We investigate the possibility that dark matter is made of heavy Dirac neutrinos with mass GeV– a few TeV] and with suppressed but non-zero coupling to the Standard Model as well as a coupling to an additional gauge boson. The first part of this paper provides a model-independent analysis for the relic density and direct detection in terms of four main parameters: the mass, the couplings to the , to the and to the Higgs. These WIMP candidates arise naturally as Kaluza-Klein states in extra-dimensional models with extended electroweak gauge group . They can be stable because of Kaluza-Klein parity or of other discrete symmetries related to baryon number for instance, or even, in the low mass and low coupling limits, just because of a phase-space-suppressed decay width. An interesting aspect of warped models is that the extra typically couples only to the third generation, thus avoiding the usual experimental constraints. In the second part of the paper, we illustrate the situation in details in a warped GUT model.
Dirac Neutrino Dark Matter
Geneviève Bélanger, Alexander Pukhov and Géraldine Servant
Laboratoire de Physique Théorique LAPTH, F-74941 Annecy-le-Vieux, France
Skobeltsyn Inst. of Nuclear Physics, Moscow State Univ., Moscow 119992, Russia
CERN Physics Department, Theory Division, CH-1211 Geneva 23, Switzerland
Service de Physique Théorique, CEA Saclay, F91191 Gif–sur–Yvette, France
firstname.lastname@example.org, email@example.com, firstname.lastname@example.org
It is well-known that a heavy ( GeV) Dirac Neutrino with Standard Model interactions is ruled out as dark matter because of its large coupling to the . On the one hand, it annihilates too strongly into to have the right thermal abundance. On the other hand, even if it had the right relic density from a non-standard production mechanism, it would scatter elastically off nuclei with a large cross section induced by the exchange and should have been seen in direct detection experiments, unless its mass is larger than several tens of TeV [1, 2]. Moreover, this type of neutrino is excluded by electroweak precision tests . In contrast, a sterile Majorana neutrino as dark matter is a possibility that has raised interest lately [4, 5]. In this case, the neutrino mass that has been considered is rather in the keV- MeV range and behaves as warm dark matter if keV. In addition, the Majorana mass scale is determined by the see-saw formula for neutrino masses.
In the present work, we are considering a different type of Dirac neutrino, denoted , corresponding to a typical cold dark matter WIMP candidate with a mass at the electroweak scale, suggesting that dark matter and the electroweak scale are somehow related (even if the mass of the dark matter particle does not come from electroweak symmetry breaking). We do not assume any particular relation between the mass scale of and that of the light standard model neutrinos. The relic density of is entirely determined by the standard thermal mechanism and therefore by its annihilation cross section. We are assuming that the coupling to the Standard Model is suppressed. There are various reasons why this can happen. A typical framework is to start with an singlet neutrino but charged under . Because the gauge bosons of are heavy, their interactions with are quite feeble and this makes behave as a WIMP. In addition, electroweak (EW) symmetry breaking typically induces a mixing between and , leading to an effective small coupling of to the . Examples of this type were studied in warped extra dimensions [6, 7, 8], and in universal extra dimensions . Note that the most promising realistic warped extra-dimensional scenarios need the EW gauge group to be extended to . In this context, Kaluza-Klein Dirac neutrinos charged under are necessarily part of these constructions, even though their stability typically requires an additional ingredient. For instance, it was shown in [6, 7] that implementing baryon number conservation in warped GUTs leads to the stability of a KK RH neutrino.
Finally, even in the absence of an additional symmetry, can be cosmologically stable if the couplings involved in its decay are very suppressed. This can happen even if the neutrino has a large annihilation cross section providing the correct relic density.
Dirac neutrino dark matter was also studied in the 4D models of Ref.  where the -coupling suppression has a different origin and results from mass mixing between gauge eigen states with opposite isospin. Ref.  did not consider the effect of a that we study in details here. In the first part of this paper, we present a model-independent analysis for the viability of Dirac Neutrino dark matter in terms of three main parameters, the mass, the coupling and the coupling of . We also discuss the effect of a coupling with the Higgs. The remaining part of the paper is a refined analysis of the dark matter candidate which arises in the warped GUT models of Ref. [6, 7]. In both studies, the computation of the relic density is performed with micrOMEGAs2.0  after implementing new model files into CalcHEP .
