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

Table of contents
- Solution c

Solution

Let

G_{5}

be the connected time-ordered five-point function. The LSZ reduction formula gives, up to one factor

Z^{- 1/2}

per external leg,

⟨ k_{1} k_{2} k_{3}, out ∣ p_{1} p_{2}, in ⟩_{c} = r = 1 \prod 3 [i \int d^{4} x_{r} e^{i k_{r} \cdot x_{r}} (□_{x_{r}} + m^{2})] \times s = 1 \prod 2 [i \int d^{4} y_{s} e^{- i p_{s} \cdot y_{s}} (□_{y_{s}} + m^{2})] G_{5} (x_{1}, x_{2}, x_{3}, y_{1}, y_{2}) .

Translation invariance factors this as

i (2 π)^{4} δ^{(4)} (p_{1} + p_{2} - \sum_{r = 1}^{3} k_{r}) M_{2 \to 3} .

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