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With the specified scalar-dependent gauge fixing, the Faddeev-Popov determinant depends on the radial Higgs mode even though the gauge group is Abelian. Besides a ghost mass squared $\xi e^2{f}^2$, the Lagrangian density contains $-2\xi e^{2}fh\bar{c}c-\xi e^2{h}^2\bar{c}c$. These Faddeev-Popov ghosts do not decouple for generic $\xi$; their interactions vanish in the $\xi \to 0$ Landau gauge limit. Free ghosts in pure Abelian Lorenz gauge rely on a field-independent determinant, rather than merely on commutativity of the gauge group.

A Higgs mode is a fluctuation in the magnitude of a complex or multicomponent order parameter, as distinct from a phase or orientational fluctuation.

Put $r^2={\varphi }_i{\varphi }_i$. Completing the square in the scalar potential givesSince $\lambda >0$, its global minima form the vacuum manifoldHere $\mu$ denotes the positive square root of the parameter ${\mu }^2$. The origin is a local maximum, not a ground state. Choose $⟨\mathbf{\varphi }⟩=(v,0,0)$. Its stabilizer subgroup is the SO(2) subgroup of rotations in the $(2,3)$ plane, so the stated SO(3) global symmetry in field theory is spontaneously broken to SO(2).

Write ${\varphi }_1=v+h$, ${\varphi }_2={\pi }_2$, ${\varphi }_3={\pi }_3$. The kinetic terms stay canonically normalized, while the scalar mass matrix isEquivalently, the quadratic scalar potential is ${\mu }^2{h}^2$. Therefore the particle masses areThe two angular Goldstone bosons match $\dim SO(3)-\dim SO(2)=3-1=2$. The radial displacement changes $r$ and costs potential energy, giving the massive Higgs mode. These are tree-level masses; the Goldstone theorem protects the angular zero-mass modes in the exact global theory.

Choose the gauge covariant derivative $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$. Write $\rho =f+h$, $r=1+h/f$ and $B_{\mu }={\partial }_{\mu }\pi /f+e{A}_{\mu }$. The polar parametrization of the Abelian Higgs field givesHence the full Lagrangian density isAssume $f>0$ and $\lambda >0$, so the minimum has $|\varphi |=f/\sqrt{2}$. In this convention the local gauge symmetry acts asThe combination ${\partial }_{\mu }\pi +ef{A}_{\mu }$, the radial field $h$, and the gauge field strength are each invariant, proving invariance of the displayed Lagrangian density.

This parametrization follows the circle of classical minima: $h$ measures distance away from the vacuum manifold, whereas $\pi$ moves along a gauge orbit. It separates the massive radial Higgs mode from the would-be Goldstone boson and makes its removal by the Higgs mechanism transparent. At quadratic order,The polar parametrization of the Abelian Higgs field is a local description near the chosen nonzero vacuum expectation value; it is singular at $\varphi =0$ and is not a global coordinate system for configurations with zeros or winding.

Unitary gauge fixes the scalar phase to zero. Locally choose $\alpha =-\pi /f$ in part b; then ${\pi }^{\prime }=0$ and $A_{\mu }^{\prime }=A_{\mu }+{\partial }_{\mu }\pi /(ef)$. The remaining fields are the radial Higgs mode and a massive vector. Its quadratic Lagrangian density is $-F^2/4+m_A^2{A}^2/2$ and its gauge-boson propagator isFor $k^2\ne 0$, introduce the transverse and longitudinal orthogonal projectors $P_{\mu \nu }^T=g_{\mu \nu }-k_{\mu }{k}_{\nu }/k^2$ and $P_{\mu \nu }^L=k_{\mu }{k}_{\nu }/k^2$. Omitting the infinitesimal in this algebraic decomposition,At large momentum in Euclidean space, the transverse coefficient falls as $k^{-2}$ but the longitudinal coefficient approaches a constant of order $m_A^{-2}$. Thus power counting applied directly to unitary gauge Feynman diagrams suggests increasingly severe ultraviolet divergences. This hides the renormalizability of physical scattering amplitudes; it does not disprove it. A finite-$\xi$ R-xi gauge exposes the better propagator scaling while preserving the fields needed to track the cancellations.

