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 τ_Π, 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.