Challenge - Theoretical Physics Question

I claim that a neutrino is truly massless. Is it possible for the neutrino to decay (into two or more fragments) in free space? Why or why not?

Short answer: No.

Long answer:

I believe that in all known (and theorised) particle decays, the total mass of the system necessarily decreases. Say you have particle $P$ which is stationary and has mass $m_0$. $P$ decays into two particles $D_1$ and $D_2$, both of mass $m_1$.

Now, in order for momentum to be conserved, both particles most have equal and opposite velocities of magnitude $v$. If both particles have velocities, that means they have kinetic energy AS WELL AS mass energy. So in order for the total energy of the system to be conserved, , or, $m_0 c^2 + m_0 \times 0 = 2 \times (m_1 c^2 + m_1 v)$

As you can see, if $m_0 = 2m_1$, this equation doesn't work unless $v = 0$, which would make the decay invalid, as the particle isn't really decaying at all.

In conclusion, if the mass of a neutrino were 0, it would be impossible for it to decay into any other particles as the mass of the system must decrease (and you can't have a negative mass!). The only other possible decay path would be into another massless particle, such as a photon or graviton, however neither of these decays would conserve lepton number.

Could you elaborate on your statement "the total mass of the system necessarily decreases"?

I ask because unfortunately this argument cannot be applied in this case. The question you need to ask to see why is:
"Does a frame exist in which the neutrino can be viewed as stationary?"

The point regarding the breaking of the conservation of lepton number still stands, however. The mass of the system could never increase without breaking the conservation of momentum, and the only other massless particles are bosons. Unless there is some way in which lepton conservation can be broken to allow a single lepton to decay into two bosons, I can't see any valid decay routes for a massless neutrino to take.

Your point raises an interesting question for me. If there exists no frame where a particle can be viewed as stationary, then how can one find a decay path in which momentum is conserved?

You said that "The mass of the system could never increase without breaking the conservation of momentum". However is this true in a relativistic sense? What is the momentum of a photon? Is it related to its rest mass? As an example look up pair production.

This is a key point to answering this problem. You must consider relativistic effects to answer it.

Why does the presence of matter allow for this "decay"?

(...)
In conclusion, if the mass of a neutrino were 0, it would be impossible for it to decay into any other particles as the mass of the system must decrease (and you can't have a negative mass!). (...)

It is said that a neutrino have a mass, but it is very small. So is it possible that a neutrino decays under the circumstances you wrote?

I believe that neutrino 'decay' has been theorised, but not yet observed. The standard model allows for a massive neutrino to spontaneously change flavour, though I've no idea on the details of the proposed mechanism.

I have also no idea what the proposed mechanism of neutrino decay is. But as I have written above (I have to corrected the comment, as I have confused radiation with decay first) the mass must be high enough to decay, otherwise the mass of neutrino were not instable enough to decay. I guess if a neutrino is able to decay, it is the same principle as the decay of an atomic nucleus.

Indeed. A tau neutrino could theoretically decay into a lighter, muon neutrino (I think).

So the tau neutrino has a higher mass than a muon one? I could not find anything about the masses out, except that both of them are small...

I think the masses are as follows
Tau > Muon > Electron (...)

It is said that a neutrino have a mass, but it is very small. So is it possible that a neutrino decays under the circumstances you wrote?