Dark top partner

Composite Higgs models with extended symmetries can feature mesonic dark matter candidates. In fundamental CHMs, the origin of dark parity can be explained in the UV theory. Combined with top partial compositeness, this leads to non-chiral Yukawa interaction connecting mesonic DM with one dark top partner and one SM top. We examine the DM phenomenology in SU(6)/SO(6) and SU(6)/Sp(6) CHMs with the presence of dark top partners. Phenomenological constraints require the mass of top partner in even parity to be of the multi-TeV order.


Introduction
With the discovery of the Higgs boson at a mass of 125 GeV at the Large Hadron Collider, the particle content predicted by the Standard Model (SM) is complete.Yet, this does not close the door to the presence of New Physics.Arguably, the presence of a Dark Matter (DM) component in the present-day universe is the most compelling evidence for new physics [1], as no part of the SM can provide a particle candidate for it.
Models of new physics based on a strong confining dynamics can explain both the naturalness of the Higgs mass and the presence of Dark Matter.In fact, both the Higgs and a set of stable new particles may emerge as composite states.The naturalness of the scale is related to its dynamical origin, in the same spirit as the QCD scale.Composite models applied to the electroweak (EW) scale are as old as the SM itself [2], where the Higgs boson can be associated to a light pseudo-Nambu-Goldstone boson (pNGB) via vacuum misalignment [3].Finally, an attractive mechanism to generate the top quark mass may be related to the presence of spin-1/2 operators with linear couplings to the elementary top fields in the SM.This leads to the idea of top partial compositeness [4].This old idea has been revamped in the 2000's thanks to the discovery of holography [5], based on the idea of walking dynamics [6].Realistic models have been constructed based on underlying gauge-fermion theories, leading to a limited number of combinations of gauge groups and fermion representations [7,8].In such scenarios, a DM candidate can emerge as an additional pNGB that accompanies the Higgs boson [9].In this direction, many symmetry breaking patterns have been examined in the literature [10][11][12][13][14][15][16][17].
Fundamental composite dynamics models [18,19], based on gauge-fermion underlying theories, however, limits the symmetry breaking patterns that can be realised.Depending on the nature of the fermions representations under the confining groups, we have: SU(N ) 2 /SU(N ) for complex representations, with minimal N = 4; SU(2N )/Sp(2N ) for pseudo-real, with minimal N = 2; SU(N )/SO(N ) for real, with minimal N = 5.In the SU(4)/Sp(4) and SU(5)/SO(5) CHMs, although the Higgs is accompanied by additional pNGBs, the CP-odd one decays via the topological anomaly [20,21].For the last two types of cosets, DM candidates emerge in the extensions of SU(6)/Sp (6) [13] and SU(6)/SO (6) [14,22].As discussed in this paper, in the real and pseudo-real realizations the dark parity of composite states naturally originate from a Z 2 parity carried by additional fundamental fermions.
In this work, we will reexamine these two cases by including the effect of top partners in the properties of the DM candidates.Hence, the results presented here complete and complement the literature.For the first time, we give the explicit embeddings of top partners in two SU (6) CHMs.In particular, we discuss the effect of non-chiral Yukawa interaction from the dark top partners on the DM relic density and the direct detection constraints.The models we consider are predictive as the possible choices of partial compositeness couplings is strongly limited by the requirement of preserving the dark parity, which keep the DM pNGBs stable.The dark top partners can also be produced at hadron colliders, like the LHC, as they will typically decay into a top quark plus a DM candidate, leading to missing transverse energy signatures.It has been shown that searches for supersymmetric tops effectively cover this signature [23], with the limits on the dark top partners mainly stemming from the larger production rates as compared to the stops.
The article is organized as follows: after discussing the basic features of the models and the origin of the dark parity in Section 2, we present the dark top partners in Section 3. In Section 4 we discuss the limits on the models stemming from electroweak precision tests, before discussing the impact on DM in Section 5. Finally, we present our conclusions in Section 6.

