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Usually we arrange the quarks into 3 generations, depending on their mass.

But for example, I can think various other ways to group the quarks. e.g. the $(charm,bottom,top)$ quarks don't seem to fit into a family since they have electric charges $Q=(+2/3,-1/3,+2/3)$ respectively. But one could postulate that they each have 2 more quantum numbers $R=(-1/3,+2/3,+2/3)$ and $S=(+2/3,+2/3,-1/3)$. And then they would form a symmetric family permuting the 3 quantum numbers $Q$ $R$ and $S$. One could do a similar thing for the $(down,up,stange)$ quarks.

Therefor perhaps an equally valid way to group the quarks would be into 2 genetations. $(up,down,strange)$ and $(charm,bottom,top)$. With a broken symmetry caused by the photon being massles and hypothetical bosons corresponding to $R$ and $S$ charges being massive.

Appart from the fact that there is no evidence (so far) of additional quantum numbers relating to the quarks. Is there any reason that we group the quarks into 3 genertions ordered by mass. (The "ordered by mass" seems very arbitrary). Or is it merely convention.

It would be possible to do a similar trick arranging the 3+3 leptons and neutrinos into 2 generations.

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    The point of the generations is to reflect (i) "carbon copies" of particle properties and particle interactions and decays, and (ii) the decays linking these copies so we can thereby order them and call them generations. – J.G. Mar 19 '21 at 17:43
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    Your 2 generation scheme obviously does not work, since it violates the quantum anomaly cancellation condition. In addition to @Cosmas Zachos's excellent answer, you can also check out a related question here: https://physics.stackexchange.com/q/611725/ – MadMax Mar 19 '21 at 22:30
  • @J.G. True, but they are only "carbon copies" until we find a new boson that acts on one but not the other. –  Mar 20 '21 at 01:11

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You may certainly arrange fermions any way that appeals to you and admire their properties, or try to discern numerological properties for their masses. But beware of theorists getting egg on their faces for three and a half decades in such efforts, which hardly discourages them...

The point you are not acknowledging is that the present arrangement of fermions in generations in popular charts is a creature of history and convenience, not a logical necessity. See here.

The generations must be such to represent triplicate superfluous replication of weak isodoublets, after suitable CKM/PMNS mixing rotations, and to expedite the obviousness of the gauge anomaly cancellations. Originally, for quarks, these mixing angles were "small", as these matrices are fatter in the diagonal (a fact exploited by Wolfenstein).

If you chose to permute the up and the charmed quarks in their generation assignments, however, nobody would cringe, provided you adjusted the CKM matrix accordingly to yield the same electroweak coupling vertices to the charged vector bosons. (But ... people would hate you for superfluously writing something down so gratuitously unmemorable. GUT desperadoes have already been there.) The lepton arrangement is already stressful in that their flavor mixings are huge, so the present arbitrary assignment by mass is the most memorable.

In fact, the linkage between specific quark generations to lepton generations is even weaker: you need not adjust any mixing matrices if you just permuted their first with their third generation, leaving quarks alone, etc...

To sum up, the present middle-school chart grew historically as the heavier particles were being discovered and joined the chart, and of course, the neutrinos in it were given their placeholder names "lightest, medium, heaviest" precisely to obviate even knowing which is which, and relegating the issue to one of PMNS matrix labelling.

Arranging fermions in different patters is as meaningful as the conclusions it motivates.

Cosmas Zachos
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    This is an excellent answer. I wish that more books on the Standard Model would make the clear. What I find interesting is that books about String Theory and such like try to say that there is something special about the way the 3 generations are organised whereas like you say it is a bit arbitrary. –  Mar 20 '21 at 01:18
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    That's interesting how they put "lightest neutrino" in there, presumably because they don't know if this will be the electron-neutrino, the muon-neutrino or the tau-neutrino! Or if it even makes sense to call them that. –  Mar 20 '21 at 01:24
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    Right... it’s meaningless to call them that, given the huge PMNS mixings. Older charts had these deeply unsound and confusing names which have done so much pedagogical damage... – Cosmas Zachos Mar 20 '21 at 02:46