Super-cool Dark Matter

In dimension-less theories of dynamical generation of the weak scale, the Universe can undergo a period of low-scale inflation during which all particles are massless and super-cool. This leads to a new mechanism of generation of the cosmological Dark Matter (DM) relic density: super-cooling can easily suppress the amount of DM to the desired level. This is achieved for TeV-scale DM, if super-cooling ends when quark condensates form at the QCD phase transition. Along this scenario, the baryon asymmetry can be generated either at the phase transition or through leptogenesis. We show that the above mechanism takes place in old and new dimension-less models.


Introduction
Dark Matter (DM) could be particles with mass M DM keV. The cosmological DM abundance Ω DM h 2 ≈ 0.110 is reproduced if their number density in units of the entropy density s is small: When DM becomes non-relativistic, thermal freeze-out at T ∼ T dec ≈ M DM /25 leaves the DM abundance n DM /s ∼ 1/(M DM M Pl σ ann ), where σ ann is the DM annihilation cross section. As well known, M DM ∼ TeV and σ ann ∼ 1/M 2 DM reproduces the desired DM abundance. Many alternative cosmological DM production mechanisms are possible, sometimes at the price of increasing model-building complexity.
We here discuss a new mechanism that can generate the desired cosmological DM abundance. The new mechanism is characteristic of models where a scale (we will consider the weak scale) is dynamically generated from a quantum field theory that only has dimension-less couplings. We assume that, in this context, all particles (in particular dark matter and the Higgs boson) remain massless until a vacuum expectation value or condensate develops. In the case of scalars, this conjecture is at odds with the usual view that attributes physical meaning to power-divergent quantum corrections, leading to the expectation that the Higgs boson should have been accompanied by new physics able of keeping its mass naturally much smaller than the Planck mass, the presumed cut-off of quantum field theories. The observation of the Higgs boson not accompanied by any new physics promoted renewed interest in dimension-less dynamics as the possible origin of the weak scale, see e.g. [1][2][3][4][5], and in attempts of building weak-scale extensions of the Standard Model valid up to infinite energy, such that no cut-off is needed [6][7][8][9]. More generically, super-cooling takes place if, for whatever reason, the mass scales in the potential are much smaller than the scale generated trough dynamical transmutation. If small scalar masses are unnatural, super-cooling is a cosmological signature of such unnaturalness.
The mechanism relies on the fact that in dimension-less models, along its thermal history, the Universe remains trapped for a while in a phase of thermal inflation during which all particles are massless, so that DM undergoes super-cooling rather than freeze-out. The formation of QCD condensates at the QCD phase transition ends this phase, leading to the electro-weak phase transition, i.e. particle mass generation. From this point particles lighter than the reheating temperature can easily thermalize, but DM will not necessarily thermalize, leading to the necessary suppression of the DM relic density, provided that its mass is at the TeV scale. In the context of freeze-out, the same coincidence is advertised as 'WIMP miracle'.
In section 2 we present the mechanism. In the next sections we consider specific models, pointing out that no ad hoc model building is needed. Indeed, in section 3 we consider the model proposed in [10] (for the non scale invariant version of this model see [11][12][13]), that extends the SM by adding a scalar doublet under a new SU(2) gauge group (its vectors are automatically stable DM candidates). We find that this model can reproduce the observed DM either through freeze-out (at larger values of the gauge coupling [10]) or through super-cooling (at smaller values of the gauge coupling). Super-cooling erases the baryon asymmetry: we will discuss how it can be regenerated at the weak scale, possibly through leptogenesis. Motivated by leptogenesis and neutrino masses, in section 4 we propose a similar model with U(1) B−L gauge group and two extra scalars. Conclusions are given in section 5.
While we focus on simple models based on weakly-coupled elementary particles, super-cool DM can also arise in more generic contexts, such as strongly coupled models with walking dynamics [14], possibly described through broken conformal symmetries and/or through branes in warped extra dimensions [15]. In such a case super-cooling can be described geometrically [16]. Furthermore, super-cool DM could arise in extensions of the Standard Model that cut quadratically divergent corrections to the Higgs mass and generate the weak scale: for example in supersymmetric models where all particles are massless in the supersymmetric limit, and where supersymmetry gets dynamically broken by some expectation value. However, the non-observation of new physics at the Large Hadron Collider casts doubts on such models. In more general terms, we expect the mechanism discussed here to be active whenever DM predominantly acquires its mass in a phase transition occurring after a significant period of super-cooling.

