Abstract
We apply the previously-developed sub-volume method to study the θ-dependence of the four-dimensional SU(2) Yang-Mills theory at finite temperature. We calculate the first two coefficients, the topological susceptibility χ and the fourth cumulant b2, in the θ-expansion of the free energy density around the critical temperature (Tc) for the confinement-deconfinement transition. Lattice calculations are performed with three different spatial sizes 243, 323, 483 to monitor finite size effects, while the temporal size is fixed to be 8. The systematic uncertainty associated with the sub-volume extrapolation is studied with special care. The sub-volume method allows us to determine the values of b2 much more accurately than the standard full-volume method, and we successfully identify the temperature dependence of b2 around Tc. Our numerical results suggest that the θ-dependence of the free energy density near θ = 0 changes from 4χ(1 − cos(θ/2)) to χ(1 − cos θ) as the temperature crosses Tc.
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C.G. Callan Jr., R.F. Dashen and D.J. Gross, The Structure of the Gauge Theory Vacuum, Phys. Lett. B 63 (1976) 334 [INSPIRE].
B. Lucini, M. Teper and U. Wenger, The deconfinement transition in SU(N) gauge theories, Phys. Lett. B 545 (2002) 197 [hep-lat/0206029] [INSPIRE].
B. Lucini, M. Teper and U. Wenger, The high temperature phase transition in SU(N) gauge theories, JHEP 01 (2004) 061 [hep-lat/0307017] [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].
R. Kitano, N. Yamada and M. Yamazaki, Is N = 2 Large?, JHEP 02 (2021) 073 [arXiv:2010.08810] [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. 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].
N. Otake and N. Yamada, θ dependence of Tc in 4d SU(3) Yang-Mills theory with histogram method and the Lee-Yang zeros in the large N limit, JHEP 06 (2022) 044 [arXiv:2202.05605] [INSPIRE].
S. Borsanyi et al., Topological features of the deconfinement transition, Phys. Rev. D 107 (2023) 054514 [arXiv:2212.08684] [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. Hirasawa et al., Determination of the CP restoration temperature at θ = π in 4D SU(2) Yang-Mills theory through simulations at imaginary θ, PoS LATTICE2023 (2024) 193 [arXiv:2401.05726] [INSPIRE].
C. Bonanno, C. Bonati, M. Papace and D. Vadacchino, The θ-dependence of the Yang-Mills spectrum from analytic continuation, JHEP 05 (2024) 163 [arXiv:2402.03096] [INSPIRE].
R. Kitano, R. Matsudo, N. Yamada and M. Yamazaki, Peeking into the θ vacuum, Phys. Lett. B 822 (2021) 136657 [arXiv:2102.08784] [INSPIRE].
G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
E. Witten, Large N Chiral Dynamics, Annals Phys. 128 (1980) 363 [INSPIRE].
G. ’t Hooft, Topology of the Gauge Condition and New Confinement Phases in Nonabelian Gauge Theories, Nucl. Phys. B 190 (1981) 455 [INSPIRE].
E. Witten, Theta dependence in the large N limit of four-dimensional gauge theories, Phys. Rev. Lett. 81 (1998) 2862 [hep-th/9807109] [INSPIRE].
M. Unsal, Theta dependence, sign problems and topological interference, Phys. Rev. D 86 (2012) 105012 [arXiv:1201.6426] [INSPIRE].
D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, Time Reversal, and Temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [INSPIRE].
R. Kitano, T. Suyama and N. Yamada, θ = π in SU(N)/ZN gauge theories, JHEP 09 (2017) 137 [arXiv:1709.04225] [INSPIRE].
M. Yamazaki and K. Yonekura, From 4d Yang-Mills to 2d ℂℙN−1 model: IR problem and confinement at weak coupling, JHEP 07 (2017) 088 [arXiv:1704.05852] [INSPIRE].
M. Yamazaki, Relating ’t Hooft Anomalies of 4d Pure Yang-Mills and 2d ℂℙN−1 Model, JHEP 10 (2018) 172 [arXiv:1711.04360] [INSPIRE].
Y. Nomura and M. Yamazaki, Tensor Modes in Pure Natural Inflation, Phys. Lett. B 780 (2018) 106 [arXiv:1711.10490] [INSPIRE].
Z. Wan, J. Wang and Y. Zheng, New higher anomalies, SU(N) Yang-Mills gauge theory and ℂℙN−1 sigma model, Annals Phys. 414 (2020) 168074 [arXiv:1812.11968] [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].
