Unitary existence of the large-N resummation of non-renormalizable interactions: the O(N) quartic theory in $4<d<6$, its borderline dimensions, and four-dimensional non-renormalizable theories

In 1975 G. Parisi showed that non-renormalizable interactions can become renormalizable outside ordinary perturbation theory: in the O(N)-symmetric scalar theory with quartic interaction, introducing an auxiliary (Hubbard-Stratonovich) field and resumming the bubble chains that dominate at large N (with $g=\lambda N$ fixed) yields an improved propagator that makes every order of the $1/N$ expansion renormalizable in $4<D<6$ dimensions, at the cost of Green functions that are not $C^{\infty}$ at $g=0$ and of a non-Gaussian ultraviolet fixed point. The construction, however, is available only for the sign of the coupling that makes the classical potential unbounded below, so no positive Hamiltonian is known; the borderline dimensions (quartic $d\to6$, trilinear and four-fermion $d\to4$) are excluded by a quadratic divergence and violate the Kallen-Lehmann representation; and no rigorous existence proof was given. Fifty years of later work -- including the modern revival of the $4<d<6$ O(N) model through the $6-\epsilon$ expansion, the conformal bootstrap, and large-charge methods, which established the formal renormalizability and quantified the metastability through instanton-induced imaginary parts of order $e^{-Nf(d)}$ -- leave a surviving open core: whether the resummed theory, or its conjectured cubic-model completion, defines a genuinely unitary continuum quantum field theory (positive Hamiltonian, Kallen-Lehmann-positive propagators) at finite N, with its fate in the borderline dimensions, and the question whether genuinely four-dimensional non-renormalizable interactions admit a predictive, local large-N resummation, as subordinate questions of the same program.

Background

In perturbation theory, a quantum field theory is called renormalizable if all ultraviolet divergences can be absorbed into redefinitions of finitely many parameters; otherwise it is non-renormalizable, and removing the cutoff requires infinitely many free counterterms, so naive perturbation theory leaves no continuum prediction. The quartic O(N)-invariant interaction $\lambda(\phi_i\phi_i)^2$ of N real scalar fields is renormalizable only for spacetime dimension $d\le4$; four-fermion interactions $(\bar\psi\psi)^2$ only for $d\le2$. In the 1970s this was a central concern for the weak interactions, then described by the non-renormalizable four-fermion (Fermi) interaction.

Parisi's 1975 construction proceeds as follows. For the O(N) scalar theory one introduces an auxiliary field $\sigma$ (Hubbard-Stratonovich transformation), a formal field with momentum-independent bare propagator coupled through the trilinear vertex $\sigma\phi_i\phi_i$, so that integrating $\sigma$ out reproduces the quartic interaction. At large N with $g=\lambda N$ held fixed, chains of one-loop bubbles dominate; summing them gives the improved $\sigma$ propagator $D(K)=1/(1+g\Pi(K))$, where $\Pi(K)$ is the standard one-loop bubble integral of two scalar propagators. This resummation is non-analytic in $g$, and in the window $4<D<6$ every order of the resulting $1/N$ expansion contains only finitely many ultraviolet-divergent integrals, so the theory becomes renormalizable in the usual sense after three subtractions. Green functions are finite but not $C^{\infty}$ at $g=0$: they contain fractional powers $g^{2/(D-4)}$ (and logarithms at special dimensions), which is why the Taylor expansion underlying naive perturbation theory fails. The resummed theory possesses an ultraviolet-stable zero $g_U$ of the Callan-Symanzik $\beta$-function -- an early instance of what is now called asymptotic safety: predictivity rescued by a non-Gaussian (interacting) UV fixed point rather than by perturbative renormalizability. The analogous construction for current-current four-fermion interactions works in $2<D<4$, with case-by-case criteria depending on whether the currents are conserved. Parisi's stated limitations, which define this problem, are: (i) for $D>4$ the resummed $D(K)$ avoids a spacelike ghost pole only for the sign of $g$ that makes the potential unbounded below, so the theory appears to lack a positive self-adjoint Hamiltonian; (ii) the method fails at the window edges (quartic $D\to6$; trilinear/four-fermion $D\to4$), where the bubble $\Pi(K)$ is quadratically divergent and the improved propagator violates the Kallen-Lehmann representation (the statement that a physical two-point function is a positive superposition of free ones); (iii) the realistic four-dimensional non-renormalizable interactions that motivated the work are outside the window; (iv) the resummation shields divergences at the propagator level, and more general interactions may require vertex-level resummations; (v) no rigorous existence proof was given.

