Current matrix elements in the quark-parton model

3 Current matrix elements in the quark-parton model 

In the SM all hadronic currents are expressed in terms of quark fields f x) where / labels the fiavour. As in Sections 1.2, 16.8 and 18.1, we often require the forward matrix elements of these currents taken between nucleon states. We shall use the quark-parton model to evaluate these in the firame S°°. 

Analogously to (16.9.15) we insert complete sets of quark states, corresponding to partons of momentum k and helicity A. 

Because the proton is polarized we need to slightly generalize the number densities nj(/c) used previously. Thus we introduce Uf K, A S) d K as the number of flavour / quarks with momentum in the range k — K -l-d/c and with helicity A inside a proton of momentmn P and covariant spin vector [see eqn (15.6.2)]. After some algebra one finds for the forward hadronic matrix elements. 

Then treating the quark-partons as free particles in 5°° one has [see eqn (Al.1.23) of Appendix 1] 

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