In 1961, the NJL model was originally developed for the study of interacting nucleons [17]. This is a pioneer chiral model for massless fermions where these particles acquire mass through the spontaneous chiral symmetry breaking. Initially, it consisted of an isospin-doublet built of nucleons, and later it was extended to describe the interaction between quarks as degrees of freedom [24]. In this work, we apply the NJL model to study the chiral dynamical behavior of a system consisting of two kind of light quarks at finite temperature and chemical potential. Our starting point for this investigation is the Lagrangian density [25], which is invariant under global chiral *SU*_{L}(2) ⊗ *SU*_{R}(2) rotations at the massless limit, and is given by

$$\begin{array}{}{\displaystyle {\mathcal{L}}_{NJL}=\overline{\psi}(i\text{\u29f8}\mathrm{\partial}-\hat{m})\psi +g\phantom{\rule{thinmathspace}{0ex}}\left[(\overline{\psi}\psi {)}^{2}+(\overline{\psi}i{\gamma}_{5}{\tau}^{a}\psi {)}^{2}\right]}\end{array}$$(2.1)
where the column vector *ψ* = (*u*, *d*) represents the quark fields with *N*_{f} flavors and *N*_{c} colors, *mˆ* is the bare quark mass matrix *mˆ* = diag(*m*_{u}, *m*_{d}), *g* is the effective coupling constant and *τ*^{a} are the Pauli matrices. We set *m* = *m*_{u} = *m*_{d} in this paper.

According to the Fierz transformation, which formulates how the direct and exchange terms are related to each other, the part of the Lagrangian, equation (2.1), which contains the four-point interaction terms (*L*_{IF}) is equivalent to [6, 20]

$$\begin{array}{}{\displaystyle {\mathcal{L}}_{IF}=\frac{g}{8{N}_{c}}[2(\overline{\psi}\psi {)}^{2}+2(\overline{\psi}i{\gamma}_{5}{\tau}^{a}\psi {)}^{2}-2(\overline{\psi}{\tau}^{a}\psi {)}^{2}}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}-2(\overline{\psi}i{\gamma}_{5}\psi {)}^{2}-4(\overline{\psi}{\gamma}_{\mu}\psi {)}^{2}-4(\overline{\psi}i{\gamma}_{\mu}{\gamma}_{5}\psi {)}^{2}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+(\overline{\psi}{\sigma}^{\mu \nu}\psi {)}^{2}-(\overline{\psi}{\sigma}^{\mu \nu}{\tau}^{a}\psi {)}^{2}]\end{array}$$(2.2)

and it leads to the following Lagrangian [26]

$$\begin{array}{}{\displaystyle \mathcal{L}=\overline{\psi}(i\text{\u29f8}\mathrm{\partial}-m)\psi +\hat{\mu}\overline{\psi}{\gamma}_{0}\psi +{G}_{s}(\overline{\psi}\psi {)}^{2}-{G}_{v}(\overline{\psi}{\gamma}_{\mu}\psi {)}^{2}}\end{array}$$(2.3)

where *G*_{s} =
$\begin{array}{}{\displaystyle \left(1+\frac{1}{4{N}_{c}}\right)}\end{array}$
*g* and *G*_{v} = *g*/2*N*_{c} are the scalar and vector couplings, respectively and *μˆ* is the chemical potential matrix *μˆ* = diag(*μ*_{u}, *μ*_{d}). We consider here only the scalar-scalar channel. A term has been introduced in the Hamiltonian density (ℋ) connected via a Legendre transformation to Lagrangian density, equation (2.1), ℋ → ℋ – *μˆ* 𝒩, where 𝒩 = *ψ*_{γ0} *ψ* = *ψ*^{†} *ψ* is the quark number density operator. Only the simple physical case *μ* = *μ*_{u} = *μ*_{d} is considered in this paper.

The NJL model is non-renormalizable due to the four-point interaction and therefore a certain regularization scheme is required to isolate divergences. We work in the 3D cut-off scheme, introducing the cut-off scale, a usual parameter in effective non-renormalizable models [27].

Using the mean-field approximation [3]

$$\begin{array}{}{\displaystyle (\overline{\psi}\mathit{\Gamma}\psi {\mathit{)}}^{\mathit{2}}\u27f6\mathit{2}\u3008\overline{\psi}\mathit{\Gamma}\psi \u3009\overline{\psi}\mathit{\Gamma}\psi \mathit{-}\u3008\overline{\psi}\mathit{\Gamma}\psi {\u3009}^{\mathit{2}}}\end{array}$$(2.4)

where *Γ* can be any one of the 4×4 matrices such as *I*, *γ*_{5}, *γ*_{μ}, *γ*_{5}*γ*_{μ}, we we take 〈*ψψ*〉 and 〈*ψ*_{γ0}〉 as mean fields in the vacuum at finite temperature and density. After this, the Lagrangian in equation (2.3), is simplified to

$$\begin{array}{}{\displaystyle {\mathcal{L}}_{MF}=\overline{\psi}(i\text{\u29f8}\mathrm{\partial}-M+{\mu}_{r}{\gamma}_{0})\psi -\frac{(M-m{)}^{2}}{4{G}_{s}}+\frac{(\mu -{\mu}_{r}{)}^{2}}{4{G}_{v}}}\end{array}$$(2.5)

where M is the effective quark mass defined as

$$\begin{array}{}{\displaystyle M=m-2{G}_{s}\u3008\overline{\psi}\phantom{\rule{thinmathspace}{0ex}}\psi \u3009}\end{array}$$(2.6)

and *μ*_{r} is the renormalized quark chemical potential

$$\begin{array}{}{\displaystyle {\mu}_{r}=\mu -2{G}_{v}\u3008{\psi}^{\u2020}\phantom{\rule{thinmathspace}{0ex}}\psi \u3009}\end{array}$$(2.7)

