A synopsis exemplifying the employment of Dyson-Schwinger equations in the calculation and explanation of hadron electromagnetic form factors and related phenomena. In particular the contribution: presents the pion form factor computed simultaneously at spacelike and timelike momenta; reports aspects of the evolution of the nucleon and Delta masses with current-quark mass and the correlation of their mass difference with that between scalar and axial-vector diquarks; describes an estimate of the s-quark content of a dressed u-quark and its impact on the nucleon's strangeness magnetic moment; and comments upon the domain within which a pseudoscalar meson cloud can materially contribute to hadron form factors.
A strongly momentum-dependent dressed-quark mass function is basic to QCD. It is central to the appearance of a constituent-quark mass-scale and an existential prerequisite for Goldstone modes. Dyson-Schwinger equation (DSE) studies have long emphasised these facts and are a natural way to exploit them.
The contemporary use of Dyson-Schwinger equations in hadronic physics is exemplified via applications to the calculation of pseudoscalar meson masses, and inclusive deep inelastic scattering with a determination of the pion's valence-quark distribution function.
We illustrate the contemporary application of Dyson-Schwinger equations using two examples: the calculation of pseudoscalar meson masses, an associated model-independent mass formula and the approach to the heavy-quark limit; and the study of nucleon observables, including a calculation of its mass, $M$, via a covariant Fadde'ev equation and an estimate of pion-loop contributions to $M$.
It is shown how to apply the Maximum Entropy Method (MEM) to numerical Dyson-Schwinger studies for the extraction of spectral functions of correlators from their corresponding Euclidean propagators. Differences to the application in lattice QCD are emphasized and, as an example, the spectral functions of massless quarks in cold and dense matter are presented.
We review applications of Dyson-Schwinger equations at nonzero temperature, T, and chemical potential, mu, touching topics such as: deconfinement and chiral symmetry restoration; the behaviour of bulk thermodynamic quantities; the (T,mu)-dependence of hadron properties; and the possibility of diquark condensation.
There have been many demonstrations of the utility of the Dyson-Schwinger equations of QCD as a systematic, phenomenological framework for describing the perturbative and non-perturbative dynamics of hadrons in terms of Euclidean Green functions of quarks and gluons. Still, there remain some unanswered questions regarding the theoretical underpinnings of the approach. I review several studies that are shedding light on how these questions might be resolved and review predictions for some exotic meson states.
In these proceedings we present a mini-review on the topic of the Dyson-Schwinger/Bethe-Salpeter approach to the study of relativistic bound-states in physics. In particular, we present a self-contained discussion of their derivation, as well as their truncation such that important symmetries are maintained.
We report on an analysis of the quark spectral representation at finite temperatures based on the quark propagator determined from its Dyson-Schwinger equation in Landau gauge. In Euclidean space we achieve nice agreement with recent results from quenched lattice QCD. We find different analytical properties of the quark propagator below and above the deconfinement transition. Using a variety of ansaetze for the spectral function we then analyze the possible quasiparticle spectrum, in particular its quark mass and momentum dependence in the high temperature phase. This analysis is completed by …
In a scalar $ϕ^2χ$ model, we exactly solve the vertex and $ϕ$ propagator Dyson-Schwinger equations under the assumption of a spatially constant (Munczek-Nemirovsky) propagator for the $χ$ field. Various truncation schemes are also considered.
We present progress in the evaluation of $γ^* N \rightarrow N^*(1535)$ transition form factors in a quark-diquark picture of these baryons. Our analysis is based upon the fully-consistent treatment of a vector $\times$ vector contact interaction, embedded in the interlaced formalism of Schwinger-Dyson and Bethe-Salpeter equations.
We obtain the QCD quark condensate from consideration of unquenched quark dynamics in Dyson-Schwinger gluon vacuum. We consider the non-local extension of the condensate and determine the quark virtuality. We also obtain the condensate-driven contribution of the non-perturbative QCD to Euler--Heisenberg Lagrangian of QED in external electromagnetic fields.
We compute the pion quark Generalized Parton Distribution H and quark Double Distributions F and G in a coupled Bethe-Salpeter and Dyson-Schwinger approach in terms of quark flavors or isospin states. We use simple analytic expressions inspired by the numerical resolution of Dyson-Schwinger and Bethe-Salpeter equations. We explicitly check the support and polynomiality properties, and the behavior under charge conjugation or time invariance of our model. We obtain analytic expressions for the pion Double Distributions and Generalized Parton Distribution at vanishing pion momentum transfer at a…
We investigate the electromagnetic form factors of the Delta and the Omega baryons within the Poincare-covariant framework of Dyson-Schwinger and Bethe-Salpeter equations. The three-quark core contributions of the form factors are evaluated by employing a quark-diquark approximation. We use a consistent setup for the quark-gluon dressing, the quark-quark bound-state kernel and the quark-photon interaction. Our predictions for the multipole form factors are compatible with available experimental data and quark-model estimates. The current-quark mass evolution of the static electromagnetic prope…
The applications of the consistently coupled Schwinger-Dyson and Bethe-Salpeter approach to the eta-eta' complex are extended to the two-photon transition form factors of eta and eta' for spacelike transferred momenta. We compare our predictions with experiment and some other theoretical approaches.
