Long Distance Contribution to $s \to dγ$ and Implications for $Ω^-\to Ξ^-γ, B_s \to B_d^*γ$ and $b \to sγ$

Abstract

We estimate the long distance (LD) contribution to the magnetic part of the $s \to dγ$ transition using the Vector Meson Dominance approximation $(V=ρ,ω,ψ_i)$. We find that this contribution may be significantly larger than the short distance (SD) contribution to $s \to dγ$ and could possibly saturate the present experimental upper bound on the $Ω^-\to Ξ^-γ$ decay rate, $Γ^{\rm MAX}_{Ω^-\to Ξ^-γ} \simeq 3.7\times10^{-9}$eV. For the decay $B_s \to B^*_dγ$, which is driven by $s \to dγ$ as well, we obtain an upper bound on the branching ratio $BR(B_s \to B_d^*γ)<3\times10^{-8}$ from $Γ^{\rm MAX}_{Ω^-\to Ξ^-γ}$. Barring the possibility that the Quantum Chromodynamics coefficient $a_2(m_s)$ be much smaller than 1, $Γ^{\rm MAX}_{Ω^-\to Ξ^-γ}$ also implies the approximate relation $\frac{2}{3} \sum_i \frac{g^2_{ψ_i}(0)}{m^2_{ψ_i}} \simeq \frac{1}{2} \frac{g^2_ρ(0)}{m^2_ρ} + \frac{1}{6}\frac{g^2_ω(0)}{m^2_ω}$. This relation agrees quantitatively with a recent independent estimate of the l.h.s. by Deshpande et al., confirming that the LD contributions to $b \to sγ$ are small. We find that these amount to an increase of $(4\pm2)\%$ in the magnitude of the $b \to s γ$ transition amplitude, relative to the SD contribution alone.