A minimal model for two-component dark matter

We propose and study a new minimal model for two-component dark matter. The model contains only three additional fields, one fermion and two scalars, all singlets under the Standard Model gauge group. Two of these fields, one fermion and one scalar, are odd under a $Z_2$ symmetry that renders them simultaneously stable. Thus, both particles contribute to the observed dark matter density. This model resembles the union of the singlet scalar and the singlet fermionic models but it contains some new features of its own. We analyze in some detail its dark matter phenomenology. Regarding the relic density, the main novelty is the possible annihilation of one dark matter particle into the other, which can affect the predicted relic density in a significant way. Regarding dark matter detection, we identify a new contribution that can lead either to an enhancement or to a suppression of the spin-independent cross section for the scalar dark matter particle. Finally, we define a set of five benchmarks models compatible with all present bounds and examine their direct detection prospects at planned experiments. A generic feature of this model is that both particles give rise to observable signals in 1-ton direct detection experiments. In fact, such experiments will be able to probe even a subdominant dark matter component at the percent level.


Introduction
Current observations [1,2] indicate that most of the matter in the Universe consists of non-baryonic dark matter, but they do not tell us what this dark matter consists of. Since the Standard Model (SM), which has been extremely successful in describing all current collider data, does not contain any dark matter candidates, the existence of dark matter provides strong evidence for physics beyond the SM. If that new physics lies at the TeV scale (the scale that is currently being probed by the LHC), the observed dark matter density can be naturally obtained via the freeze-out mechanism in the early Universe -the so-called WIMP (Weakly Interacting Massive Particle) miracle. Within this WIMP framework, several dark matter models have been studied, from those inspired by supersymmetry [3,4] or extra-dimensions [5,4] to simpler models that extend the SM in a minimal way.
The idea behind minimal models of dark matter is to consider the simplest extensions of the SM that can account for the dark matter. In these models, the SM particle content is extended by a small number of fields, and a new discrete symmetry is usually introduced to guarantee the stability of the dark matter particle. Several variations can be obtained depending on the number and type of new fields (e.g. a scalar or a fermion, a singlet or a doublet under SU (2), etc.) and on the discrete symmetry imposed (Z 2 , Z 3 , etc). They include models such as the singlet scalar [6,7], the inert doublet [8,9], the singlet fermion [10], higher scalar multiplets [11], minimal dark matter [12], and Z N models [13,14], to name a few. The main advantage of these models is that because they introduce only a small number of free parameters, they tend to be quite predictive.
Even though it is often assumed that the dark matter density is entirely explained by a single particle, this is not necessarily the case. Two or even more particles could contribute to the observed dark matter density, a situation referred to as multi-component dark matter. This possibility has already been considered in a number of published works -see e.g. [15,16,17,18,19,20,21,22,23]. In this paper we propose a new model for two-component dark matter and we analyze its phenomenological implications. The most salient feature of this model is its simplicity. Besides the two dark matter particles (one scalar and one fermion, both SM singlets), the model contains only one additional field, a singlet scalar field that slightly mixes with the SM Higgs boson, and a single Z 2 symmetry is used to stabilize both dark matter particles. The model can be seen as the union of the singlet fermionic model [10,24,25,26,27,28] and the singlet scalar model [6,7,29,30,31], but it has some new elements of its own. There are new processes affecting the relic density, including the annihilation of one type of dark matter into the other, which we study in some detail. There are also new contributions to the spin-independent direct detection cross section that can increase or decrease the predicted detection rate. We examine the detection prospects of this model and show that both particles usually produce observable signals in planned direct detection experiments. In fact, those experiments can probe even a subdominant dark matter component at the per cent level.
The rest of the paper is organized as follows. In the next section we introduce the model, discuss its main features, and determine its relevant parameter space. In section 3 we study how the relic density for both dark matter components depends on the parameters of the model. Special attention is paid to the role of dark matter conversion. Section 4 deals with the direct detection cross sections. We provide the analytical results and numerically study a new contribution present in this model. In section 5 we define a set of benchmark models that are compatible with all current bounds-including the relic density-and analyze their direct detection prospects. Finally, we present our conclusions in section 6.

