Finite-N status of Wilson's fixed points below four dimensions: the four-fermion–Yukawa equivalence and summation of the $\epsilon$ expansion

Kenneth Wilson's 1973 study of the scalar $(\phi^2)^2$ and four-fermion $(\bar\psi\psi)^2$ theories in $2<d<4$ space-time dimensions — the origin of the four-fermion (Gross–Neveu-type) fixed-point construction — left explicitly acknowledged gaps: no proof that the perturbative logarithms exponentiate for arbitrary $N$ at the scale-invariant fixed point; the conjecture, verified only at $N=\infty$, that the renormalized four-fermion theory is exactly equivalent to a Yukawa theory for all $N$; uncontrolled higher orders of the $\epsilon=4-d$ expansion at $\epsilon=1,2$; prescription-based spinor bookkeeping in noninteger dimension; and the open question of lifting the construction to four dimensions. Later literature settled parts of this at the perturbative level (high-loop $\epsilon$ expansions, standard Callan–Symanzik reasoning, dimensional regularization) and clarified the noninteger-dimensional foundations negatively (such continuations are necessarily non-unitary), but a precise finite-$N$ core survives: whether the pure Gross–Neveu critical point and the Gross–Neveu–Yukawa fixed point are one and the same conformal field theory for every finite $N$; and a proof of convergence or Borel summability of the $\epsilon$ series at $\epsilon=1,2$.

Background

A scale-invariant fixed point of a quantum field theory is a continuum limit in which correlation functions transform as pure powers of distance, with anomalous dimensions — interaction-induced shifts of the scaling powers away from the engineering (classical) dimensions of the fields. Wilson (1973) studied two families of theories in space-time dimension $2\le d\le 4$ with $N$-component fields: the scalar theory with interaction $\lambda(\phi^2)^2$ and the four-fermion theory with interaction $G(\bar\psi\psi)^2$ (the two-dimensional version of the latter is now called the Gross–Neveu model; Gross and Neveu, 1974). By naive power counting both interactions are nonrenormalizable below $d=4$, so Wilson used an unconventional renormalization: the bare coupling is scaled with a power of the ultraviolet cutoff $\Lambda$ so that a dimensionless coupling is held fixed as $\Lambda\to\infty$, and the dimensionless bare coupling must then satisfy a Gell-Mann–Low-type eigenvalue condition — a fixed-point condition whose solutions define the scale-invariant theories. He solved both theories in two limiting senses: the large-$N$ limit, where chain ("bubble") diagrams dominate and can be summed, and the small-$\epsilon$ limit ($\epsilon=4-d$), where the fixed-point coupling is weak of order $\epsilon$ and observables can be computed as power series in $\epsilon$ — the $\epsilon$ expansion introduced by Wilson and Fisher (1972). At $N=\infty$ Wilson found that the renormalized four-fermion theory in $2<d<4$ coincides with a Yukawa theory, in which the fermions couple to an elementary scalar $\sigma$ through $\bar\psi\sigma\psi$ (an auxiliary or Hubbard–Stratonovich scalar reproduces the four-fermion interaction exactly when it has no kinetic term; the nontrivial statement is that a propagating scalar emerges), the composite boson becoming weakly coupled near $d=4$. The $\epsilon$ expansion computes directly in noninteger dimension $d$, where the momentum integrals, the Dirac algebra, $\gamma_5$, the antisymmetric $\epsilon$ tensor, and the trace normalization $\operatorname{Tr}I=2^{d/2}$ are defined only by prescription; Wilson fixed these rules in an appendix and left the associated convergence questions open. Within perturbation theory, the renormalization-group or Callan–Symanzik equations (the differential equations governing the dependence of renormalized Green functions on the renormalization scale) are the standard tool implying that the logarithms generated at successive orders exponentiate into anomalous dimensions at a fixed point. In modern language the finite-$N$ successor of Wilson's equivalence conjecture is the question whether the strong-coupling critical point of the pure four-fermion (Gross–Neveu) model and the Gross–Neveu–Yukawa (GNY) fixed point — which is perturbatively accessible in $4-\epsilon$ dimensions — are the same conformal field theory (a quantum field theory whose symmetry group includes arbitrary local rescalings and rotations of the coordinate system — the maximal symmetry compatible with scale invariance, and the expected symmetry at any renormalization-group fixed point), i.e. belong to one universality class (a universality class is the set of microscopic systems sharing one critical continuum limit). This identification is load-bearing across physics: the Gross–Neveu universality classes describe, for example, semimetal–Mott-insulator and symmetry-breaking transitions in condensed matter, and the same large-$N$ and $\epsilon$-expansion techniques underlie work on dynamical symmetry breaking and asymptotic safety (the scenario in which couplings remain finite at arbitrarily high energy because the renormalization-group flow approaches a nontrivial ultraviolet fixed point rather than a free theory). Independent modern determinations include the conformal bootstrap (a numerical method that bounds, and often pins down, the scaling dimensions of a conformal field theory from the consistency of its four-point functions) and lattice simulations, while constructive field theory (the rigorous mathematical construction of continuum limits) provides the natural framework for existence questions.

Problem Statement

The problem is the finite-$N$, nonperturbative status of Wilson's fixed-point construction for four-fermion theories below four dimensions, in the form of two linked questions. (A) Equivalence: Wilson proved at $N=\infty$ that the renormalized four-fermion theory $G(\bar\psi\psi)^2$ in $2<d<4$ (with his cutoff-scaled bare coupling) coincides with a Yukawa theory of a scalar $\sigma$ coupled through $\bar\psi\sigma\psi$, and conjectured the equivalence to be exact for every $N$. In modern form: is the strong-coupling critical fixed point of the pure Gross–Neveu model identical, for every finite $N$, to the Gross–Neveu–Yukawa fixed point accessible in the $4-\epsilon$ expansion — i.e., do they define one conformal field theory (one universality class) throughout Wilson's dimension range $2<d<4$ (with the integer-dimensional case $d=3$ of primary physical interest), with matching spectra of the fermionic and $O(N)$-singlet operators? The identification may be expressed as equality of the fermionic and $O(N)$-singlet conformal data, or of Wilson's renormalized amplitudes under his cutoff renormalization, while a finite-$N$ observable at which the two fixed points differ would refute it. (B) Reliability of the $\epsilon$ expansion at physical dimensions: establish, or disprove, that the $\epsilon$ series for the critical exponents and anomalous dimensions of these theories converges, or is Borel summable (recoverable from the divergent series by the standard Borel-transform summation procedure), to the actual critical theories at $\epsilon=1$ ($d=3$) and $\epsilon=2$ ($d=2$), noting the established structural fact that the noninteger-dimensional intermediate theories are necessarily non-unitary (negative-norm states, complex anomalous dimensions already at order $\epsilon$), so the summation to an integer-dimensional unitary theory requires justification that is currently missing. Of the five limitations noted above, the noninteger-dimensional foundations are settled by dimensional regularization and the exponentiation of the perturbative logarithms is established by the renormalization-group and Callan–Symanzik literature, while the four-dimensional-applications direction is an open-ended research direction rather than a well-posed question and is not among the demands here. The two questions are the surviving open core and share one objective — securing Wilson's fixed-point construction for finite $N$ beyond the two approximations (large $N$, small $\epsilon$) in which it was derived.

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