Search

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., Comment: 11 pages, no figure

The effect that the current quark mass $M_0$ may result in nonzero-ness of chiral condensates is systematically reexamined and analyzed in a two-flavor Nambu-Jona-Lasinio model simulating Quantum Chromodynamics (QCD) at temperature $T$ and finite quark chemical potential $\mu$ without and with electrical neutrality (EN) condition and at any $T$ and $\mu$ without EN condition. By means of a quantitative investigation of the order parameter $m$, it is shown that a nonzero $M_0$ is bound to lead to nonzero quark-antiquark condensates throughout chiral phase transitions , no matter whether the order parameter $m$ varies discontinuously or continuously. In fact, a complete disappearance of the quark-antiquark condensates are proven to demand the non-physical and unrealistic conditions $\mu \,\geq$ or $\gg\, \sqrt{\Lambda^2+M_0^2}$ if $T=0$ and finite, or $T\to \infty$ if $\mu<\sqrt{\Lambda^2+M_0^2}$, where $\Lambda$ is the 3D momentum cut of the loop integrals. Theoretically these results show that when $M_0$ is included, we never have a complete restoration of dynamical (spontaneous) chiral symmetry breaking, including after a first order chiral phase transition at low $T$ and high $\mu$. In physical reality, it is the nonzero-ness of the quark-antiquark condensates that leads to the appearance of a critical end point in the first order phase transition line and the crossover behavior at high $T$ and/or high $\mu$ cases, rather than a possible tricritical point and a second order phase transition line. They also provide a basic reason for that one must consider the interplay between the chiral and diquark condensates in the research on color superconductor at zero $T$ and high $\mu$ case. The research shows that how a source term of the Lagrangian (at present i.e. the current quark mass term) can greatly affect dynamical behavior of a physical system., Comment: 16 pages, no figure. ed. A. Reimer ( Nova Science Publishers, Inc. New York, 2010 )

The color number $N_c$-dependence of the interplay between quark-antiquark condensates $<\bar{q}q>$ and diquark condensates $$ in vacuum in two-flavor four-fermion interaction models is researched. The results show that the $G_S$-$H_S$ (the coupling constant of scalar $(\bar{q}q)^2$-scalar $(qq)^2$ channel) phase diagrams will be qualitatively consistent with the case of $N_c=3$ as $N_c$ varies in 4D Nambu-Jona Lasinio model and 2D Gross-Neveu (GN) model. However, in 3D GN model, the behavior of the $G_S$-$H_P$ (the coupling constant of pseudoscalar $(qq)^2$ channel) phase diagram will obviously depend on $N_c$. The known characteristic that a 3D GN model has not the coexistence phase of the condensates $<\bar{q}q>$ and $$ is proven to appear only in the case of $N_c\leq 4$. In all the models, the regions occupied by the phases containing the diquark condensates $$ in corresponding phase diagrams will gradually decrease as $N_c$ grows up and finally go to zero if $N_c \to \infty$, i.e. in this limit only the pure $<\bar{q}q>$ phase could exist., Comment: 15 pages, 4 figures

We study a typical complete gluonic phase (LGP) in two-flavor color superconductivity (2SC) by calculating the essential cubic and quartic interfering term between the gluonic condensates $$ and $$ with a gauged NJL model. It is proven that the coefficient of the cubic interfering term and the vacuum contributions of the cubic and quartic interfering term are all equal to zeroes. The coefficients of the quartic interfering term and the $$'s quartic self-interaction term at stationary points of LOFF phase are calculated. Comparisons among the effective potentials of LGP, g2SC and LOFF phase indicate that LGP could be the genuine ground state of 2SC for some reasonable parameters., Comment: 4 pages, 3 figures

A general research on chiral symmetry restoring phase transitions at zero temperature and finite chemical potentials under electrical neutrality condition has been conducted in a Nambu-Jona-Lasinio model to describe two-flavor normal quark matter. Depending on that $m_0/\Lambda$, the ratio of dynamical quark mass in vacuum and the 3D momentum cutoff in the loop integrals, is less or greater than 0.413, the phase transition will be second or first order. A complete phase diagram of $u$ quark chemical potential versus $m_0$ is given. With the electrical neutrality constraint, the region where second order phase transition happens will be wider than the one without electrical neutrality limitation. The results also show that, for the value of $m_0/\Lambda$ from QCD phenomenology, the phase transition must be first order., Comment: 9 pages, 1 figure

