Light Dark Matter in a Minimal Extension with Two Additional Real Singlets

PHYSICAL REVIEW D 103, 015010 (2021)

Light dark matter in a minimal extension with two additional real singlets

Markos Maniatis * Centro de Ciencias Exactas and Departamento de Ciencias Básicas, UBB, Avenida Andres Bello 720, 3780000 Chillán, Chile

(Received 24 August 2020; accepted 12 December 2020; published 8 January 2021)

The direct searches for heavy scalar dark matter with a mass of order 100 GeV are much more sensitive than for light dark matter of order 1 GeV. The question arises whether dark matter could be light and has escaped detection so far. We study a simple extension of the Standard Model with two additional real singlets. We show that this simple extension may provide the observed relic dark matter density, does neither disturb big bang nucleosynthesis nor the cosmic microwave background radiation observations, and fulfills the conditions of clumping behavior for different sizes of Galaxies. The potential of one Standard Model–like Higgs-boson doublet and the two singlets gives rise to a changed Higgs phenomenology, in particular, an enhanced invisible Higgs-boson decay rate is expected, detectable by missing transversal momentum searches at the ATLAS and CMS experiments at CERN.

DOI: 10.1103/PhysRevD.103.015010

I. INTRODUCTION collisions. In particular, in this way a DM candidate may enhance the invisible decay rate of a SM-like Higgs boson. Recently it has been reported [1] that the NGC1052–DF2 Note that in the SM the only invisible decay channel of galaxy with a stellar mass of approximately 2 × 108 solar the Higgs boson (h) is via two electroweak Z bosons masses has a rotational movement in accordance with its which subsequently decay into pairs of neutrinos (ν), that observed mass. If this negative indirect search for dark is, h → ZZ → 4ν. matter (DM) is confirmed it challenges any attempt to solve The collider experiments ATLAS [15] and CMS [16] the puzzle of galaxy rotations by modifications of gravity. have measured upper limits on the spin-independent On the other hand we are facing very severe negative DM-nucleon scattering cross section [17,18]. Under the results from direct searches of DM, for instance from assumption that DM couples to the Higgs boson, these the cryogenic experiments SuperCDMS [2], CRESST [3], measurements provide the most stringent bounds available EDELWEISS [4], and the noble liquid experiments on light DM detection, far below the underground DM- ArDM [5], DarkSide [6], DEAP [7],LUX[8], PandaX nucleon direct-detection limits. However, the upper bounds [9], WARP [10], XENON [11], and ZEPLIN [12]. These for light scalar DM are orders of magnitudes less stringent experiments provide very low upper limits for the interaction than the bounds for heavy DM particles with masses cross sections of DM with nucleons. For instance, the of Oð100 GeVÞ. XENON100 [13] direct detection experiment at Gran Let us in this context recall that interactions of DM with Sasso excludes spin-independent elastic nucleon cross sec- SM particles can also not be arbitrarily small: supposing 2 10−45 2 tions down to values as tiny as × cm for DM that DM is produced dynamically in the evolution of the particles with a mass of 55 GeV at 90% confidence level. Universe, for decreasing annihilation cross sections of the In case that the Higgs boson couples to DM also collider DM particles to Standard Model particles, the annihilation experiments may detect DM; for an overview of this subject processes become more and more rare in the evolution of the we refer to [14]. Even that DM is expected not to show Universe and freeze out happens earlier—corresponding to a traces in the detector components, since it is expected to higher DM number density. Eventually this would lead to an couple only very weakly to Standard Model (SM) particles, overclosure of the Universe. it may be discovered as missing transversal momentum in One example of this is the so called Lee-Weinberg bound [19]: in case of leptonic dark matter it has been shown that *[email protected] its mass has to be greater than of the order of 2 GeV in order not to lead to an overclosure of the Universe. However this Published by the American Physical Society under the terms of bound refers to dark matter lepton candidates. If instead the the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to annihilation is not mediated by electroweak gauge bosons the author(s) and the published article’s title, journal citation, this bound does not apply—opening the window for light and DOI. Funded by SCOAP3. dark matter.

