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REVIEW 3 major objections 4 minor 87 references

Spin polarization of an expanding and rotating system

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A rotating, expanding spin-1/2 fluid's longitudinal polarization can be tracked from free streaming to the hydrodynamic regime through a closed set of moment equations.

desk verdict A careful, transparent extension of the authors' spin-kinetic framework to rotating expanding systems, with a real but addressable weakness in the closure of the moment hierarchy. read the letter →

arxiv 2412.05733 v2 pith:ZAZROX2Y submitted 2024-12-07 nucl-th hep-ph

Nora Weickgenannt , Jean-Paul Blaizot This is my paper · ORCID
classification nucl-thhep-ph
keywords spinpolarizationkinetictheorynonlocalrelaxationtimeapproximationmomentsBjorkenexpansionrelativisticheavy-ioncollisionsthermalvorticityLambda
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 claims that the longitudinal spin polarization of a rotating, longitudinally expanding fluid of massive spin-1/2 particles can be described by a finite list of spin moments—weighted averages of the spin vector with momentum angular factors—whose equations of motion close into a set of 36 linear ordinary differential equations. These equations interpolate between free streaming at early times and a collision-dominated, hydrodynamic regime at late times, so the polarization can be followed across the full evolution rather than only near equilibrium. The final expression gives the polarization as a function of the azimuthal momentum angle up to first order in the ratio of relaxation time to expansion time, and it reproduces the known late-time local-equilibrium polarization as a limiting case. The practical payoff is that a kinetic-theory problem relevant to heavy-ion collisions is reduced to a solvable finite system that includes dissipative gradient contributions to spin polarization.

What carries the argument

The argument is carried by the spin moments $G^k_{n\ell r}$ and $I^k_{n\ell r}$, defined as phase-space integrals of the spin three-vector times spherical harmonics and powers of $p/E_p$, and by the nonlocal relaxation time approximation $C[f]=-(f-f_\infty)/\tau_R$, where $f_\infty$ is the asymptotic distribution built from local equilibrium plus nonlocal gradient terms. A matching condition expresses the spin potential in terms of the total angular momentum, whose components become additional dynamical variables. The infinite moment hierarchy is closed by replacing higher moments with late-free-streaming ratios and by interpolating $r\neq 0$ moments with an exponential factor $e^{-w/2}$ between free-streaming and asymptotic values.

What would settle it

One can settle the central claim by numerically integrating the full spin Boltzmann equation with the nonlocal relaxation time approximation for the same boost-invariant, vortical flow and comparing the resulting longitudinal polarization and the moments $G^z_{000}$, $G^z_{200}$, and $G^z_{110}$ with the closed 36-equation system of App. F; sizable disagreement at moderate $w=\tau/\tau_R$ would show the closure or interpolation fails. A second check is that Eq. (61) contains only azimuthal harmonics of order 0, $e^{i\varphi}$, and $e^{2i\varphi}$ at first order in $w^{-1}$, so a measurement of a clean higher azimuthal harmonic in the longitudinal polarization would indicate the truncation misses relevant physics.

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Extended reading notes

Core claim

The paper establishes that under boost-invariant longitudinal expansion and purely vortical transverse flow, the longitudinal Pauli-Lubanski polarization receives contributions only from a selected set of spin moments up to first order in $w^{-1}=\tau_R/\tau$, and that these moments, together with the total angular momentum components, obey closed equations derived from the Boltzmann equation with a nonlocal relaxation time approximation. The nonlocal part of the collision term is what feeds fluid-velocity and temperature gradients into the polarization, so the late-time polarization contains thermal-vorticity and thermal-shear contributions beyond the local-equilibrium piece. The central deliverable is Eq. (61) for the polarization as a function of the azimuthal angle, together with the closed equations of motion in App. F, which are claimed to be valid at any time from free streaming to the hydrodynamic regime.

Load-bearing premise

The derivation assumes that free streaming drives the system to a state with $\cos\theta=0$ well before collisions become important, and that spin moments outside the kept list can be replaced by late-free-streaming ratios or by an interpolation borrowed from scalar studies; if that ordering or that closure fails, the closed equations and Eq. (61) lose their justification.

Editorial extensions

If this is right

  • The longitudinal polarization of the expanding, rotating system can be computed by solving 36 linear ordinary differential equations instead of the full Boltzmann equation.
  • The late-time limit of the result coincides with the Zubarev local-equilibrium polarization, so the framework connects the dissipative early-time history to a known equilibrium endpoint.
  • Up to first order in $w^{-1}$, the polarization depends on the azimuthal angle only through constant, $e^{i\varphi}$, and $e^{2i\varphi}$ terms, implying that higher spherical harmonics would signal higher-order corrections.
  • The equations describe how parts of an initial polarization survive the expansion and rotation, so freeze-out polarization can carry an imprint of early-time spin dynamics.
  • Gradients of the fluid velocity and temperature, entering through the nonlocal collision term, generate contributions to the polarization that are absent from ideal spin-hydrodynamic treatments.

