TheGross–Neveu model (GN) is aquantum field theory model ofDirac fermions interacting viafour-fermion interactions in 1 spatial and 1 time dimension. It was introduced in 1974 byDavid Gross andAndré Neveu^[1]as atoy model forquantum 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${\displaystyle \beta }$ function of the interaction coupling is negative, the theory has a dynamical mass generation mechanism with${\displaystyle {\mathbb{Z}}_2}$ chiral symmetry breaking, and in the large number of flavor (${\displaystyle N\to \mathrm{\infty }}$) limit, GN theory behaves as 't Hooft's large${\displaystyle N_c}$ limit in QCD.^[2]

It is made using a finite, but possibly large number,${\displaystyle N,}$ ofDirac fermionwave functions${\displaystyle {\psi }_1,{\psi }_2,\dots ,{\psi }_N}$, indexed below by Latin letter${\displaystyle a.}$

The model'sLagrangian density is

${\displaystyle \mathcal{L}={\overline{\psi }}_a\left(i\mathrm{\partial }/-m\right){\psi }^a+\frac{g^2}{2N}{\left[{\overline{\psi }}_a{\psi }^a\right]}^2,}$

where the formula usesEinstein summation notation. Each wave function${\displaystyle {\psi }^a}$ is a two component (left / right)spinor and${\displaystyle g}$ is the interaction'scoupling constant. If the mass${\displaystyle m}$ is zero, the model ischiral symmetric type, otherwise, for non-zero mass, it isclassical mass type.

This model has aU(N) globalinternal symmetry. If one takes${\displaystyle N=1}$ (which permits only onequartic interaction) and makes no attempt toanalytically continue the dimension, the model reduces to the massiveThirring model (which is completely integrable).^[3]

It is a 2 dimensional version of the 4 dimensionalNambu–Jona-Lasinio model (NJL), which was introduced 14 years earlier as a model ofdynamical chiral symmetry breaking (but noquark confinement) modeled upon theBCS theory of superconductivity. The 2 dimensional version has the advantage that the 4 fermi interaction isrenormalizable, which it is not in any higher number of dimensions.

Gross and Neveu studied this model in the large${\displaystyle N}$ limit, expanding the relevant parameters in a${\displaystyle \frac{1}{N}}$ expansion. After demonstrating that this and related models are asymptotically free, they found that, in the subleading order, for small fermion masses the bifermion condensate${\displaystyle {\overline{\psi }}_a{\psi }^a}$ acquires avacuum expectation value (VEV) and as a result the fundamental fermions become massive. They find that the mass is not analytic in the coupling constant${\displaystyle g.}$ The vacuum expectation valuespontaneously breaks the chiral symmetry of the theory.

More precisely, expanding about the vacuum with no vacuum expectation value for the bilinear condensate they found a tachyon. To do this they solve therenormalization group equations for thepropagator of the bifermion field, using the fact that the only renormalization of the coupling constant comes from thewave function renormalization of the composite field. They then calculated, at leading order in a${\displaystyle \frac{1}{N}}$ expansion but to all orders in the coupling constant, the dependence of thepotential energy on the condensate using theeffective action techniques introduced the previous year bySidney Coleman at theErice International Summer School of Physics. They found that this potential is minimized at a nonzero value of the condensate, indicating that this is the true value of the condensate. 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 amass 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${\displaystyle \frac{1}{N}}$ the 4 dimensional${\displaystyle {\varphi }^4}$ theory does not exist. They also argued that in asymptotically free theories the dynamically generated masses never depend analytically on thecoupling constants.

Gross and Neveu considered several generalizations. First, they considered a Lagrangian with one extra quartic interaction

${\displaystyle \mathcal{L}={\overline{\psi }}_a\left(i\mathrm{\partial }/-m\right){\psi }^a+\frac{g^2}{2N}({\left[{\overline{\psi }}_a{\psi }^a\right]}^2-{\left[{\overline{\psi }}_a{\gamma }_5{\psi }^a\right]}^2)}$

chosen so that the discrete chiral symmetry${\displaystyle \psi \mapsto {\gamma }_5\psi }$ of the original model is enhanced to a continuous U(1)-valued chiral symmetry${\displaystyle \psi \to e^{i\theta {\gamma }_5}\psi .}$Chiral symmetry breaking occurs as before, caused by the sameVEV. However, as the spontaneously broken symmetry is now continuous, a masslessGoldstone boson appears in the spectrum. Although this leads to no problems at the leading order in the${\displaystyle \frac{1}{N}}$ expansion, massless particles in 2 dimensional quantum field theories inevitably lead toinfrared divergences, and so there appears to be no such theory.

Two further modifications of the modified theory, which remedy this problem, were then considered. In one modification one increases the number of dimensions. As a result, the massless field does not lead to divergences. In the other modification, the chiral symmetry is gauged. Consequently, the Golstone boson is "eaten" by theHiggs mechanism as thephoton becomes massive, and so does not lead to any divergences.

• Dirac equation
• Nonlinear Dirac equation
• Thirring model
• Nambu–Jona-Lasinio model

• Quantum field theory

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