Three-body radiative decay

We study the three-body radiative decay $$B_s\rightarrow \phi \bar{K}^0 \gamma $$ by introducing the $$\phi K$$ pair distribution amplitudes (DAs) in the perturbative QCD approach. This nonperturbative input, the two-meson DAs, is very important to simplify the calculations. Besides the dominant electromagnetic penguin operator $$O_{7\gamma }$$, the subleading contributions from the chromomagnetic penguin operator $$O_{8g}$$, quark-loop corrections and annihilation type amplitudes are also considered. We find that the branching ratio for the decay $$B_s\rightarrow \phi \bar{K}^0 \gamma $$ is about $$(9.26^{+1.79+3.12+0.64}_{-1.61-3.86-0.49})\times 10^{-8}$$, which is much smaller than that for the decay $$B^0\rightarrow \phi K^0\gamma $$. It is mainly because the former decay induced by $$b\rightarrow d\gamma $$ with small CKM matrix element is proportional to $$\lambda ^3$$. The prediction for the direct CP asymmetry is $$A^{dir}_{CP}(B_s\rightarrow \phi \bar{K}^0 \gamma )=(-4.1^{+0.4+1.7+0.2}_{-0.6-1.2-0.1})\%$$, which is well consistent with the result from the U-spin symmetry approach. We also predict the $$B_s \rightarrow \phi \bar{K}^0\gamma $$ decay spectrum, which exhibits a maximum at the $$\phi K$$ invariant masss around 1.95 GeV