Bound-state decay of a charged scalar particle in an external Coulomb field

We determine the decay rate $\mathrm{\Gamma }$ of a scalar particle bound to a light nucleus, numerically and analytically. As a model system, we consider a negatively charged kaon in the ground state of a hydrogen-like ion decaying into charged and neutral scalar pions, $K^{-}\to {\pi }^{-}{\pi }^0$. All particles are assumed point-like; the nucleus is infinitely heavy. The partial wave expansion converges slowly, as in a recent analysis of the bound-muon decay. To overcome this difficulty, we introduce a subtraction method: the free ${\pi }^{-}$ contribution is evaluated separately, while the partial-wave sum is used only for the Coulomb distortion. The ratio of bound-to-free decay rates, $\mathrm{\Gamma }\mathrm{/}{\mathrm{\Gamma }}_0$, is found for light nuclei, $1\le Z\le 10$. The numerical calculation is complemented by a diagrammatic small-$\alpha Z$ expansion. We find $\mathrm{\Gamma }\mathrm{/}{\mathrm{\Gamma }}_0=1-(\alpha Z{)}^2\mathrm{/}2+17(\alpha Z{)}^4\mathrm{/}24+\mathcal{O}((\alpha Z{)}^5)$, in agreement with the all-orders numerical calculation at low $Z$.