# Evolution of Chirality-odd Twist-3 Fragmentation Functions

## TL;DR

This paper derives the one-loop evolution equations for all nine chirality-odd twist-3 fragmentation functions, highlighting their scale dependence and mixing patterns, which are crucial for understanding single transverse-spin asymmetries.

## Contribution

It provides the complete set of evolution equations for chirality-odd twist-3 fragmentation functions, including their mixing behavior and simplifications in the large-Nc limit.

## Key findings

- Three complex fragmentation functions evolve with self-mixing.
- No mixing occurs with bilinear quark operator functions.
- Large-Nc limit simplifies evolution equations to six homogeneous forms.

## Abstract

We derive the complete set of evolutions of chirality-odd twist-3 fragmentation functions at one-loop level. There are totally nine real twist-3 fragmentation functions, among which seven are independent. The renormalization-scale dependence of the nine functions has an important implication for studies of single transverse-spin asymmetries. We find that the evolutions of the three complex fragmentation functions defined by quark-gluon-quark operator are mixed with themselves. There is no mixing with the fragmentation functions defined only with bilinear quark field operators. In the large-$N_c$ limit the evolutions of the three complex fragmentation functions are simplified and reduced to six homogenous equations.