Di-boson signatures as Standard Candles for Partial Compositeness

Composite Higgs Models are often constructed including fermionic top partners with a mass around the TeV scale, with the top partners playing the role of stabilizing the Higgs potential and enforcing partial compositeness for the top quark. A class of models of this kind can be formulated in terms of fermionic strongly coupled gauge theories. A common feature they all share is the presence of specific additional scalar resonances, namely two neutral singlets and a colored octet, described by a simple effective Lagrangian. We study the phenomenology of these scalars, both in a model independent and model dependent way, including the bounds from all the available searches in the relevant channels with di-boson and di-top final states. We develop a generic framework which can be used to constrain any model containing pseudo-scalar singlets or octets. Using it, we find that such signatures provide strong bounds on the compositeness scale complementary to the traditional EWPT and Higgs couplings deviations. In many cases a relatively light scalar can be on the verge of discovery as a first sign of new physics.


I. INTRODUCTION
The LHC has entered a phase with exceptional potential for discovering new physics, and new data is being collected at an unprecedented rate during the Run-II that started last year. Not surprisingly, this fact has led to a flurry of model-building activity, with the intent of charting the landscape Beyond the Standard Model (BSM) and proposing new discovery channels.
Among the various BSM proposals, the idea that the Higgs sector of the Standard Model (SM) is dynamically generated by a confining strong dynamics is playing an important role and is being continuously tested experimentally. In particular, the models discussed in this work are four dimensional gauge theories combining the concept of the Higgs as a pseudo-Nambu-Goldstone boson (pNGB) [1] with that of partial compositeness [2], where the top quark mass arises by a linear coupling with a spin-1/2 "top-partner". Therefore, the main requirement on the underlying theory is to provide a viable Higgs sector together with the appropriate colored fermionic bound states. The construction of these models has been discussed elsewhere [3,4], and some specific examples were given in [5][6][7]. With the exception of [7], all models contain at least two species of underlying fermions belonging to different irreducible representations (irreps) under the confining hypercolor (HC) gauge group. This observation will play a crucial role in the rest of this paper. The chiral perturbation theory for these models has been recently presented in Ref. [8]. The coupling to tops has been addressed in [9].
The phenomenology of Composite Higgs models has been already extensively studied, with particular focus on the minimal symmetry breaking pattern SO(5)/SO(4) that leads to only a Higgs boson in the pNGB spectrum (see [10,11] for recent reviews). Because of the lack of additional light scalars, collider searches have focused on colored top partners, together with other indirect constraints on SM quantities. The current bounds on the masses of top partners range around 700-900 GeV [12][13][14][15][16]. However, it is very challenging to obtain the minimal scenario starting from a four dimensional fermionic theory: attempts present in the literature are either relying on supersymmetry [17] or on effective four-fermion interactionsà la Nambu-Jona Lasinio (NJL) [18].
In the class of models we consider, based on a confining gauged HC and with only fermionic matter fields 1 , the symmetry breaking patterns are determined by the representations of the underlying fermions [20,21], giving rise to non-minimal cosets with additional pNGBs. Thus, the main message we want to convey is that the first evidence of this class of models of partial compositeness may come from the discovery of the additional pNGBs rather than from the direct observation of the top partners.
The phenomenological relevance of pNGBs in composite models is not new [22][23][24][25][26]. What we aim at, on the other hand, is to directly link their presence to the mechanism of partial compositeness. To do so, instead of looking at the details of each model [4], we focus on two types of scalars that are universally present in all models: singlet pseudo-scalars associated to global U(1) symmetries [27], and a color octet arising from the presence of colored underlying fermions.
The presence of two types of fermions in the underlying theory guarantees that there is always a combination of the two U(1)'s which is non anomalous with respect to the hypercolor group. Thus, contrary to the anomalous axial current in QCD, the associated pseudo-scalar will be light. Inspired by large-N c estimates in QCD, we will also keep the anomalous U(1) scalar in the spectrum because, depending on the model, it may also be light. These two states will be denoted a and η in the mass eigenstate basis (as non-trivial mixing is present).
1 The possibility of top partners arising as bound states of a fermion and a scalar has been recently proposed in [19].
We will briefly review the salient points of these constructions, however the focus of the paper is to investigate their phenomenology, derive all constraints from up-to-date searches, point to the promising signatures and their correlations, and make concrete suggestions for their further exploration at the LHC. In particular, we will focus on the two singlets and on the color octet, commonly present in all models. Their couplings to the SM gauge bosons are generated via the Wess-Zumino-Witten [28,29] anomalous term, and are thus computable in terms of the properties of the underlying theory.
Additional couplings to tops can also appear: we prove that the singlets always couple to tops via partial compositeness, while this coupling may be absent for the octet, and we present an estimate of the couplings to tops (and other SM fermions) proportional to their mass. The calculability of the phenomenologically relevant couplings makes these three pseudoscalars standard candles for fundamental models of partial compositeness, that will shine in particular via di-boson searches at the LHC. In fact, the observation of resonances in di-boson channels would allow to extract information about the WZW couplings, which are directly related to the properties of the underlying theory.
The scalar singlet production via gluon fusion and its subsequent decay to a pair of gauge bosons, both mediated by the WZW interactions, leads to clean signatures at the LHC. Such final states are intensely searched for at the LHC, and recently the emergence of excesses in both di-boson and di-photon, now less prominent or entirely disappeared, was the source of big excitement and inspiration in the particle physics community. A pseudo-scalar decaying via WZW interactions can easily accommodate such signatures [30], and the case of the models under investigation has been already pointed out by a subset of the authors [27,31].
The paper is organized as follows: in Section II we briefly present the content of the models under consideration and their salient dynamical properties. In Section III we discuss the pNGBs of relevance for this work. We present their chiral lagrangian, their couplings and their masses. Section IV discusses their phenomenology and presents up-to-date (post ICHEP2016) bounds on their couplings in a model-independent way. We focus on the most updated constraints deriving from di-boson searches, di-top resonances and other relevant channels (like pair production in the case of the color octet). Section V confronts these bounds with the models presented in Section II. We explore two specific models and extract a combined lower bound on the decay constant of the pNGBs. Being associated to the Higgs sector, the value of such scale is a direct measure of the fine tuning involved in these models.
As a result of this study we set up the strategy and create the framework for a generic exploration of the models with di-boson, di-jet and top-quarks final state signatures. We summarize our findings and conclude in Section VI.

