the difficulty in writing down the Yukawa interactions to generate fermion masses in the Standard Model. After the LEP-I experiments, supersymmetry (SUSY) has become very popular as a natural scenario for the light weakly coupled Higgs boson. However, with the experimental bound on the lightest Higgs boson mass from the LEP-II experiments, parameters in the minimal SUSY standard model (MSSM) are required to be more and more fine-tuned, at least in the conventional scenarios.5–7 In this situation, it may be interesting to (re)consider a hybrid of technicolor and SUSY along the similar spirit of the early attempts of SUSY model building.8,9 We assume that strong dynamics breaks SUSY at the multi–TeV energy scale (which we call the scale Λ), with electroweak symmetry breaking triggered by the dynamics through direct couplings between the Higgs field and the dynamical sector. This scenario has several virtues: (1) the Yukawa interactions can be written down by ∗ Based
on the work in Ref. 1. The talk was given by R. Kitano.
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assuming an existence of elementary Higgs fields in the UV theory, which mix with (or remain as) the Higgs field to break electroweak symmetry at low energy;10–12 (2) the hierarchy problem, Λ MPl , is explained by dynamical SUSY breaking;13 (3) one can hope that the little hierarchy, mW ∼ mh Λ, is explained by either SUSY or some other mechanisms such as the Higgs boson as the pseudo-Nambu-Goldstone particle in the strong dynamics;14 (4) the cosmological gravitino problem is absent;15 (5) one can expect additional contributions to the Higgs boson mass from the SUSY breaking sector, with which the mass bound from the LEP-II experiments can be evaded;16 and (6), there is an interesting possibility that the LHC experiments can probe the SUSY breaking dynamics directly. SUSY is phenomenologically motivated from the point (1) (and also (6)) in this framework in addition to the connection to string theory. Although the TeV-scale SUSY breaking scenario is an interesting possibility, an explicit model realizing this scenario will not be attempted here. In this paper, we take a less ambitious approach and construct an effective Lagrangian for the scenario without specifying (while hoping for the existence of) a UV theory responsible for SUSY breaking and its mediation. In constructing the effective Lagrangian, we take the following as organizing principles: (1) the Lagrangian possesses non-linearly realized supersymmetry; (2) the quarks/leptons and gauge fields are only weakly coupled to the SUSY breaking sector, so that the typical mass splitting between bosons and fermions are O(100) GeV (in other words, the matter and gauge fields are introduced as superfields which transform linearly under SUSY); and (3) the Higgs boson is introduced as a nonlinearly transforming field because it is assumed to be directly coupled to the SUSY breaking sector. The Higgsino field is absent in the minimal model. In this Higgsinoless model, the Higgs potential receives quadratic divergences from loop diagrams with the gauge interactions and the Higgs quartic interaction although the top-quark loops can be cancelled by the loops of the scalar top quarks as usual. The rough estimate of the correction to the Higgs boson mass is of the order of (α/4π)Λ2 and (k/16π 2 )Λ2 with k being the coupling constant of the Higgs quartic interaction. By comparing with the quadratic term needed for electroweak symmetry breaking, m2H = kH2 /2, naturalness suggests Λ 4πH ∼ (a few) × TeV. Precision electroweak constraints, on the other hand, obtained from the LEP-II and SLC experiments do not generically allow such a low scale without fine-tuning.17 The dynamical scale may therefore have to be larger, Λ O(6 − 10 TeV). To obtain a light Higgs boson at this larger scale either requires fine-tuning, or some new weakly coupled new physics below Λ. (Or simply the Higgsino appears around a few TeV.) It is also true that the direct coupling to the dynamical sector generically gives the Higgs boson mass to be O(Λ). We may therefore need to assume that the Higgs boson is somewhat special in the dynamics, e.g., a pseudo-Goldstone boson. In this paper we simply ignore the issue because its resolution depends on the UV completion, and here we only concerned with the effective theory below the TeV scale.
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The stop potential also receives quadratic divergences, in this case from a loop diagram involving the Higgs boson (and proportional to λ2t ). This divergence is not cancelled, simply because the Higgsinos are not present in the low energy theory. One therefore expects the stops to have a mass no smaller than a loop factor below the scale Λ. The Lagrangian we construct needs to contain interaction terms among superfields and also partnerless fields such as the Higgs boson. Since these two kinds of fields are defined on different spaces – one superspace and the other the usual Minkowski space – one needs to convert the partnerless fields into superfields or vice-versa. One approach is to utilize established formulations for constructing superfields out of partnerless fields18–20 where the Goldstino field is also promoted to a superfield. In this paper, we present a simple manifestly supersymmetric formulation where we do not try to convert partnerless fields into superfields, although it is totally equivalent to the known formalisms. The essence is to prepare two kinds of spaces: the superspace and the Minkowski space, on which superfields and partnerless fields are defined. By embedding the Minkowski space into the superspace by using a SUSY invariant map, one can define a Lagrangian density on a single spacetime. By using the formalism, one can write down a SUSY invariant Lagrangian, in particular the Yukawa interactions, only with a single Higgs field. We also find that the coupling constant of the Higgs quartic interaction can be a free parameter, unrelated to the gauge coupling constant. Therefore, the Higgs boson mass can be treated as a free parameter in this model. As a related topic, a model in which the MSSM is only partly supersymmetric has been proposed in Ref.21 There SUSY is broken explicitly at the Planck scale, and only the Higgs sector is remained to be supersymmetric which is made possible by a warped extra-dimension (or a conformal dynamics). Our philosophy is opposite to that and is, relatively speaking, closer to Ref.22 by the same authors, where SUSY is broken on the IR brane (or equivalently by some strong dynamics at the O(TeV) scale). As a possible signature of the TeV-scale dynamics, we construct a model “Hidden Gravity,” which is an analogy of the Hidden Local Symmetry23–27 in the chiral Lagrangian. The Hidden Local Symmetry is a manifestly chiral symmetric model to describe the vector resonance (the ρ meson) as the gauge boson of the hidden vectorial SU(2) symmetry (the unbroken symmetry of the chiral Lagrangian). When we apply this technique to SUSY, we obtain a supersymmetric Lagrangian for a massive spin-two field which is introduced as a graviton associated with a hidden general covariance because the unbroken symmetry is the Poincar´e symmetry. One can consistently incorporate the resonance as a non-strongly coupled field for a range of parameters and small range of energy. Indeed, we show that there is a sensible parameter region where we can perform a perturbative calculation of the resonant single-graviton production cross section. At energies not far above the graviton mass the effective theory becomes strongly coupled and incalculable. If the graviton is much lighter than the cut-off scale, new physics is required to complete
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the theory up to Λ, another direction not pursued here. We discuss signatures of this graviton scenario at the LHC. 2. Non-linear SUSY and invariant Lagrangian In this section we present a method to construct a Lagrangian invariant under the non-linearly realized global supersymmetry. We will introduce the Higgs boson as a non-linearly transforming field (which we call a non-linear field) and also matter and gauge fields as superfields. We therefore need a formulation to write down a supersymmetric Lagrangian where both kinds of fields are interacting. Ro˘cek,18 Ivanov and Kapustnikov,19 and Samuel and Wess20 have established a superfield formalism of non-linear SUSY by upgrading the Goldstino fermion and other nonlinear fields to constrained superfields. (See28 for a recent work.) Although the formalism is somewhat complicated, using superfields is motivated there as a first step towards embedding the theory into supergravity. As we are not interested in supergravity in this paper, we will use a simpler formalism where the Goldstino field remains as a non-linearly transforming field. We will also use results from earlier work by Ivanov and Kapustnikov29 that establishes the correspondence between superfields and non-linear fields a . 2.1. Linear and non-linear SUSY Linearly realized SUSY is most simply formulated in the superspace. Under a group element, g = eic
a
¯ Pa +iηQ+i¯ ηQ
,
(1)
the superspace coordinate (xa , θα , θ¯α˙ ) transforms as31 xa → xa = xa + ca + ∆a (η, θ),
θα → θα = θα + ηα ,
θ¯α˙ → θ¯α˙ = θ¯α˙ + η¯α˙ , (2)
where the ∆a factor is defined by ∆a (η, ξ) ≡ iησ a ξ¯ − iξσ a η¯.
(3)
On the other hand, the non-linear SUSY transformation under g in Eq. (1) is defined by Volkov and Akulov in Ref.32 It is ˜µ = x ˜µ + cµ + ∆µ (η, λ(˜ x)), x ˜µ → x
(4)
λα (˜ x) → λα (˜ x ) = λα (˜ x) + ηα ,
(5)
¯ (˜ ¯ x) + η¯α˙ . ¯ α˙ (˜ x) → λ λ α˙ x ) = λα˙ (˜
(6)
¯ are the Goldstino fermion and its complex conjugate, respectively. The fields λ and λ 33 (See for a formalism for constructing the non-linear Lagrangian.) a The
generalization of this relationship to local supersymmetry can be found in Ref.30
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We are interested in constructing a Lagrangian where non-linear fields (such as the Goldstino) and superfields coexist and they are interacting. (See18,20,29,34–37 for constructing superfields out of non-linear fields.) However, as one can see two kinds of fields are living in different spaces x and x ˜ which we cannot identify as the same space at this stage since their SUSY transformations are different. This situation can be cured by introducing a SUSY invariant delta function: ˜µ − ∆µ (λ(˜ x), θ)), (7) 1 = d4 x˜ det Xδ 4 (xµ − x with
¯ + i∂µ λσ a θ. ¯ Xµa = ηµa − iθσ a ∂µ λ The invariant action can now be written down as S = d4 xd4 θd4 x˜ det X δ 4 (xµ − x ˜µ − ∆µ (λ(˜ x), θ)) ¯ A, X, · · · , ¯ φ(˜ ¯ ∇a λ, ∇a λ, × K Ψ(x, θ, θ), x), θ − λ, θ¯ − λ,
(8)
(9)
where Ψ and φ represents arbitrary superfields and non-linear fields, respectively. The function K must be real and scalar under the general coordinate transformation about the x ˜ coordinate. As an example of the invariant action, we can take K = ¯ which gives the supersymmetric action, Ψ(x, θ, θ) ¯ (10) S = d4 xd4 θΨ(x, θ, θ).
If we take K = δ 4 (θ − λ) · (−f 4 /2), we obtain the Volkov-Akulov action32 f4 S=− d4 x˜ det A, 2
(11)
where the matrix A is
¯ + i∂µ λσ a λ. ¯ Aµa = ηµa − iλσ a ∂µ λ
(12)
The action contains the kinetic term for the Goldstino. The parameter f is the decay constant which represents the size of the SUSY breaking. 3. Higgsinoless SUSY We are now ready to construct a Lagrangian. For quarks/leptons and gauge superfields, one can simply write down the MSSM Lagrangian in the superspace. Soft SUSY breaking terms can be written down by using delta functions δ 2 (θ − λ) and δ 4 (θ − λ): Ksoft = −δ 4 (θ − λ) · m2Ψ Ψ† Ψ
(13)
and Wsoft = −δ 2 (θ − λ) ·
m1/2 α W Wα . 2
(14)
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These are the same as the spurion method for the soft SUSY breaking terms. The appearance of the Goldstino interactions makes these terms manifestly supersymmetric. One can also add hard breaking terms by using covariant derivatives. We assume that such soft and hard breaking terms are somewhat suppressed because the quarks/leptons and gauge fields are not participating the SUSY breaking dynamics. We introduce the Higgs field as a non-linear field on the x ˜ space, h(˜ x), motivated by an assumption that the SUSY breaking dynamics at the cut-off scale Λ has something to do with the origin of electroweak symmetry breaking. The way to construct interaction terms has been discussed already in the previous subsection. The Yukawa interactions for up-type quarks are 1 2 4 ij c D x) · U Qi . (15) Kup = δ (θ − λ) yu h(˜ 2 (cov) j For down-type quarks and leptons, Kdown = δ 4 (θ − λ) 1 2 1 2 ij † −2gV c ij † −2gV c × yd h(˜ D D x) e D Qi + ye h(˜ x) e E Li . 2 (cov) j 2 (cov) j (16) Here we have used the covariant derivative: 2 ≡ e2gV D2 e−2gV . D(cov)
(17)
It is not necessary to introduce two kinds of Higgs fields for the Yukawa interactions. The A-terms can also be written down by taking x) · (Ujc Qi ) , (18) WA = δ 2 (θ − λ) Aij u h(˜
and
x)† e−2gV (Djc Qi ) + Aij x)† e−2gV (Ejc Li ) . KA = δ 4 (θ − λ) Aij e h(˜ d h(˜
The kinetic term and potential for the Higgs boson can be written as Kkin. = δ 4 (θ − λ) (Da φ(˜ x))† e−2gV Da φ(˜ x) ,
and
4
2 †
x)e Kpot. = δ (θ − λ) −m φ (˜ The derivative in the kinetic term is
−2gV
2 k † −2gV φ(˜ x) − x)e φ(˜ x) . φ (˜ 4
Da ≡ ∇a − igAa + g(∇a λα )Aα ,
(19)
(20)
(21)
(22)
where ¯ ≡ gAa (x, θ, θ)
1 ¯ 2gV De σ ¯a De−2gV , 4
(23)
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and ¯ ≡ e2gV Dα e−2gV . gAα (x, θ, θ)
(24)
Since the quartic coupling of the Higgs boson in Eq. (21) is a free parameter, the Higgs boson mass is not related to the Z-boson mass. It is not a very obvious result that we could write down a Lagrangian with a single Higgs boson with the enlarged gauge invariance. For example, in Ref.35 it has been necessary to introduce an extra Higgs boson, and that is claimed to be a general requirement for constructing a realistic model with non-linear SUSY. 4. Hidden gravity A SUSY transformation in the x ˜ space is realized as a local coordinate transformation in Eq. (4). This local translation allows us to introduce a metric in the x ˜ space having a local transformation law under the global SUSY. This provides a description of a composite spin-two fieldb in the SUSY breaking dynamics analogous to the ρ meson in QCD. We further elaborate on this comparison towards the end of this section. Specifically, we introduce a second “metric” whose transformation under g is ˜σ ∂x ˜ρ ∂ x gρσ (˜ x), (25) µ ∂x ˜ ∂x ˜ν where x˜ is given in Eq. (4). Note that this is a global SUSY transformation, and one should not be confused with the actual general coordinate transformation on the x-space. The space-time is always flat. The deviation of gµν from the Minkowski metric describes the spin-two field. The invariant action having the Fierz-Pauli form42 is x) → gµν (˜ x ) = gµν (˜
S=
where
m2 √ m2 m2 √ µν αβ
f4 det A − P gR(g) − P d x ˜ − gg g Hµα Hνβ − Hµν Hαβ 2 2 8 4
(26)
Hµν = gµν − Gµν
(27)
Gµν = Aµa Aνb ηab .
(28)
and
is a covariant tensor, defined previously. The Hµν field is therefore a SUSY covariant tensor. The scale mP is a mass parameter of O(TeV), unrelated to the −1/2 of Einstein gravity. With four-dimensional Planck mass GN gµν = ηµν + b An
2 hµν , mP
(29)
attempt to describe a spin-two resonance in QCD as a massive graviton can be found in Ref.,38 whose supergravity extension is discussed in Ref.39 A more ambitious attempt to formulate Einstein gravity as a composite of the Goldstino fermions can be found in Ref.40 The appearance of a massive bound-state graviton in open string field theory can be found in Ref.41
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one has Hµν =
2 ¯ − i∂ν λσµ λ ¯ + (µ ↔ ν) + (four-fermion terms) (30) hµν + iλσµ ∂ν λ mP
Note the relative coefficient (of −1) between the two terms appearing in the definition of Hµν is fixed by requiring that the Fierz-Pauli mass term not introduce a tadpole for the graviton. The last term in the action gives a mass m to the spin-two field in a global SUSY invariant way. In the chiral Lagrangian of QCD one can construct SU(2) vector- and axialtype 1-forms jV,A out of the pion fields. A chirally invariant Lagrangian can be constructed only out of jA , since jV transforms inhom*ogeneously under the chiral SU(2). By introducing an SU(2)V vector boson Vµ , the term Tr˜jV ˜jV , with ˜jV = jV − V , is made chirally invariant and can be added to the action. This gives a mass to the vector boson, but no kinetic term; it can be trivially integrated out. The key assumption of23–27 is that this vector boson is dynamical (and describes the ρ vector meson). The action obtained in this way coincides with the spontaneously broken SU(2)V gauge theory in the unitary gauge, which obviously can have a sensible description up to some high energy scale. The analogy of the massive spin-two field as formulated here to the Hidden Local Symmetry (HLS) of QCD can now be drawn more closely, though imprecisely. Here the Maurer-Cartan 1-forms Aµa are analogous to the jV in QCD. By introducing a spin-two field having a local and inhom*ogeneous transformation under the global symmetry, it is then possible to introduce the 1-forms into the action, in the form of the Fierz-Pauli mass term. Further assuming a Ricci scalar term in the action for the spin-two field is equivalent to the physical assumption in HLS that the gauge boson is dynamical. 4.1. Perturbative unitarity A question to be addressed here is whether this new spin-two resonance can be consistently introduced in a weakly coupled regime, in which perturbative calculations make sense at energies of O(m). We first check this by looking at the elastic scattering of two Goldstinos with the same helicity. Then we require that the spin-two field is not strongly coupled at threshold. The s-wave amplitude Mλλ for λλ → λλ receives contributions from both the Volkov-Akulov action and the action for the spin-two field. Specifically, one obtains 1 s2 16π f 4 5 m2P m2 s2 1 m2P m4 s 3m2 s − − 3 − 4 + . (31) ln 1 + 32π f8 16π f 8 s m2
M=0 λλ =
The first term in the RHS comes from the Goldstino self-interactions, while the remaining terms come from graviton exchanges. There is no parameter to control the relative sign of the two contributions. One can see that two O(s2 ) terms in
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s−wave amplitude
0.5
0.25
−0.25
−0.5
1
2
3
4
s f √ Fig. 1. The magnitude of the s-wave amplitude as a function of s/f . Upper and lower curves represent contributions from (−f 4 /2) det A and (Hµν H µν − Hµµ Hνν ) terms, and the middle curve represents the total amplitude. Here, mP = 0.7f and m = 1.2f are used as an example.
Eq. (31) always have an opposite sign, and thus the graviton contribution partially cancels the growth of the amplitude. The magnitude of the s-wave amplitude is √ plotted in Fig. 1 as a function of s/f . The upper and lower curves represent contributions from the first and second row of the RHS of Eq. (31), respectively. The middle curve represents the total amplitude. The parameters mP = 0.7f and m = 1.2f are chosen for illustration. We define the perturbative-unitarity-violation scale E∗ , by the energy where the tree level s-wave amplitude of λλ scattering reaches the value 0.5. In the case of the example depicted in Fig. 1, the pure Goldstino amplitude (i.e., the upper curve) gives E∗ ∼ 2.2f . The contribution of the spin-two particle to this amplitude has the opposite sign (lower curve). This contribution partially cancels the pure Goldstino amplitude, delaying the onset of the strong coupling regime. For the parameter values used in Fig. 1, one finds E∗ ∼ 3.4f . Combined with the cut-off scale of the massive spin-two theory, we show in Fig. 2 the parameter region in which the spin-two resonance can be consistently incorporated in the effective theory.
4.2. Phenomenological signatures We estimate the cross section for the spin-two resonance at the LHC experiments. The cross section for pp → hµν → hh are shown in Fig. 3 for a choice of parameters. The spin-two couplings to the Higgs boson and the gluon are chosen to be O(1). We note that the production cross-section is sensitive to the spin-two coupling δg to gluons (which is taken to be 0.1 in the figure), so that larger rates are possible.
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3
at E=m
m
2
mP /f
non-perturbative
E∗ >
2.5
perturbative
1.5
at E=m
1 (5) Λ∗
0.5 0
√ 2m >
0.5
d
ple
cou gly
on
str
1
1.5
2
2.5
3
m/f √ (5) Fig. 2. The contour of E∗ = m and Λ∗ = 2m in the parameter space of m/f and mP /f . In √ (5) the region to the left of the thick lines, both E∗ > m and Λ∗ > 2m are satisfied.
10
4
10
3
10
2
10
1
10
σ [fb]
(hh mode)
f=mP=m δH = 1
√ s=
-1
1.96
10
14
Te V
Te V
TeV
-2
10
δ g = 0.1
10
-3
10
500
1000
1500 2000 m [GeV]
2500
3000
Fig. 3. The cross section (hh mode) as functions of the graviton mass for 1.96, 10 and 14 TeV center-of-mass energies. Using CTEQ6M parton distribution functions43 and setting mh = 114 GeV. The relation f = mP = m is kept fixed as m is varied.
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5. Summary If SUSY is broken near the TeV scale by strong dynamics then there may be composites that can be accessed at the LHC. It is desirable to have a formalism for writing the effective theory describing the interactions between the matter or gauge fields and the composites, especially in the situation where the matter and gauge fields are not participants of these dynamics. The challenge then in this case is that the matter and gauge fields appear in linearly realized multiplets, whereas the composites do not. We have presented a formulation of non-linearly realized global SUSY in which this can be done. As an application, we consider two scenarios. In both, the Higgs boson is such a composite. No Higgsinos are present in the low-energy theory. We show that it is possible to write down SUSY invariant Yukawa couplings and A terms, despite the presence of only one Higgs boson. Next, we further suppose that the composites include a light spin-two field in addition to the Higgs boson. The construction of the SUSY invariant action is in analogy to the Hidden Local Symmetry of chiral dynamics, where the ρ meson is a massive vector boson of a hidden local SU(2)V symmetry. Here though the hidden symmetry is local Poincar´e. Thus we find that a massive graviton can naturally be incorporated in a theory with an enlarged spacetime symmetry, which in this case is global SUSY. Some phenomenological signatures are discussed. Unlike a generic Kaluza-Klein graviton, the spin-two particle does not couple to the stress-energy tensor. Instead its interactions with matter and gauge fields are constrained only by gauge, Lorentz and non-linear SUSY invariance. Its dominant decay mode is to Goldstinos (invisible), electroweak gauge bosons and the Higgs boson. Search strategies to find boosted Higgs bosons are particularly interesting for this scenario. Vector boson fusion producing the spin-two particle occurs and may be of experimental interest. Rare decays to di-photons also occur but the rate is more model-dependent. Search strategies to find the Standard Model Higgs boson are therefore simultaneously sensitive to finding the spin-two particle. This scenario has the usual low-energy SUSY experimental signatures (without Higgsinos), while in addition possessing signatures of both large44 and warped extra dimensions45 - monojets (ADD) and a spin-two resonance (RS) - even though there is no extra dimension. The discovery of SUSY particles and a single spin-two resonance is not sufficient to claim discovery of an extra (supersymmetric) dimension. It may just be due to four-dimensional strong SUSY breaking dynamics. Acknowledgments RK thanks the organizers of the SCGT 2009 workshop. He especially thanks Prof. Koichi Yamawaki for his hospitality in Nagoya. The work of MG is supported by the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. The work of RK is supported in part by the Grant-in-Aid for Scientific Research 21840006 of JSPS.
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The Latest Status of LHC and the EWSB Physics S. Asai Physics Department, University of Tokyo, Hongo Bunkyo-ku Tokyo 133-033 Japan E-mail: [emailprotected] The latest status of LHC and the performances of ATLAS and CMS detectors are summarized in the first part. Physics potential to solve the origin of the ElectroWeak Symmetry Breaking is summarized in the 2nd part, focusing especially on two major scenarios, (1) the light Higgs boson plus SUSY and (2) the Strong Coupling Gauge Theory. Both ATLAS and CMS detectors have the excellent potential to discover them, and we can perform crucial test on the ElectroWeak Symmetry Breaking. Keywords: LHC, Higgs, SUSY, technicolor.
1. Introduction The most urgent and important topics of the particle physics is to understand the electroweak symmetry breaking and the origin of ”Mass”. These are the main purpose of the Large Hadron Collider (LHC). The LHC accelerator is composed with 1232 units of the 8.4 T superconducting dipole magnet (length is 14.2 m each) and 392 units of the quadruple magnet. They have already be arranged in the 26.6 km circumference tunnel. Protons will be accelerated upto 7 TeV and collide √ each other. Thus the center-of-mass energy( s) of pp-system is 14 TeV. The first √ physics collisions have been observed in 2009 with the lower s’s of 900 GeV and 2.36 TeV. The design luminosity is 1034 cm−2 s−1 , which corresponds to 100 fb−1 per year, will be achieved at 2015. The production cross-sections are expected to be large at LHC for the various high pT and high mass elementary processes as listed in Table 1, since gluon inside partons can contribute remarkably. Furthermore, because the LHC provides the high luminosity of 10–100 fb−1 per year, large numbers of the interesting events will be observed. LHC has an excellent potential to produce high mass particles, for example, the top quark, the Higgs boson, SUSY particles. 2. Status and future plan of LHC 2.1. Incident on 19th September 2008 The LHC has started on 10th September 2008. Protons were injected with the energy of 450GeV and were RF-captured successfully to store in the LHC ring. But there were the bad connections of the splicing in the copper bar between magnet
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Event number
(pb at 14TeV)
(L=10 fb−1 )
1.7 × 105
∼ 109
t¯t
830
jj pT > 200GeV
105
∼ 107
SM Higgs (M=115GeV)
35
∼ 105
∼ 100
∼ 106
W ± → ± ν
Z0 → + −
2.5 × 104
g ˜˜ g (M=500GeV)
∼1
(M=1TeV)
∼ 108 ∼ 109
∼ 104
units. The copper bar becomes the path of intense current (∼10000 A), when the superconducting in the magnet is quenched. Since the bad connections have the resistance of ∼50µΩ, the temperature goes up quickly and the liquid He boils up. Total 6 ton of the liquid He leaks from the pipe and 53 units of the magnets have been destroyed. To repair these magnets and to add the various safety systems for the quench (improving the quench detectors, increasing the safety valves, and making the magnet support stronger) it has taken more than one year. 2.2. Restart at November 2009 On 20th November 2009, the proton beam makes a turn successfully in the LHC rings and are stored in the beam pipe with the RF power. This test have been performed for both proton beams separately, and the LHC is back to the status before the incident. √ On 23rd November, the first collision at s =900 GeV has been observed in the all detectors and the events observed at the ATLAS and CMS detectors are shown in Fig.1. These events are the soft proton-proton collision, called as ”minimum bias”. Since the cross-section of minimum bias at 900 GeV is very huge, σ ∼ 58 mb, large number of the events were observed even with the lower luminosity of 1024 cm2 s−1 . Many soft π’s whose pT less than 2 GeV are emitted in the Minimum bias events, and these are useful to check the performance of the detectors as mentioned in Sec.3. 2.3. Acceleration and collision at
√
s=2.36TeV
On 30th November, both beams are accelerated upto 1.18 TeV. Since the beam just after injected has large emittance, the first step of the acceleration is important to control beam. The beam emittance decrease after acceleration, so this success shows that LHC clears the first critical point.
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Fig. 1.
The collision events at
√
s=900 GeV observed at ATLAS and CMS
√ On 8th December, we have the first collision at s=2.36TeV, which is the highest collision energy in the world. The events observed at the ATLAS and CMS detectors are shown in Fig.2, and these events are multi-jet QCD. The production crosssection and jet-properties measured at both 900 GeV and 2.36 TeV are consistent with the Monte-Calro predictions, still preliminary.
Fig. 2.