Part I - Model-independent analysis
In this part, we consider a generic extension of the Standard Model (SM) containing a stable heavy Dirac neutrino denoted , with mass . We assume a parity symmetry (except in Section 2.2) that does not act on SM fields and is the lightest new particle charged under it. The model also contains an additional gauge boson and potentially a charged gauge boson , with masses GeV. To avoid most low energy constraints we will assume that these new gauge bosons couple only to the fermions of the third generation. is the mixing which induces the and couplings. We also introduce as the Higgs coupling to the (for instance induced via mixing with a heavy as illustrated in Part II). We assume that only one chirality of couples to the gauge bosons (in our numerical examples, we chose the right-handed chirality). The effective couplings of to , and are denoted , and respectively:
We work at the level of a low energy effective theory. We assume that the remaining new physics which makes the model more complete does not interfere much with our dark matter analysis.
1 Direct detection constraints
Direct detection constraints are very simple for a Dirac neutrino (see for instance section 3 of Ref. ). The cross-section for scattering on nucleons is governed by the -channel exchange of the , the exchange is comparatively negligible. In contrast with Majorana dark matter, the exchange contributes to the spin-independent scattering cross section. Therefore, strong constraints on , the coupling to the , are derived from direct detection experiments, in particular CDMS  and recently XENON which has now the most stringent limit . The dependence on that we observe for this constraint is related to the experimental sensitivity, which is optimal around 50–100 GeV. The theoretical prediction does not depend on , as illustrated on Fig. 1. In our plots, we use the parameter where is the SM electroweak coupling . The direct detection limit has been rescaled to take into account the fact that, for a Dirac fermion, the interaction of the with protons is suppressed by a factor (see Eq. 2) so that the scattering with nucleons is completely dominated by neutrons. When CDMS and XENON quote their limit they rather assume that protons and neutrons contribute equally. This means that the CDMS and XENON exclusion curves go up by a factor i.e 3.7 and 3.4 respectively. We should keep in mind that the bound from direct detection experiments is subject to some astrophysical uncertainties such as the velocity distribution of the WIMP. Based on Ref. we could estimate these uncertainties and allow for a factor in the interpretation of the CDMS and XENON limits, over the full 10 GeV - 1 TeV WIMP mass range even though there are actually much larger uncertainties for masses below 40 GeV. For clarity, we have not displayed this uncertainty in our plots.
It is clear from Fig. 1 that we have to impose for 400 GeV to satisfy the XENON constraint. If has a sizable coupling to the Higgs, the elastic scattering via Higgs exchange is not always negligible compared to the -exchange especially when couples weakly to the . The spin-independent elastic scattering cross section on nucleons is the sum of two contributions (when averaging over and , the negative interference term cancels):
where are taken from Ref. . For example, when GeV, and , the Higgs and contributions become comparable for elastic scattering on neutrons, as illustrated in Fig. 1, while for elastic scattering on protons, the Higgs contribution dominates.
In contrast with Majorana fermions like in the MSSM, the parameter that determines the elastic scattering of Dirac neutrino dark matter on nucleus is the same parameter that drives annihilation and determines the relic density. We now look at when Dirac neutrinos can inherit the correct thermal abundance.
2.1 Annihilation via s-channel exchange
Annihilation channels of are listed in Fig. 2. We first look at the effect of the coupling to the , , and set . For , annihilation into fermions dominate due to the resonance. For , main annihilation channels are into and (the relative contribution of increases for larger ). The contribution from top pairs is small. Figure 3a shows how decreases as function of for different couplings. In Fig. 3a, we show for comparison the prediction for a fourth generation Dirac neutrino with SM coupling to the and a Yukawa coupling to the Higgs as studied in . Since has non-standard couplings, the total annihilation cross section grows with 111This is different from the behaviour of the vector-like Kaluza-Klein neutrino studied in .. Unitarity breaks down for at the multi TeV scale and we do not show any predictions beyond these values. Figure 3b shows the effect of a coupling of with the Higgs222We neglect the contribution to the mass of due to the Yukawa coupling., in particular the resonance at . The contributions of the different annihilation channels are also displayed in Fig. 4. The channel is important if has a sizable coupling to the Higgs.
