Exact calculations of vertex $\bar{s}\gamma b$ and $\bar{s}Zb$ in the unitary gauge

Abstract

In this paper, we present the exact calculations for the vertex $\bar{s}\gamma b$ and $\bar{s}Zb$ in the unitary gauge. We found that (a) the divergent- and $\mu$-dependent terms are left in the effective vertex function ${\Gamma }_{\mu }^{\gamma }(p,k)$ for $b\to s\gamma$ transition even after we sum up the contributions from four related Feynman diagrams; (b) for an on-shell photon, such terms do not contribute et al; (c) for off-shell photon, these terms will be canceled when the contributions from both vertex $\bar{s}\gamma b$ and $\bar{s}Zb$ are taken into account simultaneously, and therefore the finite and gauge independent function $Z_0(x_t)=C_0(x_t)+D_0(x_t)/4$, which governs the semi-leptonic decay $b\to s{l}^{-}{l}^{+}$, is derived in the unitary gauge.

13 pages, 2 figures, Revtex