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The Discrete Composite Higgs Model

We describe a concrete, predictive incarnation of the general paradigm of a composite Higgs boson, which provides a valid alternative to the standard holographic models in five space-time dimensions. Differently from the latter, our model is four-dimensional and simple enough to be implemented in an event generator for collider studies. The model is inspired by dimensional deconstruction and hence it retains useful features of the five-dimensional scenario, in particular, the Higgs potential is finite and calculable. Therefore our setup, in spite of being simple, provides a complete description of the composite Higgs physics. After constructing the model we present a first analysis of its phenomenology, focusing on the structure of the Higgs potential, on the constraints from the EWPT and on the spectrum of the new particles.


Introduction
The Hierarchy Problem, in combination with the LEP data on precision ElectroWeak (EW) physics, leads rather naturally to the idea of a composite Higgs boson. This paradigm, first proposed in [1], has received considerable attention in the recent years [2][3][4][5][6][7]. The idea, in very broad terms, is the following: the Higgs boson emerges as the bound state of a new strongly-interacting sector, in particular it is the (pseudo) Nambu-Goldstone Boson (pNGB) associated with the spontaneous breaking of the global symmetry group G of the new sector. In this construction, differently from technicolor, the SM group is not broken by the strong dynamics. The unbroken subgroup H of G therefore contains the SM SU(2) L × U(1) Y and the coset space G/H is chosen such as to include at least one doublet of Higgs scalars. It is the VEV of this field that breaks spontaneously, as in the ordinary Higgs model, the SM group. The "minimal" G/H coset with these features, which also contains a custodial SO (4) symmetry that keeps δρ under control, is SO(5)/SO (4). We will restrict to this minimal choice in the following.
While the general idea is quite simple and clear, it is rather complicated to formulate explicit and predictive incarnations of the composite Higgs scenario. The problem, obviously, is modeling the dynamics of the strongly-interacting sector, a task for which no definite recipe exists. We have however robust expectations on how this composite Higgs model should look like [6], based first of all on symmetry and large-N c considerations, but also on the experience with QCD. We expect that the strong sector delivers, on top of the Goldstone boson Higgs, resonances of typical mass m ρ which interact among each other and with the Higgs with a "typical coupling" g ρ . In spite of emerging from a strongly-interacting dynamics, this coupling is not necessarily large and close to the perturbative bound of 4π. Provided we assume that the underlying dynamics of our strong sector is a large-N c gauge theory, indeed, g ρ can be made parametrically small as it scales like g ρ 4π/ √ N c . Whether g ρ is large or small becomes at this point a phenomenological question.
The answer given in ref. [2,6] is that g ρ is preferentially large because this helps the model to pass the EW precisions tests (EWPT). We can restrict, for definiteness, the g ρ coupling in the range 1 < g ρ < 4π.
The above general considerations are already sufficient to develop a comprehensive but qualitative description of the composite Higgs model, the so-called "SILH" approach [6]. 1 To be quantitative, a specific and explicit construction is needed. At present, the only candidates are the "holographic" models, formulated as gauge theories in five space-time dimensions (5d), which have been extensively explored [2][3][4]. The aim of the present paper is to describe an alternative and simpler possibility. The need of simplifying the 5d models has already been recognized in the literature [5]: their objective technical complication makes them very difficult to implement in an event generator, and therefore not really suited for the LHC collider phenomenology. This technical problem goes together with a conceptual limitation. The 5d models are unnecessarily complicated because they contain much more information than what could be tested at the LHC. Namely, they incorporate Kaluza-Klein (KK) towers of resonances, with increasing mass, and formally describe the dynamics of each of these particles. Only the very first few resonances, however, are accessible at the LHC, retaining also the heavier ones is a redundancy of the 5d construction that we aim to overcome.
Our approach is inspired by the 5d models and by the idea of dimensional deconstruction [8], which consists in discretizing the extra coordinate in a finite number of points, or sites. We therefore denote our setup as the "Discrete" Composite Higgs Model (DCHM) and, depending on the number of deconstruction sites, we will be talking about the two-site or the three-site models. The DCHM is a completely four-dimensional (4d) theory, and each site is associated to a set of 4d degrees of freedom which correspond, roughly speaking, to one level of the KK tower. The DCHM provides a simplified version of the 5d model where only few KK levels, i.e. few resonances of the strong sector, are included. Given that the first level, the "zero-modes", corresponds to the ordinary SM particles, the two-site model describes only one set of strong sector resonances; two replicas of the latter appear in the case of three sites and so on. 2 An approach similar to ours has already been applied to the 5d Higgsless models (see for instance [9]) and in the context of Little Higgs theories [10][11][12]. 3 Aiming to simplicity, as we do in the present paper, the two-site model seems the more convenient choice because already the second level of resonances is probably rather heavy and not easy to observe at the LHC. However, we want to build a model which is not only simple, but also predictive, or better as predictive as the 5d ones for what the EW and the Higgs physics is concerned. In particular, the relevant observables that we would like to be calculable in our model, like they are in the 5d ones, are the potential of the composite Higgs boson and the S and T parameters of EWPT [13]. If this was the case, we would be able to incorporate rigorously, and not only to estimate, the constraint on the parameter space that follows from the EWPT and also from the direct measurement of the Higgs mass, when it will become available. In case of discovery, one could imagine fitting the masses and couplings of the new resonances together with the Higgs mass and, if measured, with other parameters of the potential such as the trilinear Higgs coupling.
It turns out from the analysis that we will present in section 2 that in the two-site model the S and T parameters are indeed calculable, but not the Higgs potential. The best compromise among simplicity and predictability will therefore turn out not to be the two-site but the three-site DCHM, in which also the potential becomes calculable.
The paper is organized as follows. In section 2 we construct the DCHM and discuss in great detail its symmetry structure and its interpretation as a calculable effective field theory. The main practical outcome will be, as previously mentioned, to establish the calculability of the S and T parameters and of the Higgs potential. Section 2 is rather technical, the reader interested in phenomenology could skip it in a first reading and pass directly to section 3, where a selfcontained phenomenological study of the three-site DCHM is presented. In particular, we discuss the properties of the Higgs potential, the constraints coming from the EWPT and the general features of the spectrum of the strong sector resonances. We discuss our conclusions in section 4 and in Appendix A we summarize the conventions adopted for the SO(5) generators. Appendix B contains an alternative derivation of the Naive Dimensional Analysis (NDA) formula [14], which we employ extensively in section 2.

Models of composite Higgs
The best way to introduce our model is to proceed in a constructive manner, starting from the simplest possible description of the composite Higgs scenario and progressively adding the necessary ingredients. This leads to a simpler exposition, in which the required technical tools are introduced gradually, and moreover it allows us to clarify the relation of our DCHM with other 4d approaches [7,15].

The non-linear σ-model
The starting point is the non-linear σ-model of the SO(5)/SO(4) coset, which obviously constitutes the "minimal" description of a composite Higgs. This model does not contain strong sector resonances, but only the pNGB Higgs. To construct the σ-model we introduce, as usual, a Σ-field which transforms in the fundamental representation of SO (5). The fields Π a are the four NGB Higgs fields, corresponding to the four components of one Higgs doublet, and transform in the 4 of SO(4). We will denote collectively as T A = T a , T a the unbroken (T a ∈ Lie[H]) and broken (T a ) generators of SO (5) in the fundamental representation, normalized to Tr The explicit form of the generators is reported in Appendix A. At the 2-derivative order, the only SO(5)-invariant term that can appear in the Lagrangian is where the normalization has been chosen such that the Π a kinetic term is canonical.
