# Fluctuations of the gluon distribution from the small-x effective action

## TL;DR

This paper develops a new method to analyze fluctuations in the gluon distribution at small-x in high energy QCD, providing explicit effective potentials for different models and revealing distinct fluctuation behaviors.

## Contribution

It introduces a novel approach to study observable fluctuations without explicit dipole involvement, and explicitly computes the effective potential for gluon distributions in the MV model and Gaussian approximation.

## Key findings

- Fluctuation spectral shape differs between MV model and JIMWLK evolution.
- Gluon fluctuations pile up near the saturation scale in the MV model.
- JIMWLK generates scale-invariant fluctuations above the saturation boundary.

## Abstract

The computation of observables in high energy QCD involves an average over stochastic semiclassical small-x gluon fields. The weight of various configurations is determined by the effective action. We introduce a method to study fluctuations of observables, functionals of the small-x fields, which does not explicitly involve dipoles. We integrate out those fluctuations of the gluon field under which a given observable is invariant. Thereby we obtain the effective potential for that observable describing its fluctuations about the saddle point. We determine explicitly the effective potential for the covariant gauge gluon distribution both for the McLerran-Venugopalan (MV) model and for a (non-local) Gaussian approximation for the small-x effective action. This provides insight into the correlation of fluctuations of the number of hard gluons versus their typical transverse momentum. We find that the spectral shape of the fluctuations of the gluon distribution is fundamentally different in the MV model, where there is a pile-up of gluons near the saturation scale, versus the solution of the small-x JIMWLK renormalization group, which generates essentially scale invariant fluctuations above the absorptive boundary set by the saturation scale.