We derive an analytic expression for one-loop effective action of QCD+QED at zero and finite temperatures by using the Schwingers proper time method. The result is a nonlinear effective action not only for electromagnetic and chromo-electromagnetic f
ields but also the Polyakov loop, and thus reproduces the Euler-Heisenberg action in QED, QCD, and QED+QCD, and also the Weiss potential for the Polyakov loop at finite temperature. As applications of this Euler-Heisenberg-Weiss action in QCD+QED, we investigate quark pair productions induced by QCD+QED fields at zero temperature and the Polyakov loop in the presence of strong electromagnetic fields. Quark one-loop contribution to the effective potential of the Polyakov loop explicitly breaks the center symmetry, and is found to be enhanced by the magnetic field, which is consistent with the inverse magnetic catalysis observed in lattice QCD simulation.