2470-0010=2021=103(1)=015010(9) 015010-1 Published by the American Physical Society MARKOS MANIATIS PHYS. REV. D 103, 015010 (2021)

Supersymmetric models provide generically new We employ the convention that the upper component of the weakly interacting bosons and fermions, and therefore doublet is charged. The doublet gives as usual masses to the in principle these models can provide light DM particles. fermions and gauge bosons. The potential reads [33,34] However, at least the simplest supersymmetric extension DM μ2φ†φ μ2χ2 μ2η2 λ φ†φ 2 λ χ4 λ η4 of the Standard Model appears to be disfavored, since Vlight ¼ h þ χ þ η þ hð Þ þ χ þ η there is a tension of the predicted Higgs mass spectrum in † 2 † 2 2 2 þ λ χðφ φÞχ þ λ ηðφ φÞη þ λχηχ η ; ð2Þ contrast to observations—in particular the detected Higgs h h boson is too heavy in order to be a supersymmetric where we suppress the argument of the fields from here on particle in the minimal supersymmetric extension (see in most cases. Inspecting this potential we encounter apart for instance the review [20]). 2 1 from the electroweak symmetry SUð ÞL × Uð ÞY the dis- Here we want to study a simple model with two addi- Z 2 0 tional real scalars. One of theses scalars serves as a DM crete symmetries Z2 with χ transforming as χ! − χ and Z2 Z0 candidate (χ) and the other is a scalar mediator (η). Since 2 as η! − η, where we assume that the Higgs doublet as well we expect that the DM particle χ annihilates into a pair η as all Standard Model fields transform trivially. We want of scalar mediators , we are not confronted with the the potential to provide vacuum expectation values for Lee-Weinberg bound and we may have sufficiently large 0 the neutral component of the doublet, φ , as well as for the annihilation cross sections χχ → ηη. On the other hand, field η, that is, besides electroweak symmetry breaking through couplings of the mediator η to neutrinos, similar to SUð2Þ × Uð1Þ → Uð1Þ , the potential should break Z0 the studies [21,22], this annihilation cross section does L Y em 2 spontaneously. Since the symmetry Z2 is not broken, neither disturb big bang nucleosynthesis nor the observed neither explicitly nor spontaneously, we expect a stable cosmic microwave background radiation [23]. Moreover, particle χ providing DM. through the coupling of DM χ to the mediator η we The domain wall problem [36], appearing inevitably for automatically get DM self-interactions [24,25] which do a spontaneously broken discrete symmetry, is expected to not contradict structure formation simulations for different be circumvented in the usual way: we assume that the sizes of galaxies [26]. discrete symmetry Z0 is broken explicitly by an additional The potential of the Higgs boson h andthetworeal 2 cubic mass-parameter term linear in η. This additional term scalars χ, η provides couplings among these elementary however is Planck-mass suppressed and therefore only of particles. These couplings give a changed Higgs phe- relevance at very high energies and therefore negligible in nomenology compared to the SM. For instance, from our analysis here and not further taken into account. the coupling of the Higgs boson h to the DM candidate χ We want the mediator η to decay sufficiently fast in we get an enhanced invisible Higgs-decay rate at order to not disturb big bang nucleosynthesis. This can be colliders, since χ is expected to be stable and escapes achieved by a coupling of η to right-handed neutrinos. We detection. Therefore it is expected to have an enhanced arrange theses couplings by imposing the right-handed invisible decay rate at the large hadron collider at CERN neutrinos to transform under Z0 as at the experiments ATLAS and CMS. The signature 2 comes from missing transversal momentum with 0 0 0 Z2 Z2 Z2 respect to the collision direction in observations at these νR;e→ þ νR;e; νR;μ→ − νR;μ; νR;τ→ − νR;τ; ð3Þ experiments. Scalar DM has been studied, for instance in [27–31] and and add appropriate Majorana kinetic terms and mass terms references therein. In [21,32–34] models are studied with λ the scalar DM candidate accompanied by a mediator, with ij c Lν ¼ iν¯ =∂ν − ην¯ ν þ H:c: ; ð4Þ the DM particle mass of the order of 100 GeV. For a review R R;i R;i 2 R;i R;j of light DM we refer to [35]. Here we focus on the two-real- scalar extension of the SM with one light DM candidate with i; j ∈ fe; μ; τg the indices denoting the three flavors 0 accompanied by a mediator. of the neutrinos. Different choices for the Z2 symmetry behavior of the right-handed neutrinos in (3) could be II. THE MINIMAL EXTENSION WITH TWO chosen, however this assignment keeps the number of λ ADDITIONAL REAL SINGLETS parameters for the matrix ij rather restricted. The parameters should provide stability of the potential In the model we propose we have besides one Higgs- and the observed electroweak symmetry breaking, that is, boson doublet two real singlets, in this case we have a global minimum of the potential with

