Conformal Freeze-In of Dark Matter

We present the conformal freeze-in (COFI) scenario for dark matter production. At high energies, the dark sector is described by a gauge theory flowing towards a Banks-Zaks fixed point, coupled to the standard model via a non-renormalizable portal interaction. At the time when the dark sector is populated in the early universe, it is described by a strongly coupled conformal field theory. As the universe cools, cosmological phase transitions in the standard model sector, either electroweak or QCD, induce conformal symmetry breaking and confinement in the dark sector. One of the resulting dark bound states is stable on the cosmological time scales and plays the role of dark matter. With the Higgs portal, the COFI scenario provides a viable dark matter candidate with mass in a phenomenologically interesting 0.1-1 MeV range. With the quark portal, a dark matter candidate with mass around 1 keV is consistent with observations. Conformal bootstrap puts a non-trivial constraint on model building in this case.


INTRODUCTION
Microscopic nature of dark matter is one of the central open questions in fundamental physics which cannot be addressed within the Standard Model (SM). Many theoretical ideas have been suggested, and an extensive experimental effort is under way to test some of the proposals [1]. While the precise nature of the dark matter sector varies greatly among the proposed models, all of them postulate that dark matter consists of point-like particles (e.g. WIMPs or axions), their bound states (e.g. dark atoms), or particle-like extended objects (monopoles, Qballs, etc.), both today and throughout its cosmological history. However, viable extensions of the SM exist in which new physics sectors do not contain spatially localized particle-like excitations at all [2]. A well-known example is a conformal field theory (CFT) [3][4][5], where scale invariance precludes the existence of stable finitesize states. In this letter, we show how dark matter can arise from a new physics sector which is described by a CFT throughout most of its cosmological history.
An immediate objection to the idea of dark matter made out of CFT "stuff" is that conformal invariance dictates that the energy density of such stuff redshifts like radiation (ρ ∝ a −4 ), rather than non-relativistic matter (ρ ∝ a −3 ), as the universe expands. However, in any phenomenologically viable model, conformal invariance is at most approximate and must be broken to some degree. In particular, any interactions of the CFT sector with the non-conformally-invariant SM inevitably break the symmetry. Generically, such effects induce a "gap" mass scale, below which the sector is no longer conformal and its spectrum consists of spatially localized particle degrees of freedom. Below, we will discuss a scenario in which dark matter production in the early universe occurs at temperatures above the gap scale, so that throughout the production process the dark sector can be well approximated by a CFT. At the same time, the gap scale, which is induced by cosmological phase transitions in SM, is sufficiently large so that the dark sector behaves as non-relativistic matter during CMB decoupling, structure formation, and today, as required by observations.

PARTICLE PHYSICS FRAMEWORK
We extend the SM by postulating a dark sector, whose fields do not carry SM gauge charges. The dark sector is assumed to be invariant under the conformal group. It is coupled to the SM via where O SM is a gauge-invariant operator consisting only of SM fields, O CFT is an operator within the dark-sector CFT, and Λ CFT is the energy scale where the CFT is replaced by its UV completion. We will consider the regime of small Wilson coefficient λ CFT 1, where the conformal symmetry breaking introduced by Eq. (1) can be treated as a (technically natural) small perturbation. A UV completion of the CFT that naturally generates λ CFT 1 is discussed below. If O SM and O CFT have scaling dimensions d SM and d, respectively, then The dark-sector CFT may be strongly coupled, resulting in large anomalous dimensions and non-integer d.
Once the conformal symmetry is broken, the spectrum consists of particle-like excitations with masses ∼ M gap which can be thought of as bound states of the original CFT degrees of freedom. We assume that one of these excitations is stable on cosmological time scales, for example, due to a discrete symmetry. This is the particle that will play the role of dark matter (DM). Regarding the DM particle mass, we will consider two possibilities. One is that the DM particle is a generic bound state, with mass m DM = M gap (up to order-one factors). The second one is that the DM particle is a pseudo-Goldstone boson (PGB) of an approximate global symmetry spontaneously broken at M gap , similar to pions in QCD. In this case, m DM M gap is natural, with the DM mass dictated by the amount of explicit symmetry breaking.
The strongly-coupled, conformally invariant dark sector of our model can arise from a weakly-coupled, asymptotically free theory in the ultraviolet (UV). This can be a simple SU (N c ) gauge theory with N F fermion flavors Q i , which flows towards a Banks-Zaks (BZ) fixed point in the infrared [6]. The theory becomes strongly coupled, and approximately conformal, at a scale Λ CFT . The interaction with the SM starts out as where O BZ is a gauge-invariant operator in the darksector gauge theory, with a scaling dimension d BZ . For example, we may consider Since O CFT is relevant, an additional symmetry must be invoked to avoid its appearance in the CFT Lagrangian, which would lead to incalculable IR breaking of the CFT that would generically dominate the effect of Eq. (1). This may for example be a Z 2 symmetry under which O CFT is odd, explicitly broken by (1). However, the operator product expansion (OPE) of O CFT × O CFT generally contains singlet scalar operators, which are even under Z 2 [7]. Numerical CFT bootstrap provides an upper bound on the dimension of the lowest singlet scalar operator in the OPE [7,8]. Requiring that no Z 2 -even relevant operators are generated implies d > 1.61 [8]. Note that this bound is model-dependent and may be avoided if a larger symmetry is used, or if operator coefficients are fine-tuned.
Note that with the exception of the IR breaking of conformal symmetry, the field theory model considered here is identical to Georgi's "unparticle" framework [2,9]), with SM coupling to a scalar CFT operator. (For earlier works on cosmology with unparticles, see [10][11][12][13][14].) In other words, the dark sector behaves as an unparticle at energy scales above M gap and below Λ CFT .

