Classification of NLO operators for composite Higgs models

We provide a general classification of template operators, up to next-to-leading order, that appear in chiral perturbation theories based on the two flavour patterns of spontaneous symmetry breaking SU($N_F$)/Sp($N_F$) and SU($N_F$)/SO($N_F$). All possible explicit-breaking sources parametrised by spurions transforming in the fundamental and in the two-index representations of the flavour symmetry are included. While our general framework can be applied to any model of strong dynamics, we specialise to composite-Higgs models, where the main explicit breaking sources are a current mass, the gauging of flavour symmetries, and the Yukawa couplings (for the top). For the top, we consider both bilinear couplings and linear ones a la partial compositeness. Our templates provide a basis for lattice calculations in specific models. As a special example, we consider the SU(4)/Sp(4)$\cong$ SO(6)/SO(5) pattern which corresponds to the minimal fundamental composite-Higgs model. We further revisit issues related to the misalignment of the vacuum. In particular, we shed light on the physical properties of the singlet $\eta$, showing that it cannot develop a vacuum expectation value without explicit CP violation in the underlying theory.

The discovery of a Higgs-like boson [1,2] at the LHC experiments is one of the most remarkable scientific successes of the beginning of the century, as it concludes a 50-yearlong difficult quest [3]. While our knowledge of the properties of the new particle is increasing thanks to the extraordinary effort of the experimental collaborations [4][5][6], its true nature is still as elusive as ever. The lack of signals of new physics in other searches at the LHC (and other experiments) may be telling us that the Standard Model (SM) is the correct model after all, or it may be telling us that new physics may be either light and lurking in signatures that are difficult to access, or heavy and difficult to produce at the LHC. The latter possibility can be seen as an indirect support for theories where electroweak (EW) symmetry breaking is induced by a confining force at a few TeV scale.
The time-honoured idea of technicolor [7,8], in fact, predicts that new resonances besides the Nambu-Goldstone bosons (NGB) eaten by the massive EW gauge bosons should appear above a few TeV and be weakly coupled to the SM (thus difficult to produce).
While early proposals did not have a light scalar that could play the role of the 125 GeV Higgs, such a light scalar can be obtained either as an additional pseudo-NGB (pNGB) [9][10][11] or as a light resonance (whose lightness may derive from an approximate infrared conformal behaviour of the theory [12][13][14][15][16][17]). The idea of a pNGB Higgs has recently been revived via holographic realisations in extra dimensions [18], which share common traits to gauge-Higgs unification models [19][20][21].
While most of the recent progress has been based either on holography or on effective theories (see, for instance, Refs [22][23][24]), models that can be based on an underlying theory have a special role to play. On the one hand, they may truly be addressing the hierarchy problem as no scalars are present in the theory 1 . On the other hand, they can be studied on the lattice, thus providing quantitative predictions for the phenomenology of the Higgs boson. In addition, the symmetry-breaking pattern is linked to the properties of the representation of the underlying fermions [25,26]: only three cases exist, SU(N F )/Sp(N F ), SU(N F )/SO(N F ) and SU(N F ) × SU(N F )/SU(N F ) for pseudo-real, real and complex representations respectively. The minimal composite-Higgs model can be achieved for the first class with N F = 4 [27]. A simple underlying theory based on a gauged SU (2) has been proposed in Refs [28,29], and studied on the lattice [30][31][32][33][34][35][36] (preliminary results for an underlying Sp(4) theory can be found in Ref. [37]). Other theories widely studied on the lattice are the ones that feature a light CP-even scalar resonance [38,39] (where the lightness is defined by comparison to the other resonances, such as spin-1 ones). This state has been proposed as a candidate for the discovered Higgs-like boson [12,15,40], even though it is not clear if its couplings can really mimic the ones of the SM Higgs [41].
A next-to-leading order (NLO) chiral Lagrangian including the singlet has been presented in Ref. [42].
Motivated by the great progress on the lattice, in this work we focus on the construction of effective theories up to NLO, which include the effect of spurions that explicitly break the global symmetry of the theory. We limit our study to spurions in up to two-index representations of the global symmetry, and provide a complete list of template operators that can be used to construct the NLO counterterm operators once the nature of the spurions is specified. We then focus on spurions relevant for composite-Higgs models, namely a current mass for the underlying fermions, the gauging of the EW symmetry (embedded in the global symmetry), and the sources generating the Yukawa coupling for the top quark. The latter play an important role, as they usually are the most relevant spurions in the theory. There are two distinct ways to introduce such coupling: either via bilinear couplings to a scalar operator, or by linear couplings to fermionic operators. The former follows the old proposal of extended technicolor interactions [43], while the latter is based on the idea of partial compositeness [44] which was also realised in holographic models. In this work we will consider both: note that, in terms of an underlying theory, both appear as four-fermion interactions involving underlying fermions and elementary ones. Realising partial compositeness in an underlying theory often requires the presence of two distinct representations of the underlying gauge group, with chromodynamics (QCD) interactions sequestered by one and the job of EW symmetry breaking assigned to the other [45,46]. An NLO chiral Lagrangian for this situation has been constructed in Ref. [47], while preliminary lattice results for the specific model of Ref. [48] can be found in Refs [49][50][51]. The main role of the spurions for the phenomenology of the composite Higgs is to misalign the vacuum toward EW symmetry breaking.
Up to now, the global symmetry G F has been assumed to be only spontaneously broken by the condensation of the strong sector to a subgroup H F . All alignments of H F within the global symmetry are equivalent from the point of view of the confining force.
However, when explicit breaking sources external to the strong dynamics are present, one direction may be preferred. Furthermore, the sources may also break H F explicitly: the prime example is QCD where the current masses and the gauging of electromagnetism explicitly break SU(N F ) V down to U(1) EM , generating a mass for the pNGBs, i.e. the pions. In composite-Higgs models, the explicit breaking sources are crucial to misalign the vacuum with respect to the EW gauge sector and, therefore, to drive EW symmetry breaking and give mass to the Higgs (and additional pNGBs).
The misalignment between the EW preserving and physical vacua is conveniently parametrised by an angle, θ [11], and the physical vacuum, E θ , can be written as where E is an EW preserving vacuum, and U θ is a rotation matrix of G F connecting the two vacua. In the above equation, we have assumed that the underlying fermions are pseudo-real or real, in which case the vacuum is an antisymmetric or symmetric matrix.
The interpretation of the angle θ is simple, as it can be directly linked to the electroweak scale as sin θ = v/ f , f being the decay constant of the pNGBs. Thus, the limit θ 1 corresponds to a pNGB Higgs, while for θ = π/2 we have a technicolor model where v = f . The value of the angle θ (as well as the form of the EW preserving vacuum E) will be determined by the interplay between the spurions of the theory.
In general, the vacuum may be misaligned along more than one direction, and not just along the Higgs one. This can easily be implemented by rotating the vacuum E (or E θ ) with other rotations in G F parametrised by the appropriate (broken) generators. Loosely, the misalignment can be thought of as a vacuum expectation value for some of the pNGBs, even though this formalism does not respect the shift symmetry of the theory along the rotated vacuum and is thus dangerous.
The paper is organised as follow. In Sec. II, we present the chiral perturbation theory based on the two patterns of symmetry breaking: SU(N F )/SO(N F ) and SU(N F )/Sp(N F ).
We introduce generic spurions belonging to the fundamental and to the two-index representations of the flavour symmetry and classify, up to NLO, the non-derivative operators.
We then specialise to the three main explicit breaking sources in composite-Higgs models.
In Sec. III, we give a concrete example with the minimal fundamental composite-Higgs model based on SU(4)/Sp (4). We discuss the vacuum alignment when NLO contributions are included as well as the properties of the additional pNGB singlet, η. We finally present our conclusions in Sec. IV. More details about the classification of the relevant operators and a complete list of templates are given in the Appendices.

