Effective potential in ultraviolet completions for composite Higgs models

We consider a class of composite Higgs models based on asymptotically free $SO(d)$ gauge theories with $d$ odd, with fermions in two irreducible representations, and in which the Higgs field arises as a pseudo Nambu-Goldstone boson and the top quark is partially composite. The Nambu-Goldstone coset containing the Higgs field, or Higgs coset, is either $SU(4)/Sp(4)$ or $SU(5)/SO(5)$, whereas the top partners live in two-index representations of the relevant flavor group ($SU(4)$ or $SU(5)$). In both cases, there is a large number of terms in the most general four-fermion lagrangian describing the interaction of third-generation quarks with the top partners. We derive the top-induced effective potential for the Higgs coset together with the singlet pseudo Nambu-Goldstone boson associated with the non-anomalous axial symmetry, to leading order in the couplings between the third-generation quarks and the composite sector. We obtain expressions for the low-energy constants in terms of top-partner two-point functions. We revisit the effective potential of another composite Higgs model that we have studied previously, which is based on an $SU(4)$ gauge theory and provides a different realization of the $SU(5)/SO(5)$ coset. The top partners of this model live in the fundamental representation of $SU(5)$, and, as a result, the effective potential of this model is qualitatively different from the $SO(d)$ gauge theories. We also discuss the role of the isospin-triplet fields contained in the $SU(5)/SO(5)$ coset, and show that, without further constraints on the four-fermion couplings, an expectation value for the the Higgs field will trigger the subsequent condensation of an isospin-triplet field.

We consider a class of composite Higgs models based on asymptotically free SO(d) gauge theories with d odd, with fermions in two irreducible representations, and in which the Higgs field arises as a pseudo Nambu-Goldstone boson and the top quark is partially composite. The Nambu-Goldstone coset containing the Higgs field, or Higgs coset, is either SU(4)/Sp (4) or SU(5)/SO (5), whereas the top partners live in two-index representations of the relevant flavor group (SU (4) or SU (5)). In both cases, there is a large number of terms in the most general four-fermion lagrangian describing the interaction of thirdgeneration quarks with the top partners. We derive the top-induced effective potential for the Higgs coset together with the singlet pseudo Nambu-Goldstone boson associated with the non-anomalous axial symmetry, to leading order in the couplings between the third-generation quarks and the composite sector. We obtain expressions for the low-energy constants in terms of top-partner two-point functions. We revisit the effective potential of another composite Higgs model that we have studied previously, which is based on an SU(4) gauge theory and provides a different realization of the SU(5)/SO(5) coset. The top partners of this model live in the fundamental representation of SU (5), and, as a result, the effective potential of this model is qualitatively different from the SO(d) gauge theories. We also discuss the role of the isospin-triplet fields contained in the SU(5)/SO(5) coset, and show that, without further constraints on the four-fermion couplings, an expectation value for the the Higgs field will trigger the subsequent condensation of an isospin-triplet field. I.

INTRODUCTION
Among the mechanisms that have been proposed to keep the Higgs particle naturally light, the so-called composite Higgs paradigm [1,2] postulates the existence of a new strong sector, perhaps in the few TeV range, based on an asymptotically free gauge theory that we will call hypercolor. Spontaneous chiral symmetry breaking in the hypercolor theory produces a set of Nambu-Goldstone bosons (NGBs). When we couple the Standard Model and the hypercolor theory, this breaks explicitly the flavor symmetry group of the hypercolor theory to a smaller group, thereby generating an effective potential for the now pseudo Nambu-Goldstone bosons (pNGBs). The Higgs doublet is composed of four of these pNGBs, and it is assumed that minimizing the effective potential triggers electroweak symmetry breaking.
As the only fermion in the Standard Model with a mass comparable to the electroweak scale, the top quark is usually given a special role in composite-Higgs models. We will assume that the top quark couples linearly to baryons of the hypercolor sector, which we will refer to as hyperbaryons. The observed top-quark mass eigenstate is a linear superposition of the elementary top quark and the composite hyperbaryons, and this partial compositeness is responsible for its large mass [3].
The literature on composite Higgs largely leaves unspecified the details of the new dynamics, and focuses on its low-energy sector containing the pNGBs, which can be studied as a non-linear sigma model (for reviews, see Refs. [4][5][6]). Nevertheless, a number of ultraviolet completions of composite Higgs models have been proposed [7,8]. All these models are asymptotically free gauge theories with fermionic matter, sharing some additional basic features. The models are free of gauge anomalies, both the hypercolor theory by itself, and the coupled system of hypercolor together with the Standard Model, including all their gauge interactions. While ultimately only a lattice calculation can settle it, 1 analytic considerations suggest that all candidate models exist in a chirally broken phase, and are not infrared conformal.
As for the spectrum of the hypercolor theory, the Nambu-Goldstone coset must contain an SU(2) L doublet that can be identified as the Higgs field. In addition, the hyperbaryon spectrum must contain top partners, states with the same Standard-Model quantum numbers as left-handed or right-handed quarks, that can couple linearly to third-generation quarks. In this paper we will consider mass generation for the top quark only, 2 and therefore we need hyperbaryons that can couple to q L = (t L , b L ) and to t R , but not to b R . From a "low-energy" point of view, the differences between the various models are mainly in the Nambu-Goldstone coset, and in which irreps of the flavor symmetry group of the hypercolor theory the top partners live [7,8]. As we will see, different hyperbaryon spectra can give rise to very different effective potentials even when the Nambu-Goldstone coset is the same.
A further assumption with important dynamical implications is that the Standard Model gauge symmetries are embedded into the unbroken flavor symmetry group of the hypercolor theory. This gives rise to the vacuum alignment phenomenon [14][15][16]. In particular, the effective potential induced by the coupling of electro-weak gauge bosons to the hypercolor theory has its minimum at the origin for the Higgs field. As a result, the top-sector effective potential is instrumental in generating the non-trivial minimum for the Higgs field that will trigger electro-weak symmetry breaking.
In this paper we discuss mainly (but not only) composite Higgs models based on an SO(d) gauge group with d odd. Each model will contain fermions in the vector and in the spinor irreps. Since we will choose d to be odd, the spinor irrep is irreducible. The number of fermions of each irrep is just enough to accommodate the Standard Model's symmetries into the unbroken flavor symmetry group, while having pNGBs with the correct quantum numbers to be identified as the Higgs field. When the spinor irrep is pseudoreal, the symmetry breaking pattern is assumed to be [17][18][19] which corresponds to 6 Weyl (equivalently, Majorana) fermions in the (real) vector irrep, plus 4 Weyl fermions (or 2 Diracs) in the spinor irrep. U(1) A is the non-anomalous axial symmetry whose generator is a linear combination of the axial charges of the two irreps.
Demanding that the SO(d) theory will be asymptotically free allows us to choose d = 5 or d = 11 [7,8]. 3 In the case that the spinor irrep is real, the symmetry breaking pattern is which requires 5 Majorana fermions in the spinor irrep. The asymptotically free cases are d = 7 and d = 9. The Standard-Model symmetries are embedded into the unbroken subgroup H as follows. The QCD gauge symmetry SU(3) c together with (ordinary) baryon number B are embedded into the unbroken SO(6), while SU(2) L and SU(2) R are embedded into the Sp(4) subgroup of SU(4), or into the SO(5) subgroup of SU (5). For all the fields of the hypercolor theory, as well as for the quark fields that will couple to it (namely, t L , b L and t R ), the usual Standard-Model hypercharge is given by Y = T 3 R + 2B, where T 3 R is the third SU(2) R generator, and baryon number has the usual normalization with B = 1/3 for a single quark. With these conventions, the electric charge is Q = T 3 L + Y = T 3 L + T 3 R + 2B. The simplest hyperbaryons which can play the role of top partners are hypercolor singlet states made out of two SO(d) spinors and one SO(d) vector, which belong to two-index irreps of the SU(4) or SU(5) flavor group.
We comment that Sp(4) is the covering group of SO (5), and SU(4) of SO (6). 4 For the purpose of this paper it does not matter if the gauge group is Sp(4) or SO (5), and we opt for SO (5) just so that most of the gauge groups we deal with (except in Sec. V) will be SO(d) groups with d odd. 5 The symmetry-breaking cosets SU(4)/Sp (4) and SO(6)/SO (5) are isomorphic, and following Ref. [8] we opt for the former.
This paper is organized as follows. In Sec. II we introduce our notation for the SO(d) gauge theories, and construct all the dimension-9/2 hyperbaryons that can serve as top partners. In Sec. III we proceed to study the case of a pseudoreal spinor irrep. Of the 5 pNGBs in the SU(4)/Sp(4) coset, four make up the Higgs doublet, while the last one, η, is inert under all the Standard Model symmetries. We begin by listing all the possible embeddings of the quark fields q L = (t L , b L ) and t R into spurions belonging to two-index irreps of SU (4). We write down the most general four-fermion lagrangian describing the interaction of these spurions with the hyperbaryons, finding that it contains 15 independent terms. We then work out the resulting effective potential for the pNGBs. Thanks to the simplicity of the SU(4)/Sp (4) coset, this potential can be obtained in closed form. We also work out all the low-energy constants, which can be expressed in terms of hyperbaryons two-point functions. A summary of our results for this coset is given in Sec. III F.
In Sec. IV we deal with the case that the spinor irrep is real. The coset SU(5)/SO(5) contains 14 NGBs, five of which are the same as before: a (2,2) of SU(2) L × SU(2) R that constitutes the Higgs doublet, and the singlet η. The remaining nine NGBs fill up the (3,3) representation. Again there are 15 independent couplings in the four-fermion lagrangian. The presence of the isospin-triplet pNGBs makes the analysis technically more involved, and we calculate the full effective potential only to third order in the pNGB fields. As before, we also discuss the low-energy constants.
We then turn to the following important issue (Sec. IV E). The SU(5)/SO(5) effective potential will in general contain cubic terms of the form ∼ h 2 ϕ, where h is the physical Higgs, and ϕ is one of the nine new pNGBs. 6 The effective potential for ϕ takes the form where f represents the scale of the hypercolor theory, and the coefficients c 1,2 are dimensionless. When the Higgs field h condenses, the cubic term (the first term on the right-hand side of Eq. (1.3)) induces a linear term for ϕ. This, in turn, forces the subsequent condensation of the ϕ field [8]. Assuming 7 c 2 > 0 (and neglecting the O(ϕ 3 ) terms), the minimum of this potential is ϕ = −(c 1 /(2c 2 ))h 2 /f . If the coefficients c 1,2 have a similar magnitude, the ϕ expectation value will be suppressed by only one power of h/f relative to h itself. This is problematic, because ϕ transforms non-trivially under SU(2) L × SU(2) R , and an expectation value for ϕ does not preserve the custodial symmetry. This diagonal subgroup of SU(2) L × SU(2) R is needed in order to protect the ρ-parameter [22], for which there are tight experimental constraints. To shed more light on this issue we also calculate the full potential in the case that all the SU(5)/SO (5) pNGBs are turned off except for h and ϕ, and we discuss whether, and if so, how, those problematic cubic terms might be avoided. In Sec. V we revisit the SU(4) composite Higgs model that was previously studied by Ferretti in Ref. [13], and by us in Ref. [23]. In the latter work, we made rather restrictive assumptions that lead to a four-fermion lagrangian containing just two terms, and to an effective potential that is quartic in the four-fermion couplings. Here we take essentially the opposite approach, and make only the most minimal assumptions, which lead to a fourfermion lagrangian containing six terms. We find that, in general, an effective potential is then generated already at second order in the four-fermion couplings. However, as we explain in the concluding section (Sec. VI), this potential may suffer from a serious drawback. In addition, for the four-fermion lagrangian we studied in Ref. [23] we find that the effective potential contains two more terms that we overlooked in Ref. [23].
Because of the length of this paper, we have collected the main phenomenological lessons that can be drawn from all our analyses in Sec. VI. The appendices cover various technical points.

