Baryogenesis and Leptogenesis from Supercooled Confinement

We propose a framework of baryogenesis and leptogenesis that relies on a supercooled confining phase transition (PT) in the early universe. The baryon or lepton asymmetry is sourced by decays of hadrons of the strong dynamics after the PT, and it is enhanced compared to the non-confining case, which was the only one explored so far. This widens the energy range of the PT, where the observed baryon asymmetry can be reproduced, down to the electroweak scale. The framework then becomes testable with gravity waves (GW) at LISA and the Einstein Telescope. We then study two explicit realisations: one of leptogenesis from composite sterile neutrinos that realises inverse see-saw; one of baryogenesis from composite scalars that is partly testable by existing colliders and flavour factories.


Introduction
The universe contains way more matter than antimatter.Independent measurements of the number density n b of baryons (i.e.particles with a non-zero baryon number B) with the cosmic microwave background [1] and big bang nucleosynthesis [2], combined with the lack of evidence for primordial antibaryons [3,4], imply a baryon asymmetry of the universe (BAU) where s is the entropy density.The Standard Model of particle physics (SM), when combined with the standard cosmological model ΛCDM, clashes with the observed BAU (see e.g.[5]) unless one imposes it as an ad-hoc initial condition, a possibility which becomes untenable when combined with the inflationary paradigm.The BAU is then a strong observational motivation for physics beyond the SM.To possibly reproduce the observed BAU, one needs processes in the early universe which, at the same time [6]: i) violate of course B; ii) violate also C and CP , otherwise any nonzero B in their final state would be compensated by a −B in the final state of the conjugate process; iii) are out of equilibrium, otherwise the reverse processes would wash-out the baryon asymmetry.Cosmological first order phase transitions (PTs) satisfy condition iii), so that they can potentially explain the BAU if they are accompanied by processes that violate B, C and CP .While first-order PTs in the early universe are not predicted by the SM in isolation [7,8], they are by many SM extensions motivated by open theoretical problems, see e.g.[9][10][11][12][13][14][15][16].The proposal of electroweak baryogenesis, for example, relies exactly on new physics that makes the EW PT first-order, see [17] for a recent review.
First-order PTs with fast bubble walls, in particular, are currently receiving massive attention because they produce gravitational waves (GW) [18,19] observable at current [20][21][22] and foreseen [23] telescopes.In addition to detectability in GW, first-order PTs with fast walls are motivated as production mechanisms of dark matter (DM) in the form of very heavy particles [24][25][26][27][28][29] or primordial black holes [30][31][32][33].PTs with fast bubble walls offer also viable mechanisms to explain the baryon asymmetry, see e.g.[34][35][36].A particularly simple possibility is to obtain the BAU from out-of-equilibrium decays of particles that obtain their mass at a supercooled PT, as proposed in [36] and worked out so far only for PTs of weakly coupled theories.This mechanism, however, works only for scales of the PT above about 10 8 GeV (see e.g. also [37,38]), so that it is not testable in any laboratory experiment and only poorly so by GW telescopes, given that the predicted GW are at very high frequencies.
In this paper we point out that, if the supercooled PT is a confining one, then with respect to analogous weakly-coupled PTs, both the yield of the mass-gaining particles (i.e. the hadrons) is larger and wash-out processes are suppressed.We find that these ingredients allow to generate the baryon asymmetry down to PT scales in the TeV range, via hadron's decays.This opens up the possibility to generically test this framework with LISA and the Einstein Telescope.Depending on its specific realisation, we find that this also allows to test the scenario at colliders and/or to connect leptogenesis with neutrino mass generation in a new framework.
2 Framework for the generation of the Baryon Asymmetry

Supercooled confining phase transitions
As the temperature of the early universe bath drops, the vacuum state of the theory that is energetically favoured can change, triggering a PT.If the PT is first order it proceeds via nucleation of bubbles of the new ('true') vacuum in the bath, see e.g.[39,40] for reviews.
A first-order phase transition in the early universe is controlled by the interplay between the energy density of radiation and the difference ∆V between the ones of the old ('false') and new vacuum.The latter is given by where f is the vacuum expectation value (VEV) of the order parameter ϕ of the phase transition in the true vacuum (which we will later identify with the dilaton), and c vac is a dimensionless, model dependent, quantity parameterizing the vacuum energy difference, typically c vac ∼ O (1) or smaller.The radiation energy density is given by where g * counts the effective degrees of freedom of the radiation bath.The phase transition takes place around the nucleation temperature, T nuc , when the bubble nucleation rate per unit volume, Γ, becomes comparable to the Hubble parameter H, namely Γ(T nuc ) ∼ H 4 (T nuc ).The PT is said to be supercooled if it happens when the universe is dominated by the vacuum energy rather than by radiation, ρ rad < ∆V , i.e. supercooling: where T eq is defined as the temperature when ρ rad = ∆V .Following the phase transition, the order parameter ϕ undergoes oscillations and decays.We assume that its width is comparable or larger than the Hubble rate, so that the universe is reheated to a temperature where g RH is the number of degrees of freedom in the broken phase that are reheated.A supercooled PT is generically predicted by potentials V (ϕ) that are flat enough at the origin at zero temperature [41], see [42] for a recent systematic study.The needed features of V (ϕ) are automatically predicted by models that are approximate scale invariant in the UV, and that spontaneously break scale invariance in the IR, like in radiative generation of IR scales à la Coleman-Weinberg [43,44].
In this paper we will be interested in strongly coupled realisations, where confinement occurs from the condensation of an approximately conformal strongly-interacting sector, and which are dual to 5D warped models, see e.g.[9][10][11]45].The spontaneous breaking results in a Goldstone boson parametrically lighter than f , the dilaton ϕ, which can be thought of as a condensate of UV quanta of the confining theory.Its potential is generated from a small explicit breaking of scale-invariance in the UV, without which the symmetry could not be spontaneously broken, so that (see e.g.[46]) where Z is the dilaton's wavefunction renormalization and in strongly coupled theory is expected to be of order one.We clarify that ϕ then denotes the canonically normalized dilaton, and m ϕ its mass.Eq. ( 6) then implies that m ϕ and c vac are not independent parameters, but are linked via m ϕ < few×f is enough to achieve values of T nuc orders of magnitude smaller than T eq , see e.g.[26,46].Larger hierarchies between m ϕ and f , corresponding to very small values of c vac , may imply that the PT never completes.One could then envisage to complete it via the PT associated to another sector, see [47,48] for examples where the QCD PT triggers a supercooled EW PT that would otherwise not complete.Obtaining large hierarchies between m ϕ and f is also non-trivial from the theoretical point of view, see e.g.[49][50][51][52][53][54] for successful attempts.

