INTRODUCTION
1.2 Theoretical Motivation for Light Dark Matter Models
Even with the abundance of evidence for DM discussed in Sec. 1.1, we know almost noth- ing about its non-gravitational interactions with the Standard Model. This leaves a large playground for theorists to build DM models which can then be tested by comparing to experiment. A DM model must predict all of the experimentally observed properties: stable on cosmological time scales [131], ΩDM = 0.265, must be relatively cold (vχ ∼ 10−3), not interact too strongly with itself (Eq. (1.11)), and have a mass consistent with the discussion in Sec. 1.1.
The canonical DM candidate is the “weakly interacting massive particle”, or the WIMP motivated by electroweak scale supersymmetry [132]. See Ref. [133] for a recent review. This is a particle with GeV.mχ .100TeV whose production is set by the freeze-out mechanism, discussed further in Sec.1.2. The lower bound on the mass is set by the observed DM density.
If the DM was lighter then it would over populate the universe with DM, and much heavier than this requires cross sections which would contradict unitarity. As mentioned earlier, this DM candidate is being experimentally ruled out quickly, and therefore other DM candidates need to be identified.
2These bounds depend crucially on a single species of DM. If there are more species the bound re- laxes [130].
In this section we will discuss some of those alternative ideas. This is in no way meant to be exhaustive, but to represent some of the theories which will be discussed in later chapters.
We will begin by discussing some of the standard production mechanisms which generate the cosmic abundance of DM. We will then discuss a couple of standard, benchmark, light DM models which provide a good starting point to understanding some of the more complicated ones discussed in later chapters.
Production Mechanisms Freeze Out
The most well known DM production mechanism is “Freeze Out” [111]. In the early universe all of the SM species were in thermal equilibrium mediated by the photon bath. Therefore it is reasonable to think that the DM was also in thermal equilibrium, and its number density was determined thermodynamically. However since DM is known to interact weakly, at some point the interaction rate,Γ, becomes smaller than the expansion rate of the universe,Γ.H. When this happens thermal equilibrium is no longer achieved, and the DM abundance stops tracking the thermal equilibrium. The evolution of the abundance is given by the Boltzmann equations,
dY
dx =−xshσvi
H Y2−Yeq2
, (1.13)
where σ is the DM annihilation cross section, x = mχ/T, s is the entropy density of the universe, Y = nχ/s, and Yeq is the equilibrium number density per entropy density. We see that as σ/H →0, Y stops changing, indicating that the particle species has frozen out.
The final DM abundance is inversely proportional toσ; a larger interaction cross section the longer DM stays in thermal equilibrium, and the lower the final abundance. The WIMP is an attractive DM candidate since for interaction cross section the size of the weak interaction, the DM abundance is set automatically. More generally, lets assume that the cross section scales as,
σ ∼ g4
m2χ, (1.14)
where g is some DM-SM coupling constant. For g smaller than typical weak interactions, mχ can be light and still in agreement with the primordial abundance, set via σ. However since freeze out happens at T ∼ mχ, if m . MeV then DM will freeze out after neutrinos decouple, at T ∼ MeV. This can influence Neff, essentially the temperature ratio between neutrinos and photons, which is severely constrained experimentally [134, 135]. Therefore thermal freeze-out is only a valid production mechanism for mχ>MeV.
Freeze In
The freeze out mechanism discussed in Sec. 1.2 depends on the DM starting in thermal equilibrium with the Standard Model, i.e., for x 1, Y =Yeq. This need not be the case;
the DM could have been absent initially: Y = 0 for x 1. If the DM-SM coupling is strong enough then the SM would produce the DM efficiently enough to bring it to thermal equilibrium, and the universe is put back in the freeze out scenario. Therefore this difference in initial conditions is only important if the DM-SM coupling is small. In this scenario the SM will produce DM inefficiently, but over a long period of time, leading to the cosmic abundance seen today. This is known as the “Freeze In” mechanism [136, 137]. Such small couplings are naturally produced in Hidden Sector DM models, discussed in more detail in Sec. 1.2. DM which freezes in avoids the low mass constraints from BBN because the DM is never in thermal equilibrium.
Misalignment Mechanism
The previously discussed freeze out/in mechanisms rely on the thermal distribution of SM particles. These are natural guesses for DM production mechanisms by analogy with the SM thermal history. However DM production mechanisms do not have to be thermal. In fact, the existence of baryons over anti-baryons is an example of production via an asymmetry, somewhat distinct from a standard freeze out mechanism.3 Perhaps the most popular non- thermal production mechanism is the “misalignment mechanism” [145–147], used mainly in the context of axion DM, a model we will discuss further in Sec.1.2.
To understand how the misalignment mechanism we will go through a simple exercise. Con- sider a complex scalar field with the Higgs-like potential,
V(Φ) =−µ2Φ∗·Φ +λ(Φ∗·Φ)2. (1.15) By analogy with electroweak symmetry breaking we know that thermal effects at high tem- peratures keep the minimum at Φ = 0. However as the temperature lowers, Φ develops a new minima at v = p
2µ2/λ. After the symmetry breaking occurs we can describe the resultant Goldstone boson, which we will refer to as the “axion”, via a field redefinition,
Φ = veia, (1.16)
where a∈[0,2π). Since the axion is a Goldstone boson, the field is massless. Conceptually the evolution of the axion field at this point is straightforward: it simply rolls around the
3There exist models of DM which use similar production methods and go by “asymmetric DM” [11,136, 138–144]. This mechanism is appealing because it can be used to connect the DM density to the baryon density which are intriguingly only different by a factor of five.
bottom of the potential. However, imagine the U(1)symmetry is explicitly broken at a later stage, such that the potential contains a mass term for the axion.
