Why does nature come with three families of particles, like the electron, muon and tau?

Three families of what?

The observed generations each contain quarks and leptons. Their charged leptons are the electron, muon and tau, paired with neutrino types. Muons have the electron's charge but greater mass; the heavier charged leptons are unstable. Familiar atoms use electrons and first-generation quarks. CERN: particle organization.

The Standard Model describes this organization but does not explain the generation count or mass differences. A successful description need not explain its own ingredients.

Could their masses follow a simple rule?

The Koide relation is an empirical numerical relation among the three charged-lepton masses. Sumino's 2009 theoretical paper proposes a family-symmetry model that reproduces it and addresses corrections. That is a proposed mechanism with assumptions, not an experimental discovery of the mechanism. Original family-symmetry model.

Consider the difference between filling in a table and finding a rule that requires its entries. A table can record three measured values accurately while leaving their origin unanswered. A proposed rule becomes more informative if it correctly anticipates a measurement that was not used to build it.

A useful explanation must do more than summarize numbers already known. Does it predict something else that measurements can check? Which assumptions are essential? Would a revised measurement spoil the agreement? These questions distinguish a promising clue from a convenient fit. No mass relation discussed here establishes why nature has exactly three generations.

Why investigate a short-lived muon?

Fermilab explains that muons are easier to produce in large numbers and live longer than taus. That makes them useful probes even though they decay. Producing many particles gives researchers many opportunities to search for an exceptionally rare outcome. The aim is not to keep one muon alive indefinitely, but to measure what large samples do. Why Mu2e studies muons.

What would a rare conversion reveal?

Ordinary muon decay produces an electron and two neutrinos. Mu2e instead seeks conversion to an electron in a nucleus's field without those neutrinos. The nucleus can recoil, conserving momentum. The collaboration uses the outgoing electron's energy to distinguish the sought process from backgrounds. In the Standard Model with neutrino masses, this conversion is far below the experiment's reach; an accessible signal would indicate additional physics. This is a search, not a reported discovery. Mu2e measurement explanation.

Even a genuine signal would not automatically select one hidden rule: competing theories can predict rare processes. Confidence is high in the observed particle organization; its deeper explanation remains open. An independently confirmed conversion or a new mass model with successful fresh predictions would change the evidence. See what quantum effects do and do not allow.