Dynamics of the walls
Supercooled PTs generically predict that bubble walls are ultrarelativistic.As bubbles are nucleated and start to expand, the wall's boost factor γ starts growing as well because the surface tension of the wall loses against the difference in pressures.If nothing prevents their acceleration, then walls 'run away' and γ keeps growing linearly with the bubble radius R.
For the values of ∆V /ρ rad ≳ 1 of our interest, plasma effects cannot prevent the walls from becoming faster and faster [55,56].Particle physics effects however can.They come from particles obtaining a mass at the wall [57] and radiation from these particles [58], which exert a pressure on the walls and can prevent them from running away.In the context of PTs that are both supercooled and confining, the pressure arises from non-perturbative effects and reads [26] where g TC is the number of degrees of freedom, that are charged under the confining group, in the deconfined phase (see the next two subsections).The non-perturbative effects responsible for this pressure can be thought of as the strong-coupling limit of the resummed radiation at the walls in weakly coupled theories, and indeed the pressure associated to the latter effects also scales linearly in γ [59].Walls then reach a terminal constant boost, γ T as soon as ∆V = P.If this condition is never met, they run away until they collide with walls from other bubbles.The wall boost factor at bubbles collision then reads where the first entry is the runaway value at collision γ run = R coll /(3R nuc ) [59], R coll = π 1/3 /β [60] is the typical bubble radius at collision and R nuc , of order T −1 nuc , the one at nucleation.Since in the supercooled PTs of our interest R nuc ≲ T −1 nuc [26], we use R nuc = T −1 nuc /2 for definiteness (other supercooled PTs may have R nuc ≳ T −1 nuc , see e.g.[61]).We have also introduced the reduced Planck mass M Pl ≃ 2.4 × 10 18 GeV and β ≡ (dΓ/dt)Γ, where we remind that Γ is the bubble nucleation rate per unit volume, so that β/H controls the typical number of bubbles per Hubble volume and β/H between 10 and 100 is a typical range for supercooled PTs.Whether the bubble-walls stay in the runaway regime or reach a terminal velocity depends on the amount of supercooling of the PT, which we quantify by the ratio T nuc /f .Examining Eq. ( 9) it is clear that, in order to have a sufficiently large γ coll , one needs T nuc ≪ f .

Abundance of hadrons
We call hadron any composite state of the confining sector.Hadrons are expected to have a mass M hadr ≈ 4πf ≫ T nuc coming from their strong coupling to the dilaton, unless they are lighter for symmetry reasons, like the dilaton itself or other pseudo-goldstone bosons (analogous to pions in QCD).The hadrons do not exist before the PT, where the fundamental 'techniquanta' (TC) are massless (like quarks and gluons in QCD).These are in equilibrium with the thermal bath before the phase transition and their number density to entropy ratio, the Yield, is given by where g TC are the degrees of freedom of the techniquanta and n TC is their number density.What happens upon the PT was first modeled in [26] and can be summarized as follows.Techniquanta enter the bubble and form fluxtubes that point towards the wall.If techniquanta have more energy than the confinement scale in the wall frame, i.e. if γT nuc ≳ f , then these fluxtubes break forming a large number of hadrons, which on average follow the bubble walls in the plasma frame2 .Fluxtube's breaking also results in the ejection of techniquanta outside the wall, to conserve charge.Both hadrons and ejected particles have very high energies in the bath frame, of order γf .As bubbles expand and collide, hadrons and ejected techniquanta collide with those from other bubbles and with particles of the pre(heated) plasma, whose typical energies are of the order of the dilaton mass m ϕ .These collisions result in deep inelastic scatterings which ultimately convert the large center of mass energies (of order m ϕ γf in the bath frame) into hadron masses.This conversion has been found by simulations of scatterings and hadronization in [26], and its simplicity makes it pretty robust against the details of this simulation and other possible processes affecting the propagation and scatterings of hadrons and ejected techniquanta [62].This redistribution of large energies into hadron masses then enhances the yield of hadrons, with respect to that of particles that gain mass at a non-confining PT, by a factor γ coll f m ϕ /M 2 hadr .The yield of a given hadron Ψ then reads where M hadr ≃ 4πf and Br hadr → Ψ < 1 is the number of hadrons Ψ produced over all hadrons in the end of the scattering chains, which is expected to be suppressed if M Ψ ≫ f , analogously to QCD.The last factor (T nuc /T eq ) 3 T RH /T eq ∝ (T nuc /f ) 3 is the usual dilution of relics from supercooled PTs, the factor (3γ coll f m ϕ /M 2 hadr ) is its enhancement specific of confining theories discussed before Eq.(11).

Baryon asymmetry from hadrons decay
Let us now assume that a portal connects the confining sector and the SM that results in an otherwise-stable hadron Ψ to decay violating B, C and CP .Note that the portal needs not to necessarily violate C and CP , as their violation could rely on the one existing in the SM (see e.g.[63,64] for a realisation of this idea in composite Higgs models, although for slow bubble walls) or it could be a property of the strong sector.We do not need to specify this nor the UV origin of the portal in this paper, we just need to know that each of the Ψ or Ψ produces on average a net baryon number ϵ Ψ .The baryon yield then reads c vac 0.01 where K Sph is the sphaleron reprocessing factor, K Sph = 1 if the baryon asymmetry is generated when electroweak sphalerons are not active, namely when T ≲ T Sph = 132 GeV [65], and K Sph < 1 otherwise (K Sph = 28/79 in the SM [66]).In general ϵ Ψ is at most of order where by Imy 2 we mean the imaginary part of some combination of couplings with coupling dimension 2, and where we include possible enhancements from small hadron mass splittings ∆M ≪ M Ψ .Small hadron mass splittings are actually expected from confining sectors with at least two light quarks (think for example of the neutron and proton masses in QCD).
The discussion so far assumes that the population of hadrons Ψ is unaffected before they decay.First, we estimate the total decay rate of Ψ as Γ Ψ = C|y| 2 M ∆ /8π, where C is a modeldependent factor, and observe that they are much faster than the Hubble rate in the entire parameter space that we will consider.So we should compare the rate of Ψ annihilations, i.e. how fast the Ψ want to go to their thermal equilibrium abundance, with Γ Ψ and not with H. Ψ's annihilate into dilatons ϕ and possibly other particles, depending on the model.As the coupling of Ψ to dilatons is model-independent and large (of order M Ψ /(Zf )), we estimate the annihilation cross section with the analogous QCD one for p − p annihilations, σ(ΨΨ * → ϕϕ)v rel ∼ 4π/M 2 Ψ .Therefore, using Eq. ( 13), we obtain the total Ψ annihilation rate In order to not change the abundance of the Ψs before they decay and source the BAU, we require Γ annh to be below Γ Ψ = C|y| 2 M ∆ /8π, i.e. the following ratio should be smaller than one: In the regions where this is larger than one, it could still possible to source the BAU, as there is still a population of Ψs.The determination of the BAU would require additional calculations.Since in most of the parameter space of our interest Γ annh < Γ Ψ , for simplicity we do not compute the BAU there but just warn the reader whenever we have Γ annh > Γ Ψ .Parameter space where our baryogenesis framework manages to reproduce the observed BAU, and its testability in gravity waves with LISA, the Einstein Telescope and the PTA.We choose g TC = 80, g * = g SM + g TC and, for definiteness, g RH = g SM + 1 (where the +1 accounts for the dilaton), the dilaton wavefunction normalization Z = 1 and T nuc /f = 10 −3 .The white area is allowed by the PT dynamics and can accomodate the baryon asymmetry, where we saturate the baryon yield at each point, by obtaining Br hadr → Ψ from Eq. ( 13).
The excluded gray region presents values of Br hadr → Ψ > 0.1, chosen as a benchmark upper limit, hence it cannot account for the whole baryon asymmetry of the Universe.The excluded red region presents γ coll < M Ψ /T nuc , hence the hadrons cannot enter the bubble and provide the BAU.Above the cyan lines hadrons start to equilibrate with dilatons before they decay, so the computation of the BAU should be different than the one we employed.The regions delimited by the green, black and blue dashed lines are testable, respectively, by PTAs, LISA and the Einstein Telescope, where the astrophysical foregrounds have (not) been taken into account in the thicker (thinner) lines.We denote with violet stars the two points of the parameter space for which we computed the gravitational wave spectrum shown in Fig. 2. The viable parameter space will be reduced in specific realizations of our framework, by model-dependent washout effects or laboratory searches depending on the SM interactions of the hadrons Ψ, see Figures 6  and 8.
The parameter space where our framework can potentially reproduce the observed BAU is displayed in Fig. 1 for representative benchmark values of the parameters, for f ≥ MeV cause lower values are excluded by Big Bang Nucleosynthesis (BBN), see e.g.[22].Imposing Y B = Y BAU fixes the value of Br hadr → Ψ in every point of the parameter space.We require Br hadr → Ψ < 0.1 because string fragmentation is expected to give rise mostly to light hadrons, and at least one light hadron, the dilaton, exists in our picture.We also require Γ annh /Γ Ψ < 1, for definiteness for C = 1 and using |y|=3 Imy.As long as the walls are in the terminal velocity regime, Y B is independent of both f and T nuc /f .These parameters play a role as soon as collisions happen in the runaway regime, i.e. in a region of large f and small T nuc /f , where Y B gets quickly suppressed (indeed Fig. 1 displays that one cannor reproduce the BAU at large f ).
f and T nuc /f also play a role in defining the regions where the BAU can be washed out by various effects.Washout effects are not included in Eq. ( 13), nor they are visualized in Fig. 1, because they depend on the specific model realising our framework.One can still identify the following general expectations for them.Concerning inverse decays, they can be neglected as long as M Ψ ≫ T RH , because their rate is Boltzmann-suppressed at large M Ψ /T RH .M Ψ ≫ T RH is actually a consequence of T RH ≲ f (see Eq.( 5)) and M Ψ ≃ 4πf .M Ψ /f is the hadron coupling to the dilaton, the large values needed to avoid washout from inverse decays can only be obtained in strongly-coupled theories.Other washout processes that generically exist are 2 → 2 scatterings mediated by Ψ. Requiring that they are negligible imposes i) an upper bound on Imy 2 of Eq. ( 14), hence our normalization of ϵ Ψ in Eq. ( 13) to a small number, and ii) an absolute lower limit on M ψ , because they are suppressed by some power of M ψ .Condition ii) is what prevents from going to values of f much below a TeV, although it is possible that a minimal model-building effort weakens this requirement.Another obstruction to achieving f ≲ TeV arises in models where the confining sector generates a lepton asymmetry, because then one needs T RH > T Sph to transmit it to the baryon sector.
Two key features of our framework, that are different with respect to its weakly coupled analogue of [36,38], can already be grasped by our general discussion so far: ⋄ the hadron yield is enhanced by the strong dynamics and it allows for a larger BAU; ⋄ the strong coupling M Ψ /f ≫ 1 suppresses washout coming from inverse decays.
They imply that this mechanism works for scales of the PT down to (at least) the weak scale, which opens the possibility to test it with both LISA and the Einstein Telescope, which is the topic of Sec. 3. It also makes the mechanism testable in the laboratory, either directly producing the hadrons at high-energy colliders, or indirectly testing their CP and flavour violating effects.Finally, a scale of the PT down to (at least) the TeV scale also allows for interesting theoretical connections, e.g. with models explaining the origin of the weak scale, or of neutrino masses, with confining dynamics.We will explore some of these consequences in two explicit models that realise our framework, in Secs. 4 and 5.

