By Yuval Neeman
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Extra info for Algebraic Theory of Particle Physics: Hadron Dynamics in Terms of Unitary Spin Currents
025. 124), see Sec. 9. 125) Note that this result is actually scheme independent; the corresponding NS anomalous dimensions vanish to all orders in perturbation theory. As anticipated, AQ(Q2) is now Q 2 -dependent and is related to the quark spin A S and to the gluon spin Ag by A0(Q2) = AE-Nf^lAg(Q2). 127) 42 Polarised deeply-inelastic scattering where Lz is the orbital angular-momentum contribution. 125) of the singlet matrix element is a consequence of the axial anomaly (Adler, 1969; Bell and Jackiw, 1969), which relates the divergence of the axial-vector current to the gluon field.
2 The Bjorken sum rule An important sum rule, first derived by Bjorken (1966) in the context of current algebra, relates the first moments of g\ and g" to the vector (gv) and axial-vector (g^) couplings measured in nuclear /3-decay. 120a) gives, to leading order in QCD, rP r„ Tl Tl ~ _ 1 9A (. 130) which is in good agreement with the data (for a more complete discussion, see Altarelli et al, 1997). 115a, b). By combining these two relations we can extract GL dn(Q2) = 2 f dxxn~l Jo \gi(x,Q2) L + -^—g2(x,Q2) n—i n = 3,5,7,...
Consider DIS on a free quark. The antisymmetric part of the hadronic tensor is then where k and s are the quark momentum and spin, respectively. 44)—with the replacements M -> mq, P —> k and S —> s, we notice first of all that the quark must be kept massive, otherwise the factor multiplying §2 vanishes. The massless limit is to be taken afterwards. 107) are gi(x) = 5 (x - 1), g2(x) = 0. 108b) Thus, in the naive parton model, in which quarks are free objects, g2 (or gx) vanishes (for a discussion of this point, see also Anselmino and Leader, 1992).