Two-Component Dark Matter

We study an extension of the Standard Model (SM) with two interacting cold Dark Matter (DM) candidates: a neutral Majorana fermion ($\nu$) and a neutral scalar singlet ($\varphi$). The scalar $\varphi$ interacts with the SM through the"Higgs portal"coupling while $\nu$ at the tree level interacts only with $\varphi$ through Yukawa interactions. The relic abundance of $\nu$ and $\varphi$ is found by solving the Boltzmann equations numerically; for the case $m_\nu>m_\varphi$ we also derive a reliable approximate analytical solution. Effects of the interaction between the two DM components are discussed. A scan over the parameter space is performed to determine the regions consistent with the WMAP data for DM relic abundance, and with the XENON100 direct detection limits for the DM-nucleus cross section. We find that although a large region of the parameter space is allowed by the WMAP constraints, the XENON100 data severely restricts the parameter space. Taking into account only amplitudes generated at the tree level one finds three allowed regions for the scalar mass: $m_\varphi \sim 62.5$ GeV (corresponding to the vicinity of the Higgs boson resonance responsible for $\varphi\varphi$ annihilation into SM particles), $m_\varphi \simeq 130-140$ GeV and $m_\varphi \gesim 3 $TeV. 1-loop induced $\nu$-nucleon scattering has been also calculated and discussed. A possibility of DM direct detection by the CREST-II experiment was considered.


I. INTRODUCTION
Dark matter (DM) was first postulated by Oort in 1932 to account for the orbital velocities of stars in the Milky Way, and then adopted by Zwicky in 1933 to explain the orbital velocities of galaxies in clusters.The existence of DM is by now well established [1] though compelling astronomical observations, including recent ones involving Bullet cluster (1E0657-558) [2].It is also know that DM plays a central role in cosmology, affecting both the evolution of the early universe and structure formation [3].Understanding the properties of DM is one of the great current problems in modern cosmology.
Despite a wealth of observations and many experimental efforts, the nature and composition of DM remains unknown.Since the early 80's there have been continuous attempts to determine whether DM might be associated with one or more elementary particles, an idea that can be probed using both collider experiments as well as cosmological observations.The most promising possibility within this scenario is for DM to be composed of cold non-baryonic particles; in this case current measurements of the anisotropy of the cosmic microwave background (CMB) can be used to estimate the non-baryonic DM density at [4] Ω DM h 2 = 0.1138 ± 0.0045 (1) where the Ω DM = ρ DM /ρ crit is the ratio of the DM density over the critical density that corresponds to flat universe, and h is the Hubble constant in units of 100 km/(s.Mpc) (in contrast, the density of visible baryonic matter is much much smaller: Ω b h 2 = 0.02264 ± 0.00050) [4].
Unfortunately, all Standard Model (SM) particles are excluded as relevant components of DM [5], so one has to look for extensions of the SM that provide stable (or with a decay time longer than the present age of the Universe), massive, neutral particles that might play this role.An enormous amount of work has been done by theoreticians in this direction, considering many types of models, most of which contain a single particle beyond the SM that is stable and might be considered as a DM candidate.
This, however, may not be the case, so that DM could have a multi-component structure (one should remember that the rich variety of SM matter is responsible only for a fiftieth of the matter density in the Universe), and there have already been some studies of multi-component DM in the literature (see for example, [6][7][8][9][10][11][12][13].Here we would like to investigate a scenario where DM consists of two species -a singlet scalar (ϕ) and a singlet neutral Majorana fermion (ν) (that we will refer to as a "neutrino").The scalar DM field in this model interacts with the SM through the Higgs field, while the fermionic DM does not couple directly to the SM.However, the fermionic and scalars DM components do interact, so the model provides a simple 'laboratory' where the interesting issue of interactions between DM components can be studied.
The paper is organized as follows.In sec.II we introduce our specific 2-component DM model and discuss its general properties.Then, in sec.III we derive the Boltzmann equations that govern cosmological evolution of the DM components and we obtain and discuss numerical and approximate analytical solutions.In the subsequent section, sec.IV, we calculate the present DM density and find the regions in parameter space for which it is consistent with (1).In sec.V we derive the constraints on our model derived from the direct detection experiments.Sec.VI contains our conclusions.In the appendix A we collect formulae related to scalar and fermion pair annihilation.

II. MINIMAL SCALAR-FERMION MODEL OF DM
Our model contains three new particles, all SM singlets: a real scalar ϕ, and two majorana fermions ν h and ν (two fermions are required in order to generate non-trivial interactions between the DM components), only one of the fermions will contribute to the DM relic density.Though the DM sector can contain particles of any spin, the simplest possibilities correspond to the presence of fermions and scalars.
Since all DM particles are singlets under the SM gauge group, their interaction with the SM will be through terms of the form O DM O SM , where O SM is gauge invariant operator composed of SM fields; of all such terms we expect those with the lowest dimension to be the most relevant.Within the SM the lowest-dimensional scalar gauge invariant operator is H † H, where H denotes the scalar isodoublet.Restricting ourselves to renormalizable interactions, and assuming that all DM particles transform nontrivially under a symmetry group, fixes the leading O DM to be of the form ϕ 2 .Therefore, the ϕ interacts directly with the SM through the usual Higgs portal term, while (at tree-level) the fermionic dark fields communicate with the SM indirectly, through their interactions with ϕ.