The collision events at
√
s=2.36 TeV observed at ATLAS and CMS
2.4. Future plan The plan of LHC after 2010 is summarized here. There is the high risk when the √ LHC is operated at s=10 TeV without the repair of the bad connection of the splicing mentioned in Sec.2.1. Magnet units should be warm in order to repair the connections, and it takes much time more of one year. Thus the LHC schedule √ is determined as listed in Table 2. LHC will be operated at s=7 TeV at both 2010 and 2011, and the expected integrated luminosity (L) is about 1 f b−1 in the √ forthcoming two years. Physics potentials with these s and L are summarized in Sec.4.
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LHC schedule.
Year
ECM
expected Luminosity
2010
7 TeV
200 pb−1
2011 2012
7 TeV ∼ 1f b−1 Long shut down
2013
13-14 TeV
2014
14 TeV
∼ af ewf b−1 ∼ 10f b−1
3. Status and performance of ATLAS/CMS detector Two general-purpose experiments exist, ATLAS and CMS, at LHC. The ATLAS (A Toroidal LHC Apparatus) detector measures 22 m high, 44 m long, and weight 7,000 tons. The characteristics of the ATLAS detector are summarized as follows • Precision inner tracking system is constituted with pixel, strip of silicon and TRT with 2 T solenoidal magnet. Good performance is expected on the B-tagging and the γ-conversion tagging. • Liquid Argon electromagnetic calorimeter has fine granularity for space resolution and longitudinal segmentation for fine angular resolution and particle identifications. It has also good energy resolution of about 1.5% for e/γ with energy of 100 GeV. • Large muon spectrometer with air core toroidal magnet will provide a precise measurement on muon momenta(about 2% for 100 GeV-µ) even in the forward region. The CMS (Compact Muon Solenoid) detector measures 15 m high, 21 m long, and weight 12,500 tons, with the following features • Precise measurement on high pT track is performed with the strong 4 T solenoidal magnet. • PbW 04 crystal electromagnetic calorimeter is dedicated for H0SM → γγ. • High purity identification and precise measurement are expected on µ tracks using the compact muon system. Both detectors are ready to observe the collision events well and the various performance (for example, tracking efficiency, resolution of pT , resolution of the energy deposited on the calorimeters) are checked with the collision data at 900 GeV. Some performance plots are shown in Fig.3; Fig.3(a) shows the reconstructed pair of tracks with the displaced vertex for KS → π + π − , and the distribution of the invariant mass of the reconstructed tracks are shown in Fig.3(b). The clear peak is observed at Ks mass. These are obtained at the ATLAS detector. π 0 → γγ is good process to check the EM calorimeters, and the invariant mass distribution of two γ’s is shown in Fig.3(c). Clear peak is observed at π 0 mass and the good resolution of σ=10 MeV is obtained at the CMS detector, since the P bW 04 crystal scintillators are used in CMS detector. The E T is the vector opposite to the sum of the all
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energy deposited on the calorimeters and muon tracks, and is crucial variable to detect neutrino or to discovery SUSY. The ET resolution is related to the energy sum deposited on the calorimeters, and it is also checked as shown in Fig.3(d). The √ resolution is proportional to 0.5 × ΣET and the data is consistent with the MC simulation.
√ Fig. 3. (a) Event display observed at ATLAS with s=900GeV. The displaced vertex has clearly + − been reconstructed for KS → π π . (b) The invariant mass distribution of π + π − pair (ATLAS) T as a (c) The invariant mass distribution of two photons π 0 → γγ (CMS) (d) The resolution of E function of ΣET
4. EWSB scenario 1: Light Higgs and SUSY √ 4.1. s dependence of the production cross-section σ √ The studies shown in this note are based on s=10 or 14 TeV. In order to extrap√ √ olate these results into s=7 TeV, s-dependences of the production cross-section σ are shown in Figs.4 (a) Higgs (b) SUSY. The production cross-sections at 7 TeV is smaller by factor 3 than σ at 10 TeV, and by factor 5–8 than σ at 14 TeV. The difference becomes larger for the heavier particles.
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√ Fig. 4. The production cross-section as a function s: (a) The light SM Higgs whose mass 120 to 200 GeV (b) The SUSY particles (gluino, squark masses 400-480 GeV)
4.2. The Light SM Higgs boson A discovery of one or several Higgs bosons will give a definite experimental proof of the breaking mechanism of the Electroweak gauge symmetry, and detail studies of the Yukawa couplings between the Higgs boson and various fermions will give insights on the origin of lepton and quark masses. The mass of the SM Higgs boson itself is not theoretically predicted, but it’s upper limit is considered to be about 160 GeV(95%C.L.) from the Electroweak precision measurements and the direct search at the Tevatron. Thee lower limit of the Higgs boson mass is set at 114 GeV(95%C.L.) by direct searches at LEP. The Higgs boson should exist in the narrow mass range of 114–160 GeV, and lighter than 130 GeV if the Supersymmetry exists. The SM Higgs boson, H0SM , is produced at the LHC predominantly via gluongluon fusion(GF) and the second dominant process is vector boson fusion(VBF) ¯ and τ + τ − for the lighter case (130 GeV). Although H0SM decays mainly into bb its decay into γγ, via the one-loop process including top quark or W boson, is suppressed (∼ 2 × 10−3 ), this decay mode is very important at the LHC for this light case. On the other hand, it decays into W+ W− and ZZ with a large branching fraction for the heavier case (140 GeV). 4.2.1. H0SM → γγ in the GF and VBF This channel is promising for the light Higgs boson, whose mass is lighter than 140 GeV, and this mode indicates the spin of the Higgs boson candidate. Although the branching fraction of this decay mode is small and there is a large background
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processes via q¯ q → γγ, the distinctive features of the signal, high pT isolated two photons with a mass peak, allows us to separate the signal from the large irreducible background. The mass resolution of the H0SM → γγ process is expected to be 1.3 GeV(ATLAS) and 0.9GeV(CMS). Sharp peak appears at Higgs boson mass over the smooth distribution of background events as shown in Fig.5. VBF provides additional signatures in which two high pT jets are observed in the forward regions, and the only two photons from the decay of H0SM will be observed in the wide rapidity gap between these jets. The rapidity gap (no jet activity in the central region) is expected because there is no color-connection between two outgoing quarks. These signatures suppress the background contributions significantly as shown in Fig.5(b).
Fig. 5. The invariant mass distribution of γγ (ATLAS). H0SM mass is assumed to be 120 GeV. (a) No jet (mainly for GF process) (b) plus 2 jets (for VBF process) In both figures, red histogram shows the signal. Blue and green show the background contributions for real photo and for fake photon, respectively.
4.2.2. H0SM (→ W+ W− → ± ν± ν) in GF and VBF
The mass range of 130-200 GeV is well covered by the analysis of H0SM → W W → νν, both in GF and VBF process. The transverse mass, MT , is defined as 2E T PT ()(1 − cos φ), in which φ is the azimuthal angle between E T and PT (). Figure 6 shows the MT distribution and a clear Jacobian peak is observed above smooth background distributions. The main background process is W+ W− for the topology without-jet (GF) t¯t for that with two forward jets (VBF). These can be suppressed by using the azimuthal angle correlation between the dileptons because of the following reason. Since the Higgs boson is a spin zero particle, the helicities of the emitted W bosons are opposite. The leptons are then emitted preferably in the same direction due to the 100% parity violation in W decays.
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Fig. 6. The transverse mass distribution of E T and , in which M(H0SM )=170 GeV is assumed. Open blue histogram shows the Higgs signal and the red(W+ W− ) and green(t¯t) histogram show the background contribution.
4.2.3. H0SM → τ + τ − in VBF
H0SM → τ + τ − provides high pT ± , when τ decays leptonically, and it can makes a clear trigger. Momenta carried by ν’s emitted from τ decays can be solved approximately by using the ET information, and the Higgs mass can be reconstructed (collinear approximation). Figures 7 show the mass distributions of the reconstructed tau-pair for leptonic plus hadronic decays and both leptonic decays. The mass resolutions of about 10 GeV can be obtained, and the signal can be separated from the Z-boson production background. The performance of the E T is crucial in this analysis as shown in Fig.3(d). This process provides a direct information on the coupling between the Higgs boson and a fermion, the tau lepton. 4.2.4. H0SM (→ ZZ → + − + − ) in GF
The four-lepton channel (H0SM → ZZ → + − + − ) is very clean and a sharp mass peak is expected as shown in Figs. 8. Mass resolution of the four lepton system is typically 1%. As shown in Figures, the main background process is ZZ production which gives continuous distribution above 200 GeV. This channel is important for H0SM whose mass is heavier than 130 GeV. 4.2.5. Discovery potential H0SM 5σ-discovery potential and 98%C.L. exclusion of H0SM are summarized in Fig.9 as a function of the Higgs mass. Vertical axis shows the integrated luminosity for discovery and exclusion. The results at ATLAS and CMS are combined. Higgs whose mass is lighter than 130 GeV can be discovered or excluded in 2013 or 2011, respectively. Higgs, heavier 130 GeV, becomes more easy. The Crucial test for light Higgs boson scenario can be performed around 2011-2013.
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Fig. 7. The invariant mass distributions of τ + τ − (L=30 fb−1 at ATLAS). (a) One tau decays leptonically and another decays hadronically. (b) both tau decays leptonically. In both figures, the red histogram shows the signal with M(H0SM )=120 GeV, and blue(Z0 → τ + τ − ) and green(t¯t) shows background contributions.
Fig. 8. The invariant mass distribution of with luminosities of 30 fb−1 . M(H0SM )=130(a) and 150(b) GeV are assumed.
H0SM → γγ and τ + τ − channels have good potential in the mass region lighter than 130 GeV. For the heavy mass case, (≥ 130 GeV), decay to ZZ(→ + − + − ) and W+ W− have an excellent performance. After the discovery of the Higgs boson, we can perform to measure the mass and couplings between Higgs and particles with accuracies of about 0.1% and 10%, respectively. 4.3. Supersymmetry If the Higgs boson is light, some mechanism to protect Higgs mass from the radiative correction is necessary. Supersymmetric (SUSY) standard models are most promising extensions of the SM, because the SUSY can naturally explain the weak boson mass scale. Furthermore, the SUSY models provide a natural candidate of
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Fig. 9. The needed integrated luminosity as a function of Higgs mass. Red shows 5σ discover and blue shows 98%C.L. exclusion.
the cold dark matter and they have given a hint of the Grand Unification in which three gauge couplings of the SM are unified at around 2 × 1016 GeV. In SUSY, each elementary particle has a superpartner whose spin differs by 1/2 from that of the particle. Discovery of the SUSY particles should open a new epoch of the fundamental physics, which is another important purpose of the LHC project. 4.3.1. Event topologies of SUSY events g and/or q ˜ ˜ are copiously produced at the LHC. High pT jets are emitted from the decays of g˜ and q ˜. If the R-parity is conserved, each event contains two χ ˜01 ’s in the final state, which is stable, neutral and weakly interacting and escapes from the detection. This χ ˜01 is an excellent candidate of the cold dark matter. The missing ˜01 ’s, plus multiple high transverse energy(ET ), which is carried away by the two χ pT jets is the leading experimental signature of the SUSY at the LHC. Additional ¯ and τ , coming from the decays of χ ˜± leptons, h(→ bb) ˜02 and χ 1 , can also be detected in the some part of event. 4.3.2. Inclusive searches Inclusive searches will be performed with the large E T and high pT multi-jet topology (no-lepton mode). One additional hard lepton (one-lepton mode) or two same-
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sign (SS-dilepton) and opposite-sign leptons(OS-dilepton) can be added to the selections. These four are the promising modes of the SUSY searches, especially no-lepton and one-lepton modes are important for the discovery. The following four SM processes can potentially have ET event topology with jets: • • • •
W± (→ ν) + jets, Z0 (→ ν ν¯, τ + τ − ) + jets, t¯t + jets, ¯ and QCD multi-jets with mis-measurement and semi-leptonic decays of bb c¯c with jets
We require at least three jets with pT ≥ 50 GeV, and pT of the leading jets and E T are required to be larger than 100 GeV. The azimuthal angles between E T and the leading three jets are required to be larger than 0.2 for the no-lepton mode to reduce the QCD background. The excess coming from the SUSY signals can be clearly seen in the E T distributions as shown in Figs.10 for both no-lepton and one-lepton modes. t¯t is the dominant background process for the one-lepton mode, and all of the four processes listed above contribute to no-lepton mode.
Fig. 10. The E distributions of the SUSY signal and background processes with a luminosity of √ T 1 fb−1 and s=14 TeV. (a) No lepton (b) one lepton mode. In both figures, red open histogram show the SUSY signal with (m0 =100GeV,m1/2 =400GeV and tan β=10) The black show the sum of the all SM backgrounds
4.3.3. Discovery potential √ Figures 12 show 5σ-discovery potential in m0 -m1/2 plane for (a) s=10TeV √ L=200 pb−1 and (b) s=14TeV L=30 fb−1 . Figure 12(a) is corresponding data in 2011, and g˜ and q ˜ can be discovered upto ∼ 800 GeV. No lepton mode has slightly good sensitivity than one-lepton mode. With the naive SUGRA assumption, Chargino and the lightest Neutralino (dark matter candidate) can be discovered upto ∼ 250 GeV and 130 GeV, respectively. The interesting region in which ΩDM is consistent with the observation is covered in 2011 run. Figure 12(b) is corresponding data in around 2014, and g˜ and q ˜ can be discovered upto ∼ 2 TeV.
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Fig. 11. 5σ-discovery potential in m0 -m1/2 plane(tan β=10) for (a) √ (b) s=14TeV L=30 fb−1 .
√
s=10TeV L=200 pb−1 and
5. EWSB scenario 2: Strong Coupling Gauge Theory Alternative scenario of the EWSB is the Strong Coupling Gauge Theory (SCGT). The di-boson and dilepton resonance are typical event signals at LHC. 5.1. W+ W− and WZ scatter W+ W− or WZ scatters will be affected in the Strong Coupling Gauge Theory. These scatter processes will be observed in VBF processes as shown in Fig.12(a). Chiral Lagrangian model is the effective theory and two parameters of the anomalous coupling (α4 and α5 ) are useful for VV scatters. Vector(ρ) and scalar(σ) resonance masses can be written: v2 4(α4 − 2α5 )
(1)
3v2 4(7α4 + 11α5 )
(2)
M2ρ =
M2σ =
Lower mass state is scalar and higher mass is vector resonance. Non-resonance processes are also possible. The differential cross-section of the various resonances(nonresonance) are shown in Fig.12(b). The cross-section is expected to small even at √ s=14 TeV. The emitted W and Z bosons decay into quark pair or lepton pair (W± → ν 0 Z → + − ). Tri-lepton topology (WZ → ν+ − ) is the promising channel as shown in Fig.13(a), since the background contribution is expected to be small. But the branching fraction of tri-lepton is also suppressed and the tri-lepton mode is promising only for the lighter case(M ∼500 GeV). For the heavier case (M>800 GeV), dilepton+jets topology (WZ → jj+ − ) becomes promising as shown in Fig.13(b) and (c). The integrated luminosity of 100 fb−1 is necessary for M = 500–800 GeV resonance case (5σ discovery potential.) More luminosity of 200 fb−1 is necessary for M> 1 TeV resonance case and it is difficult at LHC for non-resonance case.
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Fig. 12. (a) VBF process for VV scatter (b) di-Boson invariant mass distributions for various models. Scalar and vector resonances (Mass=500,800,1100 GeV) and non-resonance case.
Fig. 13. The invariant mass distribution (a) MT mass for trilepton and missing. The red show the signal of 500GeV scalar. (b) and (c) The invariant mass of + − qq for 800 GeV scalar and 1.1 TeV vector resonance. Blue shows the signal and the red shows the background contributions of Z0 + jets
5.2. Low mass technicolor model There might be many TC fermions (ND ) to make the walking model, otherwise TC models are strictly conflict with the EW precise measurements at LEP. TC scale √ ΛT C ∼ 250GeV / ND becomes smaller and mass scale of Technicolor particles becomes light.
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Nambu-Gladstone-Boson (πT C ) is mixed with ZL and WL . Technifermion bound state, ρT C , ωT C and aT C will couple with W,Z,γ. They decay into VV or fermion pair and make the narrow resonance. Mass of these bound state is around 400800GeV and the decay modes and J CP are listed in the next table. L=10 or 5 fb−1 is necessary for di-boson decay mode and di-lepton decay mode, respectively. These luminosity is expected around 2013 or 2014. Table 3.
TC boson and decay mode.
JPC
name ρ±0 TC
Spin (1−− , I=1)
0 ωT C a±0 TC
(1−− , I=0) (1++ , I=1)
decay mode → W+ W− , WZ, ZZ → + − → γZ → γW → + −
6. Conclusion √ LHC is back now and we have real data at s=900 GeV and 2.36 TeV. The ATLAS and CMS (also ALICE and LHCb) detectors work well and the good performance √ have been obtained. The LHC will be operated at s=7 TeV for 2010 and 2011 due to the bad connection of splicing. The integrated luminosity is about 1 fb−1 in these two years. There are two major scenarios for the ElectroWeak Symmetry Breaking: (1) light Higgs boson and SUSY (2) diboson and dilepton resonance in Strong Coupling Gauge Theory They can be discovered at LHC before 2013–2015, and we can perform the crucial test.
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Continuum Superpartners from Supersymmetric Unparticles Hsin-Chia Cheng Department of Physics, University of California Davis, California 95616, USA E-mail: [emailprotected] In an exact conformal theory there is no particle. The excitations have continuum spectra and are called “unparticles” by Georgi. We consider supersymmetric extensions of the Standard Model with approximate conformal sectors. The conformal symmetry is softly broken in the infrared which generates a gap. However, the spectrum can still have a continuum above the gap if there is no confinement. Using the AdS/CFT correspondence this can be achieved with a soft wall in the warped extra dimension. When supersymmetry is broken the superpartners of the Standard Model particles may simply be a continuum above gap. The collider signals can be quite different from the standard supersymmetric scenarios and the experimental searches for the continuum superpartners can be very challenging. Keywords: Supersymmetry, unparticle, LHC.
1. Introduction The Large Hadron Collider (LHC) has started running. With its unprecedented center of mass energy, there is a high hope that it will discover new physics which revolutionizes high energy physics. However, LHC is a complicated machine with very high luminosities. In order to find new physics, we need to know what signals the new physics may give rise to, and how to search for them with the enormous Standard Model (SM) backgrounds. A major motivation for new physics at the TeV scale is the hierarchy problem – the stability of the electroweak scale under radiative corrections. Many models for TeV-scale new physics are invented to address this problem, e.g., supersymmetry (SUSY), technicolor models, large and warped extra dimensions, little Higgs models, etc. A lot of collider phenomenology studies have been devoted to these models. However, these models certainly do not cover all possible new physics that may appear at the LHC. There is no theorem that at the TeV scale we should only see the minimal models which just address the hierarchy problem. From the experimental point of view, the collider searches should be signalbased. Even though there are experimental studies targeted for some specific models, generic searches for new particles such as new gauge bosons, new quarks and leptons apply to a wide range of models. As the LHC is starting, the recent model-building efforts have shifted from “solving problems of the Standard Model” to studying
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models which can give rise to “unexpected signals” (e.g., hidden valleys,1 quirks,2 unparticles,3 etc.) irrespective of whether they are related to any particular problem of the Standard Model. It is possible that the TeV-scale physics includes some of these extra sectors in additional to the standard scenario. The presence of the extra sector may obscure the experimental signals of the standard scenario if there is mixing between them, so it is imperative to study these possibilities and their experimental consequences. In this talk, consider such a case with a (super)conformal sector softly broken at the TeV scale and mixed with the supersymmetric Standard Model. As we will see, the mixings can make the superpartners of the SM particles have continuum spectra, and hence make their experimental consequences quite different from the standard SUSY scenario. This presentation is based on the work done in collaboration with Haiying Cai, Anibal Medina, and John Terning.4 2. Unparticles We first give a brief introduction to unparticles. In Ref. [3] Georgi considered the possibility that a hidden conformal field theory (CFT) sector coupled to the SM through higher-dimensional operators, CU ΛdUU V −dU OSM OU , MUk
(1)
where MU is the scale where the interaction is induced, ΛU is the scale where the hidden sector becomes conformal, dUV = k + 4 − d(OSM ), and dU is the scaling dimension of the operator OU . There is no “particle” in a CFT. One can only talk about operators with some scaling dimensions. The spectral densities are continuous and hence Georgi called them “unparticles.” The phase space of an unparticle looks like a fractional number of particles. For a scalar unparticle, d4 P −iP x e |0|OU |P |2 ρ(P 2 ), OU (x)OU (0) = (2π)4 |0|OU |P |2 ρ(P 2 ) = AdU θ(P 0 )θ(P 2 )(P 2 )dU −2 ,
(2)
where AdU is a normalization constant. In the limit dU → 1, it becomes the phase space of a single massless particle, δ(P 2 ). For dU = 1, the spectral density is continuous. Because the unparticle spectral density is continuous down to zero, many phenomenological consequences and constraints are similar to other scenarios with very light (or massless) degrees of freedom, e.g., large extra dimensions proposed by Arkani-Hamed, Dimopoulos, and Dvali.5 A list of things to be considered containsa a There
is a huge literature on phenomenology tests and constraints on unparticles. We only list a few sample references due to the space limit. Please check the citation list of the original Gerogi’s paper for more complete references.
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• • • • • •
invisible decays, e.g., from Z, heavy mesons, quarkonia, neutrinos, . . . ,6,7 missing energy in high energy collisions,8 higher-dimensional operators induced by unparticles,8–10 astrophysics: star (SN1987A) cooling,11,12 cosmology: Big Bang nucleosynthesis (BBN),11,13 long-range forces induced by unparticles.14–16
In particular, unparticles cannot carry SM charges if the spectral density continues down to zero without any gap. Otherwise they would have been copiously produced and observed. However, as pointed by Fox, Rajaraman, and Shirman,17 coupling to the Higgs sector will break the conformal invariance of the unparticles in the infrared (IR), which will induce a gap in general. Above the gap, there are two possibilities: • If the theory confines, then it would produce QCD-like resonances. • If the theory does not confine, one then expect that the spectral density is still continuous above the mass gap. A simple ansatz to describe it is to introduce an IR cutoff m,17,18 ∆(p, m, d) ≡ d4 xeipx 0|T O(x)O† (0)|0 i Ad ∞ dM 2 . = (M 2 − m2 )d−2 2 (3) 2π m2 p − M 2 + i
Such a spectral density arises when a massive particle couples to massless degrees of freedom. For example, quark jets can be considered as unparticles.19
With a mass gap, many low-energy phenomenological constraints which result from new light degrees of freedom no longer apply. In particular, with a mass gap, one can consider the possibility that unparticles carry SM charges18 (if the gap is larger than O(100) GeV), which can be more interesting phenomenologically. This is the scenario that will be discussed in this talk. 3. Unparticles and AdS/CFT A useful tool to study the (large N ) conformal field theory is the AdS/CFT correspondence.20 The metric of a 5D anti de Sitter (AdS5 ) space in conformal coordinates is given by R2 (dx2µ − dz 2 ), (4) z2 where R is the curvature radius and z is the coordinate in the extra dimension. Using the AdS/CFT correspondence, Georgi’s unparticle scenario can be described by the Randall-Sundrum II (RS2)21 setup. The 5-dimensional (5D) bulk fields correspond to the 4-dimensional (4D) CFT operators, and the 5D bulk mass of a bulk field is related to the scaling dimension of the corresponding CFT operator. In RS2, the space is cut off by a UV brane (located at z = zUV ≡ ) but there is no IR ds2 =
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brane which means that the conformal invariance is good down to zero energy. The momentum in the extra dimension of a bulk field is not quantized. From the 4D point of view, the momentum in the extra dimension becomes the mass in 4D, so the 4D mass spectrum is continuous. SM fields are localized on the UV brane. They can interact with the bulk fields (CFT operators) through higherdimensional interactions. Such a representation allows us to perform calculations involving unparticles using the ordinary (5D) field theory.22–24 In this talk we are interested in unparticles carrying SM charges with mass gaps. Some modifications of the RS2 setup are necessary. In particular, SM fields have to propagate in the extra dimension as well and a soft wall is needed to produce the gap in the IR. Before going to these discussions, we first describe how to formulate SUSY in the AdS bulk. 3.1. SUSY in AdS bulk It is convenient to describe a 5D supersymmetric theory using the 4D N = 1 superspace formalism.25,26 The 5D N = 1 SUSY corresponds to N = 2 SUSY in 4D. A 5D N = 1 hypermultiplet contains two chiral multiplets in 4D, Φ = {φ, χ, F }, Φc = {φc , ψ, Fc }. The action of the 5D N = 1 hypermultiplet is given by26 3 R 4 S = d x dz d4 θ [Φ∗ Φ + Φc Φ∗c ] + z 3 R 1 1 R + d2 θ Φc ∂z Φ − ∂z Φc Φ + m Φc Φ + h.c. , (5) z 2 2 z It is convenient to define a dimensionless bulk mass c ≡ mR. It is related to the dimension of the corresponding operator in the 4D CFT picture.27–29 For a lefthanded CFT operator OL (which corresponds to the 5D bulk field Φ), we have ds = 3/2 − c, df = 2 − c for c ≤ 1/2, where ds and df are the scaling dimensions of the scalar and fermion operators respectively. For c < −1/2 (ds > 2), the correlator diverges as we take the UV brane to the AdS boundary ( → 0). A counter term is needed on the UV brane to cancel the divergence, which implies UV sensitivity. On the other hand, for c > 1/2 the CFT becomes a free field theory with ds = 1, df = 3/2. We will focus on the most interesting range −1/2 ≤ c ≤ 1/2 (1 ≤ ds ≤ 2). For the right-handed CFT operator, we have the similar interpretations but with c → −c. In solving the equations of motion (EOMs), we include a z-dependent mass m(z) which represents the soft wall. The EOMs for the fermions and the F -terms are 1 −i¯ σ µ ∂µ χ − ∂z ψ¯ + (m(z)R + 2) ψ¯ = 0, z 1 µ −iσ ∂µ ψ¯ + ∂z χ + (m(z)R − 2) χ = 0. z
(6) (7)
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1 3 − m(z)R φ, Fc∗ = −∂z φ + 2 z 1 ∗ 3 + m(z)R φ . F = ∂z φ∗c − 2 z c
(8) (9)
Using the F -term equations we can find the second order EOM for the scalars: 3 15 1 R µ 2 2 2 φ − (∂z m(z)) φ = 0 .(10) ∂µ ∂ φ − ∂z φ + ∂z φ + m(z) R + m(z)R − 2 z 4 z z We can decompose the 5D field into a product of the 4D field and a profile in the extra dimensions 2 3/2 z z χ(p, z) = χ4 (p) fL (p, z), φ(p, z) = φ4 (p) fL (p, z), (11) zUV zUV 2 3/2 z z fR (p, z), φc (p, z) = φc4 (p) fR (p, z), (12) ψ(p, z) = ψ4 (p) zUV zUV where p ≡ p2 represents the 4D momentum. Then we find the profiles in the odinger-like equations with the potential extra dimension fL , fR satisfy the Schr¨ determined by the mass term. For a constant mass, the potential scales as 1/z 2 for large z, so there is a continuum of solutions starting from zero energy (4D mass). We can get different behaviors if m ∼ z α for large z: • For α < 1, the potential still goes to zero for large z, so we have continuum solutions without gap. • For α > 1, the potential grows without bounds at large z, so we get discrete solutions.30 • For α = 1, the potential approaches a positive constant for large z, so we have a continuum with a gap.22 We are interested in the last possibility so we consider the case m(z) = c+µz. In the UV (small z), m ≈ c, we have the usual CFT interpretation. At large z, m(z) ∝ z, the conformal invariance in broken in the IR and a mass gap is developed. We can easily obtain the zero mode solutions for m(z) = c + µz: fL (0, z) ∼ e−µz z −c ,
fR (0, z) ∼ eµz z c .
