Accidental Composite Dark Matter

We build models where Dark Matter candidates arise as composite states of a new confining gauge force, stable thanks to accidental symmetries. Restricting to renormalizable theories compatible with SU(5) unification, we find 13 models based on SU(N) gauge theories and 9 based on SO(N). We also describe other models that require non-renormalizable interactions. The two gauge groups lead to distinctive phenomenologies: SU(N) theories give complex DM, with potentially observable electric and magnetic dipole moments that lead to peculiar spin-independent cross sections; SO(N) theories give real DM, with challenging spin-dependent cross sections or inelastic scatterings. Models with Yukawa couplings also give rise to spin-independent direct detection mediated by the Higgs boson and to electric dipole moments for the electron. In some models DM has higher spin. Each model predicts a specific set of lighter composite scalars, possibly observable at colliders.


Introduction
A striking success of the Standard Model is that all observed global symmetries are understood as accidental symmetries of the renormalizable Lagrangian. This explains in particular the stability of the proton as a consequence of baryon number conservation.
In nature, besides the proton, at least another particle should be stable to provide the necessary Dark Matter (DM) abundance required by cosmological observations. It is natural to imagine that dark matter too is stable because of accidental symmetries. This idea can be minimally realized by adding to the SM one extra multiplet that cannot have any Yukawa interaction with SM particles, and that contains a DM candidate [1].
The fact that bounds from DM searches require a successful weak-scale DM candidate to have no electric charge, no color, and almost no coupling to the Z (the vectorial coupling to the Z must be a few orders of magnitude smaller than a typical weak coupling) calls for an explanation. A simple way of explaining why DM is so dark and stable is to add to the SM (with its elementary Higgs) new fermions Ψ charged under a new technicolor interaction that confines at a scale Λ TC . Techni-quarks are assumed to lie in (possibly reducible) real representations under the SM gauge group, such that their condensates do not break the electro-weak symmetry, realising the framework dubbed 'vector-like confinement' in [2]. The renormalizable Lagrangian of the theory is where the latter term, Yukawa interactions with the Higgs doublet H, can be allowed by quantum numbers. The topological term for technicolor gauge fields is physical for non-vanishing techni-quark masses m i . We assume that when technicolor interactions confine at a scale Λ TC , the approximate global techni-flavor symmetry is broken by condensates producing light techni-pions (TCπ) and other heavier composite particles, such as techni-baryons (TCb). All these particles are splitted in mass by SM gauge interactions in such a way that the lightest stable techni-particle (charged under an accidental symmetry that keeps it stable) tends to be the 'most neutral' one.
Composite Dark Matter has been rarely considered in the literature, and mostly in models with different goals, e.g. with supersymmetry [3], with composite [4] or partially composite Higgs [5], with a mirror-SM sector [6] or quirks [7] or a fourth generation [8] as well as from a phenomenological point of view, in order to realise special situations (such as inelastic DM, asymmetric DM, strongly interacting DM, magnetic DM, etc.) often motivated by anomalies [9]. An approach similar to the present study was considered in [10,11,13]. In [10,13] bosonic techni-baryon DM in SU(4) gauge theories was studied. In [11] we began a general study of composite DM adopting a specific point of view with respect to the naturalness problem, according to which the Lagrangian does not contain any massive parameter, power divergences are unphysical, all masses arise via dimensional transmutation. The resulting assumption m i = 0 lead to very predictive models [11]. Allowing for techni-quark masses (if lighter than about 1 TeV, they do not induce unnaturally large corrections to the Higgs mass [12]) and for an order one θ TC modifies the mass spectrum of the theory, inducing electric dipole moments (EDMs) for TCb that leads to a sizeable Dark Matter direct detection signal with characteristic dependence on velocity and transferred momentum.
The issue of composite dark matter is logically independent from the point of view in [11] on naturalness. We here revisit the DM issue remaining agnostic about the explanation of smallness of the electro-weak scale: we just assume that for some reason the SM is much lighter than other unspecified new physics, such that accidental symmetries appear at low energy. We make the following simplifying assumptions: 1. We study both SU(N ) TC and SO(N ) TC techni-color gauge groups, but we restrict to techni-quarks in the fundamental representations of the TC group.
2. We consider techni-quark representations that can be embedded in SU(5)-unified models.
3. We do not consider techni-scalars, that would generate a different set of TCb, and would allow to realise partial compositeness in a fundamental theory.

The accidentally stable Dark Matter candidates
This scenario has the following accidental symmetries that lead to automatically stable composite DM candidates: • Techni-baryon number. The Lagrangian is accidentally symmetric under a U(1) TB global symmetry (sometimes broken by anomalies down to Z 2 ) that rotates the techniquarks Ψ with the same phase. This guarantees the stability of the lightest techni-baryon.
• Species Number. When the techni-quarks are in a reducible representation of the SM, each phase rotation acting individually on a Ψ i is an accidental techni-flavor symmetry of the renormalizable Lagrangian. This leads to stable techni-pions made of different speciesΨ i Ψ j . TCb made of different species can also be stable if their decay to TCπ is kinematically forbidden.
• G-parity. In models with electro-weak representations the Lagrangian can be invariant under a discrete symmetry known as G-parity [14], that acts on techni-quarks as Ψ → exp(iπT 2 )Ψ c . In SU(N ) TC theories G-parity acts on TCπ so that even (odd) isospin TCπ are even (odd) under G-parity. Standard Model states are G-parity even, so that the lightest G-parity odd TCπ is stable. This symmetry is broken by non-vanishing hypercharge.
We assume that, in a successful model, all stable particles must be good DM candidates.

Breaking of accidental symmetries
The symmetries above can be violated by various effects. First, when the quantum numbers allow for Yukawa interactions with the Higgs, this breaks both species number and G-parity while preserving techni-baryon number. States whose stability was insured by these broken symmetries will then decay with specific patterns. We assume that all allowed couplings are present and that decays are fast enough that unstable particles are not relevant for dark matter.
Second, species number and G−parity can also be broken by dimension 5 operators, The lifetime of TCπ is shorter than the age of the universe for M <M Pl ≡ 2.4 × 10 18 GeV. Third, techni-baryon and species number can be broken by operators of dimension 6 or higher depending on the quantum numbers. In the first case the lifetime is consistent with the present experimental bound from indirect searches [15] τ ∼ 8π if M is comparable toM Pl and M DM ≈ 100 TeV. For dimension 7 operators the scale M must be larger than ≈ 10 14 GeV. Any species number symmetry can also be broken by adding e.g. ad-hoc scalars with quantum numbers such that desired extra Yukawa couplings arise.
The upshot is that techni-baryon number is more robust than species number or G-parity in the framework of vector-like confinement, at least working within the standard assumptions of effective field theory. TCb are then the most promising dark matter candidate. We will focus mostly on TCb dark matter in what follows.
The paper is structured as follows. We identify successful DM models based on SU(N ) TC in section 2 and models based on SO(N ) TC in section 3. In section 4 we discuss the effect of techniquark masses and of the θ TC on the spectrum and the generation of Electric Dipole moments. In section 5 we discuss the resulting phenomenology. Conclusions are given in section 6. In the appendices we provide technical details of the techni-baryon classification and we collect models that require higher dimensional operators.

