Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 301 / 4 / d / Solution

Let $p_1,p_2$ be the incoming scalar and antiscalar momenta and $k_1,k_2$ the outgoing photon momenta, so $p_1+p_2=k_1+k_2$. Up to the common overall phase fixed by the $S$-matrix convention, the connected leading amplitude iswhereOn shell, the denominators are $-2p_1\cdot k_1=-2p_2\cdot k_2$ and $-2p_1\cdot k_2=-2p_2\cdot k_1$. Contracting with $k_{1\mu }$ therefore givesusing momentum conservation and $k_2\cdot {ϵ}_2=0$. The same calculation with the photons interchanged gives $k_{2\nu }{\mathcal{M}}^{\mu \nu }{ϵ}_{1\mu }^{\ast }=0$. The two exchange diagrams and the seagull term are all required for this Ward identity.

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