Long-distance contribution to s→dγ and implications for ${\mathrm{\Omega }}^{\mathrm{-}}$→${\mathrm{\Xi }}^{\mathrm{-}}$γ, ${\mathrm{B}}_{\mathrm{s}}$→${\mathrm{B}}_{\mathrm{d}}^{\mathrm{*}}$γ, and b→sγ

We estimate the long-distance (LD) contribution to the magnetic part of the s→dγ transition using the vector meson dominance approximation (V=ρ,ω,${\mathrm{\psi }}_{\mathit{i}}$). We find that this contribution may be significantly larger than the short-distance (SD) contribution to s→dγ and could possibly saturate the present experimental upper bound on the ${\mathrm{\Omega }}^{\mathrm{-}}$→${\mathrm{\Xi }}^{\mathrm{-}}$γ decay rate, ${\mathrm{\Gamma }}_{{\mathrm{\Omega }}^{\mathrm{-}}\to {\mathrm{\Xi }}^{\mathrm{-}}\gamma }^{\max }$≃3.7×${10}^{\mathrm{-}9}$ eV. For the decay ${\mathit{B}}_{\mathit{s}}$→${\mathit{B}}_{\mathit{d}}^{\mathrm{*}}$γ, which is driven by s→dγ as well, we obtain an upper bound on the branching ratio B(${\mathit{B}}_{\mathit{s}}$→${\mathit{B}}_{\mathit{d}}^{\mathrm{*}}$γ)<3×${10}^{\mathrm{-}8}$ from ${\mathrm{\Gamma }}_{{\mathrm{\Omega }}^{\mathrm{-}}\to {\mathrm{\Xi }}^{\mathrm{-}}\gamma }^{\max }$. Barring the possibility that the quantum chromodynamics coefficient ${\mathit{a}}_2$(${\mathit{m}}_{\mathit{s}}^2$) is much smaller than 1, ${\mathrm{\Gamma }}_{{\mathrm{\Omega }}^{\mathrm{-}}\to {\mathrm{\Xi }}^{\mathrm{-}}\gamma }^{\max }$ also implies the approximate relation 2/3${\sum }_{\mathit{ig}{\mathrm{\psi }}_{\mathit{i}}}^2$(0)/${\mathit{m}}_{{\mathrm{\psi }}_{\mathit{i}}}^2$≃1/2${\mathit{g}}_{\mathrm{\rho }}^2$(0 /${\mathit{m}}_{\mathrm{\rho }}^2$+1/6${\mathit{g}}_{\mathrm{\omega }}^2$(0)/${\mathit{m}}_{\mathrm{\omega }}^2$. This relation agrees quantitatively with a recent independent estimate of the left-hand side by Deshpande, He, and Trampetic, confirming that the LD contributions to...