Kubo formulas for spin polarization in dissipative relativistic spin hydrodynamics: a first-order gradient expansion approach

We use linear response theory to derive both the non-dissipative and dissipative effects of spin polarization for massive and massless interacting spin 1/2 particles in a relativistic fluid. We list and classify all the possible contributions up to first order in gradients of hydrodynamic fields including the axial chemical potential and the spin potential, and we obtain the corresponding Kubo formulas. We find that all the possible dissipative contributions, except those coming from the gradients of spin potential, require a chiral imbalance or parity violating interactions. In a fluid with chiral imbalance we find a chiral version of the spin Hall effect, i.e. a spin polarization is induced by the gradients of temperature and of axial chemical potential in the direction orthogonal to the momentum of the particle and to the gradients. Moreover, we identify several other new non-dissipative contributions that are not present for free fields.