Computational Physics

530 Computational methods for lattice field theories

The quark part of the action, which includes the coupling with the gluons, reads in
the case of Wilson fermions (see above):

SFermions=

∑

n

(m+ 4 r)ψ( ̄ n)ψ(n)−

∑

n,μ

[ψ( ̄ n)(r−γμ)Uμ(n)ψ(n+μ)

+ψ( ̄ n+μ)(r+γμ)U†μ(n)ψ(n)]. (15.180)

An extensive discussion of this regularisation, including a demonstration that
its continuum limit reduces to the continuum action( 15. 174 ),can be found, for
example, in Rothe’s book[43]. The lattice QCD action

SLQCD=SGauge+SFermions (15.181)

can now be simulated straightforwardly on the computer, although it is certainly
complicated. We shall not describe the procedure in detail. In the previous sections
of this chapter we have described all the necessary elements, except for updating
the gauge field, which is now a bit different because we are dealing with matrices
as stochastic variables as opposed to numbers. Below we shall return to this point.
Simulating QCD on a four-dimensional lattice requires a lot of computer time
and memory. A problem is that the lattice must be rather large. To see this, let us
return to the simpler problem of quenched QCD, where the quarks have infinite
mass so that they do not move; furthermore there is no vacuum polarisation in that
case. The Wilson loop correlation function is now defined as

W(C)=Tr

∏

(n,μ)C

Uμ(n), (15.182)

where the product is to be evaluated in apath-orderedfashion, i.e. the matrices
must be multiplied in the order in which they are encountered when running along
the loop. This is different from QED and reflects the fact that theUs are noncom-
muting matrices rather than complex numbers. This correlation function gives us
the quenched inter-quark potential in the same way as in QED. In this approxima-
tion, perturbative renormalisation theory can be used to find an expansion for the
potential at short distances in the coupling constant,g, with the result:

V(R,g,a)=

C

4 πR

[

g^2 +

22

16 π^2

g^4 ln

R

a

+O(g^6 )

]

. (15.183)

HereCis a constant. We see that the coefficient of the second term increases for
largeR, rendering the perturbative expansion suspect, as mentioned before. The
general form of this expression is

V(R,g,a)=α(R)/R, (15.184)

in other words, a ‘screened Coulomb’ interaction. Equation (15.183) can be
combined with the requirement that the potential should be independent of the