Dynamical Chiral Symmetry Breaking

We study the color confinement, the q-q pair creation and the dynamical chiralsymmetry breaking of nonperturbative QCD by using the dual Ginzburg-Landau theory, where QCD-monopole condensation plays an essential role on the nonperturbative dynamics in the infrared region. As a result of the dual Meissner effect, the linear static quark potential, which characterizes the quark confinement, is obtained in the long distance within the quenched approximation. We obtain a simple expression for the string tension similar to the energy per unit length of a vortex in the superconductivity physics. The dynamical effect of light quarks on the quark confining potential is investigated in terms of the infrared screening effect due to the q-q pair creation or the cut of the hadronic string. The screening length of the potential is estimated by using the Schwinger formula for the q-q pair creation. We introduce the corresponding infrared cutoff to the strong long-range correlation factor in the gluon propagator as a dynamical effect of light quarks, and obtain a compact formula of the quark potential including the screening effect in the infrared region. We investigate the dynamical chiral-symmetry breaking by using the Schwinger-Dyson equation, where the gluon propagator includes the nonperturbative effect related to the color confinement. We find a large enhancement of the chiral-symmetry breaking when the dual Meissner effect takes place, which supports the close relation between the color confinement and the chiral-symmetry breaking. The dynamical quark mass, the pion decay constant and the quark condensate are well reproduced by using the consistent values of the gauge coupling constant and the QCD scale parameter with the perturbative QCD and the quark confining potential. The confinement of light quarks is also investigated by the smooth extrapolation of the quark mass function to the time-like momentum region. We find the disappearance of physical poles in the light-quark propagator, and show the confinement of light quarks in the dual Ginzburg-Landau theory.

We survey contemporary studies of hadrons and strongly interacting quarks using QCD's Dyson-Schwinger equations, addressing: aspects of confinement and dynamical chiral symmetry breaking; the hadron spectrum; hadron elastic and transition form factors, from small-to large-Q 2 ; parton distribution functions; the physics of hadrons containing one or more heavy quarks; and properties of the quark gluon plasma.

Dynamical chiral symmetry breaking and confinement are two crucial features of Quantum Chromodynamics responsible for the nature of the hadron spectrum. These phenomena, presumably coincidental, can account for 98% of the mass of our visible universe. In this set of lectures, I shall present an introductory review of them in the light of the Schwinger-Dyson equations.

The ambiguities associated with the lack of gauge invariance in the non-perturbative truncations of Schwinger-Dyson equations (SDEs) are a challenging problem which has not yet been resolved in a decisive fashion. Pursuing this aim, we study dynamical chiral symmetry breaking in quantum electrodynamics in three space-time dimensions (QED3). We investigate the gauge dependence of the chiral condensate both in the quenched and the unquenched versions of the theory and emphasize the importance of taking into account the gauge covariance properties of the fermion propagator as dictated by its Landau-Khalatnikov-Fradkin transformation (LKFT). We present numerical solutions of the SDE of the fermion propagator which respect Ward-Green-Takahashi identities (WGTI) and LKFT simultaneously. As a striking consequence, we obtain a practically gauge independent chiral condensate.

It is shown that a constant magnetic field in 3+1 and 2+1 dimensions is a strong catalyst of dynamical chiral symmetry breaking, leading to the generation of a fermion dynamical mass even at the weakest attractive interaction between fermions. The essence of this effect is the dimensional reduction D → D − 2 in the dynamics of fermion pairing in a magnetic field. The effect is illustrated in the Nambu-Jona-Lasinio (NJL) model and QED. In the NJL model in a magnetic field, the low-energy effective action and the spectrum of long wavelength collective excitations are derived. In QED (in ladder and improved ladder approximations) the dynamical mass of fermions (energy gap in the fermion spectrum) is determined. Possible applications of this effect and its extension to inhomogeneous field configurations are discussed.

We introduce the dual Ginzbutg-Landau (DGL) theory ss a low energy effective theory of QCD. We study color confinement and dynamical chital symmetry breaking of nonpettutbative QCD by using the DGL theory, where color monopole and its condensation play an essential role on the nonpettutbative dynamics in the infrared region. As a result of the dual Meissnet effect, the linear static quark potential, which characterizes the quark confinement, is obtained in the long distance. We investigate then the dynamical chiial symmetry breaking by using the Schwinget-Dyson equation, where the &on propagator includes the nonpettutbative effect related to monopole condensation. We find a large enhancement of the chital-symmetry breaking when the dual Meissnet effect takes place. We study the recovery of the chital symmetry and the decominement at finite temperature in the DGL theory. We discuss then the essential assumption of the DGL theory, which is the abelian dominance for the inftated physics, in the maximal abelian (MA) gauge in lattice QCD. The lattice QCD simulation demonstrates that the non-abelian gluons have a finite rn=s of order of 1 GeV in the MA gauge. We introduce further the instanton configuration as the source of the color monopole. In the MA gauge, a monopole circles atound an instanton and with the increase of the instanton density, the monopole loop connects many instantons and a complicated monopole loop covets the whole 4 dimensional space. This study indicates that instautons may be playing an essential role even for color confinement.

