# To what extent are lepton and quark generations tied in the Standard Model?

The Standard Model of particle physics splits both the leptons and the quarks into three generations, with mass and instability going up from the first to the third generation. These are normally displayed together, on the same rows or columns of the table of fundamental particles:

This makes some sense: each charged lepton is tied to its neutrino in most Feynman vertices it appears in, and the quarks are linked to each other by their charges, if nothing else. However, I can't think of any way in which the SM formally links, say, muons, and strange quarks. Is there some explicit link with generation-specific interactions? Or is it just a coincidence that there are three rungs in both ladders with increasing mass on both?

Is there some explicit link with generation-specific interactions? Or is it just a coincidence that there are three rungs in both ladders with increasing mass on both?

In the SM, there are no generation-specific interactions, beyond color, which factors out. The generation tabulation system simply arrays states according to increasing mass, a still mysterious pattern; which is why, to date, the neutrino mass eigenstates $$\nu_{1,2,3}$$ have not been assigned to generations, yet, pending complete experimental confirmation of the normal hierarchy/ordering (the $$\nu_{e,\mu,\tau}$$ are not mass eigenstates, and were on that chart to simply confuse and abuse; mercifully, they are going away). The increasing mass pattern is not a coincidence: it is the construction principle.

The weak interactions, through weak mixing, hop across all generations, and neutrinos mix a lot, unlike quarks, whose mixing angles are small. So scrambling green columns will do nothing to physics, as long as you take care to rewrite your PMNs matrix to reflect the labelling change. Of course you do need the number of states you see on such tables to be matched, 3 rungs to 3 rungs, whatever their order, so as to prevent gauge anomalies, invalidating gauge invariance, but the assignment of states in generations, the order of the rungs, is basically arbitrary.

In speculative GUTs (like the Georgi-Glashow SU(5)) one tries to link fermion masses (cf (22), since leptons and quarks are put in common representations, 5 and 10; but, again, alternate inequivalent models wiggle the leptons around associating them to different quarks, to adapt to negative proton decay results. So, indeed, such alternative speculative models take advantage of the SM freedom to scramble green columns.

Yes, they are very strongly linked. On top of the classification in generations, and on top of writing the classical field theory action for the Standard Model, the Standard Model should be a well behaved quantum theory. In particular, quantum gauge theories with chiral (left handed) fermions lead, in general, to appearance of gauge anomalies. If they appear, they render the original theory invalid, if considered as a quantum theory. It turns out, that one full generation -- neutrino-charged lepton-up and down quarks leads to cancellation of the anomaly, while partial generation do not, rendering the theory invalid as quantum theory. See discussion at stackexchange, or an example of a lecture on this topic.

• I fear that is not the OPs question: he is asking why the lepton doublet of the 2nd generation, for instance, is tied to the quark one, and not to the quark doublet of the 3rd one, and vice versa. Anomalies cancel within each generation, but there is no intrinsic "secondness" in the second one, for instance. People have simply put these generations together out of lepton and quarks historically, so therefore ordered in mass. Feb 18, 2017 at 14:36
• Hmm. In this form it is actually wrong to say that the generations are separte at all in SM. That is, in the mass basis both quarks and leptons (neutrinos) are mixed between generations. Moreover, the mixing is rather strong, especially for neutrinos. So, the ordering of the three lepton pairs and quark pairs is indeed arbitrary for pure SM, but the existence of three of both is not. Feb 18, 2017 at 15:00
• My point exactly. Because quark mixing is small, people could, and do!, talk about weak and mass eigenstates linked, sloppily. But because neutrino mixing is huge, talking about masses of $\nu_e$, etc is "not even wrong". Feb 18, 2017 at 15:13
• @Cosmas' first comments gets the essence of my question correctly. I was unaware of quark mixing, though. Is there a similar mass/flavour mixing of the charged leptons? Feb 19, 2017 at 16:18
• @Emilio Here is my own answer to the issue: they do mix, but you need to anchor to some mass eigenstates, so we call e,μ,τ mass eigenstates , and their weak isopartners are then mixtures of neutrino mass eigenstates. When we produce charged leptons out of a neutrino mass eigenstate, such mixed states which oscillate so rapidly on acount of their high masses that, by the time they hit the detector, they have decohered to mixtures of charged lepton mass eigenstates. Feb 19, 2017 at 16:42