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

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$\rho-\omega$mixingcontributionto$A_{CP}^{$\text{exp}$}$\mathrm{.}$A_{CP}^{$\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 $\mathrm{\Delta }{A}_{CP}^{\rho \omega }$ 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 $\mathrm{\Delta }{A}_{CP}^{\rho \omega }$, and is compared with the theoretical calculations in the literature.