$\Sigma^0$ and $n$ decay I was asked to show that in the baryon octet, the $\Sigma^0$ baryon is the only particle which decays electromagnetically.
Since it is an electromagnetic decay, strangeness should be conserved but I don't really get why can't a neutron (which doesn't have strangeness) also decay ellectromagnetically into a meson plus a photon. Is there any conservation law violated? I don't really know whether the quark content has to be conserved or not...
I also was asked to show why in the baryon octet, the neutron is the only particle which can decay into leptons. In this case, I came up with another decay
$\Sigma^-$ $\rightarrow$ n + e$^-$+ $\bar{\nu_{e}}$ which doesn't seem to violate any conservation law.
Is there something that I'm missing? Any help is welcome!
 A: You may be basically asked to understand the PDG. You can't change strangeness, electromagnetically.
So you may only decay by emitting a photon and rearranging your quarks in the case where your baryon charge stays the same (!), $\Sigma ^0 \to \Lambda \gamma$, which makes the neutral Σ nine orders of magnitude shorter-lived than its isopartners. No other options are available, energetically, and this is the only spot in the octet where this (same-charge members) happens. (And, as you appreciated, you must conserve baryon number, so that option is closed.)
Now, in a different world, the neutron would prefer to decay (by rearranging quarks) to a proton and a $\pi^-$ if it could, energetically: but check that  the n-p mass difference is too small. So its only option is the usual weak decay to an electron and an antineutrino and a proton (semileptonic: some of the decay products are leptons) and it takes it forever to do that, about a quarter of an hour...
The $\Sigma^-$, however, has that option, since it is so much heavier than the neutron, to decay weakly to $n~\pi^-$, which it does most of the time. But your mode $n~e^- \bar \nu$ is also OK, except subdominant (BR ~ $10^{-3}$), and is, in fact, measured, Bourquin 1983,  providing valuable WI information.

NB on comment/good-followup-question of @Marco Villalobos
I don't know. Related, especially the Wu & Rosner 1986 reference on non-leptonic  channel enhancements... Looks like it is a just-so accepted mystery on just hyperons. As one moves on to charm and bottom, the enhancement abates. Go figure...
