Abstract
We determine the pure-gauge SU(3) topological susceptibility slope χ′, related to the next-to-leading-order term of the momentum expansion of the topological charge density 2-point correlator, from numerical lattice Monte Carlo simulations. Our strategy consists in performing a double-limit extrapolation: first we take the continuum limit at fixed smoothing radius, then we take the zero-smoothing-radius limit. Our final result is χ′ = [17.1(2.1) MeV]2. We also discuss a theoretical argument to predict its value in the large-N limit, which turns out to be remarkably close to the obtained N = 3 lattice result.
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B. Alles, M. D’Elia, A. Di Giacomo and R. Kirchner, Topology in SU(2) Yang-Mills theory, Nucl. Phys. B Proc. Suppl. 63 (1998) 510 [hep-lat/9709074] [INSPIRE].
E. Vicari, The Euclidean two point correlation function of the topological charge density, Nucl. Phys. B 554 (1999) 301 [hep-lat/9901008] [INSPIRE].
I. Horvath et al., The negativity of the overlap-based topological charge density correlator in pure-glue QCD and the non-integrable nature of its contact part, Phys. Lett. B 617 (2005) 49 [hep-lat/0504005] [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].
A. Chowdhury et al., Topological charge density correlator in Lattice QCD with two flavours of unimproved Wilson fermions, JHEP 11 (2012) 029 [arXiv:1208.4235] [INSPIRE].
JLQCD collaboration, η′ meson mass from topological charge density correlator in QCD, Phys. Rev. D 92 (2015) 111501 [arXiv:1509.00944] [INSPIRE].
L. Mazur, L. Altenkort, O. Kaczmarek and H.-T. Shu, Euclidean correlation functions of the topological charge density, PoS LATTICE2019 (2020) 219 [arXiv:2001.11967] [INSPIRE].
L. Altenkort et al., Sphaleron rate from Euclidean lattice correlators: An exploration, Phys. Rev. D 103 (2021) 114513 [arXiv:2012.08279] [INSPIRE].
M. Barroso Mancha and G.D. Moore, The sphaleron rate from 4D Euclidean lattices, JHEP 01 (2023) 155 [arXiv:2210.05507] [INSPIRE].
E. Witten, Instantons, the Quark Model, and the 1/N Expansion, Nucl. Phys. B 149 (1979) 285 [INSPIRE].
G. Veneziano, U(1) Without Instantons, Nucl. Phys. B 159 (1979) 213 [INSPIRE].
P. Di Vecchia and G. Veneziano, Chiral Dynamics in the Large n Limit, Nucl. Phys. B 171 (1980) 253 [INSPIRE].
G.M. Shore and G. Veneziano, The U(1) Goldberger-Treiman Relation and the Two Components of the Proton ‘Spin’, Phys. Lett. B 244 (1990) 75 [INSPIRE].
G.M. Shore and G. Veneziano, Renormalisation group aspects of η′ ⟶ γγ, Nucl. Phys. B 381 (1992) 3 [INSPIRE].
B. Alles, A. Di Giacomo, H. Panagopoulos and E. Vicari, Topological charge density renormalization in the presence of dynamical fermions, Phys. Lett. B 350 (1995) 70 [hep-lat/9501030] [INSPIRE].
S. Narison, G.M. Shore and G. Veneziano, Topological charge screening and the ‘proton spin’ beyond the chiral limit, Nucl. Phys. B 546 (1999) 235 [hep-ph/9812333] [INSPIRE].
V. Bernard, L. Elouadrhiri and U.-G. Meissner, Axial structure of the nucleon: Topical Review, J. Phys. G 28 (2002) R1 [hep-ph/0107088] [INSPIRE].
H. Leutwyler, Chiral dynamics, hep-ph/0008124 [https://doi.org/10.1142/9789812810458_0012] [INSPIRE].
B.L. Ioffe and A.G. Oganesian, Proton spin content and QCD topological susceptibility, Phys. Rev. D 57 (1998) 6590 [hep-ph/9801345] [INSPIRE].
