[Paper Review] The Transverse-momentum-dependent Parton Distribution Function and Jet Transport in Medium

Zuo-tang Liang, Xin-Nian Wang+1 more|arXiv (Cornell University)|Jan 2, 2008

High-Energy Particle Collisions Research4 citations

TL;DR

This paper derives a gauge-invariant expression for the transverse-momentum-dependent (TMD) parton distribution function (PDF) in nuclear matter using a transport operator formalism, showing that nuclear transverse momentum broadening arises from the path integral of the quark transport parameter $\hat{q}_F$. Under the two-gluon correlation approximation, the nuclear TMD PDF becomes a convolution of a Gaussian (with width governed by $\hat{q}_F$) and the nucleon TMD PDF, with $\hat{q}_F$ expressible via gluon distribution density in the medium. Extensions to hot QCD matter and $\mathcal{N}=4$ SYM theory reveal significant contributions from multiple gluon correlations in strongly coupled systems.

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.

Motivation & Objective

• To establish a gauge-invariant formulation of the TMD parton distribution function (PDF) in nuclear matter using a transport operator formalism.
• To derive the nuclear broadening of transverse momentum distributions from higher-twist collinear matrix elements.
• To express the mean transverse momentum broadening squared as a path integral of the quark transport parameter $\hat{q}_F$, linked to gluon distribution density in the medium.
• To extend the analysis to hot medium and compare results with the dipole model and $\mathcal{N}=4$ SYM theory in the strong coupling limit.
• To investigate the role of higher-twist gluon correlations, especially in strongly coupled systems like $\mathcal{N}=4$ SYM plasma.

Proposed method

• The TMD PDF is expressed as a sum of higher-twist collinear parton matrix elements via a transport operator, ensuring gauge invariance.
• The gauge link structure in the TMD PDF is decomposed into longitudinal and transverse components, with the transverse link encoding multiple scattering effects.
• Under the maximal two-gluon correlation approximation, the nuclear TMD PDF is derived as a convolution of a Gaussian (with width $\langle k_\perp^2 \rangle = \int \hat{q}_F \, dz$) and the nucleon TMD PDF.
• The transport parameter $\hat{q}_F$ is shown to be expressible in gauge-invariant form as $\hat{q}_F = \frac{1}{2N_c} \langle \vec{\cal F}_\perp^a \cdot \vec{\cal F}_\perp^a \rangle$, linking it to the gluon distribution density.
• Higher-twist nucleon gluon distributions are resummed under the extended adjoint two-gluon correlation approximation, leading to a transverse scale-dependent quark transport parameter.
• The formalism is extended to hot medium, and results are compared with the dipole model and $\mathcal{N}=4$ SYM theory in the strong coupling limit, revealing enhanced multiple-gluon correlation effects.

Experimental results

Research questions

• RQ1 How can the TMD parton distribution function in a nucleus be expressed in a gauge-invariant way using higher-twist collinear matrix elements and a transport operator?
• RQ2 What is the origin of nuclear transverse momentum broadening in deep inelastic scattering, and how is it related to the quark transport parameter $\hat{q}_F$?
• RQ3 How does the inclusion of higher-twist gluon correlations modify the nuclear TMD PDF, especially in strongly coupled systems?
• RQ4 To what extent do multiple gluon correlations dominate in the $\mathcal{N}=4$ SYM plasma compared to weakly coupled QCD?
• RQ5 Can the transport parameter $\hat{q}_F$ be expressed in terms of medium gluon distribution density in a gauge-invariant manner?

Key findings

• The TMD quark distribution function in a nucleus is expressed as a sum of higher-twist collinear matrix elements via a transport operator, ensuring gauge invariance.
• Under the two-gluon correlation approximation, the nuclear TMD PDF becomes a convolution of a Gaussian distribution (with width $\langle k_\perp^2 \rangle = \int \hat{q}_F \, dz$) and the nucleon TMD PDF.
• The mean transverse momentum broadening squared is given by the path integral of the quark transport parameter $\hat{q}_F$, which is expressible in gauge-invariant form as $\hat{q}_F = \frac{1}{2N_c} \langle \vec{\cal F}_\perp^a \cdot \vec{\cal F}_\perp^a \rangle$, directly linked to the gluon distribution density in the medium.
• Higher-twist nucleon gluon distributions can be resummed under the extended adjoint two-gluon correlation approximation, leading to a transverse scale-dependent quark transport parameter.
• In hot medium and especially in $\mathcal{N}=4$ SYM theory at strong coupling, multiple gluon correlations become significant, indicating a breakdown of the two-gluon approximation.
• The results show that the dipole model approximation underestimates transverse momentum broadening in strongly coupled systems, where higher-twist effects dominate.