Abstract
We compute the large-N limit of the QCD chiral condensate on the lattice using twisted reduced models, and performing controlled continuum and chiral extrapolations. We perform two different calculations: one consists in extracting the chiral condensate from the quark mass dependence of the pion mass, and the other consists in extracting the chiral condensate from the behaviour of the mode number of the Wilson-Dirac operator for small eigenvalues. We find consistency between the results of the two methods, giving a joint estimate of limN→∞ Σ(N)/N = [184(13) MeV]3 (\( \overline{\textrm{MS}} \), μ = 2 GeV, taking the square root of the string tension \( \sqrt{\sigma } \) = 440 MeV to set the scale), in remarkable agreement with the SU(3) 2-flavor FLAG result.
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L. Del Debbio, H. Panagopoulos, P. Rossi and E. Vicari, Spectrum of confining strings in SU(N) gauge theories, JHEP 01 (2002) 009 [hep-th/0111090] [INSPIRE].
B. Lucini and M. Teper, SU(N) gauge theories in four-dimensions: exploring the approach to N = ∞, JHEP 06 (2001) 050 [hep-lat/0103027] [INSPIRE].
L. Del Debbio, H. Panagopoulos and E. Vicari, θ dependence of SU(N) gauge theories, JHEP 08 (2002) 044 [hep-th/0204125] [INSPIRE].
B. Lucini, M. Teper and U. Wenger, Glueballs and k-strings in SU(N) gauge theories: calculations with improved operators, JHEP 06 (2004) 012 [hep-lat/0404008] [INSPIRE].
L. Del Debbio et al., θ-dependence of the spectrum of SU(N) gauge theories, JHEP 06 (2006) 005 [hep-th/0603041] [INSPIRE].
E. Vicari and H. Panagopoulos, θ dependence of SU(N) gauge theories in the presence of a topological term, Phys. Rept. 470 (2009) 93 [arXiv:0803.1593] [INSPIRE].
C. Allton, M. Teper and A. Trivini, On the running of the bare coupling in SU(N) lattice gauge theories, JHEP 07 (2008) 021 [arXiv:0803.1092] [INSPIRE].
B. Lucini, A. Rago and E. Rinaldi, Glueball masses in the large N limit, JHEP 08 (2010) 119 [arXiv:1007.3879] [INSPIRE].
B. Lucini and M. Panero, SU(N) gauge theories at large N, Phys. Rept. 526 (2013) 93 [arXiv:1210.4997] [INSPIRE].
G.S. Bali et al., Mesons in large-N QCD, JHEP 06 (2013) 071 [arXiv:1304.4437] [INSPIRE].
C. Bonati, M. D’Elia, P. Rossi and E. Vicari, θ dependence of 4D SU(N) gauge theories in the large-N limit, Phys. Rev. D 94 (2016) 085017 [arXiv:1607.06360] [INSPIRE].
M. Cè, M. García Vera, L. Giusti and S. Schaefer, The topological susceptibility in the large-N limit of SU(N) Yang-Mills theory, Phys. Lett. B 762 (2016) 232 [arXiv:1607.05939] [INSPIRE].
T. DeGrand and Y. Liu, Lattice study of large Nc QCD, Phys. Rev. D 94 (2016) 034506 [Erratum ibid. 95 (2017) 019902] [arXiv:1606.01277] [INSPIRE].
P. Hernández, C. Pena and F. Romero-López, Large Nc scaling of meson masses and decay constants, Eur. Phys. J. C 79 (2019) 865 [arXiv:1907.11511] [INSPIRE].
E. Bennett et al., Color dependence of tensor and scalar glueball masses in Yang-Mills theories, Phys. Rev. D 102 (2020) 011501 [arXiv:2004.11063] [INSPIRE].
T. DeGrand, Topological susceptibility in QCD with two flavors and 3-5 colors: a pilot study, Phys. Rev. D 101 (2020) 114509 [arXiv:2004.09649] [INSPIRE].
P. Hernández and F. Romero-López, The large Nc limit of QCD on the lattice, Eur. Phys. J. A 57 (2021) 52 [arXiv:2012.03331] [INSPIRE].
