TPI-MINN-00/46, UMN-TH-1922-00

McGill-00/31; IASSNS-HEP-00/83

hep-ph/0011335

The Minimal Model of Nonbaryonic Dark Matter:

A Singlet Scalar

[5mm] C.P. Burgess, Maxim Pospelov and Tonnis ter Veldhuis

[4mm] The Institute for Advanced Study, Princeton, NJ 08540, USA

[1mm] Physics Department, McGill University,

[-0.3em] 3600 University St., Montréal, Québec, Canada, H3A 2T8.

[1mm] Department of Physics, University of Minnesota

Minneapolis, MN 55455, USA

[5mm] Abstract

[5mm]

We propose the simplest possible renormalizable extension of the Standard Model — the addition of just one singlet scalar field — as a minimalist model for non-baryonic dark matter. Such a model is characterized by only three parameters in addition to those already appearing within the Standard Model: a dimensionless self-coupling and a mass for the new scalar, and a dimensionless coupling, , to the Higgs field. If the singlet is the dark matter, these parameters are related to one another by the cosmological abundance constraint, implying that the coupling of the singlet to the Higgs field is large, . Since this parameter also controls couplings to ordinary matter, we obtain predictions for the elastic cross section of the singlet with nuclei. The resulting scattering rates are close to current limits from both direct and indirect searches. The existence of the singlet also has implications for current Higgs searches, as it gives a large contribution to the invisible Higgs width for much of parameter space. These scalars can be strongly self-coupled in the cosmologically interesting sense recently proposed by Spergel and Steinhardt, but only for very low masses ( GeV), which is possible only at the expense of some fine-tuning of parameters.

## 1 Introduction

It is an amazing fact of our times that even as our understanding of cosmology progresses by leaps and bounds, we remain almost completely ignorant about the nature of most of the matter in the universe. According to recent fits to cosmological parameters [1], dark matter of some sort makes up close to 30% of the total energy density. This is much more than what is inferred from inventories of the luminous matter we can see. Moreover, the successes of big bang nucleosynthesis suggest that only a fraction of this matter can be made of ordinary baryons, like massive compact objects, faint stars, etc.. Thus, unless gravity undergoes some drastic changes at distances larger than a few kpc (which is quite improbable from several points of view), we must postulate the existence of enormous amounts of dark matter of an unknown, non-baryonic origin.

A simple argument shows why current theories of particle physics are so prolific in suggestions for what the nature of this dark matter might be [2]. The vast majority of the proposed alternatives to the Standard Model involve new particles having masses which are of order GeV, and which couple with electroweak strength to the ordinary matter which we know and love. If any of these particles is stable enough to have a lifetime as long as the age of the universe, it makes a natural candidate for dark matter. It does so because its abundance is naturally of the right order of magnitude so long as its interaction cross sections have weak-interaction strength. The abundance comes out right because it is set by the annihilation rate for particles which are initially in thermal equilibrium with ordinary matter. Cosmologically interesting abundances follow pretty much automatically for particles whose mass is of order and whose annihilation cross sections have weak (or rather milliweak) interaction strength. (We give this argument in more detail within the body of the paper.)

Better yet, supersymmetric models, which are perhaps the best motivated of the many theories which have been proposed, very often have such long-lived states, due to the natural existence there of a conserved quantum number, -parity, which keeps the lightest -odd state from decaying. These particles cry out for interpretation as dark-matter particles, and it is no surprise that these models are by far the most widely explored in the literature [3].

Best of all, this explanation of the nature of dark matter can be tested experimentally. This is the direct goal of dedicated dark-matter detectors [4], and an indirect goal of accelerator searches for events with missing energy, showing that a weakly-interacting particle has escaped the detector. If Nature smiles on us we soon may be treated to the discovery of new physics in both of these kinds of experiments. Indeed, recently the DAMA collaboration has announced the detection of a dark matter signal, as indicated by their seeing an annual modulation of the counting rate in a NaI detector [5]. However, the comparably precise data from the Ge detectors of the CDMS collaboration [6] do not support these findings. (These two experiments need not be in contradiction with each other if the spin-dependent part of the cross-section is enhanced relative to the spin-independent part [7].)

Our goal in the present paper is to present a slightly unorthodox view. Although very well motivated, supersymmetric models are very complicated and enjoy an enormous parameter space. This makes them unable to definitively predict what dark-matter detectors must see. Furthermore, unlike the extensive evidence for the existence of dark matter, the arguments in favour of supersymmetry are almost exclusively theoretical. In our opinion, with the advent of good-quality data from dark-matter detectors, it behooves theorists to propose simple models for the dark matter which are consistent with present evidence, but which make definite predictions and so are easily falsifiable. These provide benchmarks against which other models and the data can be compared. We believe that it is only by comparing the implications of such models with one another, and with supersymmetry, that one can hope to properly interpret the data.

