The Inert Zee Model

We study a realization of the topology of the Zee model for the generation of neutrino masses at one-loop with a minimal set of vector-like fermions. After imposing an exact $Z_2$ symmetry to avoid tree-level Higgs-mediated flavor changing neutral currents, one dark matter candidate is obtained from the subjacent inert doublet model, but with the presence of new co-annihilating particles. We show that the model is consistent with the constraints coming from lepton flavor violation processes, oblique parameters, dark matter and neutrino oscillation data.


II. THE MODEL
We start as in [7] by extending the SM with a second Higgs doublet, H 2 , and a charged SU(2)-singlet, S + . Within this setup, Majorana neutrino masses are generated at one-loop. In this way, the Zee model is realized in the context of the general THDM-III with tree-level HMFCNC. In the model, ten new couplings are directly related to the neutrino sector. In particular, the analysis in terms of THDM-III basis independent parameters [17] was done in [18], with further analysis in [19,20].
To avoid HMFCNC at tree-level, in the Zee-Wolfenstein model [8] was proposed the usual Z 2 symmetry in which the two doublets have opposite parity, like in Type-I or Type-II or other THDM realizations [6]. Under this symmetry, the Lagrangian relevant for the neutrino mass generation requires S ± to be Z 2 -even, and hence a Z 2 soft-breaking mass parameter needs to be introduced in the scalar sector, which, in joint with the three antisymmetric Yukawa couplings of S ± with the lepton doublets of different families, account for only four new couplings directly related to the neutrino sector. This minimal model, however, turns to be not enough to fit the observables related to neutrino oscillation data and is now excluded [9].
In this work, we want to explore the minimal realization of the T1-ii topology of [2], which is safe regarding strongly constrained lepton-flavor violation, in particular, without tree-level HMFCNC. We start by assigning a Z 2 -odd charge to both S ± and the second Higgs doublet H 2 . At this level, the resulting model would be a Type-I THDM with an extra S ± and massless neutrinos. After that, we propose one minimal extension of this setup that only involves six additional Yukawa-couplings related to neutrino physics (instead of the nine of the general Zee model without the Z 2 ). This consists of adding a Z 2 -odd pair of VL fermions: a SU (2) L -singlet, , and a doublet, Ψ. However, the Z 2 symmetry is not enough to avoid mixing of the new VL fermions with the SM leptons which could regenerate tree-level HMFCNC, as well as other lepton flavor violating processes subject to several (stringent) constraints [21][22][23][24][25]. Therefore, we impose in addition that the neutral part of H 2 does not develop a vacuum expectation value (vev). In this way, the IDM is obtained, which includes a potential scalar DM candidate. To our knowledge, the model was first proposed in the catalog of the realization of the d = 5 Weinberg operator at one-loop with DM candidates [5] and labeled there as T1-ii-A model with α = −2.
The new particle content and their charges are summarized in the Table I. A similar approach with controlled FCNC and DM was followed in [26] where the minimal supersymmetric standard model was extended with two SU(2)-singlet opposite-charge superfields.
A. The scalar sector The most general Z 2 -invariant scalar potential of the model is given by where ab is the SU (2) L antisymmetric tensor with 12 = 1, H 1 = (0, H 0 1 ) T is the SM Higgs doublet and H 2 = (H + 2 , H 0 2 ) T . The scalar couplings λ 5 and µ are taken to be real. After the electroweak symmetry breaking, the neutral scalar fields can be parametrized in the form H 0 2 = (H 0 + iA 0 )/ √ 2 and H 0 1 = (h + v)/ √ 2, with h being the Higgs boson and v = 246 GeV. Note that H 0 2 does not develop a vacuum expectation value in order to ensure the conservation of the Z 2 symmetry. The neutral scalar spectrum coincides with the one of the IDM [10,27,28], which consists of two CP-even neutral states (H 0 , h) and a CP-odd neutral state (A 0 ). The masses of the Z 2 -odd neutral The charged mass eigenstates χ 1 and χ 2 are defined by with masses The Z 2 -odd fermion spectrum also contains a neutral Dirac fermion N , with a mass m N = m Ψ . From above expression, it follows that m N = m χ1 cos 2 α + m χ2 sin 2 α, which implies the hierarchical spectrum m χ1 ≤ m N ≤ m χ2 . In other words, the neutral fermion N can not be the lightest Z 2 -odd particle in the spectrum.

