LHC constraints on ${W}^{\prime },\phantom{\rule{3.33333pt}{0ex}}{Z}^{\prime }$ that couple mainly to third generation fermions

We use the results of CMS and ATLAS searches for resonances that decay to $\tau \nu $ or tb and ${\tau }^{+}{\tau }^{-}$ or $t\overline{t}$ final states to constrain the parameters of non-universal ${W}^{\prime }$ and ${Z}^{\prime }$ gauge bosons that couple preferentially to the third generation. For the former we consider production from $c\overline{b}$ 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 $pp\to {\tau }_{h}{p}_{T}^{\mathrm{miss}}$ channel. The constraints on the latter are much stronger and arise from both $t\overline{t}$ and ${\tau }^{+}{\tau }^{-}$ production. Treated separately, we find that the weak constraints on the ${W}^{\prime }$ still permit an explanation of the $R\left({D}^{\left(\star \right)}\right)$ anomalies with a light sterile neutrino whereas the stronger constraints on the ${Z}^{\prime }$ exclude significant light sterile neutrino contributions to the $K\to \pi \nu \overline{\nu }$ rates. Within specific models the masses of ${W}^{\prime }$ and ${Z}^{\prime }$ are of course related and we briefly discuss the consequences.