Di-photon resonance and Dark Matter as heavy pions

We analyse confining gauge theories where the 750 GeV di-photon resonance is a composite techni-pion that undergoes anomalous decays into SM vectors. These scenarios naturally contain accidentally stable techni-pions Dark Matter candidates. The di-photon resonance can acquire a larger width by decaying into Dark Matter through the CP-violating $\theta$-term of the new gauge theory reproducing the cosmological Dark Matter density as thermal relic.


Introduction
The simplest and most compelling explanation of the γγ excess observed at M ≈ 750 GeV [1] is provided by an s-channel scalar resonance coupled to gluons and photons.
Theoretical analyses [2,3] find that reproducing the experimentally favoured rate might need non-perturbative dynamics. Strongly interacting models elegantly predict resonances coupled to γγ and gluons (for example, they were mentioned in eq. (95) of [4], before that the excess was found). Loop-level decays into γγ and gluons give typically a small width. Taking into account that the ATLAS fit favours a resonance with a large width Γ ∼ 0.06M (although with less than 0.5 σ improvement from the small width scenario), extra decay channels could be needed. A suggestive possibility is that the 750 GeV resonance has extra decay channels into Dark Matter (DM) particles, given that these decays are relatively weakly constrained [2] and that they allow to reproduce the observed cosmological DM abundance [2,5].
We present simple explicit models where both the 750 GeV resonance and DM are Nambu-Goldstone bosons (NGB) of a new confining gauge theory, and where can decay into DM pairs, providing a relatively large width Γ .
We will study confining gauge theories with fermions in a vectorial representation of the SM, such that the new strong dynamics does not break the SM gauge group. We assume that the Higgs is an elementary scalar particle. As in QCD, the lightest composite states are pion-like NGB arising from the spontaneous breaking of the accidental global symmetries of the new strong dynamics. 1 The anomaly structure is entirely encoded in the Wess-Zumino-Witten term of the chiral Lagrangian, giving rise to predictions for the decay rates into γγ, γZ, ZZ and gg.
The interactions among TCπ are strongly constrained by the symmetries. We will search for theories where some of the TCπ are automatically long lived due to the accidental symmetries of the renomalizable Lagrangian and provide DM candidates. 2 Two symmetries can be responsible for the stability of the DM techni-pions: • Species number. Models where TCq fill two copies X 1 , X 2 of the same representation, give rise to neutral TCπ η ± ∼ X 1X1 ± X 2X2 which undergo anomalous decays to SM gauge bosons and to neutral TCπ Π ∼ X 1X2 stable because of the accidental U(1) 1 ⊗ U(1) 2 symmetry thus providing automatic DM candidates.
• G-parity. In models where TCq fill a representation X plus its SM conjugateX, one can impose a generalised G-parity symmetry that exchanges them. As a consequence the lightest G-odd techni-meson η is a stable DM candidate [6,7]. This G-parity is not an accidental symmetry and can be broken by different mass terms. Furthermore unbroken species number keeps stable the charged TCπ ∼ XX.
The paper is structured as follows. We start in section 2 reviewing some general phenomenological aspects of the γγ excess. In section 3 we discuss the structure of the theories and present the full list of models based on two SM species. In section 4 we discuss general aspects of heavy pion DM phenomenology. Models of composite DM are discussed in section 5, considering in section 5.2 the case where DM stability results from species number, and in section 5.3 models where DM is stable thanks to a G-parity. In section 6 we present our conclusions. A technical appendix on the chiral Lagrangian in the presence of the θ angle follows. 1 In the literature they are sometimes called 'pions' or 'techni-pions' or 'hyper-pions': in order to avoid confusion and lengthy words in the text we will use TCπ for techni-pions, TCq for techni-quarks, Λ TC for the dynamical scale, where techni-color (TC) refers to the new confining gauge interaction. 2 The strong dynamics also produces accidentally stable techni-baryons that could be viable DM candidates [4]. For techni-baryons made of light fermions the thermal production requires a dynamical scale in the 100 TeV range, incompatible with the di-photon excess. This conclusion could be avoided with different production mechanisms or introducing fermions heavier than the confinement scale. We will focus on techni-pions in this work.

Phenomenology of the di-photon resonance
We will study theories where the 750 GeV resonance is a composite pseudo-scalar coupled to SM gauge bosons as described by the effective Lagrangian where for a generic vector fieldṼ µν = 1 2 µνρσ V ρσ . In models where is a NGB c B , c W , c G are anomaly coefficients fixed by group theory, proportional to N TC in SU(N TC ) gauge theories. In fact the full effect of anomalies can be encoded in the Wess-Zumino-Witten term of the chiral Lagrangian that, up to the normalization only depends on the pattern of symmetry breaking. The effective Lagrangian could also contain derivative couplings to SM fermion currents. This is for example the case in composite Higgs models with partial compositeness. In this work we focus on UV complete theories based on gauge dynamics where such terms do not appear at leading order so that it is sufficient for our analysis to focus on di-boson SM decay channels. In addition we consider the possibility that can decay in a extra channel, X, focusing on the possibility that this is DM. From the above Lagrangian, the rate in SM vector bosons is given by where κ V = 1, 8 for photons and gluons and c γ = c B + c W . More explicitely the rate into photons is and the decay widths into the other SM vectors are We assume in what follows a production cross-section, The experimental upper bounds on the other decay channels reads [2]: implying the constraints on the anomaly coefficients Assuming that the only relevant production channel is gluon fusion, as will always be the case in our models, the cross section is reproduced for [2] Γ γγ M that, combined with the latter equation of (4), gives From eq. (9) and (3) one can derive a relation between f and the coefficients c γ and c G : implying that the f is proportional to N TC . Two cases are of special interest: • Small N TC : Maximises the strong sector effective coupling g TC ∼ 4π/ √ N TC giving M /f ∼ 10. The mass of the TCπ is around its maximal value Λ TC ∼ g TC f . For example the η of QCD naturally falls into this category. Note that in this case states associated to the new strong dynamics will be nearby. This is not necessarily a problem because a large g TC shields the strong dynamics effects.
• Large N TC : Leads to a smaller strong coupling g TC , but the anomaly coefficients are enhanced by N TC . As a benchmark we can take f ∼ M , N TC ∼ 10. The new strong dynamics now lies around 2-3 TeV but it is more strongly coupled to the SM.
In what follows we will focus mostly on the first possibility. The second possibility implies a larger number of TCq, easily leading to Landau poles for SM couplings at low scales.

