Top Quark as a Dark Portal and Neutrino Mass Generation

We present a new model for radiatively generating Majorana active neutrino masses while incorporating a viable dark matter candidate. This is possible by extending the Standard Model with a single Majorana neutrino endowed with a dark parity, a colour electroweak singlet scalar, as well as a colour electroweak triplet scalar. Within this framework, the $up$-type quarks play a special role, serving as a portal for dark matter, and a messenger for neutrino mass generation. We consider three benchmark scenarios where the abundance of dark matter can match the latest experimental results, while generating neutrino masses in the milli-electronvolt range. We show how constraints from lepton flavour violation, in particular the branching fraction of $\mu\to e\gamma$, can place lower bounds on the coupling between our dark matter candidate and top quarks. Furthermore, we show that this coupling can also be constrained using collider data from the Tevatron and the LHC.


Introduction
We now have compelling evidence for the existence of three active neutrino species [1]. Radiochemical experiments such as Homestake, Gallex/GNO and SAGE [2,3,4] together with the SuperK and SNO experiments [6,7] have narrowed down the mass patterns to three possibilities: A normal or an inverted hierarchy or almost degenerate masses. Moreover, the absolute scale of neutrino masses remains unknown. Additionally, the last mixing angle, θ 13 , has been measured by several reactor experiments [8,9,10] and the T2K accelerator experiment [11]. This is a breakthrough for the standard picture of neutrino oscillations since it now paves the way towards measuring CP violation in the lepton sector. Furthermore, the evidence for neutrino masses represents one clear motivation for new physics beyond the Standard Model (SM). Within the SM, neutrinos are massless; they can be accommodated in a variety of ways such as incorporating new degrees of freedom and/or new effective interactions. Extending the SM model in this way allows us to be sensitive to new high energy scales. Take for example the Type I seesaw mechanism [12], where the SM is extended with a singlet Majorana fermion that couples to left-handed leptons through the Higgs, as with the charged leptons. This class of models generates viable neutrino masses with a Majorana mass scale 10 14 GeV and Yukawa interaction of order one. Such a high seesaw scale can arise from Grand Unified models such as SO(10) [13]. However, such a high scale for new physics makes the mechanism impossible to test. A TeV scale Majorana mass is also possible in models such as left-right symmetric models, (for a recent discussion see [14]). Models with flavour symmetries are also used to explain the neutrino masses (see [15,16] for recent reviews). Models where neutrino masses are radiatively generated have also been studied. In particular, the simplest model where neutrino masses are induced as one-loop radiative corrections was first introduced in [17]. In this class of models a charged scalar singlet under the SM gauge group couples to left-handed lepton doublets and one is able to generate active neutrino masses of the right order with a charged scalar mass scale as low as a TeV. Neutrino masses may also arise as two loop radiative corrections in extensions of the SM with an additional singlet charged scalar and a doubly charged scalar [18,19,20]. The main motivation for this class of models is that they employ new physics at the TeV scale and hence can be probed at the LHC.
The nature of the neutrino mass matrix can be accessed through data on neutrino oscillations. In the gauge basis the mass matrix can be parametrized in the following way: Another strong indicator of physics beyond the SM is the ample evidence pointing towards the existence of dark matter [22,23]. Velocity dispersion and rotation curves of galaxies suggest the existence of non-luminous matter that is not composed by any of the known SM particles. Furthermore, the most recent data from Plank estimates a cold dark matter cosmological parameter Ω DM h 2 = 0.1199 ± 0.0027 [1] or roughly 26.8% of the universe's total energy. Unfortunately, all experimen-tal evidence for dark matter is due to its gravitational properties and its identity remains unknown to date. One candidate explanation for dark matter is the existence of a weakly interacting massive particle (WIMP). Supersymmetric models are known to provide a natural WIMP candidate, usually the lightest superpartner. The abundance of these particles in the universe is determined by their self-annihilation rate in relation to the expansion of the universe. When the expansion rate dominates over the rate of annihilation, interaction among dark matter particles becomes less efficient and their density becomes a constant or "freezes out". There is, however, some possible signal regions for WIMP scattering with nuclei in direct detection experiments, most notably the DAMA/LIBRA result [24] and CRESST [25]. Experimental upper limits on the WIMPnucleon cross section have also been found by various experiments [26,27,28].