0 φþðxÞ φ χ η hφi¼ 1 ; hηi¼vη; hχi¼0: ð5Þ ðxÞ¼ 0 ; ðxÞ; ðxÞ: ð1Þ pﬃﬃ φ ðxÞ 2 vh

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We write the neutral component of the Higgs-boson doublet t-channel diagram with a mediator propagator. The and the scalar η expanded about their respective vacuum- s-channel diagram with a Higgs-boson propagator is expectation values, kinematically suppressed due to the light DM and mediator masses compared to the Higgs-boson mass. Therefore the φþðxÞ relic density of DM depends mainly on the coupling φðxÞ¼ 1 ; ηðxÞ¼vη þ η ðxÞ; ð6Þ λ η0 pﬃﬃ e parameter χη besides the masses mχ and mη0 .For lighter 2 ðvh þ hðxÞÞ than χ, the DM annihilation cross section is too large, freeze out occurs too late and the resulting relic density is with hðxÞ and ηeðxÞ real fields. The necessary tadpole conditions for stability at the vacuum yield too small. In Fig. 1 we show the values for the mediator mass mη0 2 2 2 2 λ 1 λ ημη − 2λημ 1 λ ημ − 2λ μη versus the quartic parameter χη, which provide the 2 h h 2 h h h 2 vh ¼ 2 ; vη ¼ 2 : ð7Þ observed relic density of Ω h ¼ 0.1200ð12Þ [37] (the 2 4λ λη − λ η 2 4λ λη − λ η DM h h h h value in parenthesis gives the 1-σ uncertainty). This study is For the DM mass squared we find from the potential, that done for four different values of the DM mass, that is, mχ in is, from the quadratic terms with respect to the DM field χ the range from 0.5 to 2 GeVas indicated in the figure. From after electroweak symmetry breaking, this study we see that it is possible for a light DM candidate accompanied by a slightly heavier mediator to achieve the 2 2μ2 λ 2 2λ 2 right, that is, indirectly observed relic density. We see that mχ ¼ χ þ hχvh þ χηvη: ð8Þ for increasing values of the mediator mass mη0 , the right relic density can be obtained by strongly raising the quartic We get for the mixing matrix of the excitations h and ηe coupling λχη over orders of magnitude as shown in these in the basis ðh; ηeÞ semilogarithmic figures. As mentioned above, the annihi- 2 2λ λ lation of dark matter into mediators occurs besides the vh h vhvη hη 2 : ð9Þ quartic DM mediator coupling via s- and t-channel dia- v vηλ η 2vηλη: h h grams. Varying the mediator mass we get competing contributions of the remaining different diagrams corre- This mixing matrix is diagonalized by an orthogonal matrix sponding to the different regions of the mediator mass. This with mixing angle θ, behavior is reflected by the kink structure in Fig. 1. As discussed later in the context of DM-self-interactions, v vηλ η 2θ h h tanð Þ¼ 2 2 ; ð10Þ we want to avoid a too strong DM self-interaction, therefore v λ − vηλη h h we set the quartic parameter λχ to zero and consider rather small values for the parameter λχη. The relic density is and we get the mass eigenstates h0 and η0. In order to get a numerically calculated from the solutions of the Boltzmann Higgs boson with the observed mass of about 125 GeV and equation using the package MICROMEGAS [38,39]. All other a light mediator, we see that the mixing angle, or equiv- parameters are fixed to mh0 ¼ 125 GeV, vh ¼ 246 GeV alently the parameter λhη, has to be rather tiny. (the standard electroweak parameters), vη ¼ 3 GeV, λ η ¼ Eventually, from the spontaneous breaking with respect h 1 10−7 λ 0 1 λ to η, the right-handed neutrino-mass matrix is generated, × , hχ ¼ . , a vanishing parameter χ, and we fix the nonvanishing right-handed neutrino-mixing- λ η μ τ matrix parameters (11) to λeμ ¼ λeτ ¼ 0.1. The remaining ðMνÞij ¼ ijh i;i;j¼ e; ; : ð11Þ parameters λh, λη and the sine of the Higgs-mediator mixing angle, sin θ, are then determined by the tadpole conditions (7). III. PHENOMENOLOGY OF DARK MATTER We proceed with a study of DM self-interactions; for an AND THE MEDIATOR introduction to this subject we refer to [26]. Observations Let us start with a study of the relic abundance of DM in show rather flat distributions in the cores in smaller galaxies the model with two additional real scalars. With view on the of dwarf size up to galaxies of the size of our Milky Way. In potential (2) we see that DM (χ) may annihilate into pairs of contrast, simulations of galaxy formations, assuming that mediators η0 (the mass eigenstate is denoted by η0). This DM is collisionless, predict a cusp in the density profile for annihilation arises from the quartic χ-χ-η-η coupling, as these sizes of galaxies. This mismatch between observation well as via the trilinear χ-χ-h and χ-χ-η couplings which and simulation can be resolved if DM has self-interactions come from the quartic couplings after spontaneous sym- χχ → χχ with a cross section per DM mass of the order of 2 metry breaking. At tree level we get from these interactions σ=mχ ≈ 1 cm =g [26] for smaller galaxy sizes up to our besides the direct coupling two s-channel diagrams with Milky Way and is approximately collisionless for sizes a Higgs-boson, respectively, mediator propagator and one larger than our Milky Way up to galaxy clusters: that is,

