The inert Zee model

Journal of High Energy Physics, Mar 2016

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.

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The inert Zee model

HJE The inert Zee model Robinson Longas 0 1 Dilia Portillo 0 1 Diego Restrepo 0 1 Oscar Zapata 0 1 0 Calle 70 No. 52-21, Medell n , Colombia 1 Instituto de F sica, Universidad de Antioquia 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 Z2 symmetry to avoid tree-level Higgs-mediated avor 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 avor violation processes, oblique parameters, dark matter and neutrino oscillation data. Beyond Standard Model; Dark Matter and Double Beta Decay 1 Introduction 2 The model 3 ! e The scalar sector Neutrino masses Dark matter Electroweak precision tests Numerical results and discussion Conclusions Yukawa interactions and the Z2-odd fermion sector A Free parameters B ST formulae C Loop function in the ! e 1 Introduction Neutrino masses and dark matter (DM) represent two phenomenological pieces of evidence for physics beyond the Standard Model (SM) which are solidly supported by the experimental data. If neutrino masses arise radiatively [1{3] it may be, though, that both originate from new physics at the TeV scale, and they are related to each other. In this direction, models with one-loop radiative neutrino masses and viable dark matter candidates have now a complete classi cation given in [4, 5]. There, the new elds are odd under a Z2 symmetry which ensures the stability of the DM particle, while the SM particles are even. In this work, we explore a particular model where the Z2 can be identi ed with the symmetry used to avoid tree-level Higgs-mediated avor changing neutral currents (HMFCNC) in the two Higgs doublet models (THDM) [6]. More concretely, we consider the realization of the d = 5 Weinberg operator at one-loop order [1, 2] with the topology labeled as T1-ii in [2] from which the Zee model [7] is the most straightforward realization. In the Zee model, the THDM-III with tree-level HMFCNC is extended with one extra SU(2)-singlet charged-scalar. The minimal realization with two Higgs doublets of opposite parity under a Z2 symmetry to avoid tree-level HMFCNC [8], gives rise to a diagonal-zero neutrino { 1 { the neutral fermion cannot be the lightest Z2-odd particle, and therefore, the DM candidate is still contained in the IDM sector of the model. In our setup, the imposed Z2 guarantees the absence of strongly constrained avor violating processes, relating one-loop neutrino masses with dark matter through new physics at TeV scale that can be tested at the LHC. Another example of this kind of relation arises in the well known scotogenic models. There, the SM is increased with at least two singlet [11] or triplet [12, 13] fermions and one scalar doublet which are odd under a Z2 symmetry. In another realization, the roles are interchanged with at least two scalar singlets and one VL doublet fermion, while one additional fermion singlet is required to close the neutrino mass loop [14, 15]. The role of the Z2 in the scotogenic models is to forbid tree-level contributions to the neutrino masses which are generated at one-loop level. In these models, the lightest odd particle (either scalar or fermion) can be a good DM candidate. One shared feature with the model presented here is that all the new states beyond the SM are odd under the imposed Z2. Under this assumption, and considering new fermion and scalar elds transforming as singlets, doublets or triplets of SU(2), a set of 35 non-equivalent models that can simultaneously account for DM and neutrino masses at one-loop was obtained in ref. [5].1 The model presented here is cataloged there as the T1-ii-A model with = 2. This paper is organized as follows. In section 2, we present the model with its particle content and calculate the neutrino masses. Then, we analyze the DM phenomenology and establish the requirements over the free parameters necessary to reproduce the IDM phenomenology. In section 3, we study the constraints coming from oblique parameters and present the expression for the rate of the ! e process. In section 4, we present the numerical results and discuss the collider limits on VL fermions. Finally, the conclusions are presented in section 5. In the appendices, we collect the loop functions for the calculation of the oblique parameters and the ! e process. 