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

Chapter 6. Nucleon Form Factors in a Covariant Diquark-Quark model 65
gauge approach: r (ud)0 + ≈ 1.1 r π. The Bethe-Salpeter amplitude is canonically normalised
[IZ] via:
[ ] K ∂
2 =−m 2 J
2 K µ = Π(K, Q)
P
, (6.41)
∂Q µ Q=K
∫ d 4 q
Π(K, Q) = tr
(2π) ¯Γ(q; −K) S(q + Q/2) Γ(q; K) S T (−q + Q/2). (6.42)
4
A solution of the BSE equation requires a simultaneous solution of the quark-DSE
[Mar02]. However, since we have already chosen to simplify the calculations by parametrising
S(p), we also employ that expedient with Γ JP , using the following one-parameter forms:
Γ 0+ (k; K) =
1
H a Ciγ 5 iτ 2 F(k 2 /ω 2
N 0+ 0 +) , (6.43)
t i Γ 1+
µ (k; K) = 1
H a iγ µ Ct i F(k 2 /ω 2
N 1+ 1 +) , (6.44)
with the normalisation, N JP , fixed by equation (6.41). These Ansätze retain only that
single Dirac-amplitude which would represent a point particle with the given quantum
numbers in a local Lagrangian density: they are usually the dominant amplitudes in a
solution of the rainbow-ladder BSE for the lowest mass J P diquarks [Mar02].
Diquark propagators
Calculations beyond rainbow-ladder truncation eliminate asymptotic diquark states from
the spectrum. It is apparent in reference [BHK + 04] that the behaviour of the diquark
propagator ∆ JP can be modelled efficiently by simple functions that are free-particle-like
at spacelike momenta but pole-free on the timelike axis. Hence we employ
∆ 0+ (K) =
∆ 1+
µν(K) =
1
F(K 2 /ω 2
m 2 0 +) , (6.45)
0
(
+ δ µν + K )
µK ν 1
F(K 2 /ω
m 2 1
m 2 1 2 +) , (6.46)
+ 1 +
where the two parameters m J P are diquark pseudoparticle masses and ω J P are widths
characterising Γ JP . Herein we require additionally that
( ) ∣
−1
d 1
∣∣∣∣K
F(K 2 /ω 2
dK 2 m 2 J
) = 1 ⇒ ω 2 P J
= 1 P 2 m2 J
, (6.47)
P
J P 2 =0
which is a normalisation that accentuates the free-particle-like propagation characteristics
of the diquarks within the hadron.