Second order transport coefficient from the chiral anomaly at weak coupling: Diagrammatic resummation

We compute one of the second order transport coefficients arising from the chiral anomaly in a high-temperature weakly coupled regime of quark-gluon plasma. This transport coefficient is responsible for the $CP$-odd current that is proportional to the time derivative of the magnetic field, and can be considered as a first correction to the chiral magnetic conductivity at finite, small frequency. We observe that this transport coefficient has a nonanalytic dependence on the coupling as $\sim 1/(g^4\log (1/g))$ at the weak coupling regime, which necessitates a resummation of infinite ladder diagrams with leading pinch singularities to get a correct leading log result, a feature quite similar to what one finds in the computation of electric conductivity. We formulate and solve the relevant $CP$-odd Schwinger-Dyson equation in real-time perturbation theory that reduces to a coupled set of second order differential equations at leading log order. Our result for this second order transport coefficient indicates that chiral magnetic current has some resistance to the time change of the magnetic field; this shall be called the “chiral induction effect.” We also discuss the case of color current induced by a color magnetic field.

Figure 1

Diagrams responsible for the retarded response at one loop in the $ra$ basis.

Figure 2

A generic ladder diagram that contributes to the leading log result.

Figure 3

Exemplar real time Feynman diagrams that can give leading pinch singularity.

Figure 4

The diagrams that need to be computed to obtain the retarded response function to leading log order. The effective vertex on the left includes an infinite number of ladder diagrams.

Figure 5

The Schwinger-Dyson equation for the effective vertex ${\mathrm{\Lambda }}^i(p,\mathit{k})$.

Figure 6

The three real-time diagrams with leading pinch singularity for the kernel in the Schwinger-Dyson equation depicted in Fig. 5. The effective vertex connects only to r-type endings of fermion propagators.

Figure 7

Generic ladder diagram that gives rise to a leading log contribution. The ladders can be codified in effective vertices $\mathrm{\Lambda }$, ${\mathrm{\Lambda }}^{\prime }$.

Figure 8

Real-time Feynman diagrams for the retarded fermion self-energy.