Abstract
We explore 4-dimensional SU(N) gauge theory with a Weyl fermion in an irreducible self-conjugate representation. This theory, in general, has a discrete chiral symmetry. We use ’t Hooft anomaly matching condition of the center symmetry and the chiral symmetry, and find constraints on the spontaneous chiral symmetry breaking in the confining phase. The domain-walls connecting different vacua are discussed from the point of view of the ’t Hooft anomaly. We consider the SU(6) gauge theory with a Weyl fermion in the rank 3 anti-symmetric representation as an example. It is argued that this theory is likely to be in the confining phase. The chiral symmetry ℤ6 should be spontaneously broken to ℤ2 under the assumption of the confinement, although there cannot be any fermion bilinear condensate in this theory.
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References
G. ’t Hooft, Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking, NATO Sci. Ser. B 59 (1980) 135.
C. Csáki and H. Murayama, Discrete anomaly matching, Nucl. Phys. B 515 (1998) 114 [hep-th/9710105] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, Time Reversal and Temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [INSPIRE].
Y. Tanizaki and Y. Kikuchi, Vacuum structure of bifundamental gauge theories at finite topological angles, JHEP 06 (2017) 102 [arXiv:1705.01949] [INSPIRE].
Z. Komargodski, T. Sulejmanpasic and M. Ünsal, Walls, anomalies and deconfinement in quantum antiferromagnets, Phys. Rev. B 97 (2018) 054418 [arXiv:1706.05731] [INSPIRE].
H. Shimizu and K. Yonekura, Anomaly constraints on deconfinement and chiral phase transition, Phys. Rev. D 97 (2018) 105011 [arXiv:1706.06104] [INSPIRE].
D. Gaiotto, Z. Komargodski and N. Seiberg, Time-reversal breaking in QCD 4 , walls and dualities in 2 + 1 dimensions, JHEP 01 (2018) 110 [arXiv:1708.06806] [INSPIRE].
R. Kitano, T. Suyama and N. Yamada, θ = π in SU(N)/ℤ N gauge theories, JHEP 09 (2017) 137 [arXiv:1709.04225] [INSPIRE].
M. Yamazaki, Relating ’t Hooft Anomalies of 4d Pure Yang-Mills and 2d ℂℙN − 1 Model, JHEP 10 (2018) 172 [arXiv:1711.04360] [INSPIRE].
C. Cordova, P.-S. Hsin and N. Seiberg, Global Symmetries, Counterterms and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups, SciPost Phys. 4 (2018) 021 [arXiv:1711.10008] [INSPIRE].
Y. Tanizaki, Y. Kikuchi, T. Misumi and N. Sakai, Anomaly matching for the phase diagram of massless ℤ N -QCD, Phys. Rev. D 97 (2018) 054012 [arXiv:1711.10487] [INSPIRE].
A. Cherman and M. Ünsal, Critical behavior of gauge theories and Coulomb gases in three and four dimensions, arXiv:1711.10567 [INSPIRE].
C. Córdova, P.-S. Hsin and N. Seiberg, Time-Reversal Symmetry, Anomalies and Dualities in (2 + 1)d, SciPost Phys. 5 (2018) 006 [arXiv:1712.08639] [INSPIRE].
L.-Y. Hung, Y.-S. Wu and Y. Zhou, Linking Entanglement and Discrete Anomaly, JHEP 05 (2018) 008 [arXiv:1801.04538] [INSPIRE].
P. Draper, Domain Walls and the CP Anomaly in Softly Broken Supersymmetric QCD, Phys. Rev. D 97 (2018) 085003 [arXiv:1801.05477] [INSPIRE].
M.M. Anber and E. Poppitz, Two-flavor adjoint QCD, Phys. Rev. D 98 (2018) 034026 [arXiv:1805.12290] [INSPIRE].
C. Córdova and T.T. Dumitrescu, Candidate Phases for SU(2) Adjoint QCD 4 with Two Flavors from \( \mathcal{N} \) = 2 Supersymmetric Yang-Mills Theory, arXiv:1806.09592 [INSPIRE].
