The Extreme Walking Behavior in a 331-TC Model

It is quite possible that the Technicolor problems are related to the poorly known self-energy expression, or the way chiral symmetry breaking (CSB) is realized in non-abelian gauge theories. Actually, the only known laboratory to test the CSB mechanism is QCD. The TC dynamics may be quite different from the QCD , this fact has led to the walking TC proposal making the new strong interaction almost conformal and changing appreciably its dynamical behavior. There are different ways to obtain of extreme walking (or quasi-conformal) technicolor theories, in this paper we propose an scheme to obtain this behavior based on an extension of the electroweak sector of the standard model, in the context of so called 331-TC model.


I. INTRODUCTION
The 125 GeV new resonance discovered at the LHC [1] has many of the characteristics expected for the Standard Model (SM) Higgs boson. If this particle is a composite or an elementary scalar boson is still an open question that probably will be answered in the next LHC run. In recent papers the ATLAS and CMS Collaborations [2] reported an experimental anomaly in diboson production with apparent excesses in W W , W Z and ZZ channels and this anomaly have inspired a number of theoretical papers proposing as an explanation the production of heavy weak bosons, W ′ and Z ′ .
Thus, it becomes interesting to investigate the possibility of obtaining a light scalar boson in the context of models which features contributions from new heavy weak bosons W ′ and Z ′ . In some extensions of the standard model (SM), as in the so called 3-3-1 models [3] SU(3) L ⊗ SU(3) c ⊗ U(1) X , new massive neutral and charged gauge bosons, Z ′ and V ± , are predicted. The 3-3-1 model is the minimal gauge group that at the leptonic level admits charged fermions and their antiparticles as members of the same multiplet, the predictions of the GW = SU(2) L ⊗ U(1) Y alternative models are leptoquark fermions with electric charges 5/3 and 4/3 and bilepton gauge bosons with lepton number L = ±2. The quantization of electric charge is inevitable in the G = 3m1 models [4] with three non-repetitive fermion generations brea-king generation universality and does not depend on the character of the neutral fermions.
In the Ref. [5] was suggested that the gauge symmetry breaking of a specific version of a 3-3-1 model [3] would be implemented dynamically because at the scale of a few TeVs the U(1) X coupling constant becomes strong and the exotic quark T (charge 5/3) will form a U(1) X condensate breaking SU(3) L ⊗ U(1) X to the electroweak symmetry. This possibility was explored by us in the Ref. [6] assuming a model based on the gauge symmetry SU (2) where the electroweak symmetry is broken dynamically by a technifermion condensate, that is characterized by the SU(2) T C Technicolor (TC) gauge group. The early technicolor models [8] suffered from problems like flavor changing neutral currents (FCNC) and contributions to the electroweak corrections not compatible with the experimental data, as can be seen in the reviews of Ref. [9]. However, the TC dynamics may be quite different from the known strong interaction theory, i.e. QCD, this fact has led to the walking TC proposal [10], which are theories where the incompatibility with the experimental data has been sol-ved, making the new strong interaction almost conformal and changing appreciably its dynamical behavior.
We can obtain an almost conformal TC theory, when the fermions are in the fundamental representation, introducing a large number of TC fermions (n T F ), leading to an almost zero β function and flat asymptotic coupling constant. The cost of such procedure may be a large S parameter [11] incompatible with the high precision electroweak measurements. However, this problem can be solved by assuming that TC fermions are in other representations than the funda-mental [12] and an effective Lagrangian analysis indicates that such models also imply in a light scalar Higgs boson [13]. This possibility was also investigated and confirmed through the use of an effective potential for composite operators [14] and through a calculation involving the Bethe-Salpeter equation (BSE) for the scalar state [15].
The reason for the existence of the different models (or different potentials) for a composite scalar boson, is a consequence of our poor knowledge of the strongly interacting theories, that is reflected in the many choices of parameters in the effective potentials. The possibility of obtaining a light composite scalar according to the approach discussed in Ref. [14], is that this result is a direct consequence of extreme walking (or quasi-conformal) technicolor theories, where the asymptotic self-energy behavior is described by Irregular form of TC fermions [14,15][27] In the Ref. [16] we considered the possibility of a light composite scalar boson arising from mass mixing between a relatively light and heavy scalar singlets from a see-saw mechanism expected to occur in two-scale Technicolor (TC) models and we identified that, regardless of the approach used for generating a light composite scalar boson, the behavior exhibited by extreme walking technicolor theories, is the main feature needed to produce a light composite scalar boson compatible with the boson observed at the LHC. The exotic technifermions will present the extreme walking behavior, the usual techniquarks (U, D) will present the known asymptotic self-energy behavior predicted by the operator product expansion(OPE) [7]. The exotic techniquarks will have two different energy scales and the 331-TC model corresponds to an example of two-scale Technicolor (TC) model. This article is organized as follows: In section II we present the U(1) X contribution to fermions and technifermions self-energy, in section III we compute the dynamically generated masses to heavy exotic quarks (T, S, D) and techniquarks (U ′ , D ′ ) , where we reproduce the results obtained in Ref. [5] for heavy exotic quarks. In the section IV we illustrate how to obtain the extreme walking behavior in the context of 331-TC model and Section V contains our conclusions.

