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

Chapter 4. The Quark Dyson-Schwinger Equation 41
holds (α(|p|) is the running coupling). From lattice calculations we can infer the large
distance behaviour [NS06], cf. sections 3.3 and 3.6.4. An analytic ansatz, which obeys
both limiting behaviours and which we will employ, is the Richardson potential [Ric79]
V C (q) =
12π
(11N c − 2N f ) · ln(1 + q 2 /Λ 2 ) . (4.33)
N f and N c are the number of colours and the number of flavours respectively. Here we use
the values N f = 3 and N c = 3. Λ is a parameter which we can calculate from the string
tension of lattice calculations by expanding V C for large momenta. Defining the quark
mass function as motivated in section 2.2 as the ratio
M(|p|) := B(|p|)
C(|p|)
(4.34)
and performing the q 0 -integral analytically we are able to express the equations (4.30) and
(4.31) as a single integral equation
M(|p|) =
Z 5 m + 2C 1
3π
Z (µ) + 2C 1
3π
∫ ∞
0
dq q 2 M(|q|)
√M 2 (|q|)+q 2 ∫ 1
−1 d(cosθ) 1
|p−q| 2 ·
1
ln(1+|p−q| 2 /Λ 2 )
∫ ∞
dq q
0 2 √ |q|
∫ 1 cos θ 1
d(cosθ) ·
|p| M 2 (|q|)+q 2 −1 |p−q| 2 ln(1+|p−q| 2 /Λ 2 )
, (4.35)
where θ is the angle between p and q and the abbreviation C 1 =
12π
(11N c−2N f )
was used.
Because the integral equation is singular at p = q we introduce a regulator µ IR by using
the substitution
|p − q| 2 → |p − q| 2 + µ 2 IR (4.36)
for all terms in question. In order to calculate the solution for (4.35) we have to solve
the equation for non-vanishing µ IR and take the limit µ IR → 0. An analytic proof for
the convergence of such a regularisation was already given in [Alk88]. Furthermore we
observe, that both momentum integrals in (4.35) are divergent in the ultraviolet. The
equation obviously has to be renormalised and in doing so we choose a MOM scheme and
impose the conditions
where |ν| is the renormalisation point.
B(|ν|) = m (4.37)
C(|ν|) = 1 , (4.38)
4.4 Numerical results
In figure 4.2 we display the results for the mass function in the chiral limit employing only
the time-time component of the gluon propagator. As a renormalisation point we choose