V ⊃ m2a
2 a2. (1.17)
This will tilt the axion potential, and push the axion to the new minimum. As the axion is pushed to the new minimum energy will be radiated which can set the primordial abun- dance. If the universe is in causal contact when the axion is generated, T ∼ v, then the axion abundance will be set by the value of a when it started decaying to the new minima (created from the introduction of the mass term). This is not very attractive as a production mechanism for the DM abundance, since the value is precisely tuned to the initial condition of the axion field which was previously rolling around in the flat potential.
However this is not the end of the story. Imagine that the universe wasnotin causal contact when T ∼ v. Then the axion field will essentially be a random value in different patches of the universe. In this scenario the DM abundance will then be set by the average value across all these patches. Therefore while different patches of the universe will have different DM densities, the average density can be used to generate a DM abundance which is not dependent on initial conditions. Moreover the axion relic density will be related to its mass, giving a concrete prediction for the DM mass.
The scenario discussed here is the standard misalignment case. Recently, there has been interest in expanding this idea by changing the initial conditions the axion field has before it starts to roll in the potential [148, 149]. Additionally there has been work on understanding whether this production mechanism is dominant. In a universe with many different patches there exist domain walls which can decay in to axions as well, which can also set the relic abundance [150].
Candidate Light Dark Matter Models
We now turn to discussing a couple specific light DM models, starting with the axion and then discussing hidden sector DM models.
Axion
As mentioned in Sec. 1.1, DM is not the only open problem particle physics faces. Another such problem is the “Strong CP” Problem [98–100]. SU(3) symmetry allows a term of the form,
L ⊃θGG˜ (1.18)
in the Lagrangian, where G is the SU(3) field strength tensor. Naturally, one expects that all terms allowed by symmetry are present in the Lagrangian with O(1) coefficients. If a coefficient is unnaturally small then one might expect that there is a symmetry forbidding that term to exist. No such symmetry is known to forbid the term in Eq. (1.18), yet the experimentally measured value is bounded to
θ .10−10, (1.19)
an unnaturally small number indeed.
One solution is to introduce a new axial U(1) symmetry to the Standard Model, typically referred to as a “Peccei-Quinn” symmetry for the creators of the mechanism. Imagine a new, heavy, field, Q, charged under this U(1)PQ. Performing a field transformation, Q→eiaγ5Q, where a is the axion, generates an anomalous coupling to GG. The theta term is then˜ absorbed in to a field redefinition ofaand vanishes, leaving a dynamicθterm. The minimum of the potential for this dynamic θ term is zero, thereby dynamically solving the Strong CP problem. An axion which does this is known as the QCD axion. The QCD axion can then be produced via the misalignment mechanism discussed in Sec. 1.2 and be a viable DM candidate. The ability to solve both the DM problem and Strong CP problem at once is powerful motivation for the QCD axion.
The mass of the QCD axion is generated by QCD effects and directly relates the U(1)PQ symmetry breaking scale, fa (essentially v in Sec.1.2),
ma∝ mπfπ
fa , (1.20)
where mπ is the pion mass, and fπ is the pion decay constant. The symmetry breaking scale, fa, also sets the strength of interactions between the QCD axion and the SM. If fa
is too small the interaction strength will be too large and in conflict with stellar cooling bounds [30]. This constrains the mass of the QCD axion toma.100meV.
More generally one can consider axion-like particles (ALPs) whose mass is disconnected from the symmetry breaking scale and generates a mass, coupling parameter space for ALP DM.
Searching for the QCD axion and ALPs experimentally is a large experimental program [91–
93, 151–169], and in Ch. 6 we discuss in detail how to use phonons and magnons to probe these DM models for meV .ma .100meV, a large chunk of previously unexplored model space below the stellar cooling limits.
Hidden Sector Dark Matter
Another class of light DM models is known as “Hidden Sector”, or “dark portal” DM models.
These are models in which the Standard Model and DM interact very weakly via a new force.
The new force acts as the “portal” between the Standard Model and the “hidden sector” of DM particles which are also charged under this new force. As an example we will discuss the light “dark photon” model [170] which has proven to be the most popular benchmark hidden sector model (at least within the context of direct detection) since it simply adds a new U(1)0 gauge group. Minimally the model adds two terms to the Standard Model Lagrangian,
L ⊃ mA0
2 A02+κ
2FµνFµν0 , (1.21)
whereA0 is the dark photon,mA0 is its mass, andκis known as a “kinetic mixing” parameter since it mixes the kinetic terms of the photon and dark photon. This κ can be radiatively generated through loops of heavy particles charged under U(1)×U(1)0. To understand the phenomenology of this model it is typically useful to diagonalize the mass matrix of theA, A0 system via the field transformation,
Aµ→Aµ+κA0µ. (1.22)
Note that A0µ cannot be rotated since that would generate a mass for the photon. In this rotated basis all of the Standard Model fermions which have an electromagnetic charge are now charged underA0and therefore the field acts like an additional photon field with rescaled couplings, e→κe.
The dark photon itself can be the DM [171, 172], typically one adds a particle χ which is charged under the U(1)0 and takes this to be the DM. This χ field then serves as the light DM candidate, and can give rise to a direct detection signal by scattering off a target via mediating an A0.
While the motivation for any given dark photon model is perhaps weaker than the motivation for the axion, this class of models, with an extra gauge group and a DM candidate is a very natural explanation for DM. The Standard Model is a collection of seemingly random gauge groups so it is not unfeasible to have one more gauge group.Moreover, extra gauge groups are a fairly generic prediction of Grand Unified Theories. The dark photon model serves as a useful benchmark for these more general cases since it is easier to calculate for than extensions with an extra SU(N)group, or an MSSM-like model with many new parameters.