Second-order evolution of spin shear tensor for massive spin-1/2 particles

The paper sets out to supply the missing piece of a causal second-order spin hydrodynamics: a relaxation-type evolution equation for the spin-shear tensor, the dissipative component of the spin tensor of a relativistic fluid of massive spin-1/2 particles. Working from a semi-classical kinetic theory with a relaxation-time collision kernel, the authors derive an equation of the form $\dot{\Sigma}^{\langle\lambda\nu\rangle}_{(s)} + \Sigma^{\langle\lambda\nu\rangle}_{(s)}/\tau_R = B^{\lambda\nu}_{\Pi}\,\theta + B^{\lambda\nu\alpha}_{a}\,\dot{u}_\alpha + B^{\lambda\nu\alpha\beta}_{\pi}\,\sigma_{\alpha\beta} + B^{\lambda\nu\alpha\beta\gamma}_{\Sigma}\,\nabla_\gamma\omega_{\alpha\beta} + \dots$, in which spin shear relaxes at a rate set by $\tau_R$ and is driven by expansion, acceleration, shear flow, gradients of the spin polarization tensor, and second-order couplings to the bulk pressure, shear stress, spin diffusion, fluid vorticity, and their gradients. The motivation is concrete: first-order spin hydrodynamics is acausal, and recent second-order studies indicate that the spin-shear mode is the decisive ingredient for describing the longitudinal polarization of $\Lambda$ hyperons observed in heavy-ion collisions. The paper evaluates the new transport coefficients in the equation and shows that its linear-gradient part reproduces the known first-order spin-shear correction.