The Borel-plane structure of the pure SU(N) Yang-Mills vacuum energy beyond the b=4 renormalon cut: continuation of the Borel transform, entanglement with the instanton singularity, and the confinement argument (open problems from Parisi's 1978 'Singularities of the Borel transform in gauge theories')

In a 1978 analysis of singularities of the Borel transform in four-dimensional pure SU(N) Yang-Mills theory, G. Parisi located, by restricting the functional integral to dilatation-invariant manifolds of classical configurations, a branch cut of the Borel transform B(b) of the vacuum energy starting at b=4, produced by the infrared divergence of the integral over configuration sizes whenever the classical action exceeds S=4. He left open: the determination of B(b) for b>4 and the correct analytic continuation across the cut; the disentangling of the instanton (instanton-anti-instanton) singularity, which lies on the cut at b=2A_I with instanton action A_I=11N/3, given that the instanton size integral is itself infrared divergent; the effect of the breakdown of the Gaussian (few-modes) approximation at large size on the singularity structure; a rigorous proof of the suggested confinement argument (mass gap plus a vanishing large Wilson loop); and the extension of the picture to large N and to dimensions below four. In modern language the b=4 cut is the leading infrared (gluon-condensate) renormalon of the vacuum energy, and the questions above are the still-open parts of the nonperturbative Borel-plane structure of four-dimensional Yang-Mills theory: no first-principles determination of the Borel sum beyond the leading renormalon, no semiclassical realization of infrared renormalons on $\mathbb{R}^4$, no controlled treatment of the large-instanton sector, and no proof of a mass gap and area law.

Background

Perturbation series in quantum field theory are factorially divergent. One studies them through the Borel transform: for an observable $A(\alpha)$ expanded in a coupling $\alpha$, one writes $A(\alpha) = (1/\alpha)\int_0^\infty db\, B(b)\, e^{-b/\alpha}$, where $B(b)$ is defined term by term from the perturbative coefficients; if $B(b)$ is analytic along the positive real $b$ axis and the integral converges, the series is Borel summable. Singularities of $B(b)$ on the positive real axis (the 'Borel plane') make the integral ambiguous and signal nonperturbative physics. Two semiclassical mechanisms were identified in the 1970s. First, instantons: finite-action classical solutions of the Yang-Mills equations (the BPST, Belavin-Polyakov-Schwarz-Tyupkin, instanton, with action $A_I = 8\pi^2/g^2$, which equals $11N/3$ in the coupling normalization used here); an instanton-anti-instanton pair - one instanton together with its anti-instanton partner, of combined action $2A_I$ - generates a singularity of $B(b)$ at $b = 2A_I$ (Brezin-Parisi-Zinn-Justin; Bogomolny-Fateev). Second, renormalons: singularities generated by the running of the coupling, $\beta(\alpha) = -\alpha^2$ (so $1/\alpha(p) = 1/\alpha(\mu) - \ln(p/\mu)$, with a Landau pole at $p = \Lambda$), discovered independently by 't Hooft, Lautrup, and Olesen via Dyson-Schwinger and diagrammatic methods; the term 'renormalon' first appeared in 't Hooft's 1977 Erice lectures. Infrared (IR) renormalons sit on the positive axis and their ambiguities are cancelled, in the operator product expansion (OPE) framework, by power corrections - for the vacuum energy, by the dimension-four gluon condensate $\langle G^2\rangle$, matching an ambiguity of order $(\Lambda/\mu)^4$. Parisi's 1978 preprint (unpublished in journals; reprinted as a book chapter) analyzed the vacuum energy of pure SU(N), $N\ge 2$, Yang-Mills theory by restricting the functional integral to dilatation-invariant manifolds: classical configurations of arbitrary size $\rho$ and action $S$, with Gaussian fluctuations in the transverse directions producing a factor $(\rho\mu)^S$ and a size measure $d\rho/\rho^5$. The size integral $\int d\rho\, \rho^{-5}(\rho\mu)^S$ diverges in the infrared (large $\rho$) precisely when $S > 4$, because the running coupling grows at large size; this produces a branch cut of $B(b)$ starting at $b = 4$. Since $2A_I = 22N/3 > 4$, the instanton-anti-instanton singularity lies on top of this cut, and the instanton size integral itself is infrared divergent. Physically, Parisi concluded that the vacuum is dominated by fluctuations of action near $S \approx 4$ rather than by dilute instantons, that this picture survives the large-$N$ limit (where instantons are exponentially suppressed but the cut persists), and speculated that the resulting strong long-wavelength fluctuations confine: a mass gap (no massless colored excitations in the spectrum) and a vanishing large Wilson loop $\langle \exp\oint A_\mu dx_\mu\rangle = 0$ - the Wilson loop being the trace of the transport factor of the gauge field around a closed contour, whose decay with the enclosed area (an 'area law') is the standard diagnostic of confinement. The analysis closed with the five unresolved issues that constitute this problem. Modern terminology: the $b=4$ cut is the leading IR renormalon of the vacuum energy (in lattice language, of the plaquette, with ambiguity of order $(a\Lambda)^4$); instanton-anti-instanton singularities lie farther out, by a factor of order $N$, than the renormalon cut; 'principal-value' or 'lateral' (above/below the cut) Borel summation is the standard modern prescription for defining the Borel integral across such a cut; and controlled semiclassical realizations of IR renormalons as instanton-anti-instanton 'bions' exist on the circle-compactified $\mathbb{R}^3\times S^1$, but not on $\mathbb{R}^4$.

Problem Statement

Consider four-dimensional pure SU(N), $N\ge 2$, Yang-Mills theory, with the coupling normalized so that the beta function is $\beta(\alpha) = -\alpha^2$ (then the instanton action is $A_I = 11N/3$ and an instanton-anti-instanton pair contributes a singularity of the Borel transform at $b = 2A_I = 22N/3$), and let $A(\alpha)$ be the vacuum (ground-state) energy, represented through its Borel transform $B(b)$ by $A(\alpha) = (1/\alpha)\int_0^\infty db\, B(b)\, e^{-b/\alpha}$. The common objective of this problem - the five unresolved issues left open by the 1978 analysis - is the first-principles determination of the Borel-plane singularity structure of $A(\alpha)$ and of the physical picture of the Yang-Mills vacuum that it implies. The five items below are independent sub-goals of this common objective. Specifically, beyond establishing the branch cut starting at $b=4$ (produced by the infrared divergence of the size integral $\int d\rho\, \rho^{-5}(\rho\mu)^S$ over dilatation-invariant manifolds for classical action $S>4$), determine or rigorously decide: (1) the Borel transform $B(b)$ for $b>4$ and the correct analytic continuation across the cut; (2) how the instanton contribution is to be disentangled and computed, given that the instanton-anti-instanton singularity at $b = 2A_I > 4$ lies on the cut and the instanton size integral is itself infrared divergent; (3) whether the uncontrolled corrections to the Gaussian (few-modes) approximation at large configuration size - multi-fluctuation effects and excluded-volume effects - modify the singularity structure found at leading semiclassical order; (4) a rigorous proof of the confinement argument, namely that the theory has a mass gap and that the Wilson loop vanishes on large loops, $\langle \exp\oint A_\mu dx_\mu\rangle = 0$, as a consequence of the strong long-wavelength vacuum fluctuations; and (5) the working-out of the whole picture at large $N$ and in space-time dimensions below four.

References

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