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REVIEW 6 minor 83 references

Gluon contribution to the angular momentum distribution of a dressed quark state

T0 review · 0 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The gluon's share of angular momentum in a dressed quark state has decomposition-independent spatial densities once the superpotential is included, and every decomposition satisfies the 1/2 spin sum rule.

desk verdict A solid one-loop light-front calculation with real gluon densities and a verified superpotential identity; the only real soft spot is an underdocumented bridge between the analytic densities and the Gaussian-smearing plots. read the letter →

arxiv 2411.18076 v1 pith:SKWRSKOG submitted 2024-11-27 hep-ph

Asmita Mukherjee , Sudeep Saha , Ravi Singh This is my paper · ORCID
classification hep-ph PACS 12.38.-t12.38.Bx
keywords light-frontHamiltonianQCDangularmomentumdecompositiongluonenergy-momentumtensorimpact-parameterdistributionsdressedquarkstatespinsumruleJaffe-ManoharJi
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks how the gluon's share of angular momentum is spread out inside a spin-1/2 composite state, and whether the answer depends on which decomposition of the energy-momentum tensor one chooses. Working with a quark dressed by a gluon at one loop in light-front Hamiltonian QCD, the authors compute two-dimensional impact-parameter densities for gluon spin, orbital angular momentum, and total angular momentum in the canonical (Jaffe-Manohar), kinetic (Ji), and Belinfante decompositions. Their central result is that once the gluon superpotential term is included, the total gluon angular-momentum density is the same in all decompositions, and the integrated spin sum rule of $1/2$ holds in each. This matters because boundary terms that vanish under integration are usually dropped, yet they control how spin is distributed inside the proton and whether quark and gluon densities can be compared meaningfully.

What carries the argument

The machinery is the Fock-space expansion of a quark dressed with a gluon, kept through the two-particle sector, with the two-particle light-front wave function taken from one-loop light-front Hamiltonian QCD in the $A^+ = 0$ gauge. A two-component formalism eliminates the constrained fields through the light-front constraint equations, so all matrix elements of the gluon energy-momentum tensor can be evaluated analytically. Impact-parameter densities are obtained from off-forward matrix elements in the Drell-Yan frame ($\Delta^+ = 0$, $P_\perp = 0$) by Fourier transforming in the transverse momentum transfer. The gluon superpotential $M^z_g$ is the boundary term that converts the kinetic (or Belinfante) gluon angular-momentum density into the canonical one; carrying this term through the calculation is what makes the total gluon density decomposition-independent.

What would settle it

Repeat the same one-loop calculation while keeping the three-gluon term in $T^{+k}_{\mathrm{kin},g}$ that is dropped after Eq. (C8), and check whether $J^z_{\mathrm{can},g} = J^z_{\mathrm{kin},g} + M^z_g$ still holds and the integrated sums still give $1/2$; a failure in either check would show that the truncation is load-bearing rather than benign.

Watch

Extended reading notes

Core claim

For a dressed quark state truncated to the one-quark and one-quark-one-gluon Fock sectors, the gluon part of the energy-momentum tensor produces analytic, boost-invariant angular-momentum densities in transverse impact-parameter space. The paper's key identity is $J^z_{\mathrm{can},g}(b_\perp) = J^z_{\mathrm{kin},g}(b_\perp) + M^z_g(b_\perp)$, where $M^z_g$ is the superpotential (total-divergence) term; the same equality holds with the Belinfante density in place of the kinetic one, and it is valid for every cutoff $\Lambda$. After integrating over $b_\perp$, the cutoff dependence cancels between the single-particle and two-particle Fock sectors and between quark and gluon terms, so the longitudinal spin sum rule is verified for each decomposition: $\langle L^z_{\mathrm{kin},q}\rangle + \langle S^z_{\mathrm{kin},q}\rangle + \langle J^z_{\mathrm{kin},g}\rangle = 1/2$, $\langle J^z_{\mathrm{Bel},q}\rangle + \langle J^z_{\mathrm{Bel},g}\rangle = 1/2$, and $\langle L^z_{\mathrm{can},q}\rangle + \langle S^z_{\mathrm{can},q}\rangle + \langle L^z_{\mathrm{can},g}\rangle + \langle S^z_{\mathrm{can},g}\rangle = 1/2$.

