To obtain the equation of state of quark matter and construct hybrid stars, we calculate the thermodynamic potential in the three-flavor Nambu-Jona-Lasinio model including the tensor-type four-point interaction and the Kobayashi-Maskawa-'t Hooft interaction. To construct the hybrid stars, it is necessary to impose the beta-equilibrium and charge neutrality conditions on the system. It is shown that tensor condensed phases appear at large chemical potential. Under the possibility of the existence of the tensor condensates, the relationship between the radius and mass of hybrid stars is estimated.
Within the general framework of SU(2) Nambu-Jona-Lasinio model beyond mean-field approximation, the S-wave $π- π$ scattering lengths $a_0^0$ and $a_0^2$ are calculated. The numerical results are in agreement with that analyzed from recent experiment $πN \to ππN$ data by using Chew-Low-Goebel technique, and the corresponding chiral condensate $<{\bar q}q>$ is around $-(250 {\rm MeV})^3$, which is close to the ones obtained from lattice and sum rules.
In this work we use the equation of state (EOS) of (2+1)-flavor Nambu-Jona-Lasinio (NJL) model to study the structure of the strange quark star. With a new free parameter $α$, the Lagrangian is constructed by two parts, the original NJL Lagrangian and the Fierz transformation of it, as $\mathcal{L}=(1-α)\mathcal{L}_{NJL}+α\mathcal{L}_{Fierz}$. To determine the range of $α$, we compare the binding energies in the 2-flavor and (2+1)-flavor cases. We also consider the constraints of chemical equilibrium and electric charge neutrality in the strange quark star and choose six representative EOSs wi…
A schematic NJL-type model is employed to investigate kaon and pion condensation in deconfined quark matter in the color-flavor locked (CFL) phase, explicitly referring to quark degrees of freedom. To that end we allow for non-vanishing pseudoscalar diquark condensates in addition to the scalar ones which constitute the CFL phase. Color neutrality is ensured by the appropriate choice of color chemical potentials. The dependence of the free energy in the Goldstone condensed phases on quark masses and charge chemical potentials is found to be in good qualitative -- in most cases also quantitativ…
We study the beahviour of the dual quark condensate $Σ_1$ in the Nambu-Jona-Lasinio (NJL) model and its nonlocal variant. In quantum chromodynamics $Σ_1$ can be realted to the breaking of the center symmetry and is therefore an (approximate) order parameter of confinement. The deconfinement transition is then signaled by a strong rise of $Σ_1$ as a function of temperature. However, a similar behaviour is also seen in the NJL model, which is known to have no confinement. Indeed, it was shown that in this model the rise of $Σ_1$ is triggered by the chiral phase transition. In order to shed more …
The influence of the magnetic field on the pion superfluidity and the phase structure is analyzed in the framework of the two-flavor Nambu--Jona-Lasinio(NJL) model. To do this, we first derive the thermodynamic potential from the Lagrangian density of the NJL model in the mean field approximation. Using this thermodynamic potential, we get the gap equation of the chiral condensate and the pion condensate. The effect of external magnetic field on the pion condensate is not simple promotion or suppression, which we will discuss in detail in the paper. It is shown that the tricritical point on th…
We have studied the properties of hot and dense quark matter based on the 3-flavor Nambu-Jona-Lasinio (NJL) model as well as its Polyakov-loop extension (pNJL) with scalar-isovector and vector-isovector couplings. Provided a considerable large isospin asymmetry or isospin chemical potential, isospin splittings of constituent mass, chiral phase transition boundary, and critical point for $u$ and $d$ quarks can be observed for positive isovector coupling constants but are suppressed for negative ones. The quark matter symmetry energy decreases with the increasing isovector coupling constant, and…
We study the QCD phase structure in the three-flavor Nambu-Jona-Lasinio model, incorporating the interplay between the chiral and diquark condensates induced by the axial anomaly. We demonstrate that for an appropriate range of parameters of the model, the interplay leads to the low temperature critical point in the phase structure predicted by a previous Ginzburg-Landau analysis. We also show that a Bose-Einstein condensate (BEC) of diquark molecules emerges in the intermediate density region, and as a result, a BEC-BCS crossover is realized with increasing quark chemical potential.