The equation (2.6) is the well-known NJL gap equation for the dynamical fermion mass M which is related to the quark condensate. The scalar density may be expressed in terms of the quark propagator as

$$\begin{array}{}{\displaystyle \u3008\overline{\psi}\psi \u3009=-i\text{Tr}S(0)=-i\text{Tr}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}\frac{\text{\u29f8}p+M}{{p}^{2}-{M}^{2}}}\end{array}$$(2.8)

Similarly, the quark number density is given by

$$\begin{array}{}{\displaystyle \u3008{\psi}^{\u2020}\psi \u3009=-i\text{Tr}\int \frac{{d}^{4}p}{(2\pi {)}^{4}}\frac{{\gamma}_{0}(\text{\u29f8}p+M)}{{p}^{2}-{M}^{2}}}\end{array}$$(2.9)

where the symbol Tr stands for the trace over color-, flavor- and Dirac-indices. In order to describe the system at non vanishing temperature *T* and chemical potential *μ*, momentum integrals are carried out in the imaginary time formalism

$$\begin{array}{}{\displaystyle i\int \frac{{d}^{4}p}{(2\pi {)}^{4}}f({p}_{0},\overrightarrow{p})\u27f6-T\sum _{n=-\mathrm{\infty}}^{+\mathrm{\infty}}\int \frac{{d}^{3}\overrightarrow{p}}{(2\pi {)}^{3}}f(i{\omega}_{n}+{\mu}_{r},\overrightarrow{p})}\end{array}$$(2.10)

The quark propagator is then defined at discrete imaginary energies *iω*_{n} + *μ*_{r}, where *ω*_{n} = (2*n*+1)*πT*, are the Matsubara frequencies for fermions. After performing the Matsubara sums, integrating over angular components in equations (2.8) and (2.9), and substituting into equations (2.6) and (2.7), respectively, we obtain

$$\begin{array}{}{\displaystyle M(T,\mu )=m+\frac{2{G}_{s}{N}_{f}{N}_{c}}{{\pi}^{2}}\underset{0}{\overset{\mathrm{\Lambda}}{\int}}{p}^{2}\phantom{\rule{mediummathspace}{0ex}}dp\frac{M}{E}\left(1-{f}^{+}-{f}^{-}\right)}\end{array}$$(2.11)

$$\begin{array}{}{\displaystyle {\mu}_{r}(T,\mu )=\mu +\frac{{G}_{v}{N}_{f}{N}_{c}}{{\pi}^{\mathit{2}}}\underset{0}{\overset{\mathrm{\Lambda}}{\int}}{p}^{2}\phantom{\rule{thinmathspace}{0ex}}dp\left({f}^{+}-{f}^{-}\right)}\end{array}$$(2.12)

with

$$\begin{array}{}{\displaystyle {f}^{\pm}(T,{\mu}_{r})=\frac{1}{{e}^{(E\pm {\mu}_{r})/T}+1}}\end{array}$$(2.13)

where
$\begin{array}{}{\displaystyle E=\sqrt{\overrightarrow{p}{\phantom{\rule{mediummathspace}{0ex}}}^{2}+{M}^{2}}}\end{array}$
is the quark energy, *f*^{±} are the Fermi occupation number of quarks (+) and anti-quark (–) and *Λ* is a momentum cut-off.

The scalar and the quark number susceptibilities are of particular interest in this work. It is known that variations of conserved charges are susceptible evidences of the thermal state of the medium as well as its critical behavior. The scalar susceptibility
$\begin{array}{}{\displaystyle {\chi}_{s}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{\mathrm{\partial}M}{\mathrm{\partial}m},}\end{array}$
is generally defined as the response of the constituent quark mass to changes of the bare quark mass [22]. It is related to the order parameter 〈*qq*〉 by

$$\begin{array}{}{\displaystyle {\chi}_{s}=\frac{1}{1-\frac{2{G}_{s}{N}_{c}{N}_{f}}{{\pi}^{2}}\frac{\mathrm{\partial}}{\mathrm{\partial}M}{\int}_{0}^{\mathrm{\Lambda}}{p}^{2}\phantom{\rule{thinmathspace}{0ex}}dp\frac{M}{E}\left[1-{f}^{(+)}-{f}^{(-)}\right]}}\end{array}$$(2.14)

In order to measure the first order response of quark number density to a change of the quark chemical potential, we consider the quark number susceptibility *χ*_{q} [28, 29], which is defined as

$$\begin{array}{}{\displaystyle {\chi}_{q}=\frac{\mathrm{\partial}\u3008{\psi}^{\u2020}\psi \u3009}{\mathrm{\partial}{\mu}_{r}}}\end{array}$$(2.15)

From this information and the divergences present in the susceptibilities, we determine the location and characteristics of the chiral phase transition for a system of quarks with parameters *T* and *μ*. Then, we obtain the phase diagram for the constituent quark mass *M*, as well as the critical behavior and the position of the CEP.

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