General mechanism
We consider extensions of the SM that provide a DM candidate and where all particles get mass from the vacuum expectation value of a scalar s, sometimes called 'dilaton'. In the standard freeze-out scenario, DM with mass M DM would decouple at a temperature T dec , equal to T dec ≈ M DM /25 if freeze-out reproduces the cosmological DM density.

Super-cooling
In dimension-less models, due to the absence of quadratic terms and vacuum stability (i.e. positive scalar quartics), thermal effects select, as the Universe cools down, the false vacuum where all scalars (in particular the dilaton s and the Higgs h) have vanishing vacuum expectation values. Around this vacuum, all particles are massless, including DM. The energy density of the Universe receives two contributions: from radiation, ρ rad (T ) = g * π 2 T 4 /30, and from the vacuum energy of the false vacuum V Λ > 0. We assume that the true vacuum has a nearly zero vacuum energy, as demanded by the observed small cosmological constant.
While the Universe cools, the vacuum energy starts dominating over radiation at some temperature T = T infl starting a phase of thermal inflation with Hubble constant H, such that V Λ determines T infl and H as where g * is the number of relativistic degrees of freedom just before inflation starts. We denote as a infl the scale factor of the Universe at this stage and write O(1) factors in Boltzmann approximation (correct within ±10%) given that DM might be bosonic or fermionic. During this phase DM is massless and thereby remains coupled, rather than undergoing thermal freeze-out at T ∼ T dec as e.g. in [17]. All particles, including DM, undergo supercooling: the scale factor of the Universe grows as a = a infl e Ht , and the temperature drops as T = T infl a infl /a.
Super-cooling ends at some temperature T end with a phase transition towards the true vacuum at s , h = 0 -at the end of this section we will discuss how this happens in the models of interest. During thermal inflation, the scale factor of the Universe inflates by a factor e N = T infl /T end .

Reheating
After the first-order phase transition to the true vacuum, the various particles, including DM, become massive, and the energy density V Λ stored in the scalars is transferred to particles, reheating the Universe. If the energy transfer rate Γ is much faster than the Hubble rate H, the reheating temperature is g where g RH is the number of reheated degrees of freedom. Otherwise the scalars, before decaying, undergo a period of oscillations and the reheating temperature is lower. During this period, the scalars dilute as matter. When scalars finally decay, their remaining energy density ρ sca becomes radiation with reheating temperature The final DM abundance Y DM = n DM /s (where s = 2π 2 g * T 3 /45 is the entropy density) receives two contributions: The super-cool contribution to the DM abundance is what remains of the original population of formerly massless DM in thermal equilibrium, suppressed by the the dilution due to thermal inflation The factor T RH /T infl arises taking into account that the energy stored in the oscillating inflaton dilutes as matter (rather than as radiation) between the end of inflation and reheating, when it is finally converted to radiation. The second contribution is the population of DM particles which can be produced from the thermal bath, through scattering effects, after reheating. 1 If in this way DM has the time to thermalize again, i.e. if T RH T dec , the supercooled population is erased and this second population reaches thermal equilibrium. In this case DM undergoes a usual freeze-out, leading to a relic density independent of whether there was previously a supercooling period. If instead T RH T dec the super-cool population remains basically unchanged, and the second 1 Production during pre-heating was discussed in [15]. Production during bubble collisions was discussed in [18]. population is produced with a sub-thermal abundance. This population is determined by the usual Boltzmann equation for Y DM = n DM /s in a radiation-dominated Universe, the same equation as the one that controls DM freeze-out. In non-relativistic approximation where λ = M Pl M DM σ ann v rel πg SM /45 if DM annihilations have a cross section σ ann dominated by s-wave scattering. For λ 1 freeze-out occurs at T dec ≈ M DM / ln λ; otherwise DM never thermalizes again after reheating and the term proportional to Y 2eq DM can be neglected. Integrating eq. (6) the regenerated DM population (starting from T = T RH ) is Summarising, the super-cool contribution dominates provided that T RH T dec . If DM interaction are small enough that it does not undergo kinetic recoupling, DM remains colder than in the freeze-out scenario: this has little observational implications [19]; the fact that supercool DM forms smaller structures could give enhanced tidal fluctuations possibly observable along the lines of [20]. The super-cool population of eq. (5) does not depend on DM properties (provided that initially, before supercooling, it was in thermal equilibrium), but only on the amount of super-cooling. As we will see below, eq. (5) matches the desired DM abundance provided that T end is a few orders of magnitude below T RH . If instead T RH T dec , an additional non-thermal DM population is generated after reheating which can also lead to the desired relic density.

