Effective interactions from q-deformed inspired transformations

TL;DR

This paper derives effective NJL-type contact interactions from q-deformed inspired transformations of quark fields, providing a new way to model quark interactions.

Contribution

It introduces a novel approach using q-deformed inspired transformations to generate effective NJL interactions from transformed quark fields.

Findings

Derivation of NJL contact interactions from transformed quark fields.

Discussion of model parameters mapping non-interacting to interacting quarks.

Insight into effective interactions via q-deformed transformations.

Abstract

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

Figures

Equations

$${\mathcal{L}}_q^{mass}$$

$$\Psi=\left(\begin{array}[c]{cc}\Psi_{1}&\\ \Psi_{2}&\end{array}\right)=\left(\begin{array}[c]{cc}U&\\ D&\end{array}\right)~{}.$$

$${\Psi }_1$$

$${\Psi }_2$$

$$U$$

$$D$$

$$\psi=\left(\begin{array}[c]{cc}\psi_{1}&\\ \psi_{2}&\end{array}\right)=\left(\begin{array}[c]{cc}u&\\ d&\end{array}\right)~{}.$$

$$\overline{U}U$$

$$\overline{D}D$$

$$\overline{U}U$$

$$\overline{D}D$$

$$\left(\overline{U}U+\overline{D}D\right)$$

$${\mathcal{L}}_q^{mass}=-M\overline{\Psi }\Psi =-M\left(1+2⟨{\psi }^{†}\psi ⟩\Gamma \right)\overline{\psi }\psi -\frac{M}{2}⟨\overline{\psi }\psi ⟩{\Gamma }^2\overline{\psi }\psi \overline{\psi }\psi ,$$

$$\overline{\Psi }{\gamma }^{\mu }{\partial }_{\mu }\Psi$$

$${\mathcal{L}}_{\mathrm{q}}^{\prime }$$

$${\mathcal{L}}_{\mathrm{NJL}}=\overline{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi -m_0\overline{\psi }\psi +G\overline{\psi }\psi \overline{\psi }\psi .$$

$$M=\frac{{\left(1+\Gamma {\rho }_{\mathrm{v}}\right)}^2}{\left(1+2\Gamma {\rho }_{\mathrm{v}}\right)}{m}_0,$$

$$G=-\frac{M}{2}\frac{{\rho }_{\mathrm{s}}{\Gamma }^2}{{\left(1+\Gamma {\rho }_{\mathrm{v}}\right)}^2}.$$

$${\Gamma }^2-2\alpha {\rho }_{\mathrm{v}}\Gamma -\alpha =0,$$

$$\alpha =-\frac{2G}{m_0{\rho }_{\mathrm{s}}}>0.$$

$${\Gamma }_{\pm }=\alpha {\rho }_{\mathrm{v}}\left(1\pm \sqrt{1+\frac{1}{\alpha {\rho }_{\mathrm{v}}^2}}\right).$$

$${\rho }_{\mathrm{s}}$$

$${\rho }_{\mathrm{v}}$$

$$n(\mathbf{p},T,\mu )=\frac{1}{1+\exp \left[\beta \left(E-\mu \right)\right]},$$

$$\overline{n}(\mathbf{p},T,\mu )=\frac{1}{1+\exp \left[\beta \left(E+\mu \right)\right]},$$

$$\left\{\begin{array}[c]{l}m=m_{0}-2G\rho_{\mathrm{s}}~{},\\ \mu=\mu_{0}-\frac{G}{N_{c}}\rho_{\mathrm{v}}~{}.\end{array}\right.$$

Full text

It is very common in physics to use transformations that make one particular system mathematically simpler, yet describing the same phenomena. A clear example is the use of canonical transformations in classical mechanics.

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

In hadron physics, the NJL model 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.

In recent works, 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.

The question we approach in this letter is: is it possible to obtain a 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 a simple transformation of the quark fields, inspired in the $q$-deformed quark fields of previous works [2], [3], [6].

Mass term

We start by writing a mass term for the transformed quark fields

[TABLE]

where

[TABLE]

The transformed quark fields can be written in terms of the standard fields as

[TABLE]

or

[TABLE]

and

[TABLE]

Here both components are modified in the same way, so that the above expressions are different from the ones used in [2, 3], where only one component is affected. Extending the transformation to both 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 $\Psi$ will also be different from the ones in [2, 3]. Since obtaining the new anti-commuation relations is not in the scope of this work, we focus on the effective interactions contained in the non-interacting Lagrangian.

