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 $\langle M_1M_2|\bar q_1 q_2|0\rangle^{\rm NR}$. 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 $\pi^- K^+$ 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 $\B(B^-\to f_2(1270)\pi^-\to\pi^+\pi^-\pi^-)$ 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.
Highlights
In 2013 and 2014 LHCb has measured direct CP violation in charmless three-body decays of B mesons [1,2,3] and found evidence of inclusive integrated CP asymmetries AiCnPcl in Bþ → πþπþπ− (4.2σ), Bþ → KþKþK− (4.3σ) and Bþ → KþK−πþ (5.6σ) and a 2.8σ signal of CP violation in Bþ → Kþπþπ−
It turns out that even for treedominated three-body decays, dominant nonresonant contributions originate from the penguin diagram rather than from the b → u tree process, as implied by the large nonresonant component observed recently in the π−Kþ system which accounts for one third of the B− → KþK−π− rate. (ii) The calculated branching fraction of B− → f2ð1270Þπ− → KþK−π− is smaller than the LHCb by a factor of ∼7 in its central value, but the predicted BðB− → f2ð1270Þπ− → πþπ−π−Þ is consistent with the data
A large fraction of 30% in B− → KþK−π− is entirely unexpected. This issue needs to be clarified in the future. (iv) We study final-state ππ ↔ KKrescattering and find that the rescattering contributions to both B− → KþK−π− and B− → πþπ−π− seem to be overestimated experimentally by a factor 4. (v) Using the QCD factorization (QCDF) expression for the B− → σ=f0ð500Þπ− amplitude to study the decay B− → σπ− → πþπ−π−, the resultant branching fraction and CP violation of 15% agree with experiment. (vi) CP asymmetry for the dominant quasi-two-body decay mode B− → ρ0π− was found by the LHCb to be consistent with zero in all three S-wave models
Summary
In 2013 and 2014 LHCb has measured direct CP violation in charmless three-body decays of B mesons [1,2,3] and found evidence of inclusive integrated CP asymmetries AiCnPcl in Bþ → πþπþπ− (4.2σ), Bþ → KþKþK− (4.3σ) and Bþ → KþK−πþ (5.6σ) and a 2.8σ signal of CP violation in Bþ → Kþπþπ−. A significant CP asymmetry in the ρ0 region can be seen in Fig. 2 where the data are separated by the sign of the value of cos θhel with θhel being the helicity angle, evaluated in the πþπ− rest frame, between the pion with opposite charge to the B and the third pion from the B decay (see Fig. 3) This feature which was already noticed previously in [2] indicates that CP violation close to the ρð770Þ resonance is proportional to ðm2ρ − m2lowÞ cos θhel. A large nonresonant πþπ− S-wave contribution observed by BABAR in Bþ → πþπ−πþ decays [7] was presented by the LHCb as the coherent sum of contributions of the σ and a πþπ− ↔ KþK− rescattering amplitude
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