SO(d) GAUGE THEORIES
The SO(d) gauge theories we study in this paper have fermions in the vector and spinor irreps. Since d is always odd, the spinor irrep is irreducible. The Higgs field is identified with pNGBs that arise from chiral symmetry breaking of the spinor-irrep fermions. We denote the Weyl fermions in the spinor irrep as Υ i , where i is the flavor index. There will be 4 of them when the spinor irrep is pseudoreal, and 5 when it is real. The flavor symmetry group is, correspondingly, G Υ = SU(4) or SU (5). We find it convenient to construct the hypercolor baryons in terms of 4-component fields and Here C is the four-dimensional charge-conjugation matrix, and C is the charge-conjugation matrix in d = 2n + 1 dimensions. For our notation, Dirac algebra conventions, and for the properties of the charge conjugation matrix in various dimensions, see App. A. (When the spinor irrep is real, as in Sec. IV below, the χ i are Majorana fermions.) For g ∈ G Υ , a flavor transformation acts as Υ → gΥ, Υ → Υg † , or, in terms of the 4-component fields, The infinitesimal form is with T a the hermitian generators. As we will discuss in the following sections, the SU(2) L and SU(2) R symmetries of the Standard Model are embedded into H Υ , the unbroken flavor symmetry group of the spinor-irrep fermions.
In addition, all models will contain 6 Majorana fermions in the real, vector irrep, with an assumed associated chiral symmetry breaking pattern SU(6) → SO (6). As already mentioned, the Standard Model symmetries SU(3) c and U(1) B , where B is ordinary baryon number, are both subgroups of the unbroken SO(6). We find it convenient to regroup the 6 Majorana fermions into 3 Dirac fermions, ψ Ia , ψ Ia , where I = 1, 2, . . . , d is the SO(d) vector index, while a = 1, 2, 3 indexes ordinary color. Like quarks, the baryon number of these Dirac fermions is 1/3. The baryon number of the χ fermions is zero. 8 The embedding of the Standard Model symmetries is such that the pNGBs in the SU(6)/SO(6) coset carry ordinary color, but no SU(2) L × SU(2) R quantum numbers. Since in this paper we are mainly interested in the Higgs potential, we will mostly ignore the SU(6)/SO(6) pNGBs.

A. Top-partner hypercolor baryons
We will restrict the discussion to the simplest top partners, which are created by local 3-fermion operators constructed as follows. We first assemble two SO(d)-spinor fermions into a bilinear transforming as an SO(d) vector, and then contract this bilinear with an SO(d)-vector fermion to form an SO(d)-singlet state. The resulting hyperbaryon and antihyperbaryon fields are tabulated in Table 1. Unless it forms a singlet, the SO(d)-spinor bilinear belongs to one of the two-index irreps of the flavor group G Υ , which, we recall, can be SU(4) or SU (5). When a single four-dimensional Dirac matrix (aside from the chiral projector) is sandwiched between the two χ fermions, we encounter the adjoint irrep (D), or a singlet (N). When the number of four-dimensional Dirac matrices is zero or two, the same chiral projector is applied to both of the χ fermions, and the bilinear then has definite symmetry properties on its spin index. Taking into account also the symmetry properties on the SO(d) index (see the last column of Table 5) fixes the symmetry on the flavor index. In view of Eq. (2.3), when the chiral projector is P R we encounter the two-index symmetric (S) or two-index antisymmetric (A) representations, whereas for P L we obtain the complex conjugate representations S c and A c . 9 We use the following notation. A generic hyperbaryon is denoted B r ij,X , where i and j are flavor indices, and the optional subscript X = L, R denotes the projector applied to the open Dirac index, which in turn is always carried by the ψ fermion. r labels the irrep, which can be one of D, N, S, S c , A or A c . Our notation is such that the anti-hyperbaryon of B r ij is denoted B r ji , with the flavor indices flipped. This will prove convenient when using matrix notation in flavor space.
We comment in passing that the Ferretti-Karateev list of requirements is fairly restrictive [7,8]. Models that satisfy all the requirements and have a prescribed coset structure of the effective theory are so few, that in effect, knowing the coset structure essentially fixes the model, and thus, ultimately, also the top-partner content. However, by itself, the coset structure does not tell us what will be the irreps to which the top-partners belong. For example, the models of Sec. IV and Sec. V both share an SU(5)/SO(5) coset. But in Sec. IV the hyperbaryons belong to 2-index irreps of SU(5), whereas in Sec. V they belong to the (anti)fundamental irrep. Thus, the straightforward way to find the top partners of a given model is to explicitly construct the relevant gauge invariant operators. Of course, the explicit form of the hyperbaryon operators will also be needed for the derivation of the low-energy constants.

B. CP symmetry
As a stand-alone theory, all the hypercolor theories we study in this paper are invariant under C and P . Because we couple the hypercolor theory to q L = (t L , b L ) and to t R , but not to b R , the four-fermion lagrangian can be invariant only under the combined CP transformation. 10 The CP transformation acts on a gauge field as The SO(d) gauge field is invariant under charge conjugation, so that its transformation rule stems from parity only. The SO(d)-vector Dirac fermions transform as Except for the choice of phases, which is explained in App. B, this is the usual CP transformation rule of a Dirac fermion. The χ fields transform according to in the case that the spinor irrep is real (C = C T ), whereas for the pseudoreal case (C = −C T ) their transformation rule is The induced transformation of the hyperbaryon fields is The sign choices we have made in Table 1  There are two models where the spinor irrep is pseudoreal, one based on an SO(5) gauge group and the other on SO (11). The SO(13) theory is asymptotically free as well, but according to analytic considerations it is probably inside the conformal window, and not chirally broken [8,24]. In any event, since all the relevant properties of the SO(d) theories are periodic in d modulo 8, the discussion of the SO(5) theory would carry over as is to the SO(13) case, if the latter were to be chirally broken. For previous work on the SU(4)/Sp(4) models, see Refs. [8,[17][18][19]24].
The order parameter for the spontaneous breaking of the flavor symmetry G Υ = SU(4) is the expectation value of χ i χ j . This order parameter is antisymmetric on its flavor indices. We will assume that χ i χ j ∝ ǫ 0,ij , where the 4 × 4 matrix ǫ 0 is defined in Eq. (C2). With this convention, we may take the order parameter to be χǫ 0 χ . Applying an infinitesimal flavor transformation (2.4) to the order parameter we get Of the 15 generators of SU(4), there are 10 which leave the order parameter invariant (see Eq. (C5)). They generate the unbroken group, H Υ = Sp(4). The remaining 5 generators belong to the coset G Υ /H Υ = SU(4)/Sp (4). Taking T a to be a coset generator, the variation of the order parameter gives rise to an interpolating field for one of the NGBs, Equivalently, the full NGB field iŝ where the notation tr indicates that the trace is over the Dirac and color indices, but not over the flavor indices. It readily follows that ǫ 0Π (orΠǫ 0 ) is antisymmetric on its flavor indices.
where we have used Eq. (2.2). Notice that (apart from the usual coordinates transformation) the CP transformation does not merely flip the sign ofΠ. Related, when the coset generator T a commutes with ǫ 0 , the NGB field (3.2) is a pseudoscalar, as in the familiar QCD case. But when T a anticommutes with ǫ 0 , the NGB field is a scalar. We will discuss the phenomenological significance of this result shortly. In the effective chiral theory, the NGBs of SU(4) → Sp(4) symmetry breaking are represented by an antisymmetric unitary field Σ ∈ SU(4), Σ T = −Σ. In addition, the effective theory depends on an SU(6)/SO(6) non-linear field, which we will not discuss in this paper, and a field Φ ∈ U(1) associated with the spontaneous breaking of the non-anomalous U(1) A symmetry [7,[17][18][19][23][24][25][26]. The axial transformations are Eq. (3.6a) gives the transformation rule of the spinor irrep, which sets the normalization of the non-anomalous axial transformation in the microscopic theory. Eq. (3.6b) is the transformation rule of the vector irrep, where q = −(1/3)T χ /T ψ , 11 and the group traces are T χ = 2 (d−5)/2 and T ψ = 2. Finally Eq. (3.6c) sets our normalization for the transformation rule of the corresponding effective field. The formal correspondence between the elementary and the effective fields is then As already mentioned, we will assume that the vacuum is given by Σ = ǫ 0 and Φ = 1, and parametrize the non-linear field as where f is the decay constant. The effective NGB field Π is hermitian, traceless, and satisfies ǫ 0 Π = Π T ǫ 0 , just asΠ. Flavor transformations act on the non-linear field as For g ∈ Sp(4), it follows that the effective NBG field Π transforms in the same way as the NGB field of the microscopic theory, Eq. (3.4). The transformation rule of Π under CP is defined to be the same as in Eq. (3.5). The leading-order chiral lagrangian is invariant under these transformations. The embedding of SU(2) L and SU(2) R in Sp(4) is given in Eq. (C6), and the parametrization of the effective field Π is given in Eq. (C7). Four of the NGBs are identified with the Higgs doublet, H = (H + , H 0 ), whereas the fifth, η, is a singlet under SU(2) L × SU(2) R . Using the parametrization (C7), a CP transformation acts as This correctly reproduces the CP transformation of the Higgs field in the Standard Model. The rest of this section is organized as follows. In Sec. III A we obtain all the spurion embeddings of the quark fields. In Sec. III B we write down the four-fermion lagrangian L EHC , and in Sec. III C we list all the effective top Yukawa couplings allowed by it. In Sec. III D we begin the discussion of the effective potential of the pNGBs, V eff . We group the various contributions into twelve "template" forms, and then work out all the contributions to V eff in closed form. In Sec. III E we derive the low-energy constants. We summarize our findings in Sec. III F, which also contains a simple example of a phenomenologically viable potential. Finally, we discuss spontaneous CP breaking in Sec. III G.

A. Spurions
Much like in technicolor theories, the coupling of the Higgs field to the gauge bosons of the Standard Model arises naturally when the relevant global symmetries of the hypercolor theory are gauged; but a more elaborate setup is needed to generate masses for fermions. 11 In the case of the SU (5)/SO(5) models of Sec. IV, q = −(5/12)T χ /T ψ . For more details see, e.g., Refs. [24,26,27].
Here we postulate the existence of yet another gauge symmetry, dubbed "extended hypercolor" (EHC). We assume that the EHC gauge symmetry breaks spontaneously at some scale Λ EHC which is large relative to the scale of the hypercolor theory, Λ HC . The remnant of the EHC interactions at the hypercolor scale is a set of four-fermion interactions, and we assume that these four-fermion interactions couple the third generation quark fields q L = (t L , b L ) and t R to the hyperbaryon fields constructed in Sec. II A. The EHC theory will thus generate a mass for the top quark through the mechanism of partial compositeness. We comment that this setup does not necessarily generate a mass for any other Standard Model's fermion. Their masses may have to involve some other dynamics (see Sec. VI).
Unlike the hyperbaryon fields, quark fields fit into irreps of the smaller, Standard-Model symmetry. They do not fill up any irreps of the global symmetry group of the hypercolor theory. The coupling of quark and hyperbaryon fields therefore explicitly breaks the flavor symmetry of the hypercolor theory. This will induce a potential V eff for the NGBs.
While V eff is invariant only under Standard-Model symmetries, it depends on low-energy constants that can be expressed in terms of correlation functions of the stand-alone hypercolor theory. When we derive expressions for these low-energy constants, we may benefit from the full global symmetry of the hypercolor theory, including in particular G Υ . The way to do this is to promote the quark fields to spurion fields transforming in irreps of G Υ .
In the rest of this subsection we construct the spurions explicitly. Each embedding of q L is defined by and similarly for t R , where the hatted objects are constant 4 × 4 matrices. Because the EHC theory is not known, we will allow for the most general four-fermion lagrangian which is compatible with the (spurionized) symmetries of the hypercolor theory, and with CP . In order to build the four-fermion lagrangian we have to allow for all embeddings of the quark fields into spurions belonging to two-index irreps of G Υ = SU(4) (or to a singlet), which are consistent with the embedding of SU(2) L and SU(2) R into SU(4). We begin with the spurion embeddings of q L . For the adjoint irrep of SU(4) there are two options, (3.14) Remembering that q L = (t L , b L ) is an SU(2) L doublet with T 3 R = −1/2, one can check that these spurions are consistent with the Standard-Model transformation properties of q L . To this end we use that the adjoint spurions transform as D i L → gD i L g † under g ∈ SU(4), and the embedding (C6) of SU(2) L and SU(2) R into SU (4). For the two-index antisymmetric irrep we have one embedding, and likewise for the two-index symmetric irrep, The A L and S L spurions transforms as X L → gX L g T , X ∈ {A, S}, under g ∈ SU(4), and again one can verify consistency with Standard-Model quantum numbers. The embeddings for the complex conjugate irreps A c and S c may be obtained using the rule where again X ∈ {A, S}. Let us explain this rule. We first observe that X c spurions transform under g ∈ SU(4) as X c → g * X c g † . Restricting to g ∈ Sp(4), and using Eqs. (3.17) and (C1), we have The rightmost expression involves the transformation rule of a field in the A or S irreps, and we have already verified that this correctly reproduces the Standard-Model transformation rules for the A L and S L spurions. Since SU(2) L × SU(2) R is a subgroup of Sp(4), it follows that the spurion X c defined by Eq. (3.17) will again reproduce the correct Standard-Model transformation rules. Applying Eq. (3.17) we find the explicit forms Let us move on to t R , which is a singlet of SU(2) L with T 3 R = 0 (note that t R is not required to be invariant under the full SU(2) R , but only under rotations generated by T 3 R ). In this case we have more options, starting with the singlet There are two linearly independent options for the adjoint irrep, another two for the anti-symmetric irrep, and one for the symmetric irrep, The spurion embeddings for the A c and S c irreps again follow using Eq. (3.17). Explicitly, It remains to construct the anti-spurion embeddings. Referring to the decompositions (3.11) and (3.12), we define the c-number coefficients of the anti-spurion fields viâ (3.28) The last equality follows because we have chosen all the c-number spurionsX to be real.

