The $Z_5$ model of two-component dark matter

Scenarios for multi-component scalar dark matter based on a single $Z_N$ ($N\geq 4$) symmetry are simple and well-motivated. In this paper we investigate, for the first time, the phenomenology of the $Z_5$ model for two-component dark matter. This model, which can be seen as an extension of the well-known singlet scalar model, features two complex scalar fields--the dark matter particles--that are Standard Model singlets but have different charges under a $Z_5$ symmetry. The interactions allowed by the $Z_5$ give rise to novel processes between the dark matter particles that affect their relic densities and their detection prospects, which we study in detail. The key parameters of the model are identified and its viable regions are characterized by means of random scans. We show that, unlike the singlet scalar model, dark matter masses below the TeV are still compatible with present data. Even though the dark matter density turns out to be dominated by the lighter component, we find that current and future direct detection experiments may be sensitive to signals from both dark matter particles.

Scenarios for multi-component scalar dark matter based on a single Z N (N ≥ 4) symmetry are simple and well-motivated. In this paper we investigate, for the first time, the phenomenology of the Z 5 model for two-component dark matter. This model, which can be seen as an extension of the well-known singlet scalar model, features two complex scalar fields-the dark matter particles-that are Standard Model singlets but have different charges under a Z 5 symmetry. The interactions allowed by the Z 5 give rise to novel processes between the dark matter particles that affect their relic densities and their detection prospects, which we study in detail. The key parameters of the model are identified and its viable regions are characterized by means of random scans. We show that, unlike the singlet scalar model, dark matter masses below the TeV are still compatible with present data. Even though the dark matter density turns out to be dominated by the lighter component, we find that current and future direct detection experiments may be sensitive to signals from both dark matter particles.

I. INTRODUCTION
Finding the correct extension of the Standard Model (SM) that accounts for the dark matter (DM) is one of the main open problems in fundamental physics today. Even if most of the models that have been proposed and studied implicitly assume that the observed dark matter density is explained by a single new particle, it does not have to be so [1][2][3][4][5][6][7][8][9]. Scenarios in which two or more different particles contribute to the dark matter density -multi-component dark matter modelsnot only are perfectly consistent with current observations but often lead to testable predictions in current and future dark matter experiments.
Among multi-component dark matter models, those featuring scalar fields that are simultane-arXiv:2006.14922v1 [hep-ph] 26 Jun 2020 ously stabilized by a single Z N symmetry are particularly appealing [10,11]. For k dark matter particles, they require only k complex scalar fields that are SM singlets but have different charges under a Z N (N ≥ 2k). This symmetry, in turn, could be a remnant of a spontaneously broken U (1) gauge symmetry and thus be related to gauge extensions of the SM. Recently, these scenarios were systematically analyzed [12] and it was found that, surprisingly, their dark matter phenomenology has yet to be investigated in detail. With this paper, we intend to partially fill that gap.
We study the two-component dark matter model based on the Z 5 symmetry, which serves as a prototype for all the Z N scenarios in which the dark matter particles are two complex scalars.
Above all, we want to characterize the viable parameter space of this model and to determine its detection prospects. To that end, we first examine the dark matter relic densities, identifying the types of processes that can modify them and the key parameters they depend on. Then, the viable parameter space of the model is characterized by means of random scans, which we analyze in detail. Our results indicate that the entire range of dark matter masses is allowed, that the dark matter density is always dominated by the lighter component, and that both dark matter particles may produce signals in future direct detection experiments.
The rest of the paper is organized as follows. In the next section, our notation is introduced and the Z 5 model is briefly described -further details are relegated to the appendices. Section III is devoted to the dark matter phenomenology. In particular, the effect of the different parameters on the relic densities is elucidated. Our central results are obtained in section IV. In it, we first determine, via random scans, the viable parameter space of the model and then use it to predict its detection prospects. Section V deals with possible extensions of our work whereas section VI presents our conclusions.

