26
GAUGE FIELDS AND STRINGS
the next result for any correlation function is unique—the jS-, and yambiguities compensate each other. Moreover, the first two coefficients
in the j9-function are more or less universal. This can be checked by a
change in the definition of e^{p). Suppose we chose:
e\p) = e\p) + C,e\p) + C^e\p) + ...
and
d Xogipjp)
=
+ ^2^^ + ••
(2.36)
(2.37)
On substituting (2.36) into (2.37) we get:
de^
de^
2
d log(p///) d log(p/p)
= (1 + 2Ci^2 + ..
+
+ ,..) =
+ 2CJ,)
X
+ ... = p,(e^ -
+ (^2 + 2 C,P,)e^ + ...
= P,t-\-P2e^ + ...
(2.38)
The coefficient JS3 will be changed, but up to two loop order the
j?-function is indeed universal. In the case when several leading coefficients of the ^-function vanish, this remark relates to the first nonvanishing one.f
So, our conclusion is that the Gell-Man-Low renormalization group
is a useful tool in the region where the effective charge is small. In order
to use it one has to compute the coefficients before the leading powers
of log(p/p) in physical quantities. This is usually easy in the lowest
order, but becomes increasingly tedious in the higher orders, since one
has to separate a nonleading contribution
log(p/p) from the leading
ones ^ (log(p/p))". In the asymptotically free theories the renormalization group gives order by order a small distance expansion which goes
in inverse powers of log(p/p). For example the two loop order solution
of equation (2.22) gives:
eHp) = -
1
log l o g y ) ^
)?, \og(p/A) I
Pi log(p/A)
1
\og\p/A)
p> X (2.39)
Another useful thing to remember is that the inverse correlation length
A is expressed in terms of the bare parameters A, C q as
A = const • A(Co)^^^^‘c^^^‘^o
(2.40)
t In the general case we have freedom to redefine coupling constants, leading to
Riemannian geometry in the space of coupling ‘constants’.