Dryson equations, Ward identities, and the infrared behavior of Yang-Mills theories. [Schwinger-Dyson equations, Slavnov-Taylor identities] Page: 4 of 43
This article is part of the collection entitled: Office of Scientific & Technical Information Technical Reports and was provided to UNT Digital Library by the UNT Libraries Government Documents Department.
Extracted Text
The following text was automatically extracted from the image on this page using optical character recognition software:
-3-
behaves like (1/p2)2 as p2 -* 0,
(3) To complete the Dyson equations we must specify the
vertices T and r4 in terms of D^. The longitudinal parts of
r and r4 are determined by the Slavnov-Taylor identities which
we write in the axial gauge. We note that any vertices satis-
fying these identities, when inserted into the Schwinger-Dyson
equations yield the structure for D^^tq) demanded by gauge
invariance. In particular we show that r . (q) -*• 0 as q2 •* 0,
2 2
even if there is a (1/p ) singularity in the gluon propagators
appearing in the Schwinger-Dyson expression for the vacuum
polarization tensor. This means that in Yang Mills theory there
is the possibility of a singular long range force correspond-
2 2
ing to a running coupling constant g(q ) behaving like 1/q
2 '
for small q .
(4) We present the general solution of the Slavnov-
Taylor identities for r and r4. We find that the requirement
that r and T4 contain no kinematic singularities yields trans-
verse parts of T and T4 which vanish when any one of the
external momenta vanishes. The kinematic singularity free
longitudinal parts are determined in terms of the gluon propa-
gator D3^ (q) .
liV
(5) From section (3) we know that one must use in the
Schwinger-Dyson equations vertices r and 1’4 which are exact
solutions of the Slavnov-Taylor identities in order to have
the possibility of finding a singular long rarge force. Thus
as a first attempt we neglect the undetermined transverse
-4-
parts of T and r4- We ther replace V and r4 by their kine-
matic singularity free longitudinal parts. The Schwinger-Dyson t
equat_ons ther become a closed set of equations for the gluon
propagator which may be appropriate for computing D3^(q) as
q2 -*â– 0. We discuss some properties of these equations which
we are now investigating.
1. TIE SCHWINGER-DYSON EQUATIONS FOR THE GLUON PROPAGATOR
The free gluor propagator is
6 D(0) (q)
ab uv
where the form of D , (q‘, depends upon the gauge.
In cpvariant gauges
g -^+bq-H^L
’uv 2 2
q q
where o is a constant whrch fixes the gauge.
In an axial gauge specified by the gauge condition
"X =
. :nuVnyV + q-»qv n2
(n*q)
(n»q)
Clearly (c) = 0.
Upcoming Pages
Here’s what’s next.
Search Inside
This article can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Article.
Baker, M. Dryson equations, Ward identities, and the infrared behavior of Yang-Mills theories. [Schwinger-Dyson equations, Slavnov-Taylor identities], article, January 1, 1979; United States. (https://digital.library.unt.edu/ark:/67531/metadc1098728/m1/4/: accessed March 29, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.