Near the resonance, only a weak coupling is necessary to get . We combine the relic density constraints with the direct detection constraints in Figure 5 where we show the WMAP  allowed region ()333A more conservative upper bound, 0.134, was recently advocated in . in the plane, as well as the CDMS and Xenon limits. The region satisfying both constraints corresponds to either 40–50 GeV or GeV if the Higgs coupling is negligible. For GeV, the relic density constraint on is more severe than the direct detection constraint. If the Higgs coupling to is sizable, a region around opens up.
2.2 Light neutrino scenario
For WIMP masses below 10 GeV, direct detection constraints do not apply. We concentrate on this particular case in this section. Light WIMPs also potentially offer the interesting prospect that they can be stable without the need to introduce an extra symmetry, just because there is no SM particle they can decay into, at least on a cosmological time scale. In the absence of any particular discrete symmetry, our exotic light neutrino can have a three-body decay, like the muon. For instance, if the underlying theory is 5-dimensional with gauge symmetry, there are gauge couplings between the KK mode and the zero mode of : , . They induce, via and mixings, the effective couplings ( then corresponds to ): and . If these are sufficiently suppressed, with strength of order , the lifetime of can indeed exceed the age of the universe, as illustrated in Fig. 6, using the width . For a neutrino mass in the 1–10 GeV range, these coupling should be smaller than for the neutrino to be cosmologically stable. This is actually naturally realized in Randall-Sundrum models where the zero mode neutrino is localized near the Planck brane while the KK is peaked on the TeV brane. The overlap of their wave functions is therefore very suppressed, resulting in a tiny effective 4D coupling. However, in this case, we expect the KK to have a mass in the TeV range rather than 10 GeV. The only KK fermions that can be naturally light are those belonging to the multiplet containing the top quark, as discussed in Part II. For instance, in warped GUT models, the KK belonging to the 10 of which contains the top quark (that is the only zero-mode SM particle in the 10) does not couple directly to any light SM fermion. The only SM fermion it can directly couple to is the top, via an gauge boson. Therefore, its decay has to go through a very large number of intermediate states and will be very suppressed. In other words, the boundary conditions on the different components of the multiplet containing are such that no three-body decay is allowed.
Now the question is whether such a neutrino can naturally inherit the correct abundance. The interesting aspect here is that these models typically offer the possibility that the multi-body decay is suppressed while the self-annihilation can be large. This can be explained in terms of the different localizations for the wave functions of SM light fermions on one side and KK modes (of both fermions and gauge bosons) on the other side.
It is clear from Fig. 3a that for (GeV), is needed to obtain the correct thermal relic density; this is in contradiction with the experimental constraint from the invisible decay width of the which requires . One way to open the window could be to consider that couples to a light ((GeV)) singlet scalar field that decays into SM fermions via its mixing with the Higgs. Near the resonance, , the right amount of annihilation can be obtained. Another possibility is to assume that the reheat temperature is below the freese-out temperature . For GeV, GeV, this is still well above the BBN temperature. In this case, the neutrino is produced through scattering in the plasma and the correct relic abundance may be reproduced with an appropriate choice of reheat temperature and coupling. Note that in addition to production via the , and couplings, there is production of via the couplings defined above, which is similar to production via neutrino oscillation [4, 21] but it is very suppressed given the values that we consider for cosmological stability.
Even if the lifetime of the neutrino exceeds the age of the universe, there are additional bounds on from EGRET and COMPTEL measurements of the diffuse gamma ray spectrum, that constrain the radiative decay with gamma emission . In conclusion, this (GeV) Dirac neutrino scenario would deserve a detailed analysis.
2.3 Annihilation via t-channel heavy lepton exchange
We now consider the effect of the t-channel exchange of a heavy charged lepton (potentially a Kaluza-Klein lepton, see also the toy model of Ref. ) leading to the annihilation into . The coupling is defined as
where is written here in terms of the mixing resulting from EW symmetry breaking. According to Fig. 7, quite a large coupling is needed to see the effect of these diagrams. For instance, if arises from mixing and TeV, then we need . We can make the following observations:
There is a destructive interference between the s and t channels leading to an increase in compared to the case with exchange only.
