Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 301 / 2 / e

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- Solution e

Solution

Take all external momenta incoming,

q_{1} = p_{1}

,

q_{2} = p_{2}

, and

q_{2 + r} = - k_{r}

. For a diagram whose middle leg is

a

and whose two end pairs are

{b, c}

and

{d, e}

, the Feynman rules give

i M_{a ∣ b c ∣ d e} = (- i g)^{3} \frac{i}{( q _{b} + q _{c} ) ^{2} - m ^{2} + i ϵ} \frac{i}{( q _{d} + q _{e} ) ^{2} - m ^{2} + i ϵ} .

Equivalently,

M_{a ∣ b c ∣ d e} = - \frac{g ^{3}}{[( q _{b} + q _{c} ) ^{2} - m ^{2} + i ϵ ] [( q _{d} + q _{e} ) ^{2} - m ^{2} + i ϵ ]} .

The full connected tree amplitude is the sum over the fifteen choices described in part d, multiplied by the overall momentum-conserving delta function from part c.

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