# Scale Magnetic Effect in Quantum Electrodynamics and the Wigner-Weyl Formalism

## TL;DR

This paper investigates the Scale Magnetic Effect in quantum electrodynamics, demonstrating how the induced current depends on particle mass and conformal anomaly, using the Wigner-Weyl formalism to analyze the decoupling of heavy fermions.

## Contribution

It introduces a Wigner-Weyl formalism approach to analyze the SME and shows how heavy fermions decouple, affecting the anomalous conductivity in massive electrodynamics.

## Key findings

- The SME current is proportional to the beta function of the theory.
- In massive electrodynamics, the SME persists but differs from the massless case.
- Heavy fermions decouple, causing the anomalous conductivity to vanish at large masses.

## Abstract

The Scale Magnetic Effect (SME) is the generation of electric current due to conformal anomaly in external magnetic field in curved spacetime. The effect appears in a vacuum with electrically charged massless particles. Similarly to the Hall effect, the direction of the induced anomalous current is perpendicular to the direction of the external magnetic field $\bf B$ and to the gradient of the conformal factor $\tau$, while the strength of the current is proportional to the beta function of the theory. In massive electrodynamics the SME remains valid, but the value of the induced current differs from the current generated in the system of massless fermions. In the present paper we use the Wigner--Weyl formalism to demonstrate that in accordance with the decoupling property of heavy fermions the corresponding anomalous conductivity vanishes in the large-mass limit with $m^2 \gg |e {\bf B}|$ and $m \gg |\nabla \tau|$.