II. CLASSIFICATION OF THE MODELS
The models we consider are gauge theories based on a simple HC group G HC characterized by having two distinct irreps of fermions, denoted by ψ and χ 2 . In addition to hypercolor, the ψ fermions carry electroweak (EW) quantum numbers chosen in order to generate a composite pNGB Higgs, while the χ carry QCD color and hypercharge.
The phenomenological reasons for this choice are two-fold. On the one hand, these models easily accommodate the presence of potential top partners obtained from HC neutral bound states of three fermions. To this end a non zero hypercharge Y χ has to be consistently assigned to the fermions χ. On the other hand, separating color (carried by the χ's) from the Higgs sector avoids problems with spontaneous color breaking and the presence of light colored pNGBs. Even more relevant for this work, the presence of two distinguished irreps allows for the existence of a light pNGB associated to a U (1) axial symmetry non-anomalous under HC.
The main constraints on the models under consideration which define their group stucture are: implementation of a composite Higgs mechanism compatible with custodial symmetry, the existence of top partners and the protection of the Z → b LbL branching ratio. The last requirement eliminates some possible solutions that were present in the lists [3] [27] with ψ in a complex irrep and top partners in the (2, 1) of SU (2) L × SU (2) R [32].
The Higgs mass is generated by the explicit breaking of the global symmetry of the strong sector. Typical sources of breaking are the coupling to the EW bosons and to the heavy quarks as well as possible bare masses for the hyperquarks. The measured Higgs mass is then used as an input to give one relation between these couplings, the low-energy coefficients of the strongly coupled theory (in principle computable on the lattice) and f ψ . A similar relation follows from fixing the Higgs vev. That it is possible to fix the Higgs mass and vev to their physical values (at the cost of some fine-tuning) is shown in various previous works: e.g. [33] and [4] for the cosets of interest in this paper.
Within the constraints above there are three minimal cosets SU (5)/SO (5), SU (4)/Sp (4) and SU (4) × SU (4) /SU (4) D in the case of N ψ = 5 real, N ψ = 4 pseudo-real or N ψ = 4 complex (plus its conjugate) irreps of G HC respectively coming from condensation of the fermions ψ. If this were the only condensate forming, there would be no more pNGBs in the spectrum. In particular, in this case the axial U (1) ψ rotating all ψ by the same phase would be spontaneously broken but also explicitly broken by a U (1) ψ G 2 HC Adler-Bell-Jackiw (ABJ) anomaly and thus its would-be Goldstone boson would acquire a large mass.
In the present class of models, however, the χ also condense, giving rise to new colored pNGBs associated to the cosets SU (6)/SO (6), SU (6)/Sp (6) and SU (3) × SU (3) /SU (3) D for N χ = 6 real, N χ = 6 pseudo-real or N χ = 3 complex (plus its conjugate) irreps of G HC respectively. Now there is also an additional axial U (1) χ spontaneously broken and it is possible to construct an ABJ anomaly free linear combination U (1) a by choosing the charges q ψ,χ to obey where T denotes the Dynkin index of the irrep and for complex irreps we must count both the complex and its conjugate, i.e. count the index twice. The pNGBã associated to this symmetry is naturally lighter than the typical confinement scale, while the remaining orthogonal stateη acquires a larger mass. We denote these states with a tilde because they do not yet correspond to mass eigenstates, as will be discussed in the following section.
Among the remaining states, a color octet π 8 stands out as an unavoidable prediction, independent on the type of irreps in the model. For the case SU (3) × SU (3) /SU (3) D this turns out to be the only one, for SU (6)/SO (6) and SU (6)/Sp (6) there is an additional color sextet and triplet respectively. The full list of pNGBs is given in Table II.
The relative model independence of these three pseudo-scalars (the a, η and π 8 ) and the fact that they have a direct coupling to gluons via the WZW anomaly, implying a larger cross section as compared to e.g. the pNGBs in the electro-weak sector, are the reasons why we focus on them in this work. They are indeed "standard candles" that will allow to falsify these models with the minimal number of additional assumptions.
We conclude this section by presenting in Table I Complex Real SU(4) 2 /SU(4) × SU(6)/SO (6) SO(N HC ) 4 × (Spin, Spin) 6 × F N HC = 10 −q χ /q ψ is the ratio of charges of the fermions under the non anomalous U (1) combination. F, A 2 , S 2 , Ad and Spin denote the fundamental, two-index antisymmetric, two-index symmetric, adjoint and spinorial irreps respectively. A bar denotes the conjugate irrep.
We split the table according to the reality properties of the irreps, from which the pNGB coset can be read-off. We also indicate the range of hypercolors for which the theory is likely to be outside of the conformal region 3 . In fact, the mechanism of partial compositeness relies on the fact that the theory is conformal in the UV, so that a large anomalous dimension for the operator corresponding to the fermionic bound state can, in principle, be generated. This large anomalous dimension would allow to decouple the scale of flavor symmetry breaking and the EW scale. The compositeness scale Λ then is identified with the scale where conformal invariance is broken explicitly.
One possible philosophy is to view the compositeness scale Λ as the scale in which some hyperfermions acquire a mass and the theory exits the conformal window due to the reduced number of fermionic matter. This mechanism has recently been tested on the Lattice for a multi-flavor QCD-like theory [36,37]. With this interpretation, the promising models are those which are not conformal and yet contain enough light fermions to allow for the construction of a custodial coset for EW symmetry breaking as well as top-partners. These models can then be simply brought into the conformal window by adding additional fermions of mass ≈ Λ, possibly in the same irreps already used. Another possible philosophy is to rely on the top couplings responsible for partial compositeness: the operator responsible for the linear mixing grows in the IR due to the large anomalous dimensions, thus it breaks the conformal invariance when its coefficient becomes relevant. We will however rely on the former scenario.
We would like to end this section by commenting on the possible symmetry breaking patterns for these theories. First of all, all models in Table I are "vector-like" in the sense that a gauge invariant mass term can be added for every fermion. This implies, by the Vafa-Witten argument [38], that the HC group remains unbroken and thus a ψχ condensate never forms.
As far as the condensation of each separate pair ψψ and χχ goes, there is also the logical possibility of the presence of massless composite fermions in the IR matching the 3 It is notoriously difficult to exactly characterize the conformal region of non-supersymmetric gauge theories outside of the perturbative regime. There are however some heuristic arguments and, luckily, most of the models in Table I are rather clear-cut cases [4]. There has also been intensive work on the lattice, reviewed in [34], with some more recent related contributions in [35,36], but unfortunately a universal consensus has not yet been reached.  Table I in addition to the ubiquitous a and η . The Electro-weak coset arises from the condensation of ψ while the Color one from χ. The sextets can have two possible charge assignments depending on whether the top-partners are realized as "χψχ" or "ψχψ".
't Hooft anomaly [39] of the chiral global symmetry and thus preempting the need for symmetry breaking. This possibility has been suggested as the reason behind the lightness of top partners in [40,41]. By invoking the persistent mass condition, however, we find this last scenario unlikely. In all the models classified as non-conformal in Table I, the only   possible hypercolor invariant fermionic bound states composed of at most three elementary fields must contain at least one ψ and one χ fermion. Giving a common mass to one type of fermions (e.g. ψ) renders all the fermionic bound states massive. However, the other type of fermion (e.g. χ) is still massless and with non vanishing 't Hooft anomaly. Since such an anomaly cannot be canceled by the composite states, the corresponding symmetry must be spontaneously broken. Reversing the role of the fermions we reach the same conclusion for the other coset. We point out that this argument is not rigorous. Its main weaknesses are the possible existence of phase transitions [42], invalidating the massless limit, as well as the fact that we are ignoring bound states composed by five or more fundamental fermions, which can sometimes be formed using only one fermion species. We find it however sufficiently convincing to assume that both condensates form, a necessary condition for the existence of the pNGBs considered in this work.

A. Chiral Lagrangian
In this section we discuss in detail how an effective Lagrangian formalism can be used to describe the properties of the singlets associated to the global U (1)'s. A chiral perturbation theory for the class of models of interest has been recently presented in Ref. [8], including the singlet -referred as a in our work -associated with the non-anomalous U(1). Here, we want to be more general and keep both states a and η in the low energy Lagrangian, as the mass generated for the anomalous current may be not very large.
As the model contains two fermion condensates, the chiral Lagrangian can be described in terms of two copies of the pNGB matrix Σ r and two singlets Φ r , where r = ψ, χ. The Σ r 's contain the pNGBs from the non-abelian cosets, while Φ r 's contain the singlets. Furthermore, we want to choose the normalization of the decay constants f r 's in such a way that the mass of the W (and Z) bosons can be written as where θ is an angle describing the misalignment of the vacuum [1] (thus, sin θ = 1 represents the "Technicolor" limit of the theory, where f ψ = v SM = 246 GeV). In this way, we can define the ratio as a measure of the fine tuning needed in the alignment of the vacuum. The presence of the parameter characterizes the main advantage of models of this type compared to earlier Technicolor models. The S-parameter has an additional suppression by a factor circumventing EW precision tests albeit at the price of some fine-tuning.
This notation has the additional advantage that the Higgs couplings to the vector bosons are the same for all cosets and are, in fact, the same as those of the minimal coset SO(5)/SO(4) (for which EW precision tests and Higgs couplings generically require 0.1 [10,11], or equivalently f ψ 800 GeV). However, this forces us to normalize the chiral lagrangian differently depending on the nature of the ψ irrep. To allow us to write a common expression for all cases, we introduce the quantity 1 elsewhere ; (4) in terms of which where T a r are the non-abelian generators in the fundamental irrep normalized so that Tr[T a r T b r ] = δ ab /2, f r and f ar are the decay constants for the non abelian pions and the singlets respectively. The matrix Σ 0,r is the gauge-preserving vacuum 4 .
Following this convention, the lowest order chiral Lagrangian can be written as: Notice that we chose the same normalization (driven by the nature of the ψ irrep) for both cosets, in order to simplify the notation for the abelian pNGBs later.
A few comments are in order at this stage: for the singlets, the lowest order operator simply gives a kinetic term which does not depend on f ar . However, the couplings of a r will always be generated by the couplings of the U (1) currents to the underlying fermions, which depend on an arbitrary parameter, i.e. the charge Q r of the fermions under the global U (1).
This consideration justifies why the decay constants f r and f ar are, in principle, unrelated.
In the following, we fix the decay constants by choosing Q r = 1 for r = ψ, χ. A stronger relation between the decay constants of the singlets and the non-abelian pions in each sector can only be drawn assuming that both are dominantly made of di-fermion states. In QCD, this situation is achieved in the large-N c limit [43], following from Zweig's rule, where the singlet associated to the anomalous U(1) is also expected to become light. All mesons can therefore be described by a single meson matrix Φ 2 r Σ r (the Φ 2 r comes from the fact that the condensate has charge 2). The chiral Lagrangian, then, looks like which is consistent with the above formulation for f ar = √ N r f r , N r being the dimension of the flavour matrix Σ r (N ψ = 4 for SU(4)/Sp(4) and SU(4)×SU(4)/SU(4), N ψ = 5 for SU(5)/SO (5), N χ = 6 for SU(6)/Sp (6) and SU(6)/SO (6), and N χ = 3 for SU(3)×SU(3)/SU(3)). In the following, we will be interested in cases like the large-N c limit of QCD where both singlets can be light, so that we introduce the parameters which should be equal to 1 in the large-N c limit. Note that corrections to this relation will be generated by loop corrections in the chiral Lagrangian [44,45].
Out of the 2 singlets we introduced, only one remains a pNGB because it is associated to the anomaly-free combination of U(1)'s. If q ψ and q χ are the charges associated to the anomaly-free current, defined in Eq.(1), the pNGB gauge eigenstates,ã and the anomalous η , can be defined as For later convenience, we define a single dimensionless parameter describing this basis, i.e.
an angle ζ: Note that all physical observables will only depend on ratios of the two charges q r . The values of q χ /q ψ for the various models are listed in Table I, always leading to values of tan ζ < 0 (for which we define the angle in the rage −π/2 < ζ < 0 in the rest of the paper).