We consider the low energy effective action of QCD below the chiral symmetry breaking scale, including, in Wilson's spirit, all operators of dimensionality less or equal to 6 which can be built with quark and chiral fields. The effect of the residual gluon interactions is contained in a number of coupling constants, whose running is studied. The resulting model is an extension of both the chiral quark model and the Nambu-Jona-Lasinio one. Constraints on the coefficients of the effective lagrangian are derived from the requirement of chiral symmetry restoration at energies above the chiral symmetry breaking scale, from matching to QCD at intermediate scales, and by fitting some hadronic observables. In this model two types of pseudoscalar states (massless pions and massive Π-mesons), as well as one scalar one arise as a consequence of dynamical chiral symmetry breaking. Their masses and coupling constants are studied. We also predict a number of low energy structural constants. We find out that QCD favours a low-energy effective theory which is largely dominated by the simplest chiral quark model, whereas higher dimensional operators (such as those of the Nambu-Jona-Lasinio type) can be assumed to be small.

We determine the critical coupling constant above which dynamical chiral symmetry breaking occurs in a class of QCD motivated models where the gluon propagator has an enhanced infrared behavior. Using methods of bifurcation theory we find that the critical value of the coupling constant is always smaller than the one obtained for QCD.

A closed system of equations for the propagators of Landau gauge QCD is obtained in a truncation scheme for their Dyson-Schwinger equations which implements the Slavnov-Taylor identities for the 3-point vertex functions while neglecting contributions from irreducible 4-point correlations. In the pure gauge theory without quarks, non-perturbative solutions for the gluon and ghost propagators are available in an approximation which

A nonperturbative construction of the 3-point fermion-boson vertex which obeys its Ward-Takahashi or Slavnov-Taylor identity, ensures the massless fermion and boson propagators transform according to their local gauge covariance relations, reproduces perturbation theory in the weak coupling regime and provides a gauge independent description for dynamical chiral symmetry breaking and confinement has been a long-standing goal in physically relevant gauge theories such as quantum electrodynamics (QED) and quantum chromodynamics. In this paper, we demonstrate that the same simple and practical form of the vertex can achieve these objectives not only in 4-dimensional quenched QED but also in its 3-dimensional counterpart. Employing this convenient form of the vertex ansatz into the Schwinger-Dyson equation for the fermion propagator, we observe that it renders the critical coupling in 4-dimensional quenched QED markedly gauge independent in contrast with the bare vertex and improves on the well-known Curtis-Pennington construction. Furthermore, our proposal yields gauge independent order parameters for confinement and dynamical chiral symmetry breaking in 3-dimensional quenched QED.

A comprehensive, relativistic many-body approach to hadron structure is advanced based on the Coulomb gauge QCD Hamiltonian. Our method incorporates standard many-body techniques which render the approximations amenable to systematic improvement. Using BCS variational methods, dynamic chiral symmetry breaking naturally emerges and both quarks and gluons acquire constituent masses. Gluonia are studied both in the valence and in the collective, random phase approximations. Using representative values for the strong coupling constant and string tension, calculated quenched glueball masses are found to be in remarkable agreement with lattice gauge theory.

We analyze the isospin-breaking corrections to quark condensates within oneloop SU(2) and SU(3) chiral perturbation theory including m u � = m d as well as electromagnetic (EM) contributions. The explicit expressions are given and several phenomenological aspects are studied. We analyze the sensitivity of recent condensate determinations to the EM low-energy constants (LEC). If the explicit chiral symmetry breaking induced by EM terms generates a ferromagnetic-like response of the vacuum, as in the case of quark masses, the increasing of the order parameter implies constraints for the EM LEC, which we check with different estimates in the literature. In addition, we extend the sum rule relating quark condensate ratios in SU(3) to include EM corrections, which are of the same order as the m u � = m d ones, and we use that sum rule to estimate the vacuum asymmetry within ChPT. We also discuss the matching conditions between the SU(2) and SU(3) LEC involved in the condensates, when both isospin-breaking sources are taken into account.

