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Angular momentum of glasma

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Pith's one-line read The paper argues that the beam-axis polarization of final-state hadrons is set by the glasma's local angular momentum, not by vorticity or thermal vorticity, and it identifies the exact cancellation that makes the two quantities differ.

desk verdict Carrington and Mrowczynski argue the glasma's local angular momentum, not vorticity, should set the sign of Lambda polarization; the mechanism hinges on a cancellation that may not survive at the order used in the numerics. read the letter →

arxiv 2505.07324 v3 pith:ZS4TOOFZ submitted 2025-05-12 nucl-th hep-ph

Margaret E. Carrington , Stanislaw Mrowczynski This is my paper · ORCID
classification nucl-thhep-ph PACS 25.75.-q12.38.Mh
keywords glasmaangularmomentumvorticityspinpolarizationheavy-ioncollisionspropertimeexpansionPoyntingvectorLambda
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

The paper studies the glasma, the dense gluon field created in the first instants of an ultrarelativistic heavy-ion collision, and asks where its angular momentum lives. It finds that only a small fraction of the incoming nuclei's large angular momentum is transferred to the glasma, so the idea of a rapidly rotating fireball does not apply. Its central claim is about the beam-axis component: the glasma's local angular momentum and its vorticity behave qualitatively differently, because the leading derivative term in the local angular momentum vanishes exactly. The surviving term, built from third derivatives of the Poynting vector, gives a quadrupole pattern at large impact parameter whose sign matches the measured Lambda polarization, while the vorticity has the opposite sign. For that reason the paper argues that local angular momentum, not thermal vorticity, drives the polarization of final-state hadrons.

What carries the argument

The load-bearing identity is the equation of universal flow, $T^{0x}=-\frac{1}{2}t\,\partial_x T^{00}$ and $T^{0y}=-\frac{1}{2}t\,\partial_y T^{00}$ at mid-rapidity, which relates the Poynting vector to gradients of the energy density and is exactly satisfied by the proper-time-expanded energy-momentum tensor through seventh order. Plugging this identity into the expression for the local angular momentum per unit rapidity, $dL_z/d\eta=-\tau\int d^2R\,(R_y T^{0x}-R_x T^{0y})$, kills the $\Delta^4$ derivative term that would otherwise mimic the vorticity, leaving the $\Delta^6$ term built from third derivatives of the Poynting vector. Vorticity, by contrast, is a first derivative of the velocity field $V^i=T^{0i}/T^{00}$. The different derivative order is what decouples the two observables.

What would settle it

Evolve the glasma energy-momentum tensor from $\tau\simeq 0.06$ fm to freeze-out with a matched transport or hydrodynamic model and check whether the quadrupole sign of $L_z$ at large impact parameter survives; if the sign flips or the final polarization tracks thermal vorticity instead of local angular momentum, the central claim is wrong. A purely experimental version is a measurement of the azimuthal angle dependence of beam-axis $\Lambda$ polarization as a function of impact parameter, looking for the predicted octupole-to-quadrupole crossover around $b\approx 4$ fm.

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Extended reading notes

Core claim

The central discovery is that for the glasma at mid-rapidity the equation of universal flow, $T^{0x}=-\frac{1}{2}t\,\partial_x T^{00}$ and $T^{0y}=-\frac{1}{2}t\,\partial_y T^{00}$, forces the leading $\Delta^4$ contribution to the local beam-axis angular momentum to vanish identically. As a result $L_z$ is controlled by the $\Delta^6$ term, which involves third spatial derivatives of the Poynting vector, whereas the vorticity $\omega_z$ is a first derivative of the velocity field. These two quantities therefore need not look alike, and in the computed glasma they do not: at small impact parameter $L_z$ is mostly octupole, at large impact parameter it develops a quadrupole component whose sign agrees with the measured $\Lambda$ polarization along the beam, while the glasma vorticity shows a quadrupole of the opposite sign. The paper also finds that the global angular momentum perpendicular to the reaction plane carried by the glasma is much smaller than the angular momentum of the participants, implying that the initial angular momentum stays with the valence quarks.

Load-bearing premise

The calculation is done at a single early instant, $\tau=0.06$ fm, with a proper-time expansion valid only below about 0.08 fm, and the paper's own conclusion is that it is unclear whether the early-time glasma is representative of the later system that produces the measured hadrons.

