Bernstein Polynomial Method Research Articles

DETERMINATION OF QCD RUNNING COUPLING CONSTANT IN NNLO APPROXIMATION, USING COMPLETE RG IMPROVEMENT

To obtain the improved QCD predictions for the moments of the xF3 structure function in νN scattering, three-loop Wilson coefficients with the corresponding NNLO anomalous dimension coefficients is used. A comparison between QCD predictions and the xF3 data of the CCFR collaboration has been done in which the Bernstein Polynomial method is used in the complete renormalization group improvement approach. A direct fits for [Formula: see text] over the corresponding range of Q2 with five effective quark flavours, is performed.

Direct extraction of QCD [formula omitted] from moments of structure functions in neutrino–nucleon scattering, using the CORGI approach

We use recently calculated next-to-next-to-leading (NNLO) anomalous dimension coefficients for the n=1,3,5,…,13 moments of the xF 3 structure function in νN scattering, together with the corresponding three-loop Wilson coefficients, to obtain improved QCD predictions for the moments. The Complete Renormalization Group Improvement (CORGI) approach is used, in which all dependence on renormalization or factorization scales is avoided by a complete resummation of ultraviolet logarithms. The Bernstein Polynomial method is used to compare these QCD predictions to the xF 3 data of the CCFR Collaboration, and direct fits for Λ MS (5) , with N f =5 effective quark flavours, over the range 20< Q 2<125.9 GeV 2, were performed. We obtain Λ MS (5)=202 +54 −45 MeV, corresponding to the three-loop running coupling α s ( M Z )=0.1174 +0.0043 −0.0043. Including target mass corrections as well we obtain Λ MS (5)=228 +35 −36 MeV, corresponding to α s ( M Z )=0.1196 +0.0027 −0.0031.

Bounds for an interval polynomial

We discuss the evaluation of the range of values of an interval polynomial over an interval. Several algorithms are proposed and tested on numerical examples. The algorithms are based on ideas by Cargo and Shiska [2] and Rivlin [4]. The one basic algorithm uses Bernstein polynomials. It is shown to converge to the exact bounds and it has furthermore the property that if the maximum respectively the minimum of the polynomials occurs at an endpoint of the interval then the bound is exact. This is a useful property in routines for polynomials zeros. The other basic method is based on the meanvalue theorem and it has the advantage that the degree of approximation required for a certain apriori tolerance is smaller than the degree required in the Bernstein polynomial case. The mean value method is shown to be at least quadratically convergent and the Bernstein polynomial method is shown to be at least linearly convergent.