$ρ-ω$ mixing contribution to the measured $CP$ asymmetry of $B^\pm\to ωK^\pm$

Abstract

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