# The spin-dependent quark beam function at NNLO

## TL;DR

This paper computes the polarized quark beam function at NNLO in QCD, completing the ingredients for NNLO analysis of polarized collision observables and providing a cross-check for unpolarized results.

## Contribution

It presents the first calculation of the polarized quark beam function at NNLO, filling a key gap for precise theoretical predictions in polarized high-energy collisions.

## Key findings

- First independent check of unpolarized quark beam function
- Provides detailed derivation of polarized beam function at NNLO
- Enables NNLO analysis of polarized N-jettiness observables

## Abstract

We calculate the beam function for longitudinally-polarized quarks through next-to-next-to-leading order (NNLO) in QCD perturbation theory. This is the last missing ingredient needed to apply the factorization theorem for the $N$-jettiness event-shape variable in polarized collisions through the NNLO level. We present all technical details of our derivation. As a by-product of our calculation we provide the first independent check of the previously-obtained unpolarized quark beam function. We anticipate that our result will have phenomenological applications in describing data from polarized collisions.