The combined analysis of the BABAR, Belle, and LHCb data on B → Dτν, B → D*τν and B_c → J/Ψτν decay observables shows evidence of physics beyond the Standard Model (SM). In this article, we study all the one- and two-dimensional scenarios which can be generated by adding a single new particle to the SM. We put special emphasis on the model-discriminating power of F_L(D*) and of the τ polarizations, and especially on the constraint from the branching fraction BR(B_c → τν). We critically review this constraint and do not support the aggressive limit of BR(B_c → τν) < 10% used in some analyses. While the impact of F_L(D*) is currently still limited, the BR(B_c → τν) constraint has a significant impact: depending on whether one uses a limit of 60%, 30% or 10%, the pull for new physics (NP) in scalar operators changes drastically. More specifically, for a conservative 60% limit a scenario with scalar operators gives the best fit to data, while for an aggressive 10% limit this scenario is strongly disfavored and the best fit is obtained in a scenario in which only a left-handed vector operator is generated. We find a sum rule for the branching ratios of B → Dτν, B → D*τν and Λ_b → Λ_cτν which holds for any NP contribution to the Wilson coefficients. This sum rule entails an enhancement of BR(Λ_b → Λ_cτν) over its SM prediction by (24±6)% for the current R(D^(*)) data.