Editorial extensions

If this is right

  • Spin-hydrodynamic descriptions that compute hadron polarization from thermal vorticity would need to be replaced or supplemented by a mechanism based on local angular momentum; the sign mismatch in the data is the paper's motivation.
  • The glasma phase is not a rapidly rotating fireball: the global angular momentum imparted to it is only a small fraction of the participants' angular momentum, consistent with vanishing global polarization at LHC energies.
  • The spin sign puzzle is resolved in sign at the glasma stage: the quadrupole part of $L_z$ at large impact parameter matches the measured $\Lambda$ polarization, while the vorticity has the opposite sign.
  • Because the leading derivative term in $L_z$ vanishes by universal flow, any early-time system with a boost-invariant, mostly diagonal energy-momentum tensor will show the same decoupling of vorticity from local angular momentum.

Reading between the lines

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

  • The paper's identity suggests a testable generalization: in any early-time model that satisfies the equation of universal flow, the local angular momentum should be dominated by third derivatives of the Poynting vector, so the same octupole-to-quadrupole crossover with impact parameter should appear.
  • A quantitative bridge is missing: the paper notes that no calculation connects the early-time $L_z$ to freeze-out polarization, so one could match a transport or hydrodynamic code at $\tau\simeq 0.06$ fm and evolve the sign pattern to see if it survives.
  • If the quadrupole $L_z$ is the correct driver, a clean experimental discriminator is the impact-parameter and multiplicity dependence of the azimuthal pattern of $\Lambda$ polarization: it should cross from octupole-dominated at small $b$ to quadrupole-dominated at large $b$.
  • The $\Delta^4$ cancellation may also explain why hydrodynamic models need ad hoc shear corrections to reproduce the sign: first-derivative quantities such as thermal vorticity are the wrong variable in a far-from-equilibrium system.
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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

4 major / 6 minor

Summary. This paper studies angular momentum of the glasma produced in ultrarelativistic heavy-ion collisions, using the authors' CGC-based proper-time expansion (up to eighth order) at τ=0.06 fm and mid-rapidity. It computes the global angular momentum perpendicular to the reaction plane and finds that only a small fraction of the initial nuclear angular momentum is transferred to the glasma. Its main subject is the local angular momentum along the beam axis, Lz, and the vorticity ωz of the transverse velocity field V = P/T^{00}. The authors find that Lz and ωz have qualitatively different spatial patterns: ωz is quadrupole-dominated with a sign opposite to the measured Λ polarization, while the quadrupole part of Lz at large impact parameter has the same sign as the measured Λ polarization. They explain the difference by the equation of universal flow T^{0i}=−(t/2)∂i T^{00} (Eq. (5.17)), which makes the order-Δ^4 contribution to Lz vanish, leaving a Δ^6 term built from third derivatives of the Poynting vector, whereas vorticity is a first derivative of the velocity field. They argue that local angular momentum, not thermal vorticity, may control the polarization of final-state hadrons.

Significance. If the Δ^6-dominance claim is correct, the paper establishes a qualitative distinction between vorticity and local angular momentum in the glasma that is not present in the naive rigid-body argument, and it offers a fresh perspective on the 'spin sign puzzle.' The analysis is transparent in several respects: the proper-time expansion is carried to eighth order with comparisons of sixth- and eighth-order results for Ly; the Fourier decomposition (Sec. 5B) is explicit; and the authors provide error bars based on the gradient-expansion cutoff δ. The central phenomenological statement is falsifiable in the sense that the sign of the quadrupole local angular momentum at large impact parameter is compared directly with data. However, the key mechanism — the vanishing of the Δ^4 term — rests on a relation that is exact only up to seventh order in the proper-time expansion, while the numerical results use the eighth-order tensor, and the numerical verification of Δ^6 dominance is not shown. This gap is the main obstacle to accepting the paper's central claim.

