The flavor of a minimal composite S₁ leptoquark and the \( b\to c\tau \overline{\nu} \) anomaly

Several experiments have measured a deviation in B → D^(∗) semileptonic decays, that point to new physics at the TeV scale violating lepton flavor universality. A scalar leptoquark S₁, with a suitable structure of couplings in flavor space, is known to be able to solve this anomaly modifying $b\to \mathit{c\tau }\overline{\nu }$ \( b\to c\tau \overline{\nu} \) . In the context of composite Higgs models, we consider a theory containing H and S₁ as Nambu-Goldstone bosons (NGBs) of a new strongly interacting sector, with ordinary resonances at a scale $\mathcal{O}\left(10\right)$ \( \mathcal{O}(10) \) 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 $R_{D^{\left(\ast \right)}}$ \( {R}_{D^{\left(\ast \right)}} \) and estimate the corrections to flavor observables by the presence of S₁. We find that the SM spectrum and $m_{S_1}$ \( {m}_{S_1} \) ∼ TeV can be obtained with a NGB decay constant of order ∼ 5 TeV. We obtain a robust correlation between the main corrections to $R_{D^{\left(\ast \right)}}$ \( {R}_{D^{\left(\ast \right)}} \) , $B_{K^{\left(\ast \right)}\mathit{\nu \nu }}$ \( {B}_{K^{\left(\ast \right)}\nu \nu} \) and g_τ/g_μ, that leads to a sever bound on $R_{D^{\left(\ast \right)}}$ \( {R}_{D^{\left(\ast \right)}} \) , roughly 2σ below the experimental value. Besides the bounds on the flavor observables $g_{\tau }^W$ \( {g}_{\tau}^W \) , BR(τ → μγ) and $\text{∆}{m}_{B_s}$ \( \Delta {m}_{B_s} \) are saturated, with the first one requiring a coupling between resonances g_* ≲ 2, whereas the second one demands $m_{S_1}$ \( {m}_{S_1} \) ≳ 1.7 TeV, up to corrections of $\mathcal{O}\left(1\right)$ \( \mathcal{O}(1) \) .