Electroweak baryogenesis and gravitational waves in a composite Higgs model with high dimensional fermion representations

We study electroweak baryogenesis in the $SO(6)/SO(5)$ composite Higgs model with the third generation quarks being embedded in the $\textbf{20}'$ representation of $SO(6)$. The scalar sector contains one Higgs doublet and one real singlet, and their potential is given by the Coleman-Weinberg potential evaluated from the form factors of the lightest vector and fermion resonances. We show that the resonance masses at $\mathcal{O}(1\sim 10{\rm ~TeV})$ can generate a potential that triggers the strong first-order electroweak phase transition (SFOEWPT). The $CP$ violating phase arising from the dimension-6 operator in the top sector is sufficient to yield the observed baryon asymmetry of the universe. The SFOEWPT parameter space is detectable at the future space-based detectors such as LISA.


Introduction
The baryon asymmetry of the universe (BAU) is quantitively described by the baryonto-entropy ratio η B ≡ n B /s = [0.82 ∼ 0.94] × 10 −10 [1]. The explanation of BAU necessitates the three Sakharov conditions [2]: i) baryon number non-conservation, ii) C and CP violation, and iii) departure from thermal equilibrium in the early universe. In the Standard Model (SM), although the first condition can be realized via the electroweak (EW) sphaleron [3], the last two conditions are unfortunately not met. The CP violating phase from CKM matrix is too tiny, and the SM EW phase transition (EWPT) is a smooth crossover that cannot provide an out-of-equilibrium environment [4]. Therefore, the observed BAU strongly motivates new physics beyond the SM (BSM). Among various BSM mechanisms accounting for BAU, the EW baryogenesis (EWB) receives extensive attention, especially after the 125 GeV SM-like Higgs boson was discovered at the LHC [5,6]. In the paradigm of EWB, the third Sakharov condition is provided by the strong first-order EWPT (SFOEWPT), and the corresponding BSM physics is typically testable at current or future colliders [4,7]. The gravitational waves (GWs) from SFOEWPT are also hopefully detectable at the future space-based detectors [8].
There have been a lot of researches realizing EWB in the supersymmetric or nonsupersymmetric BSM models. As one of the most plausible non-supersymmetric frameworks addressing the SM hierarchy problem, the composite Higgs model (CHM) is an attractive scenario. In this framework, the hierarchy problem is solved by identifying the Higgs doublet as the pseudo-Nambu-Goldstone bosons (pNGBs) from the spontaneous global symmetry breaking G/H of a new strong interacting sector [9][10][11]. In CHMs, the SFOEWPT can be triggered by the enlarged scalar sector, either from the dilaton of conformal invariance breaking [12,13] or from the extra pNGBs of G/H breaking [14][15][16][17][18]; and the new CP phase from the fermion sector can generate BAU [12][13][14][15][16][17].
In this work we focus on the next-to-minimal CHM (NMCHM), whose coset is G/H = SO(6)/SO (5), yielding one Higgs doublet plus one real singlet [19]. It is well-known that such a scalar sector is able to generate a SFOEWPT through the "two-step" pattern, providing the essential cosmic environment for EWB [20][21][22][23][24][25][26][27][28][29][30][31]. However, unlike the normal singlet-extended SM, the NMCHM's scalar potential is generated by the SO(6)-breaking terms, which depend on the fermion embeddings in SO (6). As the fermion contribution is dominated by the top quark due to its large mass, hereafter we refer "fermion embedding" to the q L = (t L , b L ) T and t R embeddings. It has been shown that 6 and 15 representations are hard to trigger a SFOEWPT, mainly because of the smallness of the quartic couplings [17,18] 1 . The NMCHM with q L in 6 and t R in 20 has plenty of parameter space triggering the SFOEWPT since the 20 embedding can generate fairly large quartic couplings for the scalars [17]. In this article we consider a NMCHM with q L and t R both in 20 (denoted as 20 + 20 ). We will demonstrate that a SFOEWPT can be realized by the Coleman-Weinberg potential from the form factors of the lightest composite resonances, and the dimension-6 operator consists of the scalars and top quark provides sufficient CP violation for generating BAU. We also present the study of GW searches for the SFOEWPT parameter space.
