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Physics » "Trends in Modern Cosmology", book edited by Abraao Jesse Capistrano de Souza, ISBN 978-953-51-3210-3, Print ISBN 978-953-51-3209-7, Published: June 7, 2017 under CC BY 3.0 license. © The Author(s).

Chapter 2

Constraining the Parameters of a Model for Cold Dark Matter

By Abdessamad Abada and Salah Nasri
DOI: 10.5772/intechopen.69044

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η01(4) versus m1 for very light S1, small mixing, and very small WIMP‐Higgs coupling.
Figure 1. η01(4) versus m1 for very light S1, small mixing, and very small WIMP‐Higgs coupling.
η01(4) versus m0 for small mixing, moderate m1, and WIMP‐Higgs coupling.
Figure 2. η01(4) versus m0 for small mixing, moderate m1, and WIMP‐Higgs coupling.
Elastic N−S0 scattering cross‐section as a function of m0 for moderate m1, small mixing, and small WIMP‐Higgs coupling.
Figure 3. Elastic N−S0 scattering cross‐section as a function of m0 for moderate m1, small mixing, and small WIMP‐Higgs coupling.
Elastic N−S0 scattering cross‐section as a function of m0 for moderate m1, small mixing, and small WIMP‐Higgs coupling.
Figure 4. Elastic N−S0 scattering cross‐section as a function of m0 for moderate m1, small mixing, and small WIMP‐Higgs coupling.
Elastic cross‐section σdet versus m0 for heavy S1, small mixing, and relatively large WIMP‐Higgs coupling.
Figure 5. Elastic cross‐section σdet versus m0 for heavy S1, small mixing, and relatively large WIMP‐Higgs coupling.
Typical branching ratios of ϒ1S decaying into τ’s, charged pions, and charged kaons as functions of m1. The corresponding experimental upper bounds are shown.
Figure 6. Typical branching ratios of ϒ1S decaying into τ’s, charged pions, and charged kaons as functions of m1. The corresponding experimental upper bounds are shown.
Typical branching ratios of ϒ3S decaying into muons and dark matter as functions of m1. The corresponding experimental upper bounds are shown.
Figure 7. Typical branching ratios of ϒ3S decaying into muons and dark matter as functions of m1. The corresponding experimental upper bounds are shown.
Running of the self‐couplings. η1 controls perturbativity and the Higgs coupling λ becomes negative quickly.
Figure 8. Running of the self‐couplings. η1 controls perturbativity and the Higgs coupling λ becomes negative quickly.
The running of the Higgs self‐coupling λ. It becomes negative at about 15 TeV for this set of parameter values.
Figure 9. The running of the Higgs self‐coupling λ. It becomes negative at about 15 TeV for this set of parameter values.
Running of the mutual couplings. The inclusion of the other SM particles flattens the runnings.
Figure 10. Running of the mutual couplings. The inclusion of the other SM particles flattens the runnings.
Regions of viability of the two‐singlet model (in dark grey). Physical Higgs self‐coupling λ0(4) and mixing angle θ very small.
Figure 11. Regions of viability of the two‐singlet model (in dark grey). Physical Higgs self‐coupling λ0(4) and mixing angle θ very small.
The region of viability (dark grey) is even richer for a larger mixing angle θ.
Figure 12. The region of viability (dark grey) is even richer for a larger mixing angle θ.
The physical Higgs self‐coupling λ0(4) shrinks the viability region (dark grey) as it increases.
Figure 13. The physical Higgs self‐coupling λ0(4) shrinks the viability region (dark grey) as it increases.

Constraining the Parameters of a Model for Cold Dark Matter

Abdessamad Abada1, 2 and Salah Nasri3
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This chapter aims at reviewing how modeling cold dark matter as weakly interacting massive particles (WIMPs) gets increasingly constrained as models have to face stringent cosmological and phenomenological experimental results as well as internal theoretical requirements like those coming from a renormalization-group analysis. The review is based on the work done on a two-singlet extension of the Standard Model of elementary particles. We conclude that the model stays viable in physically meaningful regions that soon will be probed by direct-detection experiments.

PACS numbers: 95.35.+d; 98.80.−k; 12.15.−y; 11.30.Qc

Keywords: cold dark matter, light WIMP, extension of Standard Model, rare decays, RGE

1. Introduction

Dark matter accounts for about 26.5% of the total mass‐energy density of the Universe [1], but we still do not know what it is. It is called dark because it is not accounted by the visible matter, the conventional baryons and leptons, which take about 4.9% of the total mass‐energy density [1]. As it clearly interacts through gravity, some argue that it could still be baryonic, in the form of massive astrophysical compact halo objects (MACHOs) which emit dim or no light [2] or some sort of huge gravitational objects like galaxy‐sized black holes. Indeed, such high concentrations of matter would bend passing light, the so‐called gravitational lensing phenomenon, including microlensing, in ways we can detect. But the amount of dark matter we know of would produce gravitational lensing with a significantly higher number of occurrences than what observation accounts for.

Neutrinos have long been thought of composing the dark matter around us. However, Standard Model neutrinos are light, and so too fast‐moving (hot) to compose the (cold) dark matter structures we see. But sterile neutrinos, non‐Standard Model particles, can be heavier, and so could be dark matter candidates. This possibility has been reignited with the recent detection of an X‐ray emission line at an energy of 3.55 keV coming from galaxy clusters, the Andromeda galaxy, the Galactic Center and the Draco dwarf spheroidal galaxy. This line is consistent with the decay of a 7.1 keV sterile neutrino [3].

