Multicomponent dark matter: the vector and fermion case
Abstract
Multicomponent dark matter scenarios constitute natural extensions of standard singlecomponent setups and offer attractive new dynamics that could be adopted to solve various puzzles of dark matter. In this work we present and illustrate properties of a minimal UVcomplete vectorfermion dark matter model where two or three dark sector particles are stable. The model we consider is an extension of the Standard Model (SM) by spontaneously broken extra \(U(1)_X\) gauge symmetry and a Dirac fermion. All terms in the Lagrangian which are consistent with the assumed symmetry are present, so the model is renormalizable and consistent. To generate mass for the darkvector \(X_\mu \) the Higgs mechanism with a complex singlet S is employed in the dark sector. Dark matter candidates are the massive vector boson \(X_\mu \) and two Majorana fermions \(\psi _\pm \). All the dark sector fields are singlets under the SM gauge group. The set of three coupled Boltzmann equations has been solved numerically and discussed. We have performed scans over the parameter space of the model implementing the total relic abundance and direct detection constraints. The dynamics of the vectorfermion dark matter model is very rich and various interesting phenomena appear, in particular, when the standard annihilations of a given dark matter are suppressed then the semiannihilations, conversions and decays within the dark sector are crucial for the evolution of relic abundance and its present value. Possibility of enhanced selfinteraction has been also discussed.
1 Introduction
The experimental data from the WMAP [1] and more recently the Planck [2] collaborations provided an independent and indisputable confirmation for the presence of dark matter (DM) in the Universe. Nevertheless, in spite of a huge theoretical and experimental effort, its nature is still unknown. Till now only gravitational interactions of DM have been detected in a series of independent observations like the flatness of rotation curves of spiral galaxies [3], gravitational lensing [4], and collision of galaxy clusters with its pronounced illustration known as the Bullet cluster [5]. All attempts to detect DM nongravitational interactions with ordinary matter have failed so far implying more and more stringent limits on DMnucleon crosssection, see e.g. LUX [6] and XENON1T [7] results. The most popular models of DM are based on the assumption that it is composed of weakly interacting massive particles (WIMPs). Unfortunately, it turns out that the WIMP scenarios suffer from various difficulties when confronted with observations on small cosmological scales. For instance, the “toobigtofail” [8, 9] and the “cuspcore” [10, 11, 12, 13] problems are widely discussed in the literature. In particular, the DM densities inferred in the central regions of DM dominated galaxies are usually smaller than expected from WIMP simulations [14, 15]. It turns out that an appealing solution to those problems is to assume that dark matter selfinteracts strongly [16]. The assumption of selfinteracting dark matter (SIDM) implies that central (largest) DM density could be reduced. Usually, selfinteracting DM scenarios require the presence of light DM and also light DM mediators.
Dark matter could also be searched for through indirect detection experiments, which assume that in regions of large DM density, its pairs are likely to annihilate. Then secondary particles released in this process, e.g. gamma rays, neutrinos, electrons, positrons, protons, and antiprotons, could be observed on Earth, which could reveal some properties of DM. Independent analysis of the FermiLAT data [17] by various groups have shown an excess of gamma ray in the energy range \(1{}3\;\hbox {GeV}\) that can be interpreted as a result of DM annihilation in the Galactic Center. Besides the 1–3 GeV excess gamma rays, there exists an observation of unidentified \(3.55\;\hbox {keV}\) Xray line found by [18] and [19]. As shown by several groups, this unknown Xray line can also be explained by DM annihilation. To explain the indirect signals relatively large DM mass is needed, e.g. \(\sim 50\;\hbox {GeV}\) in [20].
Since very different DM masses are needed to solve the smallscale problems (through selfinteraction) and to interpret the potential indirect signals, therefore in order to accommodate both observations, a multicomponent DM seems to be a natural option. Various multicomponent DM models have been proposed and studied in the literature, for instance, multiscalar DM [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], multifermion DM [34, 35, 36, 37, 38, 39, 40, 41, 42], multivector DM [43, 44], scalarfermion DM [45, 46, 47, 48, 49, 50, 51, 52, 53, 54], scalarvector DM [55, 56, 57], vectorfermion DM [58, 59], and various other generic multicomponent DM [60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77] scenarios. Models of multicomponent DM were also considered and adopted in astrophysical simulations, e.g. by [78, 79, 80, 81]. Needless to say, the dynamics of multicomponent DM is much richer than that of a simple WIMP, and therefore attractive to study by itself, even without any phenomenological direct application. In particular multicomponent DM models allow to have, besides the standard annihilations and coannihilations; conversion, semiannihilation, and decay processes which make the dark sector (thermal) dynamics much more involved and interesting. Note that most of the models mentioned above, discuss the implications of one or two of these multicomponent DM properties.
In this work, we propose a minimal UV complete vectorfermion DM model that predicts two or three stable dark states. Our model involves an extension of the SM by an Abelian dark gauge symmetry \(U(1)_X\). The model is minimal in a sense that it contains only three new fields in the dark sector; a dark gauge boson \(X_\mu \), a Dirac fermion \(\chi \), and a complex scalar S, which serves as a Higgs field in the hidden sector. They are singlets under the SM gauge group but they are all charged under the dark \(U(1)_X\) gauge symmetry and therefore they interact with each other. The complex scalar S acquires a vacuum expectation value (vev) and gives mass to the dark gauge boson \(X_\mu \) by the dark sector Higgs mechanism. It also contributes to the mass of the Dirac fermion \(\chi \) through the Yukawa coupling. Moreover, the presence of the Dirac mass for the fermion introduces a mixing of its chiral components. After diagonalization of the mass matrix, the Dirac fermion splits into two Majorana fermions \(\psi _\pm \) with mass eigenvalues \(m_\pm \). As a result, after the dark sector symmetry breaking we have three potentially stable interacting particles: a dark vector and two Majorana fermions. Their stability is ensured by a residual \(Z_2\!\times \!Z_2^\prime \) discrete symmetry, which also dictates the possible form of dark sector interactions. The communication with the visible (SM) sector proceeds only via the Higgs portal \(\kappa H^2S^2\).
Our minimal vectorfermion dark matter model has many attractive features. First of all, the very fact that in the multicomponent DM literature, the vectorfermion dark matter possibility has not been studied in detail^{1} speaks for itself and therefore the goal of this work is to provide an extensive analysis of the minimal vectorfermion scenario. Some of the interesting features of the model are: (1) the presence of a second scalar helps to achieve the stability of the SM Higgs potential even at the tree level, see e.g. [82, 83, 84], (2), a possibility of enhancing vector component self interactions, see e.g. [85], (3) a very small/large mass splitting among the dark sector states (vector and Majorana fermions) are possible without large tuning of the parameters, (4) our model is a gauged version of the model considered by Weinberg [86] for different purposes, and (5) more importantly the minimality of the model; since there is only one parameter, the dark gauge coupling coupling \(g_{\textsc {x}}\), which controls the dynamics in the dark sector, including the conversion, semiannihilation and decay processes. In this work, we are going to illustrate the relevance of conversions, semiannihilations, and decays in the vectorfermion DM model.
The paper is organized as follows. In Sect. 2 the vectorfermion (23 component) model of DM is presented. Solutions of three coupled Boltzmann equations are discussed in Sect. 3 focusing on conversion, semiannihilation, and decay processes. There we show results of a detailed scan over the parameter space of the model satisfying the observed total relic density and direct detection constraints. In Sect. 4 we focus on the region with large selfinteraction crosssection. Section 5 contains summary and conclusions. Moreover, we supplement our work with an “Appendix A”, collecting details of the derivation of Boltzmann equations, and an “Appendix B”, describing the method adopted to obtain constraints for a multicomponent DM model by direct detection experiments.
2 Vectorfermion multicomponent dark matter model
Discrete symmetries of the vectorfermion DM model
Symmetry  \(X_\mu \)  \(\psi _{\!+}\)  \(\psi _{\!}\)  \(h_i(\mathrm{SM})\) 

\(Z_2\)  −  \(+\)  −  \(+\) 
\(Z_2^\prime \)  −  −  \(+\)  \(+\) 
\(Z_2^{\prime \prime }\)  \(+\)  −  −  \(+\) 

The first case is when \(m_+ \!>\! m_+m_X\), the Majorana fermion \(\psi _{\!+}\) will decay into a stable vector \(X_\mu \) and a stable Majorana fermion \(\psi _{\!}\). This is a 2CDM case, the white area (left) in Fig. 2.

The second case is when \(m_X\!>\! m_++m_\), the vector \(X_\mu \) will decay into two stable Majorana fermions \(\psi _\pm \). This is a 2CDM case, the white area (right) in Fig. 2.

