How much can one break chiral symmetry in the linear sigma model?

Abstract

We show that the hedgehog soliton solution describing the nucleon in the SU(3)xSU(3) linear sigma model breaks down when the pion mass becomes too large. PACS: 11.30Rd; 12.50Lr The experimental value of the nucleon sigma-term aN ~60 MeV [1] is roughly twice the theoretical quark model prediction obtained by relating aN to the octetbaryon mass-splittings. It has been suggested [2] that this discrepancy points to a large strange-quark content (gs)N in the nucleon, in violation of the OZI rule: Another solution to this puzzle, advocated by Jaffe [3], is that the variation of baryonic masses with (rh-m~) (where rh and m~ are the non-strange and strange currentquark masses) is strongly non-linear, so that first-order perturbation theory does not give a reliable estimate of the octet-baryon mass-splittings. Some recent model investigations [4,5] have indeed shown evidence that the nucleon mass may depend non-linearly on the strangequark mass. A more natural possibility, which is suggested by chiral perturbation theory [2,6], is that the nucleon mass depends non-linearly on the non-strange quark mass.We showed recently [7] that, in the framework of the SU(3)x SU(3) linear sigma-model I-8], the nucleon mass does not vary significantly with the strange currentquark mass, but that its dependence on the non-strange current-quark mass rfi (proportional to. the SU(2) x SU(2) dynamical symmetry breaking parameter -cl=F,~rn~) is strongly non-linear. Because this nonlinearity has nothing to do with strangeness, it also occurs in the SU(2) x SU(2) linear sigma model [9]. Actually, we shall show that, when Cl is varied sufficiently far away from the chiral limit ca =0, something more dramatic happens in the (two-or three-flavor) linear sig-* On leave of absence from the Laboratoire de Physique Th6or-ique, Universit6 de Nice, parc Valrose, F-06034 Nice Cedex, France ma model. The nucleon is described in this model as a non-topological soliton 1-10-12]. It is well known that classically stable non-topological solitonic solutions exist only for a limited range of the model parameter values [13][14][15]. This general property, which has been well documented by numerical investigations in various field-theoretical models [16][17][18][19], has the consequence that, when -ca (or mR) is varied, all the other model parameters being fixed, the solitonic solution describing the nucleon ceases to exist above a critical value -c] (or m~).To show this, we consider the SU ( (a:= 0, ..., 8) combines a scalar nonet ~ and a pseudoscalar nonet ~b~ coupled to the quark triplet field 0, and