Improved global determination of two-meson distribution amplitudes from multi-body $B$ decays

Abstract

We improve the perturbative QCD (PQCD) formalism for multi-body charmless hadronic $B$ meson decays, such as $B\to V{P}_3\to P_1{P}_2{P}_3$, by resolving the possible discrepancy in parametrizing the contribution of the $P$-wave resonance $V$ to the two-meson distribution amplitudes (DAs) associated with the pairs $P_1{P}_2=\pi \pi ,K\pi ,KK$. The determination of the Gegenbauer moments in the two-meson DAs is then updated in the global fit of the improved PQCD factorization formulas at leading order in the strong coupling ${\alpha }_s$ to available data for branching ratios and polarization fractions of three- and four-body $B$ decays. The convergence of the Gegenbauer expansion of the resultant two-meson DAs is manifest. The satisfactory quality of the fit implies the consistency of the PQCD framework for multi-body $B$ decays and the universality of the nonperturbative two-meson DAs. In particular, the predicted longitudinal polarization fraction $f_0(B_s^0\to K^{\ast 0}{\bar{K}}^{\ast 0})=28.2_{-9.5}^{+8.8}\%$ with the updated Gegenbauer moments matches well the measurement. The observable $L_{K^{\ast 0}{\bar{K}}^{\ast 0}}=7.7_{-3.8}^{+4.9}$, defined as the ratio of the longitudinal amplitudes of the two $U$-spin related channels $B_s^0\to K^{\ast 0}{\bar{K}}^{\ast 0}$ and $B^0\to K^{\ast 0}{\bar{K}}^{\ast 0}$, accommodates the current data within errors. It is found that the direct $CP$ asymmetries ${\mathcal{A}}_{\mathrm{CP}}^{0,||,\perp }$ in the polarization states of some four-body decays $B\to V_1{V}_2\to (P_1{P}_2)(P_3{P}_4)$ might be large, but the destruction among them result in small net $CP$ violation. Our predictions can be confronted with LHCb and Belle-II data in the future.