Kubo Formulas for Second-Order Hydrodynamic Coefficients

At second order in gradients, conformal relativistic hydrodynamics depends on the viscosity η and on five additional ''second-order'' hydrodynamical coefficients τΠ, κ, λ1, λ2, and λ3. We derive Kubo relations for these coefficients, relating them to equilibrium, fully retarded three-point correlation functions of the stress tensor. We show that the coefficient λ3 can be evaluated directly by Euclidean means and does not in general vanish.