B. Couplings within the strong sector
The couplings of the singlets can only be generated by terms explicitly breaking the global symmetries. The partial gauging of the non-abelian global symmetries cannot do the job, as the gauged generators are not charged under the U(1)'s. (For recent lattice results on the case of charged pNGBs see [46]. Even more recently, a detailed analysis of the reach of the LHC in the search for the doubly charged pNGB present in the SU (5)/SO(5) models has been given in [47].) If a mass term for the underlying fermions is added, however, it necessarily carries the U(1) charge of the specific fermion. Following [8], we add the fermion masses in the Lagrangian as follows: The spurions X r are related to the fermion masses linearly where B r is a dimensional constant (that can, in principle, be calculated on the Lattice).
Note that, without loss of generality, m r is a real matrix in the non-abelian flavour space of the fermion specie r. From the above expressions, we can read off the masses of the singlets and non-abelian pions: In the limit where the condensates are aligned with the mass matrices m r = µ r Σ † 0,r , which corresponds to the EW preserving vacuum and where µ r is a common mass for all underlying fermions, the masses simplify to (all the non abelian pions having the same mass) where N r is the dimension of the matrix Σ r . We recover the result that in the large-N c limit, the masses of all mesons are equal as ξ r = 1.
We also note that Eq. (11) contains linear couplings of the singlets to the non-abelian pions: which potentially include mass mixing terms between the singlet and the non-abelian pions.
In the limit where both vacuum and mass matrices are aligned with the EW preserving direction, the expression simplifies to where the dots include terms with more fields and d abc = 2Tr[T a r {T b r , T c r }] is a fully-symmetric tensor. The presence of mixing with or couplings to other non-abelian pions depends on the coset. In the EW sector, 3 possible cosets are allowed. For the coset SU(4)/Sp(4), we found that no mixing and no coupling is possible as the trace Tr[Σ ψ X † ψ ] is real. In the SU(4)×SU(4)/SU(4) case, at leading order in v/f ψ no mixing is generated however a coupling to the triplets and to the second doublet is generated, allowing 2-body decays into these additional pions. This coupling can potentially affect the phenomenology of the singlet, if the additional pions are light enough. In the SU(5)/SO(5) case, we found that a mass mixing with all neutral pseudo-scalar is generated by the Higgs VEV at leading order.
More details on such couplings can be found in the Appendix B. Finally, in the color sector generated by the χχ condensate, we found that a coupling to 3 colored pions is present in the SU(6)/Sp(6) and SU(6)/SO(6) cases.

C. Couplings to SM fermions
The link of the strong dynamics to SM fermions is another source of explicit breaking of the global symmetries that may induce direct couplings of the singlets to fermions [9]. To generate a mass for the top, the class of models we want to investigate implements partial compositeness, where the top mass is proportional to two linear mixings of the elementary fermions to composite states 5 : where Ψ q L /t R are fermionic composite operators that have the same quantum numbers as the left-handed and right-handed tops respectively, and which contain the top partners at low energy. As such operators are made of 3 fermions, they carry charge under the two U(1)'s: the couplings of the pions can then be recovered by assigning a charge to the pre-Yukawas y L/R that matches the one of the composite operators. Without loss of generality, each spurion can be associated with a combination of pion matrices and similarly for y R . As m top ∼ y L y R , the singlets decouple from the top quark as long as the charges of the two pre-Yukawas are opposite [27,48]. However, this situation can never be realized in the class of models under consideration. If both pre-Yukawas involve the same operator in terms of fundamental states, then the charges are the same as the U(1)'s are axial. The charge assignments depend on the structure of the fermionic bound states: if the fermion is of type "ψψχ", then the possible charges of the spurions and of the top mass m top → (n Lψ + n Rψ , n Lχ + n Rχ ) ≡ (n ψ , n χ ) = (±4, 2) , (0, ±2) , (±2, 0) .
For "ψχχ", it suffices to exchange the two charges. We see that in no case the charge of the top mass can be zero for both singlets. The couplings of the singlets to tops can therefore be written as Changing basis toã andη , the couplings read where we recognize that the couplings of the pNGBã is proportional to the charge under the non-anomalous U(1). Note, however, that the reasoning above is only valid if the operators Ψ q L and Ψ t R , that mix to the top, have definite charges, i.e. they correspond to a well defined combination of hyperfermions. In general, as different operators transform in the same way under the global symmetries, mixing among operators is possible.
In this class of composite Higgs models, the matter content of the confining sector cannot accommodate enough partners to realize partial compositeness for all fermions: the Yukawa couplings of the light fermions must therefore come from a different operator. A simple possibility [49] is to introduce couplings of SM bilinearsf f with the strong sector: 6 The various assignments correspond to the following operators: (2,1) for ψψχ, (-2,1) forψψχ and (0,-1) forψψχ. Here we only focus on left-handed operators, which can be made of 3 left-handed fermions, or 2 right-handed and 1 left-handed one.
where y bil ∼ m f and the flavour scale Λ F can be much higher than the condensation scale.
While these operators are generically irrelevant, they can be large enough to reproduce light quark masses, and suppressed enough to evade flavour bounds [4,50,51]. Another possibility would be that the masses of light fermions are generated at higher scale, possibly via partial compositeness [52]. The U(1) symmetries can be formally restored promoting y bil to be a spurion only charged under U(1) ψ , and this implies a low energy coupling proportional to This coupling has the same form as the one we derived for the top, but with fixed charges n ψ = 2 and n χ = 0.

D. Masses and Mixing of the Singlets
The masses for the singlets are generated by the masses of the underlying fermions, m ψ and m χ , and the instanton related to the anomalous current. Even though couplings to tops and light fermions exist, they do not lead to corrections to the mass of the singlets. One way to see this is that all loops of fermions will be proportional to the absolute value of the spurions in order to write an operator which is gauge invariant. Thus, the dependence on the singlet pions, which comes in via exponentials, vanishes.
The mass matrix for the singlets, therefore, can be written from where M 2 A is the mass generated by instanton effects, proportional to the topological susceptibility of the hypercolor group, for the singletη associated with the anomalous combination of U (1)'s. For now, we will consider it as a free parameter, even though the topological mass is, in principle, calculable once the underlying dynamics is specified.
In the following, we want to entertain the case where the topological mass may be small, as it happens in large-N c QCD [43,53]. In fact, in many of the models we consider the number of colors is large and/or the representation of the underlying fermions is large.
Another physical consideration allows us to simplify the mixing structure: the mass of the pNGBs in the EW sector, due to the condensation of the ψ's, also contributes to the mass of the SM-like Higgs boson. Thus, its value is constrained to be small in order to minimize the fine tuning in the Higgs mass. While the details depend on the specific model, some general considerations are in order. The mass term can be used to stabilize the Higgs potential against the contribution of the top loops and obtain a small misalignment in the vacuum [54,55]. In such cases, one would expect m π ψ ∼ f ψ . Alternatively, if the top partners are light enough, their contribution to the Higgs potential is also enough to stabilize it and give the correct value of the Higgs mass [56][57][58]. In this case, therefore, one would require that the contribution of the fermion mass were small, i.e. m π ψ f ψ . This situation contrasts with the coset generated by χ: here, colored pNGBs are expected and the strong constraints from searches at the LHC require their masses to be close to the TeV scale [59].
It is thus natural to expect that m π ψ m πχ . 7 In the following, we will work under this assumption and, for simplicity, neglect the contribution of m a ψ 8 .
We will first diagonalize the mass matrix from Eq. (24), after setting m a ψ = 0. We define The mixing angle can be expressed in terms of the mass eigenvalues and the parameter ζ as Note that for m a m η (m aχ M A ), then α ∼ ζ and the mass eigenstates coincide with the pNGB and the anomalous combination, as expected.
The mass matrix depends on 3 independent parameters: 2 masses and the angle ζ. It is convenient to trade the two masses for the mass eigenvalues which have a more direct physical meaning. Thus, we can define a "physical basis" thanks to the following relations: 7 The pNGB masses are related to the hyperquark masses m ψ and m χ by the usual Gell-Mann Oakes-Renner relation, see Eq. (13). The hierarchy m π ψ m πχ can be obtained by choosing m ψ m χ . 8 To restore the dependence on m a ψ it is sufficient to replace m 2 aχ → m 2 aχ − m 2 a ψ , m 2 a → m 2 a − m 2 a ψ and m 2 η → m 2 η − m 2 a ψ in all the formulas in this section, as long as m a ψ < m aχ .
However, there are constraints on the value of the physical masses. From the positivity of the argument of the square root in the above formulas, we can derive a lower bound on the mass difference: From the equation above we can see that the two masses can be equal only in the limiting cases ζ = ±π/2 and ζ = 0, when the two U(1) pNGBs decouple: in the former, a χ is identified with the non-anomalous U(1), while in the latter it is a ψ . Note that the apparent divergence for ζ = 0 is removed by the fact that m a = 0 in that limit. The value of the lighter mass is also a monotonically increasing function of M A , thus it reaches the maximum value for M A → ∞: The above constraint has significant physical implications as, for models with low values of ζ, it implies that the mass of the lightest singlet has to be much lighter than the condensation scale f χ , as m aχ cannot be much larger than f χ without spoiling the validity of the chiral Lagrangian expansion. It is also interesting to notice that the mixing angle α is bounded between: The lower bound corresponds to the minimal splitting between the two mass eigenvalues, while α = ζ is achieved in the decoupling of η .
As already mentioned the topological mass term is in principle calculable in a given underlying theory. We can then extract a simple correlation between the mass mixing angle α and the mass of the lightest singlet m a for fixed topological mass M A From Eq. (26) we can extract the allowed range for each mass eigenvalue when m aχ ≤ M A , i.e. (cosine is taken to be positive) For m a M A we get the upper bound in Eq. (26), while for m a maximal we can saturate the lower bound. Such an additional constraint can be significant when the topological mass is not larger than a few TeV.