TIONS
The chiral perturbation theory that we introduce in this section is intended to parametrise the low-energy physics of some strongly coupled hypercolour (HC) gauge theories. We focus on the sector of the theory that is responsible for the breaking of the Note that underlying models with a different gauge group and fermionic representations may lead, at low energy, to the same chiral perturbation theory, i.e. to the same global symmetry-breaking pattern. Furthermore, the number of fermions N F is constrained by the fact that the unbroken global symmetry needs to contain the EW gauge symmetry of the SM extended to the full custodial symmetry, G EW = SU(2) L × U(1) Y ⊂ 2 We assume that the theory is vector-like with respect to the SM gauge quantum numbers, so that an EW preserving vacuum is allowed. 3 However, this class of models can no longer be considered as vector-like gauge theories [25,26]. SU(2) L × SU(2) R ⊂ H F , and a Higgs doublet candidate in the coset. Under these conditions, the minimal coset with an underlying fermionic origin is SU(4)/Sp(4) [27], which can be generated by a G HC = SU(2) gauge group with four Weyl fermions transforming as doublets [28,29]. The next-to-minimal cosets are G F /H F = SU(6)/Sp (6) [53], SU(5)/SO (5) [11,48] and SU(4) × SU(4)/SU(4) D [54] for the pseudo-real, real and complex cases respectively. We focus here on the real and pseudo-real cases as they can be described by a chiral Lagrangian of the same form, and they are associated with the smallest viable cosets. Moreover, the complex case is the one associated with QCD, and it has already been explored in great detail in the literature [55][56][57][58]. We leave the number of flavours, N F , free in order to remain as general as possible.
Besides the spontaneous breaking of G F due to the strong dynamics, the global symmetries are also explicitly broken by the interactions with the elementary states of the SM: the EW gauge interactions and the interactions giving rise to the top mass are the prime examples. Note also that, as the theories we study are vector-like with respect to the SM gauge interactions (and also non-chiral with respect to the HC interactions), a bare mass term for the fermions can (and should) always be added. The explicit breaking terms can be thought of as spurions that transform under both the global symmetry, G F , and the SM symmetries (both gauged and global). The fact that they are not dynamical fields explicitly breaks G F . They will play a crucial role for the alignment of the condensate with respect to the EW symmetries.
In the following, we first present the chiral Lagrangian associated with the real and pseudo-real cases [59,60] up to NLO in the chiral expansion. Then, we parametrise the effect of the explicit breaking interactions in the chiral perturbation theory through generic spurionic fields. Finally, we specialise to the explicit-breaking sources appearing in composite-Higgs models, namely: a current mass for the fundamental fermions, the gauging of G EW , and the linear or bilinear couplings between the elementary top quark and the strong sector.

A. Chiral Lagrangian up to NLO
In this section, we present the NLO chiral perturbation theory for the real and pseudoreal cases. Both give rise to an SU(N F ) global symmetry, N F ≥ 4, and they can, therefore, be described within a unified framework. We will parametrise the NGBs in terms of a linearly transforming matrix, Σ, which is symmetric under flavour indices of G F for the real case and antisymmetric for the pseudo-real one. We finally remind the reader that the chiral expansion is in terms of powers of the momentum p µ of the NGBs. At LO, i.e. order p 2 , the chiral Lagrangian reads: where f is related to the decay constant of the NGBs 4 , and the complex matrix χ (and the covariant derivative D µ ) contain scalar (and axial/vector) sources. Following Ref. [61], we introduce a normalisation factor c r (equal to √ 2 for real representations, and 1 for pseudoreal) so that the relation between f and the EW scale, v, is the same for all models. 5 The NGBs, GÂ, are parametrised by the matrix Σ as follows: where E is a matrix giving the orientation of the vacuum within G F , and XÂ are the corresponding broken generators. In the absence of explicit breaking of the global symmetry, all the vacua are equivalent. The broken XÂ and unbroken S A generators are defined by the following relations: and are normalised according to Tr[S A S B ] = 1/2 δ AB and Tr[XÂXB] = 1/2 δÂB.
The covariant derivative is defined as follows: where v µ and a µ are the vector and axial sources, respectively. It is convenient to define the field strength tensor j µν = ∂ µ j ν − ∂ ν j µ → gj µν g † .
Note that, apart from the NGB matrix, Σ, the other fields appearing in the Lagrangian are external sources that transform in complete representations of G F . They should not 4 By expanding the kinetic term in Eq. (2), one obtains the relation to the decay constant defined by:

Fields
Transformation  (4) and SU(5)/SO (5). The chiral counting is also given in the last column.
be confused with the spurions, that we will introduce in the next section, because they do not break the global symmetries of the strong dynamics. The transformation properties under G F of the NGB matrix and of the external sources, as well as their chiral counting, are summarised in Table I. The NLO chiral Lagrangian at order O(p 4 ) is given by [62]: where the coefficients L i and H i are low-energy constants (LEC) that only depend on the strong dynamics and can be computed on the lattice once the details of the underlying theory are specified. The above Lagrangian is expressed in a particular basis where we remove the redundant operators 6 in complete analogy with the Gasser and Leutwyler [58] 6 When the number of flavours, N F , is small, the Caley-Hamilton relations may be used to remove additional redundant operators. The equations of motion have also been used to remove two other operators.

Spurions Transformation
Convenient form shown, transforming as X → gXg † , as they allow one to easily construct the explicit operators.
The tensor product ( * ) allows one to define a two-index matrix out of a fundamental and an anti-fundamental, F * F † .
Lagrangian for the complex case.

B. Generic spurionic operators
The chiral Lagrangian can be completed by introducing explicit breaking terms of the flavour symmetry, G F : in the following, we will employ the spurion technique by defining non-dynamical spurions, Ξ, that transform as complete representations of G F . We will limit ourselves to the lowest-dimensional representations with up to two indices, so that the subscripts F, A, S and Adj indicate, in the following, the fundamental, antisymmetric, symmetric and adjoint representation, respectively. The spurions also carry quantum numbers related to the SM gauge and global symmetries. Being agnostic of their origin, we will overlook this in this section, together with their proper counting in the chiral expansion: we will, thus, classify the operators based on the number of spurions. We will then specialise to the quantum numbers and chiral counting for various models of composite Higgs in the next section. Note that, sometimes, it will be convenient to embed  the non-derivative ones as explained in App. C. We finally remark that the list of operators derived from the above templates may contain redundant operators, which need to be eliminated case by case if one wants to identify the minimal number of independent LECs in the model. As already mentioned, the chiral counting of each operator crucially depends on the physical origin of the spurions, and this will be discussed in the following section.

