Penguin-dominated $B\to \varphi K_1(1270)$ and $\varphi K_1(1400)$ decays in the perturbative QCD approach

We investigate the $CP$-averaged branching ratios, the polarization fractions, the relative phases, and the $CP$-violating asymmetries of the penguin-dominated $B\to \varphi K_1(1270)$ and $\varphi K_1(1400)$ decays in the perturbative QCD (pQCD) approach, where $K_1(1270)$ and $K_1(1400)$ are believed to be the mixtures of two distinct types of axial-vector $K_{1A}({}^3{P}_1)$ and $K_{1B}({}^1{P}_1)$ states with different behavior, however, their mixing angle ${\theta }_{K_1}$ is still a hot and controversial topic presently. By numerical evaluations with two different mixing angles ${\theta }_{K_1}\sim 33^{\circ }$ and 58° and phenomenological analysis, we find that: (i) the pQCD predictions for the branching ratio, the longitudinal polarization fraction and the direct $CP$ violation of $B^{\pm }\to \varphi K_1(1270{)}^{\pm }$ decay with the smaller angle 33° are in good agreement with the currently available data; (ii) though the central values significantly exceed the available upper limit, both pQCD predictions of $\mathrm{Br}(B^{\pm }\to \varphi K_1(1400{)}^{\pm })$ with two different mixing angles are consistent with that obtained in QCD factorization and with the preliminary data in $2\sigma$ errors. These results and other relevant predictions for the considered decays will be further tested by the LHCb and the forthcoming Super-B experiments; (iii) the weak annihilation contributions can play an important role in $B\to \varphi K_1(1270)$ and $\varphi K_1(1400)$ decays; (d) these pQCD predictions combined with the future precision measurements can examine the reliability of the factorization approach employed here, but also explore the complicated QCD dynamics and mixing angle ${\theta }_{K_1}$ of the axial-vector $K_1(1270)$ and $K_1(1400)$ system.

Figure 1

Typical Feynman diagrams contributing to the penguin-dominated $B\to \varphi K_1$ decays in the pQCD approach at leading order, in which $K_1$ stands for the axial-vector $K_1(1270)$ and $K_1(1400)$, respectively.