E. Non-abelian pions: the octet
Among the many non-abelian pions present in these models, there is a common player that appears necessarily in all cosets: a color octet from the χχ condensation. Independently on the representation of χ under the confining HC, the octet π 8 can be identified as a bound state of χ 1 χ 2 , where χ 1,2 are the fermions transforming like a QCD color triplet and antitriplet respectively. Due to its ubiquitous presence, and the fairly large production cross sections one may expect at the LHC, in the following we will consider its phenomenology and possible connections with the properties of the singlets.
As a first connection, we note that its mass can be expressed in terms of the χ-mass as where the second term comes from loop corrections from QCD, and C g > 0 is an unknown order one number (the loop contribution is cut off at a scale Λ ∼ 4πf χ ). This provides a link between the mass of the octet and the masses in the singlet sector: in fact, m aχ is related to the singlet masses by Eq. (28). We also recall that ξ χ ∼ 1, as expected in the large N c -limit in QCD. In the limit of m a m η , where the lighter singlet reaches its maximal mass m a ∼ m aχ sin ζ, we obtain The relation above shows that typically we would expect the octet to be heavier than the light singlet pNGB, even if the color corrections were small.
The octet has also the possibility to couple to tops: like in the case of singlets, the presence or not of this coupling depends on the representation of the composite top partners under the global symmetries. As the octet pNGB is associated to the bound state χχ , which is also charged under the U(1) χ , it is straightforward to find a correlation between the effective charges of the top mass and the presence of a coupling with the octet. If the top mass has a effective charge ±2 under U(1) χ , as indicated in the previous section, then the effective operator generating the mass of the top needs to be "dressed" by the appropriate pNGB matrix Φ 2 χ Σ χ . If the charge is ±4, then two matrices are needed: this can also be understood in terms of indices of the global symmetry that cannot be contracted in an invariant way (but need the breaking generated by the consensate). On the other hand, if the charge is zero, then it is not needed to couple Σ χ to the top mass term, and a coupling to the octet is not necessarily present. One can thus find a nice correlation between the charges determining the coupling of the singlets to the tops, and the presence of an octet coupling. If present, the coupling will have the form: where λ a are the Gell-Mann matrices, and we have omitted the other pNGB and singlets. For the light quarks, if their masses are generated by 4-fermion interactions then no couplings to the octet pNGB are generated.
It should also be remarked that, contrary to the case of the singlet, the presence of top couplings will also generate corrections to the masses of the octet. Those contributions are more model dependent, as they crucially depend on the representations of the top partners, and are typically of the same order as the QCD corrections but expected to be negative: we refer the reader to [59] for an example.

F. Wess-Zumino-Witten terms
The couplings of the singlets to the SM gauge bosons, generated by the WZW term, can be computed in a similar way as in QCD [44]. Following the normalization adopted in this work, the couplings can be written as where and d r is the dimension of the rep r of HC, and S a,b in the trace correspond to the gauged generators with gauge coupling α A = g 2 A /(4π). The normalization of the gauged generators depends on the global group the gauge interactions are embedded in, so that their trace is not the same as for the generators of the flavor group. Specifically, we note that, in the cases of interest Tr[S a S b ] = δ ab , for SU(5) (ψ) and SU(6) (χ) ; For completeness and comparison, the WZW term for the non-abelian pions is where for complex r, and there is an additional factor of 1/2 for real/pseudo-real representations.

Singlets
The coefficients for the anomalous couplings of the two singlets are summarized in Ta-ble III, where we recall that d ψ and d χ are the dimensions of the representation of the fermions under HC. These numbers, calculated directly from the WZW term, have a simple physical interpretation. In the EW sector described by ψ, up to a factor of 1/2, the C W (C B ) coefficients count the number of Weyl spinors transforming as SU(2) L (SU(2) R ) doublets: d ψ in the SU(4)/Sp(4) coset and 2d ψ in the other two cases. Furthermore, as the theory is symmetric under the custodial symmetry, the number of doublets is equal, leading to Similarly, in the χ sector, the anomaly of QCD color is equal to half the number of SU(3) c triplets, which is 2d χ in all cases. Furthermore, Combining the two relations above, we can see that for both a ψ and a χ , the values of the anomalous couplings always satisfy the relation: which only depends on the model-specific value of the hypercharge Y χ . This relation will also be respected by the coupling of any linear combination of the two singlets, thus also by the mass eigenstates. As Y χ = 2/3 or 1/3, all the models under consideration have anomalous couplings lying on 2 universal lines

Color octet
The anomalous coupling of the octet with the gluon field strength G a µν and the hypercharge field strength B µν can be computed from Eq. (40), and are where d abc = 1 4 Tr[λ a {λ b , λ c }] and Y χ is the hypercharge assigned to the χ fermions, in agreement with [60]. The second term, coupling the color octet to a gluon and hypercharge gauge boson, will thus induce an effective coupling with a photon and one with a Z boson.
Neglecting the mass of the Z boson and using the color factors (1/8) abc (d abc ) 2 = 5/3 and (1/8) ab (δ ab ) 2 = 1, we find the following relations between partial widths in the 3 channels Γ gg : Γ gγ : Γ gZ = 1 2 with the additional factor of 1/2 in Γ gg being due to the indistinguishability of the gluons.
This means that the ratios of branching ratios in di-boson final states only depend on the hypercharge assigned to the χ's, which has two possible assignments (see Table I). The numerical values are thus reported in Table IV, where the coupling constants are evaluated at a mass scale of 1 TeV. We see that while the decay to a Z boson is always suppressed by a tan 2 θ W factor, the decay into a photon can be sizeable, especially for Y χ = 2/3, and will lead to interesting phenomenology [61].  Another important observation is that top loops are phenomenologically relevant only for large couplings to the top, in which case one would also expect that the decay rate is dominantly into tops. In such a case, the WZW couplings, with top loop corrections, are only important for the production cross section via gluon fusion. To illustrate this fact, we focus on the octet. The correction to the amplitude for gluon fusion production from the top loops from Eq.(C1) gives: where the function f (x) is defined in Eq.(C5). This correction can be compared to the ratio of partial width in tops and gluons (not including top loops): Already from the numerical factors involved one can see that the partial width into tops dominates over the one into gluons well before the top loop corrections become relevant.
The same conclusion can be obtained for the singlet, unless the WZW amplitude is small: in this case, however, that particular channel is not relevant for the phenomenology.