SU(N ) TC Composite Dark Matter models
In this section we consider an SU(N ) TC techni-color group with N TF techni-quarks in its fundamental representation. We assume that the dynamics is as in QCD: when techni-color interactions become strong, confinement takes place and the global flavor symmetry SU(N TF ) L ⊗ SU(N TF ) R is spontaneously broken to the diagonal sub-group SU(N TF ) producing N 2 TF − 1 Goldstone bosons in the adjoint representation of the unbroken group. We assume the standard large N scaling Table 1: Techni-quarks are assumed to belong to fragments of SU(5) representations (plus their conjugates for complex representations). We give the SM decomposition, assign standard names used throughout the paper, and list the contributions ∆b i to the SM β-function coefficients (to be multiplied by the multiplicity of the techni-color representation).
where, to be definite, we denote with Λ TC the mass of the lightest vector meson, with f the Goldstone bosons decay constant, and with m B the techni-baryon mass. We consider a model as viable from the point of view of Dark Matter phenomenology, provided that all its stable states have no color, no charge and no hypercharge. This implies that dark matter should belong to a multiplet with integer isospin. As in weakly coupled theories, the neutral component within an electroweak multiplet becomes the lightest component, with a calculable splitting, of order 100 MeV, induced by electro-weak symmetry breaking [1].
We analyzed these requirements using the tools in appendix A and the package LieArt [16]. We assume an SU(5) unification scheme, so we select techni-quarks from components of the simpler SU(5) representations listed in table 1. In general for a SM representation there are two inequivalent assignments of techni-quark quantum numbers: where R N andR N transform in the fundamental of SU(N ) TC , while RN andRN in antifundamentals. Since the V , N and G representations are real under the SM gauge group, one has V =Ṽ , N =Ñ and G =G. For each SM representation, an unbroken species symmetry exists corresponding to a U(1) that rotates the (anti)fundamental of SU(N ) TC with charge +1 (−1). Because of this accidental symmetry, TCπ made by different species are stable unless the symmetry is broken e.g. by Yukawa couplings. Techni-baryon number, that guarantees the stability of the lightest TCb, is the sum of all species numbers. It is convenient to classify models in the following way: 1. Golden-class models, such that all stable states are acceptable DM candidates with just renormalizable interactions 1 . Yukawa couplings are often needed in order to break accidental symmetries, avoiding unwanted stable TCπ. All possible Yukawa couplings among the SU(5) fragments are: as well as similar interactions with H ↔ H † or x ↔x where x denotes all techni-quarks.
2. Silver-class models where non-renormalizable interactions or ad-hoc extra particles are introduced in order to break accidental symmetries that lead to unwanted stable particles.
3. Models with no DM candidates.
An important restriction on the techni-quark content arises from the requirement that SU(N ) TC with N TF flavors of techni-quarks (e.g. a singlet N contributes as N TF = 1) is asymptotically free. Defining the gauge β-function coefficients as Furthermore we demand that the SM gauge couplings do not develop Landau poles below the Planck scale: where the numerical factors have been computed assuming Λ TC ∼100 TeV, motivated by DM as a thermal relic, see section 5. Colored techni-quarks such as U or D contribute as ∆b 3 = 2N/3, while a G state gives ∆b 3 = 4N . The weak doublet L contributes as ∆b 2 = 2N/3, while for the weak triplet V we have ∆b 2 = 8N/3. Finally ∆b Y = 2 3 R dim(R)Y 2 R (e.g. a singlet E contributes as ∆b Y = 4N/3). The contributions ∆b 2,3,Y are summed over techni-quarks, and the constant terms in the β-function coefficients b 2,3,Y are the SM contributions.
Summarising, the constraints on the techni-quark content are: This implies that one weak triplet V is allowed by the constraint on ∆b 2 for N = 3 techni-colors but not for N ≥ 4. Models that contain the techni-quark G, S, X are not allowed, not even for N = 3, because of ∆b 3 or ∆b Y .

Techni-pions and techni-baryons of SU(N ) TC
Techni-pions are ΨΨ states in the adjoint representations of SU(N TF ) under the unbroken techni-flavor symmetry. Their decomposition under the SM group is given by where the sum runs over the N S species (e.g. a model with Ψ = L⊕N techni-quarks has N S = 2 species and N TF = 2 + 1 techni-flavors). SM gauge interaction generate a positive contribution to TCπ masses that can be estimated as The N S − 1 singlets under the SM gauge group do not acquire mass from gauge interactions. In our previous study [11] we assumed vanishing techni-quark masses, such that these singlet TCπ were massless in absence of Yukawa interactions, and thereby experimentally excluded because of their axion-like coupling to SM vectors. Here we allow for techni-quark masses, such that the singlets become massive avoiding phenomenological problems. The contribution from techni-quark masses to TCπ masses scales as and can be described using chiral Lagrangian techniques. Techni-pions can be stable because of G-parity or species number if they are made by different species. For example in QCD, the charged pion π + decays because species number is broken by weak interactions, while G−parity is broken by hypercharge allowing π 0 to decay through the anomaly. Among our representations, only the weak triplet V is symmetric under G-parity leading to stable TCπ.
TCb are techni-color singlets constructed with N techni-quarks. They are fermions for N odd and bosons for N even, leading to vastly different dark matter phenomenology. The SM quantum numbers of TCb multiplets are determined by group theory: the TCb fill representations of the unbroken SU(N TF ) global techni-flavor symmetry that can be decomposed under the SM. TCb wave-function is totally antisymmetric in techni-color. Furthermore, one can argue that the lighter TCb have the smallest possible spin, and the lowest possible angular momentum (fully symmetric s-wave function in space). Due to Fermi statistics, this implies that TCb must be fully symmetric in spin and techni-flavour. This determines the representation of the lighter TCb under the unbroken global techni-flavor symmetry corresponding to a Young tableau with two rows with N/2 boxes (N even) or two rows with (N + 1)/2 and (N − 1)/2 boxes (N odd) and also the spin. Explicitly for N = 3, 4, 5 they are, A fully symmetric representation is obtained by a tensor product of each techni-flavor representation with an identical spin representation: for even (odd) N we obtain spin-0 (spin 1/2) DM. The case N TF = 1 is special because flavour cannot be anti-symmetrized, TCb have spin N/2. The heavier TCb (analog of the decuplet in QCD) transform instead in the following representations and have higher spin described by an identical spin representation. The mass difference between the heavier and the lighter TCb is expected of order Λ TC . Heavier TCb usually decay into a lighter TCb and TCπ; however some heavier TCb could be accidentally stable due to species number if they are the lightest states with TCb and species number. This can happen for techni-quark masses comparable to Λ TC . An analog exists in QCD where, in absence of the weak interactions, the lightest strange baryon (Λ, with quark content uds) would be stable because its decay to kaons and nucleons is not kinematically allowed. Furthermore, the spin 3/2 baryon Ω − (1672) (quark content sss) cannot decay to Ξ 0 K − through strong interactions: its decay is allowed only by strangeness-violating weak interactions.
TCb flavour multiplets are split by SM gauge interactions, by techni-quark masses and possibly by techni-quark Yukawa interactions and by higher dimensional operators (that we neglect). While for the TCπ one can argue that in the limit of zero techni-quark masses the lightest multiplets are those with the smallest charge under the SM gauge group, the same sentence is not rigorously proved for TCb. Indeed, while the long distance gauge contribution to the energy of charged fields is proportional to their total charge, the short distance contribution is difficult to estimate. Experience with electromagnetic splitting of baryons in QCD hints however to the fact that the lightest states are indeed the ones with smaller charge. This is what we will assume in the following. We estimate, Finally, the breaking of the electro-weak symmetry induces calculable splittings within the components of each electro-weak multiplet (of order 100 MeV), with the result that the component with smallest electric charge is the lightest state. The spectrum of the theory is illustrated in fig. 1. Figure 1: Spectrum of techni-color DM models. Splitting between techni-flavor multiplets is of order the dynamical scale Λ TC splitting between different SM representation Λ TC /100 or larger and hyperfine splitting from electro-weak symmetry breaking of order 100 MeV.