A. The model
In order to ensure stability of DM candidates we will assume that the dark sector is invariant under some global symmetry group G under which all the extra fields transform non-trivially, while all SM particles are G-singlets.For simplicity we choose G = Z 2 × Z 2 and, as mentioned previously, assume that the DM sector is composed of two majorana fermions, ν h and ν, and one real scalar ϕ, which under G, We introduce the Z 2 × Z 2 symmetry to stabilize both the DM components; models with more complicated discrete symmetries will require additional particles.
The most general, gauge-and G-symmetric and renormalizable potential reads: where H is the SM SU (2) Higgs isodoublet and λ x parametrizes the 'Higgs-portal' interaction discussed above.The Lagrangian density for the scalar sector is then given by: As usual, we require that the potential breaks spontaneously the electroweak symmetry via non-zero vacuum expectation value of the Higgs doublet H = (0, v/ √ 2), v = 246 GeV.Since we also require the G symmetry to remain unbroken, we assume that µ 2 ϕ > 0, so ϕ = 0. Note that ϕ = 0 implies there is no mass-mixing between ϕ and H, so that the existing collider limits on the Higgs properties are not modified.After the symmetry breaking, the physical scalars have masses m The part of the DM Lagrangian involving fermions reads Note that the interaction between the SM and DM and the DM self-interactions are generated by just two terms: Although this model can describe a 3 component DM sector we will introduce a further simplification by assuming that M h > m ν + m ϕ , which allows the fast decay ν h → ϕν.In this case only ν and ϕ are stable and therefore can serve as realistic DM candidates.The reactions that are relevant for the evolution of DM are ϕϕ ↔ SM, SM and ϕϕ ↔ νν (we will ignore the process νν → SM, SM that occurs at one loop).We will investigate this model as a simple realization of a 2-component scalar-fermion dark sector, using it as a laboratory where the interplay of the various dark components can be studied.
It is worth noticing that the dark sector has no conserved Noetherian charges, so that all the corresponding chemical potentials vanish.This can be altered in a simple way by introducing additional fermions that can serve as Dirac partners of ν h and ν, in which case the "dark" fermion number could be conserved; we have not done so to simplify the discussion.It is also worth mentioning that the scalar singlet could be used to tame the little hierarchy problem by canceling top-quark loop induced quadratic divergences in radiative corrections to the Higgs boson mass [7,14].
In the following we will fix M h at the smallest value that ensures the fast decay of ν h , so we will effectively deal with only four parameters: m ϕ , m ν , λ x and g ν .Our goal is to constrain the parameters taking into account available restrictions: theoretical (vacuum stability, unitarity/perturbativity, triviality of the scalar sector) and experimental (DM relic abundance, direct detection experiments).

B. Theoretical constraints
In order to stabilize the vacuum we require that the scalar potential in eq. ( 3) is bounded from below.At the tree level it implies the following conditions [15] where m h denotes the Higgs mass.Amplitudes for all possible scalar-scalar scatterings will satisfy the tree-level unitarity constraints provided [16] Finally, it is sufficient to require µ 2 ϕ > 0 for the global G symmetry to remain unbroken, which leads to the very useful inequality as a consequence, light scalars (m ϕ v) must couple very weakly to the SM (λ x 1) whenever λ x > 0.
We also impose the following perturbativity limits on λ ϕ , λ x and g ν .
Separating positive and negative values of λ x , the above constraints imply that the following regions are allowed: where we have adopted in eq. 12 the Higgs mass m H = 125 GeV and the maximal value of λ ϕ consistent with unitarity (8).

III. DARK MATTER DENSITY AND THE BOLTZMANN EQUATION
In the following we will focus on the minimal model specified in sec.II A. Our goal is to determine the DM relic density and test this model against the relic density constraint derived from WMAP and the available data on direct DM detection.
We start with formulating and solving the two Boltzmann equations (BEQ) that govern the cosmological evolution of our DM candidates, the DM neutrinos (ν) and scalar singlets (ϕ).Ignoring loop corrections the relevant reactions are νν ↔ ϕϕ and ϕϕ ↔ SM SM, where the last one occurs through the Higgs portal interaction λ x H † Hϕ 2 .Therefore (at tree level) for the ν to interact with the SM, they must be first converted into ϕ pairs through Yukawa interactions ∝ g ν .The BEQs then read: where n X denote the number density of X = ν, ϕ, and n EQ X the corresponding equilibrium densities; a dot denotes a time derivative, M i→f is the amplitude for the process i → f (note that M ϕϕ→νν = M νν→ϕϕ ); ζ i , i = ϕ, ν, SM are the numbers of internal degrees of freedom (ζ ϕ = 1 and ζ ν = 2, since the ν are Majorana particles), and the matrix element squared |M | 2 contains an average over the initial and final spins together with the corresponding 1/n! factors for n identical particles in the initial and final states; H denotes the Hubble parameter.The phase space density fX and an equilibrium density f EQ X are related to corresponding number densities as follows: where, as mentioned above, the chemical potential vanishes, and ± refers to fermions and bosons, respectively.To simplify BEQs we will use the thermally averaged cross section σ XX→Y Y v , defined as: Assuming kinetic equilibrium and neglecting possible effects of quantum statistics the BEQs in eq. 13 simplify considerably: where it is important to remember that The above relation restates that there are just two independent cross sections that influence the dynamics of DM density evolution: σ ϕϕ→SM SM v and σ ϕϕ→νν v ; the first one is well known (see e.g.[7]) nevertheless it is included in the appendix A for completeness.The Feynman diagram and the corresponding cross section for the process ϕϕ → νν are also shown in the appendix.The interactions between ϕ and ν involve an exchange of a virtual heavy neutrino ν h ; if the corresponding mass M h is very large σ ϕϕ→νν v is strongly suppressed, which leads to an over abundance of ν.To remedy this we will assume M h as small as allowed by the requirement of ν h being unstable: we adopt M h = m ϕ +m ν +∆M ν , with fixed ∆M ν = 10 GeV.Then the cross sections are parameterized by four parameters: m ϕ , m ν , the Yukawa coupling g ν and the Higgs portal coupling λ x .