B. L EHC
With the top-partner hyperbaryons and the spurions at hand, the most general four-fermion lagrangian that couples them is given by where the trace is over SU(4) indices. λ 1 , . . . , λ 7 andλ 1 , . . . ,λ 8 are (dimensionful) coupling constants. We have grouped in L EHC,1 those terms where B L belongs to D or N, while B R belongs to A, A c , S or S c , and the other way around for L EHC,2 . The four-fermion lagrangian is invariant under the spurionized SU(4) symmetry. In addition, it is truly invariant under the Standard-Model gauge symmetries SU(3) c , SU(2) L , and U(1) Y , and it conserves baryon number, or, which is equivalent, the T 3 R charge. Assuming that all the coupling constants are real, the four-fermion lagrangian is also invariant under the combined CP transformation of the hypercolor theory and the Standard Model, in which the c-number spurions are inert. How CP works is best illustrated through an example. The CP rules of Sec. II B imply in particular that t R B L,ij ↔ B L,ji t R . Remembering that c-number spurions don't transform, we have where again the trace and transpose operations are applied to the flavor indices. In order to establish the CP -invariance of L EHC we have used Eq. (3.28), which in turn relies on the fact that all the c-number spurions are real. That such a choice can be made, is a special feature of the SU(4)/Sp(4) coset. (As we will see in Sec. IV, things are slightly more involved for the SU(5)/SO(5) case.) Of course, we could have chosen to multiply some c-number spurions by arbitrary phases. This would invalidate Eq. (3.28) for those c-number spurions, and, as a result, there would be fewer terms in L EHC if we wish to maintain CP invariance. However, opting to do this is arbitrary. Once again, the point is that apart from some very general assumptions, we do not know the EHC theory. Therefore, we must consider the most general four-fermion lagrangian consistent with those general assumptions. When all the four-fermion couplings are taken to be real, this requires choosing all the c-number spurions to be real as well.
As already noted, in this paper we do not study the SU(6)/SO(6) pNGBs associated with the vector-irrep fermions, and therefore we only gave the SU(3) c quantum numbers of the hyperbaryons. Requiring full SU(6) invariance will give rise to the same four-fermion lagrangian once the spurions assume their Standard Model values. Indeed, each term in Eq. (3.29) can be trivially "lifted" to an SU(6)-invariant form, as we illustrate through the following examples. For definiteness, we will refer to the hyperbaryons of the SO(5) gauge theory.
We begin with the first term on the right-hand side of Eq. (3.29b), A L B A R . Since the χ fermions play little role, for brevity we express the hyperbaryon operator as B A R,a = f (χ) A I ψ R,Ia = f (χ) A I Ψ Ia , where in the last equality we have used that the 3 Dirac fermions introduced earlier are composed of 6 right-handed vector-irrep Weyl fermions Ψ 1 , . . . , Ψ 6 according to A complete SU (6) irrep is now obtained by simply replacing the index a = 1, 2, 3, with a new indexā = 1, . . . , 6, explicitly,B A R,ā = f (χ) A I Ψ Iā , where we are using a tilde to refer to SU(6) irreps. The SU(6)-invariant interaction is thusÃ L,āB A R,ā . In order to ensure equality between the SU(3) c and SU(6) versions, we simply embed the SU(3) c spurion into the SU(6) spurion, namely, we defineÃ L,ā = A L,a forā = a = 1, 2, 3, andÃ L,ā = 0 forā = 4, 5, 6.
These examples demonstrate that there is one-to-one correspondence between the SU(3) cinvariant and SU(6)-invariant forms of L EHC . The underlying reason is that the "expectation values" of the spurions are only constrained by SM symmetries.

C. Top Yukawa couplings
Effective top-Yukawa couplings are generated by integrating out all the states of the hypercolor theory except for the pNGBs. These effective interactions are organized in a weak-coupling expansion in the four-fermion couplings, as well as according to the usual power counting of the chiral lagrangian. To second order in the four-fermion couplings, and to leading order in the chiral expansion, we find effective interactions that are either linear or bilinear in Σ or Σ * . Any effective interaction which is cubic or higher in the nonlinear field must contain additional derivatives and/or mass insertions, 12 and therefore belongs to a higher order in the chiral expansion.
We begin with effective interactions that are linear in Σ or Σ * . Each effective interaction contains one spurion and one anti-spurion, one of which must be left-handed and the other right-handed. The effective Yukawa interactions have the same symmetries as L EHC . In order to form an SU(4) singlet, the spurion must belong to A, A c , S or S c and the antispurion to D or N, or the other way around, because the effective interaction has to contain a Σ or a Σ * . It follows that the spurion and the anti-spurion must both come from L EHC,1 , or both from L EHC,2 , which explains why we have grouped the four-fermion interactions this way. The list of possible top-Yukawa effective interactions is thus where the hermitian conjugate is to be added to each operator. X L can be A L or S L , and X R can be A 1,2 R or S R . The explicit form of each effective interaction can be worked out 12 See Sec. III F below for a discussion of explicit mass terms for the fermions of the hypercolor theory.   Table 1 for the field content of the hyperbaryons. The axial charges of the hyperbaryons are listed in Table 2. Notice that the dependence on the axial charge q of the vector irrep always cancels out in the effective Yukawa interactions.
Similar considerations give rise to the list of effective interactions which are bilinear in Σ or Σ † , given by , where again the hermitian conjugate is to be added to each operator. This amounts to 18 additional possibilities, none of which vanish.
The coupling constant that multiplies a given effective top-Yukawa interaction term is obtained using the procedure that we have discussed in detail in Ref. [23]. As an example, let us consider the term Φ tr(A L ΣN R ). Denoting by y A L ,N R the coupling constant that multiplies this term in the effective theory, and using Φ = 1, we have where a, b are SU(3)-color indices, α, β are Dirac indices, and we have treated the singlet N as an SU(4) scalar with no flavor indices. In the microscopic theory, Demanding equality between the effective and microscopic theories, and using that the right-hand side of Eq. (3.34) is the leading term in a derivative expansion, we obtain In the absence of spontaneous symmetry breaking, this two-point function would evidently vanish, because B A and B N belong to two different irreps of SU(4). But the antisymmetric irrep of SU(4) contains an Sp(4) singlet (see App. C), and so this two-point function is non-zero after symmetry breaking. In terms of the elementary fermions of the hypercolor theory we have, using Table 1, where DA denotes the Haar measure for the gauge field, and µ(A) is the Boltzmann weight. Inside the gauge-field integral, the expectation values denote correlation functions of the elementary fermions in a fixed gauge-field background. There are three different ways to contract the four χ fermions into a product of two χχ propagators. In every case we will have a P R applied to both sides of one χχ propagator, which projects out an order parameter for the SU(4) → Sp(4) symmetry breaking. Expressions for all other contributions to the top-Yukawa coupling can be worked out in a similar way. It is clear that the experimental value of the top-Yukawa coupling in the Standard Model provides only one constraint on the many couplings present in L EHC .

