Charged lepton flavor violation and electric dipole moments in the inert Zee model

The inert Zee model is an extension of the Zee model for neutrino masses to allow for a solution to the dark matter problem that involves two vector-like fields, a doublet and a singlet of SU(2)L, and two scalars, also a doublet and a singlet of SU(2)L, all of them being odd under an exact Z2 symmetry. The introduction of the Z2 guarantees one-loop neutrino masses, forbids tree-level Higgs-mediated flavor changing neutral currents and ensures the stability of the dark matter candidate. Due to the natural breaking of lepton numbers in the inert Zee model and encouraged by the ambitious experimental program designed to look for charged lepton flavor violation signals and the electron electric dipole moment, we study the phenomenology of the processes leading to these kind of signals, and establish which are the most promising experimental perspectives on that matter.


I. INTRODUCTION
Neutrino oscillations [1, 2] provide a clear evidence for lepton flavor violation (LFV) in the neutral sector, pointing out to physics beyond the Standard Model (SM).However, no evidence of lepton flavor violating processes in the charged sector has been found despite the great experimental effort on searching for that violation [3,4].Indeed, the experimental searches not only have reached a great sensitivity but will also be improved in the near future by, in some cases, several orders of magnitude.For instance, the MEG collaboration has reported an upper limit on the decay branching ratio for the rare decay µ → eγ around 6 × 10 −13 [5], which will be improved soon by a factor of 10 [6].Concerning the three-body decay µ → 3e, the negative searches for rare decays in the SINDRUM experiment lead to an upper limit for the branching ratio of around 10 −12 [7], whereas the Mu3e experiment collaboration expects to reach the ultimate sensitivity to test such a decay in 10 16 muon decays [8].In addition, the neutrinoless µ-e conversion in muonic atoms is also a promising way to search for charged LFV (CLFV) signals due to the significant increase of sensitivity (up to six orders of magnitude) expected for this class of experiments [9][10][11][12][13][14][15][16].Last but not least, the future plans regarding electron electric dipole moment (eEDM) [17,18] are also in quest for New Physics signals since the expected sensitivity for these facilities will improve by two orders of magnitude the current bound |d e | < 8.7 × 10 −29 e • cm [19].This ambitious experimental program, in turn, calls for a deep phenomenological analysis of the CLFV and EDM signals in models featuring new charged lepton interactions such as those undertaking neutrino masses.
On the other hand, despite the abundant and compelling evidence for the massiveness of neutrinos [20,21], the underlying mechanism behind it remains unknown, which is not a bizarre occurrence since the particle theory responsible for the dark matter (DM) of the Universe also resists to be experimentally elucidated.Hence, it would desirable that both phenomena may have a common origin with a New Physics laying at the electroweak scale, as happens in the radiative neutrino mass models1 involving a dark matter candidate at or below the TeV scale [23][24][25][26][27][28][29].Thus, the resulting model not only would constitute a way out to two of the open questions in the SM but also would have the additional bonus that the new particles may induce potentially large rates for the CLFV and EDM processes, give rise to novel observable phemomena at the LHC, and lead to signals in direct and indirect DM experiments.
In this work we consider the inert Zee model (IZM) -a dark matter realization of the Zee model for neutrino masses [30][31][32] where the above features are present-, with the aim of pursuing a dedicated analysis of LFV processes and EDM signals 2 .As in the Zee model, neutrino masses are generated at one loop, while DM is addressed as in the inert doublet model [41][42][43][44][45][46][47] since both models share a Z 2 -odd scalar SU (2) L doublet.In addition, due to the new interactions, the IZM has a richer phenomenology than the minimal scotogenic model [48].Once we will have determined the viable parameter space consistent with dark matter, neutrino oscillation observables, lepton-flavor violating processes and electroweak precision tests, we will establish the most relevant experimental perspectives regarding LFV searches.Furthermore, since the Yukawa couplings that reproduce the neutrino oscillation data are complex, which in turn constitute new sources of CP violation, we will look into the regions in the parameter space where the prospects for the eEDM are within the future experimental sensitivity [17][18][19].Lastly, we will also study the possible connection, via lepton Yukawa interactions, that may exist between DM and the anomalous magnetic dipole moment of the muon [49][50][51][52][53].
