Notice that
did not participate in our derivation because it enters in
a gauge invariant way. As a consequence it appears only in combinations containing
(where
is the Yang-Mills field strength)
which have dimension 4 and cannot contain logarithmic divergences.
This gauge invariance is just the reflection of the arbitrariness of the
fields {e^} (remember that only n is physical).
This kind of derivation can be easily generalized to fields belonging
to an arbitrary coset space, G/H. (Above we have worked out the cases
H = I, and the case of the /i-field: ne
= SO(N -h l)/SO(N\
being
a unit sphere).
ASYMPTOTIC FREEDOM AND THE RENORMALIZATION GROUP 29
2.3 Non-Abelian Gauge Fields for ^ = 4
In this case the action has the form:
We shall follow the same procedure as before. Let us set
and expand (2.53) up to terms quadratic in a^;
- ^ 2 Tr I
■ [a^, aj}
(2.53)
(2.54)
= S - ^ T r [
?0 J
d^x {(V,a,)^ + 2f,,- K , a j - (V^«.)'
(2.55)
(Here
a j.)
We have to integrate over all possible a^-fields with momenta
A<\p\ < A. At this point we have to fix the gauge of because due to
the gauge invariance of the action the quadratic form (2.55) has zero
eigenvalues. Namely if we take
w]
(2.56)
we find that 5^"^ = 0 for an arbitrary function w. This simply means that
we have to integrate in directions in our functional space which are