[Paper Review] Topological approach to proton spin problem: decomposition controversy and beyond

S. C. Tiwari|arXiv (Cornell University)|Sep 10, 2015

Particle physics theoretical and experimental studies7 references3 citations

TL;DR

This paper challenges the gauge symmetry paradigm (GSP) proposed by Chen et al. for decomposing proton spin into quark and gluon contributions, arguing that a logically consistent, manifestly Lorentz-covariant, and gauge-invariant formulation under GSP is impossible—either trivial or equivalent to the conventional transverse-longitudinal decomposition. It advocates a nonperturbative topological approach using generalized de Rham theorems and non-Abelian Stokes theorems to treat spin as a topological invariant in QCD, suggesting new physics beyond current decomposition controversies.

ABSTRACT

Lorentz covariant and gauge invariant definitions of quark and gluon spin and orbital angular momenta continue to pose a great theoretical challenge. A major controversy on the fundamental concepts followed Chen et al proposal: the basic idea is to split the gauge potential into pure gauge and physical components motivated by the gauge symmetry. We term it gauge symmetry paradigm (GSP) to distinguish it from the well-known inertial frame dependent transverse-longitudinal decomposition (TLP). A thorough study adhering to the traditional meaning of Lorentz covariance and gauge invariance is reported; it leads to a new result: logically consistent development of GSP does not exist and Chen et al proposal turns out to be either trivial or metamorphosed into TLP. Going beyond the controversy and the spin sum rules the necessity for a nonperurbative QCD approach to address the proton spin problem is underlined. We suggest topological approach: generalized de Rham theorems for QCD, and spin as a topological invariant for baryons are discussed. Nonabelian Stokes theorem is applied to derive color flux for the closed loop in a variant of Burkardt's U-shaped path. Similarity between Chen et al decomposition and Kondo decomposition of the gauge potential is suggestive of a topological perspective on the Chen et al proposal with interesting physics

Motivation & Objective

• To critically assess the logical consistency of Chen et al.'s gauge symmetry paradigm (GSP) for proton spin decomposition.
• To resolve the long-standing controversy over manifest Lorentz covariance and gauge invariance in quark and gluon angular momentum definitions.
• To argue that the GSP proposal either reduces to triviality or becomes equivalent to the transverse-longitudinal decomposition (TLP), undermining its claimed novelty.
• To advocate for a nonperturbative, topological approach to the proton spin problem, moving beyond conventional sum rules.
• To explore the role of topological invariants and generalized de Rham theorems in non-Abelian gauge theories for understanding baryon spin.

Proposed method

• Analyzes Chen et al.'s decomposition by rigorously applying Lorentz covariance and gauge invariance, showing logical inconsistencies in the GSP framework.
• Applies generalized de Rham theorems to non-Abelian gauge fields, extending cohomological methods to QCD with SU(N) structure.
• Uses the non-Abelian Stokes theorem to compute color flux along a U-shaped path in spacetime, linking it to topological invariants.
• Compares the Chen et al. decomposition with Kondo's gauge potential decomposition, revealing a suggestive topological parallel.
• Introduces the concept of spin as a topological invariant in baryons, motivated by homology and cohomology duality in differential geometry.
• Considers the implications of metric-independent differential forms and Hodge duality in Lorentzian spacetime for QCD.

Experimental results

Research questions

• RQ1 Can a manifestly Lorentz-covariant and gauge-invariant decomposition of proton spin into quark and gluon spin and orbital angular momentum be consistently formulated?
• RQ2 Does Chen et al.'s gauge symmetry paradigm (GSP) provide genuinely new physics beyond existing decompositions like Jaffe-Manohar or Ji?
• RQ3 Is the GSP proposal logically consistent when rigorously applied, or does it reduce to triviality or effectively become equivalent to the transverse-longitudinal decomposition?
• RQ4 Can topological invariants in non-Abelian gauge theories, such as those derived from de Rham cohomology, provide a new framework for understanding baryon spin?
• RQ5 What is the role of the non-Abelian Stokes theorem and U-shaped paths in defining color flux and topological structures in QCD?

Key findings

• A logically consistent, manifestly Lorentz-covariant, and gauge-invariant formulation of the Chen et al. GSP proposal does not exist; it is either trivial or equivalent to the conventional transverse-longitudinal decomposition (TLP).
• The GSP approach, when rigorously analyzed, fails to provide a new physical framework beyond existing decompositions, undermining its claimed novelty.
• The paper identifies a nontrivial similarity between Chen et al.'s decomposition and Kondo's gauge potential decomposition, suggesting a deeper topological origin.
• A topological approach based on generalized de Rham theorems and non-Abelian Stokes theorems offers a promising path forward for nonperturbative QCD analysis of proton spin.
• Spin in baryons may be understood as a topological invariant, with implications for the global structure of QCD vacuum and nonperturbative dynamics.
• The extension of de Rham cohomology to non-Abelian gauge theories requires careful treatment of Lie algebra-valued differential forms and nonlinear field equations, posing significant mathematical challenges.