55 research outputs found
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 , with dimensions
of (mass), whose value is fixed in terms of , by the gap
equation, known as the horizon condition, {\p \Gamma \over \p \gamma} = 0,
where 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 , and show that the solution to the gap
equation, which defines the classical vacuum, spontaneously breaks some of the
symmetries . 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 .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 in SU(N) gauge theory satisfies the bound . This holds for Landau
gauge, in which case is the dimension of space-time, and for Coulomb gauge,
in which case is the dimension of ordinary space and is the
instantaneous spatial gluon propagator. This bound implies that , where is the gluon propagator at momentum , and
consequently in Landau gauge in space-time , and in Coulomb
gauge in space dimension , 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, . Here we have taken the
infinite-volume limit of lattice gauge theory at fixed lattice spacing, and the
lattice momentum componant is a continuous angle . 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
Center Vortices and the Gribov Horizon
We show how the infinite color-Coulomb energy of color-charged states is
related to enhanced density of near-zero modes of the Faddeev-Popov operator,
and calculate this density numerically for both pure Yang-Mills and gauge-Higgs
systems at zero temperature, and for pure gauge theory in the deconfined phase.
We find that the enhancement of the eigenvalue density is tied to the presence
of percolating center vortex configurations, and that this property disappears
when center vortices are either removed from the lattice configurations, or
cease to percolate. We further demonstrate that thin center vortices have a
special geometrical status in gauge-field configuration space: Thin vortices
are located at conical or wedge singularities on the Gribov horizon. We show
that the Gribov region is itself a convex manifold in lattice configuration
space. The Coulomb gauge condition also has a special status; it is shown to be
an attractive fixed point of a more general gauge condition, interpolating
between the Coulomb and Landau gauges.Comment: 19 pages, 17 EPS figures, RevTeX4; v2: added references, corrected
caption of fig. 11; v3: new data for higher couplings, clarifications on
color-Coulomb potential in deconfined phase, version to appear in JHE
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 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 in the Landau gauge
We consider the gluon propagator 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
. 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 . As a byproduct, we can
also provide the prediction obtained at
the renormalization scale .Comment: 17 pages, 10 figures, updated version, accepted for publication in
Phs.Rev.
Poisson-Lie group of pseudodifferential symbols
We introduce a Lie bialgebra structure on the central extension of the Lie
algebra of differential operators on the line and the circle (with scalar or
matrix coefficients). This defines a Poisson--Lie structure on the dual group
of pseudodifferential symbols of an arbitrary real (or complex) order. We show
that the usual (second) Benney, KdV (or GL_n--Adler--Gelfand--Dickey) and KP
Poisson structures are naturally realized as restrictions of this Poisson
structure to submanifolds of this ``universal'' Poisson--Lie group.
Moreover, the reduced (=SL_n) versions of these manifolds (W_n-algebras in
physical terminology) can be viewed as subspaces of the quotient (or Poisson
reduction) of this Poisson--Lie group by the dressing action of the group of
functions.
Finally, we define an infinite set of functions in involution on the
Poisson--Lie group that give the standard families of Hamiltonians when
restricted to the submanifolds mentioned above. The Poisson structure and
Hamiltonians on the whole group interpolate between the Poisson structures and
Hamiltonians of Benney, KP and KdV flows. We also discuss the geometrical
meaning of W_\infty as a limit of Poisson algebras W_\epsilon as \epsilon goes
to 0.Comment: 64 pages, no figure
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
Maximal Non-Abelian Gauges and Topology of Gauge Orbit Space
We introduce two maximal non-abelian gauge fixing conditions on the space of
gauge orbits M for gauge theories over spaces with dimensions d < 3. The gauge
fixings are complete in the sense that describe an open dense set M_0 of the
space of gauge orbits M and select one and only one gauge field per gauge orbit
in M_0. There are not Gribov copies or ambiguities in these gauges. M_0 is a
contractible manifold with trivial topology. The set of gauge orbits which are
not described by the gauge conditions M \ M_0 is the boundary of M_0 and
encodes all non-trivial topological properties of the space of gauge orbits.
The gauge fields configurations of this boundary M \ M_0 can be explicitly
identified with non-abelian monopoles and they are shown to play a very
relevant role in the non-perturbative behaviour of gauge theories in one, two
and three space dimensions. It is conjectured that their role is also crucial
for quark confinement in 3+1 dimensional gauge theories.Comment: 31 pages, harvmac, 1 figur
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
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