New physics in the angular distribution of $B_c^{-}\to J\mathrm{/}\psi (\to {\mu }^{+}{\mu }^{-}){\tau }^{-}(\to {\pi }^{-}{\nu }_{\tau }){\bar{\nu }}_{\tau }$ decay

In $B_c^{-}\to J\mathrm{/}\psi (\to {\mu }^{+}{\mu }^{-}){\tau }^{-}{\bar{\nu }}_{\tau }$ decay, the three-momentum ${\mathit{p}}_{{\tau }^{-}}$ cannot be determined accurately due to the decay products of ${\tau }^{-}$ inevitably include an undetected ${\nu }_{\tau }$. As a consequence, the angular distribution of this decay cannot be measured. In this work, we construct a {\it measurable} angular distribution by considering the subsequent decay ${\tau }^{-}\to {\pi }^{-}{\nu }_{\tau }$. The full cascade decay is $B_c^{-}\to J\mathrm{/}\psi (\to {\mu }^{+}{\mu }^{-}){\tau }^{-}(\to {\pi }^{-}{\nu }_{\tau }){\bar{\nu }}_{\tau }$, in which the three-momenta ${\mathit{p}}_{{\mu }^{+}}$, ${\mathit{p}}_{{\mu }^{-}}$, and ${\mathit{p}}_{{\pi }^{-}}$ can be measured. The five-fold differential angular distribution containing all Lorentz structures of the new physics (NP) effective operators can be written in terms of twelve angular observables ${\mathcal{I}}_i(q^2,E_{\pi })$. Integrating over the energy of pion $E_{\pi }$, we construct twelve normalized angular observables ${\widehat{\mathcal{I}}}_i(q^2)$ and two lepton-flavor-universality ratios $R(P_{L,T}^{J\mathrm{/}\psi })(q^2)$. Based on the $B_c\to J\mathrm{/}\psi$ form factors calculated by the latest lattice QCD and sum rule, we predict the $q^2$ distribution of all ${\widehat{\mathcal{I}}}_i$ and $R(P_{L,T}^{J\mathrm{/}\psi })$ both within the Standard Model and in eight NP benchmark points. We find that the benchmark BP2 (corresponding to the hypothesis of tensor operator) has the greatest effect on all ${\widehat{\mathcal{I}}}_i$ and $R(P_{L,T}^{J\mathrm{/}\psi })$, except ${\widehat{\mathcal{I}}}_5$. The ratios $R(P_{L,T}^{J\mathrm{/}\psi })$ are more sensitive to the NP with pseudo-scalar operators than the ${\widehat{\mathcal{I}}}_i$. Finally, we discuss the symmetries in the angular observables and present a model-independent method to determine the existence of tensor operators.