A preprint reports that pion and kaon valence-quark distributions in a covariant NJL calculation showed stronger dependence on momentum transfer than on skewness. The contrast was strongest in the lower parts of the modeled distributions, across x values from 0 to 0.8 for kaons and from 0 to 0.7 for pions.
The calculation produced a different pattern for gluons. Pion and kaon gluon distributions decreased as momentum transfer increased but showed weak dependence on both momentum transfer and skewness. The pion valence result agreed with available experimental data and the JAM analysis, while the evolved pion gluon result was consistent with JAM.
The calculation stays within a defined grid
The study calculates generalized parton distributions, or GPDs, for pion and kaon valence quarks and gluons. It varies momentum transfer, written as −t, and skewness, written as ξ, over −t values of 0, 0.11, 0.5, 1.0 and 1.5 GeV2 and ξ values of 0.05, 0.15 and 0.25.
The distributions were reported at µ2 = 4 and 27 GeV2. Next-to-leading-order DGLAP evolution equations were used to carry the modeled results to those scales before comparison with reference results.
Behind the results is a fixed numerical setup: Mu = Md = 0.4 GeV, ΛIR = 0.24 GeV, mπ = 0.14 GeV, fπ = 0.093 GeV, mK = 0.495 GeV and fK = 0.097 GeV. These values define the parameter set used for the calculation.
Momentum transfer leaves the clearest mark
At both reported scales, the pion and kaon valence distributions showed weak dependence on ξ but stronger dependence on −t. In other words, the modeled curves changed more across the tested momentum-transfer settings than across the tested skewness settings.
The finding is a comparison of relative changes inside the model’s tested grid. It does not show that skewness has no role; the reported ξ dependence is weak rather than absent.
One external check came from the forward-limit pion calculation. At µ2 = 27 GeV2, the modeled pion valence-quark distribution agreed well with available experimental data and the JAM analysis. That agreement applies to the forward-limit comparison, while the study also evaluates nonzero momentum-transfer and skewness settings.
A lower first moment and a quieter gluon signal
The study also calculated pion valence-quark Mellin moments, summary measures of the modeled distributions. The first moment was 0.119 at fixed −t = 1.5 GeV2 and ξ = 0, at µ2 = 27 GeV2, compared with 0.200 in the forward limit.
The modeled first moment was therefore lower at the reported nonzero momentum transfer than in the forward limit. No uncertainty range is reported for that comparison.
Gluons followed another pattern. Pion and kaon gluon distributions decreased with increasing −t but showed weak dependence on both −t and ξ at µ2 = 4 and 27 GeV2. After NLO DGLAP evolution, the pion gluon distribution at µ2 = 4 GeV2 was consistent with the JAM analysis.
The analysis notes that experimental and lattice-QCD results for the kaon gluon distribution are lacking. Its kaon gluon prediction therefore remains a model result without the same kind of external comparison available for the pion gluon calculation.
Form factors separate the flavors
Generalized form factors, or GFFs, provide another summary of the modeled distributions. For n = 1, 2, 3 and 4, the same ordering appeared at µ2 = 4 and 27 GeV2: kaon up-quark GFFs were lowest, pion up-quark GFFs were in the middle and kaon strange-quark GFFs were highest.
At t = 0, the relevant pion and kaon vector GFFs were normalized to unity. At µ2 = 4 GeV2, the reported pion and kaon GFFs were also in good agreement with corresponding lattice-QCD results.
The more detailed A20 ratios produced a mixed but mostly consistent comparison. The pion-up/kaon-up ratio stayed above unity and was consistent with lattice QCD, although it exceeded the lattice magnitude at larger −t. The pion-up/kaon-strange and kaon-up/kaon-strange ratios decreased with −t and were also consistent with lattice QCD.
One higher-order result stood apart. At µ2 = 4 GeV2, the pion Aπ50 prediction differed substantially from the corresponding lattice-QCD result. The analysis notes large lattice uncertainties but gives no numerical interval for them.
A model-based pattern, not a direct measurement
These results describe what this particular calculation produces, not a model-independent measurement of pion or kaon structure. The evidence is tied to the covariant SU(3)-flavor NJL setup, its Schwinger proper-time regularization, the chosen parameter set, the reported −t and ξ values and the NLO DGLAP evolution procedure.
Because the study is a modeling calculation, its comparisons concern calculated distributions and form factors rather than a conventional experimental control group. The supplied analysis reports no inferential statistical tests, confidence intervals or formal model-uncertainty analysis.
The reported kinematic grid is finite, so the study does not establish how the distributions behave outside the listed momentum transfers, skewness values and scales. Nor does it resolve the Aπ50 difference from lattice QCD.
Taken together, the findings define a clear theoretical pattern: valence distributions show stronger −t than ξ dependence, gluon distributions depend weakly on both variables and GFFs follow the kaon-up < pion-up < kaon-strange hierarchy. Several external comparisons are favorable, but the kaon gluon result and the Aπ50 discrepancy remain open tests.
The manuscript is identified as arXiv:2608.19867v1 [hep-ph] and dated 20 August 2026. The supplied metadata lists it as a preprint.
Paper data and sources
Original title: Skewness dependence of the pion and kaon generalized parton distributions
Authors: Fernando Chandra, Parada T. P. Hutauruk, Terry Mart
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text