# Gauge Invariance, Color-Octet Vector Resonances and Double Technieta Production at the Tevatron.

###### Abstract

We show that the usual vector meson dominance method does not apply directly to the mixing of a color-octet vector boson (color-octet technirho) with the gluon because of gauge invariance. We propose a gauge invariant method where one works in a physical basis with mass eigenstate fields. As a result, we show that the physical technirho does not couple to two gluons, contrary to the general belief. Consequences for the production of a pair of color-octet, isosinglet technipions (technietas) at Fermilab is analysed by means of a simulation of the signal and background, including kinematical cuts. We find that the signal is too small to be observed.

^{†}

^{†}preprint: USM-TH-102

## I Introduction

The Standard Model has enjoyed an impressive phenomenological success. Its agreement with the precision measurement from LEP has increased the general confidence on the structure of the model [1]. Nevertheless, the fundamental question of how is the electroweak symmetry broken in Nature has not been experimentally answered yet. In fact it is generally acknowledged that the standard symmetry breaking mechanism is unsatisfactory from the theoretical point of view due to the hierarchy and triviality problems. These reasons make it necessary to investigate new signatures for alternative models of electroweak symmetry breaking. In this paper we deal with dynamical electroweak symmetry breaking [2], not committing to any specific model but rather using some general features of realistic models, like non-minimal technicolor.

Many non-minimal technicolor models predict the existence of color non-singlet scalar and vector resonances. In a recent Letter we have studied in detail the single production of a color octet isosinglet technipion (technieta, ) at the Tevatron Run II [3]. In the present Letter we extend our work by considering the production of a pair of technietas. This process was studied for the first time by Eichten et al. [4] in the context of the one-family technicolor model and by Lane and Ramana [5] in the context of walking technicolor. They perform calculations for the production cross section and for the invariant mass distribution. They also correctly point out the fact that the production of a pair of technietas is enhanced by the mixing, in the s-channel, between a gluon and a color octet isosinglet vector resonance, the so called “technirho” (). Nevertheless, they do not take into account either the decay of technietas or realistic effects such as the smearing of final momenta. We also find their description of the gluon-technirho mixing incomplete as we shall see below.

In what follows we analyze critically the traditional description of the gluon-technirho mixing and we emphasize its failure with a simple example. We show how to construct a consistent gauge invariant model using physical fields in order to describe the interactions among technirhos, gluons, fermions and technietas. Using this consistent approach, we perform a detailed simulation of double technieta production at the Tevatron followed by their decay into three jets and a photon via the process . We finally perform a simulation of the irreducible QCD background and present our results.

## Ii The Traditional Approach to gluon-technirho Mixing

An important feature of technieta pair production is its enhancement due to gluon-technirho mixing through vector meson dominance, as shown in Figure 1. This approach is phenomenologically successful in describing photon-rho mixing.However, one must be careful when trying to implement the vector dominance principle in the presence of non-abelian interactions. As an illustrative example we consider technirho decay into two gluons. Traditionally this process is represented only by Fig. 2a. The technirho-gluon mixing is described by the Lagrangian

(1) |

where is the strong coupling constant, is the mass of the technirho and is the relevant coupling constant of the process . One estimates by scaling from low energy dynamics as:

(2) |

where is the number of technicolors. With these ingredients one obtains the usual result for the invariant amplitude and the decay width:

(3) |

(4) |

However, the technirho is a color octet particle and a three-point technirho vertex must exist, as shown in Fig. 2b. As we will show in the next section, this self-interaction must have the same dynamical origin as the decay and so the three-point technirho vertex must be proportional to (in fact to due to normalization). As a consequence, the invariant amplitude for the diagram shown in Fig. 2b can be written as (the extra () factor comes from the addition of an extra propagator and an extra mixing term):

(5) |

As we can see, when the technirho is on-shell () the contribution from diagram 2b, which as far as we know has not been taken into account previously, exactly cancels diagram 2a. It is an indication that a more consistent and complete model for the gluon-technirho interaction is needed.

## Iii A consistent model for gluon-technirho interaction

### iii.1 The Gluon-Technirho Interaction

We start by considering a general lagrangian which describes the mixing of two color octet vector fields

(6) |

where

(7) |

and

(8) |

We propose the following transformation laws for the fields and :

(9) |

and

(10) |

Imposing gauge invariance we find the relations:

(11) |

Notice that from equations (1),(6) and (11) we find that . On the other hand, the lagrangian (6) and the relations (11) define the following mass matrix:

(12) |

It is easy to see that the matrix is singular and so it has a null eigenvalue. The eigenstates can be written as:

(13) | |||||

(14) |

where the mixing angle is defined by . Remark that field represents a physical (massless) gluon and is a physical (and massive) technirho. It is instructive to note from equations (9,10) and equations (13,14) that the new (physical) fields transform according to

(15) | |||||

(16) |

(where we have used the useful definition and we have omitted Lorentz indexes). That means that the field transforms as a gauge field and as a matter field in the adjoint representation of SU(3).

Now we can analyze the interaction. It can be obtained from lagrangian (6) and the definition of the physical fields. The result is

(17) |

Using the fact that we can see that the coupling constant of the interaction vanishes exactly.