We explore the effect of including quantum fluctuations in the two flavor Nambu$-$Jona-Lasinio model at finite temperature. This is accomplished, in a symmetry preserving way, by including collective and noncollective modes in the one-meson-loop gap equation which originate from poles and branch cuts in the complex plane, respectively. The inclusion of a boson cutoff, $Λ_b$, is necessary to regularize the meson-loop momenta. This new parameter is used to study the influence of going beyond the usual mean field approximation in the quark condensate. As the temperature increases, chiral symmetry…
We analyze the phase diagram of a three-flavor Nambu-Jona-Lasinio model at finite temperature $T$ and chemical potentials $μ_u, μ_d, μ_s$. We study the competition of pion and kaon condensation and we propose a physical situation in which kaon condensation could be led only by light quark finite densities.
Baryon number fluctuations are believed to be good signatures of the QCD phase transition and its CP. Since the fluctuations are proportional to the various order baryon-number susceptibilities and the quark-number density is related to the dressed quark propagator, then by the two flavor NJL model we can calculate the moments of the up and down quarks. By comparing the two flavor NJL model results with the experimental data of $Sσ$ and $κσ^2$ for the net-proton distributions of Au+Au collisions by STAR, we found that the NJL results fit the experiments better at large collision energies and s…
With the isovector coupling constants adjusted to reproduce the physical pion mass and lattice QCD results in baryon-free quark matter, we have carried out rigourous calculations for the pion condensate in the 3-flavor Nambu-Jona-Lasinio model, and studied the 3-dimensional QCD phase diagram. With the increasing isospin chemical potential $μ_I$, we have observed two nonzero solutions of the pion condensate at finite baryon chemical potentials $μ_B$, representing respectively the pion superfluid phase and the Sarma phase, and their appearance and disappearance correspond to a second-order (firs…
Employing an extended three flavor version of the NJL model we discuss in detail the phase diagram of quark matter. The presence of quark as well as of diquark condensates gives raise to a rich structure of the phase diagram. We study in detail the chiral phase transition and the color superconductivity as well as color flavor locking as a function of the temperature and chemical potentials of the system.
We use a Nambu-Jona Lasinio type model to investigate the phase diagram of dense quark matter under neutron star conditions in mean field approximation. The model contains selfconsistently determined quark masses and allows for diquark condensation in the scalar as well as in the pseudoscalar channel. The latter gives rise to the possibility of K^0 condensation in the CFL phase. In agreement with earlier studies we find that this CFLK^0 phase covers large regions of the phase diagram and that the predominant part of this phase is fully gapped. We show, however, that there exists a region at ve…
We discuss the quark condensate of the vacuum inside the baryon. We analyze the 1+1 dimensional chiral bag in analogy with the realistic 3+1 dimensional one. The Nambu--Jona-Lasinio (NJL) type interaction is used to investigate the quark condensate in the chiral bag. Considering the strong meson-quark coupling, we solve the mean field solution to the scalar and pseudoscalar channels by including the chiral Casimir effects. The self-consistent equation allows a finite value of the quark condensate and hence a finite dynamical quark mass inside the bag. It is shown that an approximate Cheshire C…
We study the 2-flavor Nambu--Jona-Lasinio (NJL) model at finite temperature, $T$, and chemical potential, $μ$, in the dimensional regularization. The pion and sigma meson masses are calculated at finite $T$ and $μ$. The color superconductivity is also discussed in the extended NJL model. We obtain the phase structure of the chiral and color symmetry in the dimensional regularization and compare the results with ones obtained in the cut-off regularization.
We present results from a lattice Monte Carlo study of the Nambu - Jona-Lasinio model in 3+1 dimensions with a baryon chemical potential mu=/=0. As mu is increased there is a transition from a chirally-broken phase to relativistic quark matter, in which baryon number symmetry appears spontaneously broken by a diquark condensate at the Fermi surface, implying a superfluid ground state. Finite volume corrections to this relativistic BCS scenario, however, are anomalously large.
