The covariant equations of motion of massive spinning particles in a background Yang-Mills field

TL;DR

The paper addresses how massive spinning quarks propagate in a strong, non-Abelian glasma field, where previous Wong-based treatments neglect spin and fail to satisfy all physical constraints. It develops a covariant, constrained Hamiltonian framework (Dirac-Bergmann) that incorporates position $x^\mu$, kinetic momentum $P^\mu$, spin $S^{\mu\nu}$, and color charge $q^a$, with an on-shell condition $P^2 - 2\mu g q^a F^{a}_{\mu\nu} \\omega^\mu \pi^\nu - m^2 = 0$ and conserved classical Casimirs. The authors derive self-consistent, gauge-invariant equations of motion for all variables, show their equivalence to BMT-like spin dynamics in a non-Abelian field, and demonstrate the coupling between spin, color, and field gradients. This formalism provides a rigorous basis for studying momentum diffusion and spin polarization of hard probes in the glasma, with potential applications to heavy-quark transport and spin observables in relativistic heavy-ion collisions. The approach preserves essential symmetries while enabling systematic exploration of spin-color effects in early-time QCD matter.

Abstract

A strong classical color field, known as the glasma, is generated in the earliest stage of relativistic heavy-ion collisions and can significantly influence the momentum and spin dynamics of hard probes such as quarks and jets. Most existing studies based on the classical equations of motion in a background Yang-Mills field, such as Wong equations, may not capture the full range of effects, for example, they neglect the Stern-Gerlach force experienced by spinning particles in non-uniform glasma fields. Although several extensions of Wong equations have been proposed to include spin degrees of freedom, they generally fail to satisfy all the required conditions simultaneously, such as Lorentz covariance, allowance for an arbitrary chromomagnetic moment, and respect for the necessary physical constraints. In this work, we extend the framework of a relativistic classical spinning particle in an electromagnetic field to describe spin one-half quarks propagating in a background non-Abelian Yang-Mills field. By systematically applying the Dirac-Bergmann algorithm, we derive self-consistent equations of motion for the particle's coordinates, momenta, spin, and color charge that satisfy all these requirements. This formalism provides a more complete and physically relevant description for studying momentum diffusion and spin polarization phenomena of hard probes in the glasma.