The Higgs Doublet and Electroweak Symmetry Breaking

The Standard Model Higgs sector uses one complex weak doublet with $Y=\tfrac12$ and a renormalizable potential whose minima form a gauge orbit. Choosing a representative makes perturbation theory transparent: three angular fields supply the longitudinal polarizations of $W^\pm$ and $Z$, one radial scalar remains, and the generator $Q=T_3+Y$ leaves the representative invariant. Physical symmetry-breaking statements are the resulting spectrum and couplings, not the gauge-variant value of $H$ itself.

Required background. Electroweak gauge and matter structure fixes $Y_H$ and the covariant derivative. Elitzur’s theorem and the gauge-invariant Higgs mechanism supplies the distinction between gauge choice and physical phase information.

Helpful background. What an interacting Lagrangian does not specify supplies the stability and quantum-definition qualifications for an interacting scalar potential.

The doublet potential and its minima

With the chapter convention,

$$H=\begin{pmatrix}H^+\\H^0\end{pmatrix}, \qquad D_\mu H=\left(\partial_\mu-i gW_\mu^a\frac{\tau^a}{2} -i g'\frac12B_\mu\right)H,$$

and the tree potential is

$$V(H)=-m^2H^\dagger H+\lambda(H^\dagger H)^2, \qquad m^2>0,\quad\lambda>0.$$

Write $r^2=2H^\dagger H$. Then

$$V(r)=-\frac12m^2r^2+\frac14\lambda r^4, \qquad \frac{dV}{dr}=r(-m^2+\lambda r^2).$$

The nonzero minima satisfy

$$r^2=v^2=\frac{m^2}{\lambda}, \qquad H^\dagger H=\frac{v^2}{2}.$$

$H^\dagger H$ is gauge invariant; an orientation in the doublet space is not. A convenient local parameterization is

$$H(x)= \exp\!\left[\frac{i\xi^a(x)\tau^a}{v}\right] \frac1{\sqrt2} \begin{pmatrix}0\\v+h(x)\end{pmatrix}.$$

The normalization of $\xi^a$ can be changed by field redefinitions; canonical normalization is checked after expanding the kinetic term. In unitary gauge the exponential is removed locally. In an $R_\xi$ gauge the angular fields remain in propagators and acquire gauge-parameter-dependent masses. Neither description changes a physical pole or S-matrix element.

The electroweak doublet construction and its perturbative expansion are given in Schwartz 2014, §29.1, pp. 584–88; the appearance of a massive vector and a physical scalar in a relativistic gauge model was identified in Higgs 1964, pp. 508–9.

The unbroken electromagnetic generator

Act on the chosen representative $H_0=(0,v/\sqrt2)^{\mathsf T}$. Its lower component has $T_3=-\tfrac12$ and $Y=+\tfrac12$, so

$$QH_0=(T_3+Y)H_0=0.$$

Therefore the subgroup generated by $Q$ leaves the representative fixed. The three independent broken directions may be represented by $T^1$, $T^2$, and the neutral combination orthogonal to $Q$. They correspond to the longitudinal components of two charged massive vectors and one neutral massive vector. The $Q$ gauge field remains massless at tree level.

This statement is stronger than counting generators: the next page verifies it by diagonalizing the exact neutral mass matrix and finding its zero eigenvector.

Scalar and vector degrees of freedom

Before selecting a representative, the complex doublet has four real field components and four massless electroweak gauge fields carry $4\times2$ physical polarizations. After the Higgs phase is described perturbatively,

Sector	Before	After
scalar	four real components	one physical radial scalar
charged vectors	two massless combinations, four polarizations	$W^+$ and $W^-$, six polarizations
neutral vectors	two massless fields, four polarizations	one massive $Z$ plus one massless photon, five polarizations

The totals agree: twelve physical degrees of freedom on either side. “Three Goldstones are eaten” is shorthand for the gauge-fixed rearrangement that supplies the third polarization of each massive vector; it is not literal disappearance of physical particles from a gauge-invariant Hilbert space.

Expanding the potential at the minimum gives

$$V(h)=V_0+\frac12(2\lambda v^2)h^2 +\lambda vh^3+\frac{\lambda}{4}h^4,$$

so at tree level

$$m_h^2=2\lambda v^2=2m^2.$$

The vanishing linear term is the tree tadpole condition. Beyond tree level, $v$, $m^2$, $\lambda$, and a renormalized tadpole condition depend on the declared renormalization prescription; a physical Higgs pole is not obtained by inserting loop-corrected numbers into this tree identity without counterterms.

Gauge-invariant interpretation and checks

The perturbative representative $H_0$ is useful because weak coupling and gauge fixing expose the particle spectrum. A gauge-invariant formulation instead characterizes the phase through operators such as $H^\dagger H$, gauge-invariant composites, screened charges, and the poles and residues of physical amplitudes. These descriptions agree in their shared perturbative domain, while Elitzur’s theorem forbids interpreting an unfixed local gauge-variant expectation value as an order parameter.

Four checks make the construction auditable:

1. Stationarity: $dV/dr=0$ at $r=v$ and $d^2V/dr^2=2\lambda v^2>0$.
2. Unbroken charge: $QH_0=0$ exactly.
3. Mode count: three angular directions plus one radial direction become three vector longitudinal modes plus one scalar.
4. Gauge independence: pole positions and physical amplitudes are independent of the $R_\xi$ parameter even though Goldstone, ghost, and intermediate Green functions are not.

The construction assumes the single-doublet renormalizable potential. Additional scalar representations can change the vacuum manifold, the tree-level $\rho$ relation, custodial structure, and the number of physical scalars.

Common pitfalls

Calling gauge symmetry physically broken. A gauge redundancy is not a global symmetry acting on distinct physical states. The convenient representative breaks the manifest redundancy; the physical content is a Higgs phase with a particular spectrum and screening structure.

Minimizing only one component from the start. First minimize the invariant $H^\dagger H$, then choose an orbit representative. Otherwise the unbroken generator and mode count can be hidden by the gauge choice.

Using the tree minimum as a loop input. A loop calculation needs a tadpole prescription and a renormalized input scheme. Mixing a loop pole mass with bare or tree $v$ and $\lambda$ is inconsistent.

Handoff

The mass and interaction derivations receive

$$\left\{H\sim(\mathbf2)_{1/2},\ V=-m^2H^\dagger H+\lambda(H^\dagger H)^2, \ v^2=m^2/\lambda,\ QH_0=0,\ \text{gauge/tadpole convention}\right\}.$$

Continue to the charged and neutral gauge-boson mass matrices or, for the radial vertices, to Higgs self-interactions.

References

• Higgs, Peter W. “Broken Symmetries and the Masses of Gauge Bosons.” Physical Review Letters 13, no. 16 (1964): 508–9. DOI.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, §29.1, pp. 584–88. DOI.