A new arXiv version-1 preprint dated 26 August 2026 presents a framework for deep-inelastic scattering in the threshold region x→1. It works beyond leading power in 1−x at leading twist and focuses on an off-diagonal channel that begins at next-to-leading power.
This is a formal analytical calculation rather than an empirical study. It uses position-space soft-collinear effective theory, or SCET, to organize the DIS hadronic tensor and its factorized component functions.
Two routes into the endpoint
The central off-diagonal contribution can arise through two mechanisms in the x→1 limit. One is an anti-hardcollinear two-parton final-state process. The other is a soft-collinear process involving a quark-to-gluon transition and the remnant.
The endpoint factorization identifies the next-to-leading-power PDF as a two-scale object. Its factorized hadronic tensor contains anti-hardcollinear current contributions and soft-collinear Lagrangian contributions, but their convolutions develop endpoint divergences. The calculation therefore has to account for overlapping contributions before the separate pieces can be combined consistently.
At large x, the next-to-leading-power gluon distribution is written in terms of hardcollinear matching coefficients and a subtracted soft-collinear function. Soft rearrangement gives that function a well-defined ultraviolet anomalous dimension, but its initial condition remains non-perturbative.
Why the endpoint is difficult
At the edge of the anti-hardcollinear convolution, the composite-operator jet contribution diverges at r = 0 in four dimensions. As a result, the hard and jet functions cannot be renormalized separately before integration. A corresponding large-ω endpoint in the soft-collinear sector is identified as an overlap problem linked to the anti-hardcollinear r→0 endpoint.
The study sets out two treatments for these endpoint terms. In the d-dimensional MS approach, endpoint poles cancel when the relevant convolutions are combined. In the refactorization approach, the endpoint convolutions are rendered finite for the component functions.
The refactorization scale Λ cancels from the final result, while allowing resummed logarithmic terms to be redistributed between the A0 and B1 contributions. The allocation of those terms therefore depends on the chosen factorization convention.
The difference between the two PDF definitions is captured by a calculable conversion coefficient whose large logarithms can be summed. The paper establishes an all-order relation between the conventional MS and endpoint or refactorization schemes.
A resummation with defined limits
The calculation produces a double-logarithmic, leading-logarithmic resummation of the large-x terms in the off-diagonal DGLAP splitting function. The result sums those leading-logarithmic contributions to all orders in αs.
The authors also compute the perturbative ingredients needed for next-to-leading-power leading-logarithmic resummation, including the composite-operator jet, radiative jet, next-to-leading-power soft-collinear function and its anomalous dimension.
At leading power, the threshold quark and gluon PDFs coincide with the ξ→1 limit of ordinary MS PDFs, and MS renormalization reproduces the DGLAP kernel. That consistency result does not establish the same equivalence for all next-to-leading-power contributions.
What remains to be done
The explicit analysis is limited to the off-diagonal next-to-leading-power channel. The non-perturbative starting condition of the soft-collinear function is not numerically specified, leaving an essential input for any numerical use of the factorized gluon distribution.
The resummation reaches leading-logarithmic, or double-logarithmic, accuracy. Next-to-leading-logarithmic work remains open, so the preprint presents an analytic framework and scheme relation rather than a completed phenomenological treatment.
Paper data and sources
Original title: Next-to-leading power endpoint DIS and PDF factorization
Authors: M. Beneke, M. Schnubel, R. Szafron
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text