An arXiv:2608.25790v1 preprint dated 26 August 2026 lays out a practical framework for calculating fully inclusive deep-inelastic scattering, or DIS, and comparing the resulting cross sections with measurements used in parton distribution function (PDF) fits. Rather than offering a fresh experimental result, it focuses on the choices underneath those predictions: the scales used in the calculation, the treatment of heavy quarks, target-mass corrections and the highest perturbative order available.
The document presents itself as a highly personal selection rather than a review, and restricts the discussion to fully inclusive DIS in the collinear perturbative-QCD framework. Its stated aim is to make theoretical predictions and experimental measurements of an inclusive DIS cross section meaningfully comparable, with the broader goal of supporting PDF extraction.
In an illustrative HERA example, the narrative reports 1,145 measured fully inclusive DIS cross sections after kinematic cuts. The accompanying accounting is not entirely unambiguous, however: the table distinguishes points counted for the fit from the overall data count. The example therefore needs to be read as an illustration of the data-to-calculation interface, not as a single clean sample total.
The calculation is built in layers
At the core is a factorization framework. It combines a perturbatively calculable coefficient function with a universal PDF, a reusable description of the parton content used in the calculation. The notes build their comparison with data and their approach to PDF extraction around this split between the calculable part and the universal input.
Making that framework usable requires doing the convolution repeatedly, combining the coefficient function with a PDF across the relevant momentum fractions. The implementation uses an RSL representation to map a coefficient function containing mathematical distributions onto three functions suitable for convolution. It also stores partonic matrix elements in interpolation grids, allowing subsequent convolutions with arbitrary PDFs to run fast during repeated fits.
Uncertainty has a built-in blind spot
One of the document's clearest cautions concerns missing higher-order uncertainty. Varying the calculation's scales is presented as an approach that can be applied to any observable, but the notes acknowledge that the choice is inherently arbitrary and has no statistical interpretation. It is a practical convention for gauging sensitivity, not a calibrated probability statement.
That convention also has a firm structural limit. Scale-varied coefficient functions remain linear combinations of existing coefficient functions and cannot introduce new PDF channels. Scale variation can therefore probe uncertainty within the channels already present, but it cannot predict contributions from a channel absent from the baseline calculation.
Heavy quarks can change the result
Heavy-quark masses can be more than a small correction. The notes state that their effects in HERA neutral-current DIS can reach up to 20% in some kinematic regimes. They are practically irrelevant for HERA charged-current DIS, the document says, but become relevant again for low-Q2 neutrino DIS.
To manage the change across scales, the notes describe a general-mass variable-flavour-number scheme, or GM-VFNS. It is intended to approximate the massive fixed-flavour-number scheme near the heavy-quark scale and the resummed zero-mass variable-flavour-number scheme at much higher scale. For charm, the FONLL construction adds the massive and massless results and subtracts their overlap.
Target-mass corrections add another layer. They matter at large x or small Q2 and, to first approximation, shift the structure-function evaluation from measured Bjorken-x toward the Nachtmann variable. In other words, the kinematic point used in the calculation is not always treated as identical to the directly measured x.
A technical map, not a benchmark
The calculation also has a known ceiling. N3 LO is described as the highest currently available perturbative order, while massive coefficient functions still have only approximate expressions. The result is an incomplete higher-order uncertainty estimate.
All of these discussed features are implemented in Yadism, an open-source code that the text says has been used for PDF extractions from a wide range of available world data on fully inclusive DIS. The software is presented as a reusable bridge between the formulas and repeated fits. But no independent performance or fit-quality metric is supplied in the text, so the preprint does not quantify Yadism's accuracy, runtime benefit or improvement over other tools.
For readers trying to understand what this work establishes, the answer is narrower than a benchmark result. It provides a practical map of how inclusive DIS predictions are assembled and where choices enter, but it does not establish that one scale choice, flavour-number scheme or target-mass approximation is uniquely correct or superior. The work remains confined to fully inclusive DIS in collinear pQCD, as its own scope statement makes clear.
The preprint acknowledges support from Academy of Finland project 358090 and from the Center of Excellence in Quark Matter of the Academy of Finland, project 346326.
Paper data and sources
Original title: Deeply Inelastic Scattering Revisited
Authors: Felix Hekhorn
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text