474 Computational methods for lattice field theories
unacceptable. Another way to remove the divergences is by formulating the theory
on a discrete lattice. This is of course similar to cutting off momentum integrations,
and the lattice constantaused is related to the momentum cut-off by
a∼ 1 /
. (15.34)Such cut-off procedures are calledregularisationsof the field theory.
We must remove the unphysical cut-off dependence from the theory. The way
to do this is to allow the coupling constant and mass constants of the theory to be
dependent on cut-off and then require that the cut-off dependency of the Green’s
functions disappears.^3 There are infinitely many different Green’s functions and
it is not obvious that these can all be made independent of cut-off by adjusting
only the three quantitiesm,gandφ. Theories for which thisispossible are called
renormalisable. The requirement that all terms in the perturbative series are merely
finite, without a prescription for the actual values, leaves some arbitrariness in the
values of field scaling, coupling constant and mass. We use experimental data to
fix these quantities.
To be more specific, suppose we carry out the perturbation theory to some order.
It turns out that the resulting two-point Green’s functionG(k,−k)assumes the form
of the free-field correlation function (15.31) with a finite mass parameter plus some
cut-off dependent terms. Removing the latter by choosing the various constants of
the theory (m,g, scale of the field) in a suitable way, we are left with
G(k,−k)= 1 /(k^2 +m^2 R) (15.35)wheremRis called the ‘renormalised mass’ – this is the physical mass which is
accessible to experiment. This is not the mass which enters in the Lagrangian and
which we have made cut-off dependent: the latter is called the ‘bare mass’, which
we shall now denote bymB. The value of the renormalised massmRis not fixed
by the theory, as the cut-off removal is prescribed up to a constant. We use the
experimental mass to ‘calibrate’ our theory by fixingmR. In a similar fashion, we
use the experimental coupling constant, which is related to the four-point Green’s
function, to fix a renormalised coupling constantgR(see the next section).
The renormalisation procedure sounds rather weird, but it is certainly not some
arbitraryad hocscheme. The aim is to find bare coupling constants and masses,
such that the theory yields cut-off independent physical (renormalised) masses and
couplings. Different regularisation schemes all lead to the same physics. We need
as many experimental data as we have parameters of the theory to adjust, and
having fixed these parameters we can predict an enormous amount of new data (in
particular, all higher order Green’s functions). Moreover, the requirement that the
(^3) In addition to mass and coupling constant, the field is rescaled by some factor.