Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 301 / 2 / Solution

The interaction-picture Hamiltonian density is ${\mathcal{H}}_I=\lambda \varphi {\Phi }^{†}\Phi$. The first-order Dyson series term cannot connect the four external particles, so the leading contribution isBy the Wick theorem, the nonzero terms annihilate the incoming complex particle using the annihilation part of $\Phi$, create the outgoing complex particle using the creation part of ${\Phi }^{†}$, annihilate the incoming real particle using the annihilation part of $\varphi$, and create the outgoing real particle using its creation part. The remaining $\Phi$ and ${\Phi }^{†}$ form an internal Feynman propagator. Interchanging which vertex absorbs the incoming real scalar gives the two contractions; this factor of two cancels the Dyson factor $1/2$.

There are therefore an $s$-channel internal momentum $p+k$ and a crossed channel internal momentum $p-k^{\prime }$. With covariantly normalized external states,whereEquivalently,with $s=(p+k{)}^2$ and $u=(p-k^{\prime }{)}^2$.