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.