IV. PHENOMENOLOGY
We now turn our attention to characterizing the LHC phenomenology of the singlets a and η (that we collectively denote as π 0 in this section) and of the color octet π 8 . The experimental results coming from post ICHEP2016 data will be used to derive general constraints on the production cross sections that can be later applied to any of the specific models.
Our goal in this section is to be as model independent as possible. We thus introduce a common notation for the couplings of the various pseudo-scalars to vector bosons, with coefficients denoted by κ g , κ W , κ B , and to tops, with coefficient C t and perform the analysis with this notation. In Section V we show how to relate these coefficients with the modelspecific ones computed in Section III and obtain model-specific bounds.
A. Phenomenology of the singlet pseudo-scalars As we discussed in the previous section, the singlet pseudo-scalars couple to a pair of SM gauge bosons via the WZW anomaly terms, and to a pair of top quarks. The generic effective Lagrangian for a SM neutral pseudo-scalar π 0 can be written as which is characterized by five parameters: the mass M π 0 , the dimension-full coupling κ g /f π (coefficient of the anomalous coupling to gluons) that controls the production cross section, and the three ratios C t /κ g , κ B /κ g and κ W /κ g which dictate the branching ratios.
In the following, we will neglect the effect of top loops to the branching ratios into a pair of gauge bosons: the main rationale behind this is that, once such effects are sizeable, the decay is dominated by the tt final state, so that searches in di-boson final states become irrelevant. Thus, in this large top coupling limit, the only phenomenologically relevant effect will be on the gluon fusion production. As shown in Appendix C, the effect on gluon fusion can be encoded in a M π 0 -dependent shift of the κ g coupling. Thus, our analysis can be extended in a straightforward way.
The dominant production channel for π 0 is gluon fusion 9 . In Fig. 1 we show the production cross sections from gluon fusion as a function of M π 0 at the LHC with 8 and 13 TeV, calculated at leading order (without K-factor) using MadGraph 5 [62] and cross-checked against CalcHEP [63]. In our analysis we have used the NNPDF23LO (as_0130_qed) PDF set [64] and the QCD scale naturally chosen to be the mass of the resonance. We would like to note that although we evaluate cross sections at LO, one can re-scale them to known higher order corrections, which, for example for CP-Even Higgs boson production, are determined up to N 3 LO in QCD (see e.g. [65] for review and references there in). Since in our signal simulation we do not include correction factors for higher order QCD corrections, the estimate of the LHC potential to probe the theories under study is conservative.
In Fig. 1, the coupling to gluons is fixed to κ g /f π = 1 TeV −1 , and the production cross The singlet pseudo-scalars decay to either di-boson via the WZW interactions or to tt. 9 The only other production channels are vector boson fusion and associated production with gauge bosons or tops. However they are always subdominant.
The partial widths are related to the parameters in the Lagrangian in Eq.(49) as [30] where θ W is the Weinberg angle. Decays into other SM fermions are negligible, since they are suppressed by the fermion masses. As the couplings are typically small, we expect the total width to be always small. To give a numerical estimate, the partial widths in gluons and tops (that are typically dominant) evaluate to: It is instructive to split the decay modes into the final state tt and into di-boson final states. Furthermore, we will use ratios of branching ratios, which depend only on few of the couplings, to characterize the decay pattern of the singlets. As a starter, the ratio only depends on the ratio C t /κ g , and on the mass M π 0 : this is mainly due to the fact that the partial width in tt scales linearly with the scalar mass versus the cubic power in di-boson partial widths. Therefore, the relevance of the top final states decreases for increasing π 0 mass. We also define di-boson ratios These ratios depend on the coupling ratios κ B /κ g and κ W /κ g , while the dependence on the mass is weak and only entering through kinematic phase space due to the non-zero masses of the W and Z bosons and the logarithmic running of the couplings (in particular, the . In (f), BF tt/gg is plotted as a function of M π 0 for C t /κ g = 0.1 (BF tt/gg scales like (C t /κ g ) 2 ).
QCD one). We will thus use the ratios defined above to characterize the decay rates in a model-independent way.
For illustration and later use, in Fig. 2 (a) -(e) we show the ratios BF XY /bosons in the κ W /κ g vs. κ B /κ g plane, while Fig. 2 (f) shows the branching fraction BF tt/gg as a function of M π 0 . The first plots, (a) -(e), depend on M π 0 via the running of α s (the weak couplings are fixed to their values at the Z pole for simplicity, as their running up to the TeV scale is mild), thus the plots refer to a mass M π 0 = 1 TeV. The mass dependence can be disentangled by absorbing the running coupling in the definition of κ g , so that the ratios at a different mass can be obtained by rescaling Figs. 2 show that the gg channel dominates the di-boson branching fractions, followed by W W which becomes increasingly important for increasing κ W /κ g . The (smaller) branching fractions of ZZ, Zγ, and γγ increase along the directions |κ W + κ B tan 4 θ W |, |κ W − κ B tan 2 θ W |, and |κ B + κ W |. The magnitude of the branching fraction into tops is mainly controlled by C t /κ g . As the tt partial width scales with M π 0 while all di-boson partial widths scale with M 3 π 0 , the branching fraction into tt is reduced at high masses M π 0 2m t and kinematically suppressed near the threshold M π 0 = 2m t .

Experimental bounds from di-boson and tt resonance searches
Both ATLAS and CMS presented numerous searches for di-jet, W W , ZZ, Zγ, and diphoton resonances in the high mass region. The list of searches we include into our study is summarized in Table V. Where possible, we directly use the bounds on the production cross section times branching ratio (σ × BR) into the respective channel given by ATLAS and CMS. In several studies (in particular for di-jet searches and partially for Zγ and γγ searches), some results were presented in terms of fiducial cross sections or in terms of cross section times acceptance. In Appendix A, we summarize the assumptions made in order to extract the bounds from ATLAS and CMS studies for the model discussed in this article.
We aim at presenting collective bounds for the different di-boson and tt final states from pNGB decays. For searches in a given channel at √ s = 13 TeV, we do not perform a combination of the ATLAS and CMS searches but simply use the strongest bound obtained for a given M π 0 . To include Run I bounds, we analogously take the strongest bound at M π 0 in each channel, and rescale the cross section by a factor σ(gg → π 0 ) 13 /σ(gg → π 0 ) 8 . The resulting constraints on the σ × BR at 13 TeV for the gg, W W , ZZ, Zγ, γγ, and tt channels  are shown in Fig. 3 10 .

