If we want to examine only one loop corrections to the classical action,
we have to consider only small fluctuations of the matrix h. We can
write:
ASYMPTOTIC FREEDOM AND THE RENORMALIZATION GROUP 21
h = ^ \ (j) +
( 2.8)
(where 0 belongs to the Lie algebra of G.). Substitution of (2.8) into
(2.7) gives:
¿€q
ZCq
(2.9)
(we have omitted the term
d^(j) because of the equation of motion
d,Rl' = 0).
Now the term in the effective action which is quadratic in
will be
given by the Feynman diagram t
^ Ij'ACDj'BCD
=
- • d^p (2p + q\(2p + q),
(2ny
%p\p + q f
(2.10)
If we extract the ultraviolet divergent contribution from (2.10) the
formula takes the form:
1 d^x{Rl\x)nRl\x)r X -CXG)
'
PmPv
(2nf p*
+ finite part
c,(G)
\ 1
H - finite part
( 2. 11)
(A is a momentum cut-off).
We see that if we introduce the renormalized coupling constant by
the formula:
1 1 Q(G)
An
log A
( 2. 12)
t We have introduced the generators of the fundamental representation of the Lie
algebra of the group G: [F*, i®] =
Tr
(j) =
are the structure constants of this algebra. Notice that