Lepton Dipole Moments in Supersymmetric Low-Scale Seesaw Models

We study the anomalous magnetic and electric dipole moments of charged leptons in supersymmetric low-scale seesaw models with right-handed neutrino superfields. We consider a minimally extended framework of minimal supergravity, by assuming that CP violation originates from complex soft SUSY-breaking bilinear and trilinear couplings associated with the right-handed sneutrino sector. We present numerical estimates of the muon anomalous magnetic moment and the electron electric dipole moment (EDM), as functions of key model parameters, such as the Majorana mass scale mN and tan(\beta). In particular, we find that the contributions of the singlet heavy neutrinos and sneutrinos to the electron EDM are naturally small in this model, of order 10^{-27} - 10^{-28} e cm, and can be probed in the present and future experiments.

amplitudes of processes, such as lepton → 3 leptons and µ → e conversion, whereas photon-penguin LFV diagrams are sub-dominant and become only relevant to models with ultra-heavy neutrinos close to the GUT scale [14]. In particular, our recent analysis has shown [13] that a significant region of the ν R MSSM parameter space exists for which the branching ratios of CLFV processes are predicted to be close to the current experimental sensitivities, despite the fact that the soft SUSY-breaking scale has been pushed to values higher than 1 TeV, as a consequence of the discovery of a SM-like Higgs boson at the CERN Large Hadron Collider (LHC) [15] and the existing non-observation limits on the gluino and squark masses that were also deduced from LHC data [16].
It is therefore of particular interest to investigate here, whether the effects of low-scale heavy neutrinos and their SUSY partners, the sneutrinos, contribute in a relevant manner to other high precision observables, such as the muon anomalous MDM a µ and the electron EDM d e . We believe that the announced higher-precision measurement of a µ by a factor of 4 in the future Fermilab experiment E989 [17,18] and the expected future sensitivities of the electron EDM down to the level of ∼ 10 −31 e cm [3], renders such an investigation both very interesting and timely.
Most studies on lepton dipole moments have been devoted to SUSY models realizing a high scale seesaw mechanism [20][21][22][23]. Here instead, we consider the ν R MSSM which provides potentially significant contributions to lepton dipole moments due to low-scale neutrinos and sneutrinos, as well as new sources of CP violation. In particular, an interesting possibility emerges if there exists CP violation beyond the SM which is sourced from the singlet sector of the ν R MSSM. This new CP violation may originate from a complex soft trilinear sneutrino parameter A ν , or from a complex soft bilinear parameter B ν . In addition, one may have new CP-odd phases residing in the 3 × 3 neutrino Yukawa-coupling matrix h ν . Assuming that these are the only additional non-zero CP-odd phases in the ν R MSSM, we find that the electron EDM is testable, but naturally small, typically of order 10 −27 e cm, thereby avoiding to some extent the well-known problem of too large CP violation, from which SUSY extensions of the SM, such as the MSSM (see, e.g. [19]), usually suffer.
The outline of the paper is as follows. In Section II, we introduce our conventions and notation for the lepton dipole moments, as well as describe the new sources of CP violation that we are considering in the ν R MSSM. Section III presents our numerical estimates for the lepton dipole moments a µ and d e . To this end, we specify our input parameters, including the neutrino Yukawa matrices adopted in our numerical analysis. Section IV summarizes our conclusions. Technical details pertinent to the lepton-dipole moment formfactors are given in Appendix A.

II. MAGNETIC AND ELECTRIC DIPOLE MOMENTS
The anomalous MDM and EDM of a charged lepton l can be read off from the Lagrangian [24]: In the on-shell limit of the photon field A µ , the form factor F l defines the anomalous MDM of the lepton l, i.e. a l ≡ F l , whilst the form factor G l defines its EDM, i.e. d l ≡ eG l /m l . Given that the general form-factor decomposition of the photonic transition amplitude is given by [13] iT γll = i eα w 8πM 2 the anomalous MDM a l and the EDM d l of a lepton l are then respectively determined by Here and in the following, we adopt the notation for the couplings and the form-factors established in [13]. At the one-loop level, the EDM d l of the lepton vanishes in the MSSM with universal soft SUSY-breaking boundary conditions and no soft CP phases, adopting the convention of a real superpotential Higgs-mixing parameter µ [22]. This result also holds true, even in extensions of the MSSM with heavy neutrinos, as long as the sneutrino sector is universal and CP-conserving as well.
