Ellis-Jaffe sum rule: The estimates of the next-to-next-to-leading-order QCD corrections.

The procedure of the estimates of the higher-order perturbative QCD corrections to the physical quantities is generalized to the case when the quantities under consideration obey the renormalization group equations with the corresponding anomalous dimension functions. This procedure is used to estimate the α s-corrections to the singlet part of the Ellis-Jaffe sum rule for f = 3 numbers of flavours. ∗) On leave of absence from Institute for Nuclear Research of the Russian Academy of Sciences, Moscow 117312, Russia. CERN-TH.7333/94 July 1994 The best way of controlling the theoretical uncertainties of the perturbative QCD predictions is the direct analytical or numerical calculation of the concrete terms in the corresponding perturbative series. However, after the results of the calculations of the next-to-next-to-leading order (NNLO) corrections to the number of physical quantities became available [1]–[4], experimentalists and theoreticians started to be interested in the effects of still uncalculated higher-order terms. In the work of Ref. [5], two “optimal” methods of fixing the renormalization scheme ambiguities were used to estimate the next-after-next-to-next-to-leading order (NANNLO) corrections to certain renormalization-group invariant quantities. These methods were the principle of minimal sensitivity (PMS) [6] and the effective charges approach (ECH) [7], which is known to be a posteriori identical to the so-called scheme-invariant perturbation theory [8]. The quantities studied in Ref. [5] are the ee-annihilation ratio R(s), the τ -lepton decay ratio Rτ , and the Bjorken non-polarized and polarized sum rules. However, the quantities obeying the renormalization group equations with anomalous dimension functions were not considered in Refs. [6], [5]. In this note we will fill in this gap and apply the generalization of the ideas used in Ref. [5] to estimate the higher-order corrections to the singlet part of the Ellis-Jaffe sum rule (EJSR), recently calculated at the O(α s) order [9]. We present the concrete O(α s) estimates for f = 3 numbers of flavours in two related forms, namely in the factorization-scheme-invariant form and in the form that necessitates the application of an additional guess about the value of the unknown four-loop coefficient of the corresponding singlet anomalous dimension function. The experimental measurements [10] of the structure functions g p(n) 1 of the polarized deepinelastic lepton-nucleon scattering has stimulated number of works aimed at a theoretical study of the structure function g p(n) 1 and their first moment, namely the EJSR (see e.g., [11]–[15]). The theoretical expression for the EJSR can be presented in the following form: EJSR(Q) = ∫ 1 0 g p(n) 1 (x,Q )dx = EJNS(Q ) + EJSI(Q ) . (1) The non-singlet contribution to this sum rule is defined as