Chiral Magnetic Effect at Weak Coupling with Relaxation Dynamics

We provide a resolution of an old issue in weak coupling computation of the Chiral Magnetic Effect (CME) current, where a free chiral fermion theory gives two different results depending on the order of the two limits, $\omega\rightarrow 0$ (frequency) and $k\rightarrow 0$ (spatial momentum). We first argue based on hydrodynamics that in any reasonable interacting theory of chiral 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 diagrammatic computation with resummation of damping rate. In the latter computation, we also show that the "pinch" singularity, which would make the summation of ladder diagrams necessary as in the P-even correlation function, is absent in the relevant P-odd correlation function. The correct value of CME current is reproduced even in the presence of relaxation dynamics in both computations.