I illustrate the phenomenological application of Dyson-Schwinger equations to the calculation of meson properties observable at TJNAF. Particular emphasis is given to this framework's ability to unify long-range effects constrained by chiral symmetry with short-range effects prescribed by perturbation theory, and interpolate between them.
It is reviewed pedagogically how a very successful description of the eta-eta' mass matrix can be achieved in the consistently coupled Schwinger-Dyson and Bethe-Salpeter approach in spite of the limitations of the ladder approximation. This description is in agreement with both phenomenology and lattice results.
Pion and kaon observables are calculated using a Dyson-Schwinger Bethe-Salpeter formalism. It is shown that an infrared finite gluon propagator can lead to quark confinement via generation of complex mass poles in quark propagators. Observables, including electromagnetic form factors, are calculated entirely in Euclidean metric for spacelike values of bound state momentum and final results are extrapolated to the physical region.
The form factor for gamma pi^+ -> pi^+ pi^0 was calculated in a simple-minded constituent model with a constant quark mass parameter, as well as in the Schwinger-Dyson approach. The comparative discussion of these and various other theoretical results on this anomalous process, as well as the scarce already available data (hopefully to be supplemented by more accurate CEBAF data soon), seem to favor Schwinger-Dyson modeling which would yield relatively small low-momentum values of the constituent (dynamically dressed) quark mass function.
The description of the $η$ and $η^\prime$ mesons in the Dyson-Schwinger approach has relied on the Witten-Veneziano relation. The present paper explores the consequences of using instead its generalization recently proposed by Shore. On the examples of three different model interactions, we find that irrespective of the concrete model dynamics, our Dyson-Schwinger approach is phenomenologically more successful in conjunction with the standard Witten-Veneziano relation than with the proposed generalization valid in all orders in the $1/N_c$ expansion.
We investigate the color-spin locked (CSL) phase of spin-one color- superconducting quark matter using a truncated Dyson-Schwinger equation for the quark propagator in Landau gauge. Starting from the most general parity conserving ansatz allowed by the CSL symmetry, the Dyson-Schwinger equation is solved self-consistently and dispersion relations are discussed. We find that chiral symmetry is spontaneously broken due to terms which have previously been neglected. As a consequence, the excitation spectrum contains only gapped modes even for massless quarks. Moreover, at moderate chemical potent…
Since the baryon-number susceptibilities are correlated with the cumulant of baryon-number fluctuations in experiments, we do calculations of the susceptibilities and compare them with the experimental fluctuation data under the framework of the Dyson-Schwinger equations (DSEs) approach. We compare our results with lattice QCD and experimental data at RHIC. The fitness of the results indicates that under the framework of DSEs, we can deal with the problems of heavy ion collisions properly.
In this letter it is shown on general ground that there exist two qualitatively distinct solutions of the Dyson-Schwinger equation for the quark propagator in the case of non-zero current quark mass. One solution corresponds to the ``Nambu-Goldstone'' phase and the other one corresponds to the ``Wigner'' phase in the chiral limit.
The masses and decay constants of the light mesons are studied within a ladder-rainbow truncation of the set of Dyson-Schwinger equations using a model 2-point gluon function. The one phenomenological parameter and two current quark masses are fitted to reproduce $f_π$, $m_π$ and $m_K$. Our results for $f_K$, and for the vector mesons $ρ$, $ϕ$, and $K^\star$ are in good agreement with the experimental values.
In this work, I studied the spectrum of scalar mesons and axial-vector mesons via Dyson-Schwinger equation and Bethe-Salpeter equation approach in the rainbow-ladder approximation. An interaction model with a repulsive term added to the one used for the pseudoscalar and vector mesons is proposed. My results are consistent with the experiment data and other model results, which shows that this interaction model is effective for all the heavy, heavy-light, and light scalar and axial-vector mesons.
The combined Dyson-Schwinger and Bethe-Salpeter equations in rainbow-ladder approximation are used to search for Regge trajectories of mesons in the pseudo-scalar and vector channels. We focus on the often employed Alkofer-Watson-Weigel kernel which is known to deliver good results for the ground state meson spectra; it provides linear Regge trajectories in the $J^P= 0^-$ channel.