In fact, there is by now quasi‐consensus that dark matter ought to be understood outside the realm of conventional matter. One other scenario is that of (pseudo)scalar particles of tiny mass 1022 eV, the so‐called ultralight axions that could account for the dark matter content of the Universe. This is supported by high‐resolution cosmological simulations [4]. Axions originated in quantum chromodynamics, the theory of quarks and gluons, in relation to the axial anomaly in this theory and the strong Charge Conjugation Parity Symmetry Violation (CP violation) problem. But like anything else related to dark matter, they elude detection. The Axion Dark Matter Experiment (ADMX) may bring in answers in the near future [5].

But maybe the most popular candidate for dark matter is an electrically neutral and colorless weakly interacting massive particle (WIMP). Such a particle originated in supersymmetric (SUSY) extensions of the Standard Model. The most obvious such a candidate is the neutralino, a neutral R‐odd supersymmetric particle. Indeed, neutralinos are only produced or destroyed in pairs, thus constituting the lightest SUSY particles. However, alas, as rich, attractive and beautiful as SUSY can be, supersymmetric particles continue to elude detection at the Large Hadron Collider (LHC), at least in Run 1 experiments with a center‐of‐mass energy s=8 TeV [6]. Run 2 experiments with s=13 TeV are currently under way, targeting a final luminosity of about 100 fb1, and so are tested in more involved and less stringent formulations of supersymmetry [7].

It must be stressed that until now, we have not detected dark matter, at least not in a conclusive manner. Indeed, we know dark matter is there only because of its gravitational interactions, and this is why and how we believe it contributes about a quarter of the mass energy of the known Universe. But we still do not know whether dark matter really interacts with ordinary matter. We believe it does, even if very weakly. We believe these interactions can yield signals with enough strength so that we can detect dark matter or produce it in collisions of Standard Model particles [8].

We must also understand that a detection process relies primarily on a theory or a model. A theory like supersymmetry, which originated in the realm of elementary particle physics, is devised as an extension to the Standard Model that is based on a yet‐to‐be‐detected symmetry between fermionic and bosonic states [9]. Its DM connection came only later. In fact, in the rather long period between the Higgs mechanism proposal [10] and the detection of the Higgs particle [11], various extensions of the Standard Model were proposed in order to alleviate some of its shortcomings, the so‐called “Beyond the Standard Model” (BSM) Physics [12]. A number of these BSM models bear in them extra fields, meaning extra particles with specific properties. Until today, such particles have never been detected. With time and change in focus, the most stable of these hypothetical particles have then been proposed as candidates for dark matter, many in the form of WIMPs. The advantage of such a paradigm is clear: the calculational techniques that built strength in the realm of particle physics were ready at the service of dark matter search with little extra effort in development. But the experimental framework was also ready. Such a state of affairs could partly explain the popularity of WIMP physics, compared to other possible scenarios for dark matter.

Accordingly, many experiments have been devised specifically to detect dark matter. Each, of course, must be based on a specific scheme that is based on a specific scenario. There are experiments that try to detect dark matter directly, through missing energy momentum after a WIMP collides directly with an ordinary nucleus. The low‐background DAMA (NaI) and then DAMA/LIBRA (NaI[Ti]) experiments at Gran Sasso in Italy [13] add a twist to this by trying to detect dark matter in the galactic halo via its suggested model‐independent flux annual modulation [14]. The CoGeNT experiment [15] in Soudan (Minnesota, USA) also tries to detect this annual modulation, but in the region where the WIMP mass is 10 GeV. The CDMS I (Stanford, USA) [16], then CDMS II (Soudan, USA) [17], and now the superCDMS (Soudan, USA, then SNOLAB, Sudbury, Canada) [18] perform direct detection, measuring ionization and phonon signals resulting from a WIMP‐nucleus collision, sensitive in the low‐mass region. The XENON10 [19], then XENON100 [20], then the coming XENON1t [21], all in Gran Sasso, Italy, use liquid Xenon as a detecting medium for WIMP‐nucleon and WIMP‐electron collisions. There is also the Large Underground Xenon (LUX) experiment (South Dakota, USA) [22], as a direct‐detection experiment, and its more sensitive successor LZ experiment [23]. The CRESST experiment [24], followed by CRESST II [25], both at Gran Sasso, Italy, also try to detect dark matter directly with low mass. We also have the series of EDELWEISS experiments [26] (Modane, France), which target low‐mass WIMPs. The list is exhaustive, and could not be accounted here due to space constraints.

The above experiments are terrestrial, with instruments buried underground to reduce noise. But there are other experiments which are space borne that carry out indirect detection in cosmic rays. There is the Fermi Gamma‐Ray Space Telescope (Fermi‐LAT), which has found excess of gamma rays in the galactic center that cannot be explained by conventional sources and which is compatible with the presence of dark matter [27]. Fermi‐LAT uses what we call indirect methods, namely, collecting gamma‐ray signals and removing from these those emitted by all possible known sources. Another space‐borne experiment is the Alpha Magnetic Spectrometer (AMS) experiment at the international space station [28], collecting and analyzing signals from cosmic rays. In addition, the Payload for Antimatter Matter Exploration and Light‐nuclei Astrophysics (PAMELA) experiment [29] is a particle identifier that uses a permanent magnet spectrometer for space cosmic‐ray direct measurements.

