# Scattering from a quantum anapole at low energies

## TL;DR

This paper investigates the anapole moment of Majorana fermions through low-energy scattering calculations, linking quantum field theory results with classical toroidal current models and demonstrating the Aharonov-Bohm effect.

## Contribution

It provides a quantum field theory analysis of the anapole moment and connects it to classical toroidal current distributions, illustrating the electromagnetic properties of Majorana fermions.

## Key findings

- Anapole moment behaves like a point-like toroidal solenoid at low energies.
- Scattering cross section matches quantum mechanical models in the non-relativistic limit.
- The work offers an educational example for understanding particle electromagnetic properties.

## Abstract

In quantum field theory, the photon-fermion vertex can be described in terms of four form factors which encode the static electromagnetic properties of the particle, namely its charge, magnetic dipole moment, electric dipole moment, and anapole moment. For Majorana fermions, only the anapole moment can be nonzero, a consequence of the fact that these particles are their own antiparticles. Using the framework of quantum field theory, we perform a scattering calculation which probes the anapole moment with a spinless charged particle. In the limit of low-momentum transfer, we confirm that the anapole can be classically likened to a point-like toroidal solenoid whose magnetic field is confined to the origin. Such a toroidal current distribution can be used to demonstrate the Aharonov-Bohm effect. We find that, in the non-relativistic limit, our scattering cross section agrees with a quantum mechanical computation of the cross section for a spinless current scattered by an infinitesimally thin toroidal solenoid. Our presentation is geared toward advanced undergraduate or beginning graduate students. This work serves as an introduction to the anapole moment and also provides an example of how one can develop an understanding of a particle's electromagnetic properties in quantum field theory.