When one can be produced in association with a in the annihilation, there is a decrease in (see the kink around 500 GeV).
When is not too much heavier than , coannihilation effects can reduce the relic density. For illustration, we show the case where coannihilations more than compensate for the increase of due to the t-channel exchange. In addition, near the resonance, there is a sharp drop in the relic density.
2.4 Constraints on the
If the comes from a Left-Right model, there are strong limits on the mass and mixing assuming a manifest LR symmetry i.e. the same mixing matrix in L and R quark sectors. Typical limits on the mixing angle are approximately (assuming ), while limits on the mass are around 1 TeV. The best limit is from mixing, TeV . There are also Tevatron limits using the decay channel into electron and right-handed neutrino leading to GeV . In our analysis, we consider that only the coupling of to the third generation is non-suppressed, as well-motivated in Randall-Sundrum models. In this case, the Tevatron constraints are weakened if we restrict the decay to the channel and indirect limits from or decays do not apply. Nevertheless, if couples to quarks of the third generation (as in Randall-Sundrum models) there is an important constraint from [25, 26]. The leads to an enhancement of the amplitude by a factor . Irrespective of the mass, a limit on the mixing is [25, 26]. This assumes similar quark mixing matrices in the L and R sectors. If the coupling to leptons is similar to that of quarks, the above constraint is incompatible with the values chosen in Fig. 7, , 1 TeV, corresponding to . Relaxing the assumption that the mixing matrix is the same in the left and right sectors will not help sufficiently. In conclusion, the effects can be ignored.
2.5 Annihilation via s-channel and exchange
We now consider the combined effect of the two annihilation channels through and . We first look at the coupling to pairs, arising from - mixing:
where is the Standard Model coupling. Next we add the interaction of the with top quarks. There can be a noticeable reduction of the relic density due to this extra channel. Figure 8 shows the effect of the exchange on for different values of the parameters. It opens a region at the resonance, for , and also reduces the upper bound on from WMAP compared to the case with exchange only.
We show in Fig. 9 the contributions of the different channels to the total annihilation cross section. There is a destructive interference between the and contributions in the channel beyond the resonance. However, this does not produce a significant increase in the relic density, since the annihilation into is still important. In Fig. 9 we chose .
2.6 Constraints on the
The coupling of to the can be induced, for example, via - mixing or via mixing with another heavy neutrino which has a large coupling to the . If the - mixing comes from the Higgs vev only, then
In the absence of a custodial symmetry protecting the (or ) parameter, we have to impose the constraint . When the coupling is induced only via mixing, then and it is difficult to satisfy the constraint on the parameter if . A similar LEP constraint comes from the shift in the vectorial coupling of the to the and the , leading to . The parameter constraint can be relaxed in models with gauge symmetry which are in any case a strong motivation for Dirac neutrino dark matter. The constraint on the coupling can also be evaded (see e.g. ).
Like the limits on , most direct collider searches involve fermions of the first and second generations and if we assume that couples to the third generation only, we can tolerate as light as GeV . A which has generation-dependent couplings (like in Randall-Sundrum models) will induce tree-level FCNC. If it couples to the third generation only:
flavour non-diagonal couplings to the down-type quarks are induced
with the mixing matrix for the left-(right)-handed down-type quarks. This is turn will induce a flavour non-diagonal coupling to the due to mixing. FCNC effects due to a with non-universal couplings are analysed in Ref. [28, 29]. Constraints are model-dependent and can be avoided. They are typically weaker than the -parameter constraint.
2.7 Contour plots
We summarize our results and show the WMAP region in three different planes: in Fig. 10 and 11, in Fig. 12 and in Fig. 13. In Fig. 12 and 13, we neglect for simplicity the coupling of to the fermions. In Fig. 10 and 11, we fixed independently of the relation (5) while the two plots of Fig. 12 satisfy Eq. (5). In addition, the two right-handed plots of Fig. 12 assume that the only source of the coupling is the mixing. As a result, the coupling is suppressed and falls within the WMAP range only for a large coupling to , , or for . The situation is best summarized in Figure 11 which captures what is the allowed region of parameter space after imposing the WMAP bound and XENON constraints: There is a small mass window allowed for . There are other wider mass windows near and . Away from these resonance effects, a large region opens up for .
