Chiral magnetic effect at weak coupling with relaxation dynamics

We provide resolution of an old issue in the weak coupling computation of the chiral magnetic effect (CME) current, where free chiral fermion theory gives two different results depending on the order of the two limits, $\omega \to 0$ (frequency) and $k\to 0$ (spatial momentum). We first argue based on hydrodynamics that in any reasonable interacting theory of chiral fermions the noncommutativity 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 a 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 chiral magnetic effect current is reproduced even in the presence of relaxation dynamics in both computations.

Figure 1

The plot of $[1-F(\omega ,k=0)]$ as a function of $\omega$ near $\omega =0$ for different values of $\tau$: ${\tau }^{-1}=1$ (dashed), ${\tau }^{-1}=0.1$ (dotted), and ${\tau }^{-1}=0.01$ (solid). The unit of ${\tau }^{-1}$ and $\omega$ is arbitrary.

Figure 2

The Schwinger-Keldysh contour appropriate for computing real-time retarded response functions at finite temperature.

Figure 3

The diagrams responsible for the retarded response of current $J^{\mu }$ to one external gauge potential.