Study of the four-body decays $B_S^0\to \pi \pi \pi \pi$ in the perturbative QCD approach

In this work we analyze the $CP$-averaged branching ratios and direct $CP$-violating asymmetries of the four-body decays $B_S\to \pi \pi \pi \pi$ decay from the $S$-wave resonances, $f_0(980)$ and $f_0(500)$ and $P$-wave resonances, $\rho (770)$ by introducing the $S$-wave and $P$-wave $\pi \pi$ distribution amplitudes within the framework of the perturbative QCD approach. We also calculate branching ratios of the two-body decays $B_S^0\to {\rho }^0{\rho }^0$, $B_S^0\to {\rho }^{+}{\rho }^{-}$ from the corresponding quasi-two-body decays models and compare our results with those obtained previously using the perturbative QCD approach, the QCD factorization approach, and the factorization-assisted topological amplitude approach. It is found that the predictions are consistent with present data within errors. The branching ratios of our calculations for the four-body decays $B_S\to \pi \pi \pi \pi$ are at the order of the ${10}^{-7}$. For the $CP$-violating asymmetries, we found that $CP$-violating asymmetry can be enhanced, largely by the $\rho -\omega$ mixing resonances, when $\pi \pi$ pairs masses are in the vicinity of $\omega$ resonance.

Figure 1

Helicity angles of $({\pi }^{+}{\pi }^{-})({\pi }^{+}{\pi }^{-})$ decays, $\theta$ is defined as the polar angle of ${\pi }^{+}$ in ${\pi }^{+}{\pi }^{-}$ intermediate states and $\varphi$ represents the angle among two $\pi \pi$ pairs in the rest frame of $B$ meson.

Figure 2

The lowest order Feynman diagrams for the $B_S\to N_1{N}_2\to \pi \pi \pi \pi$ decays.