3 Collider signatures
Like in other WIMP models, the standard searches rely on pair production of the heavier exotic particles which ultimately decay into the WIMP, leading to signals with energetic leptons and/or jets and missing . We list below some signatures which are more specific to the neutrino WIMP model.
3.1 Invisible Higgs decay into
As seen previously, if has a significant coupling to the Higgs, it can account for the dark matter of the universe for corresponding to the Higgs resonance. A significant coupling can arise for instance in the model of [6, 7] (see section 9.3).
As a result, the Higgs can decay invisibly into with a significant branching fraction. The coupling could be probed at the LHC in Higgs production associated with gauge bosons , or with top quarks or in the weak boson fusion process . The weak boson fusion process seems to be the most promising. In the following we use the results of Ref.  where an analysis including a detector simulation was performed and a limit on the invisible width was obtained for various Higgs masses. In general, this limit takes into account the fact that the production cross-section could be modified relative to the SM one, here we assume that the couplings of the Higgs to quarks are the standard ones. A more recent study  performed a refined analysis of the channel, combined with the boson fusion channel. For light Higgses (GeV) the results are similar to the ones of Ref. . The partial width of the Higgs into neutrinos is
Fig. 14 shows the resulting limit that can be obtained on the coupling at LHC.
Let us consider models where belongs to a gauge multiplet, for instance an multiplet. If and its partner are nearly degenerate in mass, is long-lived and the pair production of could lead to interesting stable CHAMP444Charged Massive Particles-like signatures. This is to be contrasted with the standard scenario in which the dark matter is dominantly produced through decays of colored particle and therefore accompanied by energetic jets rather than by charged tracks. This situation was also discussed in the dark matter model of . Fig. 15 shows the region in the plane where decays outside the detector. Limits from LEP  ( GeV) and D0  are reproduced on our plot (Fig. 16) of the pair production cross section, which is dominated by the exchange since only couples to the third generation. A more likely possibility is that will decay inside the detector. In this case, the search is similar to that of sleptons with signature two leptons (see Ref. [37, 38] for LEP constraints).
4 Conclusion of Part I
In summary, Dirac neutrinos are viable dark matter candidates. The situation is best summarized in Figure 11. For a mass between 10 GeV and 500 GeV, the main requirement is that the coupling to the should be at least 100 times smaller than the SM neutrino- coupling in order to satisfy the direct detection constraint. Once this is satisfied, there is a large range of neutrino and masses as well as couplings that lead to the correct thermal abundance. The annihilation via is the dominant mechanism for masses below 100 GeV. Near , the annihilation mechanism is even too efficient. If has a large coupling to the Higgs, can lead to the correct relic density. Finally, couplings open a large spectrum of possibilities in the multi hundred GeV range up to the mass.
In this work, we have assumed that there is no primordial leptonic asymmetry in the dark matter sector (). Obviously, our predictions for the relic density could change significantly if there was such an asymmetry, like there is in the visible matter sector. In constrast, this issue does not arise with neutralino dark matter or heavy Kaluza-Klein gauge boson dark matter. We have also restricted our analysis to the case where Dirac neutrinos would constitute all the dark matter. Constraints would be relaxed if we assumed instead that they constitute only a subdominant piece of dark matter555The scenario where a fourth generation neutrino with mass near the window is a subdominant component of dark matter was studied in .. Moreover, we only studied the case where Dirac neutrinos are thermal relics. Very different conclusions can be drawn if instead, the production mechanism is non thermal, and this is left for a future project.
Except in Section 2.3, we have not considered coannihilation effects, as this is a more model-dependent issue. They will be studied in the explicit example of Part II.
Finally, we have not discussed indirect searches in this work, this was done in  for the case of the LZP model that we now present in details.
Part II - An explicit example: The LZP in warped GUTs
In Ref. [6, 7] it was shown that in models of warped extra-dimensions embedded in a GUT, the symmetry introduced to prevent rapid proton decay, a symmetry, also guarantees the stability of a light KK fermion, a KK right-handed neutrino. This particle is called the LZP and its properties have been studied in [6, 7]. A detailed analysis of the indirect detection prospects in neutrino telescopes, cosmic positron experiments and gamma ray telescopes was also presented in  and in  for antiproton experiments. Some collider signatures were discussed in  and more recently in . In this paper, we revisit the properties of the LZP and perform a complete calculation of its relic density.