The model we study in this paper was first introduced by Veltman and Yndurain [8] in a different context. Its cosmology was later studied by Silveira and Zee [9], and (with a complex scalar) by McDonald [10]. It is the absolute minimal modification of the Standard Model which can explain the dark matter. It consists of the addition of a single spinless species of new particle, , to those of the Standard Model, using only renormalizable interactions. To keep the new particle from interacting too strongly with ordinary matter, it is taken to be completely neutral under the Standard Model gauge group. Besides involving the fewest new states, the model is also just complicated enough to offer interestingly rich dark matter properties. Unlike the case if only spin-half or only spin-one singlet particles are added, it is possible for a singlet scalar to have both significant renormalizable self-interactions and renormalizable interactions with some Standard Model fields.

There is also a sense in which the model we propose is generic, should the dark matter consist of a single species of spinless particle. To this end, it is useful to ask the question of what a generic dark-matter model should look like. It is clear that the main property which one needs to ensure is the stability of the new particle, suggesting that the fields , representing these particles appears in the Lagrangian in even powers, so that its decay is forbidden. If this field is considerably lighter than the rest of the other exotic undiscovered particles, these may be integrated out, leaving an effective Lagrangian at electroweak scale which has the generic form

(1.1) |

where the kinetic, mass terms and interactions with the SM (via the set of operators ) in general would depend on the spin of . The most important couplings at low energies are those of lowest dimension, corresponding to the lowest-dimension choices for the operators . Our model also has this form, with only a single singlet scalar . In this language, our dropping of all nonrenormalizable interactions corresponds to keeping only those interactions which are consistent with (but do not require) all other exotic particles to be arbitrarily heavy compared with the weak scale. We might expect our model to therefore capture the physics of any more complicated theory whose impact on the dark matter problem is conveyed purely through the low-energy interactions of a single spinless particle.

An additional, more tentative, incentive for formulating more models
stems from recent indications of problems with subgalactic
structure formation within the non-interacting cold-dark-matter scenario
[11]. A ‘generalized’ form of cold dark matter may avoid these
problems if its self-interactions^{1}^{1}1
The required self-interactions however also lead to spherical
halo centers in clusters, which are inconsistent with the
ellipsoidal centers indicated by strong gravitational lensing data
[13].
can produce scattering cross sections
of order cm in size [12]. Within the present
context this proposal would require the masses of dark matter particles
not to exceed 1 GeV. Within the minimal model described in this paper,
we find this range of masses may be just barely possible, but requires
unnatural fine tunings due to the relationship between masses
and couplings imposed by the maintenance of the correct cosmic abundance
of dark-matter scalars.

This paper is organized as follows. In the next section we identify the three parameters which describe the model, and determine the general conditions which lead to acceptable masses and to sufficiently stable dark matter. In section 3 we calculate the annihilation cross section of -particles and give the resulting cosmic abundance as a function of masses and couplings. This calculation is similar to the analysis of Ref. [10]. We perform the numerical analysis for the most interesting part of the parameter space, with 100 GeV 200 GeV and 10 GeV 100 GeV. In section 4 we obtain the cross section for elastic scattering with ordinary matter and apply the constraints, imposed by direct and indirect searches. Section 5 computes the cross sections for the missing energy events which are predicted for colliders due to the pair production of particles. It also contains a prediction for the degradation of the Higgs boson signal at hadronic colliders, when the Higgs boson is allowed to decay into a pair of particles. Our conclusions are reserved for section 6.

## 2 The Model’s Lagrangian

The lagrangian which describes our model has the following simple form:

(2.1) |

where and respectively denote the Standard Model Higgs doublet and lagrangian, and is a real scalar field which does not transform under the Standard Model gauge group. (Lagrangians similar to this have been considered as models for strongly-interacting dark matter [14] and as potential complications for Higgs searches [15]. The same number of free parameters appears in the simplest Q-ball models [16].) We assume to be the only new degree of freedom relevant at the electroweak scale, permitting the neglect of nonrenormalizable couplings in eq. (2.1), which contains all possible renormalizable interactions consistent with the field content and the symmetry .

Within this framework the properties of the field are described by three parameters. Two of these, and are internal to the sector, characterizing the mass and the strength of its self-interactions. Of these, is largely unconstrained and can be chosen arbitrarily. We need only assume it to be small enough to permit the perturbative analysis which we present. Couplings to all Standard Model fields are controlled by the single parameter .

We now identify what constraints are implied for these couplings by general considerations like vacuum stability or from the requirement that the vacuum produce an acceptable symmetry-breaking pattern. These are most simply identified in unitary gauge, with real , where the scalar potential takes the form:

(2.2) |

and GeV are the usual parameters of the Standard Model Higgs potential.

1. The Existence of a Vacuum: This potential is bounded from below provided that the quartic couplings satisfy the following three conditions:

(2.3) | |||||

We shall assume that these relations are satisified and study the minima of the scalar potential.

