The Nonlinear Sigma Model.

A study of the large N behavior of both the O(N) linear and nonlinear sigma models is presented. The purpose is to investigate the relationship between the disordered (ordered) phase of the linear and nonlinear sigma models. Utilizing operator product expansions and stability analyses, it is shown that for 2 (LESSTHEQ) d < 4, it is the (lamda)(,R)(M) - (lamda)* (--->) O('-) ((lamda)(,R)(M) is the dimensionless renormalized quartic coupling and (lamda)* is the IR fixed point) limit of the linear sigma model which yields the nonlinear sigma model. It is also shown that stable large N linear sigma models with (lamda) < 0 ((lamda) is the bare quartic coupling) can exist (at least in the context of no tachyonic states being present). A criteria valid for all dimensionalities d, less than four, is derived which determines when (lamda) < 0 models are tachyonic free. Arguments are given showing that the d = 4 large N linear (for (lamda) > 0) and nonlinear models are trivial. This result (i.e. triviality) is well known but only for one and two component models. Interestingly enough, the (lamda) < 0 d = 4 linear sigma model remains nontrivial and tachyonic free. The initial steps toward relating the effective gluon mass and the gluon condensate are taken. The contribution of the self-energy insertion diagram (two-loop order) to the electromagnetic polarization tensor is given. Assuming that the vertex correction gives a comparable contribution, a value for the effective gluon mass of 400-700 MeV is found.