Dark Matter from self-dual gauge/Higgs dynamics

We show that a new gauge group with one new scalar leads to automatically stable Dark Matter candidates. We consider theories where the Higgs phase is dual to the confined phase: it is known that SU(2) gauge theories with a scalar doublet (like the Standard Model) obey this non-trivial feature. We provide a general criterion, showing that this self-duality holds for SU(N), SO(N), Sp(N) and G_2 gauge dynamics with a scalar field in the fundamental representation. The resulting Dark Matter phenomenology has non-trivial features that are characteristic of the group, and that we discuss case by case. Just to mention a few, SU(N) has an accidental conserved dark baryon number, SO(2N+1) leads to stable glue-balls thanks to a special parity, G_2 leads to a Dark Matter system analogous to neutral kaons. The cosmological Dark Matter abundance is often reproduced for masses around 100 TeV: all constraints are satisfied and lighter dark glue-balls can affect Higgs physics. These theories acquire additional interest and predictivity assuming that both the dark and weak scales are dynamically generated.

We show that a new gauge group with one new scalar leads to automatically stable Dark Matter candidates. We consider theories where the Higgs phase is dual to the confined phase: it is known that SU(2) gauge theories with a scalar doublet (like the Standard Model) obey this non-trivial feature. We provide a general criterion, showing that this self-duality holds for SU(N ), SO(N ), Sp(N ) and G 2 gauge dynamics with a scalar field in the fundamental representation. The resulting Dark Matter phenomenology has non-trivial features that are characteristic of the group, and that we discuss case by case. Just to mention a few, SU(N ) has an accidental conserved dark baryon number, SO(2N + 1) leads to stable glue-balls thanks to a special parity, G 2 leads to a Dark Matter system analogous to neutral kaons. The cosmological Dark Matter abundance is often reproduced for masses around 100 TeV: all constraints are satisfied and lighter dark glueballs can affect Higgs physics. These theories acquire additional interest and predictivity assuming that both the dark and weak scales are dynamically generated.

Introduction
We know that Dark Matter (DM) exists because we observed its collective gravitational interactions, but we do not know what DM is. Many theories are possible. Since gauge interactions are maximally predictive in relativistic quantum field theory, it makes sense to explore theories where gauge dynamics leads to DM. We thereby add a new 'dark' gauge group G. Its glue-balls could be DM without any interaction with the Standard Model sector. In order to thermally reproduce the cosmological DM abundance we minimally connect the dark sector to the Standard Model by adding one scalar field S charged under G. Depending on G, this leads to non-trivial accidental symmetries that imply DM stability with non-standard physics. Despite that light elementary scalars are considered as unnatural by some theorists, interesting DM matter models based on scalars have already been proposed: 1) The most minimal DM model in terms of new degrees of freedom involves just one singlet scalar S [1]. This is stable imposing an ad-hoc Z 2 symmetry S → −S and assuming that the S vacuum expectation value vanishes. Direct detection bounds excluded a significant part of the parameter space of this model [1].
2) Next, if the field S is complex, describing two scalar degrees of freedom, it can be charged under a new G = U(1) gauge group. A vacuum expectation value of S breaks U(1) to nothing and the resulting massive vector A µ is a DM candidate, stable thanks to charge conjugation, S → S * and A µ → −A µ , which is a symmetry if the U(1) has vanishing kinetic mixing with hypercharge [2,3].
3) A more interesting model where DM stability is automatically implied by the particle content has been proposed in [2,4,5], assuming that the scalar S fills the fundamental representation 2 of a new SU(2) gauge group. A vacuum expectation value of S breaks SU (2) to nothing and the DM candidates are the three SU(2) vectors, which acquire a common mass because of an accidental custodial symmetry.
The SU(2) model admits two apparently different phases: Higgs and confined. A non-trivial feature of the SU(2) model -interesting even from a purely theoretical point of view -is that the two phases give the same spectrum of asymptotic particles. The lack of a sharp distinction between the Higgs and confined phases in SU(2) theories with a scalar in the fundamental has been proved by Fradkin, Shenker et al. [6][7][8][9]. A detailed analysis of how this surprising duality applies to the Standard Model can be found in [10,11] (we now know that the SU(2) L gauge group is weakly coupled, so that in the SM this duality has no physical interest). We will find extra examples of Higgs/confinement dualities, and propose a general criterion: such a duality holds when a scalar S in a representation R can break the gauge group G to a unique sub-group H (and thereby with a Higgs phase that is unique). In these cases S admits a single quartic coupling, and the broken theory contains a single Higgs scalar, that we call s. This happens when S fills a fundamental of the SU(N ), SO(N ), Sp(N ), G 2 groups (up to equivalences). While in the original model [2,4,5] G = SU(2) gets fully broken, in our examples H has a non-trivial gauge dynamics -its own confinement -that must be taken into account. On the other hand, a scalar in the fundamental of F 4 , E 6 , E 7 , E 8 , or in a higher representation of any group, such as a spinorial of SO (10), instead has multiple quartic couplings and gives inequivalent breaking patterns, leaving extra scalars in the broken theory.
We will here study theories that satisfy the Higgs/confinement duality, and their application to DM. Such theories can be seen as extensions of those previously listed in 1), 2), 3), and give qualitatively new physics. We consider one elementary scalar S in the fundamental representation of a gauge group G with vectors G a µν in the adjoint. We consider the most generic renormalizable Lagrangian 1 with scalar potential S is complex when G = SU(N ) or Sp(N ): in such cases the the theory is invariant under an accidental U(1) global symmetry, dark baryon number, that rotates the phase of S. S is real when G = SO(N ) or G 2 : we will discuss the accidental symmetries of these theories. These minimal theories give non-trivial DM physics. If G confines, baryons made of scalars S are stable DM candidates. As we will see, their nature qualitatively depends on the group G. If S gets a vacuum expectation value, G gets broken to a subgroup H, and some massive vectors are accidentally stable DM candidates. At lower energy H confines, giving rise to various states (dark glue-balls, dark mesons, ...) and to baryonic DM, in such a way that the Higgs/confined and G-confined phases are equivalent. 2 1 The dark gauge group can have an extra topological term. In the absence of fermions, it cannot be rotated away. Such term would violate CP at non-perturbative level. The SU(N ), SO(2N ) and E 6 groups with symmetric Dynkin diagrams admit a Z 2 outer automorphism (complex conjugation) [12] that acts on vectors by flipping the sign of some vectors, as determined by the vanishing of some f abc group structure constants. More simply, the CP-even vectors are those associated to purely imaginary generators T a in some complex representation (e.g. fundamental or spinorial). 2 In order to avoid confinement, [13,14] considered non-minimal models with enough multiple scalars that SU(N ) gets broken to nothing. We accept condensation and focus on the minimal scalar content. Given that DM is the lightest stable particle, this also approximates the DM phenomenology of more general theories provided that the extra particles are heavier at least by ∆M > ∼ Λ DC , where Λ DC is the scale at which g DC becomes strongly coupled, if unbroken.
Models with a new confining gauge group G and new fermions F have been explored in [15][16][17]: in such models communication with the SM arises if F is charged also under the SM gauge group: models need to be selected such that the composite DM is neutral. In scalar models, instead, we can assume that S is neutral under the SM (resulting into a neutral DM candidate) because S interacts with the Higgs through the mixed scalar quartic λ HS .
The paper is structured as follows. In section 2 we consider the group G = SU(N ) and study both the Higgs and condensed phases, focusing on their equivalence, on the accidental symmetry that protects the stability of DM, as seen by both the dual phases, and on DM phenomenology. We then extend the analysis to the other groups for which we find that the duality holds: SO(N ) (section 3), Sp(N ) (section 4) and G 2 (section 5). Conclusions are finally given in section 6, where we summarize our main results.

