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

Chapter 6. Nucleon Form Factors in a Covariant Diquark-Quark model 63
the asymptotic behaviour familiar from perturbative analyses, cf. section 2.2, and that
behaviour is unambiguously evident for p 2 10 GeV 2 [AS01].
While numerical solutions of the quark DSE are now readily obtained, the utility of
an algebraic form for S(p) when calculations require the evaluation of numerous multidimensional
integrals is obvious. An suitable parametrisation of S(p), which exhibits the
features described above, has been used extensively in hadron studies [AS01, MR03]. It is
expressed via
¯σ S (x) = 2 ¯m F(2(x + ¯m 2 )) + F(b 1 x) F(b 3 x) [b 0 + b 2 F(ǫx)] , (6.31)
1 [
¯σ V (x) = 1 − F(2(x + ¯m 2 )) ] , (6.32)
x + ¯m 2
with x = p 2 /λ 2 , ¯m = m/λ,
F(x) = 1 − e−x
, (6.33)
x
¯σ S (x) = λ σ S (p 2 ) and ¯σ V (x) = λ 2 σ V (p 2 ). The mass-scale, λ = 0.566 GeV, and parameter
values 3 ¯m b 0 b 1 b 2 b 3
0.00897 0.131 2.90 0.603 0.185 , (6.34)
were fixed in a least-squares fit to light-meson observables [BRT96]. The dimensionless
u = d current-quark mass in equation (6.34) corresponds to
m u,d = 5.1 MeV . (6.35)
The parametrisation yields a Euclidean constituent-quark mass
M E u,d
= 0.33 GeV, (6.36)
defined as the solution of p 2 = M 2 (p 2 ), whose magnitude is typical of that employed in
constituent-quark models. In ref.[BRT96] it is shown that m s = 25 m u,d and M E s =
0.49 GeV. It is generally true that M E s − M E u,d ˆm s − ˆm u,d , where ˆm denotes the
renormalisation point independent current-quark mass. The constituent-quark mass is
an expression of dynamical chiral symmetry breaking, as is the vacuum quark condensate 4
(Λ QCD = 0.2 GeV)
−〈¯qq〉 1GeV2
0 = λ 3 3
4π 2 b 0
b 1 b 3
ln 1 GeV2
Λ 2 QCD
= (0.221 GeV) 3 . (6.37)
3 ǫ = 10 −4 in equation (6.31) acts only to decouple the large- and intermediate-p 2 domains.
4 The condensate is calculated directly from its gauge invariant definition [MR03] after making allowance
for the fact that equations(6.31) and (6.32) yield a chiral-limit quark mass function with anomalous
dimension γ m = 1. This omission of the additional ln(p 2 /Λ 2 QCD )-suppression that is characteristic of
QCD is merely a practical simplification.