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

Chapter 6. Nucleon Form Factors in a Covariant Diquark-Quark model 57
for the nucleon and ∆ that are vital for our analysis, and present the solutions. Section
6.4 describes the formulation of a Ward-Takahashi identity preserving current that is
appropriate to a nucleon represented by a solution of the Faddeev equation. With the necessary
elements thus specified, the results for the nucleons’ electromagnetic form factors
are presented and discussed in section 6.5.
6.2 Two-quark correlations
In the preceding section we pointed out that by using the concept of diquarks we effectively
assume the two-quark correlation matrix to be separable. As this approximation is mainly
borrowed from the Nambu-Jona-Lasinio (NJL) model, we will describe how diquarks arise
there and how the diquark-quark picture is realized in other models.
It is well-known that a separable two-quark correlation matrix arises from a separable
four-quark scattering kernel. In QCD already the simplest contribution in the perturbative
kernel, the gluon exchange, has a non-separable form. By introducing a δ-function
in configuration space for the gluon propagator a NJL model with pointlike four-quark
interaction is realised. This local current-current interaction can be expressed as an attractive
interaction in colour singlet meson and colour triplet diquark channels [AR]. The
quark-quark scattering kernel in the scalar channel s can be written as
(K s ) αβ,γδ = 4G s (χ s ) αβ (χ s ) γδ (6.2)
= 4G s (γ 5 Cτ 2 λ k ) αβ (C T γ 5 τ 2 λ k ) γδ . (6.3)
Here we only consider an isospin doublet of two quarks that are in a flavour antisymmetric
(τ 2 ) and a colour antitriplet state (λ k , k = 2, 5, 7) (the τ i are Pauli matrices and λ k are
Gell-Mann matrices). C denotes the charge conjugation matrix and G s regulates the
strength in the interaction channel. The scalar diquark propagator can be computed to
be
(D s ) −1 (k 2 ) = 1
4G s
− 2tr D
∫
d 4 q
γ 5 )S(q + k (2π) 4(CT 2 )(γ5 C)S T ( k − q) . (6.4)
2
After regularising the divergent integral the inverse propagator has zeros for certain values
of k 2 . This shows that in the NJL model bound scalar diquarks exist and the propagator
can be seen as a scalar propagator, which has the effect of a dressing quark loop
incorporated.
Looking at the meson spectrum the axialvector diquarks are analogously supposed to
play an important role in the diquark channel. Indeed in [WBAR93] the axialvector diquark
propagator has poles, but their appearance varies with the ratio of the coupling