Ordinary technicolor and extended technicolor cannot produce the heavy top quark unaided. We demonstrate that a flavor-universal extension of the color interactions combined with an extended hypercharge sector that singles out the third generation can provide the necessary assistance. We discuss current experimental constraints and suggest how collider experiments can search for the predicted new heavy gauge bosons.

## 1 Introduction

Generating mass through strong gauge dynamics is a challenge. While a technicolor [1] gauge sector can provide appropriate masses for the electroweak gauge bosons by breaking the chiral symmetries of technicolored fermions, explaining the masses and mixings of the quarks and leptons has proven more difficult. Extended technicolor (ETC) models [2] postulate an enlarged gauge group coupling the quarks and leptons to the technifermion condensate, enabling them to acquire mass. The simplest models of this type tend to produce large flavor-changing neutral currents [2] and, if the heavy top quark mass is generated by ETC interactions, excessive weak isospin violation [3] and contributions to [4]. Substantially raising the scale at which extended technicolor breaks to its technicolor subgroup can alleviate some of these problems – but renders the model incapable of naturally producing quark masses larger than a few GeV.

Given the large value of the top quark’s mass ( GeV [5]) and the sizable splitting between the masses of the top and bottom quarks, it is natural to wonder whether has a different origin than the masses of the other quarks and leptons. A variety of dynamical models that exploit this idea have been proposed. Key examples are the dynamical models of ‘top-mode’ mass generation in which top quark self-interactions drive all of electroweak symmetry breaking [6]. Related to those are the topcolor [7] and topcolor-assisted technicolor [8] models [9, 10] in which the top quark feels different color and hypercharge interactions than other quarks; as a consequence, a top quark condensate enhances the top quark’s mass. Finally, there are the non-commuting ETC scenarios where the top quark has weak and extended technicolor interactions different from those of other quarks [11]. The conclusion of these investigations has been that new dynamics peculiar to the top quark can certainly create a large top quark mass. It may even possible to do so while creating a model that accords reasonably well with electroweak precision data.

In this paper, we discuss a variant class of models of dynamical top quark mass generation in which the large mass comes from top-specific gauge interactions. What sets these theories apart is that the top quark differs from the other quarks only in its hypercharge interactions. The extended color interactions are flavor-universal, just as in the coloron model of [12]; the weak interactions display ordinary Cabibbo universality.

After introducing the class of models in section 2 and showing, in section 3, that the low-energy dynamics admit the possibility of top quark condensation and a large top quark mass, we focus on experimental constraints. Section 4 discusses the phenomenology of the low-energy effective theory, while section 5 explores the possibility of direct searches for the additional massive gauge bosons in the theory.

We note that the physics discussed here must be part of some larger (e.g. ETC) structure at high energies which will create the masses and mixings of the light fermions and produce condensates that break our extended gauge symmetries to their standard model subgroups. However, we focus on exploring the dominant effects of the new physics that produces the top quark mass. A discussion of higher-scale operators that break all fermion chiral symmetries, account for generational mixing and produce relevant symmetry-breaking condensates may be found in [13].

## 2 The Class of Models

Our models have a gauge structure like that of the original topcolor-assisted technicolor models [8]. Far above the electroweak scale, the gauge group is

(2.1) |

with coupling constants , , , , , and respectively. We take the first and groups to have the stronger couplings: and . The group is the technicolor gauge group.

At an energy scale , a condensate transforming under the initial symmetry group as breaks the color sector () to its diagonal subgroup () and similarly breaks the hypercharge groups in the pattern (). The gauge symmetry is reduced to that of the standard model plus the unbroken technicolor group:

(2.2) |

At the weak scale, , the technicolor force becomes strong enough to break the chiral symmetries of a set of technifermions and cause electroweak symmetry breaking . Thus the low-energy gauge boson spectrum includes the massless photon and gluons, the massive ’s and , and two additional kinds of massive states: an octet of colorons and a single .