This article is organized as follows. In Section 2 we build The 20 + 20 NMCHM and briefly discuss its phenomenological implications. We then use the form factors combining suitable Weinberg sum rules to generate the scalar potential in Section 3, and investigate the possibility of the SFOEWPT. The source of CP violation is considered in Section 4, where we also realize EWB and explain the observed BAU. Section 5 is devoted to the GW detectability of the SFOEWPT parameter space. Finally, we conclude in Section 6.

The 20 + 20 NMCHM
Below the confinement scale of the CHMs, the relevant physical degrees of freedom are the pNGBs and the composite resonances, and the effective Lagrangian can be written using the Coleman-Callan-Wess-Zumino (CCWZ) formalism [32,33]. In this section we only quote the main results, while the full expressions of the formulae can be found in Appendix A. For a nice introduction to the application of CCWZ in the CHMs, we refer the readers to Ref. [34].

The scalar and vector sectors
Symmetry breaking pattern is the crucial part of the CCWZ construction. For the NM-CHM, the SO(6) group contains 15 generators, which can be chosen as T A = {TĀ,T r 2 }, with TĀ being the 10 generators of the unbroken SO(5) andT r 2 being the 5 generators of the coset SO(6)/SO (5). For the convenience of later discussion about the SM gauge interactions, we further choose (5), whileT i 1 are the generators of the coset SO(5)/SO(4). The subscripts vary in the ranges (a = 1, 2, 3), (i = 1, ..., 4) and (r = 1, ..., 5).
The SO(6)/SO(5) breaking gives 5 pNGBs π = (π 1 , ..., π 5 ) T , which can be used to constructed the Goldstone matrix with f being the Goldstone decay constant. The building blocks of the CCWZ Lagrangian are the d and e symbols, which are defined by the Maurer-Cartan form as follows where gauge covariant derivative is The π as a 5 in SO(5) can be decomposed under the SM gauge group as 5 → 2 1/2 ⊕ 2 −1/2 ⊕ 1 0 , where 2 1/2 is the Higgs doublet and 2 −1/2 is just the charge conjugate of H, while 1 0 is the real singlet π 5 . The kinetic term of the pNGBs is constructed using the d symbol, i.e. L kin = (f 2 /4) tr [d µ d µ ]. To simplify the discussion, we adopt the unitary gauge by setting π 1,2,3 = 0 and redefining π 4,5 as [19] h f = π 4 π 2 4 + π 2 Then the Goldstone kinetic term becomes After EW symmetry breaking (EWSB), h gets the vacuum expectation value (VEV), and the W , Z bosons gain their masses. The T -parameter is zero at tree-level because of the custodial symmetry SU (2) V ⊂ SU (2) L × SU (2) R is preserved in the EW vacuum.
Another import feature of the NMCHM is the existence of the composite resonances. According to their spins, we can classify those resonances into the vector mesons (spin-1) and the fermionic top partners (spin-1/2). In the CCWZ framework, the composite objects form representations of the unbroken SO (5). We consider the the vector resonances in 10 and 5, and denote them as ρ µ = ρĀ µ TĀ and a µ = a r µT r 2 respectively. The Lagrangian is constructed using the d and e symbols where the strong sector coupling constant g ρ g, g , and the field strengths read is understood as a summation of resonances with the same quantum number but increasing masses, e.g.
and M ρ(n+1) > M ρ(n) . This short notation is also used in the Lagrangian the top partners (see the next subsection).
The ρ-and a-resonances decompose to multiplets under the SM gauge group [18] 10 whereρ D = iσ 2 ρ * D is the charge conjugate of ρ D , and similar forã D . The expressions for this decomposition is in Appendix A. Those vector resonances can be produced via Drell-Yan process or vector boson fusion at the LHC, and decay to a pair of light bosons (SM bosons or η), or fermions (SM quarks or top partners). The 139 fb −1 LHC data have constrained M ρ 4 TeV, provided the dominant branching ratio is the the SM di-boson (W ± Z, W + W − , etc) [35,36]. The bounds are released if other decay channels are also considerable. For example, if the decay to a pair of top partners kinematically opens, then it dominates the branching ratios and the bound on M ρ is weakened to ∼ 2.5 TeV [37]. The collider phenomenology of vector resonances in NMCHM can be found in Refs. [18,[38][39][40].