COSMOLOGICAL EVOLUTION
We assume that after the end of inflation, the inflaton decay reheats the SM sector to a temperature T R , but the dark sector is not reheated due to absence of a direct coupling to the inflaton. As the universe expands and cools after reheating, collisions and decays of SM particles gradually populate the dark sector. Assuming M gap T R < Λ CFT , this process occurs via production of CFT stuff ("unparticles"). The dark sector cannot be described by Boltzmann equations, since the concept of particle number density is not applicable in the CFT. However, since the dark sector has many degrees of freedom and they interact strongly among themselves, it will be in a spatially isotropic thermal state. Rotational symmetry dictates that the energymomentum tensor of this state has the form T µν = diag(ρ CFT , −P CFT , −P CFT , −P CFT ), while conformal invariance further requires P CFT = 1 3 ρ CFT . The CFT energy density is given by where T D is the temperature of the CFT sector, and A is a model-dependent constant. We will study a scenario where T D T at all times, where T is the SM plasma temperature; at the same time, T D M gap during the period when the dark sector is populated, so that the CFT description is appropriate.
On the SM side, the particle number is well-defined and the Boltzmann equations have the usual form, with collision terms describing the loss of SM particles due to collisions (SM+SM→ CFT) and decays (SM→ CFT), and their creation due to inverse processes. The evolution of the SM energy density ρ SM follows from the Boltzmann equations: where H is the Hubble expansion rate, and Γ E are energy transfer rates per unit volume. In our scenario, the CFT sector will always remain at densities far below equilibrium with the SM, and Γ E (CFT → SM) can be safely neglected. The energy transfer rate from SM to CFT is given by where the sums run over all SM degrees of freedom coupled to the CFT. The cross sections and decay rates can be evaluated using the technique of Georgi [2,9]. For example, with the Higgs portal, the Higgs decay contribution is given by , and K 2 (x) is the modified Bessel function of the second kind. The scattering contribution (when T m h ) is given by The CFT sector is populated at the time when the energy density is dominated by relativistic SM matter, P SM = 1 3 ρ SM , so that SM and CFT energy densities redshift in the same way. The total energy of the two sectors can only change due to work done against the expansion of the universe: Subtracting Eq. (7), we find that the CFT energy density evolves according to with the initial condition ρ CFT = 0 at T = T R . With minor simplifying assumptions, such as ignoring the masses of colliding SM particles and the temperature dependence of the effective number of SM degrees of freedom g * , Eq. (12) can be solved analytically [15]. The qualitative behavior of ρ CFT with temperature is dictated by the dimension d of the operator O CFT . For d above the critical dimension d * , most of the CFT energy is produced at high temperatures, close to T R , by pairannihilations of SM particles; the decay contribution, if present, is subdominant. On the other hand, for d < d * , contributions from both pair-annihilations and decays (if present) grow with decreasing T . The resulting CFT density is IR dominated and can be calculated without knowledge of UV quantities such as T R , as in the freezein scenario of Ref. [16,17] (for review of variations of freeze-in models and phenomenology, see [18]). We will focus on this case for the rest of the paper. We call this scenario Conformal Freeze-In, or COFI. The critical dimensions are d * = 5/2 for the Higgs portal, and d * = 3/2 for the quark portal.
For both portals, production of CFT energy effectively ceases soon after the SM temperature drops below the mass of the particle coupled to the CFT: T ∼ m h for the Higgs portal 1 and T ∼ Λ QCD for the quark portal. After that, ρ CFT redshifts as a −4 , until the CFT temperature drops to T D ∼ M gap . At that time, conformal symmetry is broken and a confining phase transition takes place. We assume that all of the energy stored in the CFT sector is transferred to DM particles, on a time scale short compared to Hubble at that time. If m DM M gap , the dark matter energy density will continue to redshift as radiation until its temperature drops below m DM , after which it behaves as non-relativistic matter. With these assumptions, the relic density can be estimated analytically. For example, for the Higgs portal, the relic density is dominated by the Higgs decay contribution, and is given by where g * ∼ 100 is the number of relativistic SM degrees of freedom at the weak scale. An example of the evolution of CFT/DM energy density, for the Higgs portal scenario and parameters that provide the observed DM relic density, is shown in Fig. 1. This and all figures below are based on full numerical solutions of Eq. (12).