II. THE MODEL
Let us consider a scenario with two new complex scalar fields, φ 1,2 , charged under a Z 5 symmetry.
The unique charge assignment (up to trivial field redefinitions) that allows both fields to be stable is [12] φ 1 ∼ ω 5 , φ 2 ∼ ω 2 5 ; These new fields-the dark matter particles-are assumed to be singlets of the SM gauge group whereas the SM particles are taken to be singlets under the Z 5 . The most general Z 5 -invariant scalar potential is then given by [13] where H is the SM Higgs doublet. To ensure that the model describes a two-component dark matter scenario, we assume that φ 1,2 do not acquire a vacuum expectation value and that their masses satisfy M 1 2 < M 2 < 2M 1 so that both are stable. In addition, due to the symmetry of the Lagrangian, we can take, without loss of generality, φ 2 to be heavier than φ 1 and so M 1 < M 2 < 2M 1 , which is assumed from now on. The stability conditions as well as the one-loop renormalization group equations (RGEs) for this model are given in the appendices.
Notice that the dark matter particles interact with the SM fields only through the Higgs boson. The Z 5 model is thus one example of the so-called Higgs-portal scenarios -see Ref. [14] for a recent review. In addition, the dark matter particles interact among themselves through trilinear and quartic interactions. The terms in brackets in equation (2) are interactions specific to the Z 5 symmetry we are considering, while the rest are present for any Z N (see also section V). Had we imposed a Z 2 × Z 2 instead, as often done in two-component dark matter scenarios, all the terms in brackets would be forbidden.
In total, this model contains 11 new parameters (4 dimensionful and 7 dimensionless), but two of them -λ 41 and λ 42 -are irrelevant for the dark matter phenomenology and can be ignored in our analysis. The parameters µ 2 i (i = 1, 2), on the other hand, can be conveniently traded for the physical masses M i of the scalar fields, so that the free parameters of the model may be taken to be M i , λ Si , λ 412 , µ Si , and λ 3i . The phases of φ 1,2 can be chosen so as to make µ S1 and µ S2 real, but then λ 31 and λ 32 may be complex. In the following we will stick, for simplicity, to real parameters. Our goal is to study how these nine parameters affect the relic densities, shape the viable parameter space, and determine the dark matter observables. This Z 5 model can be seen as an extension of (and shares many features with) the well-known singlet scalar model [15][16][17], which is based on the standard Z 2 symmetry and includes just one dark matter particle. This latter model is currently highly constrained, requiring dark matter masses right at the Higgs-resonance or above a TeV or so [18,19]. We would like to know, therefore, whether this restriction on low dark matter masses still holds in the Z 5 model, or if the new interactions present in it weaken such bounds and allow the dark matter particles to have masses below the TeV.

III. DARK MATTER PHENOMENOLOGY
In this model, the dark matter particles and the SM particles are connected only via Higgsportal interactions. Thus, depending on the size of the couplings λ Si , both freeze-in [20,21] and freeze-out scenarios can be envisaged for the dark matter relic densities. We will focus, in this paper, on the more compelling freeze-out realization, which has the advantage of giving rise to testable signatures in dark matter experiments.

A. The relic density
The full set of 2 → 2 processes that may contribute to the relic density in an arbitrary twocomponent dark matter scenario was listed in Ref. [13]. They can be classified in types that are denoted by four digits (each a 0, 1, or 2) indicating the sector to which the particles involved in the process belong to -0 is used for SM particles, 1 for φ 1 or φ † 1 , and 2 for φ 2 or φ † 2 . Thus, the type 2210 includes all processes with one SM particle and one φ 1 (or φ † 1 ) in the final state, and with an initial state consisting of either two φ 2 , two φ † 2 , or φ 2 and φ † 2 . Among the various types, the only ones not compatible with the Z 5 symmetry are 1110 and 2220. Table I displays all the processes that contribute to the relic densities in the Z 5 model, with their respective type.
