We present a path-integral hadronization for doubly heavy baryons. The two heavy quarks in the baryon are approximated as a scalar or axial-vector diquark described by a heavy diquark effective theory. The gluon dynamics are represented by a NJL-Model interaction for the heavy diquarks and light quarks, which leads to an effective action of the baryon fields after the quark and diquark fields are integrated out. This effective action for doubly heavy baryon includes the electromagnetic and electroweak interactions, as well as the interaction with light mesons. We also verify the Ward-Takahashi identity at the baryon level, obtain the Isgur-Wise function for weak transitions, and calculate the strong coupling constant of the doubly heavy baryon and pion. Numerical studies are also performed.
We describe baryons as quark-diquark bound states at finite temperature and density within the NJL model for chiral symmetry breaking and restoration in quark matter. Based on a generalized Beth-Uhlenbeck approach to mesons and diquarks we present in a first step the thermodynamics of quark-diquark matter which includes the Mott dissociation of diquarks at finite temperature. In a second step we solve the Bethe-Salpeter equation for the baryon as a quark-diquark bound state in quark-diquark matter. We obtain a stable, bound baryon even beyond the Mott temperature for diquark dissociation since…
In a Nambu-Jona-Lasinio model supplemented with an infrared cutoff in addition to the ultraviolet cutoff we study the issue whether diquarks are confined when the model is extended beyond the rainbow-ladder approximation. The gap equation, obtained in a truncation scheme motivated via a nontrivial quark-gluon vertex function, is solved to determine the constituent quark mass if chiral symmetry is spontaneously broken. In a second step, the Bethe-Salpeter equations for mesons and diquarks beyond ladder approximation are derived, taking care to preserve Goldstone's theorem in the pion channel. W…
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 study the nucleon and diquarks in a non-local Nambu-Jona-Lasinio model. For certain parameters the model exhibits quark confinement, in the form of a propagator without real poles. After truncation of the two-body channels to the scalar and axial-vector diquarks, a relativistic Faddeev equation for nucleon bound states is solved in the covariant diquark-quark picture. The dependence of the nucleon mass on diquark masses is studied in detail. We find parameters that lead to a simultaneous reasonable description of pions and nucleons. Both the diquarks contribute attractively to the nucleon m…
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.
The diquark and nucleon are studied in a non-local NJL model. We solve the relativistic Faddeev equation and compare the results with the ordinary NJL model. Although the model is quark confining, it is not diquark confining in the rainbow-ladder approximation. We show that the off-shell contribution to the diquark $T$ matrix is crucial for the structure of the nucleon: without its inclusion the attraction in the scalar channel is too weak to form a three-body bound state.
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…
The approach of path integral hadronization, applied to a quark-diquark toy model, is used to derive an effective chiral meson-baryon Lagrangian. In this approach the Goldberger-Treiman relation, found at the quark-meson level, is re-established at the composite hadron level. For illu- strations, electromagnetic low energy characteristics of composite nucleons are estimated.
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 study the description of nucleons and diquarks in the presence of a uniform strong magnetic field within the framework of the two-flavor Nambu-Jona--Lasinio (NJL) model. Diquarks are constructed through the resummation of quark loop chains using the random phase approximation, while nucleons are treated as bound quark-diquark states described by a relativistic Fadeev equation, using the static approximation for quark exchange interactions. For charged particles, analytical calculations are performed using the Ritus eigenfunction method, which properly takes into account the breakdown of tra…
We study the interplay of the chiral and the color superconducting phase transitions in an extended Nambu--Jona-Lasinio model with a multi-quark interaction that produces the nonlinear chiral-diquark coupling. We observe that this nonlinear coupling adds up coherently with the ω^2 interaction to produce the chiral-color superconductivity coexistence phase or cancel each other depending on its sign. We discuss that large coexistence region in the phase diagram is consistent with the quark-diquark picture for the nucleon whereas its smallness is the prerequisite for the applicability of the Ginz…
One of the confident predictions of QCD is that at sufficiently high temperature and/or density, hadronic matter should undergo a thermodynamic phase transition to a colour deconfined state of matter - popularly called the Quark-Gluon Plasma (QGP). In low energy effective theories of Quantum Chromodynamics (QCD), one usually talks of the chiral transition for which a well defined order parameter exists. We investigate the dissociation of pions and kaons in a medium of hot quark matter decsribed by the Nambu - Jona Lasinio (NJL) model. The decay widths of pion and kaon are found to be large but…
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 chiral phase transition and color superconductivity in an extended NJL model with eight-quark interactions are studied. The scalar-type nonlinear term hastens the chiral phase transition,the scalar-vector mixing term suppresses effects of the vector-type linear term and the scalar-diquark mixing term makes the coexisting phase wider.
