CP -violating observables in four-body B→ϕ(→KK¯)K*(→Kπ) decays

Abstract

We analyze the four-body $B\ensuremath{\rightarrow}\ensuremath{\phi}(\ensuremath{\rightarrow}K\overline{K}){K}^{*}(\ensuremath{\rightarrow}K\ensuremath{\pi})$ decays in the perturbative QCD approach, where the invariant mass of the $K\overline{K}$ ($K\ensuremath{\pi}$) system is limited in a window of $\ifmmode\pm\else\textpm\fi{}15\text{ }\text{ }\mathrm{MeV}$ ($\ifmmode\pm\else\textpm\fi{}150\text{ }\text{ }\mathrm{MeV}$) around the nominal $\ensuremath{\phi}({K}^{*}(892))$ mass. In addition to the dominant $P$-wave resonances, two important $S$-wave backgrounds in the selected invariant mass region are also accounted for. Angular momentum conservation allows six helicity amplitudes to contribute, including three $P$ waves, two single $S$ waves, and one double $S$ wave, in the decays under study. We calculated the branching ratio for each component and found sizable $S$-wave contributions, which coincide roughly with the experimental observation. The obtained branching ratios of ${B}^{0(+)}\ensuremath{\rightarrow}\ensuremath{\phi}{K}^{*0(+)}$ are comparable with the previous theoretical predictions and support the experimental measurements, whereas the predicted $\mathcal{B}({B}_{s}^{0}\ensuremath{\rightarrow}\ensuremath{\phi}{\overline{K}}^{*0})$ is an order of magnitude smaller than the current world average in its central value. The longitudinal polarizations are predicted to be around 0.7, consistent with previous PQCD results but larger than the world average values. Aside from the direct $CP$ asymmetries, the true and fake triple product asymmetries, originating from the interference between the perpendicular polarization amplitude and other helicity amplitudes, are also calculated in this work. In the special case of the neutral modes, both the direct $CP$ asymmetries and true triple product asymmetries are expected to be zero due to the vanishing weak phase difference. The direct $CP$ asymmetries for the ${B}^{+}$ mode are predicted to be tiny, of order ${10}^{\ensuremath{-}2}$, since the tree contributions are suppressed strongly with respect to the penguin ones. The true triple product asymmetries have shown no significant deviations from zero. In contrast, large fake asymmetries are observed in these decays, indicating the presence of significant final-state interactions. We give the theoretical predictions of the $S$-wave induced triple product asymmetries for the first time, which is consistent with current LHCb data and would be checked with future measurements from Belle and BABAR experiments if the $S$-wave components can be properly taken into account in the angular analysis.