Model-independent bounds on the singlet pseudo-scalar parameter space
The experimental constraints shown in Fig. 3 translate into bounds for the still allowed production cross section as a function of (M π 0 , κ W /κ g , κ B /κ g , C t /κ g ) via the branching fractions following from Eqs. (50)(51)(52)(53)(54)(55), as exemplified in Fig. 2. Using Fig. 1, the bound on the production cross section translates into a bound on the coupling to gluons κ g /f π .
To simplify the impact of the multi-dimensional parameter space, it is useful to split the final states into two categories: di-boson and tops (tt). The advantage is that the ratios between di-boson modes only depend on two ratios of couplings (and very mildly on the mass), while the rate of tt final states can be expressed in terms of C t /κ g . We thus define the following strategy apt to explore, in a way which is as model independent as possible, the parameter space of this class of models: -define the cross section in a specific di-boson final state as: -from the above, one can extract a bound on σ × BR bosons as a function of the mass and the two ratios of couplings κ W /κ g and κ B /κ g ; -for each value of C t /κ g , the function BF tt/gg can be used to calculate the cross section in tt final state, as matching the di-boson bound, which can be directly compared to the bound from tt searches as shown in Fig. 3.
The latter step allows to determine whether the strongest bound comes from di-boson searches, or from tt. Note, however, that this approach is only valid in the narrow width approximation, which is always true in this class of models where the couplings are small, as suppressed by a loop factor in the case of WZW interactions, or a ration m t /f π for top couplings, as shown in Eq. (56).
To map out the model parameter space, let us first consider bounds for fixed mass M π 0 .
In Fig. 4 we show the bounds on σ 13 × BR bosons in the κ W /κ g vs. κ B /κ g plane for various resonance masses. The colored regions tag the decay channel that, with current data, yields the strongest bound at a given parameter point. At κ W /κ g = κ B /κ g = 0, the branching ratios in all di-boson channels, apart from gg, are zero, thus strongest bound around the origin arises from the gg channel (in orange). For increasing |κ B,W /κ g |, the bound on σ 13 × BR bosons initially becomes marginally weaker because of a depletion in the leading gg channel. For further increased |κ B,W /κ g |, channels other than gg become the most constraining ones, at which point the bound becomes stronger again, being dominated by EW boson final states. We see that along the direction κ B ∼ κ W , it is γγ that dominates the constraints (in blue), while along the orthogonal direction, where the coupling to photons partially cancels, the W W (green) and/or Zγ (yellow) channels take over the lead. Eq. (49). As outlined in Sec.II, the models considered in this article predict SM singlets a and η for which depending on the hypercharge of χ: the two grey diagonal lines in Fig. 4 mark these model lines for reference.
As becomes clear from Fig. 4, all di-boson channels (apart from ZZ) yield the dominant constraint in some portion of the parameter space. Fig. 4 only indicates the channel setting the bound, but through Fig. 2, or equivalently Eqs. (50)(51)(52)(53)(54), and the experimental bounds shown in Fig. 3, the relevance of each decay channel at a given parameter point M π 0 , κ B /κ g , κ W /κ g can easily be obtained.
As an example of how to use these results in application to a specific model, let us consider the point ( κ B κg , κ W κg ) = (2.1, −0.55) at M π 0 = 1 TeV 11 . From Fig. 4 (top right), the constraint on σ 13 × BR bosons reads as 0.3 pb. Multiplying 0.3 pb by BF σ XY /bosons = (98%, 1%, 0.14%, 0.9%) for XY = (gg, W W, ZZ, Zγ) (extracted from Fig. 2)  as evident in Fig. 3. a signal cross sections of (290 fb, 3 fb, 0.4 fb, 2.7 fb) for the respective final states. These values are a factor of (6, 6.5, 50, 3.5) respectively lower than the cross section bound for M π 0 = 1 TeV in Fig. 3, showing how close each bound is to the limit. Following this universal recipe different models from Table III with different mass values can be easily tested using the information from Figs. 2-4.
As outlined in Sec. II, the models considered in this article predict SM singlets a and η whose couplings lie on two lines depending on the hypercharge of the χ's (shown by the grey diagonals in Fig. 4). In order to present bounds for different resonance masses than the ones given in Fig. 4, we give results along the above lines. The bound on σ 13 × BR bosons as a function of M π 0 and κ B /κ g is shown in Fig. 5, with the same color code as above.
In addition to the di-boson bounds presented in Figs. 4 and 5, tt resonant searches provide a further constraint, that depends on the precise value of the ratio of couplings C t /κ g : we present here a simple way to extract the bound on the cross section. The tt constraint dominates over the di-boson bounds if where (σ 13 × BR tt ) exp is shown in Fig. 3 (f)  The color octet π 8 , which is present in all models discussed in Sec. II, can be described by the effective Lagrangian where the covariant derivative contains QCD interactions with gluons. In the models discussed in this article (f π 8 = f χ ), matching with Eq.s (46) and (37), the coefficients are equal The octet π 8 is produced at the LHC in pairs via QCD interactions or singly via gluon fusion 12 . The production cross section at the LHC for 8 and 13 TeV are shown in Fig. 7. 12 Single production through gluon-photon fusion would require very large hypercharge of the constituent fermions, so we neglect it, here. Single production from tt fusion is also suppressed by the need of creating top pairs from gluon splittings as well as by the additional (m t /f π8 ) 2 suppression from (64). Like for the singlet, we calculated the cross section at leading order (without K-factor) using MadGraph 5 and cross-checked against CalcHEP both with the NNPDF23LO (as_0130_qed) PDF set, and the QCD scale set to the mass of the resonance. The partial widths of π 8 from the Lagrangian (64) are given by: Like for the singlets, the total width is always small, as numerically shown for the di-gluon partial width below: In all the models under study in this paper, the ratio of the two WZW couplings only depends on the hypercharge Y χ , that can take two values Y χ = 1/3 or 2/3, depending on the model. Instead of a complete model independent analysis, we will impose this constraint that fixes the ratios of decay rate in the bosons. In analogy with the singlets, we can thus Besides a mild mass dependence, the rate into a Z is suppressed by the Weinberg angle: numerical values of these ratios are reported in Table IV. The decay in tops, however, strongly depends on the ratio C t8 /κ g8 which is very dependent on the details of the model, and on the mass of the octet: In the following we will treat the ratio C t8 /κ g8 as a free parameter. For the models in Section III, this ratio is always smaller than 1 and it vanishes when the couplings to tops is absent.

Searches and bounds for pair-produced color octets
Spin zero color octets have already received some attention in the literature [102][103][104], as they arise in other models like sgluons in extended supersymmetry [105,106], and they are copiously produced at hadronic colliders. A direct comparison of cross sections can be seen in Fig. 8 where we show the pair production cross sections at 8 and 13 TeV, together with the bounds on the cross section times branching ratios in the two covered final states, (σ × BR(4g)) exp and (σ × BR(4t)) exp respectively. The bound on the 4-top final state can be directly compared to the production cross section for large C t8 /κ g8 , for which the BR in tt is nearly 100%, excluding masses below 880 GeV. On the other hand, in the absence of top couplings, the bound on 4-jet 13 For the 8 TeV search, CMS presents bounds for an inclusive search for R-parity violating decays of pair produced squarks as well as for a coloron (a scalar octet) search. We use the latter, which yields weaker bounds, such that the exclusions quoted here are conservative. We remark that the 13 TeV search [108] yields stronger bounds than both searches at 8 TeV. 14 A first search with the 13 TeV data has been published [111], however not presenting the case of the color octet. Thus, we cannot use directly these results. depends on the BR which depends on Y χ 15 : A recast of the pair production bounds as a function of C t8 /κ g8 is shown in Fig. 9. For vanishing top coupling, the lower bound on the mass from the 13 TeV search gives 650 (700) GeV for Y χ = 2/3 (1/3).
We A very interesting option is to search for (gγ)(gg), i.e. a di-jet resonance with the same invariant mass as a photon-jet resonance. As compared to the two di-jet channel (gg)(gg), the cross section of the (gγ)(gg) channel is only reduced by σ(pp → π 8 π 8 → γggg)/σ(pp → π 8 π 8 → 4g) ≈ 0.4 (0.1) for Y χ = 2/3 (1/3) 16 . However, the background of the process can be vastly reduced due to the photon in the final state.

Searches and bounds for a singly produced color octet
The color octet π 8 can be singly produced via gluon fusion, with cross section at √ s = 8 TeV and 13 TeV shown in Fig. 7. The possible final states for single-production are tt, gg, 15 We give branching ratios for a reference mass M π8 = 1 TeV, here. 16 We use the branching ratios for an octet mass of 1 TeV given in Table IV.  [59]) and the current bounds on σ × BR(pp →tttt) [109,110] and σ × BR(pp → 4j) [  and can be reused here as the kinematical differences due to the color of the resonance are subleading. The contribution of a color octet to tt production was also analyzed in [112] in the similar context of multiscale technicolor. The gγ final state offers a clean channel due to the presence of an energetic photon [61]. While dedicated searches are not available, one can easily adapt searches for excited quarks by both ATLAS [113,114] and CMS [115,116] where the gluon is replaced by a light quark jet. The bounds from these searches are summarized in (sensitive to invisible decays of the Z, Z inv j) by ATLAS [117,118] and CMS [118,119] at both 8 and 13 TeV. In Ref. [120], a recast of the 8 TeV searches in di-boson channels for a color octet scalar are presented, showing that, for the Zg/gg branching ratios in the models considered in this article, the Zg channel yields subleading bounds as compared to the gg and gγ channels. We therefore do not include the Zg channel in our analysis.
To combine the constraints on singly produced color octets, we follow a strategy similar to the one designed for the singlet pseudo-scalars in Sec. IV A: the analysis is simpler because the color octet bosonic ratios BF π 8 gγ/gg and BF π 8 gZ/gg in Eq. (72) are fixed up to a discrete choice Y χ = 1/3 or 2/3. For these two choices, we can directly translate the bounds on the gγ channel (Fig. 10) and the gg and tt channels (Fig. 3) into bounds on the π 8 production cross section as a function of the mass and the model parameters C t8 /κ g8 , as shown in Fig. 11.  FIG. 11: Bounds on the production cross section (in pb) in the C t8 /κ g8 vs. M π 8 plane for Y χ = 1/3 (left) and Y χ = 2/3 (right). The blue region is excluded by octet pair production searches. The currently strongest bounds arise from the tt channel (gray regions), gg channel (orange regions) and the gγ channel (light-cyan regions).