C. Neutrino masses
The usual lepton number (L) assignment in the Zee model corresponds to L(H 2 ) = 0 and L(S) = −2, which makes the µ term in the scalar potential the only explicit L-violating term in the Lagrangian. Hence, by keeping such assignment and charging under L the new fermion fields as L(Ψ) = L( ) = +1, in order to make the Yukawa interactions L conserving, the µ term is again the responsible for the L breaking in the model, and the subsequent neutrino Majorana masses and lepton flavor violation processes. Non-zero neutrino masses at one-loop are generated in this model thanks to the combination of the Yukawa-coupling η i and f i , the scalar mixing µ, and fermion mixing Π, as displayed in the left-panel of Fig. 1. The corresponding Majorana mass-matrix in the mass-eigenstate basis, calculated from the Feynman diagram displayed in the right-panel of Fig. 1, takes the form Here c 1 = −1, c 2 = +1 and the loop function is given by Due to the flavor structure of M ν , it has a zero determinant and, therefore, contains only two massive neutrinos. In this way, the number of Majorana phases is reduced to only one, and neutrinos masses are entirely set by the solar and atmospheric mass differences. Specifically This means that barring cancellations in the mass sector, and between Yukawa-couplings, small mixing angles and Yukawa-couplings are required. Certainly large values for the Yukawa-couplings can be obtained for smaller values of sin 2α sin 2δ or more compressed mass spectra. The neutrino mass matrix is diagonalized by the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix U PMNS [34] as which can be written in the form U PMNS = V P [35], where the matrix V contains the neutrino mixing angles and the CP Dirac phase and P = diag(1, e iα/2 , 1) carries the dependence on the CP Majorana phase. It is worth mentioning that for α = 0, ±π, ±2π, the Majorana phase does not contribute to the CP violation and in such a case the relative CP-parity of the two massive neutrinos would be λ = e ±iα = ±1. From Eq. (15) and thanks to the flavor structure of the neutrino mass matrix, given by Eq. (12), we can express five of the six Yukawa-couplings η i and f i in terms of the neutrino observables. Without loss of generality η 1 can be chosen to be the free parameter which can be restricted using other low energy observables such as µ → eγ. Thus, the most general Yuwawa-couplings that are compatible with the neutrino oscillation data are given by where we have defined In this way, it is always possible to correctly reproduce the neutrino oscillation parameters in the present model. Note that, in general, the non-free Yukawa-couplings are complex numbers. However, they become real in a CP-conserving scenario with λ = −1 and η 1 being real.

D. Dark Matter
The Z 2 symmetry renders the lightest Z 2 -odd particle stable, and if it is electrically neutral then it can play the role of the DM particle. Since m χ1 ≤ m N , doublet fermion DM can not take place in this model 4 . Therefore, only the neutral Z 2 -odd scalars, either H 0 or A 0 , can be the DM candidates. This makes this model to resemble up to some extent the IDM from the DM phenomenology point of view. Accordingly, two possible scenarios emerge depending on whether the particles not belonging to the IDM (S ± , χ 1,2 and N ) participate or not in the DM annihilation. When these particles do take part of DM annihilation, the extra (not present in the IDM) coannihilation processes are the ones mediated by the Yukawa-couplings η i , f i and ρ i , and by the scalar couplings µ and λ 6 .
For the scenario without the extra coannihilation processes, the DM phenomenology is expected to be similar to that of the IDM by assuming m κ + 2 , m χ1 m κ + 1 , a small scalar mixing angle and η i , f i , ρ i , λ 6 1. In addition, 1 must also be satisfied. In this way, the coannihilation effects of the mentioned particles with the DM particle can be neglected. Note that the requirement of having small Yukawa-couplings is also in agreement with neutrino masses and µ → eγ as it will be shown below. It follows that the viable DM mass range for this scenario (the same of the one in the IDM) is composed by two regions [27,28,33,[36][37][38][39] 5 : the low mass regime, m H 0 m h /2, and the high mass regime, m H 0 500 GeV. In the region 100 GeV m H 0 < 500 GeV the gauge interactions become large so that it is not possible to reach the observed relic density, i.e. Ω H 0 < Ω DM . In the Higgs funnel region, DM self-annihilations through the Higgs s-channel exchange provide the dominant contribution to the DM annihilation cross section, with λ L and m H 0 as the relevant parameters. LEP measurements give rise to the following constraints: GeV. On the other hand, 4 Furthermore, since N has a direct coupling to the Z gauge boson which gives rise to a spin-independent cross section orders of magnitude larger than present limits, it is excluded as a viable DM candidate. 5 Without loss of generality we assume H 0 to be the DM candidate.
