Relativistic effects in the pionium lifetime

The pionium decay width, the measurement of which allows determination of the scattering amplitude of the process ${\pi }^{+}{\pi }^{-}\to {\pi }^0{\pi }^0$ near threshold, is evaluated in the framework of chiral perturbation theory and the relativistic bound state formalism of constraint theory. Corrections of order $O\left(\alpha \right)$ are calculated with respect to the conventional lowest-order formula, in which the strong interaction scattering amplitude has been evaluated to two-loop order with charged pion masses. Strong interaction corrections from second-order perturbation theory of the bound state wave equation are found to be of the order of 0.4%. Electromagnetic radiative corrections, due to pion-photon interactions, are estimated to be of the order of $-0.1\%.$ Electromagnetic mass shift insertions in internal propagators produce a correction of the order of 0.3%. The correction due to the passage from the strong interaction scattering amplitude evaluated with the mass parameter fixed at the charged pion mass to the amplitude evaluated with the mass parameter fixed at the neutral pion mass is found to be of the order of 6.4%. The overall uncertainty to these corrections is estimated to be of the order of 1.5%. The numerical prediction for the pionium lifetime in the standard scheme of chiral perturbation theory is presented.