Maximal width
In absence of extra decay channels the di-photon signal requires Γ γγ /M ≈ 0.7 × 10 −6 , and the total width is dominated by Γ gg . The experimental bound on di-jets implies A larger decay width needs new decay channels. Let us assume that decays to γγ, to gg and into a third channel X. We have where the first equation demands that the total pp → γγ rate is reproduced while the others are the experimental bounds on decays widths into gg and X. The most favourable situation is obtained when X is DM: in such a case the experimental bound sets k X ≈ 800. The situation Figure 1: Values of Γ gg and Γ γγ needed to reproduce the total width Γ indicated on the best-fit regions, assuming that decays into gg, into hypercharge vectors, and into Dark Matter.
is summarized in fig. 1 that shows, as a function of Γ gg /Γ γγ , the value of Γ γγ needed to achieve different values of the total width Γ /M : we see that a large width can be reproduced only if Γ γγ is itself large and Γ gg /Γ γγ is not too large. These considerations are encoded in the following equation, obtained from eq. (12) by expressing Γ gg and Γ γγ in terms of the model parameters c G and c γ : Table 1: SM representations arising from the smallest multiplets of the SU(5) unified group. We assign standard names used throughout the paper.
To summarise a width of 45 GeV would require in the most optimistic case c G ∼ 0.5 and c γ ∼ 8. While the first condition could be realised we find that the second is extremely difficult to achieve in concrete models.

Confining theories for the di-photon resonance
The above phenomenological analysis applies in general to theories where the 750 GeV resonance is a NGB. In particular couplings to SM gauge bosons through anomalies depend only on the pattern of symmetry breaking up to an overall coefficient. In what follows we will study UV realisations of this framework in terms of 4 dimensional gauge theories.
We will focus on SU(N TC ) gauge dynamics with N TF techni-flavours 3 . The dynamics of this theory is well known from QCD and can be also understood in the large N TC limit: the gauge theory is asymptotically free (provided the usual bound on the number of techni-flavours is satisfied) and confines at a scale Λ TC . In order to avoid severe constraints (common to old techni-colour theories) we consider fermions that are in a vectorial representation of the SM and in the fundamental N TC of SU(N TC ) [4,8,9] where R i denotes a generic SM representation and N S is the number of species with mass below the confinement scale. For a given TCq Q i , we denote asQ i the representation obtained exchanging R i withR i : they are inequivalent if R i is complex. For simplicity we consider R i representations that can be embedded in the simplest SU(5) representations listed and named in table 1. The choice in eq. (16) ensures that the vacuum configuration of the confining sector does not break the SM SU(3) c ⊗ SU(2) L ⊗ U(1) Y symmetries. Assuming QCD-like dynamics the strong interactions confine and spontaneously break the chiral global symmetry as SU(N TF ) L ⊗ SU(N TF ) R → SU(N TF ) at the scale f given by The number of techni-flavour is given by where N S is the number of SM species. This produces NGBs, the TCπ, which are QQ composite and thereby fill the representation We denote the singlets TCπ as η. Given that each R i ⊗R i contains a singlet, any model contains at least N S η singlets. 4 Among them, the singlet associated with the generator proportional to the identity in techni-flavour space is anomalous under the SU(N TC ) gauge interactions.
Analogously to the η in QCD, it acquires a large mass that can be estimated in a large-N TC expansion [10] as while the orthogonal combination η acquires mass only from the mass terms of the TCq, m 2 η ∼ m Q Λ TC , and can be much lighter. The anomaly coefficients of the singlets η with SM gauge bosons are given by Here T a are the SU(2) L generators, T A are the SU(3) c generators, and T η is the chiral symmetry generator associated to the singlet η.
A remarkable feature of gauge theories is the existence of accidental symmetries. To each irreducible representation of fermions we can associate a conserved species number. This conserved quantum number is responsible for the accidental stability of TCπ made of different species. Discrete symmetries could also produce stable particles. In section 5 we will construct explicit examples where stable TCπ are identified with DM. GeV for N TC = 3. Models highlighted in green contain two acceptable di-photon candidates, the η and the heavier η , while in blue (red) models only the η (the η) is an acceptable candidate, while the other lighter (heavier) state is experimentally allowed. The prediction for f /N TC is computed according to eq. (10). Mixing between η and η can modify the result.

