Operator analysis of $p_T$-widths of TMDs

TL;DR

This paper investigates the process dependence of transverse momentum dependent parton distribution functions (TMDs), especially those involving products of transverse momenta, and proposes methods for their experimental and lattice QCD study.

Contribution

It extends the understanding of TMD universality by analyzing TMDs with $p_T$-widths and proposes specific quantities for lattice QCD to study their process dependence.

Findings

Analysis of TMDs with $p_T$-widths and rank two structures.

Identification of experimental observables for process dependence.

Proposal of quantities for lattice QCD calculations.

Abstract

Transverse momentum dependent (TMD) parton distribution functions (PDFs), TMDs for short, are defined as the Fourier transform of matrix elements of nonlocal combinations of quark and gluon fields. The nonlocality is bridged by gauge links, which for TMDs have characteristic paths (future or past pointing), giving rise to a process dependence that breaks universality. It is possible, however, to construct sets of universal TMDs of which in a given process particular combinations are needed with calculable, process-dependent, coefficients. This occurs for both T-odd and T-even TMDs, including also the {\it unpolarized} quark and gluon TMDs. This extends the by now well-known example of T-odd TMDs that appear with opposite sign in single-spin azimuthal asymmetries in semi-inclusive deep inelastic scattering or in the Drell-Yan process. In this paper we analyze the cases where TMDs enter multiplied by products of two transverse momenta, which includes besides the $p_T$-broadening observable, also instances with rank two structures. To experimentally demonstrate the process dependence of the latter cases requires measurements of second harmonic azimuthal asymmetries, while the $p_T$-broadening will require measurements of processes beyond semi-inclusive deep inelastic scattering or the Drell-Yan process. Furthermore, we propose specific quantities that will allow for theoretical studies of the process dependence of TMDs using lattice QCD calculations.