We calculate the leading-twist, helicity-independent generalized parton distributions (GPDs) of the proton, at finite skewness, in the Nambu--Jona-Lasinio (NJL) model of quantum chomodynamics (QCD). The NJL model reproduces low-energy characteristics of QCD, including dynamical chiral symmetry breaking (DCSB). The proton bound-state amplitude is solved for using the Faddeev equation in a quark-diquark approximation, including both dynamical scalar and axial vector diquarks. GPDs are calculated using a dressed non-local correlator, consistent with DCSB, which is obtained by solving a Bethe-Salp…
The method of path integral hadronization is applied to a local quark-diquark toy model in order to derive an effective chiral meson-baryon Lagrangian. Further generalizations to models including both scalar and axial-vector diquarks as well as nonlocal interactions are discussed.
We present results of a Monte Carlo simulation of a 2+1 dimensional Nambu - Jona-Lasinio model including diquark source terms. A diquark condensate <qq> is measured as a function of source strength j. In the vacuum phase <qq> vanishes linearly with j as expected, but simulations in a region with non-zero baryon density suggest a power-law scaling and hence a critical system for all mu > mu_c. There is no diquark condensation signalling superfluidity. Comparisons are drawn with known results in two dimensional theories, and with the pseudogap phase in cuprate superconductors. We also measure th…
We present a new model for the description of heavy-quark hadronization in relativistic heavy-ion collisions in the presence of a reservoir of lighter thermal particles with which recombination can occur leading to the formation of color-singlet clusters. Color neutralization is assumed to occur locally, within the same fluid cell occupied by the heavy quark and it proceeds via the recombination of the latter with light antiquarks (quarks) or diquarks (anti-diquarks), which are assumed to be present in the medium with thermal abundance as effective degrees of freedom around the QCD crossover t…
The recently developed Nambu-Jona-Lasinio (NJL) - Jet model is used as an effective chiral quark theory to calculate the quark fragmentation functions to pions, kaons, nucleons, and antinucleons. The effects of the vector mesons rho, K* and phi on the production of secondary pions and kaons are included. The fragmentation processes to nucleons and antinucleons are described by using the quark-diquark picture, which has been shown to give a reasonable description of quark distribution functions. We incorporate effects of next-to-leading order (NLO) in the Q^2 evolution, and compare our results …
A Bethe-Salpeter-Faddeev (BSF) calculation is performed for the pentaquark $Θ^+$ in the diquark picture of Jaffe and Wilczek in which $Θ^+$ is a diquark-diquark-${\bar s}$ three-body system. Nambu-Jona-Lasinio (NJL) model is used to calculate the lowest order diagrams in the two-body scatterings of ${\bar s}D$ and $D D$. With the use of coupling constants determined from the meson sector, we find that ${\bar s}D$ interaction is attractive while $DD$ interaction is repulsive, and there is no bound $\frac 12^+$ pentaquark state. A bound pentaquark $Θ^+$ can only be obtained with unphysically str…
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…
Dihadron Fragmentation Functions (DFF) provide a vast amount of information on the intricate details of the parton hadronization process. Moreover, they provide a unique access to the "clean" extraction of nucleon transversity parton distribution functions in semi inclusive deep inelastic two hadron production process with a transversely polarised target. The NJL-jet model has been extended for calculations of light and strange quark unpolarised DFFs to pions, kaons and several vector mesons. This is accomplished by using the probabilistic interpretation of the DFFs, and employing the NJL-jet …
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…
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.
