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

Abstract

We consider the long distance contributions to inclusive penguin processes through processes like $b\rightarrow s V$ and $b\rightarrow d V$ where $V$ are $^3S_1(c\bar c)$ 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 $b\rightarrow s\gamma$ and $b\rightarrow d \gamma$. Although the long distance contributions are small, the ratio $BR(b\rightarrow d\gamma)/BR(b \rightarrow s \gamma) = |V_{td}/V_{ts}|^2$ does not hold due to significant $u$ and $c$ loop contributions to the short distance $b\rightarrow d\gamma$ amplitudes .