The weak decay constant of positronium

TL;DR

This paper introduces a weak decay constant for positronium within a quantum field theory framework, providing insights into its electroweak interactions and internal structure, which are negligible compared to electromagnetic decay.

Contribution

It extends the composite QFT model to include weak decay processes of positronium and defines a new weak decay constant analogous to meson decay constants.

Findings

The weak decay constant of positronium is approximately 89.86 eV.

The derived expression links positronium's internal structure to its electroweak coupling.

Comparison with quarkonium systems helps determine the vertex function connecting constituents.

Abstract

The positronium, as the lightest purely leptonic bound state, provides an ideal testing ground for a quantum field--theoretical (QFT) description of composite systems. While its electromagnetic annihilation is well understood as the dominant decay channel, the weak interaction sector of positronium is strongly suppressed. In this work, we extend the composite QFT framework previously developed for the two--photon decay and introduce the concept of a weak decay constant for para--positronium. This constant, defined in analogy with those of pseudoscalar mesons, quantifies the coupling of the positronium field to the weak axial current and serves as a measure of its internal structure in the electroweak domain. Its numerical value $ f_{P} \simeq \frac{m_{e}}{2\sqrt{\pi}} \alpha^{3/2} = 89.8593~\mathrm{eV} $ is, as expected, small. Nevertheless, the resulting expression and a related comparison to quarkonium systems allow us to determine the vertex function linking the positronium to its own constituents.