Information Geometry of DGLAP Evolution: From Geodesic Tests to Emergent Small-Circle Geometry
Abstract
We investigate the geometric structure of leading-order (LO) nonsinglet DGLAP evolution on the Fisher--Rao manifold of normalized valence distributions, asking whether the QCD-selected trajectory is geodesic and how its departure from geodesic motion is organized. Through the isometric embedding Φ=2√p, the evolution is represented as a curve on a Hilbert sphere of radius two, with its velocity and acceleration determined directly by the DGLAP operator. For a GRV98-based total-valence trajectory over 2≤ Q2≤ 2000 GeV2, we find a nonzero but weakly scale-dependent geodesic curvature. An arc-length-weighted principal-component analysis shows that two affine directions capture more than 99.999% of the trajectory variance. The corresponding affine plane intersects the Fisher sphere in a small circle, whose predicted geodesic curvature closely agrees with that reconstructed from the DGLAP-generated acceleration. An x-resolved analysis further reveals that Fisher motion is governed by relative PDF reshaping rather than probability weight alone, with different momentum-fraction regions contributing to motion along the trajectory and its transverse bending. The Mellin representation provides a complementary perspective on the effective dimensional reduction through the collective evolution of the spectral hierarchy. These results establish a quantitative information-geometric characterization of the reference LO DGLAP trajectory and reveal an organized, near-small-circle structure over the finite evolution interval.
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BibTeX
@misc{kou2026information,
title = {{Information Geometry of DGLAP Evolution: From Geodesic Tests to Emergent Small-Circle Geometry}},
author = {Wei Kou and Xurong Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27886},
eprint = {2610.27886},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27886},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Information Geometry of DGLAP Evolution: From Geodesic Tests to Emergent Small-Circle Geometry
AU - Wei Kou
AU - Xurong Chen
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27886
UR - https://arxiv.org/abs/2610.27886
AB - We investigate the geometric structure of leading-order (LO) nonsinglet DGLAP evolution on the Fisher--Rao manifold of normalized valence distributions, asking whether the QCD-selected trajectory is geodesic and how its departure from geodesic motion is organized. Through the isometric embedding Φ=2√p, the evolution is represented as a curve on a Hilbert sphere of radius two, with its velocity and acceleration determined directly by the DGLAP operator. For a GRV98-based total-valence trajectory over 2≤ Q^2≤ 2000 GeV^2, we find a nonzero but weakly scale-dependent geodesic curvature. An arc-length-weighted principal-component analysis shows that two affine directions capture more than 99.999% of the trajectory variance. The corresponding affine plane intersects the Fisher sphere in a small circle, whose predicted geodesic curvature closely agrees with that reconstructed from the DGLAP-generated acceleration. An x-resolved analysis further reveals that Fisher motion is governed by relative PDF reshaping rather than probability weight alone, with different momentum-fraction regions contributing to motion along the trajectory and its transverse bending. The Mellin representation provides a complementary perspective on the effective dimensional reduction through the collective evolution of the spectral hierarchy. These results establish a quantitative information-geometric characterization of the reference LO DGLAP trajectory and reveal an organized, near-small-circle structure over the finite evolution interval.
ER -