[Paper Review] Partonic Picture of Generalized Transverse Momentum Distributions

Simonetta Liuti, Aurore Courtoy+3 more|arXiv (Cornell University)|Sep 26, 2013

Quantum Chromodynamics and Particle Interactions3 references3 citations

TL;DR

This paper argues that the orbital angular momentum (OAM) of quarks in a nucleon, long considered inaccessible in leading-twist hard scattering processes, can only be probed via twist-three contributions in generalized transverse momentum distributions (GTMDs). By analyzing parity constraints and helicity structure in quark-proton scattering, the authors show that GTMDs like $F_{14}$—previously thought to represent OAM—are not observable at leading twist but emerge only at twist three, where they connect to measurable observables in deeply virtual Compton scattering (DVCS).

ABSTRACT

We argue that due to parity constraints, the helicity combination of the purely momentum space counterparts of the Wigner distributions -- the generalized transverse momentum distributions -- that describes the configuration of an unpolarized quark in a longitudinally polarized nucleon, can enter the deeply virtual Compton scattering amplitude only through matrix elements involving a final state interaction. The relevant matrix elements in turn involve light cone operators projections in the transverse direction, or they appear in the deeply virtual Compton scattering amplitude at twist three. Orbital angular momentum or the spin structure of the nucleon was a major reason for these various distributions and amplitudes to have been introduced. We show that twist three contributions to deeply virtual Compton scattering provide observables related to orbital angular momentum.

Motivation & Objective

• To resolve the physical observability of quark orbital angular momentum (OAM) in nucleon structure, particularly in relation to GTMDs.
• To clarify the role of parity and helicity constraints in limiting the number of independent GTMDs in quark-nucleon scattering.
• To demonstrate that OAM contributions to the nucleon spin sum rule appear only at twist three, not at leading twist.
• To connect the twist-three GTMDs to measurable observables in deeply virtual Compton scattering (DVCS).
• To reconcile the Ji and Jaffe-Mandelstam decompositions of nucleon spin by showing both OAM components appear at twist three.

Proposed method

• Analyzing the helicity structure of GTMDs in the center-of-mass frame under parity, time-reversal, and Hermiticity constraints.
• Applying the Jacob-Wick formalism to reduce the number of independent helicity amplitudes from 16 to 8 in 2-body quark-proton scattering.
• Identifying that GTMDs such as $F_{14}$ and $G_{11}$, which were previously linked to OAM, violate parity and vanish at leading twist.
• Using light-cone quantization and $u$-channel quark-proton scattering to derive the kinematic structure of GTMDs with fixed skewness ($ ilde{ heta}=0$).
• Connecting twist-three contributions in the DVCS amplitude to the OAM component via the sum rule derived by Polyakov and Hatta.
• Comparing the Ji and Jaffe-Mandelstam decompositions of nucleon spin to show that both OAM terms are inherently twist-three observables.

Experimental results

Research questions

• RQ1 Why are certain GTMDs, such as $F_{14}$, considered unphysical despite being non-zero in models?
• RQ2 How do parity and helicity constraints reduce the number of independent GTMDs from 16 to 8 in quark-nucleon scattering?
• RQ3 Can the orbital angular momentum (OAM) of quarks be accessed in hard exclusive processes like deeply virtual Compton scattering?
• RQ4 What is the twist structure of the OAM contribution in the nucleon spin sum rule, and how does it relate to measurable GTMDs?
• RQ5 How do the Ji and Jaffe-Mandelstam decompositions of nucleon spin relate to twist-three distributions in GTMDs?

Key findings

• The GTMD $F_{14}$, previously proposed as a proxy for quark orbital angular momentum, is parity-odd and vanishes at leading twist, making it unobservable in standard hard scattering processes.
• Due to parity and kinematic constraints in the center-of-mass frame, only eight independent helicity amplitudes exist in quark-nucleon scattering, reducing the apparent 16 GTMDs to eight at leading twist.
• Orbital angular momentum (OAM) contributions to the nucleon spin sum rule appear exclusively at twist three, not at leading twist, and are thus not accessible through standard GPD or TMD formalisms.
• The twist-three vector component $G_2^q(x,0,0)$, as defined in the Polyakov sum rule, corresponds to the second moment of quark OAM and is measurable via DVCS experiments.
• Both the Ji and Jaffe-Mandelstam decompositions of nucleon spin require twist-three contributions for their OAM components, confirming that OAM is not accessible at leading twist.
• Recent Jefferson Lab measurements may already be sensitive to these twist-three observables, offering a pathway to experimentally probe quark OAM.