According to the number of SM particles, these processes can be divided into three kinds: annihilation processes (two SM particles), semi-annihilation processes [22] (one SM particle), and dark matter conversion processes (no SM particles). Figures 1 and 2 display representative Feynman FIG. 1: Dark matter semi-annihilation processes involving one trilinear µ S1 and one Higgs-DM and λ 412 (right). Bottom: via a µ S1 trilinear interaction (left) or Higgs-portal couplings (right).
diagrams for semi-annihilation and dark matter conversion processes respectively. Given that some processes receive contributions from more than one Feynman diagram, e.g. φ 2 + φ † 2 → φ 1 + φ † 1 , interference effects are expected to play a role in certain cases. Let us also note that while the quartic couplings, λ 3i , induce only dark matter conversion processes, the trilinear couplings, µ Si , contribute to both, semi-annihilations and conversions. The annihilations into two SM particles (not shown), on the other hand, proceed via the usual s-channel Higgs-mediated diagram, with The Boltzmann equations for the Z 5 model thus read Here n i (i = 1, 2) denote the number densities of φ i , andn i their respective equilibrium values.
By solving these equations, the relic densities of φ 1 and φ 2 -Ω 1 and Ω 2 -can be calculated. Since its version 4.1, micrOMEGAs [13] incorporated two-component dark matter scenarios, automatically including all the relevant processes for a given model and numerically solving the corresponding Boltzmann equations. It also includes the code files of the Z 5 model we are studying. We will rely on micrOMEGAs [13,23,24] to compute the relic densities and the dark matter detection observables. Keep in mind, though, that in the course of this work we found and corrected some bugs affecting the calculation of the relic density for two dark matter particles. To reproduce our results, micrOMEGAs version 5.2 or later should be used.
To estimate the relevance of the three kinds of processes-annihilations, semi-annihilations, and conversions-that can contribute to the relic density of φ 1 , it is convenient to define the following three parameters with M 1 /25 for definiteness. Each of them varies between 0 and 1 depending on how important the respective type of process is. Thus, ζ 1 semi ≈ 1 indicates that the φ 1 relic density is mostly driven by semi-annihilations. Notice that they are normalized such that ζ 1 anni + ζ 1 semi + ζ 1 conv = 1. Analogous parameters can be defined for φ 2 . If a more detailed pictured is required of how the different processes affect the relic density, micrOMEGAs has the option, since its version 5.2, to exclude from the calculation one or more types of processes via the variable Excludefor2DM. We have used this option to perform several checks on our results.
Semi-annihilation processes will play a crucial role in our analysis so it is useful to get a feeling of how they compare against the usual annihilation processes. When φ 1 annihilations are mediated by the typical Higgs portal, the thermally averaged cross section goes as The semi-annihilation processes φ 1 + φ 1 → h + φ 2 , on the other hand, feature a thermally averaged cross section Since σ 1120 v rapidly decreases with M 1 , semi-annihilations are expected to stop being efficient at high values of M 1 .
In the following section, we will impose the relic density constraint, where Ω DM is the observed value of the dark matter density. The fraction of the total dark matter density that is accounted for by each dark matter particle is then given by the parameters with ξ 1 +ξ 2 = 1. One of the main questions in two-component dark matter scenarios is determining what these fractions are (can they be comparable?), and they also affect the dark matter detection signals, as shown next.

B. Direct and indirect detection
The elastic scattering of the dark matter particles off nuclei are possible thanks to the Higgs portal interaction λ Si , just as in the singlet scalar model [15][16][17]. The expression for the spin-independent (SI) cross-section reads where µ R is the reduced mass, m p the proton mass and f p ≈ 0.3 is the quark content of the proton. But since we have two dark matter particles, the quantity to be compared against the direct detection limits provided by the experimental collaborations is not the cross section itself but rather the product ξ i σ SI i . Such direct detection limits usually provide very strong constraints on Higgs-portal scenarios like the Z 5 model we are discussing. For example, in the limit Ω 2 Ω 1 and with the new Z 5 interactions switched off-where the singlet complex scalar DM model [15][16][17] is recovered-we get in order to fulfill Ω 1 = Ω DM . Taking into account the upper limit set by the XENON1T collaboration [25] it follows that M 1 2 TeV (for a real scalar the lower limit is ∼ 950 GeV). Hence, for M 1 2 TeV the Z 5 -invariant interactions must be required in order to simultaneously satisfy the relic density constraint and current direct detection limits-a result we will numerically confirm in the next section. In our analysis, we will consider the current direct detection limit set by the XENON1T collaboration [25] as well as the projected sensitivities of LZ [26] and DARWIN [27].
