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

62 6.3. Covariant Faddeev equation
It is illuminating to note that u(P) in equation(6.19) is a normalised average of ϕ ± so
that, e.g., the proton equation is obtained by projection on the left with ϕ † +. To illustrate
this we note that equation(6.22) generates an isospin coupling between u(P) ϕ+ on the
left-hand-side (l.h.s.) of equation(6.19) and, on the r.h.s.,
√
2 A
+
ν u(P) ϕ − − A 0 ν u(P) ϕ+ . (6.25)
This is just the Clebsch-Gordon coupling of isospin-1⊕ isospin- 1 to total 2 isospin-1 and 2
means that the scalar diquark amplitude in the proton, (ud) 0 + u, is coupled to itself and
the linear combination:
√
2(uu)1 + d − (ud) 1 + u . (6.26)
Similar statements are obviously true of the spin couplings.
with
The ∆’s Faddeev equation is
∫
D λρ (k; P) u ρ (P) = 4
d 4 l
(2π) 4 M∆ λµ (k, l; P) D µσ(l; P) u σ (P) , (6.27)
M ∆ λµ = t + Γ 1+
σ (k q − l qq /2; l qq ) S T (l qq − k q )t +¯Γ1 +
λ (l q − k qq /2; −k qq ) S(l q ) ∆ 1+
σµ(l qq ). (6.28)
6.3.2 Propagators and diquark amplitudes
To complete the Faddeev equations, equations(6.19) and (6.27), one must specify the
dressed-quark propagator, the diquark Bethe-Salpeter amplitudes and the diquark propagators
that appear in the kernels. In contrast to the preceding chapters we use in this one
Landau gauge.
Dressed-quark propagator
The dressed-quark propagator can be obtained from QCD’s gap equation and the general
form of the solution is
S(p) = −iγ · p σ V (p 2 ) + σ S (p 2 ) = 1/[iγ · p A(p 2 ) + B(p 2 )] . (6.29)
The enhancement of the mass function
M(p 2 ) := B(p2 )
A(p 2 )
(6.30)
is central to the appearance of a constituent-quark mass-scale and an existential prerequisite
for Goldstone modes. The mass function evolves with increasing p 2 to reproduce