Abstract
We propose a random matrix theory for QCD in three dimensions with a Chern-Simons term at level k which spontaneously breaks the flavor symmetry according to U(2Nf) → U(Nf + k)×U(Nf− k). This random matrix model is obtained by adding a complex part to the action for the k = 0 random matrix model. We derive the pattern of spontaneous symmetry breaking from the analytical solution of the model. Additionally, we obtain explicit analytical results for the spectral density and the spectral correlation func- tions for the Dirac operator at finite matrix dimension, that become complex. In the micro- scopic domain where the matrix size tends to infinity, they are expected to be universal, and give an exact analytical prediction to the spectral properties of the Dirac operator in the presence of a Chern-Simons term. Here, we calculate the microscopic spectral density. It shows exponentially large (complex) oscillations which cancel the phase of the k = 0 theory.
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H. Leutwyler and A.V. Smilga, Spectrum of Dirac operator and role of winding number in QCD, Phys. Rev.D 46 (1992) 5607 [INSPIRE].
E.V. Shuryak and J.J.M. Verbaarschot, Random matrix theory and spectral sum rules for the Dirac operator in QCD, Nucl. Phys.A 560 (1993) 306 [hep-th/9212088] [INSPIRE].
J.J.M. Verbaarschot and I. Zahed, Spectral density of the QCD Dirac operator near zero virtuality, Phys. Rev. Lett.70 (1993) 3852 [hep-th/9303012] [INSPIRE].
J.J.M. Verbaarschot, The spectrum of the QCD Dirac operator and chiral random matrix theory: the threefold way, Phys. Rev. Lett.72 (1994) 2531 [hep-th/9401059] [INSPIRE].
J.J.M. Verbaarschot, Universal behavior in Dirac spectra, in Confinement, duality and nonperturbative aspects of QCD. Proceedings, NATO Advanced Study Institute, Newton Institute Workshop, Cambridge, U.K., 23 June–4 July 1997, pg. 343 [hep-th/9710114] [INSPIRE].
J.J.M. Verbaarschot and T. Wettig, Random matrix theory and chiral symmetry in QCD, Ann. Rev. Nucl. Part. Sci.50 (2000) 343 [hep-ph/0003017] [INSPIRE].
J.J.M. Verbaarschot, QCD, chiral random matrix theory and integrability, in Application of random matrices in physics. Proceedings, NATO Advanced Study Institute, Les Houches, France, 6–25 June 2004, pg. 163 [hep-th/0502029] [INSPIRE].
T. Kanazawa, Dirac spectra in dense QCD, Springer theses 124, Springer, Japan (2013).
G. Akemann, Random matrix theory and quantum chromodynamics, Les Houches lecture notes, Oxford University Press, Oxford, U.K. (2016) [arXiv:1603.06011] [INSPIRE].
R.D. Pisarski, Chiral symmetry breaking in three-dimensional electrodynamics, Phys. Rev.D 29 (1984) 2423 [INSPIRE].
T.W. Appelquist, M.J. Bowick, D. Karabali and L.C.R. Wijewardhana, Spontaneous chiral symmetry breaking in three-dimensional QED, Phys. Rev.D 33 (1986) 3704 [INSPIRE].
T. Appelquist, M.J. Bowick, D. Karabali and L.C.R. Wijewardhana, Spontaneous breaking of parity in (2 + 1)-dimensional QED, Phys. Rev.D 33 (1986) 3774 [INSPIRE].
P.A. Lee, N. Nagaosa and X.-G. Wen, Doping a Mott insulator: physics of high-temperature superconductivity, Rev. Mod. Phys.78 (2006) 17 [INSPIRE].
C. Nayak, S.H. Simon, A. Stern, M. Freedman and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys.80 (2008) 1083 [arXiv:0707.1889] [INSPIRE].
