Goldstone bosons and fermions in QCD

We consider the version of QCD in Euclidean Landau gauge in which the restriction to the Gribov region is implemented by a local, renormalizable action. This action depends on the Gribov parameter $\gamma$, with dimensions of (mass)$^4$, whose value is fixed in terms of $\Lambda_{QCD}$, by the gap equation, known as the horizon condition, {\p \Gamma \over \p \gamma} = 0, where $\Gamma$ is the quantum effective action. The restriction to the Gribov region suppresses gluons in the infrared, which nicely explains why gluons are not in the physical spectrum, but this only makes makes more mysterious the origin of the long-range force between quarks. In the present article we exhibit the symmetries of $\Gamma$, and show that the solution to the gap equation, which defines the classical vacuum, spontaneously breaks some of the symmetries $\Gamma$. This implies the existence of massless Goldstone bosons and fermions that do not appear in the physical spectrum. Some of the Goldstone bosons may be exchanged between quarks, and are candidates for a long-range confining force. As an exact result we also find that in the infrared limit the gluon propagator vanishes like $k^2$.Comment: 22 pages, typos corrected, improved comparison with lattice dat

Some exact properties of the gluon propagator

Recent numerical studies of the gluon propagator in the minimal Landau and Coulomb gauges in space-time dimension 2, 3, and 4 pose a challenge to the Gribov confinement scenario. We prove, without approximation, that for these gauges, the continuum gluon propagator $D(k)$ in SU(N) gauge theory satisfies the bound ${d-1 \over d} {1 \over (2 \pi)^d} \int d^dk {D(k) \over k^2} \leq N$. This holds for Landau gauge, in which case $d$ is the dimension of space-time, and for Coulomb gauge, in which case $d$ is the dimension of ordinary space and $D(k)$ is the instantaneous spatial gluon propagator. This bound implies that $\lim_{k \to 0}k^{d-2} D(k) = 0$, where $D(k)$ is the gluon propagator at momentum $k$, and consequently $D(0) = 0$ in Landau gauge in space-time $d = 2$, and in Coulomb gauge in space dimension $d = 2$, but D(0) may be finite in higher dimension. These results are compatible with numerical studies of the Landau-and Coulomb-gauge propagator. In 4-dimensional space-time a regularization is required, and we also prove an analogous bound on the lattice gluon propagator, ${1 \over d (2 \pi)^d} \int_{- \pi}^{\pi} d^dk {\sum_\mu \cos^2(k_\mu/2) D_{\mu \mu}(k) \over 4 \sum_\lambda \sin^2(k_\lambda/2)} \leq N$. Here we have taken the infinite-volume limit of lattice gauge theory at fixed lattice spacing, and the lattice momentum componant $k_\mu$ is a continuous angle $- \pi \leq k_\mu \leq \pi$. Unexpectedly, this implies a bound on the {\it high-momentum} behavior of the continuum propagator in minimum Landau and Coulomb gauge in 4 space-time dimensions which, moreover, is compatible with the perturbative renormalization group when the theory is asymptotically free.Comment: 13 page

Equivariant Symplectic Geometry of Gauge Fixing in Yang-Mills Theory

The Faddeev-Popov gauge fixing in Yang-Mills theory is interpreted as equivariant localization. It is shown that the Faddeev-Popov procedure amounts to a construction of a symplectic manifold with a Hamiltonian group action. The BRST cohomology is shown to be equivalent to the equivariant cohomology based on this symplectic manifold with Hamiltonian group action. The ghost operator is interpreted as a (pre)symplectic form and the gauge condition as the moment map corresponding to the Hamiltonian group action. This results in the identification of the gauge fixing action as a closed equivariant form, the sum of an equivariant symplectic form and a certain closed equivariant 4-form which ensures convergence. An almost complex structure compatible with the symplectic form is constructed. The equivariant localization principle is used to localize the path integrals onto the gauge slice. The Gribov problem is also discussed in the context of equivariant localization principle. As a simple illustration of the methods developed in the paper, the partition function of N=2 supersymmetric quantum mechanics is calculated by equivariant localizationComment: 46 pages, added remarks, typos and references correcte

An Infrared Safe perturbative approach to Yang-Mills correlators

We investigate the 2-point correlation functions of Yang-Mills theory in the Landau gauge by means of a massive extension of the Faddeev-Popov action. This model is based on some phenomenological arguments and constraints on the ultraviolet behavior of the theory. We show that the running coupling constant remains finite at all energy scales (no Landau pole) for $d>2$ and argue that the relevant parameter of perturbation theory is significantly smaller than 1 at all energies. Perturbative results at low orders are therefore expected to be satisfactory and we indeed find a very good agreement between 1-loop correlation functions and the lattice simulations, in 3 and 4 dimensions. Dimension 2 is shown to play the role of an upper critical dimension, which explains why the lattice predictions are qualitatively different from those in higher dimensions.Comment: 16 pages, 7 figures, accepted for publication in PR

