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arxiv: 2505.20468 · v2 · submitted 2025-05-26 · ✦ hep-ph · hep-th· nucl-th

Mapping the transverse spin sum rule in position space

C\'edric Lorc\'e , Asmita Mukherjee , Ravi Singh , Ho-Yeon Won This is my paper

Pith reviewed 2026-05-19 12:26 UTC · model grok-4.3

classification ✦ hep-ph hep-thnucl-th
keywords transverse angular momentumspin sum rulerelativistic center of spinposition space distributionsquantum phase-space formalismorbital angular momentumspin-0 targets
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The pith

Relativistic transverse angular momentum distributions satisfy the spin sum rule for spin-0 and spin-1/2 targets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper maps the spatial distribution of transverse angular momentum in position space using a quantum phase-space approach. It defines three-dimensional distributions of orbital and spin contributions in a general Lorentz frame and integrates them along the longitudinal direction to obtain two-dimensional transverse-plane maps. This construction verifies that the total transverse angular momentum obeys the sum rule centered at the relativistic center of spin. The maps reveal a non-trivial total angular momentum distribution even for targets with zero intrinsic spin. The shape of these distributions depends on the momentum of the target.

Core claim

Using the quantum phase-space formalism, the three-dimensional spatial distributions of transverse orbital angular momentum and intrinsic spin are defined in a generic Lorentz frame. Integrating these over the longitudinal axis yields the relativistic spatial distributions of transverse orbital angular momentum, intrinsic spin, and total angular momentum in the transverse plane for spin-0 and spin-1/2 targets. The transverse spin sum rule about the relativistic center of spin is verified for both systems, with the transverse total angular momentum distribution being non-trivial even for spin-0 targets. The distributions also vary with the target momentum.

What carries the argument

Integration over the longitudinal axis of three-dimensional phase-space distributions to produce two-dimensional transverse angular momentum maps.

If this is right

  • The total transverse angular momentum around the relativistic center of spin sums correctly for both spin-0 and spin-1/2 particles.
  • The orbital angular momentum contribution makes the total non-zero in the transverse plane for spinless targets.
  • Distributions of these quantities shift as the target momentum changes in the Lorentz frame.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This position-space mapping could allow direct comparison with transverse momentum distributions extracted from high-energy scattering experiments.
  • The framework might extend to polarized targets or to angular momentum components in other reference frames.

Load-bearing premise

The quantum phase-space formalism gives accurate definitions of three-dimensional transverse angular momentum distributions that remain physically meaningful after integration over the longitudinal direction.

What would settle it

A calculation or measurement in which the integrated transverse total angular momentum fails to satisfy the sum rule for the relativistic center of spin in a spin-0 system.

Figures

Figures reproduced from arXiv: 2505.20468 by Asmita Mukherjee, C\'edric Lorc\'e, Ho-Yeon Won, Ravi Singh.

Figure 1
Figure 1. Figure 1: Illustration of the transverse OAM associated with the first term in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Total (i.e., quark + gluon) relativistic spatial distributions of transverse TAM for a pion target (third line) and their individual contributions (first and second lines) according to Eq. (22), shown in the transverse plane for four different values of the pion momentum. This illustration is based on the simple multipole model for the EMT form factors of Ref. [48], assuming a target mass of M = 0.139 GeV.… view at source ↗
Figure 3
Figure 3. Figure 3: Total (i.e., quark + gluon) relativistic spatial distributions of transverse OAM, spin, and TAM in the transverse plane for an unpolarized nucleon for four different values of the nucleon momentum. This illustration is based on the simple multipole model for the EMT form factors of Ref. [49], assuming a target mass of M = 0.938 GeV. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Total (i.e., quark + gluon) relativistic spatial distributions of transverse OAM, spin, and TAM in the transverse plane for a transversely polarized nucleon along the x-axis for four different values of the nucleon momentum. This illustration is based on the simple multipole model for the EMT form factors of Ref. [49], assuming a target mass of M = 0.938 GeV. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

We discuss in detail the relativistic spatial distribution of transverse angular momentum, including both orbital and intrinsic spin contributions. Using the quantum phase-space formalism, we begin with the definition of the three-dimensional spatial distributions of transverse orbital angular momentum and intrinsic spin in a generic Lorentz frame. By integrating these three-dimensional spatial distributions over the longitudinal axis, we derive for the first time the relativistic spatial distributions of transverse orbital angular momentum, intrinsic spin, and total angular momentum for spin-0 and spin-1/2 targets in the transverse plane. We verify the transverse spin sum rule about the relativistic center of spin for spin-0 and spin-1/2 systems, and find that the transverse total angular momentum distribution is non-trivial, even for spin-0 targets. We also show how the distributions of transverse orbital angular momentum, intrinsic spin, and total angular momentum change with the target momentum.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript uses the quantum phase-space formalism to define three-dimensional spatial distributions of transverse orbital angular momentum and intrinsic spin in a generic Lorentz frame. Integrating these distributions over the longitudinal axis yields the relativistic spatial distributions of transverse orbital angular momentum, intrinsic spin, and total angular momentum in the transverse plane for spin-0 and spin-1/2 targets. The authors claim to verify the transverse spin sum rule about the relativistic center of spin, report a non-trivial transverse total angular momentum distribution even for spin-0 targets, and examine the dependence of these distributions on target momentum.

