$\mu \to e\gamma$ selecting scalar leptoquark solutions for the $(g-2{)}_{e,\mu }$ puzzles

We investigate all potentially viable scenarios that can produce the chiral enhancement required to simultaneously explain the $(g-2{)}_e$ and $(g-2{)}_{\mu }$ data with either a single scalar leptoquark or a pair of scalar leptoquarks. We provide a classification of these scenarios in terms of their ability to satisfy the existing limits on the branching ratio for the $\mu \to e\gamma$ process. The simultaneous explanation of the $(g-2{)}_{e,\mu }$ discrepancies, coupled with the current experimental data, implies that the $(g-2{)}_e$ loops are exclusively due to the charm-quark propagation, whereas the $(g-2{)}_{\mu }$ loops are due to the top-quark propagation. The scenarios where the $(g-2{)}_e$ loops are due to the top (bottom) quark propagation are, at best, approximately 9 (3) orders of magnitude away from the experimental limit on the $\mu \to e\gamma$ branching ratio. All in all, there are only three particular scenarios that can pass the $\mu \to e\gamma$ test and simultaneously create a large enough impact on the $(g-2{)}_{e,\mu }$ discrepancies when the new physics is based on the Standard Model fermion content. These are the $S_1$, $R_2$, and $S_1\&S_3$ scenarios, where the first two are already known to be phenomenologically viable candidates with respect to all other flavor and collider data constraints. We show that the third scenario—where the right-chiral couplings to charged leptons are due to $S_1$, the left-chiral couplings to charged leptons are due to $S_3$, and the two leptoquarks mix through the Standard Model Higgs field—cannot address the $(g-2{)}_e$ and $(g-2{)}_{\mu }$ discrepancies at the $1\sigma$ level due to an interplay between $K_L^0\to e^{\pm }{\mu }^{\mp }$, $Z\to e^{+}{e}^{-}$, and $Z\to {\mu }^{+}{\mu }^{-}$ data despite the ability of that scenario to avoid the $\mu \to e\gamma$ limit.

Figure 1

The available parameter space for the left-chiral couplings of $S_3$ (left) and the right-chiral couplings of $S_1$ (right) with respect to the $K_L^0\to e^{\pm }{\mu }^{\mp }$ and $Z\to {\ell }^{+}{\ell }^{-}$ constraints for two different inputs for the leptoquark masses. The $(g-2{)}_{e,\mu }$ discrepancies are addressed at the $1\sigma$ or $2\sigma$ level, as indicated.