The QCD Quark Propagator in Coulomb Gauge and - Institut für Physik

Chapter 3. Remarks on QCD in Coulomb Gauge 17
like 1/ √ a, but they cancel pairwise in each order in perturbation theory by virtue of the
r-symmetry. Therefore in the Coulomb gauge limit the correlation functions are finite,
which remains true for the renormalised correlation functions. However, one-loop graphs
are identified that vanish like √ a and that do not exist in the formal Coulomb gauge,
i.e. a = 0. These graphs cannot be neglected since they give a finite contribution when
inserted into the graphs that diverge like 1/ √ a. It is still possible that these two-loop
graphs, which are not existent in the formal Coulomb gauge, are mere gauge artefacts and
decouple from expectation values of all gauge-invariant quantities such as a Wilson loop.
However, up to now it was not possible to show this.
3.3 Approaches in Coulomb gauge
Working perturbative calculations are desirable in every gauge. However, in Coulomb
gauge no further insight has been achieved with the standard dimensional regularisation
technique. The reason for this is, that the gauge boson propagator is of the form
[
G ab
µν (q) = −iδab g
q 2 µν + n2
q 2q µq ν − q · n ]
q (q µn 2 ν + n µ q ν ) , (3.2)
where n µ = (1, 0, 0, 0). It generates loop integrals with the expression
∫
d D q
q 2 (q − p) 2 . (3.3)
The energy integral is not defined, even if one uses dimensional regularisation.
A procedure called split dimensional regularisation for this ill-defined integrals was
brought up in [LW96]. Here energy and 3-momentum integrals are separately dimensionally
regularised. Two regulating parameters are introduced by splitting the dimensionality
of space-time into two different sectors, namely, D = 4 − 2ǫ = ω + ρ and the divergences
contained in the energy integrals are expressed as poles in ρ besides the usual ones in
terms of ω. In this approach Coulomb gauge integrals were studies up to one and two-loop
level and results for the divergent part of several of them were obtained. This concept of
splitting up the dimensions is also used together with the so-called negative dimensional
integration method [SS01]. With these methods combined results at the one and two loop
level for arbitrary exponents of propagators and dimension have been achieved.
In recent years the Yang-Mills sector of QCD was examined in the Schrödinger picture
[FR04a, FR04b, RF05, ERS06]. Using canonical quantisation a combination of Coulomb
and Weyl gauge is fixed with the Faddeev-Popov method. In this approach the functional