Z ) from Dis Structure Functions

Various QCD analyzes of the world unpolarized and polarized deep-inelastic data on charged lepton–nucleon and neutrino–nucleus scattering were performed in order to measure the QCD scale ΛQCD, resp. αs(M 2 Z), from the scaling violations of the nucleon structure functions. In this note we give a brief overview on the status of these analyzes. Most of the analyzes performed in the past were of next-to-leading order, see Table 1. Here the values of αs(M 2 Z) range between 0.1171–0.1148mainly with the exception of the old, very low BCDMS value [6] and Ref. [7] obtaining αs(M 2 Z) = 0.112. The typical theory error is estimated by varying the renormalization and factorization scales between Q/4 and 4 · Q amounts to ∼5% for αs(M 2 Z), a theoretical uncertainty too large to cope with the experimental uncertainty of O(1..2%) after the completion of the HERA programme. The analyzes of polarized nucleon data still yield rather large errors [9, 10] due to the present accuracy reached for polarization asymmetries. Moreover, these analyzes include data down to Q ∼ 1GeV , which is not unproblematic w.r.t. higher twist terms, the scaling violations of which are yet unknown. The unpolarized analyzes at the present level of accuracy require rigorous cuts for potential higher twist effects, which can be achieved demanding W 2 > 12...15GeV . Furthermore, we will consider only proton and deuteron data, to avoid potential interference with nuclear effects. With the advent of the 3-loop anomalous dimensions [15] in the unpolarized case one may extend the analysis to next-to-next-to-leading order, where the remaining theory error is of O(1%) or less, see below. To cope with the present experimental errors 3–loop analyzes are mandatory. A theoretically consistent analysis can be performed at least in the nonsinglet case, where the heavy flavor effects known to O(αs), are negligibly small. 3–loop valence analyzes were performed in [7, 8]. One even may extend the non-singlet analysis to the 4–loop level [8]. A closer numerical study of the potential effect of the yet missing 4–loop anomalous dimension, performing a comparison with the recently calculated second moment in [16] shows that the overwhelming effect at 4–loops is due to the 3–loop Wilson coefficient. To see the convergence of the perturbative expansion we list the values for αs(M 2 Z) obtained in the NLO, NLO, and NLO analyzes : αs(M 2 Z) = 0.1148 → 0.1134 → 0.1142± 0.0021. (1)