A single TeV-scale scalar leptoquark in $\mathbf{SO(10)}$ grand unification and $\mathbf{B}$-decay anomalies

One of the explanations proposed for the recent rare $B$-decay anomalies is the existence of a scalar leptoquark. We investigate a grand unification scenario where a single, charge -1/3 scalar leptoquark ($S_1$) is present as the only new physics candidate at the TeV-scale. This leptoquark along with the Standard Model (SM) Higgs doublet originates from the 10-dimensional real scalar multiplet in $\mathrm{SO}(10)$ grand unification framework. An $S_1$ residing in the same representation as the SM Higgs motivates the idea that its mass is close to the electroweak scale as the peculiar mass splittings within this multiplet do not occur. Therefore, possible detection of a TeV-scale $S_1$ leptoquark, unaccompanied by any other new particles, could be interpreted in favour of $\mathrm{SO}(10)$ grand unification. We explicitly show how the gauge coupling unification is achieved with only one intermediate symmetry breaking scale at which the Pati-Salam gauge group, obtained from the $\mathrm{SO}(10)$ breaking at the unification scale, 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 still be a viable candidate to explain the $R_{D^{(*)}}$ anomalies. In order to obtain the allowed parameter space of our scenario, we consider the relevant flavour data including $R_{D^{(*)}}$ and $R_K^{\nu\nu}$ measurements, $Z\to\tau\tau$ decay and the latest $\tau\tau$ resonance search data at the LHC. We find that the LHC data strongly constrain the $S_1$ parameter space. We also find that there exist parts of parameter space where a single $S_1$ can still explain the $R_{D^{(*)}}$ anomalies without being in conflict with any of these constraints.