The third case is when \(m_+ \!+\!m_\!>\!m_X\!>\!m_+\!\!m_\), so that none of the three particle will decay and hence all are stable. This is a 3CDM case, shown as gray region in Fig. 2. Note that the boundaries (right/left) of the gray region correspond to the case when \(m_X=m_+\pm m_\).
2.1 Input parameters
3 Vectorfermion dark matter phenomenology

Conversions are present even in the absence of the dark sector selfinteractions, an existence of a mediator is the only requirement. On the other hand, the existence of semiannihilations and decays of dark particle depend on the presence a vertex with three dark states, which have different transformation rules under the dark symmetry (such that a singlet could be formed), in our model such an interaction is in Eq. (24).

When, for a given dark matter species, a standard annihilation channel is suppressed then its abundance might be very sensitive to the presence of other ingredients of the dark segment. In this case semiannihilation plays a major role, e.g. if \(\psi _\psi _ \rightarrow h_ih_i\) (or any SM states) is suppressed then \(\psi _\) can still disappear, for instance, through \(\psi _ \psi _+ \rightarrow X_\mu h_i\) followed by unsuppressed annihilation of pairs of \(X_\mu \), see also [45]. In other words, \(X_\mu \) can work as a catalyzer that enables disappearance of \(\psi _\). In this case, it is possible that the presence of other (\(\psi _+\) and/or \(X_\mu \)) dark components might be crucial for the determination of the asymptotic abundance of the major DM element. Also, decays within the dark sector may play a relevant role in the determination of the final abundance.

Standard WIMPs decouple from thermal equilibrium at \(m/T \!\sim \!20\!\!25\), which implies that the heavy states decouple earlier (large T). However, in the multicomponent scenario, it might be possible that the decoupling of a heavier dark component is delayed so that it happens later than that of a lighter one. The effect is again a consequence of interactions with remaining dark matter states.
 (A)
\({y_{\textsc {x}}\ll 1 (m_+\simeq m_)}\): Small \(y_{\textsc {x}}\) implies suppressed \(\psi _\pm \psi _\pm \) annihilation, so \(\psi _\pm \) dominates the dark matter abundance. Since the annihilation is slow therefore \(Y_{\psi _\pm }\) is controlled by semiannihilation which is sensitive to \(g_{\textsc {x}}\) and to the presence of other dark components. In order to have semiannihilation controlled exclusively by \(g_{\textsc {x}}\) one should assume \(m_++m_ > m_X+m_{h_2}\).
 (B)
\({y_{\textsc {x}}\gg 1 (m_+\gg m_)}\): In this case, one expects fast \(\psi _\pm \psi _\pm \) annihilation and so that \(X_\mu \) may dominate the dark matter abundance. If in addition \(\sin \!\alpha \ll 1\) then XX annihilation would be suppressed so that \(Y_X\) shall be controlled by semiannihilation and conversion processes which are sensitive to the gauge coupling \(g_{\textsc {x}}\) and Yukawa coupling \(y_{\textsc {x}}\). In both cases, \(X_\mu \) would be effectively replaced by \(\psi _\pm \), which then would disappear through enhanced standard annihilation.
3.1 Multicomponent cases
In the following sections we consider various interesting setups with two or three dark particles. Matrix elements squared needed for collision terms in the Boltzmann equations (30)–(32) are computed by employing the CalcHEP [90], whereas for thermal averaging and solutions of the Boltzmann equations, we adopt our dedicated C++ code.^{5}
3.1.1 2CDM: a vector and a Majorana fermion as dark matter
The right panels correspond to parameter points such that around \(m_X\!=\!200\;\hbox {GeV}\) there is a resonant enhancement of XX annihilation through the \(h_2\) schannel exchange. Of course, there is also a nonresonant contribution from \(\psi _\psi _\) annihilation. In the case of \(h_2\) resonance, in order to satisfy the relic abundance constraint, the relevant coupling constants must be small, i.e. \(g_X \sin \alpha \ll 1\), it turns out that for the region of \(\sin \alpha \) considered here the gauge coupling constants must be in the range \(g_{\textsc {x}}\!\sim \!0.030.13\).
In the plots of Fig. 7, and similar figures in the following sections, we show dark matter yields \(Y_i(x) \!\equiv \! n_i/s\) (s is the total entropy density and x is defined as \(x \!\equiv \! m_X/T\)) for different species \(i=\psi _+,\psi _\) and \(X_\mu \). Note that we plot bare values of yields, with no extra normalization adopted. Moreover, the tables in Fig. 7 and similar figures in the following section contain first two nonvanishing coefficients of thermallyaveraged crosssections (pb) expanded in powers of \(x^{1}\), given by \(\langle \sigma ^{ijkl}v_{\text {M}\varnothing \text {l}}\rangle \! =\! a_Nx^{N} + a_{N+1}x^{(N+1)} + \cdots \), and the decay width \(\langle \varGamma _{\psi _+\rightarrow X\psi _}\rangle \) (GeV). The plots in Fig. 7 illustrate solutions of the Boltzmann equations for three selected sample points. The middle panel shows solutions for parameters that reproduce correct total DM abundance and also satisfy the direct detection LUX limits, so the corresponding point is also present in the scan results shown in the upperleft panel of Fig. 6 as a black star \(\star \). In order to illustrate the relevance of the \(g_{\textsc {x}}\) coupling, the left, middle and right plots are obtained for \(g_{\textsc {x}}=0.02, 0.2\) and 1, respectively, while other parameters remain unchanged. It is clear that the dependence of the abundance for the major DM component (\(\psi _\)) on \(g_{\textsc {x}}\) is very strong. The gray dots and boxes show results obtained using the micrOMEGAs code for 2CDM [91]. As it is seen in the plots they agree very well with the solid lines which correspond to solutions obtained from our dedicated C++ code that solves the set of three Boltzmann equations (30)–(32). In order to identify the most important processes for a given parameter set, in the tables below the panels in Fig. 7 we collect the first two nonvanishing coefficients in the expansion of thermally averaged crosssections in powers of \(x^{1}\).
Let’s look closer at the middle table of Fig. 7. The crosssections shown there correspond to the point in the parameter space marked by \(\star \) which is located in the upper left panel of Fig. 6 at \(m_X = 245\) GeV. As seen from the middle upper panel of Fig. 7 the abundance is dominated by \(\psi _\), and X is a subleading component abundance of which is by nearly two orders of magnitude smaller than for \(\psi _\), while the abundance of \(\psi _+\) is absolutely negligible. Note that both \(\psi _\) and X decouple from equilibrium roughly at the same temperature, this is the first signal that there must be some correlation between annihilation mechanisms responsible for their disappearance. Since the abundance of \(\psi _+\) could be neglected the only relevant processes are \(XX \rightarrow \text {SM}\), \(\psi _\psi _ \rightarrow \text {SM}\) and \(XX \rightarrow \psi _\psi _\). Note that the ratio of crosssections for the first and the second process is \(\sim 1.8\) while their abundances differ by almost two orders of magnitude, therefore the process for additional depletion of X abundance must be \(XX \rightarrow \psi _\psi _\). This illustrates how subleading components may influence the abundance of a dominant component.
3.1.2 2CDM: two Majorana fermions as dark matter
Figure 8 shows results of a scan over \(\sin \!\alpha , \,g_{\textsc {x}},\, m_\) for fixed \(m_{h_2}, m_X\) and \(\varDelta m= 100\;\hbox {GeV}\) (left panel) or \(50\;\hbox {GeV}\) (right panel). All the points satisfy the correct relic density (for the total abundance) observed by PLANCK at \(5\sigma \) and the recent direct detection experimental bound from LUX2016 at \(2\sigma \). In this case the \(\psi _\) turns out to be the dominant DM component in most of the parameter space. The second Higgs boson mass was chosen to be \(m_{h_2}=120, 125, 130\;\hbox {GeV}\) and \(390, 400, 410\;\hbox {GeV}\) in the left and right panels, respectively. Therefore the left panel allows for partial cancellation between an exchange of \(h_1\) and \(h_2\) both for \(\psi _\pm \) annihilation diagrams and also for \(\psi _\pm \)nuclei scattering process to avoid the direct detection limits even if couplings are not small. Since \(50\;\hbox {GeV}\le m_ \le 200\;\hbox {GeV}\) therefore one can observe both a resonance behavior at \(m_\!\sim \!m_{h_1}/2 \!\sim \! m_{h_2}/2\) in \(\psi _\pm \psi _\pm \) annihilation trough schannel \(h_{1,2}\) exchange and also a threshold effect at \(m_\sim m_{h_1} \sim m_{h_2}\) for annihilation into \(h_ih_j\) final state. The vertical structure observed in Fig. 8 around \(m_\sim 60\;\hbox {GeV}\) corresponds to a domination of \(\psi _\pm \psi _\pm \rightarrow h_i^*\rightarrow VV,\bar{f} f\). As it is seen from the plot large values of \(g_{\textsc {x}}\) are needed, this is a consequence of partial cancellation between \(h_1\) and \(h_2\) exchange. On the other hand, the independence on \(g_{\textsc {x}}\) could be understood as a result of resonance enhancement around \(m_\! \sim \! m_{h_1}/2\! \sim \! m_{h_2}/2\): even a tiny change of \(m_\) can compensate large variation of \(g_{\textsc {x}}\). The other triplebranch structure that starts around \(m_ \sim 120\;\hbox {GeV}\) corresponds to a threshold for the process \(\psi _\pm \psi _\pm \rightarrow h_i h_j\). Its initial steepness represents the opening of the \(h_i h_j\) final state that must be compensated by suppression of \(g_{\textsc {x}}\) in order to generate correct dark matter abundance.
