Implications for Electric Dipole Moments of a Leptoquark Scenario for the $B$-Physics Anomalies

Vector leptoquarks can address the lepton flavor universality anomalies in decays associated with the $b \to c \ell \nu$ and $b \to s \ell \ell$ transitions, as observed in recent years. Generically, these leptoquarks yield new sources of CP violation. In this paper, we explore constraints and discovery potential for electric dipole moments (EDMs) in leptonic and hadronic systems. We provide the most generic expressions for dipole moments induced by vector leptoquarks at one loop. We find that $O(1)$ CP-violating phases in tau and muon couplings can lead to corresponding EDMs within reach of next-generation EDM experiments, and that existing bounds on the electron EDM already put stringent constraints on CP-violating electron couplings.

The combined discrepancy between the SM prediction and experimental world averages of R D and R D * is at the 3.1σ level.
The most precise measurement to date of the LFU ratio R K has been performed by LHCb [11] R K = 0.846 +0.060 −0.054

+0.016
−0.014 , for 1.1 GeV 2 < q 2 < 6 GeV 2 , with q 2 being the dilepton invariant mass squared. The SM predicts R SM K 1 with theoretical uncertainties well below the current experimental ones [12]. The above experimental value is closer to the SM prediction than the Run-1 result [13]. However, the reduced experimental uncertainties still imply a tension between theory and experiment of 2.5σ.
The result for both q 2 bins are in tension with the SM prediction [12], R SM are compatible with both the SM prediction and the LHCb results. Several papers have re-analyzed the status of the B anomalies in light of the latest experimental updates, and found preference for new physics with high significance [17][18][19][20][21][22][23].
While the anomalies detailed upon above persist, the question of the origin of the observed baryon asymmetry [24] also remains a long standing problem in cosmology. Any dynamical explanation requires sizable C-and CP-violating interactions in the early universe [25]. In light of upcoming low-energy experiments with much greater sensitivity to electric and magnetic dipole moments of elementary particles, it is interesting to ask whether solutions to the flavor anomalies may also be associated with sizable CP violating complex phases that may be probed by these experiments.
The only known viable, single-mediator explanation of all flavor anomalies is a U 1 vector leptoquark [26][27][28][29][30][31][32]. This leptoquark generically introduces new sources of CP violation in the Lagrangian in the form of complex parameters [33]. The scope of the present study is to explore, for the first time, the prospects of observing electric dipole moments (EDMs) induced by a U 1 vector leptoquark that could explain the flavor anomalies reviewed above. We additionally explore collider constraints, as well as constraints from measurements of the magnetic moments, and other flavor observables. Implications for EDMs and other CPV observables in scalar leptoquark scenarios have recently been discussed in [34][35][36][37].
This paper is organized as follows: In Sec. II, we introduce the CP violating U 1 model and discuss its effects on the B-physics anomalies. In Sec. III, we give an overview of the effects of the CP violating leptoquark on EDMs of quarks, leptons, and neutrons. We also include a discussion of the present status of the experimental searches and the prospects for future measurements. In Sec. IV, we report the main results of our paper, showing the leptoquark parameter space that can be probed by B-physics and EDM measurements. In Sec. V, we discuss the LHC bounds on our leptoquark model. Finally, we reserve Sec. VI for our conclusions.

II. THE CP VIOLATING U 1 VECTOR LEPTOQUARK MODEL
We consider the vector leptoquark U 1 = (3, 1) 2/3 (triplet under SU (3) c , singlet under SU (2) L , and with hypercharge +2/3). This model may be viewed as the low energy limit of Pati-Salam models described in Ref. [38,39] (see also [40][41][42][43][44][45][46][47]). The most general dimension-4 Lagrangian describing the vector leptoquark of mass M U 1 is (see e.g. [48] for a recent review) where U µν = D µ U ν − D ν U µ is the leptoquark field strength tensor in terms of its vector potential  [49].) The couplings λ q ij and λ d ij are in general complex and are therefore a potential source of CP violation of the model. We work in the fermion mass eigenstate basis and define the leptoquark couplings λ q ij and λ d ij in a way such that where V is the CKM matrix.
