= 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.
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2022-pion-beta-diagrams.png]
{title=Representative quark-line diagrams for the two formal pion beta transitions}
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,
$$
m_{\pi^0}=134.9768\,\mathrm{MeV},\qquad
m_{\pi^+}=139.57039\,\mathrm{MeV},\qquad m_e\simeq0.510999\,\mathrm{MeV}.
$$
Therefore the first process has <Q value>
$$
Q(\pi^0\to\pi^+e^-\bar\nu_e)=m_{\pi^0}-m_{\pi^+}-m_e
=-5.10459\,\mathrm{MeV}<0,
$$
and is forbidden for an isolated <pion>. The second has
$$
Q(\pi^+\to\pi^0e^+\nu_e)=m_{\pi^+}-m_{\pi^0}-m_e
=4.08259\,\mathrm{MeV}>0.
$$
The table therefore lists only the second, <pion beta decay>, with
$$
\boxed{B(\pi^+\to\pi^0e^+\nu_e)=(1.036\pm0.006)\times10^{-8}.}
$$
Its 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 $(1.036\times10^{-8})/(2.6033\times10^{-8}\,\mathrm s)=0.398\,\mathrm{s^{-1}}$, competing with the much faster dominant muonic decay.
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