2 The model We start as in [7] by extending the SM with a second Higgs doublet, H2, 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 Z2 symmetry in which the two doublets have opposite parity, like in Type-I or Type1A comprehensive list of the radiative seesaw literature is given in [16]. { 2 { H2 S Spin 1/2 1/2 0 0 SU(3)C ; SU(2)L; U( 1 )Y ; Z2 to be Z2-even, and hence a Z2 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 di erent families, account for only four new couplings directly related to the neutrino sector. This minimal model, however, turns to be not enough to t 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- avor violation, in particular, without tree-level HMFCNC. We start by assigning a Z2-odd charge to both S and the second Higgs doublet H2. At this level, the resulting model would be a Type-I THDM with an extra 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 Z2). This consists of adding a Z2-odd pair of VL fermions: a SU(2)L-singlet, , and a doublet, . However, the Z2 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 avor violating processes subject to several (stringent) constraints [21{25]. Therefore, we impose in addition that the neutral part of H2 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 rst 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 = 2. The new particle content and their charges are summarized in the table 1. 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 super elds. 2.1 The scalar sector The most general Z2-invariant scalar potential of the model is given by HJEP03(216) V = 21H1yH1 + 22H2yH2 + 1 (H1yH1)2 + 2 (H2yH2)2 + 3(H1yH1)(H2yH2) (2.1) 2 2 + 4(H1yH2)(H2yH1) + 5 h(H1yH2)2 + h:c:i + 2SS+S + S(S+S )2 + 6(S+S )H1yH1 + 7(S+S )(H2yH2) + ab hH1aH2bS + h:c:i ; 2 { 3 { doublet and H2 = (H2+; H20)T. The scalar couplings 5 and where ab is the SU(2)L antisymmetric tensor with 12 = 1, H1 = (0; H10)T is the SM Higgs are taken to be real. After the electroweak symmetry breaking, the neutral scalar elds can be parametrized in the form H20 = (H0 + iA0)=p2 and H10 = (h + v)=p2, with h being the Higgs boson and v = 246 GeV. Note that H20 does not develop a vacuum expectation value in order to ensure the conservation of the Z2 symmetry. The neutral scalar spectrum coincides with the one of the IDM [10, 27, 28], which consists of two CP-even neutral states (H0; h) and a CP-odd neutral state (A0). The masses of the Z2-odd neutral scalar particles read as On the other hand, the charged scalar sector involves a mixture of the singlet and doublet Z2-odd charged states which leads to the following mass matrix in the basis (H2 ; S ) 1 2 m2H0 = p v 2 p 2 m2 A ; S = ! 1 2 H2 S ! = cos sin sin cos m2 1;2 = 1 2 m2H + m2S ! ; 1 2 q sin 2 = p 2 v m2+ 2 m2+ ; 1 m2H m2 S 2 + 2 2v2 ; where m2H de ned as with the corresponding masses particles. following set (2.3) (2.4) (2.5) (2.7) = 2S + 12 6v2. The mass eigenstates 1 and With regard to the free parameters in the scalar sector, it is possible to choose the mH0 ; mA0 ; m + ; m + ; L; 6; ; 1 2 where L = 12 ( 3 + 4 + 5) controls the trilinear coupling between the SM Higgs and H0. Because the quartic couplings 2 , S and 7 are only relevant for interactions exclusively involving Z2-odd particles, they can be left apart in a tree-level analysis.2 The 2Note that at one-loop level 2 and 7 may play a main role in processes such as the DM annihilations into and Z [30, 31], DM scattering on nucleons [32] and other radiative processes [33]. with constrained from above by the requirement of having m2+ > 0. Lastly, the scalar couplings are subject to perturbativity and vacuum stability con1 straints, which imply the following conditions [10, 29]: 21 < 0; These theoretical conditions constrain the mass splittings among the Z2-odd scalar relation between the remaining scalar couplings and the scalar masses are presented in the appendix A. From eqs. (2.2) and (2.5), we can expect that for appropriate scalar couplings, H0 or A0 can be the lightest Z2-odd scalar particle in the scalar spectrum. 