Is weak hypercharge carried by $Z$ boson?

Question

In the PBS spacetime video about charge (https://www.youtube.com/watch?v=esayi49OAk4), at 10:55 he said that weak hypercharge is carried by the $Z$ boson. Is this accurate?

Answer 1

The lip-smacker misleads you by misusing "carried" for "coupling", as he did for the charged current and gauge bosons: Z couples to weak hypercharge, among other things, but does not carry it, having weak hypercharge 0.

• The Z does not carry weak hypercharge, or charge, just as the photon (its group-theoretic brother) doesn't.

This is evident from weak mixing, since they, γ & Z, are a rearrangement of the weak hypercharge gauge field, B, and the $T_3$ one, $W^0$. Since these operators commute, the respective gauge fields are neutral w.r.t. them, and the eigenvalue of $W^0$ under $T_3$ is 0, while B is Abelian, so neutral, just as the photon is such, under the U(1) of charge. So any combination thereof, like γ & Z, also has to be neutral!

Of course, γ & Z do couple to Q and Q & Y respectively, as this is their function in the model. Geeky : If you have access to a good SM introductory book, you'd see that the Z couples to both currents that generate the charges of the weak Gell-Mann Nishijima relation, $Q=T_3+Y/2$, $$ {e\over \cos\theta_W\sin\theta_W}Z^\mu (\cos\theta_W^2 J_\mu ^3- \sin^2\theta_W J^Y_\mu /2). $$ The hypercharge current, $J^Y_\mu$, has bizarre and unmemorable parity/chirality properties, and that is why it is usually supplanted by a linear combination of the isospin current, $J_\mu^3$, maximally parity violating, and the EM current, $J_\mu^{em}$, parity preserving. Don't worry about them for now...