# Scale-invariant two-component dark matter

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## Abstract

We study a scale-invariant extension of the standard model which can simultaneously explain dark matter and the hierarchy problem. In our set-up, we introduce a scalar and a spinor as two-component dark matter in addition to a scalon field as a mediator. An interesting point about our model is that due to scale-invariant conditions, compared to other two-component dark matter models, it has fewer independent parameters. Possible astrophysical and laboratory signatures of a two-component dark matter candidate are explored and it is shown that the highest contribution of observed relic density of dark matter can be determined by spinor dark matter. The detectability of these dark matter particles is studied and direct and invisible Higgs decay experiments are used to rule out part of the parameter space of the model. In addition, the dark matter self-interactions are considered and it is shown that their contributions saturate this constraint in the resonant regions.

## 1 Introduction

The standard model (SM) has been established by the discovery of the Higgs boson and it can explain almost all of experimental results obtained until now. However, there are a number of unanswered issues, either theoretical or experimental, such as the hierarchy problem, active neutrino masses, the dark matter (DM) relic abundance, the baryon asymmetry of the Universe, inflation in the early Universe, and dark energy.

The existence of DM is inferred from crucial evidence such as galactic rotation curves, gravitational lensing, observations of merging galaxies, cosmic microwave background (CMB) measurements, the large scale structure of the Universe and collisions of bullet clusters. As mentioned, there is still lack of experimental or observational evidence to precisely distinguish the correct particle physics model for DM physics.

To explain these issues a number of the SM extensions, such as supersymmetric standard model, technicolor and extra dimension theories, have been proposed. Despite the broad searches as regards beyond SM physics at LHC, null results for beyond SM theories [1, 2, 3, 4, 5, 6] show that we have enough motivation to think about alternative theories.

In almost all extended models, there are some additional particles, which usually have heavier masses than the electroweak (EW) scale. It is well known that the hierarchy problem arises from the fact that the negative Higgs mass term in the Lagrangian of the SM causes a quadratical divergent term proportional to the energy scale cut-off \(\Lambda ^2\) after including the quantum corrections. As an idea avoiding the hierarchy problem, classically scale-invariant extensions provide an attractive framework [7, 8, 9]. In this picture, it is supposed that the tree-level Higgs mass is zero and at the quantum level the Higgs scalar gains a small mass from the radiative corrections. In fact, the Higgs mass term is the only term that breaks the classical scale invariance in the SM. Note that classical scale invariance by itself does not explain the hierarchy problem; however, it can be regarded as a procedure for model building, which limits the space of Lagrangians to contain only operators with dimensionless coupling constants. The hierarchy problem then reveals itself as the absence of couplings between the Higgs and other energy scales that are dynamically generated in the UV cut-off scale [10, 11]. Therefore, by regarding this condition, one can practically remove the hierarchy problem.

In recent years, a lot of classically scale-invariant models have been studied for the solution of the hierarchy problem and the DM problem [12, 13, 14, 15, 16, 17, 18, 19, 20]. On the other hand, in order to resolve the small-scale problems (through self-interaction) and at the same time explain the potential indirect signals, one needs very different DM masses. Therefore, to interpret both observations, a multi-component DM seems to be a natural possibility. Furthermore, multi-component DM models, besides the standard annihilations and coannihilations, allow one to have conversion, semi-annihilation, and decay processes which make the dark sector (thermal) dynamics much more interesting. Therefore, the dynamics of multi-component DM is much richer than simple WIMP, and it arouses curiosity by itself. In this paper, we study scalar–spinor two-component DM, in order to have one candidate for each bosonic and fermionic particles. The possibility of other two-component models without scale invariance has been extensively considered in the literature [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. Also the two-component DM has been studied in the context of scalar WIMP-like candidates [35]. Our goal in this paper is to address the DM relic density and the hierarchy problem by an extension of the scale-invariant standard model (SISM) which contains a scalar and a spinor DM candidate.