In this work we present a new model for radiatively generating Majorana active neutrino masses while incorporating a viable dark matter candidate. This is possible by extending the SM with a single electroweak singlet Majorana neutrino, N R , to which we also assign an odd parity, referred to as dark parity (DP). We also add a colour electroweak singlet scalar, ψ, and a colour electroweak triplet scalar, χ. These are, respectively, odd and even under DP. Furthermore, all SM fields have even DP. Of the two particles that are odd under DP, we assume N R to be the lightest. Such an assignment makes N R a good dark matter candidate since it will be stable, as we engineer DP to be unbroken. It also forbids the usual coupling of N R to the SM lepton doublet and the Higgs doublet and hence no Dirac mass term is generated. The new colour scalars couple to quark fields, in particular the up-type quarks. In our framework, the up-type quarks play two roles: the first one, serving as a messenger for neutrino mass generation. This is possible given the rich structure of the Lagrangian which is used to radiatively generate masses for the left-handed neutrinos via the exchange of the exotic colour scalars at three loops. The second role is as a portal for dark matter, where the relic abundance of dark matter is reproduced through renormalizable interactions between the Majorana neutrino and up-type quarks via ψ. We study three benchmark scenarios where the abundance of dark matter can match the latest experimental results, while generating neutrino masses in the milli-electronvolt range. Our model is consistent with constraints from lepton flavour violation and collider data from the Tevatron and the LHC. The idea of using a discrete symmetry such as a Z 2 parity to forbid a Dirac mass term for the neutrinos and identify N R as a dark matter candidate was first proposed in [29]. Radiative neutrino masses are generated by the use of Higgs triplets or inert doublets. Here we explore a new avenue by making use of colour scalars which allow neutrino masses to be generated at the 3-loop level. Furthermore, the phenomenology at the LHC is richer by virtue that it is very efficient in producing new colour degrees of freedom.

Model
The model we consider in this study is an extension to the SM that incorporates a dark matter candidate and generates Majo-rana masses for the active left-handed neutrinos, radiatively and at the three loop level. Within this framework, N R couples to right handed up-type quarks through a colour electroweak singlet scalar, ψ. Furthermore, we incorporate a coupling between the electroweak lepton doublets and the up-type quark doublets through a colour electroweak triplet, χ. The new physics can be parametrized in the following way: where l = e, µ, τ and i = 1, 2, 3 is the quark family index. The coupling y i ψ denotes the strength of the interaction between N R and u i R via ψ, while λ i l the strength between the quark doublets (u i , d i ) L and (ν, l) L via χ. Throughout this work, we make use of P R/L = 1±γ 5 2 . Furthermore, unless otherwise stated, we work in the charged fermion mass basis.
The gauge covariant derivatives for the scalars are given by where The implicit sums are over the generators λ a of SU(3), a = 1, ...8, and the generators σ i of SU(2), i = 1, 2, 3. Within this framework a N R Majorana mass term, can be added. This term is even under DP, and we treat M N R as a free parameter. We further assume that M N R < m ψ which makes N R a suitable dark matter candidate 1 .
The gauge and Z 2 invariant potential is given by where H is the SM Higgs field. In order not to have a colour breaking vacuum we take m 2 χ , m 2 ψ to be positive. Since all the new physics terms we have incorporated are of dimension four, the theory remains renormalizable. Of particular interest to us is the last term of Equation (7), since this coupling would play a role in the radiative generation of the active neutrino masses. In addition, the DP remains exact even after electroweak symmetry breaking.