A. Solving BEQ Instead of a number density (n X ) it is more convenient to use the number density normalized to T 3 , so in the following we adopt f X (T ) ≡ n X (T )/T 3 (not to be confused with the phase-space density f introduced previously).The initial conditions are fixed at large temperature T ini = max(m ϕ , m ν ); we assume that the couplings λ x and g ν are large enough so that at T ini both DM components are in equilibrium with the SM (the SM is assumed to be in equilibrium); hence, f X (T ini ) = n EQ X (T ini )/T 3 ini .As the Universe cools the DM components eventually decouple from the SM when their rate of interaction becomes smaller than the rate of expansion of the universe.Since here we are looking for cold DM (CDM) candidates, we will consider only cases where this decoupling occurs when both ν an ϕ are non-relativistic.In the following, we will solve the BEQs ( 16) and determine the present, i.e. at T = T CMB = 2.37 • 10 −13 GeV, DM abundance.
The solutions can be classified according to the mass hierarchy in the dark sector: The dynamics of the DM number density evolution turns out to be very different for these two cases, as we will see.The factor K is defined in (18) If m ν > m ϕ (Case A), there is a temperature range where the ϕ do not have enough energy to create ν pairs, so the thermally averaged cross section σ ϕϕ→νν v → 0 below that temperature; on the other hand, neutrinos still have enough energy to maintain a high rate of annihilation νν → ϕϕ.This is illustrated in the left panel of fig. 1 where σ ϕϕ→νν v /K is seen to drop precipitously below 10 GeV while σ ϕϕ→SM SM v /K approaches a constant value already at T ∼ 5 GeV.σ νν→ϕϕ v /K is vanishing at T → 0 as will be discussed below.The K factor is defined as follows where g(T ) is the number of relativistic degrees of freedom, and m P l the Planck mass.K appears in the BEQs for the normalized number densities f X (T ) ≡ n X (T )/T 3 .In contrast, for m ϕ > m ν (Case B), it is σ νν→ϕϕ v /K that becomes very small at small temperatures (right panel of fig.1), while σ ϕϕ→SM SM v /K and σ ϕϕ→νν v /K tend to a constant value, we will return to this issue in sec.III A 3.
1. Case A. (mν > mϕ) -numerical solutions In terms of the normalized number densities f X (T ) defined earlier the BEQs ( 16) for case A become where f X denotes a derivative with respect to T and σ ≡ σ ϕϕ→SM SM v /K, σ A ≡ σ νν→ϕϕ v /K; note that σ, σ A have dimensions of mass −1 .Since in the non-relativistic limit σ A is vanishing (as implied by angular momentum and parity conservation) therefore solving numerically the BEQs for the case A we have approximated σ A by keeping only linear terms in the expansion of σ A in powers of x −1 ϕ where x ϕ ≡ m ϕ /T , see [17].It was assumed that σ is T-independent.The quality of this approximation can be estimated from the left panel of fig. 1.We have also verified this approximation for a number of points in the parameter space by comparing results for f ϕ (T CMB ) and f ν (T CMB ) obtained through exact numerical solution with the one obtained adopting expansion of σ A , relative errors obtained for the case A are: Examples of numerical solutions of BEQs (19)(20) for various illustrative parameter choices are shown in fig. 2. The plots on the left hand side panels correspond to case A, while case B examples are presented on the right hand side .
For case A we see that the ν (red dashed line), which are heavier, decouple from equilibrium (solid red line) before (i.e. at a higher temperature) the ϕ (black dashed line); after decoupling from the scalars the ν quickly freeze-out.Sometime later (at a lower temperature) the ϕ decouple from the SM, and since there is no communication between dark neutrinos and scalars, the latter immediately freeze-out.
It is seen from left panels of fig.2, the resulting low-temperature densities for ν and ϕ are similar (note the logarithmic scale), which is a signal that both components decouple form equilibrium roughly at the same x (∼ 20 − 30) as is typical for the standard cold DM scenario.Note also that for fixed m ϕ , the scalar decoupling temperature T ϕ f and the scalar DM relic density are insensitive to m ν , as a consequence of the early decoupling of the ν.Again this is an indication that both components evolve roughly independently.The dark neutrino decoupling temperature, T ν f grows with m ν (since m ν /T ν f is roughly constant).
The green line in fig. 2 refers to solutions for scalar DM density when the fermionic DM component is absent.One can see that in case A (the left panels) the decoupling temperature of the scalar DM in the two component scenario is roughly the same as in the one component scenario with the same m ϕ and λ x , though the relic density is usually (depending on parameters chosen) smaller in the single component case. ) for scalars (neutrinos), dashed lines are the corresponding numerical solutions of the BEQs.Green dashed lines show numerical solutions of a single BEQ for scalars without neutrinos present in the theory.

Case A. (mν > mϕ) -approximate analytical solutions
In the standard case of a single cold DM candidate, it is easy to find an approximate analytical solution of the BEQs that allows to determine the abundance of DM at low temperatures (see for example [18], [19]).The solution is often sufficiently accurate, so that one can avoid obtaining the numerical solutions of the BEQs.In this subsection we will derive an analogous approximate solution within our model of two-component DM for case A.
