Solution (source code)

= Solution

For (i), <CP symmetry> relates the negative- and positive-<pion> radiative channels to excellent accuracy. The original positive-pion table therefore predicts
$$
\boxed{B(\pi^-\to\mu^-\bar\nu_\mu\gamma)
\simeq(2.00\pm0.25)\times10^{-4}.}
$$
The comparison requires the same photon-energy and event-selection definition. The table attaches an unexplained footnote to this radiative channel; it does not specify those cuts in the supplied excerpt. An arbitrarily soft <photon> is not a separately infrared-finite exclusive observable, so the numerical value must retain the table's definition.

Part (ii) contains an apparent notation error in the original PDF as well as the TeX: it prints $e^-\nu_e e^-e^+$, rather than $e^-\bar\nu_e e^-e^+$. Taken literally, the charged leptons carry net electron-family <lepton number> $2-1=+1$, and the printed $\nu_e$ adds another $+1$. The initial <pion> has zero <lepton number>. This is a $\Delta L_e=2$ channel, not the charge conjugate of the listed positive-pion decay. In the ordinary lepton-number-conserving <Standard Model> approximation,
$$
\boxed{B(\pi^-\to e^-\nu_e e^-e^+)=0\quad\text{for the channel as printed}.}
$$
Allowing lepton-number-violating <masses> for <Majorana neutrinos> would require additional physical assumptions and does not justify assigning the tabulated number to that channel.

For the likely intended charge-conjugate process, the prediction is instead
$$
\boxed{B(\pi^-\to e^-\bar\nu_e e^-e^+)
\simeq(3.2\pm0.5)\times10^{-9}.}
$$
This distinction follows from a <lepton-number test of a decay channel>; replacing the printed <neutrino> by an <antineutrino> must be stated explicitly.