Reading between the lines

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

  • My inference: the same closure strategy could be applied to the transverse polarization with nonzero vorticity, since the transverse and longitudinal moment equations decouple from each other in this setup.
  • My inference: a numerical implementation of the 36 equations for realistic initial conditions could quantify whether the first-order dissipative terms are large enough to affect measured Lambda polarization patterns in heavy-ion collisions.
  • My inference: the exponential interpolation between free streaming and asymptotic values is the most delicate step to test; comparing the closed system against a direct numerical solution of the nonlocal relaxation-time Boltzmann equation would isolate its error.
  • My inference: the absence of harmonics above $e^{2i\varphi}$ at this order offers a clean experimental discriminator, since any robust higher harmonic in the azimuthal dependence would require physics beyond the present truncation.
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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

3 major / 4 minor

Summary. The paper studies the longitudinal spin polarization of a boost-invariant, transversely rotating fluid of massive spin-1/2 particles. Starting from spin kinetic theory with a nonlocal relaxation time approximation (NLRTA), the authors express the polarization in terms of spin moments, derive exact equations of motion for these moments, and then close the infinite hierarchy by keeping a finite set of late-time-relevant moments and using an interpolation between free-streaming and hydrodynamic behavior. The main results are Eq. (61), giving the polarization as a function of the azimuthal angle up to first order in w^{-1}=tau_R/tau, and the accompanying 36 closed equations of motion collected in App. F. The late-time limit is benchmarked against the Zubarev local-equilibrium result from Refs. [14,62].

Significance. If the closure is reliable, this is a useful, tractable model for dissipative spin polarization in heavy-ion collisions, going beyond ideal-vorticity contributions and providing explicit first-order gradient corrections. The algebraic derivation is detailed and transparent, the final late-time limit correctly reduces to known external results, and the paper delivers a concrete finite-dimensional dynamical system that can be solved numerically with modest effort. These are genuine strengths. The main uncertainty is the closure/interpolation scheme, which is imported from scalar relaxation-time studies and is not validated for the coupled spin-moment system; this bears directly on the paper's central claim of validity at any time.

major comments (3)
  1. [Sec. VII, Eqs. (39)-(40)] The central claim that the closed equations are valid at any time rests on an unvalidated truncation. Equations (39) and (40) replace all spin moments outside the list (38) by their free-streaming ratios evaluated at cos(theta)=0. This replacement is exact only in the limit cos(theta)->0 and in the late-time limit, but it is applied at all times in the equations of motion, so the early-time coupling to the neglected moments is simply dropped. No numerical comparison with the unclosed moment hierarchy, nor an estimate of the error incurred in the intermediate regime, is provided. This is load-bearing: without such a check, Eq. (61) and the 36 equations in App. F cannot be claimed to capture the full time evolution.
  2. [Sec. VII, Eq. (41)] The interpolation for r != 0 moments, G^k_{n l r} -> e^{-w/2} G^k_{n l r,o} + (1-e^{-w/2}) G^k_{n l r,infinity}, is imported from scalar RTA studies [60,61] and is not re-derived for the coupled spin-moment system. The decay rates of the total-angular-momentum components, shown in App. E, Eqs. (50)-(54), differ among the (z,x), (x,z), (z,0) and (0,z) components; there is no reason that a single exponential e^{-w/2} interpolates all of these simultaneously. Since the interpolation controls the transient coefficients in Eq. (61), the agreement with the Zubarev asymptotic result only fixes the w->infinity limit and does not validate the interpolated transient.
  3. [Sec. IV] The time-ordering assumption that cos(theta)=0 is reached well before the collision-dominated regime sets in is stated but not quantified. For massive particles the free-streaming depletion of p_z occurs on a time scale tau_0 p_z/m, which can be comparable to tau_R for realistic initial conditions. If the two regimes overlap, the late-free-streaming decay laws used in Eqs. (39)-(41) lose their justification, and the coefficients in Eq. (61) carry an uncontrolled early-time error. The authors should either state a quantitative condition under which the ordering holds, or test the sensitivity of the final polarization to this assumption.
minor comments (4)
  1. [Introduction] There are typographical errors: "A priory" should be "A priori" and "contibutions" should be "contributions".
  2. [Eq. (9)] The normalization constant reads (n+l!) in the denominator; this appears to be a typo for (n+l)!.
  3. [References] Reference [55] contains the malformed author string "N. /suppress Lygan"; this should be corrected.
  4. [Sec. IX] The conclusion states that the interpolation "has been shown to successfully reproduce the exact solution for the same type of equations of motion in Refs. [60,61]"; this is true for the scalar case, but the paper should make clear that this demonstration does not automatically cover the coupled spin-moment equations used here.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the moment equations are derived from the Boltzmann equation, and the late-time benchmark is external to the authors' own ansatz.

