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As far as I am aware, each generation of fermion varies only in mass, thus my question is why do we have only three distinct generations of fermion, instead of a continuous mass spectrum of particles from none-zero mass to very large mass? I am aware that the fermions couple to the Higgs field and therefore a gauge invariant mass term is introduced for the fermions, but I do not understand why the masses are set and not continuous.

anna v
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spacexyz
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    What sort of "why" are you looking for here? There's no way in standard QFT to have a continuous distributions of masses for elementary particles (because there's 1:1 correspondence between fields and elementary particles and we don't really have field theories with more than finitely many fields), but one might say the reason we don't have such formulations is simply because they would be of no use since we do not observe any situation with continuous distribution of masses. Roughly related: https://physics.stackexchange.com/q/122/50583 – ACuriousMind Mar 25 '22 at 00:39
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    How do you tell the difference between your "continuous mass" and 14873 families with discrete nearby masses? They'd nix asymptotic freedom, and we wouldn't have hadrons. Do you care at all? – Cosmas Zachos Mar 25 '22 at 01:05
  • @ACuriousMind So the standard model doesn't restrict us to three distinct generations of fermion, rather the generations of fermion are based upon experimental measurements only ever discovering three generations? – spacexyz Mar 25 '22 at 01:17
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    I mean, sure, the standard model gives us three generations, but the standard model is designed to replicate the results of real life observations. A model with a different (finite) number of generations wouldn't be the standard model! – ACuriousMind Mar 25 '22 at 01:19
  • maybe my answer here will help https://physics.stackexchange.com/questions/90164/physics-the-why-vs-how-question/90526#90526 – anna v Mar 25 '22 at 04:26

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There is one test or meta-analysis I know of in which particle accelerator data was plotted in such a manner that the number of particle generations was a variable. The data indicated that the number of generations could not be greater than three (which is indeed the number that is observed) and still be consistent with experiment. I believe this chart appeared in a book called The Ideas Of Particle Physics, second edition.

niels nielsen
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Since several people have focused on our modelling nature that way because it matches observations, I'll focus on another possible read of the OP's question: why would nature be that way? Questions like this are hard to answer because, if it were another way, we'd probably ponder why it's that way instead. So I'll try to relate this fact warranting explanation to similar facts, which hopefully makes it feel like it makes more sense.

We need to think about not only how generations differ, but also how they relate. Higher generations decay to lower ones; single particles can't do the reverse, as in the CM frame it would create energy. This is analogous to an excited electron in an atom emitting a photon, thereby shifting to (or a closer to) the ground state.

One key lesson of quantum mechanics is that these states are discrete; were they not so, atoms would be unstable, because an electron in a classical orbit would continually radiate energy. Similarly, if generation were a continuous parameter, particles would continually decay toward a ground state (or indefinitely if there weren't one).

But if e.g. a tau became a series of muon-tau intermediates followed by a series of electron-muon intermediates eventually culminating in an electron, it would shed mass continually in a manner that makes an "elementary particle" characterization unhelpful. Would we describe that as infinitely many species, or one species with infinitely many states? Even on the latter account, why (given the above comparable example) would we expect infinitely many states?

J.G.
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Physics is a science that has a large body of observations, and a limited number of mathematical models/theories that aim to organize and explain those observations and , very important, get validated by predicting the behavior of new observations.

Mathematical theories start with axioms and some tools that develop theorems from those axioms and then various setups can be examined. The theories are self consistent.

Physics uses mathematics as a tool, imposing extra postulates that have the strength of axioms, in order to pick up those mathematical solutions that can fit the data and be predictive.

When the quark model with its generations was experimentally discovered there were distinct generations needed, imposed by the symmetries experimentally observed. This is what is encapsulated in the standard model, in order to describe the data and be predictive.

anna v
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