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λ Ω 2 FIG. 1. Mass of the mediator mη0 versus the quartic coupling parameter χη providing the observed relic density of DMh ¼ 0.1200ð12Þ [37] (with 1-σ uncertainty). The study shows the values for four different choices of a light DM particle, that is, mχ equals 0.5 (upper left), 1.0 (upper right), 1.5 (lower left), and 2.0 GeV (lower right). The width of the line corresponds to the 1-σ variation. with self-interaction cross sections per DM mass dropping the range of 0.5 to 2.0 GeV as indicated in the figure. We to values below 0.1 cm2=g. Typical rotational velocities for have chosen a relative velocity of 10 km=s, corresponding to dwarf galaxies are 10 km=s, for Milky Way–type galaxies dwarf galaxies with a required value of σðχχ → χχÞ=mχ ¼ 200 km=s and for galaxy clusters 1000 km=s [26]. 1 cm2=g [26]. We see that it is possible to get the observed In the model considered here with two additional value for the self-interaction. In this figure we also replicate scalars, the self-interactions arise, as can be seen from the results from the DM relic density study, as shown the potential, from the quartic DM self coupling with previously in Fig. 1. The correct relic density curve (green parameter λχ,fromtheχ-χ-η-η coupling with parameter line) raises much steeper than the contours for the self- λχη,andfromtheχ-χ-h-h coupling with parameter λhχ. interaction cross section per DM mass (black line) with Since we are considering light DM, the contribution from respect to increasing values of the mediator mass mη0 . the SM-like Higgs boson h0 is suppressed. The quartic All other parameters are set as above in the study of the coupling λχ on the other hand has to be suppressed DM relic density. In particular we see that both curves also since otherwise we immediately overshoot the self- intersect, giving parameter values which on the one hand interaction cross sections. In addition to the quartic χ provide the observed DM relic density and on the other hand interaction, we encounter self-interactions from trilinear provide the correct DM-self-interaction cross section. As we couplings of χ with the Higgs boson h and the mediator η can see in this study, even for a vanishing quartic DM- in s-andt-channel diagrams. coupling parameter λχ the quartic χ-χ-η-η parameter λχη has In Fig. 2 we show a study of DM self-interactions. to be rather small. In principle multiple exchanges between The contour plots show the mediator mass mη0 versus the the DM particles could lead to Sommerfeld enhance- quartic coupling parameter λχη with respect to the DM self- ment [40] of the self-interactions, however these long- interaction cross section per DM mass. Similar to the study term interactions are for the here considered similar of the DM relic density we have varied the value of DM in masses of DM and the mediator negligible. Let us note that