The phase transition to the massive vacuum
Super-cooling ends anyway at the nucleation temperature T nuc when the rate of thermal vacuum decay becomes faster than the Hubble rate. So far nothing connects the DM mass to the weak scale. However, if this nucleation temperature lies below the QCD phase transition, supercooling in fact ends sooner, at this transition [21,22]. The temperature at which super-cooling ends, T end is approximated by Indeed, quark condensates form at T QCD cr ∼ Λ QCD . In view of its Yukawa couplings to quarks (in particular to the top) the Higgs acquires a vacuum expectation value h QCD ∼ Λ QCD , which induces a squared mass term M 2 s for the s scalar. If M 2 s is negative and bigger in modulus than the thermal s mass (which dominates the potential around the origin in dimension-less theories) s immediately starts rolling down, ending super-cooling at T QCD cr . Otherwise, s starts rolling at a lower temperature T QCD end , as soon as its thermal mass becomes smaller than |M s |. When supercooling is stopped at T end ∼ Λ QCD the amount of super-cooling that reproduces the observed DM abundance is obtained for a DM mass fixed by T end ∼ Λ QCD , leading to T end /T infl ∼ 10 −3−4 , which corresponds to TeV-scale DM. In this way the DM mass gets connected to approximately the weak scale.

The baryon asymmetry
The baryon asymmetry is washed out by the super-cooling factor e 3N . Thereby the scenario needs to be supplemented by some mechanism that regenerates the baryon asymmetry around the weak scale, after super-cooling. This can happen, provided that the three Sakharov conditions for baryogenesis are satisfied. 2 1. First, deviation from thermal equilibrium can be automatically provided by the end of super-cooling, either through a first-order phase transition or through the QCD-induced tachionic instability of s, h.

Second, violation of baryon number is automatically provided by sphalerons.
3. Third, CP violation.
The contribution from the CKM phase is too small. One possibility is to add an axion a coupled to gluons as α 3 aG a µνG a µν /8πf a : the observed baryon asymmetry can be obtained for values of the axion decay constant allowed by data, f a ∼ 10 11 GeV [25]. However, depending on the axion model, such a large scale risks conflicting with our assumption of a dimensionless theory where the weak scale is dynamically-generated [3]. It would be nice if one could generate a large effective f a from weak-scale loops, as attempted in [26]. It seems easier to devise scale-invariant models with extended interactions introduced ad hoc to violate CP: either extra Yukawa couplings or extra scalar quartics. One possibility is adding a second Higgs doublet such that the scalar potential contains one CP-violating phase that can lead to baryogengesis [27]. However this also contributes to electric dipoles, and flavour data agree with the SM with one Higgs doublet. Furthermore, detailed computations are needed to establish if the phase transition predicted by the model is enough out of equilibrium.
In the absence of a lepton asymmetry, in the above context the reheating temperature must remain below the decoupling temperature of electroweak sphalerons, T sph ≈ 132 GeV, otherwise sphalerons reach thermal equilibrium and wash-out the baryon asymmetry.
Given that observed neutrino masses anyhow demand an extension of the Standard Model, an appealing alternative possibility developed in the following is low-scale leptogenesis, where new neutrino physics generates a lepton asymmetry, converted by sphalerons into the desired baryon asymmetry.

