# Generalized Fragmentation Functions for Fractal Jet Observables

## TL;DR

This paper introduces a new class of fractal jet observables based on generalized fragmentation functions, which are capable of capturing complex, correlated properties of hadrons in jets, with applications in quark-gluon discrimination.

## Contribution

It develops a formalism for nonlinear evolution of generalized fragmentation functions for fractal jet observables, including a numerical code and phenomenological applications.

## Key findings

- Nonlinear evolution equations for generalized fragmentation functions are derived.
- A numerical code for evolution of fractal jet observables is developed.
- Fractal jet observables can effectively discriminate quark from gluon jets.

## Abstract

We introduce a broad class of fractal jet observables that recursively probe the collective properties of hadrons produced in jet fragmentation. To describe these collinear-unsafe observables, we generalize the formalism of fragmentation functions, which are important objects in QCD for calculating cross sections involving identified final-state hadrons. Fragmentation functions are fundamentally nonperturbative, but have a calculable renormalization group evolution. Unlike ordinary fragmentation functions, generalized fragmentation functions exhibit nonlinear evolution, since fractal observables involve correlated subsets of hadrons within a jet. Some special cases of generalized fragmentation functions are reviewed, including jet charge and track functions. We then consider fractal jet observables that are based on hierarchical clustering trees, where the nonlinear evolution equations also exhibit tree-like structure at leading order. We develop a numeric code for performing this evolution and study its phenomenological implications. As an application, we present examples of fractal jet observables that are useful in discriminating quark jets from gluon jets.