2.2 Yukawa interactions and the Z2-odd fermion sector The Z2-invariant Lagrangian respecting the SM gauge symmetry contains the following new terms L iLiH2 + i H2eRi + H1 + fi Lic S+ + h:c + m where Li and eRi are the lepton doublets and SU(2)-singlets respectively, the VL doublet, , i , i and fi are Yukawa-couplings controlling the new interactions, and i is the family index. As it will be shown below, the i, fi terms with the mixing terms and give rise to nonzero neutrino masses at one loop level, and along with the i term, induce lepton avor violation (LFV) processes such as ! e . Once the electroweak symmetry is spontaneously broken the term generates a mixture of the two charged Z2-odd fermions, leading to a mass matrix in the basis (E; ) given by3 + m ; (2.8) = (N; E)T is HJEP03(216) 0 m p v 2 p m The charged mass eigenstates 1 and 2 are de ned by with masses E ! = cos sin sin cos ! 1 2 m 1;2 = m + m 1 2 ! ; q (m sin 2 = p 2 v m 2 m 1 ; m )2 + 2 2v2 : (2.9) (2.10) (2.11) The Z2-odd fermion spectrum also contains a neutral Dirac fermion N , with a mass mN = m . From above expression, it follows that mN = m 1 cos2 + m 2 sin2 , which implies the hierarchical spectrum m 1 mN m 2 . In other words, the neutral fermion N can not be the lightest Z2-odd particle in the spectrum. 2.3 Neutrino masses The usual lepton number (L) assignment in the Zee model corresponds to L(H2) = 0 and L(S) = 2, which makes the 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 elds as L( ) = L( ) = +1, in order to make the Yukawa interactions L conserving, the term is again the responsible for the L breaking in the model, and the subsequent neutrino Majorana masses and lepton avor violation processes. 3For simplicity we have assumed to be real. { 5 { νC L νC L νL κ±1,2 χ1,2, χ1C,2 bination of the Yukawa-coupling i and fi, the scalar mixing , and fermion mixing , as displayed in the left-panel of gure 1. The corresponding Majorana mass-matrix in the mass-eigenstate basis, calculated from the Feynman diagram displayed in the right-panel of gure 1, takes the form [M ]ij = 1, c2 = +1 and the loop function is given by n m2 b mc2 m2a q I m2a; mb2; mc2 = m2 b m2 b mc2 ln m2a mc2 ln m2a mc2 : Due to the avor structure of M , 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 di erences. Speci cally, for normal hierarchy (NH) m1 = 0, m2 = for inverted hierarchy (IH) m1 = p q m2atm, m2 = ms2ol + m21 p m2atm and m3 = 0. ms2ol and m3 = p m2atm while On the other hand, M depends on the scalar and fermion mixing angles with vanisihing 1 2 entries for either m + = m + , or m 1 = m 2 . Thus, to have small neutrino masses a degenerate mass spectrum up to some extent could be required. By taking the trace of M we can estimate the values of the di erent quantities involved in the calculation of neutrino masses: mixing matrix UPMNS [34] as 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 2 sin 2 or more compressed The neutrino mass matrix is diagonalized by the Pontecorvo-Maki-Nakagawa-Sakata UPTMNSM UPMNS = diag(m1; m2; m3); mi 0; (2.15) { 6 { (2.12) (2.13) which can be written in the form UPMNS = V P [35], where the matrix V contains the neutrino mixing angles and the CP Dirac phase and P = diag(1; ei =2; 1) carries the dependence on the CP Majorana phase. It is worth mentioning that for = 0; ; 2 , the Majorana phase does not contribute to the CP violation and in such a case the relative CPparity of the two massive neutrinos would be = e i = 1. From eq. (2.15) and thanks to the avor structure of the neutrino mass matrix, given by eq. (2.12), we can express ve of the six Yukawa-couplings i and fi in terms of the neutrino observables. Without loss of generality 1 can be chosen to be the free parameter which can be restricted using other low energy observables such as ! e . Thus, the most general Yuwawa-couplings that are compatible with the neutrino oscillation data are given by A3= 11 0 1 11= 1 33= 3 1 33=A3 ij = Aj = Aj = q q m2Vi2Vj2 + m3Vi3Vj3; ij = m1Vi1Vj1 + m2Vi2Vj2; m2m3(V12Vj3 V13Vj2)2 + 1j eiArg( 1 ); for NH; m1m2(V11Vj2 V12Vj1)2 + 1j eiArg( 1 ); for IH; = The Z2 symmetry renders the lightest Z2-odd particle stable, and if it is electrically neutral then it can play the role of the DM particle. Since m 1 mN , doublet fermion DM can not take place in this model.4 Therefore, only the neutral Z2-odd scalars, either H0 or A0, 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 , 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 i, fi and i, and by the scalar couplings and 6 . 