  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 $\kappa>0$ be the hopping
parameter, $\lambda>0$ the four-fermion coupling and $M>0$ denote the fermion
mass; and take $s\times s$ spin matrices, $s=2,4$. We work in the $\kappa\ll 1$
regime. 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 $-\ln\kappa$. 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 $sN(sN/2-1)/4$ identical bound states have
${\cal O}(\kappa^0)$ binding energies at zero system momentum and their masses
are all equal, with value $\approx -2\ln\kappa$. Our results can be validated
to the complete model as the Bethe-Salpeter kernel exhibits good decay
properties.

High Energy Physics - Theory