# Is there a weak hypercharge : spin-3/2 analogy?

So I was not trained as a particle physicist, but I left a table of particles on a post on Hacker News based on what little I do understand about the conservation of weak hypercharge and weak isospin.

Originally I wanted to pedagogically simplify this table by mapping $$Y_\text w\mapsto \frac34 \left(Y_\text w+\frac13\right)$$ because then an electron and a down quark might be mapped to $$\pm1/2$$ which would have been symmetric with the allowed values for $$T_3$$. Filling out the table I realized that this sort of affine mapping is an unqualified mistake: it does not preserve the antimatter property that antimatter should have opposite quantum numbers. So I abandoned the effort and just stated the usual numbers by fiat: the fundamental fermions of the Standard Model have a weak hypercharge in the set $$\{-1,-\frac13,+\frac13,+1\}.$$

Now that looks arbitrary but it has a very interesting property which is that if we restrict ourselves to linear rather than affine transforms, and multiply by $$3/2$$ rather than $$3/4$$, that looks like the spectrum of a spin-3/2 particle: in other words the nearest neighbors are all separated by a fixed distance $$2/3$$ that we can map to $$1$$ without making this mistake of mine. Now I know that this screws up a certain electroweak mixing angle but I am not clear on whether that is a physical thing or just a convenient way to talk about a dimensionless number.

I was wondering if this was known to be more than just a coincidence, or else known to be a coincidence: in other words, is there an important reason that we use the set $$\{-1,-\frac13,+\frac13,+1\}$$ rather than $$\{-\frac32,-\frac12,+\frac12,+\frac32\}.$$

• I know the current understanding is that these are elementary particles, but does this in any way suggest that the electron and the quark are made up of the same constituents (if they would be one day found to be non elementary)? – Árpád Szendrei Mar 29 '19 at 20:47
• I mean that's extremely hard to say off the top of my head and I think it would require a bigger theory than just the electroweak model is going after, but yes, if you were to think about a model which coupled two hypercharge "spins" irrespective of weak isospins then you might have two $Y=1/2$ spins, say an up and a down quark, and one of them might steal the other's angular momentum to become $Y=-1/2$ and $Y=3/2$ particles, which would be a proton decay channel ($uud\mapsto u\bar u e^+$ or $uud\mapsto u\bar d\bar \nu_e$). – CR Drost Mar 29 '19 at 21:15
• Where did you get the set $\{-1,-1/3,1/3,1\}$ for hypercharge? – user178876 Mar 29 '19 at 22:53
• The normalization of hypercharge is purely convention unless you try to embed hypercharge in a larger group, as it happens in grand unified theories. So if your charges were the correct hypercharges, both choices were possible. You can rescale the standard hypercharges $\{1/6,2/3,-1/3,-1/2,-1\}$ by $3/2$ to get $\{1/4,1,-1/2,-3/4,-3/2\}$ if you feel like it and can absorb the change in a redefinition of the hypercharge coupling strength but why would you want to do that? Of course, if you embed hypercharge in SO(10), there is some connection to what you have in mind. – user178876 Mar 30 '19 at 0:45
• @marmot within one generation, 4 particles and 4 antiparticles realize all combinations of $\{-1/2,1/2\}\times\{-1,-1/3,1/3,1\}.$ I think I am only considering left-handed particles? – CR Drost Mar 30 '19 at 2:05