An Effective Action for Finite Temperature Lattice Gauge Theories with Dynamical Fermions

Dynamical fermions induce via the fermion determinant a gauge-invariant effective action. In principle, this effective action can be added to the usual gauge action in simulations, reproducing the effects of closed fermion loops. Using lattice perturbation theory at finite temperature, we compute for staggered fermions the one-loop fermionic corrections to the spatial and temporal plaquette couplings as well as the leading $Z_N$ symmetry breaking coupling. A. Hasenfratz and T. DeGrand have shown that $\beta_c$ for dynamical staggered fermions can be accurately estimated by the formula $\beta_c = \beta^{\rm pure}_c - \Delta\beta_F$ where $\Delta\beta_F$ is the shift induced by the fermions at zero temperature. Numerical and analytical results indicate that the finite temperature corrections to the zero-temperature calculation of A. Hasenfratz and T. DeGrand are small for small values of $\kappa = {1\over 2m_F}$, but become significant for intermediate values of $\kappa$. The effect of these finite temperature corrections is to ruin the agreement of the Hasenfratz-DeGrand calculation with Monte Carlo data. We argue, however, that the finite temperature corrections are suppressed nonperturbatively at low temperatures, resolving this apparent disagreement. The $Z_N$ symmetry breaking coupling is small; we argue that it changes the order of the transition while having little effect on the critical value of $\beta$.