V. IMPLICATIONS FOR COMPOSITE MODELS
After presenting general results in Section IV, we want to look back at the models introduced in Section II and study how the present searches can constrain the presence of the singlet and octet pseudo-scalars, and what are the prospects at the LHC Run 2. Instead of looking at all models, we will derive general properties before focusing on a few interesting cases. The first step is to connect the general Lagrangians used in the previous section with the couplings derived in Section III: the two scales f π and f π 8 that we used to normalize the couplings are arbitrary, thus allowing us to chose the most convenient normalization.
For the singlets, the most natural choice is to normalize f π = f ψ for both mass eigenstates, as this is the scale most directly connected to the EW symmetry breaking and thus to the fine tuning in the Higgs sector. Furthermore, the couplings will depend on the mixing angle α that defines the mass eigenbasis. For the lightest singlet a, the couplings to field strengths in Eq. (38) are mapped to the model independent parametrization Eq. (49) by Similarly, the coupling to tops is matched by where we recall that n ψ/χ are the U(1) ψ/χ charges associated to the top mass operator. The couplings of η are obtained from the same formulas with the replacement α → α + π/2.
Concerning the octet pseudo-scalar, it is convenient to normalize f π 8 = f χ , as this is the only scale directly connected to it. The couplings to the field strengths in Eq. (64)  While no Lattice data is available, we will reduce the number of unknown parameters by imposing some reasonable relations between the decay constants: 1. we impose the "large-N c " relation between the decay constants of the singlets and non-abelian pNGBs: f a ψ = N ψ f ψ and f aχ = N χ f χ ; 2. we fix the ratio of the two remaining decay constants to be equal: Regarding the second relation, this simple choice is not entirely justified on dynamical grounds 17 , and it is only chosen for its simplicity. The effect of making any other choice for this ration can be easily inferred by rescaling the couplings of the effective theory. 17 We thank Michele Frigerio for pointing out this argument to us.
One could use an argument based on the MAC hypothesis [121] to estimate the ratio of the scales where the two condensates occur [122]. This argument has been used in [6] As this arguments are semi-quantitative at best, we do not use these numbers in the paper, but only present them to show that it reasonable to expect the ratios to be of order one. Similar estimates have been performed earlier in the context of multi-scale walking technicolor [112,123].
Besides one decay constant f ψ , setting the scale of condensation, the other 3 mass parameters can be traded for the 3 mass eigenvalues m a , m η and m π 8 . The mixing angle α between the two singlets is then related to the mass eigenvalues (and the value of ζ) by Eq. (27).
A first phenomenological observation is that the mass splitting m 2 η − m 2 a is constrained by Eq. (30) and models with small | tan ζ| predict a large mass splitting and vice versa. To better quantify this effect, Eq. (30) that contains the minimum mass splitting, can be used to extract the maximal ratio of the two masses (achieved at minimal splitting with α = ζ/2): Numerical values of tan ζ and the quantity in the above equation for the 12 models under consideration, and under our ansatz on the decay constants, are reported in Table VI. In models with small tan ζ, like for instance M2, M7, M8 and M12, therefore, the light singlet tends to be substantially lighter than the second one and the octet. Another consideration is that the largest m a mass is correlated to the χ-mass by m a ≤ m aχ sin ζ: for the lighter singlet to be in the TeV range, one would thus need the mass generated by the χ to be in the multi TeV scale, implying that χ tends to behave like a heavy flavor -a fundamental fermion with sizeable mass compared to the condensation scale -and the chiral Lagrangian description needs to be modified. Those are qualitative arguments, but they tend to point towards a situation where only one of the two singlets (η ) may be relevant at the LHC. In In the following, instead of studying all the models, we will focus on two sample cases: they are chosen in such a way that the symmetries at low energy are the same, so that they can be described by the same low energy effective theory, and they have small couplings to tops. Nevertheless, they differ in the value of tan ζ, which substantially affects the spectra and couplings of the singlets. Complete tables reporting the numerical values of the couplings for all models can be found in Appendix D.

A. Two explicit examples
The two models we focus on are M8 and M9 (see Table I), where M8 was first introduced in [5] and its phenomenology partially studied in [59]. The two models are based on very different underlying gauge theories, however the global symmetry breaking pattern is the same, Sp (4) (2, 0) ψψχ (6 ⊕ 10) 2,1 1 ⊕ 5 ⊕ 10  Interestingly, in all cases the right-handed top can couple to a singlet of Sp(4) and the left-handed one to a 5-plet.
Once the gauge theory and the top partner operators are specified, the couplings of the singlets and octet can be calculated: as an example, we provide the numerical values in Table VIII, where the couplings of the singlets are provided for a mixing angle α = ζ (which corresponds to the decoupling of η ) and α = ζ/2 (corresponding to the minimal splitting). We recall that the couplings are normalised to f ψ for both singlets, and to f χ (here set equal to f ψ ) for the octet. The table clearly shows that the two models give rise to very different values of the couplings, thus providing an handle apt to distinguish the two if a signal is detected. Armed with the values of the couplings in the Table, one can go back to the plots of the previous section and reconstruct the best constraint for each mass point. In the following, we will put together all the constraints, and extract a lower bound on the decay constant f ψ .  Couplings for the two models discussed in the text in the limit α = ζ (and in parenthesis the values for α = ζ/2).
We start by discussing the octet in the two models: combining all the searches described in Section IV, one can extract a lower bound on f χ as a function of the octet mass. The final result is shown in Fig. 12, Left plot for model M8, and Right plot for model M9. The first feature we observe is that the constraint is much stronger for M9, due to the larger coupling to gluons of the octet, as shown in Table VIII f ψ = f χ . While here we consider only the octet, these two models also feature a charged sextet in the spectrum: its phenomenology has been studied in detail in [59]. The sextet mainly couples to right-handed tops, and it is expected to be slightly lighter that the octet.
It only affects searches for 4-tops, which yield a lower bound on the mass of the order of 1 TeV, thus stronger that the octet one, while the other octet final states are not affected by the presence of the sextet.
The channels that can give a direct probe of the fine tuning in the EW scale are the singlets, which can directly feel the value of f ψ . Also, for singlets, the two models appear rather different due to the value of tan ζ: in particular, for M8, the lighter singlet is always expected to be much lighter that the second one, as m a m η /5. Furthermore, due to the large coupling to tops, as shown in Table VIII, it will dominantly lead to tt final states with a weak bound on f ψ of the order of 200 GeV for a mass below 1 TeV. We thus decided to focus on the signatures generated by the heavier singlet η : in the top row of Fig. 13 we show the results for the two charge assignments, assuming decoupling limit (i.e. α = ζ and m a m η ). From the ratio values in Table VIII and Fig. 6 (bottom right), we see that for (n ψ , n χ ) = (±2, 0) the most constraining channel is γγ in the entire mass range 500 GeV < m η < 4000 GeV, while for (n ψ , n χ ) = (0, ±2), the γγ and the tt channels yield comparable bounds. Closer investigation shows that the current constraint from the tt channel dominates only in the mass regime 900 GeV < m η < 1100 GeV. Overall, the bound on f ψ is weaker for the (n ψ , n χ ) = (0, ±2) charge assignment, which has a larger coupling to tops, because the larger branching ratio into tt reduces the dominant γγ one.
The plots also show that the bounds tend to be below the EW precision test ones, except for very light masses.
In the case of model M9, we can study the constraint as a function of the two masses, as they are allowed to be close. We recall that the mixing angle α depends under study is studied on the Lattice: in this case, the ratios between the various decay constants can be calculated, and a more solid prediction can be obtained for each model.
The plots we present in this section are, therefore, just a numerical example. New plots following any Lattice input can be easily obtained following the recipe presented in this paper.

VI. CONCLUSIONS
We investigated the dynamics of a specific class of Composite Higgs Models with top partial compositeness, constructed via ordinary four-dimensional gauge theories with fermionic matter belonging to two different irreps of the hypercolor group. These models give rise to EW cosets beyond the "minimal" SO(5)/SO(4) type and thus contain additional pNGBs carrying EW charges. Furthermore, additional colored pNGBs arise from the need of introducing hyperquarks carrying ordinary color in order to construct top-partners. Two more pseudo-scalars arise from the breaking of the two chiral global U (1) symmetries associated to the two hyperquarks.
In our choice of models of this type, we were guided by the need to preserve both the custodial symmetry of the Higgs sector as well as the one protecting the Z → b LbL branching ratio. As discussed in Section II, we focused on models that are likely to lie outside of the conformal window and that can be brought into it from strong coupling at energies above the confinement scale Λ.
We identified a set of three pseudo-scalars, the two singlets a and η with respect to the SM groups and a color octet π 8 , that are present in all models in this class. Their dynamics is controlled by model-specific group-theory data and a few phenomenological parameters such as the hyperfermion masses and the pNGBs decay constants. In particular, the couplings to gauge bosons are determined by the coefficients of the WZW anomalies, which are sensitive to the microscopic details of the model.
One of the most striking signals from these preudo-scalars are di-boson signatures which is one of the main focuses of the paper. We have performed a complete analysis of the constraints from di-boson and di-top final states (also including pair production for the octet) using post-ICHEP2016 LHC data and the respective experimental results, and formulated a model independent strategy to combine known bounds and establish new limits on models under study.
Following our recipe, formulated in Section IV (with the concrete example given in Section IV.A.2) we applied the bounds to the models under consideration, giving numerical results in the case of two of them (M8 and M9 in Table I). We found that present LHC data already sets important constraints on the condensation scale which are stronger than the typical bounds from EW precision tests, thus demonstrating that the direct search for additional pNGBs with di-boson and di-top signatures in models of partial compositeness can be the first probe for such models. The fact that the couplings are predictive and sensitive to the underlying model makes these channels attractive. We should remark that the presence of these signatures is common to all models of partial compositeness based on a gauge-fermionic underlying theory.
The analysis of the post-ICHEP2016 data, and the framework we have developed in Section IV can be used in a straightforward way (including an update with the new coming data) to any model containing pseudo-scalar singlets or octets at the TeV scale.

