On general ultraviolet properties of a class of confining propagators
Eur. Phys. J. C
On general ultraviolet properties of a class of confining propagators
M. A. L. Capri 0
M. S. Guimaraes 0
I. F. Justo 0
L. F. Palhares 0
S. P. Sorella 0
0 Departamento de Física Teórica, Instituto de Física , UERJ, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier 524, Maracanã, Rio de Janeiro 20550-013 , Brazil
We study the ultraviolet properties of theories whose fundamental fields display a confining, Gribov-type, propagator. These are propagators that exhibit complex poles and violate positivity, thus precluding a physical propagating particle interpretation. We show that the properties of this type of confining propagators do not change the ultraviolet behavior of the theory, in the sense that no new ultraviolet primitive divergences are generated, thus securing the renormalizability of these confining theories. We illustrate these properties by studying a variety of models, including bosonic and fermionic confined degrees of freedom. The more intricate case of super-Yang-Mills with N = 1 supersymmetries in the Wess-Zumino gauge is taken as example in order to prove these statements to all orders by means of the algebraic renormalization set up.
1 Introduction
The quantization of non-abelian gauge theories is still an
open and rich subject for quantum field theorists. A general
framework which takes into account the non-perturbative
phenomena of gluon and quark confinement and of chiral
symmetry breaking is still lacking. Needless to say, these
issues represent a major challenge for our current
understanding of non-abelian gauge theories in the non-perturbative
infrared regime.
A successful approach to investigate these topics within
the context of the Euclidean quantum field theory is provided
by the Gribov framework1. In his seminal work [
4
],
Gribov pointed out that the Faddeev–Popov gauge-fixing
procedure is plagued by the existence of Gribov copies. In a
path-integral formulation this manifests in the fact that the
Faddeev–Popov operator develops zero modes. For example,
in the Landau gauge, ∂μ Aaμ = 0, the Faddeev–Popov
operator is given by Mab = −(∂2δab − g f abc Acμ∂μ). For
sufficiently strong coupling constant, zero modes of this operator
start to appear, rendering ill-defined the Faddeev-Propov
procedure. To deal with this issue, Gribov proposed to restrict the
domain of integration in the path integral to a region in field
space where the eigenvalues of the Faddeev–Popov
operator are strictly positive. This region is known as the Gribov
region , being defined as
It is useful to observe here that the inverse of the Faddeev–
Popov operator, (M−1)ab, yields precisely the propagator
G(k, A) of the Faddeev–Popov ghosts in the presence of an
external gauge field Aaμ [
1–4
], namely
G(k, A) = (N 2 − 1)
1
δab k|(M−1)ab|k .
This property can be employed to implement the restriction to
the Gribov region by requiring the absence of poles in the
ghost propagator for any non-zero value of the ghost external
momentum k. This requirement is known as the Gribov
nopole condition. More precisely, following [
4
], one can always
represent the exact ghost propagator in the presence of an
external gauge field as
1 For reviews on the Gribov issues, see [
1–3
] and references therein.
(1)
(2)
1
G(k, A) = k2 (1 + σ (k, A)),
where σ (k, A) is the ghost form factor. Using the general
properties of the diagrammatic expansion of quantum field
theory, we can write
1
G(k) = G(k, A) conn = k2 (1 + σ (k, A) conn)
1
1
= k2 (1 − σ (k, A) 1P I )
where “conn” stands for the connected set of diagrams and
1 P I denotes the 1-particle irreducible ones. It can be shown
that σ (k, A) 1P I is a decreasing function of k [
1–4
].
Therefore, the condition that the ghost propagator has no poles for
any non-zero value of the ghost external momentum can be
expressed as a condition for the maximum value of the ghost
form factor, i.e.
σ (0, A) 1P I = 1.
Equation (5) expresses the no-pole condition. In [
5
], it has
been shown that an exact closed expression for σ (0, A) can
be obtained, being proportional to the horizon function H ( A)
of Zwanziger’s formalism [
1
]:
.
The no-pole condition (5) is thus equivalent to
H ( A) 1P I = V 4(N 2 − 1),
which is called the horizon condition. The relevance of the
horizon function H ( A) relies on the fact that the restriction
of the domain of integration in the functional integral to the
Gribov region can be effectively implemented by adding
to the Yang–Mills action the quantity H ( A). More precisely,
it turns out that, in the thermodynamic limit [
1,6–8
], the
partition function of the theory with the cut-off at the Gribov
region is given by
Z =
=
D Aδ(∂ A) det(Mab)e−SY M
D Aδ(∂ A) det(Mab)e−(SY M +γ 4 H(A)−γ 4V 4(N 2−1)),
where the massive parameter γ is known as the Gribov
parameter [
1–4
]. It is not a free parameter, being determined
in a self-consistent way by the horizon condition (7), which
can be rewritten as a stati (...truncated)