Addressing the ${\mathbf R_{D^{(*)}}}$ anomalies with an ${\mathbf S_1}$ leptoquark from $\mathbf{SO(10)}$ grand unification

Motivated by the $R_{D^{(*)}}$ anomalies, we investigate an $\mathrm{SO}(10)$ grand unification scenario where a charge $-1/3$ scalar leptoquark ($S_1$) remains as the only new physics candidate at the TeV scale. This leptoquark along with the Standard Model (SM) Higgs doublet originates from the same 10-dimensional real scalar multiplet in the $\mathrm{SO}(10)$ grand unification framework taking its mass close to the electroweak scale. We explicitly show how the gauge coupling unification is achieved with only one intermediate symmetry-breaking scale at which the Pati-Salam gauge group is broken into the SM group. We investigate the phenomenological implications of our scenario and show that an $S_1$ with a specific Yukawa texture can explain the $R_{D^{(*)}}$ anomalies. We perform a multiparameter scan considering the relevant flavour constraints on $R_{D^{(*)}}$, $F_L{(D^*)}$, $P_\tau(D^*)$ and $R_{K^{(*)}}^{\nu\nu}$ as well as the constraint coming from the $Z\to\tau\tau$ decay and the latest $\tau\tau$ resonance search data at the LHC. Our analysis shows that a single leptoquark solution to the observed $R_{D^{(*)}}$ anomalies with $S_1$ is still a viable solution.