Simultaneous Explanation of the $R_K$ and $R_{D^{(*)}}$ Puzzles: a Model Analysis

$R_K$ and $R_{D^{(*)}}$ are two $B$-decay measurements that presently exhibit discrepancies with the SM. Recently, using an effective field theory approach, it was demonstrated that a new-physics model can simultaneously explain both the $R_K$ and $R_{D^{(*)}}$ puzzles. There are two UV completions that can give rise to the effective Lagrangian: (i) $VB$: a vector boson that transforms as an $SU(2)_L$ triplet, as in the SM, (ii) $U_1$: an $SU(2)_L$-singlet vector leptoquark. In this paper, we examine these models individually. A key point is that $VB$ contributes to $B^0_s$-${\bar B}^0_s$ mixing and $\tau \to 3\mu$, while $U_1$ does not. We show that, when constraints from these processes are taken into account, the $VB$ model is just barely viable. It predicts ${\cal B} (\tau^-\to\mu^-\mu^+\mu^-) \simeq 2.1 \times 10^{-8}$. This is measurable at Belle II and LHCb, and therefore constitutes a smoking-gun signal of $VB$. For $U_1$, there are several observables that may point to this model. Perhaps the most interesting is the lepton-flavor-violating decay $\Upsilon(3S) \to \mu \tau$, which has previously been overlooked in the literature. $U_1$ predicts ${\cal B}(\Upsilon(3S) \to \mu \tau)|_{\rm max} = 8.0 \times 10^{-7}$. Thus, if a large value of ${\cal B}(\Upsilon(3S) \to \mu \tau)$ is observed -- and this should be measurable at Belle II -- the $U_1$ model would be indicated.