The fermion sector
The boson sector is fixed by the coset SO(6)/SO(5) thus is universal for all NMCHMs. However, the fermion sector is model-dependent. Partial compositeness mechanism says the fermions should be embedded in the incomplete representation of SO(6) and mix with the strong fermionic operators linearly [11,41], but one has the freedom to choose different embeddings and build various models. As mentioned in the introduction, embeddings in 15 and lower representations are not easy to trigger a SFOEWPT, while in this article we propose a novel scenario in which q L and t R are both embedded in the high dimensional representation 20 .
There are three dimension-20 representations for SO(6) [42], while 20 is the one obtained by 6 ⊗ 6 = 1 ⊕ 15 ⊕ 20 , i.e. the traceless symmetric representation. To provide the correct hypercharge for the fermions, an additional U (1) X must be introduced and Y = X + T 3 R . To see the structure of the 20 , we list below the decomposition chain under There are two 2 1/6 inside the 20 , coming from the 14 and 5 representations of SO(5), respectively. Therefore, there are two ways to embed q L , namely The general embedding is the superposition of them On the other hand, there are three 1 2/3 in 20 , coming respectively from the 14, 5 and 1 of the SO(5) subgroup and yielding three embeddings: where σ a are the Pauli matrices. The general embedding of t R is then (2.14) We consider the top partners with X = 2/3 and in 1, 5 and 14 representations of SO (5). The Lagrangian of top partners is where Ψ 14 and Ψ 5 are respectively 5 × 5 and 5 × 1 matrices, and where c 14,5 are O(1) numbers. Those vertices imply the vector resonances can decay to a pair of top partners (if kinematically allowed). Due to the large coupling g ρ , once opened those channels will dominate branching ratio quickly [37]. The interactions between the SM quarks and top partners are connected by the Goldstone matrix, and from which we get a set of vector-like quarks (VLQ) with electric charges varying from 8/3 to −4/3 with a step size of 1. Again, the full expressions of the decomposition are given in Appendix A. While the VLQs with exotic charge 8/5, 5/3 or −4/3 are already in their mass eigenstates, the ones with charge 2/3 and −1/3 mix with the SM third generation quarks after EWSB, and mass eigenstates should be extracted by diagonalizing the mass matrices. The SM bottom quark remains massless after such a diagonalization, because we don't include b R yet in Eq. (2.18). On the other hand, b L mixes with the VLQs with charge −1/3. For example, implying the b L -N −1/3 and b L -Y −1/3 mixing after EWSB, where N −1/3 and Y −1/3 denote the charge −1/3 component of the N and Y triplet, respectively. Such a mixing changes the coupling between left-handed fermion and the Z boson, which is for a fermion with third-component weak isospin T 3 L and charge Q. the Zb LbL coupling, which is unacceptable because this vertex has been measured at the LEP at a very high accuracy [43,44]. One proper way to avoid this problem is to choose θ L = 0 in the q 20 L embedding of Eq. (2.12), and require η = 0 at zero temperature. The mixing terms in Eq. (2.21) then vanish. That means we use q 20 The top partners can be produced at the LHC either in pair via QCD or singly via EW fusion, and finally decay to a SM fermion plus boson(s) (e.g bW + , tW + , tη, etc). Searches for the pair production VLQs with charge 5/3 or 2/3 have set limits of M 5 , M 14 1.3 TeV at the LHC with an integrated luminosity of ≈ 36 fb −1 [46,47], while the bounds from single production are typically weaker [48,49]. The Ψ 1 mainly decays to tη via the term y 1 Rt R Ψ 1 η ⊂ L qΨ , the constraints can be as weak as M 1 1 TeV [50]. About the collider phenomenology of the VLQs in the CHMs, see Refs. [38][39][40][50][51][52][53] for the charge 5/3 and 2/3 ones and Ref. [54] for the charge 8/3 one (coming from the K triplet).

Calculating the scalar potential
The potential in NMCHM is generated by two kinds of SO(6)-breaking terms, i.e. the gauge interactions and the partial compositeness terms. Each kind of sources can be further classified into the IR contributions, coming from the Coleman-Weinberg potential driven by the one-loop form factors of the leading operators of the resonances in Eq. (2.7) and Eq. (2.18); and the UV contributions, coming from the local operators generated by the matching of physics at the cut-off scale [55]. The UV contributions are not calculable but only estimated by the naïve dimensional analysis (NDA) [56]. However, if the UV contributions are negligible, then the potential can be calculated by the Coleman-Weinberg mechanism and expressed as a function of the resonance masses and couplings. This is the so-called minimal Higgs potential hypothesis (MHP), which is generally adopted in the collider phenomenology studies of the CHMs [51,55,[57][58][59]. In the aspect of cosmological implications, however, NMCHMs with fermion in 15 and lower representations cannot give a SFOEWPT under the MHP, due to the small quartic couplings in the potential [18]. In this section, we demonstrate that the 20 + 20 NMCHM is able to trigger a SFOEWPT, as the quartic coefficients are enhanced in the high-dimensional representation.