C. Bonanno, C. Bonati and M. D’Elia, Large-N SU(N) Yang-Mills theories with milder topological freezing, JHEP 03 (2021) 111 [arXiv:2012.14000] [INSPIRE].
T. DeGrand, Finite temperature properties of QCD with two flavors and three, four and five colors, Phys. Rev. D 103 (2021) 094513 [arXiv:2102.01150] [INSPIRE].
A. Athenodorou and M. Teper, SU(N) gauge theories in 3 + 1 dimensions: glueball spectrum, string tensions and topology, JHEP 12 (2021) 082 [arXiv:2106.00364] [INSPIRE].
E. Bennett et al., Color dependence of the topological susceptibility in Yang-Mills theories, Phys. Lett. B 835 (2022) 137504 [arXiv:2205.09254] [INSPIRE].
C. Bonanno, M. D’Elia, B. Lucini and D. Vadacchino, Towards glueball masses of large-N SU(N) pure-gauge theories without topological freezing, Phys. Lett. B 833 (2022) 137281 [arXiv:2205.06190] [INSPIRE].
T.A. DeGrand and E. Wickenden, Lattice study of the chiral properties of large-Nc QCD, arXiv:2309.12270 [https://doi.org/10.1103/PhysRevD.108.094516] [INSPIRE].
T. Eguchi and H. Kawai, Reduction of dynamical degrees of freedom in the large N gauge theory, Phys. Rev. Lett. 48 (1982) 1063 [INSPIRE].
G. Bhanot, U.M. Heller and H. Neuberger, The quenched Eguchi-Kawai model, Phys. Lett. B 113 (1982) 47 [INSPIRE].
D.J. Gross and Y. Kitazawa, A quenched momentum prescription for large N theories, Nucl. Phys. B 206 (1982) 440 [INSPIRE].
A. Gonzalez-Arroyo and M. Okawa, A twisted model for large N lattice gauge theory, Phys. Lett. B 120 (1983) 174 [INSPIRE].
A. Gonzalez-Arroyo and M. Okawa, The twisted Eguchi-Kawai model: a reduced model for large N lattice gauge theory, Phys. Rev. D 27 (1983) 2397 [INSPIRE].
G. Aldazabal, N. Parga, M. Okawa and A. Gonzalez-Arroyo, Large N reduced models and stochastic quantization, Phys. Lett. B 129 (1983) 90 [INSPIRE].
J. Kiskis, R. Narayanan and H. Neuberger, Proposal for the numerical solution of planar QCD, Phys. Rev. D 66 (2002) 025019 [hep-lat/0203005] [INSPIRE].
R. Narayanan and H. Neuberger, Large N reduction in continuum, Phys. Rev. Lett. 91 (2003) 081601 [hep-lat/0303023] [INSPIRE].
P. Kovtun, M. Unsal and L.G. Yaffe, Volume independence in large Nc QCD-like gauge theories, JHEP 06 (2007) 019 [hep-th/0702021] [INSPIRE].
M. Unsal 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].
A. Gonzalez-Arroyo and M. Okawa, Large N reduction with the twisted Eguchi-Kawai model, JHEP 07 (2010) 043 [arXiv:1005.1981] [INSPIRE].
H. Neuberger, Quenched Eguchi-Kawai model revisited, Phys. Rev. D 102 (2020) 094503 [arXiv:2009.09539] [INSPIRE].
A. Gonzalez-Arroyo and M. Okawa, String tension for large N gauge theory, Phys. Lett. B 133 (1983) 415 [INSPIRE].
S.R. Das and J.B. Kogut, On the deconfining transition of SU(∞) gauge theory, Nucl. Phys. B 257 (1985) 141 [INSPIRE].
S.R. Das and J.B. Kogut, Evidence for a first order deconfinement transition in large N gauge theory, Phys. Rev. D 31 (1985) 2704 [INSPIRE].
R. Narayanan and H. Neuberger, Chiral symmetry breaking at large Nc, Nucl. Phys. B 696 (2004) 107 [hep-lat/0405025] [INSPIRE].
