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arxiv: 2606.30368 · v1 · pith:ANYJJWZEnew · submitted 2026-06-29 · ✦ hep-th

Positivity properties of observables in planar maximally supersymmetric Yang-Mills theory

Maximilian Haensch This is my paper

Pith reviewed 2026-06-30 05:06 UTC · model grok-4.3

classification ✦ hep-th
keywords Stieltjes propertydispersion relationsN=4 SYManomalous dimensionsWilson loopBremsstrahlung functionpositivityintegrability
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The pith

Observables in planar N=4 SYM satisfy the Stieltjes property through integral representations over positive measures.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates whether exact observables in planar N=4 super Yang-Mills, treated as functions of the 't Hooft coupling, admit once-subtracted dispersion representations with positive spectral measures. It proves this Stieltjes property analytically for the octagon anomalous dimension, the logarithm of the circular Wilson loop, the Bremsstrahlung function, and anomalous dimensions in the BMN limit by direct verification from their integral representations. Numerical evidence supports the property for the cusp and tilted cusp anomalous dimensions, while certain other quantities fail the Stieltjes test but satisfy the weaker condition of complete monotonicity. The property enables conversion of perturbative data into non-perturbative bounds and bootstrapping of series coefficients, plus recovery of strong-coupling expansions via Mellin-Barnes transforms.

Core claim

A broad class of exact observables in planar N=4 SYM possess the Stieltjes property as functions of the coupling. This is shown analytically via integral representations whose kernels or measures yield positive spectral densities, for the octagon anomalous dimension, logarithm of the circular Wilson loop, Bremsstrahlung function, and BMN-limit anomalous dimensions. Numerical checks confirm it for cusp and tilted-cusp cases. Quantities lacking the property are identified and weaker positivity is examined. The representation yields non-perturbative bounds from perturbative input, coefficient bootstrapping, and a route to strong-coupling data from weak-coupling input through Mellin-Barnes integ

What carries the argument

The Stieltjes property: the existence of a once-subtracted dispersion representation in the coupling with a positive spectral measure, verified from integral representations.

If this is right

  • Perturbative input converts directly into rigorous non-perturbative bounds on the observables.
  • Perturbative series coefficients can be bootstrapped from the positivity constraint.
  • The strong-coupling expansion including non-perturbative corrections is recoverable from the dispersion representation via a Mellin-Barnes contour integral.
  • Weak-coupling data alone can be used to estimate the strong-coupling expansion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same integral-representation technique might be applied to test positivity in other integrable gauge theories or in limits of N=4 SYM beyond those examined.
  • Failure of the Stieltjes property for specific quantities could be linked to the presence of additional branch cuts or non-integrable singularities not captured by the once-subtracted form.
  • If the positivity holds more generally, it could constrain the possible functional forms of observables in planar theories even when full integrability is absent.

Load-bearing premise

The observables possess integral representations over the coupling whose kernels permit direct verification that the associated spectral density is positive.

What would settle it

A concrete computation showing that the spectral measure extracted from the integral representation of the octagon anomalous dimension takes negative values for some range of the coupling.

read the original abstract

We study positivity properties of exact observables in planar N=4 super Yang-Mills as functions of the 't Hooft coupling. Motivated by analogous results in quantum mechanics, we ask whether such observables admit a once-subtracted dispersion representation in the coupling over a positive spectral measure. Our main result is that this property, also known as the Stieltjes property, holds for a broad class of exact observables. We prove it analytically, through integral representations, for the octagon anomalous dimension, the logarithm of the circular Wilson loop, the Bremsstrahlung function, and anomalous dimensions in the BMN limit, and we provide numerical evidence for the cusp and tilted cusp anomalous dimensions. We also identify quantities for which the Stieltjes property does not hold, and study the weaker positivity property of complete monotonicity. The Stieltjes property yields two powerful consequences: it lets us turn perturbative input into rigorous non-perturbative bounds, and bootstrap perturbative coefficients. We also show how the strong-coupling expansion and its non-perturbative corrections can be recovered from the once-subtracted dispersion representation via a Mellin-Barnes representation and outline a method to estimate the strong-coupling expansion from weak-coupling data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript claims that a broad class of exact observables in planar N=4 SYM satisfy the Stieltjes property (once-subtracted dispersion relation with positive spectral measure in the 't Hooft coupling). This is proven analytically via integral representations for the octagon anomalous dimension, logarithm of the circular Wilson loop, Bremsstrahlung function, and BMN anomalous dimensions; numerical evidence is supplied for the cusp and tilted cusp anomalous dimensions, counterexamples are identified where the property fails, and the weaker complete monotonicity property is studied. Consequences include non-perturbative bounds from perturbative data, bootstrapping of coefficients, and recovery of strong-coupling expansions via Mellin-Barnes representations.

Significance. If the central claims hold, the work is significant because it establishes positivity properties that convert perturbative series into rigorous non-perturbative bounds and enable coefficient bootstrapping in planar N=4 SYM. The analytic proofs rest on explicit integral representations whose positivity is directly verifiable, and the paper supplies both these representations and explicit counterexamples. The outlined method for estimating strong-coupling data from weak-coupling input is a concrete practical contribution.

minor comments (2)
  1. [Abstract] Abstract: the statement that numerical evidence is provided for the cusp anomalous dimensions would be strengthened by indicating the perturbative orders employed and the range of coupling values tested.
  2. The discussion of complete monotonicity for cases where the Stieltjes property fails would benefit from a short comparative table or explicit examples to clarify the distinction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, their assessment of its significance, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via independent integral representations

full rationale

The paper proves the Stieltjes property analytically by supplying explicit integral representations over the 't Hooft coupling for the octagon anomalous dimension, log of the circular Wilson loop, Bremsstrahlung function, and BMN anomalous dimensions, then directly verifying positivity of the spectral measures from the kernels. These steps are independent of fitted parameters or prior self-citations; the manuscript also supplies numerical evidence for other cases and explicit counterexamples where the property fails. The derived consequences (non-perturbative bounds, coefficient bootstrapping, strong-coupling recovery via Mellin-Barnes) follow from the established positivity without reducing to the input representations by construction. No load-bearing step collapses to a self-definition or fitted input.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of suitable integral representations for the observables that allow positivity of the spectral measure to be verified; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The listed observables possess integral representations over the coupling whose kernels permit direct positivity analysis of the spectral measure.
    Invoked to prove the once-subtracted dispersion representation with positive measure.

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