Confronting $B$ anomalies with low energy parity violation

Indirect searches have the potential to probe scales beyond the realm of direct searches. In this letter we consider the implications of two parity violating experiments: weak charge of proton $Q_W^p$ and the Caesium atom $Q_W^{Cs}$ on the solutions to lepton flavour non-universality violations (LFUV) in the decay of $B$ mesons. Working in a generic implementation of a minimal $Z^\prime$ model, we assume the primary contribution being due to the electron to facilitate comparison with the low $q^2$ parity violating experiments. We demonstrate that the conclusion is characterized by different limiting behavior depending on the chirality of the lepton current. The correlation developed in this study demonstrates the effectiveness in studying the synergy between different experiments leading to a deeper understanding of the interpretation of the existing data. It is shown that a possible future improvement in the parity violating experiments can have far reaching implications in the context of direct searches. We also comment on the prospect of addition of the muon to the fits and the role it plays in ameliorating the constraints on models of $Z'$. This offers a complimentary understanding of the pattern of the coupling of the NP to the leptons, strongly suggesting either a muon only or a combination of solutions to the anomalies.

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