A. Gonzalez-Arroyo, R. Narayanan and H. Neuberger, Large N reduction on a twisted torus, Phys. Lett. B 631 (2005) 133 [hep-lat/0509074] [INSPIRE].
J. Kiskis and R. Narayanan, Computation of the string tension in four-dimensional Yang-Mills theory using large N reduction, Phys. Lett. B 681 (2009) 372 [arXiv:0908.1451] [INSPIRE].
A. Hietanen and R. Narayanan, The large N limit of four dimensional Yang-Mills field coupled to adjoint fermions on a single site lattice, JHEP 01 (2010) 079 [arXiv:0911.2449] [INSPIRE].
A. Hietanen and R. Narayanan, Large-N reduction of SU(N) Yang-Mills theory with massive adjoint overlap fermions, Phys. Lett. B 698 (2011) 171 [arXiv:1011.2150] [INSPIRE].
B. Bringoltz, M. Koren and S.R. Sharpe, Large-N reduction in QCD with two adjoint Dirac fermions, Phys. Rev. D 85 (2012) 094504 [arXiv:1106.5538] [INSPIRE].
A. Hietanen and R. Narayanan, Numerical evidence for non-analytic behavior in the β function of large N SU(N) gauge theory coupled to an adjoint Dirac fermion, Phys. Rev. D 86 (2012) 085002 [arXiv:1204.0331] [INSPIRE].
A. Gonzalez-Arroyo and M. Okawa, The string tension from smeared Wilson loops at large N, Phys. Lett. B 718 (2013) 1524 [arXiv:1206.0049] [INSPIRE].
A. González-Arroyo and M. Okawa, Twisted space-time reduced model of large N QCD with two adjoint Wilson fermions, Phys. Rev. D 88 (2013) 014514 [arXiv:1305.6253] [INSPIRE].
R. Lohmayer and R. Narayanan, Weak-coupling analysis of the single-site large-N gauge theory coupled to adjoint fermions, Phys. Rev. D 87 (2013) 125024 [arXiv:1305.1279] [INSPIRE].
A. Gonzalez-Arroyo and M. Okawa, Testing volume independence of SU(N) pure gauge theories at large N, JHEP 12 (2014) 106 [arXiv:1410.6405] [INSPIRE].
M. García Pérez, A. González-Arroyo, L. Keegan and M. Okawa, The SU(∞) twisted gradient flow running coupling, JHEP 01 (2015) 038 [arXiv:1412.0941] [INSPIRE].
M. García Pérez, A. González-Arroyo, L. Keegan and M. Okawa, Mass anomalous dimension of adjoint QCD at large N from twisted volume reduction, JHEP 08 (2015) 034 [arXiv:1506.06536] [INSPIRE].
M. García Pérez et al., A comparison of updating algorithms for large N reduced models, JHEP 06 (2015) 193 [arXiv:1505.05784] [INSPIRE].
A. González-Arroyo and M. Okawa, Large N meson masses from a matrix model, Phys. Lett. B 755 (2016) 132 [arXiv:1510.05428] [INSPIRE].
M. García Pérez, A. González-Arroyo and M. Okawa, Perturbative contributions to Wilson loops in twisted lattice boxes and reduced models, JHEP 10 (2017) 150 [arXiv:1708.00841] [INSPIRE].
M. García Pérez, Prospects for large N gauge theories on the lattice, PoS LATTICE2019 (2020) 276 [arXiv:2001.10859] [INSPIRE].
M. García Pérez, A. González-Arroyo and M. Okawa, Meson spectrum in the large N limit, JHEP 04 (2021) 230 [arXiv:2011.13061] [INSPIRE].
P. Butti et al., Scale setting for large-N SUSY Yang-Mills on the lattice, JHEP 07 (2022) 074 [arXiv:2205.03166] [INSPIRE].
P. Butti and A. Gonzalez-Arroyo, Asymptotic scaling in Yang-Mills theory at large-Nc, PoS LATTICE2023 (2023) 381 [arXiv:2311.18696] [INSPIRE].
