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

Expanding the covariant derivatives of scalar quantum electrodynamics givesandIn four dimensions, $[A_{\mu }]=[\varphi ]=1$ and $[{\partial }_{\mu }]=1$, so $[e]=0$. Both interactions have dimension four and $e$ is a marginal coupling by power counting in quantum field theory.

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With all momenta directed into each vertex, the momentum-space Feynman rules are:

Each vertex also carries its momentum-conserving delta function; integrate every independent loop momentum and attach the appropriate external wavefunctions and photon polarization vectors.

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Three connected tree-level Feynman diagrams contribute at order $e^2$: a scalar-exchange diagram in which photon $k_1$ is emitted before $k_2$ along the charged scalar line; the crossed scalar-exchange diagram with $k_1$ and $k_2$ interchanged; and the local seagull vertex joining both incoming scalars and both outgoing photons. The exchange denominators are $(p_1-k_1{)}^2-m^2$ and $(p_1-k_2{)}^2-m^2$.

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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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