  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.
