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

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