P. Butti and A. Gonzalez-Arroyo, Asymptotic scaling in Yang-Mills theory at large-N, in preparation (2023).
L. Giusti and M. Lüscher, Chiral symmetry breaking and the Banks-Casher relation in lattice QCD with Wilson quarks, JHEP 03 (2009) 013 [arXiv:0812.3638] [INSPIRE].
JLQCD and TWQCD collaborations, Convergence of the chiral expansion in two-flavor lattice QCD, Phys. Rev. Lett. 101 (2008) 202004 [arXiv:0806.0894] [INSPIRE].
S. Borsanyi et al., SU(2) chiral perturbation theory low-energy constants from 2 + 1 flavor staggered lattice simulations, Phys. Rev. D 88 (2013) 014513 [arXiv:1205.0788] [INSPIRE].
B.B. Brandt, A. Jüttner and H. Wittig, The pion vector form factor from lattice QCD and NNLO chiral perturbation theory, JHEP 11 (2013) 034 [arXiv:1306.2916] [INSPIRE].
G.P. Engel, L. Giusti, S. Lottini and R. Sommer, Spectral density of the Dirac operator in two-flavor QCD, Phys. Rev. D 91 (2015) 054505 [arXiv:1411.6386] [INSPIRE].
P.A. Boyle et al., Low energy constants of SU(2) partially quenched chiral perturbation theory from Nf = 2 + 1 domain wall QCD, Phys. Rev. D 93 (2016) 054502 [arXiv:1511.01950] [INSPIRE].
C. Wang et al., Quark chiral condensate from the overlap quark propagator, Chin. Phys. C 41 (2017) 053102 [arXiv:1612.04579] [INSPIRE].
C. Alexandrou et al., Topological susceptibility from twisted mass fermions using spectral projectors and the gradient flow, Phys. Rev. D 97 (2018) 074503 [arXiv:1709.06596] [INSPIRE].
JLQCD collaboration, Topological susceptibility of QCD with dynamical Möbius domain-wall fermions, PTEP 2018 (2018) 043B07 [arXiv:1705.10906] [INSPIRE].
Extended Twisted Mass collaboration, Quark masses using twisted-mass fermion gauge ensembles, Phys. Rev. D 104 (2021) 074515 [arXiv:2104.13408] [INSPIRE].
J. Liang et al., Detecting flavor content of the vacuum using the Dirac operator spectrum, arXiv:2102.05380 [INSPIRE].
C. Bonanno, F. D’Angelo and M. D’Elia, The chiral condensate of Nf = 2 + 1 QCD from the spectrum of the staggered Dirac operator, JHEP 11 (2023) 013 [arXiv:2308.01303] [INSPIRE].
C. Bonanno et al., The chiral condensate at large N, in the proceedings of the 40th international symposium on lattice field theory, (2023) [arXiv:2311.03325] [INSPIRE].
T. Ishikawa and M. Okawa, \( {Z}_N^D \) symmetry breaking on the numerical simulation of twisted Eguchi-Kawai model, talk given at the Annual meeting of the physical society of Japan, Sendai, Japan, 28–31 March (2003).
W. Bietenholz, J. Nishimura, Y. Susaki and J. Volkholz, A non-perturbative study of 4D U(1) non-commutative gauge theory: the fate of one-loop instability, JHEP 10 (2006) 042 [hep-th/0608072] [INSPIRE].
M. Teper and H. Vairinhos, Symmetry breaking in twisted Eguchi-Kawai models, Phys. Lett. B 652 (2007) 359 [hep-th/0612097] [INSPIRE].
T. Azeyanagi, M. Hanada, T. Hirata and T. Ishikawa, Phase structure of twisted Eguchi-Kawai model, JHEP 01 (2008) 025 [arXiv:0711.1925] [INSPIRE].
F. Chamizo and A. Gonzalez-Arroyo, Tachyonic instabilities in 2 + 1 dimensional Yang-Mills theory and its connection to number theory, J. Phys. A 50 (2017) 265401 [arXiv:1610.07972] [INSPIRE].
M. García Pérez, A. González-Arroyo, M. Koren and M. Okawa, The spectrum of 2 + 1 dimensional Yang-Mills theory on a twisted spatial torus, JHEP 07 (2018) 169 [arXiv:1807.03481] [INSPIRE].