Dark parity in Fundamental Composite Higgs Models
In models with a fundamental composite dynamics, the global symmetry is broken due to the phase transition of a strong gauge dynamics.The condensation of Hyper-Color fermions leads to composite pNGBs and top partners.For the HC fermions ψ in the pseudo-real and real representations, no dark matter candidates are allowed in the minimal cosets of SU(4)/Sp(4) and SU(5)/SO(5) respectively.For these two types of realizations, a dark matter candidate can be generated by extending the content of HC fermions.We will consider here models containing the minimal set of fermions to generate a pNGB Higgs, and enlarge with necessary number of dark fermions.As the dark fermions are not compositions of the pNGB Higgs, they can be odd under a Z 2 parity.With an appropriate embedding, the Z 2 symmetry will be preserved after the HC fermion condensation, as long as the spurions of the electroweak gauge and Yukawa interactions are invariant under this parity.Therefore, this dark parity emerging as an EFT symmetry is in fact associated to the HC fermions in the UV theory.
In particular, the dark parity is conserved in the Wess-Zumino-Witten (WZW) topological term [20,21] in the EFT.We start with the pNGBs embedded in the coset of G/H: where U α = exp(i √ 2αX h ) with X h being Higgs generator is the rotation matrix that misaligns the vacuum and the generators are normalized as Tr X i X j = δ ij (real) and Tr X i X j = 1 2 δ ij (pseudo-real).However unlike QCD, there is a misalignment effect in the composite sector.Here we will focus on the real and pseudo-real types of realizations, where the vaccum along the EW symmetry breaking direction is defined as Σ α = U α Σ EW U T α .And in these two cases, using the differential form approach [24], the WZW Lagrangian can be derived to be: ), B µ and g a being the corresponding gauge couplings.The d ψ is the representation dimension of HC fermion ψ in the UV gauge theory.And the exact coefficients for anomaly terms in SU (6)/SO(6) CHM were first calculated in [14].Note that the dark parity needs to be a good symmetry after the vacuum misalignment, i.e.Ω DM U α Ω DM = U α .Hence under the Z 2 parity operation, the rotated Π matrix transforms as Ω DM Π(π odd )Ω DM = Π(−π odd ), where only the odd parity pNGBs change to be in the opposite sign.In addition, because the W i µ , B µ gauge bosons are even fields under this parity, the dark parity will prevent the odd parity pNGB from decaying via WZW terms.
For the complex realization of CHM, in principal one can also define the dark parity in terms of the HC fermion operators.While we can understand the Z 2 parity for the breaking pattern SU (N ) l × SU (N ) r /SU (N ) v from an EFT perspective.In this scenario, the dark parity is equivalent to exchange the flavor groups SU (N ) l ↔ SU (N ) r .Defining X i as the SU(N) generators, the parity with the property of Ω DM X i,T Ω † DM = ±X i is in fact the EW preserving vacuum that breaks SU(N) → SO(N) or Sp(N).As a concrete example, the odd parity pNGBs in SU (5) 2 /SU (5) CHM decompose as (2, 2) ⊕ (1, 3) ⊕ (3, 1) under the EW SU (2) L × SU (2) R symmetry.Due to this parity, the WZW term structure in SU (N ) 2 /SU (N ) is the same as in SU(N)/SO(N) or SU(N)/Sp(N).In the following, we will mainly illustrate the features of the resulting two minimal models with dark matter: SU(6)/SO (6) and SU(6)/Sp(6).

SU(6)/SO(6)
The minimal set of fundamental fermions consists of two doublets with opposite hypercharge and a singlet.Hence, we include a second singlet, odd under the dark parity.The fermion content is illustrated in Table 1.Upon condensation, the symmetry breaking pattern SU(6)/SO (6) emerges, leading to 20 pNGBs.A subset of them will be odd under the dark Z 2 , hence playing the role of Dark Matter candidates [14,22].We summarise here their main features, in preparation for the inclusion of top partners.The SU (2) L × SU (2) R generators that are invariant under Z 2 are: that satisfy the condition S i L/R Σ EW + Σ EW S iT L/R = 0, with the EW preserving vacuum to be: (2.4) The Π matrix in the SU(6)/SO(6) CHM can be written as where the columns and rows correspond to the fermions in Table 1.We can see that the first doublet H 1 is composed of HC fermions (ψ D1 , ψ D2 ) and ψ S1 in even parity.The matrices φ and Λ represent the triplets, transforming as a bi-triplet of the custodial SU(2) L ×SU(2) R symmetry, like in the SU(5)/SO(5) model [25].The two Higgs doublets read: where the second doublet stems from (ψ D1 , ψ D2 ) and the odd parity singlet ψ S2 , while η i , i = 1, 2, 3 are gauge singlets.Note that the Goldstone bosons G 1,2,3 inside the first doublet H 1 are eaten by W1,2 µ and Z µ respectively 1 .Following the Hyper-Color fermions in Table 1, the dark parity in the EFT is defined as that acts on the pNGB matrix Σ = U Π Σ α U T Π in the following way [14]: hence the Z 2 -odd pNGBs are the second doublet H 2 and the singlet η 2 .We will discuss the dark parity of the top partners of this model in the next section.Since the top partners normally are much heavier than the pNGBs, they mainly participate in the pNGB DM production as calculated in Section 5.

