Implications of Recent Measurements of Nonleptonic Charmless B Decays

Both B factories BELLE 1 and BABAR 2 have reported at this conference the preliminary results of B → Kπ and B → ππ (see Table I). For the unitary angle γ ∼ 60, the ratio R = B(B → Kπ)/B(B → ππ) is conventionally predicted to lie in the region 1.3−2.0, to be compared with the experimental values of 4.0± 1.6, 2.8± 2.0 and 1.3± 0.5 obtained by CLEO , BELLE and BABAR, respectively. Hence, the expected ratio R is smaller than the CLEO and BELLE results and in agreement with BABAR. Of course, more data are needed to clarify the issue. If the CLEO or BELLE data are taken seriously, we may ask the question of how to accommodate the data of Kπ and ππ simultaneously. A fit to ππ yields F 0 (0) < 0.25 for |Vub/Vcb| = 0.09 and γ = 60 . The Kπ rates will then become too small compared to the data. By contrast, a fit to Kπ modes usually implies a too large ππ rate. There are several possibilities that the CLEO or BELLE data of Kπ and ππ can be accommodated: (1) γ ∼ 60 and F 0 (0) < 0.25 with a smaller strange quark mass, say ms(mb) = 60 MeV. The idea is that the Kπ mode will receive a sizable (S − P )(S + P ) penguin contributions, while the ππ decay is not much affected. However, a rather smaller ms is not consistent with recent lattice calculations. (2) γ ∼ 60 and F 0 (0) = 0.30 with the following cases: (i) a smaller Vub, say |Vub/Vcb| ≈ 0.06, so that the π π rate is suppressed. However, this small Vub is not favored by data. (ii) a large nonzero isospin ππ phase shift difference of order, say 70, can yield a substantial suppression of the ππ mode . However, ππ will be substantially enhanced by the same strong phase. The CLEO new measurement 3 B(B → ππ) < 5.7 × 10 indicates that the strong phase cannot be too large. (iii) a large inelasticity for ππ and DD modes so that the former is suppressed whereas the latter is enhanced. (3) a large γ, say γ ∼ (110− 130), and F 0 (0) = 0.30. Several calculations 5 based on generalized or QCD improved factorization imply γ > 100. This scenario is interesting and popular, but it encounters two problems: (i) It is in conflict with the unitary angle γ = (58.5 ± 7.1) extracted from the global CKM fit. (ii) The CLEO data other than Kπ and ππ do not strongly support a large γ (see below). (4) γ ∼ 90 and F 0 (0) = 0.25 as assumed in a recent PQCD analysis . In this work, the ππ rate is small because of the small form factor F 0 (0), and Kπ rates are enhanced by large penguin effects owing to steep μ dependence of the leading-order penguin coefficients c4(μ) and c6(μ) at the hard scale t < mb/2. As shown in , the nonfactorized term is dominated by hard gluon exchange in the heavy quark limit as soft gluon contributions to χi are suppressed by orders of ΛQCD/mb. However, there is an additional chirally enhanced corrections to the spectator-interaction diagram, which are logarithmically divergent . For example, an additional (V − A)(V − A) spectator interaction proportional to the twist-3 wave function φσ will contribute to B → ππ, πK.