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

Chapter 4. The Quark Dyson-Schwinger Equation 45
4.6 Numerical results
A description of the numerical methods can be found in appendix C. Parts of it apply also
to calculations in the sections 4.3 and 5. All presented results are in the chiral limit. For the
Coulomb string tension we employ the value as in section 4.4, namely σ c = 0.5476 GeV 2 .
For non-vanishing current quark mass no converging result could be obtained. Figure 4.3
displays the propagator function B(p E , |p|) in dependence of the infrared regulator. As
already experienced in our calculation with the time-time component of the gluon propagator
only we observe a diverging behaviour. The same is true for the propagator function
C(p E , |p|). The third propagator function A(p E , |p|) deviates only very slightly from its
value when only the time-time components of the gluon propagator is kept. This already
hints that the changes in the mass function are not sizeable. Indeed in the computed
result, illustrated in the figures 4.5 and 4.6, no considerable increase in the mass function
compared to the case without transverse gluons and retardation is obtained. Also
for all displayed functions we have a variation in the frequency coordinate p E only up to
the numerical accuracy. The p-axis is for all presented functions a symmetry axis. As a
renormalisation point we choose (ν E , |ν|) = (20000GeV, 45000GeV). As in section 4.4 the
results do not vary while increasing the renormalisation point a few orders of magnitude.
An increase of the factor Z in the equations (4.43)-(4.45) on the percentage level
increases the mass function slightly but has no impact on the frequency dependence.
Evidently with the current results for the quark Dyson-Schwinger equation a realistic
calculation and description of observables is not possible. Further progress in this direction
has to come from a more realistic version of the quark-gluon vertex.