ASYMPTOTIC FREEDOM AND THE RENORMALIZATION GROUP 27
which is the consequence of the relations:
dX
dX del
d \
del
X = ^(t){el)
(2.41)
(the first is the renormalization condition and the second is just a naive
dimensional statement). The equation (2.40) indicates a consistent
limiting procedure necessary to reach the continuum limit of the theory.
For the asymptotically free case (j?i < 0) it is described by the conditions:
^0 ^ 0
A -► 00
2 -► const.
(2.42)
2.2 The/i-Fields
In this section we shall consider another important example of an
asymptotically-free system with a global symmetry. It is the theory of
the field of an N-dimensional unit vector n,n^ = \. The action is given
by:
1
S =
á^x{d.nf
In order to exploit the renormalization group, we set:
n{x) = (1 -
+ Y.
(2.43)
(2.44)
(where /lo(^) is a slowly varying vector and the {ej are orthogonal to it
and to each other; cp^ represents “fast” fluctuations with A < Ipl < A).
The vectors («q, ej obey the relations
^„»0 = Z
"
(2.45)
= Z
b
(Here
and A f are some potentials characterizing the «Q-field; (2.45)
is the consequence of «0• = 0 and e^-e^ =
Substituting (2.44)
into (2.43) gives:
S = 2elj
{(a,(l -
+ (3,9“ - Afq>” + Bl(\ -
d^x
(2.46)