Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 47 / 3 / a

Only a $t$-channel Z boson connects the muon-neutrino line to the electron line. Its endpoints are the weak neutral current vertices $Z{\nu }_{\mu }{\bar{\nu }}_{\mu }$ and $Z{e}^{-}{e}^{+}$. A W boson couples ${\nu }_{\mu }$ to a muon, so it cannot produce an additional elastic charged-current diagram with external electrons. There is no photon-neutrino vertex, and a strictly massless Standard Model neutrino has no Higgs Yukawa vertex. There is one diagram.

There are two diagrams: the $t$-channel Z boson exchange from the weak neutral current, and a crossed $u$-channel W boson exchange from the weak charged current. The charged-current vertices pair the incoming neutrino $k$ with the outgoing electron $p^{\prime }$ and the incoming electron $p$ with the outgoing neutrino $k^{\prime }$. The exchanged momentum has square $u=(k-p^{\prime }{)}^2$. With charge flow from the first vertex to the second, the exchanged boson is $W^{+}$. The corresponding panels show both topologies; no photon or Higgs diagram survives the stated neutrino assumptions.

There are two diagrams: $t$-channel Z boson exchange and $s$-channel W boson exchange. The incoming electron and electron-antineutrino can annihilate into $W^{-}$, which then produces the outgoing pair. The charged-current propagator carries momentum $k+p$, hence invariant $s$. The solid-line arrows in the diagram indicate fermion-number flow; those on antineutrino legs run opposite their physical motion.