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What are the basic differences between a Tau, Muon, and Electron? I can read the wikipedia article, but was wondering what specifically distinguished the three leptons and in what experimental contexts they show up.

Qmechanic
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sakurashinken
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3 Answers3

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They only differ in their masses. How they interact with other particles, their charge, their lepton number, everything else is the same.

In fact they appear in the SM lagrangian just as three identical replica of the basic fermionic family

$$\mathcal{L} = \sum_{i=1}^3\bar{\psi}^i(i\not D-m_i)\psi^i$$ where $i$ is the flavour number. One could ask why only $3$ and not $2$ or $4$ or $10$ or whatever. To this question we don't have a definite answer yet. But the fact that they are $3$ makes perfect sense from the theoretical standpoint since, for example, the existence of three families of leptons and quarks cancels all the anomalies in the SM gauge group.

Edit:

As suggested in the comments, we think it's also worth adding that, in reality, in the SM the difference between the various families is in the Yukawa coupling (which is fixed by experiments) that, after symmetry breaking governs both masses and interaction strength, including their decay time which is dependent on the mass again.

Moreover, as @kaylimekay suggests, and I quote

"This answer is only true in the SM, which we already know is not the full story. We are all hoping to see deviations from this structure, and maybe already have, if you believe the LFV B-decay data."

Hope this clears any misunderstanding.

Davide Morgante
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  • See also: https://physics.stackexchange.com/q/2051 – Nihar Karve Dec 29 '20 at 08:44
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    They only differ in their masses. (when you ignore their lifetime :-) – Jens Dec 29 '20 at 20:22
  • @Jens Yeah, I thought about underlying the fact that the Yukawa couplings differ from each other but then this enters again as an experimental fact rather than a theoretical one! – Davide Morgante Dec 29 '20 at 20:26
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    They also differ in lepton flavor number. To go from a muon to an electron, it is not sufficient to take away mass, you also have to take away "muonness" and put back "electronness", so to speak: $\mu \rightarrow \nu_\mu + \bar\nu_e + e$. – jdm Dec 29 '20 at 21:41
  • @DavideMorgante Isn't the mass explained via the coupling of leptons to the Higgs field? And wouldn't that coupling be on equal footing (when we try to see how $e$, $\mu$ and $\tau$ differ) as the couplings that produce the decays of muons and taus? – anonymous Dec 30 '20 at 04:53
  • @anonymous The Standard Model has 18 (potentially 26) free parameters which need to be given experimentally and, as of now, we don't have a theory that explains why these parameters are as we measure them. Nine of this 18 parameters are the Yukawa coupling constants that determine the masses of the charged quarks and leptons. One important unanswered is why the yukawa coupling of the top quark is so much bigger than the one of the other quarks, this problem goes under the name of hierarchy problem – Davide Morgante Dec 30 '20 at 08:14
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    @DavideMorgante The reason why I asked is that between all those parameters, you seemed to choose those corresponding to the masses as fundamental ("They only differ in their masses."), while (according to your answer to @Jens) the parameters involved in other couplings (I'm assuming here the weak couplings ruling $\tau$ and $\mu$ decays) are "an experimental fact rather than a theoretical one". My question, in other words, is why are the mass-related parameters different to the decay-related ones when it comes to telling the difference between electrons, muons and taus? – anonymous Dec 30 '20 at 08:37
  • @anonymous Yukawa couplings govern both mass and interaction strength. One could speak of the SM before symmetry breaking and say that, again, the structure is exactly the same for all three generations, the only thing that changes are the Yukawa couplings $Y_{ij}$. We put them as distinct since experimentally we know that they have to be different. After symmetry breaking the unbroken Yukawa parameters make up the mass and the interactions strength. Again the functional form of the theory is the same for all generations but the coupling parameters are, experimentally, different. – Davide Morgante Dec 30 '20 at 09:11
  • @DavideMorgante I'm not sure we are understanding each other. If the couplings determine both the mass and the interaction strengths, what is the meaning of the sentence They only differ in their masses. How they interact with other particles [...] is the same.? It seems to me that both masses and interactions are different, and different for the same reason. Would you agree with that? – anonymous Dec 30 '20 at 09:40
  • @anonymous Yes, I agree. The point that i was trying to make in my answer with How they interact with other particles [...] is the same is that the type of interaction is the same, not the strength. But yes, I understand that my point is misleading since even the functional form of the mass term is the same, the only difference is in the experimental values, which again differ even for the strength of the interaction. To settle this down we could say that the three families differ in their Yukawa coupling and nothing more! – Davide Morgante Dec 30 '20 at 11:23
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    Right, this answer automatically covers @Jens comment because the decay lifetimes are uniquely determined by the masses (primarily due to kinematics, but also through Yukawas when considering extremely rare Higgs mediated decays) once gauge couplings are fixed. This wouldn't be a bad addition to the answer, IMO. – kaylimekay Dec 30 '20 at 13:23
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    I think it would also be worth adding the caveat that this answer is only true in the SM, which we already know is not the full story. We are all hoping to see deviations from this structure, and maybe already have, if you believe the LFV B-decay data. – kaylimekay Dec 30 '20 at 13:27
  • @jdm I suspect technically you could emit a tau neutrino, and then recapture the same neutrino after it flips flavor? – John Dvorak Dec 30 '20 at 18:04
  • @John Dvorak Sure, I assume that would work, but would be extremely suppressed because the neutrino needs to propagate to oscillate. But it would have to happen in the same Feynman diagram so to speak. In general different "flavor changing neutral currents" are possible but only at higher order and thus very rare. – jdm Dec 31 '20 at 08:53
  • @JohnDvorak Moreover neutrino oscillations are not directly permitted in the SM since they require a mass for the neutrino. This is not present in the SM and is in fact one of the problems with it. This is solved, for example, by adding a very massive right-handed Majorana neutrino. In fact, if neutrinos turn out to be Majorana fermions, lepton number would not a conserved quantity. As per jdm answer, I'm not sure that neutrinos in the SM can undergo FCNC, being massless. But i stand to be corrected. – Davide Morgante Dec 31 '20 at 09:45
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Davide gives the theoretical side. Experimentally, in detecting them, there is a large difference.

Electrons have been detected long ago: cathode ray tubes and then the Millikan oil drop experiment established the existence of electrons.

Muons were seen in cosmic rays and it took accelerator experiments to detect them in the lab

Taus, because of their mass were discovered when specific high energy beams could reach those energies that could create taus.

The neutrinos were necessary to have energy and momentum conservation in the decay of the neutron, and this led to lepton number conservation, the antineutrino_electron.The decay of muons and the theoretical proposal of lepton number conservation allowed the study of interactions producing muons.

The electron as the lowest mass is stable, taking part in the creation of atoms. The muon and tau are unstable decaying with the weak interaction. The muon decays to electron and neutrinos, because its mass is too low to have an emergent pi0. The tau has high enough mass to decay into hadrons, again through the weak interaction.

So experimentally they are very much different.

Mithoron
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anna v
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Electrons have the least mass of any charged particle, so cannot decay. Muons, the second lightest of the three, can decay in one dominant way, by producing an electron viz. $\mu^-\to e^-+\nu_\mu+\bar{\nu}_e$. The far more massive tauon is the shortest-lived, partly because its high mass accelerates the decay by Sargent's rule, but partly also because it can decay in many ways. These include electron- and muon-producing decays, but uniquely the tauon can also produce mesons

J.G.
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