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Thermal QCD sum rules in the $ρ^0$ channel revisited

From the hypothesis that at zero temperature the square root of the spectral continuum threshold $s_0$ is linearly related to the QCD scale $Λ$ we derive in the chiral limit and for temperatures considerably smaller than $Λ$ scaling relations for the vacuum parts of the Gibbs averaged scalar operators contributing to the thermal operator product expansion of the $ρ^0$ current-current correlator. The scaling with $λ\equiv \sqrt{s_0(T)/s_0(0)}$, $s_0$ being the $T$-dependent perturbative QCD continuum threshold in the spectral integral, is simple for renormalization group invariant operators, and becomes nontrivial for a set of operators which mix and scale anomalously under a change of the renormalization point. In contrast to previous works on thermal QCD sum rules with this approach the gluon condensate exhibits a sizable $T$-dependence. The $ρ$ -meson mass is found to rise slowly with temperature which coincides with the result found by means of a PCAC and current algebra analysis of the $ρ^0$ correlator.