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Second order hydrodynamic coefficients from 3-point stress tensor correlators via AdS/CFT

We study second order relativistic viscous hydrodynamics in 4-dimensional conformal field theories. We derive Kubo-type relations for second order hydrodynamic coefficients in terms of 3-point stress tensor retarded correlators. For N=4 super Yang-Mills theory at strong coupling and at finite temperature we compute these stress tensor 3-point correlators, using AdS/CFT, by evaluating real-time cubic Witten diagrams in the AdS-Schwarzschild background. The small momentum expansion of the 3-point correlators in terms of first and second order hydrodynamic coefficients is matched with the AdS result. We arrive at the same expressions for the hydrodynamic coefficients which multiply terms quadratic in the shear and vorticity tensors in the hydrodynamic expansion of the stress tensor as did Bhattacharyya, Hubeny, Minwalla and Rangamani \cite{Bhattacharyya:2008jc}. Our method extends the results of Baier et al \cite{Baier:2007ix}, and allows for a unified treatment of hydrodynamic coefficients, which are extracted from 2-, and now, 3-point retarded stress tensor correlators in the AdS-Schwarzschild background.