The second line in Eq. (8) encodes the chromo-and hypercharge-magnetic and electric dipole moments of the U 1 leptoquark.
If the leptoquark arises from the spontaneous breakdown of a gauge symmetry, gauge invariance requires these couplings to be fixed to κ s = κ Y = 1,κ s =κ Y = 0. In more generic scenarios where U 1 is composite, the values of κ s ,κ s , κ Y ,κ Y are free parameters. Non-zero values forκ s andκ Y are an additional potential source of CP violation. However, since they do not directly influence flavor physics, we will focus our attention to CP-violation contained in λ q ij and λ d ij (even though in Sec. III we will present fully generic expressions for the EDMs, including their dependence onκ s andκ Y ).

A. Leptoquark Effects in B-meson Decays
The U 1 leptoquark can simultaneously address the hints for LFU violation in charged current decays R D ( * ) and in neutral current decays R K ( * ) 2 . Here we will use the results of a recent study [21] that identified a benchmark point in the leptoquark parameter space that gives a remarkably consistent new physics explanation of these hints. We will explore the parameter space around this benchmark point (supplemented by a few more points), focusing on the implications for dipole moments. As we discuss below, not all leptoquark couplings in (8) are required to address the anomalies.
Explaining the observed values of R D ( * ) by non-standard effects in the b → cτ ν transition is possible if the leptoquark has sizable couplings to the left-handed tau. Avoiding strong constraints from leptonic tau decays τ → ν τ ν and the B → X s γ decay is possible in a well defined parameter space around the benchmark point with λ q 33 0.7, λ q 23 0.6 with a leptoquark mass of M U 1 = 2 TeV [21]. This corresponds to the following non-standard value for R D ( * ) which is in good agreement with observations (in this equation we normalize v = 246 GeV).
The results for R K ( * ) can be accommodated by a non-standard effect in the b → sµµ transition if the couplings to the left-handed muon obey Re(λ q 22 × λ q 32 ) −2.5 × 10 −3 for M U 1 = 2 TeV [21].
The leptoquark effects for this choice of couplings are described by a shift in the Wilson coefficients of the effective Hamiltonian relevant for b → s transitions (see e.g. [21] for the precise definition) This agrees well with the best fit value for the Wilson coefficients found in [21].
The muonic couplings λ q 22 , λ q 32 (that can explain the R K ( * ) anomalies) in combination with the tauonic couplings λ q 23 , λ q 33 (that are required to explain the R D ( * ) anomalies) lead to lepton flavor violating decays. The strongest constraints arise from the decays τ → φµ and B → Kτ µ. For the λ q 33 , λ q 23 benchmark mentioned above, existing limits on those decay modes result in the bounds on the leptoquark couplings |λ q 22 | 0.16 and |λ q 32 | 0.40 for M U 1 = 2 TeV [21]. The experimental values of R K ( * ) may also be explained by new physics in the b → see transition as opposed to modifying the b → sµµ transition. Focusing on left-handed couplings, the required shifts in the relevant Wilson coefficients is [21] C bsee corresponding to the couplings Re(λ q 21 × λ q 31 ) +2.5 × 10 −3 for M U 1 = 2 TeV. The experimental bounds on the lepton flavor violating processes τ → φe and B → Kτ e are comparable to those of τ → φµ and B → Kτ µ [54][55][56]. We therefore expect that the constraints on the left-handed electron couplings |λ q 21 | and |λ q 31 | are similar to the muon couplings mentioned above, i.e. |λ q 21 | 0.16 and |λ q 31 | 0.40 for M U 1 = 2 TeV. Motivated by this discussion, in the next sections we will explore the leptoquark parameter space in the neighborhood of four benchmark scenarios: Note that in benchmark BM3, the R K ( * ) anomalies are only partially addressed. For BM3 we have 88 which is in good agreement with the latest R K measurement, but ∼ 2σ away from the measured R K * value. As we discuss below in Sec. IV B benchmark BM3 is motivated because it can accommodate the longstanding discrepancy in the anomalous magnetic moment of the muon.