P. Keith-Hynes and H.B. Thacker, Fractionally charged Wilson loops as a probe of theta-dependence in CP(N-1) sigma models: instantons vs. large N, Phys. Rev. D 78 (2008) 025009 [arXiv:0804.1534] [INSPIRE].
P. de Forcrand et al., Local topological and chiral properties of QCD, Nucl. Phys. B Proc. Suppl. 73 (1999) 578 [hep-lat/9810033] [INSPIRE].
W. Bietenholz, P. de Forcrand and U. Gerber, Topological Susceptibility from Slabs, JHEP 12 (2015) 070 [arXiv:1509.06433] [INSPIRE].
W. Bietenholz et al., The Slab Method to Measure the Topological Susceptibility, PoS LATTICE2016 (2016) 321 [arXiv:1610.00685] [INSPIRE].
LSD collaboration, Maximum-Likelihood Approach to Topological Charge Fluctuations in Lattice Gauge Theory, Phys. Rev. D 90 (2014) 014503 [arXiv:1403.2761] [INSPIRE].
P. Weisz, Continuum Limit Improved Lattice Action for Pure Yang-Mills Theory. I, Nucl. Phys. B 212 (1983) 1 [INSPIRE].
G. Cella, G. Curci, R. Tripiccione and A. Vicere, Scaling, asymptotic scaling and Symanzik improvement. Deconfinement temperature in SU(2) pure gauge theory, Phys. Rev. D 49 (1994) 511 [hep-lat/9306011] [INSPIRE].
P. Giudice and S. Piemonte, Improved thermodynamics of SU(2) gauge theory, Eur. Phys. J. C 77 (2017) 821 [arXiv:1708.01216] [INSPIRE].
P. Hasenfratz, V. Laliena and F. Niedermayer, The index theorem in QCD with a finite cutoff, Phys. Lett. B 427 (1998) 125 [hep-lat/9801021] [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].
C. Alexandrou et al., Comparison of topological charge definitions in Lattice QCD, Eur. Phys. J. C 80 (2020) 424 [arXiv:1708.00696] [INSPIRE].
APE collaboration, Glueball Masses and String Tension in Lattice QCD, Phys. Lett. B 192 (1987) 163 [INSPIRE].
P. de Forcrand, M. Garcia Perez and I.-O. Stamatescu, Topology of the SU(2) vacuum: a lattice study using improved cooling, Nucl. Phys. B 499 (1997) 409 [hep-lat/9701012] [INSPIRE].
J. Frison et al., Topological susceptibility at high temperature on the lattice, JHEP 09 (2016) 021 [arXiv:1606.07175] [INSPIRE].
M. Caselle and R. Pellegrini, Finite-Temperature Behavior of Glueballs in Lattice Gauge Theories, Phys. Rev. Lett. 111 (2013) 132001 [arXiv:1304.4757] [INSPIRE].
G. ’t Hooft, Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle, Phys. Rev. D 14 (1976) 3432 [Erratum ibid. 18 (1978) 2199] [INSPIRE].
L. Del Debbio, H. Panagopoulos and E. Vicari, theta dependence of SU(N) gauge theories, JHEP 08 (2002) 044 [hep-th/0204125] [INSPIRE].
S. Ueda et al., Development of an object oriented lattice QCD code ‘Bridge++’, J. Phys. Conf. Ser. 523 (2014) 012046 [INSPIRE].
Acknowledgments
This work is based in part on the Bridge++ code [45] and is supported in part by JSPS KAK-ENHI Grant-in-Aid for Scientific Research (Nos. 19H00689 [RK, NY, MY], 21H01086 [RK], 22K21350 [RK], 22K03645 [NY], 19K03820, 20H05860, 23H01168 [MY]), and JST, Japan (PRESTO Grant No. JPMJPR225A, Moonshot R& D Grant No. JPMJMS2061 [MY]). This research used computational resources of Cygnus and Wisteria/BDEC-01 Odyssey (the University of Tokyo), provided by the Multidisciplinary Cooperative Research Program in the Center for Computational Sciences, University of Tsukuba.
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Yamada, N., Yamazaki, M. & Kitano, R. Subvolume method for SU(2) Yang-Mills theory at finite temperature: topological charge distributions. J. High Energ. Phys. 2024, 198 (2024). https://doi.org/10.1007/JHEP07(2024)198
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DOI: https://doi.org/10.1007/JHEP07(2024)198