D. V eff
The effective potential for the pNGBs is generated by integrating out all other states of the hypercolor theory, and, in addition, the Standard-Model gauge and fermion fields. Here we will calculate the effective potential V eff for the SU(4)/Sp(4) and U(1) A pNGBs (we keep disregarding the SU(6)/SO(6) pNGBs), which is obtained by integrating out the third-generation quarks. 14 To leading order in the four-fermion couplings, the effective potential arises from correlation functions of two four-fermion vertices, where every correlation function is a convolution of a hyperbaryon two-point function with a single massless quark propagator. As a result, every term in the effective potential will be quadratic in the spurions, and both spurions will have the same handedness.
There is a large number of ways to generate an effective potential, which we organize into twelve "templates," 14 For the gauge boson's contribution to the effective potential, see Sec. III F below.
T 7 = tr(AΣ) tr(AΣ * ) , As in Sec. III C, the power of Φ in each template matches the axial charge of the associated product of hyperbaryons in the microscopic theory. The axial charge vanishes for templates T 7 , T 8 and T 12 . For the other templates it doesn't. We have normalized the axial charge such that the χ's give rise to an integer power of Φ. Templates T 1 through T 6 are sensitive also to q, the axial charge of the ψ's. In templates T 3 through T 6 , Φ −2q (Φ +2q ) corresponds to right-handed (left-handed) spurions. For T 1 and T 2 we always obtain Φ −2q from the ψ's, because only t R can be embedded into a neutral spurion (see below).
The alert reader will have noticed the similarity between templates T 1 through T 6 and the effective Yukawa interactions in Eq. (3.32), and likewise, between templates T 7 through T 12 and the effective Yukawa interactions in Eq. (3.33). The underlying reason is the similar group theoretic structure, as well as the power counting, which again allows for a maximum of two non-linear fields (Σ or Σ * ) in the leading-order effective potential. While we will shortly explain in detail how the templates encode the effective potential, already at this stage we point out several important differences. First, in the effective Yukawa interactions the quark fields are present, whereas in the effective potential they have been integrated out. Second, the two spurions in the effective Yukawa interactions are one right-handed and one left-handed, whereas here both of them have the same handedness. As a result, the pattern of axial charges in the effective Yukawa interactions and in the effective potential is different as well.
Every template from Eq. (3.39) will expand out to several terms in V eff . 15 We illustrate this using the example of T 1 . In this case the two spurions must be right-handed, because q L cannot be embedded into a singlet of SU(4). As for t R , it can be embedded into an antisymmetric spurion in two different ways. Template T 1 thus gives rise to the following two terms Each term consists of the product of three elements: a low-energy constant, a pair of coupling constants from L EHC , and an expression of the form T 1 , where we have made a particular choice for the spurions in the template T 1 . The meaning of the notation · here is the following. For the right-handed case, this is the outcome of integrating out the t R field, and the hyperbaryon fields to which it couples in Eq. (3.29). In practice, denoting the spurion and anti-spurion fields generically as X R and X R , they are traded inside the · symbol with the corresponding constant spurion matricesX t R andX t R , see Eq. (3.12). In the left-handed case, we in addition sum over the contributions of t L and b L (Eq. (3.11)). As mentioned above, each correlation function that contributes to the leading-order effective potential is built from two vertices from L EHC , and so it contains a hyperbaryon two-point function together with a single quark propagator, which, in this approximation, is a free massless propagator of a given chirality. The spurion and the anti-spurion in each template must therefore have the same chirality. In the example of Eq. (3.40), only t R can be embedded into a spurion belonging to the singlet irrep. Since there are two independent options for the embedding of t R into the antisymmetric irrep, A 1 R and A 2 R , the template expands out to two terms in V eff .
The four-fermion coupling constants together with the low-energy constant are inferred from a matching procedure that we have discussed in detail in Ref. [23], and which is similar to the one used in the previous subsection for the case of the effective top-Yukawa couplings. The four-fermion coupling constants are the two coupling constants from L EHC associated with the spurion and the anti-spurion that occur inside the · symbol. The remaining lowenergy constant is expressed in terms of a correlation function of the stand-alone hypercolor theory, which does not depend on the particular embedding of the quark fields into the spurion. Therefore, for each template T i we have just two low-energy constants C iL and C iR , one for each chirality. For the first two templates we only need C iR , because only t R can be embedded into an N spurion.
Since both the hypercolor theory and the four-fermion lagrangian are CP invariant, so will be the effective potential V eff . Using the assumed reality of the four-fermion coupling constants, one can also verify directly the CP invariance of Eq. (3.40), and of the corresponding expressions for all other templates. Because V eff is always real, it follows as a corollary that all the low-energy constants are real. Similar statements apply to the low-energy constants that multiply the effective Yukawa couplings discussed in the previous subsection.
We comment in passing that CP is only an approximate symmetry of the Standard Model, whose breaking is encoded in the Yukawa couplings. In a similar spirit, one may relax the assumption that the coupling constants in L EHC are all real, and assume, instead, that any imaginary parts of these coupling constants are parametrically small. How the EHC theory would induce this small amount of CP violation goes beyond the scope of this paper. To avoid confusion, we stress that since we have defined the low-energy constants to be independent of the four-fermion lagrangian, their reality is true regardless of whether or not the coupling constants of L EHC are real.
In the rest of this subsection we list all the contributions to V eff for the twelve templates. As explained above, the four-fermion couplings that multiply each expression are easily read off from L EHC . The low-energy constants will be derived in the next subsection. Thanks to the simplicity of the SU(4)/Sp(4) coset, it is possible to obtain the potential in closed form. Because some of the templates depend on the U(1) field Φ, in general an effective potential will be generated for the U(1) A pNGB as well.
We begin with T 1 , which gives rise to the two terms in Eq. (3.40). Using Eqs. (C7) and (C9), we have where α is given by (cf. App. C) and we wrote Φ = e iζ . (3.43) The field ζ is dimensionless, and is introduced here for the sake of brevity. For the chiral expansion, it is more natural to use instead the expansion Φ = exp(iζ/( √ 2f ζ )), where the NGB field has the appropriate canonical dimension, and f ζ is the decay constant of the U(1) A NGB [26].
Because t R is embedded into the A R and A c R spurions in the same way, each result for T 2 may be obtained from the corresponding result for T 1 by flipping the signs of the SU(4)/Sp(4) pNGBs, and multiplying ζ by −1 − 2q instead of 1 − 2q. The outcome is the same as just replacing 1 − 2q by 1 + 2q everywhere.
Considering next templates T 3 through T 6 , which also have a single non-linear field, but a D spurion instead of the N spurion, we find for T 3 For T 4 we have Φ 1+2q tr(S L ΣD 1T L ) + h.c. = 4 cos(α) cos((1 + 2q)ζ) and for T 6 , Turning to the templates with two non-linear fields, for T 7 we have The results for T 8 are the same as for the corresponding results for T 7 . The last double-trace template is T 9 , for which we obtain Moving on to the single-trace templates, for T 10 we find and for T 11 , Finally, for T 12 the non-zero results are To complete the construction of the effective potential, we need the low-energy constants. In order to fully benefit from the SU(4) symmetry of the hypercolor theory, we now expand each spurion as where η L,R (x) is a free massless Weyl field. Let us compare this with Eqs. (3.11) and (3.12).
In the latter case, the (hatted) matrices that carry the SU(4) indices are assigned a fixed numerical value that defines a particular embedding of a quark field. By contrast, we now treatX L,R andX L,R as global spurions that do not have any particular value, but, instead, transform in an irrep of SU(4). As a final preparatory step, we eliminate from L EHC the information about any specific embedding of the quark fields while keeping only the information about the SU(4) irreps, by writing, e.g., (λ 5 D 1 where D R is a global spurion in the adjoint irrep. In this process we also deliberately suppress the information about the four-fermion coupling constants. As discussed above, this information can easily be read off from the original definition (3.29). We end up re-expressing L EHC in terms of the hyperbaryon fields, the η L,R (x) field, and a pair of global spurions for each irrep: singlet N L,R , adjoint D L,R , two-index antisymmetric A L,R , two-index symmetric S L,R , and their complex conjugates A c L,R and S c L,R . In the (templates for the) effective potential, Eq. (3.39), we simply trade every spurion field with the corresponding global spurion. Each low-energy constant will be obtained by taking ordinary derivatives with respect to the global spurions, and matching the results between the microscopic and the effective theories. This matching procedure will allow us to replace the Σ field in the effective theory by its expectation value. This, in turn, simplifies considerably the calculation of the low-energy constants. Indeed, by making use of the global symmetry, we are able to extract the low-energy constants from correlation functions of the microscopic theory that do not involve any NGB asymptotic states.
We start with T 1 , whose contribution to V eff now reads We recall that we only need the right-handed low-energy constant C 1R , because the lefthanded quarks cannot be embedded into the singlet irrep. In the effective theory, where we have used that Σ = ǫ 0 and Φ = 1. In the microscopic theory we have As in Sec. III C we may express the hyperbaryon two-point function in terms of the elementary fermions. As can be seen from Table 1, while in the case of the D and N irreps the hyperbaryon fields have the same form for the SO(5) and SO(11) gauge theories, their forms for the other irreps are different in the two theories. For definiteness, we will assume in this subsection that the microscopic theory is the SO(5) gauge theory, 16 obtaining where we have used that χ k Γ J γ 5 γ ν χ k = 0, and the ellipses stand for a term that vanishes when contracted with ǫ 0,ij in Eq. (3.57). As expected, the expectation value of χ j P R Γ I χ i provides for an order parameter for SU(4) → Sp(4) symmetry breaking. Unlike the basic local order parameter (Eq. (3.7)), because of the presence of the SO(d) matrices Γ I and Γ J inside of the χ bilinears, only the two-point function as a whole is a gauge invariant (non-local) order parameter. In addition, the factor ψ J (y)γ ν γ µ P L ψ I (x) does not vanish because of the symmetry breaking SU(6) → SO (6), so the non-vanishing of the correlator B N L (y)γ µ B A Lji (x) requires both SU(4) and SU(6) to be spontaneously broken. For template T 2 , the only difference in the calculation of C 2R is that the hyperbaryon B A L is replaced by B A c L . This has the effect of replacing the P L projector inside the χ j (x)P L Γ I χ i (x) bilinear in Eq. (3.58) by a P R . For T 3 , we need an adjoint hyperbaryon instead of the neutral one. In this case both chiralities are needed, and by similar arguments we find The low-energy constants for templates T 4 , T 5 and T 6 can be similarly obtained.
In the case of template T 7 we need to do a little more work, because one can construct from the A and A spurions also a symmetry-preserving term that does not depend on the Σ field, tr(AA). Considering the left-handed case for definiteness, the relevant terms are and so (3.61) 16 The reader can easily work out the minor changes for the SO(11) case.
We may now extract C 7L by contracting this result with the fully antisymmetric fourdimensional tensor ǫ ijkℓ . By applying the same differentiations to the microscopic theory, and comparing the results, we find For C 7R , the chiral projector inside the hyperbaryon two-point function is P L . In terms of the elementary fermions, where again the ellipses denote terms that vanish when contracted with ǫ ijkℓ in Eq. (3.62). We see that from each χ propagator we pick up the part proportional to ǫ 0 in flavor space, which is non-zero in the broken phase. For template T 8 , the A and A spurions are replaced by A c and A c spurions, respectively.
The result is similar, except that, in Eq. (3.63), the chiral projectors inside the χ bilinears get flipped. For templates T 9 , T 10 and T 11 there are no Σ independent terms. For T 9 we find This time, the three possible contractions of the χ's are all non-zero in the broken phase, and contribute to the low-energy constants. For template T 10 , For template T 11 we find C 11L,R = −C 9L,R . Finally, in the case of template T 12 we once more have a symmetry preserving term, C ′ 12L,R tr(D L,R D L,R ), that we need to separate out. 17 Expanding the adjoint fields on the basis of SU(4) generators T a we have in the effective theory (omitting the chirality label) The right-hand side is proportional to (±C 12 + C ′ 12 )δ ab when T a is an unbroken, respectively, broken generator. By considering both cases we may extract the low-energy constant. In the microscopic theory (considering the left-handed spurions for definiteness) and The first term on the right-hand side of the second equality picks up the kinetic part of the χ propagator, which is symmetry preserving and proportional to δ ij in flavor space. The flavor trace therefore collapses to tr(T b T a ), which corresponds to the C ′ 12 term in Eq. (3.68). The last term picks up the symmetry breaking part of the χ propagator, which is proportional to ǫ 0,ij . This precisely corresponds to the flavor trace multiplying the C 12 term in Eq. (3.68), and therefore the low-energy constants C 12L,R are obtained by substituting this term into Eq. (3.70). This completes the derivation of the low-energy constants for this theory.

F. Summary
Collecting everything, we see that the effective potential arising from integrating out the third-generation quarks takes the form with the following nine functions and where α is given by Eq. (3.42). An interesting feature of this result is that, in general, a potential is generated not only for the Higgs doublet and for η, which are the NGBs of the SU(4)/Sp(4) coset, but also for the singlet NGB ζ. (We recall that in this paper we disregard the NGBs of the SU(6)/SO(6) coset.) The c i 's of Eq. (3.72) can be expressed in terms of the coupling constants of L EHC and the low-energy constants that we have derived in the previous subsection. The low-energy constants can be determined from a lattice calculation, which would then allow for a study of the experimental constraints on the four-fermion coupling constants. We note that experimental constraints on the effective potential alone can, of course, be studied directly in terms of the c i 's. However, if one wants to incorporate the top Yukawa coupling into this analysis, then it has to be done in terms of the four-fermion couplings, and thus, it depends on the knowledge of the low-energy constants.
For completeness, we also give the gauge-boson contribution to the effective potential, which is where Q a is to be summed over gT i L and g ′ Y = g ′ T 3 R , and where C w > 0 [15]. The expression for the low-energy constant C w may be found in Ref. [16] for the case of a real irrep. The case of a pseudoreal irrep defers only by the overall sign. However, relative to the definition of V EW given in Ref. [16], in Eq. (3.74) we have introduced an extra a minus sign on the right-hand side. This cancels out against the sign that is encountered in the derivation, so that now C w comes out positive in the pseudoreal case as well. With this, we find where f 6 is defined in Eq. (3.73). The gauge bosons contribution will therefore add up to the coefficient c 6 . As usual, taken by itself this contribution prefers the trivial vacuum H = 0, a phenomenon that goes under the name of vacuum alignment [14]. But considering V eff as a whole, there is ample room for a non-trivial minimum of the Higgs field. A final contribution to the effective potential might come from mass terms for the χ fermions. One can write down two mass terms which are invariant under the Standard Model symmetries [28]. Introducing ǫ ± 0 = ±(i/2)(1 ± τ 3 ) × τ 2 (where we are using the notation of App. C), these mass terms are where we have used Eqs. The mass term (3.77) breaks the global SU(4) symmetry explicitly to Sp(4), and the individual mass terms (3.76) further break it explicitly to the Standard Model symmetry SU(2) L × SU(2) R . From the point of view of the stand-alone hypercolor theory it may be more natural to avoid any mass terms, since this keeps the full SU(4) global symmetry intact. Having said this, we observe that explicit breaking of the flavor symmetry of the hypercolor theory, encoded in the four-fermion lagrangian (3.29), must originate from the EHC theory. Since we do not know the details of this EHC theory, we cannot rule out that it might also induce some of the mass terms discussed above. Similar statements apply to a Dirac mass term ∝ ψψ for the vector-irrep fermions, which breaks the SU(6) symmetry explicitly to SU(3) c .
The structure of the total potential is complicated. Its minimum will depend on the values of the low-energy constants, which can be determined within the hypercolor theory, and on the four-fermion couplings λ i andλ i , which arise from integrating out heavy degrees of freedom of the EHC theory. In addition, the potential depends on the electroweak couplings through Eq. (3.75), and possibly, on the mass term (3.76) or (3.77). Here we will be content with an example of a phenomenologically viable potential obtained by setting to zero by hand most of the four-fermion couplings.
Our example consists of turning on the following couplings: λ 2 , λ 7 , andλ 1 = −λ 2 , setting to zero the rest of the four-fermion couplings and the mass terms. Notice thatλ 1 andλ 2 involve the same hyperbaryon, B A L , hence the notion of a fixed ratioλ 1 /λ 2 is invariant under renormalization-group evolution. Also,λ 1 = −λ 2 implies that the spurions A 1 R and A 2 R always occur as the linear combination A 1 R − A 2 R ∝ ǫ 0 . With this choice, the only contribution that depends on ζ arises from template T 1 (see Eq. (3.41)), and is given by (3.78) We will demand that the minimum of the potential occurs for |α| < π/2, as is required for a phenomenologically viable solution. Further assuming that where a 1 = −8C 1R λ 7λ1 , a 2 = 16 C 3Rλ 2 1 , and The a 2 term arises from the contributions of right-handed spurions to T 7 , while the a 3 term arises from the gauge-bosons contribution as well as from the left-handed spurions in T 7 . The a 1 and a 2 terms have full Sp(4) invariance since they depend on H and η only through α. It follows that, if the minimum of the potential occurs for non-zero α, it will point in the H direction (i.e., H = 0 and η = 0) when a 3 < 0, and in the η direction when a 3 > 0. This conclusion is confirmed by studying the saddle-point equations. Thus, to be phenomenologically viable, the top-sector contribution to a 3 must be (negative and) large enough to overcome the positive contribution of the gauge bosons. A sufficient set of conditions to ensure a vacuum with ζ = η = 0 and H = 0 is a 1 > 0, a 3 < 0, and Values n = −1, 0, +1 of the phase transformation exp(−inη 0 ), which is to be applied to a Standard Model field, together with the spurion embeddings of q L (2nd column) and t R (3rd column) for which, for this n, the corresponding term in L EHC remains invariant when the SU (4) transformation U 0 is applied to the χ fields.
where this last condition implies that the curvature in the H direction is negative at the origin, and thus that the minimum of the potential cannot occur for α = 0. Once ζ = η = 0, the potential further simplifies. We defer further discussion of the resulting potential to the concluding section.
Returning momentarily to the EHC theory, we observe that if the four-fermion couplings arise from integrating out heavy gauge bosons, then each four-fermion term must take the form of a current-current interaction (possibly up to a Fierz rearrangement). Checking Table 1 shows that this condition is satisfied for all the four-fermion couplings that contribute to our example potential. Some other four-fermion couplings, such as, for example, the λ 1 term, cannot be brought to the form of a current-current interaction, and would thus vanish. However, if the heavy EHC degrees of freedom that have been integrated out include not only gauge bosons but also fermions (whose mass could have either an explicit or a dynamical origin), or scalars, then none of the four-fermion operators in Eq. (3.29) is ruled out. In that case we could, for example, turn off λ 2 and turn on λ 1 instead. The only change in the potential would be that C 8L λ 2 2 gets replaced by C 7L λ 2 1 .

