Abstract
We investigate the topological properties of Nf = 2 + 1 QCD with physical quark masses, at temperatures around 500 MeV. With the aim of obtaining a reliable sampling of topological modes in a regime where the fluctuations of the topological charge Q are very rare, we adopt a multicanonical approach, adding a bias potential to the action which enhances the probability of suppressed topological sectors. This method permits to gain up to three orders magnitude in computational power in the explored temperature regime. Results at different lattice spacings and physical spatial volumes reveal no significant finite size effects and the presence, instead, of large finite cut-off effects, with the topological susceptibility which decreases by 3-4 orders of magnitude while moving from a ≃ 0.06 fm towards the continuum limit. The continuum extrapolation is in agreeement with previous lattice determinations with smaller uncertainties but obtained based on ansatzes justified by several theoretical assumptions. The parameter b2, related to the fourth order coefficient in the Taylor expansion of the free energy density f (θ), has instead a smooth continuum extrapolation which is in agreement with the dilute instanton gas approximation (DIGA); moreover, a direct measurement of the relative weights of the different topological sectors gives an even stronger support to the validity of DIGA.
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References
R.D. Peccei and H.R. Quinn, CP Conservation in the Presence of Instantons, Phys. Rev. Lett. 38 (1977) 1440 [INSPIRE].
R.D. Peccei and H.R. Quinn, Constraints Imposed by CP Conservation in the Presence of Instantons, Phys. Rev. D 16 (1977) 1791 [INSPIRE].
F. Wilczek, Problem of Strong P and T Invariance in the Presence of Instantons, Phys. Rev. Lett. 40 (1978) 279 [INSPIRE].
S. Weinberg, A New Light Boson?, Phys. Rev. Lett. 40 (1978) 223 [INSPIRE].
J. Preskill, M.B. Wise and F. Wilczek, Cosmology of the Invisible Axion, Phys. Lett. B 120 (1983) 127 [INSPIRE].
L.F. Abbott and P. Sikivie, A Cosmological Bound on the Invisible Axion, Phys. Lett. B 120 (1983) 133 [INSPIRE].
M. Dine and W. Fischler, The Not So Harmless Axion, Phys. Lett. B 120 (1983) 137 [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].
T.R. Morris, D.A. Ross and C.T. Sachrajda, Higher Order Quantum Corrections in the Presence of an Instanton Background Field, Nucl. Phys. B 255 (1985) 115 [INSPIRE].
A. Ringwald and F. Schrempp, Confronting instanton perturbation theory with QCD lattice results, Phys. Lett. B 459 (1999) 249 [hep-lat/9903039] [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 [Erratum ibid. B 679 (2004) 397] [hep-lat/9605013] [INSPIRE].
B. Alles, M. D’Elia and A. Di Giacomo, Topological susceptibility in full QCD at zero and finite temperature, Phys. Lett. B 483 (2000) 139 [hep-lat/0004020] [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 ≃ T c, Nucl. Phys. B 715 (2005) 461 [hep-lat/0401028] [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].
S. Dürr, Z. Fodor, C. Hölbling and T. Kurth, Precision study of the SU(3) topological susceptibility in the continuum, JHEP 04 (2007) 055 [hep-lat/0612021] [INSPIRE].
L. Giusti, S. Petrarca and B. Taglienti, Theta dependence of the vacuum energy in the SU(3) gauge theory from the lattice, Phys. Rev. D 76 (2007) 094510 [arXiv:0705.2352] [INSPIRE].
MILC collaboration, A. Bazavov et al., Topological susceptibility with the asqtad action, Phys. Rev. D 81 (2010) 114501 [arXiv:1003.5695] [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].
MILC collaboration, A. Bazavov et al., Lattice QCD ensembles with four flavors of highly improved staggered quarks, Phys. Rev. D 87 (2013) 054505 [arXiv:1212.4768] [INSPIRE].