SU: Higgs phase
Independently of whether symmetry breaking happens dynamically, in the Higgs phase S can always be written as such that the gauge group SU(N ) gets broken to SU(N − 1), leaving one degree of freedom s in S. While S breaks dark baryon number U(1) DB (under which S has charge 1), a stable DM candidate remains thanks to an accidental global U(1) symmetry. Its generator N (1, . . . , 1, 0)/(N − 1) is the unbroken linear combination of U(1) DB and the broken U(1) gauge symmetry in SU(N ) corresponding to the generator (cf. Appendix A) Here and in the following, we normalize SU(N ) generators in the fundamental representation as Tr(T a T b ) = 1 2 δ ab . It is especially interesting to consider dynamical symmetry breaking through the Coleman-Weinberg mechanism, obtained by setting M S = 0. Assuming that λ HS is negligibly small, the scalar S dynamically acquires a vacuum expectation value w = s * e −1/4 where s * is the Renormalization Group Equation (RGE) scale µ at which the running quartic coupling λ S (µ) crosses 0, becoming negative at low energy, in view of its RGE at one loop In such a case the scalar s is known as 'scalon' [18] and its mass squared is one-loop suppressed, M 2 s = w 2 β λ S , with β λ S ≡ dλ S /d ln µ [5]. If the Higgs mass term is absent too, this model can also generate the weak scale v, where v ≈ 246 GeV is the needed Higgs vacuum expectation value. Assuming a small positive λ HS , the weak scale is generated as v ≈ w λ HS /2λ H [5]. More complicated expressions hold if λ HS is not negligibly small.
Writing the gauge bosons as the perturbative spectrum is: • the scalon s, singlet under SU(N − 1), with mass M s ; • N (N − 2) massless dark gluons A µ in the adjoint of SU(N − 1); The case N = 2 of this model was studied in [2,4]: the dark gluons do not exist, and W, Z are degenerate thanks to a custodial symmetry. We consider N > 2 such that M W < M Z < √ 2M W . The DM candidate is the W, that undergoes WW * → AA, ss, sA, sZ, AZ annihilations, while co-annihilations WZ → Ws [2] become irrelevant because Z is not DM for N > 2.

Condensation of SU(N − 1)
The case N > 2 is qualitatively different from N = 2 because the vectors A confine at a scale Λ DC that can be exponentially smaller than M W : where α DC (M W ) is the value of the dark gauge coupling at the W mass. The squared masses of dark-colored particles receive extra contributions of order Λ 2 DC . After the condensation of SU(N − 1), the spectrum of the theory contains: • dark glue-balls AA with mass M DG ∼ 7Λ DC [19]; • the dark scalon s; • the Z; • scalar mesons WW * that decay through the annihilation of their constituents; • dark baryons B ∼ W N −1 , that remain as stable DM.
For Λ DC M W the dark baryon spectrum can be computed from non-relativistic quantum mechanics. 3 In order to later address the condensed phase Λ DC ∼ M W (where the non-relativistic approximation does not hold) we here compute the dark baryon spectrum using a less usual formalism: by constructing gauge-singlet operators made of the constituent fields W I µ and their covariant derivatives that interpolate between the dark baryon and the vacuum. For N = 4 the ground state is a scalar because the anti-symmetric spin wave-function is obtained as (3⊗3⊗3) antisym = 1. The associated dark-baryon operator is, for instance, the The vector operator IJK µνρσ W I µ W J ν W K ρ excites a physical spin-1 resonance only at higher order in β.
For N = 5 the lowest-lying bound state is a spin-1 resonance, corresponding for instance to the operator B λ = IJKL µνρσ (D µ W ν ) I W J ρ W K σ W L λ . A scalar state arises at higher order in the non-relativistic limit, e.g. from the O(β) operator IJKL µνρσ W I µ W J ν W K ρ W L σ .
For N = 6 two derivatives are needed and at leading order the combination is possible, which gives a spin-1 state.
Similar considerations can be done for higher N . In each case, different resonances are split by an amount ∆M ∼ α 2 DC M W . For some values of N , bound states with different spin exist at leading order; their fine-structure mass splitting is ∆M ∼ α 4 DC M W . 3 The lightest dark baryon is obtained by minimizing the angular momentum in the spatial part of its wavefunction, compatibly with its symmetry under the exchange of two constituents. For a baryon the dark-color part is totally anti-symmetric, so the product of the spin and spatial wave-functions must also be totally antisymmetric. For N = 3, 4 the spin part alone can be anti-symmetrized, so that the ground state can have a totally symmetric s-wave. For higher values of N , the spatial wave-function cannot be symmetric, and some orbital angular momentum must be involved.