G. Spontaneous CP breaking
The Standard-Model neutral fields η and ζ are pseudoscalars, and so, at face value, their expectation values break CP spontaneously. (We are assuming that all the four-fermion couplings are real, so that CP is not broken explicitly.) Recently, it has been pointed out in Ref. [29] that this is not necessarily true, because it might be possible to shift the expectation value to zero through field redefinitions. 18 Here we address this question, first for η , and then for ζ .
Assume that at the minimum of the effective potential, η = η 0 = 0. In order to "rotate away" this expectation value we need to apply to the χ fields of the hypercolor theory the SU(4) transformation U 0 = exp(−iη 0 X/2), where we have introduced the dimensionless quantityη 0 = η 0 /( √ 2f ), and X = τ 3 × 1 is the generator associated with η (see Eq. (C7)). Indeed, if Σ = exp(iη 0 X)ǫ 0 , then U 0 Σ U T 0 = ǫ 0 . If initially both η and H have non-zero expectation values, then the U 0 transformation will set η = 0 while in general changing the expectation value of H as well.
The question now is whether we can find a matching transformation of the Standard Model fields q L and t R , such that, together with the transformation χ → U 0 χ, the total lagrangian L HC + L EHC will be invariant. If the answer is Yes, then we have achieved η = 0 via the field redefinitions, which implies that η was indeed unphysical.
In order to keep a particular term in L EHC invariant, the transformation needed for a given Standard Model field depends on its spurion embedding. Using the SU(4) transformation rules of the spurions, and applying the transformation to each spurion embedding in turn, we find that this transformation can always be realized via the multiplication of the Standard Model field by a U(1) phase exp(−inη 0 ), where the possible values of n are −1, 0, +1. We list the values of n for all spurion embeddings of q L and t R in Table 3.
The answer to the question is now clear. Consider the set of non-zero couplings in L EHC . If all of the spurion embeddings of q L belong to the same row of Table 3, and the same is true also for the embeddings of t R , then invariance of L EHC will be achieved by applying the corresponding phase transformations to q L and to t R . In this case the expectation value of η can indeed be rotated away, and is thus unphysical. But if the spurion embeddings of q L and/or t R belong to more than one row of the table, then it is not possible to maintain the invariance of L EHC . In this case η is physical, and η = 0 signifies the spontaneous breaking of CP (for an exception, see below).
A similar argument applies to ζ . The phase transformation of a Standard Model field that we now need for a particular term in L EHC is determined by the axial charge of the hyperbaryon to which it couples (see Table 2). Once again, in order to be able to rotate ζ away, the necessary and sufficient condition is that q L couples to hyperbaryons that all have the same axial charge, and that the same is true for t R .
For the example potential discussed in the previous subsection we have turned on the couplings λ 2 , λ 7 ,λ 1 andλ 2 . Only the λ 2 term is a spurion embedding of q L , so this poses no difficulty. However, the three spurion embeddings of t R associated with the remaining three couplings populate all three lines of Table 3. Therefore, the invariance of L EHC under the field redefinition χ → U 0 χ cannot be maintained, which implies that η is physical. The same is true for ζ since the axial charges of the relevant hyperbaryons are all different from each other. As a result, for η = 0 and/or ζ = 0, CP is broken spontaneously.
An exception is the special case Σ = (τ 3 × 1)ǫ 0 , which corresponds to specific non-zero values of both η and ζ . Even if both expectation values are physical, in this special case CP is not broken spontaneously, because Σ is real, and so it remains invariant under the combined sign flip of η and ζ.
Finally, we comment that an advantage of the SU(4)/Sp(4) coset is that it does not contain any isospin-triplet fields, and, as a result, the difficulties with triplet expectation values and their potential influence on the ρ-parameter do not arise.

IV. THE SU (5)/SO(5) COSET
The list of Ref. [8] includes two models in which the spinor irrep is real, based on the gauge groups SO(7) and SO (9). These models are the subject of this section. While the vector-irrep fermions ψ Ia are the same as before, χ i will now denote 5 Majorana fermions in the real spinor irrep (the relation between χ i and χ i is still given by Eq. (2.2)). In comparison with the SU(4)/Sp(4) coset we have studied in the previous section, the SU(5)/SO(5) coset is larger. Apart from the Higgs field and the singlet η, it contains nine additional NGBs that fill up the (3, 3) representation of SU(2) L × SU(2) R . For the basic features of the SU(5)/SO(5) coset, and the embedding of the 14 NGBs into the pion field, see App. D.
The order parameter χ i χ j is symmetric on its indices for a real irrep. We will assume that the vacuum state has χ i χ j ∝ δ ij . Applying the infinitesimal flavor transformation, Eq. (2.4), we see that the NGB fields are all pseudoscalars, δ a (χχ) = iχγ 5 (T a + T T a )χ . (4.1) The NGBs correspond to the 14 real symmetric generators of SU (5). For the 10 antisymmetric, imaginary generators of SU(5), we have δ a (χχ) = 0, showing that the unbroken group is SO(5). These features of the NGBs resembles QCD, and are different from what we saw in the previous section for the case of a pseudoreal irrep. As in QCD, it is easy to check that all the NGB fields flip sign under the CP transformation of the hypercolor theory, Eq. (2.5c). This creates a phenomenological problem concerning the Higgs field. The Standard Model's CP transformation, which we will denote as CP , must be different from the original CP transformation of the hypercolor theory, because the real components of H 0 and H + are even under CP , but, like all NGBs, they are odd under the CP transformation of the hypercolor theory. As it turns out, CP may be obtained as the product of the original CP and a diagonal SO(5) transformation. 19 Explicitly, The formal correspondence of the effective fields with the microscopic theory takes a similar form to Eq. (3.7), except that now the non-linear coset field Σ is a symmetric unitary 5 × 5 matrix. The pion field Π is real, symmetric, and traceless (see Eq. (D2)). Using the embedding of the Higgs field into the pion field, given in App. D, it is straightforward to check that Eq. (4.2) correctly reproduces the Standard-Model transformation rules of all components of the Higgs field.
The organization of this section is as follows. Since the methodology is the same as in the previous section, we will be brief, and focus on those features of the SU(5)/SO(5) coset that are different from the SU(4)/Sp(4) coset. As before we begin with the spurions in Sec. IV A, and write down the four-fermion lagrangian in Sec. IV B, which is then followed by the list of top Yukawa effective couplings in Sec. IV C. Turning to the effective potential for the pNGBs, we begin in Sec. IV D with the templates, which are followed by the list of low-energy constants. Because of the complexity of the SU(5)/SO(5) coset we were unable to obtain the effective potential in closed form. The expansion of V eff to second order in the pNGB fields is relegated to App. E, while in Sec. IV E and App. F we focus on the third order terms and their phenomenological role.

A. Spurions
As usual, we assume that the third-generation quark fields couple linearly to threeconstituent baryons of the hypercolor theory, via four-fermion interactions that originate from an extended hypercolor theory which is operative at an as yet much higher energy scale. In view of our ignorance of the EHC theory, we must allow for the most general form of the four-fermion lagrangian which is compatible with the symmetries of the Standard Model: the continuous symmetries SU(3) c , SU(2) L , T 3 R and B, and the discrete symmetry CP . Analogous to Sec. III, we do this by looking for all the embeddings of q L and t R into SU(5) spurions. Demanding consistency with the assignment of Standard-Model quantum numbers then yields the most general coupling between the third-generation quarks and the hyperbaryons.
We begin with the left-handed doublet q L = (t L , b L ). Introducing the 5 × 5 matrices all the spurion embeddings of q L may be constructed using Θ q and Θ T q , and all the embeddings of q L may be constructed using Θ q and Θ T q . For the adjoint irrep we have two independent embeddings, D 1 L = Θ q and D 2 L = Θ T q . For the symmetric irrep there is only one embedding S L = S c L = Θ q + Θ T q , and similarly for the anti-symmetric irrep we have Notice that while the quark content of the spurions S L and S c L is the same (and similarly for A L and A c L ), they are nevertheless different spurions, because their SU(5) transformation rules are different. For g ∈ SO(5), the transformation rules of all the two-index SU(5) irreps collapse to the common rule X → gXg T . The relative phases of different entries of Θ q and Θ q are fixed by the embedding of SU(2) L and SU(2) R as subgroups of SO(5) (see Eq. (D1)). Our choice of the overall phase of Θ q will be explained shortly.
Being an SU(2) L singlet with T 3 R = 0, the right-handed quark field t R can be embedded into a (1, 1) or into a (1, 3) of SU(2) L × SU(2) R . 20 The simplest possibility is the SU(5) singlet N R = diag(1, 1, 1, 1, 1)t R . For the adjoint irrep we again have two embeddings, D 1 R = T 3 R t R and D 2 R = diag(1, 1, 1, 1, −4)t R , which correspond to the (1, 3) and (1, 1) cases, respectively. There are two more possibilities for the symmetric irrep, S 1 R = S 1c R = diag(1, 1, 1, 1, 0)t R and S 2 R = S 2c R = diag(0, 0, 0, 0, 1)t R , both of which correspond to the (1, 1) case. Finally, there is a single embedding for antisymmetric irrep, A R = A c R = T 3 R t R , which belongs to (1,3). As for q L , we sometimes encounter the same embedding of t R for different SU(5) irreps. For example, in each of the spurions D 1 R , A R and A c R , the quark field t R is multiplied by the same constant matrix, T 3 R . Again, these are nevertheless different spurions, because of their different SU(5) transformation properties.
The c-number matrices that define the anti-spurion embeddings (recall Eqs. (3.11) and (3.12)) are always given byX where the SO(5) matrix Q is defined in Eq. (4.2). The last equality, which can be verified on a case-by-case basis, depends on the fact that all the right-handed spurion matricesX R are real, and all the left-handed spurion matricesX L were constructed using Θ q (and its transpose), which implies thatX L,ij is always real for even i + j, and imaginary for odd i + j. Of course, choosing to multiply any spurion matrix by some arbitrary phase would spoil these features. As already explained in Sec. III B, we refrain from doing this because we are after the most general four-fermion lagrangian which is consistent with the Standard Model's symmetries, including, in particular, CP .