The structure of this paper is as follows: in Sect. 2, we introduce the scale-invariant SM with two-component scalar and fermionic DM scenarios. In Sect. 3, we study perturbativity constraints on two-component scale-invariant DM. In Sect. 4, we study freeze-out solutions to the relic density constraint. In Sect. 5, we will study phenomenological aspects such as direct detection, indirect detection, self-interaction and invisible Higgs decay searches on parameter space of our model. The results are summarized in Sect. 6. The decay rate and cross section formulas for the self-interaction of the two components of DM are summarized in the appendix.

## 2 The model

In the SISM, before electroweak symmetry breaking all fields in the scale-invariant sector of potential are massless. At the quantum level these fields gain mass from radiative Coleman–Weinberg symmetry breaking [9].

*S*and the spinor \( \chi \), are assumed to be odd under a \( Z_{2} \) symmetry. This discrete symmetry guarantees the stability of the lightest odd particles. The other scalar field, \( \phi \), and all SM particles are even under \( Z_{2} \). Therefore under \( Z_{2} \) symmetry new fields transform as follows:

*H*, \(\phi \) and

*S*are the doublet Higgs, the scalon and DM scalars, respectively.

*h*and \( \phi \) so the scalar field

*S*remains stable because of the \(Z_2\) symmetry and thereby it can play the role of the DM. Therefore, we put \( S = 0 \) in Eq. (6):

*h*and \( \phi \). For non-zero

*h*and \( \phi \), Eq. (7) leads to

*H*breaks the electroweak symmetry with vacuum expectation value \( \langle H \rangle = \frac{1}{\sqrt{2}} ( {\begin{matrix} 0 \\ \nu _{1} \end{matrix}} ) \), where \( \nu _1 = 246 \, \text {GeV} \). Thus the Higgs field after spontaneous symmetry breaking is given by

*Z*gauge boson and the top quark masses. For this reason, adding the scalar field is inevitable. Moreover, in the absence of additional scalar DM, the square scalon mass could be negative. Since \( M_{H_{2}}^{2} > 0 \) and \( \lambda _{\phi H} < 0 \), Eq. (14) leads to the following constraint on \( M_{s} \):

## 3 Theoretical constraints

## 4 Relic abundance

*S*and \(\chi \) should be solved in order to compute the number density. The coupled Boltzmann equations for scalar

*S*and fermion \(\chi \) are given by

*p*denotes any SM particles. In \(\langle \sigma _{ab\rightarrow cd}\upsilon \rangle \) all annihilations are taken into account except \(\langle \sigma _{S\chi \rightarrow S\chi }\upsilon \rangle \), which does not affect the number density. By using \(x = m/T\), where

*T*is the photon temperature, as the independent variable instead of time and \(\dot{T}=-HT\), one can rewrite the Boltzmann equations in terms of the yield quantity, \(Y=n/s\):

*S*and \( \chi \), denoted by \( \xi _{S} \) and \( \xi _{\chi } \), respectively. So

*g*. For any given value of

*g*the fermionic relic density \( \Omega _{\chi } \) features a double reduction at the \( H_{1} \) and \( H_{2} \) resonances (respectively, at \( M_{\chi } = \frac{M_{H_{1}}}{2} = 62.5 ~{\mathrm{GeV}} \) and at \( M_{\chi } = \frac{M_{H_{2}}}{2} \)). There is another reduction due to the opening of the \( \chi \chi \rightarrow H_{2} H_{2} \) annihilation channel. Note that, according to Eq. (14), \( M_{H_{2}} \) itself depends on

*g*, \( M_{s} \) and \( M_{\chi } \), so it is not an independent parameter. Therefore, in our relic density plots, it varies with

*g*and DM masses. In Fig. 1, scalar relic density \( \Omega _{S} \) does not vary dramatically with \( M_{\chi } \) or

*g*. Note that \( \lambda _{s H} \) is a determinative parameter in scalar DM annihilation to SM particles. On the other hand, annihilation of scalar DM to SM particles is more favorable than its annihilation to fermionic DM, because most SM particles are lighter than fermionic DM. Therefore, \( \Omega _{S} \) mostly depends on \( \lambda _{s H} \), rather than \( \lambda _{\phi s} \). According to Eqs. (13) \( \lambda _{s H} \) is given by

*g*.

