Higgs doublet vacuum and gauge-boson masses

Choose a potential with a negative quadratic coefficient and a positive quartic coefficient. Its minimum fixes the Higgs norm, but gauge-equivalent vacuum orientations describe the same physics. In a convenient orientation the lower neutral component has expectation value v divided by square root of two. The canonically normalized real fluctuation h enters with the same divisor. Confusing v with the component expectation value changes vector masses by a square-root-of-two factor.

Expanding the potential around this minimum gives the radial scalar mass squared equal to twice lambda times v squared. The doublet contains four real components. In perturbation theory three would-be Goldstone modes account for the additional longitudinal polarizations of the charged W pair and Z, leaving one physical scalar excitation. This is degree-of-freedom bookkeeping; local gauge symmetry is a redundancy and its orientation is not itself an observable that spontaneously selects a detector direction.

Act with the covariant derivative on the vacuum. The charged weak generators rotate its lower component into the upper component, generating mass terms for W-one and W-two. Their complex charged combinations have a mass term of the form mass squared times W-plus times W-minus. The neutral generators instead act on the same lower component, producing the square of g times W-three minus g-prime times B.

For real neutral fields, compare the quadratic expression with one half of the row vector times the mass-squared matrix times the column vector. This extra one half is required when reading eigenvalues. The neutral matrix has determinant zero and positive trace, hence one zero eigenvalue and one positive eigenvalue. A null eigenvector proportional to the couplings in reversed order identifies the photon; the orthogonal combination becomes Z. Diagonalizing the matrix explains the zero without adjusting a photon mass by hand.

The vacuum's lower component has electric charge zero because its T-three and hypercharge eigenvalues cancel. Therefore the electromagnetic generator annihilates the vacuum and its gauge-field direction receives no mass from this doublet. The statement links the scalar representation, its vacuum, and the neutral null eigenvector. A different scalar representation or additional expectation values can alter vector-mass relations, so the result is not a property of every imaginable symmetry-breaking sector.

These tree-level masses describe parameters in the leading Lagrangian. Precision comparisons use pole definitions, radiative corrections, and an input scheme, with unstable-vector line shapes treated consistently. Increasing v at fixed dimensionless couplings rescales these leading masses, but is a theoretical parameter variation rather than an experimental procedure that holds every measured electroweak input fixed. The same distinction applies to the illustrative scalar mass: its quartic coupling is not predicted merely by knowing the gauge group.