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We provide a resolution of an old issue in weak coupling computation of theChiral Magnetic Effect (CME) current, where a free chiral fermion theory givestwo different results depending on the order of the two limits,$\omega\rightarrow 0$ (frequency) and $k\rightarrow 0$ (spatial momentum). Wefirst argue based on hydrodynamics that in any reasonable interacting theory ofchiral fermions the non-commutativity between the two limits should be absent,and we demonstrate this at weak coupling regime in two different frameworks:kinetic theory in the relaxation time approximation, and diagrammaticcomputation with resummation of damping rate. In the latter computation, wealso show that the "pinch" singularity, which would make the summation ofladder diagrams necessary as in the P-even correlation function, is absent inthe relevant P-odd correlation function. The correct value of CME current isreproduced even in the presence of relaxation dynamics in both computations.

Chiral Magnetic Effect at Weak Coupling with Relaxation Dynamics