Optimized perturbation theory applied to the study of the thermodynamics and BEC-BCS crossover in the three-color Nambu–Jona-Lasinio model

The Nambu–Jona-Lasinio model with two flavors, three colors, and diquark interactions is analyzed in the context of optimized perturbation theory (OPT). Corrections to the thermodynamical potential that go beyond the large-$N_c$ (LN) approximation are taken into account, and the region of the phase diagram corresponding to intermediate chemical potentials and very low temperatures is explored. The simultaneous presence of both the quark-antiquark and diquark condensates can cause the system to behave as a fluid composed of a Bose-Einstein condensate (BEC) or a color superconductor one, in the form of a Bardeen-Cooper-Schrieffer (BCS) superfluid. The BEC-BCS crossover is then studied in the nonperturbative OPT scheme. The results obtained in the context of the OPT method are then contrasted with those obtained in the LN approximation. We show that there are values for the coupling constants related to quark-quark and quark-antiquark interactions where the corrections beyond LN brought by the OPT method can influence the behavior of the diquark condensate and the effective quark mass as a function of the baryon chemical potential. These changes in the behavior of the phase structure of the model modify the location of the critical point related to the phase structure as a whole of the model. Also, when we impose the color neutrality condition, our results show that the nature of the phase transition can change as well, shifting the ratio of the quark-antiquark and quark-quark interactions to higher values in the OPT case as compared to the LN approximation.

Figure 1

The representation of the Feynman rule elements in the OPT method. (a) Fermionic propagator related to the $\delta$-dependent quark $\mathrm{\Psi }$-field. (b) Fermionic propagator when $\delta =0$. (c) Propagator related to the boson (auxiliary) $\zeta$-field. (d) Propagator related to each of three ${\pi }_i$-fields. (e) Propagator related to the diquark auxiliary field, real component ${\varphi }_R$. (f) Propagator related to the diquark auxiliary field, imaginary component ${\varphi }_I$. (g), (h), (i), (j) The vertices related to the point interactions between a bosonic and two fermionic fields at $\delta =0$. (If fermionic propagators depend on $\delta$, the lines are represented here as thick ones.)

Figure 2

One-loop diagrams in OPT expanded up to order ${\delta }^1$: The thick, continuous line represents the fermionic propagator as a function of $\delta$. Thin lines represent the propagator when $\delta =0$. The dashed line represents the $\sigma$-field propagator, while the dotted one is associated with the $\mathrm{\Delta }$-field propagator. The large black dot represents a $\delta \eta$-vertex insertion, and the white one represents a $\delta \alpha$-vertex insertion.

Figure 3

To the left of the equality, the sum of two-loop diagrams in OPT ($\propto 1/N_c^1$) capable of generating contributions of order ${\delta }^1$. To the right of the equality, sum of two-loops diagrams when expanded up to order ${\delta }^1$. All elements are defined according to the legend of Fig. 1.

Figure 4

Diagrams relevant to the calculation of the pion mass and its decay constant in the OPT expansion up to $\mathcal{O}(\delta )$. The thick, continuous line represents vacuum fermionic propagators for quarks with all colors, which are evaluated when $\mathrm{\Delta }={\mathrm{\Delta }}_c=0$; the thin line represents vacuum fermionic propagators for only quarks with colors 1 and 2; the dashed line represents the chiral bosonic scalar $\zeta$ field propagator; the dash-dotted one represents the pion $\overrightarrow{\pi }$-fields propagator; the dotted one is related to the ${\varphi }_R$ field; and the continuous-dotted line is associated with the ${\varphi }_I$ field. In addition, the vertex pairs represented by an “X” in each diagram can be ${\gamma }^5{\tau }_{i,j}$ (in the case of the pion mass equation) or ${\gamma }^5{\gamma }_{\mu ,\nu }{\tau }_{i,j}$ (in the case of the decay constant equation). All quantities are calculated when $\delta =1$.

