Geometric Scaling above the Saturation Scale

TL;DR

The paper investigates whether geometric scaling, known from saturation, extends to momenta above the saturation scale. It combines nonlinear BK evolution with linear BFKL dynamics to show extended geometric scaling in a window where $1<\ln(Q^2/Q_s^2)\ll \ln(Q_s^2/\Lambda_{QCD}^2)$, with the saturation scale $Q_s(\tau)$ determined by matching to the BK boundary. A saddle-point analysis of BFKL yields a universal slope $c\approx 4.88$ and an anomalous dimension $\gamma\approx 0.37$ at saturation, and the scaling persists for both fixed and running coupling (the latter with $Q_s^2(\tau)$ growing as $\Lambda_{QCD}^2 e^{\sqrt{2 b_0 c (\tau+\tau_0)}}$). The results highlight the central role of the BFKL saddle point in extended scaling and discuss limitations and potential phenomenological implications for DIS and heavy-ion phenomenology. Overall, the work provides a coherent framework linking saturation physics to high-energy linear evolution, with concrete predictions for the scaling window and the behavior of the saturation scale.

Abstract

We show that the evolution equations in QCD predict geometric scaling for quark and gluon distribution functions in a large kinematical window, which extends above the saturation scale up to momenta $Q^2$ of order $100 {\rm GeV}^2$. For $Q^2 < Q^2_s$, with $Q_s$ the saturation momentum, this is the scaling predicted by the Colour Glass Condensate and by phenomenological saturation models. For $1 \simle \ln(Q^2/Q_s^2) \ll \ln(Q_s^2/Λ^2_{\rm QCD})$, we show that the solution to the BFKL equation shows approximate scaling, with the scale set by $Q_s$. At larger $Q^2$, this solution does not scale any longer. We argue that for the intermediate values of $Q^2$ where we find scaling, the BFKL rather than the double logarithmic approximation to the DGLAP equation properly describes the dynamics. We consider both fixed and running couplings, with the scale for running set by the saturation momentum. The anomalous dimension which characterizes the approach of the gluon distribution function towards saturation is found to be close to, but lower than, one half.