For all benchmark scenarios we explicitly checked compatibility with the measurements of the di-lepton [57] and di-tau [58] invariant mass distributions at the LHC and searches for electronquark contact interactions at LEP [59]. In the case of the di-lepton invariant mass distributions at the LHC, the value of λ q 32 in BM3 is close to the exclusion bound. Starting with these benchmark points, in the following sections we turn on couplings to right-

III. DIPOLE MOMENTS OF QUARKS AND LEPTONS
In this section, we calculate and present new and original formulae for shifts in the electric and magnetic dipole moments of leptons and quarks induced by the leptoquark. We then estimate the size of the neutron electric dipole. Finally, we review experimental limits on the dipole moments.
The leptoquark radiatively induces dipole moments starting at one loop order as shown in Fig. 1.
After integrating out the leptoquark, effective interactions encoding the dipole moments are given where a f is the anomalous magnetic dipole moment, and d f is the electric dipole moment of SM fermion f . In the absence of right-handed neutrinos, the U 1 leptoquark does not generate dipole moments for neutrinos.
Through its coupling with the gluons, the leptoquark induces chromomagnetic,â q , and chromoelectric,d q , dipole moments of quarks

A. Leptoquark Contribution to Dipole Moments of SM Leptons and Quarks
In the large M U 1 limit, the leptoquark contribution to the anomalous magnetic moment of the muon is where Q d = −1/3, Q U = +2/3 is the leptoquark electric charge, and N C = 3. Note that if κ Y = 1 orκ Y = 0, relevant for scenarios in which the leptoquark is not a gauge boson, the dipole moment exhibits logarithmic dependence on the cut-off scale Λ U V not far above the leptoquark mass. Our formula is in agreement with [60,61] when specialized to the vector leptoquark model with κ Y = 1 andκ Y = 0. Similarly, the muon electric dipole moment is CP violation is provided either by the imaginary part of the fermion coupling combination λ q i2 λ d * i2 , or by the CP violating hypercharge couplingκ Y . Dipole moments of other charged leptons are obtained by the appropriate replacement of the muon mass, m µ , and leptoquark couplings to The bottom quark electric dipole moment induced by the leptoquark is and the chromoelectric dipole moment (cEDM) iŝ The other down-type quark (chromo-)electric dipole moments can be obtained by appropriate replacements of flavor indices.
Analogously, up-type quark (chromo-)electric dipole moments are obtained from the bottom We do not consider anomalous (chromo-)magnetic moments of the quarks as they are experimentally not constrained.

B. Connecting Quark Dipole Moments to the Neutron EDM
In the following, we determine the neutron electric dipole moment due to quark-level dipole moments. We neglect the running of quark dipole moments from the leptoquark scale to the hadronic scale, since the neglected logarithm of order α s ln M 2 U 1 /M 2 n ≈ 1.6 leads to corrections which are small compared to the relevant hadronic uncertainties discussed below.
The dominant contributions to the neutron EDM are from the short range QCD interactions involving quark EDMs, d i , and cEDMs,d i , given by where the β (k) i are the hadronic matrix elements. Estimates from quark cEDM are given by β uG n ≈ 4 +6 −3 × 10 −4 e fm and β dG n ≈ 8 +10 −6 × 10 −4 e fm [62]. The most recent lattice evaluations of the matrix elements involving the electromagnetic EDMs are [63,64] Contributions from heavy quark cEDM are estimated by integrating out the heavy quark, Q = c, b, to generate the three gluon Weinberg (gluon cEDM) operator, where the Wilson coefficient is given by [65][66][67] Contributions to cG from CP-violating leptoquark gluon interactions proportional toκ s are also present, but we do not consider them since they are unrelated to flavor anomalies. In terms of cG, the neutron EDM is given by [62] where βG n ≈ [2, 40] × 10 −20 e cm is the nucleon matrix element estimated using QCD sum rules and chiral perturbation theory [68].
