  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.
