The Gross–Neveu model (GN) is a quantum field theory model of Dirac fermions interacting via four-fermion interactions in 1 spatial and 1 time dimension.
It was introduced in 1974 by David Gross and André Neveu[1] as a toy model for quantum chromodynamics (QCD), the theory of strong interactions.
It shares several features of the QCD: GN theory is asymptotically free thus at strong coupling the strength of the interaction gets weaker and the corresponding
function of the interaction coupling is negative, the theory has a dynamical mass generation mechanism with
chiral symmetry breaking, and in the large number of flavor (
) limit, GN theory behaves as t'Hooft's large
The model's Lagrangian density is where the formula uses Einstein summation notation.
(which permits only one quartic interaction) and makes no attempt to analytically continue the dimension, the model reduces to the massive Thirring model (which is completely integrable).
The 2 dimensional version has the advantage that the 4 fermi interaction is renormalizable, which it is not in any higher number of dimensions.
After demonstrating that this and related models are asymptotically free, they found that, in the subleading order, for small fermion masses the bifermion condensate
acquires a vacuum expectation value (VEV) and as a result the fundamental fermions become massive.
The vacuum expectation value spontaneously breaks the chiral symmetry of the theory.
expansion but to all orders in the coupling constant, the dependence of the potential energy on the condensate using the effective action techniques introduced the previous year by Sidney Coleman at the Erice International Summer School of Physics.
Expanding the theory about the new vacuum, the tachyon was found to be no longer present and in fact, like the BCS theory of superconductivity, there is a mass gap.
They then made a number of general arguments about dynamical mass generation in quantum field theories.
For example, they demonstrated that not all masses may be dynamically generated in theories which are infrared-stable, using this to argue that, at least to leading order in
They also argued that in asymptotically free theories the dynamically generated masses never depend analytically on the coupling constants.
First, they considered a Lagrangian with one extra quartic interaction chosen so that the discrete chiral symmetry
of the original model is enhanced to a continuous U(1)-valued chiral symmetry
Chiral symmetry breaking occurs as before, caused by the same VEV.
However, as the spontaneously broken symmetry is now continuous, a massless Goldstone boson appears in the spectrum.
Two further modifications of the modified theory, which remedy this problem, were then considered.
Consequently, the Golstone boson is "eaten" by the Higgs mechanism as the photon becomes massive, and so does not lead to any divergences.