Figure 5

Diquark condensate $\mathrm{\Delta }$, effective mass $M_q$, and effective chemical potential ${\mu }_N$ as a function of the baryon chemical potential ${\mu }_B$ for different values of the ratio $G_d/G_s$ in the LN approximation and OPT comparatively.

Figure 6

The effective potential in the OPT at zero temperature with the vacuum energy subtracted, $V_{\mathrm{eff}}\equiv V_{\mathrm{eff}}(\sigma ,\mathrm{\Delta },\overline{\eta },\overline{\alpha },{\mu }_B)-V_{\mathrm{eff}}({\sigma }_{\mathrm{vac}},\mathrm{\Delta }=0,{\overline{\eta }}_{\mathrm{vac}},\overline{\alpha },{\mu }_B=0)$, as a function of $\mathrm{\Delta }$ for different values of baryon chemical potential ${\mu }_B$ around the critical value ${\mu }_{B,c}=0.4686\mathrm{GeV}$, for the ratio $G_d/G_s=1.4$. The evolution of the global mininum of potential when changing ${\mu }_B$ suggests a second-order phase transition in the order parameter $\mathrm{\Delta }$, as occurs in the LN case.

Figure 7

The pressure (top plot), the energy density as a function of ${\mu }_B$ (middle plot), and the equation of state (bottom plot), both for the ratio $G_d/G_s=1.4$.

Figure 8

The vacuum subtracted effective potential in the OPT at zero temperature as a function of $\mathrm{\Delta }$, for different values of the baryon chemical potential ${\mu }_B$ around the critical value ${\mu }_{B,c}=0.7323\mathrm{GeV}$ and for the ratio $G_d/G_s=1.3$. The behavior of the global minimum of potential with the variation of ${\mu }_B$ and the coexistence between the two minima suggests a first-order phase transition in the order parameter $\mathrm{\Delta }$.

Figure 9

The diquark condensate $\mathrm{\Delta }$, the effective quark mass $M_q$, and the effective chemical potential ${\mu }_N$ as a function of the baryon chemical potential ${\mu }_B$ for $G_d/G_s=1.3$, in the LN approximation and OPT comparatively. Color neutrality is considered. The thin vertical line indicates the position of the first-order transition (discontinuity) in $\mathrm{\Delta }$.

Figure 10

The vacuum subtracted effective potential in the OPT at zero temperature as a function of $\mathrm{\Delta }$ for different values of baryon chemical potential ${\mu }_B$ around the critical value ${\mu }_{B,c}=0.4653\mathrm{GeV}$, for $G_d/G_s=1.53$. The behavior of the global minimum of potential with the variation of ${\mu }_B$ suggests a second-order phase transition in the order parameter $\mathrm{\Delta }$. Color neutrality is considered.

Figure 11

Diquark condensate $\mathrm{\Delta }$, effective mass $M_q$, and effective chemical potential ${\mu }_N$ as a function of baryon chemical potential ${\mu }_B$ for different values of $G_d/G_s$ in the OPT case. Color neutrality is considered.

Figure 12

The critical chemical potentials associated with the BEC phase transition ${\mu }_{B,c}$, as a function of $G_d/G_s$, for the LN and OPT cases, in their correspondent validity range. Color neutrality is considered in both cases.

Figure 13

The width of the BEC region, defined by $({\mu }_{B,c}^{\mathrm{BEC}-\mathrm{BCS}}-{\mu }_{B,c})$, as a function of $G_d/G_s$, for the LN and OPT cases, in their correspondent validity range. Color neutrality is considered in both cases.

Figure 14

Particle dispersion relation $E_{\mathrm{\Delta },\overrightarrow{p}}^{\delta =1-}(\mu )$ in the OPT approximation for $G_d/G_s=1.53$. Color neutrality is considered.

Figure 15

The thermodynamic quantities of the system (normalized pressure $P_n$ and normalized energy density ${ϵ}_n$) are given for different values of $G_d/G_s$ as a function of the baryon chemical potential ${\mu }_B$ at zero temperature on the OPT approximation, as well as the state equation $P_n({ϵ}_n)$. Color neutrality is considered.