Symanzik's improvement program applied to the Gross-Neveu model at large N

Abstract

Symanzik's improvement program for lattice actions is applied to the large- N limit of the Gross-Neveu model in the σ-field formulation. We construct a series of actions which suppress scaling violations up to order a 2 at tree level, the l-loop level, and to all orders in the large- N coupling λ = g 2 N. The fermions are treated by Wilson's method. We investigate the gap equation for the dynamical fermion mass. For this nonperturbative quantity it is shown that the correlation between the order of improvement and the order to which powers of ln a are supressed follows the pattern expected from perturbation theory, i.e. tree improvement removes the leading logarithm, l-loop improvement removes the next leading logarithm, etc. This property is also verified for related quantities such as the mass defined by the fall-off behavior of the fermion correlation function and the fermion-antifermion scattering amplitude in the singlet channel. As far as scaling is concerned, we observe that beyond tree improvement the β-function contains non-universal terms implying a difference between scaling and asymptotic scaling which increases with the order of improvement. Comparing for the dynamical mass the gain by supressing scaling violations versus the loss in asymptotic scale behavior, we argue that in a numerical approach the tree-improved action would yield the best estimate for the dynamical mass.