Dark Matter
As mentioned in the previous section, the unbroken Z 2 symmetry stabilizes N R . Due to the interaction introduced in Equation (1), the mechanism that leads to a reduction in the relic abundance of N R is via t-channel annihilation into right-handed top and charm quarks through the exchange of the colour electroweak singlet scalar, ψ. In this work we consider three benchmark points which depict three important regions of parameter space: M N R = 80, 150, 450 GeV.
The evolution of the comoving particle density is given by the Boltzmann equatioṅ where n is the particle density at time t and n eq is the density at equilibrium, H is the Hubble expansion rate and Γ parametrizes the interaction rate, Γ = σv n eq , where σv denotes the thermally average annihilation cross section. By solving numerically the above equation one can find the temperature at which particles depart from equilibrium and freeze out. This temperature is given by where g denotes the number of degrees of freedom of the particle under consideration and g * the number of relativistic degrees of freedom at the freeze out temperature. The present day relic abundance is then given by where The thermalized cross section at temperature T can be calculated from the annihilation cross section of our dark matter candidate, N R . The thermalized cross section is given by where σ(s) is the annihilation cross section as a function of the center of mass energy squared of the interaction, and K 1 (z), K 2 (z) are Modified Bessel function of the first and second kind respectively. We calculated the relic abundance using the latest version of MicOMEGAs [30] and the model files were generated with the latest version of FeynRules [31]. We carried out a scan over three parameters, y t,c ψ , and m ψ , for the three benchmark points. The results are shown in Figures 1 and 2.

Radiative Neutrino Mass generation
The conserved DP allows us to identify N R as a candidate for dark matter and it also forbids Dirac neutrino mass terms for the active neutrinos, ν i . Therefore, the usual seesaw mechanism is not operative in this model. However, the Lagrangian of Equation (3) has enough structure to radiatively generate masses for ν i via the exchange of the exotic colour scalars. In particular, it has the novel feature of using the t R and c R quarks as a portal to communicate with the dark sector and as messengers for the neutrinos. Within this framework, the lowest order diagram for neutrino mass generation is at 3-loops. The diagram is due to exchanges of both ψ and χ fields. This is depicted in Figure 3 which gives the ℓ, ℓ ′ element of the active neutrino mass matrix M ν . This mechanism yields finite contributions to all the elements of M ν and it is best seen using the mass insertion technique. The ℓℓ ′ element of the active neutrino mass matrix is given by where i, j = u, c, t and K i j which controls the scale of neutrino masses is given by From the above equation we see that the u-quark yields a negligible contribution to the neutrino masses and we can concentrate on the t and c quarks. Furthermore, if only one type of quark is involved in the neutrino mass generation, then Equation (13) gives rise to two massless active neutrinos, excluded by experimental data. Therefore, at least two quark families must come into play. To simplify the model we assume that the top quark gives the main contribution and also demand that λ c e,µ << λ c τ , such that the c-quark contribution only modifies the 3, 3 element of M ν . These requirements are sufficient to lift the degeneracy of two massless neutrinos. Using this framework for neutrino mass generation we analyzed the parameter space consistent with 75-100% of the dark matter relic abundance, and calculated the K i j factors. In Figure 4 we show the K t,t factor as a function of y t ψ . The black region corresponds to a Majorana neutrino with M N R = 80 GeV and the grey and maroon regions correspond to Majorana neutrino masses of 150 and 450 GeV respectively. We use a colour electroweak triplet with mass m χ = 1 TeV and a scalar potential coupling between χ and ψ of ρ = 0.1. The bulk of the neutrino mass is due to K t,t since K t,t ≫ K c,c . The K t,t parameter ranges from ∼one meV to 100 eV for parameter points responsible for 75-100% of the dark matter relic abundance. This range of K t,t values can naturally provide this model with a milli-electronvolt active neutrino mass. It is easy to see why the neutrino masses are naturally small. Let us consider the tquark contribution. The 3-loop suppression yields a factor of 10 −7 . Since the LHC has not seen any new colour states we can assume that m χ > 1 TeV. A further suppression comes from ( m t m χ ) 2 ∼ 10 −2 . For M N R = 100 GeV, the factor ( M N R m χ ) 2 gives another 10 −2 suppression. Therefore, sub-eV active neutrinos are natural in this model and no fine tuning of y t,c ψ or ρ is required.