The paper is organized as follows: in Sec.II we present the generalities of the model, including neutrino masses and dark matter.In Sec.III we study the LFV processes, the EDM and the magnetic dipole moment (MDM) of charged leptons.Their phenomenology is presented in Sec.IV.Finally, we conclude in Sec.V.

II. THE MODEL
The new particle content of the model [24,54] consists of two vectorlike fermions, a SU (2) L -singlet and a SU (2) L -doublet Ψ = (N, E) T , and two scalar multiplets, a SU (2) Lsinglet S − and a SU (2) L -doublet H 2 = (H + 2 , H 0 2 ) T .All of them are odd under the Z 2 symmetry, which in turn allows us to avoid Higgs-mediated flavor changing neutral currents at tree-level, forbid tree-level contributions to the neutrino masses and render the lightest Z 2 -odd particle stable [54].It follows that the most general Z 2 -invariant Lagrangian of the model can be written as where L SM is the SM Lagrangian which includes the Higgs potential The interaction terms between the scalars are included in L 1 , where ab is the SU (2) L antisymmetric tensor with 12 = 1, and the scalar couplings λ 5 and µ are assumed real.It is worth mentioning that H 0 2 does not develop a vacuum expectation value in order to ensure the conservation of the Z 2 symmetry.Note that the scalar potential is rather similar to one of the singletdoublet scalar DM model [55][56][57][58] or two Higgs doublet models plus a scalar singlet, see e.g.
Refs. [59,60], with the main difference that our SU (2) L -singlet scalar is electrically charged (which in turn implies that S has also a null vacuum expectation value and that the charged scalars get mixed instead of neutral ones).Finally, L 2 includes the new Yukawa interaction terms3 : where L i and e Ri (i = 1, 2, 3) are the SM leptons, doublets and singlets of SU (2) L , respectively.η i , ρ i and f i are Yukawa couplings controlling the new lepton interactions, while y leads to the mixing among the Z 2 -odd charged fermions.Note that only η i and f i involve neutrino interactions, so they are the ones participating in the neutrino mass generation.
After electroweak symmetry breaking, the Z 2 -odd scalar spectrum consists of a CP-even state H 0 and a CP-odd state A 0 , and two charged states κ 1,2 .Their masses are given by where ).On the other hand, the Z 2 -odd fermion spectrum and a mixing angle given by sin 2α = ( With respect to DM in the IZM, H 0 is the DM candidate4 as long as it remains as the lightest Z 2 -odd particle in the spectrum.Hence, we expect the DM phenomenology to be similar to the one in the inert doublet model (IDM) in scenarios where the particles not belonging to the IDM (κ 1,2 , χ 1,2 and N ) do not participate in the DM annihilation processes [54].Accordingly, the viable DM mass range for this scenario (the same of the one in the IDM) is divided into two regimes [41][42][43][44][45][46][47]: the low mass regime, m H 0 m h /2, and the high mass regime, m H 0 500 GeV.Since in the latter regime the CLFV processes are quite suppressed (the corresponding rates scale as m −4 χω or m −4 κ β ), in our numerical analysis we will only consider the low mass regime.
In this model the neutrino masses are generated at one-loop thanks to the scalar and fermion mixings and to the Yukawa interactions mediated by η i and f i .From Fig. 1, the neutrino mass matrix in the mass eigenstates is given by where ζ = (sin 2α sin 2δ)/(64π Note that for a vanishing scalar or fermion mixing the neutrino masses are zero and that, due to the flavor structure of M ν , the lightest neutrino is massless.This, the masslessness of the lightest neutrino, entails several phenomenological consequences: i) there is only single Majorana CP phase since the second phase can be absorbed by a redefinition of the massless neutrino field; ii) the two remaining neutrino masses are entirely set by the solar and atmospheric mass scales: for normal hierarchy (NH) m 1 = 0, m 2 = ∆m 2 sol and m 3 = ∆m 2 atm , while for inverted hierarchy (IH) m 1 = ∆m 2 atm , m 2 = ∆m 2 sol + m 2 1 ≈ ∆m 2 atm and m 3 = 0; iii) the amplitude for neutrinoless double beta decay [61] presents a lower bound, which for the case of IH lies within the sensitivity of future facilities dedicated for that goal [62].
