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The Gross-Neveu model is a simple quantum field theory in 2 space-time dimensions that is considered a toy model for QCD, in the sense that it realizes asymptotic freedom, chiral symmetry breaking and dimensional transmutation.

My question is: does the Gross-Neveu model also realize confinement, and if yes, how so?

Details on the model and a more precise formulation of the question are given below.


The Gross-Neveu model consists in $N$ Dirac fermions with a 4-fermion contact interaction. It is given by the Lagrangian $$ \mathcal{L} = \bar{\psi}_i i \gamma^\mu \partial_\mu \psi_i + \frac{g^2}{2} \left( \bar{\psi}_i \psi_i \right)^2 \qquad i = 1, \ldots, N $$ In addition to the $U(N)$ symmetry that rotates the fermion among themselves, there is a discrete $\mathbb{Z}_2$ chiral symmetry $$ \psi_i \to \gamma^5 \psi_i $$ which in principle forbids the presence of a mass term. It turns out however that this chiral symmetry is spontaneously broken in the quantum theory, and that the fermions acquire a dynamical mass.

The easiest way to see this is to introduce an auxiliary field $\sigma$ and to rewrite the Lagrangian as $$ \mathcal{L} = \bar{\psi}_i i \gamma^\mu \partial_\mu \psi_i - \sigma \bar{\psi}_i \psi_i - \frac{1}{2g^2} \sigma^2 $$ Now the effective potential for $\sigma$ can be computed exactly at large $N$, and we find $$ V_\text{eff}(\sigma) = \frac{\sigma^2}{2g^2} + \frac{N \sigma^2}{4\pi} \left( \log \frac{\sigma^2}{M^2} - 1 \right) $$ This potential is minimized at $$ \sigma_* = \pm M e^{-\pi/Ng^2} $$ so that the fermions acquire a mass $m = |\sigma_*|$. Computing the full effective action for $\sigma$ in the symmetry-breaking vacuum, one can also verify that $\sigma$ becomes a dynamical field with mass $2m$.

So the Gross-Neveu model has a mass gap, and it realizes dimensional transmutation, like in QCD. But how far can we push the analogy? Are the fermions confined?

To answer this question, one should look at the interaction between 2 fermions. The mediator of this interaction is the field $\sigma$, and the interaction is of Yukawa type, so I would naively expect that the interaction strength decreases exponentially fast at large distances, and that there is no confinement. Is this correct?

I cannot find the answer to this question, neither in the original paper by Gross and Neveu, nor in the literature.

If there is no confinement, then I have a subsidiary question: is $\sigma$ is a stable bound state or not? At leading order in powers of $1/N$, its mass is exactly twice the mass of the fermions, so we would need to compute the effective action at the next order to answer this question.

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There is a subtle distinction between confinement and a bosonic bound state with a mass gap.

Take for example BCS theory of superconductivity. The cooper pair is a bosonic bound state consists of spin up and spin down electrons. Does it imply all electrons are confined in cooper pairs? Not necessarily so. After Bogoliubov transformation, you can arrive at a well-defined single fermion propagator, which indicates that electrons are not totally confined.

And for that matter, practically all models of effective four-fermion interactions exhibit this sort of dynamical symmetry breaking behavior devoid of confinement. These includes NJL model and the 2D Gross-Neveu model (involving chiral symmetry breaking condensation of left- and right-handed fermion pairs) you mentioned.

A home work question: The hydrogen atom is a bosonic bound state of a proton and a electron. Does it indicate confinement of protons and electrons?

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  • $\begingroup$ Thanks for your answer! If I understand you argument correctly, confinement in the sense of my question is never realized. The reason why we don't see gluons and quarks going around freely in QCD is gauge invariance (they are not physical states) while in the Gross-Neveu model the fermions are true physical states, and therefore never completely "confined". $\endgroup$ – M.Jo Feb 14 '19 at 18:08

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