The electron emits a photon, propagates as a virtual electron, then annihilates with the positron to produce the second photon. Related to the Compton s-channel by crossing.
The computational chain
From diagram to number—every step of the amplitude calculation.
The diagram
The electron emits a photon at vertex 1, propagates as a virtual electron with momentum $p - q_1$, then annihilates with the positron at vertex 2 to produce the second photon.
Feynman rules expression
$$i\mathcal{M}_t = (-ie)^2\,\epsilon_\mu^*(q_1)\epsilon_\nu^*(q_2)\,\bar{v}(p')\gamma^\nu\frac{i(\not{p}-\not{q}_1+m)}{(p-q_1)^2-m^2}\gamma^\mu u(p)$$
The structure is related to Compton scattering by crossing: replace the incoming photon with an outgoing photon and the outgoing electron with an incoming positron. The Mandelstam variables remap: what was $s$ in Compton becomes $t$ here.
Square and sum over spins and polarizations
$$\overline{|\mathcal{M}_t|^2} = \frac{1}{4}\frac{e^4}{(t-m^2)^2}\,\mathrm{Tr}\left[(\not{p}'-m)\gamma^\nu(\not{p}-\not{q}_1+m)\gamma^\mu(\not{p}+m)\gamma^{\mu'}(\not{p}-\not{q}_1+m)\gamma^{\nu'}\right]\cdot(-g_{\mu\mu'})(-g_{\nu\nu'})$$
Average over initial electron spin (1/2) and initial positron spin (1/2). Sum over final photon polarizations using $\sum_\lambda\epsilon_\mu\epsilon_\nu^* \to -g_{\mu\nu}$. The positron completeness gives $\sum v\bar{v} = \not{p}' - m$ (minus sign). The trace has the same single-fermion-line structure as Compton: two vertices and an internal propagator sandwiched between external spinors.
Evaluate the trace
$$\overline{|\mathcal{M}_t|^2} \supset -2e^4\frac{m^2 - s}{m^2 - t}$$
The trace evaluation uses the same techniques as Compton — single fermion line, internal propagator numerator appearing twice, contraction identities $\gamma^\mu\gamma_\alpha\gamma_\mu = -2\gamma_\alpha$ to reduce the gamma chain. Evaluating directly gives this result, where the propagator denominator $(m^2 - t)$ appears from the virtual electron momentum $(p - q_1)^2 = t$.
Why mass cannot be neglected
$$\text{Denominator: } (m^2 - t)$$
The propagator denominator $(m^2 - t)$ vanishes when $m \to 0$. The electron mass regulates the singularity and cannot be dropped.