A new order $α$ correction to the decay rate of pionium

Abstract

Recently, much work has been devoted to the calculation of order $α$ corrections to the decay rate of pionium, the $π^+ π^-$ bound state. In previous calculations, nonrelativistic QED corrections were neglected since they start at order $α^2$ in hydrogen and positronium. In this note, we point out that there is one correction which is actually of order $α$ times a function of the ratio $μ_r α/ m_e$, where $μ_r$ is the reduced mass of the system. When $μ_r α\ll m_e$, this function can be Taylor expanded and leads to higher order corrections. When $μ_r α\approx m_e$, as is the case in pionium, the function is of order one and the correction is of order $α$. We use an effective field theory approach to calculate this correction and find it equal to $0.4298 αΓ_0$. We also calculate the corresponding correction to the dimuonium ($μ^+ μ^-$ bound state) decay rate and obtained a result in agreement with Jentschura et al.