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⁻→ π⁻π⁰. 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/Γ₀, is found for light nuclei, 1≤ Z≤ 10. The numerical calculation is complemented by a diagrammatic small-$αZ$ expansion. We find Γ/Γ₀ = 1-(αZ)²/2 + 17(αZ)⁴/24 +O((αZ)⁵), in agreement with the all-orders numerical calculation at low Z.