Branching fractions and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:math> violation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>B</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mi>K</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>K</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>B</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:math> decays

Abstract

We present in this work a study of tree-dominated charmless three-body decays of $B$ mesons, ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}$ and ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}$, within the factorization approach. The main results are: (i) There are two distinct sources of nonresonant contributions: one arises from the $b\ensuremath{\rightarrow}u$ tree transition and the other from the nonresonant matrix element of scalar densities $⟨{M}_{1}{M}_{2}|{\overline{q}}_{1}{q}_{2}|0{⟩}^{\mathrm{NR}}$. It turns out that even for tree-dominated three-body decays, dominant nonresonant contributions originate from the penguin diagram rather than from the $b\ensuremath{\rightarrow}u$ tree process, as implied by the large nonresonant component observed recently in the ${\ensuremath{\pi}}^{\ensuremath{-}}{K}^{+}$ system which accounts for one third of the ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}$ rate. (ii) The calculated branching fraction of ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{f}_{2}(1270){\ensuremath{\pi}}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}$ is smaller than the LHCb by a factor of $\ensuremath{\sim}7$ in its central value, but the predicted $\mathcal{B}({B}^{\ensuremath{-}}\ensuremath{\rightarrow}{f}_{2}(1270){\ensuremath{\pi}}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}})$ is consistent with the data. Branching fractions of ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{f}_{2}(1270){\ensuremath{\pi}}^{\ensuremath{-}}$ extracted from the LHCb measurements of these two processes also differ by a factor of seven! Therefore, it is likely that the ${f}_{2}(1270)$ contribution to ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}$ is largely overestimated experimentally. Including $1/{m}_{b}$ power corrections from penguin annihilation inferred from QCD factorization (QCDF), a sizable $CP$ asymmetry of 25% in the ${f}_{2}(1270)$ component agrees with experiment. (iii) A fraction of 5% for the $\ensuremath{\rho}(1450)$ component in ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}$ is in accordance with the theoretical expectation. However, a large fraction of 30% in ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}$ is entirely unexpected. This issue needs to be clarified in the future. (iv) We study final-state $\ensuremath{\pi}\ensuremath{\pi}\ensuremath{\leftrightarrow}K\overline{K}$ rescattering and find that the rescattering contributions to both ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}$ and ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}$ seem to be overestimated experimentally by a factor 4. (v) Using the QCDF expression for the ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\sigma}/{f}_{0}(500){\ensuremath{\pi}}^{\ensuremath{-}}$ amplitude to study the decay ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\sigma}{\ensuremath{\pi}}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}$, the resultant branching fraction and $CP$ violation of 15% agree with experiment. (vi) $CP$ asymmetry for the dominant quasi-two-body decay mode ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{0}{\ensuremath{\pi}}^{\ensuremath{-}}$ was found by the LHCb to be consistent with zero in all three $S$-wave models. In the QCDF approach, $1/{m}_{b}$ power corrections, namely, penguin annihilation and hard spectator interactions contribute destructively to ${\mathcal{A}}_{CP}({B}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{0}{\ensuremath{\pi}}^{\ensuremath{-}})$ to render it consistent with zero. (vii) A significant $CP$ asymmetry has been seen in the ${\ensuremath{\rho}}^{0}(770)$ region for positive- and negative-helicity angle cosines. Considering the low ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ invariant mass region of the ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ Dalitz plot of $CP$ asymmetries divided into four zones, the pattern of $CP$ violation in each zone is well described by the interference between $\ensuremath{\rho}(770)$ and $\ensuremath{\sigma}(500)$ as well as the nonresonant background.

Topics & Concepts