24
GAUGE FIELDS AND STRINGS
This is the Gell-Mann-Low equation. Iterations of its solution in
log(p//i) give:
e\p) ~ e^(p) +
log(p//<) + 0(log^(p/p))
At the same time, from (2.13) we deduce:
e^(p) ~ e^(p)
C/G)
4n
e*(p) log(p/p)
(2.23)
(2.24)
Comparison gives:
4n
^{e^) = -
Solving the differential equation (2.22) we get:
e\p)
e (p) ■ ■
1 +
e\p) log|
©
(2.25)
It is easy to check that while (2.24) does not have the form (2.19), (2.25)
does (with f{x) = 47r/C„((^) x). Another useful expression for e^{p) is:
e\p) ■
4
1
Q G)
Stt ei log _2
?
(2.26)
(A is the inverse lattice spacing, p A).
What is the range of applicability of (2.25) and (2.26)? It is defined by
the fact that we have neglected all higher powers in e^(p) in the
expression for the jS-function. Therefore the condition is:
e\p) < 1
(2.27)
The real meaning of this improvement to perturbation theory, invented
by Gell-Mann and Low, is that it replaces the expansion in the bare
charge el which may not be small by the expansion in e^(p), which in
many important cases is small.
For example if we rewrite (2.25) as:
e\p) =
Sn
1
C,(G) log(p7/^)
(2.28)
we conclude that this is a true asymptotic expansion for e^(p) when
p P /. As we shall see, in this region all correlation functions can be
computed for the reason that the interaction is small. This ultraviolet
smallness is called asymptotic freedom. For p < / perturbation theory