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

We present in this work a study of tree-dominated charmless three-body decays of $B$ mesons, $B^-\to K^+K^-��^-$ and $B^-\to��^+��^-��^-$, 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 $��^- K^+$ system which accounts for one third of the $B^-\to K^+K^-��^-$ rate. (ii) The calculated branching fraction of $B^-\to f_2(1270)��^-\to K^+K^-��^-$ is smaller than the LHCb by a factor of $\sim 7$ in its central value, but the predicted $\B(B^-\to f_2(1270)��^-\to��^+��^-��^-)$ is consistent with the data. Branching fractions of $B^-\to f_2(1270)��^-$ 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 $B^-\to K^+K^-��^-$ 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 $��(1450)$ component in $B^-\to��^+��^-��^-$ is in accordance with the theoretical expectation. However, a large fraction of 30\% in $B^-\to K^+K^-��^-$ is entirely unexpected. This issue needs to be clarified in the future.
40 pages and 5 figures, discussions added on p.5 and p.9, version to appear in PRD