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FIG. 2. Contour plots of self-interacting DM cross section per DM mass σðχχ → χχÞ=mχ in the mη0 -λχη plane for four different values of the light DM mass, that is, mχ equals 0.5 GeV (upper left), 1.0 GeV (upper right), 1.5 GeV (lower left), 2.0 GeV (lower right). The legend gives the regions of different values of the DM-self-interaction per DM mass in units of cm2=g, the central black curve in the 2 figures corresponds to σðχχ → χχÞ=mχ ¼ 1 cm =g. In the calculation of the cross section per DM mass, the relative velocity of DM particles is set to 10 km=s corresponding to dwarf galaxies. The steeply raising green line replicates the results from Fig. 1 Ω 2 0 1200 12 1 σ corresponding to the right DM relic density of DMh ¼ . ð Þ [37] (with - uncertainty). the Sommerfeld enhancement is implemented in the gives parameters which not only give the right self- MICROMEGA package. interactions of DM, but also naturally drop to values in We study also the dependence of the self-interaction agreement with observations of density profiles of different cross section per DM mass on the relative velocity of DM. sizes of galaxies. From the explicit calculation of the cross section we find In the early Universe the mediators thermalize with the that the self-interacting cross section per DM mass drops right-handed neutrinos, but below a temperature of about orders of magnitude with increasing relative velocities. In the mass of the mediator, that is, of the GeVorder, the right- particular, if we arrange the parameters such that we get for handed neutrinos decouple. Since they are stable they 10 a relative velocity of vrel ¼ km=s (corresponding to contribute to the relic neutrino density. The energy density dwarf galaxies) the wanted cross section per DM mass of of radiation of neutrinos ρν with respect to photons ργ after 2 σðχχ → χχÞ=mχ ¼ 1 cm =g, then we find for larger values the time of electron-positron annihilation is [41] of the relative velocity, that is, v > 200 km=s (corre- rel ρ 7 4 4=3 sponding to galaxy sizes larger than our Milky Way) values ν ν ¼ Neff ; ð12Þ below 0.1 cm2=g. Therefore, the study as shown in Fig. 2 ργ 8 11

