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

Chapter 5. Meson Observables, Diquark Confinement and Radii 49
M(p 2 )
0,15
0,1
µ IR
= 10 -1
µ IR
= 10 -2
µ IR
= 10 -3
µ IR
= 10 -4
0,05
0
10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3
p 2
Figure 5.1: The quark mass function M(p 2 ) for four values of the infrared regulator µ IR .
All quantities are given in appropriate units of √ σ c .
diquarks. The appropriate 4-point Green’s function fulfils 1
∫
G αβγδ (k, p ′ , p) = k αβγδ (p ′ d 4 q
− p) + i
(2π) 4G τξγδ(k, p ′ , q)S ξσ
(q + k )
2
(
S ρτ q − k )
k αβρσ (p − q) . (5.4)
2
For G αβγδ of the pion we make the ansatz
(
G αβγδ (k, p ′ , p) = Γ αβ p + k 2 , p − k ) (
1
Γ
2 k 2 − m 2 γδ p ′ − k
π 2 , p′ + k )
+ . . .
2
, (5.5)
where terms, which are finite on the mass shell, were omitted. Taking only the time-time
component of the gluon propagator in the Bethe-Salpeter kernel (4.24) into account, we
attain the following BSE at k 2 = m 2 π for the pion vertex function:
(
Γ p + k 2 , p − k ) ∫
(
d 4 q
= C f (p − q)γ
2 (2π) 4g2 0 S q + k )
Γ
2
(
S q − k 2
(
q + k 2 , q − k 2
)
·
)
γ 0 D 00 (p − q) . (5.6)
Exemplarily we examine the pseudoscalar meson state in greater detail. To simplify this
expression, we consider a pion at rest, i.e. choose k = (m π , 0). In the instantaneous
approximation the pion vertex depends only on the three-momentum, Γ(p + k 2 , p − k 2 ) =
Γ(p, m π ) = Γ(p, m π ). Expanding the pion vertex function,
Γ(p, m π ) = Γ p (|p|)γ 5 + m π Γ A (|p|)γ 0 γ 5 + m π Γ T (|p|)(ˆpγ)γ 0 γ 5 (5.7)
1 This chapter applies the Minkowski space conventions A.1 .