Calculation of the QED coupling<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant="normal">α</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">^</mml:mi></mml:mrow></mml:mover></mml:mrow></mml:mrow></mml:math>(<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>Z</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>) in the modified minimal-subtraction scheme

Abstract

I calculate the QED coupling, alpha, directly in the MS-bar scheme using an\nunsubtracted dispersion relation for the three light quarks, and perturbative\nQCD for charm and bottom quarks. Compact analytical expressions are presented,\nmaking this approach particularly suitable for electroweak fits. After\nalpha^(-1) (m_tau) = 133.513 +- 0.025 is obtained in a first step, I perform a\n4-loop renormalization group evolution with 3-loop matching conditions to\narrive at alpha^(-1) (M_Z) = 127.934 +- 0.026 for alpha_s (M_Z) = 0.120. The\ncorresponding hadronic contribution to the on-shell coupling is Delta\nalpha_had^(5) (M_Z) = 0.02779 +- 0.00019. The error is mainly from m_c, and\nfrom experimental uncertainties in e^+ e^- annihilation into unflavored and\nstrange hadrons and tau decay data.\n