Physical Mathematics

The Low Lying Energy-Momentum Spectrum for the Lattice Four-Fermi Model

Abstract

Petrus H. R. dos Anjos and Paulo A. Faria da Veiga

We obtain the low-lying energy-momentum spectrum for the imaginary-time lattice four-Fermi or Gross- Neveu model in d + 1 space-time dimensions (d = 1, 2, 3) and with N-component fermions. Let 0 < ?  0 be the hopping parameter, ? > 0 the four-fermion coupling, m > 0 the bare fermion mass and take s × s spin matrices (s = 2, 4). Our analysis of the one and the two-particle spectrum is based on spectral representation for suitable two- and four-fermion correlations. The one-particle energy-momentum spectrum is obtained rigorously and is manifested by sN 2 isolated and identical dispersion curves, and the mass of particles has asymptotic value order ?ln ?. The existence of two-particle bound states above or below the two-particle band depends on whether Gaussian domination does hold or does not, respectively. Two-particle bound states emerge from solutions to a lattice Bethe-Salpeter equation, in a ladder approximation. Within this approximation, the (sN 2 ? 1)sN 4 identical bound states have O(?0) binding energies at zero system momentum and their masses are all equal, with value? ?2 ln?. Our results can be validated to the complete model as the Bethe-Salpeter kernel exhibits good decay properties. MSC 2010: 81Qxx, 81Txx, 81Vxx.