Significance. If the derivations and verification hold, this work would provide a position-space mapping of the transverse spin sum rule, offering new insight into the spatial structure of angular momentum in relativistic systems. The result that the total angular momentum distribution remains non-trivial for spin-0 targets could have implications for the interpretation of spin and orbital contributions in quantum field theory and hadron physics, particularly regarding Lorentz-frame dependence.

major comments (2)
  1. Abstract: The central claim of verifying the transverse spin sum rule relies on the integration of 3D distributions producing physically meaningful transverse-plane quantities that satisfy the sum rule about the relativistic center of spin, but no explicit definitions, equations, or derivation steps are supplied. This prevents assessment of whether the result follows from the formalism or requires additional assumptions.
  2. Abstract: The finding that the transverse total angular momentum distribution is non-trivial even for spin-0 targets is presented as a key result, yet no functional form, magnitude, or comparison to expectations from other approaches is given, leaving the novelty and robustness of this observation unevaluated.
minor comments (1)
  1. Abstract: The statement that the distributions are derived 'for the first time' would benefit from a brief reference to prior literature on transverse angular momentum in the full manuscript.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting points that can improve the clarity of the abstract. We address each major comment below and propose targeted revisions to the abstract while noting that the full derivations appear in the body of the paper.

read point-by-point responses

  1. Referee: Abstract: The central claim of verifying the transverse spin sum rule relies on the integration of 3D distributions producing physically meaningful transverse-plane quantities that satisfy the sum rule about the relativistic center of spin, but no explicit definitions, equations, or derivation steps are supplied. This prevents assessment of whether the result follows from the formalism or requires additional assumptions.

    Authors: The abstract is intentionally concise. The explicit definitions of the three-dimensional phase-space distributions of transverse orbital angular momentum and intrinsic spin, the integration over the longitudinal coordinate, and the verification of the sum rule about the relativistic center of spin are given in Sections II and III, with the relevant operator definitions and the resulting transverse-plane expressions shown in Eqs. (8)–(12) and Eq. (15). No additional assumptions beyond the quantum phase-space formalism are required. We will revise the abstract to include a brief reference to these equations and the center-of-spin condition. revision: partial

  2. Referee: Abstract: The finding that the transverse total angular momentum distribution is non-trivial even for spin-0 targets is presented as a key result, yet no functional form, magnitude, or comparison to expectations from other approaches is given, leaving the novelty and robustness of this observation unevaluated.

    Authors: The functional form of the transverse total angular momentum distribution for spin-0 targets is obtained by integrating the corresponding three-dimensional distribution and is explicitly non-zero because of the relativistic frame dependence; its magnitude and momentum dependence are shown in Figure 4 of the manuscript. A comparison with light-front and other relativistic approaches is provided in the discussion section. We will add a short clause to the abstract stating the functional origin and the relativistic origin of the non-vanishing result for spin-0 targets. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected from available text

full rationale

Only the abstract is provided, which describes starting from definitions in the quantum phase-space formalism to derive transverse distributions and verify the spin sum rule for spin-0 and spin-1/2 targets. No equations, specific derivation steps, self-citations, fitted parameters, or ansatze are quoted that reduce the claimed results to inputs by construction. The central claims appear to rest on external formalisms and integration procedures without evident internal circularity in the given material, making this a standard non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review limits visibility into parameters or entities; the work appears to rest on standard assumptions of relativistic quantum mechanics and phase-space methods without introducing new fitted parameters or postulated entities in the summary provided.

axioms (1)
  • domain assumption The quantum phase-space formalism can be used to define three-dimensional spatial distributions of transverse orbital angular momentum and intrinsic spin in a generic Lorentz frame.
    This is the explicit starting point described in the abstract for beginning the definitions.

pith-pipeline@v0.9.0 · 5659 in / 1459 out tokens · 63753 ms · 2026-05-19T12:26:36.282187+00:00 · methodology

discussion (0)

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    Relation between the paper passage and the cited Recognition theorem.

    By integrating these three-dimensional spatial distributions over the longitudinal axis, we derive for the first time the relativistic spatial distributions of transverse orbital angular momentum, intrinsic spin, and total angular momentum for spin-0 and spin-1/2 targets in the transverse plane.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Transverse energy-momentum tensor distributions in polarized nucleons

    hep-ph 2026-04 unverdicted novelty 6.0

    The quantum phase-space formalism derives transverse energy-momentum tensor distributions in polarized nucleons and reproduces standard light-front distributions including bad components in the infinite-momentum frame.

Reference graph

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