A third prong in the dark matter search enterprise is to produce it in particle colliders like the LHC [8]. There is an added difficulty here, which is that we do not know in which mass range we should look into. It could well be that the present center‐of‐mass energy that is available, 13 TeV, may not be sufficient. Nevertheless, the search for dark matter at the LHC is intense. One reason is that, experimentally, this is feasible now: small amounts of missing energy and transverse momentum can be detected now. Note that the present detectors are not built to detect dark matter directly. Rather, the latter would appear as a missing energy or missing momentum. For example, we now look at events in which a Z boson and a missing transverse momentum are produced in a proton‐proton collision at s=13 TeV. The Z boson decays into two charged leptons, a recognizable signature, and a possible missing transverse momentum, which would indicate the production of dark matter in the process. A similar search, conducted previously by the CMS Collaboration and based on data collected with s=8 TeV (Run 1), found no evidence of new physics and hence set limits on dark matter production. A recent search performed by the ATLAS Collaboration with s=13 TeV with an integrated luminosity of 3.2 fb1 also reported no evidence [30].

What should be clear by now is that interpreting signals as dark matter necessitates modeling. On the other hand, any model needs experimental results to restrict the range of its free parameters, to fine‐tune these parameters, and, ultimately, in many cases, to be eliminated. The aim of this chapter is to shed light on the main steps a phenomenologist takes when building a model for dark matter, then testing the model against experimental results. It is an attempt to look into the modeling process itself, from the “cradle to the grave,” so to speak. The discussion is based on a model proposed in Ref. [31] for cold dark matter, exposed to particle‐physics phenomenology in [32], and further restricted by internal consistency in Ref. [33]. We will see how gradually the parameters of the model are constrained, and how the region of viability is reached. To carry out the discussion smoothly, we have chosen a model which is simple enough to avoid confusion created by the often involved details of the calculations and could‐be‐complexity of the model itself, but at the same time rich enough to be able to accommodate a vast range of experimental results. The material presented in this chapter is drawn from the works just cited.

This chapter is organized as follows. After this Introduction, Section 2 motivates and then presents the model based on WIMP physics, namely, a two‐singlet extension of the Standard Model of elementary particles. We will try to avoid lengthy arguments and focus on the essentials. Section 3 shows how the measured amount of dark matter relic‐density constrains the value of the dark matter annihilation cross‐section, a constraint any model has to satisfy. We then discuss how the two‐singlet extension fits into this, and add to it a perturbativity ingredient. Section 4 takes the two‐singlet model into the arena of particle phenomenology and sees how it copes with rare meson decays. Section 5 goes back to the fundamentals and runs a renormalization‐group analysis to inquire into the sustainability of the model. Section 6 puts all these constraints together and determines the regions of viability of the model. Section 7 is left for concluding remarks.

2. A model for dark matter: motivation and parametrization

As mentioned in the Introduction, the most popular candidate for dark matter is an electrically neutral colorless weakly interacting massive particle (WIMP), and the neutralino, the lightest supersymmetric particle, is a robust fit for this role. However, as explained in Ref. [31] and references therein, it is hard to argue in favor of a neutralino when it comes to light cold dark matter, say, a WIMP mass of up to 10 GeV. In addition, up to now, we have not detected supersymmetric signatures at the LHC [34].

Therefore, with no prior hints as to what the internal structure of the WIMP might be, one adopts a bottom‐up approach, in which one extends the Standard Model by adding to it the simplest of fields, one real spinless scalar, which will be the WIMP. This field must be a Standard Model gauge singlet so that we avoid any “direct contact” with any of the Standard Model particles. It is allowed to interact with visible particles only via the Higgs field. It is made stable against annihilation by enforcing upon it the simplest of symmetries, a discrete Z2 symmetry that does not break spontaneously. This construction is called the minimal extension to the Standard Model. In view of its cosmological implication, the minimal extension has first been proposed in Ref. [35] and has been extensively studied and explored in Ref. [36]. However, this model is shown in Ref. [37] to be inadequate if we want the WIMP to be light.

In the logic of this bottom‐up approach, adding another real scalar seems the natural step forward. This field will also be endowed with a Z2 symmetry, but this one we will break spontaneously, and the reason is to open new channels for dark matter annihilation, which implies an increase in the corresponding annihilation cross‐section, which in turn would allow smaller WIMP masses, something we want to achieve. Needless to say that this auxiliary field must also be a Standard Model gauge singlet.

Therefore, we extend the Standard Model by adding two real, spinless and Z2‐symmetric fields: the dark matter field S0 for which the Z2 symmetry is unbroken and an auxiliary field for which it is spontaneously broken. Both fields are Standard Model gauge singlets and hence can interact with “visible” particles only via the Higgs doublet, taken in the unitary gauge. We must also assume all processes calculable in perturbation theory. The details of the spontaneous breaking of the electroweak gauge symmetry and the additional auxiliary Z2 symmetry are left aside [31].

The potential function that involves the physical scalar Higgs field h, the dark matter field S0, and the physical auxiliary scalar field S1 is as follows:


The quantities m0,mh, and m1 are the masses of the corresponding fields S0, h, and S1, respectively, and all the other parameters are real coupling constants. Also, the part of the Standard Model Lagrangian that is relevant to Dark matter annihilation is given in terms of the physical fields h and S1 by the following potential function:


The coupling constants in the above expression are given by the following relations, in which the quantities mf, mw, and mz are the masses of the fermion f, the W, and the Z gauge bosons, respectively:


The angle θ is the mixing angle between the fields h and S1 [31]. The quantities v and v1, both positive, are the vacuum expectation values of the Higgs and auxiliary fields, respectively.

