Non-perturbative definition of Green's functions from the non-Borel-summable perturbation series of non-asymptotically-free quantum field theories (the ultraviolet renormalon problem)

The perturbation series of a renormalizable quantum field theory with positive one-loop beta-function coefficient -- a non-asymptotically-free theory such as four-dimensional $O(N)$ $\phi^4$ or quantum electrodynamics -- diverges factorially, and its Borel transform carries ultraviolet renormalon singularities on the positive real axis. These singularities, whose positions ($b\beta_2=2,4,\dots$) and exponents are fixed by renormalization-group arguments (Parisi 1978-1979), make Borel summation ambiguous by terms of order $\exp(-2/(\beta_2 g))$, so the perturbative series alone does not define the theory's Green's functions non-perturbatively. Parisi's 1979 Les Houches lecture chapter posed the problem: how can the non-perturbative Green's functions be defined in this situation, and what is the full analytic structure of the Borel plane, including the interplay between renormalon and instanton singularities? Candidate routes discussed in the secondary literature include generalized summation, analytic continuation around the cuts, and renormalization-group arguments -- notably Parisi's own documented proposal to remove the renormalon ambiguities through exponentially small local composite-operator counterterms. Subsequent work proved parts of the singularity structure (factorization into composite-operator insertions; rigorous bounds on the Borel radius), closed only the scalar continuum-limit reading negatively (triviality of four-dimensional $\phi^4$), and produced candidate frameworks (Ecalle resurgent transseries, non-Wilsonian ultraviolet completion), but no accepted definition or impossibility theorem exists for QED-like theories or cutoff scalar theories. The problem remains open in the literature up to 2026: its precise core is to define the non-perturbative Green's functions of a non-asymptotically-free theory from its perturbative data, and to settle the disputed ultraviolet-renormalon analytic structure.

Background

The setting. In a renormalizable quantum field theory (QFT) the $n$-point Green's functions $G(x_1,\dots,x_n)$ -- vacuum expectation values of products of fields -- are the basic objects from which physical quantities are derived. At weak coupling $g$ they are computed as formal power series in $g$ (perturbation theory). These series are generically divergent, with factorially growing coefficients, so they carry no direct numerical definition of the theory.

Borel transform and Borel summation. For a series $\sum_K a_K g^{K+1}$ one defines the Borel transform $B(b)=\sum_K a_K b^K/K!$; the factorial growth is divided out and $B(b)$ typically has a finite radius of convergence. If $B(b)$ is analytic along the positive real axis and of at most exponential growth, the Borel sum $G(g)=\int_0^\infty e^{-b/g} B(b)\,db$ reconstructs a genuine function from the divergent series. The obstruction studied here is a singularity of $B(b)$ on the positive real axis, i.e. on the integration contour: the integral becomes ambiguous, different contour choices differ by terms of order $e^{-b_0/g}$ (small beyond all orders of perturbation theory), and the series alone no longer defines the Green's functions.

Renormalization group. The beta function $\beta(g)=\mu\,dg/d\mu$ describes how the renormalized coupling runs with the energy scale $\mu$; write $\beta(g)=\beta_2 g^2+\beta_3 g^3+\cdots$ in the normalization of the source. A theory is asymptotically free (AF) when the coupling decreases at short distance, which in this convention corresponds to $\beta_2<0$ (as in QCD); non-AF renormalizable theories -- four-dimensional $\phi^4$-type models and QED -- have $\beta_2>0$: the coupling grows at short distance (the Landau-pole problem). The Borel transform of a Green's function inherits renormalization-group (RG) constraints: at large Borel variable $b$ and large momentum $p$, $C(p,b)\sim r(b)\,p^{d}\,(p/\mu)^{\beta_2 b}$, and consistency of this scaling forces singularities at $b\beta_2=2,4,\dots$.

Renormalons. These RG-fixed singularities, called ultraviolet (UV) renormalons, were introduced by 't Hooft (1979) for QCD and analyzed by Parisi (1978, 1979): they arise from the factorial growth of individual renormalized diagrams with nested ultraviolet momentum integrations, rather than from the diagram ensemble as a whole. Near the first singularity the Borel transform behaves as $B(b)\sim \Gamma(1-\alpha)\,(2-\beta_2 b)^{\alpha-1}$ with exponent $\alpha=2\beta_3/\beta_2^2+\gamma^1/\beta_2$, built from the two- and three-loop beta coefficients and the anomalous dimension $\gamma^1$ of the associated local composite operator (a dimension-6 $\phi^6$-type operator in $\phi^4$ theory); correspondingly the large orders behave as $a_K\sim K!\,K^{-1+\alpha}(\beta_2/2)^K$. Parisi conjectured that renormalon contributions factorize into insertions of such local operators, i.e. act like exponentially small counterterms of order $e^{-2/(\beta_2 g)}$. In AF theories the UV renormalons lie on the negative real axis (sign-alternating, Borel-summable series) while infrared renormalons lie on the positive axis; in non-AF theories the UV renormalons move to the positive axis and obstruct Borel summation itself -- the situation of this problem.

