Scattering matrix and equivalent potentials

The equations of motion corresponding to the full electroweak Lagrangian can in general not be solved analytically. Approaches that treat the interaction between the fields perturbatively, however, have turned out to be very successful, in particular in QED (see for instance chapter 8 by Labzowsky and Goidenko [80] in the first volume of this book). 

Scattering processes are described with the aid of the S matrix, which is defined by 

We write the lowest order contributions to an S matrix element between an initial state i) and a final state f) now explicitly as 

This S matrix element is related to the scattering amplitude F via 

On the other hand, the relativistically invariant scattering amplitude F can be related to the scattering amplitude Tg, which is not Lorentz invariant, according to 

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