The underlying model is based on the Randall–Sundrum setup , where the hierarchy between the electroweak (EW) and the Planck scales arises from a warped higher dimensional spacetime. All Standard Model (SM) fields except the Higgs (to solve the hierarchy problem, it is sufficient that just the Higgs –or alternative dynamics responsible for electroweak symmetry breaking– be localized at the TeV brane) have been promoted to bulk fields rather than brane fields. EW precision constraints require the EW gauge symmetry in the 5-dimensional bulk to be enlarged to . The AdS/CFT correspondence suggests that this model is dual to a strongly coupled CFT Higgs sector . Also, the gauge symmetry in the RS bulk implies the presence of a global custodial isospin symmetry of the CFT Higgs sector, thus protecting EW observables from excessive new contributions .
In this framework, Kaluza-Klein (KK) excitations of gauge bosons of mass TeV are allowed and interestingly, light KK fermions are expected in the spectrum as a consequence of the heaviness of the top quark. The heaviness of the top quark is explained by the localization of the wave function of the top quark zero mode near the TeV brane. This is done by choosing a small 5D bulk mass (the so-called “c” parameter), guaranteeing a large Yukawa coupling with the Higgs. It is clear that if we want to embed these models into a GUT, we cannot consider but rather Pati-Salam or schemes. When the Right-Handed (RH) top quark is included in a GUT multiplet, its KK partners do not have a zero mode but their first KK excitation turn out to be light. The masses and some of the couplings of these KK fermions are determined by the -parameters (see Eq. 22 in the appendix). To have top Yukawa, the right-handed top must have . We fix the parameters associated with the top quark to be . Among all SM particles, the of the RH top quark is the smallest. As a consequence, the KK modes inside its multiplet are predicted to be light (see Eq. 22). They are likely to be the lightest KK states in these models and could be produced at the LHC [7, 41].
The main feature of unification in extra dimensions is that the SM fermions have to be split into different GUT multiplets (unless the SM fermions are localized on the Planck brane). However, this is still not enough to prevent proton decay from dangerous higher dimensional operators localized on the TeV brane. In [6, 7], the problem of baryon number violation was solved by imposing a symmetry. The consequence of the symmetry is also to provide a stable particle. In this model, for each fermion generation, there are at least three 16 multiplets. Each of them is assigned a baryon number. For instance, for the third generation:
where the particles in bold have zero modes and correspond to the SM fermions. The symmetry is
where () is the number of color indices of the (anti)particle . SM particles do not carry any charge. The Lightest charged Particle (LZP) is therefore stable.
Although there are many new fermions in this model, we focus on the lightest ones, i.e. the level one KK fermions that belong to the multiplet containing the SM (the multiplet of the third generation in Eq. 9). The LZP belongs to this multiplet. The only phenomenologically acceptable choice is that is the LZP ( couples too strongly to the ). All fermions inside a given multiplet should have the same -parameter. However, because of bulk GUT breaking effects, there can be effectively large splittings so that the -parameters of each component in the multiplet can be treated as free parameters. The -parameters of members of doublets are identical, for example for both . The model therefore includes six free parameters for the third generation KK fermions,
We ignore the KK partner of the right-handed top. has different boundary conditions to provide the zero mode of the RH top quark and its first KK mode is in the multi-TeV range.
We also consider the influence of the of the multiplet which plays a role because of the mixing induced to the LZP. We will not include this particle in the model or compute explicitly its contribution to scattering processes but we will take into account its mixing with the LZP which will influence the -LZP coupling.
Among the 45 gauge bosons of , we only consider the gauge bosons of , which neutral components recombine to give , and a leptoquark gauge boson of electric charge which belongs to a color triplet + anti-triplet invariant under ’s, . This state has number . Other gauge bosons will not directly enter the annihilation cross-sections that are relevant for us. The masses of these KK gauge bosons are taken to be all equal, .
The parameters that we expect to have an effect on the annihilation rates are: , the 4D coupling which determines the strength of the couplings (thus the coupling of the to the LZP) and can be considered as a free parameter; which determines the LZP mass; which enters the mixing thus affects the -LZP coupling; the Higgs mass which is relevant mainly when and which sets the mass of the new gauge bosons, in particular and .