B. L EHC
With all the spurion embeddings at hand, the four-fermion lagrangian is L EHC = L EHC,1 + L EHC,2 , (4.5a) where now the trace is over SU(5) indices. As usual, the invariance of L EHC under Standard-Model continuous symmetries follows from the consistency of the spurion embeddings with those symmetries. Assuming again that all the coupling constants are real, and using that all the c-number spurion matrices satisfy the algebraic property (4.4), one can verify that L EHC is also invariant under CP . As discussed above, our spurion construction ensures that L EHC is in fact the most general four-fermion lagrangian that enjoys these symmetries. As in Sec. III, one can then infer that all the low-energy constants occurring in the effective top Yukawa interactions and in the effective Higgs potential are real.

C. Top Yukawa couplings
As in Sec. III C, the leading effective top Yukawa couplings are either linear or bilinear in Σ and Σ * . For the same reason as before, those interactions that are linear in Σ or Σ * must involve a spurion and an anti-spurion that both come from L EHC,1 or both from L EHC,2 . In the former case we obtain 10 effective interactions where i = 1, 2, and in the latter case we obtain 12 more, where i, j = 1, 2. The extraction of the associated low-energy constants can be done following the example we have given in Sec. III C. The effective Yukawa couplings that are bilinear in Σ and Σ * may be read off from templates T 7 thru T 12 in Eq. (4.8) below, in the same way that the effective interactions in Eq. (3.33) are related to templates T 7 through T 12 of Eq. (3.39).

D. Low-energy constants
We now move on to the effective potential for the pNGBs, and begin by listing the templates for V eff . This time, they are given by (4.8) T 7 = tr(SΣ) tr(SΣ * ) , The main difference compared to the previous case (Eq. (3.39)), is that the roles of the A and S irreps have been interchanged, because Σ is now symmetric instead of antisymmetric. For completeness, we note that one can write down two mass terms which are invariant under the Standard model symmetries, given by B tr((m 1 M 1 + m 2 M 2 )Σ + h.c. ), where the mass matrices are M 1 = diag (1, 1, 1, 1, 0) and M 2 = diag(0, 0, 0, 0, 1). For m 1 = m 2 , the mass term is invariant under SO (5). Because of the similarity between the mass matrices M 1,2 and the symmetric right-handed spurions S 1,2 R , the explicit form of the mass terms bears resemblance to the effective potential for template T 1 . We leave the details to the reader.
The derivation of the low-energy constants is very similar to the previous section, and so we will only give the results. Also, except for T 12 , we leave it to the reader to work out the explicit expressions for the hyperbaryon two-point functions, using Table 1. In all cases it can be verified that SU(5) must break spontaneously to SO(5) for the relevant two-point function not to vanish. In some cases, SU(6) must be broken to SO(6) as well.
As in the previous section, for T 1 we only need the right-handed low-energy constant, For T 2 , B S gets replaced by B S c . For T 3 both chiralities occur in V eff , and Again the low-energy constants for templates T 4 , T 5 and T 6 can be similarly obtained. For T 7 we find The special choice of flavor indices we have made separates out the coefficient of tr(SΣ) tr(SΣ * ), which is what we need for V eff , from the coefficient of tr(SS), which is a Σ-independent effective term (for the spurion notation we use here, see Sec. III E). For T 8 , we replace B S by B S c and B S by B S c in Eq. (4.11). Next, the low-energy constants for T 9 and T 10 are obtained from the same hyperbaryon two-point function, (4.12) and differ only by the choice of flavor indices needed to project them out. For C 9L,R we set i = j = k = ℓ in Eq. (4.12), whereas for C 10L,R we set j = k = ℓ = i. For T 11 we use the same choice of flavor indices as for T 10 , so that We finally consider T 12 , where, just like in Sec. III E, we need to separate out the lowenergy constant of interest from the Σ-independent effective term C ′ 12L,R tr(D L,R D L,R ). Instead of Eq. (3.68), in the effective theory we now have (again omitting the common chirality index) (4.14) In the microscopic theory, the hyperbaryon two-point function is given by Eqs. (3.69) through (3.71) as before. But the symmetry-breaking part of χ i χ j is now proportional to δ ij , instead of to ǫ 0,ij , as it was in Sec. III. The upshot is that C 12L,R can be expressed in terms of the contraction on the last line of Eq. (3.71) in the same way as in the previous section.

E. V eff
With its nine additional NGBs, the structure of the SU(5)/SO(5) coset is richer than that of SU(4)/Sp (4), and the calculation of V eff is more difficult. We have not been able to obtain V eff in closed form. As a first step, we have worked it out to second order in the pNGBs. The results may be found in App. E.
One way to understand the extra complexity of the SU(5)/SO(5) coset is to consider the invariants of the Standard Model symmetries SU(2) L and T 3 R that can be constructed from the pNGB fields. If, in addition, such an operator (possibly together with its hermitian conjugate) is invariant also under CP , it can occur as a separate term in the effective potential. In the case of the SU(4)/Sp(4) coset, the simplest invariants that can occur in V eff were the bilinear H † H and powers of the inert pNGBs η and ζ. Moreover, the SU(4)/Sp(4) non-linear field Σ can be expressed as a linear function of the pion field Π, with coefficients that depend on the bilinears η 2 and H † H (see Eq. (C9)). This has enabled us to obtain the effective potential in closed form. By contrast, in the case of the SU(5)/SO(5) coset we also have a (3, 3)-plet of SU(2) L × SU(2) R at our disposal. There are two new invariant bilinears, given by tr(Φ 2 0 ) and tr(Φ +Φ− ) in the notation of App. D. At third order there are new invariants that depend only on the triplet fields: tr(Φ 3 0 ) and tr(Φ 0Φ+Φ− ), as well as mixed invariants that depend on both the Higgs and the triplet fields: The mixed invariants are particularly important for phenomenology. This is best illustrated through an example. We consider the contribution of q L to template T 7 , whose third-order term is (see Sec. III D for the · notation) tr(S L Σ) tr(S L Σ * ) 3rd order = 32 We see that once the Higgs field acquires an expectation value, H 0 = h/ √ 2 = 0, this induces a linear potential for Im φ − + (see Eq. (1.3)). As a result, the expectation value Im φ − + = ϕ/ √ 2 will necessarily move away from zero [8], while the expectation values of all the remaining components of the (3, 3)-plet remain zero at this order. As explained in the introduction, this is undesirable, because Im φ + − does not preserve the diagonal subgroup of SU(2) L × SU(2) R (the custodial symmetry) [22]. Therefore, this expectation value will drive the ρ-parameter away from unity.
Let us investigate this issue in more detail. While we have not been able to obtain the effective potential in closed form for arbitrary values of the pNGB fields, this can be done when only h and ϕ are turned on. The results may be found in App. F. Examining these results, we see that odd-order terms, and, in particular, the cubic term h 2 ϕ, are present in several cases. These include the contribution of q L to templates T 7 (Eq. (F9a)) and T 8 (Eq. (F10)). Similar terms are obtained for template T 12 , see Eqs. (F14a), (F14b), (F14g) and (F14h).
The question arises whether these undesirable contributions can be avoided. A simple observation is that odd-order terms would be absent if one could show that the effective potential is invariant under an "intrinsic parity" transformation that takes Σ → Σ * and Φ → Φ * , while leaving the Standard-Model quark fields unchanged. The obvious reason is that this transformation flips the sign of all the pNGB fields. 21 A case-by-case check, using the explicit forms of the spurions (and assuming the general form of the pion field, Eq. (D3)), reveals that the individual contributions to V eff are each invariant under the intrinsic parity transformation, except for the six cases we have listed above, where the cubic term h 2 ϕ is actually present.
Individual odd-order contributions can be avoided by imposing suitable constraints on the coupling constants of L EHC . For example, the contributions of Eqs. (F14a) and (F14b) cancel each other ifλ 7 = ±λ 8 [8]. The contributions from Eqs. (F14g) and (F14h) are absent if λ 5 and/or λ 6 vanish. Similarly, Eq. (F9a) is absent when λ 1 vanishes, and Eq. (F10) when λ 2 does. Interestingly, for the parametrization (F2) all the odd-order contributions happen to involve the same function of h and ϕ. In V eff , every term from App. F comes multiplied by two coupling constants from L EHC , and a low-energy constant (see Eq. (3.40)). Therefore, mathematically, the minimal requirement that would eliminate all the odd-order terms for the parametrization (F2) is a single constraint, which is bilinear in the coupling constants of L EHC , and linear in the low-energy constants.
Physically, the four-fermion couplings and the low-energy constants have an entirely different origin. The former arise from integrating out heavy gauge bosons of the EHC 21 The transformation Σ → Σ * is physically equivalent to the transformation P π considered in Ref. [8], because the difference between them is an SO(5) transformation. theory, whereas the latter only depend on correlation functions of the hypercolor theory. Therefore, it is unlikely that they will satisfy a constraint of the kind described above. Intuitively, what makes more sense is that the odd-order terms in V eff might vanish thanks to the vanishing of sufficiently many four-fermion couplings. Some new constraint in the EHC theory would have to set the proper linear combinations of the couplings λ i andλ i equal to zero. One way this might happen is if the intrinsic parity symmetry discussed above would arise from some discrete symmetry of the EHC theory. Unfortunately, we have not been able to identify such a symmetry. Having said this, it remains a possibility that integrating out the heavy gauge bosons of the EHC theory would give rise to a small set of four-fermion couplings, that happens to satisfy the needed constraints on the couplings of L EHC , at least when the heavy gauge bosons exchange is considered at tree level.

