Vacuum Polarization Effects from Dark Fermion Loops in $Z^{\mathrm{\prime }}$ Mediator Models and their Impact on Angular Observables in $B\to K^{\ast }{\mu }^{+}{\mu }^{-}$ Decays

Rare flavour changing neutral current decays such as $B\to K^{\ast }{\mu }^{+}{\mu }^{-}$ provide sensitive probes of physics beyond the Standard Model (SM), being forbidden at tree level and arising only through loop induced processes. Angular analyses yield observables sensitive to short distance dynamics. The optimized observable $P_5^{\mathrm{\prime }}$ shows a persistent $\sim 3\sigma$ deviation from SM predictions in $q^2\in [1.1,6.0]{\mathrm{G}\mathrm{e}\mathrm{V}}^2$, motivating modified Wilson coefficients. A theoretically motivated framework is investigated in which vacuum polarization of a Dirac dark fermion $\chi$ modifies the propagator of a heavy $Z^{\mathrm{\prime }}$ boson, inducing a momentum dependent contribution to $C_9(q^2)$. The loop corrected propagator is incorporated into the effective weak Hamiltonian for $b\to s{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$. Bin averaged angular observables are computed with $\mathtt{\text{flavio}}$, and $\mathrm{34,997}$ parameter points are scanned over different chiral structures of the $Z^{\mathrm{\prime }}$ couplings. A ${\chi }^2$ analysis identifies preferred regions that are vector dominated and chirally asymmetric. The best fit solution significantly alleviates the $P_5^{\mathrm{\prime }}$ tension, reducing the deviation in the central $q^2$ bin to $0.12\sigma$, an improvement of approximately $96\mathrm{\%}$ relative to the SM. This improvement is driven by tree level $Z^{\mathrm{\prime }}$ exchange, which induces $\mathrm{\Delta }{C}_9\simeq -1.2$, consistent with global fits and compatible with other angular observables and $B_s\to {\mu }^{+}{\mu }^{-}$. Dark fermion vacuum polarization adds a subleading $q^2$ dependence to $C_9$; it does not dominate the fit, but supplies a distinctive threshold structure testable with finer binning. These results show that a $Z^{\mathrm{\prime }}$ portal to a dark fermion provides a phenomenologically consistent description of the $P_5^{\mathrm{\prime }}$ anomaly and a concrete benchmark for LHCb Run~3 analyses.