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 is

M = e^{2} ϵ_{1 μ}^{*} ϵ_{2 ν}^{*} M^{μν},

where

M^{μν} = 2 η^{μν} + \frac{( 2 p _{1} - k _{1} ) ^{μ} ( 2 p _{2} - k _{2} ) ^{ν}}{( p _{1} - k _{1} ) ^{2} - m ^{2}} + \frac{( 2 p _{2} - k _{1} ) ^{μ} ( 2 p _{1} - k _{2} ) ^{ν}}{( p _{1} - k _{2} ) ^{2} - m ^{2}} .

- 2 p_{1} \cdot k_{1} = - 2 p_{2} \cdot k_{2}

and

- 2 p_{1} \cdot k_{2} = - 2 p_{2} \cdot k_{1}

. Contracting with

k_{1 μ}

therefore gives

k_{1 μ} M^{μν} e p s i l o n_{2 ν}^{*} = 2 k_{1} \cdot ϵ_{2}^{*} - (2 p_{2} - k_{2}) \cdot ϵ_{2}^{*} - (2 p_{1} - k_{2}) \cdot ϵ_{2}^{*} = 0,

using momentum conservation and

k_{2} \cdot ϵ_{2} = 0

. The same calculation with the photons interchanged gives

k_{2 ν} M^{μν} ϵ_{1 μ}^{*} = 0

. The two exchange diagrams and the seagull term are all required for this Ward identity.

Solved by gpt-5.6-sol high.

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