An improved formula for Wigner function and spin polarization in a decoupling relativistic fluid at local thermodynamic equilibrium

TL;DR

This work develops a new gradient-expansion framework for the Wigner function and spin polarization of LTE fermions emitted at decoupling in relativistic fluids, valid for hypersurfaces of arbitrary geometry. By inverting the momentum and hypersurface integrations, it isolates leading LTE corrections as a tangential-gradient expansion on the decoupling surface, naturally excluding normal-gradient contributions and providing a theoretical justification for isothermal decoupling. The main results include an upgraded expression for spin polarization of spin-1/2 fermions, where thermal vorticity and thermal shear contributions are refined and extended to account for geometry, including a sign factor and potential quantum-interference-like terms from nontrivial worldline intersections. The framework generalizes to arbitrary spin and offers a more reliable basis for numerical simulations of heavy-ion collisions, particularly in computing spin observables from LTE on complex decoupling geometries.

Abstract

We present an upgraded formula for Wigner function and spin polarization of fermions emitted by a relativistic fluid at local thermodynamic equilibrium at the decoupling which improves the one obtained in refs. [1, 2] and used in numerical simulations of relativistic nuclear collisions. By using a new expansion method, applicable to decoupling hypersurfaces with arbitrary geometry, we reproduce the known term proportional to thermal vorticity and obtain an upgraded form of the spin-shear term which captures the dependence on the geometry. The new method also includes additional contributions whose physical nature is to be assessed. The new expression also naturally excludes contributions from space-time gradients in the normal direction of the hypersurface, providing a theoretical justification for the isothermal condition previously imposed a priori. This framework can be extended to particles with arbitrary spin.

Figures (1)

• Figure 1: A slice of the decoupling hypersurface $\Sigma_{\rm D}$ (red and blue lines) on the transverse plane $x=y=0$, obtained by 3+1D viscous hydrodynamic simulations with the CLVisc model at $\sqrt{s_{\text{NN}}}=27$ GeV Pang:2018zzoWu:2021fjfWu:2022mkr. The black solid line represents the worldline of a particle, which intersects $\Sigma_{\rm D}$ at two points: one on the spacelike branch and the other one on the timelike branch.