The coupling of the NGB Higgs to the SM gauge fields W α µ (α = 1, 2, 3) and B µ is introduced by gauging the SU(2) L × U(1) Y subgroup of SO(4) 4 , i.e. by substituting the ordinary derivative in eq. (2) with the covariant one where the SU(2) L × U(1) Y generators T α L and T 3 R are defined in eq. (86). One also has to include canonical kinetic (and interaction) terms for the gauge fields where we have defined . The Lagrangian L 0 = L π + L g we have constructed is just the first of the infinite number of terms, L = L 0 + i L i , that appear in our effective field theory. As a prerequisite for any quantitative statement on the effect of these operators we need a power counting rule to estimate the size of their coefficients. We assume the NDA power counting [14], which assigns to each operator the size it receives from radiative corrections with a cut-off Λ. The contribution from a diagram with L loops takes the generic form (see Appendix B) 5 4 We momentarily ignore the presence of the extra U(1)X charge in the definition of the hypercharge and identify Y = T 3 R . The U(1)X does not play any role until we introduce the SM fermions. 5 The only diagrams included in this estimate are those constructed with the vertices of the "leading order" Lagrangian L0.
where g and W collectively denote, respectively, the g and g couplings and the gauge fields W µ and B µ . The cutoff Λ should be set to its NDA value Λ = Λ Max = 4πf corresponding to the scale at which the σ-model enters the strongly coupled regime. We have however left Λ as a free parameter because we want to use the above equation not only to read the NDA estimate of each given operator, but also its degree of divergence.
Another important aspect of eq. (5) is the presence of the (gf /Λ) 2η term, that by using Λ = 4πf can be rewritten as (g 2 /16π 2 ) η . Each operator generically receives diagrammatic contributions of different type, which lead to different values of η. Given that g < 4π, the leading contribution is the one with minimal value of η. It is shown in Appendix B that η is necessarily positive, η ≥ 0, but there might be obstructions in reaching the absolute minimum η = 0. In several cases that we will consider below, the symmetries enforce η ≥ η min > 0, which leads to a reduction of the degree of divergence by the factor (gf /Λ) 2η min and to the corresponding (g 2 /16π 2 ) η min suppression of the NDA estimate.
Several interesting aspects of the composite-Higgs scenario can be captured by the simple model we have described (see [7]), but since we want to go beyond let us focus on its limitations. First of all, it does not contain a description of the strong sector resonances, which instead provide the key ingredient for the LHC phenomenology. Secondly, and related to this, it does not allow to compute several important observables such as the S and T EWPT parameter and the Higgs mass m H . It offers, in this sense, a limited predictive power.
Let us first discuss S, which, we remind, is proportional to the derivative at zero momentum of the W 3 -B correlator. In our σ-model, this comes from local operators with two derivatives and two gauge fields such as where Π 4 v = 246 GeV in the limit of small v/f . The coefficient c S , following the NDA estimate of eq. (5), is a parameter of order unity and it has vanishing degree of divergence at one loop, which means, of course, that it diverges logarithmically. The contribution to S from the local operator O S , which we denote as "UV contribution" S UV , is given by and it is obviously incalculable within our effective theory: it can just be matched to the observations by the choice of the input parameter c S .
On top of the UV one, the S parameter also receives a calculable "IR" contribution S IR . Given that the latter first emerges at the one-loop order like the divergent UV term does, it is naively expected to be of the same order of magnitude. If this was the case, the IR term would be completely hidden by the incalculable one and the total S = S U V + S IR could not be predicted. Actually, as the explicit calculation of ref. [7] shows, there is one IR effect which is enhanced with respect to S U V . This effect is an additive running of the c S coupling from the scale Λ = 4πf where it is generated with its NDA size, down to the Higgs mass m H . This running effect results in which is parametrically dominant, with respect to the UV contribution of eq. (7), due to the logarithmic factor. Formally, therefore, the S parameter is calculable in this model, in spite of being divergent already at the leading order in the loop expansion. In practice however the accidental numerical pre-factor of 1/6 completely compensates for the parametric logarithmic enhancement and S IR never dominates in practical situations. Moreover, and more importantly, the presence of weakly coupled resonances unavoidably gives contributions to c S that are larger than the ones estimated by NDA in eq. (7) [6]. With respect to the latter, the running term of eq. (8) is numerically even more irrelevant, to the point that we will even be entitled to ignore it in the phenomenological study of our model, presented in section 3. In this sense it is correct to say that, in spite of being calculable, S cannot be reliably predicted within the σ-model.
The situation is similar for the T parameter, because it is also logarithmically divergent at one loop. This might be surprising at the first sight: T is made of zero-momentum gauge field correlators, with no derivatives, and eq. (5) predicts a quadratic one-loop divergence for a local operator with two gauge fields, no derivatives, and an arbitrary number of Higgses. This leading divergence cancels because T is forbidden by the custodial SO(3) c symmetry, which emerges from the SO(4) unbroken group of our coset. Due to the symmetry, T is not generated by the σ-model interactions, one needs insertions of the SO(4)-breaking vertices, which are the ones that involve the B µ hypercharge field. The leading divergence then contains two extra powers of g , so that η min = 1 and the degree of divergence is reduced by two, as previously explained.
The above argument can be made more rigorous and systematic by the method of spurions, which we will extensively use in the rest of the paper and that we now illustrate in this simple example. Even if all what matters for T is the SO(4) subgroup, let us be more general and consider the full non-linearly realized SO (5). The latter is an exact symmetry of the σ-model Lagrangian (2) and it is only broken by the couplings with the SM gauge fields in eq. (3). We formally restore SO(5) by introducing two spurions G and G and rewriting the gauge field A µ in eq. (3) as with the index "A" in the adjoint representation, so that the covariant derivative transforms homogeneously under SO (5). The other index, α, forms a triplet of another symmetry group, which we denote as "elementary" SU(2) 0 L group, defined as the one under which only the three W α µ rotate, while the Higgs and the B field are invariant. This group commutes with SO (5), which instead only acts on the Higgs, and is obviously also a symmetry of the gauge Lagrangian in eq. (4). The SM gauge group, under which both the W and the Higgs transform simultaneously, is given by the vector combination of SU(2) 0 L and the SU(2) L (see Appendix A) subgroup of SO (5). We also notice the existence of a further symmetry, a Z 2 parity that changes sign to G and B µ .
Of course, eq. (9) is just a rewriting of A µ , the physical values of the spurions (which we will occasionally denote as "spurion expectation values", or VEVs) are indeed which gives back eq. (3). The spurion G breaks the total SU(2) 0 L × SO(5) symmetry group down to SO(4) SU(2) L × SU(2) R , the further breaking to the SM group is due to G . The point is that before setting them to their physical value, the spurions have well-defined transformation properties under the symmetry and they must enter in symmetry-preserving combinations with the other fields. Using the symmetry we can classify the allowed local operators with two gauge fields in terms of the number of spurions they contain. With no spurions insertions, the only operator is L π in eq. (2) (with, obviously, covariant derivatives), which however does not contribute to T because of the custodial symmetry. The leading contribution to T comes from operators such as and contains, as expected, two powers of the custodial-breaking coupling g . The coefficient c T , using eq. (5), is expected to be of order unity.
Going back to the issue of calculability, the situation for T is very similar to the one for S.
Being logarithmically divergent, T is calculable because it is dominated by an additive running effect analogous to the one for S. Also in this case, however, this calculable IR contribution is numerically not very relevant, the dominant one being radiative effects due to the resonances. As for S, the T parameter is calculable but not predictable within the σ-model, a description of the resonances (in particular, of the fermionic resonances which give the dominant contribution) would be needed.
The mass m H of the composite Higgs boson is also beyond the reach of the σ-model. The entire Higgs potential, indeed, diverges quadratically at one loop and it is not calculable. Notice that, was not for symmetries, the degree of divergence would have been even higher: the power counting of equation (5) predicts a quartic divergence for an operator with no gauge fields and derivatives. But the Higgs is a NGB, and it shifts under the non-linearly realized transformations of SO (5). The shift symmetry forbids the potential, which can therefore only originate from its explicit breaking.
In the language of spurions, this means that the potential must contain powers of G or G which lower the degree of divergence from quartic to quadratic. At the leading order, g 2 or g 2 , it is easy to classify the operators: they are only two and read The method of spurions is not only useful to count the powers of the gauge couplings and to determine the NDA size of the operators, it often makes manifest more subtle implications of the underlying symmetry. In the case at hand, the spurion analysis has fixed the functional form of the Higgs potential, up to unknown c g and c g coefficients. Unfortunately, the latter diverge quadratically, and therefore are not calculable within the σ-model. Very much like c S and c T , they should be fixed by observations or matched to the predictions of a more complete UV theory.

Two and three sites
We have seen that neither the Higgs potential, nor the S and T EWPT parameters are predictable within the σ-model. In the present section we introduce the three-site DCHM in which, as anticipated in the Introduction, all these observables are on the contrary perfectly calculable. The DCHM, on top of the NGB Higgs, also describes some of the strong sector resonances.

