What are the weak hypercharge and weak isospin quantum numbers of the helicity states of the $W^\pm$ and $Z^0$ bosons?
The W boson is a spin-1 massless particle. Consequently it has three helicity states, negative (left-handed), positive (right-handed), and neutral. As discussed in this paper, these states restrict which orientations of quarks they can interact with:
The emitted $b$ quark can be regarded as massless compared to the top quark, and hence expected to be predominantly of negative helicity (left-handed), meaning that its spin points opposite to its line of flight. The emitted $W$ boson, being a massive spin-1 particle, can assume any of three helicities: one longitudinal ($W_0$) and two transverse states ($W_−$, left-handed and $W_+$, right-handed)
Int.J.Mod.Phys.A24:2899-3037,(2009), Marc-André Pleier, Review of Properties of the Top Quark from Measurements at the Tevatron
The helicity states of a particle like the electron act very much differently, and they carry different weak hypercharge and/or weak isospin quantum numbers. So really this is two questions. First, are weak hypercharge and weak isospin good quantum numbers for the helicity states of the W and Z bosons? And if they are good states, what are the quantum numbers?