Analytic structure of stress-energy response functions and new Kubo formulas

Determining the transport properties of quark-gluon plasma is one of the most important aspects of relativistic heavy ion collision studies. Field-theoretical calculations of transport coefficients such as the shear and bulk viscosities require Kubo formulas, which in turn require real-time correlation functions of stress-energy tensors. Consequently, knowing the analytic structure of these correlation functions is essential in any such studies. Using the energy-conservation laws and the results from gravity-hydrodynamics analysis, we determine the low-frequency and low-wavenumber analytic structures of all stress-energy correlation functions in the rest frame of the medium. By comparing with the diffusion and sound spectra from the second-order and third-order relativistic hydrodynamics, various new Kubo formulas are derived in the limit where the zero-frequency limit is taken first. We also show that the meaning of the Kubo formulas for relaxation times can change when higher-order terms are added to the hydrodynamics. A subtle issue of taking the zero frequency and zero wave0number limits when using skeleton diagrams is addressed as well.