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The Standard Model contains three generations of quarks, and three generations of leptons. We generally pair off these generations into the "light" generation ($e, \nu_e, u, d$), the "medium" generation ($\mu, \nu_\mu, c, s$), and the "heavy" generation ($\tau, \nu_\tau, t, b$.)

Is the reason we do this just because of the relative masses of the particles? Or is there some underlying symmetry between the leptons & quarks that requires us to associate the electron with the up and down quarks? I know you have to have complete electroweak multiplets to cancel out the anomalies, but is there any reason other than mass that we don't pair $e$ and $\nu_e$ with $t$ and $b$?

To put it another way: If all of these particles had the same mass, would there be any reason to "pair off" the lepton and quark generations the way we currently do, or would any pairing of the generations be OK?

This is probably a basic fact that I learned in my QFT classes, about [mumble mumble] years ago, but I seem to have forgotten it if I ever learned it.

Michael Seifert
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    Interesting question. I guess if you do not assume any GUT in which the quarks and leptons of a generation live together in some multiplet, there is no reason to associate a specific quark with a specific lepton. After all the only difference between different quarks of the same charge is the mass since flavor is no useful quantum number in the standard model. I thought that maybe one could associate them due to structures in the mixing matrices (i.e. CKM and PMNS) but those are not meaningful without mass differences. I'm curious if someone sees another connection than mass... – Katermickie Dec 05 '19 at 22:17
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    It looks like https://physics.stackexchange.com/q/312985/ from the sidebar is similar. But it didn't go anywhere. – dmckee --- ex-moderator kitten Dec 05 '19 at 22:21
  • @dmckee: if nothing else, knowing that Emilio asked a similar question makes me feel more confident that it's not a silly question. :-) – Michael Seifert Dec 05 '19 at 22:24
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    I suspect that near duplicate has correct answers. The grouping into generations is subjective and a plausible listing of convenience, unless one were a believer in a particular speculative framework of GUTS; but his is not a listing of known facts. CKM and PMNS mixing moots even the definition of the grouping, since the weak couplings evade mass eigenstates! Consider what would go wrong if you interchanged two lepton generations with each other: *nothing*. – Cosmas Zachos Oct 06 '21 at 16:14

2 Answers2

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There is NO reason, other than mass hierarchy, that we associate each quark generation with a particular lepton generation.

The standard generation assignment $$ (e, \nu_e, u, d)\\ (\mu, \nu_\mu, c, s)\\ (\tau, \nu_\tau, t, b) $$ is essentially arbitrary: the only rational of the above generation assignment is the relative magnitudes of masses.

If we adopt an alternative generation assignment, say $$ (\mu, \nu_\mu,u, d)\\ (\tau, \nu_\tau, c, s)\\ (e, \nu_e, t, b) $$ the standard model is still working fine as usual, including obeying the quantum chiral anomaly cancellation conditions.

The question in concern is not just an arcane topic for late night bull sessions. Different generation assignments could have tangible consequences. For example, the primary channel of proton decay in the grand unified theories (GUT) is usually "proton to electron/positron and meson", which is actually based on an unfounded assumption of $(e, \nu_e, u, d)$ being in the same generation.

Alternatively if we assume that $(\mu, \nu_\mu, u, d)$ are in the same generation and the GUT quark-lepton flipping gauge fields only interact within the same generation, then protons may be restricted to decay to muon/antimuon instead, which is of course suppressed due to large muon mass. This explains why we haven't observed proton decay yet.

