ar X iv : h ep - p h / 98 09 28 8 v 2 2 6 Ja n 19 99 Studying High Energy Final State Interactions by N / D Method

We discuss the final state interaction effects at high energies via a multichannel N/D method. We find that the 2 by 2 charge–exchange final state interactions typically contribute an enhancement factor of a few times 10−2 in the B meson decay amplitudes, both for the real and the imaginary part. We also make some discussions on the elastic rescattering effects. PACS numbers: 11.55.Fv, 11.80.Gw, 13.20.Fc, 13.20.He Typeset using REVTEX 1 There have been increasing interests in recent years in studying the final state interaction (FSI) effects in B meson hadronic decays. In Ref. [1] , a Regge pole model was suggested to estimate the FSI effects in B decays and it was found that, for B → DK decays the FSI effects are small in the sense that the corresponding partial wave S–matrix is close to unity. Especially the final state rescattering effects in charge-exchange processes are of order O(10). These results imply that for the color non-suppressed charged final states, a tree level calculation to the B decay amplitude can be a good approximation [2]. Using a similar method as in Ref. [1], the authors of Ref. [3] observed that, as a consequence of Pomeron exchanges the FSI effects do not vanish as the center of mass energy square s approaches infinity and therefore they suggest that there are sizable strong phases generated from FSIs. On the other hand, the authors of Ref. [4] pointed out that in above estimates the real part contributions of the rescattering amplitudes were not considered. They use a dispersion relation to estimate also the real part contributions of the charge–exchange processes (B → Dπ,ππ ) and find that Reggeon contributions (in charge–exchange rescattering processes) are one order of magnitude larger than those estimated in Ref. [1,2]. Since this result, if correct, leads to important phenomenological consequences [5], it is therefore worthwhile to re-examine the problem by carefully studying the FSI effects, including the real part, of charge–exchange rescattering processes, which is what this paper mainly devote to. We will also discuss the FSI effects of elastic rescatterings although there exist some uncertainties related to such rescatterings in the scheme presented in this paper. To correctly evaluate the FSI effects to the decay amplitudes, it is important to note that if the initial interactions are of short range the production amplitude and FSIs are actually factorized [6]. Assuming (quasi) two–particle unitarity we will use the multi-channel N/D method [7] to study the FSI effects. The inelasticity effects caused by those multi–particle states is presumably not large, since the pion multiplicity is rather low at s = mB as the multiplicity increases only as log(s) (see [4] and references therein) and at least part of the multi–particle final states can be classified as two–particle final states of resonances (accompanied by cascade resonance decays). 2 For a multi-channel system we can rewrite the partial wave T-matrix as,