Two sites
Before moving to the three-site model, however, let us discuss the simpler case of two sites, in which most of the ingredients of the DCHM are already present and more easily illustrated. The starting point is again a non-linear σ-model, but based on a coset different from the SO(5)/SO(4) used in the previous section. We add a second SO (5) and consider the chiral group SO(5) L × SO(5) R spontaneously broken to the vector subgroup SO(5) V . This coset, SO(5) L × SO(5) R /SO(5) V , is parametrized by an SO(5) Goldstone matrix which transforms linearly under SO ( The SM group is embedded in the unbroken SO(5) V , under which the ten Goldstones Π A transform in the adjoint representation. Under SO(4) ⊂ SO(5) V , four of these (Π a ) form a four-plet and are identified with the Higgs field while the remaining six, in the adjoint of SO(4), will be removed by gauging and will not appear in the spectrum as physical scalars.
With the threefold purpose of getting rid of these scalars, of breaking the extra SO(5) R symmetry and of adding to the model a description of the vector resonances, we gauge the SO (4) subgroup of SO(5) R by introducing six gauge fields ρ a µ . Given that SO(5) R is non-linearly realized, these gauge bosons become massive and acquire their longitudinal components by eating the Π a 's.
Since we want to interpret the massive ρ states as resonances of the strongly-interacting sector, we assign them a coupling g * of the order of the typical strong sector coupling g ρ . Their mass is given by m ρ g * f (see eq. (20)) and it is of the order of the typical strong sector mass m ρ . As described in the Introduction, m ρ is expected to be of TeV size and the coupling g ρ is "large", though not maximal. We also have to describe the EW bosons and, for this purpose, we gauge the SU(2) L × U(1) Y subgroup of SO(5) L with gauge fields W α µ and B µ . We interpret the W and B fields as elementary, their couplings g 0 and g 0 are much smaller than g * and, as we will see below, almost coincide with the SM g and g couplings.
The structure of the two-site DCHM, i.e. its field-content and the gauging that we have just described, is summarized in figure 1. This structure corresponds to a leading order Lagrangian where we have separated the σ-model term from the "gauge" part L g = L g st + L g el that only contains gauge boson fields and renormalizable interactions. The covariant derivative, which is responsible for all the interactions among the Goldstones and the gauge fields, is defined as Figure 1: Pictorial representation of the two-site DCHM. The Goldstone matrix U is depicted as a "link", i.e. a segment with vertical lines at the endpoints representing the global SO(5) L and SO(5) R groups. The SM and ρ fields are located at two different "sites", represented respectively by a gray square or circle. The first site can be interpreted as the elementary group SU (2) under which only W and B transform, the second one is the analogous group for ρ, SO (4). The corresponding gauge couplings, or better their associated spurions G, G and G are also indicated. Their location reminds the symmetry groups under which they transform.
with the elementary gauge field A µ given, in analogy to eq. (3), by and with The gauge Lagrangian of the strong sector fields is given by The Lagrangian of the elementary gauge fields, L g el , is instead given by eq. (4).
The above equations contain mass-term mixings of the elementary W/B and the composite ρ fields. The massless combination corresponds to the gauge field of the unbroken SU(2) L × U(1) Y gauge invariance and are easily obtained by diagonalizing the mass matrix. In this way we also find the masses of the heavy vector states: which correspond respectively to the resonances associated to T α L , to T 3 R and to the remaining SO(5) generators. Furthermore, the SM gauge couplings are given by the usual expressions while the couplings among the resonances are of order g * .
The DCHM is a non-renormalizable effective field theory with a cut-off Λ that can at most reach the scale Λ Max = 4πf , where the σ-model interactions in eq. (15) become non-perturbative.
We assume that Λ = Λ Max and adopt, as in the previous section, the NDA estimate for the higherorder operators in the Lagrangian. As shown in Appendix B, the NDA formula for the DCHM is basically identical to eq. (5), and can be rewritten as where "V " and "g" denote all the gauge fields (W µ , B µ and ρ µ ) and the gauge couplings (g 0 , g 0 and g * ) which are present in the theory. The dependence on the cutoff Λ, which is left as a free parameter instead of being set to Λ Max , provides the degree of divergence of the operator.
The important novelty of the two-site DCHM, with respect to the model of the previous section, is that the Higgs boson is now a NGB with respect to two symmetry groups instead of one, so that its dynamics is "doubly protected". This mechanism is denoted in the literature as "collective breaking" [16]. To see clearly what this implies, imagine setting the Higgs to its VEV, which corresponds to Π 4 = 0, with all the other components of Π A vanishing. This VEV can be eliminated from the matrix U in eq. (13) by either performing an SO(5) L or an SO(5) R transformation, in eq. (14), along the T 4 generator. This implies that the dependence on Π 4 always cancels provided that any of the two groups, or at least the subgroup generated by T 4 , is an exact symmetry of the theory. All the effects triggered by EWSB and associated to the Higgs VEV, such as the S and T parameters and the Higgs potential, can therefore only originate from the breaking of both symmetries. This leads to extra powers of the SO(5) R -breaking coupling g * and makes the latter effects, following eq. (22), further suppressed with respect to the σ-model case of the previous section where the breaking of a single SO (5) was sufficient. This extra suppression reduces the degree of divergence, S and T therefore become finite at one-loop given that they were already logarithmically divergent. The potential is instead not yet expected to be finite, because in the σ-model it was quadratically divergent.
We have seen that the broken symmetries play a crucial role in the DCHM. In order to best exploit their implications we introduce spurions G, G and G associated to, respectively, the gauging of the SU(2) L and U(1) Y subgroup of SO(5) L and to the one of the SO(4) subgroup of SO(5) R .
The first ones arise, as in the previous section, from rewriting with "A L " in the adjoint representation of SO(5) L and α in the triplet of the elementary SU(2) 0 L group. The physical value of these spurions is, similarly to eq. (10) in the previous section The "new" spurion G is introduced by replacing, in equation (16), R µ with where "A R " is in the adjoint of SO(5) R . The other index, "a" is in the adjoint of the group SO (4) which we define, similarly to SU(2) 0 L , as the group under which only the ρ a µ fields transform, the Goldstones and the EW bosons being invariant. The physical value of G is and it breaks SO(5) R × SO (4) to the diagonal SO(4) subgroup.
As a first application of the spurion method, let us classify the local operators that contribute to S and T . For what concerns S, the leading operator is which, as expected by collective breaking, contains two powers of g * . This reduces the degree of divergence and also the estimated NDA size: in comparison with the σ-model result of eq. (6) we have one more factor of ( g * /4π) 2 < 1. For T , the spurion analysis reveals that no operator of order g 2 * contributes and that the leading term is which contains four powers of g * . In this case, the spurionic classification of the operators gives a stronger suppression than what expected from collective breaking. The conclusion is unchanged, however, S and T are finite at one loop and therefore predictable within the two-site DCHM.
The Higgs potential is, on the contrary, still divergent. The divergence is associated to operators which contain two more powers of g * than the σ-model ones in eq. (12). The presence of these additional couplings, which in turn is due to the collective breaking of the Higgs shift symmetry, ρ G The same as figure 1, but for the three-site model.
reduces the degree of divergence from quadratic to logarithmic. To further reduce the divergence and make the potential finite we need to introduce one additional symmetry under which the Higgs is a Goldstone. This is achieved in the three-site DCHM, as we will now discuss.

Three sites
The central ingredient for the construction of the three-site model, schematically depicted in figure 2, is a pair of identical σ-models, based as before on the coset SO(5) L × SO(5) R /SO (5) V . These are parametrized by two SO (5) matrices U 1 and U 2 , for a total of 20 Goldstone bosons Π A 1 and Π A 2 . The Goldstone Lagrangian is given, at the leading order, by The assumption that the two σ-models are identical, which led to the choice of equal decay constants in the above equation, is equivalent to imposing a 1 ↔ 2 discrete symmetry.
