The flavor of a minimal composite $S_1$ leptoquark and the $b\to c\tau\bar\nu$ anomaly

Several experiments have measured a deviation in $B\to D^{(*)}$ semileptonic decays, that point to new physics at the TeV scale violating lepton flavor universality. A scalar leptoquark $S_1$, with a suitable structure of couplings in flavor space, is known to be able to solve this anomaly modifying $b\to c\tau\bar\nu$. In the context of composite Higgs models, we consider a theory containing $H$ and $S_1$ as Nambu-Goldstone bosons (NGBs) of a new strongly interacting sector, with ordinary resonances at a scale ${\cal O}(10)$ TeV. Assuming anarchic partial compositeness of the Standard Model (SM) fermions we calculate the potential of the NGBs that is dominated by the fermions of the third generation, we compute $R_{D^{(*)}}$ and estimate the corrections to flavor observables by the presence of $S_1$. We find that the SM spectrum and $m_{S_1}\sim$ TeV can be obtained with a NGB decay constant of order $\sim 5$ TeV. We obtain a robust correlation between the main corrections to $R_{D^{(*)}}$, $B_{K^{(*)}\nu\nu}$ and $g_\tau/g_\mu$, that leads to a sever bound on $R_{D^{(*)}}$, roughly $2\sigma$ below the experimental value. Besides the bounds on the flavor observables $g_\tau^W$, BR$(\tau\to\mu\gamma)$ and $\Delta m_{B_s}$ are saturated, with the first one requiring a coupling between resonances $g_*\lesssim 2$, whereas the second one demands $m_{S_1}\gtrsim 1.7$ TeV, up to corrections of ${\cal O}(1)$.