As a minimal departure of the above universal scenario, we assume here that only the sneutrino sector is CPviolating, due to soft CP phases in the bilinear and trilinear soft-SUSY breaking parameters:

5)
where B 0 and A 0 are real parameters determined at the GUT scale, m N is a real parameter inputed at the scale m N , and θ and φ are physical, flavour blind CP-odd phases. In addition, h ν is the 3 × 3 neutrino Yukawa matrix to be specified in the next section. The soft SUSY breaking terms corresponding to the b ν and A ν are obtained from the Lagrangian terms and respectively. Correspondingly,ν c iR ,ẽ jL , h + uL and h 0 uL denote the heavy sneutrino, selectron, charged Higgs and neutral Higgs fields. The O(3) flavour symmetry of the model for the heavy neutrinos assures that the heavy neutrino mass matrix m N is proportional to the unit matrix 1 3 with eigenvalues m N , up to small renormalization-group effects. To keep things simple, we also assume that the 3 × 3 soft bilinear mass matrix b ν is proportional to 1 3 . In the standard SUSY seesaw scenarios with ultra-heavy neutrinos of mass m N , the CP-violating sneutrino contributions to electron EDM d e scale as B 0 /m N and A 0 /m N at the one-loop level, and practically decouple for heavy-neutrino masses m N close to the GUT scale. Hence, sizeable effects on d e should only be expected in low-scale seesaw scenarios, in which m N can become comparable to B 0 and A 0 .
Following the conventions of [13], the 12 × 12 sneutrino mass matrix may be cast into the 4 × 4 block form: The entries of M 2 ν are expressed in terms of the 3 × 3 matrices: Here m 2 L , m 2 ν and A ν are 3 × 3 soft SUSY-breaking matrices associated with the left-handed slepton doublets, the right-handed sneutrinos and their trilinear couplings, respectively. We note that the bilinear soft 3 × 3 matrix b ν was neglected in Ref. [13], where the authors tacitly assumed that it was small compared to the other soft SUSY-breaking parameters in (2.9). Here, we take this term into account, but restrict the size of the universal bilinear mass parameter B 0 , such that the sneutrino masses remain always positive and hence physical.
The generation of a non-zero EDM d e results from the soft sneutrino CP-odd phases θ and φ, as well as from complex neutrino Yukawa couplings h ν . All these CP-odd phases are present in the photon dipole form factors G L,Ñ llγ and G R,Ñ llγ , whose analytical forms may be found in [13]. In fact, we noticed that d e may be generated by products of vertices that are not relatively complex conjugate to each other, such as [25] In the exact supersymmetric limit of softly-broken SUSY theories, the anomalous MDM (as well as EDM) operator is forbidden, as a consequence of the Ferrara and Remiddi no-go theorem [26]. The theorem can be verified for every particle and its SUSY-counterpart contribution to the anomalous MDM a µ . Besides the SM contribution, there are three additional contributions in the ν R MSSM, which originate from: (i) heavy neutrinos, (ii) sneutrinos and (iii) soft SUSY-breaking parameters. In the supersymmetric limit, the latter contribution (iii) vanishes. In the same limit, the heavy neutrino and sneutrino contributions read: where B lNa are the lepton-to-heavy neutrino mixings defined in the first article of Ref. [10] and in Ref. [27]. Obviously, the sum (G ll γ ) N + (G ll γ )Ñ vanishes, thereby confirming the Ferrara-Remiddi theorem. In the MSSM, the leading contribution to a l behaves as [28,30].