Load-bearing premise

The entire calculation assumes the dressed quark is well described by just the one-quark and one-quark-one-gluon Fock sectors, with the two-particle wavefunction fixed at one loop; three-gluon interactions and any higher Fock states are neglected, and if those contributed at the same order the densities and the cancellations that produce the spin sum rule would be altered.

Editorial extensions

If this is right

  • The gluon total angular-momentum density is decomposition-independent once the superpotential is included, so boundary terms cannot be ignored in position-space studies even though they vanish upon integration.
  • Within the dressed quark state, gluon OAM density is positive and gluon spin density is negative, and the gluon contribution is much smaller than the quark contribution.
  • Because in the light-front gauge the transverse gluon field is purely physical, the gauge-invariant canonical and kinetic decompositions reduce to the Jaffe-Manohar and Ji decompositions respectively, extending the result to those formulations.
  • The integrated spin sum rule holds for kinetic, Belinfante, and canonical decompositions, with the single-particle Fock-state normalization supplying the $1/2$ and all cutoff-dependent pieces canceling between quark and gluon terms.
  • The equality $J^z_{\mathrm{can},g} = J^z_{\mathrm{kin},g} + M^z_g$ is independent of the transverse-momentum cutoff, although the individual OAM and spin densities depend on it.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the same pattern survives in a state with gluon self-interactions, the density-level equality between canonical and kinetic decompositions is likely a general consequence of the superpotential structure rather than a special feature of the one-loop two-particle truncation.
  • For a proton, the superpotential term would show up as a model-independent difference between Jaffe-Manohar and Ji densities; measuring only integrated quantities would miss precisely the information this calculation exposes.
  • One could test the framework further by computing the same densities for a quark dressed with two gluons or for a quark-diquark state with a gluon, checking whether the cancellation of cutoff dependence persists beyond the one-quark-one-gluon sector.
  • The densities computed here are wave-packet-broadened by a Gaussian of width $\sigma$; comparing the $\sigma \to 0$ limit with a direct fixed-$b_\perp$ calculation would provide a numerical cross-check of the Fourier prescription.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The manuscript computes, for a quark dressed with a gluon in light-front Hamiltonian QCD, the gluonic contribution to the longitudinal angular momentum density in impact-parameter space. The authors derive analytic expressions for the canonical (Jaffe-Manohar), kinetic (Ji), and Belinfante decompositions, including the gluon superpotential density M_z^g that connects the canonical and kinetic total gluon angular momentum densities. Combining these with their earlier quark-sector results, they verify that the integrated spin sum rule holds in each decomposition, with the ultraviolet cutoff Λ cancelling explicitly among the O(g^2) terms. The paper also presents numerical plots showing the dependence of the densities on the impact parameter, the cutoff, and the width of a Gaussian wave packet.

Significance. If the results hold, this is a useful one-loop benchmark study: it is one of the few models with explicit gluonic degrees of freedom in which the density-level difference between canonical and kinetic decompositions is exhibited analytically, and in which the integrated spin sum rule is verified by direct cancellation rather than by construction. The paper's strengths are the explicit analytic formulas for the gluon densities, the demonstration that J_z^can,g = J_z^kin,g + M_z^g holds for arbitrary cutoff, and the explicit verification that M_z^g integrates to zero, so that the total gluon angular momentum density is decomposition-independent. The work complements the same group's previous quark-sector calculation and provides a concrete testing ground for light-front techniques.

