Fisher-Rao Geometry of Gluon Saturation Fronts
Abstract
How nonlinear gluon saturation erases the memory of initial conditions is a central question in high-energy QCD. We develop the Fisher-Rao information geometry of saturation fronts in the Balitsky-Kovchegov (BK) equation and its Fisher-KPP diffusive limit. Using the normalized coordinate-space front density p = ∂ N/∂ln r2, we isolate the transverse metric g⊥θθ that quantifies shape distinguishability once longitudinal propagation along the flow is factored out. Nonlinear saturation arrests diffusive broadening and freezes the Fisher speed. Transverse deformations exhibit a universal t−4 relaxation in the Fisher-KPP asymptotic analysis, with full BK evolution providing numerical evidence for the corresponding τ−4 behavior. We trace this steep power law to a relaxation mechanism: deforming the initial profile is equivalent, to leading order, to a spatial translation combined with an effective time delay of the universal relaxation flow. Extracting the translation and the delay from neighboring runs alone predicts the ratio of g⊥θθ to its shift-control value within 1% for Fisher-KPP and for BK up to τ= 15. For Fisher-KPP, the asymptotic front expansion yields the t−4 law conditionally on stated analytic hypotheses, with a universal amplitude C* that agrees with numerical extrapolations to sub-percent accuracy. For BK, the asymptotic translation information exceeds its diffusive approximation by a factor of 3.5, which we attribute to the non-diffusive Levin-Tuchin saturation region. These results provide an information-geometric characterization of the approach to geometric scaling, consistent with the persistent sensitivity of pre-asymptotic HERA fits to initial shape parameters, and motivate systematic studies of parameter sensitivity in EIC kinematics.
Related
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BibTeX
@misc{spezzano2026fisherrao,
title = {{Fisher-Rao Geometry of Gluon Saturation Fronts}},
author = {Edoardo Spezzano},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02459},
eprint = {2611.02459},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02459},
}
RIS
TY - UNPB
TI - Fisher-Rao Geometry of Gluon Saturation Fronts
AU - Edoardo Spezzano
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02459
UR - https://arxiv.org/abs/2611.02459
AB - How nonlinear gluon saturation erases the memory of initial conditions is a central question in high-energy QCD. We develop the Fisher-Rao information geometry of saturation fronts in the Balitsky-Kovchegov (BK) equation and its Fisher-KPP diffusive limit. Using the normalized coordinate-space front density p = ∂ N/∂ln r^2, we isolate the transverse metric g^⊥_θθ that quantifies shape distinguishability once longitudinal propagation along the flow is factored out. Nonlinear saturation arrests diffusive broadening and freezes the Fisher speed. Transverse deformations exhibit a universal t^−4 relaxation in the Fisher-KPP asymptotic analysis, with full BK evolution providing numerical evidence for the corresponding τ^−4 behavior. We trace this steep power law to a relaxation mechanism: deforming the initial profile is equivalent, to leading order, to a spatial translation combined with an effective time delay of the universal relaxation flow. Extracting the translation and the delay from neighboring runs alone predicts the ratio of g^⊥_θθ to its shift-control value within 1% for Fisher-KPP and for BK up to τ= 15. For Fisher-KPP, the asymptotic front expansion yields the t^−4 law conditionally on stated analytic hypotheses, with a universal amplitude C_* that agrees with numerical extrapolations to sub-percent accuracy. For BK, the asymptotic translation information exceeds its diffusive approximation by a factor of 3.5, which we attribute to the non-diffusive Levin-Tuchin saturation region. These results provide an information-geometric characterization of the approach to geometric scaling, consistent with the persistent sensitivity of pre-asymptotic HERA fits to initial shape parameters, and motivate systematic studies of parameter sensitivity in EIC kinematics.
ER -