Axial Ward identity and the Schwinger mechanism -- Applications to the real-time chiral magnetic effect and condensates

We elucidate chirality production under parity breaking constant electromagnetic fields, with which we clarify qualitative differences in and out of equilibrium. For a strong magnetic field the pair production from the Schwinger mechanism increments the chirality. The pair production rate is exponentially suppressed with mass according to the Schwinger formula, while the mass dependence of chirality production in the axial Ward identity appears in the pesudo-scalar term. We demonstrate that in equilibrium field theory calculus the axial anomaly is canceled by the pseudo-scalar condensate for any mass. In a real-time formulation with in- and out-states, we show that the axial Ward identity leads to the chirality production rate consistent with the Schwinger formula. We illuminate that such an in- and out-states formulation makes clear the chiral magnetic effect in and out of equilibrium, and we discuss further applications to real-time condensates.