SU: condensed phase
In the discussion above, we studied the Higgs phase of a theory with a SU(N ) gauge group and a scalar S in the fundamental representation. The same theory admits an apparently different confined phase, where the gauge group SU(N ) becomes strong at energies Λ DC > ∼ M S (confinement happens before Higgsing) such that a SS * condensate forms, rather than a vacuum expectation value. While the strong dynamics of scalars is mostly unknown, Fradkin et al. [6][7][8][9] claim a nontrivial theorem: in the presence of a scalar in the fundamental there is no sharp distinction between the confined phase and the Higgs phase. This means that the same asymptotic particles appear in the spectrum -a surprising result given that the two phases naively look different (for example, dark baryon number is unbroken in the confined phase). More in general, asymptotic states (even in the Higgs phase) should be described through gauge-invariant operators. This state/operator association has practical use in lattice computations and is interesting from a formal point of view as a way to describe physics in an explicitly gauge-invariant way, in particular avoiding splitting fields as a fluctuation over a vacuum expectation value.
The case with N = 2 has been explicitly discussed in [4], 4 showing that both phases lead to a real scalar s and to 3 degenerate massive vectors Z µ , W µ , W * µ . As discussed above, in the Higgs phase s is the radial part of S, and W µ , Z µ are the vectors of SU(2) DC . In the confined phase asymptotic states (bound states) are associated to gauge-invariant (singlet) operators. The same asymptotic states are recovered as [2] with M W = M Z dictated by a custodial symmetry. The W states involve contractions with the SU(2)-invariant ij tensor and thereby are replaced by S N baryons for N > 2. Indeed, N > 2 leads to a more complicated dynamics in the Higgs phase, due to the unbroken SU(N − 1) that confines at lower energy (the relevance of this confinement for the validity of the theorem was stressed in [20]). The main states found in the confined phase match those found in the Higgs phase as follows: 5 • Glue-balls, associated to G a µν G a µν ; • The scalar S † S corresponds to s; 4 The SM provides a more complicated example of a SU(2) theory with extra fermions: the equivalence of the Higgs and confined phases has been discussed in [10,11]. 5 In the case of SU(3) DC the validity of the theorem has been tested through lattice computations [21][22][23], with a puzzling result. Even in a weakly coupled theory the physics in the gauge-invariant formalism seems different from the physics found in the standard formalism, where scalars are split in a gauge dependent way as fluctuations over a vacuum expectation value. For N = 2 (for example, in the SM) the difference is claimed to be in details of cross sections; for N = 3 the difference is claimed to be already at the level of the spectrum of asymptotic states. We follow the standard procedure.
• The vector S † D µ S corresponds to Z µ ; • Baryon states B ∼ S N can be constructed as follows: in view of the contraction with the totally anti-symmetric tensor with N indices, a non-zero contribution is obtained when at least N − 1 terms contain covariant derivatives D acting on the corresponding S; the baryonic interpolating operators can thus be written as To see that the condensed baryons B ∼ S N correspond to the baryons of the Higgs phase B ∼ W N −1 discussed in section 2.1, we notice that when one component of S acquires a vacuum expectation value, the term in the square brackets in eq. (10) reduces to the W N −1 baryons formed in the Higgs phase when the residual SU(N −1) confines, after the identification of the N − 1 Goldstone bosons D µ S J ↔ W J µ . The baryons of the Higgs and confined phases are in a one-to-one correspondence, consistently with Fradkin-Shenker theorem [6-9].

SU: phenomenology
As the Higgs and confined phases contain the same asymptotic particles, we perform all computations in the Higgs phase for weak couplings.
The model has 4 extra parameters beyond the SM: g DC , λ S , λ HS , M 2 S . For simplicity, we will discuss predictions in the dimension-less limit, where the model has only 2 extra parameters beyond the SM, given that M S vanishes and that the weak scale v is generated by the dynamical scale w, such that λ HS is fixed by Furthermore, one can trade λ S for the DM mass (N − 1)M W , and determine its value by assuming that the thermal DM relic abundance reproduces the observed cosmological abundance. At the end, only g DC remains as a free parameter. The condensed phase is smoothly obtained in the limit where g DC ∼ 4π/ √ N − 1 becomes non-perturbative. The phenomenology is similar to the N = 2 model of [5] with an important difference: the presence of extra light glue-balls.

