Feynman Diagrams For Pedestrians - Herbstschule Maria Laach

• the following Feynman rules produce an expression for iT:1. draw all diagrams using propagators and interaction vertices that connectthe initial state with the final state2. assign momenta and external wave functions accordingly3. use momentum conservation at each vertex to fix the momenta of the internallines4. follow each connected fermion line against the direction of the arrows andcollect wave functions, propagators and vertices along the way5. complete iT with the remaining wave functions, propagators and vertices6. add all diagrams with the relative signs such that that sum is anti symmetricunder the exchange of identical external (anti-)fermions (and symmetric forbosons)• in diagrams with closed loops not all momenta are fixed by momentum conservation,e. g.:p − kkpk• in this case, one must integrate over these loop momenta with∫ d 4 p(2π) 4 · · · (107)• unfortunately, there are infinitely many loop diagrams for each process• fortunately, each loop comes with additional powers of the coupling constants• in weakly interacting theories, we can expand the scattering amplitudes in thecouplings constants or the number of loops3.3 Cross Section• definition in terms of observablesσ (∆Φ) = R (∆Φ)jwhere (108)∆Φ = region of phase spaceσ (∆Φ) = cross section for scattering into ∆ΦR (∆Φ) = event rate in ∆Φj = incoming flux(109a)(109b)(109c)(109d)21