The Nambu Jona-Lasinio model with a Polyakov loop is extended to finite isospin chemical potential case, which is characterized by simultaneous coupling of pion condensate, chiral condensate and Polyakov loop. The pion condensate, chiral condensate and the Polyakov loop as functions of temperature and isospin chemical potential are investigated by minimizing the thermodynamic potential of the system. The resulting $(T,μ_I)$ phase diagram is studied with emphasis on the critical point and Polyakov loop dynamics. The tricritical point for the pion superfluidity phase transition is confirmed and …
We calculate numerically the phase diagram of the three-flavor Nambu-Jona-Lasinio model at zero and finite temperature as a function of the up, down, and strange quark chemical potentials. We focus on the competition between pseudoscalar condensation and color superconductivity. We find that the two types of phases are separated by first-order transitions.
We formulate the nonlocal NJL model with a momentum dependent constituent quark mass and calculate pion light cone wave functions of twist 2 and 3. The leading twist wave function is not asymptotic and agrees well with the new CLEO data. Normalization conditions for the twist 3 wave functions are used to calculate the quark condensate. A prescription to calculate the gluon condensate is proposed. The numerical value of the gluon condensate nicely agrees with the phenomenological value, whereas the quark condensate is larger than the phenomenological value of $-(250$ MeV$)^3$. The relation betw…
Investigations of deconfined quark matter within NJL-type models are reviewed, focusing on the regime of low temperatures and ``moderate'' densities, which is not accessible by perturbative QCD. Central issue is the interplay between chiral symmetry restoration and the formation of color superconducting phases. In order to lay a solid ground for this analysis, we begin with a rather detailed discussion of two and three-flavor NJL models and their phase structure, neglecting the possibility of diquark pairing in a first step. An important aspect of this part is a comparison with the MIT bag mod…
We present results of the calculation of phase diagrams of the three-flavor NJL model as a function of temperature and different quark chemical potentials. These phase diagrams are an extension of earlier calculations in the literature in which either only the possibility of color superconductivity or the possibility of pseudoscalar condensation was investigated. This study takes into account both options and hence allows one to investigate their competition. It turns out that the color superconducting phase and the phase in which the pseudoscalar mesons condense are separated by a first order…
In this contribution, we summarize our recent studies on the chiral invariant mass and the chiral condensates in neutron star matter. We construct a unified equations of state assuming the crossover phase transition from hadronic matter described by a parity doublet model to quark matter by an Nambu--Jona-Lasinio type quark model. We first show that the chiral invariant mass is constrained to be 600 MeV $\lesssim m_0 \lesssim$ 900 MeV from recent observations of neutron stars. We then determine the density dependence of the chiral condensate in the crossover description, and show that the chir…
We investigate the phase structure of the Nambu--Jona-Lasinio model at zero temperature, allowing for a two-dimensional spatial dependence of the chiral condensate. Applying the mean-field approximation, we consider various periodic structures with rectangular and hexagonal geometries, and minimize the corresponding free energy. We find that these two-dimensional chiral crystals are favored over homogeneous phases in a certain window in the region where the phase transition would take place when the analysis was restricted to homogeneous condensates. It turns out, however, that in this regime …
We study the chiral condensates in neutron star matter from nuclear to quark matter domain. We describe nuclear matter with a parity doublet model (PDM), quark matter with the Nambu--Jona-Lasino (NJL) model, and a matter at the intermediate density by interpolating nuclear and quark matter equations of state. The model parameters are constrained by nuclear physics and neutron star observations. Various condensates in the interpolated domain are estimated from the chemical potential dependence of the condensates at the boundaries of the interpolation. The use of the PDM with substantial chiral …
In this paper we use the equation of state (EOS) of 2+1 flavors Nambu-Jona-Lasinio (NJL) model to study the structure of compact stars. To avoid the ultraviolet divergence, we employ the proper-time regularization (PTR) with an ultraviolet cutoff. For comparison, we fix three sets of parameters, where the constraints of chemical equilibrium and electric charge neutrality conditions are taken into consideration. With a certain interpolation method in the crossover region, we construct three corresponding hybrid EOSs but find that the maximum masses of hybrid stars in the three different cases d…
We present results of numerical simulations of the 3+1d Nambu - Jona-Lasinio model with a non-zero baryon chemical potential mu, with particular emphasis on the superfluid diquark condensate and associated susceptibilities. The results, when extrapolated to the zero diquark source limit, are consistent with the existence of a non-zero BCS condensate at high baryon density. The nature of the infinite volume and zero temperature limits are discussed.