S. Narison, U(1)(A) topological susceptibility and its slope, pseudoscalar gluonium and the spin of the proton, in the proceedings of the Sense of Beauty in Physics: Miniconference in Honor of Adriano Di Giacomo on his 70th Birthday, Pisa, Italy, January 26–27 (2006) [hep-ph/0601066] [INSPIRE].
S. Narison, Slope of the topological charge, proton spin and the 0−+ pseudoscalar di-gluonia spectra, Nucl. Phys. A 1020 (2022) 122393 [arXiv:2111.02873] [INSPIRE].
C. Bonati et al., Axion phenomenology and θ-dependence from Nf = 2 + 1 lattice QCD, JHEP 03 (2016) 155 [arXiv:1512.06746] [INSPIRE].
J. Frison et al., Topological susceptibility at high temperature on the lattice, JHEP 09 (2016) 021 [arXiv:1606.07175] [INSPIRE].
S. Borsanyi et al., Calculation of the axion mass based on high-temperature lattice quantum chromodynamics, Nature 539 (2016) 69 [arXiv:1606.07494] [INSPIRE].
P. Petreczky, H.-P. Schadler and S. Sharma, The topological susceptibility in finite temperature QCD and axion cosmology, Phys. Lett. B 762 (2016) 498 [arXiv:1606.03145] [INSPIRE].
C. Bonati et al., Topology in full QCD at high temperature: a multicanonical approach, JHEP 11 (2018) 170 [arXiv:1807.07954] [INSPIRE].
F. Burger, E.-M. Ilgenfritz, M.P. Lombardo and A. Trunin, Chiral observables and topology in hot QCD with two families of quarks, Phys. Rev. D 98 (2018) 094501 [arXiv:1805.06001] [INSPIRE].
C. Bonanno, G. Clemente, M. D’Elia and F. Sanfilippo, Topology via spectral projectors with staggered fermions, JHEP 10 (2019) 187 [arXiv:1908.11832] [INSPIRE].
M.P. Lombardo and A. Trunin, Topology and axions in QCD, Int. J. Mod. Phys. A 35 (2020) 2030010 [arXiv:2005.06547] [INSPIRE].
A.Y. Kotov, A. Trunin and M.P. Lombardo, QCD topology and axion’s properties from Wilson twisted mass lattice simulations, PoS LATTICE2021 (2022) 032 [arXiv:2111.15421] [INSPIRE].
A. Athenodorou et al., Topological susceptibility of Nf = 2 + 1 QCD from staggered fermions spectral projectors at high temperatures, JHEP 10 (2022) 197 [arXiv:2208.08921] [INSPIRE].
TWQCD collaboration, Topological susceptibility in finite temperature QCD with physical (u/d, s, c) domain-wall quarks, Phys. Rev. D 106 (2022) 074501 [arXiv:2204.01556] [INSPIRE].
B. Alles, M. D’Elia and A. Di Giacomo, Topological susceptibility at zero and finite T in SU(3) Yang-Mills theory, Nucl. Phys. B 494 (1997) 281 [hep-lat/9605013] [INSPIRE].
B. Alles, M. D’Elia and A. Di Giacomo, Topology at zero and finite T in SU(2) Yang-Mills theory, Phys. Lett. B 412 (1997) 119 [hep-lat/9706016] [INSPIRE].
L. Del Debbio, L. Giusti and C. Pica, Topological susceptibility in the SU(3) gauge theory, Phys. Rev. Lett. 94 (2005) 032003 [hep-th/0407052] [INSPIRE].
L. Del Debbio, H. Panagopoulos and E. Vicari, θ dependence of SU(N) gauge theories, JHEP 08 (2002) 044 [hep-th/0204125] [INSPIRE].
M. D’Elia, Field theoretical approach to the study of theta dependence in Yang-Mills theories on the lattice, Nucl. Phys. B 661 (2003) 139 [hep-lat/0302007] [INSPIRE].