We first discuss the fermion-induced part of the potential. Integrating out the top partners in Eq. (2.18), the form factors in momentum space are The mixing terms between bL and the charge −1/3 top partners from SO(4) bi-doublets (such as Q or JQ) are safe because the PLR symmetry protects the ZbLbL vertex [45].
where p is the momentum, while Π q,t 0,1,2 and M q,t 0,1,2 are p 2 -dependent form factors. Substituting Eqs. (2.12) and (2.14), Eq. (3.1) reduces to The form factors for left-handed quarks are where we have applied the θ L = 0 condition in the q 20 L embedding due to the constraint from Zb LbL . The form factors involving t R are a little bit complicated, since the t 20 R is a superposition of three orthogonal embeddings, see Eq. (2.14). To simplify the expressions we write down the form factors in terms of t and (3.5) According to Eq. (3.2), Π t LR p 2 =0 is the top quark mass. It can be read from Eq. (3.5) that only the t 20 B R embedding gives a massive top when η = 0, i.e.
Since η = 0 is needed for a SM-like Zb LbL , we conclude that the t 20 R embedding must have a non-zero t  Given the form factors, the fermion-induced Colman-Weinberg potential is where Q 2 = −p 2 is the Euclidean momentum, and N c = 3 is the SM color number. Since f is constrained to be 800 GeV by the current EW and Higgs measurements [60,61], we expect h 2 , η 2 f 2 at temperatures around and below the EW scale. Therefore, expanding Eq. (3.7) to a polynomial of h 2 /f 2 and η 2 /f 2 is a reasonable approximation. Hence we match Eq. (3.7) to 8) and the coefficients are where the basic integrals are defined as (3.10) Now the coefficients in Eq. For a QCD-like theory, the form factors can be explicitly written as a sum of the resonance poles , (3.11) and where N 1,5,14 stand for the number of top partners in corresponding representations of SO(5). In general, at large Q 2 the form factors scale as Π q,t 1,2 , M t 1,2 ∼ Q −2 . That means the in Eq. (3.10) the integrals are convergent, while the α and β integrals diverge quadratically and logarithmically, respectively. Inspired by the successful experience in QCD [41,[62][63][64][65], people apply the Weinberg sum rules to the form factor integrals in CHMs to get a finite scalar potential [18,51,55,[57][58][59] 3 . We will also adopt this assumption here. The convergence of the α and β integrals requires Π q,t 1,2 ∼ Q −6 , which can be achieved by imposing the following sum rules  1,2 ≡ 0 and hence α q,t 1,2 = 0 and β q,t 1,2,12 = 0, which, after substituting into Eq. (3.9), gives h = −µ 2 h /λ h = f / √ 2. This is obviously inconsistent with the EW measurement. The next-to-minimal (N 14 , N 5 , N 1 ) contents are also ruled out based on the following considerations: the (1, 1, 2) gives λ h = 0 thus EWSB cannot be triggered; the (1, 2, 1) implies λ η < 0 thus the potential is not bounded below; the (2, 1, 1) is very likely to have µ 2 η > 0 and the necessary condition of SFOEWPT is not satisfied. Finally, we find the next-to-next-to-minimal setup (N 14 , N 5 , N 1 ) = (2, 1, 2) has the potential to trigger a SFOEWPT. In this case, the sum rules reduce to where we denote the two top partners in 14 as Ψ 14 and Ψ 14 , with the latter being the heavier one. Similar notation also applies to Ψ 5 and Ψ 5 . Eq. (3.14) implies For the form factors we have .
