Penguin operators in nonresonantB−→MM¯π−(M=π−,K−,K0)decays

We investigate the contributions coming from the penguin operators in the nonresonant ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}M\overline{M}{\ensuremath{\pi}}^{\ensuremath{-}}(M={\ensuremath{\pi}}^{\ensuremath{-}}{,K}^{\ensuremath{-}}{,K}^{0})$ decays. The effective Wilson coefficients of the strong penguin operators ${O}_{4}$ and ${O}_{6}$ are found to be relatively larger than those for other penguin operators. We calculate the contributions arising from the ${O}_{4}$ and ${O}_{6}$ operators in the nonresonant decays ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}M\overline{M}{\ensuremath{\pi}}^{\ensuremath{-}}(M={\ensuremath{\pi}}^{\ensuremath{-}}{,K}^{\ensuremath{-}},{K}^{0})$ using a model combining heavy quark symmetry and the chiral symmetry, developed previously. We find that the CKM-forbidden nonresonant ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{0}{K}^{0}{\ensuremath{\pi}}^{\ensuremath{-}}$ decay occurs through the strong penguin operators. These penguin contributions affect the branching ratios for ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}M\overline{M}{\ensuremath{\pi}}^{\ensuremath{-}}(M={\ensuremath{\pi}}^{\ensuremath{-}}{,K}^{\ensuremath{-}})$ by only a few percent. The branching ratio for ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{0}{K}^{0}{\ensuremath{\pi}}^{\ensuremath{-}}$ is estimated to be of the order ${10}^{\ensuremath{-}6}.$