Calculation of Electroproduction to NNLO and Precision Determination of $\alpha_s$

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/dof decreases substantially when going from LO to NLO, and also from NLO to NNLO (although only a little now) to $\chi/dof=79.2/(102-12)$. The favoured value of $\Lambda$ is $$\Lambda(n_f=4,3 {\rm loop})=282.7\pm35.1 MeV,$$ corresponding to the value of the coupling at the Z mass of $$\alpha^{(\rm 3 loop)}_s(M_Z^2)=0.1172\pm0.0024. $$ 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.