Three body radiative decay $B_s\to \phi \bar K^0 \gamma$ in the PQCD approach

We study the three body radiative decay $B_s\to \phi \bar K^0 \gamma$ by introducing the $\phi K$ pair distribution amplitudes (DAs) in the perturbative QCD approach. This nonperturbative inputs, the two meson DAs, is very important to simplify the calculations. Besides the dominant electromagnetic penguin operator $O_{7\gamma}$, the subleading contributions from chromomagnetic penguin operator $O_{8g}$, quark-loop corrections and annihilation type amplitudes are also considered. We find that the branching ratio for the decay $B_s\to \phi \bar K^0 \gamma$ is about $(9.26^{+1.79+3.12+0.64}_{-1.61-3.86-0.49})\times10^{-8}$, which is much smaller compared with that for the decay $B^0\to \phi K^0\gamma$. It is mainly because that the former decay induces by $b\to d\gamma$ with small CKM matrix element being proportional to $\lambda^3$. The prediction for the direct CP asymmetry is $A^{dir}_{CP}(B_s\to \phi \bar K^0 \gamma)=(-4.1^{+0.4+1.7+0.2}_{-0.6-1.2-0.1})\%$, which is well consistent with the result from the U-spin symmetry approach. we also predict the $B_s \to\phi \bar K^0\gamma$ decay spectrum, which exhibits a maximu at the $\phi K$ invariant masss around 1.95 GeV.