Past exam of the particle and nuclear physics course of the University of Cambridge / 2022 / Section B / 6 / c / Solution

Both formal processes use one weak charged current transition, with a virtual W boson producing the lepton pair. Two representative lowest-order quark-line diagrams are shown below. They use the $d\bar{d}$ component of ${\pi }^0$; attachment to the antiquark line in the $u\bar{u}$ component gives the other contribution to the same hadronic current.

In the first diagram, $d\to u+W^{-}$ and $W^{-}\to e^{-}{\bar{\nu }}_e$, with $\bar{d}$ a spectator quark. In the second, $u\to d+W^{+}$ and $W^{+}\to e^{+}{\nu }_e$, again with a spectator $\bar{d}$. The diagram for the first process describes the coupling but cannot make an energetically forbidden on-shell decay occur.

From the original tables,Therefore the first process has Q valueand is forbidden for an isolated pion. The second hasThe table therefore lists only the second, pion beta decay, withIts tiny branching fraction combines a weak interaction with an exceptionally small available energy. An allowed semileptonic transition has a decay width scaling approximately as $G_F^2|V_{ud}{|}^2{\Delta }^5$, where $\Delta =m_{{\pi }^{+}}-m_{{\pi }^0}$, with electron-threshold and recoil corrections. The fifth power makes the few-MeV gap especially restrictive. Unlike direct ${\pi }^{+}\to e^{+}{\nu }_e$, this semileptonic channel is not forced to have an amplitude proportional to $m_e$; its primary extra suppression is the small phase space. Its approximately Cabibbo-favoured $u\leftrightarrow d$ current does not supply a further large flavour suppression. The observed branching fraction corresponds to a partial decay rate of about $(1.036\times 10^{-8})/(2.6033\times 10^{-8}\mathrm{s})=0.398{\mathrm{s}}^{-1}$, competing with the much faster dominant muonic decay.