ep - p h / 99 11 53 4 v 3 1 4 Ju n 20 00 Test of factorisation in B → Kπ decays

We analyse the B → Kπ decays using the factorisation model with final state interaction phase shift included. We find that factorisation seems to describe qualitatively the latest CLEO data. For a test of the factorisation model, we derive a relation for the branching ratios independent of the strength of the strong penguin interactions. This relation gives a central value of (0.60×10−5) for B(B̄0 → K̄π), somewhat smaller than the latest CLEO measurement, but the experimental errors are yet too big to take it as a real prediction of the factorisation model. We also find that a ratio obtained from the CP-averaged B → Kπ decay rates could be used to test the factorisation model and to determine the weak angle γ with more precise data, though the latest CLEO data seem to favor γ in the range of (90 − 120). Typeset using REVTEX 1 One of the possibilities offered by the B → Kπ decays is the determination of the CPviolating phase γ, one of the angles in the (db) unitary triangle of the Cabibbo-KobayashiMaskawa (CKM) quark mixing matrix in the standard model [1]. Infact the large CPaveraged branching ratio(B) for B → Kπ as observed by the CLEO Collaboration [3] indicate that the penguin interactions contribute a major part to the decay rates and provide an interference between the Cabibbo-suppressed tree and penguin contribution resulting in a CP-asymmetry between the B → Kπ and its charge conjugate mode. The CP-averaged decay rates depend also on the weak phase γ and give us a determination of this phase once a reliable description of the B → Kπ decays could be established [4,5]. With the latest measurement by the CLEO collaboration [3] , we have now the CPaveraged branching ratios for all the B → Kπ decay modes. In particular, the B̄ → K̄π mode is found to have a large branching ratio of (1.46 −5.1−3.3)×10 compared with a value in the range (0.5−0.74)×10−5 in the factorisation model [6,7]. The predicted values for other modes are, however, more or less in agreement with experiment. As the effective Hamiltonian for B → Kπ decays is well established with the short-distance Wilson coefficients for tree and penguin operators now given at the next-to-leading logarithms(NLL) QCD radiative corrections [7–12], the most important theoretical uncertainties would probably come from long-distance matrix elements obtained with the factorisation model and final state interaction (FSI) effects. Infact one of the main uncertainties in the penguin contributions to B → Kπ decays come from the value of the current s quark mass which is not known to a good accuracy. There are also non-factorisation terms which must be included in the form of an effective Wilson coefficients to make the amplitudes scale-independent [7,13]. Thus a more precise test of factorisation is to consider quantities which are independent of the strong penguin contributions. This is the main purpose of this paper. When all the B → Kπ decay modes are measured with good accuracy, and if the rescattering phase is known the dominant strong penguin contribution could be determined from the measured branching ratios assuming factorisation for the small tree-level and electroweak penguin terms, as will be discussed in the following. Though the present data are not yet sufficiently accurate for a 2 determination of the effective Wilson coefficients in B → Kπ decays at this time, a first step toward an understanding of B → Kπ decays is to see how well these penguin-dominated charmless B decays can be described by factorisation using the Wilson coefficients obtained from perturbative QCD. As argued in [14], for these very energetic decays, because of color transparency, factorisation should be a good approximation for B → Kπ decays if the Wilson coefficients are evaluated at a scale μ = O(mb). We could thus proceed to the test of factorisation bearing in mind that there are possible scale-dependent corrections from nonfactorisation terms to be determined with more precise data. To include FSI effects, as in [6], we assume that elastic FSI effects can be absorbed into the two ∆I = 1/2 and ∆I = 3/2 elastic πK → πK rescattering phases δ1 and δ3 taken as free parameters and include only inelastic effects coming from the charm and charmless intermediate state contributions to the absorptive part of the decay amplitudes. These inelastic contributions can be included in the Wilson coefficients of the penguin operators which now have an absorptive part and are given in [10,12,15]. We begin by first giving predictions in factorisation model for the B → Kπ decay rates and branching ratios in terms of the rescattering phase difference δ and for a typical value of the weak phase γ. As will be seen, factorisation seems to produce sufficient B → Kπ decay rates. We could thus proceed to a test of the factorization model by comparing with experiments, quantities obtained by factorisation which are independent of the strong rescattering phase difference [16]. We find that the sum of the CP-averaged branching ratios B(B− → K−π0)+B(B− → K̄π) and B(B̄0 → K−π+)+B(B̄0 → K̄π) are independent of the FSI rescattering phase. Other quantities obtained from various combination of the decay rates, for example, the quantity ∆ defined as Γ(B → K̄π)+Γ(B̄ → K−π+)−2(Γ(B− → Kπ) + Γ(B̄ → K̄π)) is independent of the strong penguin contributions and could be used to predict B(B̄0 → K̄π) in terms of the other measured branching ratios. As the main purpose of this paper is to test the factorisation model using relations independent of the strong penguin interactions, we will not discuss here a recent theoretical work on factorisation in B → ππ decays which should be completed to have all the logarithms of mb 3 under control [17]. In the standard model, the effective Hamiltonian for B → Kπ decays are given by [8,9,12], Heff = GF √ 2 [VubV ∗ us(c1O u 1 + c2O u 2 ) + VcbV ∗ cs(c1O c 1 + c2O c 2) − 10