Enhancement of Direct $CP$ asymmetry in $Z'$ models

Abstract: We consider $CP$-violating $Z'$ models to account for the anomalies in $b \to s \ell \ell$ decays. Using the updated constraints from lepton flavor universality (LFU) violating ratios, $b \to s \mu \mu $ $CP$-conserving and $CP$-violating observables, $B_s-\bar{B}s$ mixing and neutrino trident we obtain the favored parameter space of two classes of $Z'$ models generating the `1D' new physics scenarios with $C_9^{\rm NP} <0$ and $C_9^{\rm NP} = -C{10}^{\rm NP}$. We show that the predictions of direct $CP$ asymmetry $A_{CP}$ in $B^+ \to K^+ \mu \, \mu$ decays close to the $c \bar{c}$ resonance region can be used to detect the presence of new $CP$ violating phases in the $Z'$ couplings.The favored $1\,\sigma$ parameter space of $Z'$ models generating the scenario $C_9^{\rm NP} <0$ allows for an enhancement in the integrated $A_{CP}$ in $q^2 =[8,9]\,\rm{GeV^2}$ bin up to $\pm 10\%$ and in the $q^2 =[6,7]\,\rm{GeV^2}$ bin up to $\pm 5\%$. We find that for a negative value of the phase of $J/\psi$ resonance, although such an enhancement is possible in $Z'$ models generating the scenario $C_9^{\rm NP} = -C_{10}^{\rm NP}$, the favored parameter space prefers only positive values of $A_{CP}$. Hence, a future measurement of direct $CP$ asymmetry in these bins can potentially constrain the size of new $CP$ violating phases, and help in distinguishing between the two classes of $Z'$ models.