The Effective Description of Strong Electroweak Symmetry Breaking

However, the local symmetry does not tolerate explicit mass terms for gauge fields. This seems to be in contradiction with reality in all cases but the photon. Fortunately, there is a way to reconcile the massive gauge bosons, namely Z, W , W , and gauge invariance: spontaneous breaking of electroweak symmetry accompanied by the Higgs mechanism. The Higgs mechanism was first suggested to explain some collective density fluctuations in plasma which were produced by a finite range electromagnetic field (a massive photon) [3]. Later it was generalized as a relativistic field theory [4] and its renormalizability was proved [5].


Introduction
The gauge invariance as a fundamental principle for building the electroweak Lagrangian enables describing the electromagnetic and weak interactions under the same formalism [1]. The theory is commonly known as the Standard model of electroweak interactions (SM) [2]. The SM leads to the long list of predictions that have been successfully verified by experiments.
However, the local symmetry does not tolerate explicit mass terms for gauge fields. This seems to be in contradiction with reality in all cases but the photon. Fortunately, there is a way to reconcile the massive gauge bosons, namely Z, W ϩ , W Ϫ , and gauge invariance: spontaneous breaking of electroweak symmetry accompanied by the Higgs mechanism. The Higgs mechanism was first suggested to explain some collective density fluctuations in plasma which were produced by a finite range electromagnetic field (a massive photon) [3]. Later it was generalized as a relativistic field theory [4] and its renormalizability was proved [5].
Despite the great success of the SM one essential component of the theory remains a puzzle: it is the actual mechanism behind the electroweak symmetry breaking (ESB). The introduction of the Higgs complex doublet scalar field of a non-zero vacuum expectation value to the electroweak Lagrangian serves as a benchmark hypothesis. A direct consequence of this hypothesis is the presence of the Higgs boson in the particle spectrum of the SM, not observed to these days. It is not the only plausible hypothesis though.
There is a host of candidates for the suitable extensions of the SM. They range from supersymmetric theories with multiple elementary Higgs bosons in their spectra to the theories of new strong interactions, like Technicolor, which might form bound states of new elementary particles in analogy with QCD. These bound states might appear in the particle spectrum as new resonances.
In more recent theories like the Little Higgs models and the Gauge-Higgs unification models, their dual-description relation to the Heavy Composite Higgs and the No Higgs strongly-interacting models has been demonstrated (see [6] and references therein). Most of these models introduce new particles at about 1 TeV.
Facing this plethora of alternative theories it is highly desirable to describe their low-energy phenomenology in a unified way. Thus, it is very useful to exploit the formalism of effective Lagrangians. Therein the so-called hidden local symmetry (HLS) approach [7] can be used to introduce new vector resonances which might originate in some new physics beyond the SM. The new particles are not only a byproduct of a particular ESB mechanism but they would be needed to tame the unitarity if the Higgs boson were not found. The HLS formalism along with the AdS/CFT correspondence plus deconstruction is also behind the dual-description relation of the recent models mentioned above [6].
In this paper we present the modified version of the existing effective model of strong ESB. The original model is called the BESS (Breaking Electroweak Symmetry Strongly) model [8]. There -beside the known SM particles -a new SU (2) V vector boson triplet is introduced exploiting HLS. The new triplet is supposed to represent the spin-1 bound states of hypothetical new strong interactions. We take this model and modify its interactions of the vector triplet with fermions. In the BESS model there is a universal direct coupling of the triplet to all fermions of a given chirality. In our model, on the other hand, we admit direct couplings of the new triplet to top and bottom quarks only. The modification reflects speculations about a special role of the top quark

