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Covariant Wigner function and spin polarization of the Rarita-Schwinger field

This paper derives a covariant Wigner function for the spin-3/2 Rarita-Schwinger field to establish a formula for mean spin polarization induced by thermal vorticity, shear, and spin potential, demonstrating that local measurements of $\Omega^-$ baryons in heavy-ion collisions can distinguish between canonical and Belinfante pseudo-gauges.

Original authors: Matteo Buzzegoli, Aleksandar Gecic, Rajeev Singh

Published 2026-10-28

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CC BY 4.0

Original authors: Matteo Buzzegoli, Aleksandar Gecic, Rajeev Singh

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the extreme heat of a heavy-ion collision, where atomic nuclei smash together at nearly the speed of light, matter melts into a super-hot soup called the quark-gluon plasma. This state of matter existed just microseconds after the Big Bang, and today, physicists recreate it in massive particle accelerators to understand how the universe began. As this fiery plasma expands and cools, it eventually freezes into a spray of new particles, including some rare, heavy particles known as hyperons. Among these are particles with a specific property called "spin," which is a form of intrinsic angular momentum, much like a spinning top, though it is a purely quantum mechanical trait. Scientists have recently discovered that these particles do not spin randomly; instead, they align in a specific direction, a phenomenon known as global spin alignment. This alignment acts as a sensitive probe, revealing details about the fluid-like behavior and the intense swirling motion of the plasma that created them.

For years, researchers have successfully explained the spin alignment of lighter particles, such as the lambda hyperon, using established theories. However, a gap remained for heavier, more complex particles like the omega-minus hyperon, which carries a spin of three-halves. The existing mathematical tools worked well for simpler particles but had not been rigorously extended to these more complicated ones. In a new study, a team of physicists has filled this gap by developing a complete theoretical framework to describe the spin polarization of these heavy particles. They derived a precise formula that connects the microscopic quantum behavior of the particles to the macroscopic flow of the fluid, allowing them to predict how these particles should align as they emerge from the collision.

The researchers focused on a specific type of field theory used to describe particles with spin three-halves, known as the Rarita-Schwinger field. To understand how these particles behave in the chaotic environment of a heavy-ion collision, they needed to calculate a quantity called the Wigner function. Think of this function as a detailed map that shows not just where a particle is and how fast it is moving, but also how its internal spin is oriented at every point in space and time. By constructing this map for the Rarita-Schwinger field, the team could trace how the spin of the omega-minus particle is influenced by the fluid's motion. They found that the spin alignment is driven by three main factors: the swirling motion of the fluid (thermal vorticity), the stretching and squeezing of the fluid (thermal shear), and a subtle quantum force related to the particle's chemical potential.

A critical part of their work involved addressing a long-standing ambiguity in physics known as the "pseudo-gauge" problem. In theoretical physics, there are different ways to mathematically define how energy and spin are distributed within a fluid. Two of the most common definitions are called the Belinfante and the canonical pseudo-gauges. For simpler particles, these different definitions often lead to the same predictions for how spin aligns. However, the team discovered that for the heavier spin-three-halves particles, these two definitions actually predict different outcomes. Specifically, the way the fluid's stretching motion affects the spin of the omega-minus particle depends on which mathematical definition is used. In one framework, the stretching has a strong effect, while in the other, that effect is significantly weaker.

This difference is not just a mathematical curiosity; it offers a concrete way to test which theory better describes reality. The authors propose that by measuring the local spin polarization of the omega-minus hyperon in future high-precision experiments, scientists can determine which of the two mathematical frameworks is correct. If the experimental data matches the prediction from the Belinfante framework, it will confirm one view of how energy and spin interact in extreme conditions. If it matches the canonical framework, it will support the other. The study provides the necessary formulas to make these predictions, taking into account the complex geometry of the collision and the moment the particles stop interacting.

The researchers also clarified how the spin of these particles relates to the fluid's temperature and density gradients. They showed that while the swirling motion of the fluid creates a baseline alignment for all particles, the stretching of the fluid adds a unique layer of complexity for the heavier particles. This stretching effect, combined with the quantum forces, creates a specific pattern of alignment that is distinct from what is seen in lighter particles. The team's calculations confirm that the alignment scales with the particle's spin in a predictable way, but only when the fluid is in a state of local thermal equilibrium, meaning the temperature and flow are relatively stable at the moment the particles break free.

By providing these exact formulas, the paper equips experimentalists with the tools needed to interpret upcoming data from facilities like the Large Hadron Collider. The ability to distinguish between the two mathematical frameworks using the omega-minus particle is a significant step forward. It moves the field from general observations of spin alignment to a precise test of the underlying laws governing how matter behaves at the highest energies. The work does not claim to have solved the mystery of the quark-gluon plasma entirely, but it has provided a crucial piece of the puzzle, turning a theoretical ambiguity into a measurable quantity that can be tested in the laboratory.

