Apparently Noninvariant Terms of U(N) x U(N) Nonlinear Sigma Model in the One-Loop Approximation

Abstract

We show how the Apparently Noninvariant Terms (ANTs), which emerge in perturbation theory of nonlinear sigma models, are consistent with the nonlinearly realized symmetry by employing the Ward-Takahashi identity (in the form of an inhomogeneous Zinn-Justin equation). In the literature the discussions on ANTs are confined to the SU(2) case. We generalize them to the U (N ) case and demonstrate explicitly at the one-loop level that despite the presence of divergent ANTs in the effective action of the "pions", the symmetry is preserved.Subject Index: 131, 132 §1. IntroductionIt has been well known that perturbation theory for nonlinear sigma models (NLSMs) produces various types of apparently noninvariant terms (ANTs), when "pion fields" are introduced. In a previous paper, 1) we classified them into two kinds: the first kind of ANTs refers to those that are quartically divergent and do not vanish in the zero momentum limit. It is well understood that the contributions from the Jacobian cancel them. 2)-4) (In the dimensional regularization scheme, this kind of terms and the contributions from the Jacobian are both absent.)The second kind 5)-9) is more subtle. They are also divergent, but do vanish in the zero momentum limit. They appear even with dimensional regularization, but cannot be absorbed by symmetric counterterms. 5), 9) In Ref. 1), we employed lattice regularization and investigated the second-kind ANTs. We have shown that the ANTs emerge despite the manifest symmetry present in the lattice formulation and that they have nothing to do with the Jacobian, which is well defined in this formulation. The appearance of the second-kind ANTs is thus consistent with the symmetry of the NLSM. In the following, we concentrate on the second-kind ANTs and refer to them simply as ANTs.Natural questions are: why do ANTs emerge? How are they consistent with the symmetry? In the present paper, we answer these questions.In order to consider the symmetry and to investigate its consequences, it is useful * )

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