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Gluon Condensates and m_b(m_b) from QCD-Exponential Moments at Higher Orders

We test the convergence of the QCD exponential moments by including PT corrections to order α_s^3 and the NP contributions up to D=8 <G^4> condensates. Then, using the ratio of exponential sum rules where the QCD PT series is more convergent, we study the correlation between the gluon condensates <α_s G^2> and < g^3f_{abc} G^3>. From charmonium systems and using the charm quark mass as input, we deduce:< g^3f_{abc} G^3> =(8.2+-1.0)GeV^2 <α_s G^2> corresponding to < α_s G^2>=(7.5+- 2.0) 10^{-2} GeV^4. Using these results for the bottomium systems, we obtain: m_b(m_b)= 4212(32) MeV, which is slightly higher but consistent within the errrors with the ones from Q^2-moments and their ratios: m_b(m_b)= 4172(12) MeV. We are tempted to consider as a final result from the sum rules approaches, the average m_b(m_b)= 4177(11) MeV of the two previous determinations.