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

TL;DR

This paper investigates how electromagnetic fields produce chirality and condensates, clarifying differences between equilibrium and real-time dynamics, and demonstrating the chiral magnetic effect through axial Ward identities and Schwinger mechanism.

Contribution

It provides a detailed analysis of chirality production mechanisms, highlighting the role of the axial Ward identity and real-time formulations in understanding the chiral magnetic effect.

Findings

Chirality production increases with strong magnetic fields via the Schwinger mechanism.

In equilibrium, the axial anomaly is canceled by pseudo-scalar condensates regardless of mass.

Real-time analysis confirms the chirality production rate aligns with the Schwinger formula.

Abstract

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.