The NJL interaction from q-deformed inspired transformations

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 ModelsIn 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][4][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 fieldswhere Ψ 1,2 = U, D are the components of Ψ. The q-deformed quark fields can be written in terms of the standard fields asor 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 Ψ 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 fieldswhere Q = (q −1 − 1). We can re-write the above equations as followsso 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.