Regarding indirect detection, the relevant particle physics quantity switches from σv to ξ i ξ j σv ij , where σv ij is the cross section times velocity for the annihilation process of dark matter particles i and j into a certain final state. The main novelty in our model is the possible appearance of semi-annihilation processes involving two different dark matter particles, such as We will rely on the indirect detection limits and on the projected sensitivities reported by the Fermi collaboration from observations of dShps [28,29].

C. Parameter dependence
To study how the different parameters affect the relic densities of the dark matter particles, we first define a reference model in which most of these parameters are set to zero, and then switch them on, one by one, while comparing the resulting relic densities against the predictions of the reference model. The non-zero parameters of the reference model are just four: the dark matter masses (M 1 , M 2 ) and the Higgs-portal couplings (λ S1 , λ S2 ). Note that even in this very simplified panel).
framework the relic densities are coupled via the Higgs-mediated processes figure 2).
For definiteness, in this section we set λ S1 = λ S2 = 0.1, and examine two different values for the ratio M 2 /M 1 (which can vary between 1 and 2): 1.2 and 1.8. In the following figures, the predictions of the reference model are shown in solid (green) lines. First, we are going to investigate the dependence of the relic densities on the dimensionless couplings (λ 31 , λ 32 , λ 412 ) and then we move on to the dimensionful ones -µ S1 and µ S2 .

The effect of λ's
The dimensionless couplings-λ 31 , λ 32 , λ 412 -induce the dark matter conversion processes shown in the top row of figure 2. Neither semi-annihilations nor annihilations can be caused by these couplings. λ 31 , for instance, leads to the conversion processes φ 1 + φ 2 ↔ φ † 1 + φ † 1 and their complex conjugates. During the φ 2 freeze-out, they contribute to the depletion of φ 2 and should therefore reduce Ω 2 . Ω 1 , on the other hand, should hardly get modified unless M 1 ≈ M 2 , when the Figure 3 shows for both values of M 2 /M 1 . What is a bit surprising is the size of the effect. Notice, in fact, that even a value of λ 31 as small as 10 −2 can modify Ω 2 by several orders of magnitude. The reason behind this behavior is that the Boltzmann equation has a term of the form (Y i = n i /s) which exponentially suppresses the φ 2 density over a range of temperatures. Thus, even a moderate which should lead to a reduction of Ω 2 while leaving Ω 1 unaffected. From the figure, we see that Ω 2 indeed decreases with λ 32 and that the effect is pretty much independent on M 2 /M 1 -the two panels seem identical (they are not). For the values of λ 32 shown, the reduction in Ω 2 reaches at most one order of magnitude.
The last quartic coupling to be examined is λ 412 , which should naively cause a reduction of Unlike the previous processes, however, this one receives an additional contribution from a Higgs-mediated diagram, and so interference effects between the two diagrams may play role and result in either an increase or a decrease of the relic density. The Higgs-mediated amplitude is proportional to λ S1 λ S2 and its sign changes (due to the propagator) at the Higgs resonance, M 2 ∼ M h /2. Thus, the sign of λ 412 turns out to be relevant in the analysis.
To illustrate these effects, figure 5 shows the relic density for M 2 /M 1 = 1.  figure 5, the maximum variation in Ω 2 amounts to two orders of magnitude for masses below the Higgs resonance, and one order of magnitude above it. For M 2 /M 1 = 1.8, the results are essentially identical, so they are not shown.
