We compute an effective action for a composite Higgs boson formed by new fermions belonging to a general technicolor non-Abelian gauge theory, using a quite general expression for the fermionic selfenergy that depends on a certain parameter ( ), that defines the technicolor theory from the extreme walking behavior up to the one with a standard operator product expansion behavior. We discuss the values of the trilinear and quadrilinear scalar couplings. Our calculation spans all the possible physical possibilities for mass and couplings of the composite system. In the case of extreme walking technicolor theories we verify that it is possible to have a composite Higgs boson with a mass as light as the present experimental limit, contrary to the usual expectation of a heavy mass for the composite Higgs boson. In this case we obtain an upper limit for the Higgs boson mass, (M H Oð700Þ GeV for SUð2Þ TC ), and the experimental data on the Higgs boson mass constrain SUðNÞ TC technicolor gauge groups to be smaller than SUð10Þ TC .

The full solution of technicolor (TC) Schwinger-Dyson equations should include radiative corrections induced by extended technicolor (ETC) (or other) interactions. We verify that when TC is embedded into a larger theory including also QCD, these radiative corrections couple the different strongly interacting Schwinger-Dyson equations, providing a tiny mass to technifermions and changing the ultraviolet behavior of the gap equation solution. We argue about the origin of the different quark masses without appealing for different ETC boson masses, in one scenario where most of the new physics will appear in interactions with the third fermion generation and with a TC scalar boson possibly lighter than the TC characteristic scale (Λ TC ). DOI: 10.1103/PhysRevD.97.115035 The origin of fermion and gauge boson masses in the standard model (SM) of elementary particles is explained by their interaction with the Higgs boson. The discovery of this boson at the LHC [1,2] has crowned the SM; however, the data still cannot discard the possibility of this boson being a composite one. The case of a composite state, generating dynamical gauge symmetry breaking, instead of an elementary one, is more akin to the phenomenon of spontaneous symmetry breaking that originated from the Ginzburg-Landau Lagrangian. The latter can be derived from the microscopic Bardeen-cooper-schrieffer (BCS) theory of superconductivity describing the electron-hole interaction, which can be interpreted as a composite state. A similar mechanism happens in QCD where the chiral symmetry breaking is promoted by a nontrivial vacuum expectation value of a fermion bilinear operator and the Higgs role is played by the composite σ meson. In particular, the technicolor (TC) idea was the earliest attempt to build models in this direction [3,4].The main ideas about TC models were reviewed in Refs. [5,6] and recent phenomenological studies about this class of models can be seen in Refs. [7][8][9][10][11][12][13] and references therein. Despite the fact that TC models are much more complex than the ones with elementary scalar bosons, the main difficulty to build a realistic model lies in the ordinary behavior of the technifermion self-energy that is propor-where μ TC is the characteristic TC dynamical mass at zero momentum and γ the anomalous mass dimension. This self-energy leads to the known quark mass (m Q ) given by m Q ∝ μ 3 TC =M 2 E , where M E is the mass of an extended technicolor boson (ETC), which is a particle that may change flavors. In order to describe, for example, the top quark mass we need a small M E value, and this boson generates flavor changing neutral currents at one undesirable level.

We study chiral symmetry breaking in QCD-like gauge theories introducing a confining effective propagator, as proposed recently by Cornwall, and considering the effect of dynamical gauge boson mass generation. The effective confining propagator has the form 1/(k 2 + m 2 ) 2 and we study the bifurcation equation finding limits on the parameter m below which a satisfactory fermion mass solution is generated. Considering the evidences that the coupling constant and the gauge boson propagator are damped in the infrared, due to the presence of dynamically massive gauge bosons, the major part of the chiral breaking is mostly due to the confining propagator. We study the asymptotic behavior of the gap equation containing confinement and massive gauge boson exchange, and find that the symmetry breaking can be approximated at some extent by an effective four-fermion interaction generated by the confining propagator. We compute some QCD chiral parameters as a function of m, finding values compatible with the experimental data. Within this approach we expect that lattice simulations should not see large differences between the confinement and chiral symmetry breaking scales independent of the fermionic representation and we find a simple approximate relation between the fermion condensate and dynamical mass for a given representation as a function of the parameters appearing in the effective confining propagator.

Scalar composite boson masses have been computed in QCD and technicolor theories with the help of the homogeneous Bethe-Salpeter equation, resulting in a scalar mass that is twice the dynamically generated fermion or technifermion mass (m dyn ). We show that in the case of walking (or quasiconformal) technicolor theories, where the m dyn behavior with the momenta may be quite different from the one predicted by the standard operator product expansion, this result is incomplete and we must consider the effect of the normalization condition of the Bethe-Salpeter equation to determine the scalar masses. We compute the composite Higgs boson mass for several groups with technifermions in the fundamental and higher dimensional representations and comment about the experimental constraints on these theories, which indicate that models based on walking theories with fermions in the fundamental representation may, within the limitations of our approach, have masses quite near the actual direct exclusion limit.