for DM masses larger than 500 GeV the relic abundance strongly depends on the mass splittings between H 0 , A 0 and κ ± 1 . Indeed, a small splitting of at most 15 GeV is required to reproduce the correct relic density implying that coannihilations between those particles must be taken into account.
Regarding the scenario where S ± , χ 1,2 and N contribute to the DM annihilation, the extra coannihilation processes involve the following initial states: These processes might play the main role in the calculation of the DM relic density affecting in a sensible way the expectations for DM detection [40][41][42] and, therefore, modifying the viable parameter space of the model. Since a detailed analysis of the impact of these extra coannihilation channels on the relic density is beyond the scope of this work, in what follows we will no longer consider this scenario.

A. Electroweak precision tests
In the present model, the new fields may modify the vacuum polarization of gauge bosons whose effects are parametrized by the S, T and U electroweak parameters [43]. The new fermion (S F , T F ) and scalar (S S , T S ) contributions to the S and T parameters are [44][45][46] 6 : where c α = cos α, s α = sin α, c δ = cos δ, s δ = sin δ and the loop functions Θ are given in the Appendix B. From these expressions we can see that the fermion contributions to T F and S F vanish in the limiting case of α = 0, which points out to the existence of a custodial symmetry. For that reason we do not expect large deviations on S and T for a small mixing angle α. In contrast, the scalar contributions do not tend to zero for δ = 0 due to the fact that after the electroweak symmetry breaking the components of the Z 2 -odd doublet H 2 have mass splittings that are independent of δ. However, the agreement with electroweak precision tests is reached due to the small mass splitting between A 0 and κ ± 1 (H 0 , A 0 and κ ± 1 ) in the low (high) mass regime, just as it happens in the IDM.

B. µ → eγ
Lepton flavor violation processes could be a clear signal of new physics. However, due to the lack of any signal in this sector, very stringent constraints over the branching ratios for particular processes are set, with µ → eγ being one of the most constraining processes. In this model such a process is controlled by the η 1,2 , f 1,2 and ρ 1,2 Yukawa-couplings and mediated by the Z 2 -odd particles. Certainly, the interactions in Eq. (8) and the scalar mixing term allow to construct the one-loop diagram shown in Fig. 2. The branching ratio for µ → eγ process reads 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 presented in the Appendix C. Note that, due to the equation (16), the couplings η 2 , η 3 , f 1 , f 2 , f 3 are related with η 1 , hence, the only free Yukawa parameters entering in the expression for B(µ → eγ) are η 1 , ρ 1 , and ρ 2 .

IV. NUMERICAL RESULTS AND DISCUSSION
In order to illustrate the compatibility of the model with the experimental constraints, we consider the scenario without the extra annihilation channels discussed on section II. Furthermore, we set H 0 to be the DM candidate and assume a small mixing angle δ and the mass spectrum with the lightest charged scalar κ ± 1 mainly doublet 7 . For the low mass regime and without lose of generality we assume m κ + 1 , m A 0 > 100 GeV and |δ| 0.2, which implies that the remaining Z 2 -odd fields do not alter the DM phenomenology expected for the IDM in that regime. On the other hand, to quantitatively assess up to what extent the presence of the new fermion fields and κ ± 2 could affect the expected phenomenology in the high mass regime, through the opening of new (co-)annihilation channels, we have calculated the DM relic density through micrOMEGAs [48] via FeynRules [49] and make a scan (to be described below) over the free parameters of the model. For this purpose, we have set λ 2 , λ S and all the Yukawa-couplings to 10 −2 . The numerical result confirms the preliminary expectations: when m κ + 2 /m κ + 1 1.1, |δ| 0.2 and |µ|/v 10 −1 the new (co-)annihilations channels compared with those present in the IDM do not play a significant role in the determination of DM relic density.
