Difermion condensates in vacuum in 2-4D four-fermion interaction models

TL;DR

This paper investigates how scalar and pseudoscalar diquark and quark-antiquark condensates interact in vacuum within 2D, 3D, and 4D four-fermion models, revealing conditions for their coexistence and phase structures.

Contribution

It provides a comprehensive analysis of the interplay between different condensates in various dimensions, highlighting the dependence on coupling ratios and the existence of coexistence phases.

Findings

Pure quark-antiquark condensates dominate when coupling ratios exceed 2/N_c.

Coexistence of condensates depends on the coupling ratios and model dimensions.

Phase diagrams illustrate the conditions for different condensate phases.

Abstract

Theoretical analysis of interplay between the condensates $<\bar{q}q>$ and $<qq>$ in vacuum is generally made by relativistic effective potentials in the mean field approximation in 2D, 3D and 4D models with two flavor and $N_c$ color massless fermions. It is found that in ground states of these models, interplay between the two condensates mainly depend on the ratio $G_S/H_S$ for 2D and 4D case or $G_S/H_P$ for 3D case, where $G_S$, $H_S$ and $H_P$ are respectively the coupling constants in a scalar $(\bar{q}q)$, a scalar $(qq)$ and a pseudoscalar $(qq)$ channel. In ground states of all the models, only pure $<\bar{q}q>$ condensates could exist if $G_S/H_S$ or $G_S/H_P$ is bigger than the critical value $2/N_c$, the ratio of the color numbers of the fermions entering into the condensates $<qq>$ and $<\bar{q}q>$. As $G_S/H_S$ or $G_S/H_P$ decreases to the region below $2/N_c$, differences of the models will manifest themselves. Depending on different models, and also on $N_c$ in 3D model, one will have or have no the coexistence phase of the two condensates, besides the pure $<qq>$ condensate phase. The $G_S-H_S$ (or $G_S-H_P$) phase diagrams in these models are given. The results also implicate a real constraint on two-flavor QCD-analogous NJL model.

Full text

It is found that in ground states of these models, interplay between the two condensates mainly depend on the ratio $G_{S}/H_{S}$ for 2D and 4D case or $G_{S}/H_{P}$ for 3D case, where $G_{S}$, $H_{S}$ and $H_{P}$ are respectively the coupling constants in a scalar $(\bar{q}q)$, a scalar $(qq)$ and a pseudoscalar $(qq)$ channel.

In ground states of all the models, only pure $\langle\bar{q}q\rangle$ condensates could exist if $G_{S}/H_{S}$ or $G_{S}/H_{P}$ is bigger than the critical value $2/N_{c}$, the ratio of the color numbers of the fermions entering into the condensates $\langle qq\rangle$ and $\langle\bar{q}q\rangle$. Below it, differences of the models will manifest themselves.

In the 4D Nambu-Jona-Lasinio (NJL) model, as $G_{S}/H_{S}$ decreases to the region below $2/N_{c}$, one will first have a coexistence phase of the two condensates then a pure $\langle qq\rangle$ condensate phase. Similar results come from a renormalized effective potential in the 2D Gross-Neveu model, except that the pure $\langle qq\rangle$ condensates could exist only if $G_{S}/H_{S}=0$. In a 3D Gross-Neveu model, when $G_{S}/H_{P}<2/N_{c}$, the phase transition similar to the 4D case can arise only if $N_{c}>4$, and for smaller $N_{c}$, only a pure $\langle qq\rangle$ condensate phase exists but no coexistence phase of the two condensates happens. The $G_{S}-H_{S}$ (or $G_{S}-H_{P}$) phase diagrams in these models are given.

The results deepen our understanding of dynamical phase structure of four-fermion interaction models in vacuum. In addition, in view of absence of difermion condensates in vacuum of QCD, they will also imply a real restriction to any given two-flavor QCD-analogous NJL model, i.e. in the model, the derived smallest ratio $G_{S}/H_{S}$ via the Fierz transformations in the Hartree approximation must be bigger than 2/3.

I Main results

We have researched interplay between the fermion($q$)-antifermion ($\bar{q}$) condensates $\langle\bar{q}q\rangle$ and the difermion condensates $\langle qq\rangle$ in vacuum in 2D, 3D and 4D four-fermion interaction models with two flavor and $N_{c}$ color massless fermions. It is found that the ground states of the systems could be in different phases shown in the following $G_{S}-H_{S}$ and $G_{S}-H_{P}$ phase diagrams [Fig.(a)–Fig.(d)], where

$G_{S}$

— coupling constant of scalar $(\bar{q}q)^{2}$ channel

$H_{S}$

— coupling constant of scalar color $\frac{\displaystyle N_{c}(N_{c}-1)}{\displaystyle 2}\;-$plet $(qq)^{2}$ channel (4D, 2D case)

$H_{P}$

— coupling constant of pseudoscalar color $\frac{\displaystyle N_{c}(N_{c}-1)}{\displaystyle 2}\;-$plet $(qq)^{2}$ channel (3D case)

$\Lambda$

— Euclidean Momentum cutoff of loop integrals (4D,3D case)

$(\sigma_{1},0)$ — pure $\langle\bar{q}q\rangle$ phase

$(0,\Delta_{1})$ — pure $\langle qq\rangle$ phase

$(\sigma_{2},\Delta_{2})$ — mixed phase with both $\langle\bar{q}q\rangle$ and $\langle qq\rangle$

Fig.(a)-Fig.(d)   (pages 3-6)