Label
$$
e^-(p_1,s_1)+e^+(p_2,s_2)
\longrightarrow\gamma(k_1,\lambda_1)+\gamma(k_2,\lambda_2).
$$
There are two electron-exchange diagrams, distinguished by which labelled outgoing photon is emitted first along the fermion line. They are the $t$- and $u$-channel topologies. Their sum is
$$
\begin{aligned}
i\mathcal M=(-ie)^2\overline v(p_2)\bigg[&
\gamma^\nu\frac{i(\not p_1-\not k_1+m)}{(p_1-k_1)^2-m^2+i\epsilon}\gamma^\mu\\
&+\gamma^\mu\frac{i(\not p_1-\not k_2+m)}{(p_1-k_2)^2-m^2+i\epsilon}\gamma^\nu
\bigg]u(p_1)
\epsilon_\mu^*(k_1,\lambda_1)\epsilon_\nu^*(k_2,\lambda_2).
\end{aligned}
$$
The two terms are added because the external photons are identical bosons. Replacing either polarization by its momentum makes their sum vanish by the <Ward identity>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 301 / 4 / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-301/4/ii/solution