[Paper Review] Azimuthal Dependence of DIS with Spin-1 Target

Jian-Ping Ma, C. Wang+1 more|arXiv (Cornell University)|Jun 28, 2013

Physics of Superconductivity and Magnetism14 references3 citations

TL;DR

This paper derives the azimuthal dependence of deep-inelastic scattering (DIS) cross-sections when a spin-1 hadron or nucleus is polarized in an arbitrary direction, using QCD factorization with twist-2 and twist-3 operators. It shows that tensor-polarized structure functions—previously predicted to vanish in non-interacting nucleon models—can be extracted from azimuthal distributions, providing direct probes of nucleon-nucleon correlations and gluonic structure in nuclei.

ABSTRACT

We study DIS with a spin-1 hadron or a nucleus polarized in arbitrary direction. The differential cross-section in this case will have an azimuthal dependence. We derive the dependence by including contributions from twist-2- and twist-3 QCD operators. The twist-3 contribution is computed at tree-level. A spin-1 hadron or nucleus can have nonzero tensor polarization. We find that all structure functions related to the tensor-polarization of the initial hadron can be extracted by study the azimuthal dependence. A nonzero result of the structure functions of a nucleus from experiment will indicate nontrivial inner structure of the nucleus.

Motivation & Objective

• To derive the differential cross-section for DIS with a spin-1 hadron or nucleus polarized in an arbitrary direction, focusing on azimuthal angular dependence.
• To investigate how twist-2 and twist-3 QCD operators contribute to the hadronic tensor and the resulting azimuthal asymmetries.
• To demonstrate that structure functions related to tensor polarization—previously expected to vanish in non-interacting nuclear models—can be extracted from experimental azimuthal distributions.
• To provide a theoretical framework for interpreting future DIS experiments with polarized spin-1 nuclei at J-Lab and the EIC, particularly to probe nuclear correlations and gluonic structure.
• To confirm the existence and extract the form of new twist-3 structure functions, including $ ilde{F}_{1,2} $, $ ilde{q}_T $, and $ ilde{G}_T $, which are sensitive to tensor polarization and gluon content.

Proposed method

• Employing collinear QCD factorization to decompose the hadronic tensor into twist-2 and twist-3 contributions, with the latter computed at tree level.
• Defining the hadronic tensor $ W^{ ueta} $ as a convolution of coefficient functions and matrix elements of twist-2 and twist-3 operators, using light-cone formalism with $ l^ u = (1,0,0,0) $ and $ n^ u = (0,1,0,0) $.
• Introducing polarization vectors $ oldsymbol{ ho} $ and $ oldsymbol{ ho}_T $ to describe vector and tensor polarizations of the spin-1 target, with the initial hadron momentum along $ z $-axis.
• Deriving the differential cross-section in the laboratory frame with azimuthal angle $ heta $, $ heta $, and $ heta $, and expressing the result in terms of structure functions $ F_1, F_2, ilde{F}_1, ilde{F}_2, ilde{G}_T, ilde{q}_T $, and $ ilde{G}_1 $.
• Using the Bjorken limit and $ 1/Q $ expansion to separate contributions by twist, showing that twist-3 terms are suppressed by $ M/Q $ or $ 1/Q $, but still measurable via azimuthal asymmetries.
• Deriving the explicit form of $ ilde{G}_{T,Qar{Q}} $ for heavy quark pair production, showing its $ eta $-dependent structure and $ eta o 1 $ limit matching the general $ ilde{G}_T $ result.

Experimental results

Research questions

• RQ1 How does the differential cross-section in DIS with a spin-1 target depend on the azimuthal angle when the target is polarized in an arbitrary direction?
• RQ2 What are the contributions of twist-2 and twist-3 operators to the hadronic tensor in DIS with a spin-1 target, and how do they affect the azimuthal asymmetry?
• RQ3 Can structure functions related to tensor polarization—such as $ ilde{F}_1, ilde{F}_2, ilde{G}_T, ilde{q}_T $—be extracted from the azimuthal dependence of the cross-section?
• RQ4 What is the role of nucleon-nucleon correlations in the nuclear medium, as probed by non-zero tensor-polarized structure functions in DIS with a nucleus?
• RQ5 How does the gluonic tensor distribution $ g_T $ contribute to the cross-section, and can it be isolated in processes with heavy quark final states?

Key findings

• The azimuthal dependence of the DIS cross-section with a spin-1 target is fully determined by the vector and tensor polarization states of the initial hadron, with structure functions $ ilde{F}_1, ilde{F}_2, ilde{G}_T, ilde{q}_T $ contributing to the $ heta $-dependent terms.
• Twist-3 contributions to the hadronic tensor arise from two new distributions: one related to tensor polarization ($ ilde{q}_T $) and another to vector polarization ($ ilde{G}_T $), both of which are suppressed by $ 1/Q $ but measurable via azimuthal asymmetries.
• The structure function $ ilde{F}_1 $ gives a constant contribution to the azimuthal distribution, while $ ilde{F}_2 $ and $ ilde{G}_T $ contribute to $ heta $-dependent terms, with $ ilde{G}_T $ being particularly sensitive to non-perturbative gluonic structure.
• In the limit $ heta_m = 0^ ext{°} $, the azimuthal dependence vanishes, confirming that the asymmetry arises only when the target polarization is transverse to the quantization axis.
• The structure function $ ilde{G}_{T,Qar{Q}} $ for heavy quark pair production is derived explicitly as $ rac{ar{ ho}}{4ar{ ho}} ilde{G}_T $, with a non-trivial $ eta $-dependence involving logarithmic terms, and reduces to the general $ ilde{G}_T $ form in the $ m_Q o 0 $ limit.
• A non-zero value of $ ilde{F}_1 $ or $ ilde{G}_T $ in experiment would indicate non-trivial nucleon-nucleon correlations or non-additive gluonic content in the nucleus, contradicting the free-nucleon model.