Models with two species
In table 2 we give a full list of models with two TCq, that is Q = X 1 +X 2 , that can be embedded into unified representations and remain perturbative up to the unification scale These models provide 2 di-photon candidates for , the η and η . Asymptotic freedom of the SU(N TC ) gauge theory and absence of Landau poles for SM couplings below the unification scale allow only a finite list of possibilities. These models do not contain DM candidates, so that the width is dominated by Γ gg . They can be extended to contain DM candidates by adding fermions that are singlets under the SM, see section 5.
Notice that the anomaly computation is reliable for m π Λ TC : for f ∼ 100 GeV, is close to the cut-off of the effective Lagrangian and higher dimensional operators could give important contributions. In QCD the η and η decay widths are predicted with 30% precision from the anomaly computation. We therefore consider the values in table 2 as an estimate with an error of similar size.
The singlets η and η in general mix. Their mixing can be estimated from the chiral Lagrangian as, As a consequence their anomaly coefficients correspondingly mix. For N TC ∼ 3 the mass of η and η are comparable so that the mixing can be significant. For example in QCD the mixing angle between η(550) and η (958) is estimated around −15 • [11] in rough agreement with the formula above.
In the models of table 2 where the di-photon candidate is the η singlet, the value of f suggests a mass scale for the η above 750 GeV. We note however that the estimate of the mass and coupling to photons of the η is particularly uncertain away from the QCD case with 3 colours and 3 flavours. In any case, consistency with the di-photon signal can be recovered thanks to a mixing between η and η . A sizable mixing is indeed common since in order to avoid the experimental constraints on extra coloured particles, TCq masses should not be much smaller than the confinement scale, so that the η and η have comparable mass allowing them to significantly mix.
The η/η mixing can give an accidentally small c G for the 750 GeV resonance. In models with other decay channels such as DM this could allow to increase the total width as discussed in section 2.1.

Other Resonances
The phenomenology of confinement models is rich and has been discussed for example in [4,8]. Given the fermion content, quantum numbers of the resonances are predicted. In a QCD-like theory the lowest lying states are expected to be techni-pions and spin-1 resonances (TCρ) at a higher mass Λ TC .
Before discussing the techni-pions we consider the TCρ. Differently from the TCπ, the interactions and the mass scale of the TCρ are less calculable. There is however a universal feature: coupling with the SM fermions arises through the mixing with SM gauge bosons, such that the resulting strength scales as The coupling g of the techni-resonances to the SM fields is suppressed by the large value of g TC , especially for N TC = 3. Thanks to this generic fact, models with Λ TC ∼ TeV are experimentally allowed.
Let us turn to techni-pions. A colour anomaly requires the existence of fermion constituents with colour so that all models predict coloured scalars with mass around the di-photon resonance. Moreover in models with more than one specie there are extra singlets that also couple to gluons and photons through anomalies and could be singly produced at the LHC.

Extra techni-η singlets
The candidate for the di-photon resonance with mass M ≈ 750 GeV is accompanied by N S − 1 extra singlets, lighter or heavier. Their couplings to the SM vectors are again described by a Lagrangian of the same form as eq. (1). Assuming that they couple to gluons, their production cross section is where s is the collider energy and C gg (m η ) are the dimensionless partonic luminosity for the single production of a resonance with mass M = m η from gg partons in pp collisions, The numerical value of C gg as a function of the mass is shown in fig. 2a. The experimental bounds on pp → γγ are given in fig. 2b, and can be roughly approximated as (dotted curve in fig. 2b) The left-handed side can be approximated as Γ γγ /M , in models where Γ γγ Γ gg ≈ Γ η . For given anomaly coefficients c B , c W , c G , the branching ratios do not depend on the mass M and the widths scale as Γ gg ∝ M 3 (dashed curve in fig. 2b) as long as M M Z . This means that, in the simple relevant limit where Γ gg Γ γγ , the experimental bounds on c γ are about a factor of 2.5 stronger at M = 1 2 M with respect to M = M . The presence of one or two di-photon candidates and the compatibility of the pp → γγ bound distinguishes the models of table 2 in three categories, denoted with different colours. The models highlighted in green contain 2 di-photon candidates, which are both acceptable candidates for the 750 GeV resonance. The models in blue contain only one acceptable diphoton candidate (the η ) and a lighter singlet η that is compatible with the experimental bound of fig. 2b. In some models the η does not couple to gluons, so that its production is strongly suppressed. The models in red contain only one acceptable di-photon candidate (the η), and a heavier singlet η .

Extra coloured techni-pions
Techni-pions in a real representation of the SM can decay into SM vector. We consider the single production of a coloured χ = (8, 1) 0 that mainly decays to jj, with cross-section given by where the quantities are defined as in eq. (24). The interaction term gives the decay width Di-jet searches at √ s = 8 TeV [14,15] imply f /N TC > ∼ 70 GeV for χ masses between 0.5 and 1.5 TeV. We therefore consider as a safe bound 1 TeV for the mass of the colour octet, since many models will require a value of f /N TC similar to the above in order to match the diphoton rate. If composed of charged constituents, χ also decays to γj and Zj, with branching ratios suppressed by ∼ α/α 3 : these decay modes lead to weaker bounds.
Complex TCπ are mainly produced via pair production. Limits on pair produced (8,1) and (8,3) TCπ are much weaker, although they are fairly model independent since the production is determined by SM gauge interactions. A rough bound on a pair-produced colour octets decaying to pairs of jj is ≈ 450 GeV [17] (after matching to the production rate for colour octets). This bound is weaker than the one from single production, although it can be the dominant one for models with a large f /N TC . TCπ charged only under the electro-weak group have smaller production cross section at the LHC.
The experimental limits on coloured techni-pions, especially those from di-jet searches, potentially constrain some models of table 2, however the actual bounds on a concrete model depends on the details of the mass spectrum. For a detailed study of the phenomenology of a given model, see [16] where the model Q = D ⊕ L is considered.