The right panel of Fig. 8 with its three distinct branches corresponds to a vicinity of the resonance at \(m_\!\sim \! m_{h_2}/2 \!=\! (390, 400, 410)/2\;\hbox {GeV}\). In this case \(g_{\textsc {x}}\) must be small to compensate the resonance enhancement, therefore direct detection limits are easily satisfied.
3.1.3 3CDM: a vector and two Majorana fermions as dark matter
All the points presented in Fig. 10 satisfy the correct relic density (for the total abundance) observed by PLANCK at \(5\sigma \) and the direct detection experimental bound from LUX2016 at \(2\sigma \). The scan is performed over \(m_, \; g_x\) with fixed values of \(\sin \!\alpha \!=\!(0.05,0.1,0.2)\), \(m_X\!=\!(200,150,100) \;\hbox {GeV}\) and \(m_{h_2}\!=\!(200,120,50) \;\hbox {GeV}\). Note that the left and right panels of Fig. 10 are for the same dataset but for different filling style, in the left and right panel the filling corresponds to \(m_X\) and \(m_{h_2}\), respectively. Here we have tested sensitivity to \(\varDelta m\equiv m_+m_\) focusing on small \(\varDelta m\!=\!(0.1, 1, 10)\) GeV. As it is seen from the Fig. 10, for \(m_\sim m_+\gtrsim 200\) GeV relatively small \(U(1)_X\) coupling is required, \(g_{\textsc {x}}=0.51\), in order to suppress too fast \(\psi _\pm \psi _\pm \) schannel annihilation. Note that \(y_{\textsc {x}}\!=\! \varDelta m\, g_{\textsc {x}}/(2m_X)\) therefore this annihilation is already quite strongly suppressed by the small Yukawa coupling \(y_{\textsc {x}}\). An important final state is \(t{\bar{t}}\), so if \(m_\sim m_+ \lesssim m_t\) this annihilation channel closes so that even large \(g_{\textsc {x}}\) is allowed/necessary, as observed in the figure.
Sets of parameters implying limiting VDM and FDM cases with proper \(\varOmega _{DM}\). Relic densities of \(X_\mu \) and \(\psi _\) are provided, density of \(\psi _+\) is negligible. All the masses are in GeV
\(m_X\)  \(m_+\)  \(m_\)  \(m_{h_2}\)  \(g_X\)  \(\sin \alpha \)  \(\varOmega _{\textsc {x}}h^2\)  \(\varOmega _h^2\) 

VDM  
100  405  400  180  0.4  0.1  0.121  \(1.72 \times 10^{15}\) 
200  705  700  120  0.256  0.1  0.121  \(1.61 \times 10^{19}\) 
FDM  
100  2500  19  50  0.3  0.3  \(5.71 \times 10^{4}\)  0.121 
100  \(5\!\cdot \! 10^{4}\)  40  140  0.1  0.25  \(1.20 \times 10^{4}\)  0.120 
3.2 Limiting cases
In this section we are going to discuss special regions in the parameter space of the model that result in simpler, models of DM.
3.2.1 The vector dark matter (VDM) model
If mass splitting between \(\psi _+\) and \(\psi _\) is small comparing to \(m_X\), then, for fixed \(g_X\), the Yukawa coupling \(y_X\) is suppressed as \(y_X=\varDelta m/(2 v_x)=(\varDelta m/m_X)(g_X/2)\). Therefore, in this limit, since Yukawa couplings become irrelevant, the model might be reduced to the 1component VDM model (see e.g. [83, 85] where the same notation as here has been adopted). Note however that even though fermionic DM decouple from the SM, nevertheless it is still present and may influence cosmological dynamics and contribute to the observed amount of DM. In order to enable efficient \(\psi _\pm \) annihilation it is sufficient to assume that \(2 m_D > m_X+m_i\) and/or \(m_X > m_i\) (\(m_D\equiv (m_++m_)/2\)) so that at present only \(X_\mu \) contributes to the observed DM abundance and could be successfully fitted by tuning \(g_X\). Samples of parameter sets that imply proper \(\varOmega _{DM}\) and fit in the VDM limit are shown in Table. 2. In Ref. [85] the VDM has been analyzed focusing on the possibility of enhancing selfinteraction, and some regions of the parameters space where elastic XX scattering is amplified and all other constraints are satisfied have been found.
3.2.2 The fermion dark matter (FDM) model
Another interesting limit of our model is a renormalizable model of fermionic DM, see e.g. [92, 93, 94, 95]. Those models usually employ an extra singlet real scalar field that couples to the SM Higgs doublet via the Higgs portal and to a singlet dark Dirac fermion as well. In the fermionic DM limit of our model, we recover a model of a Majorana singlet DM that couples to a complex scalar S. Since our model is invariant under local \(U(1)_X\) therefore in the limit of small gauge coupling, \(g_X \ll 1\), and substantial masssplitting between \(\psi _+\) and \(\psi _\) (so for enhanced Yukawa coupling, \(y_X\)), effectively we obtain a renormalizable model symmetric under a global \(U(1)_X\) of a single Majorana dark fermion \(\psi _\) interacting with S. The scalar S controls communication between dark sector and the SM. Examples of parameter sets that imply proper \(\varOmega _{DM}\) and fit in the FDM limit are shown in Table 2. The model is slightly more restrictive than those considered earlier in [92, 93, 94, 95] since our DM is a Majorana fermion and the scalar potential is more restricted, as being invariant under the global \(U(1)_X\), however predictions of those models are similar, see [94]. In order to obtain fermionic (Majorana) DM model one should reduce \(g_X\) in order to decouple \(X_\mu \), keeping in mind that certain minimal interaction strength is necessary for \(X_\mu \) to maintain kinetic equilibrium. Note that in order to reduce the model to a single fermionic DM model, we have to remove somehow \(\psi _+\) and \(X_\mu \). The easiest way to get rid of \(\psi _+\) is to assume that it is the heaviest dark state, so that it will have a chance to decay quickly. If the following mass ordering, \(m_+> m_X > m_\) is fulfilled, then indeed the dominant DM component is the Majorana fermion \(\psi _\), while other dark components disappear. The model contains, of course, two scalar Higgs bosons that mix in the standard manner and play a role of mediators between the dark sector and the SM. Selfinteraction in the FDM model has been discussed in [96]. It turns out that for the selfinteractions to be sufficiently strong, the scalar mediator, \(h_2\), has to be very light what implies wellknown problems [97] in the early Universe if \(h_2\) is present during the era of BBN.
3.2.3 The fermion dark matter (FDM) model with a stable vector mediator
Another interesting limit of our model has been considered very recently in [98]. If, in our model, \(\varDelta m \rightarrow 0\) then the Yukawa coupling \(y_X\) vanishes and masses of Majorana fermions \(\psi _+\) and \(\psi _\) become degenerate. Then our model reduces to the model considered in [98], which is just a model of a Dirac fermion as a DM and a stable vector mediator. Their [98] mediator corresponds to our vector component of DM, \(X_\mu \), while the dark Dirac fermion is an analog of our degenerate Majorana dark fermions \(\psi _+\) and \(\psi _\). The authors of [98] show that the model can indeed predict enhanced DM selfinteraction while satisfying all existing experimental constraints if mass of the stable vector mediator is of the order of \(1\;\hbox {MeV}\). This has been also confirmed in the appropriate (\(\varDelta m \rightarrow 0\), \(m_X \sim {{\mathcal {O}}}(1) \;\hbox {MeV}\)) region of the parameter space of our model. In this case the DM abundance is dominated by massdegenerate \(\psi _\pm \), even though formally it is a 3component case (3CDM) if \(\varDelta m < m_X\).