L. Del Debbio et al., θ-dependence of the spectrum of SU(N) gauge theories, JHEP 06 (2006) 005 [hep-th/0603041] [INSPIRE].
B. Lucini, M. Teper and U. Wenger, Topology of SU(N) gauge theories at T ≃ 0 and T ≃ Tc, Nucl. Phys. B 715 (2005) 461 [hep-lat/0401028] [INSPIRE].
L. Giusti, S. Petrarca and B. Taglienti, θ dependence of the vacuum energy in the SU(3) gauge theory from the lattice, Phys. Rev. D 76 (2007) 094510 [arXiv:0705.2352] [INSPIRE].
M. Lüscher and F. Palombi, Universality of the topological susceptibility in the SU(3) gauge theory, JHEP 09 (2010) 110 [arXiv:1008.0732] [INSPIRE].
H. Panagopoulos and E. Vicari, The 4D SU (3) gauge theory with an imaginary θ term, JHEP 11 (2011) 119 [arXiv:1109.6815] [INSPIRE].
M. Cè, C. Consonni, G.P. Engel and L. Giusti, Non-Gaussianities in the topological charge distribution of the SU (3) Yang-Mills theory, Phys. Rev. D 92 (2015) 074502 [arXiv:1506.06052] [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].
C. Bonati, M. D’Elia and A. Scapellato, θ dependence in SU(3) Yang-Mills theory from analytic continuation, Phys. Rev. D 93 (2016) 025028 [arXiv:1512.01544] [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].
C. Bonati, M. Cardinali and M. D’Elia, θ dependence in trace deformed SU(3) Yang-Mills theory: a lattice study, Phys. Rev. D 98 (2018) 054508 [arXiv:1807.06558] [INSPIRE].
C. Bonati, M. Cardinali, M. D’Elia and F. Mazziotti, θ-dependence and center symmetry in Yang-Mills theories, Phys. Rev. D 101 (2020) 034508 [arXiv:1912.02662] [INSPIRE].
A. Athenodorou and M. Teper, The glueball spectrum of SU(3) gauge theory in 3 + 1 dimensions, JHEP 11 (2020) 172 [arXiv:2007.06422] [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].
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].
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].
E. Bennett et al., Sp(2N) Yang-Mills theories on the lattice: Scale setting and topology, Phys. Rev. D 106 (2022) 094503 [arXiv:2205.09364] [INSPIRE].
M. Campostrini, A. Di Giacomo and H. Panagopoulos, The Topological Susceptibility on the Lattice, Phys. Lett. B 212 (1988) 206 [INSPIRE].
M. Campostrini, P. Rossi and E. Vicari, Monte Carlo simulation of CPN−1 models, Phys. Rev. D 46 (1992) 2647 [INSPIRE].
M. Campostrini, P. Rossi and E. Vicari, Topological susceptibility and string tension in the lattice CPN−1 models, Phys. Rev. D 46 (1992) 4643 [hep-lat/9207032] [INSPIRE].
B. Alles, M. D’Elia, A. Di Giacomo and R. Kirchner, A critical comparison of different definitions of topological charge on the lattice, Phys. Rev. D 58 (1998) 114506 [hep-lat/9711026] [INSPIRE].
L. Del Debbio, G.M. Manca and E. Vicari, Critical slowing down of topological modes, Phys. Lett. B 594 (2004) 315 [hep-lat/0403001] [INSPIRE].
W. Bietenholz, U. Gerber, M. Pepe and U.-J. Wiese, Topological Lattice Actions, JHEP 12 (2010) 020 [arXiv:1009.2146] [INSPIRE].
M. Hasenbusch, Fighting topological freezing in the two-dimensional CPN−1 model, Phys. Rev. D 96 (2017) 054504 [arXiv:1706.04443] [INSPIRE].
C. Bonati and M. D’Elia, Topological critical slowing down: variations on a toy model, Phys. Rev. E 98 (2018) 013308 [arXiv:1709.10034] [INSPIRE].