Effective Lagrangian
The interactions of the TCπ can be studied using chiral Lagrangian techniques, reviewed in the appendix, to which we refer for all the details. We include in our description the η that provides a di-photon candidate in most models. Of particular relevance to the following discussion will be the hidden sector θ TC angle (see also [18]). The strong dynamics violates CP if its action includes the topological term θ TC is physical if the masses of the TCq are different from zero. We assume in what follows that the QCD strong CP problem is solved by axions in the usual way and that no axion mechanism exists for θ TC 5 . On the other hand θ TC has important effects on the spectrum and dynamics of the composite states. The main physical effects of θ TC is to induce electric dipoles for the techni-baryons [4] and CP-violating interactions for techni-pions [19]. The latter is important in the present context as it allows the decay of the 750 GeV di-photon candidate η into lighter TCπ pairs. For the present work it will be sufficient the following effective Lagrangian [19] L includes all the TCq masses, B 0 is a non perturbative constant of order O(Λ TC ) and a is related to the η mass as m 2 η ≈ N TF a/N TC + O(M Q ). For θ TC = 0 the vacuum is at Ũ = 1I and the minimization of the potential leads to the Dashen's equations, see eq. (78) in the appendix. Expanding around the vacuum one finds cubic vertices for the techni-pions whereθ TC measures the violation of CP and is related to the TCq masses and the θ TC -angle by the Dashen equations. For small fermion masses the approximate relation holds for small θ TC . Accurate formulas can be found in the appendix.
Techni-pions have also multipole couplings to SM gauge bosons that are of phenomenological relevance. This is particularly important for neutral techni-pions that do not couple to SM fields to leading order. Such couplings explicitly break the global symmetries so they have to be proportional to the mass parameters of the fundamental Lagrangian. The strong dynamics generates operators such as [20] Analogous couplings to electro-weak gauge bosons are also generated. Expanding this term one finds CP preserving interaction (TCπ) 2 G 2 and well as CP violating terms TCπ G 2 further suppressed byθ TC . The first ones, also known as Cromo-Rayleigh interactions, will play an important role in the DM phenomenology discussed in the next section [21]. They also allow double production of the di-photon candidate through gluon fusion. From the above equation the coupling can be estimated as CP-violating effects in η decays to SM gauge bosons are further suppressed, see also [18].

Phenomenology of techni-pion Dark Matter
Gauge theories automatically deliver particles stable thanks to accidental symmetries. In particular in models with several SM representations, TCπ made of different species are stable at the renormalisable level. Alternatively TCπ could be stable imposing appropriate discrete symmetries. It is tempting to identify such particles with DM. DM as a composite scalar TCπ can be charged or neutral under the SM gauge group. In the former case SM gauge interactions contribute to the DM annihilation cross section as in minimal DM models [4,22], such that, for DM masses below a TeV, the thermal relic DM abundance is smaller than the observed cosmological DM abundance. The only possible exception is N ≥ 2 copies of scalar doublets.
We focus in what follows on neutral DM candidates, that we will call Π. From a phenomenological point of view, their most relevant interactions are with gluons [21] and with the di-photon resonance η as described in the previous section. The leading terms relevant for the DM interactions are 6 The NGB nature of the particles implies restrictions on the coefficients of the effective operators. Since the above operators break the NGB shift symmetry their coefficient must be proportional to the explicit breaking effects. While for the η the coefficient c G is due to the strong interactions, for stable singlets like Π the only source of explicit breaking is given by the fermion masses so that the coefficients above must be proportional to the TCπ mass. Moreover, C ηΠΠ breaks both the shift symmetry and CP so that it is proportional to the TCπ mass and to θ TC . From eq. (32) and (35) one finds the estimates, As expected the coefficients go to zero for m Π → 0 as in this limit Π becomes an exact NGB. Our estimate differs from [7] where the coefficient was assumed to be constant. In models with lighter coloured NGB, Λ TC should be replaced by the mass of these objects as a perturbative computation shows. Coloured resonances should however be heavier than about 1 TeV. The coefficient c G and C ηΠΠ can be extracted from the leading terms of the chiral Lagrangian, while the coefficients of the Rayleigh interaction can only be estimated. Therefore, when the dominant interactions between DM and the SM are induced by the cubic CP-violating couplings, this setup is calculable.

Thermal relic abundance
Assuming that the interactions in eq. (36) dominate, the thermal relic abundance of DM can be derived in the standard way. From the s-wave annihilation cross section of a real scalar DM we obtain: If the first non-resonant contribution dominates, the observed relic abundance is reproduced for The second contribution is generically expected to be comparable and it can be resonantly enhanced if m Π ≈ 1 2 M . The situation is illustrated in figure 3, where along the solid blue curves the relic abundance is mainly reproduced due to CP-violating effects from θ TC , for the models of sections 5.2 and 5.3.
In some models (see section 5.2) an extra singlet η * is lighter than DM, or almost degenerate with it, and decays into SM vectors through anomalies. This extra light state changes the thermal relic abundance with respect to our discussion above. Interactions between DM and η * arise from the non-linearities of the kinetic term and mass terms and have the generic form The DM annihilation cross section receives an extra contribution from DM DM → η * η * scatterings, which can be estimated as σv ≈ m 2 Π /64πf 4 . When this dominates, the desired thermal relic abundance is reproduced for a DM mass m Π ∼ 50 GeV(f /300 GeV) 2 . Each model predicts a specific form for these interactions: the model in section 5.2 reproduces the thermal relic abundance along the dashed blue curve in the right panel of figure 3. Such interactions will be (somewhat improperly) named co-annihilations, given that η * is part of the DM sector, and in some limits η * itself becomes a stable DM particle.