SU(N ) TC golden-class models
In this section we present the golden-class models for SU(N ) TC strong interactions. The models are obtained scanning over techni-quarks made by combinations of the SU(5) fragments of table 1. excluding models that lead to sub-Planckian Landau poles for g Y , g 2 or g 3 . We require that the lightest stable TCb has no color, no hypercharge, and integer isospin. For example, for N = 3, the possible DM candidates are made of the following techni-quarks: where x denotes any techni-quark, any E can be substituted by a T , any V can be substituted by a N . By replacing all techni-quarks with their tilded counterparts one obtains equivalent descriptions of the same models. However, if species number is conserved, most of the models that can give rise to such TCb DM candidates also lead to extra stable TCπ with Y = 0 or color, that are thereby excluded by DM direct searches (unless their thermal abundance is small enough). In the context of renormalizable golden-class models, Yukawa couplings to the Higgs doublet determine the accidental symmetries. For example, a Yukawa coupling to the Higgs boson is allowed by gauge quantum numbers in a model containing the techni-quarks Ψ = L ⊕Ẽ. The Yukawa coupling HLẼ breaks the unwanted species number. On the contrary, no Yukawa coupling is allowed in a model with Ψ = L ⊕ E that would lead to the first TCb in eq. (16). In appendix B we present a list of silver-class models (limited for simplicity to N = 3, 4 and to the case of 1 or 2 species) where extra effects (non-renormalizable interactions or other particles) are needed to break unwanted symmetries.
The list of SU(N ) TC golden-class models presented below is summarized in table 2 2 . We start the description of golden-class models from models that only involve color-less techniquarks.
The simplest model contains the singlet N as the only techni-quark, such that the lightest DM TCb has spin N/2. Interactions with SM particles arise only adding extra states, as described below.
The model has a single specie of techni-quarks: a triplet with zero hypercharge in the adjoint of SU(2) L , such that N TF = 3. No Yukawa coupling is allowed. If N ≥ 4 the g 2 gauge coupling becomes non-perturbative below the Planck scale. Thereby this model is only allowed for N = 3. Both TCb and TCπ lie in the 8 of SU(3) TF , that decomposes as The TCπ triplet is stable because of G-parity, and the TCb triplet is stable because of technibaryon number. These are good DM candidates. This model has been already presented in [11].
The previous model can be simply extended to N S = 2 techni-quarks by adding an N (SM gauge singlet) such that N TF = 4. Again, no Yukawa coupling is allowed and the model can be considered only for N = 3 because of sub-Planckian Landau poles. TCπ lie in the 15 of SU(4) TF that decomposes as The three triplets are stable because of species number and because of G-parity. The lighter TCb lie in the 20 representation of SU(4) TF that decomposes as  The lightest TCb is a stable DM candidate, and its identity depends on the techni-quark masses. For m V m N , the triplet 3 0 (V V V ) is expected to be the lightest. For m N m V > ∼ Λ TC the extra TCb N N N * (denoted with a * and not included in the list above because it has spin 3/2) could become the stable DM candidate; at the same time the SU(4) TF classification breaks down.
In both models, enough Yukawa couplings are allowed such that only techni-baryon number is conserved and all TCπ are unstable. For the N ⊕ L (N TF = 3) and the N ⊕ L ⊕Ẽ (N TF = 4) models respectively, these are: For N = 3, the spin 1/2 TCb do not contain any DM candidate, for example in the the N ⊕ L model they are The DM candidate is the singlet N N N * , which only exists with spin 3/2. Thereby these models are viable only as long as the techni-quark masses m L and mẼ are of order Λ TC and large enough that N N N * is the lightest TCb. This state lies in the 10 of SU(3) TF in the N ⊕ L model and in the 20 of SU(4) TF in the N ⊕ L ⊕Ẽ model. The same is true for N = 4, where the only DM candidate is the singlet N N N N * that lies in the completely symmetric spin 2 representation . In the N ⊕ L model, this representation decomposes as The N ⊕ L model is allowed by perturbativity constraints up to N = 14, while the N ⊕ L ⊕Ẽ is allowed up to N = 5 (with increasing spin of the DM candidate).
Other possible extensions of the first model are Ψ = V ⊕ L and V ⊕ L ⊕Ẽ. A possible problem of these models is that, even for N = 3, the SU(2) L gauge coupling becomes non perturbative around 10 17 GeV. In view of the Yukawa couplings V LH,ẼLH, all TCπ are unstable and given by in the V ⊕ L model, and by in the V ⊕ L ⊕Ẽ model. In both models the TCb DM candidate is the V V V state that forms a weak triplet as in the Ψ = V model: the extra techni-quarks L (and possiblyẼ) does not lead to any extra DM candidates and play a minor role provided that they are heavy enough. In the V ⊕ L model, the lightest TCb multiplet is a 40 of SU(5) TF that decomposes as: As in the previous models, sub-Planckian Landau poles are avoided only for N = 3 (where g 2 becomes non perturbative around 10 17 GeV). Since L andẼ cannot enter in an hyperchargeless TCb, the DM candidates are the same of the V ⊕ N model. Unlike in the V ⊕ N model, the Yukawa couplings V LH, N LH and LẼH break all species number symmetries, such that all TCπ are unstable. In the V ⊕ N ⊕ L model (N TF = 6), the TCπ are In the V ⊕ N ⊕ L ⊕Ẽ model (N TF = 7), the list extends to The model allows two Yukawa couplings (N LH, NLH) such that there are no stable TCπ and allows for DM TCb candidates not present in the previous models. The unstable TCπ are: Sub-Planckian Landau poles are avoided for N ≤ 7. Here we discuss the TCb DM candidates for N = 3, 4. For N = 3, the lighter TCb fill a 40 of SU(5) TF that decomposes as under SU(2) L ⊗ U(1) Y , so that the TCb DM candidates are singlets made of NLL.
For N = 4, the lighter TCb are under SU(2) L ⊗ U(1) Y . The TCb DM candidates are singlets made of LLLL and LLN N .
This is a non trivial extension of the previous model, with one more Yukawa coupling allowed (LẼH), so that there are no stable TCπ. The model is allowed only for N = 3, 4, since for greater values of N the coupling g Y develops a sub-Planckian Landau pole. The unstable TCπ can be listed as: This model gives a TCb DM candidate not present in the previous models:LLẼ and NLLẼ for N = 3 and N = 4 respectively. For N = 3, the lightest multiplet of TCb decomposes under SU(2) L ⊗ U(1) Y as where TCb N ⊕L⊕L is defined in eq. (30). For N = 4 we get where now TCb N ⊕L⊕L refers to eq. (31). In each case, besides the TCb DM candidates of the N ⊕ L ⊕L model, there a singlet DM candidate made ofLLẼ or NLLẼ.
The model has N TF = 7 and for N = 3 gives ∆b Y = 12 so that hypercharge has a Landau pole around the Planck scale, so that it cannot be extended to N > 3. Thanks to the presence of N , it allows for 4 Yukawa couplings (LẼH,LEH, LN H,LN H) that break all species number symmetries. The unstable TCπ are: under SU(2) L ⊗ U(1) Y . The lightest TCb fill a 112 of SU(7) TF , that decomposes as The TCb DM candidates are those of the N ⊕ L ⊕L model, defined in eq. (30), plus the singlets LLE,LLẼ and EẼN .
We next consider models with coloured techni-quarks.
The simplest golden-class model with colored techni-quarks is Ψ = Q ⊕D, that is allowed for N = 3, 4 and gives a DM candidate only for N = 3. The model has N TF = 6 and does not lead to unwanted stable states because species number is broken by the Yukawa coupling QDH.
The model predicts a set of unstable TCπ in the 80 representation of SU(9) TF , that decomposes under the SM gauge group SU For N = 3 the multiplet of lighter TCb fills a 240 of SU(9) TF , that decomposes as The DM candidate is the neutral singlet QQD, which can be the lightest TCb.
This extension of the previous model allows for two Yukawa couplings, QHD and QHŨ , so that there are no stable TCπ. This model has N TF = 12 and is allowed only for N = 3, where ∆b 3 = 8. It predicts an extended set of unstable TCπ, that fills a 143 of SU(12) TF : The model contains two TCb DM candidates: QQD and DDŨ . The lighter TCb fill a 572 of SU(12) T F , that decomposes as The TCb DM candidates are those of the Ψ = Q ⊕D model plus a singlet made ofDDŨ .
Notice that colored techni-quarks never provide golden-class models for N ≥ 4. For example the model Ψ = G leads, for N = 4, to an acceptable TCb DM candidate; but g 3 develops a sub-Planckian Landau pole. 3 Landau poles also exclude the model Ψ = Q⊕Ũ ⊕D (two Yukawa couplings allowed, no stable TCπ) that for N = 6 provides a TCb DM candidate, QQDDDŨ .

