Ghost poles in the nucleon propagator in the linear σ model approach and its role in πN low-energy theorems

Abstract

Complex mass poles, or ghost poles, are present in the Hartree-Fock solution of the Schwinger-Dyson equation for the nucleon propagator in renormalizable models with Yukawa-type meson-nucleon couplings, as shown many years ago by Brown, Puff and Wilets (BPW). These ghosts violate basic theorems of quantum field theory and their origin is related to the ultraviolet behavior of the model interactions. Recently, Krein et.al, proved that the ghosts disappear when vertex corrections are included in a self-consistent way, softening the interaction sufficiently in the ultraviolet region. In previous studies of πN scattering using “dressed” nucleon propagator and bare vertices, did by Nutt and Wilets in the 70's (NW), it was found that if these poles are explicitly included, the value of the isospin-even amplitude A( +) is satisfied within 20% at threshold. The absence of a theoretical explanation for the ghosts and the lack of chiral symmetry in these previous studies led us to re-investigate the subject using the approach of the linear σ-model and study the interplay of low-energy theorems for πN scattering and ghost poles. For bare interaction vertices we find that ghosts are present in this model as well and that the A( +) value is badly described. As a first approach to remove these complex poles, we dress the vertices with phenomenological form factors and a reasonable agreement with experiment is achieved. In order to fix the two cutoff parameters, we use the A( +) value for the chiral limit ( m π → 0) and the experimental value of the isoscalar scattering length. Finally, we test our model by calculating the phase shifts for the S waves and we find a good agreement at threshold.