Theoretical analysis of interplay between the condensates $<\bar{q}q>$ and $$ in vacuum is generally made by relativistic effective potentials in the mean field approximation in 2D, 3D and 4D models with two flavor and $N_c$ color massless fermions. It is found that in ground states of these models, interplay between the two condensates mainly depend on the ratio $G_S/H_S$ for 2D and 4D case or $G_S/H_P$ for 3D case, where $G_S$, $H_S$ and $H_P$ are respectively the coupling constants in a scalar $(\bar{q}q)$, a scalar $(qq)$ and a pseudoscalar $(qq)$ channel. In ground states of all the models, only pure $<\bar{q}q>$ condensates could exist if $G_S/H_S$ or $G_S/H_P$ is bigger than the critical value $2/N_c$, the ratio of the color numbers of the fermions entering into the condensates $$ 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 $$ 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., Comment: 11 pages, 4 figures, General Poster 76 (Physical sciences) at 2007 AAAS Annual Meeting, San Francisco, Feb. 14-20, 2007, minor typos corrected,acknowledgment to financial supporter added

The effective potential analysis indicates that, in a 3D two-flavor Gross-Neveu model in vacuum, depending on less or bigger than the critical value 2/3 of $G_S/H_P$, where $G_S$ and $H_P$ are respectively the coupling constants of scalar quark-antiquark channel and pseudoscalar diquark channel, the system will have the ground state with pure diquark condensates or with pure quark-antiquark condensates, but no the one with coexistence of the two forms of condensates. The similarities and differences in the interplay between the quark-antiquark and the diquark condensates in vacuum in the 2D, 3D and 4D two-flavor four-fermion interaction models are summarized., Comment: 5 pages, revtex4, no figure, minor typos corrected

By means of a relativistic effective potential, we have analytically researched competition between the quark-antiquark condensates $<\bar{q}q>$ and the diquark condensates $$ in vacuum in ground state of a two-flavor Nambu-Jona-Lasinio (NJL) model and obtained the $G_S-H_S$ phase diagram, where $G_S$ and $H_S$ are the respective four-fermion coupling constants in scalar quark-antiquark channel and scalar color anti-triplet diquark channel. The results show that, in the chiral limit, there is only the pure $<\bar{q}q>$ phase when $G_S/H_S>2/3$, and as $G_S/H_S$ decreases to $2/3>G_S/H_S\geq 0$ one will first have a coexistence phase of the condensates $<\bar{q}q>$ and $$ and then a pure $$ phase. In non-zero bare quark mass case, the critical value of $G_S/H_S$ at which the pure $<\bar{q}q>$ phase will transfer to the coexistence phase of the condensates $<\bar{q}q>$ and $$ will be less than 2/3. Our theoretical results, combined with present phenomenological fact that there is no diquark condensates in the vacuum of QCD, will also impose a real restriction to any given two-flavor NJL model which is intended to simulate QCD, i.e. in such model the resulting smallest ratio $G_S/H_S$ after the Fierz transformations in the Hartree approximation must be larger than 2/3. A few phenomenological QCD-like NJL models are checked and analyzed., Comment: 7 pages, revtex4, 1 figure, minor typos corrected

We derive the effective potentials for composite operators in a Nambu-Jona-Lasinio (NJL) model at zero and finite temperature and show that in each case they are equivalent to the corresponding effective potentials based on an auxiliary scalar field. The both effective potentials could lead to the same possible spontaneous breaking and restoration of symmetries including chiral symmetry if the momentum cutoff in the loop integrals is large enough, and can be transformed to each other when the Schwinger-Dyson (SD) equation of the dynamical fermion mass from the fermion-antifermion vacuum (or thermal) condensates is used. The results also generally indicate that two effective potentials with the same single order parameter but rather different mathematical expressions can still be considered physically equivalent if the SD equation corresponding to the extreme value conditions of the two potentials have the same form., Comment: 7 pages, no figure

By critical analyses of the order parameter of symmetry breaking, we have researched the phase transitions at high density in D=2 and D=3 Gross-Neveu (GN) model and shown that the gap equation obeyed by the dynamical fermion mass has the same effectivenesss as the effective potentials for such analyses of all the second order and some special first order phase transitions. In the meantime we also further ironed out a theoretical divergence and proven that in D=3 GN model a first order phase transition does occur in the case of zero temperature and finite chemical potential., Comment: 8 pages, ReVTeX, no figure, contents of NJL model have been taken off and will be stated separately elsewhere