The Axigluon, a Four-Site Model and the Top Quark Forward-Backward Asymmetry at the Tevatron

Very recently, it has been shown by other authors that the CDF measurement of the top quark forward-backward asymmetry can be explained by means of a heavy and broad axigluon. In order to work, this mechanism needs that the axigluon coupling to the top quark must be different than the coupling to light quarks and both must be stronger than the one predicted in classical axigluon models. In this paper, we argue that this kind of axigluon can be accommodated in an extended chiral color model we proposed previously. Additionally, we show that the desired features can be derived from a simple four-site model with delocalized fermions.


Introduction
Although the High Energy Physics community is unanimous in considering the Standard Model (SM) to be incomplete for many and well known reasons, such as the Higgs mass instability under quantum corrections or the unexplained hierarchy observed in the masses of the fermionic sector; it is impossible to deny its enormous (and, to some extent, unexpected) experimental success. Indeed, after years of scrutiny and new physics search, first at the LEP and the Tevatron and now at the LHC, the SM has not been still seriously challenged by collider experiments. The only exception today may be the 3.2σ deviation recently reported by CDF in the top-quark forward-backward asymmetry (A tt F B ) [1] (see also [2] and [3]). The fact that the process under question involves the top quark makes it even more interesting since, due to its mass which is near the electroweak scale, it has been suspected for a long time that the top-quark plays a special rôle in nature.
A natural candidate to explain a deviation in A tt F B is an axigluon (other alternatives beyond the SM have been considered in [4], [5], [6], [7], [8], [9], [10] and [11]). In fact, this is an expected axigluon signal [12]. It is true that it has been argued that the present signal cannot be explained by an axigluon due to other experimental constraints over it [13]. Nevertheless, Bai, Hewett, Kaplan and Rizzo have argued that a heavy and broad axigluon can be a viable explanation of the A tt F B while remaining invisible in the dijet spectrum [14].
In their paper, they parameterize the axigluon interaction as where g s is the usual QCD coupling constant a g A measures the deviation of the axigluon coupling constant from QCD. In order to evade experimental constraint it proves useful to consider that g A may take different values for light quarks (g q A ) and for the top quark (g t A ). The analysis presented in [14] shows that in order to have a broad axigluon it must occur that |g q A |, |g t A | > 1. Additionally, in order to escape detection in contact interaction searches it must happen that |g q A | < |g t A | (this kind of non-universality has been also reported in [15] ). Finally, g q A g t A must be negative in order to reproduce correctly the sign of A tt F B . The construction of a model that can incorporate all these conditions, specially the different coupling constants for different generations, has proven to be difficult. As a solution, the authors of [14] propose a two-site model but they have to include exotic vector-like quarks in order to satisfy the |g q A | < |g t A | constrain. In this paper, we address the construction of a viable axigluon model not by adding new fermions but by extending the gauge sector of the model. In section 2 we show that an effective model, based on SU(3) L × SU(3) R but with four octet fields transforming as gauge bosons, has enough structure to accommodate all the desire features of the axigluon. In section 3, we show that the essential characteristics of the effective model can be obtained from a four-site model with delocalized fermions. Finally, in section 4, we summarize our conclusions.

