Forward-backward asymmetry induced $CP$ asymmetry of $B^{\pm }\to {\pi }^{\pm }{\pi }^{+}{\pi }^{-}$

Abstract

$CP$ violation of the decay $B^{\pm }\to {\pi }^{\pm }{\pi }^{+}{\pi }^{-}$ in the $f_0(500)-\rho (770{)}^0$ interfering region is analyzed. The forward-backward asymmetries (FBAs) and the corresponding $CP$ asymmetries FB-$CP$As are particularly investigated. To isolate the $CP$V caused by the interference of different partial wave more cleanly, we also introduce the direct-$CP$V-subtracted FB-$CP$A. Based on the LHCb data, we extract the FBAs, FB-$CP$As, direct-$CP$V-subtracted FB-$CP$A, as well as the regional $CP$As with invariant mass of the ${\pi }^{+}{\pi }^{-}$ pair in the range 0.2 GeV/$c^2<\sqrt{s_{\text{low}}}<1.8$ GeV/$c^2$. It is found that the (direct-$CP$V-subtracted) FB-$CP$As are quite large in the $f_0(500)-\rho (770{)}^0$ interfering region, which confirms that the interference of the intermediate resonances $f_0(500)$ and $\rho (770{)}^0$ plays an important role for the $CP$ violation of the three-body decay channel $B^{\pm }\to {\pi }^{\pm }{\pi }^{+}{\pi }^{-}$.

Comments: 14 pages, 5 figures, 3 tables, version accepted by PRD