is untrue and qualitatively different methods are needed. They will be
discussed in later chapters.
Let us proceed to the computation of the correlation functions. Take
as an example:
9(x-y) = {Tr(g-\x)g(ym
(2.29)
The tactic is again the following. Let us integrate over rapid fluctuations with A < |/?| < A and use the renormalization group argument. If
we write
g{x) =
(1 + (l> ix) + j(j)\x))goix)
(2.30)
and integrate over the 0 with the wavelengths in our interval we get:f
ASYMPTOTIC FREEDOM AND THE RENORMALIZATION GROUP 25
C
@(p) = ®o(P)l * “ V
X
(2.31)
From (2.31) we deduce by the repetition of the arguments leading to
the Gell-Mann-Low equation, that if we set
Slip) = d(p)/p^
then:
where
d log dip)
d log(p/p) = yie\p))
y(e^) =
-h O ie"^)
n
(2.32)
(2.33)
(2.34)
Integrating (2.33) we get:
1 /
C e^ (p )
, ,
®(p) = p M +
log(pV p^)'
,2n ^\4C2/C^
for p > X
(2.35)
which is the desired answer. Of course, the range of applicability is
again p P L
As we said before the definition of e^(p) is ambiguous. This ambiguity
leads to an ambiguity of the j?-function and of the y-function. However
tHere we put
= — C2 /. For the fundamental representation of G = SU{N):
C 2 = {N^ - 1)/2N; C,{SU{N)) = N.