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Quark Condensate from Renormalization Group Optimized Spectral Density

Our renormalization group consistent variant of optimized perturbation, RGOPT, is used to calculate the nonperturbative QCD spectral density of the Dirac operator and the related chiral quark condensate $\langle \bar q q \rangle$, for $n_f=2$ and $n_f=3$ massless quarks. Sequences of approximations at two-, three-, and four-loop orders are very stable and give $\langle \bar q q \rangle^{1/3}_{n_f=2}(2\, {\rm GeV}) = -(0.833-0.845) \barΛ_2 $, and $ \langle \bar q q \rangle^{1/3}_{n_f=3}(2\, {\rm GeV}) = -(0.814-0.838) \barΛ_3 $ where the range is our estimated theoretical error and $\barΛ_{n_f}$ the basic QCD scale in the $\rm \bar{MS}$-scheme. We compare those results with other recent determinations (from lattice calculations and spectral sum rules).