To compare the relative sizes of contributions from light and heavy quark to the neutron EDM, we take the strange and bottom quark contributions, and assume for simplicity that κ Y = 1, κ Y = 0. We also assume M U 1 ∼ 2 TeV for the leptoquark scale.
Putting together Eqs. (18) and (19) with Eq. (22), we find that the strange quark EDM contribution to the neutron EDM is The bottom quark cEDM contribution to the neutron EDM is instead given by For generic O(1) sized leptoquark couplings λ q ik and λ d ik the strange quark contribution (26) to the neutron EDM is much larger than the bottom quark contribution (27). However, in the region of parameter space we are exploring, the bottom quark contribution is typically bigger than the strange quark contribution.

C. Experimental Status and Prospects
We review here the current experimental status of dipole moments of Standard Model fermions.
The anomalous magnetic moments of the electron, a e , and the muon, a µ , are measured extremely precisely [69,70], and are predicted to similarly high precision within the SM, with new physics contributions constrained to lie within the range [71,72] (see also [73][74][75]) In addition to the long standing discrepancy in the muon magnetic moment with a significance of more than 3σ, a discrepancy in the electron magnetic moment arose after a recent precision measurement of the fine structure constant [76] with a significance of ∼ 2.4σ. Combining the expected sensitivity from the running g − 2 experiment at Fermilab [77] with expected progress on the SM prediction (see [78][79][80][81][82][83] for recent lattice efforts and [84][85][86][87][88] for recent efforts using the framework of dispersion relations) the uncertainty on ∆a µ will be reduced by a factor of a few in the coming years. Similarly, for ∆a e we expect an order of magnitude improvement in the sensitivity [89].
The anomalous magnetic moment of the tau, a τ , is currently only very weakly constrained. The strongest constraint comes from LEP and reads at 95% C.L. [90] − 0.055 < a τ < 0.013 .
Improvements in sensitivity by an order of magnitude or more might be achieved at Belle II or future electron positron colliders (see [91] for a review).
Strong experimental constraints exist for the EDM of the electron. The strongest bound is inferred from the bound on the EDM of ThO obtained by the ACME collaboration which gives at Significant improvements by an order of magnitude or more can be expected from ACME in the future [92].
Only weak constraints exist for the EDMs of the muon and the tau, d µ and d τ . Analyses by the Muon g-2 collaboration [93] and the Belle collaboration [94] give the following bounds at 95% C.L.
The proposed muon EDM experiment at PSI aims at improving the sensitivity to the muon EDM by 4 orders of magnitude, d µ 5 × 10 −23 e cm [95]. Improving the sensitivity to the tau EDM by roughly two orders of magnitude (d τ < 2 × 10 −19 e cm) might be possible at Belle II or at future e + e − colliders [96].
Turning to quarks, we note that the magnetic and chromo-magnetic dipole moments of quarks, a q andâ q , are very weakly constrained and we therefore do not consider them in this work. As discussed in the previous section, the EDMs and cEDMs of quarks, d q andd q , lead to EDMs of hadronic systems like the neutron and are therefore strongly constrained. In the following we will focus on the neutron EDM which is bounded at 95% C.L. by [97] |d n | < 3.6 × 10 −26 e cm .   Experimental sensitivities should improve by two orders of magnitude to a few 10 −28 e cm in the next decade [98].
We collect the SM predictions, the current experimental results, and expected future experimental sensitivities to the dipole moments in Table I.