ETM collaboration, K. Cichy, E. Garcia-Ramos and K. Jansen, Topological susceptibility from the twisted mass Dirac operator spectrum, JHEP 02 (2014) 119 [arXiv:1312.5161] [INSPIRE].
ALPHA collaboration, M. Bruno, S. Schaefer and R. Sommer, Topological susceptibility and the sampling of field space in N f = 2 lattice QCD simulations, JHEP 08 (2014) 150 [arXiv:1406.5363] [INSPIRE].
JLQCD collaboration, H. Fukaya, S. Aoki, G. Cossu, S. Hashimoto, T. Kaneko and J. Noaki, Topology density correlator on dynamical domain-wall ensembles with nearly frozen topological charge, PoS(LATTICE2014)323 (2014) [arXiv:1411.1473] [INSPIRE].
ETM collaboration, K. Cichy, E. Garcia-Ramos, K. Jansen, K. Ottnad and C. Urbach, Non-perturbative Test of the Witten-Veneziano Formula from Lattice QCD, JHEP 09 (2015) 020 [arXiv:1504.07954] [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].
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. Gattringer, R. Hoffmann and S. Schaefer, The Topological susceptibility of SU(3) gauge theory near T c, Phys. Lett. B 535 (2002) 358 [hep-lat/0203013] [INSPIRE].
L. Del Debbio, H. Panagopoulos and E. Vicari, Topological susceptibility of SU(N ) gauge theories at finite temperature, JHEP 09 (2004) 028 [hep-th/0407068] [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].
C. Bonati, Topology and θ dependence in finite temperature G 2 lattice gauge theory, JHEP 03 (2015) 006 [arXiv:1501.01172] [INSPIRE].
G.-Y. Xiong, J.-B. Zhang, Y. Chen, C. Liu, Y.-B. Liu and J.-P. Ma, Topological susceptibility near T c in SU(3) gauge theory, Phys. Lett. B 752 (2016) 34 [arXiv:1508.07704] [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].
R. Kitano and N. Yamada, Topology in QCD and the axion abundance, JHEP 10 (2015) 136 [arXiv:1506.00370] [INSPIRE].
S. Borsányi et al., Axion cosmology, lattice QCD and the dilute instanton gas, Phys. Lett. B 752 (2016) 175 [arXiv:1508.06917] [INSPIRE].
A. Trunin, F. Burger, E.-M. Ilgenfritz, M.P. Lombardo and M. Müller-Preussker, Topological susceptibility from N f = 2 + 1 + 1 lattice QCD at nonzero temperature, J. Phys. Conf. Ser. 668 (2016) 012123 [arXiv:1510.02265] [INSPIRE].
C. Bonati et al., Axion phenomenology and θ-dependence from N f = 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. Borsányi et al., Calculation of the axion mass based on high-temperature lattice quantum chromodynamics, Nature 539 (2016) 69 [arXiv:1606.07494] [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].
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, H. Panagopoulos and E. Vicari, theta dependence of SU(N) gauge theories, JHEP 08 (2002) 044 [hep-th/0204125] [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, S. Schaefer et al., Critical slowing down and error analysis in lattice QCD simulations, Nucl. Phys. B 845 (2011) 93 [arXiv:1009.5228] [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].
E. Vicari and H. Panagopoulos, Theta dependence of SU(N ) gauge theories in the presence of a topological term, Phys. Rept. 470 (2009) 93 [arXiv:0803.1593] [INSPIRE].
B.A. Berg and T. Neuhaus, Multicanonical ensemble: A New approach to simulate first order phase transitions, Phys. Rev. Lett. 68 (1992) 9 [hep-lat/9202004] [INSPIRE].
A. Laio and M. Parrinello Escaping free-energy minima, Proc. Nat. Acad. Sci. 99 (2002) 12562 [cond-mat/0208352].
A. Laio and F.L. Gervasio Metadynamics: a method to simulate rare events and reconstruct the free energy in biophysics, chemistry and material science, Rept. Prog. Phys. 71 (2008) 126601.