major comments (4)
  1. [Sec. 5E, Eq. (5.17)] The claim that the order-Δ^4 contribution to Lz is 'identically zero' is not established for the quantity actually computed. Equation (5.17) is stated to hold 'exactly ... up to seventh order' in the proper-time expansion, while the numerical Lz in Sec. 5D is computed from the eighth-order energy-momentum tensor. The Δ^4 coefficient is the curl ∂xT^{0y}−∂yT^{0x}; any eighth-order correction to T^{0i} contributes to it, so the cancellation is, at best, approximate in the presented calculation. The only numerical support is the sentence 'We have verified numerically that Lz(⃗ r)/Δ^6 is approximately independent of Δ,' with no data, no range of Δ, and no error estimate. Please provide a quantitative verification: e.g., a plot of Lz(⃗ r)/Δ^6 for several values of Δ (say 0.1, 0.15, 0.2, 0.25, 0.3 fm) and the separate magnitudes of the Δ^4 and Δ^6 coefficients. This check is load-bearing because the qualitative difference between Lz and vorticity, and the sign argument, rest entirely on Δ^6 dominance.
  2. [Secs. 5C and 5E] The use of Eq. (5.17) appears inconsistent with the nonzero vorticity reported in Sec. 5C. If Eq. (5.17) were exact for the T used to compute ωz, then V = −(t/2)∇ ln T^{00} and ωz would vanish identically. The resolution is that Eq. (5.17) holds only up to seventh order in the proper-time expansion; this caveat needs to be stated wherever Eq. (5.17) is invoked. Moreover, the size of the eighth-order violation of Eq. (5.17) sets the scale of the residual Δ^4 contribution to Lz. Please quantify ∂xT^{0y}−∂yT^{0x} relative to the third-derivative combinations in Eq. (5.12) to demonstrate that the residual Δ^4 term is negligible in the region and at the impact parameters used for the sign statement.
  3. [Secs. 5D, 6, Figs. 12-14] The phenomenological sign claim relies on the quadrupole component Φ_L^2 at b=6 fm, but the dominant contribution to Φ_L comes from large transverse distances where the gradient expansion is least trustworthy, as the authors acknowledge in Sec. 2 and in footnote 2. The error bars obtained by varying δ_max between 0.8 and 1.0 probe only the gradient-expansion cutoff; they do not test the radial weighting of the integral or the sensitivity to the proper-time truncation order. Please provide a radial-shell decomposition of Φ_L^2 or an equivalent sensitivity test (e.g., varying the upper limit Rmax in Eq. (5.15)) and state explicitly whether the sign of the quadrupole contribution is stable.
  4. [Abstract and Secs. 6-7] The abstract's statement that 'neither vorticity nor thermal vorticity but instead the local angular momentum controls the polarization of final-state hadrons' is stronger than what the body supports. Section 6 states 'Since our method only works at early times, it is unclear if our results have any relevance for a description of the later stages of the collision,' and Section 7 states that 'a quantitative approach which would allow one to relate the local angular momentum to polarization needs to be developed.' Please align the abstract with these caveats, e.g., by saying that the results 'suggest' or 'are consistent with' a role for local angular momentum, or by adding a concrete (even schematic) argument for how the early-time Lz pattern would survive to freeze-out.
minor comments (6)
  1. [Throughout] There are several typos: 'preceeds' (p. 2) should be 'precedes', 'determing' (Sec. 7) should be 'determining', 'emphasis' (Sec. 5D) should be 'emphasize', and 'local local' (Sec. 5E) should be 'local'.
  2. [Eqs. (5.8)-(5.9)] The symbol h_n appears without definition and the displayed formula for ωz seems to have unbalanced parentheses; please fix.
  3. [Sec. 2 and Sec. 5B] The use of δ both for the gradient-expansion parameter (Sec. 2) and for the non-radial Fourier coefficient δ_n(r) (Sec. 5B) is confusing; consider renaming one of them.
  4. [Sec. 4, Fig. 5] The discussion around Fig. 5 states that 'the curves obtained at the fourth and eighth orders show rapid growth' and that 'the second and sixth order results suggest the saturation,' which is confusing because the preceding sentence says the sixth- and eighth-order results agree for τ < 0.06 fm; please clarify which orders are being compared and why the eighth-order curve departs from the sixth-order one.
  5. [Sec. 5D] The grid geometry is not fully specified: the transverse plane dimensions and the total number of boxes are not given.
  6. [Sec. 6] The estimate ω ∼ 10^{−3} fm^{−1} relies on the assumption that ∫ d^2σ·ω is conserved; the authors flag this with 'If we assume,' but it would be helpful to state explicitly that this is an order-of-magnitude assumption with no derivation for the glasma stage.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the glasma L_z calculation is self-contained and the sign comparison with Λ polarization is an external benchmark.

