Calculation of the QED Coupling alpha (M_Z) in the Modified Minimal-Subtraction Scheme
I calculate the QED coupling, alpha, directly in the MS-bar scheme using an unsubtracted dispersion relation for the three light quarks, and perturbative QCD for charm and bottom quarks. Compact analytical expressions are presented, making this approach particularly suitable for electroweak fits. After alpha^(-1) (m_tau) = 133.513 +- 0.025 is obtained in a first step, I perform a 4-loop renormalization group evolution with 3-loop matching conditions to arrive at alpha^(-1) (M_Z) = 127.934 +- 0.026 for alpha_s (M_Z) = 0.120. The corresponding hadronic contribution to the on-shell coupling is Delta alpha_had^(5) (M_Z) = 0.02779 +- 0.00019. The error is mainly from m_c, and from experimental uncertainties in e^+ e^- annihilation into unflavored and strange hadrons and tau decay data.