Gravitational wave signal
We now detail the expected GW signal from a supercooled PT.Using the numerical results from [67], the GW energy density power spectrum as measured today, for initially thick walled, runaway bubbles is given by where h ≃ 0.67 is the dimensionless Hubble parameter as determined by Planck [1].In Eq. ( 17) the factors Ω 0 rad and (g 0 /g RH ) 3 take into account that the GW energy density redshifts as radiation from its value at the PT, Ω PT GW , until today, and Ω 0 rad and g 0 are, respectively, the energy fraction in radiation and the degrees of freedom in entropy today.α GW is the energy released as bulk motion during the PT (which we approximate as the energy difference between the false and true vacuum, ∆V ) normalized to the radiation density ρ rad , namely thus one expects α GW ≫ 1 for supercooled PTs.The shape of the spectrum is governed by the spectral function where we adopt the central values for the bubbles in the setup with the thickest walls from [67], namely {a, b} = {0.742,2.16}.The peak frequency of the signal today is given by ν = 33 µHz g RH 100 Finally, one should impose that Ω GW ∝ ν 3 at small frequencies, due to causality for superhorizon modes at GW production [68].We therefore apply a cut in the spectrum at the redshifted frequency today of below which we enforce the correct IR scaling and where a(T RH )/a 0 is the ratio of the scale factors between the PT and today.
The numerical simulations performed in [67] assume that bubbles effectively runaway until collision, however a large part of our parameter space, notably the one associated to a PT around the TeV scale, sits in the terminal velocity regime, where the vacuum energy pressure is counterbalanced by the friction produced by particles in the plasma.The vacuum energy is thus partly converted into kinetic energy of the particles in the plasma already prior to the bubble wall collision.However, the contribution from sound waves or turbulence [69,70] in supercooled PTs is not yet clearly understood.Indeed, current hydrodynamical simulations, which aim to capture the contribution of the bulk motion of the plasma to the GW signal, do not yet extend into the regime in which the energy density in radiation is subdominant to the vacuum [71] and analytical studies of shock-waves in the relativistic limit have just started [72].In this scenario one can argue that derivatives of the envelope approximation such as the so-called bulk flow model are more realistic in modelling the GW signal.
In practice, the difference between [67] and bulk-flow models like [73] is very mild, see e.g. the comparison in [61].Therefore we use the parametrization from [67], described above, for both regimes of wall velocities.Examples of GW spectra are shown in Fig. 2, for α GW = 100 and β/H = 15, both typical of supercooled PTs and for two benchmark values of c vac and M Ψ .
Our model can be probed by several current or upcoming GW interferometers, due to the wide parameter space allowed.The sensitivity of a detector to a GW signal is given by where H 0 = 100h km/s/Mpc is the present day Hubble parameter and S n is the noise spectral density [74].The signal to noise ratio SNR to a GW background is given in terms of the above quantities as [75] where ν min , ν max and t denotes the minimal and maximal frequencies accessible at the detector and the fiducial observation time, which is taken to be t= 10 years for the ET and t=3 years for LISA.Using this expression, the so called power-law-integrated (PLI) sensitivity curves are computed [75], and our calculations for the PLI sensitivity curves of, LISA [76], BDECIGO [77], LIGO-VIRGO-KAGRA [78] and Einstein Telescope (ET) [79] (configuration '2L 0 • 20km' for definiteness) are shown in Fig. 2, for a fiducial choice of SNR=10.The spectra are compared with PLI sensitivity curves, shown as red lines, with signal-to-noise ratio SNR=10 for LISA [76] and the Einstein Telescope (ET) [79].We also show PLI sensitivity curves for LIGO-VIRGO O3, LIGO-VIRGO Design A+ sensitivity [78] and BDECIGO [77].Estimated astrophysical foregrounds from binary super-massive black holes [80], galactic whitedwarf binaries [81], extragalactic white-dwarf binaries [82], binary black holes and neutron stars [78] are also shown as green shaded regions.
There are several astrophysical stochastic GW foregrounds which could mimic the GW signal coming from the PT, see Fig. 2 for details.In order to take this limitation into account we also define a foreground-limited signal to noise ratio SNR FGL , With the above conservative definition we impose that our signal is not detectable if buried under the astrophysical foreground Ω F G (ν).We show in Fig. 1 the regions of parameter space of our BAU framework that could be probed by Pulsar Timing Arrays (PTA), LISA and the ET, which we define as those where SNR FGL ≥ 5.For comparison, we also show the regions that one could probe in absence of astrophysical foregrounds, which could be relevant in case one will envisage ways to tell them apart from a GW signal from PTs.The two sensitivities are anyway very similar for LISA, due to the strong GW signal from the PT in comparison with foregrounds, while for PTAs and ET we find an appreciable difference.The sensitivity showing up for f ≲ 100 GeV is the one attainable using GWs as measured by PTA, which we compute following [83] and identifies yet another way to test our framework, in case one manages to make it work down to such small scales of the PT.If one did, then one could potentially explain the GW background observed by PTAs [84][85][86][87] with a PT [88,89] that, at the same time, explains the BAU via the framework proposed in this paper.In any case, a very large fraction of the allowed space could indeed be probed by GW interferometers.
4 Model 1: Baryogenesis from a composite scalar We have so far discussed a general framework for the generation of the baryon asymmetry and we have established that it presents some interesting features, as an enhanced production of the baryon yield with respect to the non-confining case and a gravitational wave signal potentially in reach of current and future GW interferometers.We now specify our framework to two models already discussed in the literature, compute their predictions for washout and for the BAU as well as for detection, and show explicitly that we can extend significantly their allowed parameter space for a successful baryogenesis/leptogenesis.

