Chiral conductivities and effective field theory

TL;DR

This work develops a three-dimensional effective field theory to capture the low-momentum static responses of four-dimensional QFTs with U(1) axial anomalies and a dynamical U(1) vector gauge field at finite temperature. By integrating out nonzero Matsubara modes and constructing S_eff and the thermal 1PI action Γ, it derives the structure of anomaly-induced transport and shows that radiative corrections generally modify the chiral conductivities, with two-point functions protected by symmetry. The authors perform leading one-loop computations within the spatial EFT, finding explicit corrections to parity-odd coefficients (tildeχ_0, tildeφ_AA, tildeφ_AV, tildeφ_Aa) and demonstrating that certain CPT-preserving pieces can receive loop-induced shifts, while many Chern-Simons-like constants remain unrenormalized. They then match these results to the full 4D theory (e.g., QED with N_f Dirac fermions), clarifying how UV and IR contributions arise and how zero-mode physics encodes the dominant finite-temperature corrections. The findings illuminate how dynamical gauge fields influence anomaly-driven transport (CME, CVE, CSE) and have implications for the quark-gluon plasma and early-universe cosmology, where mixed gauge-gravitational anomalies can affect transport at high temperatures.

Abstract

We construct the three-dimensional effective field theory which reproduces low-momentum static correlation functions in four-dimensional quantum field theories with U(1) axial anomalies and a dynamical vector gauge field, in thermal equilibrium. We compute radiative corrections to parity-violating chiral conductivities, to leading order in the effective theory. All of the anomaly-induced transport is susceptible to radiative corrections, except for certain two-point functions which are required by symmetry to vanish.

Figures (2)

• Figure 1: The basic one-loop diagrams correcting the various chiral conductivities. The cross denotes the P-violating vertex $\tilde{g}$ in \ref{['E:j1j2']}, and the solid dot indicates the P-preserving vertex $g$ as discussed in the text.
• Figure 2: We can associate the zero-mode contribution to the two-loop thermal diagram in four dimensions on the left with the two one-loop diagrams in the spatial effective theory on the right, where the crossed vertex represents the fermion loop. The $V_k V_0$ mixing, denoted by a solid vertex on the left, is induced by a single $g_{0k}$ metric perturbation. The three-dimensional decomposition of the spatial vector $V_k = \hat{V}_k - \frac{V_0}{2}g_{0k}$ then leads, at linear order in $g_{0k}$, to the two diagrams on the right. The metric variations compute the two-point function via an insertion of $T^{0j}$, and the diagrams on the right are then equivalent to Fig. \ref{['F:VA']}).