Past exam of the quantum field theory course of the University of Oxford / 2020 / 1 / b / iii

Table of contents
- Solution iii

Solution

Let

p_{1}, p_{3}

label the incoming and outgoing fermion, and

p_{2}, p_{4}

the incoming and outgoing antifermion. The bottom row of the figure gives the two tree-level Feynman diagrams:

The $t$ channel has an open fermion line $1 \to 3$ and an open antifermion line with fermion flow $4 \to 2$ . The scalar propagator carries $q_{t} = p_{1} - p_{3} = p_{4} - p_{2}$ .
The $s$ channel has initial particle-antiparticle annihilation into a scalar propagator with $q_{s} = p_{1} + p_{2} = p_{3} + p_{4}$ , followed by creation of the final pair.

Thus the Mandelstam variables in the two denominators are

t = (p_{1} - p_{3})^{2}, s = (p_{1} + p_{2})^{2} .

The two scattering amplitude terms have a relative fermionic sign of minus when written with spinor products

(\overset{u}{ˉ}_{3} u_{1}) (\overset{v}{ˉ}_{2} v_{4})

and

(\overset{v}{ˉ}_{2} u_{1}) (\overset{u}{ˉ}_{3} v_{4})

. The overall phase depends on the chosen ordering of the external states. No

u

channel joins an incoming fermion to an outgoing antifermion through a species-preserving Yukawa interaction.

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