DARK MATTER PHENOMENOLOGY
Since the dark matter relic density is determined by IR physics, and the dark matter mass and its interaction strength are related through Eqs. (3), the COFI scenario is remarkably predictive. In particular, there is a nearly universal relationship between the dark matter relic density and the gap scale, with only a mild dependence on other parameters. In the case of the Higgs portal, the observed relic density is reproduced for M gap ∼ 1−10 MeV, while for the quark portal, it is M gap ∼ 10 − 100 keV. If the DM particle is a generic bound state of a strongly-coupled theory with m DM ∼ M gap , its elastic self-scattering interaction cross section can be estimated as σ self ≈ 1/ 8πM 2 gap . This is far too large, in both Higgs and quark portal scenarios, to be consistent with bounds from galaxy clusters such as the Bullet cluster [19,20]. We therefore consider the case where the DM is a PGB, with mass ratio r = m DM /M gap 1. The self-scattering cross section scales as where we assumed that self-scattering is mediated by states with masses ∼ M gap (e.g. the counterparts of the ρ meson of QCD), and derivative couplings of the PGB have been taken into account. Modest values of r ∼ 0.01 − 0.1 are sufficient to avoid the self-interaction bounds for parameters with viable Ω dm . This is illustrated in Figs. 2 and 3. Another important phenomenological constraint is that the DM should be cold, i.e. remain non-relativistic during structure formation [21]. This constraint is somewhat weaker than in the case of SM sterile neutrinos, m > ∼ 5 keV, because the CFT sector is colder than the SM. Nevertheless, for the quark portal, the warm DM bound rules out a significant part of the parameter space; see Fig. 3. For the Higgs portal, the DM is heavier and this bound is irrelevant.
There are many experimental and observational constraints on the strength of CFT/DM coupling to the SM. These include LHC searches for unparticles produced in qq annihilations [22,23] or Higgs decays [24,25]; bounds on invisible meson decays involving unparticles from the B-factories [26][27][28]; supernova SN1987a energy loss and stellar cooling due to unparticle or DM emissions [10,29,30]; modification of the ionization history due to energy injection by late-time DM annihilations [31][32][33][34]; diffuse X-and Gamma-ray backgrounds [35]; and spectral distortion of the CMB blackbody distribution by early energy injection [36]. We checked that our scenario is easily consistent with all these bounds, due to a highly suppressed coupling between the dark sector and the SM. The details of these constraints will be presented in future work [15]. Note also that the Big Bang Nucleosynthesis (BBN) bound on the number of new light degrees of freedom [37] does not apply, because T D T at the time of BBN. On the theoretical side, there are several consistency conditions that may further constrain the parameter space, as shown in Figs. 2 and 3. The "naturalness" bound stems from requiring that if T R < Λ CFT , then M U < M Pl . "Bootstrap" condition, d > 1.61, is explained in the paragraph below Eq. (5). Both these bounds are model-dependent, and may be modified or eliminated by the choice of a UV completion of the CFT and symmetry charge assignment for O CFT , respectively.
We conclude that for the Higgs portal, the conformal freeze-in (COFI) dark matter scenario is easily viable, consistent with all observational and theoretical constraints. This scenario contains a novel DM candidate, a CFT bound state, whose mass is robustly predicted to be in the experimentally interesting 100 keV -1 MeV range. This candidate and its phenomenology will be studied further in [15]. For the quark portal, the situation is more constrained. A small sliver of parameter space, with d close to 1, is observationally viable, but inconsistent with the bootstrap condition in its simplest form. It is interesting that viability of the DM candidate in this case hinges upon highly non-trivial intrinsic properties of the CFT.