Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 50 / 2 / c / Solution

The Yukawa interaction has one real-scalar leg and an oriented fermion line at each vertex. At lowest connected order, two vertices exchange one scalar, so the diagrams are of order ${\lambda }^2$.

For two incoming fermions, there are two pairings of the outgoing fermions: the $t$ channel joins $p$ to $p^{\prime }$ and $q$ to $q^{\prime }$, with scalar momentum $p-p^{\prime }$; the $u$ channel joins $p$ to $q^{\prime }$ and $q$ to $p^{\prime }$, with scalar momentum $p-q^{\prime }$. Fermion number conservation excludes an annihilation channel for two particles carrying the same charge.

For an incoming fermion and antiparticle, there is a $t$-channel scalar exchange with momentum $p-p^{\prime }=q^{\prime }-q$ and an $s$-channel annihilation and recreation diagram with momentum $p+q=p^{\prime }+q^{\prime }$. Arrows on antiparticle lines run opposite to the direction of physical propagation. These are the four tree scattering in Yukawa theory diagrams shown below; all external momenta are labelled.

The crossing in the $u$-channel drawing is not a vertex. Dashed internal lines are the real scalar field; solid arrows show fermion-number flow.