The symmetries of the two σ-models, SO(5) 1 L × SO(5) 1 R and SO(5) 2 L × SO(5) 2 R , are broken by gauging. As in the two-site case, the "first" group SO (5) 1 L is broken by the couplings with the SM gauge bosons and the "last" one, SO (5) 2 R , by the couplings with ρ. We break the remaining groups, SO(5) 1 R and SO(5) 2 L , by gauging their vector combination. The 10 associated gauge fields ρ A µ , whose coupling is denoted by g * , become massive by eating 10 Goldstones and are interpreted as resonances of the strongly-interacting sector. The expressions for the masses of the composite resonances and for the gauge couplings of the massless states will be given in section 3.3.
This gauge structure, summarized in figure 2, corresponds to the covariant derivatives where R µ and L µ are actually identical, R µ = L µ = g * ρ A µ T A , and A µ is defined in eq. (17). After introducing the spurions, R µ and L µ become different and transform under different symmetries, respectively SO(5) 1 R and SO (5) 2 L . The other index carried by the two spurions, "A", is associated to transformations, which we denote as SO (5) and break SO (5) (5) ρ to the vector combination. As in the previous sectioneq. (23) and eq. (25) -we also have the spurions associated to the SM couplings, G and G , and to the one of ρ, G.
In the three-site model the Higgs dynamics is triply protected by the Goldstone symmetries.
Suppose indeed setting the Higgs to its VEV. This corresponds, generically, to a constant configuration Π 4 1,2 of the Π 4 1,2 Goldstone fields. It is easy to check that this constant configuration can always be eliminated by a symmetry transformation provided at least one of the broken symmetries is restored by setting the corresponding coupling to zero. Let us start from g * . Putting it to zero restores SO(5) 1 R and SO(5) 2 L , through which it is easy to get rid of Π 4 1 and Π 4 2 . The situation is slightly different for g * . We have to remember that even in the presence of the g * coupling the vector combination of SO(5) 1 R and SO(5) 2 L is still unbroken, and it can be used to set Π 4 1 to zero, moving the VEV entirely on Π 4 2 . Afterwards, if g * is set to zero, an SO(5) 2 R transformation can eliminate Π 4 2 as well. The same holds for the elementary couplings g 0 or g 0 . Any physical effect of the Higgs VEV, therefore, is necessarily mediated by the three couplings g * , g * and g 0 (or g 0 ).
This triple collective breaking of the Higgs shift symmetry provides a further lowering of the degrees of divergence of all those observable which are sensitive to the Higgs VEV. In particular, it makes the Higgs potential finite because it has to contain at least two powers of g * . Performing the spurion analysis we actually find no contributions of O(g 2 * ). The leading operators, like contain four powers of g * . This further lowers the degree of divergence. The power counting of eq. (22) shows that the gauge contribution to the potential, because of the eight powers of the couplings, not only is finite at one loop, but it starts diverging at the three-loop order.

Matter Sector
We now have to introduce the SM fermions and couple them with the Higgs. These states originate in our construction from elementary degrees of freedom, external to the strong sector, which however are coupled linearly to some strong sector operator. The construction is analogous to the one of the SM vector bosons that we described in the previous sections. The vectors arise from gauging the SM group embedded in the strong sector SO (4), which means writing down linear couplings where J µ α denotes the strong sector current operator. The linear coupling results in a mixing of the elementary fields with the composite resonances of the strong sector. For the vector bosons we indeed find, in our Lagrangian, terms of the form L mix g 0 /g ρ m 2 ρ W α µ ρ µ α . 6 Analogously, for the SM fermions we assume a mixing of the form L mix y f /g ρ m ρ f ψ, where y f is the coupling of the elementary field with the corresponding fermionic operator and ψ generically denotes the fermionic strong sector resonance. Because of the mixings, the lightest state which eventually describe the SM particles are linear combinations of the elementary states, W and f , with the composite ones, ρ and ψ, realizing the so-called partial compositeness scenario [2,17].
In the case of two sites, focusing for simplicity on the top quark sector, we introduce only one Dirac five-plet of fermionic resonances ψ. This transforms under the SO(5) R σ-model group 7 and is mixed to the elementary doublet q L = {t L , b L } and singlet t R . The mixing term is where Q L and T R are the "embeddings" into incomplete five-plets of SO(5) L of the q L and t R SM fermions. The 5 decomposes as 5 = (2, 2) ⊕ (1, 1), where the (2, 2) -of SO(4) -consists of two We chose to embed q L in the negativecharge doublet and t R in the singlet. Given the explicit form of the generators in Appendix A, the embeddings are given by 6 Actually, because of gauge invariance, the elementary-composite mixings arise from terms of the form f 2 (g0W α µ − g * ρ α µ ) 2 , see for instance eq. (15). 7 This choice is merely conventional because by acting with the Goldstone matrix U one can convert ψ into U ψ, which transforms in SO(5)L. Physically, ψ or U ψ equivalently describe a 5 of the unbroken SO (5)V .
If we identify the U(1) Y symmetry with the subgroup of SO(5) L generated by T 3 R , as we did in the previous sections, we do not obtain the correct hypercharges for the fermions. It is well known that this requires an extra U(1) X global symmetry, which acts on the matter fields q L , t R and ψ as a phase rotation with charge X = 2/3. The hypercharge gauge field B µ is now introduced by gauging the subgroup of SO(5) L × U(1) X corresponding to the combination Given that the Goldstones are not charged under the extra U(1) X , this change in the definition of Y does not modify their couplings to B µ and the construction discussed in the previous sections for the gauge sector is unaffected. The W α µ and the massive resonances ρ µ are included, again as before, from gauging the SU(2) L subgroup of SO(5) L and the SO(4) subgroup of SO(5) R .
In order to make clear the symmetry pattern in the fermionic sector, we now discuss how to introduce the spurions which we will then use to analyze the fermionic contribution to the Higgs potential and to the oblique parameters S and T . Due to the presence of the extra U(1) X symmetry, the group under which the elementary fields transform is enlarged to SU (2) The charge of the elementary fields under U(1) 0 R corresponds to the charge under the original subgroup of SO(5) L generated by T 3 R , thus q L has charge −1/2 while t R is neutral. The spurions are two vectors ∆ L and ∆ R in the 5 of SO(5) L which also transform under the elementary SU (2) The ∆ L spurion is in the 2 with U(1) 0 R charge 1/2 and X 0 = −2/3 (the conjugate of the q L representation), while ∆ R is in the singlet with X 0 = −2/3 and is neutral under U(1) 0 R . Using these objects we can rewrite the action in eq. (34) as where i denotes the SU(2) 0 L index, while I and J are the usual SO (5) indices. The VEV of ∆ L,R breaks the strong sector group SO(5) L × U(1) X and the elementary one (1) Y subgroup that we identify with the SM group and that we gauged by the W and B elementary gauge fields. The unbroken SU(2) L is the vectorial combination of SU(2) 0 L with the SU(2) L subgroup of SO(5) L , while the hypercharge is the combination of the U(1) 0 Y subgroup of the elementary U(1) 0 R × U(1) 0 X , whose generator is specified in eq. (36), with the analogous combination coming from SO(5) L × U(1) X .
On top of the SM group, as already discussed, we also gauge the SO(4) subgroup of SO ( Figure 3: The matter sector of the two-site DCHM. The q L and t R fermions live at the first site, which indicates that they transform under the elementary group SU (2) The resonance ψ transforms instead with the SO(5) R (×U(1) X ) σ-model group, and therefore it is placed at the right endpoint of the link. The spurions ∆ L/R and m are also indicated, their location reminds the symmetry groups under which they transform. order to introduce the ρ vector resonances. This leads to the covariant derivatives where the spurion G has been introduced in the previous section. Notice that ψ is neutral under SO(5) L , therefore its covariant derivative does not contain the W α µ gauge fields. It contains instead a B µ term because ψ is charged under U(1) X and the hypercharge is defined as in eq. (36). In terms of the covariant derivatives above, the elementary and strong sector kinetic Lagrangians read where a mass-term m = diag( m Q , m T ), different for the four-plet and the singlet components of ψ, has also been introduced. This mass-matrix m IJ is a spurion with two indices in the fundamental of SO(5) R , and breaks it to its SO(4) subgroup.
Let us pause briefly to comment on the relation of our model with the existing literature. It can be shown by a field redefinition that the fermionic sector of the two-site DCHM is identical to the one of ref. [15]. The gauge sector, on the contrary, is different because no vector resonance ρ is taken into account in ref. [15].