C. Explicit breaking sources in composite-Higgs models
Having at our disposal a complete basis of non-derivative operators involving up to four spurions (see Tab. III and App. C), we now specify the sources of explicit breaking that are relevant in the context of composite-Higgs models. We focus on the following possibilities: (i) A current mass for the underlying fermions ψ. In general, this spurion transforms in the same representation of the NGB matrix, Σ. The maximally symmetric case corresponds to a common mass with the flavour structure aligned to the EW preserving vacuum, E.
(ii) The gauging of the EW symmetry, G EW = SU(2) L × U(1) Y ⊂ G F . Note that, in general, additional gauging is allowed if the flavour symmetry, G F , is large enough: for instance, the SU(3) c of QCD may be included [63], or additional non-SM gauge symmetries. Examples of the latter are a U(1) symmetry broken on the EW-preserving vacuum, E, in the SU(4)/Sp(4) case [27], or duplicates of the SM gauge symmetries in little-Higgs models [53,64].
(iii) A SM-like bilinear coupling between the elementary top quark multiplets and the strong dynamics: Qt c couples to a scalar operator of the strong sector O Qt that has the same quantum numbers as the Higgs doublet in the SM. Note that the coset may allow for more than one doublet, so that multiple choices for O Qt within the NGB matrix are allowed.
(iv) Linear couplingsà la partial compositeness [44] between the elementary top quark multiplets and the strong dynamics: Q and t c couple separately to the fermionic operators O Q and O t , respectively.
A detailed list of all the relevant spurions can be found in Table IV. These sources of explicit breaking generate masses for the gauge bosons and SM fermions, as well as a potential for the NGBs and in particular for the Higgs boson. Four-fermion interactions among the SM fermions are also generated in the same formalism. The potential determines the alignment of the vacuum within the flavour symmetry, G F , thus allowing for a spontaneous breaking of the EW symmetry and for mass generation for some of the NGBs (that thus become pseudo-NGB, or pNGB in the following). The time-honoured result is that the top loops, associated with (iii) and (iv), have the correct sign to destabilise the Higgs potential, while the current mass and the EW gauging cannot break G EW alone.
The vacuum alignment in the presence of the above spurions and in the context of the minimal SU(4)/Sp(4) model will be discussed in detail in Sec. III C.
The underlying fundamental theory involving the hyper-fermions, ψ, dictates the form and the properties of the spurions. We start, therefore, from the fundamental interactions in order to derive the chiral counting of the spurions as well as their general properties.
These underlying properties imply that a large number of operators present in the general classification are not anymore allowed in these specific cases. In the following, we describe in detail the underlying properties of the composite-Higgs spurions. The complete NLO basis of non-derivative operators is reported in App. A and B where more details on its derivation are given.

Current mass
Let us start with the simplest source of explicit breaking, namely a current mass for the hyper-fermions. At the fundamental level, the relevant Lagrangian is given by   mass parameters proportional to other EW preserving directions in the vacuum E i , so that in general the mass term can be written as M = mE + i δm i E i . In chiral perturbation theory, the spurion associated with the mass transforms as the NGB matrix Σ (as it can be inferred from Eq. (8)), and it can be introduced as a vacuum expectation value for the scalar source, χ, that we introduced in Eqs (2) and (8). It is defined as follows with B 0 being a positive LEC.
From Tab. III, we derive the LO operator (with one spurion) involving χ. Note that this operator has the same form as the second term in Eq. (2) once we replace the scalar source with the spurion defined in Eq. (9). Expanding to second order in the Goldstone fields, we get for the pNGB mass M 2 G = 2B 0 m, and thus χ counts as O(p 2 ). The NLO operators involving the mass spurion, χ, can be derived in the same way starting from our general basis of operators. Due to the counting of χ, a great simplification appears at NLO: only the operators with two spurions need to be considered. The result is reported in App. A and is in agreement with Eq. (8) providing a first check of our procedure to derive all the non-derivative NLO operators starting from our template list.

Gauging of flavour symmetries
We now turn to the second obvious source of explicit breaking, i.e. the gauging of the EW symmetry, G EW ⊂ G F . At the fundamental level, the fermions are minimally coupled to the SM gauge bosons via a covariant derivative Note that the generators T A L and T Y are written as matrices in the G F space. However, they are not normalised as the G F generators in Eq. (5) but in order to reproduce the correct transformation properties of each of the components of ψ. For each gauged generator, thus, one can define a spurion transforming as the adjoint of G F , Ξ A Adj = gT A L and Ξ Y Adj = g T Y , that also transforms as the adjoint representations of the gauge groups. In the chiral expansion, they inherit the same counting as derivatives, i.e. O(p). It is also convenient to define a spurion that contains the gauge fields, i.e. Ξ µ Adj = gT A L W A µ + g T Y B µ , that can be introduced by replacing the vector and axial sources as j µ → Ξ µ Adj ; see Eq. (6). The LO and NLO operators containing EW gauge fields can easily be read off from Eqs (2) and (8).
The effect of the gauging also appears in non-derivative operators that can be built in terms of the spurions Ξ A Adj and Ξ Y Adj : technically, they should be thought of as counterterms necessary to regulate loops of gauge bosons. Thus, besides the counting of the chiral expansion, one needs to add loop suppression factors in order to correctly estimate the impact of such operators. The LO operators, containing two spurions and thus appearing at O(p 2 ) read where there appears a single LEC, C g , that depends on the HC dynamics. The two factors, where the Λ 2 HC ∼ (4π f ) 2 factor comes from the quadratic divergence of the loop. Thus, the loop suppression is compensated by the quadratic sensitivity to the cut-off of the effective theory, and the operators can be estimated to It is natural to expect that the contribution of the gauge bosons cannot break the gauge symmetry by misaligning the vacuum [66,67]; thus we can assume C g > 0. Following the chiral counting, NLO terms are generated with four spurions and are proportional to the gauge couplings to the fourth power. However, due to the smallness of the gauge couplings, one can realistically restrict to order g 2 and g 2 , as it is done in QCD [68,69].
Furthermore, we would like to remind the reader that the gauging of additional gauge interactions can be introduced in a similar way as done for the EW ones. A complete list of NLO non-derivative operators containing gauge spurions can be found in Tab. V in Finally, we would like to point out that the effect of the gauging of additional symmetries within G F , such as QCD or beyond-the-SM symmetries, can be included by adding appropriate terms to Eqs (11) and (13). No additional LECs are needed, as long as the masses of the additional gauge bosons are generated by the condensation itself.

Top couplings
The third source of explicit breaking relevant for composite-Higgs models that we consider is due to couplings between the elementary top quarks and the strong sector. Two main possibilities are available: couplings that are either bilinear or linear in the SM fields, with the latter realising the partial compositeness paradigm. Linear couplings, however, always need an extension of the underlying theory as, minimally, hyper-fermions charged under QCD are needed in order to generate QCD-coloured bound states. This can be done either by sequestering the QCD interactions to a sector containing a different HC representation [45,46], or by adding heavy-flavours in QCD-like theories [63]. In either case, the fermionic operators that couple linearly to tops are made of three hyper-fermions.
Another possibility to achieve partial compositeness is to add hypercoloured scalars, so that the linear couplings arise as renormalisable Yukawa couplings in the underlying theory; see Refs [70][71][72]. In all cases, the top partners always appear in a representation of G F with one or two indices, and thus we will restrict ourselves to these.

