32
GAUGE FIELDS AND STRINGS
(In this formula cp is any field and V((p) is any perturbation with
= 0). So the normal sign in (2.64) would be negative. However,
vector fields enter the action with an extra factor i (in Euclidean space
the extra term in the amplitude for a scalar particle is exp{i^^.4d/}
where C is the particle’s trajectory). This eflfect in second order gives an
i^ = — 1 factor, which changes the sign in (2.65).
From this consideration it is obvious that the spin term in (2.55) will
give a negative sign contribution to the charge renormalization. This
paramagnetic effect is readily evaluated:
c (p a ra m a g n e tic ) __ __ ____
"
2et
■ C Ù
d^q
X r ‘T ^ ^ ( a y , ( - q ) , a X ( q ) }
p + q
u « = 0
C,(G)
2e*^
Q(G)
lÓTt^
(Fiy d*x
fd V
J (2A)>let
(P + Q ) q = 0
l0gX2 (FIŸ d*x
Putting together these two effects we obtain:
1
1
11
1
(2.66)
(2.67)
Now we can repeat all the previous arguments concerning the renormalization group and we find that the effective coupling in the
nonabelian gauge theories is asymptotically free:
e\p) =
4Sn^
ncxG)\og(p^/py
p> À
( 2.68)
This is the famous result of Gross, Wilzek and Politzer.
It is not difficult to evaluate different gauge invariant Green functions
at small distances. As before, they differ from the free ones by factors of
the type (\og(p^/P)y where y depends on the type of the Green
function. However, the methods described above are absolutely inadequate for the investigation of the infrared region
which is
physically the most interesting. To accomplish this task we proceed to a
quite different approach.