The possibility of using the optimized δ expansion for studying medium effects on hadronic properties in quark or nuclear matter is investigated. The δ expansion is employed to study density effects with two commonly used models in hadron and nuclear physics, the Nambu-Jona-Lasinio model for the dynamical chiral symmetry breaking and the Walecka model for the equation of state of nuclear matter. The results obtained with the δ expansion are compared to those obtained with the traditional Hartree-Fock approximation. Perspectives for using the δ expansion in other field theoretic models in hadron 1 and nuclear physics are discussed.

We study the gauge invariance of physical observables related to confinement and dynamical chiral symmetry breaking in unquenched QED3 for a simple truncation of the corresponding Schwinger-Dyson equations in arbitrary covariant gauges. An explicit implementation of Landau-Khalatnikov-Fradkin transformations renders these observables gauge independent.

The gap equation is a cornerstone in understanding dynamical chiral symmetry breaking and may also provide clues to confinement. A symmetry-preserving truncation of its kernel enables proofs of important results and the development of an efficacious phenomenology. We describe a model of the kernel that yields: a momentum-dependent dressed-quark propagator in fair agreement with quenched lattice-QCD results; and chiral limit values, f 0 π = 68 MeV and qq = −(190 MeV) 3 . It is compared with models inferred from studies of the gauge sector.

We study the quark-meson model with two quark flavors in a strong external magnetic field B at finite temperature T and finite baryon chemical potential µB. We calculate the full renormalized effective potential to one loop order in perturbation theory. In the approximation where we treat the bosonic modes at tree level, the chiral transition is of second order in the entire µB-T plane in the chiral limit and a crossover at the physical point if we include the vacuum fluctuations. If they are excluded, the transition is first order.

n this paper we discuss a mechanism for the dynamical generation of flavor mixing, in the framework of the Nambu--Jona Lasinio model. Our approach is illustrated both with the conventional operatorial formalism and with functional integral and ensuing one-loop effective action. The results obtained are briefly discussed.

We extend a previous Landau-gauge study of subtractive renormalization of the fermion propagator Dyson-Schwinger equation (DSE) in strong-coupling, quenched QED 4 to arbitrary covariant gauges. We use the fermion-photon proper vertex proposed by Curtis and Pennington with an additional correction term included to compensate for the small gauge-dependence induced by the ultraviolet regulator. We discuss the chiral limit and the onset of dynamical chiral symmetry breaking in the presence of nonperturbative renormalization. We extract the critical coupling in several different gauges and find evidence of a small residual gauge-dependence in this quantity.

A non-perturbative construction of the 3-point fermion-boson vertex which obeys its Ward-Takahashi or Slavnov-Taylor identity, ensures the massless fermion and boson propagators transform according to their local gauge covariance relations, reproduces perturbation theory in the weak coupling regime and provides a gauge independent description for dynamical chiral symmetry breaking (DCSB) and confinement has been a long-standing goal in physically relevant gauge theories such as quantum electrodynamics (QED) and quantum chromodynamics (QCD). In this paper, we demonstrate that the same simple and practical form of the vertex can achieve these objectives not only in 4-dimensional quenched QED (qQED4) but also in its 3-dimensional counterpart (qQED3). Employing this convenient form of the vertex ansatz into the Schwinger-Dyson equation (SDE) for the fermion propagator, we observe that it renders the critical coupling in qQED4 markedly gauge independent in contrast with the bare vertex and improves on the well-known Curtis-Pennington construction. Furthermore, our proposal yields gauge independent order parameters for confinement and DCSB in qQED3.

We establish that QED3 can possess a critical number of flavours, N c f , associated with dynamical chiral symmetry breaking if, and only if, the fermion wave function renormalisation and photon vacuum polarisation are homogeneous functions at infrared momenta when the fermion mass function vanishes. The Ward identity entails that the fermion-photon vertex possesses the same property and ensures a simple relationship between the homogeneity degrees of each of these functions. Simple models for the photon vacuum polarisation and fermion-photon vertex are used to illustrate these observations. The existence and value of N c f are contingent upon the precise form of the vertex but any discussion of gauge dependence is moot. We introduce an order parameter for confinement. Chiral symmetry restoration and deconfinement are coincident owing to an abrupt change in the analytic properties of the fermion propagator when a nonzero scalar self-energy becomes insupportable.