As we have seen, a common feature of the quartic interactions is that they mostly affect the relic density of the heavier dark matter particle, Ω 2 . For the parameter values we have considered in this section, the effect on Ω 1 is negligible. Thus, the φ 1 relic density is determined by the characteristic Higgs-mediated interactions of the singlet scalar model, and it is therefore expected to be subject to the same stringent direct detection constraints as that model. The trilinear couplings, µ S1 and µ S2 , might influence Ω 1 and help relax such constraints.

The effect of µ's
The trilinear couplings, µ S1 and µ S2 , give rise to both semi-annihilation and conversion processes -see figures 1 and 2. The semi-annihilation processes involve also one Higgs-dark matter coupling, either λ S1 or λ S2 , and always feature a Higgs boson as an external particle. The conversion processes, on the other hand, depend only on µ Si and are mediated by a dark matter particle in the t-channel. To illustrate how these processes alter the dark matter relic densities, in this section we consider three possible values for µ Si : 0.3, 1, 3 TeV.
µ S1 induces the processes φ 1 + φ † 2 ↔ φ 1 + h and φ 1 + φ 1 ↔ φ † 2 + h, the former affect only Ω 2 while the latter may affect both relic densities. Figure 6 displays Ω i versus M i for different values of µ S1 and for M 2 /M 1 = 1.2. From the left panel we see that Ω 2 can be suppressed by orders of magnitude as a consequence of the exponential behaviour mentioned previously but now involving σ 1210 v . Notice also that Ω 2 increases steeply as soon as the process is kinematically open, as observed in the figure. From the right panel, we notice instead that, at intermediate values of M 1 , Ω 1 can be reduced by up to two orders of magnitude. At low masses, the process φ 1 + φ 1 → φ † 2 + h is kinematically closed during φ 1 freeze-out, so there is no effect on Ω 1 , in agreement with the figure. At high masses, it is instead the propagator that suppresses the Regarding the µ S2 -induced processes, they can affect Ω 2 at low and intermediate masses as shown in figure 8. The only process that may reduce the φ 1 number density after φ 2 freeze-out is φ 1 + φ 2 → φ 2 + h but it has a negligible effect on Ω 1 due to the small value of Ω 2 .

IV. THE VIABLE PARAMETER SPACE
As we have seen, both relic densities may be modified by the new interactions allowed by the Z 5 symmetry. Now we want to explore in detail their implications on the viable parameter space of this model and on the dark matter detection prospects. To that end, we have randomly scanned the parameter space of the model and selected a large sample of points consistent with current data. In particular, they are compatible with the limit on the invisible decays of the Higgs boson obtained from the LHC data [30], with the direct detection limits recently derived by the XENON1T collaboration [25] (we apply the corresponding recasted exclusion given by micrOMEGAs [31]) and with the dark matter density as measured by PLANCK [32]. While the PLANCK collaboration There is no appreciable effect on Ω 1 for the values considered in this figure. reports the theoretical prediction of the relic density is not expected to be that precise. In our scans, we consider a model compatible with the above value if its relic density, as given by micrOMEGAs, lies between 0.11 and 0.13, which amounts to about a 10% uncertainty. In any case, our results are robust against plausible variations in such interval.
We have performed several random scans, varying just a subset of the free parameters of the model at a time so as to make the analysis simpler. In all the scans, the dark matter masses and the Higgs-portal couplings are varied in the following ranges: 10 −4 ≤ |λ S1 | ≤ 1, If these were the only parameters different from zero, the viable points would all lie at the Higgs resonance. The interplay between the relic density constraint and the strong limits from direct detection searches would exclude the rest of the parameter space. And this conclusion still holds To bypass the direct detection bounds, the relic density of φ 1 must be reduced by the new interactions. In the previous section, we saw that the parameter µ S1 can have this effect, so in our first scan we set the dimensionless couplings as well as µ S2 to zero (λ 3i , λ 412 = 0, µ S2 = 0) and vary µ S1 between 0.1 TeV and 10 TeV. This upper limit on µ S1 is rather arbitrary but seems reasonable given that M 1 and M 2 -the other dimensionful parameters of the model-take a maximum value of 2 TeV and 4 TeV respectively.