Relic DM abundance
The thermal relic DM abundance is determined by various events.
First, the usual decoupling of free W vectors takes place at the temperature T ∼ M W /25. The relic abundance of W vectors is dictated by σv rel , the tree-level non-relativistic s-wave annihilation cross section (averaging over initial spin and gauge components, and multiplying by κ = 1/2 for complex DM particles, or κ = 1 for real DM particles). Using the Feynman rules collected in Appendix B and summing over all annihilation processes we obtain the needed cross sections.
We first write those generic cross-sections that arise whenever DM vectors W with mass M W fill a representation R under the unbroken group H with massless vectors A: In the above expressions d R is the dimension of the representation R, C R is the quadratic Casimir of R, and C adj is the one of the adjoint of H. Specializing to the case H = SU(n) with n = N − 1, we have C adj = n, C fund = (n 2 − 1)/2n, d fund = n, and κ = 1/2 for the complex fundamental, so that 6 where the last two extra cross sections involve the Z vector present in SU models. These cross sections are enhanced by order-one Sommerfeld and bound-state effects, that we neglect. If W decoupling were the only process, it would leave the present abundance Second, at the scale of dark confinement the W forms either mesons WW * that annihilate, or baryons B ∼ W N −1 that remain as DM. The mass fraction in baryons is estimated as [16] .
6 As a check of our computation, we verified that all cross sections scale as 1/s in the ultra-relativistic limit, often thanks to cancellations related to the Higgs mechanism. The WW * → ZZ process is kinematically closed. For N = 2 cross sections involving A vanish, Z becomes DM forming a degenerate triplet with W, and the result in [5] is reproduced taking into account the extra cross sections 2σv rel (WW * → ss) = σv rel (ZZ → ss) and 2σv rel (WW * → Zs) = σv rel (WZ → Ws) = 3g 4 DC /128πM 2 W .
Third, the B baryons can annihilate with B * . The cross section is enhanced by recombination and ranges between the squared Bohr radius of the ground state, πR 2 B up to 1/Λ 2 DC depending on which levels get occupied during the cosmological evolution. If such annihilations happen at temperatures above the Coloumbian binding energy of these states, the cross section gets enhanced by the thermal size of the occupied levels [24,16].
Fourth, if scalons s and/or glue-balls AA produced by DM annihilations have a long enough life-time, so that they dominate the energy density of the Universe while decaying into SM particles, the reheating effects dilutes the DM density.
As physics is complex, it is useful to show estimates that exhibit the dependencies on parameters. The thermal relic DM abundance is estimated as Y DM ≡ n DM /s ∼ 1/(T dec M Pl σ ann ), by demanding n DM σ ∼ H, where H is the Hubble rate: • If all bound states have large cross section σ ann ∼ 1/Λ 2 DC at T dec ∼ Λ DC , this phase leaves • However, DM can survive forming heavy baryons with σ ann ∼ πR 2 • If the binding energy of heavy baryons The predicted relic density is plotted in fig. 1 as a function of the two free parameters Λ DC and M W . In the red (green) regions the overall DM abundance turns out to be above (below) the cosmological value, which is reproduced on the boundary between them. We show the results both assuming a Bohr-like annihilation cross-section, as well as a fully non-perturbative 1/Λ 2 DC one. All in all, the observed cosmological abundance Ω DM h 2 = 0.11 can be obtained for DM masses of ≈ 100 TeV or higher.

Dark glue-balls
Dark Glue-balls (DG) decay through the λ HS coupling. Their life-time can be computed as follows. Since the s mass is one-loop suppressed with respect to the W mass, the one-loop effective interaction between s and light vectors A can be obtained from their one loop RGEcorrected kinetic term The the upper gray region is excluded by BBN because of too slow DG decays and we also show limits from direct [28] and indirect [25] detection (plotted assuming the cosmological DM abundance).
Expanding for s w gives the interaction which results in various decays depending on the mass ordering. If DG are heavier than the weak scale, they decay into Higgs components as  scalon is heavy, integrating out s gives the Higgs coupling to dark gluons resulting in a mixing angle 1 between the Higgs and the DG, and in the consequent DG decay into SM particles where Γ h DG is the decay width of a SM Higgs with mass M DG . More in general, if s can be as light as h, the above expression for gets replaced by where γ is the mixing angle that rotates the scalars {h, s} to the mass-eigenstates {S 1 , S 2 } with masses M S 1,2 . In terms of v ≈ 246 GeV it is given by [5] with M 2 S 2 ≈ 2v 2 β λ S λ H /λ HS and M 2 S 1 ≈ 2(λ H − λ 2 H /β λ S )v 2 , and where we fix λ H and λ HS in terms of the masses from eq. (6). In the limit M S 2 ≈ M s M h ≈ M S 1 M DG the width reduces to eq. (20). The left plot of fig. 2 shows the predicted value of sin 2 γ, which is equal to the production cross-section for the scalon s in SM Higgs cross-section units (vertical axis), and of M s (horizontal axis) as function of g DC (colored legend). Various present and projected constraints from Higgs measurements and direct searches are also shown [26,27]. In particular, it can be seen that measuring Higgs couplings with a 10 −3 precision, which can be attained at future lepton colliders, would allow to probe models where s is light for several values of N .

DM indirect detection
The cross section for BB * annihilations, that gives indirect detection signals, is enhanced by recombination. For small relative velocities v rel , it can be written as where R B is the size of the baryon, M B its mass, and E B its binding energy. In the limit Λ DC M W where the constituents are non-relativistic one has where we neglected factors of order one. Inserting the baryon mass M B ≈ nM W we get For larger Λ DC > ∼ M W the constituents are no longer non-relativistic, and the cross-section becomes σ BB * ≈ 1/Λ 2 DC . Annihilations produce scalons and dark glue-balls, that decay into SM particles.
We compare the annihilation cross-section with the Fermi-LAT limits on a gamma-ray signal in dwarf spheroidal satellite galaxies of the Milky Way [25]. The resulting bound, under the assumption that the dark baryons reproduce the full DM abundance of the Universe, is shown in fig. 1 as a function of Λ DC and M W . However, in the region excluded by indirect detection the predicted DM abundance is much smaller than the cosmological abundance. The region where the DM abundance is reproduced thermally is allowed by bounds on indirect DM detection.

