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

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