## Brazilian Journal of Physics

##
*On-line version* ISSN 1678-4448

### Braz. J. Phys. vol.37 no.1a São Paulo Mar. 2007

#### http://dx.doi.org/10.1590/S0103-97332007000100022

**The NJL interaction from q-deformed inspired transformations^{*}**

**V. S. Timóteo ^{I}; C. L. Lima^{II}**

^{I}Centro Superior de Educação Tecnológica, Universidade Estadual de Campinas, 13484-370, Limeira, SP, Brazil

^{II}Instituto de Física, Universidade de São Paulo, CP 66318, 05315-970, São Paulo, SP, Brazil

**ABSTRACT**

From the mass term for *q*-deformed quark fields, we obtain effective contact interactions of the NJL type. The parameters of the model that maps a system of non-interacting deformed fields into quarks interacting via NJL contact terms is discussed.

**Keywords:** Quantum Algebras; Hadronic Physics; Effective Models

In hadron physics, the NJL model [1, 2] is a very simple effective model for strong interactions that describes important features like the dynamical mass generation, spontaneous chiral symmetry breaking, and chiral symmetry restoration at finite temperature.

*q*-Deformed algebras provide a nice framework to incorporate, in an effective way, interactions not originally contained in the Lagrangian of a particular system [3-5].

In recent works [6-9], we have been investigating possible applications of quantum algebras in hadronic physics. In general, we observed that when we deform the underlying algebra, the system is affected with correlations between its constituents. We have studied in detail the NJL model under the influence of a quantum *su*(2) algebra.

In this work we approach the following question: is it possible to obtain an art of "canonical transformation" connecting the NJL model to a simpler non-interacting system? We verified that we can indeed obtain the same dynamics of the NJL interaction with non-interacting *q*-deformed quark fields.

We start by writing a mass term for the *q*-deformed quark fields

where Y_{1,2} = *U,D* are the components of Y.

The *q*-deformed quark fields can be written in terms of the standard fields as

or

and y_{1,2} = *u,d*. Here both components are modified in the same way, so that the above expressions are different from he ones used in [4, 5], where only one component is affected. Extending the deformation to the two components is required to obtain a set of terms that will form an interaction of the NJL type. This implies that the anti-commutation relations for the deformed fields Y will also be different from the ones in [4, 5]. Since obtaining the new anti-commutation relations is not in the scope of this work, we focus on the effective interactions contained in the non-interacting Lagrangian.

Using Eqs. (4) and (5), we can re-write the Lagragian Eq.(1) in terms of the non-deformed quark fields

where *Q* = (q^{-1}-1).

We can re-write the above equations as follows

so that we identify the contact interactions between the quarks contained in the non-interacting deformed fields Lagrangian. The six-point contact terms are depicted in Fig. 1.

We can reduce the six-point interactions of Fig. 1 to four-point contact terms in a mean field approach [2], so that we have

where áY^{}Yñ = á*u*^{}*u*ñ = á*d*^{}*d*ñ = r_{v}, áyñ = á*u*ñ = á*d*ñ = r_{s}, and *A* = *A*(*T*;*q*) has the same dimension of the condensate and will be determined later in this work. This procedure corresponds to have one fermion line closed and can be represented by the diagrams of Fig. 2.

Now we can write the mass term for the *q*-deformed quark fields

with G = *Q/A*

Accordingly, the kinetic energy term for the deformed fields, g^{m}¶_{m}Y, can be written in terms of the non-deformed ones as

By using an extreme mean field approximation, namely, substituting everywhere in the kinetic energy contribution áY^{}Yñ = á*u*^{}*u*ñ = á*d*^{}*d*ñ ® r_{v}, and áyñ = á*u*ñ = á*d*ñ c® r_{s}, we obtain

This corresponds to a usual kinetic energy with a shifted momentum *p *® *p*(1 + Gr_{v})^{2}.

The treatment of the density dependence of the kinetic energy term is rather cumbersome and will be postponed to a further contribution. We will consider the influence of this momentum dependent kinetic energy term in an effective way. Therefore, we will study a class of Lagrangians of the type

This representative of the full Lagrangian *L*_{q} = g^{m}¶_{m}Y + , when written in terms of the standard quark fields, can be identified with the NJL Lagrangian

The conditions for both Lagrangians, *L*_{NJL} and , to be equivalent for any values of *T* and *q* are

and

If we insert Eq. (14) in Eq. (15), we obtain an equation for G

where

This equation has two solutions

The mass of the *q*-deformed fermion fields, *M*, has to be positive, so we associate the two solutions G_{-} and G_{+} with the two regimes *q* < 1 and *q* > 1, respectively. The quantity *A* will be negative in both cases.

The scalar (r_{s}) and vector (r_{v}) densities were calculated from the NJL model at finite temperature:

where

and

are the fermions and anti-fermions distribution functions respectively with *E* = .

First solve the set of coupled gap equations for *m*, µ, and r_{v} (Eqs. and 20, respectively) in the NJL model at finite temperature and chemical potential

The next step is to calculate the scalar and vector densities entering in the equation for G for a given value of the deformation parameter *q*. In this way we obtain *A*(*T;q*), which in turn is used to obtain *M*. The numerical results are displayed in Figures 3 and 4, where we show the quantity *A*, in units of the condensate at zero temperature, as a function of both temperature and deformation for the *q* > 1 and *q* < 1 regimes.

It is worth to note that the mass of the *q*-deformed fermion fields, *M*, does not depend on the deformation of the algebra while its temperature dependence is shown in 5.

The quantity *A*(*T;q*) maps the simple non-interacting model into the NJL model. It represents, in an effective way, the correlations introduced by the quantum algebra, when we write the non-interacting Lagrangian in terms of the standard quark fields. These correlations, in a mean field approximation, are effectively represented by contact interactions of the NJL type. It is also important to mention that it inherits the phase transition. When the condensate and the dynamical mass vanishes with increasing *T*, the quantity A also experiences the phase transition. This is an expected behavior, since it depends on the dynamical mass. For a given temperature, *T*, and deformation, *q*, there is a value of the mapping function, *A*(*T;q*), that makes the Lagrangians Eq.(12) and Eq.(13) equivalent.

Summarizing, we have shown that it is possible to describe the dynamics of an interacting system of the NJL type with a simple non-interacting system by using a set of quantum algebra transformations and a mapping function.

**Acknowledgments**

This work was partially supported by FAPESP Grant No. 2002/10896-7. V.S.T. would like to thank FAPESP for financial support under grant number 2005/01911-0.

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Received on 29 September, 2006

* Talk given at RETINHA 18, IFUSP, São Paulo, May 2006. Dedicated to Prof. Yogiro Hama in honor of his 70th birth.