[Paper Review] Comment on "Proton Spin Structure from Measurable Parton Distributions" by Ji, Xiong and Yuan (PRL109, 152005 (2012))

Elliot Leader, Cédric Lorcé|arXiv (Cornell University)|Nov 20, 2012

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper challenges the interpretation of Ji, Xiong, and Yuan (2012) that the expression $\frac{1}{2}\int_{-1}^{1}dx\,x\left[H_q(x,0,0)+E_q(x,0,0)\right]$ measures the transverse angular momentum of quarks in a transversely polarized nucleon. It argues instead that this expression corresponds to the light-front transverse boost operator, not angular momentum, and that a true transverse angular momentum sum rule must be frame-dependent and involve dynamical operators, not leading-twist kinematic ones.

ABSTRACT

We show that the recent claim that the expression 1/2 \int x dx [ H_q (x,0,0) + E_q(x,0,0)], involving the generalized parton distributions H and E, measures the transverse angular momentum of quarks in a transversely polarized nucleon, is incorrect.

Motivation & Objective

• To challenge the claim by Ji, Xiong, and Yuan (2012) that a specific combination of generalized parton distributions (GPDs) measures transverse quark angular momentum in a transversely polarized nucleon.
• To clarify the distinction between kinematic (boost) operators and dynamical (angular momentum) operators in light-front quantization.
• To demonstrate that the expression in question corresponds to the light-front transverse boost operator $J^{+i}$, not the transverse angular momentum operator $J^{-i}$.
• To emphasize that a genuine transverse angular momentum sum rule must be frame-dependent due to the non-commutativity of boosts and rotations in special relativity.
• To reaffirm the correctness of the original longitudinal angular momentum sum rule in terms of GPDs, while rejecting its transverse analog in the same form.

Proposed method

• Analyzes the light-front formulation of the angular momentum operator using the Belinfante energy-momentum tensor and the definition $M^{\mu\rho\sigma}_q(x) = x^\rho T^{\mu\sigma}_q(x) - x^\sigma T^{\mu\rho}_q(x)$.
• Compares the light-front transverse boost operators $J^{+i} = \frac{1}{\sqrt{2}}(K^i \pm J^j)$ with the transverse angular momentum operators $J^{-i} = \frac{1}{\sqrt{2}}(K^i \mp J^j)$, showing they are distinct.
• Uses the standard decomposition of the angular momentum operator in instant-form and light-front quantization to show that $J^{+i}$ is kinematic (leading-twist), while $J^{-i}$ is dynamical (higher-twist).
• Demonstrates that the quark and gluon spin operators contribute to $J^{-i}$, not $J^{+i}$, via the $\epsilon^{\mu\rho\sigma\nu}$-coupled current in the $A^+ = 0$ gauge.
• Reviews the frame dependence of transverse angular momentum due to non-commuting boosts and rotations, contrasting it with the frame-independence of longitudinal angular momentum.
• Reaffirms the validity of the original longitudinal sum rule $J_q = \frac{1}{2}\int dx\,x\left[H_q(x,0,0)+E_q(x,0,0)\right]$ for $\langle J^z_q \rangle_L$.

Experimental results

Research questions

• RQ1 Does the expression $\frac{1}{2}\int dx\,x\left[H_q(x,0,0)+E_q(x,0,0)\right]$ correctly represent the transverse angular momentum of quarks in a transversely polarized nucleon?
• RQ2 What is the true physical interpretation of the light-front operator $J^{+i}$ in terms of angular momentum or boost generators?
• RQ3 Why is a simple partonic interpretation of the GPD combination not valid for transverse angular momentum, despite being valid for longitudinal angular momentum?
• RQ4 How does the non-commutativity of boosts and rotations affect the frame dependence of transverse angular momentum sum rules?
• RQ5 Can the GPDs $H_q$ and $E_q$ at zero virtuality and momentum transfer be used to define a dynamical transverse angular momentum operator in light-front quantization?

Key findings

• The expression $\frac{1}{2}\int dx\,x\left[H_q(x,0,0)+E_q(x,0,0)\right]$ does not measure the transverse angular momentum of quarks in a transversely polarized nucleon.
• This expression corresponds to the light-front transverse boost operator $J^{+i}$, which is a kinematic, leading-twist operator, not a dynamical angular momentum operator.
• The true transverse angular momentum operator $J^{-i}$ is a higher-twist, dynamical operator that does not admit a simple partonic interpretation in terms of GPDs at zero virtuality.
• The claim by Ji, Xiong, and Yuan conflates the boost operator $J^{+i}$ with the angular momentum operator $J^{-i}$, leading to a physically incorrect interpretation.
• A genuine transverse angular momentum sum rule must be frame-dependent due to the non-commutativity of Lorentz boosts and rotations.
• The original longitudinal angular momentum sum rule $J_q = \frac{1}{2}\int dx\,x\left[H_q(x,0,0)+E_q(x,0,0)\right]$ remains valid for $\langle J^z_q \rangle_L$, confirming its correctness in the longitudinal case.