Model and baryon asymmetry
We start by revisiting the model of baryogenesis discussed in [36], where the baryon number is produced by the CP -violating decays of a heavy scalar denoted there as ∆.In our setup ∆ is a composite scalar field originating from a dark confining sector and it is a realization of the Ψ hadron discussed above.We introduce i = 1, 2 generations of the ∆ i scalars, with We assume these fields to transform as (3 , 1 , 2/3) under the SM gauge group We also introduce a SM gauge singlet fermion χ with mass M χ and the following baryon number violating Yukawa interactions of ∆ i : where colour indices have been suppressed for simplicity of the notation.y di is antisymmetric in flavour d as a consequence of the antisymmetry of the colour indices.A necessary condition for CP violation is that the above couplings should be complex.Some, but not all the phases, even in the minimal scenario, can be removed through field rephasings f → e iα f f .To be more concrete let's examine some specific scenarios: ⋄ In the minimal scenario there is only one copy of χ and the ∆ 1,2 couple each to one up-type quark and to a pair of down-type quarks (that do not have the same flavour).This results in 4 Yukawa couplings y.One can remove 3 out of their 4 phases by rephasing the ∆ i and χ, but one phase survives since in general Arg(y d2 ) − Arg(y d1 ) ̸ = Arg(y u2 ) − Arg(y u1 ).Thus in this setup there are 4 couplings and 1 (physical) phase.
⋄ One can switch on all the allowed SM quark Yukawa couplings, keeping a single copy of χ.This setup features 12 couplings, 6 involving down-type quarks and 6 involving the up-type quarks.As in the previous case we can rephase the 3 new fields, so we're left with 9 independent new phases.
⋄ For 3 copies of χ, we find 24 couplings (6 from down-and 18 from up-type quarks).With respect to the previous setup we have the freedom to remove 2 more phases by redefining χ 2,3 , therefore we end up with 19 independent phases.
Note that in the second and third case (but not the first one) one could remove further phases from the Yukawa couplings of Eq. ( 25) by rephasing the right-handed quarks, but those phases would not be unphysical as they would show up in the quark Yukawas.Assuming M χ ≪ M ∆ i to avoid complications with phase space effects, the couplings of the Lagrangian Eq. ( 25) lead to the following tree level decay rates for the scalars ∆ i where by d ′ R we denote a right down quark of different flavour than d R , we have not summed over flavours and we have summed over final state colours for the first decay.Interference between tree-and loop-level diagrams, as those shown in Fig. 3, leads to CP violation in the decays.Focusing on the decays of ∆ 1 only, CP violation can be parametrized as Since the total decay widths of ∆ 1 and ∆ * 1 have to be equal due to the CP T theorem, it follows that ϵ d Γ 1d = −ϵ u Γ 1u .The imaginary part of the loop in Fig. 3 controls the rate asymmetry between ∆ 1 and ∆ * 1 decays and can be extracted via Cutkosky rules [90].One then finds for the rate asymmetry Im(y * d1 y u1 y * u2 y d2 ) where from now on we assume that there is no major hierarchy among the Yukawa couplings y and we take generic O(1) phases.We are also discarding resonant enhancements of the rate asymmetry, namely we are assuming that M ∆ 2 − M ∆ 1 ≫ Γ ∆ i .Finally, the baryon asymmetry is obtained by plugging ϵ ∆ of Eq. ( 29) in Y B of Eq. ( 13).

Washout processes
In our scenario the most constraining washout processes are those occurring after reheating, after the PT.The reason is that the population of the ∆ hadrons, yielding the baryon asymmetry via their decays, is mostly sourced by the deep inelastic scatterings of the hadrons (produced upon string fragmentation of incoming or ejected techniquanta) with other hadrons or bath particles.These scatterings mostly happen after bubble percolation, so that washout effects just after wall crossing, like those considered in [36], are not relevant in our framework (but see the end of the previous subsection for ∆ annihilations after percolation).
After bubbles percolate and all hadrons are produced, diquark interactions violating B − L can result in washout if T RH is too high.Off-shell 2 → 2 scattering processes mediated by ∆ that violate baryon number, e.g.d d ′ ↔ χ u, have a rate that can be estimated as where we are integrating out ∆ because T RH ≪ M ∆ in our framework.This rate is harmless for our scenario provided that it is below the Hubble rate H, namely Moreover, also inverse decays into on-shell ∆ can lead to B − L washout.The Boltzmann suppressed rate is given by This is safely below H, provided that

Proton decays and further washout from χ
So far we have not addressed the detailed nature of χ, which we assume not to be linked to the confining sector, to allow us more freedom with its phenomenology.A first idea is to take χ to be close to massless, which would be the case if the SM neutrinos gain Dirac mass via a Yukawa-type interaction y ν LHχ R , with y ν ∼ 10 −12 .However, χ allows nucleon decays via t-channel diagrams mediated by ∆, like p → K + χ, as long as M χ < m p − m + K .Taking the couplings to all the generations to be of the same order, we estimate the decay rate From the experimental bound on the partial proton lifetime, τ (p → K + ν) > 5.9×10 33 years [91], it follows that Γ(p → K + ν) < 3.5 × 10 −66 GeV, therefore a scenario in which M χ < m p − m + K would completely rule out our interesting region of the parameter space around the TeV.Considering now the case where M χ > m p − m + K , a natural assumption would be to assume χ to be Majorana, with a 1 2 M χ χ R χ c R mass term and to couple it to the SM leptons and the Higgs H via the Yukawa coupling y ν and provide SM neutrino masses with a standard type-I seesaw.In this case the decay p → K + ν would receive contribution from y ν and one can get an estimate for the decay width which again would exclude the most interesting region of our parameter space.Therefore, in order not to spoil the interesting features discussed so far, one needs M χ > m p − m + K and to disentangle χ from the generation of neutrino masses, so that the coupling y ν can be taken as small as needed to respect limits on nucleon's lifetimes, independently of the other parameters.
Moreover, the same couplings in Eq. ( 25) which produce the baryon asymmetry in our setup also lead to a partial decay width of χ given by After baryogenesis there is an excess of χ over χ, thus if χ is Dirac and M χ < M ∆ the asymmetry gets erased exactly once χ decay.Indeed the decays χ → udd ′ and χ → udd ′ would exactly erase the CP asymmetry, due to ϵ d Γ d = −ϵ u Γ u following from CP T .Note that M χ ≥ M ∆ is not a viable option, because then Γ u = 0 and so ϵ d = 0 via CP T , so one would not generate the baryon asymmetry.Instead, if χ is Majorana then both udd and udd decays are allowed and the asymmetry is preserved.However, the asymmetry could be spoiled by inverse decays: any excess of udd or udd would go back to χ, that would wash it out upon decaying Assuming no hierarchies among the Yukawas y we get the following ratio between the rate of inverse decays and Hubble: where we have evaluated H at T = M χ because that is the range of temperatures for which inverse decays are most efficient: for T ≪ M χ , Γ(udd ′ → χ) would be exponentially suppressed, for T ≫ M χ , H would be enhanced but not Γ(udd ′ → χ).
To summarize, χ does not alter our results as long as i) to avoid nucleons decays, M χ > m p − m + K and its coupling to neutrinos is small enough (so that it cannot play a role in the generation of neutrino masses) and ii) to avoid further washouts, M χ is Majorana and it is smaller enough than M ∆ to keep Γ(udd ′ → χ)/H ≪ 1, see Eq. (37).

