Currents hadronic matrix elements

The hadronic matrix element of is thus known provided one subtracts neutron and proton matrix elements. When this is done for the current A,i all the other terms on the RHS of (16.8.12) cancel out so one obtains... 

In the previous chapters we saw that in so far as deep inelastic lepton-nucleon scattering was concerned the nucleon could be visualized as a bound system of constituent quark-partons, with which the lepton interacted as if they were free particles. Our aim now is to try to give some sort of justification for such a picture, and to derive more reliable results for deep inelastic scattering in which allowance is made for the internal longitudinal and transverse (Fermi) motion of the quark-partons. The approach also allows us to evaluate the forward hadronic matrix elements of currents which appear in the sum rules discussed in Chapters 16 and 17 in terms of parton number densities. 

It is these relations together with the exact results for the hadronic matrix elements of conserved currents discussed in Appendix 3 that provide the real basis for the sum rules quoted in eqns (16.4.2) to (16.4.4). 

The four nonrelativistic operators of Eq. (4a) arise from an expansion of the lepton wave functions, while those of Eq. (4b) occur in the hadronic weak current, connecting the large and small components of nucleonic wave functions, and are relativistic. Matrix elements of different rank contribute incoherently to the decay rate. The rank R has the selection rule... 

An immediate application connects part of the neutron /3-decay matrix element to the electromagnetic form factors of protons and neutrons. Taking for J+ the vector part of the hadronic weak current and using (1.2.11), (1.3.17) yields... 

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°°. 

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