V. REVISITING THE SU (4) COMPOSITE-HIGGS MODEL
Another composite Higgs model whose low-energy sector yields the SU(5)/SO(5) coset was first studied in detail by Ferretti in Ref. [13], and later by us in Ref. [23]. In this section we revisit the effective potential induced by the coupling to third generation quarks in this model. We begin with a brief summary. The model is an SU(4) gauge theory. The matter content includes 5 Majorana fermions χ i in the 2-index antisymmetric (sextet) irrep, together with 3 Dirac fermions ψ a in the fundamental irrep. 22 The global symmetry is 23 where χ R transforms as 5 of SU (5) If we take the U(1) X charge of ψ to be 1/6, 24 it will coincide with ordinary baryon number. U(1) A is the conserved axial current. As in Ref. [23], we take the axial charge of χ R to be −1. The axial charges are then 5/3 for ψ R and −5/3 for ψ L . In Ref. [23] we studied the top-induced effective potential. Making rather restrictive assumptions, we found that the potential is quartic in the spurions (equivalently, in the four-fermion couplings), and we discussed it in some detail. In this section, as in the rest of this paper, we will instead make minimal assumptions about the four-fermion lagrangian. We begin by reconsidering the dimension-9/2 hyperbaryons that can serve as top partners, finding two more operators that can play this role, in addition to the four operators already considered in Ref. [23]. The most general four-fermion lagrangian thus contain six independent couplings. Using this lagrangian we find that, in general, an effective potential is induced already at second order in the four-fermion couplings. We also reconsider the potential that is induced by the same four-fermion lagrangian as in Ref. [23], and find that it contains two additional terms that we overlooked. We conclude with a short discussion of the phenomenological implications of our findings.
The top-partners we consider are limited to three-fermion operators of the minimal dimension, 9/2. They must transform as 3 under SU(3) c , and can belong to 5 or 5 of SU(5). Since SU(3) c is the diagonal subgroup of SU(3) × SU(3) ′ , this allows for several possibilities  Table 1 of Ref. [23] (omitting the anti-hyperbaryons), and the last two lines to Eq. (5.2). The left column is the name of the hyperbaryon in the notation used in this section. When relevant, we give for comparison the name we used for the same operator in Ref. [23] in the second column. The remaining columns list the quantum numbers. The (ordinary) baryon number of all these hyperbaryons is 1/3.
for the SU(3)×SU(3) ′ quantum numbers of the hyperbaryons. Altogether, we can construct 3 right-handed and 3 left-handed hyperbaryons that satisfy the requirements. We list them in Table 4. The first four were already introduced in Ref. [23]. The last two are given by where the subscripts A, B, . . . , are SU(4)-hypercolor indices. In this section we label the hyperbaryons by a superscript that specifies the SU(5)×SU(3)×SU(3) ′ quantum numbers. 25 Under SU(3) c , the operators in Eq. (5.2) describe a 3 and a 6, but only the 3 will couple to Standard-Model fields. The most general four-fermion lagrangian that we can construct using these hyperbaryons is (5.5) 25 We label a hyperbaryon and its anti-hyperbaryon by the same superscript.
where the constant 5-vectors arê In Ref. [23], the effective potential was O(λ 4 ), i.e., it was quartic in the coupling constants of L EHC . Correspondingly, the low-energy constant discussed in Ref. [23] was determined in terms of a hyperbaryon 4-point function. The additional terms proportional to λ 5 and λ 6 present in Eq. (5.3) allow for the generation of an effective potential already at O(λ 2 ), with low-energy constants that depend on hyperbaryon two-point functions. The O(λ 2 ) potential is given by As in the previous sections, the global spurions (the v's) result from integrating over the quark fields q L and t R . The dependence of each spurion field on the relevant global spurions is similar to Eqs. (3.11) and (3.12). It follows that in the right-handed case we simply need to substitutet R for v R . For the left-handed case, we have to sum overt L andb L , paying attention to the possible presence of the SU(3) c invariant tensor ǫ abc in the embedding of the Standard-Model fields into the spurions. Since in this section we keep track of the SU(3) × SU(3) ′ symmetry, we show in Eq. (5.7) the dependence of the potential on Ω, the nonlinear field for SU(3) × SU(3) ′ → SU(3) c symmetry breaking. Ω transforms as Ω → gΩg ′ † , with g ∈ SU(3) and g ′ ∈ SU(3) ′ , i.e., it belongs to (1, 3, 3). As for the dependence on the U(1) A nonlinear field Φ, its power in each term is given by the axial charge of the hyperbaryon two-point function occurring in the calculation of the low-energy constant. (The actual calculation of the low-energy constants is similar to Ref. [23], and is left for the reader.) In order to proceed, we will for simplicity set Ω ab = δ ab . This means that, as in the previous sections, we do not calculate the effective potential for the colored pNGBs. The result is Here we introduced the orthogonal projectors 9c) whose sum P 1 + P 2 + P 3 is equal to the 5 × 5 identity matrix. As in the previous section, we were unable to work out the dependence of V top eff on all the pNGBs in closed form. But, as before, we can obtain the potential in some special cases. First, expanding the potential to second order in all the pNGBs gives If we use the parametrization (F2), i.e., we retain only the h and ϕ fields of Eq. (F1), the potential is given by We observe that there are no odd-order terms. Indeed, it is easy to check that the potential (5.8) is invariant under the intrinsic parity transformation of Sec. IV E. The gauge bosons contribution for this parametrization is the same as in Sec. IV, see Eq. (F15).
In this section we have allowed for spurions with all possible SU(3) × SU(3) ′ quantum numbers, resulting in the four-fermion lagrangian (5.3). By contrast, in Ref. [23] we only considered top spurions with particular SU(3)×SU(3) ′ quantum numbers. This corresponds to retaining only the λ 1 and λ 2 terms in Eq. (5.3), while setting λ 3 = λ 4 = λ 5 = λ 6 = 0. In this case the O(λ 2 ) potential vanishes, and the leading potential is O(λ 4 ). Explicitly, where we have used Eq. (5.9). The C top LR term was discussed in Ref. [23], whereas the other two terms were overlooked. 26 As in the rest of this paper, the low-energy constants introduced in this section are always determined by the stand-alone hypercolor theory. Expanding this potential to second order in the pNGB fields gives while for the parametrization (F2) we obtain This result shows that there are no odd-order terms associated with C top RR , consistent with the invariance of the corresponding term in Eq. (5.12) under the intrinsic parity transformation of Sec. IV E. Cubic terms arise from the contributions associated with C top LR and C top LL . These contributions will be absent if λ 1 = 0. 27 If both λ 1 and λ 2 are non-zero, then the cubic terms will be present except in the (unlikely, because arbitrarily fine-tuned) case that λ 2 2 C top LR = λ 2 1 C top LL . In this case the sum of the two terms is proportional to which is again invariant under the intrinsic parity transformation.
The main phenomenological implications of the results of this section are discussed in the concluding section.