015010-5 MARKOS MANIATIS PHYS. REV. D 103, 015010 (2021) where the fraction 7=8 originates from the fermionic nature of neutrinos and the factor ð4=11Þ1=3 comes from the reheating of the photons from electron-positron annihilation. In the Standard Model, the number of effective SM 3 045 neutrinos has been calculated to Neff ¼ . [42] for three neutrino species. Every additional neutrino contributes to the number of effective neutrinos [43,44], g TL 4=3 Δ ð decÞ Neff ¼ R ; ð13Þ gðTdecÞ with gðTÞ the effective number of degrees of freedom at temperature T, where the respective neutrinos decouple. FIG. 3. Branching ratios of the SM-like Higgs boson h0 Since the left-handed neutrinos decouple at the MeV scale, depending on the quartic potential parameter λhχ. The decay but the right-handed neutrinos in the model considered here rates are for a pair of DM particles χχ, for a b-quark pair bb¯, decouple at the GeV scale, we find from [45] with gluons gg, leptons ll¯, c-quarks cc¯, and branching ratios of nearly gðMeVÞ¼9.5 and gðGeVÞ¼76.5 for every right-handed the same size for photons γγ and a s-quark pair ss¯. Δ 0 062 neutrino the rather small contribution, Neff ¼ . .On the other hand we get from the recent cosmic microwave background measurement from the Planck satellite experi- momentum. The invisible decay width of the SM-like ment (combined with baryon acoustic oscillation), Neff ¼ 0 χ 2 99 0 17 Higgs boson h into a pair of s at tree-level reads . . [37]. This means that the measurement of Neff is in agreement with the light DM model considered here. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 λ v βχ We conclude that no substancial change of microwave inv hχ h mχ Γ 0 ¼ ; with βχ ¼ 1 − 4 : ð14Þ background radiation, nor big bang nucleosynthesis is to be h →χχ 8π 2 mh0 mh0 expected (for the case of new scalars coupled to Standard Model particles we refer to the overview [46]). This decay width is obviously proportional to the quartic Eventually we want to study the Higgs phenomenology potential parameter λ χ squared. In Fig. 3 we study the in the model considered here. Compared to the SM, the h branching ratios of the SM-like Higgs boson h0 varying this Higgs-boson doublet is accompanied by two additional quartic coupling parameter λ χ. In this study we have scalars, χ and η. The potential (2) involves in addition to h chosen all other parameters as mentioned above with a DM the quadratic and quartic self couplings also couplings 0 5 among the three scalars. As discussed in Sec. II we mass of mχ ¼ . GeV and the mediator slightly heavier, 0 55 encounter a mixing of the neutral component of the that is, mη0 ¼ . GeV. This pair of values together with −4 doublet h with the mediator η forming a Higgs mass the parameter λχη ¼ 10 yields the correct values for the eigenstate h0.From(10) we see that the tangent of relic density of DM and for the DM self-interactions, see twice the mixing angle is proportional to the potential Fig. 2. Since the Higgs boson h0 has a tiny mediator η parameter λhη. component it could also decay into neutrinos and in Since the Higgs boson h couples on the one hand to the addition there are couplings of the Higgs boson to mediator λ DM particle χ with coupling strength λhχ and on the other particles. However for the tiny mixing parameter hη hand has the usual Yukawa couplings to the fermions, we considered here these contributions are negligible, as see that DM interacts with the hadrons of the nucleon. This shown explicitly in Fig. 3. DM-nucleon interaction is therefore expected to be detect- As we can see in this study, we find for larger values of λ able at direct detection experiments—in the Introduction the parameter hχ a dominant Higgs-decay channel into we have mentioned some direct-detection experiments DM particles χ. Depending on the parameter λhχ we find searching for DM-nucleon interactions. therefore an enhanced invisible branching ratio compared Moreover, the coupling of the DM candidate to the Higgs to the SM. Let us remind us that in the SM the Higgs boson boson opens the possibility to detect DM at collider can only decay invisibly via Z bosons, which subsequently experiments. In particular, since we are considering light decay into neutrinos, that is, h → ZZ → 4ν. DM, that is, we assume to have mh0 > 2mχ, the SM-like Since the invisible decay rate it restricted from the Higgs boson h0 can decay into a pair of DM particles χ. missing transverse momentum searches at the CMS and Since the DM particles are assumed to be stable, these ATLAS experiments at the LHC this translates into an decays do not show visible traces in the detector compo- upper bound on the coupling strength between the Higgs nents of collider experiments. However, these events can boson and the DM particle. The ATLAS collaboration has be discovered from the observation of missing transversal recently measured [17] the upper limits of the invisible

