Is the nonlinearσmodel themσ→∞ limit of the linearσmodel?

We propose to investigate whether the SO(N) nonlinear σ model is equivalent to the $m_{\sigma }$→infinity limit of the linear σ model by comparing the corresponding one-loop effective-action expansions up to the four-derivative terms and including the symmetry-breaking term. For this purpose we use a new background-field method to calculate the effective-action expansion directly. In the case of the linear σ model, the renormalization procedure is implemented carefully before the $m_{\sigma }$→∞ limit is taken. For the nonlinear σ model we introduce a new and intuitive covariant treatment for the perturbation calculation of the field theory with nonlinear constraint. We do not find any noninvariant terms in either case. We show that the divergent parts of the effective Lagrangians due to $m_{\pi }$→0, $m_{\sigma }$→∞, or N→∞ are equivalent in the two models. However, the nonleading finite parts of the effective Lagrangians are different. Therefore, the two operations, taking the $m_{\sigma }$→∞ limit and calculating the quantum corrections, do not commute. The origin of this difference may be a violation of decoupling.