Indirect lattice evidence for the Refined Gribov-Zwanziger formalism and the gluon condensate ⟨A2⟩\braket{A^2} in the Landau gauge

We consider the gluon propagator $D(p^2)$ at various lattice sizes and spacings in the case of pure SU(3) Yang-Mills gauge theories using the Landau gauge fixing. We discuss a class of fits in the infrared region in order to (in)validate the tree level analytical prediction in terms of the (Refined) Gribov-Zwanziger framework. It turns out that an important role is played by the presence of the widely studied dimension two gluon condensate $\braket{A^2}$. Including this effect allows to obtain an acceptable fit up to 1 \'{a} 1.5 GeV, while corroborating the Refined Gribov-Zwanziger prediction for the gluon propagator. We also discuss the infinite volume extrapolation, leading to the estimate $D(0)=8.3\pm0.5\text{GeV}^{-2}$. As a byproduct, we can also provide the prediction $\braket{g^2 A^2}\approx 3\text{GeV}^2$ obtained at the renormalization scale $\mu=10\text{GeV}$.Comment: 17 pages, 10 figures, updated version, accepted for publication in Phs.Rev.

The ice-limit of Coulomb gauge Yang-Mills theory

In this paper we describe gauge invariant multi-quark states generalising the path integral framework developed by Parrinello, Jona-Lasinio and Zwanziger to amend the Faddeev-Popov approach. This allows us to produce states such that, in a limit which we call the ice-limit, fermions are dressed with glue exclusively from the fundamental modular region associated with Coulomb gauge. The limit can be taken analytically without difficulties, avoiding the Gribov problem. This is llustrated by an unambiguous construction of gauge invariant mesonic states for which we simulate the static quark--antiquark potential.Comment: 25 pages, 4 figure

A refinement of the Gribov-Zwanziger approach in the Landau gauge: infrared propagators in harmony with the lattice results

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 BRST symmetry arising in the Gribov-Zwanziger approach is provided.Comment: 38 pages, 9 figures, the content of section V has been extended and adapte

Two loop MSbar Gribov mass gap equation with massive quarks

We compute the two loop MSbar correction to the Gribov mass gap equation in the Landau gauge using the Gribov-Zwanziger Lagrangian with massive quarks included. The computation involves dilogarithms of complex arguments and reproduces the known gap equation when the quark mass tends to zero.Comment: 20 latex page

Hamiltonian structures of fermionic two-dimensional Toda lattice hierarchies

By exhibiting the corresponding Lax pair representations we propose a wide class of integrable two-dimensional (2D) fermionic Toda lattice (TL) hierarchies which includes the 2D N=(2|2) and N=(0|2) supersymmetric TL hierarchies as particular cases. We develop the generalized graded R-matrix formalism using the generalized graded bracket on the space of graded operators with involution generalizing the graded commutator in superalgebras, which allows one to describe these hierarchies in the framework of the Hamiltonian formalism and construct their first two Hamiltonian structures. The first Hamiltonian structure is obtained for both bosonic and fermionic Lax operators while the second Hamiltonian structure is established for bosonic Lax operators only.Comment: 12 pages, LaTeX, the talks delivered at the International Workshop on Classical and Quantum Integrable Systems (Dubna, January 24 - 28, 2005) and International Conference on Theoretical Physics (Moscow, April 11 - 16, 2005

A study of the gauge invariant, nonlocal mass operator Tr∫d4xFμν(D2)−1FμνTr \int d^4x F_{\mu\nu}(D^2)^{-1} F_{\mu\nu} in Yang-Mills theories

The nonlocal mass operator $Tr \int d^4x F_{\mu\nu} (D^2)^{-1} F_{\mu\nu}$ is considered in Yang-Mills theories in Euclidean space-time. It is shown that the operator $Tr \int d^4x F_{\mu\nu} (D^2)^{-1} F_{\mu\nu}$ can be cast in local form through the introduction of a set of additional fields. A local and polynomial action is thus identified. Its multiplicative renormalizability is proven by means of the algebraic renormalization in the class of linear covariant gauges. The anomalous dimensions of the fields and of the mass operator are computed at one loop order. A few remarks on the possible role of this operator for the issue of the gauge invariance of the dimension two condensates are outlined.Comment: 34 page