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

We determine the decay rate $Γ$ 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, $K^-\to π^-π^0$. 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 $π^-$ contribution is evaluated separately, while the partial-wave sum is used only for the Coulomb distortion. The ratio of bound-to-free decay rates, $Γ/Γ_0$, is found for light nuclei, $1\leq Z\leq 10$. The numerical calculation is complemented by a diagrammatic small-$αZ$ expansion. We find $ Γ/Γ_0 = 1-(αZ)^2/2 + 17(αZ)^4/24 +\mathcal{O}((αZ)^5)$, in agreement with the all-orders numerical calculation at low $Z$.