Now, let US examine the perturbation theory for the partition function:
Z = (e x p
(10.33)
260
GAUGE FIELDS AND STRINGS
When we expand this expression in A we get different products of
«-operators, integrated over (^-space. These integrals are divergent
because of the singularities in the operator product (10.31). Therefore,
we have to introduce a cut-off. The renormalization groups tells us how
we should change
under a given change of the cut-off, so that Z
remains unchanged.
In order to find these conditions, we shall first of all discard all
quadratic divergences, coming from the first term in (10.31). Formal
justification of this lies in the fact that these divergences are absent if we
use a dimensional regularization scheme. The real reason is that they
are purely short-distance effect, contributing to Z but not to any
connected Green function. On the contrary, logarithmic divergences,
coming from the second term in (10.31) are physically important and
we are now going to analyse them. Let us take the Nth term in the
expansion for Z:
Z<^> = I
,...,)...(10.34)
As, say,
^ 2 this integral becomes divergent. Using (10.31) we find:
(10.35)
-H finite terms
where a is an ultraviolet cut-off. We derive from (10.35) that the change
in a: a-*^d can be compensated by a change of if we choose:
(10.36)
In other words we have computed (approximately), how
must
depend on a so as to keep Z «-independent. The answer is contained in
the Gell-Mann-Low equation:
dL
d(2n log(l/«))
= PnW =
+ 0(P)
(10.37)