Implications of New Physics on  \(D_s^+\xrightarrow {} \eta ^{(\prime )} \mu ^+ \nu _\mu \) Decays

In recent years, the flavour changing charged current (FCCC) \(b \xrightarrow {} c\bar{l}\nu _l\) transitions have gained special attention both in experiment and in phenomenological study to explore new physics (NP) beyond the standard model (SM). Recently, the BaBar, Belle and LHCb have measured lepton flavour universality (LFU) ratios \(R_{D^{(*)}}\) and the world average values of the colliders exhibited around \(3.3\sigma \) tension from their corresponding SM results [1], implies possible hint of NP. Such anomalies associated with b hadron decays raise a question. Whether the similar inconsistency can be observed in charm decays induced by \(c\xrightarrow {}s\bar{l}\nu _l\) transitions! Most of the recent experimental measurements on the pure leptonic and semileptonic D decays except \(D_{(s)}^+\xrightarrow {} \eta ^{(\prime )} \bar{l} \nu _l\) decays agree with the SM predictions. In the last few years, a lot of theoretical efforts have been made for searching NP contribution in D decays [3]. Experimental results of branching fraction for \(D_{(s)}^+\xrightarrow {} \eta ^{(\prime )} \bar{l} \nu _l\) decays show greater than \(1\sigma \) tension from their corresponding SM predictions [6] which indicates possibility of the existence of NP. Inspired by these results, we will study the \(D_{s}^+\xrightarrow {} \eta ^{(\prime )} \mu ^+ \nu _\mu \) decays in \(W^\prime \) model [2, 4, 5] to explore NP effects. Basically, \(W^\prime \) is a theoretically predicted vector boson which arises from the simplest extension of the SM electroweak gauge group by adding extra SU(2) group. In this work, we will predict the new \(W^\prime \) coupling parameters using recent experimental results of branching fractions of the above decays. Finally, the effects of NP on the branching fraction will be investigated using our new coupling parameters. Our investigation may be helpful to search \(W^\prime \) boson in future collider experiments.