Abstract
We study the phase structure of bifundamental quantum chromodynamics (QCD(BF)), which is the 4-dimensional SU(N) × SU(N) gauge theory coupled with the bifundamental fermion. Firstly, we refine constraints on its phase diagram from ’t Hooft anomalies and global inconsistencies, and we find more severe constraints than those in previous literature about QCD(BF). Secondly, we employ the recently-proposed semiclassical approach for confining vacua to investigate this model concretely, and this is made possible via anomaly-preserving T2 compactification. For sufficiently small T2 with the ’t Hooft flux, the dilute gas approximation of center vortices gives reliable semiclassical computations, and we determine the phase diagram as a function of the fermion mass m, two strong scales Λ1, Λ2, and two vacuum angles, θ1, θ2. In particular, we find that the QCD(BF) vacuum respects the ℤ2 exchange symmetry of two gauge groups. Under the assumption of the adiabatic continuity, our result successfully explains one of the conjectured phase diagrams in the previous literature and also gives positive support for the nonperturbative validity of the large-N orbifold equivalence between QCD(BF) and \( \mathcal{N} \) = 1 SU(2N) supersymmetric Yang-Mills theory. We also comment on problems of domain walls.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Y. Tanizaki and Y. Kikuchi, Vacuum structure of bifundamental gauge theories at finite topological angles, JHEP 06 (2017) 102 [arXiv:1705.01949] [INSPIRE].
A. Karasik and Z. Komargodski, The Bi-Fundamental Gauge Theory in 3 + 1 Dimensions: The Vacuum Structure and a Cascade, JHEP 05 (2019) 144 [arXiv:1904.09551] [INSPIRE].
G. ’t Hooft, Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking, in proceedings of the Cargese Summer Institute: Recent Developments in Gauge Theories, Cargese, France, 26 August–8 September 1979, Nato Advanced Study Institute 59, Springer, Boston, MA, U.S.A. (1980), pp. 135–157 [https://doi.org/10.1007/978-1-4684-7571-5_9] [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].
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].
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].
D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, Time Reversal, and Temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [INSPIRE].
Y. Kikuchi and Y. Tanizaki, Global inconsistency, ’t Hooft anomaly, and level crossing in quantum mechanics, Prog. Theor. Exp. Phys. 2017 (2017) 113B05 [arXiv:1708.01962] [INSPIRE].
Y. Tanizaki and T. Sulejmanpasic, Anomaly and global inconsistency matching: θ-angles, SU(3)/U(1)2 nonlinear sigma model, SU(3) chains and its generalizations, Phys. Rev. B 98 (2018) 115126 [arXiv:1805.11423] [INSPIRE].
C. Córdova, D.S. Freed, H.T. Lam and N. Seiberg, Anomalies in the Space of Coupling Constants and Their Dynamical Applications. Part I, SciPost Phys. 8 (2020) 001 [arXiv:1905.09315] [INSPIRE].
C. Córdova, D.S. Freed, H.T. Lam and N. Seiberg, Anomalies in the Space of Coupling Constants and Their Dynamical Applications. Part II, SciPost Phys. 8 (2020) 002 [arXiv:1905.13361] [INSPIRE].
S. Kachru and E. Silverstein, 4D conformal theories and strings on orbifolds, Phys. Rev. Lett. 80 (1998) 4855 [hep-th/9802183] [INSPIRE].
M. Bershadsky and A. Johansen, Large N limit of orbifold field theories, Nucl. Phys. B 536 (1998) 141 [hep-th/9803249] [INSPIRE].
M. Schmaltz, Duality of nonsupersymmetric large N gauge theories, Phys. Rev. D 59 (1999) 105018 [hep-th/9805218] [INSPIRE].
M.J. Strassler, On methods for extracting exact nonperturbative results in nonsupersymmetric gauge theories, hep-th/0104032 [INSPIRE].
R. Dijkgraaf, A. Neitzke and C. Vafa, Large N strong coupling dynamics in nonsupersymmetric orbifold field theories, hep-th/0211194 [INSPIRE].
P. Kovtun, M. Ünsal and L.G. Yaffe, Nonperturbative equivalences among large Nc gauge theories with adjoint and bifundamental matter fields, JHEP 12 (2003) 034 [hep-th/0311098] [INSPIRE].
P. Kovtun, M. Ünsal and L.G. Yaffe, Necessary and sufficient conditions for non-perturbative equivalences of large Nc orbifold gauge theories, JHEP 07 (2005) 008 [hep-th/0411177] [INSPIRE].
