Vector and Tensor Spin Polarization for Vector Bosons at Local Equilibrium

TL;DR

This paper derives detailed expressions for the spin polarization components of massive vector bosons at local equilibrium, including second-order effects, and provides formulas to compare with experimental data in heavy-ion collisions.

Contribution

It introduces a systematic derivation of vector and tensor polarization expressions up to second order in gradients, with a new set of Feynman rules for calculations.

Findings

Leading spin alignment arises from second-order terms due to time-reversal symmetry.

Provides analytic formulas for spin polarization components.

Predicts contributions to spin alignment observable in heavy-ion collisions.

Abstract

We derive expressions for the vector and tensor components of the spin polarization of massive vector bosons at local thermodynamic equilibrium up to second order in the space-time gradients of the thermodynamic fields pertaining to the canonical stress-energy tensor and spin tensor of the free Proca field. A set of Feynman rules is devised to calculate the Wigner function and the matrix-valued spin-dependent distribution (MVSD) functions order by order in space-time gradients. Due to constraints imposed by time-reversal symmetry, the leading contribution to spin alignment - defined as the 00-component of the tensor polarization - arises from second-order terms in MVSD, for which we provide an analytic formula. We discuss the physical meaning of different contributions to vector and tensor polarization. These formulae provide a prediction of a contribution to the spin alignment which can be compared with the observations in relativistic heavy-ion collisions.