Model with SU(2) X gauge group
The considerations above are fully relevant for basically any dimensionless model that contains a DM candidate (see e.g. ). Here we consider the model of [10] where the SM gauge group is extended adding an extra SU(2) X with gauge coupling g X , and the field content is extended adding a scalar S, doublet under the extra SU(2) X , neutral under the SM gauge group. The Yukawa interactions are those of the SM. The theory is assumed to be dimension-less, such that the tree-level scalar potential is This model generates the weak scale through Coleman-Weinberg dynamical symmetry breaking: the scalar doublets acquire vacuum expectation values and can be written as without loss of generality. The Coleman-Weinberg mechanism takes place because the quartic λ S runs as becoming negative at low energy below some scale s * , such that the one-loop potential is approximated as which has a minimum at s = w = s * e −1/4 . The SU(2) X vectors acquire a mass M X = g X w/2, and are stable DM candidates. Thereby DM has g DM = 9 degrees of freedom, including the components of S 'eaten' by the massive vectors in the broken phase. The model has only two free parameters beyond the ones of the SM: we will use g X and M X as free parameters.
We compute the other masses assuming, for simplicity, that λ HS is positive and small. Then s = w induces a Higgs vev h = v equal to v/w = λ HS /2λ H , where λ H ≈ 0.126 is the SM Higgs quartic, up to small corrections. This fixes the value of λ HS needed to reproduce the desired EW vacuum. The s mass is X /8(4π) 2 must be added to the potential such that the true vacuum at s = w has zero energy. This is the usual tuning of the cosmological constant.

Super-cooling
At finite temperature the potential receives thermal corrections V T , dominated by where )dx and Π X = 5g 2 X T 2 /6 is the thermal propagator for the longitudinal X component which accounts for re-summation of higher order ring-diagrams [56,29]. At small field values, the potential is approximated by positive thermal masses for the scalars s and h such that the thermal vacuum is s = h = 0. As the Universe cools down, a deeper true vacuum appears below a critical temperature T cr , equal to 0.31M X if g X < ∼ 0.7 such that ring diagrams can be neglected, and roughly a factor g X /0.7 larger otherwise. Given that dimensionless theories only have thermal masses and quartics, the Universe remains trapped in the false vacuum at s = h = 0 down to some temperature T end . Thermal inflation begins at the temperature T infl at which the vacuum energy starts dominating with respect to radiation. Applying eq. (2) to our model gives

End of super-cooling
During super-cooling, s and h are kept to 0 by thermal masses. The temperature T end at which thermal inflation ends has the form anticipated in eq. (8). First, we consider the possibility that thermal inflation ends through nucleation and compute T nuc . We solve numerically the bounce equation 3 s (r) + 2 r s (r) = dV ds , s (0) = 0 , lim r→∞ s(r) = 0 (16) and use it to calculate the thermal bounce action At T w the potential can be approximated as 1 2 M 2T s s 2 + 1 4 λ S (T )s 4 where λ(T ) < 0 is the quartic coupling renormalized at T , and the bounce action as S 3 /T ≈ 6.0πM T s /T |λ S | ≈ 8.2g X /|λ S | [21]. Nucleation happens at the temperature T nuc where the tunnelling rate is comparable to the Hubble rate, S 3 (T nuc )/T nuc ≈ 4 ln M Pl /M X ≈ 142. The numerical results are shown in fig. 1: T nuc is very small for small g X .
In such a case, QCD stops super-cooling earlier [21,22]. In the ordinary QCD chiral phase transition scenario where quarks are massive, this phase transition happens at T QCD cr ≈ 154 ± 9 MeV [58]. However, during super-cooling all quarks are massless, which leads to a smaller value of α 3 at low energy. Fig. 1 shows the running of the SM couplings: α 3 (μ) diverges at Λ h=0 QCD ≈ 144 MeV, with Λ MS = (89 ± 7) MeV if only α 3 is kept in the RGE [59]. Then the QCD chiral phase transition happens at a lower temperature, T QCD cr ∼ 85 MeV according to the estimate of [60,22]. When a zero-mode quark condensate forms, the Yukawa coupling y t h t L t R / √ 2 + h.c. induces a linear term in the Higgs potential, such that the Higgs acquires a T -dependent vacuum expectation value h QCD . Given that the couplings y t and λ H too run to non-perturbative values (see fig. 1) h QCD can at best be estimated. We will proceed assuming h QCD ≈ 100 MeV, up to order one factors.
Next, h induces a mass term for the s scalar, M 2 s = −λ HS h 2 /2. If λ HS < 0, the positive M 2 s delays the end of thermal inflation. If λ HS is positive (as needed to break SU(2) L at the true minimum) the negative M 2 s triggers the end of thermal inflation: s too starts rolling down as soon as its extra mass term M s becomes larger than its thermal mass M T s in eq. (14). If λ HS is large enough, this happens immediately at T end = T QCD cr ; otherwise this happens later at a lower temperature In the present model the thermal mass M T s is dominated by DM vectors, so that thermal inflation ends when their density is diluted enough.
We now have all the factors that determine T end in eq. (8). During super-cooling, the Universe inflates by a factor T infl /T end plotted in fig. 2a. The horizontal part of the contours corresponds to end of super-cooling via vacuum decay, and the vertical part to the QCDtriggered end.