SU(6)/Sp(6)
For pseudo-real realization, the minimal HC fermion content consists of one SU (2) L doublet ψ L and two singlets with opposite hypercharge, forming a doublet of the global SU(2) R1 .When one extends this type of model, the dark sector needs to consist of two additional Z 2 -odd fermions.In order to avoid pNGBs with semi-integer charges and gauge anomalies, these states must have opposite semi-integer hypercharges.Here, we follow the minimal choice as shown in Table 2.The odd and even singlets can be organized as doublets of two custodial SU(2) R symmetries, where the first acts on the SM-like Higgs, while the second acts on the odd doublet.
The pNGB in the SU (6)/Sp(6) CHM was investigated in [13] and the generators of (2.9) that obviously preserves the dark parity.In Table 2, the gauged SU (2) L group is the first flavor SU (2) 1 subgroup and the gauged hypercharge U (1) Y is defined as2 : i.e. the sum of the two diagonal generators of the global SU(2) R symmetries.Note, however, that the abelian gauging still leaves two U(1) global symmetries unbroken, U(1) R1 ×U(1) R2 .The first is broken by the Higgs vacuum expectation value, while the second remains unbroken.Hence for the EW gauge group embedding in Table 2, the discrete Z 2 parity is actually enhanced to a dark U(1) symmetry protecting the DM.The model in Table 2 generates the coset SU(6)/Sp (6), which has 14 pNGBs, which can be expressed as in Eq. (2.1), with (2.12) with the bi-doublets explicitly to be Here we can recognize the components of the SM Higgs doublet H 1 and of a second doublet H 2 , like in the previous model.Furthermore, η ± , η3 +iη 4 √ 2 in Φ are singlets that form a bi-doublet of SU(2) R1 ×SU(2) R2 .Finally, η 1 and η 2 are singlets.Note that in terms of the HC fermions condensation, H 2 is from ψ L ψ R 2 and Φ is from ψ R1 ψ R2 .Hence in the EFT, the dark parity reads: under which the odd states are the second doublet H 2 and the singlets η ± , 1 √ 2 (η 3 + iη 4 ), i.e. the only pNGBs that transform as doublets under SU(2) R2 .Due to the enhanced U (1) symmetry, the 1 √ 2 (η 3 + iη 4 ) will be a complex DM.And like in the previous case, Ω DM will also be used to classify the top partners as discussed in Section 3.

Dark top partners and partial compositeness
Top partners are spin-1/2 composite state that generate the top mass via linear mixing with the elementary top fields [4,26].Hence, the mass eigenstate corresponding to the physical top is a mixture of elementary and composite states.This mixing plays a crucial role in determining both the vacuum misalignment and the properties of the Higgs boson and preserve dark parity in the extended models.Here we will study the compositions of top partners as condensation of HC fermions and provide their embeddings in the effective Lagrangian.In particular, the properties of the odd top partners are investigated, which were not considered before.
In fundamental composite models, the top partners are determined by the microscopic dynamics of the confining interactions.In general terms, top partners are resonances associated to some spin-1/2 operators made of the HC fermions.As such, they transform coherently under the unbroken global symmetry in the confined sector.In the models of Refs [7,8,27], such operators are built of two species of fermions: the electroweak ones ψ, which we introduced in the previous section, and a new species χ, which carries QCD charges and the appropriate hypercharges.The χ sector will be extended in order to accommodate the top partners and we will follow the nomenclature of models introduced in Ref. [28].
For the two SU(6) CHMs, the top partners correspond to the operators of ψψχ, ψ ψχ and ψ ψ χ that are singlets under the HC gauge group.In Table 3, the top partners are classified according to their transformation property under the unbroken flavor groups of SO(6) × SO (6) or Sp(6) × SO (6), with the second flavor group related to the QCD sector.Depending on G HC , the first two ψψχ, ψ ψχ will either be in the 2-index symmetric S 2 or antisymmetric A 2 representations of the flavor group SU (6).While the third composition ψ ψ χ contains the adjoint and singlet representations of the global SU (6).Under the unbroken group, the adjoint decomposes into S 2 and A 2 of the SO (6) or Sp (6) global symmetries in the EW sector.In the following we will study top partners belonging to the anti-symmetric, Ψ A , leaving the others for future studies.We only accompany a singlet Ψ 1 with Ψ A in the SU (6)/Sp(6) model, as ̸ dΨ 1 is symmetric in SU(N)/SO(N).Their effective Lagrangian can be written as [29]: where the two masses M A and M 1 are naturally expected to be a few times the pNGB decay constant f .The covariant derivative reads with X to be the hypercharge carried by the χ and including the misalignment effect, The dots indicate higher order terms with the presence of pNGBs.The κ terms contain the Maurer-Cartan form aligned with the broken generators, which reads with X i G being the generators for the eaten Goldstone bosons shown in Eq.(2.6) and Eq.(2.13).As discussed in [29], the CCWZ forms E µ and d µ are universal at the leading order for a generic CHM.The covariant derivative and the κ terms parameterize the 1/f suppressed interactions with the pNGBs, as well as corrections to the EW gauge couplings due to misalignment.The effect of the latter on EW precision tests has been first discussed in Ref. [29] in a minimal model.Note that the self-conjugate κ ′ term only appears in the extended models thanks to the unbroken SO (6) or Sp (6) symmetries, but it is absent in the minimal models.
The linear mixing with the elementary top fields crucially depends on the properties of the composite operator generating the top partners.While the elementary fields need to be embedded in an incomplete representation of SU (6).In this paper we choose the spurions to be in the adjoint.The matching is performed by dressing the pNGB matrix U Π in Eq. (2.1), leading to the terms of partial compositeness (PC): where Σ α is the misaligned vacuum, and D L and D R are the spurions of the left-handed doublet and the right-handed top, respectively.Their explicit form depends on the model and will be discussed below.Finally, y L and y R parameterize the strength of the linear mixing of the top partners with the elementary fields.The Lagrangian in Eq.(3.5) gives rise to the mass matrix of spin-1/2 states and determine the Yukawa couplings at the higher order, that connect the top partners to one SM field and one pNGB.The couplings involving the dark top partners relevant to DM production are listed in Appendix B.
For G HC = SO (7), the ψψχ gives rise to a two-index anti-symmetric representation 15 in SO (6).Under the custodial SU(2) L ×SU(2) R , it decomposes as The top partners carrying a single Z 2 -odd HC fermion will be odd under the operation of dark parity.Hence, the top partner field can be written as where the various components are expressed in the following form: where the tilde fields satisfy Ω DM Ψ Ω DM = − Ψ, i.e. the bi-doublet ( T , B, TX , X) and the singlet T1 are Z 2 -odd.And the other even states can mix with the top fields, according to the chosen SM spurions.
In order to conserve the DM parity, the elementary top spurions needs to be even under the DM parity Ω DM D L,R Ω DM = D L,R .Under this condition, the adjoint representation of SU(6) allows 2 possibilities for the left-handed doublet and 3 for the right-handed singlet: and (3.10) In the above, the "A" and "S" subscript indicate if the spurion is in the anti-symmetric or symmetric part.Therefore the top and bottom spurions appearing in Eq. (3.5) are: Another constraint stems from avoiding tadpoles for the pNGBs other than h 0 , as their presence would lead to a spontaneous violation of custodial symmetry.The following conditions need to be satisfied [14]: that will be satisfied given that t L and t R are paired in opposite parts (A-S or S-A) of spurions.
In fact, the adjoint is a realistic choice to allow for a tadpole free potential.Expanding (3.5) to the leading order, we obtain in the gauge basis: Note that the spurion choice fixes the structure of DM Yukawa interaction (ref Appendix B) and the mixing pattern for the even states that has impact on the EW precision observables.
Diagonalizing the resulting mass matrix, the top quark mass is given by: (3.15)