We study the relevance of strange degrees of freedom for nucleon structure functions. For this purpose we employ the three flavor generalization of the collective quantization approach to the chiral soliton of the bosonized Nambu-Jona-Lasinio model. Contrary to many other soliton models the hadronic tensor is tractable in this model. By applying the Bjorken limit to the hadronic tensor we extract the leading twist contributions to the nucleon structure functions at the low energy scale at which the model is assumed to approximate QCD. After transforming to the infinite momentum frame and perfo…
We analyze the color-flavor-locked phase of strange quark matter modelled by the three-flavor Nambu-Jona-Lasinio (NJL) framework with and without magnetic field and discuss some additional constraints on the stability scenario when a high magnetic field is applied. We compare the results obtained by employing the MIT Bag Model and discuss the pairing gap behavior and its influence on the equation of state.
In this paper, we study the pseudo-Wigner solution of the quark gap equation with a recently proposed algorithm in the framework of the (2+1)-flavor Nambu-Jona-Lasinio (NJL) model. We find that for the current quark mass $m_{\rm u,d}=5.5$ MeV and chemical potential $μ<μ_{\rm TCP}=272.5$ MeV, the Nambu solution and the positive pseudo-Wigner solution obtained via this algorithm is consistent with the physical solution obtained with the iterative method. Furthermore, the algorithm we used can help to illustrate the evolution of the solutions of the gap equation from the chiral limit to non-chira…
In a previous work, we proposed an extended Nambu--Jona-Lasinio (NJL) model including heavy quark flavors. In this work, we will calculate strong and radiative decays of vector mesons in this extended NJL model, including light $ρ$, $ω$, $K^*$, $ϕ$ and heavy $D^*$, $D^*_s$, $B^*$, $B^*_s$.
Density screening effects in the NJL model: Chiral condensate, speed of sound, and the Critical End Point
Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo（米却肯自治大学物理与数学研究所）；Facultad de Ingeniería Eléctrica, Universidad Michoacana de San Nicolás de Hidalgo（米却肯自治大学电气工程学院）；Centro de Ciencias Exactas, Universidad del Bío-Bío（比奥比奥大学精确科学中心）；Universidad Aeronáutica en Queretaro（克雷塔罗航空大学）
We analyze the thermodynamical properties, at finite density and nonzero temperature, of the (1+1)-dimensional chiral Gross-Neveu model (the NJL_2 model), using the exact inhomogeneous (crystalline) condensate solutions to the gap equation. The continuous chiral symmetry of the model plays a crucial role, and the thermodynamics leads to a broken phase with a periodic spiral condensate, the "chiral spiral", as a thermodynamically preferred limit of the more general "twisted kink crystal" solution of the gap equation. This situation should be contrasted with the Gross-Neveu model, which has a di…
In this paper we explore the chiral phase transition in QCD within the three-flavor Nambu-Jona-Lasinio (NJL) model with a negative coupling constant in the isoscalar axial-vector channel, which is associated with a polarized instanton--anti-instanton molecule background. The QCD phase diagram described in this scenario shows a new first order phase transition around the transition temperature $T_c$ toward a phase without chiral condensates, but with nontrivial dynamic chiral chemical potentials for the light quarks, spontaneously giving rise to local $\mathcal {CP}$ violation and local chirali…
The influence of a constant uniform external chromomagnetic field $H$ on the formation of color superconductivity has been investigated. The consideration was performed in the framework of a (2+1)-dimensional Nambu--Jona-Lasinio model with two different four-fermionic structures responsible for $<\bar qq>$ and diquark $<qq>$ condensates. In particular, it was shown that there exists a critical value $H_c$ of the external chromomagnetic field such that at $H>H_c$ a nonvanishing diquark condensate is dynamically created (the so-called chromomagnetic catalysis effect of color superconductivity). …
The phase diagram at finite temperature and chemical potential for the Polyakov-NJL model with eight quark interactions is reviewed for the SU(3) flavor case.
