Long Distance Contributions to Penguin Processes $b\to s\gamma$ and $b\to d\gamma$

Abstract

We consider the long distance contributions to inclusive penguin processes through processes like $b\to sV$ and $b\to dV$ where $V$ are ${}^3{S}_1(c\bar{c})$ states ${\psi }_i$ in the former case and include $\rho$, $\omega$ for the latter case. We carefully examine vector dominance for $\bar{c}c$ states, and conclude that there is a large suppression of $\psi \sim \gamma$ transition when $\psi$ is at $q^2=0$. The long distance effects can be at most 10\% in both the amplitudes for $b\to s\gamma$ and $b\to d\gamma$. Although the long distance contributions are small, the ratio $BR(b\to d\gamma )/BR(b\to s\gamma )=|V_{td}/V_{ts}{|}^2$ does not hold due to significant $u$ and $c$ loop contributions to the short distance $b\to d\gamma$ amplitudes .

Comments: 13 pages, Revtex. Some numerical errors in the calculation of $b\to d\gamma$ are corrected. The long distance contribution to $b\to d\gamma$ is smaller than stated in the earlier version. New estimate due to u and c loop contributions to the short distance $b\to d\gamma$ amplitude are included