minor comments (6)
  1. [Section VII / Section IV] The plotted densities in Section VII are obtained with the Gaussian wave packet of Eq. (45), but the analytic formulas (50)-(54) are plane-wave impact-parameter densities with no σ dependence. The smearing factor that relates these two objects is never written down, so the numerical curves cannot be reproduced from the stated formulas. Please provide the convolution kernel or the explicit σ-dependent form used for the plots; this is a reproducibility issue even though it does not affect the analytic sum-rule verification.
  2. [Sections V and VI] The symbol x is used with two different meanings: in the gluon density formulas Eqs. (50)-(54) and in Appendix B it denotes the gluon longitudinal momentum fraction, while in Section VI (e.g., just below Eq. (61)) it is explicitly defined as the quark momentum fraction. This redefinition is natural under x → 1−x but it is not stated at the point of use. Please state the convention clearly at the start of Section V and again in Section VI, or use a different symbol for the gluon fraction.
  3. [Section IV, Eq. (45)] There is a typo in the sentence introducing the wave packet: "We a choose" should read "We choose".
  4. [Appendix A and Section VI] The x-integrations in the sum-rule evaluations are performed without specifying the integration domain or the treatment of the endpoint x=1, where log(1−x) is singular but the products with powers of (1−x) are finite. A brief statement of the integration range and limiting procedure would help the reader reproduce Eqs. (61), (63), and (65).
  5. [Figure 3 caption] The caption states "g = Cf = Nf = 1", but Nf is not defined elsewhere in the paper and the model has no flavor sum; presumably this should be g = Cf = 1.
  6. [References] Reference [58] contains a typo in the arXiv number: "hep-ph-/9705477" should read "hep-ph/9705477".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: gluon densities and sum-rule cancellations are computed from independent one-loop LFWFs, not fitted or defined into existence.

full rationale

The paper's central objects are gluon EMT matrix elements evaluated with the one-loop quark-gluon LFWF of Eq. (37), which is taken from [57] and is not constructed to enforce the angular momentum sum rules. The Fock-space truncation to one- and two-particle sectors is a controlled O(g^2) approximation; the non-diagonal overlaps vanish by gluon-number counting and the three-gluon terms are higher order (Appendix C). The quark-sector distributions used in Sec. VI are imported from the separate published calculation [61]; they are not fit parameters, and the sum-rule verification consists of explicit analytic cancellations of the cutoff-dependent terms (Eqs. 61, 63, 65) that would fail if the gluon matrix elements were incorrect. The relation J_can,g = J_kin,g + M_g follows from the definition of the superpotential (Eq. 14), but the paper computes each distribution separately from the LFWFs and uses the equality as a consistency check at density level, not as a fitted input. No parameter is tuned to force the integrated spin sum rule. The only notable gap is a reproducibility issue, not a circular one: the analytic densities of Sec. V are plane-wave expressions, while the numerical plots in Sec. VII introduce a Gaussian width sigma without displaying the smearing relation; this affects the plots but not the analytic sum-rule verification.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the one-loop dressed quark LFWF from [57], the two-component light-front formalism, and the Fock-space truncation to single- and two-particle sectors. The numerical plots use hand-chosen parameters (m=0.3 GeV, g=1, C_f=1, Lambda=2.63 GeV, sigma varied). The GIE decompositions are shown to coincide with canonical and kinetic in the light-front gauge, so only the latter two are computed. No invented entities are introduced.

