Exploring New Physics in transition $b\to s\,\ell^+\ell^-$ through different $B_c\to D_s^{(\ast)} \,\ell^+\ell^-$ observables

Inspired by the discrepancies observed in the $b\to s\ell^+\ell^-$ neutral current decays, we study the decay channel $B_c\to D_s^{(\ast)} \,\ell^+\ell^-$ ($\ell=\mu,\tau$), which is based on the same flavor changing neutral current (FCNC) transition at the quark level. The current study shows that this decay channel can provide a useful probe for physics beyond the standard model. We use the helicity formalism while employing the effective theory approach where we include the effects of vector and axial vector `new' physics (NP) operators. In this study, we have computed the branching ratio $\mathcal{B}_r$, the $D^\ast$ helicity fraction $f_L$, the lepton forward-backward asymmetry $\mathcal{A}_{FB}$, and the lepton flavor universality ratio (LFU) $R^{\tau\mu}_{D_s^*}$. In addition, as a complementary check on the LFU, we also calculate the various other LFU observables, $R_{i}^{\tau\mu}$ where $i=A_{FB}$, $f_L$. We assume that the NP universal coupling is present for both muons and tauons, while the non-universal coupling is only present for muons. Regarding these couplings, we employ the latest global fit to the $b\to s\ell^+\ell^-$ data, which is recently computed in arXiv:2304.07330. We give predictions of some of the mentioned observables within the SM and the various NP scenarios. We have found that not only are the considered observables sensitive to NP but are also helpful in distinguishing among the different NP scenarios. These results can be tested at the LHCb, HL-LHC, and FCC-ee, and therefore, a precise measurements of these observables not only deepens our understanding of the $b\to s\ell^+\ell^-$ process but also provides a window of opportunity to possibly study various NP scenarios.