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

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we obtain three coupled integral equations for Γ p , Γ A and Γ T . For a confining potential
these quantities also diverge. We therefore introduce
h(|p|) = Γ p(|p|)
ω(|p|)
(5.8)
and
where
g(|p|) = Γ p(|p|) + 2B(|p|)Γ A (|p|) + 2|p|C(|p|)Γ T (|p|)
ω 2 (|p|) − m 2 π /4 , (5.9)
ω(|p|) = √ B 2 (|p|) + |p| 2 C 2 (|p|) . (5.10)
The functions h(|p|) and g(|p|) are finite and satisfy the coupled integral equations
h(|p|)ω(|p|) = 1 ∫ [
]
d 3 qV
3π 2 C (k) h(|q|) +
m2 π
4ω(|q|) g(|q|) [ ]
g(|p|) ω(|p|) −
m2 π
= h(|p|)+
ω(|p|)
∫
1
d 3 V C (|k|)
q (M(|p|)M(|q|) + (ˆpˆq)|p||q|)g(|q|) , (5.11)
3π 2 ˜ω(|p|)˜ω(|q|)
where we introduced
˜ω(|p|) = √ M 2 (|p|) + p 2 . (5.12)
For vector mesons (and correspondingly axial-vector diquarks) the Bethe-Salpeter vertex
function has four linearly independent amplitudes. The construction of the four coupled
integral equations corresponding to the BSE is analogous to the pseudoscalar case.
The homogeneous BSE in equation (5.6) can be solved by introducing an eigenvalue
λ(P 2 = M 2 ) with M the bound-state mass. One then finds M such that λ = 1 for mesons
and λ = 2 for diquarks. The numerical solution of the BSEs can be performed as in
[OVSA02].
We studie the Bethe-Salpeter amplitudes as µ IR → 0: the results for g and h of
(5.11) are presented in figs. 5.2 and 5.3 for the pion and scalar diquark, respectively. For
convenience, the normalisation of the amplitudes has been chosen such that h(0) = 1. We
note, however, that IR-cancellations appearing in the pion case lead to a stable h as well
as ratio of g/h, which is not the case (as one would naively expect) in the diquark case:
there g/h ∼ µ IR → 0 and h ∼ 1/ √ µ IR .
The curve for λ(P 2 ) gets less inclined with smaller values of the infrared regulator µ IR ,
and its intersection point with λ = 1 stabilises in the limit µ IR → 0. As a consequence,
while the meson mass is stable, the mass eigenvalue for the corresponding diquark state
(corresponding to λ(M) = 2) increases like 1/µ IR , ultimately completely removing these