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

Table of contents
- Solution i

Solution

With all momenta incoming and Fourier convention

e^{- ik x}

, differentiating the scalar contributes

- i k_{μ}

. The sole interaction vertex is therefore

\overset{ˉ}{ψ} ψ ϕ : - λ γ^{μ} k_{μ} = - λ \neq k,

where

k

enters on the scalar line; reversing the convention reverses the irrelevant overall sign. The free internal lines use the Dirac propagator

\frac{i ( \neq p + m )}{p ^{2} - m ^{2} + i ϵ}

and the scalar Feynman propagator

i / (p^{2} - μ^{2} + i ϵ)

. Momentum is conserved at the vertex.

Figure 1.
Derivative scalar-current vertex and scalar decay cut diagram
. The scalar momentum enters a derivative vertex on an oriented fermion line. The decay amplitude and its conjugate vanish because the scalar momentum contracts the conserved on-shell Dirac current.

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