The Transverse-momentum-dependent Parton Distribution Function and Jet Transport in Medium

TL;DR

This paper develops a gauge-invariant framework for TMD quark distributions, linking them to higher-twist collinear matrix elements and deriving nuclear transverse momentum broadening in medium, with applications to hot and strongly coupled systems.

Contribution

It introduces a general expression for TMD quark distributions in terms of transport operators and derives nuclear broadening formulas under various approximations, extending to hot and strongly coupled media.

Findings

Nuclear transverse momentum distribution as a convolution of Gaussian and nucleon TMDs.

Width of Gaussian related to the path integral of the quark transport parameter.

Multiple gluon correlations are significant in strongly coupled systems like ${ m N}=4$ SYM plasma.

Abstract

We show that the gauge-invariant transverse-momentum-dependent (TMD) quark distribution function can be expressed as a sum of all higher-twist collinear parton matrix elements in terms of a transport operator. From such a general expression, we derive the nuclear broadening of the transverse momentum distribution. Under the maximal two-gluon correlation approximation, in which all higher-twist nuclear multiple-parton correlations with the leading nuclear enhancement are given by products of twist-two nucleon parton distributions, we find the nuclear transverse momentum distribution as a convolution of a Gaussian distribution and the nucleon TMD quark distribution. The width of the Gaussian, or the mean total transverse momentum broadening squared, is given by the path integral of the quark transport parameter $\hat q_F$ which can also be expressed in a gauge invariant form and is given by the gluon distribution density in the nuclear medium. We further show that contributions from higher-twist nucleon gluon distributions can be resummed under the extended adjoint two-gluon correlation approximation and the nuclear transverse momentum distribution can be expressed in terms of a transverse scale dependent quark transport parameter or gluon distribution density. We extend the study to hot medium and compare to dipole model approximation and ${\cal N}=4$ Supersymmetric Yang-Mills (SYM) theory in the strong coupling limit. We find that multiple gluon correlations become important in the strongly coupled system such as ${\cal N}=4$ SYM plasma.