Broken phase of the O(N) \phi^4 model in light cone quantization and 1/N expansion

The solution of the O$(N) \phi^4$ scalar field theory in the broken phase is given in the framework of light cone quantization and a 1/N expansion. It involves the successive building of operator solutions to the equation of motion and constraints including operator zero modes of the fields which are the LC counterpart to the equal time non trivial vacuum effects. The renormalization of the procedure is accomplished up to $2^{nd}$ order in the 1/N expansion for the equation of motion and constraints. In addition the renormalization of the divergent contributions of the 2-point and 4-point functions is performed in a covariant way. The presence of a zero modes leads to genuine non perturbative renormalization features.