= Solution
The <gauge fields> require their own <Yang-Mills Lagrangian>, $-W^a_{\mu\nu}W^{a\mu\nu}/4$. The non-Abelian field strength contains a term quadratic in the gauge potentials, so squaring it produces \b[three-vector and four-vector self-interaction vertices], with <derivative> <momentum> factors in the cubic vertex and SU(2) structure constants in both. They exist in either scalar phase and must be added to the scalar-generated vertices. A gauge-fixed quantum calculation also has gauge-fixing and, where applicable, ghost terms; those are not extra physical <boson> species.
For $\mu^2>0$, the global phase transformation $\Phi\mapsto e^{i\alpha}\Phi$ leaves both the scalar potential and the covariant <kinetic term> unchanged while leaving $W_j$ fixed. Its <Noether charge> counts scalar particles minus <antiparticles>. Thus $\phi_a\phi_b\to\phi_a\phi_b$ and $\phi_a\phi_a^*\to W_3W_3$ are compatible with this charge, whereas $\phi_a\phi_b\to W_jW_k$ is forbidden: its initial charge is two and final charge zero. Charge conservation is necessary, not sufficient, for a particular amplitude to be nonzero; other <quantum numbers> and kinematics still apply.
After adding the <hypercharge> field, the neutral-vector <mass> term becomes
$$
\frac{v^2}{8}(gW_3-g'B)^2.
$$
Its <mass> matrix has zero determinant. With $G=\sqrt{g^2+g'^2}$, the orthonormal combinations are
$$
\boxed{A=\frac{g'W_3+gB}{G},\quad m_A=0;\qquad
Z=\frac{gW_3-g'B}{G},\quad m_Z=Gv/2.}
$$
The unbroken generator is <electric charge>, $Q=T_3+Y/2$, because the lower doublet component has $T_3=-1/2$ and $Y=1$. The charged combinations $W^\pm=(W_1\mp iW_2)/\sqrt2$ keep <mass> $gv/2$.
The observed charged <weak interaction> couples left-chiral <fermions> within SU(2) doublets. In the <Standard Model>, $Q_L=(u_L,d_L)$ and $L_L=(\nu_L,e_L)$ are doublets, while $u_R,d_R,e_R$ are singlets. A vectorlike assignment would also give right-handed charged weak currents, contrary to this observed chiral structure. Once the measured <electric charges> are specified, $Y=2(Q-T_3)$ fixes
$$
\begin{array}{c|rrrrrr}
\text{multiplet}&Q_L&u_R&d_R&L_L&e_R&\Phi\\\hline
Y&1/3&4/3&-2/3&-1&-2&1
\end{array}.
$$
An optional <right-handed neutrino> is a singlet with $Y=0$. Conjugate left-chiral fields have the opposite <hypercharges> to these right-chiral fields. These assignments permit gauge-invariant <Yukawa interactions>; they are not obtained by assigning one common <hypercharge> to every <chirality>.
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