4Furthermore, 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. { 7 { For the scenario without the extra coannihilation processes, the DM phenomenology is expected to be similar to that of the IDM by assuming m + ; m 1 m + , a small scalar 1. In addition, =v = sin 2 (2m2+ also be satis ed. In this way, the coannihilation e ects of the mentioned particles with the DM particle can be neglected. Note that the requirement of having small Yukawacouplings is also in agreement with neutrino masses and ! e 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{39]:5 the low mass regime, mH0 ' mh=2, and the high mass regime, mH0 & 500 GeV. In the region 100 GeV . mH0 < 500 GeV the gauge interactions become large so that it is not possible to reach the observed relic density, i.e. H0 < DM . In the Higgs funnel region, DM self-annihilations through the Higgs schannel exchange provide the dominant contribution to the DM annihilation cross section, with L and mH0 as the relevant parameters. LEP measurements give rise to the following constraints: mH0 + mA0 > MZ , max(mH0 ; mA0 ) > 100 GeV and m + & 70 GeV. On the other hand, for DM masses larger than 500 GeV the relic abundance strongly depends on 1 the mass splittings between H0; A0 and 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 , 1;2 and N contribute to the DM annihilation, the extra coannihilation processes involve the following initial states: H20 i, H20 i , i j . These processes might play the main role in the calculation of the DM relic density a ecting in a sensible way the expectations for DM detection [40{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. 3 3.1 Constraints Electroweak precision tests In the present model, the new elds may modify the vacuum polarization of gauge bosons whose e ects are parametrized by the S, T and U electroweak parameters [43]. The new fermion (SF , TF ) and scalar (SS, TS) contributions to the S and T parameters are [44{46]:6 SF = TF = 1 " 3 5Without loss of generality we assume H0 to be the DM candidate. 6Because the U parameter is suppressed by the new physics scale U (MW = )2 T , we do not take it into account [47]. { 8 { where c = cos , s = sin , c = cos , s = sin and the loop functions are given in the appendix B. From these expressions we can see that the fermion contributions to TF and SF vanish in the limiting case of = 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 . In contrast, the scalar contributions do not tend to zero for = 0 due to the fact that after the electroweak symmetry breaking the components of the Z2-odd doublet H2 have mass splittings that are independent of . However, the agreement with electroweak precision tests is reached due to the small mass splitting between A0 and 1 (H0; A0 and 1 ) in the low (high) mass regime, just as it happens in the IDM. 3.2 ! e Lepton avor 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 ! e being one of the most constraining processes. In this model such a process is controlled by the 1;2, f1;2 and 1;2 Yukawa-couplings and mediated by the Z2-odd particles. Certainly, the interactions in eq. (2.8) and the scalar mixing term allow to construct the one-loop diagram shown in gure 2. The branching ratio for ! e process reads B ( ! e ) = F 3 em 64 m2 G2 j Lj2 + j Rj 2 ; SS = TS = 1 16 m2W s2W i i ; 1 0; m + ; mH0 + c2 1 0; m + ; mA0 + s2 0; m + ; mH0 2 + s2 0; m + ; mA0 2 (0; mA0 ; mH0 ) 2s2c2 0; m + ; m + 1 2 where em is the electromagnetic ne structure constant, GF is the Fermi constant and 1 2 s2 F1(m2 2 ; m2A0 ; m2H0 ) + c2 F1(m2 1 ; m2A0 ; m2H0 ) 1 2 hc2F2(m2+ ; m2N ) + s2F2(m2+ ; m2N )i ; R = 2 1s c m 1 G1(m2 1 ; m2A0 ; m2H0 ) m 2 G1(m2 2 ; m2A0 ; m2H0 ) m 1 2 c2 F1(m2 2 ; m2A0 ; m2H0 ) + s2 F1(m2 1 ; m2A0 ; m2H0 ) + m f1f2 hs2F2(m2+ ; m2N ) + c2F2(m2+ ; m2N )i : 2 2 { 9 { The loop functions are presented in the appendix C. Note that, due to the equation (2.16), the couplings 2 ; 3; f1; f2; f3 are related with 1, hence, the only free Yukawa parameters entering in the expression for B( . μ κi + N κi + e μ χi φ 0 γ χi e 1 4 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 2. Furthermore, we set H0 to be the DM candidate and assume a small mixing angle and the mass spectrum with the lightest charged scalar 1 mainly doublet.7 For the low mass regime and without lose of generality we assume m + ; mA0 > 100 GeV and j j . 