• Bilinear couplings
At the fundamental level, we assume that the top mass is generated by the following operators: where Λ t,i ≥ Λ HC are scales independent from the strong sector 7 , and α = 1, 2 stands for the index of an SU(2) L doublet. In the second equality, we assume that the scalar operators, O Qt,i , originating from the strong sector, are fermionic bilinears, thus leading to four-fermion operators. The projectors, P α i , select the SU(2) L -doublet components of ψ T ψ with hypercharge −1/2: in general there may be several possibilities, and for an explicit example with four independent couplings, we refer the reader to Ref. [54]. Note that one can write different types of operators where the spurion transforms as Σ, with O Qt,i = ψ †Pα i ψ * ; however the physical results are the same as the matrix Σ is always symmetric or antisymmetric.
The spurion encoding the explicit breaking is Ξ α, † A = i y t,i P α i , transforming as a doublet of SU(2) L with hypercharge −1/2, so that it always needs to appear in pairs in order to build gauge-invariant operators. Similar to what we did for the gauging, we define a single spurion including elementary fields that reads Ξ Qt, † A = i y t i P α i (Q α t c ) † . Then, the LO operators associated with the bilinear spurion are given by [27,29]: where the form of the second operator derives from the fact that it is generated by loops of elementary tops. Only two LECs are needed: one relative to the operators generating a mass for the top (the former), and one for the NGB potential (the latter). Note that 7 The scales Λ t,i need to be, at least, larger than the cut-off of the effective theory, because they correspond to additional interactions that may affect the low-energy properties of the strong dynamics. For an explicit example, see Ref. [73].
the dependence on the scales Λ t,i , which may contain a large anomalous exponent if the theory is conformal above Λ HC , can be embedded in a redefinition of the couplings y t,i without loss of information. In some cases the presence of many possible alignments of the doublet within the NGB matrix is superfluous as a transformation of G F may be used to change basis (i.e. reshuffling the hyper-fermions ψ) and write a smaller number of couplings without affecting other spurions. Assuming that the top, m t , and Higgs (pNGB) masses are naturally of the same origin, we can impose the following counting for the spurions: y t P α (Q α t c ) † ∼ O(p 2 ) and y t P α ∼ O(p). Note that the two spurions do not have the same counting contrary to the gauge ones. Similar to the gauge boson loops, the loop factor for massless tops can be approximated by Eq. (12), where colour and other factors are embedded in the LEC, C t .
At NLO, the O(p 4 ) Lagrangian contains five new operators contributing to the potential (for simplicity we omit the sums, so that i y t,i P α i → y t P α ): where the first three operators are self-hermitian. Three additional operators contain one insertion of the hyper-fermion mass spurion: Other operators involving gauge couplings are also present, and listed in Table VII in • Linear couplingsà la partial compositeness Let us now consider the second way of giving mass to the top quark by means of linear couplings of the elementary top fields to fermionic operators of the strong dynamics (partial compositeness), where the sums span over all the possible operators and, as for the bilinear case, the interactions are generated at scales Λ t,i ≥ Λ HC . We will assume that the operators are made of three underlying fermions, as it happens in all explicit examples [45,46,48,63]; the linear couplings will thus correspond to four-fermion operators. 8 As previously mentioned, the operators need to contain at least one hyper-fermion that carries QCD colour, which we denote as X, and which corresponds to a different HC representation or to heavy flavours. As a consequence, either one or two ψ's are allowed: the former case corresponds to the fundamental of G F , while the latter corresponds to two-index representations. The fundamental can also be obtained in models with scalars [70,71].
Spelling out the various cases, the linear couplings can thus be rewritten as follows: Note that with the preceding definitions the spurions have the same transformation properties as the left-handed composite operators and of the left-handed SM quark fields.
We recall that for each operator representation under G F , there may be several possibilities to embed the top partners, and thus an index i should be intended in the preceding expressions. Furthermore, for the adjoint case, the right-handed top, t c , may be associated with the singlet of G F . Finally, the case of models with scalars, S, charged under HC can be recovered by replacing XX → Λ 2 t S in the case of the fundamental. The projectors P α Q and P t select the components of the bound state that have the same quantum numbers as the elementary SM tops. In the following, for simplicity, we will assume that the new physics generating the four-fermion interactions will only generate mixing to a single representation of G F , or equivalently that the top mass is dominantly generated by a single operator. A more general case has been discussed at LO in Refs [74,75], and it leads to the presence of a plethora of operators. 8 There is also the possibility of a hyper-fermion/hyper-gluon bound state. However this is unlikely because it would require the hyper-fermion to be in the adjoint representation of HC, thus making the theory lose asymptotic freedom.
The couplings of the underlying theory in Eq. (19) generate, in the confined phase, linear mixing of the elementary tops to fermionic resonances (i.e. top partners). On top of this, effective operators are generated in terms of the spurions defined above: in the following we will assume that the leading contribution to the top mass is generated by the operators. This assumption is valid as long as the top partners are heavier than the NGB decay constant, f , and thus cannot be included as light states in the low energy chiral Lagrangian.
The LO operators contributing to the top mass, for all the choices of spurion representations, are given by the following expressions: plus hermitian conjugate. The factor of 1/4π derives from applying naive dimensional analysis (NDA) as explained in Refs [76,77]. Note that, as expected, the above operators involve both spurions y t L and y t R in order to generate the top mass, and that only case A involves two independent operators. The case of the right-handed top mixing to the singlet can be used only if the left-handed tops are in the antisymmetric representation (as that is the only case with an operator containing a single spurion; see Table III), and we do not consider it in the following because of non-minimality.
Similarly, we can construct the operators contributing to the potential for the NGBs.
At leading order, there exist operators involving only two spurions only for the case of the antisymmetric and adjoint representations, where a factor of 1/4π comes from NDA. The only consistent chiral counting that allows for these operators to appear at LO, O(p 2 ), is that the Yukawa couplings y t L/R count as p. Note that this chiral counting is consistent with the appearance of the NDA factor in Eq. (20), as the top mass operator would appear at chiral order O(p 3 ).
For the spurions in the symmetric and fundamental representations, the leading oper-ators contain at least four spurions, leading to the following expressions for the symmetric S, and for the fundamental F. 9 At NLO, many more operators are generated, as listed in App. A. For reasons of space, we will limit ourselves here to the operators generated in the case of the symmetric representation. For the potential, mixed operators involving two Yukawas with the mass spurion or the gauge couplings arise at the same level as the leading pure Yukawa ones listed above. There exist only one operator with a mass insertion, and four involving gauge couplings where we have left implicit all the possible combinations of Yukawas and gauge couplings.

III. MINIMAL SU(4)/Sp(4) MODEL
In this section, we apply the machinery developed in the previous section to the coset SU(4)/Sp(4). This is the minimal composite-Higgs framework with underlying fourdimensional fermionic realisations [27]. Models based on this coset have been studied from an effective point of view in Refs [78][79][80], and the coset has also been used to construct minimal technicolor models in Refs [27-29, 45, 46].
The most minimal underlying fermionic model is based on a confining SU(2) gauge group with four Weyl fermions transforming under the fundamental representation of the new gauge group [28,29]. Since the fundamental representation of SU(2) ∼ Sp (2) is pseudo-real, the fermion sector has an enhanced global symmetry, SU(4). The condensate forming due to the new strong dynamics then breaks this global symmetry spontaneously to Sp(4), as confirmed from lattice simulations [30,31]. The spectrum of this theory has also been extensively studied on the lattice [33][34][35][36]. Preliminary lattice studies based on a HC Sp(4) 10 have also been recently published [37].
In the following, we will revisit the operator analysis that we detailed in the previous section focusing in particular on the potential generated for the NGBs of the model.

