It is pointed out that when comparing QCD predictions with experimental data for deep inelastic structure functions it is advisable to use Bernstein moments

Wosiek, Jacek

Zalewski, K.

Acta Physica Polonica B

English

Jagiellonian Univeristy Repository

Vol. Bil (1980) ACTA PHYSICA POLONICA No 9LETTERS TO THE EDITORUSE OF BERNSTEIN MOMENTS IN QCD TESTSBy J. WosiekInstitute of Computer Science, Jagellonian University, Cracow** Address: Instytut Informatyki UJ, Reymonta 4, 30-059 Kraków, Poland.** Address: Instytut Fizyki Jądrowej, Kawiory 26a, 30-055 Kraków. Poland.and K. ZalewskiInstitute of Nuclear Physics, Cracow**(Received June 11, 1980)It is pointed out that when comparing QCD predictions with experimental data for deep inelastic structure functions it is advisable to use Bernstein moments.PACS numbers: 11.10.Jj, 11.10.Np, 13.10. + qThe purpose of this letter is to point out that Bernstein moments⅛(β2) = ∫ ⅛ω⅜, Q2)dx, (i)owhere the Bernstein polynomials(2)are ρarticularly convenient for the comparison with experiment of QCD predictions for the Q2 evolution of structure functions.Two approaches for this comparison have been popular (cf. e.g. review [1]). One is to study the moments(3)(697)698The advantage is that simple closed QCD formulae for the Q2 dependence of these moments are available. The disadvantage is that within this approach it is not possible to cut out the x-region close to x = 1, where the ρresent version of perturbative QCD predictions is probably unreliable (cf. e.g. Ref. [2]), or any other x-region, where the data would be ρoor, or nonexistent. Moreover, except for the first few, all the moments are dominated by the contributions from the x ≈ 1 region. Thus their errors are strongly correlated and the confidence levels of the fits are unclear.The other approach is, to invert the formulae for the moments and thus to obtain predictions for the Q2 evolution of the structure functions themselves. Actually, in this context the Bernstein moments have already successfully been used [3]. The inversion approach makes any desired cuts in x easy to apply, but the inversion has to be performed numerically and no closed analytic expression results.The Bernstein moments combine the advantages of both approaches. Since the mo­ments (1) are known linear combinations of the moments (3), close analytic formulae for their Q2 evolution are available. On the other hand, a Bernstein polynomial 2⅛) has a maximum at x = M∣N and a width decreasing with increasing N at fixed M∣N. There- fore, provided N is large enough, one can very well approximate a cut in x by rejecting some moments. For instance, rejecting the moments with M = N, one circumvents the controversial problem: whether or not the elastic contribution should be included. Bern­stein polynomials for M = 0, 1, ..., N are independent linear combinations of the mono- mials xM for M = 0, 1, ..., N. Thus, using all the Bernstein polynomials with a given N corresponds to the use of standard moments up to N. Since, however, the Bernstein poly­nomials differing in M peak in different regions of the x axis, the errors on the moments (1) are much less correlated than those for the moments (3).Our formulae have been written down for Coleman Norton moments. The application to Nachtmann moments is, however, also possible. Work on application of these ideas to data analysis is in progress.REFERENCES[1] A. J. Buras, Rev. Mod. Phys. 52, 199 (1980).[2] R. M. Barnett, The Impact of Higher Twist Terms on the Analysis of Scaling Violations, SLAC-PUB- -2396, September 1979.[3] F. J. Ynduróin, Phys. Lett. 74B, 68 (1978); A. Gonzalez-Arroyo, C. Lopez, F. J. Ynduriin, Nuci. Phys. B159, 161 (1979).

Use of bernstein moments in QCD tests

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Use of bernstein moments in QCD tests

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