SELF-ENERGY
As described in Refs. [5,6] the gauge symmetry breaking in 3-3-1 models can be implemented dynamically because at the scale of a few TeVs the U(1) X coupling constant becomes strong. The exotic quark T introduced will form a condensate breaking SU(3) L ⊗ U(1) X to electroweak symmetry, in this paper we will numerically determine the U(1) X contribution to dynamic mass of exotic quarks (T, S, D) and techniquarks (U ′ , D ′ ) that appear in the fermionic content of the model [6] following the same procedure described in Ref. [5].
The Schwinger-Dyson equation for quarks(techni-quarks) due to U(1) X interaction can be written as [5,6] where in the equation above we assumed the rainbow approximation for the vertex Γ µ,ν , 1) X hypercharges attributed at chiral components of the exotic quarks(techniquarks).
With the purpose of simplifying the calculations it is convenient to choose the Landau Gauge. In this case the Z ′ propagator can be written in the following form Writing the quark propagator as iS −1 F (p) = i( / p − Σ X (p 2 )), and considering the equation above, we finally can write in the euclidean space the gap equation for Σ(p where To obtain the last equation we assumed the angle approximation to transform the term The integral equation described above can be transformed into a differential equation for where M ≡ Σ X (0) is the dynamical mass of exotic quarks(or techniquarks) generated by U(1) X interaction and the respective boundary conditions for f (x) are f (0) = 1 and f ′ (0) = 0. In order to obtain the mass spectrum generated due to U(1) X interaction, we follow the same procedure described in Ref. [5] where the (Y i ) hipercharges of exotic quarks (T, S, D) were assumed according to the table 1 and we include the corresponding hipercharges of exotic techniquarks (U ′ , D ′ ) [6] that are singlets of SU(3) c .
As commented in the previous section we follow the same procedure described in Ref. [5], therefore, in order to get an estimate of the U(1) X dynamically generated mass, for exotic fermions(or technifermions), we will numerically solve the Eq. (3) imposing an ultraviolet cutoff Λ on this equation. If the gap equation accepts a M T solution(M T is mass of exotic quark (T)) [28], then the gauge symmetry SU(3) L ⊗U(1) X is dynamically broken to SU(2) L ⊗ U(1) Y and the Z ′ , V ± , and U ±± gauge bosons become massive. The Z ′ mass is given by where F Π is the pseudoscalar decay constant and we calculate it using the Pagels-Stokar approximation that is given by Assuming the set of variables described below of Eq.(4), the above equation together with the definition of M Z ′ , allows to write the Pagels-Stokar relation as where the coefficients α and β were defined in the previous section.
According to the description given in [5] the consistency requirement imposed about the solution of Eq. (5) is that the mass of the exchanged particle Z ′ (or α = M Z ′ M ) has to be equal to Z ′ mass (or α) obtained using Eq. (7). In other words the solutions of the gap equation Eq.(5) are iteratively improved by starting with a trial guess for the exchanged boson mass and then comparing it with the predicted mass obtained using the Eq. (7). In the Fig I we show the behavior of Eq. (7) assuming the numerical solution of Eq. (5) for Λ = 42T eV ,  M Z ′ = 2T eV as a function of parameters α and β. In the table II we show the results obtained for the dynamically generated masses for heavy exotic quarks (T, S, D) and exotic techniquarks (U ′ , D ′ ) , where we reproduce the results obtained in Ref. [5] for heavy exotic quarks .

