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

Chapter 4. The Quark Dyson-Schwinger Equation 39
related to the quark propagator by the non-anomalous Ward identities for the vector and
axial vector currents [Adl69, BD, IZ]
(p ′ − p) µ Γ µ (p ′ , p) = iS −1 (p ′ ) − iS −1 (p) ,
(p ′ − p) µ Γ µ5 (p ′ , p) = γ 5 iS −1 (p) + iS −1 (p ′ )γ 5 + 2mΓ 5 (p ′ , p) . (4.21)
In order to derive the renormalised version of the gap equation we make the ladder approximation
to the Bethe-Salpeter kernel
In lower order perturbation theory k(q) is given by
K(p + q, p ′ + q, q) ≈ k(q) . (4.22)
k(q) = −iC f g 2 γ µ ⊗ γ ν D µν (q) , (4.23)
which means that in the case of keeping only the time-time component in the instantaneous
approximation we are left with
k(q) = k(q) = −4πC f γ 0 ⊗ γ 0 V C (q) , (4.24)
where V C is the famous colour Coulomb potential. We point out that in the ladder approximation,
which is exclusively used in this work, k(q) does not contain any quark
propagators, and therefore there are no quark loops generated when we insert it in (4.20).
Applying the equations (4.20) and (4.21) yields
(p ′ − p) µ Γ µ5 (p ′ , p) = γ 5 iS −1 (p) + iS −1 (p ′ )γ 5 + 2mΓ 5 (p ′ , p)
∫
= Z (µ)5 γ µ γ 5 (p ′ − p) µ d 4 q
+ + q)[γ
(2π) 4(S(p′ 5 iS −1 (p + q)
+ iS −1 (p ′ + q)γ 5 + 2mΓ 5 (p ′ + q, p + q)]S(p + q))k(q)
∫
)
= γ 5
(Z (µ)5 γ µ p µ d 4 q
− Z 5 m +
(2π) 4S(p + q)k(q) (
∫
)
+ Z (µ)5 γ µ p ′µ d 4 q
− Z 5 m + + q)k(q) γ
(2π) 4S(p′ 5
2mΓ 5 (p ′ , p) . (4.25)
It has to be stressed that the vector character of the renormalised quark propagator therefore
satisfies the equation
∫
iS −1 (p) = Z (µ)5 γ 5 p µ − Z 5 m +
d 4 q
(2π) 4S(p + q)k(q) . (4.26)