Kπ decays with small elastic final state interactions Claudia

An analysis of B → Kπ decays is given, assuming a small elastic πK rescattering phase difference. Using factorization model elastic πK rescattering phase difference. Using factorization model only for the tree-level and electroweak penguin amplitudes, we show that the strong penguin amplitude, its absorptive part and the CP-violation weak phase γ can be obtained from the measured B → Kπ branching ratios. The strength of the strong penguin and its absorptive part thus obtained from the CLEO data, are found to be very close to factorization model values and suggests a current s quark mass around ms = 106MeV. The central value of γ is found to be around 76.10 , with a possible value in the range 50 − 100. CPHT-S 085.0900 September 2000 Typeset using REVTEX 1 The recent CLEO data [1] on charmless two-body nonleptonic B decays indicates that non-leptonic interactions enhanced by shortdistance QCD radiative corrections combined with factorization model seems to describe qualitatively the B → Kπ decays. In particular, 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 decay and its charge conjugate mode. Though the data are not yet sufficiently accurate to allow a determination of the weak CP-violating phase γ, they seem to favor a large γ in the range of (90 − 120), as shown in a previous analysis of B → Kπ using the factorization model with elastic rescattering phase included [2]. This large value of γ is also found by the CLEO Collaboration in an analysis of all measured charmless two-body B decays with the factorization model [3]. A large γ would also help to explain the suppression of the B̄ → ππ decays as the interference between the tree-level and penguin terms which increases the B → Kπ decay rates, becomes destructive in B → ππ decays. However, as shown in our previous analysis, a large γ would require a large πK → πK rescattering phase difference in the range (50 − 100) to account for the near-equality of the two largest B̄ → Kπ and B → K̄π branching ratios. Although a large elastic rescattering phase is not excluded by experiments, it is not expected in high energy elastic πK scattering which is dominated by the isospin-independent Pomeron exchange amplitude so that the rescattering phase difference δ = δ3/2 − δ1/2 would be small in B decays. Indeed, a recent analysis of πK elastic scattering at the B mass [4] finds δ = (17± 3). For B → Kπ, from the factorization model, if γ < 110, with some adjustment of form factors, the current s quark mass and CKM parameters it might be possible to accommodate the two largest branching ratios with a small δ. It is thus useful to study the consequence of a small δ in B → Kπ decays. Infact, as shown in Fig.1 in [2], for δ < 50, the CP-averaged B → Kπ branching ratios are practically independent of δ in this range, as the δ-dependent terms which come from the small cos δ − 1 and the sin δ term which is also very small. Since the strong penguin amplitude with its small inelastic absorptive part found in perturbative QCD [5,6] produce sufficiently the B → Kπ decay rates, it is likely 2 that, in general, the inelastic absorptive part should not be too large in B → Kπ decays. In this case, for δ < 50, the δ-dependent terms in the B → Kπ decay rates would be small and could be therefore neglected in the CP-averaged B → Kπ decay rates. In particular, for δ in this range, the B → K̄π decay rate, is essentially given by the γ-independent strong penguin contribution. Now, if we assume factorization for the small tree-level and electroweak terms, the strong penguin and its absorptive part and γ could then be obtained from the measured CP-averaged B → Kπ decay rates. With the dominant strong penguin contribution obtained from experiments, the value of γ thus obtained is subjected to very little theoretical uncertainties in contrast with the result from the factorization model for the whole amplitude as the penguin matrix elements are sensitive to the s quark mass which is not known to a good accuracy. This is the main purpose of this paper. In the following, we shall obtain the strong penguin contribution and γ from the measured B → Kπ decay rates, assuming factorization for the tree-level and electrowaek penguin terms and a small πK rescattering phase difference δ. We find agreement with factorization model for the strong penguin contribution and its absorptive part. The value of γ is found to be in the range (50 − 100), with a central value of 70. To proceed, we begin by writing down the B → Kπ decay amplitudes in terms of the decay amplitudes into I = 1/2 and I = 3/2 final states [2,7], AK−π0 = 2 3 B3e iδ3 + √ 1 3 (A1 +B1)e iδ1 , AK̄0π− = √ 2 3 B3e iδ3 − √ 2 3 (A1 +B1)e iδ1 , AK−π+ = √ 2 3 B3e iδ3 + √ 2 3 (A1 −B1)e , AK̄0π0 = 2 3 B3e iδ3 − √ 1 3 (A1 − B1)e , (1) A1 is the sum of the strong penguin A S 1 and the I = 0 tree-level A T 1 as well as the I = 0 electroweak penguin AW1 contributions to the B → Kπ I = 1/2 amplitude; similarly B1 is the sum of the I = 1 tree-level B 1 and electroweak penguin B W 1 contribution to the I = 1/2 amplitude, and B3 is the sum of the I = 1 tree-level B T 3 and electroweak penguin 3 B 3 contribution to the I = 3/2 amplitude. Using the following effective Hamiltonian for B → Kπ decays [5,8–11] Heff = GF √ 2 [VubV ∗ us(c1O u 1 + c2O u 2 ) + VcbV ∗ cs(c1O c 1 + c2O c 2) − 10