full rationale

The claimed derivation is not circular. Equations of motion for the spin moments (16), (17) and for the total angular momentum (32) are obtained by direct insertion of the NLRTA Boltzmann equation (7), with no occurrence of the final polarization (61) as an input. The closure replacements (39)-(41) are explicit truncation/interpolation assumptions imported from Refs. [60,61]; they are model inputs whose accuracy is a correctness question, not a disguised restatement of the result. The late-time limit (62) is matched to the external Zubarev results of Refs. [14,62], and the present paper expressly cites those external works rather than defining the asymptotic polarization to equal its own ansatz. The self-citations to the NLRTA [26], the spin-moment strategy [50], and the interpolation method [60,61] are real prior derivations or benchmarks with independent content; none of them is a uniqueness theorem or a fitted parameter that forces Eq. (61). The Sec. IV time-ordering assumption ('this state is reached well before the collision-dominated regime sets in') is an explicitly stated physical assumption; if it fails, the transient is inaccurate, but that is a robustness risk, not circularity. Accordingly no circular step is identified; the score of 2 reflects only the presence of minor self-citations that are not load-bearing.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The derivation rests on several model assumptions beyond standard mathematics. The most load-bearing are the NLRTA collision term, the replacement of the zeroth-order distribution by local equilibrium in the nonlocal term, the ordering assumption that cos theta = 0 is reached before the collision-dominated regime, and the interpolation/truncation closure borrowed from scalar studies. The free parameters tau_R and xi set time scales and the pseudo-gauge choice; none is fitted to data here. No new physical entities are introduced.

free parameters (2)
  • relaxation time tau_R
    Introduced in the NLRTA collision term, Eq. (3). It sets the time scale of relaxation and defines w = tau/tau_R used throughout the closed equations. No value is fitted here; it is an input model parameter.
  • nonlocal conversion parameter xi
    Controls the relative strength of the nonlocal angular-momentum-conversion term in Eq. (3), later absorbed into xi_bar = Ep/(m(Ep+m)) xi. The paper notes that xi_bar = 1/m gives the canonical pseudo-gauge, but no value is fitted.
assumptions (6)
  • standard math Standard spin integration identities, such as the integral of s_mu s_nu, and associated Legendre recurrence relations used throughout the derivation.
    Used in Apps. B and D to derive the equations of motion; these are standard mathematical results assumed without proof.
  • domain assumption Boost invariance in z and purely vortical transverse flow: p_perp dot partial f = 0, and the symmetry constraints on derivatives of beta_mu.
    Equation (1) and the symmetry constraints in Sec. II define the highly symmetric background and restrict the applicability of the result to idealised expanding rotating systems.
  • ad hoc to paper The nonlocal collision term is modeled by the NLRTA, and f^(0) is replaced by local-equilibrium f_LE^(0) in the nonlocal part.
    Section II, around Eq. (7). This is a phenomenological collision model, not derived from an underlying interaction, and it is the mechanism that produces gradient contributions to polarization.
  • domain assumption The state cos theta = 0 is reached well before the collision-dominated regime begins.
    Section IV states this explicitly. It justifies the late-free-streaming decay rates and the closure ratios; if the two regimes overlap, the early-time dynamics and the interpolation are not valid.
  • ad hoc to paper Truncation and interpolation closure: keep only the spin moments in (38), replace higher moments by free-streaming ratios (39)-(40), and interpolate r != 0 moments with e^{-w/2} as in Eq. (41).
    Section VII. This is an ansatz imported from Refs. [60,61] and is not derived from the Boltzmann equation for the coupled spin-moment system.
  • domain assumption First order in hbar and neglect of spin backreaction on the fluid background.
    Section II and the conclusions: the fluid flow is a fixed background unaffected by polarization, up to order hbar.

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Pith. "Pith review of Spin polarization of an expanding and rotating system." pith.science (2026). https://pith.science/paper/ZAZROX2Y

@misc{pith2026241205733,
  author       = {Pith},
  title        = {Pith review of: Spin polarization of an expanding and rotating system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZAZROX2Y}},
  note         = {Machine review of arXiv:2412.05733}
}
read the original abstract

We study the longitudinal spin polarization of a relativistic fluid of massive spin-1/2 particles undergoing a boost-invariant expansion in the longitudinal direction and rotating in the transverse plane. We express the polarization vector in terms of spin moments and derive closed equations of motion for the latter using spin kinetic theory with a nonlocal relaxation time approximation. These equations of motion are valid at any time of the evolution, from the free-streaming regime to the hydrodynamic regime. At late time, the polarization features contributions from gradients of the fluid velocity and of the temperature, that emerge from the nonlocal part of the collision term. Our results can be used to explore polarization phenomena in the context of heavy-ion collisions.

