QCD Sum Rule Determination of $\alpha(M_Z)$ with Minimal Data Input

Abstract

We present the results of a new evaluation of the running fine structure constant $\alpha$ at the scale of the $Z$ mass in which the role of the $e^+e^-$ annihilation input data needed in this evaluation is minimized. This is achieved by reducing the weight function $M_Z^2/(s(M_Z^2-s))$ in the dispersion integral over the $e^+e^-$ annihilation data by subtracting a polynomial function from the weight function which mimics its energy dependence in given energy intervals. In order to compensate for this subtraction the same polynomial weight integral is added again but is now evaluated on a circular contour in the complex plane using QCD and global duality. For the hadronic contribution to the shift in the fine structure constant we obtain $\Delta\alpha^{(5)}_{\rm had}=(277.6\pm 4.1)\cdot 10^{-4}$. Adding in the leptonic and top contributions our final result is $\alpha(M_Z)^{-1}=128.925\pm 0.056$.