130
GAUGE FIELDS AND STRINGS
which depends on X , Eventually we have to take the minimum in X or,
equivalently, impose the condition = 1.
Let us see how this programme works. The action for [X^^q] is given
by:
X(x) =
+ /a(x), a^ = o = 0
N
W(X) = - - Tr\ogl-d^ +
+ ia(x)]
N
,
N (\
d^q
\ r
= 2
- 2 |2 J n
( ^ - 3 J
(819)
^
+ i2 + is ) ---------------+ (2;:)“
N
,
1
= 2
- 2
d^q
{2nf
-h
1
3 l 2
d^qi
919293 ^91^92^93
^2
Ç2n)^ ^
Here:
U ( q ) =
d^k
yV\W2
1
{2nŸ
+ fi^)((q - kf + fi^)
\^2
(8.20)
..A /
etc.
We see, that the effective nonlinearity in (8.19) contains
factors,
and that perturbation theory for this jS-field has a good parameter
1/^iV. The quantity
is to be adjusted at the very end of calculation.
To give an example, we see from (8.19) that up to the order I/^Jn we
have:
,
.
1
1
1
1
,
=-iVF'^ITo + V~^W^
+ ■
- { ¿ - î
1 r d^q
2j(2ny
logn(9^) + 0(N -‘'")
(8.21)