An Effective Model
We start by recalling an effective extended chiral model we have previously proposed [16]. In this model, we consider an usual SU(3) L × SU(3) R with two pairs of vectors fields: l µ and L µ transforming as gauge fields of SU(3) L , and r µ and R µ transforming as gauge fields of SU(3) R . The gauge sector of the model (including a non-linear sigma model sector which is introduced to produce the breaking of SU(3) L × SU(3) R to SU(3) c ) is described by the following Lagrangian: where In the unitary gauge a non-diagonal mass matrix is explicitly generated. Although the resulting mass matrix can be exactly diagonalized, in order to obtain easily manipulable expressions we will assume that g ′ ≫ g and we will write our results in the first order in g/g ′ . The eigenvalues, that is, the masses of the physical states are: Thus, the physical spectrum is composed of a (exactly) massless gluon, an axigluon and two degenerate (at this level of approximation) heavy gluon and axigluon. The normalized mass eigenvectors can be written as: Because we have now two fields that transform as gauge fields for each group (l µ and L µ for SU(3) L and r µ and R µ for SU(3) R ) any combination of the form g(1 − k)l µ + g ′ kL µ and g(1 − k)r µ + g ′ kR µ , where k is a new arbitrary parameter. In principle, it can take different values in the right and left sectors but, for simplicity we will assume it is the same in both cases. With this structure it is possible to construct generalized covariant derivatives. This means that the Lagrangian describing the gauge interaction of quarks can be written as: Using Lagrangian (3) and the definition of the physical fields, we can obtain the terms of the Lagrangian that couple the gluon and the axigluon to quarks where: and Figure 1: Group structure of our simple four-site model Interestingly, the mechanism described just above is related with fermion delocalization in deconstruction theory [17,18] . Now we have an axigluon with a modified coupling to quarks. Nothing in the model prevents that the value of g A varies from one generation to another. This implies that in this framework we can easily satisfy the necessary conditions (in particular, |g q A | < |g t A | constrain) for making the axigluon compatible with existing data and, at the same time, explain the A tt F B data from CDF.
The key features of the model described in this section are the presence of two pairs of vector bosons which transform like gauge fields and delocalized fermions. These properties can be naturally implemented in a four-site model. The construction of such a model is the subject of next section.

A Simple Four-Site Model
Our simple four site model is based on the gauge group SU(3) 1 4 and three link fields (Σ 1 , U and Σ 2 ) as shown in figure  1. We assume that the scalar fields develop vacuum expectation values ( u = Σ 1 = Σ 2 and v = U ) breaking down the gauge symmetry to SU(3) c which we identify with the usual color group.
The Lagrangian for the bosonic sector can be written as: where Notice that in order to simplify our equations we have assume that all groups share the same coupling constant g.
As usual, in the unitary gauge (Σ 1 = U = Σ 2 = ½) a non-diagonal mass matrix for the gauges bosons explicitly appears. This mass matrix can be exactly diagonalized, nevertheless, in benefice of simplicity, we work in the limit where v ≪ u. In this limit, the physical spectrum is composed of a (exactly) massless gluon G, a light (in the sense that its mass is proportional to v) axigluon A and two degenerate heavy states (V 1 and V 2 ) with masses proportional to u. The original gauge fields can be written in terms of the physical fields as follows: Now we turn our attention to quarks. A crucial feature of our effective model described in the previous section was the fact that the fermions coupled to a certain linear combination of gauge bosons. In the language of deconstruction theory, it correspond to "fermion delocalization". The main idea of fermion delocalization [17,18] is that the low energy properties of the fermions (for example the color interaction of quarks) originate from the contribution of many sites of the underlying gauge group. In other words, the current couples to a linear combination of gauge fields n x n A nµ with the constrain n x n = 1 in order to preserve gauge invariance under the whole group. As shown in figure 1, we decide to couple left handed quarks to A 1µ and A 3µ (with weights x and 1 − x respectively) and right handed quarks to A 2µ and A 4µ (with weights 1 − x and x respectively). In this way, the interaction Lagrangian for quarks can be written as: where J a Lµ and J a Rµ are the quark left-handed and right-handed currents, respectively. Notice that we have chosen, for simplicity, the same delocalization parameter x for right handed and left handed quarks. In terms of the physical fields the interaction Lagrangian can be written as: where g s = g/2 is the usual strong interaction coupling constant. Our delocalization pattern has provoked the appearance of a axigluon interaction which depends on the delocalization parameter x. As in our effective model, nothing prevents that different quark generations can be differently delocalized. In this way, we can select appropriated values of x for the top quark and for the light quarks in order to consistently explain the A tt F B measurements.

Summary
In summary, we have shown that an extended chiral color model with extra color octet spin-1 fields has enough structure to describe an axigluon with appropriate coupling to quarks in such a way that it can be consistent with dijet and contact interactions searches and, at the same time, explain the A tt F B anomaly reported by CDF. The fact that this effective model can be re-obtained from a four-site model, as we have shown, is interesting from the theoretical point of view since, in principle, it may be related, via dimensional deconstruction, to five dimensional models.