We begin by defining ∆ ϕ ≡ f ϕ − f EQ ϕ , ∆ ν ≡ f ν − f EQ ν , which parameterize the deviation from equilibrium in the solutions.Then we can rewrite the BEQs (19)(20) as: where the primes denote temperature derivatives.Let's consider first the high temperature regionbefore decoupling of the DM candidates.At these temperatures f ϕ , f ν track f EQ ϕ , f EQ ν very closely, so that ∆ ϕ,ν and ∆ ϕ,ν are very small.The corresponding solution to ( 22) is obtained by neglecting ∆ ϕ,ν as well as all terms proportional to (f EQ ν /f EQ ϕ ) 2 ∝ e −2(mν −mϕ)/T (since in this case m ν > m ϕ ): We define the decoupling temperature (freeze-out temperature) 1 for scalars (T ϕ f ) and neutrinos (T ν f ) as the temperatures at which ∆ ϕ (T . This means that at decoupling temperature the number density differs from the corresponding equilibrium density roughly by a factor of few.We will later assume c ϕ (c ϕ + 2) = c ν (c ν + 2) = 1, because this choice of c ϕ , c ν will provide good agreement with numerical solutions and simplifies the analytical expressions.The freeze-out temperatures T ϕ f and T ν f are then determined by where we have substituted out choice c ν,ϕ = √ 2 − 1.In obtaining this we have assumed, consistent with the cold dark matter requirement, that the parameters are such that m ν , m ϕ T ν,ϕ f , and kept only the leading terms.Once the freeze-out temperatures T ϕ,ν f are obtained by solving (25 -26), ∆ ϕ (T ϕ f ) and ∆ ν (T ν f ) can be calculated using eqs.(23)(24).It turns out that for the choice c ν,ϕ = √ 2 − 1 our approximate equations for T ν,ϕ f reproduce the exact ones (found numerically) very well, typical errors calculated from 20 random points are 0.9% for T ν f and 1.2% for T ϕ f .After freeze-out the number densities remain much larger than their equilibrium counterparts, so that ∆ ν,ϕ f ν,ϕ and we can neglect all terms containing f EQ ϕ,ν and f EQ ϕ,ν as well as all terms proportional to (f EQ ν /f EQ ϕ ) 2 ∝ e −2(mν −mϕ)/T In this case (22) where in this we assumed σ A ∝ T , as discussed above and illustrated in fig. 1.Using now (26) we find that σ The ratio fX (TCMB) num /fX (TCMB) approx for case A for scalars (left panel) and neutrinos (right panel) as a function of scalar DM mass.500 parameter points (mϕ, mν , λx, gν ) were chosen randomly within the ranges 10 GeV < mϕ, mν < 1 TeV, 0.001 < λx < 4π and 0.1 < gν < 4π.
After freeze-out the evolution equation for ϕ becomes (25).In solving this equation we will approximate ∆ ν by its value at T CMB and σ A by its value at T ν f (we have verified the accuracy of these assumptions by comparing the analytic results with the exact numerical results in a set of randomly selected paramter points).Using these approximations the solution is easy to find: therefore in the case A, its value is typically small.Expanding (30) around r f = 0 one obtains in the leading order The above expression shows that the resulting low-temperature ϕ density is roughly the same as it would be in the case without neutrinos at all.That is also seen in the left panels of fig. 2 where dashed green lines (no neutrinos) coincides with black ones (the full system).Since x f for ν and ϕ are similar therefore so are the densities.
The accuracy of the above results can be gauged by calculating the ratio of f num X , the numerical solution, over the corresponding analytical approximate solution, f approx X , at T = T CMB , the present Universe temperature; the results are presented in fig. 3.As one can see, the approximations are often satisfactory for the chosen parameter space.In general, the result for f approx ν are more reliable and become more accurate as the splitting between the ϕ and ν masses increases (which is natural as we are neglecting terms containing (f EQ ν /f EQ ϕ ) 2 ∝ e −2(mν −mϕ)/T ).The quality of the approximation seems to be independent of λ x , both for f ϕ and f ν .

Case B. (mν < mϕ)
When m ϕ > m ν we again assume equilibrium at high temperatures.As the temperature drops, DM particles become non-relativistic and the neutrinos will no longer have enough energy to create pairs of the heavier scalars through annihilation νν → ϕϕ, so that σ νν→ϕϕ v → 0 as T → 0. On the other hand the rate of ν-pair creation, ϕϕ → νν has a non-zero limit as T → 0 (see the right panel of fig.1).
In this case the BEQs ( 16) read where As it is shown in the right panel of fig. 1, for low temperatures σ is well approximated by a constant while for σ B we used low-temperature expansion keeping linear and quadratic terms in x −1 .We have estimated the quality of the approximation by comparing the exact numerical integration of the BEQ's for a number of points in the parameter space with the one obtained with quadratic expansion of σ B , resulting errors for f ϕ (T CMB ) and f ν (T CMB ) are the following δ B ϕ = 6.3%, δ B ν = 2.6%.It is also useful to notice that the ratio of equilibrium distributions, Numerical solutions of (33,34) are shown in the right panel of fig.2, where the neutrino mass was fixed at m ν = 100 GeV for three choices of scalar mass: m ϕ = 120, 400 and 700 GeV.Note that for parameters adopted in the figure (λ x = 0.1, g ν = 2.5), ν and ϕ decouple roughly simultaneously; we have verified numerically that this is typical throughout most of the relevant region of parameter space2 .Since T ν f T ϕ f and f EQ ϕ < f EQ ν for m ν < m ϕ , the asymptotic low-temperature density will be larger for neutrinos, f ϕ (T ) < f ν (T ).Therefore, in case B, it is typical that the number density of DM at low temperatures is dominated by neutrinos.In fact, f ν domination at low temperatures can be understood intuitively since neutrinos do no couple directly to the SM, and in consequence, they annihilate into SM particles slower than scalars.