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Reference graph

Works this paper leans on

87 extracted references · 26 canonical work pages

  1. [1]

    Becattini, L

    F. Becattini, L. Csernai, and D. J. Wang, Phys. Rev. C88, 034905 (2013), [Erratum: Phys. Rev.C93,no.6,069901(201 6)], 1304.4427

    arXiv 2013

  2. [2]

    Becattini, V

    F. Becattini, V. Chandra, L. Del Zanna, and E. Grossi, Ann als Phys. 338, 32 (2013), 1303.3431

    arXiv 2013

  3. [3]

    conserves the total angular momentum J µν ≡ ∫ dΓ ( ∆ [µpν] + ℏ 2 Σ µν s ) f (28) in the collision. Note that ∆ µ is the space-like separation between the particle position and the ce nter of the collision 1 x, and therefore ∆ [µpν] is the orbital angular momentum of the particle in the collision, which ca n be converted into spin. Due to the matching cond...

  4. [4]

    Becattini, I

    F. Becattini, I. Karpenko, M. Lisa, I. Upsal, and S. Volos hin, Phys. Rev. C95, 054902 (2017), 1610.02506

    arXiv 2017

  5. [5]

    We also absorbed coefficients into ξ(Ep) by defining ¯ξ ≡ Ep (Ep + m)m ξ

    becomes ∆ j = − ℏ 2m(Ep + m) ǫjik pisk , (21) and we defined κµ 0 ≡ −Ω µνuν and ̟µν ≡ −(1/2)∂[µβν]. We also absorbed coefficients into ξ(Ep) by defining ¯ξ ≡ Ep (Ep + m)m ξ . (22) We remember that the longitudinal spin moments which are odd in pz do not contribute to the longitudinal polar- ization. For the spin moments even in pz, we find from Eq. ( 20) that ...

  6. [6]

    center of the collision

    in Eq. ( 29), it is clear that the components of Ω µν can be expressed as a function of the components of J µν with the coefficients being thermodynamic integrals. We may therefo re equivalently calculate ˜J µν ≡ ǫµναβ Jαβ to obtain Ω µν. Contracting Eq. ( 28) with ǫµναβ , we obtain ˜J µν = ℏ 4 ∫ ps 1 E2p [ m 2(Ep + m) s[µtν] + 1 m ( 1 + Ep 2(Ep + m) ) p[µs...

  7. [7]

    Y. Xie, D. Wang, and L. P. Csernai, Phys. Rev. C95, 031901 (2017), 1703.03770

    arXiv 2017

  8. [8]

    Analogously to what was done for the transverse po larization in Ref

    is obtained by expanding the distribution function in spherical harm onics, where the integral over spherical harmonics with n + ℓ odd vanishes. Analogously to what was done for the transverse po larization in Ref. [ 50], we express the longitudinal polarization in terms of the following spin moments Gk nℓr ≡ ∫ ps ( p Ep )r Y ℓ n (θ, φ)skf , I k nℓr ≡ m ∫...

Show all 87 references
  1. [9]

    Adam et al

    J. Adam et al. (STAR), Phys. Rev. C98, 014910 (2018), 1805.04400

    2018 arXiv

  2. [10]

    Acharya et al

    S. Acharya et al. (ALICE), Phys. Rev. Lett. 125, 012301 (2020), 1910.14408

    2020 arXiv

  3. [11]

    Adam et al

    J. Adam et al. (STAR), Phys. Rev. Lett. 123, 132301 (2019), 1905.11917

    2019

  4. [12]

    S. Y. F. Liu and Y. Yin, JHEP 07, 188 (2021), 2103.09200

    2021 arXiv

  5. [13]

    (F1) 33 Analogously one finds ∂wI z 000 = − 1 w (ζ00I z 00 + ξ00I z

    − 1 w ( 1 − e−w/2 )( a11 ˜J xz + ¯a11 ˜J yz ) − ( Gz 110 − λ00 11 ˜J xz − ¯λ00 11 ˜J yz ) , ∂wGz 310 = − 1 w ( ˆζ31Gz 310 + χ31Gz 110 ) − 1 w ( 1 − e−w/2 )( an1 ˜J xz + ¯a31 ˜J yz ) − G z 310 . (F1) 33 Analogously one finds ∂wI z 000 = − 1 w (ζ00I z 00 + ξ00I z

  6. [14]

    − 1 w ( 1 − e−w/2 )( a′ 00 ˜J z0 + b′ 00̟xy + c′ 00σ ) − ( I z 000 − λ01 00 ˜J z0 − κ01 00̟xy ) , ∂wI z 200 = − 1 w (ζ20I z 200 + χ20I z 000 + ξ20I z

  7. [15]

    − 1 w ( 1 − e−w/2 )( a′ 20 ˜J z0 + b′ 20̟xy + c′ 20σ ) − ( I z 200 − λ01 20 ˜J z0 − κ01 20̟xy ) , ∂wI z 220 = − 1 w (ζ22I z 220 + ξ22I z

  8. [16]

    − 1 w ( 1 − e−w/2 )( a′ 22 ˜J z0 + b′ 22̟xy + c′ 22σ ) − I z 220 , ∂wI z 400 = − 1 w ( ˆζ40I z 400 + χ40I z 200 ) − 1 w ( 1 − e−w/2 )( a′ 40 ˜J z0 + b′ 40̟xy + c′ 40σ ) − I z 400 , ∂wI z 420 = − 1 w ( ˆζ42I z 420 + χ42I z 220 ) − 1 w ( 1 − e−w/2 )( a′ 42 ˜J z0 + b′ 42̟xy + c′ ...

