The thermal chiral anomaly in the Schwinger model

TL;DR

This paper derives a closed-form expression for the thermal chiral anomaly in the Schwinger model, exploring its properties, relation to the Dirac operator index, and loop contributions, and computes the effective action at finite temperature.

Contribution

It provides the first explicit derivation of the thermal chiral anomaly in the Schwinger model, including the effective action and anomaly functional at finite temperature.

Findings

Thermal anomaly depends on the long-distance behavior of the electric field.

The anomaly does not receive higher-loop contributions, similar to the zero-temperature case.

Complete effective action and anomaly functional are derived on thermal branches.

Abstract

In the Schwinger model at finite temperature, we derive a closed form result for the chiral anomaly which arises from the long distance behavior of the electric field \cite{frenkel}. We discuss the general properties associated with this thermal anomaly as well as its relation with the "index" of the Dirac operator. We further show that the thermal anomaly, like the zero temperature anomaly which arises from the ultraviolet behavior of the theory, does not receive any contribution from higher loops. Finally, we determine the complete effective action as well as the anomaly functional on both the thermal branches in the closed time path formalism.