M.M. Anber and E. Poppitz, Anomaly matching, (axial) Schwinger models and high-T super Yang-Mills domain walls, JHEP 09 (2018) 076 [arXiv:1807.00093] [INSPIRE].
Y. Tanizaki, Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking, JHEP 08 (2018) 171 [arXiv:1807.07666] [INSPIRE].
Z. Bi and T. Senthil, An Adventure in Topological Phase Transitions in 3 + 1-D: Non-abelian Deconfined Quantum Criticalities and a Possible Duality, arXiv:1808.07465 [INSPIRE].
S.L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev. 177 (1969) 2426 [INSPIRE].
J.S. Bell and R. Jackiw, A PCAC puzzle: π 0 → γγ in the σ model, Nuovo Cim. A 60 (1969) 47 [INSPIRE].
I.I. Kogan, A. Kovner and M.A. Shifman, Chiral symmetry breaking without bilinear condensates, unbroken axial Z(N) symmetry and exact QCD inequalities, Phys. Rev. D 59 (1999) 016001 [hep-ph/9807286] [INSPIRE].
T. Kanazawa, Chiral symmetry breaking with no bilinear condensate revisited, JHEP 10 (2015) 010 [arXiv:1507.06376] [INSPIRE].
E. Poppitz and M. Ünsal, Chiral gauge dynamics and dynamical supersymmetry breaking, JHEP 07 (2009) 060 [arXiv:0905.0634] [INSPIRE].
R. Slansky, Group Theory for Unified Model Building, Phys. Rept. 79 (1981) 1 [INSPIRE].
N. Yamatsu, Finite-Dimensional Lie Algebras and Their Representations for Unified Model Building, arXiv:1511.08771 [INSPIRE].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, Princeton, NJ, U.S.A. (1992).
E. Witten, An SU(2) Anomaly, Phys. Lett. B 117 (1982) 324 [INSPIRE].
K. Fujikawa, Path Integral Measure for Gauge Invariant Fermion Theories, Phys. Rev. Lett. 42 (1979) 1195 [INSPIRE].
A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].
G. ’t Hooft, A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories, Nucl. Phys. B 153 (1979) 141 [INSPIRE].
G. ’t Hooft, Some Twisted Selfdual Solutions for the Yang-Mills Equations on a Hypertorus, Commun. Math. Phys. 81 (1981) 267 [INSPIRE].
E. Witten, Supersymmetric index in four-dimensional gauge theories, Adv. Theor. Math. Phys. 5 (2002) 841 [hep-th/0006010] [INSPIRE].
J. Wang, X.-G. Wen and E. Witten, Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions, Phys. Rev. X 8 (2018) 031048 [arXiv:1705.06728] [INSPIRE].
Y. Tachikawa, On gauging finite subgroups, arXiv:1712.09542 [INSPIRE].
C.G. Callan Jr. and J.A. Harvey, Anomalies and Fermion Zero Modes on Strings and Domain Walls, Nucl. Phys. B 250 (1985) 427 [INSPIRE].
F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, The five-loop β-function of Yang-Mills theory with fermions, JHEP 02 (2017) 090 [arXiv:1701.01404] [INSPIRE].
J. Wang, X.-G. Wen and E. Witten, A New SU(2) Anomaly, arXiv:1810.00844 [INSPIRE].
Y. Tachikawa and S. Terashima, Seiberg-Witten Geometries Revisited, JHEP 09 (2011) 010 [arXiv:1108.2315] [INSPIRE].
W.E. Caswell, Asymptotic Behavior of Nonabelian Gauge Theories to Two Loop Order, Phys. Rev. Lett. 33 (1974) 244 [INSPIRE].
T. Banks and A. Zaks, On the Phase Structure of Vector-Like Gauge Theories with Massless Fermions, Nucl. Phys. B 196 (1982) 189 [INSPIRE].
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Yamaguchi, S. ’t Hooft anomaly matching condition and chiral symmetry breaking without bilinear condensate. J. High Energ. Phys. 2019, 14 (2019). https://doi.org/10.1007/JHEP01(2019)014
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DOI: https://doi.org/10.1007/JHEP01(2019)014