SO(N ) TC Composite Dark Matter models
In this section we consider models based on SO(N ) techni-color interactions with techni-quarks in the vector representation of SO(N ) 4 . The techni-quark content is restricted by demanding that g Y,2,3 do not develop sub-Planckian Landau poles, and that SO(N ) TC is asymptotically free. Normalizing the generators in the fundamental as Tr( Considering again techni-quarks in fragments of the simplest SU(5) representations in table 1, vectorial techni-quarks Ψ are defined as: The dynamics of the theory is as follows. In the limit of negligible techni-quarks masses, the anomaly free global symmetry is SU( The spontaneous breaking produces N TF (N TF + 1)/2 − 1 pseudo-Goldstone bosons that transform in the two-index symmetric representation of the unbroken SO(N TF ) group. The condensate preserves the accidental U(1) symmetry rotating C N andC N with opposite phases, that generalises the species symmetry defined for SU(N ) TC theories. The important novelty of this class of models is that the technicolor representation is real. This has various consequences: TCπ are ΨΨ states and there is no distinction between TCb and anti-TCb. Moreover N, V, G techni-quarks lie in real representations under both G SM and SO(N ) TC and can have Majorana masses that do not arise in SU(N ) TC models.

Techni-pions and techni-baryons of SO(N ) TC
There are important differences with respect to SU(N ) TC models.
Techni-pions are now ΨΨ states, such that, if species number is conserved, TCπ made of C N C N are stable because they have species number 2. Furthermore they have quantum numbers under the SM gauge group not compatible with DM phenomenology. Real techni-quarks R N instead do not produce stable TCπ since the techni-quark condensate and masses break their species number.
The presence of at least one techni-quark in a real representation is a necessary ingredient to build viable models without unwanted stable TCπ. In fact, Yukawa couplings of the form HR N C N can break the unwanted species symmetries allowing all TCπ to decay. The allowed Yukawa interactions with the Higgs are (analogously to eq. (6)): G-parity can still be defined as in SU(N ) TC theories. However, with our choice of representations, G-parity is only conserved by the SM multiplet V that in SO(N ) TC theories only gives rise to (unstable) G-even TCπ.
Techni-baryons (TCb) are, as in SU(N ) TC theories, antisymmetric combinations of N techni-quarks. Techni-baryon number is not conserved, such that TCb cannot have an asymmetry, two TCb can annihilate and TCb can now be real particles, e.g. Majorana fermions. The lightest TCb is stable and can be a DM candidate. For N odd stability simply follows from the accidental Ψ → −Ψ symmetry. For generic N stability follows because the SO(N ) gauge theory actually has an accidental O(N ) symmetry; the quotient Z 2 = O(N )/ SO(N ) (that distinguishes orthogonal matrices according to the sign of their determinant) acts as a global symmetry group. All TCb built with the N -index anti-symmetric tensor are odd under this Z 2 symmetry, and the lightest odd state is stable.
Since the same anti-symmetric tensor with N -indices is invariant under both SU(N ) TC and SO(N ) TC , the TCb following from a given set of techni-quarks are the same. They must however be decomposed under different techni-flavor groups conserved by technicolor interactions: SU(N TF ) for SU(N ) TC , and SO(N TF ) for SO(N ) TC . Since SO(N TF ) ⊂ SU(N TF ), one can start from the TCb of SU(N TF ) and split them into SO(N TF ) multiplets. The group-theoretic decomposition rules that connect the TCb representations of SU(N TF ) and SO(N TF ) are the following: 5 N = 3 : .
(45) 5 The information contained in these SO(N TF ) Young diagrams is redundant for small N TF . Only diagrams with as many rows as the rank of the corresponding SO(N TF ) group are independent. The rank of SO(N TF ) is N TF /2 for N TF even and (N TF − 1)/2 for N TF odd.
This leads to a novel physical phenomenon: SO(N ) TC gives different masses to the TCb multiplets that were degenerate in SU(N ) TC models. For example, in ordinary QCD, if color SU (3) were replaced by SO(3) (with 3 quarks in its real fundamental representation), the 'eightfold way' would split into 'threefold way' and 'pentafold' way: with a similar decomposition for the heavier decuplet of spin-3/2 baryons: Unfortunately, QCD gives us no guidance in understanding a crucial question for composite DM phenomenology: which SO(N TF ) multiplet contains the lighter TCb, given that more representations have the same spin?
Given that composite spin-1 resonances behave as gauge vectors of the techni-flavor symmetries, and that gauging of global symmetries likely generates positive contributions to TCb masses, a plausible answer is that the lightest TCb multiplet is the one in the smallest representation of SO(N TF ) among those with lowest spin. We will make this assumption in what follows (lattice simulations could check it). This means that for N odd the lightest TCb will be in the vectorial representation of SO(N TF ) (denoted by ) with the same quantum numbers as techni-quarks Ψ itself, while for even N it will be a singlet of SO(N TF ).
Even within the assumption above, if techni-quark masses are comparable to Λ TC , it becomes possible that the lightest TCb belongs to a higher SO(N TF ) representation. For completeness, we therefore also specify the SM decomposition of the higher SO(N TF ) representations appearing in eq. (45). Notice that for N = 4, the representation coincides with the representation of the TCπ, so we only need to specify the representation. Analogously, for N = 5 we only need to decompose and .
Finally, the members of the lightest TCb SO(N TF ) multiplet are further split by SM gauge interactions and the lightest TCb is the one with the smallest SM charge.