Reduced gauge theories are theories in which while gauge fields propagate in a bulk, fermion fields are localized on a brane. We study dynamical chiral symmetry breaking on a 2-brane and a 1-brane in reduced QED 3+1 , and on a 1-brane in reduced QED 2+1. Since, unlike higher dimensional gauge theories, QED 3+1 and QED 2+1 are well defined, their reduced versions can serve as a laboratory for studying dynamics in a higher dimensional brane world. The analysis of the Schwinger-Dyson (SD) equations in these theories reveals rich and quite nontrivial dynamics in which the conformal symmetry and its breakdown play a crucial role. Explicit solutions of the SD equations in the near-critical regime are obtained and the character of the corresponding phase transition is described. 11.10.Kk, 11.30.Qc, 12.20.-m I. INTRODUCTION Dynamics in a brane world has recently attracted considerable interest. In most cases, it has been studied in higher dimensional theories (for a recent review, see Ref. [1]). The aim of this paper is to consider dynamics on a brane not in higher dimensions but in a 3+1 and 2+1 dimensional world. More precisely, we study dynamical chiral symmetry breaking in the reduced QED 3+1 and QED 2+1. The term "reduced" implies here that while massless gauge fields propagate in a (3+1 or 2+1 dimensional) bulk, fermion fields are localized on a brane. We will consider the cases of a 2-brane and a 1-brane in QED 3+1 and a 1-brane in QED 2+1. We would like also to emphasize that though we use the conventional term "QED" for a U (1) abelian gauge theory with fermions, we do not specify the origin of the gauge field: it is not necessary electromagnetic field. Motivations for considering this type of models are rather obvious. It is well known that relativistic field models can serve as effective theories for the description of long wavelength excitations in condensed matter systems [2]. The reduced QED describes the situation when while fermions are localized on a plane (say, on a Cu-O plane in a high-T c superconductor) or on a string (polymer like systems), interactions between them are provided by a bulk gauge field. Besides that, reduced QED 3+1 can be relevant for the dynamics of cosmological

It is shown that a constant magnetic field in 3+1 and 2+1 dimensions is a strong catalyst of dynamical chiral symmetry breaking, leading to the generation of a fermion dynamical mass even at the weakest attractive interaction between fermions. The essence of this effect is the dimensional reduction D → D − 2 in the dynamics of fermion pairing in a magnetic field. The effect is illustrated in the Nambu-Jona-Lasinio (NJL) model and QED. In the NJL model in a magnetic field, the low-energy effective action and the spectrum of long wavelength collective excitations are derived. In QED (in ladder and improved ladder approximations) the dynamical mass of fermions (energy gap in the fermion spectrum) is determined. Possible applications of this effect and its extension to inhomogeneous field configurations are discussed.

We explore the consequences of an equation of state (EOS) obtained in a confining Dyson-Schwinger equation model of QCD for the structure and stability of nonstrange quark stars at finite-T , and compare the results with those obtained using a bag-model EOS. Both models support a temperature profile that varies over the star's volume and the consequences of this are model independent. However, in our model the analogue of the bag pressure is (T, µ)-dependent, which is not the case in the bag model. This is a significant qualitative difference and comparing the results effects a primary goal of elucidating the sensitivity of quark star properties to the form of the EOS.

Dynamical Chiral Symmetry Breaking (DCSB) and Confinement are two crucial features of QCD which are responsible for the nature of the hadronic spectrum. A simpler model which exhibits both is quantum electrodynamics in (2+1) space-time dimensions, QED 3 . A long standing debate in this model is the existence of a critical number of fermion families, N c , above which DCSB ceases to take place. This was established from the solutions of the Schwinger-Dyson equations (SDEs), in the leading order of the 1/N expansion in the Landau gauge. Confinement has also been found to be absent in this scenario. In this work, we study the stability of the solutions to the said SDEs under a variation of gauge while still working with the bare vertex. We find that the Landau gauge is the only gauge which exhibits the above mentioned results. Away from this gauge, DCSB takes place for an arbitrarily large N and confinement is reinstated. Attempting to understand this apparent inconsistency, we argue that in order to maintain the gauge covariance of the results, full vertex has to be employed in other gauges and/or constraints like the Landau-Khalatnikov-Fradkin transformations must be employed in going from Landau gauge to other gauges.