Quark-hadron matter at finite temperature and density is studied using a two-flavor, color SU(2) (P)NJL model. The hadronic effective Lagrangian, derived by bosonization of the quark fields and renormalized using the Eguchi method, emerges in the form of an extended linear sigma model with meson and diquark-baryon fields. Chiral and diquark condensates are studied as functions of temperature and baryon density. Masses of mesons and diquark-baryons are calculated with and without the Polyakov loop effect. We investigate the equation of state of quark-hadron matter by taking into account the con…
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.
We study the structure of the nuclear force by using a path-integral hadronization approach in a chiral quark-diquark model. After the construction of the chiral quark-diquark model, we hadronize it to obtain a meson-baryon Lagrangian. The effective meson-baryon Lagrangian incorporates chiral symmetry and the composite description of the mesons and baryons. Using the effective meson-baryon Lagrangian we investigate the structure of the nuclear force in the simple case of neglecting the axial-vector diquark. It is shown that the meson-baryon Lagrangian contains two kinds of the nuclear force; t…
The NJL-jet model is extended to accommodate hadronization of a transversely polarized quark in order to explore the Collins effect within a multihadron emission framework. This is accomplished by calculating the polarized quark spin flip probabilities after a pseudoscalar hadron emission and the elementary Collins functions. The model is used to calculate the number densities of the hadrons produced in the polarized quark's decay chain. The full Collins fragmentation function is extracted from the sine modulation of the polarized number densities with respect to the polar angle between the in…
We study the effect of a large magnetic field on the chiral and diquark condensates in a regime of moderately dense quark matter. Our focus is on the inter-dependence of the two condensates through non-perturbative quark mass and strong coupling effects, which we address in a 2-flavor Nambu-Jona-Lasinio (NJL) model. For magnetic fields $eB\lesssim 0.01$ GeV$^2$ (corresponding to $B\lesssim 10^{18}$G), our results agree qualitatively with the zero-field study of Huang et al., who found a mixed broken phase region where the chiral and superconducting gap are both non-zero. For $eB\gtrsim 0.01$ G…
We study the formation of baryons as composed of quarks and diquarks in hot and dense hadronic matter in a Nambu--Jona-Lasinio (NJL)--type model. We first solve the Dyson-Schwinger equation for the diquark propagator and then use this to solve the Dyson-Schwinger equation for the baryon propagator. We find that stable baryon resonances exist only in the phase of broken chiral symmetry. In the chirally symmetric phase, we do not find a pole in the baryon propagator. In the color-superconducting phase, there is a pole, but is has a large decay width. The diquark does not need to be stable in ord…
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 revisit the recipe for hadron formation in the Lund string hadronization model. Given an incoming quark or quark-diquark pair, weights for hadron formation are updated to take hyperfine splitting effects arising from the mass difference between u, d-type and s-type quarks. We find that the procedure improves the description of hadron yields in $\mathrm{e}^+\mathrm{e}^-$ collisions and the $ϕ$ cross section in neutral current DIS. We also show results for proton collisions, and discuss the future use of this study in the context of small system collectivity.