A. Armoni, A. Gorsky and M. Shifman, Spontaneous Z2 symmetry breaking in the orbifold daughter of N = 1 super Yang-Mills theory, fractional domain walls and vacuum structure, Phys. Rev. D 72 (2005) 105001 [hep-th/0505022] [INSPIRE].
P. Kovtun, M. Ünsal and L.G. Yaffe, Can large Nc equivalence between supersymmetric Yang-Mills theory and its orbifold projections be valid?, Phys. Rev. D 72 (2005) 105006 [hep-th/0505075] [INSPIRE].
G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
Y. Tanizaki and M. Ünsal, Center vortex and confinement in Yang-Mills theory and QCD with anomaly-preserving compactifications, Prog. Theor. Exp. Phys. 2022 (2022) 04A108 [arXiv:2201.06166] [INSPIRE].
Y. Tanizaki and M. Ünsal, Semiclassics with ’t Hooft flux background for QCD with 2-index quarks, JHEP 08 (2022) 038 [arXiv:2205.11339] [INSPIRE].
L. Del Debbio, M. Faber, J. Greensite and S. Olejnik, Center dominance and Z2 vortices in SU(2) lattice gauge theory, Phys. Rev. D 55 (1997) 2298 [hep-lat/9610005] [INSPIRE].
M. Faber, J. Greensite and S. Olejnik, Casimir scaling from center vortices: Towards an understanding of the adjoint string tension, Phys. Rev. D 57 (1998) 2603 [hep-lat/9710039] [INSPIRE].
K. Langfeld, O. Tennert, M. Engelhardt and H. Reinhardt, Center vortices of Yang-Mills theory at finite temperatures, Phys. Lett. B 452 (1999) 301 [hep-lat/9805002] [INSPIRE].
T.G. Kovacs and E.T. Tomboulis, Vortices and confinement at weak coupling, Phys. Rev. D 57 (1998) 4054 [hep-lat/9711009] [INSPIRE].
J. Greensite, An introduction to the confinement problem, in Lecture Notes in Physics 821, Springer (2011) [https://doi.org/10.1007/978-3-642-14382-3] [INSPIRE].
A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].
T. Sulejmanpasic, Y. Tanizaki and M. Ünsal, Universality between vector-like and chiral quiver gauge theories: Anomalies and domain walls, JHEP 06 (2020) 173 [arXiv:2004.10328] [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 QCD4, walls, and dualities in 2 + 1 dimensions, JHEP 01 (2018) 110 [arXiv:1708.06806] [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].
Y. Tanizaki, Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking, JHEP 08 (2018) 171 [arXiv:1807.07666] [INSPIRE].
K. Yonekura, Anomaly matching in QCD thermal phase transition, JHEP 05 (2019) 062 [arXiv:1901.08188] [INSPIRE].
M.M. Anber and E. Poppitz, On the baryon-color-flavor (BCF) anomaly in vector-like theories, JHEP 11 (2019) 063 [arXiv:1909.09027] [INSPIRE].
O. Morikawa, H. Wada and S. Yamaguchi, Phase structure of linear quiver gauge theories from anomaly matching, Phys. Rev. D 107 (2023) 045020 [arXiv:2211.12079] [INSPIRE].
M. Yamazaki and K. Yonekura, From 4d Yang-Mills to 2d ℂℙN−1 model: IR problem and confinement at weak coupling, JHEP 07 (2017) 088 [arXiv:1704.05852] [INSPIRE].
Y. Tanizaki, T. Misumi and N. Sakai, Circle compactification and ’t Hooft anomaly, JHEP 12 (2017) 056 [arXiv:1710.08923] [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].
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].
G. ’t Hooft, A Property of Electric and Magnetic Flux in Non-Abelian Gauge Theories, Nucl. Phys. B 153 (1979) 141 [INSPIRE].
A. Gonzalez-Arroyo and A. Montero, Selfdual vortex-like configurations in SU(2) Yang-Mills theory, Phys. Lett. B 442 (1998) 273 [hep-th/9809037] [INSPIRE].
A. Montero, Study of SU(3) vortex-like configurations with a new maximal center gauge fixing method, Phys. Lett. B 467 (1999) 106 [hep-lat/9906010] [INSPIRE].
A. Montero, Vortex configurations in the large N limit, Phys. Lett. B 483 (2000) 309 [hep-lat/0004002] [INSPIRE].
M.M. Anber and E. Poppitz, The gaugino condensate from asymmetric four-torus with twists, JHEP 01 (2023) 118 [arXiv:2210.13568] [INSPIRE].