Instantons. The second mechanism of factorial growth is given by the saddle points (instantons) of the Euclidean functional integral (the Lipatov method, presented in the companion chapter by Zinn-Justin in the same lecture collection): $a_K\sim K!\,A^{-K}$ with $A$ the instanton action, equal to $16\pi^2$ in the normalization of the source. For the $O(N)$-symmetric $\phi^4$ theory ($N$ = number of real scalar field components, interaction $g(\sum_i \phi_i^2)^2$) the instanton singularity lies at $b_I=-16\pi^2$ and the first UV renormalon at $b_u=2/\beta_2=96\pi^2/(N+8)$, with $\beta_2=(N+8)/(48\pi^2)$. The renormalon is closer to the origin for $N>-2$ (in particular for all physical $N\ge 1$) and then dominates the large-order growth; for $N<-14$ (a formal analytic continuation in $N$) the instanton singularity would sit on the UV renormalon cut.

Candidate frameworks and obstructions. Resurgence and transseries: Ecalle's theory of resurgent functions studies formal expansions of the form $\sum_n \sigma^n e^{-n b_0/g} g^{\beta n}\Phi_n(g)$ whose lateral Borel sums have controlled discontinuities, offering a generalized (Borel-Ecalle) summation that can define functions even where ordinary Borel summation is ambiguous; this is the modern candidate tool for the present problem. Triviality: rigorous constructive results (Aizenman; Lüscher and Weisz; Aizenman and Duminil-Copin) show that four-dimensional $\phi^4$ theory has no interacting continuum scaling limit, so for scalar theories the target Green's functions must be understood at finite ultraviolet cutoff; for QED-like gauge theories no corresponding theorem exists and the Landau ghost remains a conjecture.

Problem Statement

Let $T$ be a renormalizable, non-asymptotically-free quantum field theory, i.e. one for which the coefficient of $g^2$ in the beta function $\beta(g)=\beta_2 g^2+O(g^3)$ is positive, so that the coupling grows at short distance; the standard examples are four-dimensional $O(N)$ $\phi^4$ theory (at fixed ultraviolet cutoff, given its trivial continuum limit) and quantum electrodynamics. Renormalization-group arguments (Parisi 1978-1979) place the ultraviolet renormalon singularities of the Borel transform of the renormalized Green's functions of $T$ on the positive real axis at $b\beta_2=2,4,\dots$, of the form $B(b)\sim(2-\beta_2 b)^{\alpha-1}$ with $\alpha=2\beta_3/\beta_2^2+\gamma^1/\beta_2$, where $\gamma^1$ is the leading anomalous dimension of the local composite operator associated with the first singularity; when present, these singularities make the perturbation series not Borel-summable, the ambiguity being of order $\exp(-2/(\beta_2 g))$.

(1) Definition facet: either construct the non-perturbative Green's functions of $T$ from its perturbative expansion by any principled definition principle -- such that the construction is mathematically well defined, reproduces the perturbative series as its asymptotic expansion, is consistent with the renormalization group (including the anomalous dimensions entering the exponent $\alpha$), and does not rest on an arbitrary choice of contour, lateral direction, or resummation prescription -- or else prove that no principled definition of this kind exists (an impossibility/no-go theorem); a constructive definition meeting the stated criteria and a proof of impossibility are both admissible resolutions of this facet.

(2) Analytic-structure facet: determine the analytic structure of the Borel transform of the exact renormalized series of $T$ -- the positions, types, and exponents of all singularities (renormalon, instanton, and any others) and their interplay, including the regime (formal $N<-14$ in the $O(N)$ model) where the instanton singularity at $b_I=-16\pi^2$ lies on the ultraviolet renormalon cut -- and in particular settle whether the positive-axis ultraviolet renormalon singularity is present in the exact $\phi^4_4$ series, given the competing analyticity claims in the literature.

A resolution of the problem requires both facets, which are independent parts of the question. The analytic-structure facet is resolved only when each of its independently refutable sub-claims is settled -- (i) the positions, types and exponents of every Borel-plane singularity (renormalon, instanton, and any others); (ii) their interplay, including the formal $N<-14$ regime in which the instanton singularity at $b_I=-16\pi^2$ lies on the ultraviolet renormalon cut; and (iii) whether the positive-axis ultraviolet renormalon singularity is present in the exact $\phi^4_4$ series -- and settling only some of them (for example the singularity positions without the $\phi^4_4$ presence question) leaves both that facet and the problem open.

The infrared renormalon problem of asymptotically free theories (Parisi's dual 'future work' direction, since developed into the QCD power-corrections program) is a different, developed problem and is not part of this one; likewise, renormalon analyses in two-dimensional sigma models and in large-$N_f$ limits supply evidence, not closure.

References

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