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Branching Fractions and CP Violation in $B^-\to K^+K^-π^-$ and $B^-\to π^+π^-π^-$ Decays

We present in this work a study of tree-dominated charmless three-body decays of $B$ mesons, $B^-\to K^+K^-π^-$ and $B^-\toπ^+π^-π^-$, 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 $\langle M_1M_2|\bar q_1 q_2|0\rangle^{\rm NR}$. 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 $π^- K^+$ system which accounts for one third of the $B^-\to K^+K^-π^-$ rate. (ii) The calculated branching fraction of $B^-\to f_2(1270)π^-\to K^+K^-π^-$ is smaller than the LHCb by a factor of $\sim 7$ in its central value, but the predicted $\B(B^-\to f_2(1270)π^-\toπ^+π^-π^-)$ is consistent with the data. Branching fractions of $B^-\to f_2(1270)π^-$ 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 $B^-\to K^+K^-π^-$ 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 $ρ(1450)$ component in $B^-\toπ^+π^-π^-$ is in accordance with the theoretical expectation. However, a large fraction of 30\% in $B^-\to K^+K^-π^-$ is entirely unexpected. This issue needs to be clarified in the future.