SU(6)/Sp(6)
A similar analysis can be conducted for the pseudo-real case.This scenario is realized by the confining gauge symmetry Sp(4) with ψ in the fundamental and χ in the two-index anti-symmetric and with SO (11) with ψ in the spinorial and χ in the fundamental [28].
For G HC = Sp(4), the traceless part of ψψχ corresponds to a two-index anti-symmetric 14 in Sp (6), which decompose under SU(2) L ×SU(2) R1 ×SU(2) R2 as: Explicitly, the top partners can be written in the following form with T 1 and T 2 being singlets, and three bi-doublets under the global symmetry SU (2) 3 , where ψ(2,1,2) and ψ(1,2,2) , that contain one ψ R2 in the condensations, are Ω DM -odd.And the singlet top partner involving in the interaction of T r[Ψ 14 ̸ dΨ 1 ] is: The elementary (t, b) are embedded in the spurions even under the DM parity.We can write down the options for the left-handed fields: and for the right-handed top: The general spurions are the linear combinations: Due to the DM parity, there is no tadpole for H 0 , A 0 and η 3,4 .To avoid the tadpole for η 1 , we need to impose the condition: Hence expanding Eq. (3.5) for the SU (6)/Sp(6) model, we obtain: And the top quark mass is derived to be:

Electroweak precision observables
Before discussing the DM property, we investigated the impact of the EW precision observables (EWPO) on the parameter space of the models.The leading effects can be encoded in the Peskin-Takeuchi parameters [30,31], in particular the S and T parameters, which are typically modified in all composite Higgs models by various sources.
Firstly, the vacuum misalignment via the reduced Higgs coupling to W, Z gauge bosons leads to the well-known logarithmic contribution: with Λ = 4πf .The EW triplets and inert doublet also contribute to S and T at one loop level.The S parameter from the scalars is relatively small, while the T parameter can be relatively large if there is a large mass splitting in the component fields.Both the SU(6)/SO( 6) and SU(6)/Sp(6) models feature an inert Higgs doublet, whose contribution to T can be estimated as [32]: In these CHMs, a characteristic property is that the mass splitting inside each SU (2) L multiplet is of the order of a few GeV.For example, δm 2 ∼ 10 GeV 2 leads to ∆T H ∼ 9.0 × 10 −5 , that is negligible compared with the ∆T h in the case of sin α ∼ 0.1.This also applies to the contribution from the triplet pNGB.
Another important source to EWPO comes from the top sector.First of all, all the top partners that are odd under the DM parity will not contribute to S and T .As pointed out in [29], two effects in the top sector are relevant: One stems from the rotation to the mass eigenstates, and it is usually included in the literature; The other emerges from the misalignment effect inherited in the CCWZ objects d µ and E µ , as first considered in Ref. [29].Both effects arise at the sin 2 α order.In particular, the misalignment violates the unitarity and will mix the top partners in two different multiplets.Following [29], we can simply consider a scenario where a single multiplet of top partners is involved in the partial compositeness mixing.Hence, the contribution from the basis rotation decouples from the misalignment one at the O(sin 2 α), so that two effects can be separately computed.
We investigated two mixing patterns for the top quark mass generation via partial compositeness, i.e. the bi-doublet scenario common to both SU(6) models, and the triplet (3, 1)+(1, 3) scenario in SU(6)/SO (6).For the latter, a large negative contribution ∆T ≲ −0.18 from the basis rotation pushes the top partner masses to O(10) TeV for small mass splitting in the inert doublet.Henceforth, we relegated the full results of this scenario to the Appendix C. Interestingly, the bi-doublet scenario permits masses of a few TeV order.We will first discuss the rotation effect in the bi-doublet scenario.At the zeroth order, the mixing from Eqs (3.14) and (3.25) gives: with the mass of other top partners to be M at the leading order.As the custodial symmetry is conserved up to O(sin 2 α) for the bi-doublet mixing, the T parameter received very small contributions ∆T mix = O(sin 4 α).Instead, the mixing contribution to S from a bi-doublet can be written in the universal form as follows: where the variable is and the explicit expressions of χ ± along with other well-known functions ψ ± , θ ± are presented in Appendix C. The ψ+ is a new function defined in [29]: with Y vq being the hyper-charge of the vector-like quark.In Eq. (4.5), the coefficients of the first four terms are proportional to T ∼ sin 2 α that comes from the rotation of gauge couplings, while the remaining two terms encode the contribution from the mass splitting within one representation.For the SU(6)/SO(6) coset, we find that: Instead, for the SU(6)/Sp(6) coset, only the mass difference among (T, B) changes: The misalignment effect will modify the gauge couplings as well.In contrast to the rotation effect, the S, T parameters from the misalignment receive divergent contributions in the low energy effective theory, as the consequence of unitarity violation.In the following we will focus on this effect in the bi-doublet mixing scenario.

