Next-to-next-to-leading order fits to CCFR’97 xF_{3} data and infrared renormalons

A.L. Kataev†, G. Parente ‡and A.V. Sidorov^{∗}

† Institute for Nuclear Research of the Academy of Sciences of Russia, 117312, Moscow, Russia ‡ Department of Particle Physics, University of Santiago de Compostela, 15706 Santiago de Compostela, Spain

∗ Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980, Russia

                 E-mail: kataev@ms2.inr.ac.ru, gonzalo@fpaxp1.usc.es,
                 sidorov@thsun1.jinr.ru

Abstract. We briefly summarize the outcomes of our recent improved fits to the experimental data of CCFR collaboration for xF_{3} structure function of νN deepinelastic scattering at the next-to-next-to-leading order. Special attention is paid to the extraction of α_{s}(M_{Z}) and the parameter of the infrared renormalon model for 1/Q^{2}-correction at different orders of perturbation theory. The results can be of interest for planning similar studies using possible future data of Neutrino Factories.

The most precise up to date data for νN deep-inelastic scattering cross-sections were provided in 1997 by CCFR collaboration [1]. In view of the necessity of understanding of the place of different theoretical effects in the fits to xF_{3} non-singlet structure function data, interesting from the point of view of possible future theoretical program of Neutrino Factories (for a review see Ref.[2]), it is worth to summarize some basic results of the most recent higher-order perturbative QCD analysis [3], which updates the results of the previous similar studies of Refs.[4],[5]. The analysis is based on the Jacobi polynomial technique of reconstruction of the structure function from its Mellin moments

xF_{3}(x, Q^{2}) = w(α, β) ∑_{n=0}^{N_{max}} Θ_{n}^{α,β}(x) ∑_{j=0}^{n} c_{j}^{(n)}(α, β)M_{j+2}^{T MC}(Q^{2})+\frac{h(x)}{Q^{2}}(1)

where w(α, β) = x^{α}(1 − x)^{β}, Θ_{n}^{α,β} are the Jacobi polynomials, c_{j}^{(n)}(α, β) contain α− and β− Euler Γ-functions with the Jacobi polynomial parameters α = 0.7 and β = 3 fixed following the detailed studies of Ref.[5].

The moments

M_{n}^{T MC}(Q^{2}) = M_{n}(Q^{2}) + \frac{n(n + 1)}{(n + 2)} \frac{M_{nucl}^{2}}{Q^{2}} M_{n+2}(Q^{2}) (2)

contain the leading order target mass correction, while the contribution of higher terms of order 1/Q^{4} do not affect the results of the fits [5]. Here we will fix the dynamical non-perturbative corrections using infrared renormalon (IRR) model as

\frac{h(x)}{Q^{2}} = x^{α}(1 − x)^{β} ∑_{n=0}^{N_{max}} Θ_{n}^{α,β} ∑_{j=0}^{n} c_{j}^{(n)}(α, β)M_{j+2}^{IRR}(Q^{2}) (3)

where M_{n}^{IRR}(Q^{2}) = C̃(n)M_{n}(Q^{2})A2^{′}/Q^{2}, C̃(n) = −n+4+2/(n+1)+4(n+2)+4S_{1}(n)

was calculated from the single chain of one-fermion loop insertions into the one- gluon contribution to the Born diagram in Ref.[6] and A2^{′} is the arbitrary fitted parameter.

The Mellin moments obey the following renormalization-group equation

\frac{M_{n}(Q^{2})}{M_{n}(Q_{0}^{2})} = exp[ − ∫_{A_{s}(Q_{0}^{2})}^{A_{s}(Q^{2})} \frac{γ_{F_{3}}^{(n)}(η)}{β(η)}dη]\frac{C_{F_{3}}^{(n)}(A_{s}(Q^{2}))}{C_{F_{3}}^{(n)}(A_{s}(Q_{0}^{2}))} (4)

where A_{s} = α_{s}/(4π), while M_{n}(Q_{0}^{2}) is the phenomenological quantity, defined at the initial scale Q_{0}^{2} as M_{n}(Q_{0}^{2}) = ∫_{0}^{1} x^{n−2}A(Q_{0}^{2})x^{b}(Q_{0}^{2})(1 + γ(Q_{0}^{2}))dx.

The main new ingredient of the analysis of Ref.[3] comes from the recent results of analytical calculations of next-to-next-to-leading order (NNLO) correction to γ_{F_{3}}^{(n)}(A_{s}) at odd n = 3, 5, 7, ..., 13 [7] (the NNLO corrections to β(x) [8] and C_{F_{3}}^{(n)}(A_{s}) [9] were already known for quite a long time). The switch from the NNLO non-singlet contributions to γ_{F_{2}}^{(n)}(A_{s}) at n=2,4,..,10 calculated in Ref.[10], which numerically are closely related to the identical contributions to γ_{F_{3}}^{(n)}(A_{s}), and were used in the previous NNLO fits of CCFR’97 data of Refs.[4],[5], is making our new analysis more selfconsistent. Note, however, that in spite of fixation of the number of previously existing theoretical ambiguities, our new fits are confirming the main results of Ref.[5], putting them on more solid ground.