We study dynamical chiral-symmetry breaking of non-perturbative QCD in the dual Ginzburg-Landau theory, where the QCD-monopole field is introduced as an essential field for color confinement stemming from the choice of the abelian gauge fixing a !a 't Hooft. In this theory, QCD-monopole condensation causes the dual Meissner effect, which changes the gluon propagator. The dynamical chiral-symmetry breaking is investigated using the Schwinger-Dyson equation with the modified gluon propagator in the QCD-monopole condensed vacuum. We introduce the low momentum cutoff on the modified gluon propagator by considering the effects of the q-q pair creation and/or the quarks being confined in hadrons. We find that dynamical chiral-symmetry breaking is largely enhanced by QCD-monopole condensation, which suggests the close relation between the color confinement and the chiral symmetry breaking. The dynamical quark mass, the pion decay constant and the quark condensate are reproduced consistently with the confining mechanism in this theory. § 1. Introduction

It is shown how the strong interaction dynamics of a multi-quark Lagrangian affects the catalysis of dynamical symmetry breaking by a constant magnetic field in (3 + 1) dimensions. Attention is drawn to the local minima structure of the theory.

We describe a (first, to the best of our knowledge) essentially soluble example of dynamical symmetry breaking phenomenon in a 3+1 dimensional gauge theory without fundamental scalar fields: QED in a constant magnetic field. 11.30. Rd, 11.30.Qc, Recently the magnetic catalysis of dynamical chiral symmetry breaking has been established as a universal phenomenon in 2+1 and 3+1 dimensions: a constant magnetic field leads to the generation of a fermion dynamical mass even at the weakest attractive interaction between fermions [1-3]. The essence of this effect is the dimensional reduction D → D − 2 in the dynamics of fermion pairing in a magnetic field: at weak coupling, this dynamics is dominated by the lowest Landau level (LLL) which is essentially (D − 2)-dimensional [1-3]. The effect may have interesting applications in condensed matter physics [4] and cosmology .

It is shown that in 3 + 1 dimensions, a constant magnetic field is a catalyst of dynamical chiral symmetry breaking, leading to generating a fermion mass even at the weakest attractive interaction between fermions. The essence of this effect is the dimensional reduction D → D − 2 (3 + 1 → 1 + 1) in the dynamics of fermion pairing in a magnetic field. The effect is illustrated in the Nambu-Jona-Lasinio model. Possible applications of this effect are briefly discussed.

We describe a (first, to the best of our knowledge) essentially soluble example of dynamical symmetry breaking phenomenon in a 3+1 dimensional gauge theory without fundamental scalar fields: QED in a constant magnetic field. 11.30. Rd, 11.30.Qc, Recently the magnetic catalysis of dynamical chiral symmetry breaking has been established as a universal phenomenon in 2+1 and 3+1 dimensions: a constant magnetic field leads to the generation of a fermion dynamical mass even at the weakest attractive interaction between fermions [1-3]. The essence of this effect is the dimensional reduction D → D − 2 in the dynamics of fermion pairing in a magnetic field: at weak coupling, this dynamics is dominated by the lowest Landau level (LLL) which is essentially (D − 2)-dimensional [1-3]. The effect may have interesting applications in condensed matter physics [4] and cosmology .

We investigate the charged-neutral difference in the pion self-energy at finite temperature T . Within Chiral Perturbation Theory (ChPT) we extend a previous analysis performed in the chiral and soft pion limits. Our analysis with physical pion masses leads to additional non-negligible contributions for temperatures typical of a meson gas, including a momentum-dependent function for the self-energy. In addition, a nonzero imaginary part arises to leading order, which we define consistently in the Coulomb gauge and comes from an infrared enhanced contribution due to thermal bath photons. For distributions typical of a heavy-ion meson gas, the charged and neutral pion masses and their difference depend on temperature through slowly increasing functions. Chiral symmetry restoration turns out to be ultimately responsible for keeping the charged-neutral mass difference smooth and compatible with the observed charged and neutral pion spectra. We study also phenomenological effects related to the thermal electromagnetic damping, which gives rise to corrections for transport coefficients and distinguishes between neutral and charged mean free times. An important part of the analysis is the connection with chiral symmetry restoration through the relation of the pion mass difference with the vector-axial spectral function difference, which holds at T = 0 due to a sum rule in the chiral and soft pion limits. We analyze the modifications of that sum rule including nonzero pion masses and temperature, up to O(T 2 ) ∼ O(M 2 π ). Both effects produce terms making the pion mass difference grow against chiral-restoring decreasing contributions. Finally, we analyze the corrections to the previous ChPT and sum rule results within the resonance saturation framework at finite temperature, including explicitly ρ and a1 exchanges. Our results show that the ChPT result is robust at low and intermediate temperatures, the leading resonance corrections within this framework being O(T 2 M 2 π /M 2 R ) with MR the involved resonance masses.