 Solution

ID: past-exam-of-the-quantum-field-theory-course-of-the-university-of-oxford/2021/1/b/i/solution

Label the external four-momenta by

Ψ (p_{1}) + χ (p_{2}) \to Ψ (p_{3}) + χ (p_{4})

. The only tree-level Feynman diagram exchanges a scalar field particle between the transitions

Ψ (p_{1}) \to χ (p_{4})

and

χ (p_{2}) \to Ψ (p_{3})

. Its internal four-momentum is

q = p_{1} - p_{4} = p_{3} - p_{2}

. Thus its scattering amplitude, with a consistent external-state ordering, is

i M = (- i g)^{2} [\overset{u}{ˉ}_{χ} (p_{4}) u_{Ψ} (p_{1})] \frac{i}{( p _{1} - p _{4} ) ^{2} - m ^{2} + i 0} [\overset{u}{ˉ}_{Ψ} (p_{3}) u_{χ} (p_{2})] .

It is the crossed, or

u

-channel, exchange with these labels. A direct

t

-channel scalar field exchange would need a flavour-preserving Yukawa coupling, which the Lagrangian density does not contain. There is no annihilation Feynman diagram for two incoming fermions.

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