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Constraining Spectral Functions at Finite Temperature and Chemical Potential with Exact Sum Rules in Asymptotically Free Theories

Within the framework of the operator product expansion (OPE) and the renormalization group equation (RGE), we show that the temperature and chemical potential dependence of the zeroth moment of a spectral function (SF) is completely determined by the one-loop structure in an asymptotically free theory, and in particular in QCD. Logarithmic corrections are found to play an essential role in the derivation. This exact result constrains the shape of SF's, and implies striking effects near phase transitions. Phenomenological parameterizations of the SF, often used in applications such as the analysis of lattice QCD data or QCD sum rule calculations at finite temperature and baryon density must satisfy these constraints. We also explicitly illustrate in detail the exact sum rule in the Gross-Neveu model.