Violation of the geometric scaling behaviour of the amplitude for running QCD coupling in the saturation region

TL;DR

This paper demonstrates that geometric scaling in the scattering amplitude is violated for running QCD coupling in the saturation region, revealing a new scaling behavior dependent on a combined variable, except near the saturation scale.

Contribution

It introduces a new scaling behavior of the amplitude as a function of the combined variable 6 = Y l, showing geometric scaling only near the saturation scale.

Findings

Amplitude depends on 6 = Y l, not just on r and Q_s

Geometric scaling holds only close to the saturation scale

Experimental geometric scaling may be due to limited probing or coupling freezing

Abstract

In this paper we show that the intuitive guess that the geometric scaling behaviour should be violated in the case of the running QCD coupling, turns out to be correct. The scattering amplitude of the dipole with the size $r$ depends on new dimensional scale: $\Lambda_{QCD}$, even at large values $Y = \ln(1/x)$ and $l\,=\, \ln\Lb \as\Lb r^2\Rb/\as\Lb 1/Q^2_s\Rb\Rb$. However, in this region we found a new scaling behaviour: the amplitude is a function of $\zeta = Y\,l$. We state that only in the vicinity of the saturation scale $Q_s$ ($\as(Q^2_s)\,\ln \Lb r^2 Q^2_s\Rb\,\leq \, 1$), the amplitude shows the geometric scaling behaviour. Based on these finding the geometric scaling behavior that has been seen experimentally, stems from either we have not probed the proton at HERA and the LHC deeply inside the saturation region or that there exists the mechanism of freezing of the QCD coupling constant at $r^2 \approx 1/Q^2_s$.