Contrary to naive expectation, it is remarkable that in case B and for fixed m ν the fermion freezeout temperature is strongly dependent on m ϕ (right panel of fig.2), it varies from T ν f 4 GeV for m ϕ = 120 GeV to T ν f 30 GeV for m ϕ = 700 GeV.Note that in this case x f differs from its standard value 20 − 30, for instance for m ϕ = 700 GeV and m ν = 100 GeV corresponding values are x f ϕ 23 and x f ν 3 for ϕ and ν respectively.This results in a rapid grow of low-temperature f ν with m ϕ at fixed m ν : f ν (T CMB ) ∼ 10 −8 at m ϕ = 120 GeV, to f ν (T CMB ) ∼ 10 −1 at m ϕ = 700 GeV.On the other hand, the low-temperature f ϕ (T CMB ) is roughly independent of m ϕ , even though the scalar decoupling temperature, T ϕ f varies with m ϕ .This case nicely illustrates the dramatic influence of the presence and interaction among DM components upon their thermal evolution.
Another comment is in order here.As one can clearly see in first panel on the right of fig.2, there are parameter ranges such that after decoupling from equilibrium, scalars (black dashed line) do not freeze-out immediately (in contrast to single-component DM or in case A): f ϕ deviates from equilibrium, but is still temperature dependent and only later freezes out.This happens because even below the temperature at which the ν and ϕ decouple from the equilibrium with the SM, ϕ pairs can still annihilate into ν pairs.This effect can be seen from the BEQs (33-34).After the ν decouple, we have f ν f EQ ν and the BEQ for scalars, eq.33 becomes We interpret this as follows: after neutrinos decouple, scalars approach a modified "equilibrium" distribution h ϕ shown as the blue dashed curve in fig. 4. As it is seen in the right panels of fig. 2 and in fig.4, as T decreases, f ϕ will eventually decouple also from h ϕ and freeze-out.In order to illustrate the difference between the modified evolution of scalars after the decoupling from f EQ ϕ we plot in the right panels of fig. 2 also the numerical solutions of a single BEQ for scalars without neutrinos present in the theory (green dashed lines).This behavior of f ϕ between decoupling and freeze-out is only possible in multi-component and self interacting DM scenarios and, to the best of our knowledge, has not been previously discussed in the literature.
The disappearance of scalars into neutrinos is, of course, more efficient and faster as the mass difference between ϕ and ν grows, this can also be observed in the right panels of fig. 2. It is also seen that a large mass splittings results in very large neutrino low-temperature density, while scalar density remains very small, f ϕ ∼ 10 −12 − 10 −13 .It follows that upper limits on the total DM density (implied e.g. by the WMAP data) favor small mass splitting.
Following a strategy analogous to the one we used in case A one can also find an approximate analytical solution of the BEQs in case B. Unfortunately the accuracy of the approximation is much worse in this case, because of the difficulties in including the intermediate state where the scalars have decoupled but have not yet frozen-out.For this reason in case B we will use only numerical solutions.

IV. RELIC ABUNDANCE
The total relic abundance of DM in our model is given by the sum of the neutrino and scalar abundances: The experimental data on the relic density measured at the 1σ level by WMAP [4] shown in equation (1).In order to determine parameters of our model that satisfy the limit, we have performed a random scan over the 4-dimensional parameter space of our model (m ϕ , m ν , λ x , g ν ) in a range: 1 GeV < m ϕ < 10 TeV, 1 GeV < m ν < 2 TeV, 0.001 < λ x < 4π and 0.1 < g ν < 4π.The results of the scan -points satisfying the relic abundance constraint (within 3σ) in the (λ x , g ν ) plane, are shown in fig. 5.
It is seen from fig. 5 that we did not find any points satisfying the WMAP bound for g ν < 0.92.In fact, it is easy to understand why g ν can not be very small: as is seen from fig. 6 the relic abundance of ν increases rapidly as g ν drops, since this suppresses annihilation into scalar pairs; g ν must be large enough in order to avoid overabundance of neutrinos.This reasoning is supported, in the case A, by our approximate analytical solution (28) for which f ν (T CMB ) ∝ σ −1 A ∼ g −4 ν , so that an order of magnitude change in g ν implies 4 orders of magnitude change in the abundance of neutrinos!From fig. 5 we also observe that the WMAP constraint requires that a growing m ϕ be correlated with large |λ x | and g ν , so that with increasing m ϕ our points are more and more concentrated in the upper right corner of the g ν − λ x plane.Note that in the lower panel only red squares survive, this is because for heavier scalar DM masses, only slightly heavier or degenerate neutrino DM masses, accompanied by large values of |λ x | and g ν survive the relic density constraint.This is also easy to understand: with increasing scalar DM mass, scalar relic density increases for case A and neutrino DM density increases for both case A and case B. So, to bring the number density down within the observed limit, we need large couplings to increase the annihilation rates and, in addition, the mass splitting has to be small in order to tame the neutrino DM density; see, for example, fig. 2. This is also understood from figs.7 and 8 as discussed below.