3.3 Distinguishing limiting cases

Direct detection
Contributions to DMnucleon scattering consists of the sum of standard \(\sigma _{XN}\) and \(\sigma _{\psi _\pm N}\) crosssections that are not sensitive to the presence of all the 23 DM components, rather this is a sum of contributions that exist in 1component models. There exists however a more interesting inelastic scattering process which is sensitive to the multicomponent nature of the model considered here, i.e. \(\psi _+ N\rightarrow \psi _ X N\), note that all the dark particle are involved, so that this process might provide a signature of the multicomponent scenario or perhaps some useful correlation with other observables. This process could be enhanced (and therefore efficiently constrained) for small \(m_X\) which on the other hand helps to enhance \(\psi _\pm \) selfinteraction.

Indirect detection
Similarly indirect detection experiments, besides standard \(XX\rightarrow SM\) and \(\psi _\pm \psi _\pm \rightarrow SM\) contributions receive also more interesting one \(\psi _+\psi _\rightarrow X h_i\) followed by \(h_i\) decays.

Colliders
\(e^+e^\) colliders provide a clean environment that might be used to test the multiDM scenario considered here. Namely one can investigate the process \(e^+e^\rightarrow Z^* \rightarrow h_i Z \rightarrow \chi _D\chi _D Z\) with \(\chi _D=X\) or \(\psi _\pm \). Energydistribution of Z might be adopted to gain some information on the invisible objects being produced. Initial estimation indicates that in some regions of the parameter space, for sufficiently large luminosity one should be able to disentangle 1 and 23 component scenarios.
4 Selfinteracting DM
In the Fig. 12, we show results of detailed scans focused on that region. The relic abundance was calculated using micrOMEGAs code [91] by placing \(\psi _+\) and \(\psi _\) in one dark sector and X in another. Here we assume that both fermions are kept in equilibrium with each other by the efficient exchange processes \(\psi _\pm X \leftrightarrow \psi _\pm h_2\). The scan was made over masses in the range \(m_X\in [1,15]\) MeV, \(m_D\in [1,10]\) GeV for fixed values of \(m_{h_2}\in \{1,2,5\}\) MeV and \(\sin \alpha \in \{10^{5},10^{6},10^{7}\}\). Parameter \(g_x\) was fitted imposing the condition that density of fermions satisfies relic abundance constraint whereas contribution of X is negligible. The latter is achieved by the effective annihilation \(XX\rightarrow h_2h_2\) if \(h_2\) is lighter than X. We choose the mass splitting \(m_+m_ = 10^{5}\) GeV. In this way we can ensure that both states are present with nearly the same relic abundance. Moreover as it leads to the suppression of the Yukawa coupling, therefore we can avoid the indirect detection bounds. Another strong constraint comes from the limit on the invisible Higgs decay \(h_1\rightarrow h_2 h_2\). It results in the bound \(\sin \alpha \le 10^{5}\), which on the other hand suppresses the DMnucleon scattering crosssection to the range which is in agreement with direct detection experiments.
Similar scenario was discussed in the case of Dirac fermion in [98]. Since here we focus on the small mass splitting therefore our model effectively also contains a Dirac dark fermion and a stable vector as in [98]. Therefore, our results for \(\sigma _T/m_D\) shown in Fig. 12 indeed agree with those obtained in [98]. Note however, this accordance takes place only in this particular region of the parameter space, while in general the models are quite different, for instance, by the presence of Yukawa interactions that are allowed in our model due to specific assignments of dark charges. More comprehensive analysis of our model will be presented elsewhere.
5 Summary and conclusions
Multicomponent dark matter scenarios are natural extensions of a simple WIMP dark matter. They predict more than one stable component in a dark sector and therefore they constitute a much richer dynamical structure. In this work we have presented a minimal UVcomplete vectorfermion DM model with two or three stable particles. Its dynamical properties were discussed. Our vectorfermion DM model involves a dark sector with a \(U(1)_X\) gauge symmetry. The dark matter contents are the dark gauge boson \(X_\mu \), a Dirac fermion \(\chi \), and a complex scalar S, all are charged under the dark \(U(1)_X\) gauge symmetry and are neutral under the SM gauge symmetry. Moreover, all the SM particles are neutral under the dark \(U(1)_X\) gauge symmetry. The dark sector communicates with the visible sector (SM) through the Higgs portal \(\kappa H^2S^2\). To generate the dark gauge boson \(X_\mu \) mass we have employed the Higgs mechanism in the dark sector.
After the dark sector spontaneous symmetry breaking and mass diagonalization, our vectorfermion DM scenario comprises a dark vector \(X_\mu \) and two dark Majorana fermions \(\psi _\pm \). Out of eight free parameters of the model, the SM Higgs vev \(v\!=\!246\;\hbox {GeV}\) and the SMlike Higgs mass \(m_{h_1}\!=\!125\;\hbox {GeV}\) are fixed which leaves us with six independent parameters. We have chosen the physical basis where the six independent parameters are four masses \(m_X,m_\pm , m_{h_2}\), the mixing angle \(\sin \!\alpha \), and the dark gauge coupling \(g_{\textsc {x}}\). To guarantee perturbativity we assumed \(g_{\textsc {x}}\!\le \!4\pi \). We have employed \(\sin \!\alpha \le 0.33\), which is consistent with the \(2\sigma \) constraint from current measurements of the SMlike Higgs boson couplings to the SM gauge bosons at the LHC. Our VFDM model has an exact charge conjugation symmetry and the dark gauge symmetry which result in an accidental discrete \(Z_2\!\times \!Z_2^\prime \) symmetry. The charge assignments under this \(Z_2\!\times \!Z_2^\prime \) symmetry are: \(X_\mu (,),\psi _{+}(,+), \psi _{}(+,)\) and \(h_{1,2}(+,+)\) (also all SM gauge bosons and fermions are even under both discrete symmetries). The dynamics of the dark sector is mainly controlled by the gauge coupling \(g_{\textsc {x}}\) which couples the three dark fields, i.e. \(X_\mu {\bar{\psi }}_{+}\gamma ^\mu \psi _{}\).
 (i)
\({m_+>m_X+m_}\): A twocomponent dark matter case where the stable particles are the vector \(X_\mu \) and the Majorana fermion \(\psi _\), see Sect. 3.1.1. In this case we have performed scans over \(m_X, g_{\textsc {x}}\) for different values of \(m_\pm ,m_{h_2}\) and \(\sin \!\alpha \) to search for regions in the parameter space where the dark matter total relic density and current direct detection constraints are satisfied. Importance of the presence of other dark sector states and their interactions, in particular, the semiannihilations and conversions has been manifested. Moreover, we have compared the twocomponent vectorfermion case with the singlecomponent vector dark matter and highlighted the presence of second component, the latter one is especially useful to compensate the underabundance of the singlecomponent vector dark matter.
 (ii)
\({m_X>m_++m_}\): A twocomponent dark matter case where the stable particles are the two Majorana fermions \(\psi _+\) and \(\psi _\), see Sect. 3.1.2. In this case we have performed scans over \(m_,g_{\textsc {x}}\) for different choices of \(m_X,\varDelta m,m_{h_2}\) and \(\sin \!\alpha \) which satisfy the correct total relic abundance and direct detection bounds. As in the previous case, we have highlighted the importance of the presence of more than one stable states in the dark sector and their interactions. In particular, we have illustrated effects of semiannihilations in Fig. 9, which are primarily controlled by the single coupling \(g_{\textsc {x}}\).
 (iii)
\({m_++m_>m_X>m_+m_}\): A threecomponent dark matter scenario where all three dark sector particles \((X_\mu ,\psi _+,\psi _)\) are stable, see Sect. 3.1.3. As in the twocomponent DM cases we have performed scan over \(m_, g_{\textsc {x}}\) for different choices of \(m_X,\varDelta m,m_{h_2}\) and \(\sin \!\alpha \). To demonstrate the importance of three stable dark matter particles, we have illustrated in Fig. 11 the case with two Majorana fermions nearly degenerate in mass, i.e. \(y\ll 1\), hence their standard annihilations are suppressed, due to small Yukawa couplings, and the semiannihilations are most important for their annihilations.
We have also discussed limiting cases of the model that are realized in appropriate regions of the parameter space. One of them corresponds to a model with Dirac fermion DM and stable vector mediator, this is an interesting scenario. A possibility of selfinteracting DM has also been addressed and the region of parameter space where \(\sigma _T/m_{DM}\) can be substantially enhanced has been found.