Mikulas Gintner -Josef Juran -Ivan Melo -Beata Trpisova *
The actual mechanism of electroweak symmetry breaking remains the last unsolved part of the Standard model of electroweak interactions. We review the general requirements that all possible solutions must fulfill. Based on them the effective description of the mechanism can be formulated. We offer such a description which is a modified version of the so-called BESS model and discuss its properties.
(or the third quark generation) in the mechanism of ESB. We named the modified BESS model as the top-BESS model.
In Section 2 of the paper we discuss the general requirements for the ESB mechanism which must be reflected by any ESB hypothesis. In Subsection 2.2 we briefly introduce the BESS and the top-BESS models. Besides we show the direct interaction vertices of the new vector triplet to top and bottom quarks and the differences in the couplings of the electroweak gauge bosons with these quarks. Section 3 is devoted to the phenomenology of the top-BESS model. Herein we discuss the masses and the decay widths of the new vector triplet. We also point to the forwardbackward asymmetry as a promising observable for measuring parameters of the V -to-t/b couplings. Conclusions can be found in Section 4.

General requirements
The central idea to the generating masses of the electroweak gauge bosons W Ϯ and Z is the spontaneous symmetry breaking (SSB): G → H where G is a global continuous symmetry of the Lagrangian and H is a continuous symmetry of its vacuum (ground state), H ʕ G. The Goldstone theorem [9] implies that the SSB results in appearance of massless scalar fields, the so-called Nambu-Goldstone bosons (NGB). The number of NGB is equal to dim G -dim H, i.e. the number of "broken" generators of the group G. However, once F ʚ G is made local by introducing gauge fields interacting with the NGB's the corresponding NGB's disappear from the spectrum while the G -H gauge fields become massive and acquire longitudinal degrees of freedom. This is depicted in the following figure: Now we formulate the conditions that must be fulfilled in order to build the most general effective Lagrangian describing the Higgs mechanism that provides masses for the Z 0 and W Ϯ gauge bosons [10]: G A part of the SM must exhibit the SSB pattern G → H, where G is a global symmetry and H is its subgroup. Since we need masses for three gauge bosons we require that there would be at least 3 broken generators: . G Three of broken generators must generate a local symmetry.
The massive gauge bosons must be coupled to the corresponding three NGB's. If we consider K ϭ 3 (otherwise we would have to explain where the uneaten NGB's are) the constraints listed above lead to a unique choice of the groups G and H: where SU(2) V is a diagonal subgroup of SU(2) L ϫ SU(2) R and SU(2) A is its axial 1) "subgroup". This also implies that NGB's are pseudoscalar particles.
Let us mention that the gauge invariance under a chiral symmetry group excludes explicit mass terms for fermions. However, gauge invariant mass terms for fermions can be generated through the ESB mechanism. 1) SU(2) A is not a group even though it is often called that way in the physical literature.

From BESS to top-BESS
The BESS model [8] effectively describes a Higgsless ESB mechanism accompanied by a hypothetical strong triplet of vector resonances V In the top-BESS model we modify the fermion sector by considering no direct interactions of V to fermions other than t and b. The SU(2) L symmetry does not allow us to disentangle the V-to-t L coupling from the V-to-b L one. However, it can be done in the case of right fermions. We do it and to simplify the numerical analysis of the model we turn off the direct interaction of V to b R completely. The full form of the top-BESS Lagrangian can be found in [11]. Here, we show only the part responsible for the direct interactions of t and b quarks to the new vector resonances parameterized by the b 1,2 coupling constants as well as the non-SM interactions of t and b with the SM gauge bosons where ϭ (t, b) T , and P 0 ϭ diag(1, 0). The gauge couplings g, gЈ and gЉ correspond to the SU(2) L , U(1) Y , and SU(2) V gauge symmetry groups of the model, respectively. As a consequence of turning off the V-to-b R coupling in (2) neither t R couples directly to the charged V-resonance.
The Lagrangians (1) and (2) are expressed in terms of gauge field flavor eigenstates. However, in experiments we measure the mass eigenstates. The relations between the two sets of the gauge field eigenstates are given by where , The free parameter ␣ is related to the mass of the vector triplet, degenerated in the leading order approximation: . M W and M Z are the W and Z boson masses in the leading order approximation, respectively.
The measurement of Zbb vertex constrains the V-to-t L coupling to relatively small values. Due to the disentangling of the t R interactions with V from the b R interactions with V the measurement does not limit the V-to-t R interaction. For the low-energy limits on the model's parameters, see [11,12].

