Analysis of angular observables of $\Lambda_b \to \Lambda (\to p\pi)\mu^{+}\mu^{-}$ decay in the standard and $Z^{\prime}$ models

In 2015, the LHCb collaboration has measured $\frac{d{\mathcal{B}}}{dq^2}$, the lepton- and hadron-side forward-backward asymmetries, denoted by $A^\ell_{FB}$ and $A^{\Lambda}_{FB}$, respectively in the range $15 < q^2(=s) < 20$ GeV$^2$ with 3 fb$^{-1}$ of data. Motivated by these measurements, we perform an analysis of $q^2$ dependent $\Lambda_b \to \Lambda (\to p \pi ) \mu^+\mu^-$ angular observables at large- and low-recoil in the SM and in a family non-universal $Z^{\prime}$ model. In the present study we use the recently performed high-precision lattice QCD calculations of the form factors that have well controlled uncertainties especially in $15 < s < 20$ GeV$^2$ bin. Using the full four-folded angular distribution of $\Lambda_b \to \Lambda (\to p \pi ) \mu^+\mu^-$ decay, firstly we calculate the values of these measured quantitites in the SM and compare their numerical values with the measurements in appropriate bins of $s$. In case of the possible discrepancy between the SM prediction and measurements, we try to see if these can be accommodated though the extra neutral $Z^{\prime}$ boson. In addition, the fraction of longitudinal polarization of the dimuon $F_{L}$ is measured to be $0.61^{+0.11}_{-0.14}\pm 0.03$ in $15 < s < 20$ GeV$^2$ at the LHCb. We find that in this bin the value found in the $Z^{\prime}$ model is close to the observed values. After comparing the results of these observables, we have proposed a number of other observables whose values are calculated in different bins of $s$ in the SM and $Z^{\prime}$ model. We illustrate that the experimental observations of these observables in several bins of $s$ can help to test the predictions of the SM and unravel NP contributions arises due to $Z^{\prime}$ model in these decays.