L. Balents, Spin liquids in frustrated magnets, Nature464 (2010) 199.
X.L. Qi and S.C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys.83 (2011) 1057 [arXiv:1008.2026] [INSPIRE].
T. Hansson, M. Hermanns, S. Simon and S. Viefers, Quantum Hall physics: hierarchies and conformal field theory techniques, Rev. Mod. Phys.89 (2017) 025005.
D.K. Hong and S.H. Park, Dynamical mass generation in (2 + 1)-dimensional QED with a Chern-Simons term, Phys. Rev.D 47 (1993) 3651 [INSPIRE].
K.I. Kondo and P. Maris, First-order phase transition in three-dimensional QED with Chern-Simons term, Phys. Rev. Lett.74 (1995) 18 [hep-ph/9408210] [INSPIRE].
D.K. Hong, Zero temperature chiral phase transition in (2 + 1)-dimensional QED with a Chern-Simons term, Phys. Rev.D 57 (1998) 1313 [hep-th/9708027] [INSPIRE].
T. Itoh and H. Kato, Dynamical generation of fermion mass and magnetic field in three-dimensional QED with Chern-Simons term, Phys. Rev. Lett.81 (1998) 30 [hep-th/9802101] [INSPIRE].
T. Matsuyama, H. Nagahiro and S. Uchida, A dynamical mass generation of a two component fermion in the Maxwell-Chern-Simons QED3: the lowest ladder approximation, Phys. Rev.D 60 (1999) 105020 [hep-th/9901049] [INSPIRE].
G.-Z. Liu and G. Cheng, Effect of gauge boson mass on chiral symmetry breaking in QED3, Phys. Rev.D 67 (2003) 065010 [hep-th/0211231] [INSPIRE].
C.P. Hofmann, A. Raya and S.S. Madrigal, Confinement in Maxwell-Chern-Simons planar quantum electrodynamics and the 1/N approximation, Phys. Rev.D 82 (2010) 096011 [arXiv:1010.3466] [INSPIRE].
N. Karthik and R. Narayanan, No evidence for bilinear condensate in parity-invariant three-dimensional QED with massless fermions, Phys. Rev.D 93 (2016) 045020 [arXiv:1512.02993] [INSPIRE].
N. Karthik and R. Narayanan, Scale-invariance of parity-invariant three-dimensional QED, Phys. Rev.D 94 (2016) 065026 [arXiv:1606.04109] [INSPIRE].
N. Karthik and R. Narayanan, Flavor and topological current correlators in parity-invariant three-dimensional QED, Phys. Rev.D 96 (2017) 054509 [arXiv:1705.11143] [INSPIRE].
D. Roscher, E. Torres and P. Strack, Dual QED3 at “NF = 1/2” is an interacting CFT in the infrared, JHEP11 (2016) 017 [arXiv:1605.05347] [INSPIRE].
N. Karthik and R. Narayanan, Parity anomaly cancellation in three-dimensional QED with a single massless Dirac fermion, Phys. Rev. Lett.121 (2018) 041602 [arXiv:1803.03596] [INSPIRE].
C. Vafa and E. Witten, Restrictions on symmetry breaking in vector-like gauge theories, Nucl. Phys.B 234 (1984) 173 [INSPIRE].
T. Appelquist and D. Nash, Critical behavior in (2 + 1)-dimensional QCD, Phys. Rev. Lett.64 (1990) 721 [INSPIRE].
G. Ferretti, S.G. Rajeev and Z. Yang, The effective Lagrangian of three-dimensional quantum chromodynamics, Int. J. Mod. Phys.A 7 (1992) 7989 [hep-th/9204075] [INSPIRE].
M.C. Diamantini, P. Sodano and G.W. Semenoff, Chiral dynamics and fermion mass generation in three-dimensional gauge theory, Phys. Rev. Lett.70 (1993) 3848 [hep-ph/9301256] [INSPIRE].
