Equation of State for SU(3) Gauge Theory via the Energy-Momentum Tensor under Gradient Flow

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 $\beta=6.287$--$7.500$ corresponding to the lattice spacing $a= 0.013$--$0.061\,\mathrm{fm}$. The spatial (temporal) sizes are chosen to be $N_s= 64$, $96$, $128$ ($N_{\tau}=12$, $16$, $20$, $22$, $24$) with the aspect ratio, $5.33 \le N_s/N_{\tau} \le 8$. Double extrapolation, $a\rightarrow 0$ (the continuum limit) followed by $t\rightarrow 0$ (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.