We investigate the combined effect of the repulsive vector interaction and the positive electric chemical potential on the chiral phase transition by considering neutral color superconductivity (CSC). The chiral condensate, diquark condensate and quark number densities are solved in both two-flavor and two-plus-one-flavor Nambu-Jona-Lasinio(NJL) models with the so called Kobayashi-Maskawa-'t Hooft term under the charge neutrality constraint. We demonstrate that multiple chiral critical-point structures always exist in the NJL model within the self-consistent mean-field approximation and the nu…
We discuss the structure of the nuclear force using a lagrangian derived from hadronization of a chiral quark and diquark model. A generalized trace log formula including meson and nucleon fields is expanded to the order in which relevant terms emerge. It is shown that the nuclear force is composed of long and medium range parts of chiral meson exchanges and short range parts of quark-diquark exchanges. The ranges of the scalar and vector interactions coincide well with those of sigma ($σ$) and omega ($ω$) meson exchanges if the size of the nucleon core of a quark-diquark bound state is adjust…
The hadronization effects induced by different colour connections of $c\bar{c}q\bar{q}$ system in $e^+e^-$ annihilation are investigated by a toy model where diquark fragmentation is employed based on Pythia. It is found that the correlations between the charm baryons and charm antibaryons produced via diquark pair fragmentation are much stronger, and their momentum spectra are harder than those from the standard colour connection in Pythia.
In the Nambu--Jona-Lasinio (NJL-) model baryons are considered either as quark-diquark-composites or as soliton configurations of mesons. The construction of a hybrid model, which possesses a dynamical interplay between both pictures, is presented. The Bethe-Salpeter equation for diquarks and the Faddeev equation for diquark-quark bound states in the background of a soliton configuration is solved.
We consider an extended Nambu--Jona-Lasinio model including both (q \bar q)- and (qq)-interactions with two light-quark flavors in the presence of a single (quark density) chemical potential. In the color superconducting phase of the quark matter the color SU(3) symmetry is spontaneously broken down to SU(2). If the usual counting of Goldstone bosons would apply, five Nambu-Goldstone (NG) bosons corresponding to the five broken color generators should appear in the mass spectrum. Unlike that expectation, we find only three gapless diquark excitations of quark matter. One of them is an SU(2)-si…
The mass of the nucleon is studied in a chiral quark-diquark model. Both scalar and axial-vector diquarks are taken into account for the construction of the nucleon state. After the hadronization procedure to obtain an effective meson-baryon Lagrangian, the quark-diquark self-energy is calculated in order to generate the baryon kinetic term as well as the mass of the nucleon. It turns out that both the scalar and axial-vector parts of the self-energy are attractive for the mass of the nucleon. We investigate the range of parameters that can reproduce the mass of the nucleon.
The precise measurement of the high masses of the pulsars PSR J1614-2230 (M_{1614}=1.97 +- 0.04 solar masses) and PSR J0348-0432 (M_{0348}=2.01 +- 0.04 solar masses) provides an important constraint for the equation of state of cold, dense matter and is suited to give interesting insights regarding the nature and existence of the possible phase transition to deconfined quark matter in the cores of neutron stars. We analyze the stability and composition of compact star sequences for a class of hybrid nuclear - quark-matter equations of state. The quark matter phase is described in the framework…
The masses of the low-lying scalar mesons are investigated in the three-flavor Nambu Jona-Lasinio (NJL) model by treating the scalar mesons as composite objects of a quark and an antiquark. It is known that a simple $\bar qq$ picture fails to reproduce so-called inverse mass ordering for the scalar mesons. Recently a new mechanism to reproduce the observed mass spectrum of the scalar mesons was proposed in a linear sigma model by introducing flavor symmetry breaking induced by the U(1) axial anomaly. Motivated by this proposal, we examine whether this new mechanism works also in the NJL model.…
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 discuss a three-flavor Nambu--Jona-Lasinio (NJL) type quantum field theoretical approach to the quark matter equation of state (EoS) with scalar diquark condensate, isoscalar vector mean field and Kobayashi-Maskawa-'t Hooft (KMT) determinant interaction. While often the diquark and vector meson couplings are considered as free parameters, we will fix them here to their values according to the Fierz transformation of a one-gluon exchange interaction. In order to estimate the effect of a possible change in the vacuum pressure of the gluon sector at finite baryon density we exploit a recent mo…