Collider searches
As it turns out (see Fig. 6), one can easily accomodate baryogenesis from supercooled confinement with the scalar ∆ around or below the electroweak scale, the properties of these particles can be probed thanks to their signatures in high-energy colliders.
A first lower bound on the mass of ∆ comes from the measurement of the decay width of the Z boson performed at LEP [92].Since ∆ interacts weakly due to its U (1) Y charge, it can contribute to the Z boson width, provided that M ∆ < M Z /2.We compute the following expression for the decay width of Z into a ∆∆ * pair Since this value is orders of magnitude larger than the uncertainty on the Z boson width, we exclude M ∆ < 45 GeV.
Then one can note that ∆ has the same quantum numbers of a right up-type squark in SUSY, therefore it will share its same production mechanisms at colliders.To our knowledge, the closest searches to our scenario that have been performed are those involving squark decays into diquarks, through R-parity violating (RPV) couplings.Constraints on top squarks decaying through RPV couplings were first set by the ALEPH experiment at LEP [93], excluding at 95% CL masses below 75 GeV via four-jet searches.More stringent bounds come from further searches performed by CDF at Tevatron [94] and by ATLAS [95] and CMS [96] at the LHC.The combination of these searches rules out top squarks with RPV decay in the 45 GeV < M ∆ < 770 GeV interval.All these searches aim to detect four jets in the final state associated to the decay of pair produced heavy resonances.This is the typical signature that one would expect associated with ∆, as for the allowed values of its mass and coupling constants it would decay promptly in the detector.However, as the diquark and leptoquark couplings of ∆ are expected to be of the same size, some significant branching ratio of ∆ scalar could reside in the u χ channel, such that the signal would be different than the one assumed in four-jet searches, and the limits would change.Performing a detailed collider study of the ∆'s, depending on their branching rations into two-jet and one-jets plus missing energy, is beyond the scope of our work, so for definiteness we use the constraints that assume ∆ to dominantly decay into two jets.

Neutron electric dipole moment
As we have seen in the previous section, the new fields ∆ i and χ contain new sources of CP violation and therefore give rise to new contributions to the neutron electric dipole moment (nEDM).The experimental bound set by the nEDM collaboration at the Paul Scherrer Institut is given by |d n | < 1.8 × 10 −26 e • cm [97].In this section we analyze the impact of this bound on our model and show that it does not place further significant constraints on the parameter space.
In an EFT approach the electromagnetic moments of a fermion ψ can be described by the effective Lagrangian where is the EDM of ψ and F µν is the field strength tensor of the photon.The diagrams which may give a non-vanishing contribution to the nEDM must have an imaginary component.In general the amplitude of any loop diagram can be recasted as a sum of amplitudes each coming by operators O i weighted by their coefficients C i , which can be written as C = y F .In this notation y stands for the product of the couplings of all the vertices and F is a function coming from the loop integration [98].In principle both y and F are allowed to be complex, but diagrams with the heavy ∆ and χ running in the loop cannot have all the intermediate states go on-shell.Therefore the optical theorem implies that Im(F ) = 0 and only y can be complex.Moreover, any combination of couplings entering the expression of the nEDM should be invariant under rephasing of the fields, thus we need at least 4 couplings involving the new BSM states, because Y = y * d1 y u1 y * u2 y d2 is the simplest combination of the Yukawas with at least one physical phase.As a consequence one needs to go at least to 2-loop to find a possible non-zero contribution.
The relevant topologies of our model are depicted in Fig. 4, they include the Barr-Zee diagrams [99].They all give in principle non-zero contributions to the nEDM.However, for Figure 4: Relevant topologies at two loops for the nEDM.Several contributions for each topology occur due to flavour.While diagrams in the upper row are manifestly symmetric under ∆ 1 ↔ ∆ 2 exchange, a more careful analysis is needed to show the cancellation among the diagrams in the lower row, which is non-trivial due to the different number of ∆ i propagators involved when the photon is attached to an internal ∆ leg. each of them one can always take the contribution of the 'mirrored' diagram, with complexconjugate Yukawa couplings and with the photon attached to the same type of field.One can decompose the sum of each set of conjugate diagrams as where F (i, j) is coming from the loop integration.F (i, j) is not generically a symmetric function of the scalar masses, but the asymmetry scales like )), therefore it vanishes in the k 2 = 0 limit for which the nEDM is measured.An explicit proof of this statement is given in the Appendix A. Therefore the sum in Eq. ( 40) ends up to be real and thus 2-loop diagrams do not produce a non-vanishing nEDM.
At the 3-loop level one expects the dominant contributions to the EDMs to scale as where m t is the top-quark mass, g is the weak gauge coupling and by 'CKM' we mean a combination of non-diagonal elements of the CKM matrix, which is unavoidable to ensure CP violation and is smaller than at least 0.1.Since limits on coefficients of quarks EDMs (and also CEDMs, for which the discussion is analogous) are in the ballpark of 10 −9 /TeV (see e.g.[100]) and M ∆ ≥ TeV from collider bounds, then 3-loop contributions do not give rise to observable EDMs.To conclude, the constraint set by nucleon EDMs on the mass of ∆ turns out to be much less stringent than those coming from colliders.

Flavour violation
Flavour-changing neutral current (FCNC) observables are expected to set powerful constraints on BSM models, due to the non trivial flavour structure that they imply in order to comply with experimental data.In our scenario, ∆F = 2 processes are prohibited at tree level due to the antisymmetric couplings of ∆ with the diquarks, which can be parametrized as y ij = ϵ ijk y k , where i, j, k are flavour indices.These bounds were firstly discussed in [101] and one can identify our state ∆ with the diquark denoted as VIII there.
First, it has to be pointed out that the most general flavour structure, with all the flavour off-diagonal couplings switched on, is not needed to account for baryogenesis, as a single nonzero complex coupling is sufficient to provide a physical CP violating phase.In this setup flavour bounds would be evaded completely.If one wants to stick to the general scenario, bounds coming from the mixing of the neutral K and B d,s mesons have to be taken into account.The ∆ scalars then enter the box diagrams in Fig. 5 and they generate a four-fermion effective Hamiltonian given by Observable where i and j are the external quark flavour indices, k ̸ = i ̸ = j is the flavour index of the internal quark, colour indices are contracted within the brackets and the loop function G is given by [101] Bounds on representative D = 6 effective operators contributing to ∆F = 2 transitions have recently been obtained by the UTfit collaboration [102], from a combination of CP conserving and CP violating observables.We can apply these bounds to our model, results are summarised in Table 1.Other constraints which we include in Table 1 are those associated to the ∆F = 1 processes b → sγ, b → dγ, R b and B ± → ϕπ ± .We refer to [101] for more details as the constraints reported there can be easily translated to our notation.One can note that even the most constraining observable, namely ϵ k , is compatible with ∆ diquark couplings of O(0.01) around the TeV scale, which, in the absence of major hierarchies in the Yukawa couplings, is the benchmark for our baryogenesis mechanism to work.Therefore, we don't expect flavour physics to exclude values of M ∆ that are not already excluded by collider searches.In fact, even assuming a hierarchy of the type y ui ≫ y di , in order to lower the bound on M ∆ from colliders, one would get at the same time weaker bounds from flavour.
To conclude, the scalar realisation of our baryogenesis framework may well show-up in current and near-future flavour experiments, but the non-observation of flavour or CP -violation there would only exclude some very specific choices of its parameters and not preclude the realisation of baryogenesis.