### iii.2 Introducing the Technieta

Because our main goal is to study the production of a pair of technietas, it is necessary to include in our model the gluon and technirho interactions with technietas. At this point we implement the vector dominance principle by assuming that the technieta only interacts with the field . Using this hypothesis we can write the following lagrangian:

(18) |

with

(19) |

After the diagonalization of the gluon-technirho mass matrix, we arrive at

(20) | |||||

### iii.3 Introducing Quarks

In a similar way, we assume that quarks only interact with the field so we write the lagrangian

(21) |

After the diagonalization of the gluon-technirho mass matrix, we obtain

(22) |

## Iv Double Technieta Production

Now we can study the production of a pair of technietas in a consistent manner. The relevant Feynman diagrams for this process are shown in figure 3. Note that, since the physical technirho does not couple to two gluons, the sub-process is not resonant.

In order to perform a realistic simulation, we take into account the technieta decay and the smearing of final momenta. We focus our attention on the complete process . We chose this specific process in order to maximize the signal over background ratio.

We use the MADGRAPH/DHELAS [6, 7] package in order to estimate the background. We include all the leading order QCD processes that contribute to the final state, where the jets are originated by gluons and (anti-)quarks of the first two generations only. The dominant process is . We make the convolution of the parton level processes with CTEQ structure functions [8].

During the event generation we use the following kinematical cuts:

(23) |

(24) |

(25) |

(26) |

where represents the pseudo-rapidity and is the angle between two jets.

In order to include the final momenta smearing effect, we use the following detector resolution:

(27) |

(28) |

The background cross section obtained with the cuts presented above is . The cross section obtained for the signal is shown in Table 1, which is many orders of magnitude smaller (we assume that all colored technipions have the same mass).

In order to improve our analysis we implement the following cuts:

(29) |

(30) |

and

(31) |

where is the invariant mass of the photon and a jet system.

Figure 4 shows, as an example, the expected event distribution for signal and background, with GeV and GeV for the Tevatron Run II, assuming an integrated luminosity of 2 fb. It also shows the improvement due to a better energy resolution of the detectors. In all the cases we analyzed, the resulting significance was lower than 2 and we can not expect to observe the production of a pair of technietas at the Tevatron.

## V conclusions

In this work we have performed a detailed analysis of the production of a pair of technietas. During our study we have shown that the traditional way to deal with the technirho-gluon mixing is inadequate due to the non-abelian interactions felt by the technirho. This fact led us to build a consistent model for the interactions among technirhos, gluons, technietas and quarks. Unfortunately, when we applied these results to the production of a pair of technietas we conclude that we have no hope of detecting such a process at the Tevatron.

Acknowledgments

We would like to thank Alexander Belyaev for contributions at early stages of this work. We also thank Prof. R. Casalbuoni for correspondence on the gauge invariance problem. This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) grant 300538/94-4, Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) 9706112-0, Programa de Apoio a Núcleos de Excelência (PRONEX) and a Cátedra Presidencial 1997 (Iván Schmidt)

## References

- [1] For a review, see G. Altarelli, hep-ph/0011078,
- [2] For a collection of reprints, see Dynamical Gauge Symmetry Breaking, edited by E. Farhi and R. Jackiw (World Scientific, 1982). More recent developments can be found in: K. Lane, lectures at the LNF Spring School in Nuclear, Subnuclear and Astroparticle Physics, Frascati (Rome), Italy, May 15-20, 2000-hep-ph/0007304; R. S. Chivukula, lectures presented at TASI 2000, Flavor Physics for the Millennium, June 4-30, 2000, University of Colorado, Boulder, CO - hep-ph/0011264 R. S. Chivukula, R. Rosenfeld, E. H. Simmons and J. Terning, in Electroweak Symmetry Breaking and New Physics at the TeV Scale, edited by T. L. Barklow, S. Dawson, H. E. Haber and J. L. Siegrist (World Scientific, 1996).
- [3] A. Belyaev, R. Rosenfeld and A. R. Zerwekh, Phys. Lett. B462 (1999) 150.
- [4] E. Eichten, I. Hinchliffe, K. Lane and C. Quigg, Rev. Mod. Phys. 56 (1984) 579.
- [5] K. Lane and M. V. Ramana, Phys. Rev. D44 (1991) 2678.
- [6] T. Stelzer and W. F. Long, Comput. Phys. Commun. 81 (1994) 357.
- [7] H. Murayama, I. Watanabe and K. Hagiwara, KEK report No. 91-11, unpublished.
- [8] H. L. Lai et al., CTEQ Collab., Phys. Rev. D55 (1997) 1280.

Figure Captions

Figure 1: Example of mixing. There is a similar diagram with quarks in the initial state.

Figure 2: Diagrams that contribute to technirho decay into two gluons. Traditionally only a) is taken into account.

Figure 3: Feynman diagrams relevant to the production of a pair of color-octet technietas.

Figure 4: Expected event distribution for
signal with detector resolution described in the text(solid line),
with detector resolution of
GeV and
GeV.

for jets
(dotted line)
and background (dashed line), for for photons
and

(GeV) | (GeV) | (GeV) | |
---|---|---|---|

400 | 150 | 77.4 | 0.0466 |

600 | 150 | 250.0 | 0.0172 |

600 | 250 | 71.7 | 0.0066 |

800 | 150 | 407 | 0.0015 |

800 | 250 | 248 | 0.0124 |