Spin alignment, tensor polarizabilities, and local equilibrium for spin-1 particles

TL;DR

The paper develops a Lorentz-covariant framework for spin-1 particles in heavy-ion collisions using the adjoint representation, where the spin density matrix separates into a polarization vector $\mathcal{P}^\mu$ and tensor polarizabilities $\mathcal{T}^{\mu\nu}$. It connects this covariant structure to the Wigner function, enabling expressions for the energy–momentum and spin tensors, and introduces a local-equilibrium spin density matrix that yields thermodynamically consistent relations and a unified description with spin-1/2 particles, paving the way for perfect spin hydrodynamics. The work clarifies the role of tensor polarizabilities in alignment measurements, showing that the observable alignment $\mathcal{A}$ is governed by $\mathcal{T}^{22}_*$ and can emerge in local equilibrium through the magnetic components of the spin polarization tensor $\omega^{\mu\nu}$, rather than requiring dissipation. The framework thus provides a coherent link between hyperon and vector-meson spin data and offers a practical route to analyze spin observables within a single, divergence-type theory of spin hydrodynamics.

Abstract

Different bases for the spin-1 density matrix are discussed to clarify the connection between its components and observables measured in heavy-ion collisions. The theoretical advantage of using the adjoint representation for spin matrices is emphasized. Next, the equilibrium spin density matrix and the corresponding Wigner function are introduced. With appropriate definitions of the energy-momentum and spin tensors, this framework allows for the formulation of perfect spin hydrodynamics in the same way as previously done for spin-1/2 particles. Together, these results provide a unified description of spin-1/2 and spin-1 particles.

Figures (1)

• Figure 1: Schematic diagram of the transformations between different representations, defined by the unitary matrices $U_{AB}$, whose action is given by $U^\dagger_{AB} A U_{AB} = B$, where $A,B = J,S,T$.