  9. [17]

    (F2) The coefficients in Eqs

    − 1 w ( 1 − e−w/2 )( a′ n1 ˜J xz + ¯a′ 11 ˜J yz ) − ( I z 110 − λ01 11 ˜J xz − ¯λ01 11 ˜J yz ) , ∂wI z 310 = − 1 w ( ˆζ31I z 310 + χ31I z 110 ) − 1 w ( 1 − e−w/2 )( a′ n1 ˜J xz + ¯a′ 31 ˜J yz ) − I z 310 . (F2) The coefficients in Eqs. ( F1) and Eq. ( F2) are defined in Eqs. ( A2...

  10. [18]

    − 1 w ( 1 − e−w/2 )1 2 α10 ˜J xz − ( Gx 100 − λ00 10 ˜J xz ) , ∂wGx 300 = − 1 w ( ˆa300Gx 300 + ¯b300Gx 100 ) − 1 w ( 1 − e−w/2 )1 2 α30 ˜J xz − G x 300 , ∂wGx 210 = − 1 w (¯a210Gx 210 + ¯c210Gx

  11. [19]

    − 1 w ( 1 − e−w/2 )1 2 ( α21 ˜J z0 + η21̟xy − ˜α21σ ) − ( Gx 210 − λ00 21 ˜J z0 − κ00 21̟xy + ˜κ00 21σ ) , ∂wGx 410 = − 1 w ( ˆa410Gx 410 + ¯b410Gx 210 ) − 1 w ( 1 − e−w/2 )1 2 ( α41 ˜J z0 + η41̟xy − ˜α41σ ) − G x 410 , ∂wGy 100 = − 1 w (¯a100Gy 100 + ¯c100Gy

  12. [20]

    − 1 w ( 1 − e−w/2 )1 2 α10 ˜J yz − ( Gy 100 − λ00 10 ˜J yz ) , ∂wGy 300 = − 1 w ( ˆa300Gy 300 + ¯b300Gy 100 ) − 1 w ( 1 − e−w/2 )1 2 α30 ˜J yz − G y 300 , ∂wGy 210 = − 1 w (¯a210Gy 210 + ¯c210Gy

  13. [21]

    − 1 w ( 1 − e−w/2 )1 2 ( α21 ˜J z0 + η21̟xy + ˆα21σ ) − ( Gy 210 − ¯λ00 21 ˜J z0 − ¯κ00 21̟xy − ˆκ00 21σ ) , ∂wGy 410 = − 1 w ( ˆa410Gy 410 + ¯b410Gy 210 ) − 1 w ( 1 − e−w/2 )1 2 ( α41 ˜J z0 + η41̟xy + ˆα41σ ) − G y 410 (F3) 34 and, analogously, ∂wI x 100 = − 1 w (¯a100I x 100...

  14. [22]

    − ( I x 100 − λ01 10 ˜J xz ) , ∂wI x 300 = − 1 w ( ˆa300I x 300 + ¯b300I x 100 ) − I x 300 , ∂wI x 210 = − 1 w (¯a210I x 210 + ¯c210I x

  15. [23]

    − ( I x 210 − λ01 21 ˜J z0 − κ01 21̟xy + ˜κ01 21σ ) , ∂wI x 410 = − 1 w ( ˆa410I x 410 + ¯b410I x 210 ) − I x 410 , ∂wI y 100 = − 1 w (¯a100I y 100 + ¯c100I y

  16. [24]

    − ( I y 100 − λ01 10 ˜J yz ) , ∂wI y 300 = − 1 w ( ˆa300I y 300 + ¯b300I y 100 ) − I y 300 , ∂wI y 210 = − 1 w (¯a210I y 210 + ¯c210I y

  17. [25]

    (F4) The coefficients in Eqs

    − ( I y 210 − ¯λ01 21 ˜J z0 − ¯κ01 21̟xy − ˆκ01 21σ ) , ∂wI y 410 = − 1 w ( ˆa410I y 410 + ¯b410I y 210 ) − I y 410 . (F4) The coefficients in Eqs. ( F3) and ( F4) are given in Eqs. ( A5), ( A7), and ( C10). Finally, the equations of motion for the total angular momentum are giv...

