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

Chapter 4. The Quark Dyson-Schwinger Equation 37
which leads to the identity
∫
δ(x − y) =
∫
=
d 4 z δη(y) δq(z)
δq(z) δη(x)
d 4 δ 2 Γ δ 2 W
z
δq(z)δq(y) δη(x)δη(z) . (4.12)
Together with the matrix relation
δχ −1
δA
we can rephrase (4.9) in the following way:
= −χ−1
δχ
δA χ−1 (4.13)
δ 3 W
δη(z)δη(y)δjµ(z a ′ ) = δ [ ]
δ 2 −1
Γ
(4.14)
δjµ
a δq(z)δq(y)
∫
= d 4 δA d ν
u (u [ ]
1) δ δ 2 −1
Γ
1 (4.15)
δjµ(z a ′ ) δA d ν(u 1 ) δq(z)δq(y)
∫
= − d 4 u 1 d 4 u 2 d 4 δ 2 W δ 2 W δ 3 Γ δ 2 W
u 3
δjµ a (z′ )δjν d(u 1) δη(y)δη(u 2 ) δA d ν (u (4.16)
1)δq(u 2 )δq(u 3 ) δη(u 3 )δη(z)
∫
= d 4 u 1 d 4 u 2 d 4 u 3 Dµν ad (z′ − u 1 )S(u 2 − y)Γ d ν (u 1, u 2 , u 3 )S(z − u 3 ) . (4.17)
Performing a Fourier transform and using the colour structure of the quark-gluon vertex,
which is [AS01]
Γ a µ(k, q, p) = −igt a (2π) 4 δ 4 (k + q − p)Γ µ (q, p) , (4.18)
we arrive at the quark Dyson-Schwinger equation in a linear covariant gauge:
∫
S −1 (p) = Z 2 S0 −1 (p) + g2
16π 4Z 1FC F d 4 q γ µ S(q)Γ ν (q, k)D µν (k) . (4.19)
The equation is represented conveniently in a graphical way (figure 4.1). In the next
section we derive a version of the renormalised gap equation in Coulomb gauge, which is
of an analogous structure.
4.2 The gap equation in Coulomb gauge
For a realistic potential these equations are UV divergent and have to be renormalised.
Looking at the theory of superconductivity, the gap equation is part of a more general
system of equations for the electron propagator and the electron-photon vertex, in which
the Ward identity is satisfied [Sch]. Therefore it seems natural concerning the quark DSE
to proceed in an analogous fashion, using the Ward identities to derive the renormalised