where Π 0,1 are p 2 -dependent form factors, and A µ is the gauge field defined in Eq. (2.3). The transverse projection operator is P µν T = g µν − (p µ p ν )/p 2 . The unitary gauge simplifies Eq. (3.17) to from which we can read the Higgs potential as [11] V g (h) ≈ 3 2 where Q 2 ≡ −p 2 and Π W = Q 2 + Π 0 , Π B = Q 2 + (g 2 /g 2 )Π 0 . Note that no potential for η is generated, because the gauge interactions don't break the U (1) η subgroup of SO(6) [19]. Expanding Eq. (3.19) up to h 4 level gives a very good approximation since the higher order terms are suppressed by g 2 h 2 /f 2 . Hence we can write where (3.21) Similar to the fermion-induced case, the form factors Π 0,1 are the sum of the vector resonance poles [41] where f ρ(n) ≡ M ρ(n) /g ρ(n) , and similar for f a(n) . To get a convergent µ 2 g and λ g , we impose the Weinberg first and second sum rules so that the scaling of Π 1 is changed to Q −4 . Assuming the lightest resonances dominate, i.e. N ρ = N a = 1, the rules reduce to which give , (3.25) and then the integral in Eq. (3.21) can be evaluated analytically [18,55]

The field shift for a physical Higgs boson is
where the factor involving f is due to the higher order operators in the Goldstone kinetic term, i.e. Eq. (2.6). The potential is shifted to from which we can read the tree-level physical masses of the scalars Since v 2 f 2 , the observed M h = 125.09 GeV and v = 246 GeV almost fix µ 2 h and λ h . The scalar interacting vertices are also obtained easily.
At the LHC, the singlet η can be produced by gg fusion via the SM quarks/top partners triangle loop, or from the decay of Higgs or composite resonances (e.g. ρ D , Ψ 1 , etc). The possible decay channels of η are model-dependent, including the SM di-boson (induced by the WZW anomaly [19]) and di-jet (gluon or quark). The η can even be a dark matter candidate if it has an odd Z 2 quantum number [59,[66][67][68]. Note that although our potential V (h, η) and the third generation fermion couplings are both symmetric under η → −η, a Z 2 -breaking term can generally arise from the WZW anomaly or the fermion embeddings of quarks in the first two generations or leptons. As long as M η > M h /2 so that the Higgs exotic decay h → ηη is kinematically forbidden, the direct search bounds on η are not very strong. A scalar of M η ∼ O(100 GeV) is still allowed [50,[69][70][71].

SFOEWPT
Thermal corrections to the potential can be derived using the finite temperature field theory. Since the vector and fermion resonances are at the O(TeV) scale, at temperature around EW scale they can be integrated out and we only deal with the SM degrees of freedom plus the singlet η. Keeping only the leading T 2 terms, the scalar potential at finite temperature is 4 where and g ( ) and y t are the EW gauge couplings and top Yukawa, respectively. The thermal potential in Eq. (3.30) can trigger a so-called two-step cosmic phase transition, in which the first-step is a second-order phase transition along the η direction, while the second-step is a first-order EWPT via the VEV flipping between the η-and h-axises. In short, when T decreases from EW scale to 0, the VEV ( h , η ) changes as (0, 0) → (0, w) → (v, 0). Such an EWPT scenario has been extensively studied, thus we only give the main results here, and put the details into Appendix B. The necessary condition for the two-step phase transition is [18] Once it is satisfied, we can resolve the critical temperature T c , at which the potential has two degenerate vacua (0, w c ) and (v c , 0). The first-order EWPT happens at T < T c , and completes at the nucleation temperature T n defined by where S 3 is the Euclidean action of the O(3) bounce solution [72]. If the Higgs VEV at T n further satisfies v n /T n 1, (3.34) then the EW sphaleron process is suppressed inside the bubble [73][74][75], and hence the generated baryon number is not washed out. This is essential for EWB. A first-order EWPT satisfying Eq. (3.34) is called a SFOEWPT. As Section 3.1 has expressed V (h, η) as a function of the resonance masses and couplings, realizing SFOEWPT in the 20 + 20 NMCHM is just to find the parameter space that generates a V (h, η) satisfying Eqs. and evaluate y 5 R , y 14 L,R , y 14 L,R , y 1 L,R and y 1 L,R (all treated as real numbers in this section) by the Weinberg sum rules Eq. (3.14) and the requirement of top mass M t = 150 GeV (the running mass at TeV scale [76]). Then the fermion-induced potential is calculated by performing the Q 2 integral for the form factors in Eq. (3.9). The gauge-induced part, which is determined by M ρ and M a in Eq. (3.26), is derived by requiring the Higgs and W boson masses to be the experimentally measured ones, i.e. M h ≈ 125 GeV, M W ≈ 80.4 GeV [1]. By this procedure, given a set of parameters in Eq. (3.35), one gets a scalar potential V (h, η) reproducing the SM particle mass spectrum. After that, we use the MultiNest package [77] combining with the CosmoTransitions [78] package to calculate S 3 and check whether the SFOEWPT is triggered. As shown in Fig. 1, SFOEWPT can be achieved by M η ∼ O(100 GeV), M ρ,a ∼ O(1 ∼ 10 TeV) and M 14,5,1 ∼ O(TeV). The magnitudes of the mixing parameters y L,R are smaller than 5, while 1 < g ρ < 4π. We have also checked that including the higher order expansions (e.g. h 6 , η 6 , etc) in the Coleman-Weinberg potential only gives 2% corrections to the VEVs at T c or T n . This confirms the validity of our treatment that keeps only the terms up to quartic-level.