(13)
One can see that only one of them has a normalizable zero mode. For definiteness we choose µ > 0, then only the left-handed field has a zero mode, which is identified as the SM fermion. The nonzero modes satisfy the Schr¨ odinger-like equations, µc c(c − 1) ∂2 2 2 fR + p − µ − 2 − (14) fR = 0, ∂z 2 z z2 ∂2 µc c(c + 1) fL + p2 − µ2 − 2 − (15) fL = 0. ∂z 2 z z2
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They have the same form as the radial wave equation of the hydrogen atom, ∂2 M α ( + 1) − u + 2M E + 2 u = 0, (16) ∂r2 r r2
except that c is in general a fractional number instead of an integer. We can immediately see the pattern of the solutions from our knowledge of the hydrogen atom. For cµ > 0 which corresponds to a repulsive Coulomb potential, we have continuum solutions above the gap, p2 > µ2 . For cµ < 0 on the other hand, we get discrete hydrogen-like spectrum below the gap in addition to the continuum above the gap. 3.2. Holographic boundary action One can derive the unparticle propagator by probing the CFT with a boundary source field Φ0c . The holographic boundary action can be obtained by integrating out the bulk using the bulk EOMs with the boundary condition Φc (zUV ) = Φ0c = (φ0c , ψ 0 , Fc0 ),22 0 0∗ ∗ Sholo = − d4 x[φ0∗ (17) c Σφc φc + Fc ΣFc Fc0 + ψ0 Σψ ψ0 ] where
Σφ c =
R zUV
3
p
fL , fR
Σψ =
R zUV
4
pµ σ µ f L , p fR
ΣFc =
R zUV
3
1 fL . p fR
(18)
From the CFT point of view, the right-handed superfield Φ0c is a source of the lefthanded CFT operator which correspond to Φ. Since Fc0 is the source for the scalar component, the propagator for the scalar CFT operator is ∆s (p) ∝ −ΣFc (p), with ΣFc
(19)
cµ 1 2 − p2 √ , + c, 2 µ W − µ2 −p2 2 (µ + µ2 − p2 ) , = · p2 cµ 1 2 2 W − √ 2 2 , 2 − c, 2 µ − p
(20)
µ −p
where W (κ, m, ζ) is the Whittaker function of the second kind. The fermionic and the F -component CFT correlators are simply ∆f = pµ σ µ ∆s and ∆F = p2 ∆s by SUSY relations. We are interested in the conformal limit → 0, so we expand ΣFc for small and focus on the range −1/2 < c < 1/2. After properly rescaling the correlator by a power of to account for the correct dimension of the correlator, we find ∆−1 s
p2 1−2c ≈ − 1 − 2c
cµ ) µ2 −p2
2−1+2c p2 (µ2 − p2 )−1/2+c Γ(1 − 2c)Γ(c + √ cµ ) µ2 −p2
Γ(2c)Γ(1 − c + √
.
(21)
The first term vanishes in the conformal limit for −1/2 < c < 1/2. We can see that there is a pole at p2 = 0 which represents the zero mode, and a branch cut
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1030 1024
m2 p2pole
1018 1012 106 1
0.4
0.2
0.0
0.2
0.4
c Fig. 1. Plot of m2 /p2pole vs c. Notice that as c gets closer to 1/2, for a given value of p2pole , the value of m2 decreases.
for p2 > µ2 which represents the continuum in the propagator. For c < 0, there are addition poles below the gap µ2 which correspond to the hydrogen-like bound states discussed in the previous subsection. 3.3. SUSY breaking Now we introduce SUSY breaking on the UV boundary by a scalar mass term, 3
R 1 4 (22) dz m2 zUV · φ∗ φ + h.c. δ(z − zUV ) . δS = d x 2 zUV The scalar propagator is modified due to the modified boundary conditions, Fc (zUV ) = Fc0 + m2 zUV φ∗ (zUV ),
ψ(zUV ) = ψ 0 ,
φc (zUV ) = φ0c
(23)
One can repeat the analysis of the previous subsection and finds for −1/2 < c < 1/2 in the limit → 0, 2 ∆−1 s (p )
2 1−2c
≈m
p2 1−2c − − 2c − 1
2−1+2c p2 (µ2 − p2 )−1/2+c Γ(1 − 2c)Γ(c + √ Γ(2c)Γ(1−c + √
cµ ) µ2 −p2
cµ ) µ2 −p2
.
(24)
For small m2 , the pole which was at zero mass is shifted to p2pole =
2(2c − 1)m2 (µ)1−2c . (−4c + 21+2c c)Γ(1 − 2c)
(25)
The shift is smaller for smaller (more negative) c (Fig. 1), because the zero mode wave function is localized farther away from the UV brane (fL,0 ∝ e−µz z −c ).
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Fig. 2. The spectra of the scalar and the fermion when we increase the scalar SUSY-breaking mass term on the UV brane.
0.15
Ρ p
1 TeV2
0.10
0.05
0.00 5
10
15
20
25
30
pTeV Fig. 3. Spectral function for four examples of boundary UV SUSY breaking masses m = 2 × 107 GeV (blue curve, dashed), m = 8 × 106 GeV (green curve, solid), m = 2 × 106 GeV (purple curve, dot-dashed) and m = 105 GeV (red curve, dotted). The red and purple curves correspond to zero-mode poles localized at p ≈ 50 GeV (red curve) and p ≈ 950 GeV (purple curve), that haven’t merged with the continuum. On the other hand, the green and blue curves correspond to the cases where the pole has merged into the continuum. In the examples, = 10−19 GeV −1 , µ = 1 TeV and c = 0.3. We can see how the continuum peaks to higher momenta as the SUSY breaking mass m increases, specially as the pole merges into the continuum.
For 0 < c < 1/2, as we increase m2 , the pole eventually merge into the continuum and only a continuum superpartner is left (Fig. 2). The continuum parts of the spectral functions for several choices of m2 are shown in Fig. 3. For −1/2 < c < 0, the discrete poles below the gap move towards the gap as 2 m increase (Fig. 4). However, they do not merge into the continuum for arbitrarily large m2 .
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5
1 pTeV2 c1 s,c
5
0.9970
0.9975
0.9980
0.9985
0.9990
0.9995
1.0000
pTeV
Fig. 4. Inverse correlator for two examples of boundary UV SUSY breaking masses: m = 1013 GeV (blue curve) and m = 2 × 1014 GeV (red curve, dashed). In the examples, = 10−19 GeV −1 , µ = 1 TeV and c = −0.2. We can see how the series of poles shift into the continuum with increasing values of m.
3.4. Gauge fields A 5D N = 1 vector supermultiplet can be decomposed into a 4D N = 1 vector super√ multiplet V = (Aµ , λ1 , D) and a chiral supermultiplet σ = ((Σ + iA5 )/ 2, λ2 , Fσ ). We cannot proceed as before by introducing a bulk mass term for the gauge field because of the gauge invariance. The same effect can be obtained with a dilaton profile Φ = e−2uz /g52 coupling to the gauge kinetic term, which softly breaks the conformal symmetry in the IR.23 The action for the 5D N = 1 vector supermultiplet can be written as SV =
d4 xdz ·
R z
2 1 R (σ + σ † ) 1 √ d2 θWα W α Φ + h.c. + d4 θ ∂ z V − (Φ + Φ† ) . 4 2 z 2
(26)
z We can obtain the bulk action in components after rescaling the fields, A5 → R A5 , R 3/2 R 1/2 λ1 → ( z ) λ1 , and λ2 → i( z ) λ2 . We find a z-dependent bulk gaugino mass 1/2 + uz induced by the dilaton profile. It leads to a continuum with a mass gap as we found for the matter fields, and c is fixed to be equal to 1/2 in this case because the 4D gauge field must have dimension one by gauge invariance. Adding a SUSY-breaking gaugino mass term on the UV brane will lift the gaugino zero mode. For small Majorana gaugino mass m on the UV brane, the zero mode pole is shifted to
p2pole ≈
m2 . (γE + ln(2u))2
(27)
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Fig. 5.
Extened decay chains between the gluino and squark continua.
For large gaugino mass the zero mode will merge into the continuum just as what we saw for the scalar superpartners of matter fields with c > 0.
4. Phenomenology and Conclusions We discussed a novel possibility for the supersymmetric extension of the Standard Model, where there are continuum excitations of the SM particles and their superpartners arising from conformal dynamics. The properties of the superpartners can be quite different in this type of models. The superpartner of a Standard Model particle can be either a discrete mode below a continuum, or the first of a series of discrete modes, or just a continuum. The continuum superpartners will be quite challenging experimentally. It would be difficult to reconstruct any peak or edge because of the additional smearing of the mass by the continuous spectrum. At a high energy collider, if the superpartners are produced well above the threshold of the continuum, there is a possibility of extended decay chains as shown in Fig. 5. These events are expected to have large multiplicities and more spherical shapes as a reflection of the underlying conformal theory.31–34 The experimental searches and verifications of this scenario will be very challenging. It is a topic under current investigation. The LHC will be the only high energy collider in the foreseeable future. We need to be prepared for any surprises and challenges that new physics may present to us at the LHC.
Acknowledgments I would like to thank the organizers for inviting me to give a talk at the SCGT 09 workshop. This work is supported in part by the Department of Energy Grant DE-FG02-91ER40674.
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28. R. Contino and A. Pomarol, JHEP 0411, 058 (2004) [arXiv:hep-th/0406257]. 29. G. Cacciapaglia, G. Marandella and J. Terning, JHEP 0906, 027 (2009) [arXiv:0802.2946 [hep-th]]. 30. A. Karch, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. D 74, 015005 (2006) [arXiv:hep-ph/0602229]. 31. J. Polchinski and M. J. Strassler, JHEP 0305, 012 (2003) [arXiv:hep-th/0209211]; M. J. Strassler, arXiv:0801.0629 [hep-ph]. 32. D. M. Hofman and J. Maldacena, JHEP 0805, 012 (2008) [arXiv:0803.1467 [hep-th]]. 33. Y. Hatta, E. Iancu and A. H. Mueller, JHEP 0805, 037 (2008) [arXiv:0803.2481 [hepth]]. 34. C. Csaki, M. Reece and J. Terning, JHEP 0905, 067 (2009) [arXiv:0811.3001 [hep-ph]].
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Review of Minimal Flavor Constraints for Technicolor∗ Hidenori S. f*ckano† and Francesco Sannino CP3 -Origins,
Campusvej 55, DK-5230 Odense M, Denmark
We analyze the constraints on the the vacuum polarization of the standard model gauge bosons from a minimal set of flavor observables valid for a general class of models of dynamical electroweak symmetry breaking. We will show that the constraints have a strong impact on the self-coupling and masses of the lightest spin-one resonances.
1. Introduction Dynamical electroweak symmetry breaking constitutes one of the best motivated extensions of the standard model (SM) of particle interactions. Walking dynamics for breaking the electroweak symmetry was introduced in [2]. Studies of the dynamics of gauge theories featuring fermions transforming according to higher dimensional representations of the new gauge group has led to several phenomenological possibilities [3, 4] such as (Next) Minimal Walking Technicolor (MWT) [5]. The reader can find in [6] a comprehensive review of the current status of the phase diagram for chiral and nonchiral gauge theories needed to construct sensible extensions of the SM featuring a new strong dynamics. In [7] it was launched a coherent program to investigate different signals of minimal models of technicolor at the Large Hadron Collider experiment at CERN. Whatever is the dynamical extension of the SM it will, in general, modify the vacuum polarizations of the SM gauge bosons. LEP I and II data provided direct constraints on these vacuum polarizations [8, 9]. In this talk, to be specific, we will assume that the vacuum polarizations are saturated by new spin-one states ( technivector meson and techni-axial vector meson) and show that it is possible to provide strong constraints on their self-couplings and masses for a general class of models of dynamical electroweak symmetry breaking. Our results can be readily applied to any extension of the SM featuring new heavy spin-one states. In particular it will severely limit the possibility to have very light spin-one resonances to occur at the LHC even if the underlying gauge theory has vanishing S-parameter.
∗ This
talk is based on [1] and given at 2009 Nagoya Global COE workshop “ Strong Coupling Gauge Theories in LHC Era” (SCGT 09), December 8-11, 2009. † Speaker, E-mail: [emailprotected]
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2. Minimal ∆F = 2 Flavor Corrections from Technicolor Our goal is to compute the minimal contributions, i.e. coming just from the technicolor sector, for processes in which the flavor number F changes by two units, i.e. ∆F = 2. Here we consider F to be either the strange or the bottom number. Besides the intrinsic technicolor corrections to flavor processes one has also the corrections stemming out from extended technicolor models [10] which are directly responsible for providing mass to the SM fermions. In this talk, we are not attempting to provide a full theory of flavor but merely estimate the impact of a new dynamical sector, per se, on well known flavor observables. We will, however, assume that whatever is the correct mechanism behind the generation of the mass of the SM fermions it will lead to SM type Yukawa interactions [11]. This means that we will constrain models of technicolor with extended technicolor interactions entering in the general scheme of minimal flavor violation (MFV) theories [12]. We use the effective Lagrangian framework presented in [5] according to which the relevant interactions of the composite Higgs sector to the SM quarks up and down reads: √ √ 2 mdi ∗ ¯ − 2 mui quark + Vi j · u¯ Ri π dL j − V ji · dRi π uL j + h.c. , (1) Lyukawa = v v where mui , (ui = u, c, t) and mdi , (di = d, s, b) are respectively the up and down quark masses of the ith generation. Vi j is the i, j element of the Cabibbo-KobayashiMaskawa (CKM) matrix. This is our starting point which will allow us to compute the ∆F = 2 processes in Fig. 1 a . After the computation of all amplitudes in Fig. 1,
Fig. 1. Box diagrams for ∆S = 2 annihilation processes. To obtain the ∆B = 2 process, we should simply rename s with b and d with q (q = d, s) in the various diagrams.
we obtain the effective Lagrangian describing these processes as ∆F=2 Leff =−
G2F M2W 4π2
· A(aV , aA ) · Q∆F=2 , A(aV , aA ) ≡
Xh i λi λ j · E(ai , a j , aV , aA ) . (2)
i, j=u,c,t
Here, we have expressed all the quantities by means of the following ratios aα ≡ m2α /M2W , (α = i, j) and av ≡ M2v /M2W , (v = V, A) and mi , (i = u, c, t) indicates the ui mass while MV , MA are respectively the mass of the lightest techni-vector meson and techni-axial vector one. Q∆F=2 and λi represent a Note that the contribution of the scattering process to the invariant amplitude is equal to that of the annihilation one.
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n o {Q∆F=2 , λi } = (¯sL γµ dL )(¯sL γµ dL ) , Vid Vis∗ for K0 − K¯ 0 system and {Q∆F=2 , λi } = n o (b¯ L γµ qL )(b¯ L γµ qL ) , Viq V ∗ for B0q − B¯ 0q system. Moreover, E(ai , a j , aV , aA ) keeps track ib
of the technicolor-modified gauge bosons propagators. Indicating with gEW the weak-coupling constant and g˜ the coupling constant governing the massive spin-one self interactions and by expanding up to the order in O(g4EW / g˜ 4 ) one can rewrite E(ai , a j , aV , aA ) as: E(ai , a j , aV , aA ) = E0 (ai , a j ) +
g2EW g˜ 2
∆E(ai , a j , aV , aA ) .
(3)
The SM contribution is fully contained in E0 which are consistent with the results in [13] and the technicolor one appear first in ∆E. The latter can be divided into a vector and an axial-vector contribution as follows: ∆E(ai , a j , aV , aA ) = h(ai , a j , aV ) + (1 − χ)2 · h(ai , a j , aA ) where the quantity χ was introduced first in [14] and the axialvector decay constant is directly proportional to the quantity (1 − χ)2 . The vector and axial decay constant are: fV2 = M2V / g˜ 2 , fA2 = (1 − χ)2 M2A / g˜ 2 . We also write: A(aV , aA ) = A0 +
g2EW g˜ 2
· ∆A(aV , aA ) .
(4)
Upon taking into account the unitarity of the CKM matrix and setting au → 0 one has A0 = η1 · λ2c · E¯ 0 (ac ) + η2 · λ2t · E¯ 0 (at ) + η3 · 2λc λt · E¯ 0 (ac , at ) and ∆A(aV , aA ) = ¯ c , aV , aA ) + η2 · λ2 · ∆E(a ¯ t , aV , aA ) + η3 · 2λc λt · ∆E(a ¯ c , at , aV , aA ) where η1,2,3 η1 · λ2c · ∆E(a t ¯ ¯ are the QCD corrections to E0 and ∆E. The explicit expressions for the functions E0 , ∆E , h, E¯ and ∆E¯ various expressions can be found in [1]. We recall that the absolute value of the CP-violation parameter in the K0 − K¯ 0 system is given by [15]:
G2F M2W MK × × BK fK2 × [−ImA(aV , aA )] , (5) √ 2 ∆M K exp. 12 2π and the meson mass difference in the Q0 − Q¯ 0 , Q = (K, Bd , Bs ) system is given by (|K |)full =
G2F M2W 0 · fQ2 · MQ × BQ × |A(aV , aA )| , (6) ≡ 2 · hQ¯ 0 | − L∆F=2 |Q i = eff full 6π2 where fQ is the decay constant of the Q-meson and MQ is its mass. BQ is identified with the QCD bag parameter. This bag parameter is an intrinsic QCD contribution and we assume that the technicolor sector does not contribute to the bag parameter b . The experimental values of GF , MW , fQ , MQ , ∆MQ and the bag parameter BQ are [16] GF p= 1.1664 × 10−5 GeV−2 ,pMW = 80.398 GeV, fK = 155.5 MeV, BK = 0.72±0.040, fBd BBd = 225±35 MeV, fBs BBs = 270±45 MeV, MK = 497.61 ± 0.02 MeV, MBd = 5279.5 ± 0.3 MeV, MBs = 5366.3 ± 0.6 MeV.
b This
∆MQ
is a particularly good approximation when the technicolor sector does not have techiquarks charged under ordinary color. The best examples are Minimal Walking Technicolor models.
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It is convenient to define the following quantities: δ ≡
g2EW Im∆A(aV , aA ) · ImA0 g˜ 2
,
δMQ ≡
g2EW ∆A(aV , aA ) · . A0 g˜ 2
(7)
Using these we expressions write = (|K |)SM × (|K |)full and (∆MQ )full as (|K |)full (1 + δ ) , ∆MQ = ∆MQ × 1 + δMQ where (|K |)SM and ∆MQ are the SM full SM SM expressions corresponding to the representations in Eqs.(5,6) with A = A0 and their values are (|K |)SM = (2.08+0.14 ) × 10−3 , (∆MK )SM = (3.55+1.09 ) ns−1 , ∆MBd SM = −0.13 −1.00 +0.19 (0.56−0.16 ) ps−1 , ∆MBs SM = (17.67+6.38 ) ps−1 . The experimental values are [16] −5.40 −3 |K | = (2.229 ± 0.012) × 10 , ∆MK = 5.292 ± 0.0009ns−1 , ∆MBd = 0.507 ± 0.005ps−1 , ∆MBs = 17.77 ± 0.10ps−1 . We are now ready to compare the SM values with the experimental one. We can read off the constrain on δ which is: −2 (68% C.L.) . (8) δ = 7.05+7.93 −7.07 × 10
In order to compare the corrections associate to the kaon mass ∆MK we formally separate the short distance contribution from the long distance one and write ∆MK = (∆MK )SD + (∆MK )LD . Here (∆MK )SD encodes the short distance contribution which must be confronted with the technicolor one Eq.(7) and the long distance contribution, (∆MK )LD , corresponds to the exchange of the light pseudoscalar mesons. It is difficult to pin-point the (∆MK )LD contribution [15, 17] and hence we can only derive very weak constraints from δMK . In fact we simply require that (∆MK )SD = (∆MK )SM |1 + δMK | ≤ (∆MK )exp. . This means that: |1 + δMK | ≤ 2.08
(68% C.L.) .
(9)
On the other hand the short distance contribution dominates the B0q − B¯ 0q mass difference [17] yielding the following constraints: +0.38 |1 + δMBd | = 0.91−0.24
,
|1 + δMBs | = 1.01+0.44 −0.27
(68% C.L.) .
(10)
3. Constraining Models of Dynamical Electroweak Symmetry Breaking We will now use the minimal flavor experimental information to reduce the parameter space of a general class of models of dynamical electroweak symmetry breaking. If the underlying technicolor theory is QCD like we can impose the standard 1st √ 2 Weinberg sum rule (WSR) : fV2 − fA2 = fπ2 = vEW / 2 and 2nd WSR : fV2 M2V − fA2 M2A = 0 with fV and fA the vector and axial decay constants as shown in [5, 14, 18]. Using the explicit expressions of the decay constants in terms of the coupling g˜ and vector masses provided in [5, 14, 18] and imposing the above sum rules we derive: 1/aV = (g2EW S)/(16π) − 1/aA (≥ 0), with the S-parameter [8] reading [5, h i h i 14, 18]: S ≡ 8π fV2 /M2V − fA2 /M2A = 8π/ g˜ 2 1 − (1 − χ)2 . The condition above yields the following additional constraint for g˜ by simply noting that the quantity √ ˜ induced by WSRs and (1 − χ)2 is positive: g˜ < 8π/S. The constraints on (MA , g)
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theoretical constraints are stronger, for a given S, than the ones deriving from flavor experiments and expressed in (8)-(10). This is not surprising given that in an ordinary technicolor theory the spin-one states are very heavy. The situation changes when allowing for a walking behavior. Besides the flavor constraints one has also the ones due to the electroweak precision measurements [19] as well as the unitarity constraint of WL −WL scattering [20]. We will consider all of them. As for the walking technicolor case we reduce the number of independent parameters at the effective Lagrangian level via the 1st WSR : fV2 − fA2 = fπ2 = √ 2 vEW / 2 and the 2nd modified [14] WSR : fV2 M2V − fA2 M2A = a · (16π2 fπ4 )/d(R) where a is a number expected to be positive and O(1) [14] and d(R) is the dimension of the representation of the underlying technifermions as shown in h i [5]. We have 2 2 2 2 −1/2 now MA < 8π fπ /S · (2 − χ)/(1 − χ) = 8π fπ /S · 1 + (1 − g˜ S/(8π)) . ˜ In Fig. 2, we show the allowed region in the (MA , g)-plane after having imposed the minimal flavor constraints due to the experimental values of |K | and ∆MQ ˜ M2A . obtained using Eqs. (8)-(10) together with the theoretical constraints for g, To h obtain Fig. i 2, we used the expressions for A0 and ∆A(aV , aA ) in which aV = 1 − g˜ 2 S/(8π) · aA + 2 g˜ 2 /g2EW . Given that the upper bound for δMK is always larger than the theoretical estimate in the region MA > 200 GeV we conclude that the ∆MK constraint is not yet very severe and hence it is not displayed in Fig. 2. To make the plots we need also the value of the S parameters and hence we analyzed as explicit example minimal walking technicolor models. In the case of MWT, we use for S the naive MWT estimate, i.e. S = 1/(2π) [3] while g˜ is constrained to be g˜ < 12.5. In the case of Next to Minimal Walking Technicolor (NMWT), we use the the naive S is approximately 1/π [3] and the constraint on g˜ yields g˜ < 8.89. ˜ We have plotted the various constraints on the (MA , g)-plane for MWT(NMWT) in the upper and lower left (right) panel of Fig. 2. In the upper (lower) left figure we compare the 68% C.L. (95% C.L.) allowed regions coming from the minimal flavor constraints (the darker region above the dotted line) with the ones from LEP II data (region above the dashed line). It is clear that the flavor constraints are stronger for the 68% C.L. case but are weaker for the 95% C.L. one with respect to the constraints from LEP II data. In the limit MA = MV = M and χ = 0 the effective theory acquires a new symmetry [21]. This new symmetry relates a vector and an axial field and can be shown to work as a custodial symmetry for the S parameter [21]. The only nonzero electroweak parameters are W, Y parameters. It was already noted in [19] that a custodial technicolor model cannot be easily achieved via an underlying walking dynamics and should be interpreted as an independent framework. This is so since custodial technicolor models do not respect the WSRs c . We directly compare in the ˜ coming Fig. 3 the constraints on the custodial technicolor parameter region (M, g) from LEP II and flavor constraints and find a similar trend as for the other cases. c One
can, of course, imagine more complicated vector spectrum leading to such a symmetry.
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˜ Fig. 2. The upper and lower left panels represents the allowed region in the (MA , g)-plane for MWT respectively for the 68% C.L. and 95% C.L.. A similar analysis is shown for NMWT in the right hand upper and lower panels. The region above the straight solid line is forbidden by the condition g˜ < 12.5 for MWT and g˜ < 8.89 for NMWT while the region below the solid curve (on the right corner) is forbidden by theoretical upper bound for MA . In the two upper (lower) plots the dotted lines correspond to the 68% C.L. (95% C.L.) flavor constraints while the dashed lines are the 68% C.L. (95% C.L.) from LEP II data. The flavor constraints come only from K since the ones from ∆MBq are not as strong.
g
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˜ Fig. 3. The left (right) panel represents the allowed region in the (MA , g)-plane for CT respectively for the 68% C.L. (95% C.L.). In the two upper (lower) plots the dotted lines correspond to the 68% C.L. (95% C.L.) flavor constraints while the dashed lines are the 68% C.L. (95% C.L.) from LEP II data. The flavor constraints come only from K since the ones from ∆MBq are not as strong.
