Renormalization of Twist-4 Operators in QCD Bjorken and Ellis-Jaffe Sum Rules

The QCD effects of twist-4 operators on the first moment of nucleon spin-dependent structure function g1(x,Q ) are studied in the framework of operator product expansion and renormalization group method. We investigate the operator mixing through renormalization of the twist-4 operators including those proportional to the equation of motion by evaluating off-shell Green’s functions in the usual covariant gauge as well as in the background gauge. Through this procedure we extract the one-loop anomalous dimension of the spin 1 and twist-4 operator which determines the logarithmic correction to the 1/Q behavior of the contribution from the twist-4 operators to the first moment of g1(x,Q ). KUCP-88-REV HUPD-9601-REV November 1996 JSPS Research Fellow Supported in part by the Monbusho Grant-in-Aid for Scientific Research No. C-06640392 In the last several years there has been much interest in nucleon’s spin structure functions g1(x,Q ) and g2(x,Q ), which can be measured by deep inelastic scattering of polarized leptons on polarized targets. Recent experiments on the nucleon spin structure functions carried out at CERN [1, 2] and SLAC [3, 4], have stimulated intensive theoretical studies on the nucleon spin structure functions [5]. In the deep inelastic scattering, the perturbative QCD has been tested so far for the effects of the leading twist operators, namely twist-2 operators, for which the QCD parton picture holds. Now the spin structure functions would provide us with a good place to investigate higher-twist effects. Our purpose in this paper is to study the renormalization of higher-twist operators, especially the twist-4 operators, which are relevant for the first moment of g1(x,Q ), that corresponds to the Bjorken and EllisJaffe sum rules [6, 7]. The anomalous dimension of the twist-4 operators determines the logarithmic correction to the 1/Q behavior of the twist-4 operator’s contribution to the first moment of g1. The first moment of the g1(x,Q ) structure functions for proton and neutron turns out to be up to the power correction of order 1/Q: Γ 1 (Q ) ≡ ∫ 1 0 g 1 (x,Q )dx = (± 1 12 gA + 1 36 a8)(1− αs π +O(α s)) + 1 9 ∆Σ (1− 33− 8Nf 33− 2Nf αs π +O(α s)), (1) where g p(n) 1 (x,Q ) is the spin structure function of the proton (neutron) and the plus (minus) sign is for proton (neutron). On the right-hand side, gA ≡ GA/GV is the ratio of the axial-vector to vector coupling constants. Here we assume that the number of active flavors in the current Q region is Nf = 3. Denoting 〈p, s|ψγμγ5ψ|p, s〉 = ∆qsμ, the flavor-SU(3) octet and singlet part, a8 and a0 = ∆Σ are given by a8 ≡ ∆u+∆d− 2∆s, ∆Σ ≡ ∆u+∆d+∆s, (2)