Computational Physics

15.7 Gauge field theories 531

renormalisation cut-offa

a

dV(r,g,a)

da

= 0 (15.185)

to find a relation between the coupling constantgand the lattice constanta. To see
howthisisdone,seeRefs.[ 6 , 43 , 45 ].Thisrelationreads

a= − 01 (g^2 γ 0 )γ^1 /(^2 γ

02 )

exp[− 1 /( 2 γ 0 g^2 )][ 1 +O(g^2 )]. (15.186)

This implies thatgdecreases with decreasinga, in other words, for small distances
the coupling constant becomes small. From(15.183)we then see that the potential
is screened to zero at small distances. This is just the opposite of ordinary screening,
where the potential decays rapidly for large distances. Therefore, the name ‘anti-
screening’ has been used for this phenomenon, which is in fact the asymptotic
freedom property of quarks. The constantsγ 0 andγ 1 are given byγ 0 =( 11 −
2 nf/ 3 )/( 16 π^2 )andγ 1 =( 102 − 22 nf/ 3 )/( 16 π^2 )^2 repectively (nf is the number
of flavours), and
0 is an integration constant in this derivation, which must be
fixed by experiment. Any mass is given in units ofa−^1 , which in turn is related tog
through the mass constant
0. The important result is that if we do not include quark
masses in the theory, only a single number must be determined from experiment,
and this number sets the scale for all the masses, such as the masses of glueballs,
or those of massive states composed of zero-mass quarks. Therefore, after having
determined
0 from comparison with a single mass, all other masses and coupling
constants can be determined from the theory, that is, from the simulation.
Nice as this result may be, it tells us that if we simulate QCD on a lattice, and
if we want the lattice constantato be small enough to describe the continuum
limit properly, we need a large lattice. The reason is that the phase diagram for
the SU(3) lattice theory is simpler than that of compact QED in four dimensions.
In the latter case, we have seen that there exist a Coulomb phase and a confined
phase, separated by a phase transition. In lattice QCD there is only one phase, but
a secret length scale is set by the lattice parameter for which(15.186)begins to
hold. The lattice theory will approach the continuum theory if this equation holds,
that is, if the lattice constant is sufficiently small. If we want to include a hadron in
the lattice, we need a certain physical dimension to be represented by the lattice (at
least a ‘hadron diameter’). The small values allowed for the lattice constant and the
fixed size required by the physical problem we want to describe cause the lattice
to contain a very large number of sites. Whether it is allowable to take the lattice
constant larger than the range where(15.186)applies is an open question, but this
cannot be relied upon.
In addition to the requirement that the lattice size exceeds the hadronic scale, it
must be large enough to accomodate small quark masses. The reason is that there
exist excitations (‘Goldstone bosons’) on the scale of the quark mass. The quark