4. Summary Flavor constraints are relevant for models of dynamical electroweak symmetry breaking with light spin-one resonances, in fact, any model featuring spin-one
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resonances with the same quantum numbers of the SM gauge bosons will have to be confronted with these flavor constraints. It would be interesting to combine the present analysis with the one presented in [22] in which a low energy effective theory for the ETC sector was introduced. References [1] H. S. f*ckano and F. Sannino, arXiv:0908.2424 [hep-ph]. [2] B. Holdom, Phys. Rev. D 24, 1441 (1981). ; K. Yamawaki, M. Bando and K. i. Matumoto, Phys. Rev. Lett. 56, 1335 (1986). ; T. W. Appelquist, D. Karabali and L. C. R. Wijewardhana, Phys. Rev. Lett. 57, 957 (1986). [3] F. Sannino and K. Tuominen, Phys. Rev. D 71, 051901 (2005). [4] D. D. Dietrich, F. Sannino and K. Tuominen, Phys. Rev. D 72, 055001 (2005); ibid ; D. D. Dietrich, F. Sannino and K. Tuominen, Phys. Rev. D 73, 037701 (2006); D. D. Dietrich and F. Sannino, Phys. Rev. D 75, 085018 (2007) ; T. A. Ryttov and F. Sannino, Phys. Rev. D 76, 105004 (2007) ; N. D. Christensen and R. Shrock, Phys. Lett. B 632, 92 (2006). [5] R. Foadi, M. T. Frandsen, T. A. Ryttov and F. Sannino, Phys. Rev. D 76, 055005 (2007). [6] F. Sannino, arXiv:0911.0931 [hep-ph]; 0804.0182 [hep-ph]. [7] A. Belyaev, R. Foadi, M. T. Frandsen, M. Jarvinen, F. Sannino and A. Pukhov, Phys. Rev. D 79, 035006 (2009). [8] M. E. Peskin and T. Takeuchi, ‘Phys. Rev. D 46, 381 (1992); M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964 (1990). [9] R. Barbieri, A. Pomarol, R. Rattazzi and A. Strumia, Nucl. Phys. B 703, 127 (2004). [10] S. Dimopoulos and L. Susskind, Nucl. Phys. B 155, 237 (1979) ; E. Eichten and K. D. Lane, Phys. Lett. B 90, 125 (1980). [11] R. S. Chivukula and H. Georgi, Phys. Lett. B 188, 99 (1987). [12] G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, Nucl. Phys. B 645, 155 (2002). [13] T. Inami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981) [Erratum-ibid. 65, 1772 (1981)]. [14] T. Appelquist and F. Sannino, Phys. Rev. D 59, 067702 (1999). [15] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996). [16] C. Amsler et al. [Particle Data Group], Phys. Lett. B 667, 1 (2008) and for definitiveness, we use the values of the bag parameters quoted in U. Nierste, arXiv:0904.1869 [hep-ph]. [17] J. F. Donoghue, E. Golowich and B. R. Holstein, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 2, 1 (1992). [18] Z. y. Duan, P. S. Rodrigues da Silva and F. Sannino, Nucl. Phys. B 592, 371 (2001). [19] R. Foadi, M. T. Frandsen and F. Sannino, Phys. Rev. D 77, 097702 (2008). [20] R. Foadi, M. Jarvinen and F. Sannino, Phys. Rev. D 79, 035010 (2009). [21] T. Appelquist, P. S. Rodrigues da Silva and F. Sannino, Phys. Rev. D 60, 116007 (1999). [22] M. Antola, M. Heikinheimo, F. Sannino and K. Tuominen, arXiv:0910.3681 [hep-ph].
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Standard Model and High Energy Lorentz Violation Damiano Anselmi Dipartimento di Fisica “Enrico Fermi”, Universit` a di Pisa Largo Pontecorvo 3, I-56127 Pisa, Italy and INFN, Sezione di Pisa, Pisa, Italy E-mail: [emailprotected] If Lorentz symmetry is violated at high energies, interactions that are usually nonrenormalizable can become renormalizable by weighted power counting. The Standard Model admits a CPT invariant, Lorentz violating extension containing two scalar-two fermion interactions (which can explain neutrino masses) and four fermion interactions (which can explain proton decay). Suppressing the elementary scalar fields, we can use a dynamical symmetry breaking mechanism, in the Nambu–Jona-Lasinio spirit, to generate composite Higgs bosons and masses for fermions and gauge bosons. The low-energy effective action is uniquely determined and predicts relations among parameters of the Standard Model, which allows us to make indirect experimental tests of the high-energy Lorentz violation. Keywords: Standard Model, Lorentz symmetry, Lorentz violation, renormalization.
Lorentz symmetry is a basic ingredient of the Standard Model of particle physics and one of the best tested and more precise symmetries of Nature.1 However, the possibility that it might be violated at very high energies is still open and has inspired several investigations about the new physics that could emerge. In quantum field theory, the violation of Lorentz symmetry at high energies allows us to renormalize otherwise non-renormalizable interactions,2–4 such as two scalar-two fermion vertices and four fermion vertices. Terms with higher space derivatives modify the dispersion relations and generate propagators with improved ultraviolet behaviors. A “weighted” power counting, which assigns different weights to space and time, allows us to prove that the theory is renormalizable and consistent with (perturbative) unitarity, namely that no counterterms with higher time derivatives are generated. The theory remains also local, polynomial and causal. Using these tools, we can formulate a Standard Model extension with the following properties:5 it is CPT invariant, but Lorentz violating at high energies, it is unitary and renormalizable by weighted power counting; it contains the vertex (LH)2 /ΛL , which gives Majorana masses to the neutrinos after symmetry breaking, but no right-handed neutrinos, nor other extra fields; it contains four fermion vertices, which can explain proton decay. The scale ΛL is interpreted as the scale of
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Lorentz violation. Inverse powers of ΛL multiply operators of higher dimension. At energies much smaller than ΛL Lorentz symmetry is recovered if certain parameters are appropriately fine-tuned. If neutrino masses have the origin we claim, then ΛL ∼ 1014 GeV. More generally, we expect that the scale of Lorentz violation is located below the Planck scale. The model has two “weighted” dimensions, which means that at high energies its power counting resembles the one of a two-dimensional quantum field theory. In particular, only the four fermion vertices are strictly renormalizable, while the gauge and Higgs interactions are super-renormalizable. Then at energies Λ L all gauge bosons and the Higgs field become free and decouple, and what remains is a (Lorentz violating) four fermion model in two weighted dimensions. It is then natural to inquire what physical effects are induced, at lower energies, by a dynamical symmetry breaking mechanism, in the Nambu–Jona-Lasinio spirit.6 If we suppress the elementary scalar field, we obtain a model that is candidate to reproduce the observed low energy physics, predict relations among otherwise independent parameters, and possibly predict new physics detectable at LHC.7 We assume that invariance under rotations in preserved. We decompose coordinates xµ as (ˆ xµ , x ¯µ ), where x ˆµ , or simply xˆ, denotes the time component, and µ x ¯ denote the space components. Similarly, we decompose the space time index µ as (ˆ µ, µ ¯), the partial derivative ∂µ as (∂ˆµ , ∂¯µ ), and gauge vectors Aµ as (Aˆµ , A¯µ ). The Lorentz violating theory is renormalizable by weighted power counting 2–4 in − d = 1+3/n “weighted dimensions”, where energy has weight 1 and the space components of momenta have weight 1/n. Scalar propagators have weight −2 and fermion propagators have weight −1. The model The “Standard-Extended Model” of ref. [5] has n = 3 and therefore weighted dimension 2. The lagrangian of its simplest version reads 5
L = LQ +Lkinf +LH +LY −
X 1 Yf g ¯3 g¯2 ¯ F¯ (χ (LH)2 − gD ¯I γ¯χI )+ 2 χχ ¯ χχ− ¯ F , 2 4ΛL Λ ΛL Λ2L I=1 L (1)
where LQ = Lkinf =
1X ¯ G − F G τ G (Υ)F ¯ G , 2FµˆGν¯ η G (Υ)F µ ˆν ¯ µ ¯ ν¯ µ ¯ ν¯ 4 G
3 X 5 X
a,b=1 I=1
bIab ¯ 3 ¯ χb , χ ¯aI i δ ab D /ˆ − 0 2 D / + bIab D / 1 I ΛL
2 ˆ µˆ H|2 − a0 |D ¯ 2D ¯ 2 H|2 − a2 |D ¯ µ¯ H|2 − µ2 |H|2 − λ4 g¯ |H|4 , ¯ µ¯ H|2 − a1 |D LH = | D H Λ4L Λ2L 4
LY = −¯ gΩi H i + h.c.,
Ωi =
3 X
a,b=1
a bj ji ab ¯ ai b ¯ ai `b + Y ab u Y1ab L R 2 ¯R QL ε + Y3 QL dR , (2)
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i, j are SU (2)L indices, χa1 = La = (νLa , `aL ), χa2 = QaL = (uaL , daL ), χa3 = `aR , χa4 = uaR and χa5 = daR . Moreover, ν a = (νe , νµ , ντ ), `a = (e, µ, τ ), ua = (u, c, t) P and da = (d, s, b). The sum G is over the gauge groups SU (3)c , SU (2)L and ¯ ≡ −D ¯ 2 /Λ2 , where U (1)Y , and the last three terms of (1) are symbolic. Finally, Υ L ΛL is the scale of Lorentz violation, and η G , τ G are polynomials of degree 2 and 4, respectively. Gauge anomalies cancel out exactly as in the Standard Model. 5 The “boundary conditions” such that Lorentz invariance is recovered at low energies are that bIab tend to δ ab and a2 , η G and τ G tend to 1 (four such conditions can be 1 trivially fulfilled normalizing the gauge fields and the space coordinates x ¯). Since propagators contain higher powers of momenta the dispersion relations are modified. At n = 3 a typical modified dispersion relation reads
E=
s
m2 + a¯ p2 + b
p¯2 +c Λ2L
p¯2 Λ2L
2
,
where a, b and c are constants and of course m = 0 for gauge fields. The ultraviolet behaviors are improved enough to make the theory renormalizable. Since the weight of a scalar field vanishes in d− = 2 a constant g¯ of weight 1/2 is attached to the scalar legs to ensure renormalizability. The gauge coupling g has weight 1. The weights of all other parameters are determined so that each lagrangian term has weight 2 − (= d ). We have neutrino masses ∼ v 2 /ΛL , v being the Higgs vev, assuming that all other parameters involved in the vertex (LH)2 /ΛL are of order 1. Reasonable estimates of the neutrino masses (a fraction of eV) give ΛL ∼ 1014 GeV. Alternative model An alternative model can be obtained rearranging the weight assignments to simplify the gauge sector. Specifically, we replace η G with unity and τ G with a polynomial of degree 2, which we denote by τ 0G . In a suitable “Feynman” gauge the gauge-field propagator becomes reasonably simple to be used in practical computations.8 The price is a more complicated Higgs sector, because g and g¯ get a lower weight (1/3). The simplest version of the alternative model has lagrangian 5
L = − −
L0Q
+ Lkinf +
L0H
X 1 Yf g¯2 ¯ F¯ (χ (LH)2 − gD ¯I γ¯χI ) + 2 χχ ¯ χχ ¯ + LY − 4ΛL Λ2L ΛL I=1
1 1 g ¯3 3 2 ¯ ¯ 2 χH ¯ + g¯2 χ ¯DχH + g¯χ ¯D F − 2 g¯ gχχ ¯ F¯ H − 2 g¯3 χχH Λ2L ΛL ΛL 1 ¯ 2 F¯ + g 2 F¯ 2 H † H, gD 4 ΛL
where L0Q =
1X ¯ G , 2FµˆGν¯ FµˆGν¯ − Fµ¯Gν¯ τ 0G (Υ)F µ ¯ ν¯ 4 G
(3)
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and L0H = LH − −
(2) (3) i λ4 g¯2 † ¯ g¯2 h (1) † ¯ λ4 g¯2 2 ¯ 2 2 2 |H| | D H| − |H D H| − λ (H D H) + h.c. µ ¯ µ ¯ µ ¯ 4 4Λ2L 4Λ2L 4Λ2L
λ6 g¯4 |H|6 . 36Λ2L
The QED subsector of this model is considered in ref. [8], where the high- and low- energy renormalizations are calculated at one loop and compared. It is also shown that at low energies power-like divergences in ΛL get multiplied by arbitrary (i.e. scheme-dependent) constants, therefore can be removed bypassing the hierarchy problem. The most general lagrangian is (3) plus the extra terms ¯ 2 H 6 , g¯2 D ¯ 4 H 4 , g¯ ¯ 2 F¯ H 4 , g D ¯ 4 F¯ H 2 , g 2 g¯2 F¯ 2 H 4 , g 2 F¯ 4 , g¯6 H 8 , g¯4 D g2D ¯ 2 F¯ 2 H 2 , g 3 F¯ 3 H 2 , g D ¯ 2 F¯ 3 , g¯ ¯ 2 H 2 , g 2 ε¯F˜ F¯ H 2 , ε¯F˜ D ¯ 2 F¯ , g2 D εF˜ D g¯ εF¯ F˜ F¯ ,
ˆD ¯ F¯ , ε¯F¯ D
g¯ gχχ ¯ F¯ H,
¯ 2 H, g¯χχ ¯ D
¯ 2, g¯2 χχ ¯ DH
3 g¯3 χχH ¯ ,
(4)
and those obtained suppressing some fields and/or derivatives, where ε¯ is the εtensor with three space indices. The extra terms (4) can be consistently dropped, because they are not generated back by renormalization. Scalarless model The scalarless Standard-Extended Model7 reads LnoH = L0Q + Lkinf −
5 X Yf g 1 ¯¯ g DF (χ ¯I γ¯ χI ) + 2 χχ ¯ χχ ¯ − 2 F¯ 3 , Λ2L ΛL ΛL
(5)
I=1
and is obtained suppressing the Higgs field in (3). Obviously, the gauge anomalies of (5) still cancel. We see that the simplification is considerable. If we suppress the Higgs field in (1), the only difference is that LQ appears instead of L0Q in (5). We keep the simpler model (5), but our arguments do not depend on this choice. Very-high-energy model At very high energies ( ΛL ) gauge and Higgs fields become free and decouple, because their interactions are super-renormalizable, so all theories (1), (3) and (5) become a four fermion model in two weighted dimensions, with lagrangian L4f =
3 X 5 X
a,b=1 I=1
χ ¯aI i
bIab Yf 0 3 Iab ¯ ¯ δ ∂/ − 2 ∂/ + b1 ∂/ χbI + 2 χχ ¯ χχ. ¯ ΛL ΛL ab ˆ
(6)
We have kept also the terms multiplied by bIab 1 , since they are necessary to recover Lorentz invariance at low energies. The high-energy renormalization of this model is studied in ref. [9].
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Predictivity The virtue of our approach to the dynamical electroweak symmetry breaking is that the high-energy physics of our model is unambiguous, encoded in (6). Since (6), as well as (5), (1) and (3), are renormalizable by weighted power counting, we do not need to consider other sectors of unknown physics beyond them. Thus, our model is predictive, and actually provides a viable renormalizable environment for the Nambu–Jona-Lasinio mechanism. With respect to other approaches to composite Higgs bosons, such as technicolor, or the introduction of extra heavy gauge bosons to renormalize four fermion vertices, it has the advantage of being conceptually more economic. We could worry that the dynamical symmetry breaking might reverberate the Lorentz violation down to low energies. Nevertheless, we have proved that the violation of Lorentz symmetry remains highly suppressed even when the dynamical symmetry breaking takes place. The minimum of the effective potential is Lorentz invariant and no Lorentz violating interactions are drawn down to low energies. We assume exact CPT invariance. While (in local, Hermitian) theories a CPT violation implies also the violation of Lorentz symmetry, the converse is not true, in general, except for special subclasses of terms. Thus, we have to introduce two a priori different energy scales, a scale of Lorentz violation ΛL , and a scale of CPT violation ΛCPT , with ΛCPT > ΛL . Recent bounds on Lorentz violation suggested by the analysis of γ-ray bursts, 10 which claim that the first correction E (7) c(E) ∼ c 1 − ¯ M ¯ > 1.3 · 1018 GeV. In the realm to the velocity of light involves an energy scale M
of local perturbative quantum field theory, a dispersion relation giving (7) must contain odd powers of the energy, therefore it must also violate CPT. Thus, we ¯ rather than are lead to interpret the results of ref. [10] as bounds on ΛCPT = M ΛL . It is conceivable that there exists an energy region ΛL 6 E 6 ΛCPT where Lorentz symmetry is violated but CPT is still conserved. Assuming ΛCPT > MP l , we still expect ΛL to be located below the Planck mass and that the energy region ΛL < E < ΛCPT can span four or five orders of magnitude.
Composite Higgs bosons Adapting an old suggestion due to Nambu,11 Miransky et al.12 and Bardeen et al.13 to our case, we can explore the following scenario. When gauge interactions are switched off, the dynamical symmetry mechanism produces fermion condensates h¯ q qi. The effective potential can be calculated in the large Nc limit and has a Lorentz invariant (local) minimum, which gives masses to the fermions. Massive scalar bound states (composite Higgs bosons) emerge, together with Goldstone bosons. At a second stage, gauge interactions are switched back on, so the Goldstone bosons associated with the breaking of SU (2)L × U (1)Y to U (1)Q are “eaten” by the W ± and Z bosons, which become massive.
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The low-energy effective action is Lorentz invariant and uniquely determined. It looks like the Standard Model, but its parameters are not all independent. Said differently, our mechanism predicts relations among parameters of the Standard Model. The naivest predictions are obtained in the leading order of the large Nc expansion, with gauge interactions switched off, and considering just the top and bottom quarks. In this approach, which has a good 50% of error, the composite Higgs boson has mass ∼ 2mt . Taking into account of the error in our calculation, we anyway conclude that our model favors a very heavy Higgs, say with mass above 180GeV. On the other hand, the relation between mt and the Fermi constant turns out to be in “too-good” agreement with the experimental value. The formula reads Λ2 Nc m2t 1 √ ln L2 . = GF 4π 2 2 mt
(8)
Using our estimated value ΛL = 1014 GeV, we find mt = 171.6GeV. We do not understand why this prediction is so right. Our mechanism can of course take place also in the Higgsed models (1) and (3), if the four fermion vertices are chosen appropriately. There, its effects sum to those of the elementary Higgs doublet.
References 1. V.A. Kosteleck´ y, Ed., Proceedings of the Fourth Meeting on CPT and Lorentz Symmetry, World Scientific, Singapore, 2008; V.A. Kosteleck´ y and N. Russell, Data tables for Lorentz and CTP violation, ibid. p. 308 and arXiv:0801.0287 [hep-ph]. 2. D. Anselmi and M. Halat, Renormalization of Lorentz violating theories, Phys. Rev. D 76 (2007) 125011 and arXiv:0707.2480 [hep-th]. 3. D. Anselmi, Weighted power counting and Lorentz violating gauge theories. I. General properties, Ann. Phys. 324 (2009) 874 and arXiv:0808.3470 [hep-th]. 4. D. Anselmi, Weighted power counting and Lorentz violating gauge theories. II. Classification, Ann. Phys. 324 (2009) 1058 and arXiv:0808.3474 [hep-th]. 5. D. Anselmi, Weighted power counting, neutrino masses and Lorentz violating extensions of the Standard Model, Phys. Rev. D 79 (2009) 025017 and arXiv:0808.3475 [hep-ph]. 6. Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. I, Phys. Rev. 122 (1961) 345. 7. D. Anselmi, Standard Model without elementary scalars and high-energy Lorentz violation, EPJC 65 (2010) 523 and arXiv:0904.1849 [hep-ph]. 8. D. Anselmi and M. Taiuti, Renormalization of high-energy Lorentz violating QED, arXiv:0912.0113 [hep-ph]. 9. D. Anselmi and E. Ciuffoli, Renormalization of high-energy Lorentz violating four fermion models, arXiv:1002.2704 [hep-ph]. 10. The Fermi LAT and Fermi GBM Collaborations, Fermi observations of Highenergy gamma-ray emission from GRB 080916C, Science 323 (1009) 1688 and DOI: 10.1126/science.1169101.
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11. Y. Nambu, in New Frontiers of Physics, proceedings of the XI International Symposium on Elementary Particle Physics, Kazimierz, Poland, 1988, Z. Ajduk, S. Pokorski and A. Trautman, Ed.s, World Scientific, Singapore, 1989. 12. V.A. Miransky, M. Tanabashi and K. Yamawaki, Is the t quark responsible for the mass of W and Z bosons?, Mod. Phys. Lett. A4 (1989) 1043; 13. W.A. Bardeen, C.T. Hill and M. Lindner, Minimal dynamical symmetry breaking of the standard model, Phys. Rev. D 41 (1990) 1647.
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Dynamical Electroweak Symmetry Breaking and Fourth Family Michio Hashimoto Theory Center, IPNS, KEK 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan E-mail: [emailprotected] We propose a dynamical model with a (2 + 1)-structure of composite Higgs doublets: two nearly degenerate composites of the fourth family quarks t and b , Φt ∼ t¯ R (t , b )L and Φb ∼ b¯ R (t , b )L , and a heavier top-Higgs resonance Φt ∼ t¯R (t, b)L . This model naturally describes both the top quark mass and the electroweak symmetry breaking. Also, a dynamical mechanism providing the quark mass hierarchy can be reflected in the model. The properties of these composites are analyzed in detail.
1. Introduction Repetition of the generation structure of quarks and leptons is a great mystery in particle physics. Although three generation models have been widely accepted, the basic principle of the standard model (SM) allows the sequential fourth generation (family).1 Also, the electroweak precision data does not exclude completely existence of the fourth family.2,3 Noticeable is that the LHC has a potential for discovering the fourth family quarks at early stage. If the fourth generation exists, we can naturally consider a scenario that the condensates of the fourth generation quarks t and b dynamically trigger the electroweak symmetry breaking (EWSB):4 The Pagels-Stokar (PS) formula suggests that their contributions to the EWSB should not be small, because the masses of t and b should be heavy, Mt > 311 GeV and Mb > 338 GeV, respectively.5 On the other hand, the role of the top quark is rather subtle, i.e., although the contribution of the top is obviously much larger than the other three generation quarks, b, c and etc., it is estimated around 10-20% of the EWSB scale. Recently, utilizing the dynamics considered in Ref. 6, we introduced a new class of models in which the top quark plays just such a role.7 Its signature is the existence of an additional top-Higgs doublet Φt composed of the quarks and antiquarks of the third family, Φt ∼ t¯R (t, b)L . In the dynamical EWSB scenario with the fourth family, the top-Higgs Φt is heavier than the fourth generation quark composites, Φt ∼ t¯R (t , b )L and Φb ∼ ¯bR (t , b )L . However, in general, Φt is not necessarily ultraheavy and decoupled from the TeV-scale physics. This leads to a model with three composite Higgs doublets.8 We explore such a possibility, based on Refs. 7, 8.
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As for the fourth family leptons, we assume that their masses are around 100 GeV,9 and thus their contributions to the EWSB are smaller than that of the top quark. For the dynamics with very heavy fourth family leptons, and thereby with a lepton condensation, one needs to incorporate more Higgs doublets, say, a five composite Higgs model. Also, the Majorana condensation of the right-handed neutrinos should be reanalyzed. This possibility will be studied elsewhere. 2. Model Based on the dynamical model in Ref. 7, we study a Nambu-Jona-Lasinio (NJL) type model with the third and fourth family quarks:8 L = Lf + Lg + LNJL ,
(1)
where Lg represents the Lagrangian density for the SM gauge bosons, the fermion kinetic term is (i) (i) (i) (i) (i) (i) / L + u ¯R iDu / R + / R, (2) ψ¯L iDψ d¯R iDd Lf ≡ i=3,4
i=3,4
i=3,4
and the NJL interactions are described by (4) (4) (4) (4) (3) (3) LNJL = Gt (ψ¯L tR )(t¯R ψL ) + Gb (ψ¯L bR )(¯bR ψL ) + Gt (ψ¯L tR )(t¯R ψL ) (4) (4) (4) (3) +Gt b (ψ¯ t )(¯b c iτ2 (ψ )c ) + Gt t (ψ¯ t )(t¯R ψ ) L
R
R
L
L
R
L
(3) (4) +Gb t (ψ¯L tR )(¯bR c iτ2 (ψL )c ) + (h.c.). (i)
(3) (i)
(i)
Here ψL denotes the weak doublet quarks of the i-th family, and uR and dR represent the right-handed up- and down-type quarks, respectively. As was shown in Ref. 7, the diagonal parts of the NJL interactions, Gt , Gb and Gt , can be generated from the topcolor interactions.10 Following the dynamical model in Ref. 7, we assume that the coupling constants Gt and Gb are supercritical and mainly responsible for the EWSB, while the four-top coupling Gt is also strong, but subcritical.7 The mixing term Gt t can be generated by a flavor-changing-neutral (FCN) interaction between t and t.7 On the other hand, Gt b may be connected with topcolor instantons.10 In this case, in order to produce the four-fermion type operator, an appropriate dynamical model should be chosen. Although we keep the Gb t term in a general discussion, it will be ignored in the numerical analysis. Since these four-fermion mixing terms provide the off-diagonal mass terms of the composite Higgs fields in low energy, at least two of them are required so as to evade (pseudo) Nambu-Goldstone (NG) bosons. 3. (2 + 1)-Higgs doublets 3.1. Low energy effective model In low energy, the model introduced in the previous section yields an approximate (2 + 1)-structure in the sector of the Higgs quartic couplings. Indeed, in the bubble
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Φt’
Φt’ Φb’
q’L t’R
b’R
t’R
q L’
Φt’ Φ b’ Fig. 1.
Φb’
q’L t’R
b’R
q ’L Φt’
Φb’ Φt’
q’L
b’R q L’
Φb’ Φ t’
Φb’
≡ (t , b ) . Higgs quartic couplings for Φt and Φb . We defined qL L
(4)
approximation, the composite Φt (b ) couples only to ψL ≡ qL = (t , b )L and (3) tR (bR ), while the top-Higgs Φt couples only to ψL ≡ qL = (t, b)L and tR . This leads to such a (2 + 1)-structure. (See Figs. 1 and 2.) Although the electroweak (EW) gauge interactions violate this structure, the breaking effects are suppressed, because the yukawa couplings are much larger than the EW gauge ones. Let us study the low energy effective model. It is convenient to introduce auxiliary fields at the NJL scale. In low energy these composite Higgs fields develop kinetic terms and hence acquire the dynamical degrees of freedom. The Lagrangian of the low energy model is then
L = Lf + Lg + Ls + Ly ,
(4)
Ls = |Dµ Φb |2 + |Dµ Φt |2 + |Dµ Φt |2 − V,
(5)
(4) ˜ b + yt ψ¯(4) tR Φt + yt ψ¯(3) tR Φt + (h.c.), −Ly = yb ψ¯L bR Φ L L
(6)
with
and
where V represents the Higgs potential and Φt ,b ,t are the renormalized composite ˜ b ≡ −iτ2 Φ∗ ). Taking into account the renormalization group (RG) Higgs fields (Φ b improved analysis, we study the following Higgs potential:8 V = V2 + V4 ,
(7)
with V2 = MΦ2 b (Φ†b Φb ) + MΦ2 t (Φ†t Φt ) + MΦ2 t (Φ†t Φt ) +MΦ2 t Φb (Φ†t Φb ) + MΦ2 b Φt (Φ†b Φt ) + MΦ2 t Φt (Φ†t Φt ) + (h.c.), V4 =
λ1 (Φ†b Φb )2
λ3 (Φ†b Φb )(Φ†t Φt )
+ 1 + λ5 (Φ†b Φt )2 + (h.c.) + λt (Φ†t Φt )2 . 2
+
λ2 (Φ†t Φt )2
+
(8)
λ4 |Φ†b Φt |2 (9)
The Higgs mass terms are connected with the inverse of the four-fermion couplings. While MΦ2 b and MΦ2 t are negative, the mass square MΦ2 t is positive, which
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Φt
Φt
qL tR
tR qL
Φt Fig. 2.