A. Electroweak embedding
The full custodial symmetry of the SM, SU(2) L × SU(2) R , is embedded in SU(4) by identifying the left and right chiral generators to be where σ i are the Pauli matrices. The generator of the hypercharge is then further identified with the diagonal generator of the SU(2) R group, Y = T 3 R . As discussed in Ref. [29], there are two inequivalent real vacua that leave the SM chiral group invariant, E ± , and we denote the one breaking the EW subgroup completely to the electromagnetic U(1) Q by E B . They can be explicitly written as where we chose the normalisation to be real.
In general the vacuum can be written as the superposition of the EW preserving and breaking ones, and the physical properties of the NGBs generically do not depend on the choice of the EW preserving vacuum E ± . We will see later in this section that, in some cases, the choice of the EW-preserving vacuum is related to some properties of the spurions. Following Refs [27,29], in this paper we use E − and parameterise the vacuum where the angle θ describes the misalignment of the unbroken Sp(4) with respect to the EW embedding and is generated by an SU(4) rotation U θ associated with the generator of the Higgs.
The (non-linearly-realised) scalar variable describing the dynamics of the NGBs associated with the above breaking pattern and the vacuum E θ can then be written, in the unitary gauge, as a matrix [27]: The matrix Σ transforms linearly under the flavour symmetry SU(4).
The matrix U (transforming non-linearly under SU(4)) contains the NGBs along the vacuum E θ , and the matrices Yˆ4 ,5 are two of the broken generators associated with the Higgs and additional singlet, η (while the remaining three generators are associated with the exact NGBs eaten by the W and Z bosons). Note that the normalisation we chose for the decay constant, f , is different from the one adopted in Refs [27][28][29] by a factor of 2 √ 2 as we follow the prescription defined in Eq.

B. Explicit form of the SU(4) spurions
We can now explicitly write the relevant spurions introduced in Sec. II C in the case of the coset SU(4)/Sp(4). Let us start by the current mass: this spurion does not explicitly break the SM gauge symmetry; thus it needs to be proportional to the EW preserving vacua, where we define m = (m 1 + m 2 )/2 and δm = (m 1 − m 2 )/2. In order for the EW preserving vacuum to be aligned with E − , we need to impose δm m because it is the potential generated by the mass term that will fix the preferred alignment of the vacuum. Note The spurions corresponding to the EW gauging including the elementary fields can be written as Ξ µ EW = gT A L W A µ + g T 3 R B µ with the explicit forms already given in Eq. (26). For the top bilinear spurions, transforming as A † , we have Ξ Qt = y t P α (Q α t c ) † , and there is a unique choice for the projectors P 1,2 given by [27,29] The uniqueness is due to the presence of a single (bi)doublet among the NGBs.
In the case of partial compositeness, we can write the spurions as Ξ Q = i y t i L P α Q i Q α and Ξ t = i y t i R P t i t c : the two sets of projectors, P α Q i and P t i , thus select the components of the fermionic operator of the strong dynamics that match the quantum numbers of the left-handed doublet and the right-handed singlet, respectively. We recall that an additional U(1) X charge needs to be included in order to fix the hypercharge of the top partners, so that the SM hypercharge is defined as Y = T 3 R + X. For the fundamental representation (that has X F = 1/6 11 ) there is only one choice available as clearly seen from the decomposition of the SU(4) representation under SU(2) L ×SU(2) R , i.e. 4 → (2, 1)⊕(1,2), and the projectors P 1,2 Q and P t are given by We recall that in the above case, t c belongs to an SU(2) R anti-doublet, and that the partialcompositeness couplings will violate the extended custodial symmetry needed to protect the Z coupling to left-handed bottom quarks [81]. For the antisymmetric (X A = 2/3) the decomposition reads 6 → (2,2) ⊕ (1, 1) ⊕ (1, 1), and thus there is a single choice for the doublet, but two for the singlet: Note that P t 2 − P t 1 is aligned with the vacuum E − , so it corresponds to a singlet of Sp (4) along the EW preserving vacuum, while P t 1 + P t 2 is part of a 5-plet together with the doublet. We want to stress that this assignment is relative to the choice of vacuum, as, for instance, P t 1 + P t 2 corresponds to the singlet for the E + vacuum. In general, the righthanded top will couple to a linear combination of the two spurions, i.e. with a generalised The relative phase of the two coefficients, however, can be rotated away by the use of an SU(4) transformation along the generator Xˆ5 associated with the singlet in the EW preserving vacuum E − . This corresponds to a relative phase redefinition of the two hyper-fermion doublets: therefore, only if a mass term is present can this phase have physical effects, as we will see in a later section. Noteworthy, the real parts cannot be removed without affecting the gauge spurions.
For the symmetric (X S = 2/3), the decomposition reads 10 → (2,2) + (3, 1) + (1, 3): for both doublet and singlet there is a single choice, with the singlet associated with the neutral component of the SU(2) R triplet. The projectors are similar to the P α Q and P t 1 of the antisymmetric by replacing −1 → 1.

Finally the adjoint (X Adj
and thus there are two options for both left-and right-handed tops: Note that in terms of Sp(4), the adjoint decomposes into one symmetric and one antisymmetric: we find that P Q 1 + P Q 2 and P t 1 project states in the symmetric (with t c in an SU(2) R triplet), while P Q 2 − P Q 1 and P t 2 in the antisymmetric. For both left-and right-handed tops, the projector is a superposition of the two: For the doublet combination P Q the relative phase of the two coefficients can be removed by the same SU(4) rotation (along X 5 ). For the right-handed top, the two coefficients are always physical as they mix a singlet and a triplet of SU(2) R . Note also that along the other EW-preserving vacuum E + , the role of the two combinations of doublet embeddings, P Q 1 ± P Q 2 , are reversed.

