Weak decays of magnetized charged pions in the symmetric gauge

TL;DR

This paper investigates the decay of charged pions in strong magnetic fields using the symmetric gauge, revealing gauge-independent decay widths and the absence of helicity suppression in such conditions.

Contribution

It provides a detailed analysis of angular momentum conservation in pion decay under magnetic fields and derives a gauge-independent expression for the decay width.

Findings

Decay width remains nonzero for massless leptons in strong magnetic fields.

Canonical angular momentum is conserved, mechanical angular momentum is not.

Derived decay width matches previous Landau gauge results, confirming gauge independence.

Abstract

We consider the decay $\pi^- \to l \, \bar{\nu}_l$ ($l=e^-,\,\mu^-$) in the presence of an arbitrary large uniform magnetic field, using the symmetric gauge. The consequences of the axial symmetry of the problem and the issue of angular momentum conservation are discussed in detail. In particular, we analyze the projection of both the canonical and the mechanical total angular momenta along the direction of the magnetic field. It is found that while the former is conserved in the symmetric gauge, the latter is not conserved in both the symmetric and Landau gauges. We derive an expression for the integrated $\pi^- \to l \, \bar{\nu}_l$ width that coincides exactly with the one we previously found using the Landau gauge, providing an explicit test of the gauge independence of that result. Such an expression implies that for nonzero magnetic fields the decay width does not vanish in the limit in which the outgoing charged leptons can be considered as massless, i.e. it does not exhibit the helicity suppression found in the case of no external field.