Final-state interactions in the $B^{\mp }\to K^{+}{K}^{-}{\pi }^{\mp }$ decays: traces of the $X(3700)$ state and CP asymmetry

In this work, we investigate final-state interaction effects in $B^{\mp }\to K^{+}{K}^{-}{\pi }^{\mp }$ decays above 3 GeV with a twofold objective: (i) to study $CP$ asymmetry, and (ii) to search for possible signatures of the $X(3700)$ state (also denoted as $X(3720)$). We consider both tree-level and resonant contributions, with the latter arising from a triangle mechanism $B^{\mp }\to {\bar{D}}^0{D}^{\ast \mp }(\to {\pi }^{\mp }{D}^0)\to K^{+}{K}^{-}{\pi }^{\mp }$, in which the $D\bar{D}\to K\bar{K}$ rescattering process carries the signature of the exotic $X(3700)$ state. We analyze the dependence of our results on the relative phase between the tree-level and resonant contributions, as well as on other relevant parameters. We estimate the differential mass distribution in the region where the $X(3700)$ may leave a trace, and the $CP$ asymmetry driven by final-state interactions. Our findings are compared with available data for the differential mass distributions and the $CP$ asymmetry. Taken together, these results may contribute to a better understanding of the properties of exotic states and the role of $CP$ asymmetry in $B$-meson phenomenology.