Possible Effects of Fierz Transformations on Vacua of Some Four-Fermion Interaction Models

TL;DR

This paper investigates how Fierz transformations influence the vacuum states of four-fermion interaction models across different dimensions, revealing conditions under which mixed or pure condensate phases occur based on parameters like N_c and coupling strength.

Contribution

It systematically analyzes the effects of Fierz transformations on the vacua of four-fermion models in various dimensions, highlighting conditions for phase coexistence and dominance.

Findings

Fierz transformations do not alter the pure $ar{q}$-$q$ condensate phase in certain models.

Vacuum phases depend on N_c and coupling strength, with coexistence phases emerging at larger N_c and H_S.

Similar phase behavior is observed in 2D, 3D, and 4D models.

Abstract

A theoretical research on possible effects of the Fierz transformations on the ground states (vacua) of some 2-flavor and $N_c$-color four-fermion (quark) interaction models has been systematically conducted. It has been shown that, based on the known criterions of the interplay between the antiquark-quark ($\bar{q}$-$q$) and diquark ($q$-$q$) condensates, in 4D space-time, for the given $\bar{q}$-$q$ channel couplings with chiral symmetry and from the heavy gluon exchange, the effects of the Fierz transformations are not enough to change the feature that the models' vacua could only be in the pure $\bar{q}$-$q$ condensate phases. However, for a given pure scalar $q$-$q$ channel coupling with the strength $H_S$, the Fierz transformations will lead to the nontrivial effect that the model's vacuum could be in the expected $q$-$q$ condensate phase only if $N_c<9$ and $H_S$ is small, and as the increase of $N_c$ and $H_S$, the vacuum will get first in a coexistence phase with $q$-$q$ and $\bar{q}$-$q$ condensates then in a pure $\bar{q}$-$q$ condensate phase until $N_c\rightarrow\infty$. Similar conclusions are also drawn from relevant four-fermion interaction models in 2D and 3D space-time. The general significance of the research is indicated.