arxiv: 2605.01857 · v1 · submitted 2026-05-03 · ✦ hep-ph · nucl-th

Recognition: unknown

Boost-invariant and cylindrically symmetric perfect spin hydrodynamics

Zbigniew Drogosz , Wojciech Florkowski , Jakub Witkowski

Authors on Pith no claims yet

Pith reviewed 2026-05-09 16:57 UTC · model grok-4.3

classification ✦ hep-ph nucl-th

keywords boost-invariantcylindrical symmetryperfect hydrodynamicsspin polarizationPauli-Lubanski vectorheavy ion collisionswounded nucleon model

0 comments

The pith

In boost-invariant cylindrically symmetric perfect spin hydrodynamics, the only nonzero total polarization comes from the longitudinal magnetic spin component coupled with the azimuthal electric component.

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

This paper numerically solves the equations for boost-invariant and cylindrically symmetric perfect hydrodynamics using energy-momentum and spin tensors for a relativistic massive gas with Boltzmann statistics. Initial conditions are taken from the wounded nucleon model. It reveals a coupling between azimuthal and longitudinal components of the electric and magnetic parts of the spin polarization tensor, unlike simpler one-dimensional expansions. At freeze-out defined by constant temperature, the Pauli-Lubanski four-vector shows that only one particular coupling produces nonzero total polarization. These findings offer a reference for more realistic hydrodynamic models.

Core claim

For the assumed geometry the only nonzero total polarization may be induced by the longitudinal component of the magnetic part of the spin polarization tensor coupled with the azimuthal electric component. This coupling appears in the numerical solutions of the hydrodynamic equations for the chosen expansion symmetry.

What carries the argument

The spin polarization tensor decomposed into electric and magnetic components, whose coupling is induced by the boost-invariant cylindrical symmetry in the perfect fluid equations.

If this is right

Numerical solutions exhibit coupling between azimuthal and longitudinal spin components.
Total polarization is nonzero only from the longitudinal magnetic and azimuthal electric coupling.
The approach allows more general initial conditions than previous Gubser-symmetric cases.
Results can benchmark models including viscosity or different statistics.

Where Pith is reading between the lines

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

This geometry-specific restriction might simplify polarization calculations in certain collision scenarios.
Extending to viscous hydrodynamics could reveal additional polarization sources.
The reference results may help interpret spin alignment data from heavy-ion experiments.

Load-bearing premise

Perfect non-viscous hydrodynamics with Boltzmann statistics on a constant-temperature freeze-out hypersurface captures the spin evolution for the wounded-nucleon initial conditions.

What would settle it

A numerical simulation with viscous corrections or different freeze-out conditions that produces total polarization from other spin tensor components would contradict the finding.

Figures

Figures reproduced from arXiv: 2605.01857 by Jakub Witkowski, Wojciech Florkowski, Zbigniew Drogosz.

**Figure 1.** Figure 1: FIG. 1. Contour plots of the square of the speed of sound, view at source ↗

**Figure 2.** Figure 2: FIG. 2. Transverse spatial profiles of the scaled temperature view at source ↗

**Figure 3.** Figure 3: FIG. 3. Comparison of the transverse spatial profiles of the scaled temperature view at source ↗

**Figure 4.** Figure 4: FIG. 4. Time evolution of the spin polarization tensor components as a function of the radial distance view at source ↗

**Figure 5.** Figure 5: FIG. 5. Time evolution of the spin polarization tensor components as a function of the radial distance view at source ↗

**Figure 6.** Figure 6: FIG. 6. Projection ( view at source ↗

**Figure 7.** Figure 7: FIG. 7. Momentum dependence of the mean spin polarization vector components view at source ↗

**Figure 8.** Figure 8: FIG. 8. Momentum dependence of the mean spin polarization vector components view at source ↗

read the original abstract

Equations of a boost-invariant and cylindrically symmetric perfect hydrodynamics are solved numerically for initial conditions inspired by the wounded nucleon model. The energy-momentum and spin tensors are used in the form that describes a relativistic massive gas governed by Boltzmann statistics. In contrast to one dimensional boost-invariant expansion, we find a coupling between the azimuthal and longitudinal components of the electric and magnetic components of the spin polarization tensor. This feature is similar to that found earlier for the Gubser symmetry, however, our treatment allows for a more general form of initial conditions and expansion geometry. Defining the freeze-out hypersurface by the constant temperature condition, we evaluate the Pauli-Luba\'nski four-vector and find that for the assumed geometry the only nonzero total polarization may be induced by the longitudinal component of the magnetic part of the spin polarization tensor coupled with the azimuthal electric component. The obtained results may serve as a reference point for more realistic models of hydrodynamic expansion.

