Why is this tau decay not allowed?

[unscientific]

Why is the decay $\tau^{-} \rightarrow \mu^+ + \mu^- + \mu^-$ not allowed?

Charge, lepton number are conserved. I have a feeling it is something really basic. I'm thinking in weak interactions you only go from a $l^- \rightarrow \nu_{l}$ and not 'hop' from one muon to another non-neutrino muon.

 

[The_Duck]

Why is the decay $\tau^{-} \rightarrow \mu^+ + \mu^- + \mu^-$ not allowed?

Charge, lepton number are conserved. I have a feeling it is something really basic. I'm thinking in weak interactions you only go from a $l^- \rightarrow \nu_{l}$ and not 'hop' from one muon to another non-neutrino muon.

In the standard model, electron number, muon number, and tau number are each separately conserved, except for very small effects coming from neutrino mixing.

One way of searching for physics beyond the standard model is to look for processes like $\mu^- \to e^- + \gamma$ that violate these individual lepton flavor numbers and so are negligibly rare in the SM. So a detection of a process like this would signal new physics.

 

[unscientific]

In the standard model, electron number, muon number, and tau number are each separately conserved, except for very small effects coming from neutrino mixing.

One way of searching for physics beyond the standard model is to look for processes like $\mu^- \to e^- + \gamma$ that violate these individual lepton flavor numbers and so are negligibly rare in the SM. So a detection of a process like this would signal new physics.

Got it. So lepton numbers must be separate, unlike hadronic numbers.

 

[Vanadium 50]

Lepton number is not conserved in this decay. The lepton numbers associated with the three lepton flavors are conserved separately in these interactions.

 

[mfb]

Even with neutrino mixing, this process is incredibly unlikely, with a branching fraction somewhere in the range of 10^-50 to 10^-60, completely undetectable. Comment from a theorist: "We didn't bother to make a more precise estimate".
Only new physics could lead to a detectable branching fraction.

 

[ChrisVer]

How would the neutrinos or/and their mixing have something to do with these processes? or are you talking about neutrinos changing their flavor (and so the conservation of each individual flavor's lepton number)?

 

[Vanadium 50]

Neutrino mixing means charged lepton number is only approximately conserved, but approximately means, as mentioned above, "good to a part in 10⁵⁰".

 

[mfb]

How would the neutrinos or/and their mixing have something to do with these processes? or are you talking about neutrinos changing their flavor (and so the conservation of each individual flavor's lepton number)?

Neutrino mixing is exactly the process of neutrinos changing their flavor.
That is frequent when the neutrinos can fly hundreds of kilometers or more, but as virtual particles within a decay process (necessary for tau -> 3 mu) it is extremely unlikely.

 

[ChrisVer]

(necessary for tau -> 3 mu)

that's what I meant :) So the SM process would be like $\tau \rightarrow \nu_\tau W \rightarrow \nu_\mu \bar{\nu}_\mu \mu \rightarrow Z \mu \rightarrow \mu \mu \mu^+$?

 

[mfb]

I doubt that diagram would be dominant. You can draw a box-like diagram with two W and two neutrinos forming the box, where one neutrino changes its flavor in between.

 

[ChrisVer]

would that really give 3 muons?
I think the best in odds [but only one muon as a product] would be a single loop, with the W in one (below) half and a changing tau neutrino to muon on the other (top) half...
this would lead to $\tau \rightarrow \mu$.

 

[mfb]

Sure, that would give three muons.

$\tau \rightarrow \mu$ is forbidden by energy/momentum conservation. You would need an additional photon, for example.

 

[unscientific]

And, I think lepton number is not conserved in neutrino oscillations.

 

[ChrisVer]

And, I think lepton number is not conserved in neutrino oscillations.

Neutrino mixing means charged lepton number is only approximately conserved, but approximately means, as mentioned above, "good to a part in 1050".