IV. FLAVOR ANOMALIES AND ELECTRIC DIPOLE MOMENTS
In this section, we study the impact of leptoquarks on (c)EDMs and B-physics measurements at the benchmark points presented in Sec. II A. will induce the dipole moments of the tau as in Eqs. (16) and (17), as well as transition dipole moments leading to the lepton flavor violating decay modes τ → µγ and τ → eγ. In the limit m e , m µ m τ m b , the partial width for the U 1 contribution to τ → µγ is given by This expression is in agreement with [32], when specialized to the vector leptoquark model with In addition to inducing lepton flavor violating tau decays, the λ d 33 coupling will modify the new physics contributions to charged current decays based on the b → cτ ν and b → uτ ν transitions and neutral current decays based on b → sτ τ . The decay modes that are particularly sensitive to right-handed currents are the helicity suppressed two body decays B c → τ ν [102,103], B ± → τ ν, Using the expression for the branching ratio in terms of the Wilson coefficients from [104], we find where we neglected the finite life time difference in the B s system. We use a normalization such that the SM value for the Wilson coefficient is C SM So far no direct measurement of the B c → τ ν branching ratio has been performed. We impose the bound BR(B c → τ ν) < 30% [103]. The SM branching ratio is Similarly, the B s → τ + τ − decay has not been observed so far. The first direct limit on the branching ratio was placed by LHCb [109] and is BR(B s → τ + τ − ) < 6.8 × 10 −3 , while the SM branching ratio is BR(B s → τ + τ − ) SM = (7.73 ± 0.49) × 10 −7 [110].
In Fig. 2, we show current and projected constraints on the U 1 leptoquark in the plane of the complex λ d 33 coupling divided by the leptoquark mass for BM1 and BM2 benchmark points. The figure represents both BM1 and BM2, since the shown constraints are independent of the muon couplings λ q 32 , λ q 22 and electron couplings λ q 31 , λ q 21 and changing from BM1 to BM2 does not affect our results. The most stringent constraint comes from B s → τ + τ − and is shown in gray in the figure. Constraints from B ± → τ ν, B c → τ ν, and lepton flavor violating tau decays (τ → µγ for benchmark BM1 and τ → eγ for BM2) are slightly weaker and exclude values of λ d 33 that are a factor of a few larger than those excluded by B s → τ + τ − . (In Fig. 2 we show only the strongest constraint coming from B s → τ + τ − .) Once the bounds are imposed, the allowed values of the right-handed coupling λ d 33 are sufficiently small such that they do not affect R D ( * ) , R K ( * ) in a significant way. Therefore, in all the allowed region in Fig. 2, the anomalies are satisfied. The surrounding purple bands reflect the theoretical uncertainty in the nucleon matrix element βG n . Note that the observables shown in the figure are independent of λ q 32 , λ q 22 and λ q 31 , λ q 21 , and the change from benchmark BM1 to BM2 has no effect on the exclusion curves. In addition to the tau electric and anomalous magnetic dipole moments, the U 1 leptoquark coupling, λ d 33 , will contribute to the neutron EDM, d n . The constraint from the current bound on the neutron EDM is shown by the solid purple line in Fig. 2, where the region above this line is excluded due to the leptoquark generating a contribution to the neutron EDM that is too large.
The surrounding purple bands reflect the theoretical uncertainty in the nucleon matrix element βG n .
We observe that the currend bound on the neutron EDM leads to a constraint that is weaker than B s → τ + τ − and is not yet probing the allowed parameter space. On the other hand, the projected sensitivity of future neutron EDM experiments [98] (shown by the dashed purple line) will begin probing the new physics parameter space and can lead to stronger constraints on the amount of CP violation present in the right-handed couplings of U 1 to tau leptons.