Assume $\lambda >0$ and $v>0$. The vacuum manifold is the sphere $|\varphi |=v$. Choose ${\varphi }_0=v{e}_3$. For the given generators, $t^3{\varphi }_0=0$ whereas $t^1{\varphi }_0$ and $t^2{\varphi }_0$ are nonzero and independent. The Adjoint representation of SU(2) rotates this sphere, and the full connected stabilizer subgroup of $v{e}_3$ is generated by $t^3$. Hence the classical symmetry pattern isThe central element of SU(2) acts trivially on the triplet and belongs to this stabilizer; it does not add a separate broken direction. The gauge-field-free representative has zero gauge field strength and constant scalar magnitude. Locally near this nonzero classical vacuum, unitary gauge aligns the triplet along the third axis.

First make the gauge normalization explicit. Write the gauge kinetic term as $-\mathcal{N}{F}_{\mu \nu }^a{F}^{a\mu \nu }/4$, where $\mathrm{Tr}(t^a{t}^b)=\mathcal{N}{\delta }^{ab}$ if $F=F^a{t}^a$ is the matrix used inside the printed trace. Canonical normalization of a gauge kinetic term givesFor a conventionally normalized component trace, $\mathcal{N}=1$ and $g_c=g$. If the trace is the ordinary matrix trace in the displayed three-dimensional representation, $\mathrm{Tr}(t^a{t}^b)=2{\delta }^{ab}$ and $\mathcal{N}=2$. The PDF does not specify which trace convention is intended; both are covered by this formula.

In unitary gauge, the gauge covariant derivative isExpanding this kinetic term and the scalar potential gives the complete physical-field LagrangianHere a square of a vector means its contraction with the Minkowski metric, in signature $(+---)$. The labels 1 and 2 represent massive vectors; 3 represents the surviving Abelian vector.

To display their charge and all interactions more clearly, put $a_{\mu }={\mathcal{A}}_{\mu }^3$, $W_{\mu }^{\pm }=({\mathcal{A}}_{\mu }^1\mp i{\mathcal{A}}_{\mu }^2)/\sqrt{2}$, and defineWith the sign of gauge field strength printed in the paper, ${\mathcal{F}}^3=f+i{g}_{c}C$ and $G^{\pm }=({\mathcal{F}}^1\mp i{\mathcal{F}}^2)/\sqrt{2}$. Thus the same Lagrangian becomesThe physical masses areIn particular, literal adjoint matrix trace gives $m_W=gv/\sqrt{2}$; the standard canonical component convention gives $m_W=gv$. These are descriptions with differently normalized couplings, not different physical spectra.

The complex vector pair carries opposite charges under the unbroken U(1) gauge symmetry. The gauge field strength terms contain $a{W}^{+}{W}^{-}$ cubic interactions, $aa{W}^{+}{W}^{-}$ quartic interactions, and four-vector interactions involving the charged fields. There is no pure Abelian cubic or quartic self-interaction. The Higgs mode $\eta$ has cubic and quartic scalar interactions and couples through $2g_c^{2}v\eta W^{+}{W}^{-}+g_c^2{\eta }^2{W}^{+}{W}^{-}$. It is neutral and has no tree-level $\eta aa$ interaction. This is the physical charged-vector Lagrangian for an adjoint SU2 Higgs model.

If the angular fields ${\pi }_1$ and ${\pi }_2$ were retained, their vanishing potential masses would identify the two Goldstone bosons of the ungauged triplet. In the gauge theory they mix with the broken-direction gauge fields and can be removed by gauge fixing; the Higgs mechanism uses them as the longitudinal polarizations of the two massive vectors. They are not additional physical massless scalars. The physical degrees of freedom are conserved: $3+3\times 2=9$ before rearrangement, and $1+2\times 3+2=9$ afterwards.