We review briefly some of the successes of Schwinger-Dyson (SD) approach to the physics of quarks and hadrons, primarily light pseudoscalar mesons including eta and eta'. The main purpose of this paper is to point out that SD results on eta and eta' mesons can be formulated and given also in the two-mixing-angle scheme.
In the case of nonzero fermion mass, within a range of Ansatze for the full fermion-boson vertex, we show that Dyson-Schwinger equation for the fermion propagator in QED3 has two qualitatively distinct dynamical chiral symmetry breaking solutions. As the fermion mass increases and reaches to a critical value mc, one solution disappears, and the dependence of mc on the number of fermion flavors is also given.
We present a simple description of pion-nucleon ($πN$) scattering taking into account the full complexity of pion absorption and creation on the nucleon. To do this we solve Dyson-Schwinger equations within the framework of Time-Ordered Perturbation Theory. This enables us to construct partial wave separable $ πN$ t matrices that can be useful in models of nuclear processes involving fully dressed nucleons. At the same time, our approach demonstrates features of Quantum Field Theory, like particle dressing, renormalisation, and the use of Dyson-Schwinger equations, in a non-relativistic contex…
The role of neutrality constraints for the phase structure of QCD at non-vanishing chemical potentials is studied within a self-consistent truncation scheme for the Dyson-Schwinger equation of the quark propagator in Landau gauge. We find the (approximate) color-flavor-locked phase to be energetically preferred at all potentially relevant densities and for physical values of the quark masses. We furthermore observe the impossibility to define this phase by residual global symmetries and discuss the role of chemical potentials.
We summarise a Dyson-Schwinger-equation-based calculation of an extensive range of light- and heavy-meson observables, characterised by heavy-meson leptonic decays, semileptonic heavy-to-heavy and heavy-to-light transitions, radiative and strong decays, and the rare flavour-changing neutral-current process. In the calculation the heavy-quark mass functions are approximated by constants, interpreted as their constituent mass.
We show that Dyson resummation schemes based on Baym's $Φ$-derivable approximations can be renormalized with counter term structures solely defined on the vacuum level. First applications to the self-consistent solution of the sunset self-energy in $ϕ^4$-theory are presented.
Motivated by our earlier findings of sensitive quark-flavor dependence of QCD topological susceptibility on products of current quark masses and corresponding condensates, we allow the breaking of isospin symmetry. For the purpose of future investigations of U_A(1) symmetry breaking and restoration at T > 0, we perform (at T = 0) refitting of the quark-mass parameters of a phenomenologically successful effective model of low-energy QCD. It belongs to the class of separable-interaction models within the Dyson-Schwinger approach to the quark-antiquark substructure of mesons.
We employ the Dyson-Schwinger equations for quark and gluon propagators in order to study QCD with 2+1 flavours at finite temperature and density. In a suitable truncation for these equations, we determine the position of the critical end-point as well as the deconfinement temperature at all chemical potentials. For the latter, the Polyakov-loop potential is obtained from the QCD propagators. This is possible for the first time at finite chemical potential, with implications for effective models.
We study the hadron-quark phase transition at finite temperature in the interior of protoneutron stars, combining the Dyson-Schwinger model for quark matter with the Brueckner-Hartree-Fock approach for hadronic matter. We discuss the dependence of the results on different nuclear three-body forces and on details of the quark model. %Compared to the bag model, the transition from the hadron phase %(without hyperons) to the quark phase appears at large density. A maximum mass exceeding two solar masses can be obtained with a strong three-body force and suitable parameter values in the Dyson-Schw…
In this paper we introduce an equation of state (EOS) of quark matter within the framework of Dyson-Schwinger equations (DSEs) to study the structure of compact stars. The smooth crossover from hadronic matter to quark matter in the hybrid star is studied. We compare different strategies to obtain crossover EOSs and find a new way to construct two-solar-mass hybrid stars with even a relatively soft quark EOS, while earlier works show that the quark EOS should be stiff enough to support a massive hybrid star.
We study pseudoscalar and scalar mesons using a practical and symmetry preserving truncation of QCD's Dyson-Schwinger equations. We investigate and compare properties of ground and radially excited meson states. In addition to exact results for radial meson excitations we also present results for meson masses and decay constants from the chiral limit up to the charm-quark mass, e. g., the mass of the chi_{c0} (2P) meson.
I briefly summarize the application of the Dyson-Schwinger/Faddeev approach to baryon form factors. Recent results for nucleon electromagnetic and axial form factors as well as N-Delta-Gamma electromagnetic transition form factors are discussed. The calculation of the current diagrams from the quark-gluon level enables an analysis of common features, such as the implications of dynamical chiral symmetry breaking and quark orbital angular momentum, the timelike structure of the form factors, and their interpretation in terms of missing pion-cloud effects.