M.M. Anber and E. Poppitz, Multi-fractional instantons in SU(N) Yang-Mills theory on the twisted 𝕋4, JHEP 09 (2023) 095 [arXiv:2307.04795] [INSPIRE].
M. Shifman and M. Ünsal, QCD-like Theories on R3 × S1: A Smooth Journey from Small to Large r(S1) with Double-Trace Deformations, Phys. Rev. D 78 (2008) 065004 [arXiv:0802.1232] [INSPIRE].
M.M. Anber, E. Poppitz and T. Sulejmanpasic, Strings from domain walls in supersymmetric Yang-Mills theory and adjoint QCD, Phys. Rev. D 92 (2015) 021701 [arXiv:1501.06773] [INSPIRE].
T. Sulejmanpasic, H. Shao, A. Sandvik and M. Ünsal, Confinement in the bulk, deconfinement on the wall: infrared equivalence between compactified QCD and quantum magnets, Phys. Rev. Lett. 119 (2017) 091601 [arXiv:1608.09011] [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].
T. Pantev and E. Sharpe, GLSM’s for Gerbes (and other toric stacks), Adv. Theor. Math. Phys. 10 (2006) 77 [hep-th/0502053] [INSPIRE].
S. Hellerman, A. Henriques, T. Pantev, E. Sharpe and M. Ando, Cluster decomposition, T-duality, and gerby CFT’s, Adv. Theor. Math. Phys. 11 (2007) 751 [hep-th/0606034] [INSPIRE].
S. Hellerman and E. Sharpe, Sums over topological sectors and quantization of Fayet-Iliopoulos parameters, Adv. Theor. Math. Phys. 15 (2011) 1141 [arXiv:1012.5999] [INSPIRE].
A. Cherman, T. Jacobson, Y. Tanizaki and M. Ünsal, Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD, SciPost Phys. 8 (2020) 072 [arXiv:1908.09858] [INSPIRE].
Y. Tanizaki and M. Ünsal, Modified instanton sum in QCD and higher-groups, JHEP 03 (2020) 123 [arXiv:1912.01033] [INSPIRE].
Z. Komargodski, K. Ohmori, K. Roumpedakis and S. Seifnashri, Symmetries and strings of adjoint QCD2, JHEP 03 (2021) 103 [arXiv:2008.07567] [INSPIRE].
M. Ünsal, Magnetic bion condensation: A New mechanism of confinement and mass gap in four dimensions, Phys. Rev. D 80 (2009) 065001 [arXiv:0709.3269] [INSPIRE].
M. Ünsal, Abelian Duality, Confinement, and Chiral-Symmetry Breaking in a SU(2) QCD-Like Theory, Phys. Rev. Lett. 100 (2008) 032005 [arXiv:0708.1772] [INSPIRE].
M. Ünsal and L.G. Yaffe, Center-stabilized Yang-Mills theory: Confinement and large N volume independence, Phys. Rev. D 78 (2008) 065035 [arXiv:0803.0344] [INSPIRE].
E. Poppitz, T. Schäfer and M. Ünsal, Continuity, Deconfinement, and (Super) Yang-Mills Theory, JHEP 10 (2012) 115 [arXiv:1205.0290] [INSPIRE].
G.V. Dunne and M. Ünsal, New Nonperturbative Methods in Quantum Field Theory: From Large-N Orbifold Equivalence to Bions and Resurgence, Ann. Rev. Nucl. Part. Sci. 66 (2016) 245 [arXiv:1601.03414] [INSPIRE].
D.R.T. Jones, The Two Loop beta Function for a G1 × G2 Gauge Theory, Phys. Rev. D 25 (1982) 581 [INSPIRE].
Acknowledgments
The authors thank Mithat Ünsal for useful discussion. The work of Y.T. was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant numbers, 22H01218, and by Center for Gravitational Physics and Quantum Information (CGPQI) at Yukawa Institute for Theoretical Physics. Y.H. was supported by JSPS Research Fellowship for Young Scientists Grant No. 23KJ1161
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2307.13954
Rights and permissions
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.
About this article
Cite this article
Hayashi, Y., Tanizaki, Y. & Watanabe, H. Semiclassical analysis of the bifundamental QCD on ℝ2 × T2 with ’t Hooft flux. J. High Energ. Phys. 2023, 146 (2023). https://doi.org/10.1007/JHEP10(2023)146
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2023)146