Explicit calculation on two-loop correction to the chiral magnetic effect with NJL model

Chiral Magnetic Effect(CME) is usually believed not receiving higher order corrections due to the non-renormalization of AVV triangle diagram in the framework of quantum field theory. However, the CME-relevant triangle, which is obtained by expanding the current-current correlation requires zero momentum on the axial vertex, is not equivalent to the general AVV triangle when taking the zero-momentum limit owing to the infrared problem on the axial vertex. Therefore, it is still significant to check if there exists perturbative higher order corrections to the current-current correlation. In this paper, we explicitly calculate the two-loop corrections of CME within NJL model with Chern-Simons term which ensures a consistent $\mu_5$. The result shows the two-loop corrections to the CME conductivity are zero, which confirms the non-renomalization of CME conductivity.