Gluon condensates and precise m¯c,b from QCD-moments and their ratios to order αs3 and 〈G4〉

We reconsider the extraction of the gluon condensates 〈αsG2〉, 〈g3fabcG3〉 and the MS¯ running quark masses m¯c,b from different Mn(Q2) moments and their ratios by including PT corrections to order αs3, NPT terms up to 〈G4〉 and using stability criteria of the results versus the degree n (number of Q2-derivative). We explicitly show that the spectral part of the lowest moment M1(0) depends strongly (as expected) on its high-energy (continuum) contribution, which is minimized for Mn⩾3−4(0). Using higher moments and the correlations of 〈αsG2〉 with 〈g3fabcG3〉 and 〈G4〉, we obtain 〈αsG2〉=(7.0±1.3)×10−2GeV4 and 〈g3fabcG3〉=(8.8±5.5)GeV2×〈αsG2〉, while our analysis favours a modified precise factorisation for 〈G4〉. Using the previous results, we re-determine m¯c(m¯c) and find that the commonly used M1(0) lowest moment tends to overestimate its value compared to the ones from higher moments where stable values of m¯c(m¯c) versus the variations of n and the continuum models are reached. These features can indicate that the quoted errors of m¯c,b from M1(0) may have been underestimated. Our best results from different high-n moments and their ratios are: m¯c(m¯c)=1261(16)MeV and m¯b(m¯b)=4171(14)MeV, in excellent agreement with results obtained in Narison (2010) [1] using some judicious choices of ratios of moments.