Effective field theory approach to $b\to s\mathrm{\ell }{\mathrm{\ell }}^{(\mathrm{\prime })}$, $B\to K^{(\ast )}\nu \bar{\nu }$ and $B\to D^{(\ast )}\tau \nu$ with third generation couplings

LHCb reported anomalies in $B\to K^{\ast }{\mu }^{+}{\mu }^{-}$, $B_s\to \varphi {\mu }^{+}{\mu }^{-}$ and $R(K)=B\to K{\mu }^{+}{\mu }^{-}\mathrm{/}B\to K{e}^{+}{e}^{-}$. Furthermore, BaBar, BELLE and LHCb found hints for the violation of lepton flavour universality violation in $R(D^{(\ast )})=B\to D^{(\ast )}\tau \nu \mathrm{/}B\to D^{(\ast )}\mathrm{\ell }\nu$. In this note we reexamine these decays and their correlations to $B\to K^{(\ast )}\nu \bar{\nu }$ using gauge invariant dim-6 operators. For the numerical analysis we focus on scenarios in which new physics couples, in the interaction eigenbasis, to third generation quarks and lepton only. We conclude that such a setup can explain the $b\to s\mu^+\mu^-$datasimultaneouslywith$R(D^{(*)})$ for small mixing angles in the lepton sector (of the order of $\pi \mathrm{/}16$) and very small mixing angles in the quark sector (smaller than $V_{cb}$). In these region of parameter space $B\to K^{(\ast )}\tau \mu$ and $B_s\to \tau \mu$ can be order ${10}^{-6}$. Possible UV completions are briefly discussed.