Using Eqs. (5) and (6), we can re-write the Lagragian Eq.(1) in terms of the standard quark fields

[TABLE]

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

We can re-write the above equations as follows

[TABLE]

so that we identify the contact interactions between the quarks contained in the non-interacting deformed fields Lagrangian. Figure 1 shows the six-point contact interactions contained in the mass term for the $q$-deformed quark fields.

We can reduce the six-point interactions to four-point contact terms in a mean field approach [5], so that we have

[TABLE]

where $\langle\psi^{\dagger}\psi\rangle=\langle u^{\dagger}u\rangle=\langle d^{\dagger}d\rangle=\rho_{\mathrm{v}}$, $\left\langle\overline{\psi}\psi\right\rangle=\left\langle\overline{u}u\right\rangle=\left\langle\overline{d}d\right\rangle=\rho_{\mathrm{s}}$, and $A=A(T;q)$ has the same dimension of the condensate and will be determined later in this letter. The reduction of the six-point to four-point contact terms by closing one fermion line is also shown in Figure 1.

Now we can write the mass term for the transformed quark fields

[TABLE]

with $\Gamma=Q/A$

Kinetic energy term

Accordingly, the kinetic energy term for the transformed fields, $\overline{\Psi}\gamma^{\mu}\partial_{\mu}\Psi$, can be written in terms of the standard ones as

[TABLE]

By using an extreme mean field approximation, namely, substituting everywhere in the kinetic energy contribution $\langle\psi^{\dagger}\psi\rangle=\langle u^{\dagger}u\rangle=\langle d^{\dagger}d\rangle\rightarrow\rho_{\mathrm{v}}$, and $\left\langle\overline{\psi}\psi\right\rangle=\left\langle\overline{u}u\right\rangle=\left\langle\overline{d}d\right\rangle\rightarrow\rho_{\mathrm{s}}$, we obtain

[TABLE]

This corresponds to a usual kinetic energy with a shifted momentum $p\rightarrow p\left(1+\Gamma\rho_{\mathrm{v}}\right)^{2}$.

The full Lagrangian

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

[TABLE]

This representative of the full Lagrangian $\mathcal{L}_{q}=\overline{\Psi}\gamma^{\mu}\partial_{\mu}\Psi+\mathcal{L}_{q}^{mass}$, when written in terms of the standard quark fields, can be identified with the NJL Lagrangian

[TABLE]

The conditions for both Lagrangians, $\mathcal{L}_{\mathrm{NJL}}$ and $\mathcal{L}_{\mathrm{q}}^{\prime}$, to be equivalent for any values of $T$ and $q$ are

[TABLE]

and

[TABLE]

Inserting Eq. (18) in Eq. (19), we obtain an equation for $\Gamma$

[TABLE]

where

[TABLE]

This equation has two solutions

[TABLE]

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

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

[TABLE]

where

[TABLE]

and

[TABLE]

are the fermions and anti-fermions distribution functions respectively with $E=\sqrt{\mathbf{p}^{2}+m^{2}}$.

First we solve the set of coupled gap equations for $m$, $\mu$, and $\rho_{\mathrm{v}}$ (Eqs. 27 and 24, respectively) in the NJL model at finite temperature and chemical potential

[TABLE]

The next step is to calculate the scalar and vector densities entering in the equation for $\Gamma$ for a given value of the transformation 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 2 and 3, where we show the quantity $A$ and the mass $M$ as a function of both temperature and transformation parameter in the $q>1$ and $q<1$ regimes. It is worth to note that the mass of the transformed fermion fields does not depend on the transformation parameter.

The well known results of the NJL model are mapped through $A(T;q)$ from the non-interacting transformed fermion fields Lagrangian. It is worth to note that the mass of the $q$-deformed fermion fields does not depend on the deformation of the algebra.

The quantity $A\left(T;q\right)$ maps the simple non-interacting model into the NJL model. It represents, in an effective way, the correlations introduced by the transformations, 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 transformation parameter, $q$, there is a value of the mapping function, $A(T;q)$, that makes the Lagrangians Eq.(16) and Eq.(17) 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 inspired transformations and a mapping function.

Acknowledgments

C. L. L. thanks Profs. D. Galetti and B. M. Pimentel for most helpful discussions. This work was partially supported by FAPESP Grant No. 2002/10896-7. V.S.T. would like to thank FAEPEX/UNICAMP for financial support.