The weak decay constant of positronium

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.