Direct detection
The DM dark baryon B couples to s (and thereby to h) proportionally to its mass M B ≈ (N − 1)M W : the effective interaction is 2M 2 B sB * B/w. The resulting spin-independent cross section for direct detection is (both for B of spin 0 and spin 1) where f ≈ 0.3 is a nucleon matrix element. In the limit M S 2 ≈ M s M h = M S 1 the cross section reduces to The right plot of fig. 2 shows the predicted value of σ SI (vertical axis) and of M B (horizontal axis) as function of g DC (colored legend). Fig. 1 summarizes the situation as function of two parameters, Λ DC and M W , without imposing that the cosmological DM abundance is reproduced. We again see that an allowed region exists, where all signals are significantly below present bounds. In particular the curve where the observed DM abundance is reproduced lies in the region allowed by all constraints.
Here and below, we estimated that non-perturbativity arises for g DC ≈ 4π/ C adj , equal to 4π/ √ n for SU(N ) models. At such value the Higgs phase smoothly becomes the confined phase. The DM mass that reproduces the cosmological density in the confined phase depends on uncertain strong dynamics.

A fundamental of SO(N )
The renormalizable Lagrangian is given in eq. (1), with S real (since a fundamental of SO(N ) is a real representation). Thereby there is no accidentally conserved U(1) baryon number. We normalize the SO(N ) generators in a non-standard way as Tr(T a T b ) = 2δ ab in the fundamental and Tr(T a T b ) = (2N − 4)δ ab in the adjoint, in order to keep SU(2) ∼ SO(3) manifest. Since the SO(N ) adjoint is the two-index anti-symmetric representation, the SO(N ) gauge vectors G a can be written as G ij = G a T a ij , anti-symmetric under i ↔ j. The RGE are

SO: Higgs phase
The most generic vacuum expectation value of S can be rotated to its N -th component such that SO(N ) is broken to SO(N − 1). Writing the gauge bosons as the perturbative spectrum is: • the singlet scalon s with squared mass M 2 s = w 2 β λ S ; For SO(2) = U(1) dark gluons A are absent and this symmetry reduces to U(1) charge conjugation.

Condensation of SO(N − 1)
When SO(N −1) confines, DM forms dark mesons W i W i (which annihilate), dark glue-balls AA (which decay to SM particles), and various baryons, defined as states formed contracting one i 1 ···i N −1 tensor with the fields of the theory: the heavy W µ i and the light A µν ij . The possibility of using dark gluons as valence constituents of baryons makes a qualitative difference with respect to the SU(N ) case. The lightest baryon is the state that contains the lowest possible number of heavy W and, as discussed in the next sub-section, it is a stable DM candidate: • for N even, baryons contain an odd number of W, and the lightest baryon contains one W: • for N odd, baryons contain an even number of W, and the lightest baryon contains zero W:

SO: condensed phase
We next consider the phase where g DC is non-perturbatively large, such that SO(N ) condenses forming the following singlets under SO(N ): • a meson S i S j δ ij , which is identified with the scalon s in the Higgs phase; 7 • glue-balls GG, identified with the AA glue-balls in the Higgs phase; • baryons formed with one i 1 ···i N tensor.
Differently from SU(N ) baryons, the lightest SO(N ) baryon does not need to be made of N fundamentals as i 1 ···i N S i 1 · · · S i N . Rather, gauge bosons G ij can be used to form SO(N ) baryons. Two constituents S i S j can annihilate into one dark gluon G ij . The lightest baryon presumably contains the minimal number of S, zero or one: These lightest baryons are in one-to-one correspondence with the ones in the broken phase. For N odd, when S gets a vev along its N -th component, the remaining G constituents are identified with the A's; for N even, exactly one of the G constituents is identified with the W in the broken phase due to the tensor, while the others are unbroken generators. The lightest dark baryon is stable because the theory is accidentally invariant under O(N ) rotations R ij with determinant −1 [29]. Acting on baryons B that contain the tensor, such After dividing by SO(N ) rotations one gets a Z 2 symmetry, that we dub O-parity. Dark baryons built with one N -index anti-symmetric tensor are odd under this Z 2 symmetry. O-parity is a special unusual symmetry, analogous to space parity and time inversion, in the sense that it can be written in equivalent explicit ways only after choosing an arbitrary basis in the field space, thus fixing one arbitrary rotation with determinant −1.
For N odd, O-parity is more conveniently realised as a full reflection −1 I i.e. O-parity acts as a usual Z 2 symmetry and the lightest baryon is stable because made of an odd number of S's. For N even, O-parity can be conveniently realised as a reflection under any direction, for example along the first component: η 1 = diag(−1, 1, . . . , 1). 8 As any vector can be gaugerotated to be along the 1 direction, this morally is parity. As η 1 anti-commutes with the generators T 1i (rotation along the 1i plane) and commutes with the other generators, the Lagrangian is invariant under Two baryons can annihilate. The meson SS and the glue-balls GG are even under O-parity and can decay into SM particles in view of the λ HS coupling.
For even N = 2N the theory contains an extra accidental symmetry different from O-parity. Indeed SO(2N ) admits one outer automorphism, which corresponds to complex conjugation C. Its action on vector bosons can be computed by looking at the generators in the simplest complex representation, the spinor. C acts as The consistency with the Lie algebra can be proved analogously to footnote 8. Since S i S j transforms as G ij the C symmetry does not give extra stable baryons (despite the fact that for SO(4N + 2) the 0-baryon is odd under C, whereas for SO(4N ) it is even). 9

