The twisted Polyakov loop coupling and the search for an IR fixed point

We report the nonperturbative behavior of the twisted Polyakov loop (TPL) coupling constant for the SU(3) gauge theories defined by the ratio of Polyakov loop correlators in finite volume with twisted boundary condition. Carrying out the numerical simulations, we determine the nonperturbative running coupling constant in this renormalization scheme for Nf=12 SU(3) gauge theory. According to the perturbative two loop analysis, the Nf=12 SU(3) gauge theory might have a conformal fixed point in the infrared region. However, the recent lattice studies show controversial results for the existence of the fixed point. We point out possible reasons of the controversy in previous works, and present our careful study. Finally, we find the infrared fixed point (IRFP) and discuss the robustness of the nontrivial IRFP of many flavor system under the change of the analysis method. This talk is based on the paper[1].