Reheating
After the end of inflation, the scalars oscillate around the true minimum, dissipating their energy density ρ sca with some rate Γ into radiation that acquires energy density ρ rad = g * π 2 T 4 /30. The rolling fields s and h finally settle at the true minimum. The scale factor a and the various components evolve as ρ sca + ρ rad 3 ρ sca = −(3H + Γ)ρ scȧ ρ rad = −4Hρ rad + Γρ scȧ n DM = −3Hn DM + σv ann (n eq2 DM − n 2 DM ) + σv semi n DM (n eq DM − n DM ).
Thereby ρ sca (t) = ρ sca (t end )e −Γ(t−t end ) [a(t end )/a(t)] 3 . This roughly means that the inflaton s reheats the Universe up to the temperature We need to compute Γ. The scalar equations of motion are where K = K h + K s = (ṡ 2 +ḣ 2 )/2 is the scalar kinetic energy. Given that scalar masses are much bigger than H, averaging over the fast oscillations around the minimum, one finds that the scalar energy ρ sca = K + V = 2 K red-shifts as non relativistic matter, in the limit Γ h = Γ s = 0. All masses stay positive around the true minimum, so decays are not enhanced by parametric resonances. Most of the energy is stored in s, but its decay rate Γ s can be smaller than H. On the other hand the Higgs potential energy is sub-leading, while its decay rate Γ h ≈ 4 MeV is fast. Thereby the decay rate Γ of the combined system is controlled by the rate for energy transfer from s to h due to the λ HS interaction. We compute Γ by solving the equations of motion in linear approximation around the minimum where α −v/w = λ HS /2λ H 1 is the angle that diagonalizes the mass matrix. Fig. 2b shows the numerical results for T RH . For a given M X , reheating is instantaneous provided that g X is large enough, so that T RH = T infl M X /8.5, corresponding to vertical contour lines in fig. 2b. For smaller g X the reheating temperature is suppressed. 4