Direct Detection
Integrating out the di-photon η, we obtain from eq. (36) the effective interactions relevant for low-energy direct DM detection: The first CP-conserving operator contributes to the spin-independent cross section as [21] σ SI = 9f 2 064 parameterizes the nucleon matrix element [24,25] and m N is the nucleon mass. Numerically we get which is in the interesting ballpark for future experiments.
The other CP-violating operator induces a spin dependent coupling to the nucleons, further suppressed by the small exchanged momentum δ q:

Collider constraints
The operators in eq. (41) can be also constrained by searches at the LHC. Assuming the validity of the effective operator description, namely that the mediator is sufficiently heavy, the bounds on the operator coefficients are [26] C ΠΠgg Both bounds roughly imply f 100 GeV. Figure 3 shows that it is possible to reproduce the observed DM abundance compatibly with direct detection constraints for values of the parameters favoured by the 750 GeV γγ anomaly, although the DM mass needs to be somehow near to the resonance condition, m Π < ∼ M /2. However, when co-annihilations are present the DM mass is below 100 GeV as shown by the dashed blue curve in the right panel of figure 3. This encourages us to try to build models that realise this scenario.

Confining di-photon resonance and Dark Matter
Our goal is constructing composite models where: i) the 750 GeV resonance is a composite TCπ with QCD and QED anomalies; ii) DM is another composite TCπ, Π, stable because of species number, iii) → ΠΠ * is allowed. iv) All experimental bounds are satisfied and no other TCπ is stable. Of course, these goals go beyond what is safely indicated by experiments and might be too ambitious. In section 5.1 we discuss the problems of models with two species. In section 5.2 we discuss models with three species, where DM is accidentally stable thanks to species number. In section 5.3 we discuss models where DM is stable because of G parity.

Models with two identical species?
To start, we consider models containing TCq that fill 2 identical copies of a representation X of the SM gauge group. Three kind of singlet TCπ are formed: 1) Π = X 1X2 , which is a stable DM candidate; 2) η − = X 1X1 − X 2X2 , with no anomalies; and 3) η + = X 1X1 + X 2X2 with anomalies under the SM and under the techni-colour group. TCq masses m 1 and m 2 contribute to the masses of the neutral states as Furthermore, the techni-anomaly gives a large mass term to η + . Because of the mixing both mass eigenstates η ≈ η − (lighter) and η ≈ η + (heavier) acquire anomalous couplings with SM gauge bosons. In the limit of large m η , η and Π are quasi-degenerate so that η → ΠΠ * decays are kinematically forbidden. In order to obtain → gg decays X should be coloured, leading to the following phenomenological issue. Besides the neutral singlet Π, η + , η − there are coloured TCπ (e.g. 3 ⊗3 = 1 ⊕ 8). A TCπ in the (r 3 , r 2 ) Y rep acquires the following contribution to its squared mass from SM gauge interactions: where C(r N ) is the quadratic Casimirs of the r N representation of SU(N ), equal to (N 2 − 1)/2N for the fundamental and to N for the adjoint. Numerically ∆ (1,3) 0 ≈ 0.015 Λ 2 TC for a triplet of SU(2) L , ∆ (3,1) Y ≈ 0.03 Λ 2 TC for a colour triplet and ∆ (8,1) 0 ≈ 0.07 Λ 2 TC for a colour octet. These numerical values imply that, while the coloured TCπ become unstable (decaying to gluons and uncoloured TCπ), it seems difficult to avoid conflicting with LHC bounds that roughly excluded coloured particles lighter than about 1 TeV (notice however that this bound is model dependent, although fairly correct for a large class of scenarios, as discussed in section 3.2). Furthermore, co-annihilations between the coloured and the neutral states render difficult to reproduce the cosmological DM thermal abundance for sub-TeV masses [27].
In conclusion, to build a viable model where decays into DM we need to add a third specie, which is heavier and coloured.

Dark Matter stability from species number
In view of the previous considerations, we consider models with three species, as listed in table 3. As their phenomenology is similar, we explicitly discuss the model with The TCπ transform in the adjont representation of the techni-flavour group SU(5) that, with the above embedding, decomposes under the SM as The TCπ with a net species numbers are the two colour triplets, φ i = UN i and the complex singlet Π = N 1N2 , which is the DM candidate. Stability of the triplets can be avoided by appropriate higher dimensional operators or by adding scalars H with quantum numbers such that the Yukawa interactions H U N i is allowed. In special models such as Q = Q ⊕Ũ 1,2 or Q = Q ⊕D 1,2 the role of H can be played by the SM Higgs doublet [4]. The two real singlets and the octet χ = UŪ are unstable and decay through anomalies to SM gauge bosons. The singlets can be di-photon candidates. Including the η , the TCπ matrix reads where the diagonal generators associated to the η i singlets are The accidental SU(5) global symmetry is broken by the SM gauge interactions and by the TCq mass matrix We compute the TCπ mass matrix from eq. (79) in the appendix. We find where B 0 is of order Λ TC and gauge contribution ∆ are given in eq. (47). TCπ with same quantum numbers and same species number can mix. In particular η i generically mix with η .
In the limit where η is much heavier, the mass matrix of η 1 , η 2 singlets in the basis of eq. (51) is given by The mass eigenstates are η m1 = cos θ 12 η 1 − sin θ 12 η 2 and η m2 = sin θ 12 η 1 + cos θ 12 η 2 where In the limit m U m N 1,2 , η m2 is approximately degenerate with the DM candidate due to an accidental SU(2) symmetry. From the mixing one finds the hierarchy m η m2 < m Π , but higher order terms in the chiral expansion should also be included at this order. As explained in the appendix, the θ TC -angle modifies the mass spectrum. In the limit of small θ TC , it is a second order effect. More interesting for our discussion is the fact that θ TC induces cubic couplings between techni-pions, as discussed in section 3.3.
In the limit m N 1 = m N 2 η m2 and Π become degenerate and stable with common mass 2B 0 m N 1,2 . They form a triplet, T a , under a global SU(2) symmetry that rotates N 1 and N 2 so that DM has 3 scalar components. The di-photon resonance is then identified with η 1 or η .

