# Rare fluctuations of the $S$-matrix at NLO in QCD

## TL;DR

This paper computes the rare fluctuations of the $S$-matrix at NLO in QCD, revealing a significant factor change due to gluon loop corrections that restore the importance of rare fluctuations in the saturation regime.

## Contribution

It introduces the calculation of rare fluctuations of the $S$-matrix at NLO in QCD and demonstrates their frame independence, highlighting the impact of gluon loops on the exponential factor.

## Key findings

- The exponential factor of the $S$-matrix is increased by a factor of $\sqrt{2}$ due to gluon loops.
- Gluon loop corrections partially offset the rapidity decrease caused by quark loops.
- Rare fluctuations remain significant and frame-independent at NLO in QCD.

## Abstract

We calculate the rare fluctuations of the $S$-matrix on top of the full next-to-leading order corrections in the center of mass frame. The relevant result in the saturation regime shows that the exponential factor of the $S$-matrix is $\sqrt{2}$ as large as the result which emerges when the rare fluctuation effects are taken into account. We find that the factor of $\sqrt{2}$ change of the exponential factor is induced by the gluon loop corrections which compensate part of rapidity decrease of the $S$-matrix made by quark loops and lead to the rare fluctuations becoming important again. To ensure the relevant results of the $S$-matrix are independent of the frame choice, the rare fluctuations of the $S$-matrix are also derived in a general frame. It is found that all the results are consistent with each other in both frames.