The fermionic sector of the two-site DCHM, that we have just described, is summarized in figure 3. With respect to the gauge sector presented in the previous section, the only new parameters that we have introduced are ∆ L,R and m, and these are masses, not new couplings. This makes particularly easy to generalize the NDA power-counting of eq. (22), following the methodology of Appendix B. The result is where ψ generically denotes the fermions q L , t R or ψ, while µ is any of the masses ∆ L,R or m.
The positive integer χ counts the number of mass-term insertions and it is forced by the chiral symmetry (ψ L → −ψ L , ψ R → ψ R and µ → −µ) to be even or odd depending on the chirality of the operator.
which gets generated, with its NDA size estimated by eq. (40), by loops of the ψ field. Notice that the presence of two powers of the m spurion is due to the chiral symmetry, under which m is odd while O f S is even. Another interesting feature is the fact that eq. (41) vanishes if m Q = m T ; this is a consequence of the restoration of the SO(5) 2 R invariance in the composite fermionic sector, which allows to remove the effects of the Higgs VEV from the Lagrangian. Exactly like the gauge contribution in eq. (27), O f S has one loop degree of divergence equal to −2, which means that it is finite at one loop and starts diverging at the two-loop order. As implied by the mechanism of collective breaking discussed in the previous section, the spurion analysis confirms that also the fermionic contributions to the S parameter are calculable in the two-site DCHM. For T , we find that the leading contribution comes from operators like which contains four powers of the custodial-breaking spurion ∆ L . 8 As for the gauge contribution in eq. (28), the spurion analysis reveals a larger suppression of T than what is implied by collective ψ m Figure 4: The matter sector of the three-site DCHM.
breaking, for which two powers of ∆ L would have been sufficient. Finally, again as for the gauge sector, the fermionic contributions to the Higgs potential are still logarithmically divergent. The local operators associated to the divergence are and originate, respectively, from loops of the elementary t R and q L .
To obtain a calculable Higgs potential we have to consider the three-site model. As shown in figure 4, this is constructed by introducing two five-plets of fermionic Dirac resonances ψ and ψ. The first one, ψ, transforms under the right group of the first link, SO(5) 1 R , while ψ is in the fundamental of SO (5) 2 R . The mixing Lagrangian is similar to the one of the two-site case, with the difference that the elementary fields mix now with ψ and not with ψ. Introducing the ∆ L,R spurions, we have The associated spurions, ∆ L and ∆ R , transform under both the elementary SU(2) 0 L ×U(1) 0 R ×U(1) 0 X and the SO(5) 1 L ×U(1) X group, and break the global symmetry to the SM group as explained above in the case of two sites. The new spurion, ∆, has indices in SO(5) 1 R and in SO(5) 2 L . Its physical value ∆ J I = ∆ δ J I is proportional to the identity, and therefore breaks SO(5) 1 R × SO(5) 2 L to the vector subgroup. The other terms which are present in the leading order Lagrangian are where the covariant derivatives are defined in eq. (38), and Figure 5: Schematic structure of the three-site DCHM.
Notice that the ψ mass-term m does not break any symmetry, differently from m which breaks SO(5) 2 R to its SO(4) subgroup. To write the Lagrangian in eqs. (44) and (45) we assumed that the strong sector is invariant under ordinary parity ( x → − x) and the only terms which break this symmetry are the mixings of the elementary fields with the composite states. If we do not impose this invariance a different mixing is allowed between the left-and right-handed components of ψ and the ψ field in eq. (44).
The leading local contributions to the Higgs potential comes from operators like and similarly with ∆ L . These fermionic contributions are finite at one loop, and start diverging only at two loops, differently from the gauge contributions of eq. (33) for which the divergence was postponed to the three-loop order.

Phenomenology of the three-site model
In the previous sections we established that the minimal calculable realization of the discrete composite-Higgs scenario is the model with three sites. This simple set-up already includes the description of two levels of resonances of the strong sector, thus giving rise to non-trivial phenomenological aspects which we will analyze in the present section. To keep separate the theoretical motivation of the model and the phenomenological analysis, we will make this section self contained and summarize the structure of the model neglecting the technical details that are unnecessary for the present purposes.
The main ingredient of the three-site DCHM consists of two replicas of the non-linear σ-model SO(5) L × SO(5) R /SO(5) V . This symmetry structure can be connected to the usual deconstructed scenarios by associating the global invariance to a three-site pattern, as schematically shown in figure 5. Each of the two σ-models, whose Goldstone degrees of freedom are denoted by U 1,2 , is represented by a link connecting two sites. In this way we can relate each site to a subgroup of the global invariance of the model: the first site is associated to the SO(5) 1 L invariance of the first σ-model, the middle site corresponds to the SO(5) 1 R × SO(5) 2 L subgroup and the last site to SO(5) 2 R . In order to accommodate the correct hypercharges for the fermionic sector, we also need an extra U(1) X global factor. This abelian factor is not associated to any of the sites and it acts on all the fermions of the model as we will see below. 9 The elementary gauge fields, as well as the vector resonances coming from the composite sector, are introduced by gauging suitable subgroups of the total global invariance (SO (5) (5) 1 R × SO(5) 2 L , introducing the ρ µ field with gauge coupling g * . At the last site we add the ρ µ gauge fields in the adjoint of SO(4), by gauging the corresponding subgroup of SO(5) 2 R ; its gauge coupling is denoted by g * . The Lagrangian for the Goldstone fields of the two σ-models can be read in eqs. (30) and (31).
An equivalent form of the Lagrangian, which is more suitable for the computation of the spectrum, of the Higgs potential and of the tree-level contribution to the S parameter, is obtained by going to the "holographic" gauge. 10 In this gauge, the only dependence on the Goldstone degrees of freedom appears at the first site, resulting in a particularly simple form for the Lagrangian of the composite sector. To reach the holographic gauge, first of all we set U 2 equal to the identity matrix by using a gauge transformation of the unbroken SO(5) D vector subgroup at the middle site. All the scalar degrees of freedom are now contained in the Goldstone matrix U 1 of the first σ-model, which we will denote simply with U from now on. At this point we can use the residual SO(4) gauge invariance at the last site to remove from U the unphysical degrees of freedom and 9 From the dimensional deconstruction of a 5d composite Higgs model we would have obtained a different structure. Analogously to the SO(5) groups, we would have found four replicas of the U(1)X and two extra Abelian σ-model U(1)L × U(1)R/U(1)V . Moreover, two combinations of the U(1)X 's would have been gauged and the associated gauge field, after eating the Abelian Goldstones, would have acquired a mass. Guided by minimality, we have chosen not to incorporate in our model this extra structure, and therefore not to describe the additional neutral vector resonances that would be present in the deconstructed case. 10 This terminology is inspired from the extra-dimensional holographic technique [18].
put it in the form where the Tâ generators (see Appendix A) are only the ones of the SO (5) In the holographic gauge the only dependence of the Lagrangian on the Goldstone matrix comes from the covariant derivative D µ U (see eq. (31)). This quantity can be rewritten by using the identity where A corresponds to a gauge transformation of the elementary fields (see eq. (17)) given by With the above definitions, the Lagrangian in eq. (30) can be written as The above expression is the right starting point to derive an effective "holographic" Lagrangian for the Goldstone boson Higgs by integrating out the massive gauge resonances. The calculation is particularly easy if we are only interested in the two-derivative term. This can be obtained by integrating out the resonances ρâ µ corresponding to the generators of the SO(5)/SO(4) coset. The equations of motion at zero momentum are g * ρâ µ = 1 2 A (U t ) â , substituting back into eq. (50) we We can immediately read the Goldstone decay constant f π = f / √ 2 and check that the hâ fields in eq. (48) have canonical normalization in the holographic Lagrangian.
The fermionic sector of the model contains the elementary states corresponding to the SM chiral fermions, which are introduced at the first site, and two sets of composite resonances added at the other two sites. We will only focus on the third quark generation, which is the one most directly connected with EWSB. Incorporating the light families and the flavor structure is beyond the scope of the present paper. These features could however be included by proceeding analogously to the extra-dimensional case [19].