This model has nine free parameters to start with, three mass parameters and six coupling constants [31]. As already mentioned, perturbativity is assumed, which means all the original coupling constants are small. The dark matter self‐coupling constant η0 in Eq. (1) will not enter the lowest‐order calculations we will consider, and so this parameter stays free for the time being and we are left with eight parameters. The spontaneous breaking of the electroweak and Z2 symmetries for the Higgs and auxiliary fields, respectively, introduces the two vacuum expectation values v and v1. The value of v is fixed experimentally to be 246 GeV [38] and for the present discussion, we fix the value of v1 at the order of the electroweak scale, say, 100 GeV. In addition, the Higgs mass is now known [11], mh=125 GeV. Hence, five free parameters remain. Three of these are chosen to be the two physical masses m0 (dark matter) and m1 (S1 field), plus the mixing angle θ between S1 and h. The two last parameters we choose are the two physical mutual coupling constants λ0(4) (dark matter—Higgs) and η01(4) (dark matter—S1 particle), see Eq. (1).

3. Constraints from cosmology and perturbativity

Any model of dark matter has to comply with astrophysical observations. Indeed, dark matter is believed to have been produced in the early Universe. A most popular paradigm for this production is the so‐called “freeze‐out scenario” by which dark matter, thought of as a set of elementary particles, interacts with ordinary matter, weakly but with enough strength to generate common thermal equilibrium at high temperature. However, as the cosmos is cooling down, at some temperature Tf, the rate of expansion of the Universe becomes higher than the rate of dark matter particle annihilation, which forces dark matter to decouple from ordinary matter, and hence a “freeze‐out”—Tf is thus called the freeze‐out temperature. The DM relic density ΩDM is essentially the one we measure today [1]:


where h¯ is Hubble constant in units of 100 km×s1×Mpc1.

In a model where dark matter is seen as WIMPs that can annihilate into ordinary elementary particles, the relic density ΩDM can be related to the annihilation DM cross‐section σann. Indeed, in the framework of the standard cosmological model, one can derive the following relation [39]:

ΩDMh¯21.07×109xfgmPlσannv GeV;xfln0.0038mPlm0σannvgxf.

The quantity mPl=1.22×1019GeV is the Planck mass, m0 is the dark matter mass, xf=m0/Tf, and g is the number of relativistic degrees of freedom with a mass less than Tf. The quantity σannv is the thermally averaged annihilation cross‐section of a pair of two dark matter particles multiplied by their relative speed in their center‐of‐mass reference frame. Solving (4) with the current value (5) for ΩDM with xf between 19.2 and 21.6 [40], we obtain the following constraint on the annihilation cross‐section:


This is one major constraint any WIMP model like the one we discuss here has to satisfy. Indeed, the quantity σannv is calculable in perturbation theory, and so, the implementation of (6) will induce an admittedly complicated but important relation between the free parameters of the model, hence reducing their space of freedom, reducing their number by one. Also, the constraint induced by (6) can be used to examine aspects of the theory like perturbativity. To implement perturbativity in the present two‐singlet model, we use (6) to obtain the mutual coupling constant η01(4) (coupling between the DM field S0 and auxiliary field S1) in terms of the dark matter mass m0 for given values of λ0(4) (coupling between S0 and Higgs) and study its behavior to tell which dark matter mass regions are consistent with perturbativity. It should be mentioned that once the two mutual coupling constants λ0(4) and η01(4) are small, all the other physical coupling constants will be small.

The quantity σannv is calculated in perturbation theory using all possible annihilation channels the model allows for [31]. As the model has many parameters, the behavior of the mutual coupling constant η01(4) is bound to be rich. Sampling is therefore necessary. In this review, we briefly comment on the behavior of η01(4) for two sets of the parameters (θ,m1,λ0(4)). A more substantial discussion can be found in Ref. [31].

The first set of parameters is a small mixing angle θ=10o, a weak mutual S0‐Higgs coupling constant λ0(4)=0.01, and a S1‐mass m1=10 GeV. The corresponding behavior of η01(4) versus m0 is shown in Figure 1. The range of m0 displayed is from 0.1 to 200 GeV. In this regime, the first feature we see is that the relic‐density constraint on dark matter annihilation forbids WIMP masses m01.3 GeV. Furthermore, just about m01.3 GeV, the c‐quark threshold, the S0S1 mutual coupling constant η01(4) starts at about 0.8, a value, while perturbative, that is roughly 80‐fold larger than the mutual S0 Higgs coupling constant λ0(4). Then as the DM mass increases, η01(4) decreases, steeply first, more slowly as we cross the τ mass toward the b mass. Just before m1/2, the coupling η01(4) hops onto another solution branch that is just emerging from negative territory, gets back to the first one at precisely m1/2 as this latter carries now smaller values, and then jumps up again onto the second branch as the first crosses the m0 axis down. It goes up this branch with a moderate slope until m0 becomes equal to m1, a value at which the S1 annihilation channel opens. Just beyond m1, there is a sudden fall to a value η01(4)0.0046 that is about half the value of λ0(4), and η01(4) stays flat till m045 GeV where it starts increasing, sharply after 60 GeV. In the mass interval m0 ≃ 66–79 GeV, there is a “desert” with no positive real solutions to the relic‐density constraint, hence no viable dark matter candidate exists. Beyond m079 GeV, the mutual coupling constant η01(4) keeps increasing monotonously, with a small notch at the W mass and a less noticeable one at the Z mass. As it increases, its values remain perturbative.


