THE LARGE N EXPANSION
129
In the large N limit all ^ 7« ^ 0 integrations are irrelevant (we shall prove
this later in this Section) and the integral over A has the form:
Z =
d/o exp^ V\
d^p
2gl -
i .
2 J (2«)'
|p|
(8.16)
which is obtained by the substitution A(x) = Xq = const, into (8.3); V
here is the volume of the system. In the complex plane of the Xq variable
we have a cut from the logarithm in (8.16) which goes from — 00 to zero,
and a saddle point when gl > ^o,cr- For gl < gl^^^ this saddle point
disappears under the cut. Therefore, in order to obtain the dominant
contribution to the integral, our contour for gl > gl must be taken to
pass through the saddle point and the integrand is strongly concentrated near this saddle point. This situation we have analysed above.
For gl < gl the contour can be deformed so that it goes around the
cut. In the limit of infinite volume the dominant contribution comes
from the origin of the cut at Xq = 0. In order to estimate the most
important values of Xq, let us notice that the singular part of the integral
in (8.16) is given by:
d^p log (p^ + A) = const. X^'^ -H regular terms
(8.17)
From this we conclude that the essential Xq is defined by:
NV X^'^ ~ 1
- (Nvy^!^
0
K-^00
(8.18)
The conclusion we reach is that for gl < g l the quantity Xq has to be
zero in the infinite volume limit, and we have massless particles in our
system. That implies that the symmetry has been broken as we go to
gl < gl ,,, and these massless particles are Goldstone’s. We postpone
direct verification of this fact and proceed to consider corrections to the
naive saddle point picture.
We have to take into account that the field X(x) acquires a vacuum
expectation value in the phase with unbroken 0(N) symmetry (the only
existing one for ^ < 2). In the leading approximation it is given by
(8.10). Successive approximations will change this value by an amount
^ l/N. In order to develop a general formalism we must not fix
= X from the beginning, but keep it arbitrary while performing
integrations over
After that we shall obtain an effective action
129
In the large N limit all ^ 7« ^ 0 integrations are irrelevant (we shall prove
this later in this Section) and the integral over A has the form:
Z =
d/o exp^ V\
d^p
2gl -
i .
2 J (2«)'
|p|
(8.16)
which is obtained by the substitution A(x) = Xq = const, into (8.3); V
here is the volume of the system. In the complex plane of the Xq variable
we have a cut from the logarithm in (8.16) which goes from — 00 to zero,
and a saddle point when gl > ^o,cr- For gl < gl^^^ this saddle point
disappears under the cut. Therefore, in order to obtain the dominant
contribution to the integral, our contour for gl > gl must be taken to
pass through the saddle point and the integrand is strongly concentrated near this saddle point. This situation we have analysed above.
For gl < gl the contour can be deformed so that it goes around the
cut. In the limit of infinite volume the dominant contribution comes
from the origin of the cut at Xq = 0. In order to estimate the most
important values of Xq, let us notice that the singular part of the integral
in (8.16) is given by:
d^p log (p^ + A) = const. X^'^ -H regular terms
(8.17)
From this we conclude that the essential Xq is defined by:
NV X^'^ ~ 1
- (Nvy^!^
0
K-^00
(8.18)
The conclusion we reach is that for gl < g l the quantity Xq has to be
zero in the infinite volume limit, and we have massless particles in our
system. That implies that the symmetry has been broken as we go to
gl < gl ,,, and these massless particles are Goldstone’s. We postpone
direct verification of this fact and proceed to consider corrections to the
naive saddle point picture.
We have to take into account that the field X(x) acquires a vacuum
expectation value in the phase with unbroken 0(N) symmetry (the only
existing one for ^ < 2). In the leading approximation it is given by
(8.10). Successive approximations will change this value by an amount
^ l/N. In order to develop a general formalism we must not fix
integrations over
After that we shall obtain an effective action