where M SUSY is a typical soft SUSY-breaking mass scale, tan β = v 2 /v 1 is the ratio of the neutral Higgs vacuum expectation values, and M 1,2 are the soft gaugino masses associated with the U(1) Y and SU(2) gauge groups, respectively. As we will see in the next section, the MSSM contribution (2.13) to a µ remains dominant in the ν R MSSM as well. From (2.13) and (2.3), one naively expects d l to behave at the one-loop level as where φ CP is a generic soft SUSY-breaking CP-odd phase. Nevertheless, beyond the one-loop approximation [22,29], other dependencies of d l on tan β are possible in the MSSM. However, we show that in the ν R MSSM at the one loop level the tan β dependence is linear.

III. NUMERICAL RESULTS
In our numerical analysis, we adopt the procedure established in [13]. As a benchmark model, we choose a minimally extended scenario of minimal supergravity (mSUGRA), in which we allow for the bilinear and trilinear soft SUSYbreaking terms, B ν and A ν , to acquire at the GUT scale overall CP-violating phases denoted as θ and φ, respectively. In addition, we choose the sign of the µ-parameter to be positive. As for the neutrino Yukawa coupling matrix h ν , we consider the approximate U(1)-and A 4 -symmetric models introduced in [31] and [32], respectively. In these two scenarios, h ν can be expressed in terms of the real parameters a, b and c and CP-odd phases that might be relevant for leptogenesis. Explicitly, the neutrino Yukawa-coupling matrix h ν in the U(1)-symmetric model is given by [31] and in the model based on the A 4 discrete symmetry by [32] It should be noted that the choices of the neutrino Yukawa matrices (3.1) and (3.2) both lead to massless light neutrinos for any value of the heavy neutrino mass scale m N , as these are protected by the U (1) and A 4 symmetries. The observed light neutrino masses and mixings can be obtained by introducing small symmetry-breaking parameters δ ij (with i, j = 1, 2, 3), such that δ ij ≪ a, b, c. Since lepton dipole moments remain practically unaffected by these small symmetry-breaking parameters, we do not consider them here in detail. For definiteness, our numerical analysis in this section is based on the following baseline scenario: where m 0 , M 1/2 and A 0 are the standard universal soft SUSY-breaking parameters. All mass parameters except m N are defined at the GUT scale and m N is intaken at m N scale. It is understood that those parameters not explicitly quoted in the text assume their default values stated in (3.3). Likewise, unless it is explicitly stated otherwise, our default scenario for h ν is the one given in (3.2), with the specific choice a = b = c = 0.05 as given in (3.3).
In the following the ν R MSSM contributions to the muon MDM not present in the SM are denoted by a µ . We investigate the dependence of a µ and d e on several key theoretical parameters, by varying them around their baseline value given in (3.3), while keeping the remaining parameters fixed. In doing so, we also make sure that the displayed parameters can accommodate the LHC data for a SM-like Higgs boson with mass m H = 125.5 ± 2 GeV [15] and satisfy the current lower limits on gluino and squark masses [16], i.e. mg > 1500 GeV and mt > 500 GeV. In the following, we present numerical results first for a µ and then for d e .

A. Results for aµ
Our numerical estimates for a µ exhibit a direct quadratic dependence on the muon mass m µ . In fact, we find that for the same set of soft SUSY-breaking parameters m 0 , M 1/2 and A 0 , the ratio a µ /a e remains constant to a good approximation, i.e. a µ /a e ≈ m 2 µ /m 2 e ≈ 42752.0. In order to understand this parameter dependence, we have to carefully analyze the soft SUSY-breaking contributions to the form-factors: where the different terms that occur in (3.4) and (3.5) are defined in [13] and are also explicitly given in Appendix A.
