Effects of neutral $Z^{\mathrm{\prime }}$ boson in $B_s\to \varphi {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ decay with polarized $\varphi$ and the unpolarized and polarized $CP$ violation asymmetry

The effects of new neutral $Z^{\mathrm{\prime }}$ boson in $B_s\to \varphi {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$, when $\varphi$ is longitudinal or transverse polarized, are studied. In addition, the implications of $Z^{\mathrm{\prime }}$ boson on the unpolarized and polarized $CP$ violation asymmetries, with reference to lepton, are also presented. It is observed that the branching ratio with polarized $\varphi$ are quite sensitive to the $Z^{\mathrm{\prime }}$ contributions which are coming through the modification of Wilson coefficients $C_9^{eff}$ and $C_{10}$. Moreover, the off-diagonal elements of the chiral $Z^{\mathrm{\prime }}$ couplings contain new weak phase ${\varphi }_{sb}$ that provide a new source of $CP$ violation. Keeping in view that in the FCNC transitions the $CP$ violation asymmetries are highly suppressed in the SM, we have studied the unpolarized and polarized $CP$ violation asymmetries in $B_s\to \varphi {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ decays. Our results indicate that these $CP$ violation asymmetries are remarkably significant and can attribute the any new physics coming through the $Z^{\mathrm{\prime }}$ boson. It is hoped that the accurate measurements of these asymmetries will not only help us to establish NP but will also give a chance to determine the precise values of the coupling parameters of the $Z^{\mathrm{\prime }}$ boson.