A. Laio, G. Martinelli and F. Sanfilippo, Metadynamics surfing on topology barriers: the CP N−1 case, JHEP 07 (2016) 089 [arXiv:1508.07270] [INSPIRE].
P.T. Jahn, G.D. Moore and D. Robaina, χ top(T ≫ T c) in pure-glue QCD through reweighting, Phys. Rev. D 98 (2018) 054512 [arXiv:1806.01162] [INSPIRE].
P. Weisz, Continuum Limit Improved Lattice Action for Pure Yang-Mills Theory. 1., Nucl. Phys. B 212 (1983) 1 [INSPIRE].
G. Curci, P. Menotti and G. Paffuti, Symanzik’s Improved Lagrangian for Lattice Gauge Theory, Phys. Lett. B 130 (1983) 205 [Erratum ibid. B 135 (1984) 516] [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.A. Clark, A.D. Kennedy and Z. Sroczynski, Exact 2+1 flavour RHMC simulations, Nucl. Phys. Proc. Suppl. 140 (2005) 835 [hep-lat/0409133] [INSPIRE].
M.A. Clark and A.D. Kennedy, Accelerating dynamical fermion computations using the rational hybrid Monte Carlo (RHMC) algorithm with multiple pseudofermion fields, Phys. Rev. Lett. 98 (2007) 051601 [hep-lat/0608015] [INSPIRE].
M.A. Clark and A.D. Kennedy, Accelerating Staggered Fermion Dynamics with the Rational Hybrid Monte Carlo (RHMC) Algorithm, Phys. Rev. D 75 (2007) 011502 [hep-lat/0610047] [INSPIRE].
Y. Aoki et al., The QCD transition temperature: results with physical masses in the continuum limit II., JHEP 06 (2009) 088 [arXiv:0903.4155] [INSPIRE].
S. Borsányi et al., The QCD equation of state with dynamical quarks, JHEP 11 (2010) 077 [arXiv:1007.2580] [INSPIRE].
S. Borsányi, Z. Fodor, C. Hölbling, S.D. Katz, S. Krieg and K.K. Szabo, Full result for the QCD equation of state with 2+1 flavors, Phys. Lett. B 730 (2014) 99 [arXiv:1309.5258] [INSPIRE].
P. Di Vecchia, K. Fabricius, G.C. Rossi and G. Veneziano, Preliminary Evidence for U(1) — A Breaking in QCD from Lattice Calculations, Nucl. Phys. B 192 (1981) 392 [INSPIRE].
P. Di Vecchia, K. Fabricius, G.C. Rossi and G. Veneziano, Numerical Checks of the Lattice Definition Independence of Topological Charge Fluctuations, Phys. Lett. B 108 (1982) 323 [INSPIRE].
M. Campostrini, A. Di Giacomo and H. Panagopoulos, The Topological Susceptibility on the Lattice, Phys. Lett. B 212 (1988) 206 [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].
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.L. Laursen, G. Schierholz, M. Muller-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].
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].
K. Cichy et al., Comparison of different lattice definitions of the topological charge, PoS(LATTICE2014)075 (2014) [arXiv:1411.1205] [INSPIRE].
Y. Namekawa, Comparative study of topological charge, PoS(LATTICE2014)344 (2015) [arXiv:1501.06295] [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].
C. Alexandrou et al., Comparison of topological charge definitions in Lattice QCD, arXiv:1708.00696 [INSPIRE].
B.A. Berg and D.A. Clarke, Deconfinement, gradient and cooling scales for pure SU(2) lattice gauge theory, Phys. Rev. D 95 (2017) 094508 [arXiv:1612.07347] [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].
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].
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Bonati, C., D’Elia, M., Martinelli, G. et al. Topology in full QCD at high temperature: a multicanonical approach. J. High Energ. Phys. 2018, 170 (2018). https://doi.org/10.1007/JHEP11(2018)170
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DOI: https://doi.org/10.1007/JHEP11(2018)170