Quark family/generation quantum numbers

Question

Shouldn't strangeness and charm be essentially the same quantum number, the eigenvalue of the family operator for family #2 of the quarks, and similarly for top and bottom, for family #3? It appears that the differing names are just a convenience in distinguishing them due to the widely differing masses between the two different weak isospin states in the same family. Family number should be the same (for the particle, and opposite for the anti-particles, of course) for both isospin projection states -- it's the weak isospin z-projection eigenvalue that distinguishes them in a complete eigenvalue set. All the other eigenvalues should be the same. The Particle Data Group doesn't seem to think so though, since their tables distinguish charm from strangeness and bottomness from topness.

Answer 1

The PDG does things just right: there is no other way. Your scheme might have some arguable merit in the absence of weak mixing, the 500 lb gorilla in the room, but, as it stands, it is meaningless.

It appears that the differing names are just a convenience in distinguishing them due to the widely differing masses between the two different weak isospin states in the same family. Family number should be the same (for the particle, and opposite for the anti-particles, of course) for both isospin projection states -- it's the weak isospin z-projection eigenvalue that distinguishes them in a complete eigenvalue set.

The basis of your error is misappreciation of the differing roles of flavor (mass eigenstates) and weak isospin states. The six flavors used, upness, down-ness, charm, strangeness, top and beauty, correspond to mass eigenstates, the six quarks we know. In the lepton sector, the three charged leptons and the three neutrino mass eigenstates, (hopefully) temporarily dubbed 1, 2, and 3.

By the sharpest of contrasts, the members of the weak isodoublets are distinctly not these flavor eigenstates; in your case, people have chosen the convention of parameterizing the quark doublets as (c,s'), where s' is a linear combination of all three generations' down quarks, d,s,b. Because the mixing in the quark sector is de facto small, one pretends that $s'\approx s\cos\theta -d\sin\theta \approx s$, where θ is the largest mixing angle, the "Cabbibo" one, so one blinks and thinks of the weak isodoublet as being (c,s), confident that this reversible metaphor would not impede their thinking at the end of the day.

But, at the end of the day, one also needs the 6 flavor names to peg down the real propagating degrees of freedom, and contrast them to the weak isodoublet members to describe this mixing in the first place.

This easy confusion is completely avoidable in the lepton sector, where the mixing is big, so as to make assignment of the neutrinos to generations ambiguous, or arbitrary. That is, before the ordering/hierarchy of the neutrino masses is confirmed experimentally, one does not even dare assign them to generations firmly. The convention is you just throw them into the 3 generations in increasing mass order, presently unconfirmed. But if you have marveled at the supreme confusion of popsicle graphs or the centrality of barycentric coordinates for them you appreciate the sheer definitional slop involved.