µ → eγ
From Equation 3 one can see that the colour electroweak triplet scalar states will give rise to lepton flavour violating decays. In particular, the decay µ → eγ can be used to place a lower bound on the y t ψ coupling. In our framework, the branching fraction of µ → eγ is given by Given that K t,t ≫ K c,c , we see that we have no sensitivity to λ c µ λ c e in the definition of M ν . In this work we have maximized the contribution to the branching fraction in the limit where λ c µ λ c e ∼ λ t µ λ t e . We then extract the value of λ t µ λ t e using the results from Figure 4 together with the latest values of m eµ [32] and the current experimental upper bound on Br(µ → eγ) ≤ 2.4 × 10 −12 [33]. In the analysis, we have used the best fit range for m eµ assuming a normal hierarchy of active neutrino masses, |m eµ | = 1.5 − 8.8 meV [32]. We have also fixed the colour electroweak triplet mass to m χ = 1 TeV. The branching fraction can then be written in the following way: Our results are shown in Figure 5, where we plot the normalized branching fraction, ξ(µ → eγ) = Br(µ → eγ)/Br(µ → eγ) exp , as a function of y t ψ using the lower limit on m eµ ; and in Figure 6 using the upper limit on m eµ . The black region corresponds to M N R = 80 GeV while the grey and maroon regions to M N R = 150, 450 GeV respectively. We see that the lower bound on y t ψ increases with decreasing M N R . This behaviour is due to the fact that the branching fraction is proportional to M 2 N R while it is inversely proportional to (y t ψ ) 4 . In particular, we find an upper bound of y t An important fact to note is that the constraints placed on y t ψ using the current experimental bound on Br(µ → eγ) are not at all sensitive to the mass of the colour electroweak singlet scalar. This scalar plays an important role in mediating the annihilation of the Majorana neutrinos. As we will see below, bounds on the mass of this scalar as well as upper bounds on the y t ψ can be obtained using collider data.

Collider constraints
This model is also highly constrained by data from high energy colliders such as the Tevatron and the LHC. In particular, our model yields two very distinct signatures for which very stringent bounds exist. We used Madgraph 5 [34] to calculate the parton-level signal prediction and implemented the initial and final state radiation using Pythia [35]. Our signal acceptances were are calculated with the PGS detector simulation implementing the cuts in the corresponding LHC and Tevatron analyses.
One constraint is due to dijet plus missing energy (MET) searches at the Tevatron. The latest bounds on this process were carried out by the CDF collaboration using pp collisions at a center of mass energy of √ s = 1.96 TeV and 2. fb −1 of integrated luminosity [36]. Within our framework, two channels can lead to this final state. The first one is tt production followed by a three body decay of the top quark into two Majorana neutrinos and a charm quark, t → N R N R c. This channel is open as long as N R has a mass below ∼ 86 GeV. The second channel is through pair production of two colour electroweak singlets, followed by the decay ψ → N R c. These two channels are sensitive to y t,c ψ and m ψ . In order to generate exclusions on all three parameters of our model we implemented the experimental sample with tight kinematic thresholds of MET > 100 GeV and H T > 225 GeV, where H T denotes the scalar sum of the two jet transverse energies: The second constraint is due to top squark pair production in pp collisions with a center of mass energy of Within our framework, the colour electroweak singlet, ψ, has the same gauge quantum numbers as the top squark but additional decay modes, in particular ψ → N R c. We apply the CMS constraint using their cut based analysis for three different MET cuts: > 150, 200, 300 GeV.