Techni-pion interactions with SM vectors
Using eq. (21) we compute the anomaly coefficient in the interaction basis. The colour octet χ decays dominantly to gg as well as into γg, Zg, as already discussed in section 3.2. The anomaly coefficients for the singlets η 1 and η are collected in table 3 for a sample of models. The combination corresponding to η 2 has no anomalies because in the limit m N 1 = m N 2 it becomes stable. In presence of two possible di-photon candidates, we need to check the experimental bound presented in section 3.2. In models highlighted in red (blue) only the η 1 (η ) singlet is a viable di-photon candidate.
For m N 1 = m N 2 , the mass eigenstate η m2 inherits anomalous couplings from the mixing with η 1 and η . The lighter η m2 has anomaly coefficients equal to those of η m1 , but suppressed by tan θ 12 , and thereby is compatible with data for small enough mixing θ 12 , see section 3.2. The signal rate is where we allowed for a branching ratio of η m1 to lighter TCπ to which we now turn.

CP-violating interactions among techni-pions
Given that TCπ are pseudo-scalars, cubic interactions among them are possible if the θ TC -term of techni-strong interaction violates CP. Using the formalism described in the appendix, from eq. (80) we find the following cubic terms 1 2 (C ηXX η + C η XX η ) X 2 in the interaction basis: as well as C η 2 ΠΠ * = C η 2 η 1 η 1 = 0 where a is a non-perturbative constant of order ∼ Λ 2 TC . Taking into account mixing effects (55), the decay widths for the kinematically allowed processes      Table 3: Anomaly coefficients for the η and η 1 singlets and their decay widths in various models, computed in the interaction basis. In red (blue) models, only the η 1 (η ) singlet can be identified with the 750 GeV resonance. The last two columns show the maximum value of the total width Γ allowed by extra decays into DM and the corresponding minimal f , computed following eq. (13) and (10). The mixing between the singlets can affect these conclusions. become where s 12 = sin θ 12 and c 12 = cos θ 12 .
The parameterθ TC is determined by the θ TC angle and by the TCπ masses, as dictated by the Dashen equations (78).θ TC is small when θ TC is small or any TCq is much lighter than Λ TC . A simple result for the reference model U ⊕ N 1 ⊕ N 2 is obtained in the limit where the η is heavy and m U m N 1 ≈ m N 2 : where m 2 Π,η 2 is the mass of the lightest almost degenerate TCπ and the formula is valid for θ TC 1. The decay rate of the 750 GeV di-photon candidate η m1 into DM is where we chose f 100 GeVN TC to match the di-photon rate, a small mixing θ 12 1 and the limiting case of eq. (59). The maximal width is obtained for r ≈ 0.90, but still it is more than one order of magnitude below the width favoured by ATLAS. We also checked that adding a larger number of light singlets TCq does not help in achieving a larger width. The difficulty in getting a large width from CP-violating decays to DM can be understood from the Dashen equations, eq. (78). They imply |aθ TC /N TC | < min i |2B 0 m i |, therefore when one TCq becomes light the size of CP-violation diminishes. This is the region where a DM candidate is lighter than M /2. Furthermore, η m1 can decay into η m2 η m2 , which, in turn, decays to SM gauge bosons thanks to the anomaly acquired via its mixing with η 1 , giving rise to a final state with 4 SM vectors. The rate of this process is Γ η m1 →η m2 η m2 /Γ γγ times the di-photon rate, assuming a dominant branching ratios to di-jet for η m2 . Searches for pairs of di-jets set a limit of 2 ÷ 3 pb at 8 TeV for pair produced di-jet resonances with mass ≈ 300 GeV [17]. Imposing the di-photon constraint and rescaling to 13 TeV, we get the limit Γ η m1 →η m2 η m2 10 3 Γ γγ . In the present scenario this constraint is satisfied since For smaller m η m2 the limits degrade quickly. We can also have final states with photons, but they are suppressed at the level of ≈ 10 −2 fb for jjγγ at 13 TeV, due to the relative suppression Γ γγ /Γ gg as from table 3.

Regimes for Dark Matter in models with species number
With the interactions derived above two different regimes for DM can be realised in this model. For m N 1 = m N 2 , from the mass diagonalization there is always a state lighter than DM, η m2 , decaying into SM gauge bosons. Annihilation induced by ΠΠ * → η m2 η m2 scattering easily dominates in the regime m Π > m η m2 . This scenario is illustrated in the right panel of figure 3. The dashed blue curve reproducing the observed relic abundance is consistent with the required di-photon rate for DM masses below about 100 GeV 7 . The tree-level co-annihilations dominate over other interactions and the relic abundance is reproduced for m Π ∼ 50 GeV. 7 In the present case the cross section times velocity for the TCπ scattering ΠΠ * → η m2 η m2 is Notice that everything can be expressed in terms of m Π and m ηm2 (and the mass of the di-photon resonance).
In the limit m N 1 = m N 2 Π, Π * and η m2 form a degenerate triplet, and η m2 = η 2 becomes an extra stable DM candidate due to the enhanced SU(2) symmetry of the fundamental Lagrangian. This case is depicted in the right panel of figure 3 (solid blue curve): Π and η m2 have in this case the correct thermal abundance for masses close to M /2, and the annihilation cross section is mainly determined by CP-violating interactions.
To conclude let us discuss the main differences between the Q = U ⊕ N 1 ⊕ N 2 model considered so far and models such as is again a neutral state, candidate to be Dark Matter. The electro-magnetic anomaly needed to achieve → γγ receives extra contributions from E 1,2 . Furthermore, Γ( → gg) can be reduced by assuming that U , the colored TCq, has a mass m U above the confinement scale: in such a case only the TCπ made of E 1,2 remain light. As discussed in section 2.1 this allows to reproduce the di-photon excess with a larger Γ . Another difference concerns techni-baryons: in the U ⊕ N 1,2 models the stable lightest techni-baryon can be neutral state, being made of N i , while this does not happen in models where N i are replaced by charged states.

