The Gross–Neveu Model and Dynamical Mass Generation

The Gross–Neveu model shows how an asymptotically free four-fermion interaction in $1+1$ dimensions can generate a fermion mass without inserting one into the Lagrangian. A Hubbard–Stratonovich scalar makes the large-$N$ saddle transparent. In the discrete-chiral model, its two signs label symmetry-related vacua and the saddle value equals the leading fermion pole mass. In a continuous-chiral variant, infrared phase fluctuations invalidate a finite-$N$ condensate even though a large-$N$ amplitude scale can remain useful.

Required background. The Dirac field fixes spinor and bilinear conventions, while running couplings and dimensional transmutation supplies the RG interpretation. Helpful background. Vector models and auxiliary large-N saddles supplies determinant power counting, and Goldstone counting and low-dimensional exceptions supplies the Coleman constraint.

Discrete-chiral model and auxiliary field

Take $N$ Dirac fermions in $1+1$ dimensions with Lorentzian Lagrangian

$$\mathcal L_{\mathrm{GN}} =\bar\psi_i\,i\partial\!\!\!/\ \psi_i +\frac{g_0^2}{2N} \left(\bar\psi_i\psi_i\right)^2, \qquad i=1,\ldots,N .$$

The repeated flavor index is summed. With no bare mass, the theory is invariant under the discrete chiral transformation

$$\psi_i\longmapsto\gamma^5\psi_i, \qquad \bar\psi_i\longmapsto-\bar\psi_i\gamma^5, \qquad \bar\psi_i\psi_i\longmapsto-\bar\psi_i\psi_i.$$

Introduce a real auxiliary scalar:

$$\mathcal L_{\mathrm{aux}} =\bar\psi_i(i\partial\!\!\!/-\sigma)\psi_i -\frac{N}{2g_0^2}\sigma^2 .$$

Its algebraic equation of motion is

$$\sigma =-\frac{g_0^2}{N}\bar\psi_i\psi_i.$$

Substitution recovers the four-fermion interaction. The transformation $\sigma\mapsto-\sigma$ makes the discrete symmetry manifest. This is an exact rewriting of the regulated functional integral up to the field-independent Gaussian normalization; treating $\sigma$ as dynamical comes only after fermion fluctuations generate its momentum-dependent effective action.

Effective potential and gap equation

For a constant Euclidean $\sigma$, integrating the $N$ fermions gives

$$\frac{V_{\mathrm{eff}}(\sigma)}{N} =\frac{\sigma^2}{2g_0^2} -\int_{\lvert p\rvert<\Lambda} \frac{\mathrm d^2p}{(2\pi)^2} \ln(p^2+\sigma^2),$$

up to a $\sigma$-independent constant. For $\sigma\ne0$, stationarity gives

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

Writing the nonzero minimum as $\sigma=\pm m$,

$$m =\frac{\Lambda} {\sqrt{e^{2\pi/g_0^2}-1}} \underset{g_0^2\ll1}{\sim} \Lambda e^{-\pi/g_0^2}.$$

The two signs are exchanged by the discrete chiral symmetry. The exponent is the dimensional-transmutation result in this sharp-cutoff and coupling convention. Gross and Neveu derived the large-$N$ vacuum, asymptotic freedom, and spectrum in the original model Gross and Neveu 1974, §§ II–IV.

Define a renormalized coupling by

$$\frac{1}{g_R^2(\mu)} =\frac{1}{g_0^2} -\frac{1}{\pi}\ln\frac{\Lambda}{\mu}.$$

Then the continuum leading saddle is

$$m =\mu\exp\!\left[-\frac{\pi}{g_R^2(\mu)}\right].$$

At this order,

$$\beta(g_R) \equiv\mu\frac{\mathrm dg_R}{\mathrm d\mu} =-\frac{g_R^3}{2\pi},$$

so the displayed $m$ is RG invariant. Finite-$N$ corrections begin at subleading order in the $1/N$ expansion. A finite scheme change rescales the associated RG scale. The exponential mechanism is robust; its regulator-level prefactor is not. The gap and RG round trips are reproduced in the chapter’s benchmark.

