= Solution
The initial <strangeness> is $-2$. The $\Lambda\pi^0$ final state has <strangeness> $-1$, so its allowed nonleptonic amplitude is weak. One representative topology is $s\to uW^{-*}$ followed by $W^{-*}\to d\bar u$; spectator $u,s$ combine with the produced $d$ into the $\Lambda$, while $u\bar u$ contributes to the neutral <pion>. The available rest <energy> is $1314.8-1115.7-135.0=64.1\,\mathrm{MeV}$.
The $\Lambda\bar K^0$ state would conserve <strangeness> and could have a strong vertex, but its threshold exceeds the parent mass by $1115.7+497.6-1314.8=298.5\,\mathrm{MeV}$. \b[It is kinematically forbidden.] Coupling strength cannot rescue a closed decay channel.
The radiative channel also changes <strangeness>. It is a <weak radiative hyperon decay>, not an ordinary electromagnetic decay between states of the same flavour. A representative two-quark contribution is W exchange $su\to ud$ with a <photon> attached to a charged line; the other allowed attachments, hadronic intermediate states and short-distance loop contributions must be included for a complete amplitude. Its extra electromagnetic coupling and hadronic dynamics suppress it relative to the nonleptonic channel. Consequently
$$
\boxed{\Gamma(\Lambda\pi^0)\gg\Gamma(\Lambda\gamma)>\Gamma(\Lambda\bar K^0)=0}.
$$
<NA48 Collaboration> measured a radiative <branching fraction> of about $1.16\times10^{-3}$, corroborating the suppression rather than a purely electromagnetic interpretation.<Measurement of the Xi0 to Lambda gamma Decay Asymmetry and Branching Fraction by NA48 Collaboration>[abstract and branching-fraction result, 2004]{ref}
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2012/paper-1-hyperon-and-rho.png]
{title=Hyperon and radiative meson topologies}
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