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

Use the Feynman rules and Feynman gauge convention from part a. Put $s=q^2$, $t=r_t^2$, $u=r_u^2$, the Mandelstam variables, and define the three-boson tensor with incoming four-momenta $(q,-k_1,-k_2)$:The contributions, with external Dirac spinors included and both external gauge boson propagators amputated, areFor example, $i{\mathcal{M}}_t$ contains $(+ig{)}^{2}i=-i{g}^2$. In the annihilation channel, the current vertex, propagator, and three-boson vertex give $(+ig)(-i)g=g^2$, so division by the defining overall $i$ gives $-i{g}^2$. These relative signs are essential for the cancellation in part d. The antisymmetric colour factor makes the last expression symmetric under simultaneously exchanging $(a,\mu ,k_1)$ and $(b,\nu ,k_2)$, as required for identical outgoing gauge bosons. The on shell external Dirac spinors obey $(/p_1-m)u(p_1)=0$, $\bar{v}(p_2)(/p_2+m)=0$.