In Ref. [54] it was shown that, using Eq. ( 9) and the diagonalization condition5 U T M ν U = diag(m 1 , m 2 , m 3 ) with U = V P and P = diag(1, e iφ/2 , 1) [63], it is possible to express five of the six Yukawa couplings η i and f i in terms of the neutrino low energy observables.

III. CHARGED LEPTON PROCESSES
Once lepton flavor violation is allowed via neutrino Majorana masses, LFV processes involving charged leptons such as i → j γ, i → 3 j and µ − e conversion in nuclei are unavoidable.In the IZM model such processes are generated at one-loop level involving the η i , f i and ρ i Yukawa interactions (see Eq. ( 5)), which in turn implies that they are mediated FIG. 2: One-loop diagrams leading to eEDM, anomalous muon MDM, and i → j γ decays when i = j = 1, i = j = 2 and i = j, respectively.Here φ 0 denotes the two Z 2 -odd neutral scalars A 0 and H 0 .by, both charged and neutral, Z 2 -odd fermions and scalars.It is worth mentioning that ρ i may enhance the rates for the CLFV processes and EDMs since it is not subject to the neutrino oscillation constraints (only η i and f i enter in the neutrino mass generation).
The triangle diagrams leading to i → j γ processes are displayed in Fig. 2. The corresponding branching ratios (neglecting lepton masses at final states) are given by where α em is the electromagnetic fine structure constant, G F is the Fermi constant and Σ L , Σ R are given by The loop functions are reported in the Appendix.Note that, in contrast to neutrino masses, these branching ratios are not double suppressed by the mixing of Z 2 -odd particles.This implies that, as is expected, the rare decays i → j γ do not depend on whether neutrino masses are zero or not.On the other hand, note that the IZM has an additional contribution (right diagram) to i → j γ with respect to the minimal scotogenic model [48].Indeed, in 3: One-loop diagrams contributing to i → j ¯ j j decays, where φ 0 denotes the two Z 2 -odd neutral scalars A 0 and H 0 .that diagram precisely enter ρ i , the Yukawa couplings that are not affected by neutrino oscillation constraints.
Concerning the i → j ¯ j j there are two class of diagrams (see Fig. 3) leading to such processes: the γ-and Z-penguin diagrams (top panels) and the box diagrams (bottom panels).There is also a contribution from Higgs-penguin diagrams which, nevertheless, is suppressed for the first two charged leptons generations due to their small Yukawa couplings.The contribution of those processes involving tau leptons is not negligible but the corresponding limits are less restrictive.Therefore, the i → j ¯ j j processes contain four kind of contributions: the photonic monopole, photonic dipole, Z-penguin and boxes.In contrast, the photonic dipole contribution is the only one present in i → j γ processes.
Finally, the µ − e conversion diagrams are obtained when the pair of lepton lines attached to the photon and Z boson in the penguin diagrams (see top panels of Fig. 3) are replaced by a pair of light quark lines 7 .There are no box diagrams since the Z 2 -odd particles do not couple to quarks at tree level.Accordingly, the photonic non-dipole and dipole terms along the Z−penguin are the only terms that contribute to the µ − e conversion processes.
Since the calculations of the i → j ¯ j j and µ − e rates are quite involved, we implement the FlavorKit code [64] to use the corresponding full expressions.
Since the parameters β ij and A j in Eqs. ( 11) and ( 12) are in general complex, it follows that the Yukawa couplings required for neutrino oscillation data constitute new sources of CP violation in the lepton sector 8 .Thus the IZM brings with it new contributions to the EDM of charged leptons with the distinctive feature that the leading contribution arises at one-loop level (see right top diagram of Fig. 2 with i = j = 1), in constrast with the one obtained in the minimal scotogenic model where it arises a two loops [39].Accordingly, large values for the eEDM may be expected.The analytical expression reads where the loop function is given in the Appendix.It is clear then that d i becomes suppressed for small fermion mixing angles.On the other hand, note that the left diagram of Fig. 2 does not contribute to the EDM because the neutral Dirac fermion, N , only has one single chiral coupling to either i or c i , i.e., such a diagram conserves CP.Yukawa interactions also lead to contributions to anomalous magnetic dipole moments of the charged leptons.These contributions can be cast as where with a F i being the contribution involving charged fermions and neutral scalars, and a S i the contribution involving charged scalars and neutral fermions.

