REVIEW
Relativistic spatial distributions of transverse angular momentum
High Energy Physics - Phenomenology · Submitted 8 Sep 2026 · 87 references
Pith's verdict: A careful, genuinely incremental formalism paper: the boost sector and pivot dependence are new, the sum rules check out, and the only real gap is an asserted (not shown) cancellation in the light-front TAM matching relation.
T0 · AI review
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Referee report
The paper
The paper claims
Measured about different relativistic centers — energy, mass, or spin — a nucleon's transverse angular momentum and boost occupy different regions of space; this paper derives all three maps, verifies their sum rules, and connects them to…
The reading
What carries the argument
The load-bearing object is the pivot-shift identity at the level of two-dimensional distributions: $J^a_X(b_\perp) = J^a_c(b_\perp) - \tfrac{1}{2}\epsilon^{ab}\sum_{s''}R^b_X(s',s'')P^3(b_\perp;s'',s)$ plus its symmetrized partner, with the boost analogue built from the energy density $E(b_\perp)$ instead of the longitudinal momentum density $P^3$. Here $R_X$ is the position vector of one of the three relativistic centers — the center of energy, the center of mass, and the canonical center of spin, the only one whose components commute and which therefore admits localized position eigenstates. The whole construction lives in the quantum phase-space formalism, where the hadron wave packet is factored into a Wigner distribution and the divergent center-of-wave-packet contribution is discarded by setting the average position $R=0$. For spin-1/2 targets the spinor matrix elements are dressed by the Wigner rotation angle between initial and final states in a generic frame with nonzero longitudinal momentum, which is what assembles the multipole amplitudes out of the form factors $A$, $J$, $D$, $\bar C$, $G_A$, and $G_P$ displayed in Eqs. (40)–(46).
Extended reading notes
Core claim
The paper's central claim is that the transverse spatial distributions of angular momentum and boost relative to the relativistic centers of energy, mass, and spin are obtained from the canonical-center distributions by a systematic subtraction of the external contribution, built from the chosen center's position vector and the spatial distributions of longitudinal momentum and energy. For spin-1/2 targets it gives the full multipole decomposition of these distributions in terms of energy-momentum tensor form factors and the Wigner rotation, and verifies the sum rules $J^a_E = (M/E_P)\,\sigma^a/2$, $J^a_M = (E_P/M)\,\sigma^a/2$, and $J^a_c = \sigma^a/2$ (with $\sigma^a$ the Pauli matrices acting on the target spin), together with $K^a_E = 0$, $K^a_M = (\sigma\times P)^a/2M$, and $K^a_c = (\sigma\times P)^a/2(E_P+M)$. Only the center-of-mass quantities transform as an antisymmetric rank-2 Lorentz tensor, and only at the integrated level, not at the level of the spatial maps themselves. In the light-front formalism, the scaled transverse boost distribution is recovered as the infinite-momentum limit of the scaled instant-form boost about the energy or spin center, while the light-front total angular momentum requires a combination of instant-form total angular momentum and boost because it involves a 'bad' component. The same construction is carried out for the pion, where the transverse TAM distribution vanishes at rest while the boost distribution survives, and the two become equal in magnitude and opposite in sign in the infinite-momentum limit.
Load-bearing premise
Everything that distinguishes the energy-, mass-, and spin-centered distributions rests on the assumption that replacing the full intermediate-state sum by a single on-shell one-nucleon projector in Appendix A is exact, meaning multi-particle and disconnected contributions to the momentum-operator matrix element are absent; if they are not, the subtracted external angular momentum — and with it the pivot-dependent parts of the maps — would shift.
Editorial extensions
If this is right
- The three transverse spin sum rules $J_E=(M/E_P)\sigma/2$, $J_M=(E_P/M)\sigma/2$, and $J_c=\sigma/2$ — previously in tension — are shown to be simultaneously correct statements about different pivots, reproduced at the level of the 2D distributions rather than imposed by hand.
- Transverse boost (energy dipole moment) distributions integrate to zero about the center of energy and to $(\sigma\times P)/2M$ about the center of mass, giving the boost sum rules a clean interpretation as moments of the energy density.
- Only the center-of-mass angular momentum and boost follow the Lorentz transformation law of an antisymmetric rank-2 tensor, and only after integration over the transverse plane; the spatial maps themselves do not transform as tensor components.
- The light-front boost distribution is exactly the infinite-momentum limit of the scaled instant-form boost about the energy or spin center, while the light-front total angular momentum is matched by a combination of instant-form total angular momentum and boost, since a 'bad' light-front component is involved.