Electroweak baryogenesis
Previous sections have demonstrated that the 20 +20 NMCHM can trigger the SFOEWPT for a large range of parameter space. In this section, we study the CP non-conservation sources and calculate the BAU. In Section 3.2, while deriving the parameter space for SFOEWPT we treated the couplings (e.g. y 14 L,R ) as real numbers. However, in general they can be complex. Omitting the complex phases in the fermion couplings is valid for the SFOEWPT study because CP violation only has a minor impact on the phase transition dynamics. But in the study of BAU, those phases are crucial. In Eq. At the EW scale, after integrating out the top partners, the CP phases manifest themselves as the complex Wilson coefficients of the operators, are complex numbers. For later convenience, we parametrize Eq. (4.1) as where y t = √ 2M t /v is the top Yukawa coupling, and ρ t and φ 1,2 are real numbers derived from the y L,R coefficients. The phase φ 1 can always be absorbed by the redefinition of t R , while φ 2 is the physical phase that characterizes the magnitude of CP violation. In this scenario, the CP non-conservation comes from the dimension-6 operator ihη 2t γ 5 t where the constraints from the electric dipole moment (EDM) measurements are weak due to no mixing between h and η at tree or loop level. This is different from the dimension-5 operator ihηtγ 5 t in previous studies [14][15][16][17] where the mixing between h and η arises after integrating out the top quark, and then the CP phase suffers from server constraints from EDM measurements [4], especially the measure of electron EDM by ACME [79].
During the SFOEWPT, h and η are treated as spacetime-dependent background fields. In the rest frame of the bubble wall, the profiles of the scalars are just the bounce solutionŝ h(r) andη(r) (see Appendix B for the details), with r being understood as the distance to the center of the vacuum bubble. Typically, the profiles have a kink shape with a wall width L w ∼ 3/T n in the 20 + 20 NMCHM. When the bubble radius R grows up to L w , near the wall one can treat the profile as a one dimensional problem with the the coordinate origin being stabilized at the wall center, and the z axis perpendicular to the wall. The bubble profiles are thenĥ(z) andη(z).
The bubble wall is "thick" in the sense that L w p −1 z , where p z ∼ T n is the typical magnitude of the z-component momentum of particles in the thermal bath. The CP violating interactions nearby the bubble wall create a chiral asymmetry, which is then converted into a baryon asymmetry via the EW sphaleron process, and swept into the bubble when the wall passes by. Inside the bubble, the sphaleron process is frozen by v n /T n 1, thus the baryon asymmetry survives, yielding the observed BAU [4]. This is the non-local EWB mechanism proposed by Refs. [80,81], and we will apply it to the 20 + 20 NMCHM case in this work 5 .
Technically, we adopt the framework of Ref. [83] to calculate the BAU 6 . First, we substitute the bounce solutions and rewrite Eq. (4.3) to the following "complex mass" form L qΨ ⊃ −m tt e iγ 5 θt t, where m t and θ t are z-dependent functions determined by the bounce solutions  Figure 2. The bubble profiles and BAU from benchmarks B1 and B2. The gray band in the right panel stands for the observed BAU from the Big Bang nucleosynthesis [1].
The excess of t L against t R is calculated by a set of coupled Boltzmann equations, see Ref. [83,85]. The BAU is generated by integrating over the region in the EW unbroken phase [83,86] where Γ ws ≈ 18 α 5 W T n is the EW sphaleron rate outside the bubble [3], g * ∼ 100 is the number of relativistic degrees of freedom at T n , µ B L (z) is the chemical potential of the left-handed quarks (all three generations), and v w is the bubble expansion velocity relative to the plasma just in front of the bubble wall. Due to the lack of a detailed simulation of the hydrodynamics in the plasma, we use v w = 0.01 and 0.1 as two benchmarks.