SO(N ) TC golden-class models
As discussed above, avoiding unwanted stable TCπ implies that the model must contain at least one real V , N , G state with Majorana mass. This leads to real DM states, with important consequences for DM phenomenology discussed in section 5.2. With the assumption that the lightest TCb multiplet is the one in the smallest representation of SO(N TF ) among those with lowest spin, table 3 lists the golden-class models discussed below. These are the models that give a DM candidate without unwanted stable particles. In appendix B we will present the silver-class models that need extra assumptions to break accidental symmetries in order to avoid unwanted stable states.
The 3 0 is stable because of species number, giving a TCπ DM candidate. For N = 3 the lightest TCb DM candidate lives in the 4-dimensional representation of SO(4) TF that is composed by a singlet N V V and a triplet made by a linear combination of V N N and V V V . For N = 4 the TCb DM candidate is a singlet linear combination of V V V V , V V N N . The remaining heavier TCb for N = 3 are TCb : As explained before, for N = 4 it is enough to specify the following decomposition TCb : to describe all possible TCb.
For N = 4 this model with N TF = 8 avoids a sub-Planckian Landau pole for g 3 and, at the same time, techni-color is asymptotically free, b TC = −2. The model leads to the following colored TCπ, that undergo anomalous decays to gluons: The TCb DM candidate is the SM singlet GGGG and the remaining heavier TCb are: To specify the complete set of TCb, we need the following decompositions TCb : under SU(2) L ⊗ U(1) Y . Taking into account the Yukawa couplings, in this model and in the following models the TCb mix giving real eigenstates which are all good DM candidates, with a peculiar phenomenology discussed in section 5.2. In the limit m N Λ TC the N state can be integrated out realizing nicely the silver-class model Ψ = L presented in appendix B.
The strong coupling g TC is asymptotically free for N ≥ 4 and g 2 avoids a sub-Planckian Landau pole for N ≤ 4 (with N = 5 slightly excluded). For N = 4 the TCb DM candidate is the SM singlet (LL + V V ) 2 and the remaining heavier TCb decompose under SU(2) L ⊗ U(1) Y as: This model with N TF = 7 and 2 Yukawa couplings HLN and HLE predicts the following unstable TCπ The model exists for N = 4, 5. For N = 4 the DM candidate is a singlet, then to fully specify the complete set of TCb we need to decompose the multiplet: TCb : = 168 = 5 0 ⊕ 4 ±3/2,2×(±1/2) ⊕ 3 ±3,2×(±2),5×(±1),5×0 This model with N TF = 9 and 2 Yukawa couplings HLV and HLE, gives rise to the set of unstable TCπ: The model exists and gives a singlet TCb DM candidate for N = 4. The multiplet of the remaining heavier TCb is: This model has N TF = 8 and 2 Yukawa couplings (HLV , HLN ) are allowed, so that all TCπ decay: The model is allowed only for N = 4 and gives a singlet TCb DM candidate. The complete set of TCb contains the multiplet: This model with N TF = 10 and 3 Yukawa couplings HLV , HLN , HLE predicts the following unstable TCπ:

Techni-quark masses and the θ TC angle
In [11] we considered composite dark matter theories in the limit of massless techni-quarks.
With masses (such that also the CP-violating θ TC angle becomes physical) the theory has a few more free parameters, that significantly affect its phenomenology. From a phenomenological point of view, we are mostly interested in checking that a successful TCb DM candidate is indeed the lightest TCb and in computing its interactions. The main new feature relevant for DM direct detection is that DM TCb fermion has magnetic and electric dipoles with moments We estimate A magnetic moment with order 1 gyro-magnetic ratio is typical of composite states. The smaller electric dipole is generated when CP is violated by a non-zero θ TC . For θ TC ∼ O(1) EDM could give striking effects in direct detection as we will see in section 5. Chromo-dipoles are generated in models with colored constituents.

A QCD-like example
To illustrate the effects of the θ TC angle, assumed to be large unlike the QCD θ-angle, we work out in detail the silver-class model with SU(3) TC and Ψ = L ⊕ E techni-quarks, described in section B.1. In this scenario the techni-strong dynamics is identical to QCD with three flavors and therefore we can rescale QCD data to make definite predictions. For this choice of quantum numbers no Yukawa couplings are allowed, such that charged TCπ are stable at renormalizable level. We assume that non-renormalizable operators break species number symmetry leading to unstable TCπ, and that DM is the singlet neutral TCb. The TCπ in the adjoint of SU(3) TF and the anomalous U(1) singlet are described by the hermitian matrix This is as in QCD, but with different charges for the isospin doublets. The effective TCπ Lagrangian described in [18] reads  The VEV U is determined dynamically by minimising the potential. One can conveniently look for a solution of the form The extrema of the potential are determined by the Dashen's equations: where we defined χ 2 E,L ≡ −2m E,L B 0 . It is easy to check that U = 1 1 when θ TC = 0 and techni-quark masses are different from zero. A non-vanishing θ TC modifies the TCπ spectrum such that m E,L → m E,L cos φ E,L in the mass formulae and generates CP violating interactions among the TCπ. In the limit m η m π (corresponding to χ 2 L,E a) and neglecting gauge contributions one finds Since cos φ E,L can be negative the effect of θ TC cannot be entirely reabsorbed by redefining the techni-quark masses (for example, in real world QCD, the measured pion spectrum is compatible with θ = 0 but not with θ = π [19]).
The spectrum of TCb can be computed with similar techniques. The octet contains where B 2 and B 2 are the analog of the nucleon and the Ξ doublet respectively, B 3 of the triplet Σ and B 0 1 of the singlet Λ. The effective lagrangian for the TCb can be found in [18]. It contains the following terms relevant to the present discussion: where m B is the common TCb mass generated by the strong interactions and M θ is the techniquark mass matrix that depends on θ TC angle through eq. (69) The second line of eq. (73) describes the CP violating interactions induced by θ TC relevant for the computation of electric dipoles and the third line contains derivative interactions with the TCπ. Dots stand for non-linear terms irrelevant for the present discussion. All the parameters of the effective lagrangian are determined by rescaling QCD data in terms of the dynamical scale, where J is the isospin of the TCπ multiplet. From the first line of eq. (73) the mass splittings between TCb due to techni-quark masses reads The LLE states, corresponding to the triplet B 3 and the singlet B 1 have zero hypercharge. Therefore they can be viable DM candidates if they are the lightest TCb. Using the QCD values of b 1 and b 2 ,we find that techni-quark masses always favor B 2 or B 2 to be the lightest TCb. The neutral LLE state can be the lightest TCb when the mass splitting due to SM gauge interactions is more important than the mass splitting due to techni-quark masses. This can be realised in the symmetric limit χ L = χ E ≡ χ where techni-quark masses respect the techni-flavor symmetry and the singlet B 0 1 (analog of the Λ) is most likely the lightest TCb. In the limit χ L = χ E a we can solve Dashen's equations analytically. The solution has multiple branches labelled by the integer n [32], The solution with minimum energy has a discontinuity at θ TC = π where it jumps from n = 0 to n = 1. This is necessary to restore the periodicity in θ TC .

