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Within the framework of the operator product expansion (OPE) and therenormalization group equation (RGE), we show that the temperature and chemicalpotential dependence of the zeroth moment of a spectral function (SF) iscompletely determined by the one-loop structure in an asymptotically freetheory, and in particular in QCD. Logarithmic corrections are found to play anessential role in the derivation. This exact result constrains the shape ofSF's, and implies striking effects near phase transitions. Phenomenologicalparameterizations of the SF, often used in applications such as the analysis oflattice QCD data or QCD sum rule calculations at finite temperature and baryondensity must satisfy these constraints. We also explicitly illustrate in detailthe exact sum rule in the Gross-Neveu model.Comment: 26 pages in RevTeX 3.0 plus 7 figures in a separate uuencoded  postscript fil

Constraining spectral functions at finite temperature and chemical potential with exact sum rules in asymptotically free theories