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Building upon the fundamental partial compositeness framework, we provide consistent and potentially complete composite extensions of the Standard Model. These are used to determine the effective operators emerging at the electroweak scale in terms of the standard model fields upon consistently integrating out the heavy composite dynamics. We exhibit the first effective field theories matching these composite theories of flavor and analyze their physical consequences for the third generation quarks. Relations with other approaches, ranging from effective analyses for partial compositeness to extra dimensions as well as purely fermionic extensions, are briefly discussed. Our methodology is applicable to any composite theory of dynamical electroweak symmetry breaking featuring a complete theory of flavor.

Since the earliest proposals of new composite dynamics (aka Technicolor—TC) as the underlying theory of electroweak symmetry breaking

In models of fundamental partial compositeness (FPC) the SM is extended with a new TC sector featuring new elementary fermions and scalars charged under a new gauge group

The TC Lagrangian before introducing the electroweak sector reads

Assuming

As the TC-scalars transform according to the same representation as the TC-fermions with respect to the new gauge group, no Yukawa interactions among the TC-fermions and TC-scalars can be written (except for a few exceptions

A straightforward realization for this model is obtained choosing

The fundamental matter fields of the theory appear in the first two lines of the table, both transforming according to the fundamental representation of TC. The last three lines correspond to the bilinear composite TC singlet states. The number of states counts the Weyl fermions or real scalars.

When adding the electroweak (EW) sector we embed it within the

In

In addition, we envision two possibilities for the TC-scalars: the formation of a condensate

We now turn our attention to the SM fermion mass generation. The presence of TC-scalars in FPC models allow for a new type of Yukawa interactions interfacing the TC and the SM sectors. In fact each new Yukawa operator involves a TC-fermion, a TC-scalar and a SM fermion and the new fundamental Yukawa Lagrangian to replace the SM one reads

Besides the SM fermions and Yukawas, the underlying theory contains two more spurions that explicitly break the flavor symmetries, that is the masses of the TC-fermions and scalars,

We are now ready to determine the effective operators emerging at the EW scale in terms of the SM fields upon consistently integrating out the heavy TC dynamics aside from the pNGB excitations. De facto we provide the first effective field theory that matches to a concrete and complete example of a composite theory of flavor. In turn, this allows for investigating its impact on electroweak observables and low energy flavor physics.

We structure the work as follows. In Sec.

Having spelled out the underlying fundamental dynamics we now move to determine the effective operators at the EW scale. We start with a brief summary of the chiral Lagrangian for the TC sector. We then list the effective operators in terms of the SM fields generated by explicit realizations of partial compositeness. This is achieved by coherently matching the operators to the underlying composite flavor dynamics. This allows us, for the first time, to build in a controlled manner the full effective field theory. All operators will then appear in the Lagrangian

To organize the expansion of the EFT we adopt the counting of chiral dimension

For the underlying model to be fundamental, it must be possible to run a perturbative Yukawa coupling from the scale of strong gravity down to the scale of compositeness where it should become strong. The leading order beta function for the fundamental Yukawa coupling,

This result differs from the analogous Eq. (32) in Ref.

The effective low-energy limit of the model may be described by a nonlinearly realized chiral Lagrangian, incorporating the Goldstone modes of the spontaneously broken symmetry

As discussed in the previous section the SM gauge symmetries are embedded into the global symmetries. Parts of these are therefore promoted to local symmetries leading to the introduction of the covariant derivative

We now turn to the effective operators in terms of the SM fermion fields starting with the bilinear ones. They can be neatly organized according to their chiral dimension, starting with the lowest one which reads

At the next order we have the operator,

At next order again we find the dipole operators,

The naming of these operators are loosely inspired by the corresponding operators in the SM effective field theory

We now construct a consistent basis of four-fermion operators starting with five independent operators featuring two left-handed spinors

It is useful to represent each of the ten operators

Representative Feynman diagrams corresponding to the operators

Representative Feynman diagrams corresponding to the operators

The case in which the masses of the scalars are much heavier than

Loops of the elementary fermions are crucial in generating a potential for the pNGBs that includes the Higgs boson. As in other pNGB Higgs models, the potential contains radiative corrections that violate the global symmetries of the model once the spurionic Yukawa couplings assume their constant value. Accordingly, they play an important role in determining the vacuum alignment of the models. The simplest way to write down the fermion loop generated operators is to separate the Yukawa couplings

At leading order in the chiral expansion, and quadratic order in the spurions, two operators might appear

The former is due to the fact that the combination of Yukawas has the quantum numbers of mass terms for the SM fermions. Thus, the only term that may survive is proportional to the Majorana mass of right-handed neutrinos.

In contrast to the lack of operators at quadratic order in the spurions, there is a plethora of operators at quartic order. They involve loops of two SM fermions, each contracting the SM indices of two spurions

Representative Feynman diagrams corresponding to the operators

As in any other composite Higgs model there are contributions to the pNGB potential stemming from SM gauge bosons. At lowest order this is due to the operator,

At NLO in the chiral expansion one finds corrections to the pNGB kinetic terms. We find a total of 21 such operators involving four

Assuming couplings to all SM fermions and right-handed neutrinos with fundamental Yukawa couplings as given in Eq.

