Combining nonperturbative transverse momentum dependence with TMD evolution

TL;DR

This paper develops a method to integrate nonperturbative models into TMD factorization, connecting phenomenological approaches with QCD evolution to better understand hadron structure across different energy scales.

Contribution

It introduces a recipe for incorporating nonperturbative models into TMD factorization and constrains their parameters using evolution and integral relations.

Findings

Clarifies differences between forward and backward TMD evolution.

Shows how integral relations constrain nonperturbative g-functions.

Provides a practical approach for merging models with TMD phenomenology.

Abstract

Central to understanding the nonpertubative, intrinsic partonic nature of hadron structure are the concepts of transverse momentum dependent (TMD) parton distribution and fragmentation functions. A TMD factorization approach to the phenomenology of semi-inclusive processes that includes evolution, higher orders, and matching to larger transverse momentum, is ultimately necessary for reliably connecting with phenomenologically extracted nonperturbative structures, especially when widely different scales are involved. In this paper, we will address some of the difficulties that arise when phenomenological techniques that were originally designed for very high energy applications are extended to studies of hadron structures, and we will solidify the connection between standard high energy TMD implementations and the more intuitive, parton model based approaches to phenomenology that emphasize nonperturbative hadron structure. In the process, we will elaborate on differences between forward and backward TMD evolution, which in the context of this paper we call "bottom-up" and "top-down" approaches, and we will explain the advantages of a bottom-up strategy. We will also emphasize and clarify the role of the integral relations that connect TMD and collinear correlation functions. We will show explicitly how they constrain the nonperturbative "$g$-functions" of standard Collins-Soper-Sterman (CSS) implementations of TMD factorization. This paper is especially targeted toward phenomenologists and model builders who are interested in merging specific nonperturbative models and calculations (including lattice QCD) with TMD factorization at large $Q$. Our main result is a recipe for incorporating nonperturbative models into TMD factorization, and for constraining their parameters in a way that matches to perturbative QCD and evolution.