# Conservation of Lepton Number

Hello there!

I am learning about the conservation of lepton number. I understand that the lepton number is composed of different sorts of lepton number,

$$L_{\mu}$$ and $$L_{\tau}$$ and $$L_{e}$$.

And these are conserved in all interactions.

I am confused however about the lepton number for anti-particles. Are the lepton numbers negative for anti-particles?

Because in the muon decay;

$$\mu^{-} \rightarrow e^{-} + \overline{\nu_{e}} + \nu_{\mu}$$

I can see that $$L_{\tau}$$ is conserved (0) and $$L_{\mu}$$ is conserved, but $$e^{-}$$ would have $$L_{e}=1$$ and then $$\overline{\nu_{e}}$$ would have $$L_{e}= -1$$ which isn't conserved.

:-S

Cheers

bcrowell
Staff Emeritus
Gold Member
Are the lepton numbers negative for anti-particles?
Yes.

And these are conserved in all interactions.
Probably not directly relevant here, but I believe they're not actually perfectly conserved, and this is related to the fact that neutrinos have mass.

but $$e^{-}$$ would have $$L_{e}=1$$ and then $$\overline{\nu_{e}}$$ would have $$L_{e}= -1$$ which isn't conserved.

It is conserved in this decay. The initial value of $$L_e$$ is zero, and so is the final value (+1-1=0).

Aaaaah!

It's 0 to begin with! Woops! I see.

Thanks! :-)

Probably not directly relevant here, but I believe they're not actually perfectly conserved, and this is related to the fact that neutrinos have mass.

It is relevant if lepton number is not always conserved.

The electron lepton family has mass and yet for these (e,mu,tau) massed particles, lepton number is always conserved.

So is it really because the neutrino has mass?

There is a difference between saying only if neutrinos have mass is there the possibility that they can change from one family member to another, and saying its because they have mass.

As far as I know, all neutrino measurements where the neutrino changes generation also involves the neutrino going through matter (which may be true source of the generational neutrino fluctuation/change and potentially lepton number is actually conserved).

If there is a neutrino experiment involving generational change measurement that doesn't travel through matter, please give me the reference.

There is a difference between saying only if neutrinos have mass is there the possibility that they can change from one family member to another, and saying its because they have mass.

Of course, even if you want to be pedantic, no one in this thread ever said "its [sic] because they have mass". Yes, it would certainly be theoretically possible for neutrinos to have a mass and not oscillate or violate lepton number. On the other hand, a neutrino mass is necessary for these effects (which have been observed), which is quite accurately described by the statement "they're not actually perfectly conserved, and this is related to the fact that neutrinos have mass". An open question is whether total lepton number conservation can be violated (as in a neutrinoless double beta decay process, which is only possible if they are majorana particles), rather than just violation of individual lepton family which is violated in neutrino oscillations (e.g., electron number, muon number, etc). All of this requires a non-zero neutrino mass, however.

As far as I know, all neutrino measurements where the neutrino changes generation also involves the neutrino going through matter (which may be true source of the generational neutrino fluctuation/change and potentially lepton number is actually conserved).

If there is a neutrino experiment involving generational change measurement that doesn't travel through matter, please give me the reference.

That's quite a tall order you've got there. I think it makes more sense for you to first explain how it would be possible to have neutrino oscillations with all massless neutrinos--in either matter or vacuum. I'm certainly no expert, but I don't see how that would be possible.

**Edit: I just realized my response probably came off as exceedingly snarky. I apologize for that, but I would appreciate it if you could back up your claim that lepton number might actually be conserved. **

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Btw, no mass implies no oscillation is very simple: if there's no mass, then they travel at the speed of light, and so there's no passing of proper time, and so literally no time to oscillate in.