2 Long distance contributions in B → K ∗ l + l − decays with polarized

We use momentum correlations as physical observables in B → K∗l+l− decays with K polarized to study the long distance contributions. We show that these observables are sensitive to the scenarios of the long distance parametrizations. We find that the T-odd observable is directly related to the nonfactorizable effect in the standard model. PACS 13.20.He, 11.30.Er, 12.40.-y Typeset using REVTEX 1 The study of flavor changing neutral currents (FCNCs) in B decays has an enormous progress since the CLEO observation [1] of b → sγ. Recently, the process of B → Kll has been also observed [2] at the Belle detector in the KEKB ee storage ring. It is known that the radiative b → sγ and semileptonic b → sll FCNC decays [3] in the standard model (SM) provide us with information on not only the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [4] but also physics beyond the SM. Moreover, for b → sll, new operators such as those from the box and Z-penguin diagrams can escape the strict constraint from b → sγ and, therefore, the new physics effect could be sizable. In addition to the short-distance (SD) contributions, the long-distant (LD) contributions to b → sγ (sll), arising from the charm (c) quark pair bound states, should be taken into account. It is known that the LD effect in b → sγ is only a few percent and negligible, whereas it is the main part to the decay rate in b → sll. However, the parametrization of the LD contributions is not unique and has an uncertainty of about 20% for the decay branching ratios (BRs) of b → sll [5]. In order to test the SM and find new physics, it is of important to extract such theoretical uncertainty. To distinguish various theoretical parametrizations, it is interesting to see if we can find some measurable physical observables which are dominated by the LD parts. In this paper, we will study the LD effects by considering the exclusive B → Kll decays with the polarized K meson. We will define some useful observables by the momentum correlations, especially those related to T-odd operators. In a three-body decay, it is known that the simplest T-odd operator is the triple correlations given by ~s · (~ pi×~pj) where ~s is the spin vector of an outgoing particle and ~pi and ~ pj denote any two independent momentum vectors. In terms of the CPT invariant theorem, T violation (TV) implies CP violation (CPV). Therefore, studying of T-odd observables could help us to understand the origin of CPV. We note that the T-odd observables such as the triple correlations are only associated with the imaginary parts of relevant dynamical variables. That is, even there is no weak CP phase, these observables may not vanish if a strong phase or absorptive part exists. In the SM, since the CKM matrix element of VtbV ∗ ts involved in the process of B → Kll contains no phase, the T-odd observables can be only generated through the LD effects. 2 Hence, these observables can be used to test the parametrizations of LD effects. In the decays of B → Kll (l = e, μ, and τ), the spin s can be the polarized lepton, sl, or the K meson, ǫ(λ). For the polarized lepton, since the T-odd transverse lepton polarization flips the helicity and thus it is always associated with the lepton mass, we expect that this type of T violating effects is suppressed and less than 1% for the light lepton modes [6]. Such effect is also negligible for the τ mode due to the small decay branching ratio. In this paper, we will concentrate on the light lepton modes with only K polarized and set the lepton masses to be zero, i.e., ml = 0. We start by writing the effective Hamiltonian for b → s l l as [7] H = GFαVtbV ∗ ts √ 2π [