Electric dipole of the DM candidates
We parameterize dipole moments in terms of gyromagnetic factors g M,E as Following [20], to leading order the dipole moments are proportional to the electric charge, where α and β are properties of the strong dynamics that, for the QCD-like model, can be extracted from the measured magnetic moments of baryons in QCD. What is different in our context is the charge matrix Q = diag(0 , −1 , 1). Plugging in the equation above we estimate g B 1 M ∼ 2.8. The same argument applies to the EDMs. To estimate the coefficient we proceed as in [18] for the computation of the neutron EDM. The CP violating vertices from the mass terms in eq. (73) generate one-loop graphs that contribute to the EDM. The dominant contributions are given by the logarithmically divergent diagrams represented in fig. 2. Similarly to the computation of the neutron EDM we obtain the estimate, For θ TC < ∼ 1 using the numerical values in (75) we obtain The discussion above can be easily generalised to other models. For example the model Ψ = V for N = 3 has again the same dynamics as QCD. From eq. (79) one can see that the magnetic and electric dipole moments of the TCb dark matter candidate (the neutral component of an isospin triplet) are zero.
For different N and N TF the relevant dynamics can be parametrized in terms of few unknown parameters that could in principle be extracted from lattice simulations. For TCπ the discussion is identical to eq. (68) with a number of Dashen's equation equal to the number of SM representations of the model. TCb are in general described by a tensor of SU(N TF ) B i 1 i 2 ...i N with the symmetry of Young tableaux as in (13). Their effective lagrangian is constructed writing all possible techni-flavor invariant combinations of the techni-baryon fields B andB with the techni-quark mass matrix M transforming in the adjoint representation of SU(N TF ).
For N odd there are two non-trivial invariants: Since the TCπ are in the adjoint representation, other two invariants can be written with derivative interactions that do not break the global symmetries. For N even, a single invariant can be written down: group theory uniquely fixes the mass splitting among TCb up to its overall coefficient. For example, in the model with N = N TF = 4 we predict equal mass differences between the TCb.

Phenomenology of Composite Dark Matter
We here briefly outline the phenomenology of the scenarios with TCb dark matter 6 .
This crucially depends on the TCb mass. Cosmology singles out two special values: TeV if DM is a complex state with a TCb asymmetry [15].
In the first case, the cosmological relic abundance is determined by the non-relativistic annihilation cross-section of TCb, that annihilate into TCπ through strong interactions and to SM states through gauge interactions. We can neglect the second sub-dominant effect. Rescaling the measured pp annihilation cross-section one finds [11] that the thermal DM abundance is reproduced for M DM ∼ 200 TeV.

Direct detection of complex Dark Matter
In various models, the DM candidate is a complex state with Y = 0 in the triplet or quintuplet representation of SU(2) L . Its weak interactions lead to a direct-detection cross section characteristic of Minimal Dark Matter, which is too small to be observed in the present context where the DM mass is around 100 TeV, if DM is a thermal relic. Moreover in various models DM is a SM singlet, such that even this cross section is absent. The main hope for direct detection of thermal TCb DM relies on the fact that composite DM made of charged constituents can have special interactions with the photon, leading to significant rates of low-energy scatterings. Scalar DM S can only have the dimension 6 interaction (S * i∂ µ S)∂ ν F µν or higher, which does not lead to interesting rates. Fermionic DM Ψ instead can have dipole interactions as in eq. (65) leading to the following cross section for direct detection [21,22]: where v is the relative DM/nucleus velocity and E R is the nucleus recoil energy. For simplicity, we here assumed a nucleus N with A, Z 1, mass M N ≈ Am N , a recoil energy E R M N v 2 , 6 If TCπ are stable due to accidental symmetries their mass should not exceed few TeV not to overclose the universe. The TCπ DM in this case likely dominates and behaves as the minimal dark matter candidates studied in [1]. and approximated nuclear form factors with their unit value that holds at small enough E R . In the same approximation, this cross section can be compared to the standard approximation used in searches for spin-independent DM interactions: We see that the dipole cross sections has a characteristic testable enhancement at low recoilenergy E R , arising because the DM/matter scattering is mediated by the massless photon. Furthermore, the magnetic-dipole cross section has a characteristic suppression at small v > v min = M N E R /2µ 2 , which could be tested relying on the seasonal variation in the average v 2 .
We parameterize the dipole moments in terms of their gyro-magnetic and gyro-electric constant g M and g E as in eq. (78). Composite DM generically predicts an order one gyromagnetic factor g M , and a possibly sizeable gyro-electric factor g E ∼ θ TC Min[m Ψ ]/M DM as discussed in section 4.
This means that for M DM ≈ 100 TeV and g M ∼ 1 the magnetic effect is 3 orders of magnitude below the experimental limit, and at the level of the neutrino background, see also [23]. The electric effect is comparable to the present LUX bound for g E ≈ 0.01 and M DM ≈ 100 TeV, as illustrated in fig. 3a. In some models DM has chromo-dipoles, that lead to a similar scattering rate with e 4 Z 2 /E R replaced by g 4 3 /Λ QCD times a nuclear form factor, which is strongly suppressed at energies below Λ QCD . Thereby chromo-dipoles do not compete with electric dipoles.
Some composite DM models predict that DM is a TCb with higher-spin. For example a spin-1 TCb B µ could have a dimension-4 interaction B µ B * ν F µν with a photon. Even when the lighter TCb is mostly composed of neutral SM singlets N , it also contains a small component of charged heavier techni-quarks with a momentum asymmetry (an effect analogous to the strange momentum asymmetry in nucleons [24]).

Direct detection of real Dark Matter
Techni-baryon DM in SO(N ) TC gauge theories has novel interesting features compared to SU(N ) TC models: there is no techni-baryon number conservation, so DM is a real state with no techni-baryon asymmetry. In most golden-class models, the techni-quarks have Yukawa couplings to the Higgs. As a consequence the DM candidates TCb with Y = 0 mix with TCb with Y = 0 after electro-weak symmetry breaking. The resulting lightest TCb is a Majorana fermion for N odd, a real scalar for N even. To illustrate this point, let us consider for example where the TCb Higgs couplings y can, in principle, be derived from the Yukawa couplings among techni-quarks. The dots refers to other TCb states that are expected to be heavier but could still be relevant if they mix significantly. The mass matrix is analogous to one of the bino and higgsino in supersymmetry. Furthermore, in our scenario TCb have a common mass m B generated by strong dynamics and are mildly split by techni-quark masses and gauge interactions: thereby the spectrum resembles the case known as 'well tempered neutralinos' [25]. Due to the mixing with Majorana states, the lightest DM TCb is a Majorana fermion. This significantly changes the phenomenology of direct detection: a Majorana fermion can neither have vector couplings to the Z, avoiding the severe constraints from spin independent cross section, nor dipole moments, removing the signals discussed in section 5.1. However, Majorana technibaryon DM χ can have an axial coupling to the Z, that leads to a spin dependent cross-section with the nuclei. Using the present LUX bound [26] σ n SD < 1.7 10 −39 M DM / TeV, one finds The situation is illustrated in fig. 3b. This is a significant constraint only if the mixing angle among states of different hyper charge is large so that g A ∼ O(1). This situation is achieved for Even assuming negligible techni-quark masses, SM gauge interactions split singlets and doublets by a few per cent: For a TCb mass around 100 TeV the condition (90) is unlikely to be realised: in the opposite regime ∆m yv the lightest TCb has suppressed coupling to the Z, Another effect of phenomenological relevance can arise if m 2 1/2 m 1 0 . In this case the lighter complex doublet splits into two real states, with a mass difference ∆m 2 1/2 ≈ y 2 v 2 /∆m. The Z gives a tree level coupling between the real mass eigenstates, becoming irrelevant for direct DM searches if ∆m 2 1/2 > ∼ 100 keV. A smaller mass difference can be obtained for y ∼ 10 −3 and gives rise to inelastic DM phenomenology [27].

Higgs-mediated direct detection of Dark Matter
In both cases (real and complex DM) many golden-class composite DM models contain Yukawa couplings to the Higgs in order to break species number symmetries that would lead to unwanted stable particles. Such Yukawa couplings give rise to an extra Higgs-mediated contribution to the spin-independent cross section for direct DM searches, given by Table 4: Techni-pion content of color neutral golden-class composite DM models.
for DM with any spin. Here f N ≈ 0.3 is a nuclear form factor, v ≈ 174 GeV is the Higgs vev, and g DM is the dimension-less coupling of the TCb DM candidate with mass M DM (h) to the higgs, defined as and roughly given by the Yukawa couplings of the Higgs to techni-quarks. The size of these Yukawa couplings is unknown. The LUX bound on σ SI implies g DM < M DM /75 TeV.

