$\rho -\omega$ mixing contribution to the measured $CP$ asymmetry of $B^{\pm }\to \omega K^{\pm }$

Abstract

We study the $CP$ asymmetry of $B^{\pm }\to \omega K^{\pm }$ with the inclusion of the $\rho -\omega$ mixing mechanism. It is shown that the $CP$ asymmetry of $B^{\pm }\to \omega K^{\pm }$ experimentally measured ($A_{CP}^{\text{exp}}$) and conventionally defined ($A_{CP}^{\text{con}}$) are in fact different, which relation can be illustrated as $A_{CP}^{\text{exp}}=A_{CP}^{\text{con}}+\Delta A_{CP}^{\rho \omega }$, with $\Delta A_{CP}^{\rho \omega }$ the $\rho -\omega$ mixing contribution to $A_{CP}^{\text{exp}}$. $A_{CP}^{\text{exp}}$ is in fact the regional $CP$ asymmetry of $B^{\pm }\to {\pi }^{+}{\pi }^{-}{\pi }^0{K}^{\pm }$ when the invariant mass of the three pions lies in the vicinity of the $\omega$ resonance. The numerical value of $\Delta A_{CP}^{\rho \omega }$ is extracted from the experimental data of $B^{\pm }\to {\pi }^{+}{\pi }^{-}{K}^{\pm }$ and is found to be comparable with $A_{CP}^{\text{exp}}$, hence, nonnegligible. The conventionally defined $CP$ asymmetry, $A_{CP}^{\text{con}}$, is obtained from the values of $A_{CP}^{\text{exp}}$ and $\Delta A_{CP}^{\rho \omega }$, and is compared with the theoretical calculations in the literature.

Comments: 8 pages, version accepted by JHEP