At low energies, the mass eigenstate fields in the color sector (colorons and gluons ) are related to the original gauge fields (denoted ) via [14]:

(2.3) |

Similar relations hold in the hypercharge sector. The familiar QCD and hypercharge gauge coupling constants are related to the high energy couplings by

(2.4) |

and their respective fine-structure constants are and . The tree level masses of the colorons and are

(2.5) |

Note the dependence of the mass on the charges of the condensate .

The gauge transformation properties of the quarks and leptons, which are summarized in Table 1, are significantly different from those in topcolor-assisted technicolor [8]. In the color sector, all quarks transform only under the stronger group, as in the flavor-universal coloron model [12]. In the hypercharge sector only the third family of fermions transforms under the stronger and the first two families transform under the weaker (all of them with standard model hypercharge assignments). All of the quarks and leptons have the same weak charge assignments as in the standard model. Each generation of ordinary fermions forms an anomaly-free representation of the gauge group (2.1).

As we shall explore in more detail, this set of gauge charge assignments for the fermions still allows natural dynamical generation of a large mass for the top quark (and only the top quark). Yet it leads to a phenomenology differing from that of topcolor-assisted technicolor [8, 10].

I | 1 | SM | 1 | SM | 0 | SM |
---|---|---|---|---|---|---|

II | 1 | SM | 1 | SM | 0 | SM |

III | 1 | SM | 1 | SM | SM | 0 |

## 3 Low energy effective theory

Below the symmetry-breaking scale, , for the extended color and hypercharge sectors, the interactions among quarks and leptons that arise from exchange of the massive colorons and are well-approximated by effective four-fermion interactions

(3.1) |

(3.2) | |||||

where is any quark, is a quark or lepton whose subscript
indicates its generation, the are the octet of Gell-Mann
matrices, and Y is the standard model hypercharge generator^{1}^{1}1We
use the convention .. The coefficients
and are defined as

(3.3) |

Note that where is the angle by which the original color (i=3) and hypercharge(i=1) gauge eigenstates were rotated to form the mass eigenstates.

The extended gauge interactions are ultimately responsible for the large mass of the top quark. The principle contributions to the dynamical mass come from the four-fermion contact interactions (3.1) and (3.2), which we can study using the gap equation in the Nambu–Jona-Lasinio (NJL) approximation [15]. The dynamical mass of fermion is the solution to:

(3.4) |

where the coefficients are

and () is the hypercharge of (). In solving (3.4), we take the cut-off for the gap equation to be of order the coloron and masses: ; corrections due to unequal values for the coloron and masses are small in the region of physical interest. Applying this to the top quark, one finds if

(3.5) |

More generally, however, we need to include contributions to the gap
equation from gluon and hypercharge boson exchange^{2}^{2}2Since the
bosons couple only to left-handed fermions, they do not
contribute here.; in effect, we are studying a gauged NJL model
[16]. As discussed in [17], this modifies the
criticality conditions for the .

Applying the gauged NJL gap equations to all the standard model fermions, we seek solutions with non-zero (i.e. formation of a top condensate ) and no mass for any other fermion (i.e. for ). Such solutions exist provided that and satisfy a set of inequalities, of which the following three are the most stringent:

(3.6) |

(3.7) |

(3.8) |

Inequality (3.6) leads to top quark condensation (). Note how including the effects of gluon and hypercharge boson exchange modifies the right-hand-side expression compared to the original NJL result (3.5). Inequality (3.7) implies (i.e., no charm quark condensation). In our class of models, this is a stronger constraint than the inequality ensuring ; in a Top-color I model [10], the latter would be the relevant constraint. Inequality (3.8) is related to the lack of condensation; it will be superseded by other constraints later in our discussion.

As inequalities (3.6) – (3.8) can be simultaneously
satisfied, our models do admit the possibility that only the top
quark condenses and receives an enhanced mass. The values of the
couplings and for which this happens fall within
the ‘gap triangle’^{3}^{3}3Due to the non-linearity of (3.7),
it is only approximately a triangle. lying to the right of curve (1),
to the left of curve (2) and below curve (3) in Figure 1 (by analogy
with results for Top-color I models [10]). Solutions to the
gauged-NJL gap equation [18] for GeV and particular
values of the cut-off lie on curves
parallel to curve (1); a few examples for ranging from 0.7 TeV
to 5 TeV are shown and labeled (A) through (D). Curves like these will
be used in calculating phenomenological limits in the next section.