Before discussing the elementary states in detail, let us consider the fermions belonging to the composite sector. At the middle site (see figure 4) we introduce two multiplets ψ u and ψ d , which transform in the fundamental representation of the vector group SO(5) D and have U(1) X charges 2/3 and −1/3 respectively. Two more multiplets in the fundamental representation of SO(5) 2 R are introduced at the last site ψ u ∈ 5 2/3 and ψ d ∈ 5 −1/3 . Given that, at the last site, the SO(5) 2 R invariance is broken, it is useful to introduce also a notation for the decomposition of the ψ u,d multiplets in representations of the unbroken SO(4) SU(2) L × SU(2) R subgroup. The fundamental representation of SO(5) decomposes as 5 = (2, 2) ⊕ (1, 1), thus where Q u,d ∈ (2, 2), while T and B are the two singlets. The Lagrangian for the composite states ψ u and ψ u in the holographic gauge is given by In the above expression we included a breaking of the SO(5) 2 R group through the mass terms for Q u and T which preserve only the SO(4) subgroup. The mixing in the last line of eq. (53) comes from the term ∆ u ψ u U 2 ψ u + h.c., which appears in the original non-gauge-fixed Lagrangian (see eq. (44)). The Lagrangian for the ψ d and ψ d fields can be obtained from the expression in eq. (53) with obvious substitutions.
The elementary states at the first site are the SM chiral states: the q L doublet in the 2 1/6 representation of SU(2) L × U(1) Y , the t R singlet in the 1 2/3 and the b R in the 1 −1/3 . The terms which involve the elementary fermions, again in the holographic gauge, are given by where Q u L and T R correspond to the embedding of the q L and t R elementary fields into the 5 2/3 representation of SO(5)×U(1) X as given in eq. (35), while Q d L and B R correspond to the embedding of q L and b R into the 5 −1/3 representation (see eq. (90) of Appendix A). The second and third lines of eq. (54) are mixings of the elementary states with the composite particles. These mixings implement the scenario of partial compositeness in our set-up; the y u,d L,R couplings determine the degree of compositeness of the associated SM fields.

The Higgs potential
We have shown in the previous sections that in our model the Higgs effective potential is finite at one loop. It can be computed by applying directly the Coleman-Weinberg formula or by first deriving the holographic action, with a technique borrowed from the extra-dimensional theories [18].
The result will not be reported here for brevity. As usual in composite Higgs models, we find that the fermion contributions dominate the potential and trigger EWSB by inducing a negative mass term for the Higgs. Given that the ψ d and ψ d multiplets are only needed to generate the bottom quark mass, their mixing with the elementary states is typically much smaller than the ones of ψ u and ψ u , which are responsible for the generation of the top mass. This implies that the leading contribution to the Higgs potential comes from the u multiplets while the one from the d fields can be safely neglected.
By expanding the Higgs potential at the leading order in the elementary mixings y u L,R f , we find that it has the structure 11 where c L,R are dimensionless functions of the microscopic parameters, typically of order unity, and N c = 3 is the number of QCD colors. The above formula follows from symmetries and could be derived without relying on the explicit calculation by performing a spurion analysis based on the tools introduced in section 2 (see also [20]). As a consequence of ordinary parity invariance, which we imposed as a symmetry of the composite sector (see section 2.3), and of the SO(5) group structure, we also find the exact relation which implies that the form of the Higgs potential at the leading order is practically fixed. 11 The expansion parameter in eq. (55) is given by (y u L,R /gρ) 2 , as can be seen explicitly from the rewriting in eq. (58). For typical configurations one finds that the expansion parameter (y u L,R /gρ) 2 ∼ yt/gρ is reasonably small. If the leading terms in eq. (55) dominate the Higgs potential, the only possible minima are the symmetry-preserving vacuum h = 0 and the configuration with maximal breaking h /f π = π/2. Both these possibilities are not acceptable, in particular the configuration with maximal breaking would lead to massless fermionic states in the spectrum and would be in strong conflict with the electroweak precision measurements, as we will see later on. In order to obtain a realistic EWSB we must have that h /f π v/f π is somewhat smaller than one, for which we need to rely on the effect of the higher order contributions. The mechanism is simple: the functional form of the (y u L,R ) 4 terms is different from the sin 2 ( h /f π ) of the leading order ones, and this allows to cancel only the Higgs mass term and not the quartic coupling, obtaining a small VEV. But for this to happen, first of all, the two leading terms in the potential in eq. (55) must cancel to a good accuracy, so that their residual contribution is comparable to the higher order terms. The situation is exactly the same in the warped models, in particular in the MCHM 5 . Using eq. (56), we see that to enforce the cancellation we need This relation implies that, in our model, the mixings of the elementary states q L and t R with the composite sector must have comparable sizes. The result in eq. (57) has been checked numerically showing that, on the points with correct EWSB, the relation between y u L and y u R is verified with a few % accuracy in a large part of the parameter space as can be seen from fig. 6.
We can get further information on the Higgs potential by estimating the size of the (y u L,R ) 4 contributions. Following [6,20] we have where y 2 denotes generically (y u L ) 2 or (y u R ) 2 . The first term in the equation above, of order y 2 , corresponds to eq. (55). As discussed before, the mass term coming from the y 2 and the y 4 contributions cancel between each other, but not the quartic, which then can be estimated as 12 From this expression we can extract the value of the Higgs mass where we denoted by y 4 max the maximum between y 4 L and y 4 R . In our case, due to eq. (57), y L 2y R = y max . On the other hand, the top mass can be estimated as usual by Comparing the above equation with the Higgs mass in eq. (60) we find The result is that, for typical values of the composite sector couplings, the Higgs is relatively heavy, usually above the top mass. A scan on the parameter space of the model shows that the ratio m H /m t follows an approximately linear growth in g ρ in fair agreement with the estimate in eq. (62), but only if no light fermionic resonance is present (see figure 7). If on the contrary light states arise from the composite sector, eq. (62) is usually violated and smaller Higgs masses are obtained. Notice that it is not surprising that our estimates are violated in this case because we assumed a common size m ρ for all the strong sector particle masses. 12 Notice that the gauge contribution to the Higgs potential, which is parametrically smaller than the fermionic one, can sizably affect the tuning when h /f is very small. The gauge contribution, however, is always smaller than each of the (y u L ) 2 and (y u R ) 2 leading order fermionic contributions to the potential in eq. (55), so that the cancellation mechanism which leads to eq. (57) remains valid. Furthermore it is easy to see that the gauge contribution to the quartic term in eq. (59) is always negligible.

The electroweak precision measurements
One of the most stringent bounds on the parameter space of the model is expected to come from the S parameter, which originates, after EWSB, from the tree-level mixing of the new strong resonances with the SM gauge fields. This contribution is easily computed and gives a positive shift to S, whose analytic expression is Given that we are not performing a complete analysis of the EWPT, we will only consider as a rough estimate the bound S 1.5 × 10 −3 . Assuming that the gauge couplings in the composite sector are all of the same order g * ∼ g * ∼ g ρ , we can extract an upper bound for the value of The bound on the S parameter can be also directly translated in a bound on the masses of the gauge resonances as we will discuss in subsection 3.3 (see eq. (78)).
The other oblique EW parameter which can put constraints on the model is T . Thanks to the SU(2) L × SU(2) R custodial symmetry, which is left unbroken in the composite sector, the T parameter vanishes at tree-level. Its first corrections arise at one-loop level due to fermion and gauge loops. The size of these contributions can be estimated to be of order [6] T ∼ N c 16π 2 (65) In our model, using eqs. (57) and (61), we find that the mixing of the elementary fermions satisfies the relation y u L ∼ y u R ∼ √ y t g ρ , where y t is the top Yukawa coupling. Thus we have (66) The experimental constraints on the T parameter imply a bound on v/f π which is comparable with the one found from S. It has been shown that, in similar models (see for instance [15]), the contributions to T from the composite fermion loops can be positive and in many cases can help to ameliorate the compatibility with the EW precision measurements. The above estimate shows that T is sizable and should be included in a complete analysis of the EW constraints, a detailed study is however beyond the scope of the present paper and we will leave it for a future analysis.
Another possibly relevant constraint comes from the corrections to the Zb L b L coupling. At tree-level the the deviation of this coupling comes from two effects, the first one is the mixing at zero momentum of the b L quark with the heavy fermionic resonances induced after EWSB and the second is the distortion of the vertex at non-zero momentum due to higher order terms in the effective Lagrangian.