Φt
Higgs quartic coupling for Φt . We defined qL ≡ (t, b)L .
reflects a subcritical dynamics of the t quark. Note that the top-Higgs Φt acquires a vacuum expectation value (VEV) due to its mixing with Φt . On the other hand, the quartic couplings λ1–5,t are induced in low energy as schematically shown in Figs. 1 and 2, and hence these values are dynamically determined. The structure of the mass term part V2 is general. On the other hand, the V4 part is presented as the sum of the potential for the two Higgs doublets Φt and Φb , and that for the one doublet Φt , i.e., it reflects the (2 + 1)-structure of the present model. In passing, when we consider general Higgs quartic couplings for the three Higgs, there appear 45 real parameters.8 3.2. Mass spectra of the quarks and the Higgs bosons Let us analyze the mass spectra of the quarks and the Higgs bosons. Note that the number of the physical Higgs bosons in our model are three for the CP even Higgs bosons (H1 , H2 and H3 with the masses MH1 ≤ MH2 ≤ MH3 ), two for the CP odd Higgs (A1 and A2 with the masses MA1 ≤ MA2 ), and four for the charged ones (H1± and H2± with the masses MH ± ≤ MH ± ). It turns out that 1 2 the heavy Higgs bosons, H2± , A2 , and H3 , consist mainly of the components of the top-Higgs Φt . The VEV’s vt ,b ,t for the Higgs fields Φt ,b ,t are approximately determined by 1 2 (10) λ2 + (λ3 + λ4 + λ5 ) cot β4 vt2 −MΦ2 t − MΦ2 t Φb cot β4 , 2 1 (11) λ1 + (λ3 + λ4 + λ5 ) tan2 β4 vb2 −MΦ2 b − MΦ2 t Φb tan β4 , 2 −MΦ2 t Φt −MΦ2 b Φt v + vb , (12) vt t MΦ2 t MΦ2 t where we defined the ratios of the VEV’s by tan β4 ≡ vt /vb . Note that the relation v 2 = vb2 + vt2 + vt2 holds, where v 246 GeV.
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Mt’(GeV)
400
tanβ4=1.0
380
340
400
tanβ4=1.0 tanβ4=1.1
360
380 tanβ4=0.9
tanβ4=1.1
2
360 340
tanβ4=0.9
320 300
420
Mb’(GeV)
320
2.2 2.4 2.6 2.8
3
3.2 3.4 3.6 2
2.2 2.4 2.6 2.8
Λ(4) (TeV)
3
3.2 3.4 3.6
300
Λ(4) (TeV)
Fig. 3. Mt and Mb . The bold and dashed curves are for Λ(3) /Λ(4) = 1, 2, respectively. The dotted lines correspond to the lower bounds for the masses of t and b at 95% C.L., Mt > 311 GeV and Mb > 338 GeV, respectively.
On the other hand, the masses of the CP odd and charged Higgs bosons are approximately given by MA2 1 −2MΦ2 t Φb (1 − tan2 β34 ),
(13)
MA2 2 MΦ2 t (1 + 2 tan2 β34 ) + MA2 1 tan2 β34 ,
(14)
2 2 2 2 2 MH ± ≈ MA + 2(mt + mb )(1 − tan β34 ), 1
(15)
2 2 2 2 2 MH ± ≈ MA + 2(mt + mb ) tan β34 , 2
(16)
1
2
where we took tan β4 = 1 and defined tan β34 ≡ vt / vt2 + vb2 . The mass formulae for H1,2,3 are quite complicated because of the 3 × 3 matrices. In order to calculate the mass spectra more precisely, we employ the RGE’s with the compositeness conditions:11 yt2 (µ = Λ(4) ) = ∞,
yb2 (µ = Λ(4) ) = ∞,
yt2 (µ = Λ(3) ) = ∞,
for the yukawa couplings, and λ1 λ2 λ3 λ4 = 4 = 2 2 = 2 2 = 0, yb4 µ=Λ(4) yt µ=Λ(4) yb yt µ=Λ(4) yb yt µ=Λ(4)
(17)
λt = 0, (18) yt4 µ=Λ(3)
for the Higgs quartic couplings, where Λ(4) and Λ(3) denote the composite scales for the fourth generation quarks and the top, respectively. The RGE’s are given in Ref. 8. For consistency with the (2 + 1)-Higgs structure, we ignore the one-loop effects of the EW interactions. As for the Higgs loop effects, although they are of the 1/N -subleading order, they are numerically relevant and hence incorporated. Notice that the initial NJL model contains six four-fermion couplings: The EWSB scale and the pole (MS) mass of the top quark are fixed to v = 246 GeV and Mt = 171.2 GeV (mt = 161.8 GeV), respectively. We further convert the NJL
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(4)
Higgs masses (GeV)
Λ = 2 TeV 900 800 700
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MA2(input)
900 800
MH2±
MH2 MH1± MH1
300
700 600 500 400 300
100 150 200 250 300 350 400 450 500 550 100 150 200 250 300 350 400 450 500 550 MA1 (GeV) MA1 (GeV)
Fig. 4. Mass spectrum of the Higgs bosons for Λ(4) = 2, 3 TeV . We took Λ(3) /Λ(4) = 1.5 and tan β4 = 1. MA2 = 800 GeV is the input.
couplings into more physical quantities, MA1 , MA2 and tan β4 . As for MΦ2 b Φt , we fix MΦ2 b Φt = 0. Numerically, it is consistent with Gb t ≈ 0. For the numerical calculations, we use the QCD coupling constant α3 (MZ ) = 0.1176.9 The results are illustrated in Figs. 3 and 4. The masses of t and b are essentially determined by the value of Λ(4) , where we converted the MS-masses mt and mb to the on-shell ones, Mt (b ) = mt (b ) [1 + 4αs /(3π)]. As is seen in Fig. 3, their dependence on Λ(3) /Λ(4) (= 1–2) is mild. When we vary tan β4 in the interval 0.9–1.1, the variations of Mt and Mb are up to 10% (see Fig. 3). The Higgs masses are relatively sensitive to the value of Λ(4) (see Fig. 4), while their sensitivity to Λ(3) /Λ(4) (= 1–2) is low. The Higgs mass dependence on tan β4 is also mild, at most 5% for tan β4 = 0.9–1.1, and Λ(4) = 2–10 TeV. Since at the compositeness scale the yukawa couplings go to infinity, there could in principle be uncontrollable nonperturbative effects. By relaxing the compositeness conditions, we estimated such “nonperturbative” effects around 10 %. Since the loop effects of the EW interactions are expected to be much smaller, the uncertainties of the “nonperturbative” effects are dominant. The 2σ-bound of Rb yields MA2 ≥ 0.70, 0.58, 0.50 TeV for Λ(4) = 2, 5, 10 TeV. Following the (S, T ) analysis a la LEP EWWG, we found that our model is within the 95% C.L. contour of the (S, T ) constraint, when the fourth family lepton mass difference is Mτ − Mν ∼ 150 GeV.3 We can introduce the CKM structure in our model.7,8 Since the mixing between the fourth family and the others is suppressed, |Vt d | ∼ |Vus |mc /mt ∼ O(10−3 ) and ¯0 |Vt s | ∼ |Vt b | ∼ mc /mt ∼ O(10−2 ), the contributions of the t -loop to the B 0 –B ¯ mixing, b → sγ and Z → bb are negligible. Note also that the effects of the charged Higgs bosons are suppressed, because their masses are relatively heavy. As for the tree FCNC and FCCC, they are highly suppressed in the first and second families, because of the assumption that the top-Higgs is responsible for the top mass and does not couple to the other quarks, in the spirit of the (2 + 1)-Higgs structure.8
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4. Summary We have studied the (2 + 1) composite Higgs doublet model. It describes rather naturally both the top quark mass and the EWSB. We can incorporate the dynamical mechanism for the quark mass hierarchy and the CKM structure into the model.7,8 It would be interesting to embed the present model into an extra dimensional one.12 The signature of the model is clear, i.e., as shown in Fig. 4, the masses of the four resonances are nearly degenerate and also the heavier top-Higgs bosons appear. A noticeable feature is that due to the t and b contributions, the gluon fusion production of H1 is considerably enhanced. For example, for Λ(4) = 3 TeV, Λ(3) /Λ(4) = 1.5, tan β4 = 1, MA1 = 0.50 TeV, and MA2 = 0.80 TeV, we obtain Mt = Mb = 0.33 TeV and MH1 = 0.49 TeV. In this case, σgg→H1 Br(H1 → ZZ) is enhanced by 5.1, where the relative H1 ZZ and H1 tt¯ couplings to the SM values are 0.86 and 2.0, respectively. Similarly, the CP odd Higgs production via the gluon fusion process should be enhanced. Also, the multiple Higgs bosons can be observed as tt¯ resonances at the LHC. Detailed analysis will be performed elsewhere. Acknowledgments The research of M.H. was supported by the Grant-in-Aid for Science Research, Ministry of Education, Culture, Sports, Science and Technology, Japan, No. 16081211. References 1. 2. 3. 4.
5. 6. 7. 8. 9. 10. 11. 12.
P. H. Frampton, P. Q. Hung and M. Sher, Phys. Rept. 330, 263 (2000). G.D.Kribs, T.Plehn, M.Spannowsky and T.M.P.Tait, Phys. Rev. D 76, 075016 (2007). M. Hashimoto, arXiv:1001.4335 [hep-ph]. B. Holdom, Phys. Rev. Lett. 57, 2496 (1986) [Erratum-ibid. 58, 177 (1987)]; Phys. Rev. D 54, 721 (1996); JHEP 0608, 076 (2006); C. T. Hill, M. A. Luty and E. A. Paschos, Phys. Rev. D 43, 3011 (1991). A. Lister [CDF Collaboration], arXiv:0810.3349 [hep-ex]; T. Aaltonen et al. [The CDF Collaboration], arXiv:0912.1057 [hep-ex]. R. R. Mendel and V. A. Miransky, Phys. Lett. B 268, 384 (1991); V. A. Miransky, Phys. Rev. Lett. 69, 1022 (1992). M. Hashimoto and V. A. Miransky, Phys. Rev. D 80, 013004 (2009). M. Hashimoto and V. A. Miransky, arXiv:0912.4453 [hep-ph], to appear in PRD. C. Amsler et al. [Particle Data Group], Phys. Lett. B 667, 1 (2008). C. T. Hill and E. H. Simmons, Phys. Rept. 381, 235 (2003) [Err. ibid. 390, 553 (2004)]. W. A. Bardeen, C. T. Hill and M. Lindner, Phys. Rev. D 41, 1647 (1990). M. Hashimoto, M. Tanabashi and K. Yamawaki, Phys. Rev. D 64, 056003 (2001); ibid. 69, 076004 (2004); V. Gusynin, M. Hashimoto, M. Tanabashi and K. Yamawaki, ibid 65, 116008 (2002); M. Hashimoto and D. K. Hong, ibid. 71, 056004 (2005).
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Holmorphic Supersymmetric Nambu–Jona-Lasino Model and Dynamical Electroweak Symmetry Breaking∗ Dong-Won Jung1 , Otto C. W. Kong1 and Jae Sik Lee2 1 Department
of Physics and Center for Mathematics and Theoretical Physics National Central University, Chung-li, 32054, Taiwan 2 Physics Division, National Center for Theoretical Sciences, Hsinchu, 300, Taiwan We analyze the model of the dynamical electroweak symmetry breaking based on the supersymmetrized Nambu–Jona-Lasinio model with holomorphic dimension-five operators. The Minimal Supersymmetric Standard Model is derived as the low energy effective theory with both Higgs superfields as composites. A renormalization group analysis is performed, including the prediction of the Higgs mass range. Keywords: Dynamical symmetry breaking, supersymmetry, Nambu–Jona-Lasinio model.
1. Introduction Dynamical symmetry breaking has been a fascinating idea for symmetry breaking. In 1961, Nambu and Jona-Lasinio (NJL)2 introduced the dimension-six four-fermion operators for the chiral symmetry breaking. They showed that bi-fermion condensate formed a scalar composite, and its non-trivial vacuum signified chiral symmetry breaking. In the setup of Nambu and Jona-Lasinio, large coupling is required to make the fermions condensate, which is interpreted as the large Yukawa coupling. With this philosophy, top quark condensate models were proposed with the Higgs as an auxiliary field.3–5 These models predicted the larger top quark mass than the recent observation. Some extenstions were also suggested, for instance, including bottom quark condensate6 and one more fundamental scalar.7 Supersymmetrization of this idea started from the imbedment of the dimensionsix four-fermion operator.8–11 In the purely supersymmetric case, non-trivial vacuum is not possible because of the cancelations forced by supersymmetry,so the introduction of the soft supersymmetry breaking terms is inevitable. In the realistic analysis, infrared quasi fixed point nature of the renormalization group equation (RGE) was used, but the models failed to fit the recent observation of top quark and bottom quark masses. ∗ Based
on the work by D.W. Jung et. al.1
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So we propose a new scheme for the supersymmetric Nambu–Jona-Lasinio (SNJL) models.1 Instead of dimension-six operators, we introduce the holomorphic dimension-five operators in the superpotential. In this case, scalar superpartners form condensates unlike the previous approaches. Moreover, it reproduces the minimal supersymmetric standard model (MSSM) as an effective low energy theory with both top and bottom quark Yukawa couplings. We provide the RGE analysis to show the phenomenological viabilities including Higgs mass spectrum. 2. Nambu–Jona-Lasino (NJL) Model In this section, we provide the brief review of the NJL model follwoing the Ref. [9]. We start from the Lagrangian with the dimension-six four-fermion operator, L = i∂m ψ+ σ m ψ + + i∂m ψ− σ m ψ − + g 2 ψ+ ψ− ψ + ψ − .
(1)
With the auxiliary field φ, this can be rewritten L → L − (φ − gψ + ψ − )(φ − gψ+ ψ− ) m
m
(2) ∗
= i∂m ψ+ σ ψ + + i∂m ψ− σ ψ − + φ φ + gφψ+ ψ− + gφψ + ψ − ,
where φ = gψ+ ψ − from the equartion of motion. From the one-loop effective potential for φ and v = 16π 2 ΛV4 , the symmetry breaking condition is 1 ∂v 2g 2 1 − 1 + η ln + 1 = 0, (3) = φ ∂φ∗ Λ2 α η 2
where η = Λg 2 φ∗ φ, α = solution for η,
g2 Λ2 8π 2 .
In order for the above equation to have the non-zero g 2 Λ2 > 1, 8π 2
(4)
should be satisfied. calculating the twoBy expanding around the vacuum expectation value φ0 and point function, we can obtain the effective action with φ(x) = 12 (σ(x) + iπ(x)), 1 1 Γ = Z d4 x σ(x) − 4m2 σ(x) + π(x)π(x) , (5) 2 2
g2 Λ2 where Z = 16π ln m . So if the coupling constant is large enough, the 2 2 + O (1) chiral symmetry is broken by the condensation of fermions and the system can be described as effective Yukawa-like theory in the low energy. Scalar particles with masses 2m and zero appear, which can be interptreted as Higgs particle and Goldstone boson each. 3. Supersymmetric NJL (SNJL) Model The supersymmetrization of NJL model was started in the 1980s. Let’s first look into the simplest toy setup briefly. The basic model Lagrangian with chiral
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superfields is L=
¯ + Φ+ + Φ ¯ − Φ− + d4 θ Φ
¯ +Φ ¯ − Φ+ Φ− . d4 θ g 2 Φ
(6)
In componet field, one can check that it contains dimension-six operator, Lψ = iψ¯+ σ µ ∂µ ψ+ + iψ¯− σ µ ∂µ ψ− + g 2 ψ¯+ ψ¯− ψ+ ψ− .
(7)
To apply auxilary field technique as in the previous section, two chiral superfields are necessary. The resulting Lagrangian is ¯ + Φ+ + Φ ¯ − Φ− )(1 − m2 θ2 θ¯2 ) + Φ ¯ 1 Φ1 L= d4 θ (Φ + d2 θ [ µΦ2 (Φ1 + gΦ+ Φ− ) ] + h.c. (8)
The equation of motion for Φ2 gives Φ1 as a composite, Φ1 = −g Φ+ Φ− , which ¯ +Φ ¯ − Φ+ Φ− Note that soft supersymmetry breaking terms are ¯ 1 Φ1 = g 2 Φ yields Φ important, since the potential is zero if the supersymmetry is exact, which means no non-trivial vacuum. In 1991, Carena et. al.11 provided the most realistic model based on this idea. They used the action ΓΛ =
Γ Y M 2VT ¯ ¯ c e−2VB B c 1 − ∆2 θθ¯2 + dV Qe Q + T c e−2VT T¯ c + B 2 ¯2 + dV H¯1 e2VH1 H1 1 − MH θθ − dS m0 H1 H2 1 + B0 θ2 − gT H2 QT c 1 + A0 θ2 + h.c.
(9)
Even though there is no kinetic term for H2 now, it is generated from the one-loop, ZH2 dV H¯2 e2VH2 H2 1 + A0 θ2 + A0 θ¯2 + 2∆2 + A20 θ2 θ¯2 , (10) 2 gT Nc Λ2 2 where ZH2 = 16π / ZH2 , we obtain 2 ln µ2 . So after rescaling H2 → H2 1 − A0 θ the action which looks almost like the MSSM,
ΓΛ =
Γ Y M 2VT ¯ + dV Qe QT c e−2VT T¯ c + B c e−2VB 1 − ∆2 θθ¯2 2 ¯2 + dV H¯1 e2VH1 H1 1 − MH θθ − dS mH1 H2 1 + B0 θ2 − hT H2 QT c 1 + A0 θ2 + h.c. + dV H¯2 e2VH2 H2 1 + 2∆2 θ2 θ¯2 ,
where m = m0 / ZH2 , hT = gT / ZH2 .
(11)
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Starting from the above Lagrangian they performed the RGE analysis. In the RGE language, symmetry breaking or condensation condition translate into the boudary conditions at the compositenss scale Λ; hT fixed. (12) m Though the Lagrangian looks almost like the MSSM, it lacks the bottom quark Yukawa interaction in reality. In addtion, only very small tan β is allowed generally, which is almost excluded by the LEP II expriment. m, hT → ∞ while keeping
4. Holomorphic SNJL (HSNJL) Model So we consier an alternative for the SNJL, which utilizes dimension-six four-fermion operators. The holmorphic dimension-five operators in the superpotential is examined: G 4 2 ¯ ¯ Φ+ Φ− Φ+ Φ− + h.c. L = d θ Φ+ Φ+ + Φ− Φ− − d θ (13) 2
With the auxiliary field Φ0 , 4 ¯ + Φ+ + Φ ¯ − Φ− )(1 − m2 θ2 θ¯2 ) L= d θ (Φ 1 √ √ √ G √ + d2 θ ( µΦ0 + GΦ+ Φ− )( µΦ0 + GΦ+ Φ− ) − Φ+ Φ− Φ+ Φ− + h.c. 2 4 2 ¯ + Φ+ + Φ ¯ − Φ− )(1 − m2 θ2 θ¯2 ) = d θ (Φ µ
+ d2 θ (14) Φ20 + µGΦ0 Φ+ Φ− + h.c.. 2 So the superpotential reduces to µ W = − Φ20 − yΦ0 Φ+ Φ− , (15) 2
√ with y = µG. Equation of motion for Φ0 yields Φ0 = − G/µ Φ+ Φ− i.e. Φ0 as composite of two chiral superfields as the case of Φ1 in the dimension-six model. Coleman-Weinberg effective potential can be calculated with the inclusion of the soft supersymmetry breaking terms, W → W (1 + ζ), ζ = Bθ2 , Vef f = Vtree + V1−loop ,
(16)
with Mf = |yφ0 |2 , Ms = |yφ0 |2 + m2 ± |Byφ0 |. Tree-level potential is Bµ 2 φ + c.c.. (17) 2 0 One-loop effective potential can be computed easily, but we drop its concrete form here because of its complexity. Due to the supersymmetry, the potential is zero when the soft supersymmetry breaking terms are absent and one-loop effective potential goes to zero as φ0 becomes large. Unfortunately, this potential is not bounded from below, so the symmetry breaking cannot be achieved. In the case of gauged model, there are always effective D-terms which produce the quartic terms of the Higgs Vtree = −
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Ms = 10 TeV Ms = 1 TeV
70
Ms = 200 GeV 65
tanβ
60 55 50 45 40 35 4 10
106
108
1010 Λ [GeV]
1012
1014
1016
Fig. 1. We have√included a SUSY threshold correction b of value −0.01 in the running of yb with [(1 + b tanβ) = 2mb /(yb v cosβ)].
fields. Those can make the potential bounded. In this way, we can construct the realistic model. Let’s construct the realistic model. Just for the third generation, W = G εαβ Q3αa U3c a Q3βb D3c b (1 + Aθ2 ).
(18)
Two auxilaiary Higgs superfields are introduced, then W = −µ(Hd − λt Q3 U3c )(Hu − λb Q3 D3c )(1 + Aθ2 )
= (−µHd Hu + yt Q3 Hu U3c + yb Hd Q3 D3c )(1 + Aθ2 ),
(19)
where µλt = yt , µλb = yb , µλt λb = G. Equation of motion for Hu yields Hd = λt Q3 U3c while that for Hd yields Hu = λb Q3 D3c . It is exaltly same superpotential with the MSSM , with the boundary conditions for Yukawa couplings and µ at the compositeness scale Λ, yt yb = G. (20) yt , yb , µ → ∞, while keeping µ Since the effective Lagragian at low energy is exactly the MSSM, we can analyse it with the MSSM RGEs. 5. Renormalization Group Equation Analysis The RGE anaysis is done with the boundary conditions of Eq. (29). The phenomenological constraints are mt = 171.3 ± 1.6GeV,
mb =
4.20+0.17 −0.07 GeV.
(21) (22)
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5 tanβ = 57.8 Λb = 104 GeV
4.5 4
yb
tanβ = 42.8 Λb = 1010 GeV
yt
3.5 yt, yb
3 2.5 2 1.5 1 0.5 3 10
Fig. 2.
104
105
106
107 µ [GeV]
108
109
1010
1011
Illustration of yb and yt runnings for a couple of cases.
For the numerical analysis we introduce the Λt and Λb for the compositen scale where yt, b2 /4π = 1, respectively. We look for admissible cases of Λt and Λb values with the now precisely determined top and bottom masses implemented. Note that for large value of tanβ, the bottom Yukawa yb is big. We find that there is always a window of tanβ value giving admissible solution, for any Λb we take (with supersymmetry scale MS from 200 GeV to 10 TeV), as given in Fig. 1. In Fig. 2, we show the yt and yb runnings for a couple of typical cases. Note that yb plays the more important role reaching the perturbative limit before yt . We have Λt ∼ 3Λb , so we can say that compositeness scale is roughly arund Λb,t . It is also important to note √ that we have included a supersymmetry threshold correction
b [cf. (1 + b tanβ) = 2mb /(yb v cosβ)] of value −0.01 in the running of yb . The negative value is important. For b > 0, solution is possble only with uncomfortably large tanβ. Even though its exact value is dependent on the details of the particle spectrums, we use | b | = αs /3π ∼ 0.0113 here as a reasonable estimate. We provide the Higgs mass prediction in our model in Fig. 3. We determine the lightest Higgs mass as a function of MS for MA > 100 GeV. The value loses sensitivity to MA as the latter get bigger. The result shows little sensitivity to Λb , which reflects the infrared quasi-fixed point nature of the MSSM RGEs.12 We can conclude that the Higgs mass predicted in our model is admissible.14 Further analysis can be done with the particle spectrums specified. 6. Summary In this literauture we study the alternative for the conventional SNJL model. Instead of usual dimension-six four-fermion operartors, in our HSNJL model with dimesionfive operators, bi-scalar vacuum condensates play the role of Higgs particles which
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mA ≥ 100 GeV mA = 140 GeV
125
mA = 130 GeV
120
mA = 120 GeV mA = 110 GeV
mh [GeV]
115
mA = 100 GeV
110 105 100 95 90
103 Ms [GeV]
200
Fig. 3.
104
Prediction for the lightest Higgs mass.
achieve the electroweak symmetry breaking. It provides the complete MSSM as low energy effective theory, which was not in the conventional SNJL approaches. The analysis with the RGEs shows that both top and bottom quark masses can be fit together. In this case, bottom Yukawa coupling drives the blow-up of top Yukawa coupling. Threshold correction is important, including sign. For MS < 1.5 TeV, the Higgs mass is on the low side, but compatible with the MSSM Higgs search limits. References 1. D. W. Jung, O. C. W. Kong and J. S. Lee, Phys. Rev. D 81, 031701(R) (2010) arXiv:0906.3580 [hep-ph]. 2. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345. 3. W. A. Bardeen, C. T. Hill and M. Lindner, Phys. Rev. D 41 (1990) 1647. 4. V. A. Miransky, M. Tanabashi and K. Yamawaki, Phys. Lett. B 221 (1989) 177. 5. W. J. Marciano, Phys. Rev. Lett. 62 (1989) 2793. 6. M. A. Luty, Phys. Rev. D 41 (1990) 2893. 7. B. Chung, K. Y. Lee, D. W. Jung and P. Ko, JHEP 0605 (2006) 010 [arXiv:hepph/0510075]. 8. W. Buchmuller and S. T. Love, Nucl. Phys. B 204 (1982) 213. 9. W. Buchmuller and U. Ellwanger, Nucl. Phys. B 245 (1984) 237. 10. T. E. Clark, S. T. Love and W. A. Bardeen, Phys. Lett. B 237 (1990) 235. 11. M. S. Carena, T. E. Clark, C. E. M. Wagner, W. A. Bardeen and K. Sasaki, Nucl. Phys. B 369 (1992) 33. 12. C.D. Froggatt et.al., Phys. Lett. B 298, 356 (1993). 13. L. Hall et.al., Phys. Rev. D 50, 7048 (1994). 14. S. Schael et.al., Eur. Phys. J. C 47, 547 (2006); see also A. Djouadi, Eur. Phys. J. C 59, 389 (2009).
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Ratchet Model of Baryogenesis Tatsu Takeuchi∗ Physics Department, Virginia Tech, Blacksburg, VA 24061, USA Azusa Minamizaki and Akio Sugamoto ¯ Physics Department, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan We propose a toy model of baryogenesis which applies the ‘ratchet mechanism,’ used frequently in the theory of biological molecular motors, to a model proposed by Dimopoulos and Susskind. Keywords: Baryogenesis, ratchet mechanism, Dimopoulos-Susskind model.
1. Introduction The ratio of baryon-number to photon-number densities in our universe has been established via Big-Bang Nucleosynthesis (BBN) [1–3] and WMAP [4, 5] to be η =
nB ≈ 6 × 10−10 . nγ
(1)
The more precise numbers are η10 (BBN : D/H) = 5.8 ± 0.3 , η10 (WMAP: 7yr) = 6.18 ± 0.15 ,
(2)
where η10 = 1010 η, and the BBN value is determined from the deutron abundance reported in Ref. [6, 7]. As we can see, the agreement is very good. The objective of baryogenesis is to explain how the above number can come about from a universe initially with zero net baryon number. Since the pioneering work of Sakharov [8], very many proposals have been made as to what this baryogenesis mechanism could be.a Among the early ones was a model by Dimopoulos and Susskind [15] in which baryon number is generated via the coherent semi-classical time-evolution of a complex scalar field. Similar mechanisms have been employed by Affleck and Dine [16], Cohen and Kaplan [17], and Dolgov and Freese [18], of which the Affleck-Dine mechanism has been popular and intensely studied due to its natural implementability in SUSY models. Two of us have also considered the ∗ Presenting author. a For recent reviews,
see Refs. [9–14].
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application of the Dimopoulos-Susskind model to the cosmological constant problem [19]. In this talk, I will discuss the Dimopoulos-Susskind model, how it satisfies Sakharov’s three conditions for baryogenesis, in particular, how it uses the expansion of the universe to satisfy the third, and then propose the ‘ratchet mechanism’ [20–22] as an alternative for driving the model away from thermal equilibrium. 2. The Dimopoulos-Susskind Model Consider the action of a complex scalar field given by Z i √ h S = d4 x −g g µν ∂µ φ† ∂ν φ − V (φ, φ† ) .