C. Bonanno, C. Bonati and M. D’Elia, Topological properties of CPN−1 models in the large-N limit, JHEP 01 (2019) 003 [arXiv:1807.11357] [INSPIRE].
M. Berni, C. Bonanno and M. D’Elia, Large-N expansion and θ-dependence of 2dCP N−1 models beyond the leading order, Phys. Rev. D 100 (2019) 114509 [arXiv:1911.03384] [INSPIRE].
M. Berni, C. Bonanno and M. D’Elia, θ-dependence in the small-N limit of 2dCPN−1 models, Phys. Rev. D 102 (2020) 114519 [arXiv:2009.14056] [INSPIRE].
C. Bonanno, M. D’Elia and F. Margari, Topological susceptibility of the 2D CP1 or O(3) nonlinear σ model: Is it divergent or not?, Phys. Rev. D 107 (2023) 014515 [arXiv:2208.00185] [INSPIRE].
A. Di Giacomo, Topology and the U(1) problem from lattice, Nucl. Phys. B Proc. Suppl. 23 (1991) 191 [INSPIRE].
G. Briganti, A. Di Giacomo and H. Panagopoulos, A lattice determination of the slope of the topological susceptibility at q2 = 0, Phys. Lett. B 253 (1991) 427 [INSPIRE].
A. Di Giacomo, E. Meggiolaro and H. Panagopoulos, A lattice determination of the slope at q2 = 0 of the topological susceptibility in su (3) yang-mills theory, Phys. Lett. B 291 (1992) 147 [INSPIRE].
G. Boyd, B. Alles, M. D’Elia and A. Di Giacomo, Topology in QCD, in the proceedings of the 1997 Europhysics Conference on High Energy Physics, Jerusalem, Israel, August 19–26 (1997), p. 1028–1033 [hep-lat/9711025] [INSPIRE].
Y. Koma et al., Momentum dependence of the topological susceptibility with overlap fermions, PoS LATTICE2010 (2010) 278 [arXiv:1012.1383] [INSPIRE].
C. Bonanno, Lattice determination of the topological susceptibility slope χ′ of 2d CPN−1 models at large N , Phys. Rev. D 107 (2023) 014514 [arXiv:2212.02330] [INSPIRE].
M. Creutz, Overrelaxation and Monte Carlo Simulation, Phys. Rev. D 36 (1987) 515 [INSPIRE].
M. Creutz, Monte Carlo Study of Quantized SU(2) Gauge Theory, Phys. Rev. D 21 (1980) 2308 [INSPIRE].
A.D. Kennedy and B.J. Pendleton, Improved Heat Bath Method for Monte Carlo Calculations in Lattice Gauge Theories, Phys. Lett. B 156 (1985) 393 [INSPIRE].
N. Cabibbo and E. Marinari, A New Method for Updating SU(N) Matrices in Computer Simulations of Gauge Theories, Phys. Lett. B 119 (1982) 387 [INSPIRE].
S. Necco and R. Sommer, The Nf = 0 heavy quark potential from short to intermediate distances, Nucl. Phys. B 622 (2002) 328 [hep-lat/0108008] [INSPIRE].
R. Sommer, Scale setting in lattice QCD, PoS LATTICE2013 (2014) 015 [arXiv:1401.3270] [INSPIRE].
P. Di Vecchia, K. Fabricius, G.C. Rossi and G. Veneziano, Preliminary Evidence for UA(1) Breaking in QCD from Lattice Calculations, Nucl. Phys. B 192 (1981) 392 [INSPIRE].
B. Berg, Dislocations and Topological Background in the Lattice O(3) σ Model, Phys. Lett. B 104 (1981) 475 [INSPIRE].
Y. Iwasaki and T. Yoshie, Instantons and Topological Charge in Lattice Gauge Theory, Phys. Lett. B 131 (1983) 159 [INSPIRE].
S. Itoh, Y. Iwasaki and T. Yoshie, Stability of Instantons on the Lattice and the Renormalized Trajectory, Phys. Lett. B 147 (1984) 141 [INSPIRE].