The first effect can be described by dimension-6 operators of the form (H † D µ H)q L γ µ q L , whose correction to the Zb L b L vertex can be estimated to be where y L denotes a generic mixing of the q L doublet with the composite fermions. Without any suppression mechanism, for typical values of the parameters the above correction would put strong bound on v/f π . As explained in [21], a left-right Z 2 symmetry P LR ensures that the tree-level corrections to the Zb L b L vertex at zero momentum are absent provided that the b L field is embedded in a component of an SO(5) multiplet with T L = T R = −1/2. In our model, this is the case for the q L component which lives in the ψ u multiplet, whereas it is not true for the component in the ψ d multiplet. The corrections can thus be estimated as Since the ψ d multiplet is only introduced to give mass to the b field, as previously explained, the value of y d L is typically small, resulting in a suppressed contribution to Zb L b L . In the case in which y d L ∼ y d R , obtaining the correct b mass requires (y d L ) 2 ∼ y b g ρ , hence (69) The second tree-level correction to the Zb L b L coupling is induced at non-zero momentum by operators of the form ∂ µ F µν q L γ µ q L . The estimate of this correction at momentum equal to the Z mass is (70) In this case the contribution coming from the ψ u multiplet is not forbidden and gives the largest correction. Using the relation y u L ∼ y u R ∼ √ y t g ρ , we get (71) These results show that the constraints coming from the tree-level corrections to the Zb L b L vertex are usually comparable or less severe than the ones coming from the S parameter. The above estimates for the Zb L b L deviation have also been verified by a numerical analysis.
In addition to the tree-level distortions, the Zb L b L coupling is also modified by one-loop effects. These are potentially relevant because they give corrections controlled by y u L,R which are not protected by the P LR symmetry. These contributions can be estimated as where the first (y u ) 2 factor comes from a massive resonance circulating in the loop, and we used y u L ∼ y u R ≡ y u . Using the estimate y u ∼ √ y t g ρ , we get (73) The 1-loop correction is therefore sizable and can be the most relevant contribution to the deviation of the Zb L b L vertex. The numerical analysis performed in similar models (see for example [15]) show results in agreement with our estimates and confirms that the constraints on v/f π coming from the Zb L b L coupling deviation are comparable with the ones coming from the S parameter.

The mass spectrum: gauge fields
The presence of new strong-sector resonances, provided they are light enough, is the simplest and most accessible manifestation of the composite Higgs scenario at the LHC. It is then important to discuss the properties of the spectrum of the massive states which are present in our model. This analysis is the subject of the present and of the next subsection.
To study the spectrum of the resonances it is useful to consider the limit in which the couplings of the elementary fields to the composite ones vanish. This approximation is very good in the gauge sector, given that we assume g 0 , g 0 g * , g * . In this limit, given that the full global invariance at the first site is restored in the gauge sector, the Higgs VEV can be removed from the gauge Lagrangian by an SO (5) (4) subgroup is gauged. We find that the heavy gauge spectrum includes two adjoints of SO(4) with masses which correspond to the unbroken SO(5) generators, and a 4-plet in the fundamental representation which corresponds to the broken generators.
Finally we need to consider the elementary gauge fields, which acquire mass after electroweak symmetry breaking thanks to the mixing to the composite sector. First of all we report the expressions for the gauge couplings corresponding to the SU(2) L ×U(1) Y gauge symmetry (compare with eq. (21)) 1 By using a leading-order expansion in the gauge couplings, g and g , we can compute the masses of the W and Z bosons: The bound coming from S can be immediately translated into a lower bound on the masses of the gauge resonances. By using the expressions for the masses of the gauge fields given in eqs. (74), (75) and (77), we can rewrite the result given in eq. (63) as From the above expression we find the bound m 6± , m 4 2 TeV.

The mass spectrum: fermions
Similarly to what we did for the gauge fields, we can analyze the fermionic spectrum. If the mixing of the elementary sector vanishes, the two sets of fields {ψ u , ψ u } and {ψ d , ψ d } give rise to independent towers of states organized in representations of the unbroken group SO (4). In the following we will only consider the fields ψ u and ψ u , an analysis of the d fields can however be obtained proceeding along the same lines.
The spectrum of the ψ u and ψ u fields contains two SU(2) L × SU(2) R bidoublets with masses Each of these multiplets contains two massive top resonances and a bottom resonance, moreover it includes one exotic state with electric charge +5/3. The exotic state does not mix with the elementary fermions and does not couple to the Higgs, so its mass is exactly given by eq. (79) and receives no corrections. An interesting feature is that m u bid− becomes small if the relation is satisfied. The above choice of the parameters leads to a light resonance.
The {ψ u , ψ u } tower also contains two SU(2) L × SU(2) R singlets with electric charge 2/3, whose masses are Analogously to the case of the bidoublets, the mass equation for the singlets predicts the presence of a light state if Let us now analyze the corrections due to the mixing of the elementary states and to the EW symmetry breaking. The elementary-composite mixings break the SO(4) symmetry to SU (2)  contribution proportional to (y u L ) 2 , which, at leading-order in y u , is In the same way, the singlets get a positive mass shift induced by the mixing with the t R elementary field. The approximate expression for the shift is given by To complete the analysis we need to consider the corrections due to the breaking of the EW symmetry. Due to the Goldstone nature of the Higgs, the effects of EWSB modify the spectrum in a very specific way. If the mixing of the elementary fermions to the composite states vanishes, then the composite fermionic sector has an unbroken SO(5) symmetry, which can be used to remove the Higgs VEV from the fermion Lagrangian. This means that the EWSB effects can modify the spectrum only through the elementary-composite mixings y u L,R , so that the induced shift in the mass is of order (y u L,R ) 2 v 2 . The size of the splitting induced by the elementary mixing in eqs. (83) and (84) are instead of order (y u L,R ) 2 f 2 . Given that in our model y u L ∼ y u R , the splitting due to EWSB are all suppressed by a factor v 2 /f 2 . This has an interesting consequence on the spectrum of the bidoublet: its states are organized in two SU(2) L doublets and the splittings between the two states in each doublet are much smaller than the mass separation between the two doublets.
This pattern can be clearly seen in the spectrum of the sample points shown in the right panel of fig. 8. Notice that this structure is a general feature of the composite Higgs scenario and it is due to the pNGB nature of the Higgs, therefore it is not specific of our set-up.
Another interesting property of the spectrum can be found by considering the distribution of the masses of the resonances obtained by a scan on the parameter space of the model. As can be seen from the left panel of fig. 8, one generically finds that the t and the exotic state with charge 5/3 are significantly lighter than the typical scale of the mass parameters in the composite sector. This means that eq. (80) and eq. (82) hold accurately enough to lead to light states in a large portion of the parameter space. The presence of these light particles, the "top-partners", has been noticed already in the context of the 5d holographic models and constitutes the most visible manifestation of the composite Higgs scenario. We have seen that they arise in the DCHM as well. Notice that there is typically no light b state because its mass is lifted by corrections of order (y u L ) 2 f 2 . Finally, we can derive the formula for the top mass at the leading order in the mixing of the elementary fermions with the composite sector: The denominator in eq. (85) vanishes if one of the relations (80) or (82)

Conclusions
In this paper we have described the DCHM, a novel, concrete incarnation of the general paradigm of composite Higgs. In comparison with the standard approach, which is based on 5d gauge theories, the most obvious advantage of our model is simplicity. The DCHM, indeed, is formulated as a standard 4d theory with a relatively small number of fields and of free parameters. Differently from the 5d one, the DCHM is suited for a detailed search program at the LHC because it is simple enough to be implemented in an event generator. The DCHM also overcomes another limitation of the 5d approach. Any 5d model hides one extra assumption, whose implications on the phenomenology are difficult to quantify. This assumption is the choice of the metric of the 5d space which in these constructions, that ignore gravity, is basically arbitrary. 13 In the DCHM, interpreted as the deconstructed version of the 5d model, the ambiguity associated with the metric is encoded in the free parameters.
Notice that our model is not exactly the three-site deconstruction of a 5d one because it does not describe all the particles that would be present in that case. In particular, a deconstructed model would contain massive vectors associated to the color SU(3) c (the KK gluons) and to the U(1) X local invariance in five dimensions, as mentioned in section 3 (see footnote 9). These states are unnecessary in our model, they could however be straightforwardly included.
