Thermodynamics for pure SU(2) gauge theory using gradient flow

Abstract

We study the equation of state of pure SU(2) gauge theory using Monte Carlo simulations. The scale-setting of lattice parameters has been carried by using the gradient flow. We propose a reference scale t 0 for the SU(2) gauge theory satisfying t 2 E | t=t0 = 0.1, which is fixed by a natural scaling-down of the standard t 0 -scale for the SU(3) case based on perturbative analyses. We also show the thermodynamic quantities as a function of T /T c , which are derived by the energy-momentum tensor using the small flow-time expansion of the gradient flow. consistent with the most perfect-liquid property rather than the gas [10]. A theoretical large-N c analysis based on AdS/CFT correspondence gives the lower bound for η/s [11], while the 1/N c correction terms to η/s have not yet been determined even for its sign in the finite N c [12]. Although the determination of transport coefficients using the lattice calculations has been developing [13][14][15][16], it is still a challenging subject. The measurement of the correlation function of the energy-momentum tensor (EMT) is the first step to obtain the viscosities, and there are at least three difficulties: (i) the small signal-to-noise ratio of the correlator of EMT, (ii) the definition of the "correct" renormalization of EMT as a conserved quantity on the lattice, (iii) solving an inverse-problem to obtain the spectral function from the correlation function. In fact, the recent study [15] has used more than 6-million configurations to obtain the shear viscosity for one set of lattice parameter in the pure SU(3) gauge theory.Based on these situations, we focus on the pure SU(2) gauge theory, which is a good testing ground for the SU(3) gauge theory since the numerical cost is lower than the one for the SU(3) because of the smallness of the matrix size, nevertheless it has almost the same properties as the SU(3) gauge theory. In addition, the study of SU(2) gauge theory will provide the larger signal of the correction term of 1/N c to η/s because of the smaller N c .Several works have obtained the thermodynamic quantities for the SU(2) gauge theory using the integral method [17-20] on the lattice. A recent work, which mainly focuses on T < T c , shows the consistency with the massive free glueball model [17]. Another one [20], which utilizes the improved gauge action, presents the thermodynamic quantities without taking continuum limit. It is reported that, in intermediate temperature (T c T 5T c ), the scaling law of the trace anomaly of the SU(2) gauge theory has the different behavior from the SU(N c ) gauge theories with N c ≥ 3 as shown in Fig. 1, where the N c -dependence of the trace anomaly (∆/T 4 ) normalized by the pressure in the Stefan-Boltzmann (SB) limit (P SB /T 4 = π 2 (N 2 c − 1)/45) for the SU(N c ) theories is plotted. The results for the SU(2) gauge theories present as the circle (red) symbols, which we precisely determine in this work, and the triangle (cyan) symbols obtained in Ref. [20]. The dashed lines present the interpolating functions of the...

Hybrid static potentials in SU(3) lattice gauge theory at small quark-antiquark separations

“…We relate the lattice spacing a to the scale t 0 using a parametrization of ln t 0 /a 2 determined in Ref. [55] via the gradient flow. Physical units are then introduced by setting √ 8t 0 = 0.3010 fm, which corresponds to r 0 = 0.5 fm.…”

“…10) and (18), where the self energy as well as lattice discretization errors at tree-level and proportional to a 2 are removed (using Fit 1), for the Σ + g , Π u and Σ − u static potentials in units of GeV (physical units are introduced by setting r 0 = 0.5 fm). ensemble β a in fm [55] (L/a)…”

“…In Ref [27]. hybrid static potentials were computed for gauge group SU(2) at very small lattice spacing a ≈ 0.022 fm (when setting the scale as in Ref [55]…”

Effective models of a semi-quark gluon plasma

“…( 2) contributes to the pressure ∼ T 2 . In the pure glue theory, that the leading power law correction to the pressure of an ideal gas is proportional to T 2 has been found to be valid, to a good approximation, in both 2 + 1 [53][54][55] and 3 + 1 dimensions [56][57][58][59][60][61][62][63][64][65][66][67][68]. Since it has dimensions of mass (squared), this term is manifestly nonperturbative.…”

Thermodynamics for SU(2) pure gauge theory using gradient flow

“…This work is the first application of the numerical calculation of the pure SU(2) gauge theory using the gradient flow method. The details of the numerical results are reported in our paper [3].…”