Abstract
We discuss consequences of the ’t Hooft anomaly matching condition for Quantum Chromodynamics (QCD) with massless fundamental quarks. We derive the new discrete ’t Hooft anomaly of massless QCD for generic numbers of color Nc and flavor Nf , and an exotic chiral-symmetry broken phase without quark-bilinear condensate is ruled out from possible QCD vacua. We show that the U(1)B baryon number symmetry is anomalously broken when the \( {\left({\mathrm{\mathbb{Z}}}_{2{N}_{\mathrm{f}}}\right)}_{\mathrm{A}} \) discrete axial symmetry and the flavor symmetry are gauged. In the ordinary chiral symmetry breaking, the Skyrmion current turns out to reproduce this ’t Hooft anomaly of massless QCD. In the exotic chiral symmetry breaking, however, the anomalous breaking of U(1)B does not take the correct form, and it is inconsistent with anomaly matching. This no-go theorem is based only on symmetries and anomalies, and thus has a wider range of applicability to the QCD phase diagram than the previous one obtained by QCD inequalities. Lastly, as another application, we check that duality of \( \mathcal{N}=1 \) supersymmetric QCD with Nf ≥Nc + 1 satisfies the new anomaly matching.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. I, Phys. Rev. 122 (1961) 345 [INSPIRE].
Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. II, Phys. Rev. 124 (1961) 246 [INSPIRE].
S. Weinberg, Dynamical approach to current algebra, Phys. Rev. Lett. 18 (1967) 188 [INSPIRE].
J.S. Schwinger, Chiral dynamics, Phys. Lett. B 24 (1967) 473 [INSPIRE].
S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177 (1969) 2239 [INSPIRE].
C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247 [INSPIRE].
G. ’t Hooft, Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking, in the proceedings of Developments in gauge theories, August 26-September 8, Cargese, France (1980) [INSPIRE]
Y. Frishman, A. Schwimmer, T. Banks and S. Yankielowicz, The axial anomaly and the bound state spectrum in confining theories, Nucl. Phys. B 177 (1981) 157 [INSPIRE].
S.R. Coleman and B. Grossman, ’t Hooft’s consistency condition as a consequence of analyticity and unitarity, Nucl. Phys. B 203 (1982) 205 [INSPIRE].
A. Kapustin and R. Thorngren, Anomalies of discrete symmetries in three dimensions and group cohomology, Phys. Rev. Lett. 112 (2014) 231602 [arXiv:1403.0617] [INSPIRE].
A. Kapustin and R. Thorngren, Anomalies of discrete symmetries in various dimensions and group cohomology, arXiv:1404.3230 [INSPIRE].
R. Stora, Continuum gauge theories, in New developments in quantum field theory and statistical mechanics Cargèse 1976 , M. Lévy and P. Mitter eds., Nato Advanced Study Institutes Series volume 26, Plenum Press, U.S.A. (1977) [INSPIRE].
R. Stora, Algebraic structure and topological origin of anomalies, in Progress in Gauge Field Theory, Springer, Germany (1983) [INSPIRE].
B. Zumino, Chiral anomalies and differential geometry, in the proceedings of the 40th Summer School of Theoretical Physics, June 27-August 4, Les Houches, France (1983) [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].
J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95 [INSPIRE].
E. Witten, Global aspects of current algebra, Nucl. Phys. B 223 (1983) 422 [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].
A. Vishwanath and T. Senthil, Physics of three dimensional bosonic topological insulators: surface deconfined criticality and quantized magnetoelectric effect, Phys. Rev. X 3 (2013) 011016 [arXiv:1209.3058] [INSPIRE].
X.-G. Wen, Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders, Phys. Rev. D 88 (2013) 045013 [arXiv:1303.1803] [INSPIRE].
G.Y. Cho, J.C.Y. Teo and S. Ryu, Conflicting Symmetries in Topologically Ordered Surface States of Three-dimensional Bosonic Symmetry Protected Topological Phases, Phys. Rev. B 89 (2014) 235103 [arXiv:1403.2018] [INSPIRE].
J.C. Wang, Z.-C. Gu and X.-G. Wen, Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond, Phys. Rev. Lett. 114 (2015) 031601 [arXiv:1405.7689] [INSPIRE].