MadMax
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    Fair, but once you have a GUT there is a good reason to assign the generations the way they are, namely that some subset of the quarks and leptons in a generation will get mass the same way. If you paired $e$ with $t$ but $\tau$ with $u$, you would have to explain why $m_e / m_t \sim 10^{-6}$ is so different from $m_\tau / m_u \sim 10^3$, when you would naively expect the ratio to be similar within each generation. – knzhou Dec 05 '19 at 23:09
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    Because in simple GUT models there are fewer independent Higgs Yukawa couplings. Since there are fewer parameters than in the SM, where all masses are independent, you get relations between the masses of fermions within each generation. For the simplest GUTs these relations are not very accurate -- but in your proposal the relations would be wrong by many orders of magnitude. – knzhou Dec 05 '19 at 23:18
  • @knzhou, is there any solid rationale that the mass ratios should be of the similar order? – MadMax Dec 05 '19 at 23:19
  • Of course you can always give yourself more independent parameters by, e.g. adding in more Higgs fields. But this defeats the entire point of a GUT, which is to "unify", which implies having fewer parameters. – knzhou Dec 05 '19 at 23:19
  • @knzhou, most (if not all) GUTs involve larger gauge groups unifying fermions WITHIN a given generation, not across generations. Is there any compelling reason that the mass ratios should be of the similar order between generations? – MadMax Dec 05 '19 at 23:31
  • And for that matter, if you invoke generational democracy, then why 1st and 2nd quark generations mix more than the third and hence the offdiagonal CKM elements are not of the 'right' ratio. Does this call for different mass generation mechanism for the third quark generation and violate your cherished 'unified' Higgs mechanism? – MadMax Dec 05 '19 at 23:44
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How did the $SU(3)\times SU(2)\times U(1)$ come to be the standard model of particle physics? After years and years of experiments that showed up the quark model, which lead to the standard model. example of symmetries in the data:

meson octet

The meson octet.

Note that even though the symmetry splits according to charge and strangeness, there is a mass dependence in the plot.These symmetries have to come out in the standard model, which uses the elementary particles in the table . It is validated by all the preexisting data.

To put it another way: If all of these particles had the same mass,

This is the basic hypothesis for the standard model before symmetry breaking, that all the elements entering have zero mass. The Higgs mechanism breaks the symmetry , finally the particles acquire mass. The group structure exists before and after symmetry breaking.

So the answer to the title

Is there a reason, other than mass hierarchy, that we associate each quark generation with a particular lepton generation?

It comes out of the symmetries of the standard model groups, after symmetry breaking and the acquisition of mass. The hierarchy is given by the symmetries of the group structure. That we call electron and down quark the lowest mass items in the group symmetries is because that is what is used in identifying the data to be checked. So it depends on the total representation of the theory.

anna v
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    How does the meson octet relate to leptons? "It comes out of the symmetries of the standard model groups, after symmetry breaking and the acquisition of mass. The hierarchy is given by the symmetries of the group structure." could you elaborate more detailed on this. I don't see how the gauge structure is related to the hierarchy of the different generations. – Katermickie Dec 06 '19 at 06:54
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    The meson octet is an example of how group symmetries appear, which do not depend on mass. In the Su(3)x(Su(2)xU(1) groups all elementary particles according to their quantum numbers will be in a group representation. When all masses are zero, the structure still exists, which will resolve in what we identify as electron and quark at our energy.. The quantum numbers are what make representations, and since these are the symmetries experiments give us, that is the way the correspondence is made.. https://en.wikipedia.org/wiki/Particle_physics_and_representation_theory . – anna v Dec 06 '19 at 07:41
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    I understand the negative votes as the divide between the basic philosophy of people on this blog. There are the "platonists" ; Theory forms reality and thus any explanations are in theory and the realists: *nature exists and theories are mathematical models describing nature" . This answer follows the second statement, stating "that is what is being observed and modeled" . – anna v Dec 07 '19 at 05:52
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    The model has niches where the observed particles can "axiomatically" be identified by their quantum numbers, and it happens, in the present standard model, that their masses form a higherarchy of the SU(2) group representations . Maybe this hierarchy in a higher SU(n) model could come out, and the axiomatic table would be "explained", but this is where we are at the moment, is what I am saying above. – anna v Dec 07 '19 at 05:55