Inquiry for the \pi^+ \pi^- bound state conversion in two \pi^0 as being due to the Weinberg pion-pion-interaction lagrangian

Abstract

In the work presented, the decay of the pionium, that is the \pi^+ \pi^- bound state, into two \pi^0 is studied, the \pi\pi-interaction causing this transition being described by the underlying Weinberg lagrangian. The calculation with such a \pi\pi-lagrangian being carried out, the \pi-meson size r_{0}-emerges to be allowed for and occurs in the final result. The bound \pi^+\pi^- system itself is presumed to be due to the instantaneous Coulomb interaction and is treated consistently nonrelativistic, the Bethe-Solpeter equation being utilized. Along calculating, the terms of the lowest order in fine structure constant \alpha and the terms \sim \ln(r_{0}) are retained. The obtained pionium lifetime \tau is thought to be compatible with the conceivable future experimental data. The dependence of the results on the effective lagrangian parameters is visualized. The investigation carried out persuades us that just the whole form of the genuine \pi-interaction determines the pionium lifetime, but not simply the \pi\pi scattering lengths. The inquiry into pionium decaying promotes to specify the validity of various \pi\pi-interaction descriptions.