Electric Dipole Moments as Probes of $B$ Anomaly

The measurements of the lepton flavor universality (LFU) in $\mathcal{B}({\bar{B}} \to D^{(\ast)} l \bar{\nu})$ indicate a significant deviation from the standard model prediction at a 3-4$\sigma$ level, revealing a violation of the LFU ($R_{D^{(\ast)}}$ anomaly). It is known that the $R_{D^{(\ast)}}$ anomaly can be easily accommodated by an $SU(2)_L$-singlet vector leptoquark (LQ) coupled primarily to third-generation fermions, whose existence is further motivated by a partial gauge unification. In general, such a LQ naturally leads to additional $CP$-violating phases in the LQ interactions. In this paper, we point out that the current $R_{D^{(\ast)}}$ anomaly prefers the $CP$-violating interaction although $\mathcal{B}({\bar{B}} \to D^{(\ast)} l\bar{\nu})$ are $CP$-conserving observables. The $CP$-violating LQ predicts a substantial size of the bottom-quark electric dipole moment (EDM), the chromo-EDM, and also the tau-lepton EDM. Eventually at low energy, the nucleon and electron EDMs are radiatively induced. Therefore, we conclude that the $R_{D^{(\ast)}}$ anomaly with the $SU(2)_L$-singlet vector LQ provides unique predictions: neutron and proton EDMs with opposite signs and a magnitude of $\mathcal{O}(10^{-27})\,e$cm, and suppressed electron EDM. Furthermore, we show that a similar EDM pattern is predicted in an $SU(2)_L$-doublet scalar LQ scenario that can accommodate the $R_{D^{(\ast)}}$ anomaly as well. These EDM signals could serve as crucial indicators in future experiments.