We study the hadron-quark phase transition in the interior of neutron stars. For the hadronic sector, we use a microscopic equation of state involving nucleons and hyperons derived within the Brueckner-Hartree-Fock many-body theory with realistic two-body and three-body forces. For the description of quark matter, we employ the Dyson-Schwinger approach and compare with the MIT bag model. We calculate the structure of neutron star interiors comprising both phases and find that with the Dyson-Schwinger model, the hadron-quark phase transition takes place only when hyperons are excluded, and that…
We solve Schwinger-Dyson (SD) equation for the color superconducting (CS) gap in nonlocal gauge which justifies the ladder approximation in subleading order calculation. The gap prefactor increases about 1.6 times to the leading order gap determined in Landau gauge.
Any practical application of the Schwinger-Dyson equations to the study of $n$-point Green's functions of a field theory requires truncations, the best known being finite order perturbation theory. Strong coupling studies require a different approach. In the case of QED, gauge covariance is a powerful constraint. By using a spectral representation for the massive fermion propagator in QED, we are able to show that the constraints imposed by the Landau-Khalatnikov-Fradkin transformations are linear operations on the spectral densities. Here we formally define these group operations and show wit…
The effects of higher order corrections of ring diagrams for the quark condensate are studied by using the bare vertex Nuclear Schwinger Dyson formalism based on $σ$-$ω$ model. At the high density the quark condensate is reduced by the higher order contribution of ring diagrams more than the mean field theory or the Hartree-Fock .
A globally converging numerical method to solve coupled sets of non-linear integral equations is presented. Such systems occur e.g. in the study of Dyson-Schwinger equations of Yang-Mills theory and QCD. The method is based on the knowledge of the qualitative properties of the solution functions in the far infrared and ultraviolet. Using this input, the full solutions are constructed using a globally convergent modified Newton iteration. Two different systems will be treated as examples: The Dyson-Schwinger equations of 3-dimensional Yang-Mills-Higgs theory provide a system of finite integrals…
The anomalous processes, γ\to 3 πand γπ\to ππ, are investigated within the Dyson-Schwinger framework using the rainbow-ladder approximation. Calculations reveal that a complete set of ladder diagrams beyond the impulse approximation are necessary to reproduce the fundamental low-energy theorem for the anomalous form factor. Higher momentum calculations also agree with the limited form factor data and exhibit the same resonance behavior as the phenomenological vector meson dominance model.
The ladder-rainbow truncation of the set of Dyson-Schwinger equations is used to study the pion and kaon electromagnetic form factors and the $γ^\star π^0 γ$ transition form factor in impulse approximation. With model parameters previously fixed by the pseudoscalar meson masses and decay constants, the obtained form factors are in good agreement with the data.
We study the two-point correlator of an O(N) scalar field with quartic self-coupling in de Sitter space. For light fields in units of the expansion rate, perturbation theory is plagued by large logarithmic terms for superhorizon momenta. We show that a proper treatment of the infinite series of self-energy insertions through the Schwinger-Dyson equations resums these infrared logarithms into well defined power laws. We provide an exact analytical solution of the Schwinger-Dyson equations for infrared momenta when the self-energy is computed at two-loop order. The obtained correlator exhibits a…
We study the properties of two-flavor quark matter in the Dyson-Schwinger model and investigate the possible consequences for hybrid neutron stars, with particular regard to the two-solar-mass limit. We find that with some extreme values of the model parameters, the mass fraction of two-flavor quark matter in heavy neutron stars can be as high as 30 percent and the possible energy release during the conversion from nucleonic neutron stars to hybrid stars can reach $10^{52}\;$erg.
The Schwinger-Dyson equations for the nucleon and meson propagators are solved self-consistently in an approximation that goes beyond the Hartree-Fock approximation. The traditional approach consists in solving the nucleon Schwinger-Dyson equation with bare meson propagators and bare meson-nucleon vertices; the corrections to the meson propagators are calculated using the bare nucleon propagator and bare nucleon-meson vertices. It is known that such an approximation scheme produces the appearance of ghost poles in the propagators. In this paper the coupled system of Schwinger-Dyson equations f…
We study rapidly rotating hybrid stars with the Dyson-Schwinger model for quark matter and the Brueckner-Hartree-Fock many-body theory with realistic two-body and three-body forces for nuclear matter. We determine the maximum gravitational mass, equatorial radius, and rotation frequency of stable stellar configurations by considering the constraints of the Keplerian limit and the secular axisymmetric instability, and compare with observational data. We also discuss the rotational evolution for constant baryonic mass, and find a spinup phenomenon for supramassive stars before they collapse to b…