Three strands of later work frame the modern problem. First, on the stable (positive-coupling) side of the quartic theory, Aizenman proved in 1981 that for $d\ge5$ the continuum limit is Gaussian (triviality) -- so the wrong sign is not a removable defect but the essential price of the construction, and the existence question cannot be bypassed by flipping the sign. Second, on the rigorous side, Gawedzki and Kupiainen constructed in 1985, via a rigorous Wilson renormalization-group analysis, the continuum limit of a perturbatively non-renormalizable fermionic model (a Gross-Neveu variant -- the N-flavor four-fermion model with interaction $(\bar\psi\psi)^2$ -- with bare propagator $1/|p|^{2-\epsilon}$) governed by a non-Gaussian ultraviolet fixed point -- proving that non-renormalizable theories with non-Gaussian UV fixed points can exist, but only for that fermionic model class. Third, since 2014 the $4<d<6$ O(N) model has been revived: the critical theory in $d=6-\epsilon$ is described by a cubic model of N+1 massless scalars with interactions $\sigma\phi_i\phi_i+\sigma^3$ (real fixed point only for $N>N_{\rm crit}\approx1038$ at one loop, with the three-loop analysis giving $N_{\rm crit}(\epsilon)=1038.266-609.840\epsilon-364.173\epsilon^2+O(\epsilon^3)$, so that the critical N drops as $d$ decreases from 6), analyzed also by the conformal bootstrap (numerical implementation of the consistency constraints of crossing symmetry and unitarity on conformal correlators), by exact renormalization-group methods (functional flow equations for the full Wilsonian effective action), and by the large-charge expansion (an expansion in the eigenvalue of a selected operator under a fixed O(N) rotation). Giombi, Huang, Klebanov, Pufu and Tarnopolsky showed in 2019-2020 that the formally unitary $4<d<6$ fixed point acquires, at large N, imaginary parts of order $e^{-Nf(d)}$ induced by instantons (finite-action saddle points of the Euclidean path integral that dominate the decay of an unstable vacuum), where $f(d)$ equals the sphere free energy of a conformal scalar in $d-2$ dimensions: the fixed point is metastable and is best regarded as a complex conformal field theory (a formal CFT whose couplings and observables carry small imaginary parts). The function $f(d)$ diverges at both window edges, $f(6-\epsilon)\sim1/(90\epsilon)$ and $f(4+\epsilon)\sim1/(3\epsilon)$ -- there the complex CFT becomes a free theory and the instanton effects are most suppressed -- while the suppression is weakest in the interior of the window ($f(5)\approx0.0638$); the resummation itself degenerates at the edges for the independent reason that the bubble $\Pi(K)$ becomes quadratically divergent and spectral positivity fails there. In parallel, solvable or partially solvable large-N four-dimensional models (Romatschke's PT-symmetric continuation -- analytic continuation of the coupling through a complex contour that keeps correlation functions real and finite; exactly solvable fermion models; four-fermion theories extended by higher-derivative interactions) and Kazakov's self-consistent S-matrix program have explored the $d=4$ borderline and higher dimensions without constructing the plain four-fermion or borderline theories as local continuum QFTs.

Problem Statement

Let $\phi_i$, $i=1,\dots,N$, be real scalar fields in $d$ Euclidean dimensions with the O(N)-invariant interaction $\frac{\lambda}{8}(\phi_i\phi_i)^2$, and hold $g=\lambda N$ fixed as $N\to\infty$. Introduce the Hubbard-Stratonovich field $\sigma$ and sum the bubble chains to obtain the improved propagator $D(K)=1/(1+g\Pi(K))$, where $\Pi(K)$ is the one-loop bubble integral of two scalar propagators; for $4<d<6$ this yields, order by order in $1/N$, a renormalizable theory with a non-Gaussian ultraviolet fixed point, but the required sign of $g$ makes the classical potential unbounded below. The problem is to settle the existence and unitary status of this resummation. Concretely, the core question is: for any fixed finite $N$ and $d$ in the window $4<d<6$ (with $d=5$ the primary integer case), determine whether the resummed fixed point corresponds to a genuine unitary continuum quantum field theory -- either by constructing a continuum quantum field theory (in the constructive sense: a limit of cutoff theories satisfying Osterwalder-Schrader reflection positivity, the Euclidean positivity condition that guarantees a unitary Lorentzian continuation; the two-point functions then necessarily admit a Kallen-Lehmann representation with non-negative spectral density, but this is a necessary condition only and is not by itself a construction) whose large-N expansion reproduces the bubble-resummed fixed point, or by proving rigorously that no such theory exists (for example, that the fixed point is necessarily complex for all finite N, or that it exists only above a critical N below which the interacting fixed point disappears). An answer may be $N$-dependent and $d$-dependent, but it must state explicitly the regimes of $N$ and $d$ for which the construction or the no-go is established; a result at a single dimension or for a single $N$ settles only that case. The core question also includes whether the conjectured cubic completion ($\sigma\phi_i\phi_i+\sigma^3$ with a positive potential, or an equivalent construction) defines a genuine unitary continuum theory in $4<d<6$ and whether its equivalence to the resummed quartic theory holds beyond the overlapping-expansion evidence (large N, $6-\epsilon$, large charge).

Two further questions belong to the problem's context -- they correspond to limitations stated in the original construction -- and are optional extensions rather than required parts of a complete answer: settling either one alone does not answer the core existence/unitarity question. First, the borderline dimensions: whether any predictive local continuum limit exists for the quartic theory as $d\to6$ or for the trilinear/four-fermion analogues at $d=4$, where the bubble integral is quadratically divergent and the improved propagator violates the Kallen-Lehmann representation. Here the instanton exponent $f(d)$ (the sphere free energy of a conformal scalar in $d-2$ dimensions, controlling imaginary parts of order $e^{-Nf(d)}$) diverges at both edges, $f(6-\epsilon)\sim1/(90\epsilon)$ and $f(4+\epsilon)\sim1/(3\epsilon)$, so the obstruction at the borders is the divergence structure and spectral positivity rather than unsuppressed instantons; the instanton imaginary parts are least suppressed in the interior of the window ($f(5)\approx0.0638$), where at finite N they raise the separate question of whether they can be resummed into a real, well-defined theory. Second, the physically motivating case of genuinely four-dimensional non-renormalizable interactions: whether any large-N or fixed-point resummation can yield a local, predictive, unitary continuum theory, given that the leading large-N limit of the plain 4D four-fermion theory is nonlocal and that current 4D constructions modify the theory (complex couplings, higher-derivative extensions). In all cases, a proposed answer must address the theory as posed rather than a modified model class. The scientific goal is deciding whether Parisi's mechanism turns the non-renormalizable $4<d<6$ quartic interaction into a genuine unitary continuum quantum field theory, with the level of rigor (constructive proof or impossibility result) clearly identified.

References

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