SO: phenomenology 3.3.1 Relic DM abundance
As in SU(N ) models, the thermal relic DM abundance is determined by various cosmological events. Specializing eq. (12) to the unbroken H = SO(n) group, so that κ = 1, d fund = n = N − 1, C fund = n−1 and C adj = 2(n−2) in our normalization, we obtain the annihilation cross sections relevant for the usual decoupling of free W vectors When SO(N − 1) confines, we need to distinguish two cases: 9 It might seem surprising that C acts non-trivially on S (a real representation). This becomes intuitive for SO(2) = U(1): two real scalars are seen as one complex scalar S 1 + iS 2 . For larger N , the C symmetry similarly reduces to the usual charge conjugation within the SU(N ) subgroup of SO(2N ). Indeed, since U(N ) = SO(2N ) ∩ Sp(2N ), the SU(N ) subalgebra of SO(2N ) has the form of eq. (61), with σ k = σ 2 . Since 1 I 2 is diagonal, the first set in eq. (61) is even under C of SO(2N ); since T (2) αβ is imaginary, this is even also under C of SU(N ): T imag → T imag , T real → −T real . Analogously for the remaining two sets in eq. (61), since σ 2 is off-diagonal and T (1) αβ is real. • For odd N roughly all W's end up in 2-baryons that decay to 0-baryons, so that the W abundance negligibly contributes to the final DM abundance, approximated by Y DM ∼ Λ DC /M Pl for particles with mass of order Λ DC . Given that in this case the cross-section σ ann ≈ 1/Λ 2 DC cannot be extrapolated from a perturbative calculation, the overall coefficient has to be assumed.
The relic abundance is plotted in fig. 3 as a function of Λ DC and M W . Notice that, since DM never contains more than one heavy constituent, the relevant annihilation cross-section is always taken to be of order 1/Λ 2 DC . Again, the correct DM abundance is obtained for masses of order 100 TeV or heavier.

DM indirect detection
The cross section for BB annihilations, that gives indirect detection signals, is where the O(1) factor is ∼ 100 in the QCD proton case. DM annihilations produce dark glueballs, that decay into SM particles. The Fermi-LAT bound is plotted in fig. 3 as a black line, assuming that the dark baryons constitute the totality of DM. While the indirect-detection signal is larger than in SU, also in this case the region where DM is a thermal relic is not probed yet.

DM direct detection
For even N , DM contains one heavy W constituent and the direct detection cross section is for SO(N ) with even N .
with M B M W . For odd N , DM is a glue-ball that contains zero W constituents ('odd-ball'), and its direct detection is qualitatively different. The odd-ball coupling to s -that mixes with the Higgscan be computed by extending soft theorems. One-loop RGE running for E < M W < E 0 can be written as From this we compute how, at fixed high-energy value of the gauge coupling, α DC (E 0 ), the scale E = Λ DC at which α DC becomes non-perturbative depends on M W and thereby s, finding: Then the odd-ball (or glue-ball) mass term M 2 B B 2 /2 with M B ∝ Λ DC gets promoted to 10 The resulting direct detection cross section is Fig. 3 shows the final results. Again, direct detection constraints are not relevant in the region where DM is a thermal relic. 10 The factor in parenthesis can be rewritten in terms of α DC : this computation is equivalent to using the usual Higgs soft theorem of eq. (17) (that becomes −(A a µν ) 2 7sα DC /8πw with SO factors) and using the scale anomaly to compute the baryonic matrix element of such operator.

Special cases
To conclude, we mention some special cases.
For N = 3 one has SO(3) → SO (2), which is the same as SU(2) → U(1) with a scalar in the adjoint. The U(1) does not confine, leaving long range interactions among DM particles. Furthermore, topological defects are possible. The viability of such models will be discussed in a separate publication.
For N = 4 the identity SO(4) ∼ = SU(2) 2 holds and the vector W lies in the adjoint 3 of the unbroken SO (3).
For N = 6 an extra model with S in the spinorial of SO(6) breaks univocally to SU(3) leaving the scalon as only scalar; in view of the identity SO(6) ∼ = SU(4) this model has been already discussed as SU (4).
For N = 8, the group SO(8) has three representations with dimension 8 (the fundamental and two spinorials) related by a S 3 outer automorphism: all 8 representations are real and break SO (8) to SO(7).

A fundamental of Sp(N )
The group Sp(N ) is defined for even N as the transformations that leave invariant the tensor γ N ≡ 1 I N /2 ⊗ , where ij is the 2-dimensional anti-symmetric tensor. The fundamental representation of Sp(N ) is pseudo-real. We thereby introduce a complex N -dimensional scalar S. The Lagrangian, given in eq. (1), conserves an accidental baryon U(1), by virtue of S T γS = 0. The adjoint is the trace-less symmetric representation with dimension N (N + 1)/2. The RGE are

Sp: Higgs phase
Again the RGE gives that S can radiatively acquire a vacuum expectation value as in eq.
obtained combining the original U(1) with the diagonal generator of the broken Sp (2). Writing the gauge bosons as the perturbative spectrum is: • the scalon singlet s; • one real Z with mass M Z = g DC w/2 and zero dark baryon charge; • one complex W, with mass M W = M Z and dark baryon charge 2.
For N = 2 this is the Sp(2) = SU(2) model of [2,4,5] where W and Z are co-stable DM candidates. For N ≥ 4 the spectrum contains extra particles: • N − 2 complex massive vectors X in the fundamental representation of Sp(N − 2) with mass M X = M W / √ 2 and dark baryon charge 1; • the massless vectors A of Sp(N − 2).
The Z boson decays into A's. At perturbative level the W and X are DM candidates, co-stable thanks to accidental baryon number conservation. The cubic vector vertices are

Condensation of Sp(N − 2)
When the theory becomes strongly coupled, Sp(N − 2) confines giving the following spectrum of asymptotic states: • The scalon s, the Z and W bosons, and dark glue-balls AA.
• Two kinds of dark mesons: the unstable X † X and X † D µ X , which have the same quantum numbers as s and Z, and M µ = X T γ N −2 D µ X , with dark baryon number 2 as W. Only one linear combination of M and W appears among the stable asymptotic states, while the other corresponds to a resonance. A similar situation holds for s and X † X , and for Z and X † DX .
• Dark baryons B (defined as states formed with one i 1 ···i N −2 tensor) are not stable because the tensor can be decomposed as i 1 ···i N −2 = γ i 1 i 2 · · · γ i N −3 i N −2 + permutations [29]. This means that B splits into N /2 − 1 mesons M.
Both the W and the mesons M carry charge 2 under conserved U(1) baryon number.