Summary of the results
We summarize our results in the plots of Fig. 6.We have chosen two different benchmark values for y = {0.01,0.1} and T nuc /f = {10 −3 , 10 −4 }, M ∆ 1 = 4πf and the dilaton wavefunction normalization Z = 1.We compute at each point the boost factor at bubbles collision γ coll using Eq. ( 9) and we set M ∆ 2 = 1.2 M ∆ 1 , g TC = 80, g RH = 107.75(= g SM + 1, where the +1 accounts for the dilaton) and g * = g TC + g SM .We saturate the baryon yield at each point, Y B /Y obs B = 1, by obtaining Br hadr → ∆ from Eq. ( 13).As anticipated in Sec. 2 we see that Y B gets suppressed in the runaway regime, hence the allowed parameter space closes at large f and/or small T nuc /f .Values of f lower than the weak scale, for large c vac , cannot be reached due to washout effects.∆ annihilations before decays instead turn out to not constrain our parameter space, for the values of the parameters that we have chosen.
We also display in Fig. 6 collider bounds, discussed in Sec.4.4.1, and the expected LISA and ET sensitivities to the GW signal from the PT, discussed in Sec. 3, where to avoid clutter we display only those that include astrophysical foregrounds.By comparing our plots with Fig. 6 of [36], we see that our scenario extends the allowed parameter space by many order of magnitudes towards smaller values of M ∆ , allowing the PT to occur down to the electroweak Figure 6: Parameter space for the composite scalar realisation of our framework, which realises baryogenesis.We choose g TC = 80, g * = g SM + g TC , g RH = g SM + 1 and Z = 1 for definiteness.The white area is allowed and accommodates the observed baryon asymmetry, the red and gray regions are excluded by the PT dynamics, as in Fig. 1.We find that ∆ annihilations are slower than ∆ decays in the entire parameter space we plot, hence our computation of the BAU is always valid.A value of y smaller than unity is needed to avoid washout.Off-shell and on-shell washout regions are shaded to exclude parameter space where, respectively, 2 → 2 interactions Eq. ( 31) and inverse decays Eq. ( 33) are in equilibrium.Constraints from collider experiments include LEP searches, excluding values of M ∆ < 75 GeV and four-jet searches performed at Tevatron and at the LHC, excluding ∆ in the 50 GeV < M ∆ < 770 GeV range.We also show the parameter space testable by the Einstein Telescope (ET) and LISA using α GW = 100, β/H = 15, where the astrophysical foregrounds have been taken into account.We denote with red stars the two points of the parameter space for which we computed the gravitational wave spectrum shown in Fig. 2. scale, potentially in reach of LISA and the Einstein Telescope, and making the mechanism testable at the LHC.
5 Model 2: Leptogenesis from a composite sterile neutrino

Model, neutrino masses and the baryon asymmetry
We now apply our framework for the generation of the baryon asymmetry, based on a supercooled confining PT, to a different model where the observed BAU is sourced via leptogenesis [103] through the decays of composite sterile neutrinos, which in addition provide mass to the SM neutrinos via a seesaw mechanism.This way we both substantiate the statement that our framework is more general than its specific realisations, and we find results that are interesting for leptogenesis per-se.
We introduce the sterile neutrinos N a (and denote their two two-component spinors as N aL and N aR ) as hadrons of the confining sector, that is responsible for their Dirac mass M Na ∼ O(πf ).We then assume that a portal between the confining sector and the SM breaks the SM lepton number, generating inverse seesaw [104,105] (ISS) as where L are the left-handed SM leptons, α = {e, µ, τ }, y is the Yukawa coupling matrix between active and sterile neutrinos, µ is a lepton number breaking Majorana mass matrix, where we further denote by µ a (µ) its complex diagonal (off-diagonal) entries.The presence of µ induces at low energies the d = 5 Weinberg operator [106] where H = iσ 2 H * and Upon electroweak symmetry breaking, the mass matrix m ν for the light neutrinos is given by one can thus express µ = µ(f ) for a fixed value of the Yukawa coupling y, assuming m ν to be of the order of the atmospheric neutrino mass, m ν ∼ 0.05 eV.We now assume the heavy Dirac states N a to come in two generations, which is the minimum number needed to comply with neutrino oscillations data.Without loss of generality, we choose a basis where M N is diagonal and real.In the non confining ISS the parameter µ is technically small but the hierarchy µ ≪ M N is not set by any concrete theoretical justification.The latter is instead naturally justified in our setup of composite sterile fermions.One can indeed envisage the possibility that, while the Dirac mass M N originates from the confining sector, the Majorana one µ originates from physics external to the confining sector, offering a natural way to accomodate µ ≪ M N .An explicit realization is provided in [107], where the strong dynamics is that of a conformal sector deformed by a relevant operator so that it confines in the IR (giving rise to the Dirac masses M N ), and by another operator with different scaling dimension and a small coefficient that gives rise to µ ≪ M N .As anticipated in Sec. 2, our framework also leads to the expectations ∆M ≡ |M N 2 − M N 1 | ≪ M N 1 , because the N 's are all hadrons of the same sector, as well as y aα ≪ 1, because the Yukawas originate from a portal.Therefore, under the natural assumption that µ a ∼ µ ∼ µ ≪ M N 1,2 , ∆M , one can diagonalize the sterile neutrinos mass matrix to first order in δ a = µ/M Na .Defining four Majorana states Ñi , i = 1, 2, 3, 4, with masses M i , we have Therefore we can see from Eq. ( 49) that the heavy neutrinos form pseudo-Dirac pairs with small Majorana splitting controlled by µ/M N i .
The CP asymmetry generated by the decays of the sterile neutrino N a then reads where we summed over the lepton flavour α in the final state and includes contributions given by vertex and self-energy corrections to the decay, which are given by [109] and [110] where Γ i ∼ (yy † ) ii M i /16π is the decay width of N i and we assume no hierarchies in masses or couplings among the singlet generations and the SM flavours.
Let us now focus on the ϵ 1 and ϵ 2 contributions to Eq. ( 51): Due to the pseudo-Dirac nature of the ( Ñ1 , Ñ2 ) pair, it follows that As a consequence, when considering ϵ 1 +ϵ 2 , the f 12 and f 21 terms will dominate the sum as those involving f i3 and f i4 turn out to be of higher order in δ and δ when summed up.An analytical  56), for a fixed reference value of the Yukawa coupling y = 0.002, a degeneracy M/∆M = 10 and µ = µ(f ) fixed by neutrino masses.One can note that an asymptotic value for the produced asymmetry is reached in the µ ≥ Γ regime.In the yellow region at large M N our derivation of Eq. ( 56) breaks down, because it assumed µ ≪ ∆M .parametric expression for ϵ in the ISS was obtained by the authors of [108], ϵ ∼ µ 2 /(M N Γ), where we dropped the family indices for µ and Γ to show only the parametric dependence.However, the above expression is assuming µ ≪ Γ, which is not strictly required by the ISS mechanism.We then derive a more general relation that assumes no hierarchy among µ and Γ.In fact as long as µ ≪ M N , ∆M and Γ ≪ M N , ∆M are satisfied, the dominant one-loop contribution to the rate asymmetry is given by the self energy correction of N , because one has f V 12 − f V 21 ∼ δ 1 , therefore the terms involving f self 12 and f self 21 ≃ −f self 12 dominate in ϵ 1 + ϵ 2 .We then obtain the total rate asymmetry ϵ ISS where the factor of 2 includes the contribution from ϵ 3 + ϵ 4 , which has the same parametric dependence as the one from ϵ 1 + ϵ 2 .We plot ϵ ISS as a function of M N ∼ M N 1,2 in Fig. 7.One can observe from Eq. ( 56) that an enhancement of the rate can be obtained in the presence of a quasi-degeneracy among the Dirac masses M Na .When the two generations a = 1, 2 are nearly degenerate, the CP asymmetry is enhanced compared to the non-degenerate case by a factor M N /2∆M , as follows from Eq. ( 50).We stress that Eq. ( 56) holds for µ ≪ ∆M , hence we exclude regions of parameter space where this does not hold from our study.A generalization to the case µ ≳ ∆M is beyond the scope of this work.
As the baryon asymmetry is now sourced by a lepton asymmetry, one has to require that the reheating temperature, Eq. ( 5), is larger than the sphalerons decoupling temperature, T RH ≥ T sph ≃ 132 GeV, in order for the sphalerons to be in thermal equilibrium and therefore to transfer the lepton asymmetry to the baryon sector.Finally, the baryon asymmetry is obtained by plugging ϵ ISS of Eq. ( 56) in Y B of Eq. (13).