To summarize, the absence of any direct, indirect or collider signatures of dark matter suggests a direction that leads beyond the single component WIMPlike dark matter. In particular, multicomponent dark matter scenarios offer very rich dynamical structures which could solve current dark matter puzzles. In this work, we have presented a minimal renormalizable vectorfermion dark matter model where the presence of gauge symmetry and charge conjugation in the hidden sector implies the existence of two or three stable (vector and/or Majorana fermions) dark matter particles. The dynamics of the dark sector in our model is primarily controlled by a single parameter, the dark gauge coupling \(g_{\textsc {x}}\) through the interaction \(\big ({\bar{\psi }}_{\!+} \gamma ^\mu \psi _{\!}  {\bar{\psi }}_{\!} \gamma ^\mu \psi _{\!+}\big ) X_\mu \) which connects all dark sector states \(X_\mu \), \(\psi _+\) and \(\psi _\). Such an interaction allows semiannihilation and decay processes within the dark sector. We have explored the parameter space of our two/threecomponent VFDM scenarios requiring the correct total relic density and compliance with the current direct detection bounds.
Footnotes
 1.
 2.
Note that because of the \(U(1)_X\) symmetry, \(v_{\textsc {x}}\) can be chosen to be real and positive. Therefore the charge conjugation (6) remains unbroken.
 3.
For generic discussion on \(Z_N\) discrete symmetries as the residual of an Abelian gauge symmetry see [63].
 4.
Since we have assumed \(m_\!<\!m_+\), hence there are only two 2CDM cases.
 5.
Note that the presence of three stable dark matter components is quite generic in models with two interacting stable states. In this case, even the 2component version of micrOMEGAs [91] is not applicable as the code assumes there are at most two DM sectors within which particles remain in equilibrium. Therefore for the case of 3component DM, we adopt our dedicated code which employs the full set of three Boltzmann equations.
 6.
Unfortunately micrOMEGAs is limited to at most two dark matter components.
 7.
With a factor \(1/S_f\), where the final state symmetry factor \(S_f=\varPi _{n=1}^{n=N}m_n !\) accounts for N groups of identical final state particles of multiplicity \(m_n\).
Notes
Acknowledgements
We would like to thank Da Huang for many useful discussions. The research of AA has been supported by the Cluster of Excellence Precision Physics, Fundamental Interactions and Structure of Matter (PRISMAEXC1098) and grant 05H12UME of the German Federal Ministry for Education and Research (BMBF). MD acknowledges support of the National Science Centre (Poland), research project no. 2017/25/N/ST2/01312. This work has been supported in part by the National Science Centre (Poland) under research projects 2014/15/B/ST2/00108 and 2017/25/B/ST2/00191.
References
 1.W.M.A.P. Collaboration, G. Hinshaw et al., NineYear Wilkinson Microwave Anisotropy Probe (WMAP) observations: cosmological parameter results. Astrophys. J. Suppl. 208, 19 (2013). https://doi.org/10.1088/00670049/208/2/19. arXiv:1212.5226
 2.Planck Collaboration, P.A.R. Ade et al., Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys. 594, A13 (2016). https://doi.org/10.1051/00046361/201525830. arXiv:1502.01589
 3.Y. Sofue, V. Rubin, Rotation curves of spiral galaxies. Ann. Rev. Astron. Astrophys. 39, 137–174 (2001). https://doi.org/10.1146/annurev.astro.39.1.137. arXiv:astroph/0010594
 4.M. Bartelmann, P. Schneider, Weak gravitational lensing. Phys. Rept. 340, 291–472 (2001). https://doi.org/10.1016/S03701573(00)00082X. arXiv:astroph/9912508
 5.D. Clowe, A. Gonzalez, M. Markevitch, Weak lensing mass reconstruction of the interacting cluster 1E0657558: direct evidence for the existence of dark matter. Astrophys. J. 604, 596–603 (2004). https://doi.org/10.1086/381970. arXiv:astroph/0312273
 6.LUX Collaboration, D.S. Akerib et al., Results from a search for dark matter in the complete LUX exposure. Phys. Rev. Lett. 118(2), 021303 (2017). https://doi.org/10.1103/PhysRevLett.118.021303. arXiv:1608.07648
 7.XENON100 Collaboration, E. Aprile et al., Dark matter results from 225 live days of XENON100 data. Phys. Rev. Lett. 109, 181301 (2012). https://doi.org/10.1103/PhysRevLett.109.181301. arXiv:1207.5988
 8.M. BoylanKolchin, J.S. Bullock, M. Kaplinghat, Too big to fail? The puzzling darkness of massive Milky Way subhaloes. Mon. Not. R. Astron. Soc. 415, L40 (2011). https://doi.org/10.1111/j.17453933.2011.01074.x. arXiv:1103.0007
 9.S. GarrisonKimmel, M. BoylanKolchin, J.S. Bullock, E.N. Kirby, Too big to fail in the local group. Mon. Not. R. Astron. Soc. 444(1), 222–236 (2014). https://doi.org/10.1093/mnras/stu1477. arXiv:1404.5313
 10.B. Moore, Evidence against dissipationless dark matter from observations of galaxy haloes. Nature 370, 629 (1994). https://doi.org/10.1038/370629a0 ADSCrossRefGoogle Scholar
 11.R.A. Flores, J.R. Primack, Observational and theoretical constraints on singular dark matter halos. Astrophys. J. 427, L1–L4 (1994). https://doi.org/10.1086/187350. arXiv:astroph/9402004
 12.S.H. Oh, C. Brook, F. Governato, E. Brinks, L. Mayer, W.J.G. de Blok, A.Brooks, F. Walter, The central slope of dark matter cores in dwarf galaxies: simulations vs. THINGS. Astron. J. 142, 24 (2011). https://doi.org/10.1088/00046256/142/1/24. arXiv:1011.2777
 13.M.G. Walker, J. Penarrubia, A method for measuring (slopes of) the mass profiles of dwarf spheroidal galaxies. Astrophys. J. 742, 20 (2011). https://doi.org/10.1088/0004637X/742/1/20. arXiv:1108.2404
 14.M. Rocha, A.H.G. Peter, J.S. Bullock, M. Kaplinghat, S. GarrisonKimmel, J. Onorbe, L.A. Moustakas, Cosmological simulations with selfinteracting dark matter I: constant density cores and substructure. Mon. Not. R. Astron. Soc. 430, 81–104 (2013). https://doi.org/10.1093/mnras/sts514. arXiv:1208.3025
 15.D.H. Weinberg, J.S. Bullock, F. Governato, R. Kuzio de Naray, A.H.G. Peter, Cold dark matter: controversies on small scales. Proc. Nat. Acad. Sci. 112, 12249–12255 (2014). https://doi.org/10.1073/pnas.1308716112. arXiv:1306.0913 [Proc. Nat. Acad. Sci. 112, 2249 (2015)]
 16.D.N. Spergel, P.J. Steinhardt, Observational evidence for selfinteracting cold dark matter. Phys. Rev. Lett. 84, 3760–3763 (2000). https://doi.org/10.1103/PhysRevLett.84.3760. arXiv:astroph/9909386
 17.FermiLAT Collaboration, W.B. Atwood et al., The large area telescope on the Fermi Gammaray space telescope mission. Astrophys. J. 697, 1071–1102 (2009). https://doi.org/10.1088/0004637X/697/2/1071. arXiv:0902.1089
 18.A. Boyarsky, O. Ruchayskiy, D. Iakubovskyi, J. Franse, Unidentified line in XRay spectra of the andromeda galaxy and perseus galaxy cluster. Phys. Rev. Lett. 113, 251301 (2014). https://doi.org/10.1103/PhysRevLett.113.251301. arXiv:1402.4119