Temperature dependence of the quark condensate is studied in the Nambu--Jona-Lasinio model with meson loops. Substantial differences are found compared to the results with quark loops only.
We consider the color spin locking (CSL) phase of two-flavor quark matter at zero temperature for nonlocal instantaneous, separable interactions. We employ a Lorentzian-type form factor allowing a parametric interpolation between the sharp (Nambu-Jona-Lasinio (NJL) model) and very smooth (e.g. Gaussian) cut-off models for systematic studies of the influence on the CSL condensate the deviation from the NJL model entails. This smoothing of the NJL model form factor shows advantageous features for the phenomenology of compact stars: (i) a lowering of the critical chemical potential for the onset …
The Nambu--Jona-Lasinio (NJL) model has been extensively studied by many researchers. In previous work we have generalized the NJL model to include a covariant model of confinement. In the present work we consider further modification of the model so as to reproduce the type of Euclidean-space momentum- dependent quark mass values obtained in lattice simulations of QCD. This may be done by introducing a nonlocal interaction, while preserving the chiral symmetry of the Lagrangian. In other work on nonlocal models, by other researchers, the momentum dependence of the quark self-energy is directl…
We discuss the net quark and isovector fluctuations as well as off-diagonal quark flavor susceptibilities along the chiral phase transition line in the Nambu--Jona-Lasinio (NJL) model. The model is formulated at non-zero quark and isospin chemical potentials with non-vanishing vector couplings in the iso-scalar and iso-vector channels. We study the influence of the quark chemical potential on the quark flavour susceptibilities in detail and the dependence of the results on model parameters as well as on the quark mass. The NJL model findings are compared with recent lattice results obtained in…
By comparing the two- and three-flavor Nambu--Jona-Lasinio (NJL) models, we demonstrate that the naively expected vacuum superconductivity (VSC) in constant magnetic field ${\bf B}=B\hat z$ is disfavored due to the splitting magnetic catalysis effect (MCE) to chiral condensates with different quark flavors. Based on the simple two-flavor NJL model, we illuminate, in the lowest Landau level approximation, the similar origins of $π^0$ and $\barρ^+_1$ ($ρ^+$ meson with spin $S_z=1$) mass reductions with smaller $B$ and their different features at larger $B$. With the full Landau levels, the two-f…
We study how the charge neutrality affects the phase structure of three-flavor PNJL model. We point out that, within the conventional PNJL model at finite density the color neutrality is missing because the Wilson line serves as an external ``colored'' field coupled to dynamical quarks. In this paper we heuristically assume that the model may still be applicable. To get color neutrality one has then to allow non vanishing color chemical potentials. We study how the quark matter phase diagram in $(T,m_s^2/μ)$-plane is affected by imposing neutrality and by including the Polyakov loop dynamics. …
Nambu-Jona-Lasinio-type models are often employed as low-energy models for the theory of the strong interaction to analyze its phase structure at finite temperature and quark chemical potential. In particular at low temperature and large chemical potential, where the application of fully first-principles approaches is currently difficult at best, this class of models still plays a prominent role to guide our understanding of the dynamics of dense strong-interaction matter. In this work, we consider a Fierz-complete version of the Nambu-Jona-Lasinio model with two massless quark flavors and stu…
We study the phase structure of the two-flavor Nambu--Jona-Lasinio (NJL) model in the chiral limit, extending a previous study of the competition of an inhomogeneous chiral phase and a two-flavor color-superconducting (2SC) phase [1, 2]. There, an analytic expression for the dispersion relations for quasiparticle excitations in the presence of both a particular inhomogeneous chiral condensate, the so-called chiral density wave (CDW), and a homogeneous 2SC condensate was found. In this work we show how to determine the dispersion relations for arbitrary modulations of the chiral condensate in t…
We discuss three applications of NJL- and PNJL-like models to assess aspects of the QCD phase diagram: First, we study the effect of mesonic correlations on the pressure below and above the finite temperature phase transition within a nonlocal PNJL model beyond the mean-field approximation. Second, we reconstruct the phase boundary of an NJL model from a Taylor expansion of the chiral susceptibility about $μ= 0$ and compare the result with the exact phase boundary. Finally, we demonstrate the realization of the "non-standard scenario" for the critical surface in a three-flavor PNJL model with …
We study the two flavour non-local Nambu\textemdash Jona-Lasinio (NJL) model in the presence of a magnetic field and explore the chiral crossover in presence of a non-local form of the 't Hooft determinant term. Its coupling is governed by a dimensionless parameter $c$. This term is responsible for the explicit breaking of $U(1)_A$ symmetry. We have attempted a systematic analysis of the model parameters by fitting to self-consistent lattice QCD calculations. Three parameters of the model are fixed by $eB=0$ results from published lattice QCD on the chiral condensate, the pion decay constant (…
We study the QCD phase structure in the three-flavor Nambu--Jona-Lasinio model, incorporating the chiral-diquark interplay due to the axial anomaly. We demonstrate that for a certain range of model parameters, the low temperature critical point predicted by a Ginzburg-Landau analysis appears in the phase diagram. In addition, we show that the axial anomaly presents a new scenario for a possible BEC-BCS crossover in the color-flavor locked phase of QCD.
