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

Chapter 3. Remarks on QCD in Coulomb Gauge 31
It is possible to show that a confining Coulomb potential is necessary for the Wilson
potential to be confining. We proof this statement with the help of group theory, following
the derivation in [Zwa03]. To this end we examine the energy of the static qq state in
Coulomb gauge. Explicitly it is
E qq = 〈Φ qq |H|Φ qq 〉 − 〈0|H|0〉 , (3.79)
where H is the Coulomb gauge Hamiltonian (3.64). The quark and antiquark of the qq
pair live in the fundamental representation N and N of the gauge group SU(N). Our trial
states in terms of the wave functionals read
Φ qq [A] = Ψ s qqΨ 0 [A] . (3.80)
The state Ψ 0 [A] is the wave functional of the vacuum state in the absence of external quarks
and the state Ψ s qq belongs to the singlet part of the decomposition N × N = 1 + (N 2 − 1).
For the following considerations we use the convenient notation |Φ qq 〉 = |s〉|0〉 A for the
trial state and 〈...〉 A = ∫ DA... for the expectation value. The Coulomb term H Coul in
the Hamiltonian contains with the colour-charge density J 0 the external quark fields. The
infinitely massive quark and antiquark are treated as pointlike particles sitting at x and
y respectively. Their colour charge density is given as
J0 a (z) = λa q δ(z − x) + λa qδ(z − y) , (3.81)
where λ a q and λq a can be expressed through the Hermitian generators λ a N of the gauge group
SU(N) in the fundamental representation. Since the total colour charge density is given
as the sum ρ = ρ gl + J 0 , the Coulomb term of the Hamiltonian can be written as
H Coul = H gl + H gl qu + H qu qu , (3.82)
where H gl is the Hamiltonian without external quarks, H gl qu is linear in J 0 and the last
term is
H qu qu = g0
2 1
2
∑
∫
a,b
dz 1 dz 2 J a 0(z 1 )V ab
Coul(z 1 ,z 2 )J b 0(z 2 ) . (3.83)
The first term is responsible for the vacuum energy without external quarks,
〈Φ qq |H gl |0〉 A = E 0 . (3.84)
Because of 〈s|λ a q,q |s〉 = 0 (|s〉 is a colour-singlet state) the second term vanishes. The third
term gives the most intriguing contribution, namely
〈Φ qq |H qu qu |Φ qq 〉 = ∑ ∑
〈s|λ a i λb j |s〉〈0|Vab Coul (x,y)|0〉 A . (3.85)
i,j∈{q,q}
a,b