## 4 Low-energy constraints

We now consider how several types of physics constrain the allowed region of the plane. We first look at the parameter and decays to tau leptons. Next, we discuss the implications of a strong coupling. Finally, we comment on flavor-changing neutral currents (FCNC).

Current measurements of the parameter are already sensitive to the presence of the low energy contact interactions (3.1) and (3.2). The main contribution to from the coloron sector of our model is [19] single coloron exchange across the top and bottom quark loops of and vacuum polarization diagrams. Applying the results of [19] to our models, we have

(4.1) |

where is the weak mixing angle and is the analog of for the top-condensate, i.e. (in the NJL approximation) [15, 20]

(4.2) |

In the sector, the main contribution to arises from mixing. Adapting the results of [21] to our models, we have

(4.3) |

Requiring [19] excludes the region to the right of curve (4) in Figure 2. This curve connects the points on the lines of constant mentioned earlier. Note how the constraint narrows the allowed region of the plane.

Another constraint comes from the partial decay width of the Z boson to tau leptons:

(4.4) |

where is the Fermi constant [5] and () is the coupling of () to the boson. Due to mixing, [21], the couplings and in our model are altered from those in the standard model (i.e. + ) by

(4.5) |

yielding a non-standard prediction for .
Including QED corrections to eq. (4.4) and requiring our
predicted value to be consistent with the experimental [5] value
MeV at 95% c.l.
excludes the region to the right of curve^{4}^{4}4This curve was
constructed by the same procedure as curve (4). (5) in Figure 2.

The asymptotic UV behavior of the strongly-coupled yields
another important, albeit elastic, constraint^{5}^{5}5We thank
R.S. Chivukula for emphasizing the relevance of this constraint.
Similar considerations apply in any model in which a gauge
interaction is used to align the vacuum [22]. on .
Combining expressions (2.4) and (3.3) shows that

(4.6) |

Applying the renormalization group equation to

(4.7) |

(with ) and considering just the contribution from the standard model particles (i.e., taking ) allows us to estimate the position of the Landau pole for a given low-energy value of . Our results are summarized in Figure 3. If the Landau pole is to lie at least an order of magnitude above the symmetry-breaking scale, , then must be of order 1 or smaller. This defines curve (6a) in Figures 2 and 3. Similarly, requiring the Landau pole to lie two or five orders of magnitude above produces curves (6b) and (6c) in Figures 2 and 3.

Finally, we turn to flavor-changing neutral currents. Because the color sector is flavor-universal, the low-energy effective interactions (3.1) cause no flavor-changing neutral currents. In other words, the low-energy effective theory now includes not just top-pions [8], but a complete set of “q-pions” strongly coupled to all flavors of quarks. To first approximation, the q-pion masses and couplings are flavor-symmetric and they make no contribution to hadronic FCNC processes like neutral meson mixing or . This is in contrast to the potentially large (but avoidable) hadronic FCNC exhibited by Top-color I models [23, 10]. The flavor symmetry among the q-pions will be modified at sub-leading level by non-universal U(1) effects; this can re-introduce hadronic FCNC at a smaller, less dangerous rate.

Because the hypercharge interactions (3.2) distinguish among
generations, they also cause semi-leptonic flavor-changing decays of
and mesons, which are the same as those in Top-color I models
[10]. As discussed in ref. [10], current data on , , , and
also^{6}^{6}6While this process involves no FCNC, it would be similarly
affected by the boson. set no limits,
but future experiments may be sensitive to the presence of the
additional interactions. For the process , ref. [10] found that the ratio of amplitudes was
roughly , so squaring this and dividing by the number of neutrino
species gives an estimate of the relative branching ratios: . Subsequently, evidence has been
published for a event that is consistent
with branching ratio
[24], as compared with a standard model branching ratio of
order . This process is therefore still able to accommodate a
in the allowed parameter space of our models (i.e., and ); future data from the E787
Collaboration may provide further constraints.