The resulting viable parameter space is shown in figure 9. Notice that the viable points cover the entire spectrum of dark matter masses, from the Higgs resonance up to the maximum value considered in the scan. This is one of our main results. From the top-left panel, we see that Had we considered higher values of µ S1 , semi-annihilations would have remained significant to larger dark matter masses. The Higgs-portal couplings are shown in the bottom panels. |λ S1 | can vary over orders of magnitude while semi-annihilations are relevant, M 1 < 1 TeV, but from then on annihilations become important and |λ S1 | is therefore restricted to a narrow band, reaching Regarding the contributions of the two dark matter particles to the total density, we see, from the right panel of figure 10, that φ 1 always gives the dominant contribution. It accounts for more than 70% of the dark matter density and in most points for more than 95% of it. In numerous cases Ω 2 turns out to be several orders of magnitude smaller than Ω 1 . The fact that the lighter dark matter particle usually accounts for the bulk of the dark matter density is one of our most important results.
At first sight, this distribution of the dark matter densities may seem to imply that the Z 5 model effectively becomes, at present, a one-component dark matter model -that φ 2 , having a small density, can be ignored. But this is not so. From figure 11 we see that either dark matter particle may be observed in future direct detection experiments. The solid line shows the current limit from XENON1T while the dashed and dotted lines correspond to the expected sensitivities of LZ [26] and DARWIN [27] respectively. What is happening with φ 2 is that its smaller density can be compensated by its larger coupling to the Higgs (see figure 9), resulting in a sizable signal. The feasibility of detecting a subdominant component of the dark matter has been noted before [33,34], but it seems to have been largely forgotten. In the Z 5 model, this possibility arises naturally. With respect to indirect detection, the most relevant dark matter annihilation channels are displayed in figure 12 with their respective scaled cross sections. For comparison, the current limits [28] for certain final states are also shown (solid lines) as well as the projected sensitivity [29] for bb (dotted line). The semi-annihilation process φ 1 + φ 1 → φ 2 + h turns out to be the most relevant one, with a cross section that can reach 10 −25 cm 3 /s. The experimental limit on such a process will depend also on M 2 and has not been derived in the literature. A related process which has been considered is φ 1 + φ 1 → φ 1 + h, whose limit is shown in the figure as a solid black line [35].
Since M 2 > M 1 , the limit on φ 1 + φ 1 → φ 2 + h should be weaker. Due to the ξ 2 suppression and its higher mass, the indirect detection signals involving φ 2 are less promising. Indirect detection experiments, therefore, do not constrain the viable parameter space of this model.
Let us summarize what we have found with the scan for µ S1 = 0: i) the model becomes viable over the entire range of dark matter masses, M 1 < 2 TeV; ii) φ 1 , the lighter dark matter particle, accounts for most of the dark matter density; iii) direct detection experiments offer great prospects to test this model, including the possibility of observing signals from both dark matter particles.
As we wil see, ii) and iii) are actually generic features of the viable parameter space of the  FIG. 12: Dark matter annihilation rates for the viable models in the scan with µ S1 = 0. The solid-green (solid-yellow) line shows current limit of φ 1 self-annihilation into bb (W + W − ) reported by the Fermi collaboration from 6 years of observation and 15 dwarf spheroidal galaxies (dSphs) [28], while the dotted-green line represents the projected sensitivity for 45 dSphs and 15 years of observation [29] which serves as an estimate of the corresponding W + W − sensitivity since both bounds for 6 years as 15 dSphs are similar at high DM masses. Moreover, for comparison purposes the upper limit on the semi-annihilation process φ 1 φ * 1 → φ 1 h [35] is also displayed.