Dark Matter stability from G-parity
We re-analyse the model presented in [7]. In our notation it corresponds to the choice Q = D ⊕D ⊕ L ⊕L, which allows to impose a generalised G-parity symmetry that exchanges L ↔L and D ↔D. This implies m D = mD and m L = mL and that techni-pions are classified as even or odd under this G-parity: the lightest G-odd techni-pion is stable (see [6] for the first discussion of techni-pion DM with G-parity).
The model has SU(10) techni-flavour symmetry. The SU(5) generators in SU(10) are T a = diag(t a , −(t a ) * ) where t a are in the fundamental of SU(5). One can define a G-parity transformation that combines charge-conjugation and a rotation R = exp(iπJ ) where J = iτ 2 ⊗ I 5 , that acts on the 10 TCq. The gauge interactions are G-parity invariant since R † t a R = −(t a ) * . However this G-parity is not an accidental symmetry: one has to impose that TCq masses respect it: The 99 TCπ decomposes under SU(5) as where we have indicated the G-parity of each multiplet. The complex representations r transform as r → −r under G-parity. In implicit notation, we can schematically write the TCπ matrix as Π = 24 + + 24 − 10 + 15 10 + 15 24 where the singlets corresponds to diagonal generators, in particular the G-even state corresponds to the η with generator T η = 1I N TF / √ 2N TF . With a further decomposition of SU (5) under the SM we have the classification of TCπ in terms of SM multiplets 8 . The G-even states in the 24 + are associated to the generators T a + = diag(t a , t a ), while the stable G-odd 24 − are associated to the generators T a − = diag(t a , −t a ). Using eq. (79) in the appendix we compute the mass spectrum of the TCπ. For charged ones, To compute the masses of singlet TCπ we must take into account that states with equal Gparity and equal quantum numbers can mix: the 1 − can mix with the singlet from 24 − , and the η with the even singlet η in the 24 + . Choosing the following basis of generators the only mixing arises between η and η . The mass matrix for the singlets ( It follows that for m L < m D the DM candidate 1 − A can be lighter than M /2. The η/η mixing is sizeable when m 2 η ∼ 10a/N TC is comparable to the other TCπ masses. 8 The standard composition is the following, 24 = (8, 1) 0 (3, 2) −5/6 (3, 2) 5/6 (1, 3) 0 + (1, 1) 0 ,  Table 4: Anomaly coefficients for the unstable singlets η and η for the model presented in section 5.3. Because of the experimental bound from σ(pp → γγ), only the scenario in which the singlet η is identified with the 750 GeV resonance is allowed. The last columns show the maximum value of the total width Γ allowed by extra decays into DM and the corresponding minimal f , computed following eq. (10) and (13). The mixing between the singlets can modify this scenario.

Interactions of the techni-pions
The G-even states in real representation of the SM can decay to SM gauge bosons via anomalies. The anomaly coefficients for the unstable singlets η and η , as defined in eq. (21), are given in table 4, together with the ratios between the widths into γZ, ZZ, W W , gg and the width to γγ. Following the discussion of section 3.2, we identify the lighter η singlet with the diphoton resonance. Actually, because of the η/η mixing, the anomaly coefficients of the mass eigenstates are linear combinations of those reported in table 4.
TCπ acquire CP-violating cubic interactions in the presence of the θ TC term. From eq. (80), we can extract the cubic couplings defined as before eq. (57), obtaining: The relative weights in different channels are given by: We can now discuss the phenomenology of the model. For simplicity we assume that m η m η so that the mass of the di-photon candidate is M 2 = 2B 0 ( 2 5 m L + 3 5 m D ). This constrains the possible mass range for the two G-odd stable singlets 1 − A,B . Notice however that differently from the model of the previous section, the lightest TCπ in the spectrum is automatically one between 1 − A and 1 − B . Defining z = m L /m D [16], the masses for the DM candidates are Not all the parameter space is allowed. For large z > ∼ 17/2 the DM candidate 1 − B becomes lighter than M /2 and also the coloured TCπ become lighter; in particular the mass of the colour octet is m (8,1) where in the second step we have imposed the di-photon rate (reproduced for f /N TC ≈ 80 GeV), and used the relation Λ TC ≈ 4πf / √ N TC . For 1 − B in the resonant region, we therefore expect a large N TC to comply with bounds from direct searches for coloured states.
We are then led to consider the case where 1 − A is the dominant DM component and we work in the limit where the coloured states are at ≈ 1 TeV. In this regime the interactions of 1 − A with the SM are mainly mediated by the η, in particular we do not find strong constraints for the scenario where 1 − A is lighter than where, in the last step, we used the relation aθ TC /N TC ∼ θ TC m 2 The main annihilation channel is mediated by the di-photon resonance and it originates from CP violation in the composite sector, see the left panel of figure 3. Co-annihilations to heavier states are negligible, the states closer in mass being (1, 1) 1 , which does not contribute to co-annihilation as long as Λ TC > 5m 1 − A , which is natural for the typical masses of the DM candidate. The other heavier stable particle 1 − B annihilates efficiently into other (unstable) TCπ via TCπ scattering, depleting its relic density which can be roughly estimated as

