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

Abstract

In this paper, we present the exact calculations for the vertex $\bar{s}γb$ and $\bar{s} Z b$ in the unitary gauge. We found that (a) the divergent- and $μ$-dependent terms are left in the effective vertex function $Γ^γ_μ(p,k)$ for $b \to s γ$ 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 $\bar{s}γb$ and $\bar{s} Z b$ 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.