The noncommutativity of the static and homogeneous limit of the axial chemical potential in chiral magnetic effect

TL;DR

This paper investigates the order-dependent noncommutativity of limits in the axial chemical potential's role in the chiral magnetic effect, revealing fundamental issues related to singularities and anomaly nonrenormalizability.

Contribution

It demonstrates the noncommutativity of zero energy-momentum limits in CME and clarifies the underlying mechanisms involving singularities and Ward identities.

Findings

The CME vanishes when static limit is taken before homogeneous limit.

Dressed axial-vector vertex singularity replaces free fermion propagator singularity.

Nonrenormalizability of chiral anomaly explains nonvanishing CME in the opposite limit.

Abstract

We study the noncommutativity of different orders of zero energy-momentum limit pertaining to the axial chemical potential in the chiral magnetic effect. While this noncommutativity issue originates from the pinching singularity at one-loop order, it cannot be removed by introducing a damping term to the fermion propagators. The physical reason is that modifying the propagator alone would violate the axial-vector Ward identity and as a result a modification of the longitudinal component of the axial-vector vertex is required, which contributes to CME. The pinching singularity with free fermion propagators was then taken over by the singularity stemming from the dressed axial-vector vertex. We show this mechanism by a concrete example. Moreover, we proved in general the vanishing CME in the limit order that the static limit was taken prior to the homogeneous limit in the light of Coleman-Hill theorem for a static external magnetic field. For the opposite limit that the homogeneous limit is taken first, we show that the nonvanishing CME was a consequence of the nonrenormalizability of chiral anomaly for an arbitrary external magnetic field.