The results for the scan with µ S2 = 0 are shown in figure 13. In this case, we set the dimensionless couplings as well as µ S1 to zero (λ 3i , λ 412 = 0, µ S1 = 0) and vary µ S2 between 0.1 TeV and 10 TeV. Three crucial differences are observed with respect to the results from the µ S1 scan. (bottom panels). In particular, a significant fraction of models predict detectable signals from both dark matter particles (yellow points). Discriminating such signals would, however, become more challenging in this case due to the degeneracy between the dark matter particles.
In another scan we allowed the dimensionless couplings to independently vary within the range while settting µ Si = 0. Semi-annihilations are absent in this case so the only new process that can reduce the φ 1 relic density is the conversion As indicated by the value of ζ 1 conv , it is the above mentioned conversion process that renders such region consistent with current data. The top-right panel shows that φ 1 essentially accounts for the total dark matter density over the entire new viable region. The contribution of φ 2 amounts to less than 2%. In spite of this, either particle could be observed in future direct detection experiments, as illustrated in the bottom panels.
We also did additional scans, including one in which all the free parameters of the model are simultaneously varied, and the results are essentially identical to what we found in the three scans already analyzed. It is fair to conclude, therefore, that our scans reveal the genuine viable parameter space of the Z 5 model.
In our analysis so far we have always assumed that M 1 < M 2 because, as already mentioned in section II, the symmetry of the Lagrangian allows us to make this simplification. The results for the case M 2 < M 1 can be obtained from ours by simply swapping the corresponding quantities: 32 , Ω 1 ↔ Ω 2 , etc. Thus, we have actually studied the full range of dark matter masses possible in this model -M 1 /2 < M 2 < 2M 1 .
In this section, the most important results of our work were derived -we characterized the viable parameter space of the Z 5 model and determined its detection prospects. Let us review our main findings: 1. It is possible to satisfy the relic density constraint and current direct detection limits over the entire range of dark matter masses we considered (M 1 < 2 TeV). In particular, the low mass region M 1 TeV, which is excluded in the singlet scalar model, is perfectly compatible with present bounds thanks to the new interactions allowed by the Z 5 symmetry.
2. The dark matter density is always dominated by the lighter dark matter particle. In our show the projected sensitivity of LZ [26] and DARWIN [27] experiments. Yellow points indicate that both DM particles lay within the sensitivity region of DARWIN.
scans, the heavier dark matter particle never accounts for more than 40% of the total density, and often contributes significantly less than that.
3. Either dark matter particle may be detected in future direct detection experiments. And in a sizable fraction of models both particles are predicted to be detectable, providing a way to differentiate this model from the usual scenarios with just one dark matter particle.
Hence, besides being simple and well-motivated, the Z 5 model turns out to be a consistent and verifiable framework for two-component dark matter.
We have seen that the new interactions allowed by the Z 5 symmetry render this model viable over a wide range of dark matter masses. This result stands in sharp contrast to what is found in similar models based on Z 2 symmetries. In the scenario with one complex scalar singlet stabilized by a Z 2 symmetry, the dark matter mass necessarily lies either at the Higgs-resonance or around 2 TeV, as a consequence of the interplay between the relic density constraint and current direct detection limits. And a similar outcome is obtained in a two-dark matter scenario where the two singlet scalars are stabilized with a Z 2 × Z 2 symmetry. The Z 5 model can be seen as a natural extension of these scenarios and has the advantage of remaining viable at low masses and of being testable via direct detection experiments.
The Z 5 symmetry used in our model is the lowest Z N compatible with two dark matter particles that are complex scalar fields [12]. Even if other Z N symmetries, with N > 5, can be imposed to simultaneously stabilize two dark matter particles [13], the Z 5 model serves as a prototype for all the two-component scenarios where the dark matter particles are complex scalars. That is, our results can be applied rather straightforwardly to other Z N frameworks, as explained next.
