Kubo Formulas for Second-Order Hydrodynamic Coefficients

At second order in gradients, conformal relativistic hydrodynamics depends on the viscosity eta and on five additional second-order'' hydrodynamical coefficients tau(Pi), kappa, lambda(1), lambda(2) and lambda(3). 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 lambda(3) can be evaluated directly by Euclidean means and does not in general vanish.