THE LARGE N EXPANSION
to construct such terms:
143
= Pl^(P2^)-P2|i(Pl^)
If we substitute this into (8.72) we obtain:
(8.74)
r^^XPl, P2) - 0(6,^PlaP2^) p
— ^^oißPlaPlß
(8.75)
Presence of the r^^-containing terms in the two point function indicates
breakdown of CP-invariance. As we have discussed before, the analogous phenomenon in QCD creates some unsolved problems.
Let us finally discuss to what extent the topological effects described
above can be attributed to instantons. A naive estimate of the instanton
contribution to Z would give a contribution of the order of e"^. This
happens because the instanton action for the CP^~^ action is finite and
independent of N. At the same time, the coupling constant el scales as
l/N. Hence we obtain an exponentially small contribution. However,
this naive argument is wrong. The one loop calculation of the determinant near a multi-instanton configuration reduces the partition function of the CP^~^ model to one of some generalized plasma, in the same
way as in CP^ (see Chapter 6). The correlation length which is
established in this plasma is such that the entropy, coming from the
fluctuations, compensates the smallness of the classical contribution.
Roughly speaking the following happens. The one instanton contribution to the partition function has the form:
K -' log
dp
/
2nN
(8.76)
This formula is strictly correct only for pp 1, thus indicating the
exponential damping described above. However, if we consider the
plasma and not a dilute gas approximation, the infrared divergence in
(8.76) gets cut off at p ^
It is true that at this point, where
instantons dissociate (just as in the 0(3) case of Chapter 6) the one loop
approximation is not applicable, and we have to deal with a strong
to construct such terms:
143
= Pl^(P2^)-P2|i(Pl^)
If we substitute this into (8.72) we obtain:
(8.74)
r^^XPl, P2) - 0(6,^PlaP2^) p
— ^^oißPlaPlß
(8.75)
Presence of the r^^-containing terms in the two point function indicates
breakdown of CP-invariance. As we have discussed before, the analogous phenomenon in QCD creates some unsolved problems.
Let us finally discuss to what extent the topological effects described
above can be attributed to instantons. A naive estimate of the instanton
contribution to Z would give a contribution of the order of e"^. This
happens because the instanton action for the CP^~^ action is finite and
independent of N. At the same time, the coupling constant el scales as
l/N. Hence we obtain an exponentially small contribution. However,
this naive argument is wrong. The one loop calculation of the determinant near a multi-instanton configuration reduces the partition function of the CP^~^ model to one of some generalized plasma, in the same
way as in CP^ (see Chapter 6). The correlation length which is
established in this plasma is such that the entropy, coming from the
fluctuations, compensates the smallness of the classical contribution.
Roughly speaking the following happens. The one instanton contribution to the partition function has the form:
K -' log
dp
/
2nN
(8.76)
This formula is strictly correct only for pp 1, thus indicating the
exponential damping described above. However, if we consider the
plasma and not a dilute gas approximation, the infrared divergence in
(8.76) gets cut off at p ^
It is true that at this point, where
instantons dissociate (just as in the 0(3) case of Chapter 6) the one loop
approximation is not applicable, and we have to deal with a strong