Techni-pions at colliders
As explained in eq. (83), cosmology suggests two possible values for M DM : 100 TeV or 3 TeV depending on whether DM has a TCb asymmetry. In both cases TCb DM is out of reach from LHC. Furthermore, DM production at colliders gives missing energy signals which, especially at hadron colliders, can be undetectably below the neutrino background.
Composite DM models predict a richer collider phenomenology: a general prediction is the existence of many resonances of various spin charged under the SM and with quantum numbers that can be determined from the ones of the constituents. Techni-pions are the lightest states in the theory so they are the most promising particles to be produced at colliders 7 . The anomalous coupling of some TCπ with SM vectors V gives rise to single production of TCπ, V * → πV and V * V * → π. Techni-pions can also be produced in pairs via their SM gauge interactions, with cross sections determined by their gauge quantum numbers and summarised e.g. in [28]. SM gauge interactions and techni-quark masses determine TCπ masses as in eq. (11) and (12). For an electro-weak triplet 3 0 the gauge contribution alone is The two values for M DM , 100 TeV or 3 TeV, correspond to M 3 0 ≈ 10 TeV (significantly above LHC capabilities) or M 3 0 ≈ 300 GeV (observable at LHC).
The only exception to the rule above is TCπ SM singlets η that do not receive mass from SM gauge interactions. Their mass is entirely determined by the constituent techni-quark masses, such that these TCπ could be very light. Usually such singlets undergo decays into pairs of SM gauge bosons through chiral anomalies [2]; when present their axion-like couplings to photons provides a mild constraint on their mass (that need to be larger than a keV) and a production mechanism at colliders. Each composite DM model predicts a distinctive set of TCπ, as summarised in table 4.
The collider TCπ phenomenology can in principle discriminate golden-class from silver-class models [29]. In both cases TCπ without species number undergo anomalous decays into pairs of SM weak vectors, π 1 0 , π 3 0 , π 5 0 → W W, ZZ, γγ (models with coloured TCπ, omitted from table 4, also predict anomalous decays into gluon pairs). In models with G-parity (Ψ = V ) the π 3 0 is stable. Techni-pions made of different species decay via couplings that violate species number. In silver-class models such couplings are provided by higher dimension operators involving SM particles (for example 4-fermion operators), giving decays into such SM particles. If these operators are suppressed by a large scale, the decay is slow leading to displaced vertices or apparently stable particles on collider length scales, see [2] for a detailed discussion.
In golden-class models, species number and G-parity can be broken by Yukawa couplings with the SM Higgs boson. As a consequence, TCπ made of different species undergo decays into same specie TCπ (possibly off-shell) emitting one or more Higgs doublets H. For example a doublet with Y = 1/2 and a singlet with Y = 1 can decay as and π 1 0 in turn decays into SM bosons through anomalies. Thereby, unlike in silver-class models, the SM fermions exhibit peaks in their invariant-mass distributions at the h, W, Z masses (the Goldstone components of the Higgs doublet become the longitudinal components of the W, Z vectors) [29].

Electron electric dipole
Many models contain Yukawa couplings of techni-quarks with un-eliminable complex phases, generating electric dipole moments for light SM fermions. Let us consider for example the model Ψ = L + V with SO(N ) TC . The techni-quark Lagrangian contains schematically 8 , It contains one physical CP-violating phase corresponding to arg [m L m V y * L y * R ]. Ignoring technicolor interactions, an EDM is generated through the diagrams in the left panel of fig. 4, giving, in leading log approximation [30], For the electron one finds to be compared with the experimental bound d e < 8.7 × 10 −29 e cm at 90% C.L [31]. However, the approximation of neglecting technicolor interactions is only reliable for m L,V > Λ TC . In the more interesting regime m L,V < Λ TC techni-color effects cannot be neglected and the loops will be dominated by the hadrons of the theory, as depicted in the right-handed panel of fig. 4. A detailed study will appear in [29]. .

Unification of SM gauge couplings
Throughout the paper we assumed that techni-quarks belong to fragments of unified SU (5) representations. We here study if they can improve unification of SM gauge couplings. The large number of independent masses allows for considerable freedom; we make the extra assumption that the missing members of the unified SU(5) multiplets have a common mass M X , below the GUT scale and above the TC scale Λ TC . Furthermore we make the rough assumption that the strong dynamics does not contribute to the running of the SM gauge couplings below the Λ TC ∼ 100 TeV, ignoring threshold effects including those of TCπ. With this mass ordering, in 1-loop approximation the running of gauge couplings is given by where ∆b i is the contribution from techni-quarks (listed in table 1), and ∆b is the contribution from the full SU (5) (5) group theory [33], and the arrows are the contributions to (∆b 32 , ∆b 21 ) from the fragments of SU(5) representations listed in table 1.
The total β-function coefficient in any given model is obtained summing the contributions of each techni-quark taking into account their techni-color multiplicity N . We see that models that can provide successful unification must contain a V or a Q in order to obtain the desired sign of ∆b 21 . For example: • The golden-class SU(3) TC model Ψ = Q ⊕D, with techni-quarks coming from unified 5 ⊕ 10 + h.c. multiplets of SU (5), provides successful unification having assumed Λ TC ≈ 100 TeV. The running of the couplings is shown in fig. 5b. Such a low unification scale would be excluded by proton decay. However, given the large uncertainties (we performed a one-loop analysis, ignoring threshold effects that could be sizeable at the technicolor scale, in view of the light TCπ) such a model could still be viable.