Again, for completeness, we note the SM gauge corrections to pNGB kinetic term. From one propagating gauge bosons, there are two operators which contribute to the

Furthermore, also at NLO in the chiral expansion the operator

We now specialize to the most minimal model

Fundamental technicolor states with their gauge quantum numbers and global symmetries. The table includes the third generation quarks too and the charge assignment under the baryon number

At the fundamental Lagrangian level the new Yukawa couplings with the SM fields read

The implicit QCD color indices of the quarks are embedded as part of Sp(6).

Note thatThe operator

A potential for the Higgs boson, and the other pNGB, generated by loops of top and bottom, is encoded in the operators in Eqs.

The potential also receives contributions from the gauge interactions, encoded in Eq.

We now turn to the operator in Eq.

For all our numerical estimates we have used

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Please note that all bounds found here, are on the effective rather than the fundamental Yukawa parameters.

Note that our normalization for the pre-Yukawa couplings differs from the one usually considered in EFT realizations, see Sec.

A possible concern for the model is that the fundamental Yukawa couplings may become large enough that they cause an unwanted condensate,

The effective Lagrangian for EW physics contains four fermion interactions which are induced by the underlying strong dynamics. In Sec.

The phenomenologically relevant operators involve four tops, as they are directly probed at the LHC in four top final states, such as

In addition to the four fermion interactions, the operators

The bounds come from the 95% C.L. limits on the SM EFT operator coefficients

The fundamental Lagrangian can be expanded to include all three generations of quarks and leptons. The minimal strategy

Fundamental technicolor states and SM fermions with their SM gauge quantum numbers. The table also includes the charge assignments under the baryon and lepton number

The complete Yukawa interactions now read

Note that the scalars are in the conjugate representation of

Just as in the case of the top and bottom, the Yukawa interactions can be written in the more compact form from Eq.

The only exception is given by the physics of the right-handed neutrinos that might have Majorana masses and order-1 fundamental Yukawa couplings. Note that the presence of both Yukawas

Representative diagram for the contribution to the Higgs potential of a right-handed neutrino with an elementary Majorana mass, symbolized here by a blue cross.

In this section we sketch the connection between our analysis, and other approaches used in the literature to study partial compositeness. We first address effective approaches, based either on the construction of an EFT or on extra dimensional implementations. Finally, we comment on the possible applicability of our results to purely fermionic underlying theories featuring partial compositeness.

The most popular approach to composite Higgs models in the literature has been to construct EFTs simply based on the symmetry breaking patterns (see Refs.

This approach might be the only available one if the underlying theory is conformal, in which case it can only be defined in terms of operators and their conformal dimensions.

As a consequence, to implement partial compositeness, the choice of the representation under which the top partners transform has been arbitrary. Furthermore, top partners in the EFT approach have been assumed to be the main driving force in the stabilization of the vacuum alignment along the small-In the case under study in this work, the representation of the top partners is fixed to be the fundamental of the global symmetry SU(4). This choice has been considered problematic in the literature, as it typically leads to large corrections to the

Another effective approach to partial compositeness relies on extra dimensions: it is mainly based on adapting the conjectured correspondence of anti–de Sitter (AdS) space-time with four-dimensional conformal field theories

Traditional approaches hope to achieve partial compositeness via pure underlying gauge-fermion realisations. In this case the new composite fermion operators

The remaining challenge is to build a theory that actually generates the operator

As noted in

We can use group theory to investigate related theories. For example, from Table I of

We built consistent extensions of the standard model of fundamental partial composite nature and determined their electroweak effective theories in terms of the standard model fields. The bases of effective operators of different mass dimensions were built and constrained using the symmetries of the underlying theories. Our results can now be used as a stepping stone to undertake studies both in the lepton and quark flavor observables within a controlled theory of composite dynamics.

To elucidate the power of our approach, we focused on the most minimal theory of fundamental partial compositeness. We analyzed the physical consequences for the composite Higgs sector as well as the third generation quarks. Here we discovered new contributions to the Higgs potential generated from the left-handed mixing of top and bottom. Intriguingly, we also discovered that right-handed neutrinos with TeV scale composite Majorana masses can affect the Higgs potential with relevant consequences for the vacuum alignment of the theory. We show that constraints on the top and bottom sectors can be naturally abided. Our effective operators are ready to be deployed for full scale analyses of composite lepton and light quark flavor physics.

Finally, we provided relations with other approaches. The overall methodology can be employed to derive effective operators stemming from related underlying composite theories of dynamical electroweak symmetry breaking able to give masses to the standard model fermions.

We thank C. Englert and M. Russell, and the

Whenever we write an invariant of an

To construct the spurion field

For the definition of the

Here we determine all possible four fermion operators respecting the symmetries of the model. The operators must be singlets under

We start by noting that the operators must have the general form

Thus one finds that

We note that the basis for the self-conjugate 4-fermion operators follows similarly, by noticing that any any Lorentz structure reduces to the forms

In this appendix we list the remaining NLO operators for the chiral kinetic term, arising through loop corrections from SM fermions. All these operators contain two derivatives of the pNGB field and some symmetry breaking parameter(s). In the list we have ignored all the terms on the form

The above consideration leave just one nontrivial,

With two insertions of

There are four real operators with two EW gauge insertion

We now list all the four-fermion operators found in the model containing only top and bottom SM fermions. These are found by expanding the operators

Operators with four left-handed quarks,