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Constraints on hard spectator scattering and annihilation corrections in $B_{u,d}$ ${\to}$ $PV$ decays within QCD factorization

In this paper, we investigate the contributions of hard spectator scattering and annihilation in $B\to PV$ decays within the QCD factorization framework. With available experimental data on $B\to πK^{\ast}$, $ρK$, $πρ$ and $Kϕ$ decays, comprehensive $χ^2$ analyses of the parameters $X_{A,H}^{i,f}(ρ_{A,H}^{i,f},ϕ_{A,H}^{i,f})$ are performed, where $X_A^f$ ($X_A^i$) and $X_H$ are used to parameterize the endpoint divergences of the (non)factorizable annihilation and hard spectator scattering amplitudes, respectively. Based on $χ^2$ analyses, it is observed that (1) The topology-dependent parameterization scheme is feasible for $B\to PV$ decays; (2) At the current accuracy of experimental measurements and theoretical evaluations, $X_H=X_A^i$ is allowed by $B\to PV$ decays, but $X_{H}\neq X_A^f$ at $68%$ C. L.; (3) With the simplification $X_H=X_A^i$, parameters $X_A^f$ and $X_A^i$ should be treated individually. The above-described findings are very similar to those obtained from $B\to PP$ decays. Numerically, for $B\to PV$ decays, we obtain $(ρ_{A,H}^i,ϕ_{A,H}^i[^{\circ}]) =(2.87^{+0.66}_{-1.95}, -145^{+14}_{-21})$ and $(ρ_A^f,ϕ_A^f[^{\circ}]) = (0.91^{+0.12}_{-0.13}, -37^{+10}_{-9})$ at $68%$ C. L.. With the best-fit values, most of the theoretical results are in good agreement with the experimental data within errors. However, significant corrections to the color-suppressed tree amplitude $α_2$ related to a large $ρ_H$ result in the wrong sign for $A^{dir}_{CP}(B^- \to π^0 K^{{\ast}-})$ compared with the most recent BABAR data, which presents a new obstacle in solving "$ππ$" and "$πK$" puzzles through $α_2$. A crosscheck with measurements at Belle (or Belle II) and LHCb, which offer higher precision, is urgently expected to confirm or refute such possible mismatch.