Conclusions
Extensions of the SM with new strong interactions are interesting from the point of view of Dark Matter. First, they naturally provide new stable particles, thanks to accidental symmetries analogous to baryon number that guarantees the stability of the proton within the SM: DM could be the lightest techni-baryon (TCb) or techni-pion (TCπ). Second, the lightest among the many TCb tends to be the one with least SM gauge interactions, thereby explaining why DM has no color, no electric charge, and at most a small hypercharge. The models that we propose are compatible with all present bound from collider and precision experiments because, with techni-quarks in a real representation of the SM gauge group, the new strong interactions do not break the electroweak symmetry. The Higgs doublet is elementary and we do not address the hierarchy problem here. We use the old name 'techni-color' in order to emphasize that we do not postulate desired good properties of effective Lagrangians. On the contrary, we propose fundamental theories where all the good properties follow from an appropriate choice of the quantum numbers: a concrete 'techni-color' gauge group and a concrete set of techni-quarks.
In the simplest 'golden-class' of models, everything follows from a renormalizable Lagrangian. In 'silver-class' models, mild assumptions on non-renormalizable interactions are needed in order to break accidental symmetries and get rid of unwanted stable particles. The list of 'golden-class' models is meant to be exhaustive, within some assumptions: no techni-scalars, only techni-fermions that transform in the fundamental representations of the technicolor gauge group, and in representations of the SM gauge group which are compatible with SU(5) unification. We found successful models with both SU(N ) TC and SO(N ) TC techni-color groups. We did not explore exceptional groups.
In SO(N ) TC theories DM is a TCb, stable thanks to a Z 2 = O(N )/ SO(N ) symmetry: there is no conserved techni-baryon number, such that DM is a real particle (a Majorana fermion for odd N , a real scalar for even N ) with no TCb asymmetry, no magnetic nor electric dipole. Assuming that its cosmological abundance comes from thermal freeze-out of technistrong annihilations into TCπ, the DM mass is expected to be around 100 TeV. TCb mix once the Higgs boson acquires its vacuum expectation value (somehow analogously to the Wino/Bino/Higgsino system in supersymmetry), giving the following phenomenology: in some regions of the parameter space DM can have an axial coupling to the Z, detectable in directdetection signals; in other regions of the parameter space it behaves as inelastic DM.
In SU(N ) TC theories, the lightest TCb is a complex particle, stable thanks to conservation of an accidental U(1) TC techni-baryon number. The DM mass could again be around 100 TeV: a Dirac fermion however can give sizable magnetic and electric dipole moments, giving directdetection cross-sections enhanced in a characteristic way at low recoil energy with respect to the case of a standard spin-independent cross section. A large θ TC -angle of the new strong sector can give an electric dipole such that direct detection is just below present bounds; while a magnetic dipole cross section (suppressed at low DM velocities) is within the capabilities of future direct detection experiments. Alternatively, the cosmological DM abundance could be due to a TCb asymmetry, with a DM mass around 3 TeV.
In both cases, successful DM models often need Yukawa couplings with the Higgs boson in order to break unwanted techni-flavor symmetries, leading to extra spin-independent direct detection signals. CP-violating phases also lead to a possibly detectable electric dipole moment for the SM particles, such as the electron.
In some models composite DM has spin 1 or higher.
Concerning collider experiments, each model predicts a distinctive set of techni-pions, summarised in table 4, which are at most a factor 10 lighter than DM itself, than techni-baryons and than other vector composite resonances. Some techni-pions undergo anomalous decays into SM vectors (and can be singly produced via the inverse process), others decay into lighter techni-pions (and can be doubly produced via their SM gauge interactions) emitting one or more Higgs doublets (i.e. h, W, Z), or, in silver-class models, emitting other SM particles.
We next decompose each component on the right hand side of eq. (105) under the SM gauge group. This can require non-trivial group theory computations: for example a techni-quark V (triplet under SU(2) L ) lies in the fundamental representation of techni-flavor SU(3) TF : TCb lie in higher representations of SU(3) TF than need to be decomposed under SU(2) L . In general, we need to decompose a given representation with K boxes of SU(n c n L ) under SU(n c ) × SU(n L ), where the fundamental of SU(n c n L ) is now embedded as (n c , n L ). This can be done writing all the representations of SU(n c ) and SU(n L ) with K boxes. From group theory we know that each tableau is associated with a representation of the permutation group S K with a given symmetry. Then (D 1 , D 2 ) appears in the decomposition if the product of D 1 and D 2 representations contains a component with the S K symmetry of the initial representation. Here is the decomposition of the two-index symmetric and antisymmetric tensors under SU(n c ) and SU(n L ): To be concrete, consider the techni-quark Ψ = (3,2). The decomposition of the two index tensors above under SU(3) c × SU(2) L become: If with respect to any SM group factor the techni-quarks transform in a representation n i higher than the fundamental, we can embed it into the fundamental of SU(n i ) and decompose representations of this larger group under the SM group. For example in the Ψ = V model, the techni-quark is a vector of SU(2) L : we can think of the two-index symmetric 3 of SU(2) L as the fundamental of SU(3) into which SU(2) is embedded symmetrically. With simple group algebra we find: from which we get the decomposition rules 6 = 5 ⊕ 1 and 8 = 5 ⊕ 3 for the SU (3) representations under the SU(2) group. After this step, each state in eq. (105) is labeled by the quantum numbers (n c1 , n L1 , n c2 , n L2 ) Y . To obtain the final representation under the SM group we have to identify SU(3) c and SU(2) L factors, taking the tensor product n c1 ⊗ n c2 and n L1 ⊗ n L2 .
For SU(4) TC , we can proceed analogously. First, we decompose the lightest TCb multiplet of then we decompose each representation under the SM group and identify the SU(3) c and SU(2) L factors respectively.
As discussed in section 3.1, the SO(N ) TC theories can be analyzed starting from the results of the SU(N ) TC models.

B Silver-class composite DM models
We here list silver-class SU(N ) TC and SO(N ) TC models restricted for simplicity to N = 3, 4 technicolors and N s ≤ 2 species of techni-quarks. These models satisfy TC asymptotic freedom and do not give rise to sub-Planckian Landau poles. But, besides to acceptable DM candidates, they give rise to unwanted stable states, that are TCπ with hypercharge or color, stable because of accidental symmetries such as species number or G-parity. They can be made unstable with extra model building, for example adding higher dimension operators that break the accidental symmetries, as explained in section 1.
TCπ made by both techni-quarks have non zero hypercharge and are stable because of species number.
If we want to make the model phenomenologically viable, we need to break species number by ad hoc assumptions. For N = 3, the lightest TCb live in the fundamental of SU(2) TF , that is The DM candidate is the spin 3/2 singlet N N N * that belongs to the symmetric representation of SU(2) TF . It can be the lightest TCb if m N m E . The same conclusion is valid for N = 4, where the DM candidate is the spin 2 singlet N N N N * that lives in the symmetric representation of SU(2) TF .

SU(N ) TC model Ψ = E ⊕Ẽ
This model with N TF = 2 can give rise to a neutral TCb for N = 4. It presents a Landau pole for g Y slightly above the Planck scale and gives rise to the following TCπ The 1 ±2 TCπ made by both species are stable, so that we need to break species number. The model gives only one lighter TCb, that is a SM singlet made by EEẼẼ and is a good DM candidate.

SU(N ) TC model Ψ = L ⊕L
The TCπ of this model with N TF = 4 are: where states with hypercharge are stable, unless the species number symmetry is broken. Analogously to the previous model, it can provide a DM candidate for N = 4, where the lighter TCb fill a 20 of SU(4) F , that decomposes as The list contains two singlets LLLL that are good DM candidates.
This model, studied in detail in section 4.1, has N TF = 3 and for N = 3 gives rise to the successful DM candidate LLE. In this case, both TCπ and TCb live in the adjoint of SU(3) TF , that decomposes as  For the Ψ = D(U ) ⊕ N models we get the following TCπ: Analogously, for the Ψ = Q ⊕ N (N TF = 7) model we get: Because of species number symmetry, we have stable TCπ with color and/or hypercharge, so that we need to break this accidental symmetry to avoid the strong experimental bounds. As in the other models containing the singlet N , the TCb DM candidate is an higher spin state made only by N techni-quarks. For N = 3 and N = 4 it has spin 3/2 and 2 respectively and it can be the lightest if the other techni-quark is sufficiently heavier.
As before, we study together the two models defining Y = 1/3, −2/3 for D and U respectively. Because of the presence of the techni-quark V , these models are allowed only for N = 3. They have N TF = 6 and give rise to the following states: TCπ made by LL andLL have non zero hypercharge and are stable because of accidental U(1) species symmetry. The extra physics needed to avoid unwanted stable TCπ can be nicely realised considering the golden-class model Ψ = L ⊕ N in the limit where m N Λ TC , such that the (LH) 2 effective operator is generated at low energy.
TCb can contain a DM candidate for N even. For N = 4, this is the singlet (LL) 2 . The other TCb that need to be specified are They give rise to an extended list of TCπ: where the extra state made by N and D or U are unwanted stable particles. For N = 4 the lightest TCb DM candidate is again a singlet, made by DD(DD + N N ) or UŪ (UŪ + N N ).