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We provide a general classification of template operators, up to next-to-leading order, that appear in chiral perturbation theories based on the two flavor patterns of spontaneous symmetry breaking

The discovery of a Higgs-like boson

While most of the recent progress has been based either on holography or on effective theories (see, for instance, Refs.

This statement is, of course, incomplete unless a theory that generates the coupling of fermions is also specified.

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 fermionsMotivated 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

Up to now, the global symmetry

The misalignment between the EW preserving and physical vacua is conveniently parametrized by an angle,

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

The paper is organized as follows. In Sec.

The chiral perturbation theory that we introduce in this section is intended to parametrize the low-energy physics of some strongly coupled hypercolor (HC) gauge theories. We focus on the sector of the theory that is responsible for the breaking of the EW sector of the SM with the aim of providing a dynamical symmetry breaking and solving the hierarchy problem of the Higgs mass. Thus, the matter content consists of fermions, and their representation under the HC interactions completely determines the pattern of global symmetry breaking

We assume that the theory is vectorlike with respect to the SM gauge quantum numbers, so that an EW preserving vacuum is allowed.

and the globalHowever, this class of models can no longer be considered as vectorlike gauge theories

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

Besides the spontaneous breaking of

In the following, we first present the chiral Lagrangian associated with the real and pseudoreal cases

In this section, we present the NLO chiral perturbation theory for the real and pseudoreal cases. Both give rise to an

By expanding the kinetic term in Eq.

In the technicolor limit,

The NGBs,

The covariant derivative is defined as follows:

Note that, apart from the NGB matrix,

Properties of the NGB matrix,

The NLO chiral Lagrangian at order

When the number of flavors,

The chiral Lagrangian can be completed by introducing explicit breaking terms of the flavor symmetry,

Spurions in representations up to two

Transformations under

Operators with one or two spurions

Explicit models can contain more than one spurion transforming under the same

Having at our disposal a complete basis of nonderivative operators involving up to four spurions (see Table

A current mass for the underlying fermions

The gauging of the EW symmetry,

A SM-like bilinear coupling between the elementary top quark multiplets and the strong dynamics:

Linear couplings

A detailed list of all the relevant spurions can be found in Table

Spurions parametrizing the explicit breaking sources appearing in composite-Higgs models. The representation of the partial-compositeness spurions depends on the trilinear baryon involved in the linear couplings, i.e. on the flavor representation of

The underlying fundamental theory involving the hyperfermions,

Let us start with the simplest source of explicit breaking, namely a current mass for the hyperfermions. At the fundamental level, the relevant Lagrangian is given by

From Table

We now turn to the second obvious source of explicit breaking, i.e. the gauging of the EW symmetry,

The effect of the gauging also appears in nonderivative operators that can be built in terms of the spurions

Finally, we would like to point out that the effect of the gauging of additional symmetries within

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 realizing the partial compositeness paradigm. Linear couplings, however, always need an extension of the underlying theory as, minimally, hyperfermions charged under QCD are needed in order to generate QCD-colored bound states. This can be done either by sequestering the QCD interactions to a sector containing a different HC representation

At the fundamental level, we assume that the top mass is generated by the following operators:

The scales

The spurion encoding the explicit breaking is

At NLO, the

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),

There is also the possibility of a hyperfermion/hypergluon bound state. However, this is unlikely because it would require the hyperfermion to be in the adjoint representation of HC, thus making the theory lose asymptotic freedom.

As previously mentioned, the operators need to contain at least one hyperfermion that carries QCD color, which we denote asSpelling out the various cases, the linear couplings can thus be rewritten as follows:

The couplings of the underlying theory in Eq.

The LO operators contributing to the top mass, for all the choices of spurion representations, are given by the following expressions:

Similarly, we can construct the operators contributing to the potential for the NGBs. At leading order, there exist operators involving only two spurions for the case of the antisymmetric and adjoint representations,

For the spurions in the symmetric and fundamental representations, the leading operators contain at least four spurions, leading to the following expressions:

For simplicity we assumed that the LECs are the same for operators that only differ on the type of spurion insertion,

At NLO, many more operators are generated, as listed in Appendix

In this section, we apply the machinery developed in the previous section to the coset

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

Purely fermionic underlying theories of partial compositeness need at least a Sp(4) hypercolor gauge symmetry.

have also been recently publishedIn 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.

The full custodial symmetry of the SM,

As discussed in Ref.

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

The (nonlinearly realized) scalar variable describing the dynamics of the NGBs associated with the above breaking pattern and the vacuum

We can now explicitly write the relevant spurions introduced in Sec.

The spurions corresponding to the EW gauging including the elementary fields can be written as

For the top bilinear spurions, transforming as

In the case of partial compositeness, we can write the spurions as

This charge assignment refers to the partner of

For the symmetric (

Finally the adjoint (

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,

For simplicity, let us discuss first the LO effects of each explicit breaking source independently. As discussed in Sec.

The challenge in composite-Higgs models is to generate a small misalignment (

A potential generated only by the gauge and top explicit breaking interactions such that the current masses are set to zero, and we have

A potential generated by gauge and top spurions as well as a nonzero current mass. In this case, it is enough to restrict to the LO contributions, and we thus assume

In case (i), the minimization of the potential in Eq.

