# Connecting Different TMD Factorization Formalisms in QCD

## TL;DR

This paper bridges various TMD factorization formalisms in QCD, demonstrating their equivalence and connecting them through perturbative calculations and effective theory, thereby unifying different approaches for Drell-Yan processes.

## Contribution

It provides a method to relate different TMD factorization schemes using known perturbative results and derives new connections with SCET formalisms, including updated higher-order inputs.

## Key findings

- Perturbative coefficients can be transferred between schemes.
- Non-perturbative functions are scheme-independent.
- Derived new results for quark form factor factorization.

## Abstract

In the original Collins-Soper-Sterman (CSS) presentation of the results of transverse-momentum-dependent (TMD) factorization for the Drell-Yan process, results for perturbative coefficients can be obtained from calculations for collinear factorization. Here we show how to use these results, plus known results for the quark form factor, to obtain coefficients for TMD factorization in more recent formulations, e.g., that due to Collins, and apply them to known results at order $\alpha_s^2$ and $\alpha_s^3$. We also show that the "non-perturbative" functions as obtained from fits to data are equal in the two schemes. We compile the higher-order perturbative inputs needed for the updated CSS scheme by appealing to results obtained in a variety of different formalisms. In addition, we derive the connection between both versions of the CSS formalism and several formalisms based in soft-collinear effective theory (SCET). Our work uses some important new results for factorization for the quark form factor, which we derive.