Two-flavor Nambu Jona-Lasinio Model Research Articles

The finite volume effects on the critical endpoint of chiral phase transition in Nambu–Jona-Lasinio model with different regularization schemes

In this paper, we study the influence of different regularization schemes on the critical endpoint (CEP) of chiral phase transition within a cubic box with volume [Formula: see text]. A two-flavor Nambu–Jona-Lasinio model at finite temperature [Formula: see text] and chemical potential [Formula: see text] is adopted as the effective model of the strong interacting matter. Due to the finite volume of the box, the momentum integral in gap equation is replaced by discrete summation, and an anti-periodic boundary condition for quark field is applied. We employ the Schwinger’s proper time and the Pauli–Villars regularization (PVR) schemes, respectively. It is found that the first-order phase transition line displays an intriguing “staircase” behavior, and eventually disappears as [Formula: see text] increases. In particular, there is no existence of the CEP for both regularization schemes in infinite volume limit [Formula: see text]. However, for the finite volume, the locations of the CEPs with proper time and PVR are determined, respectively.

Enhancement of dilepton production rate and electric conductivity around the QCD critical point

Abstract We investigate whether the soft mode that becomes massless at the QCD critical point (CP) causes an enhancement of the dilepton production rate (DPR) and electric conductivity around the CP through modification of the photon self-energy using the two-flavor Nambu–Jona-Lasinio model. The modification is described by the so-called Aslamazov–Larkin, Maki–Thompson, and density of states terms, which have been taken into account in our previous study on the DPR near the color-superconducting phase transition, with a replacement of the diquark modes with the soft mode of the QCD CP. We show that the coupling of photons with the soft modes brings about an enhancement of the DPR in the low-invariant-mass region and the conductivity near the CP, which would be observable in relativistic heavy-ion collisions.

Phenomenological study of the anisotropic quark matter in the two-flavor Nambu–Jona–Lasinio model

Phenomenological study of the anisotropic quark matter in the two-flavor Nambu–Jona–Lasinio model

Anomalous enhancement of dilepton production as a precursor of color superconductivity

Abstract We compute the modification of the photon self-energy due to dynamical diquark fluctuations developed near the critical temperature of two-flavor superconductivity, which is one of the color superconducting phases, through the Aslamasov–Larkin, Maki–Thompson, and density of states terms, which are responsible for the paraconductivity in metals at vanishing energy and momentum. The analysis is performed through an investigation of the diquark fluctuations within the normal phase determined by the mean-field approximation in the massless two-flavor Nambu–Jona-Lasinio model. It is shown that the rate has a significant enhancement in low-invariant-mass regions over a rather wide range of temperatures in the normal phase. This enhancement is worth exploring in the relativistic heavy-ion collisions, which may thereby reveal the significance of the diquark fluctuations in dense quark matter.

Magnetized pole-mass of neutral ρ meson within full RPA evaluation

In this work we calculate the pole-mass of the ρ0 meson with different spin projections sz=0,±1 in the context of the magnetized two-flavor Nambu–Jona–Lasinio model. Making use of the mean field approximation to obtain the effective quark mass as a function of the magnetic field, we apply the random phase approximation (RPA) to the vector channel in order to calculate the polarization function for each spin component. We adopt the magnetic field independent regularization (MFIR) in our evaluations as a method of separating divergences and the Pauli-Villars regularization for the vacuum contributions. The ρ0 meson mass with spin projection sz=±1 is always catalyzed with the magnitude of the magnetic field, showing good agreement with Lattice QCD results. The mass projection sz=0 has a non-monotonic behavior, decreasing until the minimum at eB≲0.15 GeV2, which is in contrast with available LQCD data.

Exploring hybrid star EOS with constraints from tidal deformability of GW170817

With an interpolation method on the P-μ plane, a hybrid equation of state is explored. The quark phase is described by a self-consistent two-flavor Nambu–Jona-Lasinio model. It retains the contribution from the vector channel in the Fierz-transformed Lagrangian by introducing a weighting parameter α. We study the dependence of hybrid EOS and mass-radius relation on α. It is found that increasing α makes the hybrid EOS softer. We can get stellar mass larger than 2M⊙. Further, we calculate the tidal deformability Λ˜ for binary stars and compare it with some theoretical analysis results on GW170817.

Do we need to use regularization for the thermal part in the NJL model? * *Supported by the Natural Science Foundation of Jiangsu Province (BK20140523), the start-up funding of Jiangsu University (15JDG042, 4111190010), the National Natural Science Foundation of China (11735007)

The Nambu–Jona-Lasinio (NJL) model is one of the most useful tools for studying non-perturbative strong interactions in matter. Because it is a nonrenormalizable model, the choice of regularization is a subtle issue. In this paper, we discuss one of the general issues regarding regularization in the NJL model, which is whether we need to use regularization for the thermal part by evaluating the quark chiral condensate and thermal properties in the two-flavor NJL model. The calculations in this work include three regularization schemes that contain both gauge covariant and invariant schemes. We found that, regardless of the regularization scheme we choose, it is necessary to use regularization for the thermal part when calculating physical quantities related to the chiral condensate and to not use regularization for the thermal part when calculating physical quantities related to the grand potential.

Dependence of the crossover zone on the regularization method in the two-flavor Nambu–Jona-Lasinio model

AbstractIn this article, we study the two-flavor Nambu and Jona-Lasinio (NJL) phase diagrams on the T–μ plane through three regularization methods. In one of these, we introduce an infrared three-momentum cutoff in addition to the usual ultraviolet regularization to the quark loop integrals and compare the obtained phase diagrams with those obtained from the NJL model with proper time regularization and Pauli–Villars regularization. We have found that the crossover appears as a band with a well-defined width in the T–μ plane. To determine the extension of the crossover zone, we propose a novel criterion, comparing it to another criterion that is commonly reported in the literature; we then obtain the phase diagrams for each criterion. We study the behavior of the phase diagrams under all these schemes, focusing on the influence of the regularization procedure on the crossover zone and the presence or absence of critical end points.

Self-consistent mean field approximation and application in three-flavor NJL model

In this study, we apply a self-consistent mean field approximation of the three-flavor Nambu–Jona-Lasinio (NJL) model and compare it with the two-flavor NJL model. The self-consistent mean field approximation introduces a new parameter, , that cannot be fixed in advance by the mean field approach itself. Due to the lack of experimental data, the parameter, , is undetermined. Hence, it is regarded as a free parameter and its influence on the chiral phase transition of strong interaction matter is studied based on this self-consistent mean field approximation. affects numerous properties of the chiral phase transitions, such as the position of the phase transition point and the order of phase transition. Additionally, increasing will decrease the number densities of different quarks and increase the chemical potential at which the number density of the strange quark is non-zero. Finally, we observed that affects the equation of state (EOS) of the quark matter, and the sound velocity can be calculated to determine the stiffness of the EOS, which provides a good basis for studying the neutron star mass-radius relationship.

Charged pion condensation in anti-parallel electromagnetic fields and nonzero isospin density * * J.Y.C. is supported by the NSFC under Grant number: 11605254 and Major State Basic Research Development Program in China (2015CB856903). M.H. is supported by the NSFC under Grant No. 11725523, 11735007 and 11261130311(CRC 110 by DFG and NSFC). A.R. is supported by the CAS President's international fellowship

The formation of charged pion condensate in anti-parallel electromagnetic fields and in the presence of the isospin chemical potential is studied in the two-flavor Nambu–Jona-Lasinio model. The method of Schwinger proper time is extended to explore the quantities in the off-diagonal flavor space, i.e. the charged pion. In this framework, are treated as bound states of quarks and not as point-like charged particles. The isospin chemical potential plays the role of a trigger for charged pion condensation. We obtain the associated effective potential as a function of the strength of the electromagnetic fields and find that it contains a sextic term which possibly induces a weak first order phase transition. The dependence of pion condensation on model parameters is investigated.

Magnetic catalysis effect prevents vacuum superconductivity in strong magnetic fields

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 $\pi^0$ and $\bar{\rho}^+_1$ (${\rho}^+$ 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-flavor NJL model is found to be invalid to study the magnetic field effect to $\bar{\rho}^+_1$ meson with physical vacuum mass $775~{\rm MeV}$. Then, restricted to $\rho$ meson mass below two-quark threshold in vacuum, that is $m_\rho^v<2m_q^v$, it is found that $\pi^0$ mass decreases and then increases with $B$ slowly, and $\bar{\rho}^+_1$ mass vanishing point is delayed to larger $B$ compared to the point particle result. In the more realistic three-flavor NJL model, all the quark masses split in strong magnetic field as a combinatorial result of their different current masses and electric charges. By choosing a vacuum mass closer to the physical one, $\bar{\rho}^+_1$ meson mass is found to be consistent with the LQCD results semi-quantitatively in smaller $B$ region but increase in larger $B$ region. These features are mainly outcomes of the interplay between the $S_z-B$ coupling effect and splitting MCE to the composite $u$ and $d$ quarks, which definitely disfavors VSC when the latter dominates. Furthermore, mesonic flavor mixing is modified by $B$ among the neutral pseudoscalars: $\pi^0,\eta_0$ and $\eta_8$, which is very important to suppress the mass enhancement of the effective mass eigenstates at large $B$.

Spontaneous magnetization under a pseudovector interaction between quarks in high density quark matter

Spontaneous magnetization and magnetic susceptibility originated from the pseudovector-type four-point interaction between quarks are calculated in quark matter with zero temperature and finite quark chemical potential by using the two-flavor Nambu–Jona-Lasinio model. It is shown that both the chiral condensate and spin polarized condensate coexist in a narrow region of the quark chemical potential. And then, it is also shown that, in this narrow region, the spontaneous magnetization appears. Also, the magnetic susceptibility due to quarks with the positive energy is evaluated in the spin polarized phase.

NJL model with the modified quark-dependent coupling strength G

In this paper, the coupling strength G of the Nambu–Jona-Lasinio (NJL) model is modified by incorporating quark’s feedback into the gluon propagator. The modified two-flavor NJL model with the quark-dependent coupling strength is explored. The quark condensate in this framework has a conspicuous agreement with the lattice quantum chromodynamics (QCD) results at finite temperature. Then, it is compared with the original NJL model in both zero (chiral limit) and nonzero current quark mass. The QCD phase diagram and susceptibilities are investigated in the temperature–chemical potential [Formula: see text] plane. Therefore, the pseudo-critical temperature [Formula: see text] and the critical end point (CEP) are worked out and compared with original NJL model or lattice QCD results. In addition, the pion mass and decay constant are studied at finite temperature.

The two-flavor NJL model with two-cutoff proper time regularization

In this paper, we use the two-flavor Nambu–Jona-Lasinio model together with the proper time regularization that has both ultraviolet and infrared cutoffs to study the chiral phase transition at finite temperature and zero chemical potential. The involved model parameters in our calculation are determined in the traditional way. Our calculations show that the dependence of the results on the choice of the parameters are really small, which can then be regarded as an advantage besides such a regularization scheme is Lorentz invariant.

Regions of different critical behavior of the two-flavor Nambu–Jona-Lasinio model

We study the chiral phase diagram in the $T$-$\ensuremath{\mu}$-${m}_{0}$ space of the two-flavor Nambu-Jona-Lasinio model with the approach of the Landau theory of phase transitions. The tricritical point (TCP) is an endpoint of a line of triple points as well as a crosspoint of three $\ensuremath{\lambda}$-transition lines. These double characters bring the connective and consistent problem of a critical phenomenon at the TCP. There is a crossover region from normal critical to tricritical behavior near the TCP, no matter whether the current quark mass is zero or not. The critical exponents need to be renormalized when one approaches the TCP along the first-order phase-transition line.

Influence of strong magnetic fields and instantons on the phase structure of the two-flavor Nambu–Jona-Lasinio model

Both in heavy-ion collisions as in magnetars very strong magnetic fields are produced, which has its influence on the phases of matter involved. In this paper we investigate the effect of strong magnetic fields (B = 5 m_pi^2 /e = 1.7 x 10^19 G) on the chiral symmetry restoring phase transition using the Nambu-Jona-Lasinio model. It is observed that the pattern of phase transitions depends on the relative magnitude of the magnetic field and the instanton interaction strength. We study two specific regimes in the phase diagram, high chemical potential and zero temperature and vice versa, which are of relevance for neutron stars and heavy-ion collisions respectively. In order to shed light on the behavior of the phase transitions we study the dependence of the minima of the effective potential on the occupation of Landau levels. We observe a near-degeneracy of multiple minima with differing occupation numbers, of which some become the global minimum upon changing the magnetic field or the chemical potential. These minima differ considerably in the amount of chiral symmetry breaking and in some cases also of isospin breaking.

Electrical Neutrality and Symmetry Restoring Phase Transitions at High Densityin a Two-Flavor Nambu–Jona-Lasinio Model

A general research on chiral symmetry restoring phase transitionsat zero temperature and finite chemical potentials under electricalneutrality condition has been conducted in a Nambu–Jona-Lasiniomodel to describe two-flavor normal quark matter. Depending on whetherm0/Λ, the ratio of dynamical quark mass in vacuum and the 3Dmomentum cutoff in the loop integrals, is less or greater than 0.413,the phase transition will be of the second or first order.A complete phase diagram of u quark chemical potential versus m0is given. With the electrical neutrality constraint, the region wherethe second order phase transition happens will be wider than the onewithout electrical neutrality limitation. The results also show that,for the value of m0/Λ from QCD phenomenology,the phase transition must be of the first order.

Interplay Between Quark-Antiquark and DiquarkCondensates in Vacuum in a Two-Flavor Nambu–Jona–LasinioModel

By means of a relativistic effective potential, weanalytically research competition between the quark-antiquarkcondensates ⟨q̄q⟩ and the diquark condensates⟨qq⟩ in vacuum in ground state of a two-flavorNambu–Jona–Lasinio (NJL) model and obtain the GS-HS phasediagram, where GS and HS are the respective four-fermioncoupling constants in scalar quark-antiquark channel and scalarcolor anti-triplet diquark channel. The results show that, in thechiral limit, there is only the pure ⟨q̄q⟩phase when GS/HS>2/3, and as GS/HS decreases to2/3>GS/HS⩾0 one will first have a coexistence phase of thecondensates ⟨q̄q⟩ and ⟨qq⟩ andthen a pure ⟨qq⟩ phase. In non-zero bare quarkmass case, the critical value of GS/HS at which the pure⟨q̄q⟩ phase will transfer to the coexistencephase of the condensates ⟨q̄q⟩ and ⟨qq⟩ will be less than 2/3. Our theoretical results,combined with present phenomenological fact that there is nodiquark condensates in the vacuum of QCD, will also impose a realrestriction to any given two-flavor NJL model which is intended tosimulate QCD, i.e. in such model the resulting smallest ratioGS/HS after the Fierz transformations in the Hartreeapproximation must be larger than 2/3. A few phenomenologicalQCD-like NJL models are checked and analyzed.

The phases of isospin-asymmetric matter in the two-flavor NJL model

We investigate the phase diagram of isospin-asymmetric matter at T=0 in the two-flavor Nambu–Jona-Lasinio model. Our approach describes the single nucleon as a confined quark–diquark state, the saturation properties of nuclear matter at normal densities, and the phase transition to normal or color superconducting quark matter at higher densities. The resulting equation of state of charge-neutral matter and the structure of compact stars are discussed.