Assuming that the 125 GeV particle observed at the LHC is a composite scalar and responsible for the electroweak gauge symmetry breaking, we consider the possibility that the bound state is generated by a non-Abelian gauge theory with dynamically generated gauge boson masses and a specific chiral symmetry breaking dynamics motivated by confinement. The scalar mass is computed with the use of the Bethe-Salpeter equation and its normalization condition as a function of the SU (N ) group and the respective fermionic representation. If the fermions that form the composite state are in the fundamental representation of the SU (N ) group, we can generate such light boson only for one specific number of fermions for each group. We address the uncertainties underlying this result, when considering the strong dynamics in isolation.

When technicolor (TC), QCD, extended technicolor (ETC) and other interactions become coupled through their different Schwinger-Dyson equations, the solution of these equations are modified compared to those of the isolated equations. The change in the self-energies is similar to that obtained in the presence of four-fermion interactions, but without their ad hoc inclusion in the theory. In this case TC and QCD self-energies decrease logarithmically with the momenta, which allows us to build models where ETC boson masses can be pushed to very high energies, and their effects will barely appear at present energies. Here we present a detailed discussion of this class of TC models. We first review the Schwinger-Dyson TC and QCD coupled equations and explain the origin of the asymptotic self-energies. We develop the basic ideas of how viable TC models may be built along this line, where ordinary lepton masses are naturally lighter than quark masses. One specific unified TC model associated with a necessary horizontal (or family) symmetry is described. The values of scalar and pseudo-Goldstone boson masses in this class of models are also discussed, as well as the value of the trilinear scalar coupling, and the consistency of the models with the experimental constraints.

The technicolor (TC) Schwinger-Dyson equations (SDE) should include radiative corrections induced by extended technicolor (ETC) interactions when TC is embedded into a larger theory including also QCD. These radiative corrections couple the different strongly interacting Dyson equations. We discuss how the boundary conditions of the coupled SDE system are modified by these corrections, and verify that the ultraviolet behavior of the self-energies are described by a function that decreases logarithmically with momentum. *

Recently it has been pointed out that no limits can be put on the scale of fermion mass generation ͑M͒ in technicolor models, because the relation between the fermion masses (m f ) and M depends on the dimensionality of the interaction responsible for generating the fermion mass. Depending on this dimensionality it may happen that m f does not depend on M at all. We show that exactly in this case m f may reach its largest value, which is almost saturated by the top quark mass. We make a few comments on the question of how large a dynamically generated fermion mass can be. DOI: 10.1103/PhysRevD.68.077702 PACS number͑s͒: 12.60.Nz, 12.10.Dm The mechanism that breaks the electroweak gauge symmetry SU(2) L ϫU(1) Y down to the gauge symmetry of electromagnetism U(1) em is still the only obscure part of the standard model. It is known that up to the scale of 1 TeV some sign of this mechanism has to become manifest in future experiments. In the same way that an upper bound on the scale of electroweak symmetry breaking has been put forward ͑the 1 TeV scale͒, it was thought that the scale of fermion mass generation also had an upper bound, and that bound would be within the reach of the next generation of accelerators ͓1͔. Recently it was shown that no upper bound can be put on the scale of fermion mass generation beyond that on the scale of electroweak symmetry breaking ͓2͔. This result was obtained by considering the scattering of the same helicity fermions into a large number of longitudinal weak vector bosons in the final state, and it was also obtained in a more involved way in Ref. ͓3͔. This result is important and at the same time disappointing, because an upper bound on the scale of fermion mass generation would provide a target for future accelerators in order to understand the origin of fermion masses.The scale related to the origin of fermion masses cannot be bounded but the fermion mass itself is bounded. The bound on the fermion masses comes out from the upper limit on the Yukawa coupling ( y рͱ8) ͓4͔. In the standard scenario this is not very interesting because it also leads to a bound on the fermion masses of the order of 1 TeV. Therefore there is still space for a heavy new family ͑respecting the constraints provided by the high precision experiments͒. This problem becomes much more interesting in theories with dynamical symmetry breaking such as technicolor theories, where, in principle, some of the free parameters of the standard model are calculable as long as we know the symmetries of the underlying theory that is responsible for the mass generation. Let us recall some of the arguments about the nonexistence of a bound on the scale for fermion mass generation in technicolor ͑tc͒ models ͓2͔. In these models the fermion mass is given bywhere c is a constant and ͗ tc tc ͘ is the technifermion con- which gives the bound on the mass scale responsible for fermion mass generation. Nowadays it is known that composite operators like ͗ tc tc ͘ ͑and the technifermion selfenergy͒ may have a large anomalous di...