  18. [26]

    Weickgenannt and J.-P

    N. Weickgenannt and J.-P. Blaizot (2024), 2409.11045

    2024 arXiv

  19. [27]

    Wang and S

    Z. Wang and S. Lin (2024), 2411.19550

    2024 arXiv

  20. [28]

    Becattini, G

    F. Becattini, G. Inghirami, V. Rolando, A. Beraudo, L. De l Zanna, A. De Pace, M. Nardi, G. Pagliara, and V. Chandra, Eur. Phys. J. C75, 406 (2015), [Erratum: Eur. Phys. J.C78,no.5,354(2018)], 1501.04468

    2015 arXiv

  21. [29]

    Florkowski, B

    W. Florkowski, B. Friman, A. Jaiswal, R. Ryblewski, and E. Speranza, Phys. Rev. D97, 116017 (2018), 1712.07676

    2018 arXiv

  22. [30]

    Karpenko and F

    I. Karpenko and F. Becattini, Eur. Phys. J. C77, 213 (2017), 1610.04717

    2017 arXiv

  23. [31]

    L.-G. Pang, H. Petersen, Q. Wang, and X.-N. Wang, Phys. Re v. Lett. 117, 192301 (2016), 1605.04024

    2016 arXiv

  24. [32]

    Hattori, M

    K. Hattori, M. Hongo, X.-G. Huang, M. Matsuo, and H. Taya , Phys. Lett. B795, 100 (2019), 1901.06615

    2019 arXiv

  25. [33]

    Adamczyk et al

    L. Adamczyk et al. (STAR), Nature 548, 62 (2017), 1701.06657

    2017 arXiv

  26. [34]

    Singh, G

    R. Singh, G. Sophys, and R. Ryblewski, Phys. Rev. D 103, 074024 (2021), 2011.14907

    2021 arXiv

  27. [35]

    Montenegro and G

    D. Montenegro and G. Torrieri, Phys. Rev. D 102, 036007 (2020), 2004.10195

    2020 arXiv

  28. [36]

    ( 18) in Eq

    between the longitudinal spin moments and the total angular mome ntum are obtained by using Eq. ( 18) in Eq. ( 15) and then inserting Eqs. ( 35), Gz 000,∞ = − ℏ m ∫ d3p E2 p ˜Ω z0f0 + 2ℏ ∫ p ¯ξǫij0z pjpν(Ω iν + ∂iβν)f (0) LE = − ℏ m ∫ d3p E2 p ˜Ω z0f0 − 2ℏ ∫ p ¯ξ[p2 y(Ω xy + ∂...

  29. [37]

    Fukushima and S

    K. Fukushima and S. Pu, Phys. Lett. B 817, 136346 (2021), 2010.01608

    2021 arXiv

  30. [38]

    B. Fu, S. Y. F. Liu, L. Pang, H. Song, and Y. Yin, Phys. Rev. Lett. 127, 142301 (2021), 2103.10403

    2021 arXiv

  31. [39]

    Becattini, M

    F. Becattini, M. Buzzegoli, and A. Palermo, Phys. Lett. B 820, 136519 (2021), 2103.10917

    2021 arXiv

  32. [40]

    Becattini, M

    F. Becattini, M. Buzzegoli, G. Inghirami, I. Karpenko, and A. Palermo, Phys. Rev. Lett. 127, 272302 (2021), 2103.14621

    2021 arXiv

  33. [41]

    Weickgenannt, X.-L

    N. Weickgenannt, X.-L. Sheng, E. Speranza, Q. Wang, and D. H. Rischke, Phys. Rev. D100, 056018 (2019), 1902.06513

    2019 arXiv

  34. [42]

    Gao and Z.-T

    J.-H. Gao and Z.-T. Liang, Phys. Rev. D100, 056021 (2019), 1902.06510

    2019 arXiv

  35. [43]

    Hattori, Y

    K. Hattori, Y. Hidaka, and D.-L. Yang, Phys. Rev. D100, 096011 (2019), 1903.01653. 35

    2019 arXiv

  36. [44]

    Z. Wang, X. Guo, S. Shi, and P. Zhuang, Phys. Rev. D 100, 014015 (2019), 1903.03461

    2019 arXiv

  37. [45]

    Weickgenannt, E

    N. Weickgenannt, E. Speranza, X.-l. Sheng, Q. Wang, and D. H. Rischke, Phys. Rev. Lett. 127, 052301 (2021), 2005.01506

    2021 arXiv

  38. [46]

    Y.-C. Liu, K. Mameda, and X.-G. Huang, Chin. Phys. C 44, 094101 (2020), [Erratum: Chin.Phys.C 45, 089001 (2021)], 2002.03753

    2020 arXiv

  39. [47]

    Weickgenannt, E

    N. Weickgenannt, E. Speranza, X.-l. Sheng, Q. Wang, and D. H. Rischke, Phys. Rev. D 104, 016022 (2021), 2103.04896

    2021 arXiv

  40. [48]

