The Effective Fine Structure Constant at TESLA Energies

We present a new estimate of the hadronic contribution to the shift in the fine structure constant at LEP and TESLA energies and calculate the effective fine structure constant. Substantial progress in a precise determination of this important parameter is a consequence of substantially improved total cross section measurements by the BES II collaboration and an improved theoretical understanding. In the standard approach which relies to a large extend on experimental data we find $\Delta \al_{\rm hadrons}^{(5)}(\mz) = 0.027896 \pm 0.000395$ which yields $\alpha^{-1}(\mz) = 128.907 \pm 0.054$. Another approach, using the Adler function as a tool to compare theory and experiment, allows us to to extend the applicability of perturbative QCD in a controlled manner. The result in this case reads $\Delta\alpha^{(5)}_{\rm had}(M_Z^2) = 0.027730 \pm 0.000209$ and hence $\alpha^{-1}(\mz) = 128.930 \pm 0.029$. At TESLA energies a new problem shows up with the definition of an effective charge. A possible solution of the problem is presented. Prospects for further progress in a precise determination of the effective fine structure constant are discussed