- For the pion, the transverse TAM distribution vanishes at rest while the boost distribution does not, and both grow without bound in the infinite-momentum limit with equal magnitudes and opposite signs — a clean discriminating signature between the two densities.
Reading between the lines
- The one-nucleon intermediate-state insertion that makes the external-angular-momentum subtraction exact is the step most worth stress-testing: a two-body or continuum Fock-space calculation of $\langle p',s'|\hat P^a|p,s\rangle$ would reveal whether multi-particle corrections enter the pivot-dependent terms and the sum rules in Eqs. (48)–(54).
- Because the pivot distortions are built from the longitudinal momentum and energy densities, measuring or computing those densities — on the lattice, in spectator models, or eventually at an electron-ion collider — directly tests the predicted differences between the energy-, mass-, and spin-centered maps.
- The matching relations in Eqs. (89)–(90) could be inverted to reconstruct instant-form transverse distributions from light-front (or light-front-lattice) calculations, effectively using the infinite-momentum limit as a bridge between the two formalisms.
- The same pivot recipe should apply to other spatial densities, such as charge, axial charge, or gluon momentum, suggesting a general dictionary for translating Wigner phase-space maps between the three relativistic centers.
Original abstract
In our previous work [C. Lorc\'e et al., Phys. Lett. B 868 (2025) 139792], we investigated the 2D spatial distributions of transverse total angular momentum, including orbital angular momentum and intrinsic spin, relative to the canonical center (or center of spin). In the present work, we extend this analysis in two directions. First, we study the corresponding transverse boost distributions relative to the canonical center. Second, since the definition of generalized angular momentum density depends crucially on the choice of pivot, we analyze how the spatial distributions of transverse total angular momentum and boost are modified when they are defined relative to different relativistic centers, viz.~the relativistic centers of mass, energy, and spin. Considering spin-1/2 targets, we derive the corresponding 2D spatial distributions in the transverse plane, and further investigate how the spatial patterns evolve under longitudinal Lorentz boosts. Additionally, we provide the corresponding light-front distributions in the transverse plane, establishing a clear connection between the instant-form and light-front descriptions of transverse angular momentum and boost.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
In our previous work [C. Lorc\'e et al., Phys. Lett. B 868 (2025) 139792], we investigated the 2D spatial distributions of transverse total angular momentum, including orbital angular momentum and intrinsic spin, relative to the canonical center (or center of spin). In the present work, we extend this analysis in two directions. First, we study the corresponding transverse boost distributions relative to the canonical center. Second, since the definition of generalized angular momentum density depends crucially on the choice of pivot, we analyze how the spatial distributions of transverse total angular momentum and boost are modified when they are defined relative to different relativistic centers, viz.~the relativistic centers of mass, energy, and spin. Considering spin-1/2 targets, we derive the corresponding 2D spatial distributions in the transverse plane, and further investigate how the spatial patterns evolve under longitudinal Lorentz boosts. Additionally, we provide the corresponding light-front distributions in the transverse plane, establishing a clear connection between the instant-form and light-front descriptions of transverse angular momentum and boost.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In our previous work [C. | background 0.99 | 0.11 | 0.01 | unclear 0.99 |
| s1 | Lorc\'e et al., Phys. Lett. B 868 (2025) 139792], we investigated the 2D spatial distributions of transverse total angular momentum, including orbital angular momentum and intrinsic spin, relative to the canonical center (or center of spin). | background 0.99 | 0.37 | 0.04 | asserted 0.99 |
| s2 | In the present work, we extend this analysis in two directions. | contribution 0.94 | 0.36 | 0.03 | asserted 0.85 |
| s3 | First, we study the corresponding transverse boost distributions relative to the canonical center. | contribution 1.00 | 0.87 | 0.04 | asserted 0.91 |
| s4 | Second, since the definition of generalized angular momentum density depends crucially on the choice of pivot, we analyze how the spatial distributions of transverse total angular momentum and boost are modified when they are defined relative to different relativistic centers, viz.~the relativistic centers of mass, energy, and spin. | contribution 1.00 | 0.69 | 0.10 | asserted 0.97 |
| s5 | Considering spin-1/2 targets, we derive the corresponding 2D spatial distributions in the transverse plane, and further investigate how the spatial patterns evolve under longitudinal Lorentz boosts. | contribution 0.93 | 0.96 | 0.12 | asserted 1.00 |
| s6 | Additionally, we provide the corresponding light-front distributions in the transverse plane, establishing a clear connection between the instant-form and light-front descriptions of transverse angular momentum and boost. | contribution 0.94 | 0.95 | 0.17 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:2609.08902 by the authors.
References
Discussion
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