The model
The model we propose is an extension of the Standard Model (SM) by three additional fields, two scalars and one fermion, all singlets under the gauge sym-metry. Two of these new fields, one fermion (χ) and one scalar (S), are assumed to be odd under a Z 2 symmetry that guarantees the stability of the lightest odd particle. All the SM fields as well as the other scalar (φ) are instead even under the Z 2 . Remarkably, in this setup the heavier odd particle turns out to be also stable as there are no allowed interaction terms in the Lagrangian including both odd fields. In other words, the model has an accidental symmetry that stabilizes the heavier odd particle. Consequently, the model contains two dark matter particles.
The mass and interaction terms involving the dark matter fermion (χ) are given by where φ is new the scalar field even under the Z 2 and g s , g p are respectively the scalar and pseudoscalar couplings of χ. Notice that, as anticipated, it is not possible to write interaction terms involving both χ and S that are invariant under the gauge and the Z 2 symmetry. The scalar potential of this model can be written as where S is the scalar dark matter particle and H is the SM Higgs doublet that breaks the electroweak symmetry after acquiring a vacuum expectation value, Concerning φ, it is always possible to choose a basis (by shifting the field) in such a way that φ = 0, and so µ 3 1 = −µv 2 /2. In the following, we will always work in that basis.
Due to the µ term in (2), h (the SM Higgs) and φ mix with each other giving rise to two scalar mass eigenstates, H 1 and H 2 , defined as where α is the mixing angle. We assume this mixing angle to be small so that we can identify H 1 with the SM-like Higgs observed at the LHC [32,33] with a mass of about 125 GeV. The other scalar, H 2 , we take to be heavier, M H2 > M H1 = 125 GeV. Even though the model introduces 13 new free parameters, not all of them are important to our discussion. The quartic couplings, λ φ and λ S , for instance, are irrelevant to the dark matter phenomenology. And we can take µ 3 = 0 and λ 4 = 0 without missing any critical effects. The remaining free parameters can be chosen to be M χ , M S , M H2 , g s , g p , sin α, λ, µ φSS , λ φφSS .
Most of these parameters can be associated with either the scalar or the fermionic dark matter sectors. M χ , g s , and g p affect only the fermionic sector whereas M S , λ, and λ φφSS concern only the scalar sector. Both sectors are influenced by M H2 , sin α, and µ φSS . In the next sections, we will study how the dark matter phenomenology of this model depends on these parameters. As stated before, this model can be seen as the union of the singlet fermionic model and the singlet scalar model, both of which have been previously studied -e.g. in [10,24,25,26,27,28] and [6,7,29,30,31]. It reduces to the singlet fermionic model in the absence of S and to the singlet scalar model in the absence of χ and φ. It contains, however, new terms and novel features not present in any of those two models. It is precisely these new features which are the main focus of this paper.

The relic density
Since the model contains two dark matter particles, χ and S, we need to simultaneously follow their abundances in the early Universe. The Boltzmann equations are given by [13] where E denotes any even particle (SM fermions or gauge bosons as well as H 1 and H 2 ), σ AA→BB v is short for the thermally averaged annihilation cross section times velocity for the process AA → BB, and x = m T with T the temperature and m = MS +Mχ 2 . In these equations Y , M P l and g 1/2 * stand respectively for the equilibrium value of Y , the Planck mass and the degrees of freedom parameter. Besides the usual term accounting for dark matter annihilation into even particles (the first term), these equations describe also the conversion of one dark matter particle into the other, χχ ↔ SS. Notice that, since σ χχ→SS v and σ SS→χχ v are determined by the same squared matrix element, they are not independent and are related to each other by These dark matter conversion processes are mediated by H 1 and H 2 as illustrated in figure 1. Since the coupling of χ to H 1 is suppressed by sin α, it usually is the H 2 -mediated diagram that gives the dominant contribution -provided In this figure we set gs = 0.5, sin α = 0.1 and MS = 800 GeV. All other parameters were taken to be zero.