Figure 1.

η01(4) versus m1 for very light S1, small mixing, and very small WIMP‐Higgs coupling.

The second set of parameters we feature is still a small Higgs S1 mixing angle θ=10o, an increased S0‐Higgs mutual coupling constant λ0(4)=0.2, and a moderate S1 mass m1=20GeV. The behavior of the S0S1 mutual coupling constant η01(4) versus the DM mass m0 is displayed in Figure 2. Here too, no viable DM masses exist below roughly 1.4 GeV, at which value η01(4) starts at 1.95. It decreases with a sharp change of slope at the b‐quark threshold, then makes a sudden dive at about 5 GeV, a change of branch at m1/2 down till about 12 GeV where it jumps up back onto the previous branch just before going to cross into negative territory. It drops sharply at m0=m1 and then increases slowly until m043.3 GeV. Then, no viable WIMP masses exist, a desert. As we see, for this set of parameters (θ,λ0(4),m1), the model constrains the dark matter mass inside the interval 1.6 GeVm043.3 GeV, with perturbative coupling constants.


Figure 2.

η01(4) versus m0 for small mixing, moderate m1, and WIMP‐Higgs coupling.

With the same mixing angle θ=10o and mutual coupling constant λ0(4)=0.2, larger masses m1 yield roughly the same behavior, but with values of η01(4) that could be nonperturbative. For example, when m1=60 GeV, the mutual coupling η01(4) starts very high (85) at m01.5 GeV, and then decreases rapidly. There is a usual change of branches and a desert starting at about 49 GeV, a behavior that is peculiar in a way because the desert starts at a mass m0<m1, that is, before the opening of the S1 annihilation channel. In other words, the dark matter is annihilating into the light fermions only and the model is perturbatively viable in the range of 20–49 GeV.

4. Constraints from direct detection

Perhaps the most known constraints on a WIMP model are those coming from direct‐detection experiments like the many we have cited in the introductory section. In such experiments, the signal sought for would typically come from the elastic scattering of a WIMP off a nonrelativistic nucleon target. However, as mentioned in the Introduction, until now, none of these direct‐detection experiments have yielded an unambiguous dark matter signal. Rather, with increasing precision from one generation to the next, these experiments put increasingly stringent exclusion bounds on the dark matter‐nucleon elastic‐scattering total cross‐section σdet in terms of the dark matter mass m0, and because of these constraints, many models can get excluded.

Therefore, a theoretical dark matter model like the two‐singlet extension we discuss here has to satisfy these bounds to remain viable. For this purpose, we calculate σdet as a function of m0 for different values of the parameters (θ,λ0(4),m1) and compare its behavior against the experimental bounds. The calculation is carried out with sufficient details in Ref. [31], and the total cross‐section for non‐relativistic S0‐nucleon elastic scattering is given by


In this relation, mN is the nucleon mass and mB is the baryon mass in the chiral limit. The mutual coupling constants λ0(3) and η01(3) are defined in Eq. (1). The relic‐density constraint on the dark matter annihilation cross‐section (6) has to be imposed throughout. In addition, we require now that the coupling constants be perturbative, and we do this by imposing the additional requirement 0η01(4)1.

Generically, as m0 increases, the detection cross‐section σdet starts from high values, slopes down to minima that depend on the parameters, and then picks up moderately. There are features and action at the usual mass thresholds, with varying sizes and shapes. Regions coming from the relic‐density constraint and new ones originating from the additional perturbativity requirement are excluded.

For the purpose of illustration, we choose three indicative sets of values for the parameters (θ,λ0(4),m1). We start first with a Higgs‐S1 mixing angle θ=10o, a weak mutual S0‐Higgs coupling λ0(4)=0.01, and an S1 mass m1=20 GeV. The behavior of σdet versus m0 is shown in Figure 3. There, we see that for the two mass intervals 20–65 GeV and 75–100 GeV, plus an almost singled‐out dip at m0=m1/2, the elastic scattering cross‐section is below the sensitivity of SuperCDMS. However, XENON1T should probe all these masses, except m058 and 85 GeV.


Figure 3.

Elastic NS0 scattering cross‐section as a function of m0 for moderate m1, small mixing, and small WIMP‐Higgs coupling.

Increasing m1 has the effect of closing possibilities for very light dark matter and thinning the intervals as it drives the predicted masses to larger values. Indeed, in Figure 4, where m1=40 GeV, in addition to the dip at m1/2 that crosses SuperCDMS but not XENON1T, we see acceptable masses in the ranges of 40–65 GeV and 78 GeV up. The intervals narrow as we descend, surviving XENON1T only as spiked dips at 62 GeV and around 95 GeV.


Figure 4.

Elastic NS0 scattering cross‐section as a function of m0 for moderate m1, small mixing, and small WIMP‐Higgs coupling.

On the other hand, a larger mutual coupling constant λ0(4) has the general effect of squeezing the acceptable intervals of m0 by pushing the values of σdet up, and it may even happen that at some point, the model has no predictability. This case is shown in Figure 5, where θ=10o,λ0(4)=0.4, and m1=60 GeV. In this example, the effects of increasing the values of both λ0(4) and m1. As we see, the model cannot even escape Cryogenic Dark Matter Search II (CDMSII).


Figure 5.

Elastic cross‐section σdet versus m0 for heavy S1, small mixing, and relatively large WIMP‐Higgs coupling.