From sigma saddle to physical channels

At a chosen minimum, the leading fermion inverse propagator is

$$S_F^{-1}(p) =p\!\!\!/-\sigma_0.$$

Therefore the discrete model has

$$M_F=\lvert\sigma_0\rvert=m \qquad (N=\infty)$$

for the fermion pole at leading order. This is a direct observable match because $\psi_i$ creates a physical global-symmetry multiplet, not a gauge-redundant excitation.

The $\sigma$ fluctuation is different. Expanding $\sigma=\sigma_0+N^{-1/2}s$ gives

$$S_{\mathrm{eff}}^{(2)} =\frac12\int\frac{\mathrm d^2p}{(2\pi)^2}\, s(-p)\, \Gamma_{\sigma\sigma}^{(2)}(p)\, s(p),$$

where $\Gamma_{\sigma\sigma}^{(2)}(p)$ is a fermion bubble plus the local auxiliary term. Its zeros and branch cuts determine the scalar-channel pole or threshold. At $N=\infty$, evaluating the bubble with the gap equation shows that the inverse auxiliary-field propagator vanishes at the two-fermion branch point

$$s_{\mathrm{thr}}=4m^2, \qquad \sqrt{s_{\mathrm{thr}}}=2m.$$

This threshold statement holds at leading order Gross and Neveu 1974, §§ III–IV. The associated branch-point singularity is not an isolated stable pole below the continuum, and its subleading analytic structure requires a separate calculation. In particular, the number $\sigma_0$ is not the mass of the $s$ fluctuation.

This is the same observable-identification test developed on Gap Equations, Dimensional Transmutation, and Physical Mass. The model’s distinct control and symmetry boundary are shown in the strong-coupling laboratory map and regime comparison.

Discrete symmetry can break in 1+1 dimensions

Coleman’s theorem excludes spontaneous breaking of a continuous internal symmetry under its standard assumptions Coleman 1973, pp. 259–264; it does not exclude a discrete symmetry. In the infinite-volume discrete Gross–Neveu model, the two minima $\sigma=\pm m$ can define distinct vacua, and kink configurations interpolate between them.

At finite spatial length, tunneling mixes the two semiclassical vacua and the exact ground state can be even under the symmetry, with a small splitting from the odd state. To define the ordered limit, add $-h\int\mathrm d^2x\,\sigma$ to the Euclidean action, so $h>0$ favors the $+m$ saddle. Then

$$\lim_{h\to0^\pm}\lim_{L\to\infty} \langle\sigma\rangle_h =\pm m.$$

In the regulated Hubbard–Stratonovich formulation, the algebraic relation gives

$$\left\langle\bar\psi_i\psi_i\right\rangle_{\mathrm{bare}} =-\frac{N}{g_0^2}\langle\sigma\rangle.$$

A renormalized bilinear requires a specified composite-operator scheme, so its numerical value should not be inserted into the order-parameter limit without that definition. Vacuum degeneracy, kink sectors, and physical mass ratios are sharper statements.

Continuous-chiral Gross–Neveu is not the same claim

The chiral Gross–Neveu or NJL$_2$ variant contains both scalar and pseudoscalar channels,

$$\mathcal L_{\mathrm{cGN}} =\bar\psi_i i\partial\!\!\!/\ \psi_i +\frac{g_0^2}{2N} \left[ (\bar\psi_i\psi_i)^2 +(\bar\psi_i i\gamma^5\psi_i)^2 \right].$$

Introduce $\sigma+i\pi=\rho e^{i\vartheta}$. A leading large-$N$ saddle can fix $\rho=m$, but the phase $\vartheta$ is the would-be Goldstone direction. At every finite $N$, its long-wavelength fluctuations prevent a nonzero continuous-chiral order parameter in infinite volume. Correlators of $\sigma+i\pi$ can show algebraic phase fluctuations rather than approach a constant.

Consequently:

• the discrete model may have two symmetry-breaking vacua and kinks;
• the continuous model cannot have a finite-$N$ continuous condensate under Coleman’s assumptions;
• a large-$N$ amplitude saddle can still organize a massive non-Abelian sector, but it does not by itself establish a simple fermion pole or a broken $U(1)$ in the exact infrared theory.

