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

Chapter 4. The Quark Dyson-Schwinger Equation 43
4.5 Adding transverse gluons and retardation
In this section we improve the approximations, which led to (4.35), in two respects. We add
transverse components of the gluon propagator and overcome the instantaneous approximation
by taking into account retardation for these components. Explicitly we employ
D ij (k 0 , |k|) = 1 g 2 ·
) (δ ij − ki k j −iZ(k 0 ,k)
·
k 2 k0 2 − ωg(|k|) 2 + iǫ , (4.39)
where for simplicity in this exploratory calculation we choose Z(k 0 ,k) = 1, and the function
ω g is adjusted to calculations in the Hamiltonian approach [FR04b],
ω g (|k|) = Λ2
+ |k| , (4.40)
|k|
where Λ is the same parameter as in the Richardson potential. This ansatz is furthermore
in accordance with lattice calculations for the transverse equal-time gluon propagator
[CZ02a]. In the present approximation the simple ansatz in equation (4.28) for the quark
propagator has to be extended,
S −1 (p) = −i · (γ 0 p 0 · A(p 0 , |p|) − γ · p C(p 0 , |p|) − B(p 0 , |p|) + iǫ) . (4.41)
Solving the gap equation for the propagator functions and performing a Wick rotation
∫ ∫
p 0 → ip E , dq 0 → dq E i , q 0 → iq E , (4.42)
yields (we define k := p − q and k E := p E − q E ):
p E A(p E , |p|) = Z (µ) p E + C F
(2π) 3 ∫
[
4πV C (|k|) +
∫∞
dq E
0
2Z(k E,k)
−k 2 E − ω2 g(k)
∫1
d|q| · |q| 2
−1
d(cosθ)
]
qE A(q E , |q|)
denom(q E , |q|)
(4.43)
B(p E , |p|) = Z 5 m −
C ∫
F
(2π) 3
[
4πV C (|k|) −
∫∞
dq E
0
2Z(k E,k)
−k 2 E − ω2 g(k)
∫1
d|q| · |q| 2
]
−1
d(cosθ)
B(q E , |q|)
denom(q E , |q|)
(4.44)