E.H. Lieb, T. Schultz and D. Mattis, Two soluble models of an antiferromagnetic chain, Annals Phys. 16 (1961) 407 [INSPIRE].
I. Affleck and E.H. Lieb, A Proof of Part of Haldane’s Conjecture on Spin Chains, Lett. Math. Phys. 12 (1986) 57 [INSPIRE].
M. Oshikawa, Commensurability, excitation gap, and topology in quantum many-particle systems on a periodic lattice, Phys. Rev. Lett. 84 (2000) 1535 [cond-mat/9911137].
M.B. Hastings, Lieb-Schultz-Mattis in higher dimensions, Phys. Rev. B 69 (2004) 104431 [cond-mat/0305505] [INSPIRE].
C. Csáki and H. Murayama, Discrete anomaly matching, Nucl. Phys. B 515 (1998) 114 [hep-th/9710105] [INSPIRE].
E. Witten, Fermion path integrals and topological phases, Rev. Mod. Phys. 88 (2016) 035001 [arXiv:1508.04715] [INSPIRE].
N. Seiberg and E. Witten, Gapped boundary phases of topological insulators via weak coupling, PTEP 2016 (2016) 12C101 [arXiv:1602.04251] [INSPIRE].
E. Witten, The “parity” anomaly on an unorientable manifold, Phys. Rev. B 94 (2016) 195150 [arXiv:1605.02391] [INSPIRE].
Y. Tachikawa and K. Yonekura, On time-reversal anomaly of 2 + 1d topological phases, PTEP 2017 (2017) 033B04 [arXiv:1610.07010] [INSPIRE].
Y. Tachikawa and K. Yonekura, More on time-reversal anomaly of 2 + 1d topological phases, Phys. Rev. Lett. 119 (2017) 111603 [arXiv:1611.01601] [INSPIRE].
D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, time reversal and temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [INSPIRE].
C. Wang, A. Nahum, M.A. Metlitski, C. Xu and T. Senthil, Deconfined quantum critical points: symmetries and dualities, Phys. Rev. X 7 (2017) 031051 [arXiv:1703.02426] [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, A. Sharon, R. Thorngren and X. Zhou, Comments on Abelian Higgs models and persistent order, arXiv:1705.04786 [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].
G.Y. Cho, S. Ryu and C.-T. Hsieh, Anomaly manifestation of Lieb-Schultz-Mattis theorem and topological phases, Phys. Rev. B 96 (2017) 195105 [arXiv:1705.03892] [INSPIRE].
H. Shimizu and K. Yonekura, Anomaly constraints on deconfinement and chiral phase transition, Phys. Rev. D 97 (2018) 105011 [arXiv:1706.06104] [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. Kikuchi and Y. Tanizaki, Global inconsistency, ’t Hooft anomaly and level crossing in quantum mechanics, PTEP 2017 (2017) 113B05 [arXiv:1708.01962] [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].
J. Gomis, Z. Komargodski and N. Seiberg, Phases of adjoint QCD 3 and dualities, SciPost Phys. 5 (2018) 007 [arXiv:1710.03258] [INSPIRE].
Y. Tanizaki, T. Misumi and N. Sakai, Circle compactification and ’t Hooft anomaly, JHEP 12 (2017) 056 [arXiv:1710.08923] [INSPIRE].
Y. Tanizaki, Y. Kikuchi, T. Misumi and N. Sakai, Anomaly matching for the phase diagram of massless ℤN -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].
M. Yamazaki, Relating ’t Hooft Anomalies of 4d pure Yang-Mills and 2d ℂℙN− 1 model, arXiv:1711.04360 [INSPIRE].
M. Guo, P. Putrov and J. Wang, Time reversal, SU(N) Yang-Mills and cobordisms: interacting topological superconductors/insulators and quantum spin liquids in 3 + 1D, Annals Phys. 394 (2018) 244 [arXiv:1711.11587] [INSPIRE].
G.V. Dunne, Y. Tanizaki and M. Ünsal, Quantum distillation of Hilbert spaces, semi-classics and anomaly matching, JHEP 08 (2018) 068 [arXiv:1803.02430] [INSPIRE].