This is not the Standard Model electroweak interaction. It has three original gauge bosons, leaving two massive charged vectors and one massless neutral vector, with no massive neutral Z boson. The Standard Model instead has $SU(2{)}_L\times U(1{)}_Y$, a complex Higgs doublet, and three massive vectors plus the photon. Its charge is $Q=T_3+Y$, not just the surviving $T_3$. Adding fermions cannot supply the missing gauge generator or turn the surviving neutral vector into both a photon and a Z boson. In particular, the usual right-handed fermions in the Standard Model are SU(2) singlets; they would have zero charge if only $T_3$ were available, instead of the charges produced by hypercharge.

The scalar potential can be written asIts vacuum manifold is the sphere ${\varphi }_a{\varphi }_a=v^2$. The Adjoint double cover from SU(2) to SO(3) rotates any nonzero vacuum expectation value to $(0,0,v)$, with $v>0$. A local unitary gauge removes the two angular fluctuations, leaving $\varphi =(0,0,v+\eta )$. This choice holds in a neighbourhood of the nonzero vacuum; a single global rotation cannot align arbitrary position-dependent fields, and topologically nontrivial configurations can obstruct a global unitary gauge.

The generator $t^3$ fixes the vacuum expectation value, while $t^1,t^2$ do not. Thus the Higgs mechanism in this SU(2) gauge theory with an adjoint Higgs field givesThe two angular Goldstone bosons become the longitudinal polarizations of two massive gauge bosons. Define physical fieldsThese labels describe the fields of this model; they do not identify it with the Standard Model. With the stated adjoint covariant derivative,To express all Yang-Mills theory terms in physical fields, introduce the gauge field strengthsThen $F_{\mu \nu }^3=f_{\mu \nu }+ig{Q}_{\mu \nu }$ and $(F_{\mu \nu }^1\mp i{F}_{\mu \nu }^2)/\sqrt{2}={\mathcal{W}}_{\mu \nu }^{\pm }$. Dropping the constant vacuum energy, the Lagrangian density isTaking the positive gauge coupling convention $g>0$, the canonically normalized mass terms giveThe residual U(1) gauge symmetry makes $W^{\pm }$ oppositely charged. The Yang-Mills theory terms supply interactions of $A$ with $W^{\pm }$ and four-vector interactions. The Higgs mode has cubic and quartic self-interactions, together with $2g^{2}v\eta W^{+}\cdot W^{-}$ and $g^2{\eta }^2{W}^{+}\cdot W^{-}$ couplings; it is neutral under the surviving U(1) gauge symmetry.

Coupling fermions permits a massless electromagnetic gauge boson and massive charged mediators of the weak interaction; fermion couplings with chirality can produce parity-violating weak charged currents. However, this model has no massive neutral Z boson, and its only charge generator is $t^3$. A fundamental doublet has opposite charges, so it cannot reproduce the observed doublet charge assignments through an independent hypercharge. The Standard Model instead has $SU(2{)}_L\times U(1{)}_Y\to U(1{)}_Q$, a complex Higgs doublet, three absorbed Goldstone bosons, a massive Z boson and a weak mixing angle.

The minima of the scalar potentialsatisfyThe SU(2) group acts transitively on this three-sphere of vacua, and the stabilizer of a nonzero fundamental doublet is trivial. A global transformation may therefore choose ${\varphi }_0=(0,v{)}^T/\sqrt{2}$. Because the symmetry is gauged, unitary gauge removes all three angular Goldstone bosons, leaving only the real radial Higgs mode $h$:Substitution into the gauge-covariant kinetic term giveswithThe interaction terms produce $h^3$, $h^4$, $hBB$, $hhBB$, and the cubic and quartic non-Abelian gauge-boson vertices shown below.

The three broken generators supply the three longitudinal polarizations of the equally massive gauge bosons. No physical massless Goldstone boson remains, and because the unbroken subgroup is trivial there is no massless gauge boson either. The remaining physical spectrum has $3\times 3+1=10$ degrees of freedom, equal to the original $3\times 2+4=10$.

