Renormalization of the σ-model

Abstract

We consider the renormalization of the σ-model in the absence of the baryon fields. The model is characterized by three parameters. In order that the current algebra and the partially conserved axial vector current condition are valid in the renormalized σ-model, it is necessary that all divergences of the theory may be absorbed in two constants, the mass and the coupling constant, and no new ad-hoc parameters be introduced to remove divergences. A perturbation method is presented, in which the lowest order gives rise to the usual results of the current algebra (or chiral dynamics). It is shown that indeed all divergences can be absorbed into the redefinitions of the mass parameter and the one coupling constant. This is achieved by demonstrating the existence of an intermediate renormalization (wherein the counter terms are precisely those of a chiral symmetric theory) which makes the theory finite. As a consequence, the mass difference between the scalar and pseudoscalar particles is finite and all the three point and four point couplings are expressible in terms of one parameter. The vacuum expectation value of the scalar σ field is given by a non-linear equation. A consistent treatment of this equation is outlined in detail, and the equation is shown to be finite and well-defined after the intermediate renormalization. The possibility of the spontaneous breakdown of the chiral symmetry, through the non-vanishing vacuum expectation value of the σ-field even when the Lagrangian is chiral invariant, is described and the Goldstone theorem is verified up to the second order in perturbation expansion. The treatment of the axial vector current - pion vertex (such as appears in the pion decay), which contains a linear divergence not encountered in the S-matrix expansion, is detailed. We relate the pion decay constant to the vacuum expectation value of the renormalized σ field in all orders of perturbation theory.