Witten’s analysis of the chiral model explains how the $1/N$ expansion and low-dimensional infrared fluctuations must be reconciled Witten 1978, §§ 2–5.

Common pitfalls

Applying Coleman’s theorem to a discrete symmetry. The theorem’s massless fluctuations arise from a continuous order-parameter direction. A $\mathbb Z_2$ symmetry can break in $1+1$ dimensions.

Using the discrete saddle for the continuous model. Adding the pseudoscalar channel creates a compact phase direction whose finite-$N$ infrared fluctuations change the symmetry claim.

Calling sigma an elementary scalar mass. $\sigma$ is introduced algebraically. Its propagating spectrum is determined by the fermion bubble and can contain a threshold rather than a pole at $\lvert\sigma_0\rvert$.

Exercises

1. Complete the square in $\sigma$ to verify the Hubbard–Stratonovich identity at the action level.

Solution

Let $B=\bar\psi_i\psi_i$. Then

$$-\sigma B-\frac{N}{2g_0^2}\sigma^2 =-\frac{N}{2g_0^2} \left( \sigma+\frac{g_0^2}{N}B \right)^2 +\frac{g_0^2}{2N}B^2.$$

The shifted Gaussian integrates to a field-independent factor, leaving the original four-fermion term.

2. Evaluate the gap integral and show that the nonzero solution has two signs.

Solution

The integral is

$$2\int_{\lvert p\rvert<\Lambda} \frac{\mathrm d^2p}{(2\pi)^2} \frac{1}{p^2+m^2} =\frac{1}{2\pi} \ln\!\left(1+\frac{\Lambda^2}{m^2}\right).$$

The stationarity equation depends only on $\sigma^2$, so both $\sigma=+m$ and $\sigma=-m$ solve it. They are exchanged by the discrete chiral transformation. The symmetric stationary point $\sigma=0$ requires separate infrared treatment and is not the large-$N$ minimum once the transmuted scale is generated.

3. Differentiate the renormalization condition to obtain the leading beta function and verify that the transmuted mass is RG invariant.

Solution

At fixed bare data,

$$\mu\frac{\mathrm d}{\mathrm d\mu}\frac1{g_R^2} =\frac1\pi.$$

Since $\mathrm d(1/g_R^2)=-2\,\mathrm dg_R/g_R^3$,

$$\beta(g_R)=-\frac{g_R^3}{2\pi}.$$

Consequently

$$\mu\frac{\mathrm d}{\mathrm d\mu} \ln\!\left(\mu e^{-\pi/g_R^2}\right) =1+\frac{2\pi}{g_R^3}\beta(g_R)=0.$$

The equality holds at the same leading large-$N$ order as the gap equation.

4. The large-$N$ saddle gives $M_F=m$ and a scalar-channel threshold at $2m$. Explain why neither statement means that $\sigma_0$ is the mass of an elementary scalar.

Solution

$\sigma$ was introduced algebraically, so it has no independent microscopic kinetic term. The background value $\sigma_0$ enters the fermion inverse propagator and therefore gives the leading physical fermion pole $M_F=m$. Fluctuations of $\sigma$ acquire their momentum dependence from a fermion bubble. Its leading inverse propagator vanishes at the two-fermion branch point $s=4m^2$, not at $s=m^2$, and that threshold singularity is not an isolated elementary-particle pole below the continuum.

Continue

The Schwinger Model, Screening, and Bosonization provides an exact mass-generation mechanism controlled by anomaly and bosonization rather than a $1/N$ saddle.

References

• Coleman, Sidney. “There Are No Goldstone Bosons in Two Dimensions.” Communications in Mathematical Physics 31 (1973): 259–264. DOI.
• Gross, David J., and André Neveu. “Dynamical Symmetry Breaking in Asymptotically Free Field Theories.” Physical Review D 10 (1974): 3235–3253. DOI.
• Witten, Edward. “Chiral Symmetry, the 1/N Expansion, and the SU(N) Thirring Model.” Nuclear Physics B 145 (1978): 110–118. DOI.