Masses and widths of the new vector resonance triplet
The masses of the SU(2) V neutral and charged resonances depend on the parameters ␣ and gЉ as shown in Fig. 1. In all calculations depicted here and below we consider the fixed values of the Z and W masses as taken from experiment. To compensate for the non-SM corrections to M Z and M W the SM parameters , g, and gЈ vary slightly with gЉ.
The vector resonances decay dominantly to the SM gauge bosons and/or to the third generation of quarks. In Fig. 2 we show the total decay widths of the vector resonances [13]. The plots of the partial decay widths are shown in Fig. 3. Finally, we compare the BESS and the top-BESS models in terms of the total decay    widths of V 0 and V Ϯ in Fig. 4 [13]. In all cases we assume M V 0 ϭ ϭ 1 TeV.

Experimental signatures
It would seem natural to search for the signs of the new resonances in the processes where the electroweak gauge bosons and/or the t and b quarks are involved. At the Large Hadron Collider (LHC) this is often a difficult case due to the large backgrounds to the processes with b and t in the final state and/or due to the negligible b and t luminosities in pp collisions.
In the BESS model the new SU(2) V vector bosons mix with the electroweak gauge bosons. While the mixing complicates theoretical analysis of the model it might provide some advantages in experimental searches. The mixing generates interactions of the new resonances with the fermions of lighter generations. Although these interactions are suppressed by the mixing factors, the LHC processes enabled by them have the advantage of higher parton luminosities and more favorable final state topologies. In [12] we have calculated sensitivities of the pp → tt X, bb X, tb X, W ϩ W Ϫ X, W Ϯ ZX processes with subsequent decays to leptons and hadronic jets at the LHC without considering backgrounds. At this level of analysis the investigated processes have displayed promising potential to serve as probes of new physics under consideration. Nevertheless, further and more detailed investigation of the issues is necessary.
New ESB-responsible physics can also be studied at a future e ϩ e Ϫ collider under preparation which is known under its generic name of ILC (International Linear Collider). While the LHC will serve as a discovery machine the ILC will provide much cleaner signal to measure parameters of new physics. In [14] we have analyzed three ILC processes -e ϩ e Ϫ → tt , e ϩ e Ϫ → bb , and e ϩ e Ϫ → W ϩ W Ϫ -at the collision energies 800-1200 GeV as probes of the top-BESS model with M V 0 ϭ 1 TeV.
The above mentioned investigations focused on discovering and distinguishing mass peaks of the V-resonances in the invariant mass distributions of the V decay products. The observation of a mass peak would result in determining the mass and, perhaps, the width of the observed resonance. However, here we would like to point to yet another ILC observables which might be suitable for determining the values of b 1 and b 2 parameters. Namely, they are the forward-backward assymmetries of top and bottom quarks. This is illustrated in Figs. 5 and 6 for processes e ϩ e Ϫ →tt and e ϩ e Ϫ → bb , respectively, with M V 0 ϭ 1 TeV. The curves for the forward-backward asymmetries depend significantly on the values  of the b 1,2 parameters. They also differ significanly from the SM curve in the region around the mass of the V resonance.
Regarding hadron colliders the forward-backward asymmetry is not applicable at the LHC because the symmetry of the colliding proton beams wipes up the forward-backward asymmetry. On the other hand, the Tevatron collider is free of this problem because it collides protons with antiprotons. However, its collision energy is probably too low to probe 1 TeV resonances.

Conclusions
In this paper we have presented the status of our ongoing study of the top-BESS model which is a well-motivated modifica-tion of the original BESS model. We had introduced the modifications to obtain the effective description of a certain class of theories which aim at explaining the mechanism of ESB. These are the theories with the top quark playing a special role in the ESB mechanism.