B. Probing the Parameter Space Using Muon Measurements
Next we focus on the BM1 and BM3 benchmarks, and investigate the impact of the leptoquark couplings to right-handed muons, λ d 32 , while setting the right-handed tau and electron couplings (λ d 33 and λ d 31 , respectively) to zero. The coupling λ d 32 will lead to a shift in the anomalous magnetic moment of the muon, ∆a µ , in the muon EDM, d µ , and in the EDM of the bottom quark given in Eqs. (16), (17), and (18), as well as the lepton flavor violating decay mode τ → µγ given in Eq. (33) with |λ q 32 λ d * 33 | 2 → |λ d 32 λ q * 33 | 2 . In the presence of the coupling λ d 32 , the muon dipole moment enjoys a sizable chiral enhancement by m b /m µ .
In addition, the coupling λ d 32 can also give sizable non-standard effects in the B s → µ + µ − decay. The corresponding expression is analogous to the one for the B s → τ + τ − decay given in Eq. (36) The terms that contain both left-handed and right-handed couplings are chirally enchanced by a factor m 2 Bs /(m µ m b ). The branching ratio BR(B s → µ + µ − ) has been measured at LHCb, CMS and ATLAS [111][112][113][114].
We use the average of these results from [21], that, combined with the SM prediction [110,115], which is in slight tension (∼ 2σ) with the SM prediction. Interestingly enough, in the region of parameter space where the couplings to left-handed muons λ q 22 , λ q 32 provide an explanation of R K ( * ) , the tension in B s → µ + µ − is largely lifted.
In Fig. 3 we show the current and projected constraints on the U 1 leptoquark for BM1 (left) and BM3 (right) in the plane of the complex coupling λ d 32 divided by the leptoquark mass. For both benchmarks, the most stringent constraint arises from B s → µ + µ − . The region that is excluded at the 95% C.L. is shaded in gray. Once the constraints from B s → µ + µ − are imposed, the allowed values of λ d 32 are sufficiently small that they do not affect R K ( * ) in a significant way. The region that is shaded in red is the region of parameter space that is able to address the anomaly in the In the left plot of Fig. 3 we observe that, once the constraints from B s → µ + µ − is imposed, the BM1 benchmark cannot address the a µ anomaly. We conclude that the U 1 leptoquark can not explain the B anomalies and the (g − 2) µ anomaly simultaneously with the parameters fixed to those of BM1. This is mainly due to limits on lepton flavor violating decays τ → φµ and B → Kτ µ that impose stringent constraints on the size of the left-handed muonic couplings λ q 32 and λ q 22 (see discussion in Sec. II A).
In order to avoid these constraints, we can instead set the U 1 couplings to left-handed tau leptons, λ q 33 and λ q 23 , to zero as in BM3 in (13c). The decay rates τ → φµ, B → Kτ µ, and τ → µγ mediated by U 1 then go to zero, allowing the muonic couplings λ q 32 and λ q 22 to have larger values. However, by switching off λ q 33 and λ q 23 we forgo an explanation of R D ( * ) .
In the right plot of Fig. 3 we show that, for BM3, the region of parameter space that can address the a µ anomaly (the red shaded region) overlaps with the region of parameter space that is allowed by B s → µ + µ − , and the U 1 leptoquark can therefore address both the (g − 2) µ anomaly and (at least partially, cf. discussion in Sec. II A) the R K ( * ) anomalies. Finally, we notice that, for this benchmark, projected sensitivities to the neutron EDM might start to probe the viable parameter space.
We also explored the region of parameter space with nonzero λ d 22 instead of λ d 32 . In this case, for BM1 and BM3, the neutron EDM is dominated by the strange quark contribution (26), so its projected sensitivity covers larger region of parameter space. However in this case, we did not find any viable region of parameter space explaining the anomaly in a µ .
C. Probing the parameter space using electron measurements Instead of muon specific couplings that address the discrepancies in the LFU ratios R K ( * ) by new physics that suppresses the b → sµµ transitions, one can also entertain the possibility that new physics addresses the anomaly by enhancing the b → see transitions. This can be achieved with the leptoquark couplings λ d 31 , λ d 21 as given in Eq. (12) and by our benchmark points BM2 and BM4.
