Recent studies of QCD Green's functions and their applications in hadronic physics are reviewed. We briefly discuss the issues of gauge fixing, BRS invariance and positivity. Evidence for the violation of positivity by quarks and transverse gluons in the covariant gauge is collected, and it is argued that this is one manifestation of confinement. 
We summarise the derivation of the Dyson-Schwinger equations (DSEs) of QED and QCD. The influence of instantons on DSEs in a 2-dimensional model is mentioned. Solutions for the Green's functions in QED in 2+1 and 3+1 dimensions provide tests of various schemes to truncate DSEs. We discuss possible extensions to QCD and their limitations. Truncation schemes for DSEs of QCD are discussed in the axial gauge and in the Landau gauge. We review the available results from a systematic non-perturbative expansion scheme established for Landau gauge QCD. Comparisons to related lattice results, where available, are presented. 
The applications of QCD Green's functions to hadron physics are summarized. Properties of ground state mesons are discussed on the basis of the Bethe-Salpeter equation for quarks and antiquarks. The Goldstone nature of pseudoscalar mesons and mechanisms of diquark confinement are reviewed. We discuss some properties of ground state baryons based on their description as Bethe-Salpeter/Faddeev bound states of quark-diquark correlations in the quantum field theory of confined quarks and gluons.

The Infrared Behavior of QCD Green's Functions - Confinement, Dynamical Symmetry Breaking, and Hadrons as Relativistic Bound States

The infrared behaviour of QCD Green's functions ☆: Confinement, dynamical symmetry breaking, and hadrons as relativistic bound states

Recent lattice data have reported an infrared suppressed, positivity violating gluon propagator which is nonvanishing at zero momentum and a ghost propagator which is no longer enhanced. This paper discusses how to obtain analytical results which are in qualitative agreement with these lattice data within the Gribov-Zwanziger framework. This framework allows one to take into account effects related to the existence of gauge copies, by restricting the domain of integration in the path integral to the Gribov region. We elaborate to great extent on a previous short paper by presenting additional results, also confirmed by the numerical simulations. A detailed discussion on the soft breaking of the Becchi-Rouet-Stora-Tyutin symmetry arising in the Gribov-Zwanziger approach is provided.

Refinement of the Gribov-Zwanziger approach in the Landau gauge: Infrared propagators in harmony with the lattice results

An account is given of the new insight into the theory of magnetic monopoles originating from the work of 't Hooft (1976) and Polyakov (1974). Their magnetic monopole, associated with the conventional electromagnetic gauge group U(1), occurs as a finite-energy smooth soliton solution to an SU(2) gauge theory. A precise picture of its internal structure, the values of its magnetic charge and its mass are obtained. These new developments bring together previously unrelated fields of study, namely the Dirac monopole (with point structure) and the Sine-Gordon soliton in two-dimensional space-time. Properties of more general monopoles, associated with large gauge groups now thought to be relevant in physics, are discussed. Based on this new viewpoint, conjectures can be made about a future quantum theory of monopoles.

/pdf/magnetic-monopoles-in-gauge-field-theories-3qb5n4wmxr.pdf

Magnetic monopoles in gauge field theories

During the last few years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wavefunction statistics. In the first part of this review we give a detailed introduction to the spectral theory of quantum graphs and discuss exact trace formulae for the spectrum and the quantum-to-classical correspondence. The second part of this review is devoted to the spectral statistics of quantum graphs as an application to quantum chaos. In particular, we summarize recent developments on the spectral statistics of generic large quantum graphs based on two approaches: the periodic-orbit approach and the supersymmetry approach. The latter provides a condition and a proof for universal spectral statistics as predicted by random-matrix theory.

https://www.physik.fu-berlin.de/en/einrichtungen/ag/ag-von-oppen/papers/nlin_CD_0605028.pdf

Quantum graphs: Applications to quantum chaos and universal spectral statistics

TheL
2 topology is introduced on the space of gauge connectionsA and a natural topology is introduced on the group of local gauge transformationsGT. It is shown that the mappingGT×A→A defined byA→A
g=g*Ag+g*dg is continuous and that each gauge orbit is closed. The Hilbert norm of the gauge connection achieves its absolute minimum on each gauge orbit, at which point the orbit intersects the region bounded by the Gribov horizon.

/pdf/every-gauge-orbit-passes-inside-the-gribov-horizon-44odgfj00z.pdf

Every gauge orbit passes inside the Gribov horizon

Ellipsoidal bound on the Gribov horizon contradicts the perturbative renormalization group

En utilisant des techniques de renormalisation de formes quadratiques singulieres, on etudie des Hamiltoniens pour systemes de N particules avec interactions de portee nulle, en dimension deux. Si on utilise la meme methode en dimension trois, on obtient des formes quadratiques qui, pour N = 3, ont une borne inferieure seulement si l'on impose des conditions de symetrie, et qui, pour N assez grand, n'ont, en tout cas, pas de borne inferieure

/pdf/hamiltonians-for-systems-of-n-particles-interacting-through-198m9xd3p3.pdf

Hamiltonians for systems of N particles interacting through point interactions

Thirring model in terms of currents: solution and light cone expansions.

We consider a quantum mechanical three-particle system made of two identical fermions of mass one and a different particle of mass m, where each fermion interacts via a zero-range force with the different particle. In particular we study the unitary regime, i.e., the case of infinite two-body scattering length. The Hamiltonians describing the system are, by definition, self-adjoint extensions of the free Hamiltonian restricted on smooth functions vanishing at the two-body coincidence planes, i.e., where the positions of two interacting particles coincide. It is known that for m larger than a critical value m
 ∗ ≃ (13.607)−1 a self-adjoint and lower bounded Hamiltonian H
 0 can be constructed, whose domain is characterized in terms of the standard point-interaction boundary condition at each coincidence plane. Here we prove that for m ∈ (m
 ∗,m
 ∗∗), where m
 ∗∗ ≃ (8.62)−1, there is a further family of self-adjoint and lower bounded Hamiltonians H
 0,β
 , β ∈ ℝ, describing the system. Using a quadratic form method, we give a rigorous construction of such Hamiltonians and we show that the elements of their domains satisfy a further boundary condition, characterizing the singular behavior when the positions of all the three particles coincide.

Gianfausto Dell’Antonio

Papers

Every gauge orbit passes inside the Gribov horizon

Ellipsoidal bound on the Gribov horizon contradicts the perturbative renormalization group

Hamiltonians for systems of N particles interacting through point interactions

Thirring model in terms of currents: solution and light cone expansions.

A Class of Hamiltonians for a Three-Particle Fermionic System at Unitarity