(3)
If the potential V (φ, φ† ) is invariant under the global change of phase φ† → e−iξ φ† ,
φ → eiξ φ ,
then the corresponding conserved current is ↔ √ Bµ = −g i φ ∂µ φ† .
(4)
(5)
If we identify B0 with the baryon number density, then adding to the action a potential which is not invariant under the above phase change, such as V0 (φ, φ† ) = λ φ + φ† α φ3 + α∗ φ†3 , |α| = 1 , (6)
would lead to baryon number violation. Furthermore, unless α = ±1, this potential also violates C and CP since φ transforms as C
φ(t, ~x) −→ φ† (t, ~x) , CP φ(t, ~x) −→ ±φ† (t, −~x) , where the sign under CP depends on the parity of φ. (P is not violated.) In Ref. [15], Dimopoulos and Susskind subject φ to the potential n Vn (φ, φ† ) = λ φ φ† φ + φ† α φ3 + α∗ φ†3 .
(7)
(8)
The purpose of the factor (φ φ† )n is simply to give the coupling constant λ a negative √ mass dimension. Setting φ = φr eiθ / 2, the baryon number density becomes √ (9) nB = B0 = −g φ2r θ˙ , which shows that to generate a non-zero baryon number nB , one must generate a ˙ The potential in the polar representation of φ is non-zero θ. 2 n φr Vn (φr , θ) = λ φ4r cos θ cos(3θ + β) , (10) 2 where we have set α = eiβ . The θ-dependence of this potential for fixed φr is shown in Fig. 1 for the case β = π/2. Note that under B, C, and CP , the phase θ
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-Π
Fig. 1.
Π
Π 2
-2
Π
Θ
θ-dependence of the B, C, and CP violating potential, Eq. (10), for the case β = π/2.
transforms as B
θ(t, ~x) −→ θ(t, ~x) + ξ , C θ(t, ~x) −→ −θ(t, ~x) , CP θ(t, ~x) −→ −θ(t, −~x) .
(11)
If the parity of φ is negative, then θ will also be shifted by π under CP . So in terms of θ, the violation of B is due to the loss of translational invariance, and the violation of C and CP are due to the loss of left-right reflection invariance which happens when β 6= 0, π. The question is, can the asymmetric force provided by this ˙ For potential make θ flow in one preferred direction thereby generate a non-zero θ? that, one must move away from thermal equilibrium. In the original Dimopoulos-Susskind paper [15], this shift away from thermal equilibrium is accomplished by the expansion of the universe. Consider a flat expanding universe with the Friedman-Robertson-Walker metric: 2 a(t) 2 2 d~x2 . (12) ds = dt − a0 During a radiation dominated epoch, the scale factor evolves as √ a(t) ∼ 2t . a0 √ Introducing the conformal variable τ = 2t, the line-element becomes ds2 = τ 2 dτ 2 − d~x2 , while the action simplifies to Z 1 ˆ φˆ† ) + · · · . S = d3 ~x dτ ∂µ φˆ† ∂ µ φˆ − 2n Vn (φ, τ
(13)
(14)
(15)
Here, the scalar field has been rescaled to φˆ ≡ τ φ, and the ellipses represent total ˆ divergences and terms that depend only on |φ|. ˆ is such At this point, a simplifying assumption is made that the dynamics of |φ| that it is essentially constant and does not evolve with τ , leaving only the phase
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√ of φˆ as the dynamic variable.b Setting φˆ = eiθ / 2, the action within a domain of spatially constant θ becomes " # Z 2 1 1 dθ 3 − 2n Vn (1, θ) . S = d ~x dτ (16) 2 dτ τ The equation of motion for θ within that domain is then 1 ∂Vn d2 θ + 2n = 0. 2 dτ τ ∂θ
(17)
ˆ is To this, a friction term, which is assumed to come from the self-interaction of φ, added by hand as d2 θ 1 ∂Vn λ2 dθ + + = 0, (18) dτ 2 τ 2n ∂θ τ 4n dτ where the coefficient of dθ/dτ has been fixed simply by dimensional analysis. If n > 0, both force and friction terms vanish in the limit τ → ∞, and it is possible to show that a non-zero nB ∼ dθ/dτ survives asymptotically, its final value depending on the initial value of θ. This initial value is expected to vary randomly from domain to domain, resulting in different asymptotic baryon numbers in each, and when summed results in an overall net baryon number. On the other hand, if n = 0, which would make the self-interactions of φ renormalizable, the friction term will eventually bring all motion to a full stop. 3. The Ratchet Mechanism A striking feature of the Dimopoulos-Susskind model is its similarity with the problem of biased random walk one encounters in the modeling of biological motors [20–22]. An example of a biological motor is the myosin molecule which walks along actin filaments. This molecule is modeled as moving along a periodic sawtoothshaped potential, similar to that shown in Fig. 1. Thermal equilibrium inside a living organism is broken by the presence of ATP (adenosine triphosphate) whose hydrolysis into ADP (adenosine diphosphate) and P (phosphate) provides the energy required to fuel the motion: ATP → ADP + P + energy .
(19)
This is often modeled as a randomly fluctuating temperature of the thermal bath: the molecule is excited out of a potential well during periods of high-temperature, allowing it to diffuse into the neighboring ones, and then drops back into a well during periods of low-temperature. Due to the asymmetry of the potential, this sequence can lead to biased motion depending on the depth and width of the repeating potential wells, and the height and frequency of the temperature fluctuations. b This assumption that |φ| ˆ is constant would require the magnitude of the unscaled field |φ| to √ evolve as 1/τ = 1/ 2t.
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Analogy with such ‘temperature ratchet’ models suggests a possible way to drive the evolution of θ in the Dimopoulos-Susskind model without relying on the nonrenormalizability of the self-interaction of φ, or the expansion of the universe directly. Let us assume the existence of ATP- and ADP-like particles A and B which interact with φ via the reaction A+φ ↔ B+φ+Q,
(20)
where Q is the energy released in the reaction. A and B are assumed to be stable (or highly meta-stable) states that have fallen out of thermal equilibrium at an earlier time in the evolution of the universe. Though they interact with φ, giving or taking energy away from it, their masses are such that the decay A → B + φ + φ¯
(21)
is kinematically forbidden. In order to isolate the effect of the presence of a bath √ of these particles, we neglect the expansion of the universe and subject φ = φr eiθ / 2 to the n = 0 renormalizable Dimopoulos-Susskind potential V0 (φr , θ). We again adopt the simplifying assumption that the evolution of φr is suppressed. Though the interactions between φ and the A and B particles occur randomly, we model their effect by a periodically fluctuating kinetic energy of θ [20] : h i2 K(t) = K0 1 + A sin(ωt) . (22) This function oscillates between Kmin = K0 (1 − A)2 and Kmax = K0 (1 + A)2 = Kmin + Q. Therefore, Q = 4K0 A .
(23)
Then, the equation of motion of θ in our model will be given by the Langevin equation φ2r θ¨ = −
p ∂V0 − η θ˙ + 4ηK(t) ξ(t) , ∂θ
(24)
where η is the coefficient of friction, and ξ(t) is Gaussian white noise: hξ(t)i = 0 ,
hξ(t)ξ(s)i = δ(t − s) .
(25)
The above Langevin equation is equivalent to the following Fokker-Planck equation governing the evolution of the probability density p(θ, t) and the probability current j(θ, t): ∂p(θ, t) ∂j(θ, t) + , ∂t ∂θ 1 ∂V0 ∂p(x, t) j(θ, t) = − p(θ, t) + 2K(t) . η ∂θ ∂x 0=
(26)
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J 0.001 5´10-4 2´10-4 1´10-4 5´10-5 2´10-5 0.5
Fig. 2.
1.0
5.0 10.0
Ω
50.0 100.0
ω-dependence of J for the case β = π/2, Kmin = 0.5, Q = η = φr = λ = 1.
The quantity of interest for baryon number generation is the period-averaged probability current Z 1 T J = j(x, t) dt , (27) T 0 which is asymptotically independent of x and approaches a constant, a non-zero value signifying a non-zero baryon number. For the sake of simplicity, we set β = π/2, and φr , λ, and η all equal to one. We then solved these equations numerically for various values of Kmin , ω, and Q, and have found that non-zero J can be generated for a very wide range of parameter choices. As an example, we show the ω-dependence of J for the case Kmin = 0.5 and Q = 1 in Fig. 2. Further details of our analysis can be found in Ref. [23]. 4. What is the ATP-like Particle? Whether the ratchet mechanism we are proposing here can be embedded into a realistic scenario remains to be seen. Of particular difficulty may be maintaining a sufficiently large population of the ATP-like particles to drive the ratchet. But what can these ATP-like particles be? Several possibilities come to mind: First, it could be the inflaton at reheating, transferring energy to the φ field via parametric resonance. Second, they could be heavy Kaluza-Klein (KK) modes in some extra-dimension model. And third, perhaps they could be technibaryons transferring energy to technimeson φ’s. Finally, regardless of what their actual identities are, if the ATP-like particles are highly stable and still around, they may constitute dark matter, thereby connecting baryogenesis with the dark matter problem. These, and other possibilities will be discussed elsewhere [24]. Acknowledgments We would like to thank Philip Argyres and Daniel Chung for helpful suggestions. T.T. is supported by the U.S. Department of Energy, grant DE–FG05–92ER40709, Task A.
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References [1] S. Weinberg, “The First Three Minutes. A Modern View of the Origin of the Universe,” (1977). [2] G. Steigman, Ann. Rev. Nucl. Part. Sci. 57, 463 (2007) [arXiv:0712.1100 [astro-ph]]. [3] B. D. Fields and S. Sarkar, in the “Review of Particle Physics,” J. Phys. G 37, 075021 (2010). [4] E. Komatsu et al., arXiv:1001.4538 [astro-ph.CO]. [5] D. J. Fixsen, Astrophys. J. 707, 916 (2009) [arXiv:0911.1955 [astro-ph.CO]]. [6] J. M. O’Meara, S. Burles, J. X. Prochaska, G. E. Prochter, R. A. Bernstein and K. M. Burgess, Astrophys. J. 649, L61 (2006) [arXiv:astro-ph/0608302]. [7] M. Pettini, B. J. Zych, M. T. Murphy, A. Lewis and C. C. Steidel, Mon. Not. Roy. Astron. Soc. 391, 1499 (2008) [arXiv:0805.0594 [astro-ph]]. [8] A. D. Sakharov, JETP Letters, 5, 24 (1967) [9] A. Riotto and M. Trodden, Ann. Rev. Nucl. Part. Sci. 49, 35 (1999) [arXiv:hepph/9901362]. [10] M. Dine and A. Kusenko, Rev. Mod. Phys. 76, 1 (2004) [arXiv:hep-ph/0303065]. [11] J. M. Cline, arXiv:hep-ph/0609145. [12] W. Buchm¨ uller, arXiv:0710.5857 [hep-ph]. [13] M. Shaposhnikov, J. Phys. Conf. Ser. 171, 012005 (2009). [14] S. Weinberg, “Cosmology,” Oxford University Press (2008). [15] S. Dimopoulos and L. Susskind, Phys. Rev. D 18, 4500 (1978). [16] I. Affleck and M. Dine, Nucl. Phys. B 249, 361 (1985). [17] A. G. Cohen and D. B. Kaplan, Phys. Lett. B 199 (1987) 251. [18] A. Dolgov and K. Freese, Phys. Rev. D 51, 2693 (1995) [arXiv:hep-ph/9410346]. [19] A. Minamizaki and A. Sugamoto, Phys. Lett. B 659, 656 (2008) [arXiv:0705.3682 [hep-ph]]. [20] P. Reimann, R. Bartussek, R. H¨ außler, P. H¨ anggi, Phys. Lett. A 215 (1996) 26. [21] F. Julicher, A. Ajdari and J. Prost, Rev. Mod. Phys. 69, 1269 (1997). [22] P. Reimann, Phys. Rept. 361, 57 (2002) [arXiv:cond-mat/0010237]. [23] A. Minamizaki, Ph.D. Thesis, Ochanomizu University, 2010 (in Japanese). [24] A. Minamizaki, A. Sugamoto, and T. Takeuchi, in preparation.
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Classical Solutions of Field Equations in Einstein Gauss-Bonnet Gravity P. Suranyi, C. Vaz and L. C. R. Wijewardhana Department of Physics, University of Cincinnati, Cincinnati, Ohio, USA In this lecture we review recent analysis of the thermodynamic properties of classical black hole and brane solutions in Einstein Gauss-Bonnet (EGB) gravity. Spherically symmetric black hole solutions of EGB gravity in five dimensions have a lower bound for its mass below which the solution fails to exist. As the mass approaches the minimum value the Hawking temperature approaches zero making the configuration semiclassically stable. Uniform black brane solutions defined in a 4+d dimensional space with d compactified torroidal dimensions also have a minimum mass. As the brane approaches the minimum mass the temperature approaches a non zero value making these configurations semiclassically unstable. Unlike these black branes, black holes caged in a compact space would still have a minimum mass and leave stable remnants under evaporation. Such remnants could constitute one of the components of dark matter.
1. Gauss Bonnet Black Holes and Branes Quantum theories of gravity like string theory give rise to additional higher curvature correction terms to the Einstein action.1 Some higher order curvature terms when taken in isolation and quantized can potentially lead to problematic high energy behavior, such as the presence of negative norm states (ghosts). The simplest non trivial higher curvature term avoiding these problems is the Gauss-Bonnet (GB) term which makes a non trivial contribution to dynamics in space-time dimensions greater than or equal to five. When expanded about a flat background the equation of motion of Einstein Gauss-Bonnet (EGB) gravity contains at most second derivative terms avoiding the problem with negative norm states. In this lecture we report an analysis of the critical behavior of a class of classical solutions to EGB gravity. Such a term arises in the low energy gravity action in string theory when corrections to zero slope limit are calculated.2 The equation of motion of EGB gravity is given by 1 Gab ≡ Rab − gab R + αLab = 0, 2 where α is the coupling, the Lanczos tensor is 1 Lab = − gab R2 − 4Rab Rab + K + 2RRab − 4Rac Rbc − 2Racde Rbcde + 4Racdb Rcd , 2 (1) and K is the Kretschmann scalar, K = Rabcd Rabcd .
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Spherically symmetric black hole solutions in EGB gravity in dimensions greater than or equal to 5 were found by Boulware and Deser.3 Thermodynamics of such black holes were formulated by Myers and Simon and by Wiltshire.4 It was seen that for α > 0, in five dimensions , spherically symmetric black hole solutions do not exist below the mass Mmin =
3πα 4G5
and the Hawking temperature goes to zero as this mass is approached, r 1 rH 8G5 M = − 2α T ' 8πα 8πα 3π Thus at the end of Hawking evaporation such black holes would leave behind stable remnants. If our world is contained in a higher dimensional bulk objects that appear as black holes in 4 dimensions would be black branes in the higher dimensional spacetime. Do such objects leave stable remnants when they evaporate if higher curvature correction terms of the Gauss Bonnet type are included in the gravitational action? We have investigated this question in a couple of papers and here we summarize our conclusions. In conventional Einstein gravity a D = 4 Schwarzschild black hole can be trivially extended to a black string (D = 5) or black brane solutions (D > 5). Solutions become more complicated if higher derivative terms are added to the Einstein action, such as the GB term. Though asymptotically Minkowski black hole solutions have been found in D ≥ 5 theories,3 no exact black string (brane) solutions are known in Einstein-Gauss-Bonnet (EGB) gravity. Yet the investigations of black strings (branes) is more important because with compact extra dimensions all but the lightest static objects must be black strings (branes). Two alternative techniques have been used to investigate black brane solutions. Kobayashi and Tanaka5 solved the 5 dimensional Einstein equations modified by Lanczos tensor (1) (which is the variation of the GB term), numerically. Using an expansion around the event horizon they also found an exact lower bound for the √ black string mass, Mc ∼ α, where α is the coupling constant of the GB term. For M < Mc the solutions do not have a horizon, they represent naked singularities. In a subsequent work,6 following,5 we used a horizon expansion to investigate black strings in D > 5 EGB theory. We found that in every dimension, D ≥ 5, a lower bound, similar to that in D = 5, exists for the mass of the black string. We also investigated the thermodynamics of the black string but the fifth order expansion in α, employed in our paper, was insufficient to get a definitive answer for the thermal behavior of the system when the mass of the black string approached the lower limit. In the paper 7 we used improved numerical and expansion methods to investigate black string solutions. We used the following ansatz for the metric of a uniform
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string, made dimensionless by factoring out rh2 , as ds2 = −f (ρ)dt2 +
g(ρ) 2 dρ + (ρ + 1)2 dΩ2 + h(ρ)dw2 , f (ρ)
ρ defined by ρ=
r − rh , rh
to replace the radial coordinate, r where rh is the radius of the horizon. Here dΩ2 is the metric on the three sphere and w is the coordinate of the fifth dimension, which assumed to be periodic with a period L. Our discussion could easily be generalized to more than one extra dimensions compactified on a torus. In our discussion we used a dimensionless GB coupling defined as α (2) β =1−8 2. rh We find two critical points, βcrit = 0 and 3, where the condition β = 0 determines the critical radius. The critical radius determines a nonzero critical mass. We paid particular attention to the singularities at the endpoints of range, R, 0 ≤ β < 3, where β is defined in (2). No black string solutions exists outside R. In particular, we found a critical solution at β = 0 which, in contrast to solutions at β > 0, has a horizon expansion that includes half-integer powers of the radial horizon variable ρ. Metric components and all scalars, including the Ricci scalar and the Kretschmann √ scalar are finite, but have β type singularities at β = 0. At β > 0 the metric components are singular inside the horizon, at ρ = −3β/7. Unless β = 0 it is always possible to continue the metric to a region inside the horizon. To further study the β = 0 limit of solutions we introduced EddingtonFinkelstein coordinates p Z g(ρ) u = t − dρ f (ρ) Then the metric takes the form p ds2 = −f (ρ)du2 + 2 g(ρ)du dρ + (ρ + 1)2 dΩ2 + h(ρ)dw2 ,
regular everywhere including a neighborhood of the horizon a . The singularity at −3β/7 moves to the horizon when β → 0. We have not been able to construct a metric inside this singular point. We have also been unable to continue the critical metric, with β = 0, inside the horizon. Since Kruskal coordinates usually represent the maximal non-singular extension of the space-time, we define the Kruskal coordinates for the critical string such that the coefficient of dU dV is −1. They can be a Note
of β.
that g(ρ) > 0 on the interval −3β/7 < ρ < ∞ for the whole range, R, of admissible values
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obtained in a asymptotic expansion from the Eddington-Finkelstein coordinates u and p Z g(ρ) v = t + dρ . f (ρ) We find ds2 = −dU dV + (ρ + 1)2 dΩ2 + h[ρ]dw2 , where ρ has the following expression by the Kruskal coordinates U and V √ ρ = −U V (1 + c U V + · · · ), where c is a non-vanishing constant. It is obvious that we cannot cross the light like surfaces U = 0 or V = 0, which represent the horizon. However, we have shown that a geodesic of a massive probe falling toward the horizon reaches it in finite proper time (at τ = 0), but it cannot pass the horizon because its radial coordinate turns complex at τ > 0 even though the curvature invariants are finite at that point. The meaning of this is unclear to us at this time. Either there is a different choice of coordinates in which the space-time may be extended through the horizon, but they are not the Kruskal coordinates and we have not been able to find them, or the critical solution is pathological and should be excluded by a censorship principle. We used numerical techniques to calculate the solutions for the whole range R. We calculated the ADM mass, tension, Hawking temperature, and entropy. While the Hawking-temperature tends to a finite value at β = 0, it diverges at β = 3. The ADM mass and tension are modified only by a finite amount compared to pure Einstein gravity and bounded from above and below over the whole range, R. The entropy is a monotonically decreasing function of β, vanishing in the limit β = 3. This is not surprising in view of the divergence of the Hawking temperature. We also performed a stability analysis by considering linear perturbations about the black brane solutions. We found that for α > 0 and small enough compactification size the solutions reach the critical mass before encountering a Gregory-Laflamme 8 type instability. There are two unresolved problems concerning the fate of black strings. If α > 0 they are driven by Hawking radiation toward the critical state at β = 0. That state, however is very problematic. Though gtt has a linear zero and all scalars are finite at that zero, test particles that reach the ρ = 0 point in finite time have nowhere to go because ρ(τ ) becomes complex as a function of proper time. As finding solutions for black holes in compactified spaces is even more difficult than finding black string solutions we have no way of comparing the entropy and relative stability of these solutions. This is one of the reasons why we are not able to answer what happens with black strings that reach the end of their lives at the β = 0 line. Since the limit of their Hawking temperature is not zero, as a study based on numerical approximations and 1/ρ expansions lead us to believe, so they
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are certainly not frozen at that point. Their ADM mass is √ 8αL Mc ' 1.23 . 2G They cannot go into a black string state with a smaller ADM, mass, however, because such black string states do not exist. It seems that the only possibility is that some other, yet undiscovered, type of higher entropy static or possibly dynamic state exists, into which the black string can morph. Another possible solution, at least for D ≥ 6, is that the inclusion of higher order Lovelock terms would cure this anomaly. The fate of black stings at and beyond β = 0 is one of the most important questions we would like to investigate in the future. Other questions include a possible numerical study of black holes in compactified spaces. This work used a Kaluza-Klein type compactification. It would be more difficult, but perhaps more interesting to repeat this work in a Randall-Sundrum9 type antide Sitter space. This work is funded in part by the U.S. Department of Energy under the grant DE-FG-02-84ER40153. References 1. C. Callen, E. Martinec, M. Perry and D. Friedan, Nucl. Phys. B262, 593 (1985). 2. Rafael I. Nepomechie, Phys.Rev. D 32, 3201 (1985); B. Zwiebach, Phys. Lett. 156B, 315 (1985); B. Zumino, Phys. Rep. 137, 109 (1986). 3. D. Boulware and S. Deser, Phys. Rev. Lett. 55, 2656 (1985). 4. R. Myers and J. Simon, Phys. Rev. D 38, 2434 (1988); D.L. Wiltshire, Phys. Rev. D 38, 2445 (1988). 5. T. Kobayashi, T.Tanaka, Phys. Rev. D 71, 084005 (2005). 6. C. Sahabandu, P. Suranyi, C. Vaz, and L.C.R. Wijewardhana, Phys. Rev. D 73, 044009 (2006). 7. P.Suranyi, C.Vaz and L.C.R.Wijewardhana, Phys.Rev.D79:124046,2009. e-Print: arXiv:0810.0525 [hep-th] 8. R. Gregory and R. Laflamme, Phys. Rev. Lett. 70, 2837, 1993. 9. L. Randall and R. Sundrum, Phys. Rev. Lett. 83:3370-3373, L. Randall and R. Sundrum, Phys. Rev. Lett. 83:4690-4693, 1999.
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Black Holes Constitute All Dark Matter Paul H. Frampton Department of Physics and Astronomy, University of North Carolina, Chapel Hill, USA and Institute for the Mathematics and Physics of the Universe, University of Tokyo, Japan E-mail: [emailprotected], [emailprotected] The dimensionless entropy , S ≡ S/k, of the visible universe, taken as a sphere of radius 50 billion light years with the Earth at its ”center”, is discussed. An upper limit (10 112 ), and a lower limit (10102 ), for S are introduced. It is suggested that intermediate-mass black holes (IMBHs) constitute all dark matter, and that they dominate S. Keywords: Dark matter, black hole, entropy, halo.
1. Introduction Two references useful for further information about the material of this talk are: (1) P.H.F. and T.W. Kephart. Upper and Lower Bounds on Gravitational Entropy. JCAP 06:008 (2008) and (2) P.H.F. Identification of All Dark Matter as Black Holes. arXiv:0905.3632 [hep-th]. JCAP 0910:016 (2009).
2. The Entropy of the Universe As interest grows in pursuing alternatives to the Big Bang, including cyclic cosmologies, it becomes more pertinent to address the difficult question of what is the present entropy of the universe? Entropy is particularly relevant to cyclicity because it does not naturally cycle but has the propensity only to increase monotonically. In one recent proposal, the entropy is jettisoned at turnaround. In any case, for cyclicity to be possible there must be a gigantic reduction in entropy (presumably without violation of the second law of thermodynamics) of the visible universe at some time during each cycle. Standard treatises on cosmology address the question of the entropy of the universe and arrive at a generic formula for a thermalized gas of the form S=
2π 2 g ∗ VU T 3 45
(1)
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where g∗ is the number of degrees of freedom, T is the Kelvin temperature and VU is the volume of the visible universe. From Eq.(1) with Tγ = 2.70 K and Tν = Tγ (4/11)1/3 = 1.90 K we find the entropy in CMB photons and neutrinos are roughly equal today Sγ (t0 ) ∼ Sν (t0 ) ∼ 1088 .
(2)
Our topic here is the gravitational entropy, Sgrav (t0 ). Following the same path as in Eqs. (1,2) we obtain for a thermal gas of gravitons Tgrav = 0.910 K and then (thermal) Sgrav (t0 ) ∼ 1086
(3)
This graviton gas entropy is a couple of orders of magnitude below that for photons and neutrinos. This graviton gas entropy is a couple of orders of magnitude below that for photons and neutrinos. But there are larger contributions to gravitational entropy from elsewhere!!!
3. Upper Limit on the Gravitational Entropy We shall assume that dark energy has zero entropy and we therefore concentrate on the gravitational entropy associated with dark matter. The dark matter is clumped into halos with typical mass M (halo) ' 1011 M where M ' 1057 GeV ' 1030 kg is the solar mass and radius R(halo) = 105 pc ' 3 × 1018 km ' 1018 rS (M ). There are, say, 1012 halos in the visible universe whose total mass is ' 1023 M and -corresponding Schwarzschild radius is rS (1023 M ) ' 3 × 1023 km ' 10Gpc. This happens to be the radius of the visible universe corresponding to the critical density. This has led to an upper limit for the gravitational entropy is for one black hole with mass MU = 1023 M . Using SBH (ηM ) ' 1077 η 2 corresponds to the holographic principle for the upper limit on the gravitational entropy of the visible universe: (HOLO) Sgrav (t0 ) ≤ Sgrav (t0 ) ' 10123
(4)
which is 37 orders of magnitude greater than for the thermalized graviton gas in Eq.(3) and leads us to suspect (correctly) that Eq.(3) is a gross underestimate. Nevertheless, Eq.(4) does provide a credible upper limit, an overestimate yet to be refined downwards below, on the quantity of interest, Sgrav (t0 ). The reason why a thermalized gas of gravitons grossly underestimates the gravitational entropy is because of the ’clumping’ effect on entropy. Because gravity is universally attractive its entropy is increased by clumping. This is somewhat counter-intuitive since the opposite is true for the familiar ’ideal gas’. It is best illustrated by the fact that a black hole always has ’maximal’ entropy by virtue of the holographic principle.