It is instructive to look into various projections of the scan points shown in fig. 7. The left panels are for λ x > 0, so the limits (11) are imposed, while the right ones are for λ x < 0 in which case only Points that satisfy WMAP bound within 3σ range projected into (λx, mϕ) (upper), (gν , mϕ) (middle) and (mν , mϕ) (lower) planes.Green circles -case A points, dark green squares -case B points.Red triangles and purple diamonds -points for which the XENON100 limit is separately satisfied, respectively by ϕ and ν.The consistency limit on λx (11) and the stability limit (12) for λϕ = 8π are satisfied.
the limit ( 12) applies.In each case we show, as a function of m ϕ , all the remaining parameters, λ x , g ν and m ν for which the WMAP bound is satisfied.The plots in fig.7 differentiate between case A (green circles) and B (dark green squares) of which the former are much more plentiful Note that the m ϕ − m ν projections (lower panels) show that in case B the WMAP restriction can be satisfied only for m ϕ m ν , as dark green squares are located just below the diagonal line.In fig.8 we illustrate the effects of the m ϕ -m ν splitting on the neutrino abundance Ω ν for case B; we can see that the WMAP bound can be met only when the masses are close enough.This can be understood from the right panels of fig. 2 from which it is clear that in order to reduce f ν , the dominant low-temperature component of DM, the splitting between m ϕ and m ν must be small; for large splittings the DM decouples and freezes-out early (T ν f ∼ O(10) GeV), and the neutrinos do not have enough time to disappear into SM particles, leading to an unacceptably large DM relic abundance.When the mass splitting is small the neutrino annihilation into scalars (followed by scalar annihilation into SM particles) is still sufficiently efficient to yield an acceptable relic abundance.Summarizing, in case B the WMAP bound can be met only if i) the neutrinos freeze-out relatively late, and ii) m ν m ϕ .
We also include in fig.7 points that satisfy direct detection limits from XENON100 (red triangles) and CREST-II (blue diamonds) (direct detection of DM will be discussed in detail in sec.V).It is important to note already at this point that there exist three regions of ϕ mass which are consistent with XENON100: m ϕ m h /2, m ϕ 130 − 140 GeV and heavy mass region m ϕ 3 TeV.
It is also worth discussing more quantitatively the degeneracy m ν m ϕ required for the case B. As it is seen in the upper and middle panels of fig.2, if the mass splitting ∆m = m ϕ − m ν is not too large, then the decoupling from equilibrium occurs in a range of temperatures where the "distance" between distributions, ∆f (T ) ≡ [log 10 f EQ ϕ (T ) − log 10 f EQ ν (T )], is approximately T -independent, and depends mainly on ∆m.Changing the coupling constants alters the decoupling temperature of both DM particles, but ∆f (T ) remains unaltered.Since scalars and neutrinos decouple roughly simultaneously ∆f (T CMB ) is also a function of ∆m only.It follows that, if m ϕ m ν , the difference between the ϕ and ν contributions to Ω DM , ∆Ω = (log 10 Ω ϕ − log 10 Ω ν ), is roughly a function of ∆m only.The minimal abundance found within the numerical scans is Ω ∼ 10 −8 .In order to reach the WMAP range of abundance (Ω ∼ 0.1), the maximal value of ∆Ω should be ∼ 7. From fig. 8 we can estimate that this value of ∆Ω corresponds to ∆m 40 GeV.This very rough estimate agrees with our numerical scans where we find that (in case B) the maximal allowed splitting found is ∆m MAX NUM = 29.8GeV.The top panel of fig.7 clearly shows the resonance region m ϕ m h /2 ∼ 62.5 GeV in the (λ x , m ϕ ) plane; λ x must be small otherwise the resonant graph with a Higgs boson in the s-channel yields too large annihilation rate and consequently too small ϕ abundance.The intermediate mass consistent with XENON100, m ϕ 130 − 140 GeV requires λ x < 0 that causes a destructive interference between diagrams contributing to the annihilation rate so that the annihilation rate could be suppressed even with substantial λ x .The high scalar mass region consistent with XENON100 requires large λ x .In the middle panel of fig.7 we again observe that usually large values of g ν are allowed by the WMAP data.In fig. 9 we present allowed region in the (λ x , m ϕ ) plane for both Ω ϕ > Ω ν and Ω ϕ < Ω ν ; it is worth noting that points that are close to the lower edge of the WMAP allowed region generally correspond to Ω ϕ > Ω ν (dark orange squares).When Ω ϕ dominates ϕ annihilation rate must be sufficiently suppressed in order to keep the ϕ abundance at the WMAP level.The edge corresponds to the result for λ x = λ x (m ϕ ) obtained for one singlet DM case investigated in [7], (see fig. 7 in that reference).Fig. 10 illustrates the manner in which the DM abundance is split between ϕ and ν.

V. DIRECT DETECTION
In this section we discuss constraints imposed on the model by searches for direct signals of DM particles scattering off a nuclei.We focus here on constraints obtained by the XENON100 experiment [20]  as they impose strongest limits on DM -nucleon scattering cross-section σ DM−N in the mass range of our interest.We will also comment on results obtained by the CREST-II experiment [21].
In our model, at the tree level, scattering of DM off nuclei originates from the interaction with the scalar DM component.Neutrino's leading contribution to the scattering appears at the one-loop level.However, as it has been multiply illustrated, the DM is often dominated by dark neutrinos.Therefore, even though ν nucleon scattering is loop induced, it might be relevant.Therefore, the dominant contributions to the scattering of DM are described by the two Feynman diagrams in fig.11.We start with ϕ nucleon scattering, the corresponding cross section is the following where the sum runs over all quark flavors q, m n is the nucleon mass and f N q are the nucleon form factors as defined in [25] and µ ≡ m ϕ m N /(m ϕ + m N ).