5. Constraints from particle phenomenology

If a dark matter model based on WIMP physics is not killed already by the constraints coming from cosmology, perturbativity, and direct detection, it has to undergo the tests of particle phenomenology. To see how this works, we discuss here the constraints on our two‐singlet model that come from a small selection of low‐energy processes, namely, the rare decays of ϒ mesons. The forthcoming discussion is based on work done in Ref. [32]. There, the interested reader will find a fuller account of this study, together with relation to Higgs phenomenology. Note that the dark matter relic‐density constraint in Eq. (6) and the perturbativity requirement 0<η01(4)<1 are implemented systematically. Also, as in Ref. [32], we will restrict the discussion to light cold dark matter.

We therefore look at the constraints that come from the decay of the meson ϒ in the state nS (n=1,3) into one photon γ and one particle S1. For m18 GeV, the branching ratio for this process is given by the relation:


In the above expression, xn(1m12/mϒns2) with the mass of ϒ1(3)S given by mϒ1(3)S=9.46(10.355) GeV, the branching ratio Br(μ)Br(ϒ1(3)Sμ+μ)=2.48(2.18)×102 [41], α is the QCD coupling constant, αs=0.184 the QCD coupling constant at the scale mϒnS, the quantity GF is the Fermi coupling constant, and mb is the b‐quark mass [38]. The function f(x) incorporates the effect of QCD radiative corrections given in [42] and the step function is denoted by Θ(x). However, a rough estimate of the lifetime of S1 indicates that the latter is likely to decay inside a typical particle detector, which means we should take into account its most dominant decay products. We first have a process by which S1 decays into a pair of pions, with the following decay rate:

Γ(S1ππ)GFm142πsin2θ[m1227(1+11mπ22m12)2 ×(14mπ2m12)12Θ(m12mπ)(2mKm1)] +3(Mu2+Md2)(14mπ2m12)32Θ(m12mK)].

Here, mπ is the pion mass and mK is the kaon mass. Also, chiral perturbation theory is used below the kaon pair production threshold [43, 44], and the spectator‐quark model above up to roughly 3 GeV, with the dressed u and d quark masses Mu=Md0.05 GeV. Note that this rate includes all pions, charged and neutral. Above the 2mK threshold, there is the production of both a pair of kaons and η particles. The decay rate for K production is


In the above rate, Ms0.45 GeV is the s‐quark mass in the spectator‐quark model [45, 46]. For η production, replace mK by mη and 9/13 by 4/13.

The particle S1 also decays into c and b quarks (mainly c). Including the radiative QCD corrections, the corresponding decay rates are given by


The dressed quark mass m¯qmq(m1) and the running strong coupling constant α¯sαs(m1) are defined at the energy scale m1 [47]. There is also a decay into a pair of gluons, with the rate


Here, αs=0.47 is the QCD coupling constant at the spectator‐quark model scale, between roughly 1 and 3 GeV.

We then have the decay of S1 into leptons, the corresponding rate given by


where m is the lepton mass. Finally, S1 can decay into a pair of dark matter particles, with a decay rate:


The coupling constant η01(3) is given in Eq. (1). The branching ratio for ϒnS decaying via S1 into a photon plus X, where X represents any kinematically allowed final state, will be


In particular, XS0S0 corresponds to a decay into invisible particles.

The best available experimental upper bounds on 1S‐state branching ratios are (i) Br(ϒ1Sγ+ττ)<5×105 for 3.5 GeV<m1<9.2 GeV [48]; (ii) Br(ϒ1Sγ+π+π)<6.3×105 for 1 GeV<m1 [49]; (iii) Br(ϒ1Sγ+K+K)<1.14×105 for 2 GeV<m1<3 GeV [50]. Figure 6 displays the corresponding branching ratios of ϒ1S decays via S1 as functions of m1, together with these upper bounds. Also, the best available experimental upper bounds on ϒ3S branching ratios are: (i) Br(ϒ3Sγ+μμ)<3×106 for 1 GeV<m1<10 GeV; (ii) Br(ϒ3Sγ+invisible)<3×106 for 1 GeV<m1<7.8 GeV [51]. Typical corresponding branching ratios are shown in Figure 7.


Figure 6.

Typical branching ratios of ϒ1S decaying into τ’s, charged pions, and charged kaons as functions of m1. The corresponding experimental upper bounds are shown.


Figure 7.

Typical branching ratios of ϒ3S decaying into muons and dark matter as functions of m1. The corresponding experimental upper bounds are shown.

If we perform a systematic scan of the parameter space, we find that the main effect of the Higgs‐dark matter coupling constant λ0(4) and the dark matter mass m0 is to exclude, via the relic‐density and perturbativity constraints, regions of applicability of the model. This is shown in Figures 6 and 7, where the region m11.4 GeV is excluded. Otherwise, these two parameters have little effect on the shapes of the branching ratios themselves. The onset of the S0S0 channel for m12m0 abates sharply the other channels, and this one becomes dominant by far. The effect of the mixing angle θ is to enhance all branching ratios as it increases, due to the factor sin2θ. The dark matter decay channel reaches the invisible upper bound already for θ15o, for fairly small m0, say, 0.5 GeV. The other channels find it hard to get to their respective experimental upper bounds, even for large values of θ. There are further constraints that come from particle phenomenology tests. The interested reader may refer to [32] for further details.