Sp: condensed phase
Confinement of Sp(N ) gives rise to the following bound states: • S † S corresponding to s and X † X (not distinguished by any quantum number); • S † D µ S corresponding to Z µ and X † D µ X ; • S T γ N D µ S corresponding to W µ and X T γ N −2 D µ X ; • dark glue-balls GG, corresponding to AA.
The condensed phase coincides with the Higgs phase, in agreement with our generalization of the Fradkin-Shenker theorem.

Sp: phenomenology 4.3.1 Relic DM abundance
At large N the dominant perturbative annihilation cross-section is with d fund = 2(N − 2), C fund = (N − 1)/4, C adj = N /2. Annihilation and semi-annihilation cross sections of the V = W, Z vectors into ss and V s are as in the SU(2) model [2]. As X annihilate more than W, the latter can have a larger relic abundance. The two sectors (W, Z and X , A) are however coupled by AW ↔ X X processes, that thermalize their relative abundances, so that the lighter X would get a larger abundance than the heavier W. The final abundance depends on which process decouples earlier: a detailed computation would be needed.
When Sp(N − 2) condenses, about half X form stable mesons M. The cubic vertex X X W * becomes a MW mass mixing. In the limit Λ DC w their masses are M M = √ 2M W so that M decays to W and glue-balls (before that MM * annihilations with σ ann ∼ πR 2 B deplete the M abundance), leaving W as the DM candidate. We estimate that the final DM relic density is approximated by perturbative freeze-out abundance of W (up to the suppression present if glue-balls decay slowly when they dominate the energy density). Hence, differently from the previous cases, in these models dark glue-balls can be so light that they can be probed by collider experiments, for example by measuring Higgs properties.

Indirect detection
The indirect detection cross section is given by the perturbative expressions (13) with n = 1, given that the W is neutral under the unbroken Sp(N − 2), analogously to the SU(2) model. The final results are shown in fig. 4 (left panel) for N = 4. We find that also in this case the parameter space where DM is a thermal relic is not yet probed by indirect detection.

Direct detection
The direct detection cross section is as in the SU(2) model [2], and the region where DM is thermally produced is fully allowed, as shown in the left panel of fig. 4.

A fundamental of G 2
We consider G 2 because it is the only exceptional group that is broken by its fundamental in an unique way, leaving the scalon and no extra scalars, such that the Higgs phase is expected to be equivalent to the condensed phase. G 2 has 14 generators and a real fundamental with dimension 7. The invariant tensors of G 2 are δ ij , i 1 ···i 7 , and O ijk , the anti-symmetric tensor that defines octonion multiplication e i e j = −δ ij + O ijk e k [30]. The most generic renormalizable Lagrangian with a real scalar S i in the fundamental has one non-vanishing quartic (S · S) 2 , while S 3 cubic interactions vanish. Such Lagrangian enjoys a S → −S accidental symmetry.
Normalizing G 2 generators in the fundamental as Tr(T a T b ) = δ ab , and generators in the adjoint as Tr(T a T b ) = 4δ ab , the RGE are • the scalon s; • the 8 massless vectors A in the adjoint of SU(3); • vectors W with mass M 2 W = g 2 DC w 2 /3 in the 3 ⊕3 of SU (3).
Notice that complex W's emerge from a real theory. It is useful to compare G 2 to SU(4), that has 15 generators and 8 scalar degrees of freedom in its complex fundamental 4. The G 2 → SU(3) theory differs from the SU(4) → SU(3) theory because of the absence of the Z vector and because of the presence of αβγ W α W β W γ gauge interactions.

Condensation of SU(3)
Taking into account the confinement of SU(3) gives the following singlets: • mesons M = W α W α which decay to glue-balls; • AA glue-balls; • baryons αβγ W α W β W γ and αβγ W α W β W γ constructed contracting with the SU (3) antisymmetric tensor .
The WWW and the WWW decay to the same final state, in such a way that the decay amplitude cancels for an appropriate combination Re WWW = (WWW + WWW)/ √ 2, similarly to what happens for neutral kaons (one combination is long lived, one combination is short lived). Stability can be understood in terms of the charge conjugation symmetry C of the Lagrangian: where A real (A imag ) are the SU(3) vectors with real (imaginary) generators. Interactions dictated by SU(3) gauge invariance respect this symmetry because it reduces (up to a phase) to the usual SU(3) complex conjugation; one can check that this symmetry is respected also by the extra WWW, AWWW interactions. Then, the Re WWW baryon is stable being odd under the C symmetry, while WWW −WWW is C-even and decays through the WWW interactions.

G 2 : condensed phase
In the confined G 2 -invariant phase, the spectrum is described by • M = S i S i mesons which decay in glue-balls; • GG glue-balls; • O ijk S i S j S k baryons built with the rank-3 invariant antisymmetric tensor O and with derivatives (not shown); • ijklmns S i G jk G lm G ns baryons built with the invariant antisymmetric tensor. The G 2 vectors G ij = T a ij G a are anti-symmetric in ij (like SO vectors) and do not fill the most generic anti-symmetric matrix (unlike SO vectors).
The two baryons have the same spin and mix [31]. In the broken theory they are comparably heavy because GGG gets its AWW * component. The lightest baryon is stable, because of the S → −S accidental symmetry.
which agrees with eq. (53). The compatibility of this symmetry with the Lie algebra is checked as follows: f abc = 0 if one or three indices correspond to odd generators. This symmetry is an inner automorphism of the real group G 2 that when restricted to its SU(3) subgroup acts as complex conjugation, which is the outer automorphism of SU (3) (56) coincides with the Goldstone part of O ijk S i S j S k , by virtue of eq. (55).
In conclusion, the same spectrum is obtained in the Higgs and condensed phases of a G 2 gauge theory with a scalar in its fundamental. The equivalence is more sophisticated because of the breaking of a real group to a complex subgroup: • the meson S T S corresponds to the scalon s; • the operator S T D µ S does not give rise to a Z µ due to the anti-symmetry of the generators (see footnote 7); • dark glue-balls in the condensed phase correspond to dark glue-balls in the Higgs phase; • the lightest baryon, admixture of SSS and SGGG, corresponds to the baryon Re WWW of the Higgs phase; • the C-even baryon Im WWW, that mixes with the scalon and with SU(3) glue-balls, corresponds to resonances of the mesons and glueballs of G 2 .

