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

Chapter 2. Chiral Symmetry Breaking in QCD 13
to get further insight we have to phrase the symmetry breaking term in a group theoretical
language. As an example we show how to do this with flavour SU(3) (the generalisation
is trivial). At first one defines the N 2 f
= 9 scalar quark densities
u r = qλq , r = 0, 1, ..., , 8 , (2.40)
where λ r are the famous Gell-Mann matrices and
⎛ ⎞
√ 1 0 0 2
λ 0 ⎜ ⎟
= ⎝0 1 0⎠ . (2.41)
3
0 0 1
With this definition one can recast the mass terms in the Lagrangian to yield
m u uu + m d dd + m s ss , (2.42)
where the symmetry breaking parameters are linear combinations of the quark masses,
c 0 = 1 √
6
(m u + m d + m s ) (2.43)
c 3 = 1 2 (m u − m d ) (2.44)
c 8 = 1
2 √ 3 (m u + m d − 2m s ) . (2.45)
For the twofold commutator in (2.37) only the symmetry breaking term in the Hamiltonian
contributes. With the well-known algebra of the Gell-Mann matrices this commutator
can be calculated easily. Taking into consideration that for three flavours the twofold
commutator in (2.37) is not diagonal in flavour space, one arrives at
f 2 π m2 π = 1 2 (m u + m d )〈Ω|uu + dd|Ω〉
f 2 K m2 K = 1 2 (m u + m s )〈Ω|uu + ss|Ω〉 (2.46)
f 2 η m2 η = 1 6 (m u + m d )〈Ω|uu + dd|Ω〉 + 4m s
3 〈Ω|ss|Ω〉 .
For simplicity it was assumed that f π is diagonal. From the definition of the decay constant
(2.29) and the assumption
〈Ω|uu|Ω〉 = 〈Ω|dd|Ω〉 = 〈Ω|ss|Ω〉 (2.47)
we conclude that all decay constants are equal. Thereby we derive with (2.46) not only
the famous, phenomenologically discovered mass relation of Gell-Mann and Okubo,
4m 2 K = 3m2 η + m2 π , (2.48)