Electroweak mixing and symmetry breaking

Take a scalar doublet with $Y=1/2$ and the convention $Q=T_3+Y$. In a convenient gauge write $H=(0,(v+h)/\sqrt2)^T$. The term $(D_\mu H)^\dagger D^\mu H$ then contains a quadratic neutral combination proportional to $(gW^3-g'B)^2$. Both couplings matter: a generic neutral gauge field is neither already a photon nor already a Z. The charged combinations of $W^1$ and $W^2$ acquire their own common mass.

The neutral matrix has zero determinant. Its null eigenvector defines the photon, while the orthogonal eigenvector defines the Z boson. This zero is not a numerical accident: the neutral Higgs expectation value is invariant under the generator $Q$. Local gauge symmetry is a redundancy of the description, so the usual phrase "symmetry breaking" should not be interpreted as directly observing a gauge-dependent vacuum orientation. Physical statements concern the spectrum and gauge-invariant observables.

Define $\tan\theta_W=g'/g$. The orthogonal rotation gives $A=\sin\theta_W W^3+\cos\theta_W B$ and $Z=\cos\theta_W W^3-\sin\theta_W B$. Substituting the photon combination into the covariant derivative makes its coupling proportional to $T_3+Y$, so $e=g\sin\theta_W=g'\cos\theta_W$. The angle aligns the mass and interaction descriptions; it is not a physical oscillation between photons and Z bosons in empty space.

At tree level $m_W=gv/2$ and $m_Z=v\sqrt{g^2+g'^2}/2$. Their ratio is $\cos\theta_W$ for this scalar representation. The neutral current also contains a coupling to $T_3-\sin^2\theta_W Q$, explaining why different chiral fermions interact differently with the Z. The simple mass relation is a useful organizing result, but adding different scalar representations or higher-dimensional interactions can modify the relation between the parameters and measured masses.

A complex scalar doublet contains four real field components. In the perturbative description, three would-be Goldstone components supply the longitudinal polarizations of the massive W and Z bosons, and one scalar degree of freedom remains as the Higgs excitation. This bookkeeping explains the spectrum without claiming that a particle literally swallows another observable particle. Gauge-fixing terms redistribute unphysical fields in intermediate calculations while leaving physical scattering probabilities consistent.

Beyond tree level, several useful definitions of a weak mixing angle coexist. An on-shell angle constructed from measured masses and a running subtraction-scheme angle need not have the same numerical value. Precision tests compare a complete specified prediction with correlated inputs and uncertainties, including loop contributions. A small discrepancy with a tree-level formula is therefore not immediately evidence for a new interaction; the parameter convention and the omitted Standard Model corrections must first be made explicit.