Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 301 / 2 / iv

Every scalar-fermion vertex contracts the scalar momentum with the Dirac current. Between on-shell external spinors,

(p^{'} - p)_{μ} \overset{u}{ˉ} (p^{'}) γ^{μ} u (p) = \overset{u}{ˉ} (p^{'}) (\neq p^{'} - \neq p) u (p) = 0,

(1)

and the analogous particle-antiparticle identity also vanishes. Equivalently, the field redefinition in the previous part turns the theory into a free theory. Consequently every putative tree channel for

ψ \overset{ˉ}{ψ} \to ψ \overset{ˉ}{ψ}

has zero amplitude and

\overline{∣ M ∣^{2}} = 0, σ (ψ \overset{ˉ}{ψ} \to ψ \overset{ˉ}{ψ}) = 0.

(2)

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Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 301 / 2 / iv

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