Effects of Final State Interactions in Pure Annihilation Decay of $B^{0}_{s}\rightarrow\pi^{0}\pi^{0}$
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Effects of Final State Interactions in Pure Annihilation Decay of Bs0 → π 0 π 0 Behnam Mohammadi · Hossein Mehraban
Received: 28 August 2012 / Accepted: 5 February 2013 / Published online: 16 February 2013 © Springer Science+Business Media New York 2013
Abstract We investigate the effects of final state interactions (FSI) contributions in the nonleptonic two body Bs0 → π 0 π 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 K¯ 0 (K 0∗ K¯ 0∗ ) are produced for intermediate states, in this case the π 0 , π − (ρ − ), K −(∗) and 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 Bs0 → π 0 π 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 . Keywords QCD factorization · Final state interaction · B meson · Absorptive part · Intermediate states
1 Introduction In the Bs0 → π 0 π 0 decay the Bs0 meson has s quark and b¯ antiquark, the π 0 mesons have uu¯ and d d¯ parts, so no form factor can be produced between the Bs0 meson and final states π 0 mesons, hence this process is pure annihilation type of the decay. In the case of uu¯ part of the π 0 meson is considered the contributions of current-current annihilation, penguin and electroweak annihilations can be used in the amplitude and according to d d¯ part only √ pen¯ 2, so guin share can be applied. As long as the structure of the π 0 meson is (uu¯ − d d)/ B. Mohammadi () · H. Mehraban Physics Department, Semnan University, P.O. Box 35195-363, Semnan, Iran e-mail: [email protected] H. Mehraban e-mail: [email protected]
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Int J Theor Phys (2013) 52:2363–2378
the contributions of the penguin annihilation cancel each other and current-current and electroweak penguin annihilations stay in the amplitude. We selected the leading order Wilson coefficients at the scale mb [1, 2] and calculated the Bs0 → π 0 π 0 decay according to the QCDF method and obtained a tiny branching ratio with the amount of 0.69 × 10−8 while the experimental result is less than 2.1 × 10−4 [3]. It seems that due to have a tiny branching ratio so far this decay has not been seriously analyzed. These may be some hints for the need of FSI in Bs0 → π 0 π 0 decay. FSI effects are non-perturbative in nature [4]. In many decay modes, the FSI may play a crucial role [5]. In this way, the CKM matrix elements and color factor are suppressed and the CKM’s most favored two-body intermediate states are the only ones that have been taken
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