Calculation of electroproduction to NNLO and precision determination of $α_s$

{Abstract}We use the known values of the two loop Wilson coefficients and the three loop anomalous dimension matrix $\gamma(n)$ to perform a next-to-next-to leading order (NNLO) calculation of $ep$ deep inelastic scattering. Because $\gamma(n)$ is only known for a few values of $n$, the method of average reconstruction has to be used, which leaves 102 effective experimental points, for 12 parameters: the QCD mass $\Lambda$, and 11 initial values for the moments of the structure functions. The data points spread in the range of momenta $2.5 GeV^2 \leq Q^2\leq 230 GeV^2$. The $\chi^2/dof$ decreases substantially when going from LO to NLO, and also from NLO to NNLO (although only a little now) to $\chi^2/dof = 79.2/(102-12)$. The favoured value of $\Lambda$ is $\Lambda(n_f=4,3 loop) = 283 \pm 35 MeV$, corresponding to the value of the coupling at the Z mass of $\alpha^{(3 loop)}_s(M_Z^2) = 0.1163 \pm 0.0023$. The calculation, which constitutes a very precise test of QCD, includes target mass corrections; the error takes into account experimental errors and higher twist effects, among other estimated theoretical errors.