Equation of state for SU(3) gauge theory via the energy-momentum tensor under gradient flow

Abstract

The energy density and the pressure of SU(3) gauge theory at finite temperature are studied by direct lattice measurements of the renormalized energy-momentum tensor obtained by the gradient flow. Numerical analyses are carried out with β = 6.287-7.500 corresponding to the lattice spacing a = 0.013-0.061 fm. The spatial (temporal) sizes are chosen to be Ns = 64, 96, 128 (Nτ = 12,16,20,22,24) with the aspect ratio, 5.33 ≤ Ns/Nτ ≤ 8. Double extrapolation, a → 0 (the continuum limit) followed by t → 0 (the zero flow-time limit), is taken using the numerical data. Above the critical temperature, the thermodynamic quantities are obtained with a few percent precision including statistical and systematic errors. The results are in good agreement with previous high-precision data obtained by using the integral method.

Energy-momentum tensor on the lattice: recent developments

“…The validity of the formula (6.13) for the lattice regularization has been examined numerically for the thermodynamics of the SU(3) pure Yang-Mills theory [25,28,31,32,34] and of the N f = 2 + 1 QCD [33,35] with very encouraging results. For the SU(3) pure Yang-Mills theory, two-point correlation functions of EMT has been computed [42,43] which even indicate the conservation law of EMT.…”

“…10 In Ref. [34], this double limit is literally taken and very encouraging results are obtained. 30) where γ i0 denotes the one-loop coefficient of the anomalous dimension γ i (g), γ i = γ i0 g 2 + O(g 4 ) (the one-loop coefficient in β , b 0 , is given by Eq.…”

“…3 and 4 of Ref. [34]. Then, to reduce the slope to the a → 0 extrapolation, the improvement ideas [49,50,51] will be very useful.…”

“…Universal formula for EMT [24,25,26,27,28,29,30,31,32,33,34,35] 6.1 Small flow time representation of EMT In this section, I explain an alternative approach [24,26] to EMT on the lattice, which is based on the UV finiteness of composite operators constructed by the gradient flow (5.1) and the flow for the fermion field [40]:…”

“…In Sect. 6, I will explain an alternative approach which does not rely on the representation (3.10); it employs the gradient flow to construct EMT itself [24,25,26,27,28,29,30,31,32,33,34,35].…”