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 pseudoscalar term only. We demonstrate that, in a real-time formulation with in and out states, the axial Ward identity with an in-in expectation value leads to a chirality production rate consistent with the Schwinger formula, while the axial anomaly with an in-out expectation value is canceled by the pseudoscalar condensate for any mass. We illuminate that such an in-and out-state formulation clarifies subtleties in the chiral magnetic effect in and out of equilibrium, and we discuss further applications to real-time condensates.