C. Vacuum alignment
We study the vacuum alignment induced by the breaking terms that have been discussed previously. The purpose is to isolate cases where the misalignment angle, θ, is sufficiently small, but non-zero, to comply with composite-Higgs models. The most general form (up to NLO) of the potential can be inferred from the tables in App. A and takes the following form: We use here and in the following the short-hand notations s The challenge in composite-Higgs models is to generate a small misalignment (θ 1) in order to have a small hierarchy between the EW and the compositeness scale, v f . To depart from the EW preserving vacuum (θ = 0) and from the technicolor limit (θ = π/2), one needs to consider several explicit breaking sources at the same time. To this end, let us focus on two simplified scenarios: (i) A potential generated only by the gauge and top explicit breaking interactions such that the current masses are set to zero, and we have c 3 = c 4 = 0. In that case, the breaking of the EW symmetry is driven by the coefficient c 2 , and one needs to include the NLO contributions to the potential. This scenario is commonly used in composite models with partially composite tops based on holography [22,82].
(ii) A potential generated by gauge and top spurions as well as a non-zero current mass.
In this case, it is enough to restrict to the LO contributions, and we thus assume This scenario is well known, and we refer to Ref. [27] for details. Here we just briefly outline this scenario for comparison.
In case (i), the minimisation of the potential in Eq. (37) leads to: Setting aside the limit where the EW symmetry remains unbroken (θ = 0) as well as the technicolor limit (θ = π/2), the third extremum corresponds to s 2 . This extremum is the global minimum of the potential only if c 2 > 0 and c 1 < 0 (as expected, see Ref. [27]). Moreover, a small misalignment angle requires |c 1 | |c 2 |. As we will see, this requirement can be obtained in several ways depending on the top coupling representation.
For the case (ii), the minimisation of the potential leads to Focusing again on the EW breaking vacuum alignment (θ 1), the potential is extremised Moreover, for the extremum to be the global minimum, one needs |c 3 | |2c 1 |, where c 1,3 < 0, or c 1 < 0 and c 3 > 0.
Let us now explore in details how the scenario (i) could be realised when NLO contributions are taken into account. In practice this requires obtaining |c 1 | |c 2 | in a natural way.
• Hierarchy between the LECs (|C t /C t | 1) This case relies on the usual hypothesis that the top loops are the dominant contributions to the coefficients c 1 and c 2 and, for some reason, the strong dynamics leads to |c 1 /c 2 | 1. In other words, the LECs associated with the operators generating c 1 need to be suppressed.
For simplicity, one can neglect the gauging of the SM as its effect is negligible in comparison to the top quark contributions. Moreover, let us consider a bilinear coupling as an example.
The potential takes the following form: where the positive coefficients C t and C t are functions of the different LECs associated with the operators in Tab. VII. Note that the discussion can also be applied to all the linear couplings as they also generate the coefficients C t and C t (for reference to the vast literature on this topic we refer the reader to the reviews in Refs [22][23][24]). To get a small misalignment requires |C t /y 2 t C t | 1; i.e. some cancellation should happen at LO making that contribution comparable to if not smaller than the NLO one. In models inspired by holography this is achieved by assuming that the main contribution to the LECs comes from top and top partner loops and that other UV effects are negligible [83]. We remark, however, that this is a very specific assumption, and not all models (especially with an underlying gauge-fermion theory) will respect it.
• Linear coupling in the symmetric representation (y t L y t R ) Choosing a symmetric representation for the left-and right-handed top couplings, one finds that the LO contributions generate c 1 and c 2 at the same order in the chiral expansion.
This is due to the fact that the Goldstone matrix is antisymmetric (pseudo-real case) such that the LO operators involve four top spurions (see Tab. X).
For simplicity, let us first consider operators of the general form Tr The corresponding potential is given by the operators in Eq. (22) such that c 1 = C tS,1 f 4 (y 2 t R y 2 t L −2y 4 t R )/(16π 2 ) and c 2 = C tS,1 f 4 (y 4 t L +y 4 t R −y 2 t R y 2 t L )/(16π 2 ). Achieving c 1 < 0 and c 2 > 0 is fairly easy as long as y t L √ 2y t R . Note that a small misalignment angle is achieved by tuning the value of y t L close to the upper bound. Using the constraint on the top mass coming from Eq. (20), we can express the Higgs mass as a function of the two relevant LECs and the misalignment angle as follows: while the singlet remains massless [79] (a mass can easily be generated by adding current masses). We see that a small enhancement in C yS , or an order 1/10 suppression in C tS,1 , is sufficient to achieve the measured value of the Higgs mass. The second type of operators that follow the template Tr[Ξ S Σ † Ξ S Ξ † S ΣΞ † S ] provide an additional term in the potential proportional to s 2 θ , C tS,2 f 4 (4π) 2 which adds up to c 1 and might help relieve the tension in the alignment and Higgs mass if C tS,2 < 0.
For completeness, we also report the expression for the top mass and linear couplings to the NGBs, where we remark the presence of a coupling of the pseudo-scalar singlet η to tops.

D. Masses and couplings of the pNGBs
The general potential presented in Eq. (37) can be further expanded to obtain the masses for the pNGBs (Higgs and η) as well as the couplings among them. We find that, if all the coefficients and couplings are real, the four terms correspond to universal functions of the fields: The The trilinear couplings in Eqs (50) and (51)  half the Higgs mass, this coupling will contribute to non-standard decays of the Higgs, as shown in the left panel of Figure 2. Bounds on this branching ratio are obtained from global fits of the Higgs properties, independently on the decay modes of the singlet: the current bound from the Higgs data combination after Run-I is at 34% [86], and thus is unable to probe the parameter space, while projections for the high luminosity phase with a data set of 3 ab −1 estimate the reach to 10% [87]. We remark that dedicated searches for h → ηη may give stronger bounds, but depend on the final states the singlets decay into.
• ηtt coupling: A coupling of the singlet η to tops may be generated from the same operator that generates the top mass, as we have seen in Eq. (44). This coupling is phenomenologically very important as it opens new decay modes for the singlet, besides the di-boson final states from the Wess-Zumino-Witten anomaly [27,29], and induces gluon fusion at one loop thus enhancing its production at hadron colliders [84]. The ηtt coupling is not present at LO for bilinear top couplings nor in the case of linear coupling with fundamental top-partner representation. However, it appears at NLO in mixed operators involving mass and top spurions. For instance, for the bilinear case we have [29] (52) and similarly for the other mixed operators. Note that the coupling is proportional to the Sp(4) violating current mass. Interestingly, couplings that do not need such violation are generated by higher-order operators containing all types of spurions [84].
The situation is different for linear couplings to two-index representations. For the symmetric, we already found in Eq.
where we see that only the component of the spurion aligned with the Sp(4) singlet (along the vacuum E − ) contributes. From the same operator, we derive the couplings, where we see that the coupling of the singlet is only generated by the component of the right-handed top aligned with the singlet inside the Sp(4) 5-plet. Via the same mechanism, couplings of the single to a top partner and a top are also generated [88]. A similar situation occurs for the adjoint, where for both doublet and singlet two possible embeddings are allowed. Using the spurions in Eq. (36), the top mass given by the third line in Eq. (20) reads: The above result can easily be interpreted: when the right-handed top is aligned with the singlet of SU(2) R (i.e. in the 5-plet of Sp(4) of E − ) the doublet is projected on the 10-plet, while when the right-handed top is in the SU(2) R triplet (which is part of the 10-plet) the doublet is projected on the 5-plet. The couplings acquire the form: where we see that a coupling to the singlet η is not generated only when the left-and right-handed tops are in different Sp(4) representations.
• Vacuum expectation value of η, CP-violation, and the choice of vacuum: So far we have only considered a vacuum misaligned along the direction of the Higgs.
However, in general, we should also consider a misalignment along the direction of the singlet η. This can be done by rotating the vacuum with an SU(4) transformation along Xˆ5: where α is related to the vacuum expectation value of the singlet. Remarkably, it appears as a phase, in accordance to the fact that η is a pseudo-scalar. This corresponds to a change in the relative phase of the two hyper-fermion doublets, and it will affect the phase associated with the current mass, if present. Thus, the presence of the phase in the vacuum is correlated to a phase in the current mass. One can always make the simplifying assumption of real masses and thus start with a real vacuum. As a consistency check, one can verify that a tadpole for η is generated by the current mass spurion if a phase is present: Once other spurions are included, it is always the phase of the current mass term that generates a tadpole for η: this is clearly seen as the gauge couplings are real, while the two Yukawas y t L and y tR can be made real 13 by choosing the phase of the elementary quark fields.
The situation is different in cases, such as partial compositeness with tops in the antisymmetric or adjoint representations, where more than one embedding is possible for the same SM elementary field: physical phases may remain as not all couplings can be made real by a phase shift of the fermion fields. We will first consider in detail the case of the antisymmetric. As before, we parametrise the spurion for the right-handed top following Eq. (34), allowing for a phase between the two coefficients. The potential generated by the LO operator in Eq. (21) gives, up to linear terms in the fields, From the equation above, we clearly see that a tadpole for the singlet is present only if a relative phase between the two coefficients A and B is present. As already commented above, such a phase can be removed by the SU(4) rotation U α , and in the absence of a current mass one can use this to remove it from the Lagrangian. In other words, the vacuum expectation value of the singlet η is not physical as it is associated with an arbitrary phase that can be removed from the theory (this point was missed in the discussion in Ref. [79]). The only situation where a tadpole for η could be physical is when both a current mass and a phase in the right-handed top spurion are present. As a misalignment of the vacuum along the singlet would imply the presence of a CP violating phase in the vacuum, this result shows that the only way to achieve this is to add a CP-violating phase in the underlying theory. Thus, no spontaneous CP-violation via the vacuum misalignment, or pNGB vacuum expectation value, is possible. We checked that the same conclusion can be drawn for the adjoint representation: the tadpole reads thus it is again proportional to the only phase that can be removed by U α . A mass mixing is also present and proportional to the same phase. Another related point is the presence of a mixing between the Higgs boson, h, and the singlet, η, in the potential: we checked that the mixing is also proportional to the same phase generating the tadpole. This mixing, which is only physical in theories with explicit CP-violation, has been used in Ref. [89] to reduce the fine-tuning in the Higgs mass.
Another case where a misalignment along the singlet direction is needed is when the potential generates a negative mass squared for η in the absence of a tadpole. This situation can occur for real coefficients; however, it is a diagnostics that the initial choice of the EW preserving vacuum is not correct. As an example, we reconsider the case of partial compositeness with the antisymmetric representation. As mentioned in Ref. [88], if the right-handed top is mostly aligned with the Sp(4) 5-plet, i.e. B + A > B − A in our notation, the singlet may develop a vacuum expectation value via a negative squared mass (for real coefficients). However, this situation can be inverted by changing the EW preserving vacuum from E − to E + by use of a U α transformation with α = π/2 (plus an overall phase shift). This shows that the vacuum expectation value of the singlet (that generates α) is unphysical in this case too, as it corresponds to an inappropriate choice of the vacuum.
It is well known from QCD [57,90] that CP violation can also occur spontaneously Foundation, grant number DNRF90.

