LHC constraints on $$W^\prime ,~Z^\prime $$ that couple mainly to third generation fermions

We use the results of CMS and ATLAS searches for resonances that decay to $$\tau \nu $$ τ ν or tb and $$\tau ^+\tau ^-$$ τ + τ - or $$t{\bar{t}}$$ t t ¯ final states to constrain the parameters of non-universal $$W^\prime $$ W ′ and $$Z^\prime $$ Z ′ gauge bosons that couple preferentially to the third generation. For the former we consider production from $$c\bar{b}$$ c b ¯ annihilation and find very weak constraints on the strength of the interaction and only for the mass range between 800 and 1100 GeV from the $$pp \rightarrow \tau _h p_T^{\mathrm{miss}}$$ p p → τ h p T miss channel. The constraints on the latter are much stronger and arise from both $$t{\bar{t}}$$ t t ¯ and $$\tau ^+\tau ^-$$ τ + τ - production. Treated separately, we find that the weak constraints on the $$W^\prime $$ W ′ still permit an explanation of the $$R(D^{(\star )})$$ R ( D ( ⋆ ) ) anomalies with a light sterile neutrino whereas the stronger constraints on the $$Z^\prime $$ Z ′ exclude significant light sterile neutrino contributions to the $$K \rightarrow \pi \nu \bar{\nu }$$ K → π ν ν ¯ rates. Within specific models the masses of $$W^\prime $$ W ′ and $$Z^\prime $$ Z ′ are of course related and we briefly discuss the consequences.