SU(6)/SO(6)
In the SU (6)/SO(6) CHM, the misalignment is encoded in Eq.(A.2-A.3).Note that the d µ term will generate gauge interactions connecting the bi-doublet with the triplets.Substituting the zeroth order rotation of (t, T ) and (b, B) into Eq.(3.1), the custodial symmetry is broken at O(sin 2 α), and we can derive the misalignment contribution to T : where the second line is related to the divergence ( Differently from the bi-doublet case of the minimal SU(4)/Sp(4) coset, the coefficient of divergent term is definite positive for sin α ̸ = 0. Using the same approach from Ref. [29], the ∆S mis turns out to be: with standing for any top partner other than (T, B) and ∆S div including all logarithmic terms: (4.11) In the limit of sin ϕ L = 0 ( y T = y M ), similarly to the minimal SU(4)/Sp(4) case, the misalignment contribution can be largely simplified to: that vanishes for κ ′ ≃ 2/ cos α 2 .

SU(6)/Sp(6)
As we can see from Eq.(A.6-A.7), the κ and κ ′ terms in the SU(6)/Sp(6) case generate the same interactions between the components fields in the bi-doublet and one singlet T 2 (in ψ A ) or T 3 (in ψ 1 ).For simplicity, we take the degenerate limit m T 2 = m T 3 = M , which gives: This contribution enjoys the same structure as in the minimal SU(4)/Sp(4).For ∆S mis , we obtain: Figure 1: Regions satisfying the S, T precision constraints at 99% C.L. for the bi-doublet mixing scenario, with the left one for SU (6)/SO( 6) and the right one for SU (6)/Sp (6).The cyan region is for sin ϕ L = 0.8 and the region between the two red lines are for sin ϕ L = 0.3. with

.15)
In analogy to the SU(6)/SO(6) case, ∆S mis can be estimated in the limit of sin ϕ L = 0 as: which is consistent with Eq. (4.12) with the coefficient difference originating from the representation of the top partners.Note that the divergence of ∆T mis and ∆S mis simultaneously vanish at the point

EWPT bounds
We performed a χ 2 analysis for the parameter space spanned by (κ, κ ′ , m T , sin ϕ L , sin α) in the two SU(6) models, using the EW precision data provided in the Particle Data Group (2022) [33].In the bi-doublet mixing scenario, the positive contribution from ∆T mis can be large enough to compensate the negative one from ∆T h .The plots in Figure 1 display the regions permitted by the S and T bounds by assuming U = 0.The cyan band for sin ϕ L = 0.8 The couplings relevant to DM annihilation in Model A (SU (6)/SO(6) CHM) and Model B (SU (6)/Sp(6) CHM) for the bi-doublet mixing scenario, with . In the 3rd row, the partner is a Z 2 -odd heavy quark that mediates the interaction with a scalar DM and SM tops.and the region between the two red lines for sin ϕ L = 0.3 are allowed by EWPO at 99% C.L..In fact, the patterns in two plots are mainly determined by ∆T mis as the κ ′2 and (κ ′2 + 2κ 2 ) contributions have opposite signs in the two models.We find that, in order to allow κ ′ = 0 in the region of m T ∼ a few TeV, a small symmetry breaking angle is preferred for the SU(6)/SO(6) model.Instead, in the SU(6)/Sp(6) model, the sin α can be of order 0.1 and smaller under the EWPO constraint.In Figure 1, we projected the EW bounds either on the (κ ′ , m T ) plane (left panel) or on the ( √ κ ′2 + 2κ 2 , m T ) plane (right panel).Both plots show that with a larger sin ϕ L ∼ 0.8, the lower bound for the top partner can be relieved.