Appendix A: Spurionic operators involved in the composite-Higgs potential
In this appendix, we classify all the operators, up to NLO, that contribute to the NGB potential at tree level. 14  (iii) Some operators contain traces made only with spurionic fields (no NGB matrix, Σ).
We will neglect in general these subleading effects as we focus in this analysis on the general form of the couplings. A simple example is provided by the two following operators associated with the bilinear spurion: where Λ 2 HC /(16π 2 ) f 2 . Thus, despite the large number of operators present in the general classification of App. C, only a smaller set that depends on the spurions 14 Sticking to the spirit of our analysis, we do not consider operators containing elementary SM fields that may contribute to the NGB potential at one-loop level. 15 For simplicity, we assume that no explicit CP-violation is present in the underlying theory (see Sec. III D), i.e. that all couplings are real. under consideration is relevant in practice. On the other hand, several spurions may transform in the same representation of flavour increasing the number of operators compared to the ones listed in App. C.

Current mass and gauge spurions
Let us start with the operators containing only the mass and gauge spurions. These operators are already well known in QCD and can then be used to check the completeness of our classification.

• Current mass
We first consider the operators containing only the mass spurion, Ξ A = χ. The latter enters in the chiral expansion at O(p 2 ), resulting in the three following classes of operators up to NLO: (iii) One spurion Ξ A or Ξ † A and two derivatives (class χD 2 ).
The corresponding operators are displayed in Tabs. V and VI. Note that the derivative operators (see App. C) as well as the contact terms (with traces made of spurions only) have been included in order to check the completeness of our basis with Eq. (8).

• Gauge spurion
We now include both the mass and gauge spurions, Ξ A Adj = gT A L and Ξ Adj = g T Y . The latter enter in the chiral expansion at order O(p) and the resulting four classes (including the mixed operators involving both mass and gauge spurions) of operators correspond to: (i) Two spurions Ξ 2 Adj (classes g 2 and g 2 ).
−2B 0 g 2 m 2 cos θ + . . . As already mentioned, in order to check the consistency of our classification, the derivative operators as well as contact terms (see Refs [68,69]) have also been included. All of the operators corresponding to the above classes are reported in Tabs. V and VI, while those associated with the gauging of U(1) Y can be obtained from g → g and T A L → T Y . Furthermore, the expansion of the operator gives the same result as for the SU(2) spurion but with an additional factor of 1/3, except when explicitly listed in the tables.

General form
Operator

Top quark spurions
We now discuss the spurions generating the top mass: in the following, we consider a bilinear coupling as well as linear couplingsà la partial compositeness. For the linear top coupling cases with antisymmetric and adjoint spurions there are more than one possible spurion embedding, and we use the general linear combinations defined in Eqs (34) and (36).

• Bilinear coupling
The four classes of non-derivative operators involving the top bilinear spurion Ξ α, † A = y t P α correspond to: (i) Two top spurions (Ξ A Ξ † A ) (class y 2 t ).
where the two last classes involved mixed operators with two different spurions.

TABLE VII. Non-derivative operators up to NLO involving the top bilinear spurion Ξ α, †
A = y t P α and contributing to the scalar potential. Also shown are the mixed operators involving the top bilinear spurion and the gauge or the mass spurion. When not explicitly written, the U(1) Y contributions are obtained by the following replacements g → g and T A L → T Y in the third column and similarly in the last column with a factor of three less. When several orderings of the spurions lead to different operators, only one is shown for each general form of operators as the others can easily be inferred from the table.
• Linear coupling in the fundamental representation The three classes of operators involving the linear spurions in the fundamental representation Ξ α F = y t L P α q and Ξ F = y t R P t correspond to: (i) Four top spurions (Ξ F Ξ † F ) 2 (classes y 2 t L y 2 t R and y 4 t L,R ).
The operators, belonging to the three above classes are listed in Tab. VIII.

TABLE VIII. Same as in Tab. VII but for linear spurions in the fundamental representation, namely
Ξ α F = y t L P α Q and Ξ F = y t R P t .

• Linear coupling in the adjoint representation
The four classes of operators involving the linear spurions in the adjoint representation Ξ α Adj = y t L P α q and Ξ Adj = y t R P t correspond to: (i) Two top spurions Ξ 2 Adj (classes y 2 t L,R ).
(ii) Four top spurions Ξ 4 Adj (classes y 2 t L y 2 t R and y 4 t L,R ).
The operators, belonging to the above classes are listed in Tab. IX.

• Linear coupling in the symmetric representation
The three classes of operators involving the linear spurions in the symmetric representation Ξ α S = y t L P α q and Ξ S = y t R P t correspond to: (i) Four top spurions (Ξ S Ξ † S ) 2 (classes y 2 t L y 2 t R and y 4 t L,R ).
The operators, belonging to the three above classes are listed in Tab. X.

• Linear coupling in the antisymmetric representation
Finally, the four classes of operators involving the linear spurions in the antisymmetric representation Ξ α A = y t L P α q and Ξ A = y t R P t correspond to: (i) Two top spurions (Ξ A Ξ † A ) (classes y 2 t L,R ).
(ii) Four top spurions (Ξ A Ξ † A ) 2 (classes y 2 t L y 2 t R and y 4 t L,R ).
The operators, belonging to the above classes are listed in Tab. XI.