    Sheng, N

    X.-L. Sheng, N. Weickgenannt, E. Speranza, D. H. Rischk e, and Q. Wang, Phys. Rev. D 104, 016029 (2021), 2103.10636

    2021 arXiv

  41. [49]

    Wagner, N

    D. Wagner, N. Weickgenannt, and D. H. Rischke, Phys. Rev . D 106, 116021 (2022), 2210.06187

    2022 arXiv

  42. [50]

    Wagner, N

    D. Wagner, N. Weickgenannt, and E. Speranza (2023), 230 6.05936

    2023

  43. [51]

    Daher, W

    A. Daher, W. Florkowski, R. Ryblewski, and F. Taghinava z, Phys. Rev. D 109, 114001 (2024), 2403.04711

    2024 arXiv

  44. [52]

    Wagner, M

    D. Wagner, M. Shokri, and D. H. Rischke (2024), 2405.005 33

    2024

  45. [53]

    Florkowski, B

    W. Florkowski, B. Friman, A. Jaiswal, and E. Speranza, P hys. Rev. C97, 041901 (2018), 1705.00587

    2018 arXiv

  46. [54]

    Wagner (2024), 2409.07143

    D. Wagner (2024), 2409.07143

    2024

  47. [55]

    Florkowski, R

    W. Florkowski, R. Ryblewski, and A. Kumar, Prog. Part. N ucl. Phys. 108, 103709 (2019), 1811.04409

    2019 arXiv

  48. [56]

    Montenegro and G

    D. Montenegro and G. Torrieri, Phys. Rev. D 100, 056011 (2019), 1807.02796

    2019 arXiv

  49. [57]

    S. A. Voloshin (2004), nucl-th/0410089

    2004 arXiv

  50. [58]

    Bhadury, W

    S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, and R. R yblewski, Phys. Lett. B 814, 136096 (2021), 2002.03937

    2021 arXiv

  51. [59]

    Becattini, F

    F. Becattini, F. Piccinini, and J. Rizzo, Phys. Rev. C77, 024906 (2008), 0711.1253

    2008 arXiv

  52. [60]

    Jaiswal, J.-P

    S. Jaiswal, J.-P. Blaizot, R. S. Bhalerao, Z. Chen, A. Ja iswal, and L. Yan, Phys. Rev. C 106, 044912 (2022), 2208.02750

    2022 arXiv

  53. [61]

    (A8) Here we defined Inℓ ≡ ∫ d cos θ P ℓ n(cos θ) (A9) Note that Inℓ = 0 for n + ℓ odd

    are given by ℵn ≡ Nn0In0(1 − gn0) − N(n+2)0I(n+2)0h(n+2)0 − N(n−2)0I(n−2)0i(n−2)0 = δn0 2 3 − δn2 12 35 , ¯ℵn ≡ Nn0In0gn0 + N(n+2)0I(n+2)0h(n+2)0 + N(n−2)0I(n−2)0i(n−2)0 = δn0 1 3 + δn2 12 35 , ˜ℵn ≡ Nn0In0d+ n0 + N(n+2)0I(n+2)0e+ (n+2)0 + N(n−2)0I(n−2)0f+ (n−2)0 ℶ n ≡ Nn1In1(...

  54. [62]

    A. D. Gallegos, U. G¨ ursoy, and A. Yarom, SciPost Phys.11, 041 (2021), 2101.04759

    2021 arXiv

  55. [63]

    S. Li, M. A. Stephanov, and H.-U. Yee, Phys. Rev. Lett. 127, 082302 (2021), 2011.12318

    2021 arXiv

  56. [64]

    D.-L. Wang, S. Fang, and S. Pu, Phys. Rev. D 104, 114043 (2021), 2107.11726

    2021 arXiv

  57. [65]

    Hu, Phys

    J. Hu, Phys. Rev. D 105, 076009 (2022), 2111.03571

    2022 arXiv

  58. [66]

    Hongo, X.-G

    M. Hongo, X.-G. Huang, M. Kaminski, M. Stephanov, and H. -U. Yee, JHEP 11, 150 (2021), 2107.14231

    2021 arXiv

  59. [67]

    Daher, A

    A. Daher, A. Das, W. Florkowski, and R. Ryblewski (2022) , 2202.12609

    2022 arXiv

  60. [68]

    Weickgenannt, D

    N. Weickgenannt, D. Wagner, E. Speranza, and D. H. Risch ke, Phys. Rev. D 106, 096014 (2022), 2203.04766

    2022 arXiv

  61. [69]

    Weickgenannt, D

    N. Weickgenannt, D. Wagner, and E. Speranza, Phys. Rev. D 105, 116026 (2022), 2204.01797

    2022 arXiv

  62. [70]

    A. D. Gallegos, U. Gursoy, and A. Yarom (2022), 2203.050 44

    2022

  63. [71]

    Z. Cao, K. Hattori, M. Hongo, X.-G. Huang, and H. Taya (20 22), 2205.08051

    arXiv

  64. [72]

