A Minimal Model For Two-Component FIMP Dark Matter: A Basic Search

In the multi-component configurations of dark matter phenomenology, we propose a minimal two-component configuration which is an extension of the Standard Model with only three new fields; one scalar and one fermion interact with the thermal soup through Higgs portal, mediated by the other scalar in such a way that the stabilities of dark matter candidates are made simultaneously by an explicit $Z_2$ symmetry. Against the most common freeze-out framework, we look for dark matter particle signatures in the freeze-in scenario by evaluating the relic density and detection signals. A simple distinguishing feature of the model is the lack of dark matter conversion, so the dark matter components act individually and the model can be adapted entirely to both singlet scalar and singlet fermionic models, separately. We find dark matter self-interaction as the most promising approach to probe such feeble models. Although the scalar component satisfies this constraint, the fermionic one refuses it even in the resonant region.


Introduction
Weakly Interacting Massive Particle (WIMP) is the most populated framework where the solution of dark matter (DM) puzzle takes place [1]. In this TeV scale (LHC one) new physics, DM particles follow the thermal scenario in which they reach thermal equilibrium as well as the chemical one with the bath particles but lose it at freeze-out temperature (which is around m DM /20) and experience decoupling from the Universe plasma. The WIMP candidates are well-found in theories such as MSSM and UED which are known as neutralino [2] and Kaluza-Klein [3], respectively, and also in other extensions of the Standard Model (SM) such as singlet scalar [4] (or fermionic [5]) DM and etc. In spite of its popularity, the WIMP particles have not been yet detected in direct experiments.
The other viable and well-motivated hypothesis to explain the DM problem is embedded in such a feeble interaction by which DM particles can never be abundant enough to thermalize. In this so-called freeze-in mechanism [6,7], Feebly Interacting Massive Particles (FIMPs) have been slowly produced in the early Universe through the collisions or decays of the bath particles. FIMP candidates are motivated in various extensions of the SM [8,9,10] and a well-known example which arises from the neutrino physics is sterile neutrino [11]. It is difficult to detect FIMP particles because of their small couplings with the SM. As for the indirect search, depending on the type of DM candidate, i.e. scalar [12], fermion [10] or etc, some experiments have parameter space to survive but are very borderline.
Although, a lot of attentions have been dedicated to single-particle DM models, some studies have considered DM models with the contribution of more particles in the observed DM density (multi-component DM [13]).
The simplest and the most common case is the union of the singlet scalar (fermionic) and the singlet fermionic (scalar) models which are employed in both freeze-in [12,14] and freeze-out [15,16,17] solutions (or intermediate case [18]). Nevertheless, it remains a mystery whether DM is the single particle or the multi-component one.
In this paper, we analyze whether the freeze-in approach can be properly used to produce the observed DM density in our Universe. To this aim, we choose a minimal two-component DM model in such a way that both of the components are FIMP particles. Following our hypothesis, we consider a singlet scalar and a Dirac fermion where an accidental symmetry guarantees their stabilities and a Higgs portal enables them to interact with the SM particles. The most striking feature of our model is its simplicity in which the two candidates of DM particles do not couple with each other and the model has separate overlaps with both the singlet scalar model [8] and the singlet fermionic one [9]. In our work, all contributing processes to the relic density are assumed and supplementary phenomenological aspects are also concerned.
Some promising possible signatures of FIMPs which are found to be most reliable in previous works are the γ-ray excess observed from the Galactic center (GC) [14,18], the X-ray line at 3.55 KeV [14,18] and DM selfinteraction [12,18]. Actually, to generate the gamma ray excess both components have to couple to Higgs, directly. In other words, both components should be scalar [14], but for the fermionic component its pseudoscalar coupling to the mediator should be in the freeze-out regime [18]. An X-ray signal with E γ = 3.55 KeV from XMM-Newton telescope and the similar signal at 3.52 KeV from the Andromeda galaxy (M31) and Perseus Cluster all could be interpreted by the decay [19] or the annihilation [20] of DM.
However, this requires a definite decay rate and annihilation cross section which is out of reach for our scenario. Therefore, we continue our probe relying only on the DM self-interaction. This non-gravitational interaction is a well-motivated indirect search as it solves the tensions between observations and simulations of small-scale structure of DM.
Following the aforementioned setup, our paper is organized as follows.
After introducing the construction of our model and identifying its parameter space in section 2, we solve two independent Boltzmann equations in the following section (section 3), in order to reach the observed relic density measured by WMAP and Planck experiments [21]. In section 4, we study the phenomenological implications in both direct and indirect experiments, and summarize our results in section 5.

