Branching Fractions and CP Violation in $B^{-}\to K^{+}{K}^{-}{\pi }^{-}$ and $B^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{-}$ Decays

Abstract

We present in this work a study of tree-dominated charmless three-body decays of $B$ mesons, $B^{-}\to K^{+}{K}^{-}{\pi }^{-}$ and $B^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{-}$, within the factorization approach. The main results are: (i) There are two distinct sources of nonresonant contributions: one arises from the $b\to u$ tree transition and the other from the nonresonant matrix element of scalar densities $⟨M_1{M}_2|{\bar{q}}_1{q}_2|0{⟩}^{\mathrm{NR}}$. It turns out that even for tree-dominated three-body decays, dominant nonresonant contributions originate from the penguin diagram rather than from the $b\to u$ tree process, as implied by the large nonresonant component observed recently in the ${\pi }^{-}{K}^{+}$ system which accounts for one third of the $B^{-}\to K^{+}{K}^{-}{\pi }^{-}$ rate. (ii) The calculated branching fraction of $B^{-}\to f_2(1270){\pi }^{-}\to K^{+}{K}^{-}{\pi }^{-}$ is smaller than the LHCb by a factor of $\sim 7$ in its central value, but the predicted $\text{B}(B^{-}\to f_2(1270){\pi }^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{-})$ is consistent with the data. Branching fractions of $B^{-}\to f_2(1270){\pi }^{-}$ 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 $B^{-}\to K^{+}{K}^{-}{\pi }^{-}$ is largely overestimated experimentally. Including $1/m_b$ power corrections from penguin annihilation inferred from QCD factorization (QCDF), a sizable CP asymmetry of 32\% in the $f_2(1270)$ component agrees with experiment. (iii) A fraction of 5\% for the $\rho (1450)$ component in $B^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{-}$ is in accordance with the theoretical expectation. However, a large fraction of 30\% in $B^{-}\to K^{+}{K}^{-}{\pi }^{-}$ is entirely unexpected. This issue needs to be clarified in the future.

Comments: 40 pages and 5 figures, discussions added on p.5 and p.9, version to appear in PRD