Hydrodynamic effective field theory and the analyticity of hydrostatic correlators

We study one-loop corrections to retarded and symmetric hydrostatic correlation functions within the Schwinger-Keldysh effective field theory framework for relativistic hydrodynamics, focusing on charge diffusion. first consider simplified setup with only diffusive density fluctuations, then augment it momentum fluctuations in a model where sound modes can be ignored. show that loop corrections, which generically induce non-analyticities long-range effects at finite frequency, non-trivially preserve analyticity of spatial due KMS constraint, as manifestation thermal screening. For purposes this analysis, we develop an interacting seen limit hydrodynamics absence temperature longitudinal velocity fluctuations.