Regarding the electroweak precision test, we have performed a numerical analysis for the two DM mass regimes mentioned above. For the high mass regime, we have considered the following ranges for the free parameters: 7 It is worth mentioning that when the lightest state κ ± 1 is mainly singlet, the relic density cannot be obtained without considering the coannihilation processes with κ ± 2 unless that m κ ± 1 300 GeV and m H 0 m h /2, in which case the relic density is independent of m κ ± 2 .
The scalar and fermion contributions to S and T are shown in Fig. 3, where the constraints coming from the DM phenomenology mentioned above have been taken into account. The black, blue and green ellipses represent the experimental constraints at 68% CL, 95% CL and 99% CL, respectively [50] 8 . It is worth to mention that contrary to the IDM, in our model the S and T parameters are not negligible in the high mass regime because the fermion contributions are already present. However, the constraints are easily satisfied for a small fermion mixing angle |α| 0.2 (red points in the left-panel). On the other hand, by allowing arbitrary values for the mixing angle, α, the contributions to S and T are kept within the 2σ level as long as m χ2 − m χ1 400 GeV (red points in the right-panel). Regarding the low mass regime we have varied the free parameters as follows: 60 GeV < m H 0 < 80 GeV, 100 GeV < m A 0 , m κ + 1 < 1000 GeV, m κ + 1 < m κ + 2 < m χ1 < 1000 GeV, and the same ranges in the Eq. (27) for the mixing angles and scalar couplings. The fermion contributions to S and T are satisfied by imposing either |α| 0.1 or m χ2 − m χ1 200 GeV. In this case, the scalar contributions are not kept within the 2σ level by just imposing the DM phenomenology of the IDM. This occurs because in the low mass regime there is always a non-negligible mass splitting between the DM particle and the other scalars. Fig. 4 shows the allowed values for the masses m A 0 and m κ + 1 that satisfy the S, T parameters at 68% CL (red points), 95% CL (green points) and 99% CL (blue points) respectively. We have taken |α| 0.1 in order to suppress the fermion contribution. Note that if m A 0 is increased, m κ + 1 will have to be increased. However, from the unitary constraints given in Eq. (6) an upper limit is obtained on the scalar masses, which leads to that they should be nearly degenerate at 800 GeV.
Concerning to the LFV constraints, we have focused on the current strongest bound, which is provided by µ → eγ process. We have made a scan over the free parameters of the model for the CP-conserving scenario (the CP Dirac phase is fixed to zero) with a normal hierarchy and choosing λ = −1. For this purpose, we have varied the free parameters within the ranges given in Eq. (27), in addition to η 1 , ρ 1 , ρ 2 , ∈ [10 −4 , 1]. The results are shown in Fig. 5. All the points satisfy the current bound [51] and only a minority will be probed by future searches [52]. We have taken |α| 0.1, m κ + 2 /m κ + 1 1.1, |δ| 0.2 and |µ|/v 10 −1 in order to satisfy the oblique parameters and preserve the DM phenomenology expected for the IDM. Note that the B(µ → eγ) limit can be easily satisfied imposing ρ 1 ρ 2 4 × 10 −2 and η 1 5 × 10 −2 . On the other hand, for the low mass regime we obtain similar results to those in the high mass regime. Remember that, in order to satisfy the oblique parameter we need to impose small mixing angles as well as a nearly degenerate masses between A 0 and κ + 1 . Finally, we turn the discussion to collider searches. The high-mass region of the IDM is quite difficult to probe at the LHC. However, the low mass region can be probed by searching for dilepton plus missing transverse energy signal [53][54][55] and trilepton plus missing transverse energy signal [56,57] with a sensitivity in the parameter region with κ + 1 , A 0 100-180 GeV. A similar sensitivity could be expected for κ + 2 . Concerning VL fermions, the searches performed at LEPII impose a limit of m χ1 > 100 GeV [58]. At the LHC, the larger exclusion for VL fermion is expected for large mass splittings, 100% branching ratios to electron or muons, and higher fermions SU (2) L representations. In our case, it corresponds to a higgsino-like VL fermion production without final state taus. For example, if a higgsino-like charged fermion is the next to lightest Z 2 -odd particle and choosing the Yukawa-couplings such that we have a dilepton plus missing transverse energy signal from Since the cuts for this kind of signal at the LHC (in both ATLAS and CMS) do not depend in angular distributions between the final states, the corresponding excluded cross sections are insensitive to the spin of the produced particles. Currently, they are interpreted in terms of slepton pair production. A recast of the excluded cross section for slepton pair production pp →l +l− → l + l −χ0 1χ 0 1 , studied in Ref. [59] 9 , allows to exclude higgsino-like charged fermions up to 510 GeV [15]. 9 Where the lightest neutralino,χ 0 1 , is the dark matter candidate.
