Spin relaxation of heavy quarks from momentum diffusion in the quark-gluon plasma

Heavy-quark spin dynamics in the quark-gluon plasma provides a potential probe of the interaction between heavy quarks and the medium. We develop a stochastic framework in which the random momentum kicks responsible for heavy-quark diffusion also induce spin rotation through Thomas precession. Starting from the Thomas-Bargmann-Michel-Telegdi equation and a Langevin equation for heavy-quark momentum, we derive a novel coupled spin-momentum Fokker-Planck equation. In the heavy-quark limit, where momentum equilibrates faster than spin, the momentum degrees of freedom can be integrated out, yielding a rotational Fokker-Planck equation for the spin orientation. This establishes a direct relation between the Thomas precession contribution to the spin-relaxation time ${\tau }_s$ and the heavy-quark momentum-diffusion coefficient $\kappa$. In the non-relativistic limit, we find ${\tau }_s=2{\tau }_p(m_Q\mathrm{/}T{)}^2$, with ${\tau }_p$ being the momentum relaxation time. Using recent determinations of $\kappa$ from lattice QCD, strong-coupling estimates, $T$-matrix calculations, and heavy-flavor phenomenology, we obtain the corresponding Thomas precession contribution to spin relaxation times for charm and bottom quarks. Relativistic kinematics modify the result by only a modest amount over the temperature range relevant for heavy-ion collisions, while the uncertainty is dominated by that of $\kappa$. We emphasize that the result represents the Thomas precession contribution to heavy-quark spin relaxation; direct coupling to the chromomagnetic field provides an additional channel.