In Figure 7, we show the parameter regions excluded for an 80 GeV Majorana neutrino from the four experimental observables mentioned at the beginning of this section. On the top, we plot the excluded region in the y t ψ − m ψ plane for y c ψ = 0.25. The region labeled 1 corresponds to regions of parameter space excluded by the CMS observable with MET > 200 GeV, while the regions labeled 2 and 3 correspond to MET > 150 and > 300 GeV respectively. The plot at the bottom corresponds to a value of y c ψ = 0.5. For this value of y c ψ the excluded region is smaller since the branching fraction of ψ → N R t is reduced, and thus, the CMS analysis is less sensitive to our model. From the plots in Figure 7 we also see that no region is excluded by the CDF experiment for m ψ > 300 GeV. This is not true for masses below 300 GeV, where the CDF experiment rules out the entire model for M N R = 80 GeV.
In Figure 8, we show the regions of parameter space excluded for a 150 GeV Majorana neutrino. The plot on the top corresponds to y c ψ = 0.25 while the plot on the bottom to y c ψ = 0.5. For this benchmark point, the CDF observable is sensitive to regions where m ψ lies above 200 GeV but it is not able to exclude any of that region of parameter space. Therefore, the only relevant observable is the CMS analysis, which is able to exclude a region of parameter space where 320 m ψ 550 for y t ψ 0.4 and y c ψ = 0.25. Again, the excluded region is sig-nificantly smaller for larger values of y c ψ , since the branching fraction of ψ → N R t is suppressed.
The above collider constraints were also applied to a Majorana neutrino with M N R = 450 GeV. We found that these constraints were not strong enough to rule out any of the parameter space consistent with 75 − 100% of the dark matter relic abundance. Furthermore, we found that for Majorana neutrinos with masses below 20 GeV, the CDF data on dijet+MET was enough to exclude it as a viable dark matter candidate.

Discussion
In this study, we have investigated the possibility of extending the Standard Model with an electroweak singlet Majorana neutrino, stabilized by a new Z 2 symmetry, to explain the abundance of the dark matter in the universe. In this model, we coupled the dark matter candidate to up-type quarks via a new colour electroweak singlet scalar. Throughout the study we considered three benchmark scenarios: M N R = 80, 150, 450 GeV. We found that the main annihilation channels were into right-handed top and charm quarks depending on the Majorana neutrino mass. Furthermore, we found that when all annihilation channels were open, we were able to generate 75 − 100% of the dark matter relic abundance with a wide range of couplings, y t,c ψ , and scalar masses, m ψ . This however was not the case for M N R = 80 GeV, where the only available annihilation channel was into charm quarks. In this case we found a very clear dependence of the coupling y c ψ on m ψ . We have also investigated the possibility of radiatively generating Majorana masses for the active neutrinos of the Standard Model by incorporating a colour electroweak triplet scalar in addition to the colour electroweak singlet scalar. This setup allowed us to generate active neutrino masses at three loops. We found that the neutrino mass was mostly sensitive to the y t ψ coupling, and that for points consistent with 75 − 100% of the dark matter relic abundance, neutrino masses in the meV to 100 eV range are natural, with data favouring the lower values.
We have considered two types of constraints. The first one arising from the lepton flavour violating decay, µ → eγ. We found that the current experimental bound on the branching fraction placed lower bounds on the coupling y t ψ independent on the colour electroweak singlet mass, m ψ . This lower bound was also higher for lighter Majorana neutrinos. The second constraint was due to two different collider searches. We found that these constraints place upper bounds on the coupling y t ψ . These constraints were also dependent on m ψ and y c ψ ; the latter responsible for the size of the excluded region, since this coupling modifies the branching fraction of ψ → N R t.
Our framework offers an attractive avenue that naturally generates small active neutrino masses while providing a large range of masses for a viable dark matter candidate. The model we presented here is a minimal one as only couplings to t and c quarks are employed. The model also predicts new colour degrees of freedom which lie below the TeV scale, and are now being probed at the LHC. Further signatures at the LHC, such as rare top quarks decays, monotop production and effects on the LHC Higgs signals, will be reported elsewhere.