Conversely, in the case of χ ± 1 nearly degenerate with H 0 (compressed spectra), the bounds on m χ1 are ∼ 100 GeV for ∆m = m χ1 − m H 0 < 50 GeV [60] 10 . If, in addition, the Yukawa-couplings are such that then B χ ± → τ H 0 ≈ 1, and the exclusion limits are worse due to the larger τ misidentification rates. Recently, an extended analysis of the LHC Run-I data have been presented by ATLAS [61] with new searches for compressed spectra and final state taus. In particular, by using multivariate analysis techniques, the 95% excluded cross section for pp →τ + 1χ 0 1 is given for several neutralino masses. As expected, and in contrast to the selectron and smuon pair production, there is no sensitivity to left-or right-stau pair production. By using the same strategy than in [62], we focus in the excluded cross section plot presented in Fig. 12 of Ref. [61] for a DM particle of 60 GeV, since it is a representative value in the case of the IDM to account for the proper relic density. Because of the larger cross section for pair produced higgsinos decaying into two taus plus missing transverse energy, we are able to exclude higgsino-like charged fermions in the range 115 < m χ + 1 /GeV < 180 by using the theoretical cross section calculated to next-to-leading order in [15].
Another attempt to circumvent both problems have been made recently in Ref. [60] of the CMS collaboration, by implementing the vector boson fusion topology to pair produce electroweakinos [63]. There, supersymmetric models with bino-like χ 0 1 and wino-like χ 0 2 and χ ± 1 are considered in the presence of a light stau. Assuming B χ ± 1 → ν τ ± → ντ ± χ 0 1 = 1 and B χ 0 2 → τ ± τ ∓ → τ ± τ ∓ χ 0 1 = 1, they are able to find some supersymmetric scenarios where the LEPII constraint can be improved. We could expect that a similar analysis for the higgsino-like charged VL fermion may allow to close the previous gap until around 115 GeV. A detailed recast of this CMS analysis, will be done elsewhere. In summary, we expect an exclusion for the higgsino-like charged VL fermions of the model around 180 GeV. On the other hand, searches in the di-tau plus missing transverse energy signature have been studied in Ref. [64]. There, it was shown that the high luminosity LHC of 3000 fb −1 can exclude SU (2) L -singlet charged VL fermion up to m χ1 ∼ 450 GeV.

V. CONCLUSIONS
We have considered an extension of the Zee model which involves two vector-like leptons, a doublet and a singlet of SU (2) L and the imposition of an exact Z 2 symmetry. This symmetry, under which all the non-Standard Model fields are odd, avoids tree-level Higgs-mediated flavor changing neutral currents and ensures the stability of the lightest neutral component inside the second scalar doublet and, therefore, allowing to have a viable dark matter candidate. We have shown that under some conditions the well-known DM phenomenology of the IDM is recovered. As in the Zee model, neutrino masses are generated at one loop, leading to either a normal mass hierarchy or a inverted mass hierarchy. However, due to the flavor structure of the neutrino mass matrix, one neutrino remains massless. Moreover, such a flavor structure always allows to reproduce the correct neutrino oscillation parameters and to have only four free Yukawa-couplings (of a total of nine), which can be constrained using the µ → eγ lepton flavor violation process. In particular, we have found that ρ 1 ρ 2 10 −2 and η 1 10 −2 in order to fulfill that constraint. On the other hand, the oblique parameters impose |α| 0.2 and m χ2 − m χ1 400 GeV for the high mass regime while |α| 0.1 and m χ2 − m χ1 200 GeV for the low mass regime. Finally, we argued that in general, the collider limits for vector-like leptons are not so far from the limit imposed by LEPII.