Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2024 / 3 / b / Solution

There are three tree-level Feynman diagrams: two ordered fermion-exchange diagrams and one s-channel gauge boson exchange diagram. Write the outgoing labels as $(k_1,a,\mu )$ and $(k_2,b,\nu )$.

Following the fermion flow from $u(p_1)$ to $\bar{v}(p_2)$, the first diagram emits boson 1 and then boson 2; its internal four-momentum is $r_t=p_1-k_1$ and its ordered colour factor is $t^b{t}^a$. The second emits boson 2 and then boson 1, giving $r_u=p_1-k_2$ and $t^a{t}^b$. The third annihilates into one virtual gauge boson of four-momentum $q=p_1+p_2=k_1+k_2$ and uses the three-boson interaction vertex. There is no two-fermion two-boson contact interaction vertex for the minimally coupled Dirac field, and no tree ghost contribution with these external states.