In order to save space, we will avoid discussions of the number of technical steps of the analysis of Ref.[3] and will concentrate ourselves on the presentation of the main physical results.

order/N_{max} Q_{0}^{2} =  5 Gev^{2}  20 GeV^{2}  100 GeV^{2}
LO/9  Λ(4) ; \overline{MS}  447±54  443±54  439±56
χ^{2}/nep  79.8/86  80.1/86  79.6/86
A2^{′}  −0.340±0.059 −0.337±0.059 −0.335±0.059
NLO/9  Λ(4) ; \overline{MS}  379±41  376±39  374±42
χ^{2}/nep  78.6/86  79.5/86  79.0
A2^{′}  −0.125±0.053 −0.125±0.053 −0.124±0.053
NNLO/9  Λ(4) ; \overline{MS}  331±33  332±35  331±35
χ^{2}/nep  73.1/86  75.7/86  76.9/86
A2^{′}  −0.013±0.051 −0.015±0.051 −0.016±0.051

Table 1. The results of the fits to the CCFR’97 xF_{3} data with Q^{2} > 5 GeV^{2} with related statistical uncertainties. The cases of different Q_{0}^{2} are considered. The values of Λ(4) ; \overline{MS} are given in MeV, while A2^{′} is “measured” in GeV^{2}.

In Table 1 we present the results of the fits taking into account leading order (LO), next-to-leading order (NLO) and NNLO perturbative QCD corrections. Due to the incorporation of the recently calculated NNLO corrections to γ_{F_{3}}^{(n)}(α_{s}) [7] into our new NNLO fits of Ref.[3] the NNLO fitted parameters turn out to be rather stable to the variation of Q_{0}^{2}-scale from 20 GeV^{2}, used in Ref.[5], to the lower initial scale Q_{0}^{2} = 5 GeV^{2}. It should be stressed, that the tendency |A2^{′}|_{LO} > |A2^{′}|_{NLO} > |A2^{′}|_{NNLO} demonstrates the minimization of the role of the power corrections for the description of xF_{3} data. Moreover, at the NNLO the value of the IRR model parameter A2^{′} is effectively minimized and within statistical uncertainties is comparable with zero. This property was already observed in the process of our previous NNLO analysis of CCFR’97 data [4, 5] and in the NNLO fits to the combined data of SLAC,NMC and BCDMS collaboration for F_{2} structure function of deep-inelastic scattering of charged leptons on nucleons [11].In all these fits the QCD coupling constant was closed to its world average value α_{s}(M_{Z}) ≈ 0.118. For example, in our analysis of Ref.[3] we got

NLO α_{s}(M_{Z}) = 0.120 ± 0.002(stat) ± 0.005(syst)  (5)
± 0.002(thresh.)_{−0.006}^{+0.010}(scale)
NNLO α_{s}(M_{Z}) = 0.119 ± 0.002(stat) ± 0.005(syst)  (6)
± 0.002(thresh.)_{−0.002}^{+0.004}(scale)

The latter result agree, within errors, with the results of the NNLO fits to CCFR’97 data, performed with the help of Bernstein polynomial technique using different theoretical representations (compare the original publication of Ref.[12] with the recent work of Ref.[13]). Note, however, that if one gets the “small” NNLO value of α_{s}(M_{Z}), namely α_{s}(M_{Z}) ≈ 0.114, like in the work of Ref.[14], the 1/Q^{2} contributions to F_{2} remain visible even at the NNLO [14]. Thus, we can conclude, that there is some correlation between twist-4 terms and NNLO perturbative QCD corrections and NNLO value of α_{s}(M_{Z}). One can hope, that the possibility to detect higher-twist terms at the NNLO will be clarified in future using drastically more precise data for F_{2} and xF_{3} structure functions, which can be obtained at Neutrino Factories in the lower Q^{2}-region.

Acknowledgments

The work of two of us (ALK and AVS) was supported by RFBR Grants N 00-02-17432 and N 02-01-00601. GP acknowledges the support of Xunta de Calicia ( PGIDT00PX20615PR) and CICYT (AEN99-0589-C02-02). AVS is grateful to INTAS (INTAS Call 2000-project N 587) for the financial support. One of us (ALK) wishes to thank the OC of NuFact’02 Workshop for hospitality in London.

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