IV. RESULTS AND DISCUSSION
In order to obtain the particle spectrum and low energy observables we have used SPheno [72,73] and the FlavorKit [64] of SARAH [74,75], and MicrOMEGAS [76] to calculate the DM relic abundance.The set of free parameters of the model relevant for our analysis has been varied as 9   10 Since we perform the numerical analysis only for the low DM regime, the mass of the DM candidate has been fixed at m H 0 = 60 GeV and the scalar coupling λ L ∼ 3 × 10 −4 , in order to reproduce the DM relic density measurement reported by the Planck collaboration [77] 10 .
We also ensure that the extra coannihilation processes do not modify the expected DM relic 9 The expressions for the relations between some of the scalar potential parameters and the scalar masses are given in Ref. [54]. 10 We have checked that the variation of m H 0 within the allowed mass range does not significantly affect the charged lepton observables.
density and that the S, T and U oblique parameters remain within the 3σ level [78].The LEP II constraints on the masses of the charged Z 2 -odd fermions and scalars [79,80] are also automatically taken into account in the scan definition.The limits from LHC Run I dilepton searches do not further constrain the parameter space under study since those apply for low scalar masses [81,82].However, since the contribution of the Z 2 -odd charged scalars and fermions to the Higgs diphoton decay may induce large deviations from the LHC Run 2 measurement [83,84], we discard those benchmark points that deviate beyond 2σ from the central value reported by the CMS and ATLAS collaborations.Regarding the low energy neutrino parameters we consider both normal and inverted hierarchies for the neutrino mass spectrum and take the current best fit values reported in Ref. [21].This, along with the scan values, allows us to calculate the set of remaining Yukawa couplings through Eq. (10).
For this set of couplings and for the fermion mixing parameter we assume the following  [86].Note that the strongest limit is set for the eEDM, and for that reason we do not display the muon and tau EDM results 11 .In regard to future eEDM searches, it is worthwhile mentioning that the ACME collaboration will increase by two orders of magnitude the current bound [19,87].
Lastly, the parameters entering in the scalar potential are subject to the following perturbativity and vacuum stability constraints 12 : Furthermore, we also impose the upper limits set by the perturbative unitarity of the S- matrix [58,60,88], Here Λ i correspond to the three real eigenvalues of the matrix In Fig. 5 we show the correlation between the B(µ → eγ), B(µ → 3e) and R µe observables, and the impact of the future searches associated to these observables over the parameter space considered.The correlation between the three observables is remarkable in a large portion of the parameter space.It follows that a large fraction of the current viable parameter space will be tested in the future experiments, with all the electron-muon observables being within the reach of the MEG and Mu3e experiments, and with the R µe observable being the most promising.
The results for the eEDM as a function of the Yukawa coupling product |ρ 1 η 1 | are displayed in Fig. 6.Our results show that eEDM future searches [19] (dashed lines) may test regions beyond the reach of experimental sensitivity of CLFV searches.Indeed, the most recent result on the eEDM from ACME collaboration [89] has begun to test those regions (solid lines in Fig. 6).It turns out that these regions are precisely formed by benchmark points (for both neutrino mass spectrum) with B(µ → 3e)/B(µ → eγ) > 1 and R µe (Ti)/B(µ → eγ) 10, that is, where the dipole contribution is not the dominant one.This means that, if it is experimentally observed that the above ratios are below the mentioned bounds, the parameter space we are considering would be disfavoured.
Finally, the contribution to the anomalous magnetic moment of the charged leptons are not so relevant since they are below the current bounds [20].For completeness purposes we display in Fig. 7 the results for the anomalous magnetic moment of the muon.In particular, h being the Higgs boson and v = 246 GeV.L F and L S comprise, respectively, the kinetic and mass terms for the new fermions, and the kinetic, mass and self-interacting terms of the new scalars,

)FIG. 6 :
FIG. 6: Expected values for the eEDM as a function of |η 1 ρ 1 | for NH (left panel) and IH (right panel).The red points constitute the current viable parameter space while the blue points are beyond the reach of future CLFV searches.