that µ φSS (which determines the SSH 2 vertex) is not too small. If χ and S are close in mass the conversion can take place in both directions, χχ → SS and SS → χχ, but if that is not the case only the conversion of the heavier particle into the lighter one is relevant. To solve the above Boltzmann equations numerically, we have implemented the model into micrOMEGAs [34] (via LanHEP [35]) and have used two different algorithms to integrate them. The first one is explained in [13] and has already been incorporated into a new version of micrOMEGAs (not yet public) suited for models with two dark matter particles. The second one is of our own making and is based on the DarkSUSY [36] routines for the solution of the evolution equation in the case with only one dark matter particle. Even in this second case we relied on micrOMEGAs for the calculation of the relevant σ v 's. We found that both procedures lead to the same values for the relic densities.
In this paper, we will only be concerned with freeze-out solutions to the relic density constraint. Freeze-in solutions also exist [37], as both dark matter particles are SM singlets, but they require very small couplings and consequently do not give rise to any observable signals in dark matter experiments. Moreover, freeze-in was already studied both in the singlet scalar model [38] and in the singlet fermionic model [39] and we do not expect significant modifications to those results in our model. Thus, in the following we only examine regions in the parameter space of this model where the dark matter particles have couplings large enough to reach thermal equilibrium in the early Universe so that their relic densities are the result of a freeze-out process.
Since we have two stable particles, the dark matter constraint in this model reads according to the data by PLANCK [1] and WMAP [2]. A useful related quantity is the fraction of the dark matter density that is due to χ and S, respectively In this figure we set λ = 0.1, sin α = 0.1, λ φφSS = 0.5 and Mχ = 800 GeV. All other parameters were taken to be zero.
denoted by ξ χ and ξ S . We have In this section, however, we will study the dependence of the relic density on the different parameters of the model without imposing this constraint. It will be taken into account in section 5, where the detection prospects will also be examined.
To begin with let us examine the fermion and scalar relic densities in the limit where dark matter conversion processes (χχ ↔ SS) are negligible. To that end we set µ φSS = 0 so that the H 2 mediated diagram in figure 1 is suppressed. Figure 2 shows the fermion relic density, Ω χ h 2 , as a function of M χ for different values of M H2 . For any given value of M H2 the relic density features a double dip at the H 1 and H 2 resonances (respectively at M χ ∼ 62.5 GeV, M H2 /2) and a marked decrease around M χ ∼ M H2 due to the opening of the χχ → H 2 H 2 annihilation channel. At high dark matter masses, the relic density is seen to increase with M χ and to become independent of M H2 . Regarding the final states from dark matter annihilation, they are dominated by gauge bosons (W + W − and Z 0 Z 0 ) for M χ M H2 and by H 2 H 2 for M χ M H2 . For the range of parameters illustrated in the figure, the relic density is seen to vary between 10 3 at low dark matter masses and 10 −4 at the H 2 resonance.
The scalar relic density, Ω S h 2 , is shown in figure 3 for the same values of M H2 . In this figure, λ and sin α were set equal to 0.1. A novelty in this model with respect to the singlet scalar is the existence of annihilations of the type SS → H 2 H 2 . To illustrate their possible effect on the relic density we have set the parameter λ φφSS to 0.5. In the figure we see that the relic density is very suppressed at the Higgs resonance (M S ∼ 62.5 GeV) but not so much at the H 2 resonance, in agreement with the fact that the SSH 2 coupling is suppressed by sin α. Notice also the significant decrease in the relic density at M S ∼ M H2 , indicating that SS → H 2 H 2 becomes the dominant annihilation process in that region (at lower masses it is instead SS → W + W − ). It is important to stress though that, in contrast to the fermionic case, this final state is not always dominant for M S M H2 . Had we taken a small value of λ φφSS , the only difference between the four lines would have been the position of the H 2 resonance. Regarding the dependence on the mass, from the figure we see that the scalar relic density is largest at small masses and smallest at the H 1 resonance, and that it increases with M S at large masses.