 19.E. Bulbul, M. Markevitch, A. Foster, R.K. Smith, M. Loewenstein, S.W. Randall, Detection of an unidentified emission line in the stacked Xray spectrum of galaxy clusters. Astrophys. J. 789, 13 (2014). https://doi.org/10.1088/0004637X/789/1/13. arXiv:1402.2301
 20.F. Calore, I. Cholis, C. McCabe, C. Weniger, A Tale of Tails: dark matter interpretations of the fermi GeV excess in light of background model systematics. Phys. Rev. D 91(6), 063003 (2015). https://doi.org/10.1103/PhysRevD.91.063003. arXiv:1411.4647
 21.B. Grzadkowski, J. Wudka, Pragmatic approach to the little hierarchy problem: the case for Dark Matter and neutrino physics. Phys. Rev. Lett. 103, 091802 (2009). https://doi.org/10.1103/PhysRevLett.103.091802. arXiv:0902.0628
 22.B. Grzadkowski, J. Wudka, Naive solution of the little hierarchy problem and its physical consequences. Acta Phys. Polon. B 40, 3007–3014 (2009). arXiv:0910.4829 ADSGoogle Scholar
 23.A. Drozd, B. Grzadkowski, J. Wudka, Multiscalarsinglet extension of the standard model—the case for Dark Matter and an invisible Higgs Boson. JHEP 04, 006 (2012). https://doi.org/10.1007/JHEP04(2012)006, https://doi.org/10.1007/JHEP11(2014)130. arXiv:1112.2582 [Erratum: JHEP 11, 130 (2014)]
 24.B. Grzadkowski, P. Osland, J. Wudka, Pragmatic extensions of the standard model. Acta Phys. Polon. B 42(11), 2245 (2011). https://doi.org/10.5506/APhysPolB.42.2245
 25.G. Belanger, K. Kannike, A. Pukhov, M. Raidal, Impact of semiannihilations on dark matter phenomenology—an example of \(Z_N\) symmetric scalar dark matter. JCAP 1204, 010 (2012). https://doi.org/10.1088/14757516/2012/04/010. arXiv:1202.2962
 26.I.P. Ivanov, V. Keus, \(Z_p\) scalar dark matter from multiHiggsdoublet models. Phys. Rev. D 86, 016004 (2012). https://doi.org/10.1103/PhysRevD.86.016004. arXiv:1203.3426
 27.K.P. Modak, D. Majumdar, S. Rakshit, A possible explanation of low energy \(\gamma \)ray excess from galactic centre and fermi bubble by a Dark Matter Model with two real scalars. JCAP 1503, 011 (2015). https://doi.org/10.1088/14757516/2015/03/011. arXiv:1312.7488
 28.M. Aoki, J. Kubo, H. Takano, Twoloop radiative seesaw mechanism with multicomponent dark matter explaining the possible \(\gamma \) excess in the Higgs boson decay and at the Fermi LAT. Phys. Rev. D 87(11), 116001 (2013). https://doi.org/10.1103/PhysRevD.87.116001. arXiv:1302.3936
 29.A. Biswas, D. Majumdar, A. Sil, P. Bhattacharjee, Two component Dark Matter: a possible explanation of 130 GeV \(\gamma \) Ray line from the galactic centre. JCAP 1312, 049 (2013). https://doi.org/10.1088/14757516/2013/12/049. arXiv:1301.3668
 30.A. Drozd, B. Grzadkowski, J. Wudka, Cosmology of multisingletscalar extensions of the standard model. Acta Phys. Polon. B 42(11), 2255 (2011). https://doi.org/10.5506/APhysPolB.42.2255. arXiv:1310.2985
 31.A. Biswas, D. Majumdar, P. Roy, Nonthermal two component dark matter model for FermiLAT \(\gamma \)ray excess and 3.55 keV Xray line. JHEP 04, 065 (2015). https://doi.org/10.1007/JHEP04(2015)065. arXiv:1501.02666
 32.S. Bhattacharya, P. Poulose, P. Ghosh, Multipartite interacting scalar Dark Matter in the light of updated LUX data. JCAP 1704(04), 043 (2017). https://doi.org/10.1088/14757516/2017/04/043. arXiv:1607.08461
 33.S. Bhattacharya, P. Ghosh, T.N. Maity, T.S. Ray, Mitigating direct detection bounds in nonminimal Higgs portal scalar Dark Matter models. arXiv:1706.04699
 34.Q.H. Cao, E. Ma, J. Wudka, C.P. Yuan, Multipartite dark matter. arXiv:0711.3881
 35.J.H. Huh, J.E. Kim, B. Kyae, Two dark matter components in dark matter extension of the minimal supersymmetric standard model and the high energy positron spectrum in PAMELA/HEAT data. Phys. Rev. D 79, 063529 (2009). https://doi.org/10.1103/PhysRevD.79.063529. arXiv:0809.2601
 36.H. Fukuoka, D. Suematsu, T. Toma, Signals of dark matter in a supersymmetric two dark matter model. JCAP 1107, 001 (2011). https://doi.org/10.1088/14757516/2011/07/001. arXiv:1012.4007
 37.M. Cirelli, J.M. Cline, Can multistate dark matter annihilation explain the highenergy cosmic ray lepton anomalies? Phys. Rev. D 82, 023503 (2010). https://doi.org/10.1103/PhysRevD.82.023503. arXiv:1005.1779
 38.J. Heeck, H. Zhang, Exotic charges, multicomponent Dark Matter and light sterile neutrinos. JHEP 05, 164 (2013). https://doi.org/10.1007/JHEP05(2013)164. arXiv:1211.0538
 39.G. Belanger, J.C. Park, Assisted freezeout. JCAP 1203, 038 (2012). https://doi.org/10.1088/14757516/2012/03/038. arXiv:1112.4491
 40.Y. Kajiyama, H. Okada, T. Toma, Multicomponent dark matter particles in a twoloop neutrino model. Phys. Rev. D 88(1), 015029 (2013). https://doi.org/10.1103/PhysRevD.88.015029. arXiv:1303.7356
 41.P.H. Gu, Multicomponent dark matter with magnetic moments for FermiLAT gammaray line, Phys. Dark Univ. 2, 35–40 (2013). https://doi.org/10.1016/j.dark.2013.03.001. arXiv:1301.4368
 42.N.F. Bell, Y. Cai, A.D. Medina, Coannihilating Dark Matter: effective operator analysis and collider phenomenology. Phys. Rev. D 89(11), 115001 (2014). https://doi.org/10.1103/PhysRevD.89.115001. arXiv:1311.6169
 43.C. Gross, O. Lebedev, Y. Mambrini, NonAbelian gauge fields as dark matter. JHEP 08, 158 (2015). https://doi.org/10.1007/JHEP08(2015)158. arXiv:1505.07480
 44.A. Karam, K. Tamvakis, Dark Matter from a classically scaleinvariant \(SU(3)_X\). Phys. Rev. D 94(5), 055004 (2016). https://doi.org/10.1103/PhysRevD.94.055004. arXiv:1607.01001
 45.F. D’Eramo, J. Thaler, Semiannihilation of Dark Matter. JHEP 06, 109 (2010). https://doi.org/10.1007/JHEP06(2010)109. arXiv:1003.5912
 46.Y. Daikoku, H. Okada, T. Toma, Two component Dark Matters in \(S_4 \times Z_2\) flavor symmetric extra \(U(1)\) model. Prog. Theor. Phys. 126, 855–883 (2011). https://doi.org/10.1143/PTP.126.855. arXiv:1106.4717
 47.M. Aoki, M. Duerr, J. Kubo, H. Takano, Multicomponent Dark Matter systems and their observation prospects. Phys. Rev. D 86, 076015 (2012). https://doi.org/10.1103/PhysRevD.86.076015. arXiv:1207.3318
 48.S. Bhattacharya, A. Drozd, B. Grzadkowski, J. Wudka, Twocomponent Dark Matter. JHEP 10, 158 (2013). https://doi.org/10.1007/JHEP10(2013)158. arXiv:1309.2986
 49.S. Bhattacharya, A. Drozd, B. Grzadkowski, J. Wudka, Constraints on twocomponent Dark Matter. Acta Phys. Polon. B 44, 2373–2379 (2013). https://doi.org/10.5506/APhysPolB.44.2373. arXiv:1310.7901
 50.K.J. Bae, H. Baer, E.J. Chun, Mixed axion/neutralino dark matter in the SUSY DFSZ axion model. JCAP 1312, 028 (2013). https://doi.org/10.1088/14757516/2013/12/028. arXiv:1309.5365
 51.M. Aoki, J. Kubo, H. Takano, Multicomponent Dark Matter in radiative seesaw model and monochromatic neutrino flux. Phys. Rev. D 90(7), 076011 (2014). https://doi.org/10.1103/PhysRevD.90.076011. arXiv:1408.1853
 52.S. Esch, M. Klasen, C.E. Yaguna, A minimal model for twocomponent dark matter. JHEP 09, 108 (2014). https://doi.org/10.1007/JHEP09(2014)108. arXiv:1406.0617