IV. THE EXTREME WALKING BEHAVIOR IN A 331-TC MODEL
Theories with large anomalous dimensions (γ m > 1) are quite desirable for technicolor phenomenology [9,18], it is known for a long date that four-fermion interactions are responsible for harder self-energy solutions in non-Abelian gauge theories (γ m ∼ 2) [18,19].
In the model considered in this work we show that due to the U(1) X interaction only the exotic techniquarks (U ′ , D ′ ) will acquire an dynamical mass . The result is that at this energy scale a bare mass appears in the (TC) Schwinger-Dyson equation assigned to exotic techniquarks, (U ′ , D ′ ) , what leads to a very "hard" self-energy, or a self-energy of the irregular type, Eq.(1), only for exotic techniquarks (U ′ , D ′ ). In this section considering a four-fermion approximation for U(1) X interaction associated to these techniquarks we will show that the results for M U ′ = M D ′ = O(T eV ) are of the same order as obtained in the previous section, in this case F Π equation is given by  (5) and (7) as function of parameters α and β to Λ = 42T eV , M Z ′ = 2T eV . To g X = 2.576(β = 0.042) we get M T = 5.73T eV (α = 0.349) which corresponds to the first value obtained for M T described in Ref. [5].
where M X = M U ′ = M D ′ or M T . As in the previous section, the equation above together with the definition of M Z ′ , allows us to write For a similar choice of parameters, used in the previous section, for example Λ = 42 TeV, β = 0.042 and M Z ′ = 2.5 TeV we obtain M T ≈ 6.3TeV and M U ′ = M D ′ ≈ 0.6TeV so that the contribution due U(1) X to the mass of exotic techniquarks can be approximated by a four-fermion interaction and the exotic techniquarks exhibit a self-energy behavior of the irregular type.
To illustrated the extreme walking behavior exhibited only by the exotic technifermions we consider the full gap equation for the "exotic techniquark U'" that contains the sum of two contributions, the U(1) X interaction and TC interaction, and we consider the presence of dynamically massive technigluons. The problems for chiral symmetry breaking in this case have been discussed recently, where confinement may play an important role [21,23,24]. In this work we consider that technigluons acquire a dynamical mass along the line of QCD as proposed by Cornwall [22] many years ago and the dynamical technigluon mass behaves as [23,24] with m tg (0) ≈ Λ T C and the technifermion dynamical mass will be given by where is the dynamical techniquark mass, and C 2 is the Casimir operator for the technifermionic representation with effective TC couplingḡ tc , given bȳ in this expression b T C is the first β T C function coefficient and we consider the Landau gauge. Similarly to the previous section the integral equation described by Eq. (11) can be transformed into a differential equation for h(x) introducing the new variables x = p 2 The full gap equation for the "exotic techniquark U'" can be written as in the Fig.(2a)[Blue curve] we show the behavior of M T C (x) = Ū U Λ 2 , that corresponds to the dynamical mass generated for the the usual techniquarks (U, D) obtained with Eq. (13). In the Fig.(2b)[Red curve] we include the contribution of U(1) X effective four-fermion interaction and we show the behavior of the dynamical mass generated for the exotic techniquarks

V. CONCLUSIONS
In the Refs. [14][15][16]25] we discussed the possibility of obtaining a light composite TC scalar boson, this result is a direct consequence of extreme walking (or quasi-conformal) technicolor theories, where the asymptotic self-energy behavior is described by Eq.(1). The extreme walking technicolor can be obtained in three different ways and in this paper we propose an scheme to obtain the quasi-conformal behavior based on an extension of the electroweak sector of the standard model, in the so called 331-TC model (SU(N) T C ⊗ SU(3) L ⊗SU(3) c ⊗U(1) X ), where the exotic quark T introduced will form a U(1) X condensate ( T T ) breaking SU(3) L ⊗ U(1) X to electroweak symmetry that is broken by an usual technifermion condensate. As the comment presented in the introductory section, 3-3-1 models predicted new massive weak gauge bosons, Z ′ and V ± , and this model has the necessary requirements to explain the reported diboson anomaly. Following the same procedure described in Ref. [5], the solutions of the gap equation, Eq.(5), were iteratively improved by starting with a trial guess for the exchanged boson mass and then comparing it with the predicted mass obtained using the Eq. (7). In the table II we show the results obtained for the dynamically generated masses of heavy exotic quarks (T, S, D) and exotic techniquarks (U ′ , D ′ ) , where we reproduce the results obtained in Ref. [5] for heavy exotic quarks. In Ref. [26] we discuss a mechanism for the dynamical mass generation, including the mass generation for the t quark, in the case of grand unified theories that incorporate quarks and techniquarks. We expect that a similar mechanism to the one described in [26] can be developed and incorporated in the present model. (see table I). The exotic technifermions have two different energy scales and the 331-TC model corresponds to an example of two-scale Technicolor (TC) model, in the Ref. [16] we considered the possibility of a light TC composite boson arising from mass mixing between a relatively light and heavy composite scalar singlets from a see-saw mechanism expected to occur in two-scale TC model. We emphasize that the see-saw mechanism expected to occur in this model is not exactly the same one described in the Ref [16], in the model proposed the extreme walking behavior displayed by exotic techniquarks is due to their strong U(1) X interaction and in this case in Eq. (1), µ ≈ Λ 2 = Λ 331 , and Λ 2 it is not generated by the TC sector. In order to provide a example we consider the SU(2) T C case with the technifermionic content showed in [6], where n T F = 6. With this we obtain m φ 1 ∼ 124 GeV, m φ 2 ∼ 8.3TeV to Λ 331 = 3T eV and Λ ET C ≈ 90TeV. Therefore, in this scenario it is possible to obtain a light composite TC scalar boson and include the necessary requirements to explain the diboson anomaly, the determination of the scalar spectrum for composite Higgs bosons of this model will be presented in detail in a future work.
Acknowledgments I thank A. A. Natale for useful discussions. This research was partially supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) by grant