Controlled nonperturbative treatment of dynamical symmetry breaking in asymptotically free field theories: the unresolved remainder of the Gross-Neveu program

Gross and Neveu (1974) showed that two-dimensional fermion models with quartic interactions generate a fermion mass dynamically via the $1/N$ expansion, and posed five follow-up questions: the fate of the continuous-chiral-symmetry phase at higher $1/N$ orders, the renormalizability of the $2+\epsilon$ continuation, the extension to four-dimensional asymptotically free theories, the nonperturbative meaning of the Borel-summable essential singularities, and the accuracy of $1/N$ at finite $N$. Later literature resolved most of the two-dimensional questions: exact Bethe-ansatz solutions established that continuous chiral symmetry is unbroken at finite $N$ with a decoupled massless excitation; multi-loop, epsilon-expansion and rigorous large-$N$ constructions settled perturbative renormalizability above two dimensions; exact mass gaps and trans-series analyses partially decoded the nonperturbative structure of the $1/N$ expansion. The surviving open core is the question that motivated the source: whether a controlled, quantitatively reliable nonperturbative treatment of dynamical chiral symmetry breaking exists in four-dimensional asymptotically free gauge theories (the dynamical-Higgs mechanism underlying technicolor), where no analytic small parameter is available; together with two concrete residuals of the same program, namely the complete resurgent (trans-series) characterization of the Gross-Neveu $1/N$ expansion for general observables, and a rigorous constructive existence proof at finite $N$ above two dimensions.

Background

The Gross-Neveu (GN) model is a quantum field theory of $N$ species of massless Dirac fermions in two spacetime dimensions interacting through a quartic four-fermion coupling (plus pseudoscalar channels in the chiral version). In two dimensions the coupling is dimensionless and the model is asymptotically free (the interaction weakens at short distances). Gross and Neveu (1974) solved it in the $1/N$ expansion: keeping the combination $g^2 N$ fixed while $N$ grows, fermion bubbles (chains of fermion-loop diagrams) resum into an effective scalar propagator, and the theory is summed explicitly to all orders in the coupling. The composite scalar develops a vacuum expectation value (dynamical symmetry breaking), the fermion acquires a mass that depends on the ultraviolet cutoff exponentially rather than polynomially, and the theory is left with no free dimensionless parameter (dimensional transmutation). The original model breaks only a discrete chiral symmetry; the chiral Gross-Neveu model has a continuous chiral symmetry whose leading-order breaking would produce a massless Goldstone boson, the massless particle that must accompany spontaneous breaking of a continuous symmetry. Coleman's theorem (1973) forbids this in two dimensions: massless particles generate untameable infrared divergences, so continuous symmetries cannot be spontaneously broken there. Gross and Neveu acknowledged the tension, noting that higher orders in $1/N$ contain infrared divergences originating from massless bound states, and discussed two escape routes: continuation to $d = 2+\epsilon$ dimensions (where the interaction becomes perturbatively nonrenormalizable by power counting but a Wilson-type nontrivial renormalization-group fixed point, that is, an interacting scale-invariant theory, exists), and gauging the chiral symmetry so that the would-be Goldstone boson becomes a dynamical (composite) Higgs particle. For the $2+\epsilon$ theory they verified renormalizability at leading and next-to-leading order in $1/N$ but explicitly did not investigate higher orders, leaving renormalizability open. They also showed that next-to-leading $1/N$ corrections generate essential singularities at zero coupling of ultraviolet origin, which are Borel summable: the divergent series can be resummed through its Borel transform, though singularities of that transform (renormalons, produced by low- or high-momentum regions of Feynman integrals) can render the resummation ambiguous and require nonperturbative completion. A trans-series is the formal sum of a perturbative expansion and all its exponentially small nonperturbative corrections; resurgence is the web of relations connecting them. The thermodynamic Bethe ansatz (TBA) is an exact integral-equation method yielding ground-state energies and mass gaps in integrable models, that is, models with enough exact conservation laws that their scattering is exactly determinable. Constructive quantum field theory provides rigorous existence proofs for such models. The paper's stated motivation was four-dimensional: the only asymptotically free theories there are non-Abelian gauge theories, which do not lend themselves to any simple approximation, and whether their dynamical symmetry breaking (the dynamical-Higgs mechanism underlying technicolor, the proposal that electroweak symmetry is broken dynamically by a new strong interaction rather than by a fundamental Higgs scalar) can be computed in a controlled way has remained the driving question.

Problem Statement

The unresolved remainder of the question posed by Gross and Neveu (1974) has the objective of replacing the leading-order $1/N$ picture of dynamical symmetry breaking by fully controlled nonperturbative descriptions. Its central surviving component is: can one give a controlled, quantitatively reliable analytic nonperturbative treatment of dynamical chiral symmetry breaking in a four-dimensional asymptotically free non-Abelian gauge theory (the dynamical-Higgs mechanism underlying technicolor)? This requires a systematically improvable scheme with a demonstrable control parameter and an explicit error budget for the order parameter (the chiral condensate, the vacuum expectation value of the fermion mass operator that signals chiral symmetry breaking) and the resulting mass spectrum, validated against exact results where available and against lattice determinations (numerical simulations of the discretized theory), going beyond uncontrolled ladder truncations of the Schwinger-Dyson equations (the exact tower of integral equations for the Green's functions). Two secondary residuals of the same program also remain open, each a self-contained target in its own right: (i) the complete resurgent (trans-series) characterization of the $1/N$ expansion of the Gross-Neveu and chiral Gross-Neveu models for general observables, currently established only for conserved charges and known to involve nonperturbative corrections not decodable by standard resurgent analysis of the perturbative series; and (ii) a rigorous constructive existence proof for Gross-Neveu-type models at finite $N$ in dimensions above two, where the existing rigorous constructions hold either exactly in two dimensions or at large $N$. The open problem thus comprises the central component together with the two residuals, (i) and (ii), each an independent target. The two-dimensional finite-$N$ symmetry question, the perturbative renormalizability of the $2+\epsilon$ theory beyond next-to-leading order in $1/N$, and the Borel-summability question for the mass gap are considered resolved by the literature cited in refs. 2-15, 18-20 and 22-23 and are not part of what remains open.

References

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