The dark matter abundance
The DM candidates are the SU(2) X vectors with mass M X . In the usual scenario where they are thermal relics, the observed DM abundance is reproduced for g 2 X ≈ M X / TeV [10] and super-cooling is negligible. Smaller values of g X lead to super-cooling, realising the novel DM production mechanism proposed here: the DM abundance is the sum of the super-cool population, plus the sub-thermal population, Eq. (4).
We compute the super-cool DM population specializing eq. (1) and eq. (5) to the present model. We find that the super-cool abundance reproduces the observed DM abundance when the end of super-cooling is triggered by the QCD phase transition as T end = T QCD end (eq. (18)), and when reheating is instantaneous. Thereby T RH = T infl ≈ M X /8.5 and the DM abundance simplifies to It does not depend on g X , giving rise to the vertical contours in the (M X , g X ) plane in the left panel of fig. 3. As anticipated, this is not the end of the story: one needs to take into account the effects of thermal scatterings after reheating. One needs to evolve the Boltzmann equation in eq. (19) 〈h〉 QCD =50 MeV In the extreme case where the reheating temperature is larger than the DM decoupling temperature, the super-cool population is erased and substituted by the usual thermal relic population. Otherwise, the super-cool population is negligibly suppressed, and complemented by the additional sub-thermal population of eq. (7). For instantaneous reheating (z RH ≈ 8.4) this evaluates to Ω DM h 2 | sub−thermal ≈ 0.110(g X /0.00020) 4 , giving rise to the horizontal part of the contour at g X ≈ 10 −4 in the (M X , g X ) plane of fig. 3. At larger M X reheating is no longer instantaneous, giving rise to the oblique part of the contour in fig. 3a, which shows the complete numerical results for the DM density. Finally, for even larger masses M X 300 TeV the super-cool abundance reproduces the observed DM relic density with g X ∼ O(1), so that super-cooling is ended by nucleation, leading to a DM abundance that depends mainly on g X and not on M X . Therefore, DM masses can be even PeV-scale or larger, higher than those allowed with freeze-out and perturbative couplings. At M X ≈ 100 TeV one has M s M h , so that a resonance-like feature in the mixing angle α, and consequently in T RH and Ω DM , appears.
In the region of the parameter space relevant for the present work, the Spin-Independent cross section for DM direct detection is dominantly mediated by s and simplifies to where f ≈ 0.295 is the nucleon matrix element and m N is the nucleon mass. So σ SI < 1.5 10 −45 cm 2 (M X / TeV) [64] for M X > 0.88 TeV. Fig. 3a also shows the existing bounds and future discovery prospects, coming both from searches of dark matter via direct detection, and from collider searches for s. The region shaded in orange is excluded by direct searches at XENON1T [64] and the dashed vertical lines denote the future sensitivity of XENONnT [65], LZ [66] and DARWIN [67]. The direct detection cross section is suppressed by two powers of the small g X coupling and enhanced by the exchange of the s scalar state, which is light, see fig. 2b. As a result the direct detection constraint is significant. The region shaded in blue is excluded by collider searches for s: the dominant collider bounds come from s → ee, µµ at CHARM and B → K * µµ at LHCb, as summarized in [68]. The dashed blue curves indicates the future sensitivity of SHiP [69,68].
Finally, fig. 3b shows a sample example of the evolution of the DM density, of entropy, of the energy density in scalars: the latter dominate during super-cooling, while n DM and s decrease equally. At reheating almost all of this energy is transferred to entropy, and only a small fraction goes to massive DM.