pNGB Dark Matter
Due to the existence of a dark parity, the odd pNGBs (S, H 2 ) and the top partners T in the composite inert Higgs model constitute the dark sector.In this paper, we consider the scenario of m T ≫ m H 2 > m S , so that the singlet S is the DM candidate.The rationale is the inert odd H 2 would most likely be excluded by direct detection bounds, due to a large Z boson coupling.The parity-odd scalars H 2 and S were kept in a thermal equilibrium with the SM particles in the early universe and froze out as the temperature drops to T f ∼ m S /25.When the mass difference between the doublet H 2 and singlet S is large enough, the co-annihilation effect will be suppressed by e −(m H 2 −m S )/T f .Hence for m H 2 > 1.2 m S , it is adequate to consider just the singlet annihilation.The Lagrangian relevant to the DM phenomenology is: with a t ≃ 1 − O(sin 2 α), g v,q = g 2 2 cos θ W T 3 L − 2Q q sin 2 θ W and g a,q = g 2 2 cos θ W T 3 L .Note that the DM is a real scalar singlet S = η 2 in SU(6)/SO(6) and a complex one S = 1 √ 2 (η 3 + iη 4 ) in SU(6)/Sp (6).The Z coupling to DM only exists for the complex singlet, and it is proportional to (1 − cos α) via the EW misalignment effect.The S † S tt term is connected to the generation of top quark mass and derived by expanding the trace part in Eq.(3.5) till the 1/f 2 order.In the mass basis, the coefficient of this quartic term can be expressed in the form of λ t mt f 2 , that matches the structure derived from an EFT approach in [22].In fact we can comprehend this operator as the effect of integrating out the even top partners in Ψ A that mix with SM tops.In Table 4, we report the dark Yukawa couplings ỹL,R from the bi-doublet mixing scenario and the quartic couplings (λ t , a V ) in the two models.For the right-handed top spurion in SU(6)/SO(6), we set R S1 = 0, R S2 = 1 for a ỹR of O(1).Since m t is generated collectively by the left and right handed pre-Yukawa spurions, two couplings ỹL,R are present in the Lagrangian to couple the DM with the dark top partners.This feature extends the simplified vector-like quark portal DM model, where only one chiral (either left or right) Yukawa coupling is taken into account [34][35][36].
The Higgs portal coupling C hS is generated by the effective potential for the pNGBs and it depends on all the parameters in the underlying theory that break the global symmetries of the strong sector.In principle, the numerical values of this coupling need to be consistent with the non-tachyon condition (i.e.m 2 pNGB > 0) and with the stability of the dark parity (i.e. the absence of a VEV for the dark pNGBs) [14,22].As this exercise is strongly dependent on the details of the model, general and assumption-free bounds cannot be obtained and we will consider C hS as a free positive parameter with a reminder that over large or small values may be unphysical.Assuming the shift symmetry is dominantly broken with dark top partners running in the loop, one would expect C hS to be of the same order of the Higgs quartic times its VEV, i.e.C hS ∼ λ h 4 v ∼ 20 GeV, as the Higgs mass is generated by the same spurions.However, this value should not be too large to conflict with DM direct detection bounds.Nevertheless, the DM candidate is not strictly correlated to the Higgs dynamics.With the cancellation from different contributions, values of C hS can be in a few GeV and even smaller in corners of the parameter space, that agrees with Ref. [11] for instance.
In this paper we will consider the DM annihilation S † S → tt, W + W − , ZZ, that is justified when the mass region of interest is 2m S < m t + m T .The main focus of this work is on the tt channel, where the top partners directly contribute.The W W and ZZ channels are included as a comparison and to prove that the tt channel dominates.The hh channel is expected to contribute at the same order as W W and ZZ ones, however it receives model-dependent contributions from the pNGB potential.Hence, to keep the focus on the top partners, we do not include it explicitly and consider its effect as an O(1) uncertainty on the W W + ZZ channels.Note that the (κ, κ ′ ) terms are neglected as well because they only entail the derivative couplings of DM at the leading order.But non-zero (κ, κ ′ ) can make the parameter space more natural under the EWPO constraint and have impact on the fractions of DM annihilation into tt and di-bosons.

Relic density computation
The DM relic density is determined by the thermal average of the annihilation cross section times the velocity ⟨σv⟩, expanded in terms of v rel .We can first calculate the un-averaged σv by summing over all the kinematically allowed channels: with where S ij is a symmetric factor for identical final states and s = E 2 tot is the Mandelstam variable, with E tot being the total energy in the center of mass frame.Due to the derivative coupling of Z-S † -S, its cross section is proportional to (p . Hence in the s-wave limit, the Z-portal interaction does not contribute to relic density.We will focus on the top Yukawa and Higgs mediating channels.Under the s-wave approximation, the thermal average of ⟨σv⟩t t in the two SU(6) models is: with β S = 2 for a real scalar (S † = S) and β S = 1 for a complex scalar, where the difference shows up in the Higgs mediating channel.From Table 4 we can see that ỹL ỹ * R can be of the same order as mt M |ỹ R | 2 , while the mt M |ỹ L | 2 term is negligible.Note that for C hS = 0 and in the limit of M ≫ m S ≫ m f , our results agree with the calculation in Ref. [37].In particular, two dark top partners ( X 2 3 , Ỹ 2

