A preprint study of low-x proton structure reports a lower overall fit statistic for the model that includes absorptive corrections, the paper’s screening treatment in the evolution of parton distributions. In the combined deep-inelastic-scattering and exclusive-production fit, global χ² per degree of freedom was 1.25 for linear evolution and 1.18 for non-linear evolution.
The work is a computational comparison of fitted PDF variants. It asks how absorptive corrections and exclusive J/ψ and Υ data affect low-x PDFs at low and moderate scales, while the NLO fits compare PDFs with and without non-linear corrections and select R from the minimum of a fit-quality profile.
What was compared
For NLO, the listed input set contained 1,181 points, including 10 exclusive J/ψ points at 13 TeV and three exclusive Υ points at 7 and 8 TeV. The exclusive measurements therefore made up a small part of the overall set.
Non-linear DGLAP evolution was implemented in APFEL++ by extending its native x-space evolution machinery. The NLO comparison then used the linear and non-linear PDF variants while choosing R from the fit-quality profile’s minimum.
For DIS-only fits, the reported total χ² per degree of freedom was 1.20 with linear evolution and 1.18 with non-linear evolution. The lower non-linear value was the same as the global value reported for the combined fit.
A different low-x gluon picture
Beyond fit quality, the paper reports a larger low-x input gluon distribution with absorptive corrections over 10⁻⁵ < x < 0.1. The two predictions were closer at a higher evolution scale: at Q² = 22 GeV², the linear result was about 10% higher, while at Q² = mZ² the difference was about 1% to 3%.
In the combined comparison, the J/ψ data were reported at χ² per degree of freedom of 12.6/10 and the Υ data at 2.9/3. Those figures correspond to the 10 J/ψ and three Υ points listed in the NLO input set.
A model-derived spatial scale
The fitted transverse parameter R was also translated into a hot-spot scale. Starting from R = 3.6 GeV⁻¹ and assuming three hot spots, the calculation gave a derived hot-spot radius of 2.57 GeV⁻¹.
That radius is a model-derived output of the stated conversion, not a separate direct measurement. The NLO procedure selected R from the minimum of a fit-quality profile, so the spatial scale is tied to the fit setup.
An approximate NNLO check
The study also tested an approximate NNLO* treatment. It rescaled the effective gluon points with an approximately constant K-factor of K = 1.2 ± 0.1 and treated that uncertainty as a correlated systematic across the points.
Without the optional A′g component, the NNLO* non-linear fit reported χ² = 1,416 for 1,166 degrees of freedom, against χ² = 1,495 when non-linear effects were neglected. The best-fit R was about 3.3 GeV⁻¹, compared with about 3.6 GeV⁻¹ at NLO.
Adding A′g under non-linear evolution changed total χ² only from 1,416 to 1,412, which the analysis describes as a marginal improvement in fit quality.
What the comparison does and does not show
Taken together, the reported comparisons point to a better numerical fit for the non-linear variant when these exclusive points are included. They do not establish a causal physical effect: the analysis compares fitted evolution schemes against selected collider data.
The NNLO* result should likewise be read as an approximate fit test using the stated K-factor. The authors say the non-linear evolution implementation and the corresponding PDF sets will be made publicly available through xFitter and in LHAPDF format. The manuscript is arXiv:2608.25724v1, dated 26 August 2026.
Paper data and sources
Original title: Low-$x$ parton densities accounting for absorptive effects and exclusive $J/ψ$ and $Υ$ data from the LHC
Authors: V. Bertone, C. A. Flett, A. D. Martin, M. G. Ryskin
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text