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

Chapter 2. Chiral Symmetry Breaking in QCD 11
As mentioned above the quantitative considerations of this subsection are all valid in
Landau gauge. Nevertheless in Coulomb gauge we define a mass function in the same way
as above and are able to show, that it exhibits a behaviour as the one in Landau gauge
for low momenta.
2.3 The pion as a Goldstone boson and PCAC
From the fact that the vacuum expectation value 〈Ω|ΨΨ|Ω〉 does not vanish we can conclude
by considering the proof of the Goldstone theorem that the axial currents A r µ (x)
couple the Goldstone bosons to the vacuum. If we denote the one particle states of the
Goldstone bosons with momentum p as |π s (p)〉, this is expressed as [Ynd, CL, Pok]
〈Ω|A r µ(x)|π s (p)〉 = if rs p µ e ipx , r, s = 1, 2, 3 , (2.29)
where the f rs are nonvanishing constants. If we assume that the SU(N f ) isospin symmetry
is unbroken, they may be written as
f rs = δ rs f π . (2.30)
For N f = 2 f π is the pion decay constant. It can be measured in weak π decays, since the
matrix element (2.29) enters there. For instance for the decay π → µν we have
G 2 m 2
Γ = fπ
2 µ (m2 π − m2 µ )2
cos 2 θ
4πm 2 c , (2.31)
π
where G is the the weak decay constant and θ c the Cabibbo angle. From experiment
f π ≈ 92 MeV is known. For three flavours the kaon decay constant is approximately
f K ≈ 1.2f π .
Applying the four divergence to equation (2.29) and using the Klein-Gordon equation
one can derive
〈Ω|∂ µ A r µ (x)|πs (p)〉 = δ rs f π m 2 π eipx . (2.32)
If chiral symmetry of the QCD Lagrangian was exact, we could infer m π = 0 or f π = 0 from
the conservation of the axial current. This is true in the Nambu-Goldstone and Wigner-
Weyl realisation of chiral symmetry. Because both quantities do not vanish, it follows
immediately that chiral symmetry is explicitly broken. Since on the other hand the pions
are sizeably lighter than the rest of the mesons, the current quark masses of up and down
quarks, which are responsible for this explicit breaking, are quite small compared to the
typical energies of strong interaction. For the relevant case of explicit symmetry breaking