 53.A. Dutta Banik, M. Pandey, D. Majumdar, A. Biswas, Two component WIMP—FIMP dark matter model with singlet fermion, scalar and pseudo scalar. Eur. Phys. J. C 77(10), 657 (2017). https://doi.org/10.1140/epjc/s100520175221y. arXiv:1612.08621
 54.N. Khan, Neutrino mass and the Higgs portal dark matter in the ESSFSM. arXiv:1707.07300
 55.L. Bian, R. Ding, B. Zhu, Two component HiggsPortal Dark Matter. Phys. Lett. B 728, 105–113 (2014). https://doi.org/10.1016/j.physletb.2013.11.034. arXiv:1308.3851
 56.L. Bian, T. Li, J. Shu, X.C. Wang, Two component dark matter with multiHiggs portals. JHEP 03, 126 (2015). https://doi.org/10.1007/JHEP03(2015)126. arXiv:1412.5443
 57.G. Arcadi, C. Gross, O. Lebedev, Y. Mambrini, S. Pokorski, T. Toma, Multicomponent Dark Matter from gauge symmetry. JHEP 12, 081 (2016). https://doi.org/10.1007/JHEP12(2016)081. arXiv:1611.00365
 58.A. DiFranzo, G. Mohlabeng, Multicomponent Dark Matter through a Radiative Higgs Portal. JHEP 01, 080 (2017). https://doi.org/10.1007/JHEP01(2017)080. arXiv:1610.07606
 59.E. Ma, Inception of selfinteracting Dark Matter with dark charge conjugation symmetry. Phys. Lett. B 772, 442–445 (2017). https://doi.org/10.1016/j.physletb.2017.06.067. arXiv:1704.04666
 60.K.M. Zurek, Multicomponent Dark Matter. Phys. Rev. D 79, 115002 (2009). https://doi.org/10.1103/PhysRevD.79.115002. arXiv:0811.4429
 61.J.L. Feng, J. Kumar, The WIMPless miracle: DarkMatter particles without weakscale masses or weak interactions. Phys. Rev. Lett. 101, 231301 (2008). https://doi.org/10.1103/PhysRevLett.101.231301. arXiv:0803.4196
 62.S. Profumo, K. Sigurdson, L. Ubaldi, Can we discover multicomponent WIMP dark matter? JCAP 0912, 016 (2009). https://doi.org/10.1088/14757516/2009/12/016. arXiv:0907.4374
 63.B. Batell, Dark discrete gauge symmetries. Phys. Rev. D 83, 035006 (2011). https://doi.org/10.1103/PhysRevD.83.035006. arXiv:1007.0045
 64.D. Feldman, Z. Liu, P. Nath, G. Peim, Multicomponent Dark Matter in supersymmetric hidden sector extensions. Phys. Rev. D 81, 095017 (2010). https://doi.org/10.1103/PhysRevD.81.095017. arXiv:1004.0649
 65.K.R. Dienes, B. Thomas, Dynamical Dark Matter: I. Theoretical overview. Phys. Rev. D 85, 083523 (2012). https://doi.org/10.1103/PhysRevD.85.083523. arXiv:1106.4546
 66.K.R. Dienes, B. Thomas, Dynamical Dark Matter: II. An explicit model. Phys. Rev. D 85, 083524 (2012). https://doi.org/10.1103/PhysRevD.85.083524. arXiv:1107.0721
 67.D. Chialva, P.S.B. Dev, A. Mazumdar, Multiple dark matter scenarios from ubiquitous stringy throats. Phys. Rev. D 87(6), 063522 (2013). https://doi.org/10.1103/PhysRevD.87.063522. arXiv:1211.0250
 68.C.Q. Geng, D. Huang, L.H. Tsai, Imprint of multicomponent dark matter on AMS02. Phys. Rev. D 89(5), 055021 (2014). https://doi.org/10.1103/PhysRevD.89.055021. arXiv:1312.0366
 69.K.R. Dienes, J. Kumar, B. Thomas, D. Yaylali, DarkMatter decay as a complementary probe of multicomponent dark sectors. Phys. Rev. Lett. 114(5), 051301 (2015). https://doi.org/10.1103/PhysRevLett.114.051301. arXiv:1406.4868
 70.C.Q. Geng, D. Huang, L.H. Tsai, Cosmic ray excesses from multicomponent Dark Matter decays. Mod. Phys. Lett. A 29(37), 1440003 (2014). https://doi.org/10.1142/S0217732314400033. arXiv:1405.7759
 71.C.Q. Geng, D. Huang, C. Lai, Revisiting multicomponent dark matter with new AMS02 data. Phys. Rev. D 91(9), 095006 (2015). https://doi.org/10.1103/PhysRevD.91.095006. arXiv:1411.4450
 72.M.A. BuenAbad, G. MarquesTavares, M. Schmaltz, NonAbelian dark matter and dark radiation. Phys. Rev. D 92(2), 023531 (2015). https://doi.org/10.1103/PhysRevD.92.023531. arXiv:1505.03542
 73.C. Lai, D. Huang, C.Q. Geng, Multicomponent dark matter in the light of new AMS02 data. Mod. Phys. Lett A30(35), 1550188 (2015). 10.1142/S0217732315501886ADSCrossRefGoogle Scholar
 74.C.Q. Geng, D. Huang, C. Lai, Multicomponent dark matter. Int. J. Mod. Phys. A 30(28 & 29), 1545009 (2015). https://doi.org/10.1142/S0217751X15450098
 75.K.R. Dienes, J. Fennick, J. Kumar, B. Thomas, Randomness in the Dark Sector: Emergent mass spectra and dynamical dark matter ensembles. Phys. Rev. D 93(8), 083506 (2016). https://doi.org/10.1103/PhysRevD.93.083506. arXiv:1601.05094
 76.J.A. Dror, E. Kuflik, W.H. Ng, Codecaying Dark Matter. Phys. Rev. Lett. 117(21), 211801 (2016). https://doi.org/10.1103/PhysRevLett.117.211801. arXiv:1607.03110
 77.K.R. Dienes, F. Huang, S. Su, B. Thomas, Dynamical Dark Matter from stronglycoupled dark sectors. Phys. Rev. D 95(4), 043526 (2017). https://doi.org/10.1103/PhysRevD.95.043526. arXiv:1610.04112
 78.M. Malekjani, S. Rahvar, D.M.Z. Jassur, Two component BaryonicDark Matter structure formation in tophat model. New Astron. 14, 398–405 (2009). https://doi.org/10.1016/j.newast.2008.11.003. arXiv:0706.3773
 79.V. Semenov, S. Pilipenko, A. Doroshkevich, V. Lukash, E. Mikheeva, Dark matter halo formation in the multicomponent dark matter models. arXiv:1306.3210
 80.M.V. Medvedev, Cosmological simulations of multicomponent cold Dark Matter. Phys. Rev. Lett. 113(7), 071303 (2014). https://doi.org/10.1103/PhysRevLett.113.071303. arXiv:1305.1307
 81.M. Demianski, A.G. Doroshkevich, Cosmology beyond the standard model: Multicomponent dark matter model. Astron. Rep. 59(6), 491–493 (2015). https://doi.org/10.1134/S1063772915060128
 82.O. Lebedev, On stability of the electroweak vacuum and the Higgs portal. Eur. Phys. J. C 72, 2058 (2012). https://doi.org/10.1140/epjc/s1005201220582. arXiv:1203.0156
 83.M. Duch, B. Grzadkowski, M. McGarrie, A stable Higgs portal with vector dark matter. JHEP 09, 162 (2015). https://doi.org/10.1007/JHEP09(2015)162. arXiv:1506.08805
 84.M. Duch, B. Grzadkowski, M. McGarrie, Vacuum stability from vector dark matter. Acta Phys. Polon. B 46(11), 2199 (2015). https://doi.org/10.5506/APhysPolB.46.2199. arXiv:1510.03413
 85.M. Duch, B. Grzadkowski, D. Huang, Strongly selfinteracting vector dark matter via freezein. JHEP 01, 020 (2018). https://doi.org/10.1007/JHEP01(2018)020. arXiv:1710.00320
 86.S. Weinberg, Goldstone bosons as fractional cosmic neutrinos. Phys. Rev. Lett. 110(24), 241301 (2013). https://doi.org/10.1103/PhysRevLett.110.241301. arXiv:1305.1971
 87.ATLAS, CMS Collaboration, G. Aad et al., Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC pp collision data at \( \sqrt{s}=7 \) and 8 TeV. JHEP 08, 045 (2016). https://doi.org/10.1007/JHEP08(2016)045. arXiv:1606.02266
 88.S. Hofmann, D.J. Schwarz, H. Stoecker, Damping scales of neutralino cold dark matter. Phys. Rev. D 64, 083507 (2001). https://doi.org/10.1103/PhysRevD.64.083507. arXiv:astroph/0104173
 89.T. Bringmann, S. Hofmann, Thermal decoupling of WIMPs from first principles. JCAP 0704, 016 (2007). https://doi.org/10.1088/14757516/2007/04/016, https://doi.org/10.1088/14757516/2016/03/E02. arXiv:hepph/0612238 [Erratum: JCAP 1603(03), E02 (2016)]
