Lifetime of $(\pi^+\pi^-)$ Atom: Analysis of the Role of Strong Interactions

The expression for the $(\pi^+\pi^-)$ atom lifetime is derived within the Bethe-Salpeter approach. First-order perturbative corrections due to the contribution of strong interactions are taken into account. It is demonstrated that the atom lifetime can be expressed in terms of the solutions of the Coulombic problem (the wave function of the $1S$ state at the origin $\Psi_1(0)$, the binding energy of the $S$-state $E_1$), the difference of the $S$-wave $\pi\pi$ scattering lengths and the energy shift $\Delta E_1$ of the level due to the strong interactions: $ 1/(\tau_{1})\sim (a^0_0-a^2_0)^2 |\Psi_1(0)|^2 (1-(9\Delta E_1)/(8E_1))$