VI. DISCUSSION AND CONCLUSIONS
The composite Higgs approach is often discussed taking the low-energy, non-linear sigma model as a starting point. In this paper we studied in detail several concrete realizations (ultraviolet completions) of this approach as an asymptotically free gauge theory with fermionic matter. In this concluding section, we discuss the lessons that can be drawn from our findings.
We begin with a simple technical observation about the Higgs potential. It is a generic feature of composite Higgs models that, if we turn off all the pNGBs except for h = √ 2Re H 0 , then the coset field Σ describes a rotation matrix by an angle α ∝ h in some generalized space. In other words, the non-zero entries of Σ depend linearly on cos(α) or sin(α). This is true in particular for the two cosets discussed in this paper. 28 For an effective potential that is at most quadratic in Σ and/or Σ * , it follows that the effective potential is then a secondorder polynomial in cos(α) and sin(α). Furthermore, SU(2) L invariance requires that, when 26 In the conventions of Ref. [23], λ 2 1 λ 2 2 C top LR corresponds to y 2 C top . In Ref. [23] we argued that C top LR dominates over the gauge bosons contribution in a certain large-N framework. Unfortunately, it is not possible to incorporate C top RR and C top LL into the same large-N framework in a meaningful way. 27 Notice, however, that in order to generate a mass for the top quark, at least two four-fermion couplings must be non-zero, e.g., λ 1 and λ 2 [23]. 28 For the SU (4)/Sp (4)  all triplet fields are turned off, the potential must be an even function of the Higgs field H, and this remains true when we retain h = √ 2Re H 0 only. The form of the resulting effective potential is very restricted. It depends on just two trigonometric functions of α, and we may take it to be [6,23,32] V eff = const. − A cos(α) + B cos 2 (α) . (6.1) The solutions of the saddle-point equation are sin(α) = 0 or which is the symmetry-breaking solution of interest. 29 We may rewrite this solution as Current experimental constraints suggest h 2 /f 2 < ∼ 0.1 as a figure of merit [5,6,33]. 30 Thus, for the right-hand side of Eq. (6.3) to be small, an "irreducible fine-tuning" at a similar level of the coefficients A and B is needed.
The effective potential receives contributions from two different sources. First, there are O(g 2 , g ′2 ) terms, arising from the interaction between the electro-weak gauge bosons and the pNGBs. The form of these terms is constrained by gauge invariance, and they depend on a single low-energy constant C LR . The other source of an effective potential arises from integrating out the third generation quark fields. This is the prime focus of this paper. In order to explain the four-fermion lagrangian that couples the quark fields to three-fermion states of the hypercolor theory, we have to postulate the existence of an "extended hypercolor" theory. This new dynamics is operative at a yet higher energy scale, Λ EHC , and requires the existence of new heavy gauge bosons that can transform an ordinary quark into one of the fermion species of the hypercolor theory. 31 The leading contributions to the effective potential from this sector are O(λ 2 ), where we use λ as a generic name for a four-fermion coupling. In the case of the model of Sec. V, for reasons that we explain below, we are also interested in O(λ 4 ) contributions.
Having a minimum of the effective potential with h 2 /f 2 < ∼ 0.1 thus requires balancing between O(g 2 , g ′2 ) effects, which depend on the gauge couplings of the Standard Model, and O(λ 2 ) effects (or, in special circumstances, O(λ 4 ) effects), which depend on the dynamics of the EHC theory, and can generically arise from several distinct four-fermion couplings. If the effects of the third-generation quarks dominates over the gauge bosons, then the balancing has to happen between the contributions coming from different four-fermion couplings. We have studied an example potential in Sec. III F. However, it remains an open question how the four-fermion couplings originating from the EHC theory can be arranged to give the desired result. We note that we did not make any ad hoc assumptions about the EHC sector. It turns out that, in all cases considered here, the most general form that the four-fermion lagrangian may take is quite complicated, leading to many possibilities for the low-energy effective theory (both the induced Higgs potential and the Yukawa couplings). New ideas will be needed to simplify the situation, but those would necessarily address the specific form of the EHC sector, and are beyond the scope of this paper.
Let us briefly touch on another basic difficulty, which is the inherent tension between fermion masses and flavor constraints. Traditionally, fermion masses are generated in technicolor models via four-fermion couplings that are induced by an extended technicolor (ETC) dynamics, of which our extended hypercolor (EHC) dynamics is a close cousin. The main difference is the following. If we generically use ψ to denote a Standard Model fermion field, and Ψ for a fermion of the new strong dynamics (be it technicolor or hypercolor), then ETC requires four-fermion interactions of the generic form ψψΨΨ, whereas the EHC interactions are assumed to have the form ψΨΨΨ. The ETC four-fermion interactions induce a fermion mass term, ψψ ΨΨ , once the operator ΨΨ acquires an expectation value. By contrast, the EHC four-fermion interaction ψΨΨΨ allows for a linear coupling of a Standard Model fermion to a hyperbaryon, thereby giving rise to a partially composite state. 32 The basic problem is that the same ETC or EHC dynamics that gives rise to the desired four-fermion interactions can, generically, also give rise to four-fermion interactions ∼ ψψψψ, namely interactions that involve four Standard Model fermions. These interactions will trigger flavor-changing processes that, if too strong, will be in conflict with experiment. According to naive power counting, fermion masses in ETC are suppressed relative to the technicolor scale Λ T C by z 2 T C , where z T C = Λ T C /Λ ET C , with Λ ET C being the ETC scale. Because of the flavor constraints Λ ET C must be quite large, making the ratio z T C small. The resulting fermion masses, of order Λ T C z 2 T C , are then too small in many cases. A partial solution may be provided by walking technicolor, where the technicolor dynamics is assumed to be nearly conformal. Taking quantum effects into account, the induced fermion mass in walking technicolor is ∼ Λ T C z 2−γm T C , where γ m is the (approximately constant) mass anomalous dimension of the technifermion Ψ. Ideally, a very large anomalous dimension γ m < ∼ 2 would wipe out entirely the suppression factor z 2−γm T C . But various theoretical considerations suggest that such large values of γ m are unlikely [4][5][6]. Lattice calculations in various models find that γ m does not exceed 1 (see the review articles [9,11]). If indeed γ m < ∼ 1 then the induced fermion mass can only be as large as Λ T C z T C , i.e., still suppressed by one power of z T C . Thus, while near-conformality together a large γ m help in generating larger fermion masses, it remains very difficult to generate a mass as large as that of the top quark. As an illustration, according to Ref. [6], Λ ET C cannot be smaller than about 10 5 TeV, 33 so that z T C cannot be larger than ∼ 10 −4 . With γ m ∼ 1 this might have allowed for generating the ∼ 1 GeV mass of the charm quark, but certainly not the top-quark mass.
If, instead, the top quark receives its mass via the partial compositeness mechanism, this mass will be naively of order Λ HC z 4 HC , where z HC = Λ HC /Λ EHC , because, when measured in units of the hypercolor theory, each four-fermion coupling is naively of order z 2 HC , and two four-fermion couplings are needed to generate a mass for the top: the top must transform into a hyperbaryon, and then back into a top. At tree level, the case for partial compositeness is thus worse than traditional ETC. Of course, one has to take into account quantum effects. If again the theory is nearly conformal, the induced top mass is of order Λ HC z 4−2γ ′ HC , where γ ′ is the (again, approximately constant) anomalous dimension of the 32 In principle, a given EHC theory may induce both ψψΨΨ and ψΨΨΨ type four-fermion interactions, in which case both mechanisms for fermion mass generation will be operative (see, e.g., Ref. [13]). 33 In the notation of Ref. [6], Λ ET C is Λ UV . relevant four-fermion operators. Once again, the suppression factor z 4−2γ ′ HC would be wiped out when γ ′ < ∼ 2. The popularity of partial compositeness stems from the fact that there are no theoretical considerations against such large values of γ ′ . Thus, at least in principle, one could end up with a suppression by a very small power of z HC [4][5][6]. 34 We stress that in order to achieve a large enhancement, be it in the context of extended technicolor or in the context of a partially composite top, the anomalous dimension must be approximately constant, and large, over many energy decades. This requires the dynamics to be nearly conformal. In contrast, if the gauge dynamics is QCD-like, then this mechanism is unlikely to be effective. The reason is that as we increase the energy scale, the gauge coupling quickly becomes perturbative. Existing perturbative calculations of the anomalous dimension of various four-fermion operators always find small values [27,34]. It remains an open question whether a realistic top-quark mass can be achieved by invoking a strong near-conformal dynamics. Lattice calculations of γ ′ in candidate hypercolor theories could help shed light on this important issue.
An alternative approach would be to assume that, while the top quark receives its mass through partial compositeness from an extended hypercolor dynamics, yet some other dynamics (or, more generally, some additional high scales), are involved in mass generation of all other Standard Model fermions. 35 This approach is, obviously, less economic, but eventually it might be forced upon us by the tension between flavor-changing processes and quark masses. In a way, in this paper we are following this approach, because we study the interaction between the third-generation quarks and the hypercolor theory, while disregarding the rest of the fermions of the Standard Model. In particular, we are in effect allowing for the extended hypercolor scale Λ EHC to be close enough to the hypercolor scale Λ HC , so that the four-fermion couplings will be large enough to generate phenomenologically viable mass for the top quark and effective potential for the pNGBs.
In this paper we studied two SO(d) gauge theories with d = 5, 11, where chiral symmetry breaking gives rise to pNGBs in the SU(4)/Sp(4) coset (Sec. III); and three models where the coset is SU(5)/SO (5), two are again based on an SO(d) gauge theory with d = 7, 9, and have a similar set of top partners (Sec. IV), while the third is an SU(4) gauge theory with a rather different set of top partners (Sec. V). Each model contains fermions in two different irreps, leading to a non-anomalous abelian axial symmetry, U(1) A , with an associated pNGB, ζ, which is inert under all the Standard-Model gauge interactions. For each theory we first listed all the dimension-9/2 hyperbaryons that can serve as top partners, and wrote down the most general four-fermion lagrangian that couples them to t L , b L and t R . We then worked out the resulting effective potential for the multiplet of pNGBs containing the Higgs field together with the U(1) A pNGB.
We started with the SU(4)/Sp(4) coset. Its structure is simpler in that, besides Higgs doublet H, this coset contains only one additional pNGB, η, which is inert under the Standard Model gauge interactions, like the U(1) A pNGB ζ. We worked out the O(λ 2 ) potential in closed form. We found that it consists of a linear superposition of nine functions of the variables H † H, η and ζ (cf. Sec. III F). Thus, in general, a potential is generated for all the pNGBs, including the U(1) A pNGB. Each coefficient c i consists of a sum of terms, where each term is the product of a low-energy constant and two four-fermion couplings. The effective potential generated by the electro-weak gauge bosons also depends on one of these 34 For a calculation of γ m and γ ′ in a gauged Nambu-Jona-Lasinio model, see Ref. [17]. 35 See, for example, Refs. [35][36][37].
functions, f 6 , and so it contributes only to its coefficient c 6 . Finally, there is an additional contribution to the effective potential if a mass term for the χ fermions is turned on in the hypercolor theory. By itself, experimental constraints on the effective potential can be studied directly in terms of the c i 's. But, if one wants to incorporate also the top Yukawa coupling into this analysis, then it has to be done in terms of the four-fermion couplings, and requires knowledge of the low-energy constants. The latter can, in principle, be calculated on the lattice.
Studying the minima of the full effective potential as a function of all the relevant parameters is challenging. Generically, minimizing the potential might give rise to the condensation of not just the Higgs field, but also the "inert" fields ζ and η. Since these fields are pseudoscalars, the expectation values η or ζ break CP spontaneously, and will thus be constrained by experiment. We have discussed the conditions that these expectation values are physical, and cannot be rotated away (Sec. III G). Our discussion of the effective potential was limited to a simple example, in which most of the four-fermion couplings are turned off by hand (Sec. III F). For this potential η and ζ are both physical. We wrote down the conditions needed to have η = ζ = 0, at which point the potential reduces to the familiar form of Eq. (6.1). 36 The low-energy constants of the SO(d) models depend on two-point functions of the hyperbaryons. While, as we have explained above, one can sometimes by-pass the calculation of the low-energy constants by studying directly the c i coefficients in Eq. (3.72), the correct form of the effective potential cannot be determined without the knowledge of the dimension-9/2 hyperbaryons. In other words, if one starts directly from the non-linear sigma model it is just not possible to determine the correct effective potential. One can, of course, determine the structure of the effective potential for a given set of spurion fields. But the spurions must match the top-partner hyperbaryons. An ad-hoc list of spurions could amount to arbitrarily setting some of the four-fermion couplings to zero, in a manner that cannot be reproduced by any extended hypercolor theory.
Next let us discuss the models that yield the SU(5)/SO(5) coset. In addition to the pNGBs that are present in the SU(4)/Sp(4) case, there are nine additional pNGBs that fill a (3,3)-plet of SU(2) L × SU(2) R . Because of this more complicated structure we were not able to obtain the full potential in closed form. Instead, we studied the potential in various simplified cases. First, we obtained the potential to second order in all the pNGBs. Some useful constraints can already be obtained from this result because, ideally, we would like the curvature at the origin to be negative in the direction of the Higgs field, and to be positive in the direction of the triplet fields, to prevent any triplet from condensing. 37 We also considered third order terms. These terms arise because one can construct invariants of both SU(2) L and the U(1) generated by T 3 R from a pair of Higgs fields and one triplet field. For a concrete example, see Eq. (4.15). As we explained in the introduction, and in more detail in Sec. IV E, these terms are especially dangerous for phenomenology. If the potential contains cubic terms, then, once the Higgs field acquires an expectation value, this induces a term linear in the triplet field. This, in turn, will necessarily drive the expectation value of the triplet field away from zero. The resulting triplet expectation value is different from the one that preserves the custodial symmetry [22], and so it will drive the ρ-parameter away from unity. The magnitude of this triplet expectation value is thus tightly constrained by experiment.
Studying this issue further, we have worked out the full effective potential in the case that only h = √ 2Re H 0 and ϕ = √ 2 Im φ − + are turned on. We checked which "templates" for the effective potential can give rise to odd-order terms, and, in particular, to the cubic term h 2 ϕ, finding that such contributions are possible in all the SU(5)/SO(5) models. 38 We then raised the question how likely it is that all cubic terms (or, more generally, all odd-order terms) will be absent from the effective potential thanks to cancellations.
As we explained in Sec. IV E, if all the four-fermion couplings are non-zero, the vanishing of the coefficient of a particular (cubic) term in the effective potential requires a "conspiracy" between the four-fermion couplings and the low-energy constants. What might be more natural is that the cubic terms will vanish thanks to the vanishing of suitable (linear combinations of) four-fermion couplings. The intrinsic parity transformation introduced in Sec. IV E is a convenient device to determine which linear combinations of the four-fermion couplings should vanish. Unfortunately, we were unable to conceive of any obvious symmetry at the level of the EHC theory that would induce the intrinsic parity symmetry at the level of the low-energy effective theory. Still, one should remember that the four-fermion couplings must be induced by integrating out the heavy degrees of freedom of an EHC theory, and a good candidate EHC theory will conceivably induce only a small number of four-fermion couplings.
The SU(4) model of Sec. V was already studied in detail previously [13,23]. We found that if we allow for the most general four-fermion lagrangian, an effective potential is induced already at O(λ 2 ). While this potential contains no cubic terms, it does have another serious phenomenological drawback. If we set to zero all the pNGB fields except for h, then the contribution from the O(λ 2 ) potential is proportional to cos(α). Because the gauge bosons also contribute to the same term, we would end up with the situation that A = 0 but B = 0 in Eq. (6.1). This appears to be incompatible with the requirement of having small h/f . A possible way out that we have discussed above is that, when turning on also the inert pNGBs η and ζ, this would reveal new minima of the potential.
An alternative is that only a smaller subset of the four-fermion couplings is actually induced by the EHC, and, as a result, the O(λ 2 ) potential vanishes. We rederived the potential in the case that only the two four-fermion couplings we considered in Ref. [23] are non-zero, finding two more terms that we overlooked in Ref. [23]. Like the other SU(5)/SO(5) models, this O(λ 4 ) potential will generically have the undesired cubic terms ∝ h 2 ϕ, so that, as explained above, further constraints must be satisfied in order to achieve a phenomenologically viable minimum.
In this paper we discussed the non-linear field Σ associated with the SU(4)/Sp(4) or SU(5)/SO(5) coset, and the field Φ that describes the pNGB of the non-anomalous U(1) A symmetry. We did not discuss the other non-linear field containing the colored pNGBs, which is associated with the SU(6)/SO(6) coset in the case of the SO(d) theories of Sec. III and Sec. IV, or with SU(3) × SU(3) ′ /SU(3) c in the case of the SU(4) model of Sec. V. While our results and conclusions are valid by themselves, a more complete analysis that includes the potential for the remaining non-linear effective field would allow for a more detailed study of the phenomenological consequences. The obvious additional constraint on the complete potential is that the colored pNGBs are not allowed to condense. where σ 4 = 1, and σ µ is equal to −iσ k for µ = k = 1, 2, 3, where σ k are the Pauli matrices. Also, γ 5 = diag(1, 1, −1, −1), and, as usual, P R = (1 + γ 5 )/2 and P L = (1 − γ 5 )/2. The charge conjugation matrix is then where ǫ = iσ 2 .

Appendix B: Discrete symmetries
Here we discuss the discrete symmetries C, P and CP in SO(d) gauge theories. We first recall the familiar case of an SU(N) gauge theory with Dirac fermions in the fundamental irrep. Charge-conjugation symmetry acts as Writing A µ = A µa T a we infer the transformation rule of the individual components, which is A µa → ∓A µa if T T a = ±T a . Because all SU(N) irreps may be constructed from tensor products of the fundamental irrep, these transformation rules remain valid for Dirac fermions in any irrep.
We take parity to act as where the presence of C in Eq. (B3) compensates for the fact that the SO(d) gauge field is invariant (note Eq. (A3)). In the case of a real irrep, the same rules (Eqs. (B1b)  We define CP by first applying P and then C. The resulting transformation rules are given in Sec. II B. The rules for the gauge field, and for the Dirac and Majorana fermions that we will encounter, follow from the transformation rules we have already discussed above.
In the case of the SU(4)/Sp(4) coset we have 4 Weyl fermions in the pseudoreal spinor irrep. The discrete symmetries can be approached in two ways. First, we may assemble the 4 Weyl fermions into 2 Dirac fermions. In this case, P acts in the usual way, while C acts as described above. However, the Dirac formulation has the disadvantage that it obscures the SU(4) flavor symmetry of the pseudoreal Weyl fermions. 39 The alternative we choose in this paper is to work in terms of the 4-component fields χ i and χ i introduced in Eqs. (2.1) and (2.2), also for the pseudoreal case. The advantage is that the flavor symmetry is manifest. The separate P and C transformations will look more complicated in terms of χ i and χ i , but, because of the properties of the four-fermion lagrangian (Sec. II B), we only need the explicit form of the combined CP transformation, which we can derive as follows. We start from the observation that the Weyl action is invariant under CP symmetry where the SO(d) gauge field transforms as described above, and In terms of the four-component fields χ i and χ i , the transformation (B5) takes the form of Eq. (2.5d) when the fermions belong to a pseudoreal irrep. For a real irrep, we recover Eq. (2.5c).
To avoid confusion, we recall that in the case of a real irrep, the action (B4) may be rewritten as S = 1 2 where the Majorana fermions are defined by Eqs. (2.1) and (2.2). But if we keep using the same 4-component fields for a pseudoreal irrep, then the right-hand side of Eq. (B6) will vanish identically. Of course, for both real and pseudoreal irreps we may recover Eq. (B4) from Eq. (B6) by inserting 2P L between / D and χ i . and 5 generators for the coset SU(4)/Sp(4), where τ i are the Pauli matrices, and 1 stands for the 2 × 2 identity matrix. These generators satisfy ǫ 0 T a = −T T a ǫ 0 , Sp(4) generators , +T T a ǫ 0 , SU(4)/Sp(4) generators . (C5) The tensor product of two fundamental SU(4) irreps contains the six-dimensional antisymmetric, and the ten-dimensional symmetric irreps. Under the reduction SU(4) → Sp(4), the 10 remains irreducible, whereas the 6 reduces to a 5 and a singlet. If A ij = −A ji transforms in the 6 of SU(4), the singlet is tr(ǫ 0 A), and the 5 is formed by A + 1 4 ǫ 0 tr(ǫ 0 A). The effective NGB field Π introduced in Eq. (3.8) transforms in the 5 of Sp(4).
Appendix E: V eff at second order for SU (5)/SO (5) In this appendix we list all the contributions to V eff , truncated to second order in the pNGB fields. We use the expansion of the coset field Σ given in App. D, and the expansion of the singlet NGB field Φ given in Eq. (3.43). The · notation is explained in Sec. III D, and the list of templates for V eff may be found in Eq. (4.8).