 90.A. Belyaev, N.D. Christensen, A. Pukhov, CalcHEP 3.4 for collider physics within and beyond the Standard Model. Comput. Phys. Commun. 184, 1729–1769 (2013). https://doi.org/10.1016/j.cpc.2013.01.014. arXiv:1207.6082
 91.G. Belanger, F. Boudjema, A. Pukhov, A. Semenov, micrOMEGAs4.1: two dark matter candidates. Comput. Phys. Commun. 192, 322–329 (2015). https://doi.org/10.1016/j.cpc.2015.03.003. arXiv:1407.6129
 92.Y.G. Kim, K.Y. Lee, S. Shin, Singlet fermionic dark matter. JHEP 05, 100 (2008). https://doi.org/10.1088/11266708/2008/05/100. arXiv:0803.2932
 93.M.M. Ettefaghi, R. Moazzemi, Annihilation of singlet fermionic dark matter into two photons. JCAP 1302, 048 (2013). https://doi.org/10.1088/14757516/2013/02/048. arXiv:1301.4892
 94.S. Baek, P. Ko, W.I. Park, Search for the Higgs portal to a singlet fermionic dark matter at the LHC. JHEP 02, 047 (2012). https://doi.org/10.1007/JHEP02(2012)047. arXiv:1112.1847
 95.S. Baek, P. Ko, J. Li, Minimal renormalizable simplified dark matter model with a pseudoscalar mediator. Phys. Rev. D 95(7), 075011 (2017). https://doi.org/10.1103/PhysRevD.95.075011. arXiv:1701.04131
 96.K. Kainulainen, K. Tuominen, V. Vaskonen, Selfinteracting dark matter and cosmology of a light scalar mediator. Phys. Rev. D 93(1), 015016 (2016). https://doi.org/10.1103/PhysRevD.95.079901, https://doi.org/10.1103/PhysRevD.93.015016. arXiv:1507.04931 [Erratum: Phys. Rev. D 95(7), 079901 (2017)]
 97.T. Bringmann, F. Kahlhoefer, K. SchmidtHoberg, P. Walia, Strong constraints on selfinteracting dark matter with light mediators. Phys. Rev. Lett. 118(14), 141802 (2017). https://doi.org/10.1103/PhysRevLett.118.141802. arXiv:1612.00845
 98.M. Duerr, K. SchmidtHoberg, S. Wild, Selfinteracting dark matter with a stable vector mediator. arXiv:1804.10385
 99.P. Ko, H. Yokoya, Search for Higgs portal DM at the ILC. JHEP 08, 109 (2016). https://doi.org/10.1007/JHEP08(2016)109. arXiv:1603.04737
 100.T. Kamon, P. Ko, J. Li, Characterizing Higgs portal dark matter models at the ILC, Eur. Phys. J. C 77)(9), 652 (2017). https://doi.org/10.1140/epjc/s1005201752408. arXiv:1705.02149
 101.B. Grzadkowski, M. Iglicki, Distinguishing fermion and vector dark matter in progress Google Scholar
 102.A.A. de Laix, R.J. Scherrer, R.K. Schaefer, Constraints of selfinteracting dark matter. Astrophys. J. 452, 495 (1995). https://doi.org/10.1086/176322. arXiv:astroph/9502087
 103.M. Vogelsberger, J. Zavala, A. Loeb, Subhaloes in selfinteracting galactic dark matter Haloes. Mon. Not. R. Astron. Soc. 423, 3740 (2012). https://doi.org/10.1111/j.13652966.2012.21182.x. arXiv:1201.5892
 104.J. Zavala, M. Vogelsberger, M.G. Walker, Constraining selfinteracting dark matter with the Milky Way’s Dwarf spheroidals. Mon. Not. R. Astron. Soc. 431, L20–L24 (2013). https://doi.org/10.1093/mnrasl/sls053. arXiv:1211.6426
 105.A.H.G. Peter, M. Rocha, J.S. Bullock, M. Kaplinghat, Cosmological simulations with selfinteracting dark matter II: Halo shapes vs. observations. Mon. Not. R. Astron. Soc. 430, 105 (2013). https://doi.org/10.1093/mnras/sts535. arXiv:1208.3026
 106.M. Kaplinghat, S. Tulin, H.B. Yu, Dark Matter Halos as particle colliders: unified solution to smallscale structure puzzles from dwarfs to clusters. Phys. Rev. Lett. 116(4), 041302 (2016). https://doi.org/10.1103/PhysRevLett.116.041302. arXiv:1508.03339
 107.S. Tulin, H.B. Yu, Dark Matter selfinteractions and small scale structure. Phys. Rept. 730, 1–57 (2018). https://doi.org/10.1016/j.physrep.2017.11.004. arXiv:1705.02358
 108.M. Markevitch, A.H. Gonzalez, D. Clowe, A. Vikhlinin, L. David, W. Forman, C. Jones, S. Murray, W. Tucker, Direct constraints on the dark matter selfinteraction crosssection from the merging galaxy cluster 1E065756. Astrophys. J. 606, 819–824 (2004). https://doi.org/10.1086/383178. arXiv:astroph/0309303
 109.S.W. Randall, M. Markevitch, D. Clowe, A.H. Gonzalez, M. Bradac, Constraints on the selfinteraction crosssection of dark matter from numerical simulations of the merging galaxy cluster 1E 065756. Astrophys. J. 679, 1173–1180 (2008). https://doi.org/10.1086/587859. arXiv:0704.0261
 110.F. Kahlhoefer, K. SchmidtHoberg, M.T. Frandsen, S. Sarkar, Colliding clusters and dark matter selfinteractions. Mon. Not. R. Astron. Soc. 437(3), 2865–2881 (2014). https://doi.org/10.1093/mnras/stt2097. arXiv:1308.3419
 111.D. Harvey, R. Massey, T. Kitching, A. Taylor, E. Tittley, The nongravitational interactions of dark matter in colliding galaxy clusters. Science 347, 1462–1465 (2015). https://doi.org/10.1126/science.1261381. arXiv:1503.07675
 112.Y. Zhang, Selfinteracting dark matter without direct detection constraints. Phys. Dark Univ. 15, 82–89 (2017). https://doi.org/10.1016/j.dark.2016.12.003. arXiv:1611.03492
 113.S. Tulin, H.B. Yu, K.M. Zurek, Beyond collisionless dark matter: particle physics dynamics for dark matter halo structure. Phys. Rev. D 87(11), 115007 (2013). https://doi.org/10.1103/PhysRevD.87.115007. Xiv:1302.3898
 114.P. Gondolo, G. Gelmini, Cosmic abundances of stable particles: improved analysis. Nucl. Phys. B 360, 145–179 (1991). https://doi.org/10.1016/05503213(91),904384 ADSCrossRefGoogle Scholar
 115.S. Dodelson, Modern Cosmology (Academic Press, Amsterdam, 2003)Google Scholar
 116.K.R. Dienes, J. Kumar, B. Thomas, Direct detection of dynamical dark matter. Phys. Rev. D 86, 055016 (2012). https://doi.org/10.1103/PhysRevD.86.055016. arXiv:1208.0336
 117.J. HerreroGarcia, A. Scaffidi, M. White, A.G. Williams, On the direct detection of multicomponent dark matter: sensitivity studies and parameter estimation. arXiv:1709.01945
 118.C.Q. Geng, D. Huang, C.H. Lee, Q. Wang, Direct detection of exothermic dark matter with light mediator. JCAP 1608(08), 009 (2016). https://doi.org/10.1088/14757516/2016/08/009. arXiv:1605.05098
 119.J.D. Lewin, P.F. Smith, Review of mathematics, numerical factors, and corrections for dark matter experiments based on elastic nuclear recoil. Astropart. Phys. 6, 87–112 (1996). https://doi.org/10.1016/S09276505(96),000473 ADSCrossRefGoogle Scholar
 120.XENON100 Collaboration, E. Aprile et al., Likelihood approach to the first dark matter results from XENON100. Phys. Rev. D 84, 052003 (2011). https://doi.org/10.1103/PhysRevD.84.052003. arXiv:1103.0303
 121.LUX Collaboration, D. S. Akerib et al., Improved limits on scattering of weakly interacting massive particles from reanalysis of 2013 LUX Data. Phys. Rev. Lett. 116(16), 161301 (2016). https://doi.org/10.1103/PhysRevLett.116.161301. arXiv:1512.03506
 122.LUX Collaboration, D.S. Akerib et al., Lowenergy (0.774 keV) nuclear recoil calibration of the LUX dark matter experiment using D–D neutron scattering kinematics. arXiv:1608.05381
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP^{3}