Question on particle decay process, conservation of energy Why is $$\Sigma^0 \rightarrow \Lambda +\pi^0$$ not a possible process?
Charge and baryon number both are conserved. There's no issue with strangeness that I can tell. The masses in $\frac{Mev}{c^2}$ are approximately $$1193 \rightarrow 1116 + 135.$$ This gives me a red flag... It doesn't seem energetically favorable, but then again, couldn't the $\Sigma^0$ have a lot of kinetic energy prior to decay?
A different example, $e^- + e^+ \rightarrow \mu^-+ \mu^+$, clearly has less mass beforehand than after but is said to occur. I'm a bit confused here about how and when to apply energy conservation.
 A: You should always apply energy conservation, and it ought to hold in all reference frames, including the frame in which the sigma is at rest. In the sigma's rest-frame,
$$
E_{\text{initial}} = E_\Sigma = m_\Sigma
$$
and 
$$
E_{\text{final}} = E_\Lambda + E_\pi \ge m_\Lambda + m_\pi
$$
Thus we have that,
$$
E_{\text{initial}} < E_{\text{final}} 
$$
The process is forbidden by energy conservation (it must be the case that $E_{\text{initial}} = E_{\text{final}}$).
In the case of $ee\to\mu\mu$, there is no frame in which both the electrons are at rest - in every reference frame, the initial state has more energy than the sum of the electron masses,
$$
E_{\text{initial}} > 2m_e.
$$
Thus, $ee\to\mu\mu$ may conserve energy, if the pair of electrons has sufficient energy.
A: Energy conservation always applies.
Your mistake is in thinking that adding masses will solve the problem instead of clarifying some aspects. 
In the case of the sigma-zero the decay at rest allows to see that the sum of the constituent masses is larger than the mass of the sigma-zero. For a decay to happen there should be energy left over to go to the kinetic energy of the decay products, thus, correctly it is a no go from energy conservation process.
In the case of the e+e- one is talking of scattering, i.e. the electron and the positron have kinetic energy and are  defined by  a four vector each. The addition of the two four vectors gives a large invariant mass, as happened in the colliders at SLAC and LEP. This large invariant mass leaves a lot of energy to appear as kinetic energy in the interaction  products.