T. Sulejmanpasic and Y. Tanizaki, C-P-T anomaly matching in bosonic quantum field theory and spin chains, Phys. Rev. B 97 (2018) 144201 [arXiv:1802.02153] [INSPIRE].
C. Córdova, T.T. Dumitrescu and K. Intriligator, Exploring 2-group global symmetries, arXiv:1802.04790 [INSPIRE].
K. Aitken, A. Cherman and M. Ünsal, Dihedral symmetry in SU(N) Yang-Mills theory, arXiv:1804.05845 [INSPIRE].
R. Kobayashi, K. Shiozaki, Y. Kikuchi and S. Ryu, Lieb-Schultz-Mattis type theorem with higher-form symmetry and the quantum dimer models, arXiv:1805.05367 [INSPIRE].
Y. Tanizaki and T. Sulejmanpasic, Anomaly and global inconsistency matching: θ-angles, SU(3)/U(1)2 nonlinear σ-model, SU(3) chains and its generalizations, arXiv:1805.11423 [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, On two-flavor QCD(adj), arXiv:1805.12290 [INSPIRE].
M.M. Anber and E. Poppitz, Anomaly matching, (axial) Schwinger models and high-T super Yang-Mills domain walls, arXiv:1807.00093 [INSPIRE].
T.H.R. Skyrme, A nonlinear field theory, Proc. Roy. Soc. Lond. A 260 (1961) 127 [INSPIRE].
T.H.R. Skyrme, A unified field theory of mesons and baryons, Nucl. Phys. 31 (1962) 556 [INSPIRE].
J. Stern, Light quark masses and condensates in QCD, Lect. Notes Phys. 513 (1998) 26 [hep-ph/9712438] [INSPIRE].
J. Stern, Two alternatives of spontaneous chiral symmetry breaking in QCD, hep-ph/9801282 [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].
A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].
P. van Baal, Some results for SU(N) gauge fields on the hypertorus, Commun. Math. Phys. 85 (1982) 529 [INSPIRE].
T. Banks and N. Seiberg, Symmetries and strings in field theory and gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
L. Álvarez-Gaumé and P.H. Ginsparg, The structure of gauge and gravitational anomalies, Annals Phys. 161 (1985) 423 [Erratum ibid. 171 (1986) 233] [INSPIRE].
K. Yonekura, Dai-Freed theorem and topological phases of matter, JHEP 09 (2016) 022 [arXiv:1607.01873] [INSPIRE].
K. Fujikawa, Path integral measure for gauge invariant fermion theories, Phys. Rev. Lett. 42 (1979) 1195 [INSPIRE].
K. Fujikawa, Path integral for gauge theories with fermions, Phys. Rev. D 21 (1980) 2848 [Erratum ibid. D 22 (1980) 1499] [INSPIRE].
K. Fujikawa and H. Suzuki, Path integrals and quantum anomalies, Clarendon Press, Oxford U.K. (2004).
J.A. Harvey, C.T. Hill and R.J. Hill, Standard model gauging of the Wess-Zumino-Witten term: anomalies, global currents and pseudo-Chern-Simons interactions, Phys. Rev. D 77 (2008) 085017 [arXiv:0712.1230] [INSPIRE].
S.R. Coleman and E. Witten, Chiral symmetry breakdown in large N chromodynamics, Phys. Rev. Lett. 45 (1980) 100 [INSPIRE].
N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
K.A. Intriligator and N. Seiberg, Lectures on supersymmetric gauge theories and electric-magnetic duality, Nucl. Phys. Proc. Suppl. 45BC (1996) 1 [hep-th/9509066] [INSPIRE].
M.G. Alford, Color superconducting quark matter, Ann. Rev. Nucl. Part. Sci. 51 (2001) 131 [hep-ph/0102047] [INSPIRE].
K. Fukushima and T. Hatsuda, The phase diagram of dense QCD, Rept. Prog. Phys. 74 (2011) 014001 [arXiv:1005.4814] [INSPIRE].
F. Sannino, A Note on anomaly matching for finite density QCD, Phys. Lett. B 480 (2000) 280 [hep-ph/0002277] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1807.07666
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Tanizaki, Y. Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking. J. High Energ. Phys. 2018, 171 (2018). https://doi.org/10.1007/JHEP08(2018)171
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2018)171