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

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