Let us next examine how dark matter conversion can affect the predicted relic density. As discussed before, these processes are determined by the parameter µ φSS and are typically relevant in only one direction: the heavier dark matter particle annihilating into the lighter dark matter particle. Figure  S (χχ → SS) may decrease Ω χ h 2 by more than two orders of magnitude.
If we now exchange the roles of χ and S we obtain figure 5, which displays the scalar relic density as a function of M S for different values of M χ . All other parameters take the same values as in figure 4. The relic density is smallest for M χ = 60 GeV (solid line), when SS → χχ can take place over most of the M S range. For M χ = 400 GeV (dash-dotted line), we observe a sudden decrease in the relic density for M S ∼ 400 GeV due to the opening of the SS → χχ channel. Another effect caused by the processes SS ↔ χχ is the small difference observed at low M S between the relic densities for M χ = 400 GeV and M χ = 800 GeV (dotted line). Naively one would not expect a heavy particle to affect the relic density of a lighter one, but, as illustrated in the figure, this seems to be the case. What happens in this example is that when M χ = 800 GeV, χ freezes-out before S and with a larger abundance. While decoupled, χ has some residual annihilations into S, increasing its relic density. If instead M χ = 400 GeV, χ freezes-out after S and with a smaller abundance, and, therefore, the effect of residual annihilations is negligible. From the figure we see that the effect of SS → χχ is significant over the entire mass range and it becomes more pronounced close to the H 2 resonance.
The dependece of the χ relic density on µ φSS is illustrated in figure 6. In it we display Ω χ h 2 as a function of M χ for µ φSS = 100, 50, 10 , 1 GeV. M S was set to 60 GeV and the other parameters were taken as in figure 4. As expected, the larger µ φSS the larger the χχ → SS annihilation rate and therefore the smaller the relic density. In the figure we observe that the value of Ω χ can differ by more than two orders of magnitude between µ φSS = 100 GeV and µ φSS = 1 GeV. Notice that for µ φSS = 100 GeV (solid line) the process χχ → SS is so dominant that the behavior of the relic density does not even change at M χ ∼ M H2 = 700 GeV, where the annihilation channel relic density does get reduced around M χ ∼ 700 GeV, implying that the H 2 H 2 annihilation channel becomes relevant. As a result, for M χ 700 GeV the variation of the relic density for different values of µ φSS is not so large.
To summarize, the main novelties of this model with respect to the singlet scalar and the singlet fermionic models regarding the dark matter relic densities are the presence of a new resonance (at M S ∼ M H2 /2) and of a new final state (H 2 H 2 ) that can affect the relic density of the scalar, and the conversion of one dark matter particle into the other, which may reduce the relic density of the heavier and increase that of the lighter, modifying in a significant way the viable regions. Next, we will briefly review how dark matter detection is modified in this model.

Direct detection
Direct detection is probably the most promising way of detecting dark matter. On the theoretical side, it is subject to fewer astrophysical uncertainties than indirect detection and so the predictions tend to be more reliable. On the experimental side, direct detection experiments such as XENON100 [40] and LUX [41] have already made outstanding progress during the last few years and have set strong bounds on the dark matter spin-independent direct detection cross section. In addition, planned 1-ton experiments such as XENON1T, which should start taking data next year, are expected to improve the current sensitivity by about two orders of magnitude. Thus, they have great chances of discovering the dark matter particle.
In our model, both spin-independent direct detection cross sections tend to be rather large, as evidence by the fact that in the singlet scalar model and in the singlet fermionic model significant regions of the parameter space are already excluded by present bounds [28,31]. Most of these bounds change little when the two models are combined, as we explain in the following. Since χ and S account only for a fraction ξ χ and ξ S of the dark matter density, the quantities that are actually constrained by direct detection experiments are ξ χ σ χ,SI and ξ S σ S,SI , where σ χ,SI and σ S,SI are the usual spin-independent cross sections for the fermion and the scalar dark matter particles. This does not mean, however, that the detection rates are suppressed by ξ. As noted sometime ago [15], the direct detection rate does not strongly depend on ξ. The reason is simple to understand: a smaller fraction of the relic density would generally imply a larger dark matter coupling (say a larger λ or g s ) that would in turn translate into a larger spin-independent cross section, exactly canceling out the ξ suppression in the direct detection rate. On the other hand, the indirect detection rate, σv, has to be multiplied by ξ 2 , which does leave a ξ suppression after taking into account the larger coupling required to reduce Ω. Thus, models with multi-component dark matter generally have better detection prospects in direct detection experiments.
