Long Distance Contribution to $s\to d\gamma$ and Implications for ${\Omega }^{-}\to {\Xi }^{-}\gamma ,B_s\to B_d^{\ast }\gamma$ and $b\to s\gamma$

Abstract

We estimate the long distance (LD) contribution to the magnetic part of the $s\to d\gamma$ transition using the Vector Meson Dominance approximation $(V=\rho ,\omega ,{\psi }_i)$. We find that this contribution may be significantly larger than the short distance (SD) contribution to $s\to d\gamma$ and could possibly saturate the present experimental upper bound on the ${\Omega }^{-}\to {\Xi }^{-}\gamma$ decay rate, ${\Gamma }_{{\Omega }^{-}\to {\Xi }^{-}\gamma }^{\mathrm{MAX}}\simeq 3.7\times 10^{-9}$eV. For the decay $B_s\to B_d^{\ast }\gamma$, which is driven by $s\to d\gamma$ as well, we obtain an upper bound on the branching ratio $BR(B_s\to B_d^{\ast }\gamma )<3\times 10^{-8}$ from ${\Gamma }_{{\Omega }^{-}\to {\Xi }^{-}\gamma }^{\mathrm{MAX}}$. Barring the possibility that the Quantum Chromodynamics coefficient $a_2(m_s)$ be much smaller than 1, ${\Gamma }_{{\Omega }^{-}\to {\Xi }^{-}\gamma }^{\mathrm{MAX}}$ also implies the approximate relation $\frac{2}{3}{\sum }_i\frac{g_{{\psi }_i}^2(0)}{m_{{\psi }_i}^2}\simeq \frac{1}{2}\frac{g_{\rho }^2(0)}{m_{\rho }^2}+\frac{1}{6}\frac{g_{\omega }^2(0)}{m_{\omega }^2}$. This relation agrees quantitatively with a recent independent estimate of the l.h.s. by Deshpande et al., confirming that the LD contributions to $b\to s\gamma$ are small. We find that these amount to an increase of $(4\pm 2)\%$ in the magnitude of the $b\to s\gamma$ transition amplitude, relative to the SD contribution alone.

Comments: 16 pages, LaTeX file