Two-Component FIMP DM
Beyond the SM, we employ three new fields to furnish our model: two scalars (χ and S) and one Dirac fermion (ψ), which all are assumed to be singlet under the SM gauge groups. A discrete Z 2 symmetry is applied such that it reads the SM fields and the S-scalar even, and the other two fields (χ and ψ) odd. This symmetry guarantees the stability of both odd particles in a way that we do not have any terms involving both fields ψ and χ. In this way, the decays of odd particles to one another are prevented. Therefore, we can have two DM candidates in our setup by an accidental symmetry.
The framework of our model is constructed by: where we introduced the scalar and pseudoscalar interactions with the couplings g s and g p , respectively, and inserted the scalar interactions in the After spontaneous symmetry breaking, the SU (2) Higgs doublet is parametrized where v H = 246 GeV, but for the mediator we assume that it does not acquire vacuum expectation value, i.e. < S >= 0, which minimalizes our model too. Now, due to the interaction terms in Eq. (2), h and S mix with each other and form a mass matrix with the following eigenstates and the eigenvalues as where θ is the mixing angle between h 1 and h 2 such that tan θ = y According to the definition of the mixing angle θ, h 1 can be considered as the SM-like Higgs observed at the LHC with a mass of about 125 GeV.
Concerning our parameters, vacuum stability implies that the scalar potential in Eq. (2) must be bounded from below. On the other hand, perturbativity does not allow the model parameters to be too large. Eventually, these theoretical conditions can be satisfied if one has and where λ H is the quartic coupling of H. Extending the SM with the new fields ψ, χ and S, embeds 19 parameters in addition to the SM ones. However, the model constraints remain 11 independent parameters in which the relevant ones to the relic abundance are and for indirect searches, λ H and λ χ are required. Couplings α s and λ 2 could be taken zero without any ambiguities, however, we consider α s = 0 for future application. In the following, we will probe our model parameter space with experimental constraints coming from the relic density, direct and indirect detections.

Boltzmann Equation
Since our model contains two DM candidates, its relic density has contributions from both fields ψ and χ. Therefore, we have to solve two Boltzmann equations for particles which will not reach equilibrium in the freeze-in mechanism where we follow the solution in Ref. [10] (following [7]). The time evolution of number density, dn DM /dt, for the fermionic DM is given by and for the scalar DM, it reads Here H is the Hubble constant, K 1 is the modified Bessel function of order 1 and s is the center of mass energy squared. All contributions to the DM relic density are considered in the corresponding cross sections and decay widths in two above equations. Our analytical results for the cross sections and decay widths are presented in Appendix. As it is well-known from the freeze-in mechanism of production, the two DM candidates in the present model have negligible initial densities (individually), which are much less than their equilibrium values, i.e. n ψ ≪ n eq ψ and n χ ≪ n eq χ . Therefore, putting n χ = n ψ = 0, the Boltzmann evolutions take the following forms and As can be derived from Eqs. (12) and (13), the process of DM conversion, i.e. χχ ↔ψψ, does not contribute to the total relic abundance and is suppressed in our next calculations. On the other hand, each of the DM candidates, independent of the other one, can be produced or annihilated in the Universe.
By solving Eqs. (12) and (13), one can obtain the number densities (n ψ , n χ ) scaled to the entropy of Universeŝ, i.e. Y ψ = n ψ /ŝ and Y χ = n χ /ŝ, as and where M pl is the Plank mass, g * s and g * ρ are the effective numbers for degrees of freedom.