3
) participate in the annihilation of the complex scalar DM in SU(6)/Sp (6), which is equivalent to the t and u channels for the real scalar DM in SU(6)/SO (6).Thus in our models, there is no 1  4 factor difference among the real and complex cases in the top partner mediating cross section as pointed out by [37].
Similarly for the di-boson annihilation channels, at the leading order, the thermal average of ⟨σv⟩ V V with V = W, Z is: and (5.6) Without the Higgs portal interaction, the di-boson annihilation rate is roughly proportional to m 2 S and becomes larger for heavier DM.Assuming C hS = 0 and m S ∼ 200 GeV, we obtain ⟨σv⟩ W W + ⟨σv⟩ ZZ ∼ a 2 V β 2 S × 2.0 × 10 −23 cm 3 /s.For a light DM and a V < 10 −2 , the di-boson channel alone can not saturate the relic density.
From the perspective of the Boltzmann equation, the freeze-out epoch is fixed by the condition that the DM particles stop to track the equilibrium distribution.By defining x f = m S T f , this freeze-out temperature is determined as: GeV and g * ∼ 100.Since g S counts the internal DM degrees of freedom, we set g S = 2 for a complex scalar and g S = 1 for a real scalar.For a total cross section ⟨σv⟩ tot ∼ 2 × 10 −26 cm 3 /s and m S ∼ 10 2 − 10 3 GeV, the freeze-out condition gives x f ≃ 20 − 25.The relic density can be estimated using the following formula: (5.8)

Direct detection constraint
To evaluate the spin-independent DM-nucleon scattering, we can start with the effective Lagrangian in terms of gluons and quarks: where the vector interaction term is obtained by integrating out the Z gauge boson with and the vector-axial interaction is not listed as its contribution to the spin-independent cross section is velocity suppressed.For the gluon operator in the first line of Eq.(5.9), the C q term is generated from the Higgs portal triangle diagram with heavy quarks (c, t, b) running in the loop, while C g S comes from the top partner box diagram.These Wilson coefficients in the first line are given by [38][39][40]: where β S takes into account the difference of mediators in two SU(6) models.Then the coupling f p,n for DM interacting with protons or neutrons are factorized as the Wilson coefficients    The colored regions provide the correct relic density Ωh 2 = 0.12 ± 0.002, and satisfy the direct detection constraint, with the upper panels for SU(6)/SO (6) and the lower panels for SU(6)/Sp (6).The red line is the EWPO bound at 99% C.L. with κ = κ ′ = 0. in Eq.(5.9) times the nucleon matrix elements.Expanding in the non-relativistic limit, the spinor structures inside ⟨N |qγ µ q|N ⟩, ⟨N |qq|N ⟩ and ⟨N |G A µν G Aµν |N ⟩ are the same.Hence the operators from Eq.(5.9) contribute to f p,n in an interference way: where the last term is from the Z portal in the complex scalar DM scenario, with the sign ± standing for the scattering of a particle or anti-particle DM.Note that m N is the nucleon mass and the form factors are = 0, Eq.(5.12) goes back to the pure Higgs portal scenario [22].And the matrix elements related to Higgs portal are defined by [41]: Operating the nucleon states on the trace of the stress energy tensor gives the following relation [42]: .14)where is the quark mass fraction that can be calculated by lattice simulation or chiral perturbation theory, and we will use the values listed in [40].Finally, the spin-independent cross section is (5.15) As indicated in Eq.(5.12), the Z-portal interaction of particle and anti-particle DMs is either constructive or deconstructive with the Higgs portal one in the DM-nucleon coupling f p,n .
For the complex DM in SU(6)/Sp(6) CHM, we need to average over this two effects, i.e. σ tot SI = 1 2 (σ SI + σSI ).The measurement of σ SI at the underground detectors [43][44][45][46][47] will impose constraint on (m S , f p,n ).With m T > 2 TeV and m S ∈ (0.2, 1.5) TeV, the dark top partner induced σ SI normally is < 10 −48 cm 2 , several orders of magnitude smaller than the upper limit.Hence, the direct detection mainly constrains the parameters related to the Higgs or Z-portal interactions, i.e. (C hS , m S ) in SU(6)/SO(6) CHM and (C hS , m S , sin α) in SU(6)/Sp(6) CHM.For the complex DM, the Z-portal interaction places a stringent upper bound on sin α.Using the latest LUX-ZEPLIN measurement [47], we find with C hS = 0 and m S ∈ (0.2, 1.5)TeV, the upper limit of sin α in SU(6)/Sp(6) CHM falls in the range of (0.024, 0.04).Adding the Higgs coupling C hS will further reduce the allowed sin α value.The benefit to include odd top partners is that the dark Yukawa interaction brings in sufficient additional annihilation rate that compensates the smallness of Higgs portal one, while is not sensitive to the direct detection in the preferred (m T , m S ) parameter space.Therefore, the dark top partners in fact relieve the tension in a Higgs portal DM model.