Conclusions
A natural explanation of di-photon excess is provided by new confining gauge theories that generate singlet Nambu-Goldstone bosons coupled to photons and gluons through anomalies in complete analogy with pions in QCD. While such theories do not protect the Higgs squared mass from quadratically divergent corrections -the Higgs and the SM particles are elementary -they are not in tension with bounds on new physics [8] and have been proposed in the past for various purposes including explaining the stability of dark matter [4] and as a source for the electro-weak scale [28].
In this note we have given a general survey of the scenarios that reproduce the di-photon excess with a composite techni-pion. The models under consideration are extremely predictive. Couplings to SM gauge bosons are determined by anomalies that are in turn fixed by the fermion constituents. The new sector should contain new fermions that carry colour and electro-weak charges. As a consequence new resonances with SM quantum numbers are predicted. Coloured particles in particular will be within the reach of the LHC. The phenomenology depends in a crucial way on the existence of a non-zero θ angle in strong sector. Among other effects, CP-violation can induce tree-level decays of the 750 GeV resonance into lighter techni-pions, increasing the width. We find however that these models can only reproduce a small width, at least unless the number of techni-colours is so large that SM gauge couplings develop sub-Planckian Landau poles.
In various models such lighter techni-pions can be neutral Dark Matter candidates, stable thanks to accidental symmetries or G-parity. Their couplings to the di-photon resonance can reproduce the observed Dark Matter relic abundance thermally for masses around 300 GeV, while if co-annihilations are effective, masses lower than 100 GeV are favoured.
If the di-photon excess will be confirmed, with more data from the LHC we will learn the coupling of to SM gauge bosons. This will allow to infer the quantum numbers of its TCq constituents and to sharpen the possible connection with Dark Matter. Given the simplicity and predictivity of composite models, we might soon be able to sort out the right theory.

Acknowledgments
This work was supported by the ERC grant NEO-NAT. AT is supported by an Oehme fellowship and MR by the MIUR-FIRB grant RBFR12H1MW. We thank Roberto Franceschini for useful discussions.

A Effective Lagrangian for techni-pions
We review the main ingredients of the effective chiral Lagrangian for TCπ (see [19] for a comprehensive review). We focus on the explicit breaking of the techni-flavour symmetry coming from TCq masses, gauge interactions and the axial anomaly. The NGBs are parametrised by the unitary matrix U (x) = exp(−2iΠ(x)/f ), with where T a are the generators of SU(N TF ) in the fundamental representation, normalised as Tr(T a T b ) = 1 2 δ ab . The effective Lagrangian in terms of the field U can be written as [19] L eff = f 2 4 Tr D µ U (D µ U ) † + Tr 2B 0MQ (U + U † ) + (76) where f is the TCπ decay constant, B 0 is a dimensional coefficient of O(Λ TC ) andM Q contains the TCq mass matrix that can be chosen diagonal. The axial anomaly induces the terms proportional to a/N TC where a has dimensions of a mass squared. The factor 1/N TC is expected in a large-N TC expansion [10] and manifestly shows that the axial anomaly disappears in the large-N TC limit. The parameterθ TC is defined asθ where θ TC is the analogue of the QCD θ-angle and ϕ j are the phases that appear in the minimization equations of the potential energy. They are the solutions of the so-called Dashen equations 2 m i B 0 sin ϕ i = a N TC (θ TC − j ϕ j ) i, j = 1, ..., N TF , with m i the TCq masses. Notice thatθ TC is zero if any of the TCq masses are zero. In eq. (76) the NGBs are fluctuations around the vacuum selected by the Dashen equations. In this basis, the effects of the axial anomaly are also present in the mass matrix that can be written as M Q = diag(m i cos ϕ i ). The mass terms for the NGBs can be extracted from the second and the third term of eq. (76), Notice that even in the chiral limit (m i = 0), the singlet η 1 acquires a mass induced by the axial anomaly m 2 η 1 ≈ N TF a/N TC . If a/N TC m i , the η 1 is much heavier than the other TCπ (similarly to the QCD case) and can be decoupled.
The axial anomaly also leads to CP-violating interactions among the techni-pions. These terms come from the last term of eq. (76)

Effects of θ TC in an explicit model
We present some analytic formulae for the U ⊕ N 1 ⊕ N 2 model considered in section 5.2. In order to study the effects induced by the θ TC -angle on the mass spectrum and techni-pions interactions, we need to solve the Dashen equations (78). For general values of the TCq masses and of a/N TC , those cannot be solved analytically. In order to get analytic results, let us consider the limit that is also relevant for the phenomenology discussed in section 5. In this limit a simple and exact solution for the Dashen equations is [29]: The θ TC -angle modifies the techni-pions mass spectrum with the substitution m N i → m N i cos ϕ N i : where the contributions ∆ from gauge interactions are defined in eq. (47). In the same way, the mixing (squared) mass matrix between the singlets η 1 and η 2 becomes The CP-violating trilinear couplings of eq. (57) are parametrized by theθ TC parameter, that is related to θ TC and to the TCq masses by the Dashen equations. The solution (82) corresponds to a N TCθ TC = 2B 0 m N 1 m N 2 sin θ TC