Gluon Condensates and m_{c,b} from QCD-Moments and their ratios to Order   alpha_s^3 and <G^4>
  We reconsider the extraction of the gluon condensates <alpha_s G^2>, <g^3f_{abc} G^3> and the MS running quark masses m_{c,b} from different M_n(Q^2) Moments and their Ratios by including PT corrections to order alpha_s^3, NPT terms up to <G^4> and using stability criteria of the results versus the degree n (number of Q^2-derivative). We explicitly show that the spectral part of the lowest moment M_1(0) depends strongly (as expected) on its high-energy (continuum) contribution, which is minimized for M_{n> 3-4}(0). Using higher moments and the correlations of <alpha_s G^2> with <g^3f_{abc} G^3> and < G^4>, we obtain <alpha_s G^2>=(7.0+- 1.3)10^{-2} GeV^4 and <g^3f_{abc} G^3>=(8.8+- 5.5) GeV^2 <alpha_s G^2>, while our analysis favours a modified factorisation for <G^4>. Using the previous results, we re-determine m_c(m_c) and find that the commonly used M_1(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 M_1(0) may have been underestimated. Optimal results from different high-n moments converge to the accurate (artithmetic) mean values: m_c(m_c)=1261(16) MeV and m_b(m_b)=4171(14) MeV, in excellent agreement with results obtained in [1] using some judicious choices of ratios of moments. 
Stephan Narison
hep-ph hep-lat