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

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