The spin-independent direct detection cross section of χ is determined by H 1 and H 2 mediated diagrams and is the same as in the singlet fermionic model, where m r is the reduced mass and Thus, the only model parameters that enter into its evaluation are sin α, g s , M χ and M H2 . For the scalar, the spin-independent cross section has a new H 2 -mediated contribution (see figure 7) not present in the singlet scalar model. The total cross section is given as Hence, it is determined by the parameters M S , λ, sin α, M H2 and µ φSS . The second term, in fact, is proportional to µ φSS , the same parameter that controls dark matter conversion (χχ ↔ SS), as we saw in the previous section. Interestingly, the two terms may cancel against each other, giving a suppressed cross section, when µ φSS takes the following value which does not depend on the mass of the dark matter particle. Let us now study numerically this spin-independent cross section and, in particular, the cancellation effect. Figure 8 shows σ S,SI as a function of M H2 for different values of µ φSS . The other parameters were chosen as M S = 350 GeV, λ = 0.05 and sin α = 0.1. When µ φSS = 0 (solid line), the dependence of σ S,SI on M H2 disappears -see equation (12)-and we simply obtain a constant. This constant is the reference value against which we are going to compare the behavior of σ S,SI for µ φSS = 0. If µ φSS = 50 GeV (dash-dotted line), for instance, we see that we obtain a smaller value of σ S,SI over the entire range of M H2 . For µ φSS = 100 GeV (dotted-line) we observe a strong suppression of σ S,SI at M H2 ∼ 200 GeV due to the above mentioned cancellations. For µ φSS = 200, 300 GeV, this cancellation occurs instead for M H2 around 150 GeV. In addition, notice that for those two values of µ φSS , σ S,SI is actually enhanced over most of the M H2 range. Thus, a non-zero value of µ φSS can lead either to an increase or a decrease of the spin-independent cross section.
In figure 9 we display instead σ S,SI as a and M H2 = 500 GeV (dash-double dotted line), the value of σ S,SI is practically the same and it is larger than that for M H2 = 250 GeV (dotted line) and M H2 = 300 GeV (dashed line). We also notice that, as expected from equation 12, σ S,SI decreases with M S . As we have seen, the spin-independent cross section of χ is identical to that found in the singlet fermionic model, whereas that of S receives a new contribution that can increase or decrease the detection rate. To compare the predicted spin-independent cross sections with the sensitivity of current and future experiments, we must first impose the dark matter constraint so that they are calculated only for models consistent with the observed dark matter density. That is what we do in the next section.

Detection prospects
To assess the dark matter detection prospects, we have selected a sample of five benchmarks models (or parameter space points) that are consistent with all phenomenological and cosmological bounds and that illustrate some of the interesting possibilities that can occur in this model. For definiteness, in this section we set g p = 0 (parity conserving case) and λ φφSS = 0. Our benchmarks are listed in table 1. For each model, we include the input parameters, the predicted relic densities, the spin-independent cross section, and, for completeness, the indirect detection rate σv. They are also displayed, in the plane (Mass, ξσ SI ), in figure 10, where we also compare their expected direct detection rates with the current bound from the LUX experiment [41] and with the expected sensitivity of XENON1T [42]. Let us now describe in detail each of these benchmark points.   Figure 10: The location of the five benchmark models in the plane (ξσ, M ). The solid line shows the current bound from the LUX experiment [41], while the dashed line shows the expected sensitivity of XENON1T [42].
