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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.

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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.

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    $\begingroup$ They only differ in their masses. (when you ignore their lifetime :-) $\endgroup$
    – Jens
    Dec 29 '20 at 20:22
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    $\begingroup$ 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: $\mu \rightarrow \nu_\mu + \bar\nu_e + e$. $\endgroup$
    – jdm
    Dec 29 '20 at 21:41
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    $\begingroup$ @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? $\endgroup$
    – anonymous
    Dec 30 '20 at 8:37
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    $\begingroup$ 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. $\endgroup$
    – kaylimekay
    Dec 30 '20 at 13:23
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    $\begingroup$ 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. $\endgroup$
    – kaylimekay
    Dec 30 '20 at 13:27
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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.

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    $\begingroup$ +1 There are always two sides for the same coin ;-) $\endgroup$ Dec 29 '20 at 13:49
  • $\begingroup$ Muons created in the upper athmosphere can be easily detected in the lab, no accelerator needed. $\endgroup$
    – lalala
    Dec 30 '20 at 16:37
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    $\begingroup$ @lalala Yes, the tracking in a cloud chamber will be there, but to identify it as a particle with the mass of the muon needs special experiments. $\endgroup$
    – anna v
    Dec 30 '20 at 17:19
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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

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