Relic Abundance
The most important constraint which should be satisfied in models describing the DM is to obtain the observed relic density. As Planck experiments have measured the current amount of DM [21], our first experimental constraint is described as where h is the Hubble parameter scaled in units of 100 km/s.Mpc. Using the yield calculated in the previous section (Eqs. (14) and (15)), we can obtain the relic density as The behavior of the relic density Ω χ h 2 regarding different Higgs-scalar DM coupling, λ χH , is depicted in Fig. 1a. In addition to the fixed values of relevant parameters λ 3 , m h 2 and sin θ, we have adopted λ χH as 10 −12 , 5 × 10 −13 and 10 −13 as we fixed λ 4 = 10 −13 . Through Fig. 1a, it is obvious that the resonance occurs at m χ ∼ m h 2 /2. For masses below the resonance, the relic density of the scalar component increases linearly but for larger values (m χ > m h 2 /2) it seems that the relic density is independent of the mass m χ . The difference between these two regions is due to the process h 2 → χχ which is allowed in the region below the resonance.
The complement analysis of the scalar component is plotted in Fig. 1b, where we have chosen λ 4 = 10 −12 , 5 × 10 −13 and 10 −13 . Regarding the resonance occurred at m χ ∼ m h 2 /2, as in Fig. 1a, it can be seen that for the region below the resonance, the relic density grows when the scalar mass m χ increases. This part of the graph seems to be independent of the λ 4 -value and it is enhanced by the h 2 → χχ process. After a significant falling at m χ ∼ m h 2 /2, the relic density seems to be independent of DM mass for the region over the resonance. It is mainly influenced by changing quartic coupling λ 4 . Finishing our investigation of the scalar component, it should be noted that this analysis has a good overlap with a singlet scalar model [8]. We continue our investigation in parallel by turning our attention to the fermionic DM. As before, we consider two classes of variations defined by the effect of scalar (parameterized by g s ) and pseudoscalar (parameterized by g p ) interactions of ψ (see Fig. 2). We first look at the coupling g s so its best effects are formed for the values of 10 −8 , 10 −9 and 10 −10 (Fig. 2a) 4 Phenomenological Implications

Direct Search
In this section, we search for signals inspired from XENON100 [22] and LUX [23] in spin-independent elastic scattering of DM off nuclei. Our intended process includes the fundamental interaction of DM-quark which occurs via and where Here, the parameter µ m = m N m DM m N + m DM is the reduced mass of DM-nucleon and m χ = (m 2 0χ + λ χH v 2 H ) 1/2 is the physical mass of the scalar DM. Cancellation effect [15] could occur when the two terms in Eq. (19) cancel against each other, giving a suppressed cross section which is not appropriate for our consideration. Generally, as was mentioned earlier, a necessary condition for our DM candidates to be nonthermal is owning extremely small couplings

Invisible Higgs Decay
Since the ATLAS and CMS have recorded the signature of SM Higgs [25], new searches have been prepared for DM phenomenology. This is actually done by considering the branching ratio of Higgs, especially for decaying into the light DM candidates which is translated here as below: Regarding the experimental upper bound 0.23 for Br(h 1 → Invisible) [26], we see that the decays of Higgs to both DM ψ and χ are suppressed due to small couplings g s and g p in the former case, and small λ 4 and λ χH in the latter one. However, another constraint comes from the decay of our Higgs to h 2 (if kinematically is possible, i.e. m h 2 < m h 1 /2) whose decay rate could be calculated as where e 3 is the relevant vertex factor which is presented in Appendix, see Eq.