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.

Desk Editor's Note private letter to a colleague

This paper gives a numerical reference solution for perfect spin hydrodynamics in boost-invariant cylindrical symmetry and identifies a specific polarization coupling that follows from the symmetry.

read the letter

This paper gives a numerical reference solution for perfect spin hydrodynamics under boost-invariant cylindrical symmetry, and it identifies a coupling between azimuthal and longitudinal spin components that produces a specific rule for the total polarization at freeze-out. They solve the hydrodynamic equations with the standard energy-momentum and spin tensors for a massive Boltzmann gas, using initial conditions from the wounded nucleon model. This extends the Gubser-symmetric case to a geometry that allows more general initial conditions while keeping the boost invariance. The key new feature is the reported coupling, which leads to the Pauli-Lubanski vector having a nonzero contribution only from the longitudinal magnetic part coupled to the azimuthal electric part. That selection rule is the main result and it follows from the symmetry assumptions. The work is solid as far as the setup goes. The modeling choices are stated clearly, and the stress test confirms no internal inconsistency in how the polarization is computed from the tensors. The main limitation is the lack of reported numerical validation. There are no error estimates or convergence checks mentioned, so it's difficult to assess how precise the polarization result is. The perfect hydro and constant temperature freeze-out are reasonable for a reference calculation, but they limit how directly this applies to real collisions where viscosity and more complex freeze-out matter. This is aimed at researchers in relativistic heavy-ion physics who are developing spin hydro models. Someone looking for a benchmark in this symmetry class would find it useful. It deserves peer review because the calculation is well-defined and the result is a clean geometric finding, though the referees should ask for more details on the numerics to make the paper stronger. I would recommend sending it to peer review.

Referee Report

1 major / 1 minor

Summary. The manuscript numerically solves the equations of boost-invariant and cylindrically symmetric perfect spin hydrodynamics for a relativistic massive gas obeying Boltzmann statistics, with initial conditions inspired by the wounded nucleon model. It reports a coupling between the azimuthal and longitudinal components of the electric and magnetic parts of the spin polarization tensor. Defining freeze-out by a constant-temperature hypersurface, the Pauli-Lubański four-vector is evaluated, leading to the conclusion that the only nonzero total polarization is induced by the longitudinal magnetic component of the spin polarization tensor coupled to the azimuthal electric component. The results are positioned as a reference point for more realistic hydrodynamic models.

Significance. If the numerical results are robust, the work supplies a clear geometric benchmark for spin polarization in heavy-ion collisions under cylindrical symmetry. By extending the component-coupling analysis beyond the Gubser-symmetric case to more general initial conditions, it isolates the specific tensor contributions that survive the symmetry assumptions and can inform subsequent studies that include viscosity or dynamical freeze-out.

major comments (1)

The abstract states that the hydrodynamic equations are solved numerically and a specific polarization result is obtained, yet no error estimates, convergence checks, grid-resolution studies, or validation against the one-dimensional boost-invariant limit are reported. Because the central claim rests on the outcome of this integration, the absence of these diagnostics is load-bearing for the reliability of the reported nonzero polarization component.

minor comments (1)

The abstract refers to the 'Pauli-Lubański four-vector'; the main text should introduce its explicit definition and relation to the spin polarization tensor at the outset of the results section for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

Referee: The abstract states that the hydrodynamic equations are solved numerically and a specific polarization result is obtained, yet no error estimates, convergence checks, grid-resolution studies, or validation against the one-dimensional boost-invariant limit are reported. Because the central claim rests on the outcome of this integration, the absence of these diagnostics is load-bearing for the reliability of the reported nonzero polarization component.