P.H. Damgaard, U.M. Heller, A. Krasnitz and T. Madsen, A quark-anti-quark condensate in three-dimensional QCD, Phys. Lett.B 440 (1998) 129 [hep-lat/9803012] [INSPIRE].
N. Karthik and R. Narayanan, Bilinear condensate in three-dimensional large-Nc QCD, Phys. Rev.D 94 (2016) 045020 [arXiv:1607.03905] [INSPIRE].
J.J.M. Verbaarschot and I. Zahed, Random matrix theory and QCD in three-dimensions, Phys. Rev. Lett.73 (1994) 2288 [hep-th/9405005] [INSPIRE].
T. Nagao and K. Slevin, Nonuniversal correlations for random matrix ensembles, J. Math. Phys.34 (1993) 2075.
P.H. Damgaard and S.M. Nishigaki, Universal massive spectral correlators and QCD in three-dimensions, Phys. Rev.D 57 (1998) 5299 [hep-th/9711096] [INSPIRE].
G. Akemann and P.H. Damgaard, Microscopic spectra of Dirac operators and finite volume partition functions, Nucl. Phys.B 528 (1998) 411 [hep-th/9801133] [INSPIRE].
J. Christiansen, Odd flavored QCD3 and random matrix theory, Nucl. Phys.B 547 (1999) 329 [hep-th/9809194] [INSPIRE].
U. Magnea, The orthogonal ensemble of random matrices and QCD in three-dimensions, Phys. Rev.D 61 (2000) 056005 [hep-th/9907096] [INSPIRE].
U. Magnea, Three-dimensional QCD in the adjoint representation and random matrix theory, Phys. Rev.D 62 (2000) 016005 [hep-th/9912207] [INSPIRE].
C. Hilmoine and R. Niclasen, The microscopic spectral density of the Dirac operator derived from Gaussian orthogonal and symplectic ensembles, Phys. Rev.D 62 (2000) 096013 [hep-th/0004081] [INSPIRE].
T. Nagao and S.M. Nishigaki, Massive random matrix ensembles at β = 1 and β = 4: QCD in three-dimensions, Phys. Rev.D 63 (2001) 045011 [hep-th/0005077] [INSPIRE].
R.J. Szabo, Microscopic spectrum of the QCD Dirac operator in three dimensions, Nucl. Phys.B 598 (2001) 309 [hep-th/0009237] [INSPIRE].
R.J. Szabo, Finite volume gauge theory partition functions in three dimensions, Nucl. Phys.B 723 (2005) 163 [hep-th/0504202] [INSPIRE].
S. Deser, R. Jackiw and S. Templeton, Topologically massive gauge theories, Annals Phys.140 (1982) 372 [Erratum ibid.185 (1988) 406] [Annals Phys.281 (2000) 409] [INSPIRE].
Z. Komargodski and N. Seiberg, A symmetry breaking scenario for QCD3 , JHEP01 (2018) 109 [arXiv:1706.08755] [INSPIRE].
D. Gaiotto, Z. Komargodski and N. Seiberg, Time-reversal breaking in QCD4 , walls and dualities in 2 + 1 dimensions, JHEP01 (2018) 110 [arXiv:1708.06806] [INSPIRE].
A. Armoni and V. Niarchos, Phases of QCD3 from non-SUSY Seiberg duality and brane dynamics, Phys. Rev.D 97 (2018) 106001 [arXiv:1711.04832] [INSPIRE].
M. Mariño, Chern-Simons theory, matrix integrals and perturbative three manifold invariants, Commun. Math. Phys.253 (2004) 25 [hep-th/0207096] [INSPIRE].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, Matrix model as a mirror of Chern-Simons theory, JHEP02 (2004) 010 [hep-th/0211098] [INSPIRE].
M. Tierz, Soft matrix models and Chern-Simons partition functions, Mod. Phys. Lett.A 19 (2004) 1365 [hep-th/0212128] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys.306 (2011) 511 [arXiv:1007.3837] [INSPIRE].