full rationale

The glasma angular-momentum calculation is self-contained. Lz is computed directly from Eq. (3.8) by integrating the eighth-order energy-momentum tensor components over small transverse boxes; no parameter of the calculation is fitted to the Λ-polarization data. The vanishing of the Δ^4 contribution is explained through the universal-flow relation (5.17), which the paper attributes to Ref. [41] and states was derived for the glasma in the authors' Ref. [28]. Although [28] is a self-citation, the relation is a parameter-free statement about the proper-time-expanded energy-momentum tensor, not a fit to the observable being explained, and the paper additionally reports a direct numerical check that Lz/Δ^6 is approximately independent of Δ. The sign comparison of the quadrupole part of Lz with measured Λ polarization is an external benchmark rather than an input. The Sec. 6 admission that the method works only at early times is a limitation about time evolution, not a circular step. The possible objection that Eq. (5.17) is exact only through seventh order while the numerics use eighth order is a technical correctness concern about the size of the Δ^4 coefficient; it does not make the derivation circular, since the numerical Δ-independence check, if reproducible, would test that coefficient directly. Overall no load-bearing step reduces by definition or by fitted input to the conclusion.

Assumptions & free parameters 9 free parameters · 8 assumptions · 0 invented entities

The calculation rests almost entirely on the authors' own CGC proper-time expansion framework (Refs 22-29), including the Glasma Graph approximation and the equation of universal flow from Ref. [28]. No new entities are postulated. The main free parameters are the saturation scale, coupling, infrared cutoff, evaluation time, gradient expansion cutoff, box size, and truncation order; the last two enter the local angular momentum directly.

free parameters (9)
  • Saturation scale Qs = 2 GeV
    Sets the momentum scale of the CGC calculation and the color charge density via mu_bar = Qs^2/g^4; chosen, not fitted.
  • Gauge coupling g = 1
    Coupling fixed to 1, standard for this type of glasma calculation.
  • Infrared cutoff m = 0.2 GeV
    Regulates long-distance behavior of the correlators; chosen by hand.
  • Proper time tau = 0.06 fm
    Evaluation time, near the upper end of the eighth-order expansion validity (tau less than about 0.08 fm).
  • Gradient expansion cutoff delta_bar = 0.9 (varied 0.8 to 1.0)
    Determines the integration region; error bars derived from varying it.
  • Box size Delta = 0.2 fm
    Size of the square integration region for local angular momentum; results scaled by Delta^6.
  • Proper-time truncation order = 8
    Series truncated at eighth order; convergence shown for Ly at early times but no error bar from this choice for Lz or vorticity.
  • Woods-Saxon radius parameter r0 = 1.25 fm
    Nuclear radius parameter, standard value.
  • Woods-Saxon skin thickness a = 0.5 fm
    Nuclear skin thickness, standard value.
assumptions (8)
  • domain assumption CGC effective theory with Gaussian color charge distribution in each nucleus.
    Section 2; basis for computing gluon fields and averaging over color configurations.
  • domain assumption Glasma Graph approximation: Wick's theorem applied to gauge potentials rather than color charges.
    Section 2; defines the correlator (2.4) used for all observables.
  • domain assumption Proper-time expansion ansatz (2.1) for the gauge potentials in the forward light cone.
    Eq. (2.1); assumed form of the solution, solved iteratively in powers of tau Qs.
  • domain assumption Infinitely contracted nuclei, width w -> 0 in the boundary conditions.
    Eqs. (2.2)-(2.3); standard ultrarelativistic approximation.
  • domain assumption Boost invariance and evaluation at mid-rapidity eta = z = 0.
    Section 2; restricts the calculation to mid-rapidity, where t = tau.
  • domain assumption Equation of universal flow (5.17): T^{0x} = -(1/2)t d_x T^{00}, T^{0y} = -(1/2)t d_y T^{00}.
    Taken from Ref. [28], where it is shown to hold exactly for the proper-time expanded energy-momentum tensor up to seventh order; used in Sec. 5E to prove the Delta^4 term vanishes.
  • standard math Belinfante improved energy-momentum tensor is symmetric and gauge invariant.
    Section 3, Eq. (14) of Ref. [22]; used to define angular momentum.
  • domain assumption Woods-Saxon distribution (2.5) for nuclear density with r0 = 1.25 fm, a = 0.5 fm.
    Section 2; standard nuclear profile generating the color charge density.