In Fig. 2 DM relic densities are plotted versus \( M_{\chi } \) for different values of \( M_{s} \). Similarly, for the given values of \( M_{s} \) the fermionic relic density again features a double reduction at the \( H_{2} \) and \( H_{1} \) resonances (respectively, at \( M_{\chi } = \frac{M_{H_{2}}}{2} \) and \( M_{\chi } = \frac{M_{H_{1}}}{2} = 62.5 \, \text {GeV} \)). Obviously, in this plot \( M_{H_{2}} \) at the first resonance is lighter than \( M_{H_{1}} = 125 \, \text {GeV} \). For the scalar relic density, according to Eq. (42), a larger \( M_{s} \) leads to a larger \( \lambda _{s H} \) and therefore the DM–SM interaction gets stronger, which leads to a smaller scalar relic density. Furthermore, now for \( M_{s} = 500 ~{\mathrm{GeV}} \), the second term of Eq. (42) can compete with the first term, and with the growth of \( M_{\chi } \), \( \lambda _{s H} \) will decrease. Due to this reduction, the scalar DM–SM interaction becomes weaker and therefore \( \Omega _{S} \) increases with \( M_{\chi } \). For larger \( M_{s} \) (for example \( M_{s} = 700 ~{\mathrm{GeV}} \) again the first term of Eq. (42) dominates and \( \Omega _{S} \) increases less with \( M_{\chi } \).

Figures 3 and 4 depict relic densities versus \( M_{s} \). In Fig. 3, for \( M_{\chi } = 50 ~{\mathrm{GeV}} \) there is a single reduction in fermionic relic density around \( M_{s} = 700 ~{\mathrm{GeV}} \). This reduction corresponds to \( M_{H_{2}} = 2 M_{\chi } = 100 ~{\mathrm{GeV}} \) which is a resonance case. According to Eq. (42), \( \lambda _{s H} \) increases with \( M_{s} \) and the scalar DM–SM interaction becomes stronger. Therefore, \( \Omega _{S} \) decreases with \( M_{s} \). In addition, for the given parameters, since the first term of Eq. (42) dominates, \( \lambda _{s H} \), and therefore \( \Omega _{S} \), is nearly independent of \( \lambda _{\phi s} \). In this figure, only for small \( M_{s} \) a little dependency of \( \Omega _{S} \) to \( \lambda _{\phi s} \) can be realized.

Finally, in Fig. 4 we display the fermionic relic density as a function of \(M_s\) for different values of \( M_{\chi } \). Therefore, we have different resonance cases, corresponding to \( M_{H_{2}} = 2 M_{\chi } \) for each value of \( M_{\chi } \). For the given parameters, the scalar relic density is not sensitive to different values of \( M_{\chi } \), because as mentioned before \( \Omega _{S} \) is mostly determined by \( \lambda _{s H} \), and, again according to Eq. (42), the second term can be neglected in comparison with the first term. Thus, for the given values of Fig. 4, only the first term, which is independent of \( M_{\chi } \), affects the scalar DM relic density so that on growing \( M_{s} \), \( \lambda _{s H} \) increases and consequently \( \Omega _{S} \) decreases.

*g*, \( M_{s} \), and \( M_{\chi } \).

## 5 Phenomenological aspects

### 5.1 Direct detection

*g*are independent and have been defined in the previous section. It is remarkable that the two terms in Eq. (44) may cancel against each other, giving a suppressed cross section. In Fig. 5, we display the direct detection cross section as a function of the mass of scalar and fermion DM. As seen in Fig. 5a, \(\sigma _S\) has a minimum in the value of \( M_s\) at which cancellation takes place. For scalar DM, the direct detection cross section depends on the scalar DM mass, \(\lambda _{\phi s}\),

*g*and \(M_{\chi }\). The fermionic DM direct detection cross section does not depend on \(\lambda _{\phi s}\). However, as mentioned in the previous section, \(m_{H_2}\) is not an independent parameter and depends on three independent parameters of our model, \(M_s\),

*g*and \(M_{\chi }\). Also \(m_{H_2}\) may be very small and so the contribution of its propagator to the direct detection cross section can be very large. For this reason, a large portion of parameter space is excluded by this observable. In order to show the allowed region in parameter space, we display scatter points in Fig. 6. Figure 6a–c depict the allowed regions in

*g*, \(\lambda _{\phi s}\) and \(M_s\) for scalar DM and Fig. 6d depicts the allowed regions in

*g*and \(M_{\chi }\) for fermionic DM, which are consistent with experimental measurements of \(\sigma _{\mathrm{Xenon100}}\) and \(\sigma _{\mathrm{LUX}}\).