The DCHM is not only a simplified version of the 5d approach, it can also be regarded as an alternative to it. For three sites or more the DCHM has indeed the same predictive power as the 5d one, at least for what the most relevant observables ( S, T and the Higgs potential) are concerned.
Our model offers, in this sense, a complete description of the composite Higgs scenario. To further illustrate this aspect a comparison with the "Simplified" model of ref. [5] is probably useful. The Simplified model is a 4d description of the first resonances and, similarly to ours, incorporates important aspects of the composite Higgs paradigm such as the idea of partial compositeness. The main difference with our approach is that the Higgs is not described as a pNGB, but instead as an ordinary scalar field. This is a serious limitation, which makes the Simplified model fail whenever the Goldstone boson nature of the Higgs becomes important. For example, consider the peculiar spectrum of the fermionic resonances that we encountered in section 3.4. This follows from the Goldstone nature of the Higgs and it is correctly reproduced in our set-up while it would be missed in the Simplified model. Because of these considerations and also because the Higgs potential is 13 The 5d holographic models are typically constructed in the truncated AdS5 geometry because of the Randall-Sundrum paradigm [22]. However, models with a different metric constitute a perfectly legitimate and phenomenologically viable possibility [4,23]. not calculable, the Simplified model is not really a complete description. The DCHM, instead, is both simple and complete.
Other models to which it is useful to compare our construction are the ones of ref. [10] and [11], in the context of Little Higgs theories. Roughly speaking, the model of ref. [10] is similar to the three-site DCHM, while the one of ref. [11] resembles the model with two sites. There are however several important differences, which follow from the different model-building philosophy of the DCHM. In ref. [10], for example, only an SU(2) × U(1) subgroup of SO (5) 2 R is gauged at the third site, rather than an SO(4) as in our case. This leaves first of all 2 extra uneaten Goldstone bosons in the spectrum, leading to an extended Higgs sector. Moreover, this breaks the custodial SO (4) and potentially induces large tree-level contribution to the T parameters. The latter are avoided in ref. [10] thank to T-parity, an option which however is not viable in the composite Higgs scenario.
An even more radical difference of the DCHM is in the matter sector, where both ref. [10] and [11] follow different (and seemingly more complicated) approaches. None of these approaches seems to capture the general paradigm of partial compositeness which we instead adopted as a crucial ingredient of the DCHM construction, as explained in sect. 2.3.
On general grounds, it is important to remind the reader that the DCHM is not a Little Higgs theory because no mechanism is present (or is assumed) for generating a parametrically large quartic Higgs coupling, and therefore a parametrically small v/f . The DCHM only aims to describe the pNGB Higgs and the associated resonances, the Little Higgs mechanism of generation of the quartic would be an additional ingredient added on top of our construction.
Our work needs to the extended in two directions. One is to implement the model in an event generator, the other is to refine our calculation of the EWPT, including the potentially relevant radiative contributions to T and to the Zb L b L vertices which we have only estimated in section 3.2.
We plan to present these developments in a future publication. regulated by the physical cutoff of the effective theory. The NDA also provides a prescription for the value of the cutoff, which should be given by the energy scale Λ Max where some interaction becomes strong and invalidates the perturbative expansion. It seems rather complicated to implement the above definition and estimate the size of the operators because we should consider completely generic loop diagrams, constructed with any of the infinite local vertices of our effective field theory. One can however proceed as follows. Instead of considering all the diagrams, let us restrict to a definite "leading order" Lagrangian L 0 , and to loop diagrams which only involve these leading operators. This subset of diagrams, at the price of going high enough in the loop expansion, already contributes to all the local operators of the theory, and can be used to estimate their NDA size.
We should however worry about the effect of the "new" vertices, the ones which do not appear in L 0 , and check that inserting them into the loops does not change the estimate. Provided they are included in the Lagrangian with their NDA size, this is obviously the case because each insertion of a new vertex can be replaced by the loop diagram that generated it, and provided the estimate of its size. Thus the contribution from the new vertices is automatically of the same order of the one from L 0 loops, and can be ignored.
To see how this works let us consider, for definiteness, the model described in section 2.1, which consists of an SO(5)/SO(4) σ-model coupled to the SM gauge fields. The leading order Lagrangian is L 0 = L π + L g , with L π and L g which are given, respectively, in eqs. (2) and (4). We want to estimate the size of the one-particle irreducible (1PI) diagrammatic contribution, with L loops, to a generic operator with d derivatives, E π external NGB states and E W external gauge fields. We use a generic regulator Λ for the UV divergences, remembering that in order to obtain the NDA estimate Λ must be eventually set to the physical cutoff. In the theory under consideration, the cutoff is Λ Max = 4πf , the energy scale at which the non-linear σ-model interactions in eq. (2) become non-perturbative.
The simplest way to obtain the result is to use dimensional analysis, counting, however, not only the energy dimension, but also the dimension in units of the Planck constant . To see how this works, we must first of all reintroduce by replacing L → L/ , and remember that acts as the loop-counting parameter. Given that each propagator leads to one and each vertex to −1 , indeed, we have that the amputated diagrams with E external states and L loops scales like φ . . . φ / φφ E ∝ ( ) L−1 , where L − 1 = I − V where I and V are, respectively, the number of internal lines and of vertices.
What we are interested to compute are however the amplitudes of normalized fields We conclude that the 1PI diagrams we want to compute must carry an " -dimension" of E/2+L−1.
In the basis of normalized fields, i.e. after the above rescaling, the only dependence on is carried by the vertices and it can be reabsorbed in the couplings, which become in this way the onlydimensionful objects. Imposing the -dimensions to match constrains the powers of the couplings that can appear in the diagram. When combined with the ordinary analysis of energy dimensions this constraint will fix the estimate of the amplitude completely.
Going back to the model of section 2.1, with Lagrangian L 0 = L π + L g , we immediately see that, after the rescaling to normalized fields, all the dependence can be reabsorbed in the decay constant f and the gauge couplings g and g . The latter ones are therefore the only couplings (i.e. -dimensionful parameters) that are present in our theory and they must be combined such as to give the correct -dimension. In particular, 1/f , g and g all have the dimension of 1/2 . Our amplitude takes the generic form where p collectively denotes the external momenta and the correct energy dimension, 4 − E π − E W , has been restored with powers of the cutoff Λ. Matching the -dimensionality requires which leads to The above equation already coincides with the final result, reported in eq. (5), it is however still to be shown that the exponent of the last term, denoted as 2η = B − E W in eq. (5), is necessarily positive end even, as mentioned in the text. In order to establish this point, one needs to look more closely to where the "B" powers of g could have originated from. They could have came from the vertices in L π (in eq. (2), with the covariant derivative in eq. (3)), in a number which equals the total number of W lines present in the vertices or from the renormalizable gauge vertices in L g (eq. (4)). In this second case, their number is where the sum runs now only on the vertices in L g , whose total number we denote as V g . Summing the two contributions, and using the relation where I W is the number of internal W lines, we finally obtain Eq. (94) follows, as usual, from the fact that each W line either connects two vertices or one vertex and one external state. We have now shown that η = I W − V g is an integer, it is not difficult to realize that it must be also be positive. This is because each "g" vertices only involves external W lines, so that in order to form part of an 1PI diagram it must be attached to two W propagators at least. This implies I W ≥ V g .
The generalization to the other models considered in the text is straightforward, consider for instance the two-site DCHM described in section 2.2.1. Also in this case we have a set of gauge couplings (g 0 , g 0 and g) of zero energy dimension, plus the σ-model dimensionful coupling 1/f . Equation (91), with the condition (92), therefore holds true in this case as well. Also, most of the considerations that led to eq. (95) remain valid. We can still count the number of gauge couplings by separating vertices from the σ-model (L π in eq. (15)) and from the renormalizable gauge interactions (L g = L g el + L g st , see eq. (19)). The only difference is that extra powers of the gauge coupling can now arise from the denominators of massive gauge field propagators. By expanding a divergent loop integral one has indeed the structure Λ D + m 2 ρ Λ D−1 + . . ., with m ρ g ρ f . These additional powers of g invalidate eq. (95), they are however unavoidably positive and even, so that the conclusion that B is positive and even remains unchanged.