The QCD up-and down-quark masses are determined from an optimized QCD Finite Energy Sum Rule (FESR) involving the correlator of axial-vector current divergences. In the QCD sector this correlator is known to five loop order in perturbative QCD (PQCD), together with non-perturbative corrections from the quark and gluon condensates. This FESR is designed to reduce considerably the systematic uncertainties arising from the hadronic spectral function. The determination is done in the framework of both fixed order and contour improved perturbation theory. Results from the latter, involving far less systematic uncertainties, are:m u (2 GeV) = (2.6 ± 0.4) MeV,m d (2 GeV) = (5.3 ± 0.4) MeV, and the summ ud ≡ (m u +m d )/2, ism ud (2 GeV) = (3.9 ± 0.3 )MeV. arXiv:1809.07042v5 [hep-ph]

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The light quark masses are determined from a QCD finite energy sum rule, using the pseudoscalar correlator to six-loop order in perturbative QCD, with the leading vacuum condensates and higher order quark mass corrections included. Both the fixed order perturbation theory (FOPT) method and contour improved perturbation theory (CIPT) method are explored. The results in the latter framework exhibit good convergence and stability in the window s 0 = 3.0 5.0 GeV2 for the strange quark and s0 = 1.5 4.0 GeV2 for the up and down quarks; where s 0 is the radius of the integration contour in the complex s-plane. The results are: m ¯ s ( 2  GeV ) = 91.8 ± 9.9 MeV , m ¯ u ( 2  GeV ) = 2.6 ± 0.4 MeV , m ¯ d ( 2  GeV ) = 5.3 ± 0.4 MeV , and the sum m ¯ u d ≡ ( m ¯ u + m ¯ d ) / 2 is m ¯ u d ( 2 GeV ) = 3.9 ± 0.3 MeV . These proceedings critically explore how the current results - computed precisely in a modern computer language, Mathematica compare to past determinations in literature.

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