Computational Physics

15.7 Gauge field theories 529

the hadron state is very complicated! If we take the lattice size in the time direction
large enough, the system will find the hadron state ‘by itself’ because that is the
ground state, so that this problem does not occur.
The QCD action has the following form (i=1, 2, 3 denotes the colour degree of
freedom of the quarks,fthe flavour):

SQCD=

∫

d^4 x







1

4

Fμνa Faμν+

∑

f

∑

ij

ψ ̄fiγμ

(

δij∂μ+ig

Aaμ

2

λaij

)

ψjf

+

∑

f

∑

i

mfψ ̄fiψfi







. (15.174)

The matricesλaare the eight generators of the group SU(3) (they are the Gell–Mann
matrices, the analogue for SU(3) of the Pauli matrices for SU(2)), satisfying

Tr(λaλb)=δa,b. (15.175)

Themfare the quark masses, andFμνa is more complicated than its QED counterpart:

Fμνa =∂μAνa−∂νAaμ−gfabcAbμAcν; (15.176)

the constantsfabcare the structure constants ofSU( 3 ), defined by

[λa,λb]=2i

∑

c

fabcλc. (15.177)

The parametergis the coupling constant of the theory; it plays the role of the charge
in QED. A new feature of this action is that thefabc-term in(15.176)introduces
interactions between the gluons, in striking contrast with QED where the photons
do not interact. This opens the possibility of having massive gluon bound states,
the so-called ‘glueballs’.
When we regularised QED on the lattice, we replaced the gauge fieldAμby
variablesUμ(n)=eieAμ(n)living on a link from sitenalong the direction given by
μ. For QCD we follow a similar procedure: we put SU(3) matricesUμ(n)on the
links. They are defined as

Uμ(n)=exp

(

ig

∑

a

Aaμλa/ 2

)

. (15.178)

The lattice action is now constructed in terms of these objects. The gauge part of
the action becomes

SGauge=

1

4

Fμνa Faμν→−

1

g^2

Tr[Uμ(n)Uν(n+μ)Uμ†(n+ν)Uν†(n)

+Hermitian conjugate]. (15.179)