To compare the prediction for the direct detection cross section obtained within our 2-component DM scenario with experimental results from the XENON collaboration one has to take into account that the standard limits on DM direct detection assume all DM particles to be interacting with SM with the same rate.In our case, this is not true as we have two components of DM and their number densities are in general different.Therefore, to compare with the data, we need to rescale the ϕN cross section by a factor that accounts for the fact that two DM components are present: In fig. 12 we plot the rescaled cross section σ ϕ DM−N as a function of m ϕ calculated for points satisfying the WMAP bounds for cases A and B. It follows from this figure that in the resonance region m ϕ m h /2 and in the middle mass region m ϕ 130 − 140 GeV direct detection constraints favor m ν < m ϕ (case B).However for the heavy scalars solution m ϕ 3 TeV it turns out that m ν > m ϕ (case A) is required.The right panel of fig. 12 illustrates the correlation between σ ϕ DM−N and the relative abundance of ϕ and ν.We observe that in the resonance region and in middle mass region Ω ϕ < Ω ν (more neutrinos) while for the large mass Ω ϕ > Ω ν (more scalars).
As seen from fig.
12 the majority of points lie above (i.e. are excluded by) the XENON100 lower limit.This is easy to understand.Using ( 37) and ( 38) we find that In order to minimize σ ϕ DM−N one should (for a given m ϕ ) choose λ x and f ϕ (T CMB )/(f ϕ (T CMB ) + f ν (T CMB )) as small as possible.These factors, however, are correlated.For a conservative estimate of the m ϕ dependence we choose the lower edge of the allowed (λ x , m ϕ ) region from the upper panel of fig. 7, and the lower edge of the f ϕ (T CMB )/(f ϕ (T CMB ) + f ν (T CMB )) region found in fig.10.From fig. 7 we find that for 100 GeV < m ϕ < 1000 GeV Since there exist solutions in the resonance region it is important to calculate the Higgs-boson-decay branching ratio to ϕϕ, as those points could be excluded by measurements of the invisible Higgsboson width.It turns out that for most of those solutions the BR(h → ϕϕ) is typically small and in agreement with the present data [22].It is worth noting that even though the XENON100 data excludes σ DM−N 10 −40 − 10 −44 cm 2 , other experiments, e.g.CREST-II [21], claim an observation of DM scattering with cross sections σ DM−N ∼ 10 −40 − 5 • 10 −43 cm 2 (significantly above the XENON100 limits) and for DM mass range 10 − 60 GeV.It is not our intention here to fit our model parameters to the CREST-II data, however few remarks are here in order.First, we have verified that our model could accommodate CREST-II 2σ data, though in that region of σ ϕ DM−N ,large λ x is necessary; and since m ϕ < m h /2, the solutions that agree with CREST-II inevitably imply BR(h → ϕϕ) ∼ 1, which is in conflict with the present collider data.Note however that, since the CREST-II 2σ region is close to the threshold for h → ϕϕ, therefore a modest (∼ 3σ) extension of the region towards the threshold allows us to find acceptable points above the threshold for which BR(h → ϕϕ) = 0 since the decay is kinematically forbidden; a sample of those is shown in tab.I.It is also worth noticing from the middle right panel of fig.7 that the corresponding Yukawa couplings could be smaller, g ν 4, than those that are needed to satisfy the XENON100 limit (red triangles) in the resonance region.3 -42.7 -42.6 -42.6 -42.6 TABLE I: Points with BR(h → ϕϕ) = 0 that satisfy WMAP bound within 3σ range and for which the cross section σ ϕ DM−N is within a 3σ range of the CREST-II region M1 and with a mϕ that is not more than 10 GeV above the maximal (2σ) mass range for CREST-II.
In fig.13 we illustrate temperature evolution of number densities (normalized such that at T CMB they coincide with relict abundances) for a sample of points that are below XENON100 limit in fig.12.
As it has already been mentioned large abundance of dark neutrinos ν may imply that their contribution, although suppressed at the level of an amplitude, may be relevant after taking into account their relative number density: Results for the cross section σ ν DM−N as a function of ν for points satisfying the WMAP are confronted with the XENON100 bound in fig.14.It is seen that the case A points are mostly in agreement with the bound, while the case B points leads to too large cross section.It is also worth to notice that points below the XENON100 limit correspond to Ω ϕ > Ω ν in agreement with our intuition.
So far we have been comparing separately ϕ and ν cross sections with experimental data.However one should take into account the fact that we do have two component DM.That is not quite straightforward if masses of the two components are different or their contributions are of the same order.Fortunately, it turns out that in almost all cases of interest it is meaningful to compare with the experimental limits.The reason is that for all points of interest either ϕ and ν are almost degenerate, or the abundance is dominated by ϕ, and both these cases are well described by plotting σ DM−N vs. m ϕ .These results are presented in fig.15.The dark green squares stand for the case B points, so with , therefore in the first approximation we may compare σ DM−N for those points with the limits.On the other hand, it turns out that light green circles correspond to points for which the cross section is dominated by scalars, so again those points might be compared with single-component DM limits.

VI. CONCLUSIONS
We have discussed the main features of a two-component cold Dark Matter model composed of a neutral Majorana fermion (ν) and a neutral real singlet (ϕ).The Boltzmann equations for number densities of ν and ϕ were solved numerically and, for the case m ν > m ϕ , an approximate analytical solution for the present DM abundance of both components was found.In order to determine a region of parameter space that is consistent both with WMAP and XENON100 data a scan over 4-dim parameter space was performed.