Observe that the neutralino vertices induce a term which is not manifestly proportional to the charged lepton mass, but to the neutralino mass. However, a closer inspection of the products of the mixing matricesṼ 0ℓR lmaṼ 0ℓR * lma and V 0ℓR lmaṼ 0ℓL * lma reveals [30] that these last expressions are by themselves proportional to the charged lepton mass m l . The latter provides a non-trivial powerful check for the correctness of the results presented here.
In addition, our numerical analysis shows that the muon anomalous MDM a µ is almost independent of the neutrino-Yukawa parameters a, b and c, the heavy neutrino mass m N and the soft trilinear parameter A 0 . Hence, our results are almost insensitive to a particular choice for a neutrino Yukawa texture, e.g. as given in (3.1) and (3.2), and also independent of the CP-odd phases θ and φ.
In Fig. 1, we give numerical estimates for a µ , as functions of the key theoretical parameters: tan β, M 1/2 , m 0 and m N . In the frame (a) of this figure, we see that a µ depends linearly on tan β, as expected from (2.13). Likewise, we have investigated in Fig. 1 the dependence of a µ on the soft SUSY-breaking parameters m 0 and M 1/2 , for different kinematic situations, and obtained results consistent with the scaling behaviour of 1/M 2 SUSY in (2.13). In the pannel (e) of Fig. 1, we observe that the effect of the heavy right-handed neutrinos (N ) and sneutrinos (Ñ ) on a µ is negative, but small, in agreement with our discussion above. The size of their contributions alone to a µ ranges from −10 −12 to −4.8 × 10 −15 , for m N = 0.5 -10 TeV. On the other hand, the left-handed sneutrino contributions to a µ are approximately independent of the heavy Majorana mass m N , reaching values ≈ 8.5 × 10 −11 . The soft SUSY-breaking contributions are also approximately independent of the heavy Majorana mass m N and have values ≈ 1.1 × 10 −12 . Note that the light sneutrino contribution to the anomalous magnetic moment is the largest in magnitude, and it is already present in the MSSM contributions to a µ . Finally, we have checked the dominance of the MSSM contributions by looking at the dependence of the parameter: (3.6) The difference δa µ of the predictions for a µ within the ν R MSSM and the MSSM divided by a µ is evaluated, and the absolute values of the results are displayed in the pannel (f) of Fig. 1, as a function of m 0 = M 1/2 . The largest deviation from the MSSM is found for largest allowed parameter value, m 0 = 3600 GeV, in which case δa µ /a MSSM µ is as large as 6.2 × 10 −2 .

B. Results for de
We now study the dependence of the electron EDM d e on several key model parameters, such as m 0 , M 1/2 , B 0 , A 0 , tan β, θ and φ. The predictions for d µ may be obtained by using the naive scaling relation: d µ ≈ (m µ /m e ) d e ≈ 205 d e . We have found this scaling behaviour is numerically satisfied very well. The maximal numerical values for d e we obtained are of the order ∼ 10 −27 e cm. Therefore predicted values for d µ are always found to be less than ∼ 10 −25 e cm, which is several orders of magnitude below the present experimental upper bound: d µ = 0.1 ± 0.9 × 10 −19 e cm [1].