A. Armoni and V. Niarchos, QCD3 with two-index quarks, mirror symmetry and fivebrane anti-BIons near orientifolds, Phys. Rev.D 98 (2018) 114009 [arXiv:1808.07715] [INSPIRE].
S.R. Das, A. Dhar, A.M. Sengupta and S.R. Wadia, New critical behavior in d = 0 large N matrix models, Mod. Phys. Lett.A 5 (1990) 1041 [INSPIRE].
G.M. Cicuta and E. Montaldi, Matrix models and marginal operators in the planar limit, Mod. Phys. Lett.A 5 (1990) 1927 [INSPIRE].
H. Ueda, Regularization of quantum gravity in the matrix model approach, Prog. Theor. Phys.86 (1991) 23 [INSPIRE].
S. Sawada and H. Ueda, Nonperturbative effect of a modified action in matrix models, Mod. Phys. Lett.A 6 (1991) 3717 [INSPIRE].
G.P. Korchemsky, Matrix model perturbed by higher order curvature terms, Mod. Phys. Lett.A 7 (1992) 3081 [hep-th/9205014] [INSPIRE].
F. David, A scenario for the c > 1 barrier in noncritical bosonic strings, Nucl. Phys.B 487 (1997) 633 [hep-th/9610037] [INSPIRE].
M. Kieburg, J.J.M. Verbaarschot and S. Zafeiropoulos, Spectral properties of the Wilson Dirac operator and random matrix theory, Phys. Rev.D 88 (2013) 094502 [arXiv:1307.7251] [INSPIRE].
J. Gasser and H. Leutwyler, Thermodynamics of chiral symmetry, Phys. Lett.B 188 (1987) 477 [INSPIRE].
L. Álvarez-Gauḿe, S. Della Pietra and G.W. Moore, Anomalies and odd dimensions, Annals Phys.163 (1985) 288 [INSPIRE].
H. Leutwyler, Dirac operator and Chern-Simons action, Helv. Phys. Acta63 (1990) 660 [INSPIRE].
A.N. Redlich, Parity violation and gauge noninvariance of the effective gauge field action in three-dimensions, Phys. Rev.D 29 (1984) 2366 [INSPIRE].
S. Deser, L. Griguolo and D. Seminara, Effective QED actions: representations, gauge invariance, anomalies and mass expansions, Phys. Rev.D 57 (1998) 7444 [hep-th/9712066] [INSPIRE].
A.M. Halasz and J.J.M. Verbaarschot, Effective Lagrangians and chiral random matrix theory, Phys. Rev.D 52 (1995) 2563 [hep-th/9502096] [INSPIRE].
A.C. Bertuola, O. Bohigas and M.P. Pato, Family of generalized random matrix ensembles, Phys. Rev.E 70 (2004) 065102 [math-ph/0411033].
A.Y. Abul-Magd, Random matrix theory within superstatistics, Phys. Rev.E 72 (2005) 066114 [cond-mat/0510494].
G. Akemann and P. Vivo, Power law deformation of Wishart-Laguerre ensembles of random matrices, J. Stat. Mech.0809 (2008) P09002 [arXiv:0806.1861] [INSPIRE].
T. Kanazawa, Heavy-tailed chiral random matrix theory, JHEP05 (2016) 166 [arXiv:1602.05631] [INSPIRE].
R.D. Pisarski, Effective theory of Wilson lines and deconfinement, Phys. Rev.D 74 (2006) 121703 [hep-ph/0608242] [INSPIRE].
J.C. Myers and M.C. Ogilvie, New phases of SU(3) and SU(4) at finite temperature, Phys. Rev.D 77 (2008) 125030 [arXiv:0707.1869] [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].
G. Akemann, D. Dalmazi, P.H. Damgaard and J.J.M. Verbaarschot, QCD3 and the replica method, Nucl. Phys.B 601 (2001) 77 [hep-th/0011072] [INSPIRE].