Combined analysis
There are 5 independent parameters in our DM models, i.e. (C hS , m S , m T , sin α, sin ϕ L ), where the last 3 ones are related to EWPOs.A comprehensive analysis is conducted to find the viable parameter space.However we will not include the bound from indirect detection, e.g. the gamma rays observed in dwarf spheroidal galaxies or the Galactic Center.The signals normally impose constraint on the low DM mass scenario (≲ 100 GeV) [48,49].A recent interpolation pushed up the bounds near several hundred GeV for the bb, W W , ZZ channels, but lack of tt information [50].Note that our analyses in both CHMs are restricted to a small sin α and this results in SS → tt playing a dominant role with more than 90% contribution to the DM annihilation.For the DM mass in (0.2, 1.5) TeV, the upper limit on ⟨σv⟩t t is around 5.0 × 10 −26 cm 3 s −1 from the 11 years of Fermi-LAT observations [49], with almost no constraint on the parameter space.
In Figure 2, we require the scalar DM to accommodate the correct relic density 0.118 < Ωh 2 < 0.122 [51] and simultaneously satisfy the constraints from the direct detection and EWPOs.In addition, collider experiments impose direct constraints on the dark top partners and the (T X , X) in the Z 2 -even bi-doublet [52][53][54].Henceforth, the points in the plots need to satisfy M = m T cos ϕ L > 1.3 TeV.The upper panels refer to SU(6)/SO(6): In Figure 2(a), we set C hS = 2 GeV, and the relic density dominantly comes from the top partner channel.We can see that the DM mass is constrained to be 200 ≲ m η 2 ≲ 255 GeV for m T ≤ 8 TeV.A lower DM mass from this region will demand for smaller sin ϕ L and m T , thus is excluded by EWPOs.Corresponding results for SU(6)/Sp (6) are displayed in the lower panels.For this case, the allowed value of sin α derived from the direct detection strictly constrains the lower limit of DM mass, while the bound from EWPO is relatively loose.In Figure 2(c

Conclusion
Extending the global symmetries of composite Higgs models allow to accommodate for stable pNGBs, which can play the role of Dark Matter.In fundamental CHMs with real and pseudo-real realizations, this amounts to add the HC fermions that are odd under a Z 2 dark parity.In return, odd resonances as condensations of HC fermions appear in all sectors of the theory.However for the complex realization of CHM, a simple Z 2 for the HC fermions is not adequate as the conjugate operation plus internal flavor rotation might be involved.Partial compositeness for the top quark mass generation plays a crucial role in this class of models.On one hand, the couplings of the elementary top fields must preserve the dark parity, hence imposing non-trivial constraints on the UV completions of the models.On the other hand, the presence of odd top partners relieves the tension of Higgs portal coupling with the direct detection.
We have considered in detail the role played by the dark top partners in two models based on fundamental dynamics, where the composite Higgs boson and Dark Matter stem from the cosets SU(6)/SO (6) and SU(6)/Sp (6).Following Ref. [29], we considered the impact of electroweak constraints on the models.Furthermore, we studied the properties of the pNGB DM in the presence of the dark top partners.They enter as mediators for the annihilation of the DM candidate in the early Universe and at direct detection experiments.This effect can dominate over the Higgs mediated processes, with non-trivial interference effects taken into account.As a result, phenomenological constraints require the mass of the even top partner to be in the multi TeV regime, with viable masses above 4 TeV for SU(6)/SO(6) and lower to 2 TeV for SU(6)/Sp (6).These masses can only be directly probed at future high-energy hadronic colliders.

C Mixing with (3, 1) + (1, 3) top partners
The calculation of oblique parameters in the bi-doublet mixing scenario is given in [29].Here we will show the relevant detail for the triplet mixing scenario.When the top quark mass is generated by the mixing among the SM (t L , b L ) and t R and top partners in (3, 1) + (1, 3) representations, the fermions can be arranged into up and down sectors:   which will recover Eq.(4.12) for sin ϕ R = 0, just like the bi-doublet mixing scenario.Note that the loop functions we used in the oblique parameters are defined in [55]: Another loop function can appear in the S parameter originating from a pure rotation effect: that is generalized for the VLQ in a non-standard doublet or triplet representation [29].

2 ≲
) with C hS = 2 GeV, sin α = 0.02, the DM mass populates in a wide range 200 ≲ m η 3 +iη 4 √ 1500GeV.Increasing sin α will cut off the lower mass region.The impacts of C hS for two models are visualized in Figure2(b) and 2(d), with its value limited by the direct detection bound.From Eq.(5.4), we find that a positive C hS makes the Higgs portal deconstructive with the top partner channel, hence requires a lighter top partner.The narrow bands from the variation of C hS indicate that the Higgs portal only offers a subleading contribution.

Table 1 :
Hyper -Color fermions in the case of real realization, and their quantum numbers.The two fermions ψ D1 and ψ D2 transform as bi-doublet under the SU (2) L × SU (2) R with U (1) Y ⊂ SU (2) R .The last two columns indicate the global symmetries, i.e. the custodial SU(2) R and the dark parity.

Table 3 :
(6) partners in two SU(6)CHMs are classified with respect to the unbroken flavor subgroups and G HC is the Hyper-Color gauge group.