A neutral superfluid can select a phase of its complex order parameter, breaking a global $U(1)$ symmetry. A spatially uniform change of that phase costs no energy. Slowly varying changes therefore produce a gapless Goldstone mode. For a dilute weakly repulsive Bose gas, the Bogoliubov quasiparticle dispersion isso $E_k\simeq \hslash c_{s}k$ at long wavelengths, with $c_s=\sqrt{g_B{n}_0/m_B}$. Density and phase fluctuations are canonically coupled components of this sound branch, rather than two independent positive-frequency modes. Superfluid helium and dilute atomic condensates supply neutral examples, although the dilute-gas formula is not quantitatively valid for strongly interacting liquid helium.

In a superconductor, the pair field is charged and its phase couples to the electromagnetic gauge field. Only combinations such as $\nabla \theta -q\mathbf{A}/\hslash$ are physically meaningful. The Anderson-Higgs mechanism in a superconductor produces electromagnetic screening and removes the neutral sound mode as an independent gapless charged-density oscillation. In a homogeneous three-dimensional continuum, long-range Coulomb forces give a longitudinal plasma gap, approximately ${\omega }^2={\omega }_p^2+c_s^2{k}^2$ at small $k$. The static transverse response gives the Meissner effect andwhen $n_s$ counts superconducting electrons of effective mass $m$. Counting pairs instead uses density $n_s/2$, charge $-2e$ and mass $2m$, giving the same stiffness. The condensate amplitude may also have a gapped Higgs mode, but that amplitude oscillation is distinct from the electromagnetic Higgs mechanism. Local gauge transformations remain a redundancy; this is not spontaneous breaking of an observable local symmetry in the same sense as the neutral global phase symmetry.

Let $v=\sqrt{-\ln \lambda /\lambda }>0$ and write $\varphi =v+\chi$. ThenSince $V^{\prime }(v^2)=0$ and $V^{\prime \prime }(v^2)=\lambda$, the quadratic potential isThereforeThe constant can be dropped from the fluctuation equations. For canonical real fields, use $\chi =(h+i\pi )/\sqrt{2}$:These are the radial and angular masses of a phase-invariant scalar potential. The radial Higgs mode changes the field magnitude and costs energy to second order. The angular Goldstone boson changes the phase along the vacuum circle without a quadratic restoring force. Its linear dispersion is ${\omega }^2=|\mathbf{k}{|}^2$, whereas the radial mode has ${\omega }^2=|\mathbf{k}{|}^2+m_h^2$. The zero angular mass follows the continuous broken global symmetry, not from an accidental cancellation in a particular coordinate choice.

For a real triplet potential $\lambda (|\varphi {|}^2-v^2{)}^2/8$ and canonical gauge coupling $g_c$, unitary gauge gives one neutral Higgs mode, two oppositely charged massive vectors and one massless Abelian vector. The scalar kinetic contribution is $\frac{1}{2}(\partial \eta {)}^2+g_c^2(v+\eta {)}^2{W}^{+}{W}^{-}$, and its potential is $\lambda v^2{\eta }^2/2+\lambda v{\eta }^3/2+\lambda {\eta }^4/8$. This potential convention is distinct from using $\lambda (|\varphi {|}^2-v^2{)}^2/4$.

Near a nonzero vacuum expectation value, the radial variable $h$ describes the Higgs mode and the angular variable $\pi$ describes the would-be Goldstone boson. For $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$, their local gauge transformation is $h\mapsto h$, $\pi \mapsto \pi +f\alpha$, and $A_{\mu }\mapsto A_{\mu }-{\partial }_{\mu }\alpha /e$. The scalar kinetic term is $(\partial h{)}^2/2+(1+h/f{)}^2(\partial \pi +efA{)}^2/2$. These coordinates simplify gauge orbits but obscure ordinary power counting through higher-dimensional derivative vertices and fail at $\varphi =0$.