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Pith. "Pith review of Angular momentum of glasma." pith.science (2026). https://pith.science/paper/ZS4TOOFZ

@misc{pith2026250507324,
  author       = {Pith},
  title        = {Pith review of: Angular momentum of glasma},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZS4TOOFZ}},
  note         = {Machine review of arXiv:2505.07324}
}
read the original abstract

The earliest phase of an ultrarelativistic heavy ion collision can be described as a highly populated system of gluons called glasma. We study some glasma characteristics related to the system's angular momentum. The first one is the global angular momentum perpendicular to the reaction plane, which is spanned by the beam axis and impact parameter vector. We show that only a small fraction of the enormous initial angular momentum of the colliding ultrarelativistic nuclei is transferred to the glasma. Our main focus is the glasma angular momentum directed along the beam axis. This quantity has a local character and results from the inhomogeneous velocity field generated in the glasma. We calculate the vorticity and local angular momentum which show noticeably different behaviour. The results are analyzed in detail and discussed in the context of existing experimental data. We argue that neither vorticity nor thermal vorticity but instead the local angular momentum controls the polarization of final-state hadrons.

Figures

Figures reproduced from arXiv: 2505.07324 by the authors.

Figure 1
Figure 1. FIG. 1. The red (solid), green (dashed) and blue (dot-dashed) curves show the colour charge [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The value of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The part of the transverse plane for which [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Elliptic flow generates vorticity. The red arrows represent the collective flow and the [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Vorticity in the transverse plane expressed in fm [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Angular dependence of the glasma vorticity from Fig. [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The vorticity from the glasma velocity field (left panel) and from the Fourier components [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The Fourier coefficients of the glasma velocity field: [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The quantities [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p021_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Angular dependence of [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Φ [PITH_FULL_IMAGE:figures/full_fig_p023_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. The left panel shows vorticity expressed in fm [PITH_FULL_IMAGE:figures/full_fig_p024_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. The fields [PITH_FULL_IMAGE:figures/full_fig_p025_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Fourier decomposition of [PITH_FULL_IMAGE:figures/full_fig_p026_17.png]

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

  1. Analytic and Approximate Solutions to Color Glass Condensate in the Classical Weak-Field Limit

    nucl-th 2026-07 accept novelty 6.5 of 10

    In the weak-field CGC limit the glasma energy-momentum tensor has universal late-time scaling ε,PT∼1/τ and PL∼1/τ³, with closed Meijer-G forms in the MV model and controlled series in an improved Gaussian model.

Reference graph

Works this paper leans on

57 extracted references · 41 canonical work pages · cited by 1 Pith paper

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    INTRODUCTION Observations of spin polarization phenomena in relativistic heavy-ion collisions opened new possibilities to study the collision dynamics and properties of the strongly interacting matter produced in these collisions. Global polarization of hyperons and vector mesons transverse to the reaction plane (defined as the plane spanned by the beam a...

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    SUMMAR Y OF THE COMPUT A TIONAL METHOD The glasma angular momentum and vorticity are obtained from components of the energy-momentum tensor which is expressed in terms of chromodynamic fields and aver- aged over colour configurations of the colliding nuclei. To calculate the energy-momentum tensor we use a proper time expansion with a method introduced in...

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    FORMULAS FOR ANGULAR MOMENTUM In our previous work [23] we derived an expression for the angular momentum of the glasma per unit rapidity 1. We repeat the basic steps of this calculation below. We define the tensor Mµνρ =TµνRρ−TµρRν, (3.1) 1 Our method is similar to that of Ref. [34] where a slightly inconsistent procedure is used. In that paper the autho...

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    The angular momentum carried by the nucleons which will participate in the collision is of order 105 at maximum RHIC energies [38, 39] and even larger at LHC energies

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    ANGULAR MOMENTUM ALONG THE BEAM DIRECTION As explained in the introduction, our main motivation is to understand the origin of the local polarization in the direction of the beam. There is a simple argument [7, 8] illustrated in the left panel of Fig. 6 that for a non-central collision the collective elliptic flow generates vorticity ωz and local angular ...

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