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

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 chiral magnetic effect (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 nonrenormalization of chiral anomaly for an arbitrary external magnetic field.

Figure 1

The AVV triangle diagram. There is a second diagram with the photon four-momenta and polarization as well as CTP indices interchanged.

Figure 2

The axial-vector vertex is attached to an internal fermion loop with $n>1$ vector vertices.

Figure 3

Subdiagrams with a modified axial-vector vertex and full fermion propagators, but bare vector vertices. There is a second set of subdiagrams with the photon four-momenta and polarization indices interchanged.

Figure 4

An axial-vector vertex not considered in Fig. 3.