DM Self-Interaction
Among different DM models, collision-less Cold DM (CDM) paradigm has been successful in explaining the large scale structure of the Universe. However, there are discrepancies between the CDM predictions and observations on smaller scales. Self Interacting DM (SIDM) paradigm has the potential to solve these issues (for a review of SIDM, see Ref. [27]). Although, such interactions can not be detected in experiments, we can infer bounds on σ DM /m by evaluating the trajectory of DM in colliding galaxy clusters [28,29]. Previous studies of astrophysical systems parametrize the aforementioned cross section as [29,30] σ DM /m ∼ 1.0 − 1.5 cm 2 g −1 .
In this section, we analyze this constraint to see if it can put new limits on the parameter space of our model. First, we start with the scalar component which has been studied in a singlet FIMP scalar model in [31]. Following [31], we obtain the self-interaction cross section per mass m χ , as where λ χ is the quartic self-coupling of the scalar DM (see Eq. (2)). Following the theoretical constraints in Eqs. (7) and (8), we obtain an experimental upper bound of about 0.1 GeV on the mass of the scalar DM which is depicted in Fig. 4. Going back to Figs. 1a and 1b, we observe that this range of scalar mass can produce proper total relic density along with the contribution of the fermionic component. Another significant point which we would like to clarify in this work, is the self-interaction of a singlet fermionic FIMP DM. In general, we have two concerns. First, it should be noted that significant self-scattering at dwarf scales requires the mediator masses smaller than 100 MeV [32]. Second, if the mediator couples to the SM through a Higgs portal, one should make sure that the mediator decays before the start of the Big Bang Nucleosynthesis (BBN) so the decay products do not affect the BBN. Eventually, we require a mediator with a lifetime ∼ 1s. One way to alleviate the second constraint in DM models with extremely weak interactions is to open a new decay channel for the mediator so it can decay faster. This is done in [33] and [34] by coupling the mediator to a light sterile neutrino (it should be noted that this new coupling does not affect the relic density). However, the first constraint (light mediator) is in conflict with our mediator of mass 100 GeV (and other usual two-component models). A promising solution seems to be working at the resonance region to minimize this mass constraint [35].
Due to the small coupling g p (pseudoscalar interaction type), and the fact that there is an energy (velocity) dependent correction to the width in the resonance region (as explained in [35]), we conclude that the resonance DM self-interaction scenario does not work in fermionic FIMP models.

Conclusions & outlook
We constructed a minimal two-component model to analyze the implications of multi-component DM in the Universe. Using the freeze-in mechanism we calculated the relic abundance predicted by our model and compared it to the observed relic abundance of DM. We started our investigation by proposing two DM particles: a real scalar and a Dirac fermion. Furthermore, a scalar mediator between the dark sector and the SM sector is added. The couplings for this interaction are assumed to be small as we are utilizing the freeze-in mechanism. We solved two independent Boltzmann equations in order to obtain the observed relic density with the contributions of both DM components. It should be noted here that at the time of finishing this work, a new version of micrOMEGAs [36] is presented which can compute the relic abundance of FIMP candidates. In the following, using theoretical constraints, we probed the model parameter space and compared our results with the relevant singlet models. Although, it is difficult to probe FIMP particles, we looked for astrophysical probes in which direct detection was the first. We considered the scattering of DM particles off nuclei and free electrons. As we explained, it is impossible to see this direct signature for our FIMP model.

Acknowledgment
We are particularly grateful to Yonit Hochberg for giving us insights into the direct probes, and we would like to thank Takashi Toma, Madhurima Pandey and Anirban Biswas for their useful discussions.

Appendix: DM production cross sections and decay rates
Here, we present our calculation of the ferminonic DM production crosssections which contribute to the relic density of our model where where the auxiliary parameters and coupling constants have the following The scalar component will account for the DM phenomenology by the following annihilation cross sections where we have employed parameter y i and function F (y i ) as and also for coupling constants, we have the following parameters Finally, the decay rates of scalars h i (with i = 1, 2) into fermionic and scalar DM particles are given as where, we have defined S 1 θ = sin θ and S 2 θ = cos θ.