The parton model including transverse motion

We now go to the reference frame in which the impulse approximation is supposed to hold. In this frame we shall calculate the inelastic nucleon tensor W (N) in terms of the partonic tensors W° k,j) for a lepton scattering on a quark j and turning it into a quark of type k. Although the impulse approximation is a simple idea, there are subtleties in the derivation so we shall go through the analysis in some detail. 

When P = P + q ) S m, so that the states X) contain many hadrons, it is permissible to replace the sum over X) by a sum over all possible parton states i), i 2)j i 2K3)) with the same total four-momentum as X). This is equivalent to stating that there is unit probabiilty for the partons to transmute into hadrons. 

The are one-body operators, i.e. they act only on partons of type j and the partons scatter under the action of in a point-like fashion. 

The nucleon can be viewed as a superposition of states of different numbers of partons which are essentially free during the time of the interaction. 

When we substitute for P) a superposition over states with n partons, it is clear that since the J do not change the number of partons present we shall end up with a sum for (16.9.13) of the form 

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