Nonlinearσmodel as an effective Lagrangian

We study the limit when the $\sigma$ mass is very large compared to other scales in the renormalized light-particle-irreducible Green's functions of the linear $\sigma$ model with $\mathrm{SO}\mathrm{}(N+1\mathrm{})\mathrm{}$ symmetry. This is a strong-coupling limit. The resulting theory is summarized by an effective Lagrangian, from which Green's functions, not just the $S$-matrix elements, can be reproduced to $O\mathrm{}\left(1\right)\mathrm{}$. At the one-loop level, the effective Lagrangian is explicitly constructed and the relationship between the infinities of the nonlinear model and the heavy-mass effects is clarified. We then discuss the construction of the effective Lagrangian to all orders via the method of oversubtraction. This investigation can either be looked upon as a systematic way of organizing the heavy-Higgs-boson effects or it may be interpreted as a method of defining a renormalization program for the nonlinear $\sigma$ model.