Given the bubble profiles and the CP phase φ 2 , η B is evaluated straight forward using the equations in Ref. [83]. We confirm that the observed BAU can be reached using the SFOEWPT parameter points derived in Section 3.2. To illustrate this, we select two benchmarks as listed in Table 1. The bubble profiles of the benchmarks are plotted in the left panel of Fig. 2, while the generated BAU are plotted in the right panel as functions of the φ 2 for different v w choices. We see that φ 2 0.15 gives the observed BAU, and increasing v w enhances the baryon asymmetry.

Gravitational waves
An important consequence of the SFOEWPT is the stochastic GWs. For a SFOEWPT that happens at T n ∼ 100 GeV, the frequency of the GW signal peak is typically mille-Hz after the cosmological redshift [87], within the sensitive signal region of a set of nearfuture space-based GW detectors, such as LISA [88] and its possible successor BBO [89], TianQin [90,91], Taiji [92] or DECIGO [93,94]. The phase transition GWs result from three sources, i.e. collision of the vacuum bubbles, sound waves in the fluid, and the turbulence in plasma. The spectrum of the GWs is described by where ρ c is the critical energy density in the present universe. For the GWs induced by the first-order cosmic phase transition, the spectra can be written in numerical functions of three parameters [87,95]: 1. α, the ratio of EWPT latent heat to the energy density of the universe at T n : here "∆" denotes the difference between the true and false vacua. Larger α produces stronger GWs.
2. β/H n , where β −1 is the time duration of the EWPT, while H n is the Hubble constant at T n , i.e.
with t n being the cosmic time at T n . The smaller β/H n is, the longer EWPT lasts and the stronger GWs are produced.
3.ṽ w , defined as the wall velocity with respect to the plasma at infinite distance. Note thatṽ w can be significantly different from v w [96], which is the relative wall velocity to plasma in front of the wall (defined in Section 4). v w is relevant for baryogenesis, whileṽ w is important in the GWs strength calculation. We adoptṽ w = 0.1, 0.3 and 0.5 as benchmarks.
Using the numerical results in Ref. [95], we can express the GW signal strengths in terms of α, β/H n andṽ w . For the benchmarks we consider, the dominant source of the GWs is the sound waves [95] 7 . We calculate α and β/H n for each parameter points with SFOEWPT, and show the results in Fig. 3. The nucleation temperature T n is shown in color. To investigate the sensitivity of LISA to the GWs, we evaluate the signal-to-noise ratio (SNR) defined as follows [95] The detailed studies on the sound waves from a SFOEWPT can be found in Refs. [97,98].
�� �� �� �� ��� Figure 3. Distributions of α, β/H n and T n for parameter points with SFOEWPT. The LISA reach is shown at differentṽ w benchmarks. The EWB benchmarks in Section 4 are highlighted as stars.
where Ω LISA is the sensitive curve of the LISA detector [88], and T is the data-taking duration, which is taken to be 75% × 4 years, i.e. 9.46 × 10 7 s [99]. Following Ref. [95], we adopt SNR = 10 as the detection threshold of LISA and plot the bounds for different benchmarks ofṽ w in Fig. 3. Largeṽ w yields higher reach, as expected. The parameters points detectable within LISA have T n 85 GeV. This is because lower T n typically yields larger α, which is helpful for the detection. TianQin and Taiji may provide a search complementary to LISA, and we leave the quantitive study of those two detectors to a future work.

