Vertex Sensitivity in the Schwinger-Dyson Equations of QCD Page: 2 of 12
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Vertex Sensitivity in the SDEs of QCD
The non-perturbative gluon and ghost propagators may be obtained from their Schwinger-
Dyson Equations (SDEs). They are interesting quantities that provide important information when
studying confinement and dynamical chiral symmetry breaking in Quantum Chromodynamics
(QCD). They are necessary inputs for studying bound states and amongst other things, can be
used to obtain non-perturbative predictions for the running coupling as. A number of early stud-
ies [1, 2, 3, 4, 5] found various singularities in the vanishing p2 limit for the non-perturbative
propagator dressings. However, more recent results from a variety of methods have found the
dressings to be finite [6, 7, 8, 9, 10, 11, 12].
In a recent study , the gluon and ghost propagator dressings, in the absence of quarks,
were studied. The SDEs were solved self-consistently using a range of approximations for the
vertices. Emphasis was given to how well these quantities are determined - particularly in the
physical region around 0.1 and 1 GeV. It turns out that the vertices are indeed very important if
numerically precise results are desired. D
Landau gauge is used, since for this kind of problem it is the most appealing theoretically.
The non-perturbative corrections to the ghost-gluon vertex are simplest in Landau gauge and state-
ments regarding confinement are typically formulated in Landau gauge also. Furthermore, there
are several lattice results to which we may compare.
The dressing functions that we solve for are related directly to the propagators, which for the
Wf~p2) (g -p ppy
Dyy(p)= (yy - (1.1)
and similarly for the ghost,
D(p) = - ,)(1.2) 1
where Dyy(p) is the full gluon propagator in Landau gauge and D(p) is the ghost propagator. The
dressing functions Wf and Wh contain all of the non-perturbative physics of these two Green's
functions. The simple transverse dressing of the gluon in Eq. (1.1) is due to the Slavnov-Taylor
identity for the propagator and is an important feature of the gauge invariance of the theory.
First we present the solutions of the ghost equation that are obtainable using a fixed gluon
input. These solutions then provide a natural starting point for investigating the coupled system
where simultaneous solutions of both propagator dressings are found. During the course of this
work it became apparent that many simple vertices do not admit self-consistent solutions. We test a
range of vertices with a range of solutions. A preferred system motivated by theoretical arguments
is then selected. We conclude by comparing to the extant lattice data.
2. The Ghost Equation
Using a fixed gluon input we may solve the Ghost SDE alone and investigate its sensitivity to
a range of input vertices. This is useful because this type of SDE is very simple to solve, and it
teaches us what to expect when solving the more complicated, coupled equations self-consistently.
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David J. Wilson, Michael R. Pennington. Vertex Sensitivity in the Schwinger-Dyson Equations of QCD, article, January 1, 2012; Newport News, Virginia. (https://digital.library.unt.edu/ark:/67531/metadc832376/m1/2/: accessed March 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.