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

Recently, much work has been devoted to the calculation of order $alpha$ corrections to the decay rate of pionium, the $pi^+ pi^-$ bound state. In previous calculations, nonrelativistic QED corrections were neglected since they start at order $alpha^2$ in hydrogen and positronium. In this note, we point out that there is one correction which is actually of order $alpha$ times a function of the ratio $mu_r alpha / m_e$, where $mu_r$ is the reduced mass of the system. When $mu_r alpha ll m_e$, this function can be Taylor expanded and leads to higher order corrections. When $mu_r alpha approx m_e$, as is the case in pionium, the function is of order one and the correction is of order $alpha$. We use an effective field theory approach to calculate this correction and find it equal to $0.4298 alpha Gamma_0$. We also calculate the corresponding correction to the dimuonium ($mu^+ mu^-$ bound state) decay rate and obtained a result in agreement with Jentschura et al.