Gross–Neveu Model

The Gross–Neveu model shows how a classically massless four-fermion theory can generate a physical mass scale. The version considered here has a discrete chiral symmetry and $N$ Dirac flavors in $1+1$ dimensions. Its large-$N$ limit turns the mass-generation problem into a controlled saddle-point calculation.

Four-fermion interaction and discrete symmetry

Choose the normalization

$$\mathcal L=\sum_{i=1}^N\bar\psi_i i\gamma^\mu\partial_\mu\psi_i +\frac{g_0^2}{2N}\left(\sum_i\bar\psi_i\psi_i\right)^2 .$$

Since $[\psi]=1/2$, the four-fermion coupling $g_0$ is dimensionless in two dimensions. The discrete transformation that changes the sign of $\bar\psi_i\psi_i$ leaves its square invariant and forbids a bare mass. A continuous-chiral four-fermion model is a different theory with additional infrared issues.

Introduce a real auxiliary field $\sigma$:

$$\mathcal L_{\mathrm{aux}} =\sum_i\bar\psi_i(i\gamma^\mu\partial_\mu-\sigma)\psi_i -\frac{N\sigma^2}{2g_0^2}.$$

Its algebraic equation is $\sigma=-(g_0^2/N)\sum_i\bar\psi_i\psi_i$. Substitution gives $+g_0^2(\sum_i\bar\psi_i\psi_i)^2/(2N)$, checking both the sign and factor of two. No elementary scalar kinetic term has been added.

The large-N mass scale

For a constant Euclidean $\sigma$ and a circular momentum cutoff $\Lambda$, the leading saddle equation at $\sigma=\pm M$ is

$$\frac1{g_0^2} =2\int_{\lvert p\rvert<\Lambda} \frac{\mathrm d^2p}{(2\pi)^2}\frac1{p^2+M^2} =\frac1{2\pi}\log\left(1+\frac{\Lambda^2}{M^2}\right).$$

Consequently $M\simeq\Lambda e^{-\pi/g_0^2}$ at weak bare coupling. A logarithmic ultraviolet integral has produced an exponentially small mass. The two saddle signs are exchanged by the discrete symmetry; choosing one is a thermodynamic vacuum choice. This controlled large-$N$ construction follows Gross and Neveu 1974, §§II–IV.

Observables and qualifications

At leading order, $\lvert\sigma\rvert$ is the fermion pole mass. Scalar-channel correlations instead describe a composite bilinear, whose fluctuations acquire dynamics from fermion loops. The auxiliary field’s background is not automatically the mass of an independent elementary boson.

The large-$N$ approximation controls corrections in $1/N$; a weak bare coupling alone does not justify ignoring infrared effects. The exponential form depends on the stated coupling normalization, and its cutoff prefactor changes under a finite scheme change. The displayed saddle is at zero temperature on infinite space. A continuous-chiral variant cannot use this calculation to claim a finite-$N$ continuous condensate in two dimensions.

Develop the calculation

The Gross–Neveu Model and Dynamical Mass Generation derives the effective potential and physical channels. Compare the exact dimensionful-coupling mass in the Schwinger Model and the geometry-driven running of the O(N) Nonlinear Sigma Model.

References

• Gross, David J., and André Neveu. “Dynamical Symmetry Breaking in Asymptotically Free Field Theories.” Physical Review D 10 (1974): 3235–3253. DOI.