The baryon asymmetry
The reheating temperature T RH in the SU(2) X model is shown in fig. 2b and can be either smaller or larger than T sph , see fig. 3a. In the first case cold baryogenesis might be a viable option (with extra CP-violation), while leptogenesis is a clear option in the second case.
Leptogenesis can in particular be achieved if one adds right-handed neutrinos N , with Yukawa couplings Y N N LH, and an extra real scalar singlet S , with quartic potential couplings and a Yukawa coupling y S S N 2 /2 that breaks lepton number. This induces a mass M N for N if S acquires a vev.
Barring fine-tunings in the structure of Yukawa couplings, a sizeable lepton asymmetry needs quasi-degenerate right-handed neutrinos, at the per-million level. The time-variation of their masses (while scalars relax to their minimum) relaxes the amount of quasi-degeneracy, a scenario we will not further explore here.
The first possibility, resonant CP-violating decays, mostly produces an asymmetry for T ∼ M N and thereby needs T RH > ∼ M N . Indeed the right-handed neutrino mass must be sizeably above the electroweak scale to allow N → HL decays and to produce efficiently the asymmetry before T sph . In the super-cool DM production scenario above, the reheating temperature can be above a TeV, see fig. 2b. This longer period of electroweak symmetry breaking restoration makes this scenario easily viable. This is shown in fig. 4. Successful leptogenesis implies a lower bound on the reheating temperature, depending on M N , which implies a lower bound on M X .
The second possibility of low scale leptogenesis, right-handed neutrino oscillations in Lconserving processes, requires lighter N , around the GeV scale. However, it is in general fully operational at temperatures orders of magnitudes larger than the electroweak scale. Thus, except in special situations, in our context it is suppressed by the low reheating temperature.
The third possibility, L-violating Higgs decay (which requires right-handed neutrino mass between a GeV up to the Higgs mass), produces dominantly the baryon asymmetry at temperatures just above the sphaleron decoupling temperature T sph . Therefore, one only needs T RH > ∼ T sph , which can be realised in the allowed parameter space of fig. 3. This mechanism explains why in fig. 4, which combines the leptogenesis contributions from L-violating N and H decays, leptogenesis is viable for masses below the Higgs boson mass and T RH ≥ T sph . We solved Boltzmann equations taken from [75] in the single-flavour approximation for the SM leptons. These do not take into account the reheating temperature suppressed purely-flavoured ARS contribution.
In all cases (including the model discussed in section 4 below), leptogenesis is significantly facilitated by the super-cool mechanism, since the required gauge couplings are small and do not dilute the asymmetry as they do, instead, in the WIMP regime [77,78], where successful leptogenesis would hardly be possible.

Model with U(1) B−L gauge group
We here study a different super-cool DM model. Given that leptogenesis seems the most plausible option for baryogenesis, a natural possibility is gauging B − L, such that right-handed neutrinos become necessary for anomaly cancellation. This makes SU(2) X no longer necessary for dynamically breaking scale invariance: the role of SU(2) X can be played by U(1) B−L , in a similar way. The scalar doublet S of SU(2) X is replaced by a complex scalar, S, charged under U(1) B−L . The gauge coupling g B−L can drive the scalar quartic λ S to run negative around the weak scale, such that the scalar S again acquires a vacuum expectation value. As a result, the B − L gauge boson Z acquires a mass, eating the would-be Majoron. The weak symmetry is again broken thanks to a −λ HS |SH| 2 term in the potential with λ HS > 0.
Assuming that it has a B − L charge equal to 2, the S scalar can be identified with the field that gives mass to right handed neutrinos N , through a Y S SN 2 /2 Yukawa interaction.
A disadvantage of this model (compared to the SU(2) X model) is that the Z cannot be DM: it decays into SM fermions, as they are charged under B − L; furthermore B − L can have a kinetic mixing with hypercharge. We thereby add one extra scalar singlet φ, with no hypercharge and B − L charge q φ chosen such that it is stable: for simplicity we assume q φ = 1. This makes DM absolutely stable due to the fact that φ is odd under the Z 2 ⊂ U(1) B−L symmetry, which remains unbroken because S has B − L charge 2. 5 Summarising, the model is described by gauge-invariant kinetic terms, plus the dimensionless scalar potential plus a constant plus the Yukawa interactions After symmetry breaking the scalar fields can be written as 5 This B − L gauge group has been considered in its scale invariant version in [79,22] as a model for neutrino masses (without DM) and in [41], with a DM scalar particle which has no B − L charge (in order to avoid direct detection bounds) and is stabilised adding an extra Z 2 symmetry. Super-cool DM, instead, requires small values of the B − L gauge coupling, such that we can assume that DM is charged under B − L, and thus automatically stable, compatibly with direct detection constraints.
At one loop, the λ S quartic runs as becoming negative at low energy below some scale s * , such that its one-loop potential is approximated as which develops a minimum at s = w = s * e −1/4 . This generates as well as electro-weak symmetry breaking and neutrino and DM masses, We neglected the contribution of λ Hφ to M φ . Electroweak precision data imply the bound M Z /g B−L > ∼ 7 TeV [80,81], up to corrections due to kinetic mixing. DM has g DM = 2 degrees of freedom and is not destabilised by the symmetry breaking of the various gauge groups provided that λ Sφ , λ Hφ , λ φ > 0 such that the DM scalar φ does not acquire a vacuum expectation value. The condition λ φ > 0 is easily satisfied in view of the smaller g 4 B−L contribution to its running: The vacuum energy vanishes for V Λ ≈ β λ S w 4 /16 ≈ (3M 4 Z /8 + M 4 DM /4)/(4π) 2 such that T infl ≈ (M 4 Z +2M 4 DM /3) 1/4 /11. The s thermal potential is V T ≈ T 4 [3f (M Z /T )+2f (M DM /T )]/2π 2 and its thermal mass is M 2T s = (g 2 B−L + λ Sφ /12)T 2 , so that T QCD end = h QCD λ HS /(2g 2 B−L + λ Sφ /6). The Spin-Independent cross section for DM direct detection receives two unavoidable contributions, from Z mediation, and from λ Sφ (via the small mixing α −v/w between h and s), as well as a contribution from λ Hφ : Finally, we need the DM pair production cross section that is at the origin of the sub-thermal population. In the non-relativistic limit, DM is produced in pairs in a s wave way from two Z , from two scalars (possibly through a Z ) while the production from two fermions via a Z is p-wave suppressed. We get   where Γ * h,s are the decay width into SM particles of a virtual h, s with mass 2M DM . 6 We neglected the s/h interference. These cross sections are similar to the ones in the DM scalar singlet model [82,83].
We now have all the ingredients to compute the DM density. In fig. 5 we plot it in the (M Z , g B−L ) plane, similarly to fig. 3 for the SU(2) X model. However the U(1) B−L has a few extra free parameters, most importantly the DM mass. In fig. 5 we thereby consider a few different values of the DM mass, and assume that the extra free parameters are in ranges which give neither enhancements nor cancellations in the various equations above. An important difference with respect to the previous model is that constraints from direct detection (in orange) are weaker. Baryogenesis through leptogenesis needs T RH > ∼ T sph : in the plotted parameter region this is satisfied when DM has a sizeable sub-thermal contribution, in addition to the super-cool contribution. 6 Longitudinal components of W ± , Z enhance Γ * h 3M 3 DM /4πv 2 at M DM M h , such that σ(φφ * → h * → W + W − , ZZ) 3σ(φφ * → hh) as demanded by SU(2) L invariance.