6. Internal constraints

Further constraints on a field‐theory dark matter model come from internal consistencies. Indeed, one must ask how high in the energy scale the model is computationally reliable. To answer this question, one investigates the running of the coupling constants as a function of the scale Λ via the renormalization‐group equations (RGE). One‐loop calculations are amply sufficient. A detailed study of the RGE for our two‐singlet model was carried out in Ref. [33]. The brief subsequent discussion is drawn from there, and the reader is referred to that article for more details.

In an RGE study, there are two standard issues to monitor, namely, the perturbativity of the scalar coupling constants and the vacuum stability of the theory. Imposing these two latter as conditions on the model will indicate at what scale Λm it is valid. As mentioned in the Introduction, it has been anticipated that new physics, such as supersymmetry would appear at the LHC at the scale Λ1 TeV. Present results from ATLAS and CMS indicate no such signs yet. One consequence of this is that the cutoff scale Λm may be higher. In this model, the RGE study suggests that it can be 40TeV. As ever, the DM relic‐density constraint is systematically imposed, together with the somewhat less stringent perturbativity restriction 0η01(4)4π.

Remember that the model is obtained by extending the Standard Model with two real, spinless, and Z2‐symmetric SM‐gauge‐singlet fields. The potential function of the scalar sector after spontaneous breaking of the gauge and one of the Z2 symmetries is given in Eq. (1). The potential function before symmetry breaking is the one we need in this section. It is given in Eq. [31]:


The field S0 is still the WIMP with unbroken Z2 symmetry, and χ1 is the auxiliary field before spontaneously breaking its Z2 symmetry. Both fields interact with the SM particles via the Higgs doublet H. The masses m˜02, μ2, and μ12 as well as all the coupling constants are real positive numbers.1

A one‐loop renormalization‐group calculation yields the following β‐functions for the above scalar coupling constants [33]:


As usual, by definition βgdg/dlnΛ, where Λ is the running mass scale, starting from Λ0=100GeV. Note that the DM self‐coupling constant η0 has so far been decoupled from the other coupling constants, but not anymore in view of Eq. (17) now that the running is the focus. However, its initial value η0(Λ0) is arbitrary and its β‐function is always positive. This means η0(Λ) will only increase as Λ increases, quickly if starting from a rather large initial value, slowly if not. Therefore, without losing generality in the subsequent discussion, we fix η0(Λ0)=1. Hence, here too we still effectively have four free parameters: λ0(4), θ, m0, and m1.

Furthermore, the constants g, g', and gs are the SM and strong gauge couplings, known [52] and given to one‐loop order by the expression:


where aG=1996π2,4196π2,716π2 and G(Λ0)=0.65, 0.36,1.2 for G=g,g,gs, respectively. The coupling constant λt is that between the Higgs field and the top quark. To one‐loop order, it runs according to Ref. [52] the following expression:


with λt(Λ0)=mt(Λ0)v=0.7, where v is the Higgs vacuum expectation value and mt is the top mass. Note that we are taking into consideration the fact that the top‐quark contribution is dominant over that of the other fermions of the Standard Model.

After the two spontaneous breakings of symmetry, we end up with the two vacuum expectation values: v=246 GeV for the Higgs field h, and v1 for the auxiliary field S1. In this section, we take v1=150 GeV. Above v, the fields and parameters of the theory are those of (16). Below v1, the fields and parameters are those of Eq. (1). We take the values of the physical parameters at the mass scale Λ0=100 GeV. The initial conditions for the coupling constants in (16) in terms of these physical free parameters are as follows:


Note that, normally, as we go down the mass scale, we should seam quantities in steps: at v, v1, and Λ0. However, the corrections to (20) are of one‐loop order times lnvv1 or lnv1Λ0, small enough for our present purposes to neglect. The perturbativity constraint we impose on all dimensionless scalar coupling constants is G(Λ)4π. Also, vacuum stability means that G(Λ)0 for the self‐coupling constants η0,λ, and η1, and the conditions:


for the mutual couplings λ0,η01, and λ1.

Figure 8 displays the behavior of the self‐couplings under RGE for θ=10o,λ0(4)=0.01,m1=110GeV, and m0=55GeV. The dramatic effect is on the Higgs self‐coupling constant λ which quickly gets into negative territory, at about 15TeV, thus rendering the theory unstable beyond this mass scale. This is better displayed in Figure 9, where the Renormalization Group (RG) behavior of λ is shown by itself. Such a negative slope for λ is expected, given the negative contributions to βλ in (17). The coupling constant η1 is dominant over the other couplings and controls perturbativity, leaving its region much later, at about 1600 TeV. This seems to be a somewhat general trend: the non‐Higgs SM particles seem to flatten the runnings of the scalar couplings.


Figure 8.

Running of the self‐couplings. η1 controls perturbativity and the Higgs coupling λ becomes negative quickly.


Figure 9.

The running of the Higgs self‐coupling λ. It becomes negative at about 15 TeV for this set of parameter values.

The runnings of the mutual coupling constants for the same set of parameters’ values are displayed in Figure 10. They also get flattened by the other SM particles, but they stay positive. They dwell well below the self‐couplings. Increasing m0 and m1 will raise the mutual coupling η01 and not the two others, higher than η1 in some regions.


Figure 10.

Running of the mutual couplings. The inclusion of the other SM particles flattens the runnings.

Raising λ0(4) will also make the self‐couplings η1 and η0 run faster while affecting very little λ. It will also make the mutual coupling η01 starts higher, and so demarked from λ0 and λ1. By contrast, the effect of θ is not very dramatic: the self‐couplings are not much affected and the mutuals only evolve differently, without any particular boosting of η01. Details and further comments are found in [33].

