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

Chapter 6. Nucleon Form Factors in a Covariant Diquark-Quark model 61
where (I D ) rs = δ rs , ˆl 2 = 1, ˆP 2 = −1. In the nucleon rest frame, s 1,2 describe, respectively,
the upper, lower component of the bound-state nucleon’s spinor. Placing the same
constraint on the axial-vector component, one has
A i ν(l; P) =
6∑
p i n(l; P) γ 5 A n ν(l; P) , i = +, 0, −, (6.16)
n=1
where (ˆl ⊥ ν = ˆl ν + ˆl · ˆP ˆP ν , γ ⊥ ν = γ ν + γ · ˆP ˆP ν )
A 1 ν = γ · ˆl ⊥ ˆPν , A 2 ν = −i ˆP ν , A 3 ν = γ · ˆl ⊥ ˆl⊥ ,
A 4 ν = iˆl ⊥ µ , A5 ν = γ⊥ ν − A3 ν , A6 ν = iγ⊥ ν γ · ˆl ⊥ − A 4 ν . (6.17)
Finally, requiring also that D νρ (l; P) be an eigenfunction of Λ + (P), one obtains
D νρ (l; P) = S ∆ (l; P) δ νρ + γ 5 A ∆ ν (l; P) l⊥ ρ , (6.18)
with S ∆ and A ∆ ν given by obvious analogues of equations(6.15) and (6.16), respectively.
One can now write the Faddeev equation satisfied by Ψ 3 as
[
] ∫
[
]
S(k; P) u(P) d 4 l
A i µ (k; P) u(P) = −4
(2π) M(k, l; P) S(l; P) u(P)
4 A j ν (l; P) u(P) , (6.19)
where one factor of “2” appears because Ψ 3 is coupled symmetrically to Ψ 1 and Ψ 2 , and
the necessary colour contraction has been evaluated: (H a ) bc (H a ) cb ′
in equation (6.19) is
with
M(k, l; P) =
[
M00 (M 01 ) j ν
(M 10 ) i µ (M 11 ) ij
µν
]
= −2 δ bb ′. The kernel
(6.20)
M 00 = Γ 0+ (k q − l qq /2; l qq ) S T (l qq − k q ) ¯Γ 0+ (l q − k qq /2; −k qq ) S(l q ) ∆ 0+ (l qq ) , (6.21)
where: 2 l q = l+P/3, k q = k+P/3, l qq = −l+2P/3, k qq = −k+2P/3 and the superscript
“T” denotes matrix transpose; and
(M 01 ) j ν = tj Γ 1+
µ (k q − l qq /2; l qq )
×S T (l qq − k q ) ¯Γ 0+ (l q − k qq /2; −k qq ) S(l q ) ∆ 1+
µν(l qq ) , (6.22)
(M 10 ) i µ = Γ0+ (k q − l qq /2; l qq )
×S T (l qq − k q )t i ¯Γ1 +
µ (l q − k qq /2; −k qq ) S(l q ) ∆ 0+ (l qq ) , (6.23)
(M 11 ) ij
µν = t j Γ 1+
ρ (k q − l qq /2; l qq )
×S T (l qq − k q )t i ¯Γ1 +
µ (l q − k qq /2; −k qq ) S(l q ) ∆ 1+
ρν (l qq) . (6.24)
2 This choice is explained by equation(6.55) and the discussion thereabout.