Summary
We presented a new mechanism that can reproduce the observed cosmological DM abundance when DM is a weak-scale particle. The mechanism arises in models where the weak scale is dynamically generated. The Universe remains trapped in a false vacuum where all particles are massless and undergoes a phase of thermal inflation during which all particles get diluted. This phase can be ended by the QCD phase transition or by vacuum decay to the true vacuum, where particles are massive. Light particles are regenerated in the subsequent reheating phase, but the DM abundance can remain suppressed, with a quite low temperature, due to supercooling. Fig. 3 exemplifies the possible cosmological evolution. When super-cooling ends at T ∼ Λ QCD , the desired DM abundance is obtained for weak-scale DM.
In section 3 we have shown that super-cool DM is produced in a simple model proposed in [10], where dynamical generation of the weak scale and DM stability is obtained adding to the SM a new SU(2) X gauge group and a new scalar doublet. In section 4 we studied a model where the new gauge group is U(1) B−L . In both models DM is reproduced dominantly through super-cooling for DM masses of about 500 GeV and for DM couplings of order 10 −4 -smaller than in the freeze-out scenario, such that the simplest models of super-cool DM are still allowed by direct detection.
The U(1) B−L gauge structure (and the scalar that breaks it), in addition of dynamically generating the symmetry breaking and of stabilizing DM, also gives rise to neutrino masses and to leptogenesis. This is a welcome feature, given that super-cooling erases a possibly preexisting baryon asymmetry, which needs to be regenerated after reheating. Depending on the model and on its parameter space, the reheating temperature can be either larger or smaller than the decoupling temperature of weak sphalerons, such that the baryon asymmetry can be regenerated either through leptogenesis or through cold baryogenesis, possibly during the phase transition that ends super-cooling.