7. All constraints together: viability regions

The above RGE analysis taught us two lessons: (i) The two couplings η1 and η01 control perturbativity. (ii) The change of sign of λ controls vacuum stability. Equipped with these indicators, we can try to systematically locate the regions in parameter space in which the model is viable. We have by now a number of tools at our disposal. First, the DM relic‐density constraint (6), which has been and will continue to be applied throughout. We have the RGE analysis of the previous section. We will require both η1(Λ) and η01(Λ) to be smaller than 4π, and λ(Λ) to be positive. From the phenomenological implications we deduced in Section 5, we will retain only two: the mixing angle θ and the physical self‐coupling λ0(4) are to be chosen small. Last, we want the model to comply with the experimental direct‐detection upper bounds. The condition we impose is that σdet of Eq. (7) be within the XENON 100 upper bounds [20]. We will vary λ0(4) and θ and track the viability regions in the (m0,m1) plane. The relevant mass range for m0 and m1 is 1–160 GeV. This is because there are no reliable data to discuss below the GeV and beyond 160GeV takes us outside the perturbativity region.2

One important issue must be addressed before we proceed: How far do we want the model to be perturbatively predictive and stable? The maximum value Λm for the mass scale Λ should not be very high. One reason, more conceptual, is that we want to allow the model to be intermediary between the current Standard Model and some possible higher structure at higher energies. Another one, more practical, is that a too high Λm is too restrictive for the parameters themselves. From the results of the RGE analysis [33], a reasonable compromise is to set Λm40TeV.

With all this in mind, Figure 11 displays the regions (blue) for which the model is viable when λ0(4)=0.01 and θ=1o. The mass m1 is confined to the interval 116–138 GeV while the DM mass is confined mainly to the region above 118GeV, the left boundary of which having a positive slope as m1 increases. In addition, m0 has a small showing in the narrow interval 57–68 GeV. The effect of increasing the mixing angle θ is to enrich the existing regions without relocating them. This is displayed in Figure 12 for which θ is increased to 15o. As θ increases, the region between the narrow band and the larger one to the right gets populated. This means more viable DM masses above 60GeV, but m1 stays in the same interval.


Figure 11.

Regions of viability of the two‐singlet model (in dark grey). Physical Higgs self‐coupling λ0(4) and mixing angle θ very small.


Figure 12.

The region of viability (dark grey) is even richer for a larger mixing angle θ.

By contrast, increasing the Higgs‐DM mutual coupling λ0(4) has the opposite effect, that of shrinking existing viability regions. To see this, compare Figure 13, for which λ0(4)=0.1 and θ=15o, with Figure 12. We see indeed shrunk regions, pushed downward by a few GeVs, which is not a substantial relocation. This effect should be expected because increasing λ0(4) raises η01(Λ0), well enough above 1 so that we leave perturbativity sooner. Increasing λ0(4) is also caught up by the relic‐density constraint, which tends to shut down such larger values of λ0(4) when m0 is large. The direct‐detection constraint has also a similar effect. Further comments can be found in [33].


Figure 13.

The physical Higgs self‐coupling λ0(4) shrinks the viability region (dark grey) as it increases.

8. Concluding remarks

The purpose of this chapter was to help the reader understand how modeling cold dark matter evolves from motivating the model itself to constraining the space of its parameters. We took as prototype a two‐singlet extension to the Standard Model of elementary particles within the paradigm of weakly interacting massive particles.

The first set of constraints the model had to undergo came from cosmology and perturbativity. The model had to reproduce the known relic density of cold dark matter while being consistent with perturbation theory. The second set of tests came from direct detection, in the form of the total elastic cross‐section of a WIMP scattering off a non‐relativistic nucleon that had to satisfy bounds set by several direct‐detection experiments. We have seen that the model is capable of satisfying all the existing bounds and will soon be probed by the coming XENON1t experiment. The third set of constraints came from particle phenomenology. We have seen how ϒ rare decays constrain the predictions of the model for light cold dark matter. The fourth set of constraints came from internal consistency of the model, in the form of viability and stability under running coupling constants via a renormalization‐group analysis. We have concluded that the model can still make sound predictions in important and useful physical regions. We then have investigated the regions in the space of parameters in which the model is viable when all these four sets of constraints are applied together with a maximum cutoff Λm40 TeV, a scale at which heavy degrees of freedom may start to be relevant. We have deduced that for small λ0(4) and θ, the auxiliary field mass m1 is confined to the interval 116–138 GeV, while the DM mass m0 is confined mainly to the region above 118GeV, with a small showing in the narrow interval 57–68 GeV. Increasing θ enriches the existing viability regions without relocating them, while increasingλ0(4) has the opposite effect, that of shrinking them without substantial relocation.

There is one aspect of the study we have not touched upon in this review, and that is the connection with and consequences from Higgs physics. This has been analyzed in Refs. [32, 33]. This aspect is important, of course, too important maybe to be just touched upon in this limited space. Such an analysis also needs to be reactualized in view of the many advances made in Higgs physics [53].

Despite all our efforts, dark matter stays elusive. Many models that tried to understand it have failed. The fate of the two‐singlet model may not be different. But this will not be a source of disappointment. On the contrary, failure will only fuel motivation to try and explore new ideas.


1The mutual couplings can be negative as discussed below, see (21).

2In practice, m0 is taken up to 200GeV, but there are no additional features to report.


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