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

60 6.3. Covariant Faddeev equation
where: the spinor satisfies
(iγ · P + M) u(P) = 0 = ū(P) (iγ · P + M) , (6.10)
with M the mass obtained by solving the Faddeev equation, and it is also a spinor in
isospin space with ϕ + = col(1, 0) for the proton and ϕ − = col(0, 1) for the neutron. The
momenta are given as K = p 1 + p 2 =: p {12} , p [12] = p 1 − p 2 , l := (−p {12} + 2p 3 )/3. ∆ 0+
is a pseudoparticle propagator for the scalar diquark formed from quarks 1 and 2, and
Γ 0+ is a Bethe-Salpeter-like amplitude describing their relative momentum correlation.
And S, a 4 × 4 Dirac matrix, describes the relative quark-diquark momentum correlation.
These objects are discussed in more detail in secion 6.3.2. The colour antisymmetry of
Ψ 3 is implicit in Γ JP , with the Levi-Civita tensor, ǫ c1 c 2 c 3
, expressed via the antisymmetric
Gell-Mann matrices. Explicitly we define
{H 1 = iλ 7 , H 2 = −iλ 5 , H 3 = iλ 2 } , (6.11)
and then we have ǫ c1 c 2 c 3
= (H c 3
) c1 c 2
. (See equations(6.43), (6.44).)
The axial-vector component in equation(6.7) is
N 1+ (p i , α i , τ i ) = [t i Γ 1+
µ (1 2 p [12]; K)] τ 1τ 2
α 1 α 2
∆ 1+
µν (K) [Ai ν (l; P)u(P)]τ 3
α 3
, (6.12)
where the symmetric isospin-triplet matrices are
t + = 1 √
2
(τ 0 + τ 3 ) , t 0 = τ 1 , t − = 1 √
2
(τ 0 − τ 3 ) , (6.13)
with (τ 0 ) ij = δ ij and τ 1,3 the usual Pauli matrices, and the other elements in equation(6.12)
are straightforward generalisations of those in equation(6.9).
The general form of the Faddeev amplitude for the spin- and isospin-3/2 ∆ is complicated.
However, isospin symmetry means one can focus on the ∆ ++ with it’s simple
flavour structure, because all the charge states are degenerate, and consider
D 1+
3 = [t + Γ 1+
µ (1 2 p [12]; K)] τ 1τ 2
α 1 α 2
∆ 1+
µν (K) [D νρ(l; P)u ρ (P) ϕ + ] τ 3
α 3
, (6.14)
where u ρ (P) is a Rarita-Schwinger spinor, equation(A.12).
The general forms of the matrices S(l; P), A i ν (l; P) and D νρ(l; P), which describe the
momentum space correlation between the quark and diquark in the nucleon and the ∆,
respectively, are described in reference[OHAR98]. The requirement that S(l; P) represent
a positive energy nucleon; namely, that it be an eigenfunction of Λ + (P), equation(A.9),
entails
S(l; P) = s 1 (l; P) I D +
(iγ · ˆl − ˆl · ˆP
)
I D s 2 (l; P) , (6.15)