Model I In this model both dark matter particles are very light, M S = 65 GeV, M χ = 75 GeV, and give about the same contribution to the dark matter density. Such low mass models are usually highly constrained-if not excluded altogether-by direct detection bounds [28]. What allows this model to evade those constraints is the high value of µ φSS (400 GeV), which permits the efficient annihilation of χχ into SS-so that g s does not have to be large to satisfy the dark matter constraint-and at the same time suppresses σ S,SI via the cancellation effect studied in the previous section. Indeed, had we set µ φSS = 0, the fermion relic density, Ω χ h 2 , would have increased to about 10 and the scalar spin-independent cross section would have reached 10 −8 pb. The suppression of σ S,SI is so effective that the scalar cross section lies well below the expected sensitivity of XENON1T. The fermion instead should be easily detected by future experiments.
Model II In this model the dark matter particles have masses in the intermediate range (200 GeV and 180 GeV) and each gives about a 50% contribution to the dark matter density. Since M H2 = 150 GeV < M χ , the fermion relic density is obtained mainly via annihilation into H 2 H 2 . Due to the closeness between M H2 and M H1 , a slight suppression in the fermion spin-independent cross section is expected -see equation (10). From the figure we see that the scalar cross section is about one order of magnitude larger for the scalar than for the fermion and that both dark matter particles could be detected in the next generation of experiments.
Model III In this model both dark matter particles are relatively heavy (300 GeV and 400 GeV) and each accounts for 50% of the dark matter density. The annihilation of χ is dominated by the H 2 H 2 final state and that of S by the usual W + W − . Since µ φSS = 0, dark matter conversion processes play no role in this case. The resulting spin-independent cross sections are between 10 −9 and 10 −10 pb, being slightly larger for the scalar, and are well within the expected sensitivity of XENON1T.
Model IV In this model the dark matter density is dominated by the scalar (97%), which has a spin-independent cross section close to the current LUX bound. The fermion has a small relic density-it accounts only for about 3% of the dark matter-due to the enhancement in its annihilation rate that occurs close to the H 2 resonance, M χ ∼ M H2 /2. Even though χ gives a small contribution to the dark matter density, it may be observed in 1-ton direct detection experiments. That is, such experiments can probe not only the dominant dark matter component but also a subdominant one at the percent level.
Model V In this case, both dark matter particles have intermediate masses (220 GeV and 280 GeV) with the fermion accounting for 92% of the dark matter and the scalar for the remaining 8%. The suppression in the S relic density is obtained simply by increasing the value of λ. In the figure we see that both dark matter particles have a spin-independent cross section well within the expected sensitivity of planned experiments. Again, even the subdominant dark matter component will be probed by such experiments. Notice also from table 1 that the indirect detection rate of the fermion, σv χ , is always very small, even before taking into account the additional ξ 2 suppression. This is due to the fact that the annihilation rate is velocity dependent (σv ∼ v 2 ) and v ∼ 10 −3 for dark matter particles in the Galactic halo. σv S , on the other hand, is not highly suppressed but once multiplied by the corresponding factor ξ 2 S it always gives a value below the thermal one (3 × 10 −26 cm 3 /s). Hence, as anticipated, the indirect detection prospects in this model are not promising.

Conclusions
We have proposed and analyzed a new minimal model for two-component dark matter. The model is an extension of the SM by three fields, one fermion and two scalars, that are singlets under the gauge group. Two of these fields, one fermion and one scalar, are odd under a Z 2 symmetry that automatically renders stable the lightest of the two. A nice feature of this model is that the heavier odd particle is also stable due to an accidental symmetry so that both particles contribute to the observed dark matter density. The model can be seen as the union of the singlet fermionic model and the singlet scalar model, but it contains interesting new features of its own. The relic density, for example, is affected by the conversion of one dark matter particle into the other (SS ↔ χχ), an effect we examined in some detail. There is also a new contribution to the scalar spin-independent cross section that could lead either to an enhancement or to a reduction of the predicted cross section. To assess the dark matter detection prospects in this model, we selected five benchmark points compatible with all phenomenological and cosmological constraints and computed their spinindependent cross sections and their indirect detection rates. We found that in most cases, both dark matter particles could be detected in planned direct detection experiments such as XENON1T.