### 5.2 Indirect detection

Indirect detection experiments hunt for the self-annihilation (or decay) products of DM particles in regions of high DM density (e.g., the center of our galaxy). Two dark matter particles could annihilate to produce gamma rays or SM particle–antiparticle pairs. Indirect detection experiments may confirm DM annihilation through an excess of gamma rays (e.g., the Fermi-LAT experiment [43]), positrons (e.g., the PAMELA experiment [44]), antiprotons (e.g., the AMS experiment [45]) or neutrinos (e.g., the IceCube experiment [46]). A major difficulty is that various astrophysical sources can resemble closely the signal expected from DM. Therefore, multiple signals are required for a conclusive discovery.

### 5.3 Self-interaction

In the next step, we consider the DM self-interacting cross section for scalar *S* and fermion \(\chi \) DM. The DM self-interactions include the processes \(SS\longrightarrow SS\), \(\chi \chi \longrightarrow \chi \chi \), \(SS\longrightarrow \chi \chi \), \(\chi \chi \longrightarrow SS\) and \(S\chi \longrightarrow S\chi \). Figure 9 shows Feynman diagrams for DM self-interactions.

The main contributions to \(\sigma /M_{s}\) for scalar annihilation (processes \(SS\longrightarrow SS\)[51] and \(SS\longrightarrow \chi \chi \)) are given in appendix. For the process \(SS\longrightarrow SS\)[51], \(\sigma /M_{s}\) is proportional to \(1/M^3_s\) and after imposing the constraint \(M_s>310~{\mathrm{GeV}}\), we find that this situation does not saturate the upper bound on the self-interaction cross section. Indeed, to obtain a reasonably strong scalar DM self-interaction, the mass of the scalar must be very small, \(M_ s<1~{\mathrm{GeV}}\). Since in the non-relativistic regime \(s\sim 4M_s^2\), \(\sigma (SS\rightarrow \chi \chi )/M_s\) will be larger than \(\sigma (SS\rightarrow SS)/M_s\), This feature is depicted in Fig. 10a. As is seen in this figure, the self-interaction for scalar DM is very much smaller than the upper bound. However, it is possible to obtain an upper bound on the self-interaction cross section for scalar DM if we consider self-interaction in the vicinity of resonance \(M_s\simeq M_{H_2}/2\). Note that according to Eq. (15), the mass of scalar DM cannot be equal to half of the SM Higgs mass. For the resonance regime (\(M_s\simeq M_{H_2}/2\)), the s-channel \(H_2\) exchange diagram in Fig. 9 dominates and the scalar DM self-interaction may exceed the experimental bound. Achieving the observed scalar DM self-interaction cross section requires that \(M_s\) be severely tuned such that \(|M_s-M_{H_2} /2|< 1~ {\mathrm{MeV}}\) (while \(M_s>310 ~{\mathrm{GeV}}\)). However, since the main contribution of the observed relic density was obtained from fermionic DM, and scalar DM has a small contribution to the relic density, we expect that this process is very rare in the center of the Milky Way.

For the process \(\chi \chi \longrightarrow SS\) in the non-relativistic limit \(s < 4M^2_s\) and so this process is forbidden. For the processes \(\chi \chi \longrightarrow \chi \chi \), since in the non-relativistic regime \(s\simeq 4M_{\chi }^2\), the self-interaction of fermionic DM is much smaller than the experimental bound (it is shown in Fig. 10b). It also turns out that to bring about a reasonably strong fermionic DM self-interaction (similar to scalar DM), we should consider self-interaction in the near resonance \(M_{\chi }\simeq M_{H_2}/2\) or \(M_{H_1}/2\). Notice that for fermionic DM, fine tuning should be stronger than for scalar DM due to the smaller self-interacting cross section for fermionic DM.

