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

74 6.5. Nucleon electromagnetic form factors: results
The new elements in these diagrams are the couplings of a photon to two dressed-quarks
as they either separate from (diagram 5) or combine to form (diagram 6) a diquark correlation.
As such they are components of the five point Schwinger function which describes the
coupling of a photon to the quark-quark scattering kernel. This Schwinger function could
be calculated, as is evident from the recent computation of analogous Schwinger functions
relevant to meson observables [CM02, CM03]. However, such a calculation provides valid
input only when a uniform truncation of the DSEs has been employed to calculate each
of the elements described hitherto. We must instead employ an algebraic parametrisation
[OPS00], which for Diagram 5 reads
Xµ JP
(k, Q) = e 4k µ − Q
[
]
µ
by Γ JP (k − Q/2) − Γ JP (k)
4k · Q − Q 2
4k µ + Q
[
]
µ
+ e ex Γ JP (k + Q/2) − Γ JP (k) , (6.86)
4k · Q + Q 2
with k the relative momentum between the quarks in the initial diquark, e by the electric
charge of the quark which becomes the bystander and e ex , the charge of the quark that is
reabsorbed into the final diquark. Diagram 6 has
4k
¯X µ JP
µ − Q µ
(k, Q) = −e by
[¯ΓJ P (k + Q/2) −
4k · Q − Q ¯Γ
]
JP
(k)
2
4k µ + Q µ
− e ex
[¯ΓJ P (k − Q/2) −
4k · Q + Q ¯Γ
]
JP
(k) , (6.87)
2
where ¯Γ JP (l) is the charge-conjugated amplitude, equation(A.11). These terms vanish if
the diquark correlation is represented by a momentum-independent Bethe-Salpeter-like
amplitude, i.e. the diquark is pointlike.
It is naturally possible to use more complicated ansätze. However, like equation(6.62),
equations.(6.86) and (6.87) are simple forms, free of kinematic singularities and sufficient
to ensure the nucleon-photon vertex satisfies the Ward-Takahashi identity when the composite
nucleon is obtained from the Faddeev equation.
6.5 Nucleon electromagnetic form factors: results
6.5.1 Remarks
In order to place the calculation of baryon observables on the same footing as the study
of mesons, the proficiency evident in [MR03, CM03] will need to be applied to every line
and vertex that appears in figure6.1. This is a feasible but tedious task. In the meantime,
herein we present a study whose merits include a capacity to explore the potential of the