Quasi-two-body decays $B_{(s)}\to P{f}_2(1270)\to P\pi \pi$ in the perturbative QCD approach

In this work, we calculate the $CP$-averaged branching ratios and direct $CP$-violating asymmetries of the quasi-two-body decays $B_{(s)}\to P{f}_2(1270)\to P\pi \pi$ with the two-pion distribution amplitude ${\mathrm{\Phi }}_{\pi \pi }^{\mathrm{D}}$ by using the perturbative QCD factorization approach, where $P$ represents a light pseudoscalar meson $K$, $\pi$, $\eta$ and ${\eta }^{\prime }$. The relativistic Breit-Wigner formula for the $D$-wave resonance $f_2(1270)$ is adopted to parameterize the timelike form factor $F_{\pi }$, which contains the final state interactions between the pions in the resonant regions. The consistency of theoretical results with data can be achieved by determining the Gegenbauer moments of the $D$-wave two-pion distribution amplitudes. The decay rates for the considered decay modes are generally in the order of ${10}^{-9}$ to ${10}^{-6}$. The integrated direct $CP$ asymmetries for the charged modes agree with the BABAR and Belle measurements. As a by-product, we extract the branching ratios of $B_{(s)}\to P{f}_2(1270)$ from the corresponding quasi-two-body decay modes, which still need experimental tests at the ongoing and forthcoming experiments.

Figure 1

Typical Feynman diagrams for the quasi-two-body decays $B\to P(f_2(1270)\to )\pi \pi$, where $B$ stands for the $B^{\pm }$, $B^0$ or $B_s$ meson and $P$ denotes $K$, $\pi$, $\eta$ or ${\eta }^{\prime }$. With $\alpha =a-d$ and $\beta =e-h$, the diagrams ($\alpha 1$) for the $B\to f_2(1270)\to \pi \pi$ transition and the diagrams ($\beta 1$) for annihilation contributions.

Figure 2

Typical Feynman diagrams for the quasi-two-body decays $B\to P(f_2(1270)\to )\pi \pi$, where $B$ stands for the $B^{\pm }$, $B^0$ or $B_s$ meson and $P$ denotes $K$, $\pi$, $\eta$ or ${\eta }^{\prime }$. With $\alpha =c-d$ and $\beta =e-h$, the diagrams ($\alpha 2$) for the $B\to P$ transition and the diagrams ($\beta 2$) for annihilation contributions.

Figure 3

(a) Differential branching ratios for the $B^{\pm }\to K^{\pm }{f}_2(1270)\to K^{\pm }{\pi }^{+}{\pi }^{-}$ decays. (b) Differential distribution of ${\mathcal{A}}_{cp}$ in $\omega$ for the decay modes $B^{+}\to K^{+}[f_2\to ]{\pi }^{+}{\pi }^{-}$, $B^{+}\to {\pi }^{+}[f_2\to ]{\pi }^{+}{\pi }^{-}$, $B^0\to {\eta }^{\prime }[f_2\to ]{\pi }^{+}{\pi }^{-}$ and $B^0\to \eta [f_2\to ]{\pi }^{+}{\pi }^{-}$.