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We study the $B_s \to \pi^+ \pi^-$ and $B_d \to K^+ K^-$ decays in thestandard model and the family non-universal $Z^\prime$ model. Since none of thequarks in final states is the same as the initial quark, these decay modes canoccur only via power-suppressed annihilation diagrams. Despite the consistenceof the standard model prediction with the available data, there is a survivingroom for a light $Z^\prime$ boson. Taking into account the $Z^\prime$contribution, we find theoretical results for branching fractions can betteraccommodate the data. With the relevant data, we also derive a constraint onthe parameter space for the $Z^\prime$. Moreover, for the $B_d \to K^+ K^-$,both the direct and the mixing-induced $CP$ asymmetry are sensitive to thecouplings between $Z^\prime$ and fermions in the parameter spaces constrainedby data. The measurements at future experimental facilities, including theLHC-b, Belle-II and the proposed high energy $e^+e^-$ collider, will provide ususeful hints for direct searching for the light $Z^\prime$ boson.

Impact of Family Non-universal $Z^\prime$ Boson on Pure Annihilation $B_s \to \pi^+ \pi^-$ and $B_d \to K^+ K^-$ Decays