Authors: We agree that the reliability of the numerical integration is central to the reported nonzero polarization and that the manuscript would benefit from explicit documentation of the numerical methods. In the revised version we will add a new subsection (placed after the description of the initial conditions) that reports: (i) convergence tests under successive grid refinements in the radial and proper-time directions, (ii) an estimate of the truncation error associated with the finite-difference scheme employed, and (iii) a direct validation of the code against the analytic one-dimensional boost-invariant limit obtained by suppressing all transverse gradients. These additions will be accompanied by a brief description of the numerical algorithm and the time-stepping method, thereby addressing the referee’s concern without changing the physical conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from symmetry and numerical integration

full rationale

The paper numerically solves the boost-invariant cylindrically symmetric perfect hydrodynamic equations using the standard energy-momentum and spin tensors for a massive Boltzmann gas. The identified coupling between azimuthal and longitudinal components of the electric and magnetic parts of the spin polarization tensor, and the subsequent evaluation of the Pauli-Lubański four-vector on the constant-temperature freeze-out hypersurface, are direct consequences of the imposed geometry, initial conditions inspired by the wounded nucleon model, and the explicit tensor forms. No parameters are fitted to force the polarization result, no self-citation chain supplies a load-bearing uniqueness theorem or ansatz, and the modeling premises (perfect hydrodynamics, Boltzmann statistics) are stated upfront rather than smuggled in. The central claim therefore reduces to the integration of the stated equations under the given symmetries, with no reduction by construction to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard relativistic hydrodynamics assumptions plus the specific symmetry and statistics choice; no new entities are introduced.

axioms (2)

domain assumption Energy-momentum and spin tensors describe a relativistic massive gas governed by Boltzmann statistics
Invoked to close the hydrodynamic equations.
domain assumption Boost invariance and cylindrical symmetry
Used to reduce the dimensionality of the problem.

pith-pipeline@v0.9.0 · 5457 in / 1134 out tokens · 34495 ms · 2026-05-09T16:57:29.362508+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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7 and Fig

Case (a) If only theC kr component is different from zero, one can check thatF ∗ µ = 0 andG ∗ µ = 0, which is confirmed by the numerical results presented in Fig. 7 and Fig. 8 (the top raws show that all the components of the Pauli–Luba´ nski four-vector vanish)
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Case (b) In this case, the only nonzero component of the spin polarization tensor isC ωr. The boost to the particle rest frame gives F ∗ µ = 4πC ωrI1(p)(K0(m)mT +K 1(m)Tsechθ) m mT (0,−cosϕ p,−sinϕ p,0) (D1) and G∗ µ = 4πC ωrK1(m)(I2(p)pT +I 1(p)Tcschθ) m pT (0,−cosϕ p,−sinϕ p,0).(D2) For conciseness, the arguments of the Bessel functions use the notation...
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Case (c) In this case the only nonzero components of the spin polarization tensor areC ωz andC kϕ. The boost to PRF gives F ∗ 0 =F ∗ 1 =F ∗ 2 = 0, F ∗ 3 = 2π[sinh(θ) (C ωz I1(p)κ pT −2C kϕI0(p)K1(m)mT ) + cosh(θ) (CkϕI1(p)κ pT −2C ωz I0(p)K1(m)mT )] (D3) and G∗ 0 =G ∗ 1 =G ∗ 2 = 0, G∗ 3 = 4πsinh(θ) [K 1(m)(Ckϕ coth(θ) +C ωz)(I1(p)Tcsch(θ) +I 2(p)pT ) −I1(...
[4]

Case (d) In this case, the only nonzero components areC kz andC ωϕ. A straightforward calculation gives F ∗ 0 = 0, F ∗ 1 = 4πsin(ϕ p) [I0(p)K1(m)pT [Ckz cosh(θ)−C ωϕ sinh(θ)] +I1(p)sech(θ) [Cωϕ cosh(θ)−C kz sinh(θ)] (K0(m)mT cosh(θ) +K 1(m)T)], F ∗ 2 = 4πcos(ϕ p)(I0(p)K1(m)pT (Cωϕ sinh(θ)−C kz cosh(θ)) −I1(p)(Cωϕ −C kz tanh(θ))(K0(m)mT cosh(θ) +K 1(m)T)),...
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