Technical Summary: Covariant Wigner Function and Spin Polarization of the Rarita-Schwinger Field

Problem Statement
The experimental observation of global spin alignment and polarization of hyperons in relativistic heavy-ion collisions has opened a new avenue for investigating the quantum properties of strongly interacting matter. While theoretical models for spin-1/2 particles (Dirac field) rigorously connect the mean spin polarization vector to the covariant Wigner function, a complete field-theoretic derivation for spin-3/2 particles has been lacking. Previous phenomenological studies relied on extending the global thermal equilibrium formula for spin-3/2 particles directly to local thermal equilibrium (LTE). However, such an approach fails to capture out-of-equilibrium effects essential for interpreting local measurements, such as contributions from thermal shear and the spin Hall effect, which vanish in global equilibrium. This work addresses the gap by providing a rigorous quantum field-theoretic (QFT) derivation for the massive spin-3/2 Rarita-Schwinger field, establishing the link between the covariant Wigner function and the mean spin polarization vector, and calculating the resulting polarization in LTE across different pseudo-gauges.

Methodology
The authors employ the framework of relativistic quantum statistical mechanics and field theory. The core methodology involves:

Field Formulation: Defining the free, massive spin-3/2 field via the Rarita-Schwinger vector-spinor operator $\Psi_\mu(x)$ and deriving the canonical energy-momentum and spin tensors using Noether's theorem.
Pseudo-Gauge Transformations: Utilizing pseudo-gauge transformations to map the canonical tensors into the symmetrized Belinfante form (vanishing spin tensor) and the de Groot–van Leeuwen–van Weert (GLW) form (non-vanishing conserved spin tensor).
Covariant Wigner Function: Defining the covariant Wigner operator as the Fourier transform of the two-point function of the Rarita-Schwinger field. The authors derive its properties, including its decomposition into particle, antiparticle, and spacelike parts, and its on-shell behavior upon integration over a hypersurface.
Spin Polarization Derivation: Deriving an exact relation between the mean spin vector and the Wigner function by expressing the spin density matrix in terms of the Wigner function components. This involves projecting the Pauli-Lubanski operator onto the momentum subspace and utilizing the embedding matrices that connect the $(0, 3/2)$ irreducible representation to the 16-dimensional vector-spinor representation.
Local Thermal Equilibrium (LTE) Calculation: Applying linear response theory to the LTE statistical operator. The authors calculate the Wigner function as a sum of a global equilibrium term and linear response corrections driven by gradients of the inverse temperature ( $\beta$ ), the spin potential ( $S$ ), and the reduced chemical potential ( $\zeta$ ).
Hypersurface Integration: Employing recent methods to integrate over the complex, topologically non-trivial decoupling hypersurface $\Sigma_D$ encountered in heavy-ion collisions, accounting for both spacelike and timelike branches.

Key Contributions and Results

General Polarization Formula: The paper derives the exact formula (Eq. 110) relating the mean spin polarization vector $S^\mu(p)$ to the covariant Wigner function for the spin-3/2 field. Unlike the spin-1/2 case, this formula includes an additional contribution from the "dual axial" part of the Wigner function, arising from the tensor structure of the vector-spinor field.
GLW Pseudo-Gauge: In the GLW gauge, the authors find that spin polarization is driven solely by the spin potential $S_{\rho\nu}$ and gradients of the chemical potential (spin Hall effect). Notably, there is no spin polarization induced by gradients of $\beta$ (thermal vorticity or thermal shear) in this gauge for spin-3/2 fields.
Belinfante and Canonical Pseudo-Gauges:
- In the Belinfante gauge, where the spin tensor vanishes, the polarization is generated by thermal vorticity $\varpi_{\rho\nu}$ , thermal shear $\xi_{\rho\nu}$ , and chemical potential gradients.
- In the Canonical gauge, the polarization depends on the spin potential, thermal vorticity, and thermal shear.
Discrimination between Pseudo-Gauges: A critical result is the difference in thermal-shear-induced polarization between the Belinfante and Canonical gauges. While these gauges yield identical predictions for the Dirac field (spin-1/2), they diverge for the Rarita-Schwinger field (spin-3/2). Specifically, in the Canonical gauge, the thermal-shear contribution is suppressed by a factor of $1/3$ compared to the Belinfante gauge, even when the spin degrees of freedom are equilibrated ( $S = \varpi$ ).
Spin Hall Effect Scaling: The paper finds that the spin Hall effect (polarization induced by chemical potential gradients) scales differently with the spin $S$ compared to thermal vorticity and shear. For spin-3/2, this results in an extra $1/3$ factor in the scaling relation.

Significance and Claims
The authors claim that this work provides the first rigorous QFT derivation connecting the mean spin polarization to the covariant Wigner function for spin-3/2 particles. The derived formulas (Eqs. 161, 172, 175) serve as the key theoretical input for extracting spin polarization predictions from relativistic hydrodynamics or transport models.

The primary significance lies in the potential to discriminate between pseudo-gauge choices using experimental data. The authors state that the difference in thermal-shear-induced polarization between the Canonical and Belinfante gauges allows for the use of local spin polarization measurements of the $\Omega^-$ baryon (a spin-3/2 hadron) to determine which pseudo-gauge better describes heavy-ion collisions. This is distinct from the $\Lambda$ hyperon (spin-1/2), where both gauges yield identical predictions for thermal-shear effects. The paper posits that once bulk viscosity is constrained using $\Lambda$ data, the competing predictions for $\Omega^-$ polarization can be tested against experimental data to validate the appropriate theoretical framework.

The paper concludes by outlining future work to generate quantitative predictions for $\Omega^-$ hyperon spin polarization using ideal spin hydrodynamics in the GLW formalism and standard relativistic hydrodynamics in the Belinfante gauge.

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