Scaling in SU(3) pure gauge theory with a renormalization-group-improved action

We study the scaling properties of the static quark potential and the ratio of the critical temperature $T_c$ to the square root of the string tension $\sigma$ in the SU(3) pure gauge theory using a renormalization-group-improved action. We first determine the critical coupling ${\beta }_c$ on lattices with a temporal extension $N_t=3\mathrm{}$, 4, and 6, and then calculate the static quark potential at the critical couplings on lattices at zero temperature. We note that the static quark potentials obtained are rotationally invariant with errors of at most 1–2 % in all three cases, and that the potential $V\left(R\right)\mathrm{}$ in physical units scales in the whole region of $R$ investigated. The values of $T_c/\sqrt{\sigma }$ for the three cases in the infinite volume limit are identical within errors. We estimate the value in the continuum limit to be $T_c/\sqrt{\sigma }=0.656\mathrm{}\left(4\right)\mathrm{}$, which is slightly larger than the value in the continuum limit from the one-plaquette action, 0.629(3).