E.I. Bribian and M. Garcia Perez, The twisted gradient flow coupling at one loop, JHEP 03 (2019) 200 [arXiv:1903.08029] [INSPIRE].
A. Gonzalez-Arroyo, Yang-Mills fields on the four-dimensional torus. Part 1. Classical theory, in the proceedings of the Advanced summer school on nonperturbative quantum field physics, (1997), p. 57 [hep-th/9807108] [INSPIRE].
ETM collaboration, Non-perturbative test of the Witten-Veneziano formula from lattice QCD, JHEP 09 (2015) 020 [arXiv:1504.07954] [INSPIRE].
L. Castagnini, Meson spectroscopy in large-N QCD, Ph.D. thesis, Regensburg U., Regensburg, Germany (2015) [https://doi.org/10.5283/epub.32024] [INSPIRE].
G. Martinelli et al., A general method for nonperturbative renormalization of lattice operators, Nucl. Phys. B 445 (1995) 81 [hep-lat/9411010] [INSPIRE].
A. Skouroupathis and H. Panagopoulos, Two-loop renormalization of scalar and pseudoscalar fermion bilinears on the lattice, Phys. Rev. D 76 (2007) 094514 [Erratum ibid. 78 (2008) 119901] [arXiv:0707.2906] [INSPIRE].
A. Skouroupathis and H. Panagopoulos, Two-loop renormalization of vector, axial-vector and tensor fermion bilinears on the lattice, Phys. Rev. D 79 (2009) 094508 [arXiv:0811.4264] [INSPIRE].
Flavour Lattice Averaging Group (FLAG) collaboration, FLAG review 2021, Eur. Phys. J. C 82 (2022) 869 [arXiv:2111.09849] [INSPIRE].
B. Lucini, M. Teper and U. Wenger, Properties of the deconfining phase transition in SU(N) gauge theories, JHEP 02 (2005) 033 [hep-lat/0502003] [INSPIRE].
C. Alexandrou et al., Renormalization constants of local operators for Wilson type improved fermions, Phys. Rev. D 86 (2012) 014505 [arXiv:1201.5025] [INSPIRE].
A. Athenodorou, H. Panagopoulos and A. Tsapalis, The lattice free energy of QCD with clover fermions, up to three-loops, Phys. Lett. B 659 (2008) 252 [arXiv:0710.3856] [INSPIRE].
Acknowledgments
It is a pleasure to thank Carlos Pena for useful discussions. This work is partially supported by the Spanish Research Agency (Agencia Estatal de Investigación) through the grant IFT Centro de Excelencia Severo Ochoa CEX2020-001007-S, funded by MCIN/AEI/10.13039/501100011033, and by grant PID2021-127526NB-I00, funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. We also acknowledge support from the project H2020-MSCAITN-2018-813942 (EuroPLEx) and the EU Horizon 2020 research and innovation programme, STRONG-2020 project, under grant agreement No 824093. P. B. is supported by Grant PGC2022-126078NB-C21 funded by MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe”. P. B. also acknowledges support by Grant DGA-FSE grant 2020-E21-17R Aragon Government and the European Union - NextGenerationEU Recovery and Resilience Program on ‘Astrofísica y Física de Altas Energías’ CEFCA-CAPA-ITAINNOVA. K.-I. I. is supported in part by MEXT as “Feasibility studies for the next-generation computing infrastructure”. M. O. is supported by JSPS KAKENHI Grant Number 21K03576. Numerical calculations have been performed on the Finisterrae III cluster at CESGA (Centro de Supercomputación de Galicia). We have also used computational resources of Oakbridge-CX at the University of Tokyo through the HPCI System Research Project (Project ID: hp230021 and hp220011).
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Bonanno, C., Butti, P., Peréz, M.G. et al. The large-N limit of the chiral condensate from twisted reduced models. J. High Energ. Phys. 2023, 34 (2023). https://doi.org/10.1007/JHEP12(2023)034
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DOI: https://doi.org/10.1007/JHEP12(2023)034