We study the phase structure of the unpolarized and polarized two-flavor quark matters at zero and finite temperatures within the Nambu--Jona-Lasinio (NJL) model. We focus on the region, which includes the coexisting phase of quark-antiquark and diquark condensates. Generalizing the NJL model so as to describe the polarized quark matter, we compute the thermodynamic potential as a function of the quark chemical potential ($μ$), the temperature ($T$), and the polarization parameter. The result heavily depends on the ratio $G_D / G_S$, where $G_S$ is the quark-antiquark coupling constant and $G_…
An important first step in the program of hadronization of chiral quark models is the bosonization in meson and diquark channels. This procedure is presented at finite temperatures and chemical potentials for the SU(2) flavor case of the NJL model with special emphasis on the mixing between scalar meson and scalar diquark modes which occurs in the 2SC color superconducting phase. The thermodynamic potential is obtained in the gaussian approximation for the meson and diquark fields and it is given the Beth-Uhlenbeck form. This allows a detailed discussion of bound state dissociation in hot, den…
Octet and Decuplet baryons are described within the hadronized NJL model as diquark--quark states, which are bound by quark exchange. Including scalar and axial--vector diquark correlations, we project the previously obtained relativistic Faddeev equation in a Poincaré invariant fashion onto good spin, using a static approximation to the exchanged quark. The resulting equations for the spin 1/2 octet and the spin 3/2 decuplet are solved numerically.
To study the high energy nuclear physics and the associated phenomenon, as the QGP/hadronic matter phase transition, the Nambu and Jona-Lasinio model (NJL) appears as an interesting alternative to the Quantum Chromodynamics, which is not solvable at the considered energies. Indeed, the NJL model allows describing the quarks physics, at finite temperatures and finite densities. Furthermore, in order to try to correct a limitation of the NJL model, i.e. the absence of confinement, it was proposed to couple the quarks/antiquarks to a Polyakov loop: it forms the PNJL model. The objective of this t…
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…
Using the model of Nambu and Jona-Lasinio to provide a microscopic description of both the structure of the nucleon and of the quark to hadron elementary fragmentation functions, we investigate the transverse momentum dependence of the unpolarized quark distributions in the nucleon and of the quark to pion and kaon fragmentation functions. The transverse momentum dependence of the fragmentation functions is determined within a Monte Carlo framework, with the notable result that the average $P_\perp^2$ of the produced kaons is significantly larger than that of the pions. We also find that $<P_\…
The spectrum of meson and diquark excitations of dense color neutral cold quark matter is investigated in the framework of a 2-flavored Nambu--Jona-Lasinio type model, including a quark $μ$- and color $μ_8$ chemical potential. It was found out that in the color superconducting (2SC) phase, i.e. at $μ>μ_c=342$ MeV, $μ_8$ aquires rather small values $\sim$ 10 MeV in order to ensure the color neutrality. In this phase the $π$- and $σ$ meson masses are evaluated around $\sim$ 330 MeV. The spectrum of scalar diquarks in the color neutral 2SC phase consists of a heavy ($\rm SU_c(2)$-singlet) resonan…
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 …
The hadronization effects induced by various colour connections of the four quark system in $e^{+}e^{-}$ annihilation are briefly reviewed. A special colour connection of four heavy quark system without colour-separation favours the production of doubly heavy baryons. For the related three-jet case, the corresponding hadronization has not been considered. We argue that it can be effectively described as two string fragmentation besides the leading heavy diquark fragmentation. The production rate and the properties of final state hadron systems are discussed. Emphasis is on the string effect as…
A quantum field theoretical approach to the thermodynamics of dense Fermi systems is developed for the description of the formation and dissolution of quantum condensates and bound states in dependence of temperature and density. As a model system we study the chiral and superconducting phase transitions in two-flavor quark matter within the NJL model and their interrelation with the formation of quark-antiquark and diquark bound states. The phase diagram of quark matter is evaluated as a function of the diquark coupling strength and a coexistence region of chiral symmetry breaking and color s…