2. Desirable Symmetry Breaking Pattern: We demand the minimum of to have the following two properties: It must spontaneously break the electroweak gauge group, ; and it must not break the symmetry , so . The first of these is an obvious requirement in order to have acceptable particle masses, while the second is necessary in order to ensure the longevity of in a natural way. ( particles must survive the age of the universe in order to play their proposed present role as dark matter.)

The configuration and is a stationary point of if and only if , in which case the extremum occurs at . This is a local minimum if and only if

(2.4) |

A second local minimum, with and , can also co-exist with the desired minimum if and . This second minimum is present so long as and . Even in this case, the minimum at and is deeper, and so is the potential’s global minimum, provided that

(2.5) |

Throughout the rest of this paper, the above conditions are assumed to hold, so that the model is in a phase having potentially acceptable phenomenology. It is therefore convenient to shift by its vacuum value, , so that represents the physical Higgs having mass . The -dependent part of the scalar potential then takes its final form

(2.6) |

and the mass is seen to be . Our prejudice in what follows is that this mass lies in the range from a few to a few hundred GeV, in which case the resulting dark matter will be cold.

## 3 Constraints from Cosmological Abundance

We next sharpen the cosmological constraints on the model by demanding the present abundance of particles to be close to today’s preferred value of . This imposes a strong relationship between the parameters and , which we now derive.

We start by assuming that the particles are in thermal equilibrium with ordinary matter for temperatures of order and above. This is ensured so long as the coupling is not too small. Just how small must be is determined by the following argument. Thermal equilibrium requires the thermalization rate, , to be larger than the universal Hubble expansion rate, . The constraint on comes from the demand that this be true throughout the thermal history of the universe, down to temperatures . But and vary differently with time as the universe expands, because they differ in their temperature dependence. On one hand, in the radiation-dominated epoch which is of primary interest to us, , where denotes the Planck mass. On the other hand, the thermalization rate varies as for , and as for . These temperature dependences imply the ratio is maximized when , taking the maximum value . particles are therefore guaranteed to remain thermalized (or get thermalized) down to the electroweak epoch, , if this maximum ratio is required to be of order one or larger, implying

(3.1) |

Once thermalization is reached (and in the absence of decays), as we shall henceforth assume, the primordial abundance is determined by the particle mass and its annihilation cross section. This cross section depends very strongly on the unknown Higgs mass, and on which annihilation channels are kinematically open.

An independent, interesting issue is
the fate of a scalar condensate that might
survive from the inflationary epoch. After the Hubble rate drops below
, coherent time oscillations of the singlet field begin. These
oscillations can be regarded as the oscillations of a Bose condensate
of particles
which is not in thermal equilibrium with other matter.
The fate of the condensate depends on the initial value of the
field and two possibilities must be distinguished. If the initial value
of the condensate is sufficiently small so that
the energy density in the oscillations, ,
is smaller than the energy density of radiation, , then
the thermalization of this condensate occurs exponentially fast. The
rate is given by or , whichever
is larger. When the
initial value of the condensate is of the order of the electroweak
v.e.v.,
the condensate will therefore completely disappear if
or is larger
then , just as in the thermalization condition (3.1).
The situation is quite different when the initial value of the field
is very large (, for example). The condensate then
dominates the energy density in the Universe and it behaves exactly as the
inflaton
condensate. The absence of the direct decay of particles
in this case may prevent the universe from reheating [17]
^{2}^{2}2We thank Lev Kofman for pointing out this possibility.
In this paper we assume that field does not drive inflation, and we
limit ourselves to the first possibility.

Since the temperature domain for which annihilation is most important is , it is the nonrelativistic annihilation cross section which is relevant. In our model the expression for the annihilation rate depends on which phase within which it occurs. If it occurs within the Higgs phase, i.e. if is low enough so that it occurs after the electroweak phase transition, the result is given by evaluating the tree-level graph of fig. (1) for -channel annihilation, ,which in the nonrelativistic limit gives

(3.2) | |||||

Here is the total Higgs decay rate, and denotes the partial rate for the decay, , for a virtual Higgs, , whose mass is . Eq. (3.2) also assumes that , so that direct (contact) annihilation to a pair of physical Higgses via the interaction term is forbidden.

Of particular interest are the large- and small- limits. The small- limit of eq. (3.2) – – implies the asymptotic behaviour:

(3.3) |

and the coefficient of proportionality depends strongly on the accessibility of certain decay channels ( or , and so on).

For large the dominant contributions to the annihilation cross section come from , and final states. (The latter originates from the interaction term, whose contribution must be summed with (3.2)). Neglecting terms which are in the result, we find the large- behavior of the annihilation cross section to be

(3.4) |

These asymptotic forms are useful in what follows for understanding what the cosmological abundance constraint implies for the coupling in the limit where is very large or very small. Our results for the annihilation cross section agree with the calculation of ref. [10].