LFU and CP violation with $S_3$

We introduce the CP violating scalar leptoquark $S_3$ to explain the measured values of the lepton universality ratios $R_{K^{(\ast )}}$. We derive constraints on the CP-even and CP-odd components of the leptoquark Yukawa couplings stemming from effects in $b\to s\mu \mu$ and $B_s$ mixing. For the $b \to s \mu \mu$processesweimpose$R_{K^{(*)}}$,$$\mathcal{B}$(B_s \to \mu^+ \mu^-)$, as well as CP-sensitive angular asymmetries $A_{7,8,9}$, whereas in the $B_s$ mixing sector $\mathrm{\Delta }{M}_s$ and $S_{\psi \varphi }$ are considered. Combining the constraints within the $S_3$ model reveals that a large CP phase with a definite sign is perfectly viable for a leptoquark of mass below a few TeV. For larger mass of the $S_3$ leptoquark the CP phase is suppressed due to the observables pertaining to the $B_s$ system. We provide predictions of direct and mixing-induced CP asymmetries in $B\to K\mu \mu$ that could reveal the presence of the novel CP phase.