Scattering of two fermions due to Z exchange

For us the most relevant process is the scattering of two fermions due to Z° exchange, for which the lowest order contribution is obtained in second order perturbation theory  

Here and in the rest of this section 0 shall explicitly represent the quantised fermionic field and also the normal order product AB is written explicitly, which implies that all annihilation operators are to the right of all creation opeartors. All boson operators within an normal ordered product are treated as if they would commute, while all fermion operators are treated as if they would anticommute (see appendix). After applying Wick s theorem a number of terms arise that describe different scattering 

The four-dimensional delta distribution arises as a result of four-momentum conservation. If the momentum transfer g = p2 — P2 is small compared to mz, then 

Note that if the interaction between the two fermions is described with the following effective interaction Hamilton density 

From the S matrix element (83) one obtaines with the equations (60) and (62) in the nonrelativistic limit the following expression for the scattering amplitude T  

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