Abstract
We study the θ-dependence of the string tension and of the lightest glueball mass in four-dimensional SU(N) Yang-Mills theories. More precisely, we focus on the coefficients parametrizing the \( \mathcal{O}\left({\theta}^2\right) \) dependence of these quantities, which we investigate by means of numerical simulations of the lattice-discretized theory, carried out using imaginary values of the θ parameter. Topological freezing at large N is avoided using the Parallel Tempering on Boundary Conditions algorithm. We provide controlled continuum extrapolations of such coefficients in the N = 3 case, and we report the results obtained on two fairly fine lattice spacings for N = 6.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Coleman, Aspects of Symmetry: Selected Erice Lectures, Cambridge University Press, Cambridge, U.K. (1985).
R. Jackiw, Introduction to the Yang-Mills Quantum Theory, Rev. Mod. Phys. 52 (1980) 661 [INSPIRE].
D.J. Gross, R.D. Pisarski and L.G. Yaffe, QCD and Instantons at Finite Temperature, Rev. Mod. Phys. 53 (1981) 43 [INSPIRE].
T. Schäfer and E.V. Shuryak, Instantons in QCD, Rev. Mod. Phys. 70 (1998) 323 [hep-ph/9610451] [INSPIRE].
A. D’Adda, M. Lüscher and P. Di Vecchia, A 1/n Expandable Series of Nonlinear Sigma Models with Instantons, Nucl. Phys. B 146 (1978) 63 [INSPIRE].
E. Witten, Instantons, the Quark Model, and the 1/N Expansion, Nucl. Phys. B 149 (1979) 285 [INSPIRE].
T.G. Kovacs, E.T. Tomboulis and Z. Schram, Topology on the lattice: 2d Yang-Mills theories with a theta term, Nucl. Phys. B 454 (1995) 45 [hep-th/9505005] [INSPIRE].
C. Bonati and P. Rossi, Topological susceptibility of two-dimensional U(N) gauge theories, Phys. Rev. D 99 (2019) 054503 [arXiv:1901.09830] [INSPIRE].
C. Bonati and P. Rossi, Topological effects in continuum two-dimensional U(N) gauge theories, Phys. Rev. D 100 (2019) 054502 [arXiv:1908.07476] [INSPIRE].
D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, Time Reversal, and Temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [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].
P. Di Vecchia and G. Veneziano, Chiral Dynamics in the Large n Limit, Nucl. Phys. B 171 (1980) 253 [INSPIRE].
G. Grilli di Cortona, E. Hardy, J. Pardo Vega and G. Villadoro, The QCD axion, precisely, JHEP 01 (2016) 034 [arXiv:1511.02867] [INSPIRE].
A. Boccaletti and D. Nogradi, The semi-classical approximation at high temperature revisited, JHEP 03 (2020) 045 [arXiv:2001.03383] [INSPIRE].
C. Bonati et al., Axion phenomenology and θ-dependence from Nf = 2 + 1 lattice QCD, JHEP 03 (2016) 155 [arXiv:1512.06746] [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].
J. Frison, R. Kitano, H. Matsufuru, S. Mori and N. Yamada, 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].
C. Bonati, M. D’Elia, G. Martinelli, F. Negro, F. Sanfilippo and A. Todaro, 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].
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].
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].
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].
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].
E. Vicari and H. Panagopoulos, Theta dependence of SU(N) gauge theories in the presence of a topological term, Phys. Rep. 470 (2009) 93 [arXiv:0803.1593] [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].
C. Bonati, M. D’Elia, H. Panagopoulos and E. Vicari, Change of θ Dependence in 4D SU(N) Gauge Theories Across the Deconfinement Transition, Phys. Rev. Lett. 110 (2013) 252003 [arXiv:1301.7640] [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].
E. Berkowitz, M.I. Buchoff and E. Rinaldi, Lattice QCD input for axion cosmology, Phys. Rev. D 92 (2015) 034507 [arXiv:1505.07455] [INSPIRE].
S. Borsanyi et al., Axion cosmology, lattice QCD and the dilute instanton gas, Phys. Lett. B 752 (2016) 175 [arXiv:1508.06917] [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].
E. Witten, Current Algebra Theorems for the U(1) Goldstone Boson, Nucl. Phys. B 156 (1979) 269 [INSPIRE].
G. Veneziano, U(1) Without Instantons, Nucl. Phys. B 159 (1979) 213 [INSPIRE].
K. Kawarabayashi and N. Ohta, The Problem of η in the Large N Limit: Effective Lagrangian Approach, Nucl. Phys. B 175 (1980) 477 [INSPIRE].