Appendix A: Interpretation of ATLAS and CMS searches used in this article
In this appendix, we summarize the experimental searches used in this article in order to constrain the diboson channels, the tt channel, and the octet pair production channels and detail the assumptions made in order to extract the bounds for the models discussed. 61-63% and 52-56% [66]. In the absence of a full recast of the search for our pseudo-scalar resonance, we assume a slightly lower acceptance of 50%, leading to a conservative estimate of the constraint of the folded cross section. dominant single-production mechanism is gluon fusion, for which the ratio between the production cross section at 13 TeV and 8 TeV as a function of mass is determined by the cross sections given in Fig. 1. In order to compare the bounds from run I searches to those from run II searches, we rescale the bounds on the production cross section times branching ratios into the the di-boson and tt final states by σ(gg → π 0 ) 13 /σ(gg → π 0 ) 8 and show the resulting bounds in Fig. 3, labeled at "8 TeV".
[74], Fig. 7a (spin-0 resonance, narrow width approximation). The study gives bounds on the fiducial cross section. To obtain bounds on the full cross sectionfollowing the information on the fiducial volume given in Ref.
[74] -we divide the bounds given by a fiducial volume function which is 54% for a mass of 200 GeV, linearly extrapolated to 61% at 700 GeV, and 61% above. The combined bound used in this article is obtained by taking the strongest bound at each resonance mass m π 8 . In order to compare the bounds from run I searches to those from run II searches, we rescale the bounds on the production cross section times branching ratios into gγ by σ(gg → π 8 ) 13 /σ(gg → π 8 ) 8 and show the resulting bounds in Fig. 10 The combined bound is obtained by taking the strongest bound at each resonance mass m π 8 . The resulting bound is shown in Fig. 10, labeled "13 TeV".
Appendix B: Additional mass mixing (and couplings) of the singlets In this Appendix, we briefly discuss the possible presence of mixing of the U(1) singlets a ψ and a χ with pNGBs from the non-abelian flavor symmetries. These mixing terms can only arise from spurions explicitly breaking the flavor symmetries, and in particular from the mass terms of the fermions which also generate masses for the pNGBs. In addition to mass mixing, couplings to two non-abelian pNGBs may also be generated thus opening the case for additional decay channels. We will discuss each case individually, as the physics involved is very different.
Coset SU(4)/Sp (4) This coset, generated by ψ in the pseudo-real representation, contains 5 pNGBs, which transform under the custodial symmetry as a singlet η and a bi-doublet H (which plays the role of the BEH field).
The underlying theory consists of 4 Weyl spinors: two transforming as a doublet of SU(2) L , and the other 2 as a doublet of SU(2) R , thus one can write down two independent mass terms, m L and m R respectively. We will parametrize the two masses as where µ ψ is the common mass used in Section III B and δ is a parameter describing the deviation from Universality: δ is required to be small in order to preserve the stability of the vacuum.
Following the normalization adopted in this work, we find that: Plugging this expression in Eq. (15), we see that a linear mixing with the non-abelian singlet η is generated, proportional to the universality breaking parameter δ: We also notice that no mixing nor coupling can be generated in the Universal case: this fact can be easily understood in terms of symmetries. In fact, there exists a symmetry acting on the pNGBs under which only η is odd, provided that SU(2) L ↔ SU(2) R are exchanged [26].
Following the CP properties of the scalar fields, and invariance under Sp(4), the possible linear couplings of the a ψ singlet to the non abelian pNGBs need to have the following form: where n and m are integers.
The odd power on η derives from CP-invariance. This coupling, however, is odd under the η-parity described above, unless the coefficient is odd, i.e. proportional to δ ∼ m L − m R .
Coset SU(4)×SU(4)/SU (4) This coset also arises in the EW sector when ψ is complex. The 15 pNGBs transform as a singlet η, two bi-doublets H 1,2 , an SU(2) L triplet ∆ and an SU(2) R triplet N . Like in the previous case, the 4 ψ's (and their conjugates) transform like doublets of the custodial symmetry, thus we can write down two mass terms m L and m R . We can parametrize then as above, in Eq.(B1).
The vacuum structure of this model is more complex that the previous one, and it has been discussed in detail in Ref. [33]. It is easier to describe the theory in the EW preserving vacuum, and think of the misalignment in terms of VEVs assigned to the pNGBs. The only pNGBs that can develop a VEV are: where v = v SM . It has been shown in [33] that the real VEV of the second doublet can always be rotated away without loss of generality, thus we do not consider it here further.
Also, v 2 violates CP, and for simplicity we set it to zero here: if present, however, it will generate a tadpole for a ψ proportional to the product of the 3 VEVs. Note also that the triplet VEV is bound to be small as it does violate custodial invariance, thus we will neglect it in the following for simplicity.
We then study the mass mixing by expanding the Lagrangian terms in Eq. (15) up to the 3rd power in the pNGB matrix, thus capturing effects up to quadratic order in the VEVs.
We find the following mass terms: We see then that a mixing to the singlet η is generated in presence of Universality violation, as in the SU(4)/Sp(4) case. Additionally, a mixing to the second doublet is generated if the small triplet VEV is present.
Couplings to two pNGBs can also be generated via the SM Higgs VEV v: The terms above can generate decays of the singlets to two pNGBs: note in particular the presence of couplings to the second doublet and triplets which are not suppressed by δ.
As these pNGBs may be odd under a conserved parity (see [33]), this opens the way to interesting invisible decay modes. Nevertheless, such couplings are proportional to the mass m π ψ which we assume being small, thus for the sake of simplicity we will neglect them in our study.
Coset SU(5)/SO (5) This coset arises when the ψ's are in a real representation of Hypercolor. The 14 pNGBs transform like a singlet η, a bi-doublet H playing the role of the BEH field, and a bitriplet, that can be decomposed into a real SU(2) L triplet φ 0 and a complex one φ + with hypercharge +1. The 5 fermions ψ transform like a bidoublet of the custodial symmetry plus a singlet, thus one can assign 2 independent masses m and m 0 = m(1 + δ) respectively.
Besides the VEV of the Higgs field, the vacuum structure also allows the triplets to develop a VEV, so that in principle one can define 4 independent parameters: However, the triplets VEVs would violate custodial invariance unless a relation among them is imposed [126]: v 1 = 0 and v 2 = −v 0 . Plugging this vacuum structure in Eq.(15), however, we see that a tadpole for the singlet a ψ is generated unless v 0 = v 2 = 0: this is easily understood, as the two VEVs correspond to CP-odd fields. Thus, in order to both preserve custodial invariance and avoid tadpoles for the singlets, the triplet VEVs must vanish: in the following, also for the sake of simplicity, we will impose this cancellation.
Expanding the mixing Lagrangian up to third order in the pNGB matrix, we find that the following mixing terms are generated: Like in the SU(4)×SU(4)/SU(4) coset, we notice a mixing with the singlets proportional to the violation of the mass Universality, however at order v 2 /f 2 ψ mixings with the singlets and the two CP-odd neutral components of the triplets are generated also in the Universal limit.
Similarly, the following couplings to two pNGBs are generated at the leading order in v: In the present framework of pNGB composite Higgs model top mass is induced via partial compositeness. This explicit breaking of the global flavor symmetry introduces direct couplings between the pNGBs and top. The octet coupling is model dependent but the coupling of the singlets is always present, as explained in Sec. III. Such new interactions will induce loop corrections to the anomalous WZW terms, via a triangle fermionic loop. In this appendix we summarize the main results of the top loop contribution.
For a pseudo-scalar, the amplitude is simply proportional to an epsilon tensor and other gauge invariant tensor structures vanish due to the CP invariance 19 . For a generic coupling of the pNGBs to tops, i.e. iγ 5 C t m t /f π (were π can be either π 0 or π 8 ), the amplitude of the process is given by: with R i = For the W + W − final state only one fermion flow contributes, this multiplicity is also taking into account in the c 1,2 parameters. In Table IX we summarize these coupling combinations.

(C8)
Appendix D: Couplings and mixing in models M1 -M12 In this Appendix we present numerical values for all models, M1 to M12, assuming f ψ = f χ and normalising all couplings with f ψ . The couplings of the singlets are shown for the two extreme values of the mixing angle α: α = ζ obtained when m η → ∞, and α = ζ/2 obtained in the limit of minimal splitting. We checked that the couplings run approximately linearly with α. In Table X we show models whose top partners are made of 2 ψ's and one χ, while in Table XI the cases with 1 ψ and 2 χ's.   with Scouting Data," CMS-PAS-EXO-14-005.