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4. Lower Limit on Gravitational Entropy It is widely believed that most, if not all, galaxies contain at their core a supermassive black hole with mass in the range 105 M to 109 M with an average mass about 107 m . Each of these carries an entropy SBH (supermassive) ' 1091 . Since there are 1012 halos this provides the lower limit on the gravitational entropy of Sgrav (t0 ) ≥ 10103
(5)
which, by now, provides an eight order of magnitude window for Sgrav (t0 ). The lower limit in Eq.(5) from the galactic supermassive black holes may be largest contributor to the entropy of the present universe but this seems to us highly unlikely because they are so very small. Each supermassive black hole is about the size of our solar system or smaller and it is intuitively unlikely that essentially all of the entropy is so concentrated. Gravitational entropy is associated with the clumping of matter because of the long range unscreened nature of the gravitational force. This is why we propose that the majority of the entropy is associated with the largest clumps of matter: the dark matter halos associated with galaxies and cluster. 5. Intermediate Mass Black Holes If we consider normal baryonic matter, other than black holes, contributions to the entropy are far smaller. The background radiation and relic neutrinos each provide ∼ 1088 . We have learned in the last decade about the dark side of the universe. WMAP suggests that the pie slices for the overall energy are 4% baryonic matter, 24% dark matter and 72% dark energy. Dark energy has no known microstructure, and especially if it is characterized only by a cosmological constant, may be assumed to have zero entropy. As already mentioned, the baryonic matter other than the SMBHs contributes far less than (SU )min . This leaves the dark matter which is concentrated in halos of galaxies and clusters. It is counter to the second law of thermodynamics when higher entropy states are available that essentially all the entropy of the universe is concentrated in SMBHs. The Schwarzschild radius for a 107 M SMBH is ∼ 3 × 107 km and so 1012 of them occupy only ∼ 10−36 of the volume of the visible universe. Several years ago important work by Xu and Ostriker showed by numerical simulations that IMBHs with masses above 106 M would have the property of disrupting the dynamics of a galactic halo leading to runaway spiral into the center. This provides an upper limit (MIM BH )max ∼ 106 M . Gravitational lensing observations are amongst the most useful for determining the mass distributions of dark matter. Weak lensing by, for example, the HST shows the strong distortion of radiation from more distant galaxies by the mass of the dark matter and leads to astonishing three-dimensional maps of the dark matter trapped
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within clusters. At the scales we consider ∼ 3 × 107 km, however, weak lensing has no realistic possibility of detecting IMBHs in the forseeable future. Gravitational microlensing presents a much more optimistic possibility. This technique which exploits the amplification of a distant source was first emphasized in modern times (Einstein considered it in 1912 unpublished work) by Paczynski. Subsequent observations found many examples of MACHOs, yet insufficient to account for all of the halo by an order of magnitude. These MACHO searches looked for masses in the range 10−6 M ≤ M ≤ 102 M . The time t0 of a microlensing event is given by rE (6) t0 ≡ v where rE is the Einstein radius and v is the lens velocity usually taken as v = 200 km/s. The radius rE is proportional to the square root of the lens mass and numerically one finds 1/2 M (7) t0 ' 0.2y M so that, for the MACHO masses considered, 2h ≤ t0 ≤ 2y. 6. Cosmological Entropy Considerations The cosmological entropy range 102 ≤ log10 SU ≤ 112
(8)
is the first of two interesting windows which are the subject. Conventional wisdom is SU ∼ (SU )min = 10102 . 7. Intermediate Mass Black Holes and Microlensing Longevity log10 nmax
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heavier mass IMBHs in the halo. If this can be achieved, and it seems a worthwhile enterprise, then the known entropy of the universe could be increased by more than two orders of magnitude. There exists interesting other analyses pertinent to existence of massive halo objects: J. Yoo, J, Chanam´e and A. Gould, Astrophys. J. 601, 311 (2004). astro-ph/0307437. It is this entropy argument based on holography and the second law of thermodynamics which is the most compelling supportive argument for IMBHs. If each galaxy halo asymptotes to a black hole the final entropy of the universe will be ∼ 10112 as in Eq.(8) and the universe will contain just ∼ 1012 supergigantic black holes. Conventional wisdom is that the present entropy due entirely to SMBHs is only ∼ 10−10 of this asymptopic value. IMBHs can increase the fraction up to ∼ 10−8 , closer to asymptopia and therefore more probable according to the second law of thermodynamics. There are several previous arguments about the existence of IMBHs and they have put upper limits on their fraction of the halo mass. The entropy arguments are new and provide additional motivation to tighten these upper bounds or discover the halo black holes. One observational method is high longevity microlensing events. It is up to the ingenuity of observers to identify other, possibly more fruitful, methods some of which have already been explored in a preliminary way. 9. Post-conference update Since the SCGT09 conference took place, the paper: Paul H. Frampton, Masihiro Kawasaki, Fuminobu Takahashi and Tsutomu Yanagida IPMU-09-0157 (December 2009). Primordial Black Holes as All Dark Matter arXiv:1001.2308 [hep-ph]. shows that it is possible to form black holes in the early universe with mass 105 M and with sufficient abundance to provide all of the dark matter. Acknowledgments This work was supported in part by the World Premier International Research Center Initiative (WPI initiative), MEXT, Japan and by U.S. Department of Energy Grant No. DE-FG02-05ER41418.
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Electroweak Precision Test and Z → bb in the Three Site Higgsless Model∗ Tomohiro Abe Department of Physics, Nagoya University Nagoya, 464-8602, Japan E-mail: [emailprotected] The three site Higgsless model is a highly-deconstructed Higgsless model with only three sites. In this model, we show that the KK gauge boson mass, KK fermion mass, and the KK gauge boson couplings with light quarks and leptons are severely constrained by the electroweak precision data. Especially we find that perfect fermiphobity of KK gauge boson is ruled out by the precision data. We also compute the flavor dependent chiral logarithmic corrections to the decay Z → bb. We find that the phenomenological constraints on this model arising from measurements of Z → bb are relatively mild, requiring only that the heavy Dirac fermion be heavier than 1 TeV or so, and are satisfied automatically in the range of parameters allowed by the electroweak precision data.
1. Introduction The standard model (SM) describes the phenomenology among the elementary particles very well. But the Higgs boson has not observed yet even though it is needed for the electroweak symmetry breaking (EWSB) in the SM. So there is a possibility the EWSB can be occurred without any scalar bosons. To pursue this possibility we need to take into account for the perturbative unitarity of the longitudinal gauge bosons’ scattering. To keep a theory perturbative, some new particles have to appear below 1TeV and contribute to those scattering. One of the possibilities is a Higgsless model.1 This is based on a gauge theory in five-dimension which means an infinite number of the gauge bosons, KK gauge bosons, exist in four-dimension. These KK gauge bosons keep the perturbative unitarity. An infinite number of the gauge bosons imply an infinite number of gauge symmetries in four-dimension.2–5 In fact a gauge theory in five-dimension can be described as a gauge theory in four-dimension by a method called deconstruction.6,7 For the low energy phenomenology, a finite number of gauge bosons are sufficient. So the roughly deconstructed Higgsless model is suitable for the phenomenological analysis. The three site Higgsless model8 is a highly deconstructed Higgsless model. We study the constraints to it from the electroweak precision test (EWPT).9,10 ∗ This
report is based on Ref. 9 and Ref. 10.
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2. Three site Higgsless model The three site Higgsless Model has SU (2)0 × SU (2)1 × U (1)2 gauge symmetry. The particles in this model are SM particles, except the Higgs boson, and their KK partners. In the gauge sector, there are W ± and Z in addition to W ± and Z. To push up the perturbative unitarity bound, W ± and Z have to be lighter than 1TeV and be coupled with W ± and Z. We can get a lower bound of the W ± and Z mass, MW , from the constraint to WWZ coupling constant MW > 380 GeV. As a result we can expect W ± and Z might be found at the LHC experiment. 3. Electroweak precision test Using the S and T parameters11,12 we can get the lower bound of KK fermion mass, MF , and the constraint for the coupling among W ± and the standard model fermions, gW f f . In general, Higgsless models have a large correction to the S parameter at the tree level. If the SM fermions are not coupled with the KK gauge bosons, however, the correction to the S parameter becomes small. In this model, therefore, the gW f f is proportional to the S parameter at the tree level and should be negligible small. This estimate is qualitative. For quantitative estimation, we need one-loop calculations. The results are following. αS = −4s2
MW gW f f MW gW f f
M2 Λ2 α MW gW f f M2 α α ln W2 + ln 2 ln W2 − (1) 24π MW gW f f MF 24π MF 12π MHref √ 4 2 3α Λ2 MW 2GF Mt4 MW 1 αT = − + ln 2 ln 2 (2) 2 − 4 g 64π 2 MT2 MW 32πc2 MHref MHref W W ff 1− M MW gW f f −
where Λ is a cutoff scale of this model. By the naive dimensional analysis, we can estimate Λ < 4.3 TeV. Comparing them with experimental constraint,13 we can find gW f f /gW f f ∼ O(10−2 ) (Fig.1). This means we need the high luminosity for the Drell-Yan production of W ± and its decay into the SM fermions at LHC experiment. The lower bound of KK fermion mass can be found at 1.8 TeV (Fig.2). This implies that KK fermions are rarely produced at the LHC experiment. 4. Zbb coupling Zbb coupling also tells us a lower bound of KK fermion mass. We use the constraint to Rb . So the flavor dependent correction to the Zbb coupling is sufficient. We bb as follows, parametrize δgL 1 1 2 bb gZbb = gZ − + δgL + sin θW , (3) 2 3
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3500 3000 2500 2000 1500 350
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Fig. 2. Allowed region of MW and KK fermion mass. The upper region is the allowed region. The region below the line is excluded.
and we find this correction in this model at the one-loop level is 2 Λ m2t f12 f22 bb δgL = log 1+ , (4) 2 2 2 2 2 (4π) v 2(f1 + f2 ) MF2 √ √ where 1/v 2 = 2GF . We take f1 ∼ f2 ∼ 2v for simplicity. From the experimental constraint to Rb , finally we find that MF > 1 TeV. So the constraint from Zbb coupling is relatively mild and automatically satisfied with EWPT. References 1. C. Csaki, C. Grojean, H. Murayama, L. Pilo and J. Terning, Phys. Rev. D 69, 055006 (2004) [arXiv:hep-ph/0305237]. 2. M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett. 54, 1215 (1985). 3. M. Bando, T. Kugo and K. Yamawaki, Prog. Theor. Phys. 73, 1541 (1985). 4. M. Bando, T. Kugo and K. Yamawaki, Nucl. Phys. B 259, 493 (1985). 5. M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988). 6. N. Arkani-Hamed, A. G. Cohen and H. Georgi, Phys. Rev. Lett. 86, 4757 (2001) [arXiv:hep-th/0104005]. 7. C. T. Hill, S. Pokorski and J. Wang, Phys. Rev. D 64, 105005 (2001) [arXiv:hepth/0104035]. 8. R. S. Chivukula, B. Coleppa, S. Di Chiara, E. H. Simmons, H. J. He, M. Kurachi and M. Tanabashi, Phys. Rev. D 74, 075011 (2006) [arXiv:hep-ph/0607124]. 9. T. Abe, S. Matsuzaki and M. Tanabashi, Phys. Rev. D 78, 055020 (2008) [arXiv:0807.2298 [hep-ph]]. 10. T. Abe, R. S. Chivukula, N. D. Christensen, K. Hsieh, S. Matsuzaki, E. H. Simmons and M. Tanabashi, Phys. Rev. D 79, 075016 (2009) [arXiv:0902.3910 [hep-ph]]. 11. M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964 (1990). 12. M. E. Peskin and T. Takeuchi, Phys. Rev. D 46, 381 (1992). 13. C. Amsler et al. [Particle Data Group], Phys. Lett. B 667, 1 (2008).
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Chiral Symmetry and BRST Symmetry Breaking, Quaternion Reality and the Lattice Simulation Sadataka Furui School of Science and Engineering, Teikyo University Utsunomiya, 320-8551, Japan E-mail: [emailprotected] We discuss that the deviation of the Kugo-Ojima color confinement parameter u(0) from -1 in the case of quenched lattice simulation and the consistency with -1 in the case of full QCD simulation could be attributed to the boundary condition defined by fermions inside the region of r < 1fm. By using the domain wall fermion propagator in lattice simulation, we show that the chiral symmetry breaking in the infrared can become manifest when one assumes that the left-handed fermion on the left wall and the righthanded fermion on the right wall are correlated by a self-dual gauge field. The relation between the infrared fixed point of the running coupling measured in lattice simulations, the prediction of the BLM renormalization theory, the conformal field theory with use of the t’Hooft anomaly matching condition in non-SUSY supersymmetric theory and the quaternion real condition are discussed. Keywords: Infrared fixed point, quaternion, triality, critical flavor number.
1. Introduction The infrared (IR) QCD is characterized by the color confinement and the chiral symmetry breaking. We performed lattice simulations1,2 using the gauge configuration produced with domain wall fermion3 which preserves the chiral symmetry in the zero - mass limit and compared with results of staggered fermion. In these lattice simulations and in comparison with other works, we observed qualitative differences between quenched and unquenched simulations. 1.1. BRST symmetry The Kugo-Ojima color confinement attracted renewed interest by a conjecture of possible BRST (Becchi-Rouet-Stora-Tyutin) symmetry breaking in the infrared.4,5 These authors pay attention to the restriction of the gauge configuration to the fundamental modular region defined by Zwanziger to solve the Gribov problem. Kondo5 parametrized the horizon function abc b c def e f ad T 2 2 qµ qν 2 2 (1) (gf Aµ c )(gf Aν c¯ )k = −δ δµν u(q ) + F (q ) 2 (u(q ) + w(q )) q
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and assuming w(0) = 0, obtained u(0) = −2/3. There is an argument against this result8 showing that w and u are not multicatively renormalizable. On the lattice, T T u(q 2 ) is δ ab δµν S(Uµ )u(q 2 ) however the first term of the r.h.s of eq.(1.1) i.e. δ ab δµν A /2
µ in the log−U definition and S(Uµ )ab = where S(Uµ ) is S(eAµ ) = tanhA µ /2 † Uµ +Uµ tr λ†a 21 , λb |traceless p in the U −linear definition. 2
The expectation value of S is proportional to e and e/d is about 0.95 in 564 lattice quenched SU(3) log−U definition but about 0.88 in the U −linear definition.6 When S(0) ∼ 0.85 instead of 1 as in DSE, a solution of w(0) = 0 and u(0) = − 23 is possible. 1.2. QCD running coupling Recently Appelquist9 claimed by performing lattice simulation of running coupling in the Schr¨ odinger functional method that there is a critical number of flavors Nfc below which both chiral symmetry breaking and confinement set in. He assigned 8 ≤ Nfc ≤ 12 and in the case of Nf = 8 the running coupling monotonically increase as β decreases and in the case of 164 lattice and β ∼ 4.65, g¯2 (L)/4π ∼ 1.7, and that there is no sign of infrared fixed point at Nf < 8. However, applicability of the Schr¨ odinger functional method in the region of q below and around Λ ∼ 213±40MeV is not clear. Brodsky et al,10 argue that in the DSE q < Λ or r > 1fm, confinement is essential and the quark-gluon should not be treated as free. In our lattice simulation of Nf = 2 + 1 DWF, αs (q) ∼ 1.7 at q ∼ 0.6GeV and the deviation of the running coupling from 2-loop perturbative calculation is significant below q ∼ 3GeV due to A2 condensates. The running coupling data of JLab7 suggest presence of the infrared fixed point and that Nf = 3 is close to the Nfc . 1.3. Quaternion real condition and proton form factor The infrared fluctuation of the gluon propagator could be attributed to the chiral anaomly. t’Hooft discussed that the anomaly cancellation on the color triplet and color sextet components in the 3 quark baryon sector,11 and formulations in non-SUSY supersymmetric theory were discussed by Sannino.12 In standard DWF analysis, the limit of the distance between the left domain wall and the right domain wall approaching infinity corresponds to the continuum limit. I take a specific gauge that the fermion on the left domain wall and the right domain wall are correlated by a self-dual gauge field (instanton) that satisfy quaternion real condition.13 Then I calculate the ratio of the three point function and the two point function of fermions using the SU(6) proton wave function. As shown in Fig.2, at zero momenrtum the form factor comes exclusively from the left-handed fermion. A product of quaternions makes an octonion, whose automorphism is the exceptional lie group G2 which has 3+6+5 dimensional stable manifolds and posesses the triality symmetry. The spontaneous supersymmetry breaking of massless fermions
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qf qnq
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Fig. 1. The running coupling of the domain wall fermion. Coulomb gauge gluonghost coupling of mu = 0.01/a(square), 0.02/a(diamond), and quark-gluon coupling of mu = 0.01/a(large disks). Small disks are the αs,g1 derived from the spin structure function of the JLab group7 and the solid curve is their fit.
0.8 0.6 0.4
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qGeV Fig. 2. The form factor of a proton using the DWF mu = 0.01/a gauge fixed such that the fermion on the left wall and the right wall are correlated by self-dual gauge field, and the dipole fit with M 2 = 0.71GeV2 . At zero momentum left-handed(LH) fermion dominates the form factor, and at finite momentum, LH and RH contribute almost equally.
on 5 dim manifolds and ghost, gluon on 3 dim and 6 dim color space, and the problem of the large critical Nf for the conformality by about a factor of 3, would be explained by the triality symmetry of the G2 group of the octonion.14 Whether the large critical Nf is an artefact, and how sensitive the results on the boundary conditions are further to be investigated. Acknowledgments I thank Dr. F. Sannino, Prof. S. Brodsky for helpful information and discussion , and Dr. A. Deur for sending me the JLab experimental data. Numerical calculation was done at KEK, YITP Kyoto Univ. and RCNP Osaka Univ. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
S. Furui, arXiV:0912.5796 and references therein. S. Furui, Few-Body Syst. 45, 51(2009), Erratum:DOI 10.1007/s00601-009-0053-4. C. Allton et al., Phys. Rev. D76,014504 (2007); arXiv:hep-lat/0701013. D. Dudal,et al, Phys. Rev. D79,121701(R)(2009). K-I. Kondo, Phys. Lett. B678,322(2009). H. Nakajima and S. Furui, Nucl. Phys. B(Proc.supl.) 141, 34(2005), arXiv:heplat/0408001. A. Deur, V. Burkert, J.P. Chen and W. Korsch, Phys. Lett. B665,349 (2008). Ph. Boucaud et al., arXiv:hep-lat/0909.2615. T. Appelquist, Prog. Theor. Phys.(Kyoto)180,72(2009). S.J. Brodsky and R. Shrock, Phys. Lett. B666,95(2008),arXiv:0806.1535[hep-th]. G.’t Hooft, in ”Recent Developments in Gauge Theories”, p.135 Plenum Press (1980). F. Sannino, Phys. Rev. D80,065011(2009); arXiv:0911.0931[hep-th]. E.Corrigan and P. Goddard, Comm. Math. Phys.80, 575(1981). ´ Cartan, ”The theory of Spinors”, Dover Pub. (1966) E.
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Holographic Techni-Dilaton, or Conformal Higgs∗ Kazumoto Habaa , Shinya Matsuzakib and Koichi Yamawakia a Department b Department
of Physics, Nagoya University, Nagoya, 464-8602, Japan of Physics, Pusan National University, Busan 609-735, Korea
We study a holographic model dual to walking/conformal technicolor (W/C TC) deforming a hard-wall type of bottom-up setup by including effects from techni-gluon condensation. We calculate masses of (techni-) ρ meson, a1 meson, and flavor/chiral-singlet scalar meson identified with techni-dilaton (TD)/conformal Higgs boson, as well as the S parameter. It is shown that gluon contributions and large anomalous dimension tend to decrease specifically mass of the TD. In the typical model with S ' 0.1, we find mTD ' 600 GeV, while mρ , ma1 ' 4TeV.
1. A holographic technicolor model with techni-gluon condensation The origin of the mass is the most urgent issue of the particle physics today and is to be resolved at the LHC experiments. In order to account for the origin of mass dynamically without introducing the ad-hoc Higgs boson, here we consider the Walking/Conformal Technicolor (W/C TC)1,2 having large anomalous dimension γm ' 1 due to the strong coupling which stays almost non-running (“walking”) over wide energy range near the “conformal fixed point”. In contrast to a folklore that TC is a “higgsless model” in analogy with the QCD, there actually exists “techni-dilaton’ (TD)’ 1 as a pseudo Nambu-Goldstone boson of the approximate conformal symmetry, which is relatively light compared with other bound states like techni-ρ and techni-a1 , etc. Here we shall estimate the mass of TD in the bottom-up holographic W/C TC by newly introducing a flavor-singlet bulk field corresponding to the techni-gluon condensate. Details are given in the forthcoming paper.3 Following a bottom-up approach of holographic-dual of QCD4 and that of walking/conformal (W/C) TC,5,6 we consider a five-dimensional gauge model possessing the SU (Nf )L × SU (Nf )R gauge symmetry. The model is defined on the five-dimensional anti-de-Sitter space with the curvature radius L, which is de2 scribed by the metric ds2 = gM N dxM dxN = (L/z) ηµν dxµ dxν − dz 2 with ηµν = diag[1, −1, −1, −1]. The fifth direction z is compactified on an interval extended from ultraviolet (UV) and infrared (IR) branes, ≤ z ≤ zm . In addition to the bulk left- (LM ) and right- (RM ) gauge fields, we introduce a bulk scalar field Φ which transforms as bifundamental representation under the SU (Nf )L × SU (Nf )R ∗ Talk
presented by K. Haba.
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gauge symmetry so as to deduce the information concerning the chiral condensationoperator T¯T . According to the holographic dictionary, the mass-parameter mΦ is related to γm , the anomalous dimension of T¯T , as m2Φ = −(3 − γm )(1 + γm )/L2 , where γm ' 0 corresponds to QCD and QCD-like TC and γm ' 1 to the W/C TC. Here we newly introduce an additional chiral-singlet and massless bulk scalar field ΦX dual to techni-gluon condensate hαG2µν i, where α is related to the TC gauge 2 couping gTC by α = gTC /(4π). We adopt a “dilaton-like” coupling for interaction terms involving ΦX in such a way that all the fields couple to ΦX in the exponential form like eΦX (z) . (ΦX is not identified with techni-dilaton in this talk.) Thus the five-dimensional action takes the form: Z Z zm 1 p 2 1 4 d z −detgM N 2 ecg5 ΦX (z) − Tr LM N LM N + RM N RM N S5 = d x g 4 5 1 +Tr DM Φ† DM Φ − m2Φ Φ† Φ + ∂M ΦX ∂ M ΦX , (1) 2 where DM Φ = ∂M Φ + iLM Φ − iΦRM , g5 denotes the gauge coupling in fivedimension, and c is the dimensionless coupling constant. It turns out that requiring the present model to reproduce high-energy behaviors in the underlying theory leads to (L/g52 ) = NTC /(12π 2 ) and c = −NTC /(192π 3 ): Our model exactly reproduces the high-energy behaviors for vector and axial-vector current correlators up till the terms of 1/Q8 expected from the OPE. For details, see Ref.3. We ignore Kaluza-Klein (KK) modes of ΦX (including the lowest mode) which are identified with massive glueballs with mass of order O(ΛTC ) which is much larger than the electroweak scale, ΛTC Fπ , in the case of W/C TC with γm ' 1. The techni-dilaton, a flavor-singlet scalar bound state of techni-fermion and antitechni-fermion, will be identified with the lowest KK mode in the KK decomposition of Φ, Φ(x, z) = v(z) + σ (1) (x)σ1 (z) + · · · , but not of ΦX . Following the holographic recipe, we obtain formulas for hαG2µν i, hT¯T i, Mρ , Ma1 , Mσ1 (≡ MTD ), Fπ , and the S parameter, where Mρ and Ma1 are the masses of the lowest KK modes for the vector and axial-vector mesons identified with the techni-ρ and-a1 mesons. Once γm is specified, in the continuum limit → 0 those quantities√involve only three undetermined parameters: zm , and dimensionless 2 quantities ξ = 2LhΦi|z=zm and G = hecg5 ΦX (z) i|z=zm −1 which are related to hT¯T i and hαG2µν i, respectively. Some dimensionless quantities such as Sˆ ≡ S/(Nf /2) (S parameter per each techni-fermion doublet) and Mρ,a1 ,TD /Fπ (ratios of meson masses to Fπ ) are expressed as a function of only ξ and G when γm fixed. 2. Mass of techni-dilaton We first present a generic analysis of the effects of the gluon condensate and the anomalous dimension on Mρ,a1 ,TD /Fπ . The gluon contribution may be evaluated 1/4 1 1 2 4 2 4 hαG i/F hαG i/f / . through a quantity Γ = Γ(γm , ξ, G) ≡ µν π µν π QCD π π
Fig. 1 shows plots of Mρ,a1 ,TD /Fπ as a function of γm and Γ and indicates that MTD /Fπ rapidly decreases as γm and/or Γ become larger in contrast to Mρ,a1 /Fπ .
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We now consider a typical model of W/C TC, based on the Caswell-BanksZaks infrared fixed point in the large Nf QCD where we use the estimate Nf ' 4NTC = 12(NTC = 3) from the two-loop beta function and ladder SchwingerDyson equation. In the W/C TC dynamics the techni-gluon condensate is not on order of the intrinsic scale ΛTC but of the mass of techni-fermions m( ΛTC ) through the conformal anomaly associated with the mass generation, ∂ µ Dµ = θµµ = N N 4θ00 = β(α)/(4α2 ) · hαG2µν i and hθ00 i = − fπ4TC m4 , where the nonperturbative beta function behaves as β(α) → 0 when ΛTC /m → ∞ (conformal fixed point). This implies that the gluon condensate hαG2µν i (or Γ) becomes large in that limit and hence mTD → 0 as is seen in Fig. 1. Indeed ΛTC is identified with the Extended TC 4 5 (ETC) scale p ΛETC : ΛTC = ΛETC ∼ (10 − 10 )Fπ where Fπ (= O(m)) is given by Fπ = 246/ Nf /2 GeV. Using the phenomenological input S = 0.1, we have Γ ' 7. Then we find MTD ' 550–680 GeV in contrast to Mρ ' Ma1 ' 3.8–3.9 TeV. This work was supported in part by the Global COE Program “Quest for Fundamental Principles in the Universe” and the Daiko Foundation. S.M. was supported by the Korean Research Foundation Grant (KRF-2008-341-C00008). References 1. K. Yamawaki, M. Bando and K. Matumoto, Phys. Rev. Lett. 56, 1335 (1986); M. Bando, K. Matumoto and K. Yamawaki, Phys. Lett. B 178, 308 (1986). 2. T. Akiba and T. Yanagida, Phys. Lett. B 169, 432 (1986); T. W. Appelquist, D. Karabali and L. C. R. Wijewardhana, Phys. Rev. Lett. 57, 957 (1986). See also B. Holdom, Phys. Lett. B 150, 301 (1985). 3. K. Haba, S. Matsuzaki, and K. Yamawaki, in preparation. 4. J. Erlich, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 95, 261602 (2005); L. Da Rold and A. Pomarol, Nucl. Phys. B 721, 79 (2005). 5. D. K. Hong and H. U. Yee, Phys. Rev. D 74, 015011 (2006); M. Piai, arXiv:0608241[hepph]. 6. K. Haba, S. Matsuzaki and K. Yamawaki, Prog. Theor. Phys. 120, 691 (2008).
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Phase Structure of Topologically Massive Gauge Theory with Fermion Yuichi Hoshino Kushiro National College of Technolofy, Otanoshike Nishi 2 -32-1, Kushiro City Hokkaido 084-0916, Japan E-mail: [emailprotected] Using Bloch-Nordsieck approximation fermion propagator in 3-dimensional gauge theory with topological mass is studied. Infrared divergence of Chern-Simon term is soft,which modifies anomalous dimension. In unquenched QCD with 2-component spinor anomalous dimension has fractional value,where order parmeter is divergent. Keywords: Topologically massive gauge theory, parity.
