The penguin operators in nonresonant B − →

We investigate the contributions coming from the penguin operators in the nonresonant B → MM̄π (M = π, K, K) decays. The effective Wilson coefficients of the the strong penguin operators O4,6 are found to be relatively larger. We calculate the contributions arising from the O4 and O6 operators in the nonresonant decays B → MM̄π (M = π, K, K̄) using a model combining heavy quark symmetry and the chiral symmetry, developed previously. We find that the forbidden nonresonant B → KK̄π decay occurs through the strong penguin operators. These penguin contributions affect the branching ratios for B → MM̄π (M = π, K) by only a few percent. The branching ratio for B → KK̄π is estimated to be of the order 10. There is considerable interest in understanding the decay mechanism of the nonleptonic charmless three body decays of B mesons [1, 2, 3, 4]. The importance of penguin operators in three body decays of charged B mesons has recently been questioned [1]. In the analysis of the Dalitz plot for B → πππ the authors of [1] have assumed that the nonresonant decay amplitude is flat, having no dependence on the Dalitz variables. They also assumed that the contributions of the penguin operators can amount to as much as 20% of the dominant decay amplitude. Others have made predictions for the branching ratios of decays [2, 3, 4] motivated in part by the CLEO limits on some of the nonresonant decays of the type B → hhh [6]. CLEO found the upper limits on the branching ratios BR(B → πππ) ≤ 4.1 × 10 and BR(B → KKπ) ≤ 7.5 × 10. In addition there is hope that the CP violating phase γ can be measured from the asymmetry in charged B meson charmless three body decays [2, 4, 5]. Motivated by the need to understand whether the nonresonant decay amplitudes for B → MM̄π (M = π, K) involve significant effects due to the penguin operators we have investigated the contributions coming from the penguin operators [7] [12] in these nonresonant decay amplitudes. The decay B → KK̄π is CKM forbidden [13, 14]. However, we found that B → KK̄π occurs through penguin operators. A measurements of this rate would allow one to extract the product of the CKM matrix elements VtbV ∗ td. In our analysis we will use of the factorization approximation in which the main contribution to the nonresonant B → MM̄π amplitudes come from either the product < MM̄ |(ūb)V−A|B > < π|(d̄u)V−A|0 > or < πM̄ |(d̄b)V−A|B > < M |(ūu)V−A|0 >, where (q̄1q2)V−A denotes q̄1γμ(1−γ5)q2. For the calculation of the matrix element< MM̄ |(ūb)V −A|B > we extend the results obtained in [15], where the nonresonant D → Kπlν decay was analyzed. The experimental result for the branching ratio of the nonresonant D → Kπlν decay was successfully reproduced within a hybrid framework [15] which combines the heavy quark effective theory (HQET) and the chiral Lagrangian (CHPT) approach. The combination of heavy quark symmetry and chiral symmetry has also been quite successful in other analyses of D meson semileptonic decays [16] [22]. Heavy quark symmetry is expected to be even better for the heavier B mesons [19, 20]. However, CHPT might be less reliable in B decays due to the large energies of light mesons in the final state. It is really only known that