An arXiv preprint reports a sharp split in a high-energy gravitational-scattering calculation. Its full expansion in the rescaled coupling alpha1 has zero radius of convergence, meaning the power series has no nonzero range around its starting point in which it converges as written. Yet the unexpanded integral representation is finite for every positive real value of alpha1. The paper also gives analytic evidence that the divergent series is Borel summable, a separate summation procedure examined as a route to a finite result.
That finding applies to a particular rescaling, not automatically to the usual expansion in G. In the high-energy setup, the couplings are defined as alpha_i = Gs/(m_i b), combining G, the large variable s, each particle's mass m_i and the impact parameter b. The alpha_i are held fixed while s becomes large, and the calculation is expanded in powers of m2/s.
Within that setup, the modeled outcome is narrow: the paper analyzes the minus component of the momentum kick, Delta p-1, received by particle 1. Other components are described as analogous rather than explicitly analyzed. The stated research question is whether the post-Minkowskian expansion converges or diverges in this regime.
A result tied to one scaling limit
To test the series, the authors start with a convergent Taylor expansion of sin-squared(zu/2) and then integrate the resulting terms one by one. For the transverse master integral, they use dimensional regularization away from two dimensions or replace the logarithm with a power limit. This reduces the integral to a massless bubble integral, the form used for the subsequent coefficient analysis.
That route leads to the central mathematical result: the full alpha1 expansion has zero radius of convergence. The authors describe it as asymptotic and then examine whether its information can be handled through Borel summation. The Borel transform has reported branch-cut endpoints at t = plus or minus 2i.
Those endpoints correspond to a reported Borel radius of length two, within which the Borel sum converges. The paper identifies that Borel sum with the integral I(alpha1), but says the identification is a priori valid only when alpha1 is below two. Separately, the integral is finite for every alpha1 on the positive real axis. That last fact does not by itself establish physical validity beyond the stated boundary.
What the coefficients reveal
Beyond convergence, the paper tracks the structure of its coefficients. Transcendentality is used here as a label for the mathematical constants appearing in a term: the nth expansion term has maximum transcendentality 2n - 1, and the leading zeta term has a nonzero coefficient for every n. This is a statement about the formula's all-order pattern.
Against a cited expansion carried to eleventh order, the authors report agreement in the specified X2n terms except for one correction. At ninth order in alpha1, one coefficient is reported as 630 rather than 640. The material does not independently resolve that discrepancy, so the change is best described as a reported correction rather than a settled error in the earlier work.
The authors also report that the approximation passed tests against existing fixed-order post-Minkowskian calculations expanding in G. That check supports the construction at the orders tested, but it does not widen the result's scope: the calculation remains a high-energy, rescaled-coupling limit focused explicitly on one momentum-kick component, while the others are only described as analogous.
Where the result stops
The authors interpret the combination of an asymptotic series, evidence for Borel summability and a finite positive-axis integral as suggestive of a regime change around alpha1 = 1. They also leave open possible non-perturbative terms of the form exp(-A/alpha1). Both ideas are presented as possibilities, not as established physical results.
Several boundaries remain. The analysis does not establish physical validity of the finite integral for every positive alpha1, does not show that the possible non-perturbative terms exist, and does not provide strong-coupling predictions for alpha1 at or above one. It also leaves open whether the same convergence and Borel properties apply to the other momentum-kick components or to broader post-Minkowskian sectors.
The document is an arXiv preprint identified as arXiv:2608.19941v1 [hep-th] and dated 20 August 2026. Its result is therefore best read narrowly: a divergent alpha1 expansion is analytically connected to a finite positive-axis integral when alpha1 is below two, while the integral's behavior outside that range remains a question of physical interpretation.
Paper data and sources
Original title: A Note on a High-Energy Limit of Gravitational Perturbation Theory
Authors: Poul H. Damgaard, Ludovic Plante
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text