Effects of Final State Interactions in Pure Annihilation Decay of $B^{0}_{s}\rightarrow\pi^{0}\pi^{0}$

Abstract

We investigate the effects of final state interactions (FSI) contributions in the nonleptonic two body $B^{0}_{s} \rightarrow \pi^{0}\pi^{0}$ decay. The short distance interaction amplitude is calculated by using the annihilation diagrams and a tiny branching ratio is obtained, then the long distance amplitude is considered and calculated within FSI effects. For contributions of FSI, the ρ 0 ρ 0, π + π −(ρ + ρ −), K + K −(K +∗ K −∗) and $K^{0}\bar{K}^{0}(K^{0*}\bar{K}^{0*})$ are produced for intermediate states, in this case the π 0, π −(ρ −), K −(∗) and $\bar{K}^{0(*)}$ mesons are exchanged. The absorptive part of the diagrams is directly calculated and the dispersive part of the rescattering amplitude can be obtained from the absorptive part via the dispersion relation. The imaginary and real parts of the amplitudes are summed over all intermediate states. The predicted branching ratio of $B^{0}_{s} \rightarrow \pi^{0}\pi^{0}$ is 0.69×10−8 in the absence of FSI effects and it becomes 1.86×10−4 when FSI contributions are taken into account, while the experimental result is less than 2.1×10−4.