Violation of Conservation Laws: Can an Electron Decay into Two Neutrinos?

[bjgawp]

Homework Statement

A claim has been made that an electron decays into 2 neutrinos traveling in different directions. Which conservation laws would be violated by this decay and which would be obeyed?

Homework Equations

Momentum: p = mv
Mass-energy: E = mc²

Electron
Baryon number: 0
Lepton Electron number: +1
Letpon muon number: 0
Lepton tau number: 0


Neutrino
Baryon number: 0
Lepton Electron number: +1
Letpon muon number: 0
Lepton tau number: 0

The Attempt at a Solution

From the question, I gather that the claim states:

e --> Ve + Ve​

e = electrons, Ve = a neutrino

The baryon, lepton muon, and lepton tau numbers are conserved but not the lepton electron number (+1 --> +2). Electric charge is also not conserved since the electron has a charge of -1 while the neutrinos are ... well, neutral.

Momentum, a little bit trickier, seems to be conserved only if the electron is at rest:

Pe = Pve1 + Pve2
0 = Pve1 + (-Pve1)
0 = 0​

However, if the electron is initially moving, we run into a problem:

Pe = Pve1 + Pve2
MeVe = MVve1 + (-mVve1)
MeVe = 0​

This contradicts what I initially said - that the electron was moving.

Now, this leaves me with the mass-energy conservation. I'm not exactly sure about this one: E = mc²
How do I take into account that when the electron decays, the neutrinos have kinetic energy as well as their new masses (which are almost neglible and probably won't add up to the mass of an electron).

Any help will be appreciated :)

 

[vanesch]

The baryon, lepton muon, and lepton tau numbers are conserved but not the lepton electron number (+1 --> +2). Electric charge is also not conserved since the electron has a charge of -1 while the neutrinos are ... well, neutral.

correct.

You might also consider spin. There is a problem with a spin-1/2 particle decaying into two spin-1/2 particles, because the two spin-1/2 particles will combine in a total spin of spin 0 or 1, and not spin-1/2. Orbital angular momentum is always integer, so this cannot compensate for the 1/2 missing.
You can normally never have a fermion decaying into two fermions. But this is a subtle issue.

Momentum, a little bit trickier, seems to be conserved only if the electron is at rest:

Pe = Pve1 + Pve2
0 = Pve1 + (-Pve1)
0 = 0​

However, if the electron is initially moving, we run into a problem:

Pe = Pve1 + Pve2
MeVe = MVve1 + (-mVve1)
MeVe = 0​

Here, you're confused. If momentum is conserved in one reference frame, it is of course conserved in all reference frames !

Now, this leaves me with the mass-energy conservation. I'm not exactly sure about this one: E = mc²
How do I take into account that when the electron decays, the neutrinos have kinetic energy as well as their new masses (which are almost neglible and probably won't add up to the mass of an electron).

Any help will be appreciated :)

You should consider the simple problem of a particle of mass m decaying into two particles of mass m' (m' < 1/2 m). Hint: work in the center of mass system of the initial particle, and use momentum and energy conservation, and work out the final velocities of the particles.

 

[bjgawp]

Here, you're confused. If momentum is conserved in one reference frame, it is of course conserved in all reference frames !

Hmm ... but according to my second calculations, I get the final result of the electron at rest even when I stated that it had a velocity of Ve at the very beginning.

You should consider the simple problem of a particle of mass m decaying into two particles of mass m' (m' < 1/2 m). Hint: work in the center of mass system of the initial particle, and use momentum and energy conservation, and work out the final velocities of the particles.

You mean like an explosion - calculating the momentum of each 'piece'?
Well, I'll see what I can do for now...

 

[vanesch]

Hmm ... but according to my second calculations, I get the final result of the electron at rest even when I stated that it had a velocity of Ve at the very beginning.

What makes you think that the two neutrinos have the same velocity ?

You mean like an explosion - calculating the momentum of each 'piece'?
Well, I'll see what I can do for now...

Yes. It was only to make you understand what is kinetically possible and what not...

 

[bjgawp]

What makes you think that the two neutrinos have the same velocity ?

Hmm, now that leaves me a bit confused. Is it even possible for the two neutrinos to travel in opposite directions when the original electron was initially moving in a given direction? But then again, this is all hypothetical ...

And so, I'm having a bit of difficulty since there are too many variables to come to a conclusion at the moment.
Momentum of electron = Momentum of Neutrino1 + Momentum of Neutrino2
MeVe = (MnVn)1 + (MnVn)2

All I can gather is that for the electron to decay, the two masses of the neutrino should add up to the mass of the electron but that isn't the case if we compare actual masses.

Gah, too confusing.

 

[OlderDan]

Hmm, now that leaves me a bit confused. Is it even possible for the two neutrinos to travel in opposite directions when the original electron was initially moving in a given direction? But then again, this is all hypothetical ...

And so, I'm having a bit of difficulty since there are too many variables to come to a conclusion at the moment.
Momentum of electron = Momentum of Neutrino1 + Momentum of Neutrino2
MeVe = (MnVn)1 + (MnVn)2

All I can gather is that for the electron to decay, the two masses of the neutrino should add up to the mass of the electron but that isn't the case if we compare actual masses.

Gah, too confusing.

Yes it is possible for the neutrinos to travel in opposite directions. For low speed electrons, that is the only possibility. Surely in the limit of zero electron velocity (rest frame of the electron) momentum conservation demands it.

When you say the neutrino masses must add up to the electron mass, but don't, you are apparently mixing up rest mass and relativistic mass. If you are talking about relativistic mass, then they must add up (conservation of mass-energy). If you are talking about rest masses, then they must not add up, or else the two neutrinos must move off stuck together with the original electron velocity. The difference in rest masses accounts for the change in kinetic energy of the system.

 

[bjgawp]

Hmm ... then that wouldn't leave me with much to work with if the electron is initially moving because we don't know the velocities of the neutrinos after it decays. I also don't understand how to go about seeing if the mass-energy conservation law is violated. Am i missing a connection here ...

Thanks for the help. Very much appreciated

 

[OlderDan]

Hmm ... then that wouldn't leave me with much to work with if the electron is initially moving because we don't know the velocities of the neutrinos after it decays. I also don't understand how to go about seeing if the mass-energy conservation law is violated. Am i missing a connection here ...

Thanks for the help. Very much appreciated

The problem you were given is not asking you to do a numerical comparison. The momentum and mass-energy parts of the question are intended to see if you understand the important concepts of conservation of momentum and mass-energy in particle interactions. Electron neutrinos are nearly massless particles, so rest mass is clearly not conserved in this decay, but energy and momentum are conserved in all such particle interactions. If you had an initial electron velocity in any frame of reference, the total momentum and total energy of the two neutrinos in this hypothethetical decay would be the same as that of the electron before the decay.

If you assume the neutrinos to be massless particles they still have energy and momentum (just like photons) and travel with velocity c. You could compute their momenta and energies from momentum and energy conservation principles. One way to do that is to look at the decay in the center of mass frame and find the equal and opposite momenta. From that you can find the frequency of each neutrino, and then use the relativistic doppler shift to find the frequecies in any frame where the electron had initial velocity. The result would be momentum and energy conservation in any frame. As massless particles they will be moving opposite one another at speed c in all frames, but their individual momenta and energies will be different in different reference frames.