### 5.4 Invisible Higgs decay

*g*coupling. In this figure, we suppose \(M_{\chi }<M_{H_1}/2\) and assign other parameters such that \(M_{H_2}<M_{H_1}/2\). By using the ATLAS upper limit for invisible Higgs decay, we display the allowed range of parameter space in Fig. 12b in our model. Note that the main contribution to \(Br(H_1\rightarrow {\mathrm{Invisible}})\) in the portion of parameter space which is consistent with the experimental limits arises from \(\Gamma (H_1\rightarrow H_2H_2)\). This feature is shown in Fig. 13. This figure separately depicts the contribution of \(Br(H_1\rightarrow {\mathrm{Invisible}})\) as a function of the fermionic DM mass for \(Br(H_1\rightarrow \chi \chi )\), \(Br(H_1\rightarrow H_2H_2)\) and \(Br(H_1\rightarrow {\mathrm{total}})\). Comparing Fig. 13a, b implies for small values of

*g*, which is consistent with experimental limits, that the main contributions of \(Br(H_1\rightarrow {\mathrm{Invisible}})\) are coming from \(Br(H_1\rightarrow H_2H_2)\). In our model, \(M_{H_2}\) generally depends on

*g*, \(M_{\chi }\) and \(M_s\). Since \(\Gamma (H_1\rightarrow H_2H_2)\) depends on \(M_{H_2}\), in the allowed region of parameter space, we expect that the branching ratio of the invisible Higgs decay also depends on \(M_s\). In Fig. 12b, we have shown for larger values of \(M_s\) that the allowed area shrinks in the \(g_s\) and \(M_{\chi }\) plane.

*g*coupling, which are consistent with the observed relic density, are shown. Comparing Figs. 14 and 12b shows that the allowed regions for invisible Higgs decay and the DM relic density do not overlap with each other. Since the highest contribution of the DM relic density arises from fermionic DM, for a small value of the

*g*coupling, the annihilation of DM to SM particles will be suppressed. This means for a portion of the parameter space, which is consistent with invisible Higgs decay, the relic density exceeds the value of the Planck measurement. Therefore, in order to evade invisible Higgs constraints, one should assume that the fermionic DM mass is larger than \( \frac{M_{H_{1}}}{2} \).

## 6 Concluding remarks

Motivated by the DM and hierarchy problems, we presented a scale-invariant extension of the SM. In order to have a scale-invariant version of the SM with scalar DM, at least two more scalars must be added to the theory. Moreover, in the absence of additional fermionic fields, the model has a small number of independent parameters, which complicates the options of satisfying all theoretical and phenomenological constraints. Given these conditions, we added a scalon field \(\phi \), a scalar field *S* and a fermionic field \(\chi \) as two-component DM to SM. To summarize, the main novelty of this model, with respect to other two-component DM models, is the much smaller number of independent parameters due to the scale-invariant conditions.

In this analysis, the relic density of two-component DM was computed. We have shown that the highest part of the contribution of the DM relic density arises from fermionic DM. We have discussed the allowed regions in parameter space of our model consistent with the observed relic density.

We have also taken into account the constraints of indirect detection and direct detection of DM. In order to constrain the parameter space of our model, we also checked the limits from the self-interaction of DM. It is shown that the former analysis cannot put a constraint on the model in a large portion of parameter space. Only in the vicinity of the resonances in \(M_s\simeq M_{H_2}/2\) for scalar DM and \(M_{\chi }\simeq M_{H_2}/2\) or \(M_{H_1}/2\) for fermionic DM, the self-interaction scenario constrains the model.

Finally, we probed the limits from the invisible decay width of the Higgs. We have found that the viable regions in parameter space are in agreement with the upper limit on the invisible Higgs decay branching ratio. We compared the consistent region in parameter space for invisible Higgs decay with the relic density of the fermionic DM and show that in order to satisfy invisible Higgs constraints, the fermionic DM mass should be larger than \( M_{H_{1}}/{2}\).

## Notes

### Acknowledgements

The authors would like to thank S. Paktinat for careful reading of the manuscript and the useful remarks.

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