$\bar B \to D^{(*)} \ell \bar{X}$ decays in effective field theory with massive right-handed neutrinos

We calculate the complete differential decay distributions for the $B$ meson decays, $\bar B \to D^{(*)} \ell \bar{X}$, to a massive right-handed (RH) neutrino in the low-energy effective field theory (LEFT) framework. We find that a massive RH neutrino does not introduce any new angular structures compared to the massless case, but can cause significant distortions in angular observables. We study the phenomenology of low-energy four-fermion operators permitted by the standard model effective field theory (SMEFT) extended with RH neutrinos (SMNEFT). We show that to explain the positive value of the difference in forward-backward asymmetries, $\Delta A_{\text{FB}}\equiv A_{\text{FB}}^\mu-A_{\text{FB}}^e$, tentatively inferred from Belle data, the RH neutrino must be massive. We also make predictions for $q^2$ dependent angular observables to motivate future measurements.