Vanishing Polyakov Loop for QCD with Twelve Massless Quarks

We consider continuum-formulation QCD in four dimensions with twelve massless fundamental quark flavors. Splitting the SU($N$) gauge field into background and fluctuation parts, we use well-developed techniques to calculate the one-loop effective action for the theory. We find that for constant self-dual background field-strength tensor the notorious infrared divergences of the effective action cancel between gauge and matter sectors if the number of massless quark flavors is exactly . The ultraviolet divergencies of the effective action are non-perturbatively renormalized with a $\beta$-function that matches the known perturbative result in the high energy limit. The resulting UV- and IR-finite effective action possesses a non-trivial saddle which has lower free energy than the perturbative vacuum, and for which the expectation value of the Polyakov loop vanishes. Inclusion of finite temperature effects points to the presence of a first-order phase transition to the perturbative vacuum with a calculable critical temperature.