We note that heavy singlet neutrinos N do not contribute to d e , even if the soft SUSY-breaking CP-odd phases φ and θ are non-zero. On the other hand, soft SUSY-breaking and right handed neutrino effects induce non-vanishing d e , if either θ or φ are non-zero. If both φ = 0 and θ = 0, lepton EDMs d l numerically vanish. Therefore, the complex products of vertices (2.11) emerging in the ν R MSSM do not induce the CP violation at one loop level, in accord with the result of Ref. [22] obtained in the MSSM with a high-scale seesaw mechanism. In Fig. 2, we present numerical estimates of d e on the ν R MSSM parameters tan β, m 0 , M 1/2 and m N , for maximal A 0 phase, φ = π/2. We also set θ = 0, since the dependence of d e on B 0 is weaker than the dependence on  In Fig. 3, we show the predicted numerical values for d e , as functions of the soft SUSY-breaking parameters A 0 and B 0 , and their corresponding CP phases φ and θ. In all pannels except the pannel (c), where φ = 0 and θ is a variable, φ assumes value π/2 or it is a variable and θ is taken to be equal zero. In the pannel (a) of Fig. 3, the soft trilinear parameter A 0 is constrained by the LHC data pertinent to Higgs, gluino and squark masses. The electron EDM d e is a complicated function of |A 0 | that slowly rises for |A 0 | between 1.8 TeV and 4.5 TeV, slowly decreases for where x assumes values between 2/3 and 1, and f (m 0 ) is roughly proportional to the function −1 − 2.4 TeV/m 0 + 6.3TeV 2 /m 2 0 . The last factor in Eq. (3.7) corresponds to the scaling factor 1/M 2 SUSY in the naive approximation (2.14), and in the approximate expressions for lepton EDM derived in [22].

IV. CONCLUSIONS
We have systematically studied the one-loop contributions to the muon anomalous MDM a µ and the electron EDM d e in the ν R MSSM. In particular, we have paid special attention to the effect of the sneutrino soft SUSY-breaking parameters, B ν and A ν , and their universal CP phases, θ and φ, on a µ and d e . To the best of our knowledge, lepton dipole moments have not been analyzed in detail before, within SUSY models with low scale singlet (s)neutrinos.
For the anomalous MDM a µ of the muon, we have found that the heavy singlet neutrino and sneutrino contributions to a µ are small, typically one to two orders of magnitude below the muon anomaly ∆a µ . Instead, left-handed sneutrinos and sleptons give the largest effect on ∆a µ , exactly as is the case in the MSSM. The dependence of a µ on the muon mass m µ , tan β and the soft SUSY-breaking mass scale M SUSY have been carefully analyzed and their scaling behaviour according to (2.13) has been confirmed. Finally, the dependence of a µ on the universal soft trilinear parameter A 0 , the neutrino Yukawa couplings h ν and the heavy neutrino mass m N are negligible. Furthermore, we have analyzed the electron EDM d e in the ν R MSSM. The heavy singlet neutrinos do not contribute to d e , and soft SUSY-breaking and sneutrino terms contribute only, if the phases φ and/or θ have a nonzero value. The contribution from the possible CP violating terms arising from the relatively complex products of the vertices exposed in (2.11) is numerically shown to be equal zero. On the other hand, the contribution due to a non-zero value of φ is the largest and may give rise to values for the electron EDM d e comparable to its present experimental upper limit. The effect of the CP-odd phase θ on d e is approximately one to two orders of magnitude smaller than that of φ. The size of d e increases with tan β and mass of the lepton m ℓ , it is approximatively independent of A 0 and B 0 , but it generically decreases, as functions of the soft SUSY-breaking parameters m 0 , M 1/2 . Based on our numerical results, we have also derived approximate semi-analytical expressions, which differ from those presented in the existing literature for SUSY models realizing a GUT-scale seesaw mechanism. Specifically, the flavour blind CP-odd phases lead to a scaling of the lepton EDM d l ∝ m l tan β/m y N , where 2/3 < y < 1. Further d l generally decreases with M SUSY , but that cannot be described with a simple scaling law. The dependences on SUSY breaking parameters A 0 and B 0 are weak in the largest part of the parameter space. The linear dependence on tan β and the dependence on heavy neutrino mass are new results of this paper. In comparison the tan β depedence in Ref. [22] is, depending on its magnitude, either cubic or constant. Given the current experimental limits on d e , we identified a significant portion of the ν R MSSM parameter space with maximal CP phase φ = π/2, where the electron EDM d e can have values comparable to the present and future experimental sensitivities. The effect of sneutrinosector CP violation on the neutron and Mercury EDMs is expected to be suppressed, which is a distinctive feature for the class of the ν R MSSM scenarios studied in this paper.