E. Witten, Large N Chiral Dynamics, Ann. Phys. 128 (1980) 363 [INSPIRE].
M. Campostrini and P. Rossi, 1/N expansion of the topological susceptibility in the CPN−1 models, Phys. Lett. B 272 (1991) 305 [INSPIRE].
L. Del Debbio, G.M. Manca, H. Panagopoulos, A. Skouroupathis and E. Vicari, Theta-dependence of the spectrum of SU(N) gauge theories, JHEP 06 (2006) 005 [hep-th/0603041] [INSPIRE].
P. Rossi, Effective Lagrangian of CP N−1 models in the large N limit, Phys. Rev. D 94 (2016) 045013 [arXiv:1606.07252] [INSPIRE].
E. Vicari, Monte Carlo simulation of lattice CPN−1 models at large N, Phys. Lett. B 309 (1993) 139 [hep-lat/9209025] [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 2d CPN−1 models beyond the leading order, Phys. Rev. D 100 (2019) 114509 [arXiv:1911.03384] [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].
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].
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].
D. Vadacchino, A review on Glueball hunting, in the proceedings of the 39th International Symposium on Lattice Field Theory, Bonn, Germany, 8–13 August 2022, arXiv:2305.04869 [INSPIRE].
B. Alles, G. Boyd, M. D’Elia, A. Di Giacomo and E. Vicari, Hybrid Monte Carlo and topological modes of full QCD, Phys. Lett. B 389 (1996) 107 [hep-lat/9607049] [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].
ALPHA collaboration, Critical slowing down and error analysis in lattice QCD simulations, Nucl. Phys. B 845 (2011) 93 [arXiv:1009.5228] [INSPIRE].
M. Hasenbusch, Fighting topological freezing in the two-dimensional CPN−1 model, Phys. Rev. D 96 (2017) 054504 [arXiv:1706.04443] [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].
R. Brower, S. Chandrasekharan, J.W. Negele and U.J. Wiese, QCD at fixed topology, Phys. Lett. B 560 (2003) 64 [hep-lat/0302005] [INSPIRE].
S. Aoki, H. Fukaya, S. Hashimoto and T. Onogi, Finite volume QCD at fixed topological charge, Phys. Rev. D 76 (2007) 054508 [arXiv:0707.0396] [INSPIRE].
M. Campostrini, A. Di Giacomo and H. Panagopoulos, The Topological Susceptibility on the Lattice, Phys. Lett. B 212 (1988) 206 [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, M. Laursen, G. Schierholz, M. Müller-Preussker and H. Schiller, 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 2D CPN−1 models on the lattice: A Critical comparison of different cooling techniques, Phys. Rev. D 62 (2000) 094507 [hep-lat/0001027] [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].
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].
M. Lüscher and P. Weisz, Perturbative analysis of the gradient flow in non-Abelian gauge theories, JHEP 02 (2011) 051 [arXiv:1101.0963] [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].
G. Bhanot and F. David, The phases of the O(3) σ-model for imaginary θ, Nucl. Phys. B 251 (1985) 127 [INSPIRE].
V. Azcoiti, G. Di Carlo, A. Galante and V. Laliena, New proposal for numerical simulations of theta vacuum-like systems, Phys. Rev. Lett. 89 (2002) 141601 [hep-lat/0203017] [INSPIRE].
B. Alles and A. Papa, Mass gap in the 2D O(3) non-linear sigma model with a θ = π term, Phys. Rev. D 77 (2008) 056008 [arXiv:0711.1496] [INSPIRE].
M. Imachi, M. Kambayashi, Y. Shinno and H. Yoneyama, The θ-term, CPN−1 model and the inversion approach in the imaginary-θ method, Prog. Theor. Phys. 116 (2006) 181 [INSPIRE].
S. Aoki et al., The Electric dipole moment of the nucleon from simulations at imaginary vacuum angle theta, arXiv:0808.1428 [INSPIRE].
B. Alles, M. Giordano and A. Papa, Behavior near θ = π of the mass gap in the two-dimensional O(3) non-linear sigma model, Phys. Rev. B 90 (2014) 184421 [arXiv:1409.1704] [INSPIRE].
M. D’Elia and F. Negro, θ dependence of the deconfinement temperature in Yang-Mills theories, Phys. Rev. Lett. 109 (2012) 072001 [arXiv:1205.0538] [INSPIRE].