The effective quark interactions that break explicitly the chiral SU(3)_L X SU(3)_R and U_A(1) symmetries by current-quark mass source terms are considered in NLO in $N_c$ counting. They are of the same order as the 't Hooft flavor determinant and the eight quark interactions that extend the LO Nambu-Jona-Lasinio Lagrangian, and complete the set of non-derivative and spin 0 interactions relevant for the $N_c$ scheme. The bosonized Lagrangian at meson tree level describes accurately the empirical ordering and magnitude of the splitting of states in the low lying pseudoscalar and scalar meson no…
Our understanding of the dynamics and the phase structure of dense strong-interaction matter is to a large extent still built on the analysis of low-energy models, such as those of the Nambu-Jona-Lasinio-type. In this work, we analyze the emergence of the latter class of models at intermediate and low energy scales from fundamental quark-gluon interactions. To this end, we study the renormalization group flow of a Fierz-complete set of four-quark interactions and monitor their strength at finite temperature and quark chemical potential. At small quark chemical potential, we find that the scala…
We give an introduction to the dynamical symmetry breaking with large anomalous dimension. This is the basis of tightly bound composite Higgs models such as walking technicolor, strong ETC technicolor and top quark condensate, which are all characterized by the large anomalous dimension, γ_m \simeq 1 (walking technicolor), 1<γ_m<2 (strong ETC technicolor) and γ_m \simeq 2 (top quark condensate) due to nontrivial short distance dynamics of the gauged Nambu-Jona-Lasinio (NJL) models (gauge theories plus four-fermion interactions). Particular emphasis will be placed on the top quark condensate in…
In this paper, the NJL-type model is used to investigate the color superconductivity. The four-fermion interactions of the NJL-type model are Fierz-transformed into two different classes, i.e., the quark-antiquark and the quark-quark interaction channels, associated with the chiral symmetry breaking and color superconductivity respectively. We conclude that the weighting factor between quark-antiquark and quark-quark interaction channels has significance on the phase structure when the mean-field approximation is employed, and the baryon number density gives a tight constraint on the weighting…
The general quasiparticle propagator in dense quark matter is derived for equal mass quarks. Specialized to an NJL model, this propagator includes one new condensate, $Δ_3$, in addition to the usual CFL condensate, $Δ_1$. The gap equation is solved in two NJL models and the dependence on the quark mass of the condensates and the gap is presented. Analytic approximations for the condensates are obtained and compared to exact numerical solutions. The results are shown to differ from those obtained by neglecting $Δ_3$, especially for smaller values of $Δ_1$. The two different NJL models presented…
We study pion condensation and the phase structure in a two-flavor Nambu-Jona-Lasinio model in the presence of baryon chemical potential $μ$ and isospin chemical potential $μ_I$at zero and finite temperature. There is a competition between the chiral condensate and a Bose-Einstein condensate of charged pions. In the chiral limit, the chiral condensate vanishes for any finite value of the isospin chemical potential, while there is a charged pion condensate that depends on the chemical potentials and the temperature. At the physical point, the chiral condensate is always nonzero, while the charg…