Washout processes
Following analogous steps to those discussed in the model with the ∆ scalar, we now have to take into account washout effects of the lepton number produced at T RH .The 2 → 2 scattering rate ℓH ↔ ℓH * , mediated by the sterile fermion N , violates lepton number.Therefore its rate must vanish in the µ → 0 limit and it can be estimated from the coefficient of the Weinberg operator in the ISS, Eq. ( 46): which presents a µ 2 /M 2 N suppression with respect to the standard Type-I Seesaw one, where one has only one scale M N ∼ µ.We also included a factor of 2 in Eq. ( 57) to account for the contributions given by the degenerate neutrinos.Moreover we assume these washouts to be efficient for T RH ≥ m H /3, when a sizable population of H is still present.Anyway the precise value of T RH for the decoupling of washout effects is not crucial as these processes turn out to play a negligible role in our parameter space.In fact this rate does not spoil the lepton number asymmetry produced by the decays of N provided that it is smaller than H(T RH ), i.e. if where the dependence of Γ W O on y, µ and M N has been absorbed into m ν by using Eq. ( 47).Also the inverse decays processes ℓH → N , ℓH * → N can lead to a lepton number washout, if they feel lepton number violation.Indeed, if they do, then they distinguish the initial ℓ states from l ones and convert more ℓ than l back into N 's.We can assume that the lepton asymmetry produced by N decays does not get erased provided that the inverse-decays rate Γ ID ≤ H at T RH .It was found by the authors of [111] that the small mass splitting between the two states among the heavy pseudo-Dirac N i leads to a destructive interference between their contributions to Γ ID , which has the effect of suppressing the washouts.This can be parametrized by the suppression factor of the decay width where ρ ∼ µ/Γ and c ∼ (Γ/M N ) 2 .The quadratic suppression in µ can be again understood from the fact that inverse decays cannot wash out a lepton asymmetry in the lepton conserving limit, hence they should depend on some power of µ in addition to powers of the Yukawa couplings.One can thus estimate Γ ID as where the factor of 2 comes from the fact that both N 1 and N 2 contribute to washout.This means that the rate of the net washout processes ℓH → N , ℓH * → N at T RH is suppressed by both the Boltzmann factor and the smallness of µ.Therefore, washouts of the lepton asymmetry turns out to be very suppressed.
To summarize we find that, in the model of leptogenesis from inverse-see-saw and composite neutrinos, 2 → 2 washout effects are never the leading constraint on the parameter space of our interest, while inverse decays can become an important constraint only if one lifts the Boltzmann suppression by choosing M N < 4πf .

Composite sterile neutrinos at colliders and discussion
We are also interested in possible signatures of the sterile fermions N at colliders, coming from their mixing with the active neutrinos.The requirement T RH > T Sph plus M N ∼ O(πf ) implies M N ≳ TeV, making a future high-energy muon collider [112,113] the leading proposed collider to test this picture.The authors of [114]  Figure 8: Parameter space for the composite singlet fermion realisation of our framework, which constitutes a variation on inverse see-saw leptogenesis.We choose g TC = 80, g * = g SM + g TC , g RH = g SM + 1 and Z = 1 for definiteness.The white area is allowed and accomodates the observed baryon asymmetry, the red and gray regions are excluded by the PT dynamics and above the cyan line the N 's start to equilibrate with the dilaton before they decay and our computation of the BAU should be changed, as in Fig. 1.The dashed red line is obtained from the projected sensitivity of a 10 TeV muon collider on the active-sterile neutrino mixing, see [114].The green region is excluded from requiring electroweak sphalerons to be out of equilibrium for reheating temperature T RH < 132 GeV.In the orange exclusion regions the onshell washout processes, i.e. inverse decays of Eq. ( 60), are in equilibrium at T RH .The yellow region is excluded by requiring µ < ∆M for our expression of the rate asymmetry Eq. ( 56) to hold.We also show the parameter space testable by the Einstein Telescope (ET) and LISA using α GW = 100, β/H = 15, where the astrophysical foregrounds have been taken into account.
We denote with red stars the two points of the parameter space for which we computed the gravitational wave spectrum shown in Fig. 2.
The discussion above however holds only at tree level.Starting from the pure LSS, a Majorana mass term µ for the N a is generated at the one-loop level, of the size µ ∼ M N yy ′ /(16π 2 ), thus one expects a general LSS+ISS model to arise from a LSS tree level Lagrangian [108].We may then wonder if a new interesting region of the parameter space opens up in the region where the interference between ISS and LSS contributions produces the relevant term for the rate asymmetry: it turns out that the neutrino mass matrix formula is dominated by the LSS term, while the rate asymmetry by the ISS piece and it reads The y coupling needed for leptogenesis, y ∼ O(10 −4 − 10 −5 ) for M N ∼ TeV, is way too small to provide any signatures at colliders, so we will not explore in more detail this model within our supercooled confinement BAU framework.