0:2, which implies that the remaining Z2-odd elds 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 elds and 2 could a ect 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 2 ; S and all the Yukawa-couplings to 10 2. The numerical result con rms the preliminary expectations: when m + =m + & 1:1, j j . 0:2 2 1 and j j=v . 10 1 the new (co-)annihilations channels compared with those present in the IDM do not play a signi cant 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: 500 GeV < mH0 < 1 TeV ; 2 1 m + = m + + [0:1; 1000] GeV ; m 2 = m 1 + [0:1; 1000] GeV ; 1 2 mA0 ; m + = mH0 + [0:1; 10] GeV ; m 1 = m + + [0:1; 1000] GeV ; ; 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 7It is worth mentioning that when the lightest state 1 is mainly singlet, the relic density cannot be obtained without considering the coannihilation processes with 2 unless that m mH0 ' mh=2, in which case the relic density is independent of m 2 . 1 & 300 GeV and .0 S .03 .02 .01 .01 .02 .03 .04 .03 .02 .01 .01 .02 .03 .04 .0 S the high mass regime. The left panel shows the S, T contributions for any mass splitting m 2 while the right panel shows S, T contributions for any value of the mixing angle . 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 satis ed for a small fermion mixing angle j j . 0:2 (red points in the left-panel). On the other hand, by allowing arbitrary values for the mixing angle, , the contributions to S and T are kept within the 2 level as long as m 2 m 1 . 400 GeV (red points in the right-panel). Regarding the low mass regime we have varied the free parameters as follows: 60 GeV < 1 1 mH0 < 80 GeV, 100 GeV < mA0 ; m + < 1000 GeV, m + < m + < m 1 < 1000 GeV, and the same ranges in the eq. (4.1) for the mixing angles and scalar couplings. The fermion 1 2 contributions to S and T are satis ed by imposing either j j . 0:1 or m 2 m 1 . 200 GeV. In this case, the scalar contributions are not kept within the 2 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. Figure 4 shows the allowed values for the masses mA0 and m + that satisfy the S, T parameters at 68% CL (red points), 95% CL (green points) and 99% CL (blue points) respectively. We 1 have taken j j . 0:1 in order to suppress the fermion contribution. Note that if mA0 is increased, m + will have to be increased. However, from the unitary constraints given in eq. (2.6) 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 ! e process. We have made a scan over the free parameters of the model for the CP-conserving scenario (the CP Dirac phase is xed to zero) with a normal hierarchy and choosing = 1. For this purpose, we have varied the free parameters within the ranges given in eq. (4.1), in addition to 1; 1; 2; 2 [10 4; 1]. The results are shown in gure 5. All the points satisfy the current bound [51] and only a minority will be probed by future searches [52]. We have taken j j . 0:1, m + =m 1+ & 1:1, j j . 0:2 and j j=v . 10 1 2 8The experimental deviations from the SM predictions in the S and T parameters for mh = 126 GeV, mt = 173 GeV and U = 0 are S = 0:06 0:09 ; T = 0:10 0:07 where the correlation factor between S and T is 0:91 [50]. current ! e constraint. Note that the correlation between 1 2 and interference between the two 1 2 contributions and/or by the 1 2 and eqs. (25) and (26)), since it is possible to obtain low values of B( ! e ) (color code) with relative large values of 1 2 10 2. For a inverted hierarchy and = 1 the numerical results are similiar to those for the normal hierarchy: 1 2 0:08 and 1 0:3 for the current bound. can be spoiled by the 2 1 contributions (see in order to satisfy the oblique parameters and preserve the DM phenomenology expected for the IDM. Note that the B( and 1 . 5 10 2. On the other hand, for the low mass regime we obtain similar results to ! e ) limit can be easily satis ed imposing 1 2 . 