These couplings will also lead to shifts in the anomalous magnetic moment of the electron, ∆a e , and, in the presence of CP violation, induce an electron EDM, d e , (see Eqs. (16) and (17) [110]. Experimentally, the B s → e + e − branching ratio is bounded at the 90% C.L. by [116] BR(B s → e + e − ) < 2.8 × 10 −7 .
The plots in Fig. 4 show the current and projected constraints on the U 1 leptoquark in the plane of the complex coupling λ d 31 divided by the leptoquark mass for BM2 (left) and BM4 (right). In both panels the gray region is excluded by the bound from B s → e + e − , while the red shaded region is the region of parameter space that can address the 2.4σ anomaly in the electron magnetic moment, a e . The blue solid (dashed) lines are the current constraint (projected sensitivity) of the electron electric dipole moment, d e . In the right panel, the dashed purple line and the surrounding For BM2 (left plot of Fig. 4) we observe that the region of parameter space that is able to address the anomaly in a e is excluded by constraints from B s → e + e − and a simultaneous explanation of all the B anomalies and a e is not possible. This is due to stringent constraints on the size of λ q 31 from the lepton flavor violating decays τ → φe and B → Kτ e (see discussion in Sec. II A). Constrains from the τ → eγ are slightly weaker.
To avoid the stringent constraints from lepton flavor violating decays, we can set all the U 1 couplings to tau leptons to zero. Then, the τ → φe and B → Kτ e rates as well as the τ → eγ rate go to zero, and the left-handed couplings to electrons can be larger. However, by setting λ q 33 and λ q 23 to zero, we forgo an explanation of R D ( * ) . This scenario is given by BM4, and the resulting constraints are shown in the right plot of Fig. 4. We observe that the smaller value of λ q 21 = 0.005 in BM4 leads to weaker constraints on λ d 31 from B s → e + e − . In addition, the larger value of λ q 31 = 0.5 generates a larger contribution to the electron magnetic moment necessary to explain the slight tension in a e . In moving from BM2 to BM4 the bound from B s → e + e − opens up a wide region in parameter space favorable for the electron magnetic moment, a e . We conclude that BM4 can address the anomalies in both R K ( * ) and a e .
We also investigated the region of parameter space with nonzero λ d 21 instead of λ d 31 . We find in BM2 and BM4 that sensitivity to d e is reduced because it is chirally enhanced by m s rather than m b in Eq. (17). We also find no region of parameter space where the U 1 leptoquark explains the tension of the measured a e with theory. The two main production mechanisms are single production in association with a lepton (gq → U 1 ), and pair production (gg, qq → U 1 U 1 ). For a recent review see [123]. Once produced, the leptoquark will decay into a pair of SM fermions. The interactions of the U 1 leptoquark with SM quarks and leptons in Eq. (9) generate the decays of U 1 into an up-type quark and a neutrino, or a down-type quark and a charged lepton. In the limit where M U 1 is much larger than the masses reintepreted SUSY searches exist. The parameter β denotes the branching ratio of the leptoquark to a quark and a charged lepton. We do not report the bounds on the decays of the LQ to down-type quarks and a neutrino since these decays do not exist in our model. of the decay products, the partial widths of U 1 are given by where i, j = 1, 2, 3 label the three generations. are less sensitive to our benchmark models than the searches for pair produced leptoquarks. In the following, we will discuss in some details the bounds from searches of pair produced leptoquarks in all benchmarks.
For BM1 and BM2, the dominant non-zero couplings of U 1 are couplings involving tau leptons (λ q 33 , λ q 23 ) and the dominant decay modes are U 1 → bτ, sτ, tν τ , cν τ . At small values of λ d 33 (see Fig.  2), the branching ratios of the bτ and τ ν τ decay modes are similar in value (∼ 0.25) and dominate over the sτ and cν τ decays modes, which themselves have similar branching ratios (∼ 0.18). For values of λ d 33 near the border of the region allowed by B s → τ + τ − (see Fig. 2), the decay into bτ becomes the dominant decay mode with BR(U 1 → bτ ) ∼ 0.4.
