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

Chapter 6. Nucleon Form Factors in a Covariant Diquark-Quark model 69
6.4.1 Coupling to the quark
In this section we deal with diagram 1. The one-body term is expressed as
(
)
Jµ qu = S(p q )ˆΓ qu
µ (p q; k q )S(k q ) ∆ 0+ (k s ) + ∆ 1+ (k s ) (2π) 4 δ 4 (p − k − ˆηQ) , (6.54)
qu
where ˆΓ µ (p q; k q ) = Q q Γ µ (p q ; k q ), with Q q = diag[2/3, −1/3] being the quark electric
charge matrix, and Γ µ (p q ; k q ) is given in equation(6.57).
Here and in the diagrams 2 and 4 the denotation
k q = ηP + k , p q = ηP ′ + p ,
k d = ˆηP − k , p d = ˆηP ′ − p ,
(6.55)
with η + ˆη = 1 is used. The results reported were obtained with η = 1/3, which provides
a single quark with one-third of the baryon’s total momentum, but, as our approach is
manifestly Poincaré covariant, the precise value is immaterial. Nevertheless, numerical
results converge more quickly with this natural choice.
This represents the photon coupling directly to the bystander quark and is obtained
explicitly from the equations (6.52) and (6.54). It is a necessary condition for current
conservation that the quark-photon vertex satisfy the Ward-Takahashi identity:
Q µ iΓ µ (l 1 , l 2 ) = S −1 (l 1 ) − S −1 (l 2 ) , (6.56)
where Q = l 1 − l 2 is the photon momentum flowing into the vertex. Since the quark
is dressed the vertex is not bare; i.e., Γ µ (l 1 , l 2 ) ≠ γ µ . It can be obtained by solving an
inhomogeneous Bethe-Salpeter equation, which was the procedure adopted in the DSE
calculation that successfully predicted the electromagnetic pion form factor [MT00]. However,
since we have parametrised S(p), we use the Ball-Chiu construction of the vertex
[BC80]
iΓ µ (l 1 , l 2 ) = iΣ A (l 2 1, l 2 2) γ µ + 2k µ
[
iγ · kµ ∆ A (l 2 1, l 2 2) + ∆ B (l 2 1, l 2 2) ] ; (6.57)
with k = (l 1 + l 2 )/2, Q = (l 1 − l 2 ) and
Σ F (l 2 1, l 2 2) = 1 2 [F(l2 1) + F(l 2 2)] , ∆ F (l 2 1, l 2 2) = F(l2 1 ) − F(l2 2 )
l 2 1 − l2 2
, (6.58)
where F = A, B, i.e. the scalar functions in equation(6.29). It is critical that Γ µ in
equation (6.57) satisfies equation (6.56) and very useful that it is completely determined
by the dressed-quark propagator. This ansatz has been used fruitfully in many hadronic
applications [AS01]. Its primary defect is the omission of pion cloud contributions. But
since one of our goals is to draw attention to consequences of that omission, this fault is
herein a virtue.