Let us denote the two dark matter particles charged under a Z N by φ i , φ j (with i < j ≤ N/2 and j = N/2 for N even [12]), where φ k gets a factor e i2πk/N upon a Z N transformation. For 5 < N ≤ 10, the complete set of possibilities for the two dark matter particles is: • (φ 1 , φ 2 ): all Z N symmetries allow the µ S1 φ 2 1 φ * 2 term and forbid the µ S2 φ 1 φ 2 2 and λ 31 φ 3 1 φ 2 terms while the Z 7 is the only one that allows λ 32 φ 1 φ 3 2 . This means that for the scenario with M 1 < M 2 the viable M 1 range can extend up to 2 TeV while for M 2 < M 1 the maximum value that M 2 can reach is 1 TeV.
As in the previous item the Z 10 model forbids both cubic (µ Si ) and quartic λ 3i interactions, which means there is no new viable DM regions.
• (φ 2 , φ 4 ): the Z 9 only allows the µ S2 φ 2 2 φ * 4 interaction, which implies a viable mass range up to 2 TeV (1 TeV) for M 2 < M 4 (M 4 < M 2 ). The case of the Z 10 model is rather special since it features an analogous Lagrangian to the Z 5 model which means it allows both cubic (µ Si ) and quartic interactions λ 3i . Therefore the results presented in this work apply to the Z 10 model with (φ 2 , φ 4 ) as DM fields. This analysis demonstrates that the Z 5 model is the most general Z N model with two complex fields, from which the DM properties for other models with a higher Z N symmetry can be deduced to a large extent. By the same token, it is the Z 7 model with (φ 1 , φ 2 , φ 3 ) that serves as a prototype for scenarios with three dark matter particles.
Finally, let us comment on possible extensions of the Z 5 model. A simple one is to embed the Z 5 symmetry within an spontaneously broken U (1) gauge symmetry [12]. In that case, the µ S1 term would still be allowed whereas the µ S2 would require an additional vacuum expectation value.
Higher gauge symmetries can also be envisioned. Another option is to introduce extra fields so as to explain neutrino masses. By including additional vectorlike fermions, Majorana masses for the neutrinos can be generated at two-loops, as in the Z 3 -based models studied in [36][37][38][39]. The minimal extra fermion content turns out to be two SU (2) L doublets and one SM singlet, both having the same Z 5 charge (either w 5 or w 2 5 ) to admit a mixing term via the Higgs doublet. It follows that φ 1 and φ 2 become the loop mediators as in the scotogenic models and continue playing the role of DM particles as long as their decays into the new fermions are kinematically closed.
Moreover, in certain regions of parameter space it may be possible to realize a scenario with 3 DM particles (two scalars plus a fermion) without additional symmetries. A phenomenological study of these interesting alternatives lies, however, beyond the scope of the present paper and will be left for future work.
We investigated the phenomenology of the two-component dark matter model based on a Z 5 symmetry, which serves as an archetype for other Z N (N > 5) models with two complex scalar dark matter particles. After describing the model, we studied in detail how the relic density depends on the new parameters allowed by the Z 5 symmetry. In order to characterize the viable parameter space, we did several random scans and analyze their implications. We found that it is possible to satisfy the dark matter constraint and direct detection limits over the entire range of dark matter masses considered, M 1 2 TeV. The key parameter turned out to be the trilinear coupling associated to the lighter dark matter particle (e.g. µ S1 for M 1 < M 2 ), which, via semi-annihilations, renders the model viable without requiring a mass degeneracy between the dark matter particles.
At low dark matter masses (M i < 1 TeV), the other trilinear coupling as well as a quartic coupling (e.g. µ S2 and λ 31 for M 1 < M 2 ) may also play a role, but only if the dark matter particles are at least mildly degenerate. We found that the dark matter density is dominated by the lighter particle for all the viable models and that a significant fraction of the viable parameter space can be probed by future direct detection experiments. Remarkably, both dark matter particles could give rise to observable signals in such experiments, providing a way not only to test this model but also to differentiate it from more conventional dark matter scenarios.