M. D’Elia and F. Negro, Phase diagram of Yang-Mills theories in the presence of a θ term, Phys. Rev. D 88 (2013) 034503 [arXiv:1306.2919] [INSPIRE].
C. Bonanno, M. D’Elia and L. Verzichelli, The θ-dependence of the SU(N) critical temperature at large N, JHEP 02 (2024) 156 [arXiv:2312.12202] [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].
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].
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].
J.L. Dasilva Golán, C. Bonanno, M. D’Elia, M. García Pérez and A. Giorgieri, The twisted gradient flow strong coupling with parallel tempering on boundary conditions, PoS LATTICE2023 (2024) 354 [arXiv:2312.09212] [INSPIRE].
C. Bonanno, J.L. Dasilva Golán, M. D’Elia, M. García Pérez and A. Giorgieri, The SU(3) twisted gradient flow strong coupling without topology freezing, arXiv:2403.13607 [INSPIRE].
M. Lüscher and S. Schaefer, Lattice QCD without topology barriers, JHEP 07 (2011) 036 [arXiv:1105.4749] [INSPIRE].
B. Berg and A. Billoire, Glueball Spectroscopy in Four-Dimensional SU(3) Lattice Gauge Theory. Part 1, Nucl. Phys. B 221 (1983) 109 [INSPIRE].
M. Teper, An Improved Method for Lattice Glueball Calculations, Phys. Lett. B 183 (1987) 345 [INSPIRE].
M.J. Teper, SU(N) gauge theories in (2 + 1)-dimensions, Phys. Rev. D 59 (1999) 014512 [hep-lat/9804008] [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].
B. Blossier, M. Della Morte, G. von Hippel, T. Mendes and R. Sommer, On the generalized eigenvalue method for energies and matrix elements in lattice field theory, JHEP 04 (2009) 094 [arXiv:0902.1265] [INSPIRE].
B. Lucini, A. Rago and E. Rinaldi, Glueball masses in the large N limit, JHEP 08 (2010) 119 [arXiv:1007.3879] [INSPIRE].
E. Bennett et al., Glueballs and strings in Sp(2N) Yang-Mills theories, Phys. Rev. D 103 (2021) 054509 [arXiv:2010.15781] [INSPIRE].
P. de Forcrand, G. Schierholz, H. Schneider and M. Teper, The String and Its Tension in SU(3) Lattice Gauge Theory: Towards Definitive Results, Phys. Lett. B 160 (1985) 137 [INSPIRE].
C. Bonanno, The topological susceptibility slope χ′ of the pure-gauge SU(3) Yang-Mills theory, JHEP 01 (2024) 116 [arXiv:2311.06646] [INSPIRE].
M. Caselle, A. Nada and M. Panero, Hagedorn spectrum and thermodynamics of SU(2) and SU(3) Yang-Mills theories, JHEP 07 (2015) 143 [Erratum ibid. 11 (2017) 016] [arXiv:1505.01106] [INSPIRE].
E. Trotti, S. Jafarzade and F. Giacosa, Thermodynamics of the glueball resonance gas, Eur. Phys. J. C 83 (2023) 390 [arXiv:2212.03272] [INSPIRE].
D.J. Gross and E. Witten, Possible Third Order Phase Transition in the Large N Lattice Gauge Theory, Phys. Rev. D 21 (1980) 446 [INSPIRE].
I. Bars and F. Green, Complete Integration of U(N) Lattice Gauge Theory in a Large N Limit, Phys. Rev. D 20 (1979) 3311 [INSPIRE].
Acknowledgments
The work of C. Bonanno is supported by the Spanish Research Agency (Agencia Estatal de Investigación) through the grant IFT Centro de Excelencia Severe Ochoa CEX2020-001007-S and, partially, by grant PID2021-127526NB-I00, both funded by MCIN/AEI/10.13039/501100011033. C. Bonanno also acknowledges 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. The work of D. Vadacchino is supported by STFC under Consolidated Grant No. ST/X000680/1. Numerical calculations have been performed on the Galileo100 machine at Cineca, based on the project IscrB_ITDGBM, on the Marconi machine at Cineca based on the agreement between INFN and Cineca (under project INF22_npqcd), and on the Plymouth University cluster.
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: 2402.03096
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
Bonanno, C., Bonati, C., Papace, M. et al. The θ-dependence of the Yang-Mills spectrum from analytic continuation. J. High Energ. Phys. 2024, 163 (2024). https://doi.org/10.1007/JHEP05(2024)163
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2024)163