Conclusion
In this paper, we studied EWB in the SO(6)/SO(5) CHM, i.e. the NMCHM. The scalar sector contains one Higgs doublet H and one real scalar η, and the concrete form of potential depends on the fermion embeddings in SO (6). In this work we considered the third generation quarks q L = (t L , b L ) and t R both in the 20 . According to the decomposition of SO(6) × U (1) X → SU (2) L × U (1) Y , there are three and two ways to embed q L and t R , respectively. To protect the Zb LbL vertex, the specific embedding q The scalar potential V (h, η) is derived using the one-loop Coleman-Weinberg potential of the form factors from the lightest resonances ρ, a and Ψ 14,5,1 . Making use of the Weinberg sum rules, the form factor integrals are convergent and a finite V (h, η) is evaluated as a function of the resonance masses and couplings. With the help of numerical tools, we found a lot of parameter points that give the SM particle spectrum and the SFOEWPT. The real singlet mass is O(100 GeV), while the vector and fermion resonance masses are typically O(1 ∼ 10 TeV), thus they are hopefully probed at the LHC. To our best knowledge, this is the first composite Higgs model that succeeds to trigger the SFOEWPT completely via the Coleman-Weinberg potential contributed from the resonances. At the EW scale, the new CP violating phase φ 2 arises from the complex Wilson coefficient of a dimension-6 operator ihη 2t γ 5 t in the top sector. The observed BAU can be explained by suitable value of φ 2 using the non-local EWB mechanism. Also, a considerable fraction of the SFOEWPT points give detectable GW signals at the near-future LISA detector. and e 1 Rµ = and Now we turn to the resonances. The full expressions of the vector resonances decomposition in Eq. (2.9) are [18] where K (3,3) , J (2,2) and T are in (3,3), (2, 2) and (1, 1) of SO(4), respectively. Under the SM gauge group, K (3,3) further decompose to three SU (2) L triplets with hypercharges 5/3, 2/3 and −1/3, while J (2,2) decomposes to two SU (2) L doublets with hypercharges 7/6 and 1/6. Explicitly, they are are the multiplets with SM quantum number 3 5/3 , 3 2/3 and 3 −1/3 respectively; and where are the multiplets with SM quantum number 2 7/6 and 2 1/6 respectively. Another top partner Ψ 5 is written as , (A. 13) and one singlet T with hyper charge 2/3 are present.

B Conditions for SFOEWPT
In this appendix we give a brief description for the two-step phase transition. Similar discussions can also be found in Refs. [17,18]. We starts with the thermal potential V T (h, η) in Eq. (3.30). The bounded below condition is By tracking T -dependence of the vacuum, we can describe the cosmic phase transition and find the corresponding parameter space. At the beginning, when T the EW scale, the vacuum should be ( h , η ) = (0, 0). This requires c h > 0 and c η > 0. When T decreases to below some temperature T η , the mass term for η becomes negative while for h is still positive, and hence a VEV exists along the η direction, i.e. (0, w T ). This means which requires the coefficients to satisfy Keep lowering T , and when it is below −µ 2 h /c h , the Higgs mass term flips sign and the h direction also acquire a local minimum (v T , 0). At first, V T (h, η) at (0, w T ) is deeper than it at (v T , 0), so that the vacuum is still at the η-axis. However, when T reaches the critical temperature T c , the two local minima have the same depth, and we get the two degenerate vacua separated by a barrier. This can happen if and At T < T c , the h-vacuum is the deeper one that it becomes the true vacuum. The universe then decays from (0, w T ) to (v T , 0) via a first-order EWPT. After the EWPT, the vacuum keep shifting until T → 0 and (v T , 0) → (v, 0), where v = −µ 2 h /λ h = 246 GeV. Note the η direction still acquires a local extremum (0, w) where w = −µ 2 η /λ η . To make sure (v, 0) is the true vacuum, an additional condition is attached All the inequalities derived above can be summarized in a compact form in Eq. (3.32), which is the necessary condition for the two-step phase transition: the first step is a second-order transition along the η direction, happens around T η 8 ; while the second step is a first-order EWPT via the VEV flipping between the η-and h-axises, happens around T c . Eq. (3.32) is not sufficient for a first-order EWPT. Below T c , the universe starts to decay to the true vacuum via bubble nucleation, i.e. the bubbles containing true vacuum emerge in the universe and expand, collide and merge, until finally fulfill the whole universe. The nucleation rate per volume is [72] Γ/V ∼ T 4 S 3 (T ) 2πT where M Pl = 1.22 × 10 19 GeV is the Planck scale and g * ∼ 100. Eq. (B.13) is dominated by the exponential factor and the Planck scale. Numerically, it is approximately S 3 (T n )/T n ∼ 140 for T n ∼ 100 GeV, as stated in Eq. (3.33). We treat Eq. (3.33) as the critical condition of a first-order EWPT, namely if a T n can be solved from the equation then the EWPT can happen, otherwise the nucleation is not efficient compare to the cosmic expansion and the universe will stay in the false vacuum. Normally, T n is a little bit smaller than T c .