Calculation of Moments of Structure Functions ∗

Deep Inelastic Scattering (DIS) experiments, such as eN → eX or νN → μ−X form an important basis for our knowledge of the structure of hadrons. In these processes the current probe (either a neutral current, γ/Z, or charged current, W+/W−) with large space-like momentum −q2 ≡ Q breaks-up the nucleon. The (inclusive) cross section is then determined by the structure functions F1, F2 when summing over beam and target polarisations and, in addition, F3 when using neutrino beams, and g1, g2 when both the beam and target are suitably polarised. The structure functions are functions of the Bjorken variable x (0 ≤ x ≤ 1) and Q. (Another class of structure functions – the transversity h1 – can be measured, in principle, from Drell-Yan type processes or in certain semi-inclusive processes [2].) While the original pioneering discoveries were made over thirty years ago at SLAC, more recently experiments with polarised beams have been reported and the field remains very active. Recent experiments and proposals, [3,4], include H1 and Zeus at DESY (unpolarised F2 at small x, [5] and F3, [6]), Hermes at DESY (polarised g1 and g2, [7]), E155 at SLAC (polarised g1, g2, ∗Plenary talk by R. Horsley at Lat02, Boston, U.S.A. 1A more complete set of structure functions available from DIS processes is given, for example, in [1]. [8]), Jefferson lab (structure functions in the resonance region, [9,10]), COMPASS at CERN (polarised gluon distribution, h1, Λ matrix elements, [11]), CCFR at Fermilab (unpolarised F3, [12]) and RHIC (spin physics, [13]). Recent results are given in the DIS conference series, [14]. A direct theoretical calculation of the structure functions seems not to be possible (but see [15– 17]); however using the Wilson Operator Product Expansion (OPE) we may relate moments of the structure functions to matrix elements of certain operators in a twist or Taylor expansion in 1/Q. Thus if we define