It has been shown that the agreement with the WMAP data requires that neutrinos cannot be substantially lighter than scalars, i.e. consistent solutions are found only for m ν m ϕ .In the region where m ν m ϕ we observe interesting and strong implications of the presence (and interactions) of two components of DM, in particular, the thermal evolution of their number densities could be dramatically altered.
It has been shown that in a majority of the parameter space ν constitute the dominant component of the present energy density of DM.This observation agrees with a naive intuition: since ν's do not interact with the SM directly therefore they annihilate slower than ϕ's that couple directly to Higgs bosons.In order to enhance the annihilation rate for ν, large values of the ν − ϕ coupling g ν 1 − 12 are favored by the WMAP data.One could safely generalize the above observations and conclude that in the multi-component DM models the generic difficulty is an overabundance of the DM components that have no direct couplings to the SM.Another remark is that when the scalars ϕ are relatively heavy (100 m ϕ 1000 GeV) their annihilation rate into SM particles must be amplified in order to maintain agreement with the WMAP data, that implies the Higgs portal (∝ H † Hϕ 2 ) coupling λ x must grow with m ϕ .
The XENON100 upper limit in DM-nucleon cross section, σ DM−N , turns out to be a very restrictive condition on the model.Let's first focus on the case with σ DM−N dominated by the ϕ-N scattering.Naively one could expect that the prediction for σ ϕ DM−N could be reduced below the XENON100 limit by increasing m ϕ .However there exist two obstacles that prohibit suppression of σ ϕ DM−N by enlarging the scalar mass (in the range m ϕ 100 GeV): (i) in order to meet the WMAP constraint data on the present DM abundance the minimal value of the Higgs portal coupling constant λ x must grow rapidly with m ϕ , and (ii) the minimal relative scalar density f ϕ (T CMB )/[f ϕ (T CMB ) + f ν (T CMB )] also increases rapidly with m ϕ .The two factors imply that the WMAP constraint restrict parameters to those for which σ ϕ DM−N is a approximately a constant function of m ϕ ; in particular, a large m ϕ does not help to suppress σ ϕ DM−N .Nevertheless for m ϕ 3 TeV σ ϕ DM−N starts to be consistent with the XENON100 data since the bound becomes weaker at large m ϕ .For those points Ω ϕ > Ω ν .We have also found consistent solutions for m ϕ m h /2 and m ϕ 130 − 140 GeV corresponding to Ω ϕ < Ω ν .
The ν-DM cross section, σ ν DM−N , that appears at the 1-loop level was also calculated and its contribution was confronted with the XENON100 data.It has been shown that in the case A (m ν > m ϕ ) points that satisfy the WMAP constraint are mostly in agreement with the XENON100 bound, while in the case B (m ν < m ϕ ) the cross section is usually too large.It is also worth to notice that points below the XENON100 limit correspond to Ω ϕ > Ω ν .
When both ν-N and ϕ-N cross sections are taken into account, it turns out only solutions with m ϕ m h /2 and m ϕ 3 TeV survive.
It has been noticed that, since the CREST-II 2σ region is close to the threshold for h → ϕϕ, therefore a moderate (∼ 3σ) extension of the region towards the h → ϕϕ threshold allowed us to find points consistent with the WMAP data with vanishing invisible decay width.
As a final remark we note that such a model is difficult to test at the Large Hadron Collider (LHC).The leading new effect would be production of scalar DM pairs, with a signature of missing energy associated with one or more jets.Such an analyses lie beyond the scope of this work.
The diagrams contributing to the scalar ϕϕ annihilation into SM particles are shown in fig.16.The corresponding cross sections are available in the literature(e.g.[23] and [24]); we have verified the results of [24]: where F (ξ) = ArcTanh(ξ)/ξ, ξ = (s − 4m 2 h )(s − 4m When m h > 2m ϕ the decay h → ϕϕ is allowed and one has to modify the h width accordingly: where θ H is the Heaviside step function (we also ignore 1-loop corrections to Γ h that might include a contribution from h → ν h ν h ).

FIG. 9 :FIG. 10 :
FIG.9: Points that satisfy WMAP bound within 3σ range projected into (λx, mϕ) plane.Orange circles -points where Ωϕ < Ων , dark orange squares -points where Ωϕ > Ων .The left panel corresponds to the solutions for positive λx, while the right panel is for negative λx.Blue dashed line is the consistency limit on λx(11), while the black horizontal dashed line is the stability limit λϕ = 8π.

2 FIG. 14 :
FIG.14: Plot of the cross section σ ν DM−N as a function of ν for points satisfying the WMAP data within 3σ; the other parameters are randomly chosen in the ranges defined in the text (including both signs of λx).Left panel: green circles (dark green squares) correspond to case A (case B) solutions.Right panel: orange circles (dark orange squares) correspond to Ωϕ < Ων ( Ωϕ > Ων ).The red line shows the XENON100 data, and the two islands in blue indicate 1 and 2σ CRESST-II results.

2 FIG. 15 :
FIG.15: Plot of the cross section σDM−N = σ ϕ DM−N + σ ν DM−N as a function of ϕ for points satisfying the WMAP data within 3σ; the other parameters are randomly chosen in the ranges defined in the text (including both signs of λx).Green circles (dark green squares) correspond to case A (case B) solutions.The red line shows the XENON100 data, and the two islands in blue indicate 1 and 2σ CRESST-II results.