free parameters (5)
  • quark mass m = 0.3 GeV (numeric; kept symbolic in analytic results)
    Input quark mass for the dressed quark state; only affects numerical curves, not the analytic sum-rule cancellations.
  • quark-gluon coupling g = 1 (numeric)
    Overall coupling strength; results scale as g^2 C_f.
  • color factor C_f = 1 (numeric; SU(3) value would be 4/3)
    Overall color factor scaling; the choice of 1 is arbitrary for plots.
  • UV cutoff Lambda = 2.63 GeV (numeric)
    Transverse momentum cutoff; individual densities depend on it, but the total spin sum rule is cutoff-independent.
  • Gaussian wave packet width sigma = 0.05 to 0.2 GeV (varied)
    Wave packet width used for plotting smooth distributions; widths are varied in Fig. 3.
assumptions (5)
  • domain assumption The dressed quark state is expanded in Fock space keeping only single-quark and quark-gluon two-particle sectors (Eq. 34).
    Truncation of the Fock space; sufficient at one-loop order in the coupling.
  • domain assumption The two-particle LFWF is taken from one-loop light-front Hamiltonian QCD [57] (Eq. 37).
    The LFWF is an external result from the literature that the calculation relies on.
  • domain assumption Three-gluon interactions and higher-order terms are neglected in the EMT matrix elements (Appendix C, before Eq. C8).
    Keeps the calculation at O(g^2). This is a stated approximation.
  • domain assumption Light-front gauge A^+=0 and the two-component formalism, where constrained fields are eliminated via equations of constraint (Section IV).
    Foundational framework for the calculation, from [57,59].
  • domain assumption The Gaussian wave packet with width sigma (Eq. 46) is used to render the impact-parameter distributions and plots.
    Regularization and plotting choice from [55,56,69,70].

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Cite this review

Pith. "Pith review of Gluon contribution to the angular momentum distribution of a dressed quark state." pith.science (2026). https://pith.science/paper/SKWRSKOG

@misc{pith2026241118076,
  author       = {Pith},
  title        = {Pith review of: Gluon contribution to the angular momentum distribution of a dressed quark state},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SKWRSKOG}},
  note         = {Machine review of arXiv:2411.18076}
}
abstract

We compute the contribution of the gluonic component of the energy-momentum tensor (EMT) to the angular momentum density in various decompositions. We use the light-front Hamiltonian technique, and a two-component formalism in light-front gauge, where the constrained degrees of freedom are eliminated. Instead of a nucleon, we consider a simple composite spin-$1/2$ state, namely a quark dressed with a gluon. We present two dimensional light-front distributions in transverse impact parameter space, and compare the different angular momentum decompositions at the density level. Incorporating also the contribution coming from the quark part of the EMT, we verify the spin sum rule for such a state.

Figures

Figures reproduced from arXiv: 2411.18076 by the authors.

Figure 1
Figure 1. FIG. 1: Plots of longitudinal angular momentum distribution of gluons as a function of [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Plot of the dependence of different components of AM distribution on transverse momentum UV cutoff. Top-left: [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Plots showing the dependence of different components of AM distribution on the width of the Gaussian wave-packet, [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

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Reference graph

Works this paper leans on

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    We just state the final result here for completion ⟨Lz kin,q⟩(b⊥) = i Z d2∆⊥ (2π)2 e−ib⊥·∆⊥ " ∂⟨T +1 kin,q⟩LF ∂∆(2) ⊥ − ∂⟨T +2 kin,q⟩LF ∂∆(1) ⊥ # DY

    Quark OAM distribution The distribution of kinetic quark OAM has already been derived in the previous work. We just state the final result here for completion ⟨Lz kin,q⟩(b⊥) = i Z d2∆⊥ (2π)2 e−ib⊥·∆⊥ " ∂⟨T +1 kin,q⟩LF ∂∆(2) ⊥ − ∂⟨T +2 kin,q⟩LF ∂∆(1) ⊥ # DY . = g2CF 72π2 Z d2∆⊥ (2π)2 e−ib⊥·∆⊥ −7 + 6 ω 1 + 2m2 ∆2 log 1 + ω −1 + ω − 6 log Λ2 m2 , (C1) where ...

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    Gluon total AM distribution The operator structure of gluon EMT in the kinetic decomposition is given by Eq.(7) T +k kin,g = − 2Tr G+λGk λ =2Tr 1 2 G+−G+k + G+⊥Gk⊥ = 1 2 G+− a G+k a + G+⊥ a Gk⊥ a (C7) = ∂+Ak a ∂⊥A⊥ a + ∂+A⊥ a ∂kA⊥ a − ∂+A⊥ a ∂⊥Ak a | {z } D + 2g ∂+Ak a 1 ∂+ ξ†taξ | {z } ND The notations D and ND in the above expression denote the diagonal...

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Pith tools

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