$CP$-violating observables of four-body $B_{(s)}\to (\ddot{I}€\ddot{I}€)(K\bar{K})$ decays in perturbative QCD

Abstract

In this work, we investigate six helicity amplitudes of the four-body $B_{(s)}\to (\ddot{I}€\ddot{I}€)(K\bar{K})$ decays in the perturbative QCD (PQCD) approach. The $\ddot{I}€\ddot{I}€$ invariant mass spectrum is dominated by the vector resonance $\ddot{I}(770)$ together with scalar resonance $f_0(980)$, while the vector resonance $\ddot{I}•(1020)$ and scalar resonance $f_0(980)$ are expected to contribute in the $K\bar{K}$ invariant mass range. We extract the two-body branching ratios $\mathcal{B}(B_{(s)}\to \ddot{I}\ddot{I}•)$ from the corresponding four-body decays $B_{(s)}\to \ddot{I}\ddot{I}•\to (\ddot{I}€\ddot{I}€)(K\bar{K})$. The predicted $\mathcal{B}(B_s^0\to \ddot{I}\ddot{I}•)$ agrees well with the current experimental data within errors. The longitudinal polarization fractions of the $B_{(s)}\to \ddot{I}\ddot{I}•$ decays are found to be as large as $90\%$, basically consistent with the previous two-body predictions within uncertainties. In addition, the triple-product asymmetries (TPAs) of the considered decays are also presented for the first time. Since the $B_s^0\to \ddot{I}{}^0\ddot{I}•\to (\ddot{I}{€}^{+}\ddot{I}{€}^{-})(K^{+}{K}^{-})$ decay is induced by both tree and penguin operators, the values of the ${\mathcal{A}}_{\mathrm{dir}}^{\mathrm{CP}}$ and ${\mathcal{A}}_{\text{T-true}}^1$ are calculated to be $(21.8_{-3.3}^{+2.7})\%$ and $(-10.23_{-1.56}^{+1.73})\%$ respectively. While for pure penguin decays $B^0\to \ddot{I}{}^0\ddot{I}•\to (\ddot{I}{€}^{+}\ddot{I}{€}^{-})(K^{+}{K}^{-})$ and $B^{+}\to \ddot{I}{}^{+}\ddot{I}•\to (\ddot{I}{€}^{+}\ddot{I}{€}^0)(K^{+}{K}^{-})$, both the direct $CP$ asymmetries and ``true" TPAs are naturally expected to be zero in the standard model (SM). The ``fake" TPAs requiring no weak phase difference are usually none zero for all considered decay channels. The sizable ``fake" ${\mathcal{A}}_{\text{T-fake}}^1=(-20.92_{-2.80}^{+6.26})\%$ of the $B^0\to \ddot{I}{}^0\ddot{I}•\to (\ddot{I}{€}^{+}\ddot{I}{€}^{-})(K^{+}{K}^{-})$ decay is predicted in the PQCD approach, which provides valuable information on the final-state interactions.Our predictions can be tested by the future experiments.

24 pages, 2 figures. arXiv admin note: text overlap with arXiv:2308.12543, arXiv:2204.01092, arXiv:2107.10684