We have in addition, , parametrizing the amount of bulk GUT breaking, and the cut-off scale of the effective 4D theory. Both enter the expression of the mixing (see Eq. 21) thus the LZP coupling to . The mass of should be important when it can contribute to a coannihilation process near a resonance. Finally, masses of KK fermions should be mostly relevant when they are near and contribute to coannihilation.
All the formulae for the calculation of masses and couplings needed for the implementation of the model into CalcHEP  are listed in the appendix.
The coupling to the LZP is induced via mixing as well as via the mixing with , the LH KK neutrino that belongs to the multiplet that contains the SM LH top. The coupling to the LZP is of order 1 and proportional to . The mixing therefore increases with and the resulting induced LZP- coupling goes as . The mixing goes approximately like thus is large when the the left-handed neutrino is light. This corresponds to . Furthermore, the mixing increases significantly when the mass of the LZP approaches that of the . Figure 17 shows how the two components of the -LZP coupling vary with . The component which arises from mixing clearly dominates, and steeply increases when TeV.
The Higgs coupling to the LZP is also suppressed by a KK mass. Explicitly, in the limit ,
where and the term under the square root is there only if . Exact expressions used in the code can be found in the Appendix. This coupling increases with so the largest contribution of the Higgs exchange is expected for a heavy LZP. On the other hand, the Higgs contribution is important for .
5 Relic density
We include all annihilation and coannihilation channels involving level 1 KK fermions of the third generation (with the exception of the KK partner of the right-handed top that is heavy) that belong to the multiplet. We include as well the exchange of level 1 KK gauge bosons, and vector leptoquarks, .
We exactly solve the Boltzmann equation for the LZP number density. Specifically, we do not rely on the non-relativistic approximation (which fails near the resonances).
We achieve this using micrOMEGAs 2.0. We have rewritten the model in the CalcHEP notation, specifying the new particles and their interactions. All details can be found in the appendix. Since we have a symmetry rather than a R-parity, special care had to be taken to specify the particles that could coannihilate with the LZP. We included all level one KK fermions. The diagrams that contribute to each (co)-annihilation process are chosen automatically according to the interactions specified in the model file. We have also written a module for the direct detection cross-section. This only includes the dominant contribution that arises from the -exchange diagram. The value of , Eq. (12), in the LZP model is indeed typically too small for the Higgs exchange to contribute, according to the analysis of Part I.
Unless otherwise noted, we consider the following range of values for the
free parameters of the model:
3 TeV TeV, GeV, , ,
The central value is fixed by the zero mode of the LH top quark and we allow deviations from bulk breaking effects (). The lower value for is constrained by EW precision tests and direct detection experiments. The upper value is set arbitrarily to concentrate on models that are relevant to low energy phenomenology.
A light LZP could contribute to the invisible width of the . We have taken this constraint into account, MeV, and found that it plays a role only for the near maximal value of , and TeV.
5.1 LZP annihilation
We first look at self-annihilation before analysing the impact of various fermion coannihilation channels.
We consider the range which leads to a LZP mass ranging from 1 (2.4) GeV to 2 (4) TeV, for =3 (6)TeV and . Figure 18 shows the behaviour of as a function of the LZP mass for two extreme values of the coupling, and = 3,6, 10 TeV. We see respectively the effects of the , the Higgs and resonances. At large LZP mass, the main annihilation channels are into pairs, heavy fermions () through exchange and into top quarks through t-channel exchange of .
5.1.1 Annihilation into top
Models with Pati-Salam gauge group instead of share most of the properties that we have discussed. As far as dark matter annihilation is concerned, the main difference comes from the absence of the gauge boson. For this reason, we estimate separately the contribution from the t-channel diagram with exchange. It is particularly important near the threshold where the s-channel contribution of neutral gauge bosons is not so large. In Fig. 20, we compare with and without the contribution from . The difference can reach an order of magnitude. As one moves in the TeV region, the shift is more modest as the annihilation into pairs becomes much more important. The net impact of ignoring the contribution of would be to increase the lower bound on from 250 to 600 GeV for this choice of parameters.