Transient spin modes from relaxational axial kinetic theory

Abstract

We study the dynamics of spin mode by solving the axial kinetic equations under the relaxation time approximation in the presence of dissipative sources. We find transient spin modes in response to electric field with spacetime inhomogeneity, fluid acceleration and shear. To the lowest order in spatial momentum $k$, we find the responses to electric field and acceleration can be interpreted as retarded response to temporal variations of magnetic field and vorticity respectively. The response to shear occurs at $O(k^2)$ and can be reduced to retarded response to spatial variation of vorticity. Beyond lowest order, the responses to all three sources are non-local with branch cut in the dispersions. We argue that the non-locality is a consequence of the quasi-particle picture underlying the kinetic description. We also analyze spin transport equation taking into account spin response to temporal and spatial variations of vorticity. We find the corrections turn the original first order spin transport equation into a third order one (or a second order one in the homogeneous limit). The change in order of transport equation is a consequence of non-local nature of the responses, suggesting possible breakdown of gradient expansion in spin hydrodynamics for microscopic theories with quasi-particles.