For case (ii), the minimization of the potential leads to

Let us now explore in detail how the scenario (i) could be realized when NLO contributions are taken into account. In practice this requires obtaining

This case relies on the usual hypothesis that the top loops are the dominant contributions to the coefficients

Choosing a symmetric representation for the left- and right-handed top couplings, one finds that the LO contributions generate

For simplicity, let us first consider operators of the general form

For completeness, we also report the expression for the top mass and linear couplings to the NGBs,

The general potential presented in Eq.

The trilinear couplings in Eqs.

Right panel: trilinear Higgs coupling normalized to the SM value as a function of

Right panel: Higgs coupling to

A coupling of the singlet

The situation is different for linear couplings to two-index representations. For the symmetric, we already found in Eq.

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

Note that the phase appearing in the Cabibbo-Kobayashi-Maskawa matrix does not play any role here, as we are dealing with overall phases carried by the Yukawas.

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 parametrize the spurion for the right-handed top following Eq.

Another case where a misalignment along the singlet direction is needed is when the potential generates a negative mass squared for

It is well known from QCD

To date, composite-Higgs models remain a valid alternative to the SM and to supersymmetric models in describing the physics of the discovered Higgs boson. One of the tools we have to explore the physics of composite Higgses is the construction of effective theories. In this work, we offer an exhaustive classification of template operators that can be used to construct effective Lagrangians, up to NLO in the chiral expansion, for models based on the symmetry breaking patterns

After a general discussion, we specialized our results to the simplest case based on

T. A. acknowledges partial funding from a Villum foundation grant when part of this article was being completed. N. B. and G. C. acknowledge partial support from the Labex-LIO (Lyon Institute of Origins) under Grant No. ANR-10-LABX-66 and FRAMA (FR3127, Fèdèration de Recherche “André Marie Ampère”), and support from the “Institut Franco Danois.” The CP3-Origins center is partially funded by the Danish National Research Foundation, Grant No. DNRF90.

In this appendix, we classify all the operators, up to NLO, that contribute to the NGB potential at tree level.

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.

We specialize the classification of the generic spurionic operators (see Sec.For simplicity, we assume that no explicit

Several points are worthwhile to remember at this point:

Once the spurions are specified, their chiral counting is fixed such that the general basis of nonderivative operators involving up to four spurions contains operators that appear beyond NLO. For instance all of the operators involving three or four mass spurions

The underlying fundamental theory (see Sec.

Some operators contain traces made only with spurionic fields (no NGB matrix,

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.

We first consider the operators containing only the mass spurion,

Only one spurion

Two spurions

One spurion

The corresponding operators are displayed in Tables

Nonderivative operators up to NLO that contain the mass spurion

Same as in Table

We now include both the mass and gauge spurions,

Two spurions

Four spurions

Two spurions

One spurion

We now discuss the spurions generating the top mass: in the following, we consider a bilinear coupling as well as linear couplings

The four classes of nonderivative operators involving the top bilinear spurion

Two top spurions

Four top spurions

Two top spurions

Two top spurions

Nonderivative operators up to NLO involving the top bilinear spurion

The three classes of operators involving the linear spurions in the fundamental representation

Four top spurions

Two top spurions

Two top spurions

Same as in Table

The four classes of operators involving the linear spurions in the adjoint representation

Two top spurions

Four top spurions

Two top spurions

Two top spurions

Same as in Table

The three classes of operators involving the linear spurions in the symmetric representation

Four top spurions

Two top spurions

Two top spurions

Same as in Table

Finally, the four classes of operators involving the linear spurions in the antisymmetric representation

Two top spurions

Four top spurions

Two top spurions

Two top spurions

Same as in Table

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

In addition to the points outlined in Appendix

The top quark spurions containing elementary fermions generate corrections to their kinetic term. For instance, the generic operator

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:

The number of operators is again drastically reduced compared to those present in the generic classification of Appendix

For a bilinear top spurion, we get four different classes of operators that contribute at tree level to the top mass:

Only one top spurion

Three top spurions

One top spurion

One top spurion

Nonderivative operators up to NLO involving the top bilinear spurions

For a linear top coupling transforming in the fundamental representation, the operators contributing at tree level to the top mass organize as follows:

Two top spurions

Four top spurions

Two top spurions

Two top spurions

Same as in Table

For a linear top coupling transforming in the adjoint representation, the operators contributing at tree level to the top mass organize as follows:

Two top spurions

Four top spurions

Two top spurions

Two top spurions

Same as in Table

For a linear top coupling transforming in the symmetric representation, the operators contributing at tree level to the top mass organize as follows:

Two top spurions

Four top spurions

Two top spurions

Two top spurions

Same as in Table

Finally, for a linear top coupling transforming in the antisymmetric representation, the operators contributing at tree level to the top mass organize as follows:

Two top spurions

Four top spurions

Two top spurions

Two top spurions

Same as in Table

The purpose of this appendix is to provide details about the general classification discussed in Sec.

This general set of operators can then be used, once the explicit breaking sources are specified, to construct all the operators up to NLO that explicitly break

A concrete application to composite-Higgs models is presented in Sec.

Except for the partial-compositeness spurions with no elementary fields where

To simplify the classification, instead of considering the two-index spurions that transform differently under

Nonderivative operators involving three two-index spurions. As explained in the text, the operators divide into the three following classes: