QCD-like theories possess a positively definite fermion determinant at finite baryon chemical potential $μ_{\text B}$ and the lattice simulation can be successfully performed. While the chiral perturbation theories are sufficient to describe the Bose condensate at low density, to describe the crossover from Bose-Einstein condensation (BEC) to BCS superfluidity at moderate density we should use some fermionic effective model of QCD, such as the Nambu-Jona-Lasinio model. In this paper, using two-color two-flavor QCD as an example, we examine how the Nambu-Jona-Lasinio model describes the weakly interacting Bose condensate at low density and the BEC-BCS crossover at moderate density. Near the quantum phase transition point $μ_{\text B}=m_π$ ($m_π$ is the mass of pion/diquark multiplet), the Ginzburg-Landau free energy at the mean-field level can be reduced to the Gross-Pitaevskii free energy describing a weakly repulsive Bose condensate with a diquark-diquark scattering length identical to that predicted by the chiral perturbation theories. The Goldstone mode recovers the Bogoliubov excitation in weakly interacting Bose condensates. The results of in-medium chiral and diquark condensates predicted by chiral perturbation theories are analytically recovered. The BEC-BCS crossover and meson Mott transition at moderate baryon chemical potential as well as the beyond-mean-field corrections are studied. Part of our results can also be applied to real QCD at finite baryon or isospin chemical potential.
We investigate the rapid rise of the dressed Polyakov loop in the Nambu--Jona-Lasinio (NJL) model as a function of temperature. In QCD such a behaviour is interpreted as a confinement- deconfinement phase transition. However, we demonstrate that in the NJL model this is simply a remnant of the chiral transition.
We point out that adding higher dimension operators to the Nambu-Jona-Lasinio model can stabilize chiral symmetry breaking and resolve the issue raised by Kleinert and Van den Bossche that chiral symmetry breaking cannot occur for $N_c \le N_c^{\rm cr}$ due to large quantum fluctuations of the composite fields.
We study the chiral phase transition of the two flavor Nambu-Jona-Lasinio (NJL) model in a sphere with the MIT boundary condition. We find that the MIT boundary condition results in much stronger finite size effects than the antiperiodic boundary condition. Our work may be helpful to study the finite size effects in heavy-ion collision in a more realistic way.
In the frame of flavor SU(2) Nambu--Jona-Lasinio model with $U_A(1)$ breaking term we found that, the structure of two chiral phase transition lines does not exist at low isospin density in real world, and the critical isospin chemical potential for pion superfluidity is exactly the pion mass in the vacuum.
We discuss the 2+1 flavor Polyakov loop enhanced Nambu--Jona-Lasinio model in a finite volume. The main objective is to check the volume scaling of thermodynamic observables for various temperatures and chemical potentials. We observe the possible violation of the scaling with system size in a considerable window along the whole transition region in the $T-μ_q$ plane.
We discuss the lightest scalar resonances, $f_0(600)$, $κ(800)$, $a_0(980)$ and $f_0(980)$ in the extended Nambu Jona-Lasinio model. We find that the model parameters can be tuned, but unnaturally, to accommodate for those scalars except the $f_0(980)$. We also discuss problems encountered in the K Matrix unitarization approximation by using $N_c$ counting technique.
We investigate strong and radiative meson decays in a generalized Nambu--Jona-Lasinio model. The one loop order calculation provides a satisfactory agreement with the data for the mesonic spectrum and for radiative decays. Higher order effects for strong decays of $ρ$ and $K^*$ are estimated to be large. We also discuss the role of the flavour mixing determinantal interaction.
Using the Nambu--Jona-Lasinio (NJL) model we study responses of the pion and kaon masses to changes in the chemical potential, $dm/dμ$, at zero and finite chemical potential. We find that the behavior of $dm/dμ$ for the pion is quite different from that for the kaon. Our results can give a clue for future studies of $dm/dμ$ on the lattice.
We report the first model study of the net charge fluctuations in terms of D-measure within the framework of the Polyakov-Nambu-Jona-Lasinio model. Net charge fluctuation is estimated from the charge susceptibility evaluated using PNJL model. A parameterization of the freeze-out curve has been used to obtain D as a function of sqrt(s). We have discussed our results vis-a-vis recent experimental findings from ALICE collaboration.
Effect of magnetic field on chiral symmetry breaking in a 3-flavor Nambu Jona Lasinio (NJL) model at finite temperature and densities is considered here using an explicit structure of ground state in terms of quark and antiquark condensates. While at zero chemical potential and finite temperature, magnetic field enhances the condensates, at zero temperature, the critical chemical potential decreases with increasing magnetic field.
We investigate pion superfluidity in the frame of two flavor Nambu--Jona-Lasinio model beyond mean field approximation. We calculate the thermodynamics to the next to leading order in an expansion in the inverse number of colors, including both quark and meson contributions at finite temperature and baryon and isospin density. Due to the meson fluctuations, the Sarma phase which exists at mean field level is washed away, and the Bose-Einstein condensation region at low isospin density is highly suppressed.
We present an analysis of two-current correlations for the pion in the Nambu--Jona-Lasinio model, with Pauli--Villars regularization. We provide explicit expressions in momentum space for two-current correlations corresponding to the zeroth component of the vector Dirac bilinear in the quark vertices, which has been evaluated on the lattice. The numerical results show a remarkable qualitative agreement with recent lattice data. The factorization approximation into one-body currents is discussed.
We study a Nambu-Jona-Lasinio (NJL) type model with two-flavor quark matter in $β$-equilibrium. It turns out that the system develops instabilities in the dispersion relations of the diquark fields, i.e., the velocity squared $v^2$ becomes negative in a certain region of the electron chemical potential. The critical point is the same as that of the chromomagnetic instability. The results imply the existence of spatially inhomogeneous diquark condensates in the genuine vacuum. We also discuss gauge equivalent classes between the inhomogeneous diquark condensates and gluon condensates.
We investigate the QCD phase diagram at zero chemical potential and finite temperature in the presence of an external magnetic field within the three flavor Polyakov-Nambu-Jona-Lasinio and entangled Polyakov-Nambu-Jona-Lasinio models looking for the inverse magnetic catalysis. Two scenarios for a scalar coupling parameter dependent on the magnetic field intensity are considered. These dependencies of the coupling allow to reproduce qualitatively lattice QCD results for the quark condensates and for the Polyakov loop: due to the magnetic field the quark condensates are enhanced at low and high …
In this review, we summarize recent progress on the possible phases of quantum chromodynamics (QCD) in the presence of a strong magnetic field, mainly from the views of the chiral effective Nambu--Jona-Lasinio model. Four kinds of phase transitions are explored in detail: chiral symmetry breaking and restoration, neutral pseudoscalar superfluidity, charged pion superfluidity and charged rho superconductivity. In particular, we revisit the unsolved problems of inverse magnetic catalysis effect and competition between the chiral density wave and solitonic modulation phases. It is shown that usef…
We calculate the shear $η(T)$ and bulk viscosities $ζ(T)$ as well as the electric conductivity $σ_e(T)$ and heat conductivity $κ(T)$ within the Nambu-Jona-Lasinio model for 3 flavors as a function of temperature as well as the entropy density $s(T)$, pressure $P(T)$ and speed of sound $c_s^2(T)$. We compare the results with other models such as the Polyakov-Nambu-Jona-Lasinio (PNJL) model and the dynamical quasiparticle model (DQPM) and confront these results with lattice QCD data whenever available. We find the NJL model to have a limited predictive power for the thermodynamic variables and v…
Given the phenomenological success of the Nambu-Jona-Lasinio model in describing the meson physics in the low energy limit, it is tempting to find the fully relativistically structured nucleon solution in the same model under the similar approximation employed in the mesonic sector. To achieve this goal we need to solve a relativistic Faddeev equation. The factorizability of the two-body T-matrix reduces the three-body Faddeev equation to a tractable two-body Bethe-Salpeter equation. The reduced equation is then solved numerically. Our result indicates that the nucleon consists of three loosel…
We calculate the shear $η(T)$ and bulk viscosities $ζ(T)$ as well as the electric conductivity $σ_e(T)$ and heat conductivity $κ(T)$ within the Nambu-Jona-Lasinio model for 3 flavors as a function of temperature as well as the entropy density $s(T)$, pressure $P(T)$ and speed of sound squared $c_s^2(T)$. We compare the results with other models such as the Polyakov-Nambu-Jona-Lasinio (PNJL) model and the dynamical quasiparticle model (DQPM) and confront these results with lattice QCD data whenever available. We find the NJL model to have a limited predictive power for the thermodynamic variabl…
The dual quark condensate $Σ^{(n)}$ proposed recently as a new order parameter of the spontaneous breaking of the ${\mathbb Z}_3$ symmetry are evaluated by the Polyakov-loop extended Nambu--Jona-Lasinio model, where $n$ are winding numbers. The Polyakov-loop extended Nambu--Jona-Lasinio model well reproduces lattice QCD data on $Σ^{(1)}$ measured very lately. The dual quark condensate $Σ^{(n)}$ at higher temperatures is sensitive to the strength of the vector-type four-quark interaction in the Polyakov-loop extended Nambu--Jona-Lasinio model and hence a good quantity to determine the strength.
The off--shell electromagnetic vertex function of pions and kaons is studied in a bosonized Nambu--Jona-Lasinio model with a gauge--invariant proper--time cutoff. The slope of the pion form factor with respect to the pion 4--momentum is found to be equal to the on--shell pion charge radius in the chiral limit. The off--shell slope of the $K^0$ form factor is zero, that of the $K^\pm$ about 15\% smaller than that of the pion. We compare our results with those of a recent calculation in chiral perturbation theory.
Using the Nambu-Jona-Lasinio model to describe the nucleon as a quark-diquark state, we discuss the stability of nuclear matter in a hybrid model for the ground state at finite nucleon density. It is shown that a simple extension of the model to simulate the effects of confinement leads to a scalar polarizability of the nucleon. This, in turn, leads to a less attractive effective interaction between the nucleons, helping to achieve saturation of the nuclear matter ground state. It is also pointed out that that the same effect naturally leads to a suppression of ``Z-graph'' contributions with i…
It is shown that the difference between the magnetic moment results in the quark model with null instantons and semibosonized Nambu-Jona-Lasinio model lies in the description of the magnetic moment of the $ Λ$-hyperon.
The QCD phase diagram at zero chemical potential and finite temperature subject to an external magnetic field is studied within the three flavor Nambu--Jona-Lasinio (NJL) model and the NJL model with the Polyakov loop (PNJL). A scalar coupling parameter dependent on the magnetic field intensity is considered. The scalar coupling has been fitted so that the lattice QCD pseudocritical chiral transition temperatures are reproduced and in the limit of large magnetic field decreases with the inverse of the magnetic field intensity. This dependence of the coupling allows to reproduce the lattice QCD…
We have investigated shear viscosity of quark matter in presence of a strong uniform magnetic field background where Nambu-Jona-Lasinio model has been considered to describe the magneto-thermodynamical properties of the medium. In presence of magnetic field, shear viscosity coefficient gets split into different components because of anisotropy in tangential stress of the fluid. Four different components can be merged to two components in limit of strong field, where collisional width of quark becomes much lower than its synchrotron frequency. A simplified contact diagram of quark-quark interac…
We have estimated the susceptibilities of conserved charges for 2-flavor strongly interacting matter with varying system sizes, using the Polyakov loop enhanced Nambu Jona-Lasinio model. The susceptibilities for vanishing baryon densities are found to show a scaling with the system volume in the hadronic as well as partonic phase. This scaling breaks down for a temperature range of about 30-50 MeV around the cross-over region. Simultaneous measurements of the various susceptibilities may thus indicate how close to the cross-over region the freeze-out occurs for the fireball created in heavy-io…
In this work we study the electrized quark matter under finite temperature and density conditions in the context of the SU(2) and SU(3) Nambu--Jona-Lasinio models. To this end, we evaluate the effective quark masses and the Schwinger quark-antiquark pair production rate. For the SU(3) NJL model we incorporate in the Lagrangian the 't Hooft determinant and we present a set of analytical expressions more convenient for numerical evaluations. We predict a decrease of the pseudocritical electric field with the increase of the temperature for both models and a more prominent production rate for the…
The real-time formalism at finite temperature and chemical potential for the nonlocal Nambu--Jona-Lasinio model is developed in the presence of a Gaussian covariant regulator. We construct the most general thermal propagator, by means of the spectral function. As a result, the model involves the propagation of massive quasiparticles. The appearance of complex poles is interpreted as a confinement signal, and in this case we have unstable quasiparticles with a finite decay width. An expression for the propagator along the critical line, where complex poles start to appear, is also obtained. A g…
Nambu--Jona-Lasinio-type models have been used extensively to study the dynamics of the theory of the strong interaction at finite temperature and quark chemical potential on a phenomenological level. In addition to these studies, which are often performed under the assumption that the ground state of the theory is homogeneous, searches for the existence of crystalline phases associated with inhomogeneous ground states have attracted a lot of interest in recent years. In this work, we study the Polyakov-loop extended Nambu--Jona-Lasinio model and find that the existence of a crystalline phase …
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 $D^*$, $D^*_s$, $B^*$, $B^*_s$.
We investigate the possible existence of spin polarization and color superconductivity in the Nambu--Jona-Lasinio model with a tensor-type interaction at finite density and temperature. The thermodynamic potential is calculated by the functional integral method. Numerical results indicate that at low temperature and quark chemical potential the chiral condensed phase exists, and at intermediate chemical potential the color superconducting phase appears. In addition, depending on the magnitude of the tensor coupling, at large chemical potential and low temperature, a color superconducting phase…
The momentum-space bosonization method is extended to the case of a Nambu--Jona-Lasinio type model with vector and axial-vector mesons. The method presented gives the possibility of deriving any meson vertex function to all orders in momenta and to the leading order in $1/N_c$. Two-point functions, which describe one-particle transitions to the hadronic vacuum, and meson self-energies are considered. We find new relations which generalize the well-known KSFR relation and both the first and the second Weinberg sum rules. These result from a consistent treatment of higher order terms in the mome…
In present report we review some preliminary results of investigation of higher orders of mean-field expansion for Nambu--Jona-Lasinio model. We discuss first results of investigation of next-to-next-to-leading order of mean-field expansion equations for four-quark and three-quark Green functions. We have considered equations for Green functions of Nambu--Jona-Lasinio model in mean-field expansion up to third order.
Chiral phase properties of finite size hadronic systems are investigated within the Nambu--Jona-Lasinio model. Finite size effects are taken into account by making use of the multiple reflection expansion. We find that, for droplets with relatively small baryon numbers, chiral symmetry restoration is enhanced by the finite size effects. However the radius of the stable droplet does not change much, as compared to that without the multiple reflection expansion.
The polarization effects of the processes $τ\rightarrow K^{-} π^{0} ν_τ$ are described in the framework of the Nambu--Jona-Lasinio model. The intermediate vector meson $K^{*}(892)$ is taken into account. The contribution of these effects to the differential decay width is obtained.
Nambu-Jona-Lasinio model used to investigate low energy phenomena is nonrenormalizable, therefore the results depend on the regularization parameter in general. A possibility of the finite in four-dimensional limit and even the regularization parameter (this is dimension in the dimensional regularization scheme) independent analysis is shown in the leading order of the 1/Nc expansion.
We review the effect on vacuum properties and magnetic catalysis, of adding the most general 0-spin 8-quark (8q) interactions to the combined SU(3) flavor Nambu--Jona-Lasinio (NJL) and 6-quark (6q) 't Hooft interaction Lagrangian.
The electromagnetic dipole and quadrupole polarizabilities of the neutral and charged pions are calculated in the Nambu--Jona-Lasinio model. Our results agree with the recent experimental analysis of these quantities based on Dispersion Sum Rules. Comparison is made with the results from the Chiral Perturbation Theory.
The survival of \bar q q bound states at temperatures higher than the chiral restoration temperature, T_c, recently observed in lattice QCD, is discussed in the framework of the Nambu Jona-Lasinio model. The perturbative determination of the spectral function provides an indication of a pseudogap phase above T_c.
The full and differential widths of the decay $τ\rightarrow K^{0}K^{-}ν_τ$ are calculated in the framework of the extended Nambu-Jona-Lasinio model. The contributions of the subprocesses with the intermediate vector mesons $ρ$(770) and $ρ$(1450) are taken into account. The obtained results are in satisfactory agreement with the experimental data.
The vacuum structure for a Nambu-Jona -Lasinio type model is studied using the effective potential approach. The relevant degrees of freedom are taken to be two different sets of static, auxiliary fields with different symmetry properties, related by Fierz transformations in the action. The results are compared with the analysis of the Dyson-Schwinger equations in the one-loop approximation.
It is shown that in the one-flavour NJL model the vector and axial-vector quasiparticles described by the antisymmetric tensor field are generated. These excitations have tensor interactions with quarks in contrast to usual vector ones. Phenomenological applications are discussed.
We investigate the phase diagram on temperature-chemical potential plane in the Nambu-Jona-Lasinio model with the dimensional regularization. While the structure of the resulting diagram shows resemblance to the one in the frequently used cutoff regularization, some results of our study indicate striking difference between these regularizations. The diagram in the dimensional regularization exhibits strong tendency of the first order phase transition.
The decay width of the process eta -> pi^0 gamma gamma is calculated in the framework of the Nambu-Jona-Lasinio model. The momentum dependence of quark loops is taken into account. Three types of diagrams are considered: quark box, scalar(a0) and vector(rho, omega) pole diagrams. The obtained estimations are in satisfactory agreement with recent experimental data.
Using the description of the subprocess $γγ^*\to S(P)$ in terms of local Nambu-Jona-Lasinio model we calculate the cross sections of photoproduction of scalar and pseudoscalar mesons in high energy photon-lepton collisions processes. The dependence on the transversal momentum and the total cross sections in Weizsaecker-Williams approximation are presented.
We investigate the compositeness condition in four dimensions at the non-trivial order (next-to-leading order in $1/N$ with the number $N$ of the colors), taking the Nambu-Jona-Lasinio model as an example. In spite of its non-renormalizability, the compositeness condition can be successfully derived and solved in compact forms. The resultant problems are discussed.
Recently, it has been shown that the standard Nambu-Jona-Lasinio (NJL) model is not able to reproduce the correct QCD behavior of the gap equation at large density, and therefore a different cutoff procedure at large momenta has ben proposed. We found that, even with this density dependent cutoff procedure, the pure quark phase in neutron stars (NS) interiors is unstable, and we argue that this could be related to the lack of confinement in the original NJL model.
The behavior of the quark condensates at zero chemical potential and finite temperature subject to an external magnetic field is studied within the three flavor Nambu--Jona-Lasinio model with Polyakov loop (PNJL) and its extension, the so-called entangled PNJL model (EPNJL). A comparison with recent lattice QCD data is performed and it is shown that at $T=0$ MeV the light quark condensates are in quantitative agreement. At finite temperature, although there is an overall reasonable agreement with several lattice results, it is shown that in the lattice calculations the effect due to the electr…
We present a study of the 1+1 flavor system of strongly interacting matter in terms of the Polyakov$-$Nambu$-$Jona-Lasinio model. We find that though the small isospin symmetry breaking brought in through unequal light quark masses is too small to affect the thermodynamics of the system in general, it may have significant effect in baryon-isospin correlations and have a measurable impact in heavy-ion collision experiments.
We argue that the spontaneous breakdown of symmetry in the chirally symmetric Nambu--Jona-Lasinio model which was supposed to illustrate the origin of the low mass of pions in hadron physics does not occur due to strong fluctuations in the sigma - pi field space. Although quarks acquire a constituent mass, sigma and pi turn out to have equal heavy masses of the order of the constituent quark mass.
The Nambu-Jona-Lasinio model is the special case of the renormalized Yukawa model with the compositeness condition. We use it to calculate the effective coupling constants in terms of the compositeness sale (momentum cut off) at the next-to-leading order in 1/N. The next-to-leading correction is too large in the model of scalar composite, while that in the induced gauge theory is reasonably suppressed due to the gauge cancellation of the leading divergences.
The nonet meson properties are studied in the Nambu-Jona-Lasinio model at finite temperature and chemical potential using dimensional regularization. This study leads to the reasonable description which is mainly similar to one obtained in the model with the cutoff regularization. However, remarkable differences between the two regularizations are observed in the behavior of the chiral phase transition at finite chemical potential.
The effective dimensional reduction of fermion-antifermion pairing dinamics in Nambu-Jona-Lasinio model in external chromomagnetic field is considered. It is shown that such a field causes the reduction of the space dimension by one unit in contrast to an ordinary magnetic field which reduces it by two units. The enhancement of the dynamical chiral symmetry breaking due to the reduction is discussed.
In the Nambu - Jona-Lasinio model the decay widths $τ\to (π, π'(1300)) ν_τ$ were calculated. For the calculation of the decay width $τ\to πν_τ$ the standard NJL model was used with taking into account the intermediate axial-vector meson and the extended Nambu - Jona-Lasinio model with the intermediate $a_1(1260)$ and $a_1'(1647)$ mesons. These results are in good agreement with each other. For the calculation of the decay width $τ\to π' ν_τ$ extended Nambu - Jona-Lasinio model was used. The results are in satisfactory agreement with the experimental data. Numerical estimates for the constant $…
We argue that the extended Nambu-Jona-Lasinio model which introduces four-quark interactions to describe hadron physics at low energies can be confronted with fundamental quantum chromodynamics by means of spectral sum rules. While there arise rather restrictive bounds on the strength of the effective four-quark interaction in the vector channel, introduction of the four-quark interaction in the pseudoscalar channel resolves a long standing puzzle of the sum rules.
It is shown that in the Nambu -- Jona - Lasinio model at nonzero chemical potential there are two different massive phases with spontaneously broken chiral symmetry. In one of them particle density is identically zero, in another phase it is not equal to zero. The transition between phases is a phase transition of the second order.
We propose an extended Nambu--Jona-Lasinio model to include the heavy mesons with the heavy quark symmetry. The quark current-current interaction is generalized to include the heavy quark currents. In order to comply with the heavy quark spin symmetry at the heavy quark limit, the quark mass dependence of the interaction strength is introduced. The light and heavy pseudo-scalar and vector meson, their masses and the weak decay constants, are calculated in the unified frame.
We present the results for the pion electromagnetic quadrupole polarizabilities, calculated within the Nambu--Jona-Lasinio model. We obtain the sign and magnitude in agreement with the respective experimental analysis based on the Dispersion Sum Rules. At the same time the dipole polarizabilities are well reproduced. Comparison is also made with the results from the Chiral Perturbation Theory.
The polarizability of a charged pion is estimated in the framework of the nonlocal chiral quark model of the Nambu--Jona-Lasinio type. Nonlocality is described by quark form factors of the Gaussian type. It is shown that the polarizability in this model is very sensitive to the form of nonlocality and choice of the model parameters.
Scalar meson contributions in chiral quark condensate are calculated for analytically regularized Nambu -- Jona-Lasinio model in framework of mean-field expansion in bilocal-source formalism. Sigma-meson contribution for physical values of parameters is found to be small. Pion contribution is found to be significant and should be taken into account at the choice of the parameter values.