4 10 2 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 + A0 and 1 Finally, we turn the discussion to collider searches. The high-mass region of the IDM is quite di cult to probe at the LHC. However, the low mass region can be probed by searching for dilepton plus missing transverse energy signal [53{55] and trilepton plus Since the cuts for this kind of signal at the LHC (in both ATLAS and CMS) do not depend in angular distributions between the nal 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 ~01 ~01, studied in ref. [59],9 allows to exclude higgsino-like charged fermions up to 510 GeV [15]. Conversely, in the case of 1 nearly degenerate with H0 (compressed spectra), the bounds on m 1 are 100 GeV for m = m 1 Yukawa-couplings are such that mH0 < 50 GeV [60].10 If, in addition, the missing transverse energy signal [56, 57] with a sensitivity in the parameter region with 1+; A0 100{180 GeV. A similar sensitivity could be expected for 2 + . Concerning VL fermions, the searches performed at LEPII impose a limit of m 1 > 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 nal state taus. For example, if a higgsino-like charged fermion is the next to lightest Z2-odd particle and choosing the Yukawa-couplings such that we have a dilepton plus missing transverse energy signal from pp ! 1+ 1 ! l+l H0H0 ; l =e ; : 3 1; 2; i ; then B ! H0 1, and the exclusion limits are worse due to the larger misidennext-to-leading order in [15]. 1 ti cation rates. Recently, an extended analysis of the LHC Run-I data have been presented by ATLAS [61] with new searches for compressed spectra and nal state taus. In particular, by using multivariate analysis techniques, the 95% excluded cross section for + pp ! ~R;L ~R;L ! + ~01 ~01 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 gure 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 + =GeV < 180 by using the theoretical cross section calculated to 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 e01 and wino-like e02 and 9Where the lightest neutralino, ~10, is the dark matter candidate. 10In our case, the low mass region of the IDM with mH0 = 70 GeV combined with the LEPII constraint on VL fermions, imply that m > 30 GeV. and B e1 are considered in the presence of a light stau. Assuming B e1 ! e ! = 1, they are able to nd 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 1 450 GeV. 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 Z2 symmetry. This symmetry, under which all the non-Standard Model elds are odd, avoids tree-level Higgsmediated avor 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 avor structure of the neutrino mass matrix, one neutrino remains massless. Moreover, such a avor 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 ! e lepton avor violation process. In particular, we have found that 1 2 . 10 2 and 1 . 10 2 in order to ful ll that constraint. On the other hand, the oblique parameters impose j j . 0:2 and m 2 m 1 . 400 GeV for the high mass regime while j j . 0:1 and m 2 m 1 . 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. Acknowledgments We are very gratefully to Federico von der Pahlen for illuminating discussions. D. R. and O. Z. are supported by UdeA through the grants Sostenibilidad-GFIF, CODI-2014361 and CODI-IN650CE, and COLCIENCIAS through the grants numbers 111-556-934918 and 111-565-842691. D. P. and R. L. are supported by COLCIENCIAS. R. L. acknowledges the hospitality of Universidade Federal do ABC in the nal stage of this work. A Free parameters Some of the scalar potential parameters can be written in terms of physical scalar masses using the relations in (2.2) and (2.5): 2 1 v2 3 = m2+ cos2 1 v 2 5 = m2H0 m2A0 ; + m2+ sin2 2 m2H0 + v2 L ; 4 = 2 L 3 T (m1; m2) = (p2; m1; m2) = (0; m1; m2) = 2 9 + Z 1 0 ST formulae parameters, Here, we present the analytical loop functions used for the analysis of the S and T S(m1; m2) = + (m21 + m22)(m41 4m21m22 + m42) + 6m31m32 log 4m21m22 Here, we present the analytical loop functions used for the analysis of the ! e constraint, G1(m2a; mb2; mc2) = F1(m2a; mb2; mc2) = G 1 m2 b 1 2m2a m2a m2 b F m2 b m2a G 1 mc2 + F m2a mc2 mc2 m2a ; ; F2(m2a; mb2) = F1(m2a; mb2; mb2); where F (x) = 2x3 + 3x2 6x + 1 6x2 log (x) 6 (x 1)4 ; G (x) = x 2 4x + 3 + 2 log (x) 2 (x 1)3 : (C.4) Open Access. 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Robinson Longas, Dilia Portillo, Diego Restrepo. The inert Zee model, Journal of High Energy Physics, 2016, 162, DOI: 10.1007/JHEP03(2016)162