1. Introduction The Lagrangeans of Topologically massive gauge theory with fermion are 1 1 1 1 (1) L = Fµν F µν + θµνρ Fµν Aρ + ψ(iγ · (∂ − ieA) − m)ψ + (∂ · A)2 , 4 4 2d θ 2 1 L = 2 tr(Fµν F µν ) − 2 µνρ tr(Fµν Aρ − Aµ Aν Aρ) 4g 4g 3 1 + ψ(iγ · (∂ − ieA) − m)ψ + (∂ · A)2 , (2) 2d where θ = (g 2 /4π)n, (n = 0, ±1, ±2...) with γ matrix for 4-comonent fermion.2 In comparison with massless QED3 , QCD3 topological mass term seems to soften the infrared divergence of massive fermion near its on-shell. In Minkowski metric we have 5 γ matrices {γµ , γν } = 2gµν , (µ = 0, 1, 2) 0I 0 −iI σ1,2 0 σ3 0 , γ4 = , γ5 = , , γ1,2 = −i γ0 = I0 iI 0 0 −σ1,2 0 −σ3 −i τ≡ [γ4 , γ5 ] = diag (I, −I) . (3) 2 There are two redundant matrices γ4 and γ5 which anticommutes with other three γ matrices. There exists two kind of chiral transformation ψ → exp(iαγ4 )ψ, ψ → exp(αγ5 )ψ. The matrices {γ4 , γ5 , I4 , τ } generate a U (2) chiral symmetry containing massless spinor fields. This U (2) symmetry is broken down to U (1) × U (1) by a spinor mass term me ψψ and parity violating mass mo ψτ ψ, where ψτ ψ is a spin density. Here after we take 4-component spinors to study chiral symmetry breaking by me ψψ in pure QED3 , where ψτ ψ is invariant under chiral transformation.After that the effects of Chern-Simon term will be studied by 2-component spinors.
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2. Fermion spectral function The spectral function of 4-component fermion and photon propagator3 are defined as √ exp(−m −x2 ) √ SF (x) = SF0 (x) exp(F (x)), SF0 (x) = −(iγ · ∂ + m) . (4) 4π x2 DF0 (k) = −i(
gµν − kµ kν /k 2 − iθµνρ k ρ /k 2 kµ kν ) + id 4 , k 2 − θ2 + i k
(5)
where F is an O(e2 ) matrix element |T1 |2 for the process electron (p+k) → electron (p) + photon (k) as µ (k, λ) γ µ exp(i(p + k) · x)U (p, s), γ · (p + k) − m X 1 (d − 1) γ · p e2 m γ · p + m 2 m2 T1 T1 = − e [ + + ] − , 2m (p · k)2 p·k k2 m 4θ p · k T1 = −ie
(6) (7)
λ,S
F =
Z
X d3 k T1 T1 . exp(ikx)θ(k0 )δ(k 2 − θ2 ) 2 (2π)
(8)
λ,S
Here we use the retarded propagator to derive the function F √ Z exp(−θ −x2 ) d3 k 2 2 √ D+ (x) = exp(ik · x)θ(k )δ(k − θ ) = 0 i(2π)2 8πi −x2
(9)
The function F is evaluated by α integration for pure QED3 .3,4 Z ∞ Z ∞ ∂ F = ie2 m2 αdαDF (x + αp) − e2 dαDF (x + αp) − i(d − 1)e2 2 DF (x, θ2 ) ∂θ 0 0 e2 exp(−θ|x|) − θ|x|E1 (θ|x|) E1 (θ|x|) (d − 1) exp(−θ |x|)) [ − + ]. (10) = 8π θ m 2θ
It is well known that function F has linear and logarithmic infrared divergence with respect to θ where θ is a bare photon mass. Here we notice the followings. (1) exp(F ) includes all infrared divergences. (2) quenched propagator has linear and logarithmic infrared divergences. Linear divegence is absent in a special gauge. In unquenched of exp(F ) is modified by dressed boson spectral R case θ dependence √ function to dsργ (s) exp(F (x, s), where πργ (s) = −=(s − Π(s))−1 . 3. Phase strucutures For short and long distance we have the approximate form of the function F FS ∼ A − θ|x| + (D + C|x|) ln(θ |x|)) −
(d + 1 − 2γ)e2 |x| , (θ|x| 1), FL ∼ 0, (1 θ|x|). 16π
(11)
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From the above formulae we have exp(F ) = A(θ|x|)D+C|x| (θ|x| 1), A = exp(
e2 (1 + d) e2 γ + ), 16πθ 8πm
e2 e2 ,D = , (12) 8π 8πm where m is the physical mass and γ is an Euler’s constant.Here see D acts to
we change the power of |x|. For D = 1, SF (0) is finite and we have ψψ 6= 0.4,6 Thus if we require D = 1, we obtain m = e2 /8π. In the same way we add the conrtribution of Chern-Simon term. For simplicity we consider the 2-component spinors in (7). In the condensed phase we have the modified anomalous dimension for θ > 0 C=
0 e2 e2 e2 e2 e2 e2 e2 + = 1, m = /(1 − ), ∆m = /(1 − ). (13) 8πm 32πθ 8π 32πθ 8π 32πθ 32πθ In unquenched case there are parity even and odd spectral function of gauge boson by vacuum polarization.1 In this case we separate D into parity even and odd contribution De = e2 /8πm, DO = e2 /32πθ. For Topologically Massive QCD, θ is quantized with n(0, ±1, ±2, ..). Thus we have D = e2 /(8πm) + 1/8n = 1 for e 2 O quenched and D = e /8πm = 1, D = 1/(8n) for unquenched case, which
case, leads to ψψ = ∞ for any n 6= 0. In the 4-component spinor free fermion propagator is decomposed into chiral representation
D=
SF (p) =
1 (γ · p + m+ )τ+ (γ · p + m− )τ− = 2 + 2 . m I + m O τ − γ · p p − m2+ + i p − m2− + i
(14)
The difference in two spectral functions is a opposite sign of each Chern-Simon contribution for τ± . In Toplogically massive gauge theory dynamical mass is parity even and Chern-Simon term shifts mass of different chirality
with opposite sign. However shifted mass may not strongly depend on θ but ψψ is proportional to θ(12).5 Our approximation is convenient for unquenched case by the use of gauge boson spectral function.6 References 1. S. Deser, R. Jackiw & Templeton, Annals of Physics 281, 409-449 (2000). 2. C. J. Burden, Nuclear Physics B387 (1992) 419-446, Kei-ichi Kondo, Int. J. Mod. Phys. A11; 777-822, 1996. 3. R. Jackiw, L. Soloviev, Phys. Rev. 173.5 (1968) 1485. 4. Yuichi Hoshino, in CONTINUOUS ADVANCES IN QCD 2008; 361-372. 5. Toyoki Matsuyama, Hideko Nagahiro, Mod. Phys. Lett. A15 (2000) 2373-2386. 6. C. S. Fisher, Reinhart Alkofer, T. Darm, P. Maris, Phys. Rev. D70: 073007, 2004.
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New Regularization in Extra Dimensional Model and Renormalization Group Flow of the Cosmological Constant Shoichi Ichinose Laboratory of Physics, School of Food and Nutritional Sciences University of Shizuoka, Yada 52-1, Shizuoka 422-8526, Japan E-mail: [emailprotected] Casimir energy is calculated for 5D scalar theory in the warped geometry. A new regularization, called sphere lattice regularization, is taken. The regularized configuration is closed-string like. We numerically evaluate Λ(4D UV-cutoff), ω(5D bulk curvature, warp parameter) and T (extra space IR parameter) dependence of Casimir energy. 5D Casimir energy is finitely obtained after the proper renormalization procedure. The warp parameter ω suffers from the renormalization effect. We examine the cosmological constant problem.
1. Introduction In the quest for the unified theory, the higher dimensional (HD) approach is a fascinating one from the geometrical point. Historically the initial successful one is the Kaluza-Klein model, which unifies the photon, graviton and dilaton from the 5D space-time approach. The HD theories , however, generally have the serious defect as the quantum field theory(QFT) : un-renormalizability. The HD quantum field theories, at present, are not defined within the QFT. In 1983, the Casimir energy in the Kaluza-Klein theory was calculated by Appelquist and Chodos.1 They took the cut-off (Λ) regularization and found the quintic (Λ5 ) divergence and the finite term. The divergent term shows the unrenormalizability of the 5D theory, but the finite term looks meaningful2 and, in fact, is widely regarded as the right vacuum energy which shows contraction of the extra axis. In the development of the string and D-brane theories, a new approach to the renormalization group was found. It is called holographic renormalization. We regard the renormalization flow as a curve in the bulk. The flow goes along the extra axis. The curve is derived as a dynamical equation such as Hamilton-Jacobi equation. It originated from the AdS/CFT correspondence. Spiritually the present basic idea overlaps with this approach. 2. Casimir Energy of 5D Scalar Theory In the warped geometry, ds2 = 1 µ ν +dz2 ) , we consider the 5D massive scalar theory with m2 = −4ω 2 . ω 2 z 2 (η √µν dx dx 1 A L = −G(− 2 ∇ Φ∇A Φ − 12 m2 Φ2 ) . The Casimir energy ECas is given by Z Z X 1 5 −T −4 ECas = DΦ exp{i d XL} e = exp {− ln(p2E + Mn2 )} , (1) 2 Euclid n,p
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where Mn is the eigenvalues of the following operator. ˆ z + Mn 2 }ψn (z) = 0 , {s(z)−1 L
2 ˆz ≡ d 1 d − m , L 3 dz (ωz) dz (ωz)5
(2)
1 where s(z) = (ωz) 3 . Z2 parity is imposed as: ψn (z) = −ψn (−z) for P = − ; ψn (z) = ψn (−z) for P = +. The expression (1) is the familiar one of the Casimir energy. It is re-expressed in a closed form using the heat-kernel method and the propagator. First we can express it, using the heat equation solution, as follows (ω/T = eωl ). Z Z ∞ d4 p 1 dt −T −4 ECas −4 0 e = (const) × exp T 2 Tr Hp (z, z ; t) , (2π)4 0 2 t Z 1/T ∂ ˆ z − p2 )}Hp (z, z 0 ; t) = 0 . (3) Tr Hp (z, z 0 ; t) = s(z)Hp (z, z; t)dz , { − (s−1 L ∂t 1/ω
The heat kernel Hp (z, z 0 ; t) is formally solved, using the Dirac’s bra and ket vec−1 ˆ 2 0 tors (z|, |z), as Hp (z, z 0 ; t) = (z|e−(−s Lz +p )t ). We here introduce the posiR |z ∞ ∓ ∓ 0 tion/momentum propagators Gp : Gp (z, z ) ≡ 0 dt Hp (z, z 0 ; t). They satisfy the following differential equations of propagators. ( ˆ (z)(z 0 )δ(|z| − |z 0 |) P= − 1 0 ˆ z − p2 s(z))G∓ (L (4) p (z, z ) = ˆ δ(|z| − |z 0 |) P=1 G∓ p can be expressed in a closed form. Taking the Dirichlet condition at all fixed points, the expression for the fundamental region (1/ω ≤ z ≤ z 0 ≤ 1/T ) is given by 0 G∓ p (z, z ) = ∓
where p˜ ≡
p
p˜ pz) ∓ K0 ( ωp˜ )I0 (˜ pz)}{I0 ( Tp˜ )K0 (˜ pz 0 ) ∓ K0 ( Tp˜ )I0 (˜ pz 0 )} ω 3 2 0 2 {I0 ( ω )K0 (˜ z z , 2 I0 ( Tp˜ )K0 ( ωp˜ ) − K0 ( Tp˜ )I0 ( ωp˜ )
(5)
p2 , p2 ≥ 0. We can express Casimir energy as,
Λ,∓ −ECas (ω, T ) =
p
Z
Z 1/T Z Λ 2 d4 pE ∓ ∓ ˜ G∓ (z, z)dk ˜, k dz F (˜ p , z), F (˜ p , z) = k (2π)4 p≤Λ (ωz)3 p˜ 1/ω ˜
(6)
where p˜ = p2E . The momentum symbol pE indicates Euclideanization. Here we introduce the UV cut-off parameter Λ for the 4D momentum space. 3. UV and IR Regularization and Evaluation of Casimir Energy The integral region of the above equation (6) is displayed in Fig.1. In the figure, we introduce the regularization cut-offs for the 4D-momentum integral, µ ≤ p˜ ≤ Λ. For simplicity, we take the following IR cutoff of 4D momentum : µ = Λ · Tω = Λe−ωl . Importantly, (6) shows the scaling behavior for large values of Λ and 1/T . Λ,− 2π 2 From a close numerical analysis, we have confirmed : (4A) ECas (ω, T ) = (2π) 4 × h i Λ5 5 −0.0250 T . The Λ -divergence, (4A), shows the notorious problem of the higher dimensional theories. We have proposed an approach to solve this problem and given
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UV
p =
’ ’
IR
z
z 1/T
Fig. 1. Space of (z,˜ p) for the integration. The hyperbolic curve was proposed.3
1/T
Fig. 2. Space of (˜ p,z) for the integration (present proposal).
a legitimate explanation within the 5D QFT.4,5 See Fig.2. The IR and UV cutoffs change along the etra axis. Their S 3 -radii are given by rIR (z) and rU V (z). The 5D volume region bounded by BU V and BIR is the integral region of the Casimir energy ECas . The forms of rU V (z) and rIR (z) can be determined by the minimal 00 r d2 r dr , r00 ≡ dz = 0, r0 ≡ dz area principle: 3 + z4 r0 r − rr02 +1 2 , 1/ω ≤ z ≤ 1/T . We have confirmed, by numerically solving the above differential eqation (Runge-Kutta), those curves that show the flow of renormalization really appear. The results imply the boundary conditions determine the property of the renormalization flow. 4. Weight Function and the Meaning We consider another approach which respects the minimal area principle. Let us introduce, instead of restricting the integral region, a weight function W (˜ p, z) in the (˜ p, z)-space for the purpose of suppressing UV and IR divergences of the Casimir Energy. Z 4 Z q d pE 1/T ∓ W ∓ −ECas (ω, T ) ≡ dz W (˜ p , z)F (˜ p , z) , p ˜ = p24 + p21 + p22 + p23 , (2π)4 1/ω −1 −(1/2)p˜2 /ω 2 −(1/2)z 2 T 2 ≡ W1 (˜ p, z), N1 = 1.711/8π 2 elliptic suppr. (N1 ) e 3 −1 −pzT ˜ /ω (N2 ) e ≡ W2 (˜ p, z), N2 = 2 Tω 3 /8π 2 hyperbolic suppr.1 (7) (N )−1 e−1/2(p˜2 /ω2 +1/z2 T 2 ) ≡ W (˜ 2 reciprocal suppr.1 8 8 p, z), N8 = 0.4177/8π
where F ∓ (˜ p, z) are defined in (6). They (except W2 ) give, after normalizing the factor Λ/T , only the log-divergence. W ECas /ΛT −1 = −αω 4 (1 − 4c ln(Λ/ω) − 4c0 ln(Λ/T )) ,
(8)
where the numerical values of α, c and c0 are obtained depending on the choice of the weight function.6 This means the 5D Casimir energy is finitely obtained by the ordinary renormalization of the warp factor ω. In the previous work,5 we have presented the following idea to define the weight function W (˜ p, z). In the evaluation (7), the (˜ p, z)-integral is over the rectangle region shown in Fig.1 (with Λ → ∞ and µ → 0). Following Feynman,7 we can replace the
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integral by the summation over all possible paths p˜(z). W −ECas (ω, T ) =
Z
Dp˜(z)
Z
1/T
dz S[˜ p(z), z], S[˜ p(z), z] =
1/ω
2π 2 p˜(z)3 W (˜ p(z), z)F ∓ (˜ p(z), z). (2π)4
(9)
There exists the dominant path p˜W (z) which is determined by the minimal principle: ∂ ln(W F ) F) dp˜ δS = 0. Dominant Path p˜W (z) : / ( p3˜ + ∂ ln(W ). Hence it is dz = − ∂z ∂ p˜ fixed by W (˜ p, z). On the other hand, there exists another independent path: the r 00 r = 0, minimal surface curve rg (z). Minimal Surface Curve rg (z) : 3 + z4 r0 r − r02 +1 1 1 ω ≤ z ≤ T . It is obtained by the minimal area principle: δA = 0 where xa xb dxa dxb ) ≡ gab (x)dxa dxb , A = ds = (δab + (rr0 )2 ω 2 z 2 2
Z
√ 4 g d x=
Z
1/T 1/ω
p r0 2 + 1 r3 dz. ω4 z4
(10)
Hence rg (z) is fixed by the induced geometry gab (x). Here we put the requirement:5 (4A) p˜W (z) = p˜g (z), where p˜g ≡ 1/rg . This means the following things. We require the dominant path coincides with the minimal surface line p˜g (z) = 1/rg (z) which is defined independently of W (˜ p, z). W (˜ p, z) is defined here by the induced geometry gab (x). In this way, we can connect the integral-measure over the 5D-space with the geometry. We have confirmed the coincidence by the numerical method. In order to most naturally accomplish the above requirement, we can go to a new step. Namely, we propose to replace the 5D space integral with the weight W , (7), by the following path-integral. We newly define the Casimir energy in the higher-dimensional theory as follows. −ECas (ω, T, Λ) =
Z
1/µ 1/Λ
" Z # p Z r(ω −1 ) Y 1/T r0 2 + 1 r3 1 a −1 dz , Dx (z)F ( , z) exp − dρ = r(T ) r 2α0 ω 4 z 4 1/ω a,z =ρ
(11) −1
where µ = ΛT /ω and the limit ΛT → ∞ is taken. The string (surface) tension parameter 1/2α0 is introduced. (Note: Dimension of α0 is [Length]4 . ) F (˜ p, z) is defined in (6) and shows the field-quantization of the bulk scalar (EM) fields. 5. Discussion and Conclusion When c and c0 are sufficiently small we find the renormalization group function for the warp factor ω as ωr = ω(1 − c ln(Λ/ω) − c0 ln(Λ/T )) , β ≡
∂ ωr ln = −c − c0 . ∂(ln Λ) ω
(12)
No local counterterms are necessary. Through the Casimir energy calculation, in the higher dimension, we find a way to quantize the higher dimensional theories within the QFT framework. The quantization with respect to the fields (except the gravitational fields G AB (X)) is done in the standard way. After this step, the has the summation over the 5D R expression Q a space(-time) coordinates or momenta dz a dpR . We have proposed that this sumQ a mation should be replaced by the path-integral a,z Dp (z) with the area action
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R√ (Hamiltonian) A = det gab d4 x where gab is the induced metric on the 4D surface. This procedure says the 4D momenta pa (or coordinates xa ) are quantum statistical operators and the extra-coordinate z is the inverse temperature (Euclidean time). We recall the similar situation occurs in the standard string approach. The space-time coordinates obey some uncertainty principle.8 Recently the dark energy (as well as the dark matter) in the universe is a hot subject. It is well-known that the dominant candidate is the cosmological term. matter ,S = constant Rλ appears as: (5A) Rµν − 21 gµν R − λgµν = Tµν RThe4 cosmological √ 1 4 √ d x −g{ GN (R + λ)} + d x −g{Lmatter }, g = det gµν . We consider here the 3+1 dim Lorentzian space-time (µ, ν = 0, 1, 2, 3). The constant λ observationally takes the value : (5B) G1N λobs ∼ GN R1cos 2 ∼ m4ν ∼ (10−3 eV )4 , λobs ∼ R12 ∼ cos
4×10−66 (eV )2 , where Rcos ∼ 5×1032 eV−1 is the cosmological size (Hubble length), mν is the neutrino mass. On the other hand, we have theoretically so far : (5C) 4 1 1 28 4 GN λth ∼ GN 2 = Mpl ∼ (10 eV ) . We have the famous huge discrepancy factor λth 2 : (5D) λobs ∼ NDL , NDL ≡ Mpl Rcos ∼ 6 × 1060 , where NDL is the Dirac’s large number. If we use the present result, we can obtain a naturalR choice of T, ω and Λ as √ follows. By identifying T −4 ECas = −α1 ΛT −1 ω 4 /T 4 with d4 x −g(1/GN )λob = ω4 Λ 2 2 2 1 Rcos (1/GN ), we obtain the following relation: (5E) NDL = Rcos GN = −α1 T 5 . The warped (AdS5 ) model predicts the cosmological constant negative, hence we have interest only in its absolute value. We take q the following choice for Λ and ω : Mpl 1 = ∼ mν ∼ 10−3 eV. (5F) Λ = Mpl ∼ 1019 GeV, ω ∼ √ 4 R 2 GN Rcos
cos
As shown above, we have the standpoint that the cosmological constant is mainly made from the Casimir energy. We do not yet succeed in obtaining the value α1 negatively, but succeed in obtaining the finiteness of the cosmological constant and its gross absolute value. The smallness of the value is naturally explained by the renormalization flow. Because we already know the warp parameter ω flows (12), 2 the λobs ∼ 1/Rcos ∝ ω 4 , says that the smallness of the cosmological constant comes from the renormalization flow for the non asymptotic-free case (c + c0 < 0 in (12)). The IR parameter T , the normalization factor Λ/T in (8) and the IR cutoff µ = −1 (NDL )1/5 ∼ 10−20 eV, TΛ = (NDL )4/5 ∼ 1050 , µ = Λ Tω are given by : (5G) T = Rcos −3/10 Mpl NDL ∼ 1GeV ∼ mN , where mN is the nucleon mass. The degree of freedom 4 6/5 Mpl 4 ω4 74 of the universe (space-time) is given by : (5H) Λ ∼ ( mN ) . µ4 = T 4 = NDL ∼ 10 References 1. 2. 3. 4. 5. 6. 7. 8.
T. Appelquist and A. Chodos, Phys. Rev. Lett. 50 (1983) 141 S. Ichinose, Phys. Lett. 152B (1985), 56 L. Randall and M.D. Schwartz, JHEP 0111 (2001) 003, hep-th/0108114 S. Ichinose and A. Murayama, Phys. Rev. D76 (2007) 065008, hep-th/0703228 S. Ichinose, Prog. Theor. Phys. 121 (2009) 727, ArXiv:0801.3064v8[hep-th] S. Ichinose, ArXiv:0812.1263[hep-th] R.P. Feynman, Statistical Mechanics, W.A. Benjamin, Inc., Massachusetts, 1972 T. Yoneya, Prog. Theor. Phys. 103 (2000) 1081
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Spectral Analysis of Dense Two-Color QCD T. Kanazawa∗ , T. Wettig† and N. Yamamoto∗ ∗ Department
of Physics, The University of Tokyo Tokyo 113-0033, Japan † Department of Physics, University of Regensburg 93040 Regensburg, Germany E-mails: [emailprotected], [emailprotected], [emailprotected] We study spectral properties of two-color QCD with an even number of flavors at high baryon density. We construct the low-energy effective Lagrangian for the NambuGoldstone bosons, derive Leutwyler-Smilga-type spectral sum rules and construct a suitable random matrix theory. Our results can in principle be tested in lattice QCD simulations. Keywords: Chiral symmetry breaking, diquark condensate, nonzero chemical potential, chiral random matrix theory.
1. Introduction Understanding the phase structure of Quantum Chromodynamics (QCD) at low temperature (T ) and nonzero quark chemical potential (µ) is an important subject relevant to many areas of physics,1 but the severe fermion sign problem in firstprinciple lattice simulations is hindering progress in this field considerably. In this report we analyse dense two-color QCD with an even number of flavors. Our principal motivation comes from the fact that two-color QCD can be simulated on the lattice even at µ 6= 0. At large µ, two-color matter exists in the form of a BCS superfluid, which is the genuine counterpart of the color superconducting phase for three-color QCD. Below, we will construct the low-energy effective Lagrangian for the Nambu-Goldstone (NG) bosons, define a new ε-regime, obtain exact spectral sum rules for the Dirac eigenvalues, and introduce a chiral random matrix theory (ChRMT) for the new ε-regime.2,3 Testing of our results on the lattice would provide the first signature of BCS pairing and give the magnitude of the fermion gap ∆. For the corresponding work in dense three-color QCD, see Ref. 4. 2. Low-energy effective theory At µ ΛQCD , perturbative one-gluon exchange indicates that the color antisymmetric channel is attractive, implying instability of the Fermi surface. A gap ∆
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appears in the spectrum of quasiquarks near the Fermi surface, while the diquark condensate forms in the color- and flavor-antisymmetric channel, breaking chiral symmetry spontaneously according to SU(Nf )L × SU(Nf )R × U(1)B × U(1)A → Sp(Nf )L × Sp(Nf )R ,
(1)
where we neglected the U(1)A -anomaly (since µ is large) and assumed that Nf is even and ≥ 4. (See Ref. 2 for Nf = 2.) The NG fields originating from (1), ΣL ∈ SU(Nf )L /Sp(Nf )L ,
ΣR ∈ SU(Nf )R /Sp(Nf )R ,
V ∈ U(1)B ,
A ∈ U(1)A ,
are gapless in the chiral limit and govern the low-energy physics near the Fermi surface. From symmetry principles and weak-coupling calculations, the chiral Lagrangian valid at energy scales below ∆ is given by 2 o Nf f 20 n o f2 n η 2 |∂i V |2 + |∂0 A|2 − vη20 |∂i A|2 L = H |∂0 V |2 − vH (3) 2 2 o n o 2n 2 3∆ f 2 T † A Tr(M Σ M Σ ) + c.c. , + π Tr |∂0 ΣL |2 − vπ2 |∂i ΣL |2 + (L ↔ R) − R L 2 4π 2 where M is a generalized Nf × Nf quark mass matrix. The absence of an O(M ) term in the chiral Lagrangian is a consequence of the Z(2)L × Z(2)R symmetry of the diquark pairing. As V and the gluon fields decouple from the other NG modes, they will be neglected in the following. 3. Partition function at finite volume Next, we consider two-color QCD in a Euclidean box of size L4 (≡ V4 ) and let mNG denote the masses of NG fields. The dynamics simplifies drastically in the regime 1 1 L ∆ mNG
(4)
since the zero modes of the NG fields dominate the physics.2,5 Thus the partition function in the new ε-regime (4) is simply given by Z Z Z 3 2 2 T † dΣR exp dΣL Z(M ) = dA V4 ∆ Re[A Tr(M ΣR M ΣL )] . 2π 2 U(1)A SU(Nf )L /Sp(Nf )L SU(Nf )R /Sp(Nf )R
(5) 4. Spectral sum rules for the Dirac operator Let us denote the complex Dirac eigenvalues by iλn . Starting from the microscopic Lagrangian of two-color QCD, we find the normalized partition function to be Y 0 M †M , (6) Z(M ) = det 1 + λ2n n where h· · · i represents expectation values with respect to the measure in the chiQ0 P0 ral limit. n (and later n ) denotes the product (sum) over all eigenvalues with
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Re λn > 0. Equating the above two expressions for Z(M ), we can derive a number of novel spectral sum rules, e.g., X X X X 0 0 1 0 1 0 1 1 9(V4 ∆2 )2 . (7) = = = 0, = λ2n λ2 λ2 λ6n λ4n 4π 4 (Nf − 1)2 m