015010-6 LIGHT DARK MATTER IN A MINIMAL EXTENSION WITH TWO … PHYS. REV. D 103, 015010 (2021) branching ratio and finds 0.13 at the 95% confidence level right-handed neutrinos. This process is sufficiently fast, in agreement with the Standard Model. The CMS collabo- that is, occurs before big bang nucleosynthesis takes place, ration has published [18] an upper limit of the invisible and in particular neither DM nor the mediator decay into branching ratio of 0.24 compared to the Standard Model Standard Model particles. Therefore we neither expect expectation of 0.23. From the calculation of the branching displaced vertices in colliders, decays into muons, nor ratios as shown in Fig. 3 we get for the more restrictive changed meson decays. upper bound of 0.13 for the invisible branching ratio the The objective of this study is to investigate the agreement constraint λhχ < 0.01 for the potential parameter. Let us of the simple light DM model with different observations in also mention the context of the coupling parameter λhχ to telescopes, underground experiments, and collider experi- the spin-independent DM-nucleon cross section, which ments with respect to DM. The relic density in this two- reads [47] real-scalar extension has been computed, and it has been shown that, for a dark matter mass in the range of 0.5 to 2 4 2 λ χ m f 2 GeV and a mediator slightly heavier, we can get the σSI h N N χN ¼ 2 : ð15Þ observed relic density. Also we have studied dark matter πm ðmχ þ m Þ h N self-interactions which may explain the observation of rather flat DM density profiles in cores of dwarf galaxies Here mN is the nucleon mass and fN a nucleon form factor. Based on phenomenological and lattice-QCD calculations, up to galaxies of the size of our Milky Way. In the model considered here the interaction of DM with the mediator a Higgs-nucleon form factor of fN ¼ 0.308ð18Þ has been provides these self-interactions quite naturally. Moreover, calculated [48]. The coupling parameter λhχ, which appears in the DM-nucleon cross section (15) can now, using (14), due to the nature of these interactions transmitted by the mediator, these self-interactions automatically drop for be replaced by the invisible decay rate. Therefore, the — measured upper bound on the invisible decay rate in Higgs- larger relative velocities in galaxy clusters in agreement boson decays translates into an upper bound for the DM- with simulations of galaxy formations. nucleon cross section [47]. The upper limits on the spin- Since the mediator decays into right-handed neutrinos independent DM-nucleon scattering cross section are which are stable, these neutrinos contribute to the neutrino compared to the direct detection limits in [18].Itisshown background radiation. Indirectly this background could that the CMS and ATLAS results from the measurement of change big bang nucleosynthesis or the cosmic microwave the invisible decay of the Higgs boson are orders of background. This effect is encoded in the number of magnitudes more sensitive than direct detection experi- effective neutrinos which is measured at the Planck satellite 2 99 0 17 ments for scalar DM masses up to 7 GeV. These are experiment yielding Neff ¼ . . [37], whereas in SM 3 045 currently the most stringent limits available—assuming that the SM the calculation gives Neff ¼ . [42] in agree- there is a Higgs-DM coupling. ment with the one sigma deviation of the observation. In the two-singlet model we calculate that every right- Δ 0 062 handed neutrino contributes Neff ¼ . to the effective IV. CONCLUSIONS number of neutrinos. This originates from the fact that the The rotational movement of stars in galaxies is typically right-handed neutrinos decouple earlier as the left-handed faster than expected from the visible mass. Attempts to neutrinos in the evolution of the Universe. The effective modify gravity appear disfavored due to the observation number of neutrinos is therefore in agreement with of exceptional galaxies showing no mismatch. New, yet observation. undiscovered, elementary dark matter particles could solve Eventually we have studied the Higgs-boson phenom- the puzzle. However, direct detection experiments of dark enology compared to the Standard Model. The inter- matter–nucleon interactions have not revealed any dark actions of the Higgs boson with the DM candidate and the matter particles. The lowest bounds in these experiments mediator give on the one hand a mixing of the Higgs come from searches of dark matter particles with a mass of boson with the mediator and on the other hand provides the order of 100 GeV. Therefore it appears quite natural to interactions of the Higgs boson with the DM particle. look for light DM particles. In case the light DM candidate These latter interactions in turn give rise to interactions of is leptonic with a mass below 2 GeV—the so-called Lee- DM with nucleons—observable at underground DM- Weinberg bound—this would lead to an overclosure of the nucleon detection experiments, but also detectable at Universe. Here we have discussed a simple model with collider experiments: the Higgs bosons produced at two additional scalars, odd under discrete symmetries, not LHC in the CMS or ATLAS detectors may decay into restricted by the Lee-Weinberg bound. The scalar potential stable dark matter particles, detectable as events with of the model provides interactions among the DM candi- missing transverse momentum. Upper bounds on such date, a mediator, and a SM-like Higgs boson. These searches have been published by both experiments and interactions provide DM annihilation into pairs of are here translated into upper bounds on the correspond- mediators. The mediators in turn decay into pairs of ing DM-Higgs coupling parameter.

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Eventually we would like to mention that it is quite ACKNOWLEDGMENTS striking that a simple extension of the Standard Model with The work is supported, in part, by the Grants, two real singlets complies with the different types of DM Universidad del Bio-Bio Materia obscura y los bosones searches discussed and also shows no contradiction with de Higgs, No. DIUBB 193209 1/R and FONDECYT our understanding of the evolution of the Universe. regular with No. 1200641.

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