G 2 : phenomenology
The theory is similar to the SU(4) → SU(3) theory, up to the absence of the Z boson and to the presence of WWW interactions. The perturbative WW * → AA, As, ss DM annihilation cross sections are thereby equal to those given in eq. (12). Furthermore there are extra WW → W * semi-annihilations, as in DM models with an ad-hoc Z 3 symmetry [32]. The perturbative W relic density is thereby similar to the density in the SU(4) → SU(3) model. WWW interactions give an extra difference at non-perturbative level: when the WWW and WWW baryons form, only half of them survive in the stable C-odd component, analogously to a K 0 beam after the decay of the short-lived K 0 S . As DM is now real, indirect detection is enhanced by a order one factor, while direct detection is as in the SU(4) → SU(3) model (after taking into account the slightly different RGE and thereby scalon mass). Figure 4 (right) summarizes our final results.

Conclusions
We have studied models with a new dark gauge group G and a new dark scalar S, selected such that the Higgs phase (where S gets a vacuum expectation value, breaking G to a sub-group H) is dual to the confined phase (where G gets strongly interacting). Fradkin, Shenker and others proved that this happens for G = SU(2) with a scalar S in its fundamental. We argued that the correspondence of the two phases holds whenever the scalar S breaks G to a unique sub-group H. 11 In these cases S admits a single quartic self-coupling, and the broken theory contains a single Higgs scalar, that we call s. This happens when S fills a fundamental of the SU(N ), SO(N ), Sp(N ), G 2 groups. Table 1 summarizes how the Higgs/confinement duality is realized in each model.
We studied such models from the point of view of DM phenomenology. When presenting final results, we further restricted the parameter space assuming that: • the cosmological DM abundance is reproduced thermally; • the G → H symmetry breaking occurs dynamicallyà la Coleman-Weinberg; • the S vacuum expectation value also induces the observed Higgs mass. 11 Indeed, in case the sub-group H were not unique, there would be different spectra of asymptotic states associated to each possible breaking. Hence, the condensed phase of G, which is presumably unique since it is dominated by gauge interactions, cannot be equivalent to the Higgs phases.

Higgs phase
Condensed phase Thanks to these extra assumptions, DM phenomenology is described by one free parameter, the dark gauge coupling g DC of G. The confined phase is obtained smoothly for g DC ∼ 4π/ √ N . Smaller perturbative g DC correspond to the Higgs phase. As strong interactions (either of G or H) are often involved, cosmology often selects the DM mass typical of strong interactions: about 100 TeV. Such DM is heavy enough that the considered models are experimentally allowed. Of course, some of the above assumptions can be relaxed, giving more general phenomenology. The various DM candidates are listed in table 2 for each case, together with their main features.
Our main results can be summarized as follows: • In section 2 we considered G = SU(N ) with S in its complex fundamental representation. In both phases the theory admits an unbroken accidental U(1) dark-baryon number, leading to DM stability. DM is the baryon made by N scalars S. In the Higgs phase G is broken to H = SU(N − 1), and one of the N scalars gets replaced by its vacuum expectation value S , so that DM is made by N − 1 heavy vectors. H confines at a lower scale Λ DC , giving a strong suppression of the cosmological DM relic density. Being made by heavy constituents, the size of DM (and thereby its cross sections in cosmology and in indirect detection) is set by its Bohr-like radius. Higgs soft theorems allowed to compute DM direct detection. Fig. 1   • In section 4 we considered G = Sp(N ) with S in its pseudo-real fundamental. DM is stable thanks to an accidental U(1) dark baryon number. In the Higgs phase G is broken to H = Sp(N − 2), giving two co-stable vector DM candidates W (neutral under H and with dark baryon number 2) and X (charged under H and with dark baryon number 1), with masses M X = M W / √ 2. When H confines at a lower scale Λ DC , two X 's form a meson and their cosmological DM relic density gets strongly suppressed. DM remains as W with cosmological relic density approximately not suppressed by H confinement. Because of this, dark glue-balls can be especially light in Sp models. Up to the presence of dark glue-balls, DM phenomenology is similar to the SU(2) = Sp(2) model. Fig. 4 (left) summarizes DM phenomenology, showing that all experimental bounds are satisfied.
• In section 5 we considered the exceptional group G 2 with S in its real fundamental. In the confined phase, the SSS and SGGG baryons remain stable thanks to an accidental S → −S symmetry. In the Higgs phase G 2 is broken to H = SU(3) and the theory contains massive W ⊕ W vectors in the 3 ⊕3. The theory contains WWW gauge interaction characteristic of G 2 , which give WW → W * processes. As a result the Im WWW baryon decays, while the Re WWW remains as a stable DM candidate, thanks to an inner automorphism of G 2 that reduces to charge conjugation of SU(3). Stability arises as a quantum mechanical interference phenomenon, analogous to how the neutral kaons split into long-lived and short-lived eigenstates. DM size is set by the Bohr-like radius. Fig. 4 (right) summarizes DM phenomenology, showing that all experimental bounds are satisfied.
As we sometimes relied on approximations, various aspects of each model can be more precisely computed. Furthermore, it will be interesting to see if other choices of scalar representations that do not satisfy the Higgs/confinement duality lead to DM candidates with distinct phenomenology.
A.4 G 2 G 2 has 14 generators which can be written in terms of the 7-dimensional matrices T (2) αβ defined in eq. (57b) as [33]