Solution

ID: past-exam-of-the-advanced-quantum-field-theory-course-of-the-university-of-oxford/2022/2/b/ii/solution

The two Bhabha scattering tree-level Feynman diagrams are the two panels of Solution with a photon mediator: -channel annihilation with internal momentum , and -channel exchange with internal momentum . Each interaction vertex preserves the electron species. There is no independent -channel Feynman diagram, since the outgoing electron and Positron are distinguishable.
Removing the color charge factor from Solution and replacing by gives
Crossing symmetry interchanges and for the exchange Feynman diagram, giving
Squaring the sum of the two scattering amplitudes also produces the interference term . The given third term must be this cross term: it has both propagator denominators , whereas a squared individual Feynman diagram has or . With the correct relative fermionic sign, its value is . Thus
The coefficient of the last displayed fraction is positive, but and in the physical scattering region, so this interference is destructive. The relative sign originates in the two possible external fermion contractions; changing an external-state phase changes both scattering amplitudes together.[ref]

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