    Weickgenannt, D

    N. Weickgenannt, D. Wagner, E. Speranza, and D. H. Risch ke, Phys. Rev. D 106, L091901 (2022), 2208.01955

    2022 arXiv

  65. [73]

    Biswas, A

    R. Biswas, A. Daher, A. Das, W. Florkowski, and R. Ryblew ski (2023), 2304.01009

    2023 arXiv

  66. [74]

    Weickgenannt, Phys

    N. Weickgenannt, Phys. Rev. D 108, 076011 (2023), 2307.13561

    2023 arXiv

  67. [75]

    Weickgenannt and J.-P

    N. Weickgenannt and J.-P. Blaizot, Phys. Rev. D 109, 056019 (2024), 2312.05917

    2024 arXiv

  68. [78]

    Drogosz, W

    Z. Drogosz, W. Florkowski, and M. Hontarenko (2024), 24 08.03106

    2024

  69. [80]

    Drogosz, W

    Z. Drogosz, W. Florkowski, N. /suppress Lygan, and R. Ryblewski (2024), 2411.06154

    2024

  70. [81]

    Liang and X.-N

    Z.-T. Liang and X.-N. Wang, Phys. Rev. Lett. 94, 102301 (2005), [Erratum: Phys. Rev. Lett.96,039901(2006 )], nucl- th/0410079

    2005

  71. [83]

    B. Betz, M. Gyulassy, and G. Torrieri, Phys. Rev. C 76, 044901 (2007), 0708.0035

    2007 arXiv

  72. [86]

    Weickgenannt and J.-P

    N. Weickgenannt and J.-P. Blaizot, Phys. Rev. D 109, 056012 (2024), 2311.15817

    2024 arXiv

  73. [87]

    Buzzegoli, Phys

    M. Buzzegoli, Phys. Rev. C 105, 044907 (2022), 2109.12084

    2022 arXiv

  74. [200]

    − 1 w ( 1 − e−w/2 )( a00 ˜J z0 + b00̟xy + c00σ ) − ( Gz 000 − λ00 00 ˜J z0 − κ00 00̟xy ) , ∂wGz 200 = − 1 w (ζ20Gz 200 + χ20Gz 000 + ξ20Gz

  75. [300]

    − 1 w ( 1 − e−w/2 )1 2 Kx 10,∞ − ( Gx 100 − G x 100,∞ ) , ∂wGx 300 = − 1 w ( ˆa300Gx 300 + ¯b300Gx 100 ) − 1 w ( 1 − e−w/2 )1 2 Kx 30,∞ − G x 300 , ∂wGx 210 = − 1 w (¯a210Gx 210 + ¯c210Gx

  76. [310]

    − 1 w ( 1 − e−w/2 )( Kz 11,∞ + Dx 11,∞ + ¯Dy 11,∞ ) − ( Gz 110 − G z 110,∞ ) , ∂wGz 310 = − 1 w ( ˆζ31Gz 310 + χ31Gz 110 ) − 1 w ( 1 − e−w/2 )( Kz 31,∞ + Dx 31,eq + ¯Dy 31,eq ) − G z 310 . (42) 10 Here we defined the following quantities, which depend only on Ω µν and thermodyn...

  77. [400]

    − 1 w ( 1 − e−w/2 )( a20 ˜J z0 + b20̟xy + c20σ ) − ( Gz 200 − λ00 20 ˜J z0 − κ00 20̟xy ) , ∂wGz 220 = − 1 w (ζ22Gz 220 + ξ22Gz

  78. [410]

    A and we defined the asymptotic quantity Kx nℓ,∞ ≡ 2 ∫ ps sx p2 E2p cos2 θ P ℓ n(cos θ)f∞

    − 1 w ( 1 − e−w/2 )1 2 Kx 21,∞ − (Gx 210 − G x 210,∞) , ∂wGx 410 = − 1 w ( ˆa410Gx 410 + ¯b410Gx 210 ) − 1 w ( 1 − e−w/2 )1 2 Kx 41,∞ − G x 410 , (45) where the coefficients are given in App. A and we defined the asymptotic quantity Kx nℓ,∞ ≡ 2 ∫ ps sx p2 E2p cos2 θ P ℓ n(cos θ)f...

    2022

  79. [420]

    − 1 w ( 1 − e−w/2 )( a22 ˜J z0 + b22̟xy + c22σ ) − G z 220 , ∂wGz 400 = − 1 w ( ˆζ40Gz 400 + χ40Gz 200 ) − 1 w ( 1 − e−w/2 )( a40 ˜J z0 + b40̟xy + c40σ ) − G z 400 , ∂wGz 420 = − 1 w ( ˆζ42Gz 420 + χ42Gz 220 ) − 1 w ( 1 − e−w/2 )( a42 ˜J z0 + b42̟xy + c42σ ) − G z 420 , ∂wGz 1...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.