M. Teper, Instantons in the Quantized SU(2) Vacuum: A Lattice Monte Carlo Investigation, Phys. Lett. B 162 (1985) 357 [INSPIRE].
E.-M. Ilgenfritz et al., First Evidence for the Existence of Instantons in the Quantized SU(2) Lattice Vacuum, Nucl. Phys. B 268 (1986) 693 [INSPIRE].
M. Campostrini, A. Di Giacomo, H. Panagopoulos and E. Vicari, Topological Charge, Renormalization and Cooling on the Lattice, Nucl. Phys. B 329 (1990) 683 [INSPIRE].
B. Alles, L. Cosmai, M. D’Elia and A. Papa, Topology in 2DCPN−1 models on the lattice: A Critical comparison of different cooling techniques, Phys. Rev. D 62 (2000) 094507 [hep-lat/0001027] [INSPIRE].
C. Bonati and M. D’Elia, Comparison of the gradient flow with cooling in SU(3) pure gauge theory, Phys. Rev. D 89 (2014) 105005 [arXiv:1401.2441] [INSPIRE].
C. Alexandrou, A. Athenodorou and K. Jansen, Topological charge using cooling and the gradient flow, Phys. Rev. D 92 (2015) 125014 [arXiv:1509.04259] [INSPIRE].
M. Lüscher, Trivializing maps, the Wilson flow and the HMC algorithm, Commun. Math. Phys. 293 (2010) 899 [arXiv:0907.5491] [INSPIRE].
M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, JHEP 08 (2010) 071 [Erratum ibid. 03 (2014) 092] [arXiv:1006.4518] [INSPIRE].
APE collaboration, Glueball Masses and String Tension in Lattice QCD, Phys. Lett. B 192 (1987) 163 [INSPIRE].
C. Morningstar and M.J. Peardon, Analytic smearing of SU(3) link variables in lattice QCD, Phys. Rev. D 69 (2004) 054501 [hep-lat/0311018] [INSPIRE].
C. Bonanno et al., Sphaleron rate from a modified Backus-Gilbert inversion method, Phys. Rev. D 108 (2023) 074515 [arXiv:2305.17120] [INSPIRE].
C. Bonanno et al., Sphaleron rate of Nf = 2 + 1 QCD, arXiv:2308.01287 [INSPIRE].
Extended Twisted Mass Collaboration (ETMC) collaboration, Probing the Energy-Smeared R Ratio Using Lattice QCD, Phys. Rev. Lett. 130 (2023) 241901 [arXiv:2212.08467] [INSPIRE].
Particle Data Group collaboration, Review of Particle Physics, PTEP 2022 (2022) 083C01 [INSPIRE].
Flavour Lattice Averaging Group (FLAG) collaboration, FLAG Review 2021, Eur. Phys. J. C 82 (2022) 869 [arXiv:2111.09849] [INSPIRE].
G.S. Bali et al., Mesons in large-N QCD, JHEP 06 (2013) 071 [arXiv:1304.4437] [INSPIRE].
M.G. Pérez, A. González-Arroyo and M. Okawa, Meson spectrum in the large N limit, JHEP 04 (2021) 230 [arXiv:2011.13061] [INSPIRE].
Acknowledgments
I am grateful to M. D’Elia and M. García Peréz for useful discussions and for reading this manuscript. This work is supported by the Spanish Research Agency (Agencia Estatal de Investigación) through the grant IFT Centro de Excelencia Severo Ochoa CEX2020- 001007-S and, partially, by grant PID2021-127526NB-I00, both funded by MCIN/AEI/ 10.13039/501100011033. I 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. Numerical calculations have been performed on the Finisterrae III cluster at CESGA (Centro de Supercomputación de Galicia).
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Bonanno, C. The topological susceptibility slope χ′ of the pure-gauge SU(3) Yang-Mills theory. J. High Energ. Phys. 2024, 116 (2024). https://doi.org/10.1007/JHEP01(2024)116
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DOI: https://doi.org/10.1007/JHEP01(2024)116