## 5 Direct Searches for the Colorons and

The colorons in this class of models are identical to those introduced in the flavor-universal coloron model of ref. [12]. As discussed in [25], searches in dijet final states should be the most powerful way of locating heavy colorons. Searches in and offer no particular advantage in searching for the flavor-universal colorons in our class of models. This is in contrast to the case of the topgluons of topcolor [7] and topcolor-assisted technicolor [8].

As discussed earlier, constraints on the low-energy effective theory for our class of models limit the value of coupling to lie quite close to the critical value . This means that the coloron cannot be very light: if we estimate the minimum coloron mass for by requiring the coloron contribution (4.1) to to be less than , we find TeV; including the contributions to would only strengthen the bound. A coloron of this large a mass lies above the reach of published searches for new particles decaying to dijets [26]. Moreover, the large value of implies that the coloron’s width

(5.1) |

is approximately twice its mass. Future searches for narrow resonances will not be appropriate for finding these colorons. A more promising approach would employ the strategies of compositeness searches, which focus on high- enhancement of single-jet inclusive and dijet spectra [27] or alteration of the dijet angular distributions [28]. At energies well below , the effects of coloron exchange on hadronic scattering are approximated by those of the color-octet quark contact interaction (3.1). If experiment set a limit TeV on a color-octet contact interaction

(5.2) |

with the usual convention , this would imply a limit TeV for our class of models in which .

Existing limits on the mass of the boson are not very stringent. For example, Tevatron bounds [29] on new contributions to the dilepton ( or ) mass spectrum from interactions like (3.2) set no useful limit on our class of models because the coupling to first generation fermions is so small. The strongest limits are derived in ref. [21] by considering the contributions to electroweak observables of a like the one in our class of models (called an “optimal” in [21]). These calculations set a 95% c.l. lower bound of GeV on the for . For other values of , the must be heavier; a mass less than a TeV is allowed for

Future experiments measuring production of third-generation fermions (, , ) have the greatest potential to find signs of the boson. Consider, for example, looking for an excess in in 50 fb of NLC data taken at GeV. Because the boson’s decay width

(5.3) |

is a large fraction of its mass (e.g., for ), we use the -dependent width in the cross-section; this renders our results insensitive to the exact value of . Assuming a 50% efficiency for identifying tau pairs and requiring a excess over the standard model prediction for of , the effects of a 2.7 TeV boson with could be visible. At a 1.5 TeV NLC with 200 fb of data, the reach in extends to 6.6 TeV.

## 6 Conclusions

We have examined the low-energy effective theory and phenomenology of a class of technicolor models with flavor-universal extended color interactions and a generation-distinguishing extended hypercharge sector. Such models are found to be capable of dynamically producing a top quark condensate that preferentially enhances the mass of the top quark. Moreover, flavor-changing neutral currents are less dangerous here than in models where the color sector couples differently to the third generation. Constraints from -pole physics and triviality single out the region of coupling-constant parameter space where and for further study. Electroweak physics presently constrains the boson in these models to weigh at least 290 GeV, while the octet of flavor-universal colorons must have a mass of at least 1.6 TeV. Future studies of jet physics at hadron colliders have the potential to uncover evidence of the colorons, while data on pair-production of third-generation fermions at machines can help discover the .

Acknowledgments

We thank R.S. Chivukula, N. Evans, C.T. Hill, K.D. Lane, and T. Rizzo for useful discussions and comments on the manuscript. E.H.S. is grateful for the hospitality of the Aspen Center for Physics during the inception of this work and that of the Theoretical Physics Group at Fermilab during its completion. E.H.S. acknowledges the support of the Faculty Early Career Development (CAREER) program and the DOE Outstanding Junior Investigator program. This work was supported in part by the National Science Foundation under grant PHY-9501249, and by the Department of Energy under grant DE-FG02-91ER40676.

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