The reinterpreted SUSY search for pair production of vector leptoquarks decaying to tν [122] and the CMS search for leptoquarks decaying to bτ [121] are the most sensitive searches. We find that these searches yield a similar lower bound on the mass of U 1 at around 1.2 TeV in the region of parameter space with small λ d 33 . The exact bound varies by at most ∼100 GeV in the region allowed by B s → τ + τ − .
In BM3, U 1 couples dominantly to 2nd generation leptons and the main decay modes are U 1 → bµ, sµ, tν µ , cν µ , with the bµ and tν µ decays modes being the dominant ones since λ q 32 λ q 22 , BR(U 1 → tν µ ) ∼ BR(U 1 → bµ) ∼ 0.5. The most stringent LHC constraint on this benchmark comes from the search for pair produced leptoquarks in final states with two muons and two jets in [119] 3 . This search leads to the bound m U 1 1.9 TeV. This bound is valid in the entire parameter space shown in the right panel Fig. 3, since λ d 32 is constrained to be very small, and therefore does not affect the leptoquark branching ratios.
Finally, in BM4, U 1 couples dominantly to 1st generation leptons and the main decay modes are U 1 → be and U 1 → tν e . In particular, at small values of λ d 31 (see Fig. 4), the branching ratios of these decay modes are very similar in value (∼ 0.5). At larger values of λ d 31 , the branching ratio into be becomes the dominant one, with BR(U 1 → be) ∼ 0.7 at the border of the allowed region for λ d 31 , as shown in the right plot of Fig. 4. The search for pair produced leptoquarks decaying in an electron and a jet in [118] provides the strongest constraint on the mass of U 1 and gives a lower bound of ∼ 1.8 TeV at small values of λ d 31 . The exact bound varies by at most ∼100 GeV in the region allowed by B s → e + e − .

VI. CONCLUSIONS
In this study, we focused on the possible, and quite likely, existence of new sources of CP violation if the flavor anomalies in b → c and b → s decays are due to new physics, specifically in the case where the new physics consists of a U 1 vector leptoquark. The underpinning of our study is that the U 1 vector leptoquark is one of the only (if not the only) new physics scenarios known to us that can provide a simultaneous explanation of the anomalies observed in lepton flavor universality ratios in b → c ν and b → s decays, R D ( * ) and R K ( * ) . Since the couplings of the U 1 to quarks and leptons are generically CP violating, they are expected just as generically to produce potentially observable electric dipole moments (EDMs) in leptonic and hadronic systems. Here, we have first provided new, original, and complete formulae for the calculation of the relevant EDMs, and carried out a phenomenological study of a few benchmark cases of how EDMs can constrain the U 1 leptoquark interpretation of the anomalies.
We note that the expressions we provided are the most general expressions for dipole moments induced by vector leptoquarks at one loop level, accounting for the most generic set of leptoquark couplings, which can accomodate scenarios for which the leptoquark may be composite.
We explored the parameter space of the U 1 leptoquark in the vicinity of 4 benchmark points that explain the R D ( * ) and R K ( * ) anomalies (or a subset of them). We identified viable regions of parameter space where the existing discrepancies in the anomalous magnetic dipole moments of the electron a e and the muon a µ can be explained in addition to R ( * ) K . However, we concluded that a simultaneous explanation of all three classes of discrepancies (R D ( * ) , R K ( * ) , a e,µ ) is not possible.
We found that, in the presence of non-zero CP-violating phases in the leptoquark couplings, EDMs play an important role in probing the parameter space of the model. Existing bounds on the electron EDM already exclude large parts of parameter space with CP violating leptoquark couplings to electrons. The expected sensitivities to the neutron EDM can probe into motivated parameter space and probe imaginary parts of leptoquark couplings to taus and muons.