TABLE XI. Same as in Tab. VII but for linear spurions in the antisymmetric representation, namely
Ξ α A = y t L P α Q and Ξ A = y t R P t .

Appendix B: Spurionic operators generating the top-quark mass
In this appendix we list the operators up to NLO that contribute to the top mass at tree level. We consider a bilinear top coupling as well as linear couplings in the fundamental, adjoint, symmetric, or antisymmetric representations. These operators also generate the top quark couplings to the pNGBs and in particular, the ηtt coupling as discussed in Sec. III D for the SU(4)/Sp (4) , We do not include these kinds of operators in our analysis.
(ii) In the same way, four-fermion operators are in general generated. Using the same generic operator as before, we obtain in the bilinear case the following operator: Again, we do not include these kinds of operators in our analysis.
The number of operators is again drastically reduced compared to those present in the generic classification of App. C.

• Bilinear coupling
For a bilinear top spurion, we get four different classes of operators that contribute at tree level to the top mass: (i) Only one top spurion Ξ A (class y t ).
(iii) One top spurion Ξ A and one mass spurion Ξ A or Ξ † A (class y t χ).
(iv) One top spurion Ξ A and two gauge spurions Ξ 2 Adj (classes y t g 2 , y t g 2 ).
The operators belonging to the above classes are displayed in Tab. XII.

Class
General form Operator and possibly Ξ α, † A = y t P α and contributing to the tree-level top mass. Also shown are the mixed operators involving the top bilinear spurion and the gauge spurions or the mass spurion. The U(1) Y contributions are obtained by the following replacements g → g and T A L → T Y in the third column.

• Linear coupling in the fundamental representation
For a linear top coupling transforming in the fundamental representation, the operators contributing at tree level to the top mass organise as follows: (i) Two top spurions Ξ 2 F (class y t L y t R ).
(ii) Four top spurions Ξ 2 F (Ξ F Ξ † F ) (classes y 3 t L y t R , y t L y 3 t R ).
(iii) Two top spurions Ξ 2 F and one mass spurion Ξ A or Ξ † A (class y t L y t R χ).
(iv) Two top spurions Ξ 2 F and two gauge spurions Ξ 2 Adj (classes y t L y t R g 2 , y t L y t R g 2 ).
They are listed in Tab. XIII.

Class
General form Operator y t L y t R g 2 , y t L y t R g 2 TABLE XIII. Same as in Tab. XII but for the linear spurions in the fundamental representation, namely Ξ F = y t L P α Q Q † α and Ξ F = y t R P t t c † and possibly the spurions Ξ α F = y t L P α Q and/or Ξ F = y t R P t .

• Linear coupling in the adjoint representation
For a linear top coupling transforming in the adjoint representation, the operators contributing at tree level to the top mass organise as follows: (i) Two top spurions Ξ 2 Adj (class y t L y t R ).
(ii) Four top spurions Ξ 4 Adj (classes y 3 t L y t R , y t L y 3 t R ).
(iii) Two top spurions Ξ 2 Adj and one mass spurion Ξ A or Ξ † A (class y t L y t R χ).
(iv) Two top spurions Ξ 2 Adj and two gauge spurions Ξ 2 Adj (classes y t L y t R g 2 , y t L y t R g 2 ).
They are listed in Tab. XIV.

Class
General form Operator TABLE XIV. Same as in Tab. XII but for the linear spurions in the adjoint representation, namely Ξ Adj = y t L P α Q Q † α and Ξ Adj = y t R P t t c † and possibly the spurions Ξ α Adj = y t L P α Q and/or Ξ Adj = y t R P t .

• Linear coupling in the symmetric representation
For a linear top coupling transforming in the symmetric representation, the operators contributing at tree level to the top mass organise as follows: (i) Two top spurions Ξ 2 S (class y t L y t R ).
(ii) Four top spurions Ξ 2 S (Ξ S Ξ † S ) (classes y 3 t L y t R , y t L y 3 t R ).
(iii) Two top spurions Ξ 2 S and one mass spurion Ξ A or Ξ † A (class y t L y t R χ).
(iv) Two top spurions Ξ 2 S and two gauge spurions Ξ 2 Adj (classes y t L y t R g 2 , y t L y t R g 2 ).
They are listed in Tab. XV.

Class
General form Operator y t L y t R g 2 , y t L y t R g 2 Tr[Ξ S Σ † Ξ Adj ] 2 + h.c. y t L y t R g 2 Tr[P α TABLE XV. Same as in Tab. XII but for the linear spurions in the symmetric representation, namely Ξ S = y t L P α Q Q † α and Ξ S = y t R P t t c † and possibly the spurions Ξ α S = y t L P α Q and/or Ξ S = y t R P t .

• Linear coupling in the antisymmetric representation
Finally, for a linear top coupling transforming in the antisymmetric representation, the operators contributing at tree level to the top mass organise as follows: (i) Two top spurions Ξ 2 A (class y t L y t R ).
(ii) Four top spurions Ξ 2 A (Ξ A Ξ † A ) (classes y 3 t L y t R , y t L y 3 t R ).
(iii) Two top spurions Ξ 2 A and one mass spurion Ξ A or Ξ † A (class y t L y t R χ).
(iv) Two top spurions Ξ 2 A and two gauge spurions Ξ 2 Adj (classes y t L y t R g 2 , y t L y t R g 2 ).
They are listed in Tab. XVI.

Class
General form Operator TABLE XVI. Same as in Tab. XII but for the linear spurions in the antisymmetric representation, namely Ξ A = y t L P α Q Q † α and Ξ A = y t R P t t c † and possibly the spurions Ξ α A = y t L P α Q and/or Ξ A = y t R P t .

Appendix C: Generic classification of spurionic operators
The purpose of this appendix is to provide details about the general classification discussed in Sec. II B. We derive a complete set of non-derivative operators involving up to four spurions in a two-index representation (Ξ S/A and Ξ Adj ) of the flavour symmetry.
An the end of the appendix, we outline how the discussion can be extended to derivative operators.  To simplify the classification, instead of considering the two-index spurions that transform differently under G F , we construct objects (see Tab. II) transforming in the same way as X i → gX i g † where X i = {Ξ S/A Σ † , ΣΞ † S/A , Ξ Adj , ΣΞ T Adj Σ † }. As explained in 16 Except for the partial-compositeness spurions with no elementary fields where y t L P α Q and y t R P t appear at O( √ p). However, as discussed in Sec. II C 3 , one can still restrict to four spurions. Sec. II B, we restrict to the pseudo-real case (coset SU(N F )/Sp(N F )) since the real case Finally, let us discuss how we can extend the above basis of non-derivative operators to derivative ones. As by definition the covariant derivatives transform like the fields themselves, it is trivial to construct objects such as D µ X i or D 2 X i with the desired properties of transformations. From these objects, one can follow the procedure described previously.
In general, we get a large number of operators and some of them are redundant. They can be eliminated [92] using Tr[(D µ A 1 )A 2 · · · A n + · · · + A 1 A 2 · · · (D µ A n )] = ∂ µ Tr[A 1 A 2 · · · A n ] = 0 , where A 1 · · · A n → gA 1 · · · A n g † . Note that it is enough to restrict to two derivatives in order to get all the NLO operators in composite-Higgs models.