Pionium lifetime and \pi\pi scattering lengths in generalized chiral perurbation theory

The relationship between the pionium lifetime and the $\pi \pi$ scattering lengths is established, including the sizable electromagnetic corrections. The bound state formalism that is used is that of constraint theory which provides a covariant three-dimensional reduction of the Bethe–Salpeter equation. The framework of generalized chiral perturbation theory allows then an analysis of the lifetime value as a function of the $\pi \pi$ scattering lengths, the latter being dependent on the quark condensate value.