G. Akemann and G. Vernizzi, Characteristic polynomials of complex random matrix models, Nucl. Phys.B 660 (2003) 532 [hep-th/0212051] [INSPIRE].
E. Strahov and Y.V. Fyodorov, Universal results for correlations of characteristic polynomials: Riemann-Hilbert approach, Commun. Math. Phys.241 (2003) 343 [math-ph/0210010] [INSPIRE].
K. Splittorff and J.J.M. Verbaarschot, Factorization of correlation functions and the replica limit of the Toda lattice equation, Nucl. Phys.B 683 (2004) 467 [hep-th/0310271] [INSPIRE].
T. Kanazawa and M. Kieburg, Symmetry crossover protecting chirality in Dirac spectra, JHEP11 (2018) 205 [arXiv:1809.10602] [INSPIRE].
Harish-Chandra, Invariant differential operators on a semisimple Lie algebra, Proc. Nat. Acad. Sci.42 (1956) 252.
C. Itzykson and J.B. Zuber, The planar approximation. 2, J. Math. Phys.21 (1980) 411 [INSPIRE].
K.A. Andréief, Note sur une relation les intégrales définies des produits des fonctions (in French), Mém. Soc. Sci. Bordeaux2 (1883) 1.
I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products, seventh ed., Elsevier, The Netherlands (2007).
G. Akemann, J.C. Osborn, K. Splittorff and J.J.M. Verbaarschot, Unquenched QCD Dirac operator spectra at nonzero baryon chemical potential, Nucl. Phys.B 712 (2005) 287 [hep-th/0411030] [INSPIRE].
J.C. Osborn, K. Splittorff and J.J.M. Verbaarschot, Chiral symmetry breaking and the Dirac spectrum at nonzero chemical potential, Phys. Rev. Lett.94 (2005) 202001 [hep-th/0501210] [INSPIRE].
G. Akemann, T. Kanazawa, M.J. Phillips and T. Wettig, Random matrix theory of unquenched two-colour QCD with nonzero chemical potential, JHEP03 (2011) 066 [arXiv:1012.4461] [INSPIRE].
T. Kanazawa, T. Wettig and N. Yamamoto, Singular values of the Dirac operator in dense QCD-like theories, JHEP12 (2011) 007 [arXiv:1110.5858] [INSPIRE].
J.J.M. Verbaarschot and T. Wettig, Dirac spectrum of one-flavor QCD at θ = 0 and continuity of the chiral condensate, Phys. Rev.D 90 (2014) 116004 [arXiv:1407.8393] [INSPIRE].
D. Dominici, Asymptotic analysis of the Hermite polynomials from their differential-difference equation, J. Diff. Eq. Appl.13 (2007) 1115 [math.CA/0601078].
M.E. Peskin, The alignment of the vacuum in theories of technicolor, Nucl. Phys.B 175 (1980) 197 [INSPIRE].
A. Armoni, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, Metastable vacua in large-N QCD3, arXiv:1905.01797 [INSPIRE].
H.-J. Sommers, Superbosonization, Acta Phys. Polon.B 38 (2007) 4105 [arXiv:0710.5375].
P. Littelmann, H.J. Sommers and M.R. Zirnbauer, Superbosonization of invariant random matrix ensembles, Commun. Math. Phys.283 (2008) 343 [arXiv:0707.2929].
M. Kieburg, H.-J. Sommers and T. Guhr, A comparison of the superbosonization formula and the generalized Hubbard-Stratonovich transformation, J. Phys.A 42 (2009) 275206 [arXiv:0905.3256].
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Kanazawa, T., Kieburg, M. & Verbaarschot, J.J.M. Random matrix approach to three-dimensional QCD with a Chern-Simons term. J. High Energ. Phys. 2019, 74 (2019). https://doi.org/10.1007/JHEP10(2019)074
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DOI: https://doi.org/10.1007/JHEP10(2019)074