Conclusions and Outlook
In this paper we have proposed a new framework for the generation of the observed baryon asymmetry of the universe (BAU), based on a first order supercooled phase transition (PT) of a confining sector.In our framework B − L violating processes are driven by the decays of some composite states (hadrons) of the confined phase.The main advantages with respect to the analogous non-confining mechanism [36] are that in our setup the rate asymmetry is enhanced due to the deep inelastic scatterings occurring upon bubble percolation, which increase the population of the decaying hadrons, and due to the suppression of washout processes because of the hierarchy between the hadron masses and the reheating temperature.Independently of its specific realization, our framework allows to reproduce the observed BAU down to the weak scale, and potentially down to the MeV one.A first important consequence of this result is that, contrarily to its weakly coupled realisations, the framework is testable in gravitational waves (GWs) not only by the Einstein Telescope but also by lower frequency ones like BDECIGO and LISA.Based on the current understanding of astrophysical foregrounds, the absence of a signal in these experiments will exclude the framework down to the TeV scale, see Fig. 1.
We have then studied two specific models within our proposed framework, one realising baryogenesis from the decays of composite scalars, charged under the SM, one realising leptogenesis from the decays of composite fermions, singlets under the SM gauge.This allows to include effects that could potentially washout the BAU, to determine other experimental signals, and to possibly connect our framework with open SM problems in addition to the BAU.
⋄ Concerning baryogenesis from composite scalars, we have found that washout effects prevent PT scales f lower than a few GeV, see Fig. 6.We have then studied collider searches for the composite scalars, finding they exclude values of f lower than the weak scale, and that future LHC searches, like 4-jet ones, will test new open territory of this model.The model could also manifest itself in flavour and CP -violating observables, although the size of the effects depends on further details of the model and an absence of deviation from the SM would not exclude it.Finally, we find that the dominant contribution to the neutron EDM arises at 3 loops and the model is not expected to show up there.
⋄ Concerning leptogenesis from composite sterile neutrinos, it offers a natural implementation of inverse and linear see-saw scenarios (ISS and LSS), that also explains the small SM neutrino masses.Focusing on the ISS for definiteness, we found that we can reproduce the observed BAU down to scales of the PT f ∼ TeV, see Fig. 8.A degeneracy at the level of 10 −2 or more is required between the masses of two composite neutrinos, but that is actually expected from a confining sector, while in the standard thermal leptogenesis one generally needs tuning.The obstacle to reach f below the electroweak scale comes from the requirement of a reheating temperature larger than about 130 GeV, so that sphalerons are active.Washout effects are further suppressed, with respect to a generic case, by the insertion of an extra parameter to break lepton number, and only inverse decays can lead to appreciable washout in case M N < 4πf .We find that the model is barely testable by N searches at a future muon collider.Our results concerning this model are also interesting from the ISS and LSS points of view, because we enlarge the parameter space where these leptogenesis models work.
Our study opens several interesting avenues of further investigation, both on the modelbuilding and on the phenomenology sides.Concerning the former, it would be for example interesting to connect our framework for the generation of the BAU with solutions to the hierarchy problem of the Fermi scale, given that it works down to the electroweak scale.Concerning phenomenology, it would be interesting to find a specific model realising our framework which allows to go down to the MeV scale for the PT, so to possibly explain the GWs observed by pulsar timing arrays [84][85][86][87].We leave these and other directions for future work.
A Cancellation of the 2-loop contributions to the nEDM Let us show explicitly that the sum of the diagrams shown in Fig. 9 with in addition those with ∆ 1,2 swapped provides a vanishing contribution to the nEDM.The starting point is given by the structure of the electromagnetic nucleon vertex in presence of CP-violation, which is given by Here k = (p − q) is the momentum of the photon, m N is the nucleon mass and the EDM of the nucleon is defined as d n = F D (0)/(2m N ).Using the Gordon-decomposition of the vector-axial current, the CP violating ( ¨C P ) part of the above expression reads Thus we need to concentrate on the (p + q) • ϵ terms in the computation of the EDM, where ϵ is the photon polarization.
In order to find whether there is a cancellation among the diagrams, we can factor out the upper loop, as it is symmetric under ∆ 1,2 exchange.Let's focus thus on the lower loop and denote by Y the product of all the Yukawa couplings entering the diagram (a).The relevant amplitude reads where m is the mass of the internal quark and M 1,2 the masses of ∆ 1,2 .Since the ∆ i live at much higher scales than those probed by the EDM we can assume M 2 1,2 ≫ k 2 , p 2 , m 2 .The integrand is dominated by ℓ 2 ∼ M 2 1,2 , thus one can expand the propagators as with analogous expressions for the other propagators.One obtains iM = Y u(q)γ α P R u(p) The denominator is now an even function of the loop momentum ℓ, thus we need to focus on terms with an even number of ℓ at numerator.At O(ℓ 2 ) there are two kind of terms which might contribute to the nEDM, • p α ℓ µ ℓ β p β ϵ µ ∼ ℓ 2 p α (p • ϵ) • p α ℓ µ ℓ β k β ϵ µ ∼ ℓ 2 p α (k • ϵ) = 0 and as the non-zero one appears also in diagram (b) the sum of the contribution of (a) and (b) to the nEDM would cancel.Let us consider now O(ℓ 4 ) terms.There are three types of terms which might contribute: The second term appears also in the diagram (b) with coefficient Y * and so it is harmless.The third one receives two contributions from diagram (a), from ∆ 1 and ∆ 2 , while (b) contains only the ∆ 1 contribution, as the momentum flowing in ∆ 2 doesn't depend on k.Therefore the sum of diagrams (a) and (b) doesn't provide a non-zero nEDM.However, the surviving contribution stems from an integral with a 1/(ℓ 2 − M 2 1 ) 2 (ℓ 2 − M 2 2 ) 2 factor and is cancelled by the couple of conjugate diagrams with ∆ 1,2 exchanged.We can thus conclude that our model allows new physics contributions to the nEDM to occur only starting at the three loop level.

- 3 LISA SNRFGL > 5 ET SNRFGL > 5 LISA SNR > 5 ET SNR > 5 PTA SNR > 5 PTA SNRFGL > 5 1 Figure 1 :
Figure1: Parameter space where our baryogenesis framework manages to reproduce the observed BAU, and its testability in gravity waves with LISA, the Einstein Telescope and the PTA.We choose g TC = 80, g * = g SM + g TC and, for definiteness, g RH = g SM + 1 (where the +1 accounts for the dilaton), the dilaton wavefunction normalization Z = 1 and T nuc /f = 10 −3 .The white area is allowed by the PT dynamics and can accomodate the baryon asymmetry, where we saturate the baryon yield at each point, by obtaining Br hadr → Ψ from Eq. (13).The excluded gray region presents values of Br hadr → Ψ > 0.1, chosen as a benchmark upper limit, hence it cannot account for the whole baryon asymmetry of the Universe.The excluded red region presents γ coll < M Ψ /T nuc , hence the hadrons cannot enter the bubble and provide the BAU.Above the cyan lines hadrons start to equilibrate with dilatons before they decay, so the computation of the BAU should be different than the one we employed.The regions delimited by the green, black and blue dashed lines are testable, respectively, by PTAs, LISA and the Einstein Telescope, where the astrophysical foregrounds have (not) been taken into account in the thicker (thinner) lines.We denote with violet stars the two points of the parameter space for which we computed the gravitational wave spectrum shown in Fig.2.The viable parameter space will be reduced in specific realizations of our framework, by model-dependent washout effects or laboratory searches depending on the SM interactions of the hadrons Ψ, see Figures6 and 8.

Figure 3 :
Figure 3: Example of the diagrams which interfere and lead to CP violation.The intermediate loop particles are kinematically able to go on-shell.

Figure 7 :
Figure7: Rate asymmetry ϵ ISS in the inverse see-saw mechanism, obtained from Eq. (56), for a fixed reference value of the Yukawa coupling y = 0.002, a degeneracy M/∆M = 10 and µ = µ(f ) fixed by neutrino masses.One can note that an asymptotic value for the produced asymmetry is reached in the µ ≥ Γ regime.In the yellow region at large M N our derivation of Eq. (56) breaks down, because it assumed µ ≪ ∆M .

Figure 9 :
Figure 9: Two-loop conjugate diagrams contributing to the nEDM.In order to obtain a cancellation one needs to take into account also the pair of diagrams where ∆ 1 and ∆ 2 are interchanged.

Table 1 :
Bounds in units of M ∆ /TeV on antisymmetrically coupled diquarks.