Comment on: Topological Invariants, instantons, and the chiral anomaly on spaces with torsion

TL;DR

This paper clarifies the form of the chiral anomaly in Riemann-Cartan spacetimes with torsion, showing that only the Pontrjagin term appears, not the Nieh-Yan term, contrary to recent claims.

Contribution

It demonstrates through renormalization group arguments that the chiral anomaly in torsioned spacetimes includes only the Pontrjagin form, correcting previous misconceptions.

Findings

Only the Pontrjagin four-form appears in the anomaly.

The Nieh-Yan term does not contribute to the anomaly.

Clarifies the role of torsion in chiral anomalies.

Abstract

In Riemann-Cartan spacetimes with torsion only its axial covector piece $A$ couples to massive Dirac fields. Using renormalization group arguments, we show that besides the familiar Riemannian term only the Pontrjagin type four-form $dA\wedge dA$ does arise additionally in the chiral anomaly, but not the Nieh-Yan term $d^\star A$, as has been claimed in a recent paper [PRD 55, 7580 (1997)].