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For a Higgs doublet with hypercharge $1/2$ and vacuum expectation value $(0,v/\sqrt{2}{)}^T$, its gauge-covariant kinetic term gives the neutral gauge-boson mass matrixin the $(W^3,B)$ basis. Its eigenvalues are zero and $(g^2+g^{\prime 2})v^2/4$, and the Weinberg angle rotation gives the photon and Z boson. The two charged electroweak gauge bosons have $m_W^2=g^2{v}^2/4$. The zero eigenvalue expresses the unbroken electric-charge gauge symmetry.

With positive canonical kinetic terms and nonzero coupling $g$, expansion of the covariant scalar kinetic term gives the displayed gauge-boson mass matrix. It is a Gram matrix: for real $c_a$, $c^T{M}_A^{2}c=g^2|{\sum }_a{c}_a{T}_a{\varphi }_0{|}^2$. Its kernel is the Lie algebra of the stabilizer subgroup of the vacuum. Thus its rank is $\dim G-\dim H$, and exactly that many gauge-field combinations acquire nonzero squared masses. Removal of these gauge-orbit directions does not remove accidental zero eigenvalues of the scalar mass matrix on the transverse complement.

This cannot be the sole symmetry-breaking sector of the Standard Model. Its gauge-boson mass matrix has two zero eigenvalues, so besides the electromagnetic photon it predicts an extra massless neutral gauge boson. Every linear combination of $W^3$ and $B$, including the combination normally called $Z$, is massless. The observed massive Z boson is a concrete contradiction. Choosing $g,g^{\prime },m,\lambda$ differently cannot change these two zero eigenvalues.

A real scalar triplet can occur as an additional sector, but a further vacuum expectation value, for example that of a Higgs doublet, is then necessary to obtain the observed neutral gauge boson mass pattern. This does not rescue the triplet-only model posed here.

Let $g$ and $g^{\prime }$ be the two gauge couplings. In unitary gauge, $\Phi =(0,0,v+h{)}^T$ and the weak hypercharge term in the gauge covariant derivative vanishes. Using $(T^a{)}_{bc}=-i{ϵ}_{abc}$, the kinetic term becomesThus, in the basis $(W^1,W^2,W^3,B)$, the gauge-boson mass matrix and its eigenvalues areHere $W^{\pm }=(W^1\mp i{W}^2)/\sqrt{2}$. The two massless neutral gauge bosons may be rotated to any orthonormal basis; no neutral mass term selects a unique mixing angle.

Expanding the same kinetic term to first order in $h$ givesFunctional differentiation gives the Feynman rulesand, in the real basis, $h{W}_{\mu }^a{W}_{\nu }^b:2i{g}^{2}v{\delta }^{ab}{g}_{\mu \nu }$ for $a,b\in \{1,2\}$. All such three-point Feynman vertices involving $W^3$ or $B$ vanish. The factor two in the real basis comes from the two identical gauge boson fields; in the charged basis it is already explicit in ${\mathcal{L}}_{hVV}$.

A stable symmetry-breaking minimum requires $\lambda >0$ and $m^2<0$. Put $v^2=-m^2/\lambda >0$. The common scalar potential can then be written, up to a constant, asIts minima form the sphere $\Phi \cdot \Phi =v^2$. Choose the vacuum direction ${\Phi }_0=(0,0,v{)}^T$. If instead $m^2>0$ with positive $\lambda$, the minimum is at the origin and the assumed nonzero vacuum expectation value is not a stable vacuum of these models.

In the ungauged theory, the internal symmetry is global $O(3)$; its connected part is $SO(3)$. Constant orthogonal transformations preserve both the kinetic term and the potential. The chosen vacuum is invariant under $O(2)$ acting on its first two components, so the continuous breaking is $SO(3)\to SO(2)$ with two broken generators. The vacuum orientations are genuinely different degenerate global vacua.

The global triplet scalar symmetry breaking is visible directly in the fluctuation masses. Write $\Phi =({\pi }_1,{\pi }_2,v+h{)}^T$. The potential Hessian isThe quadratic Lagrangian is thereforeThere is one radial scalar of mass $m_h^2=2\lambda v^2=-2m^2$ and two massless Goldstone bosons. The two Goldstone modes describe motion tangential to the vacuum sphere. They remain physical, since a global rotation has only constant parameters and cannot remove arbitrary spacetime-dependent fluctuations. All three original real-scalar degrees of freedom remain. Interactions persist: the shifted potential is $V-V_{\min }=\frac{\lambda }{4}(2vh+h^2+{\pi }_1^2+{\pi }_2^2{)}^2$, containing cubic and quartic terms. Masslessness of the angular modes is protected by the exact continuous symmetry through Goldstone's theorem.

In the gauged theory, the three-component real field is the vector representation of local $SO(3)$, equivalently the Adjoint representation of $SU(2)$. The center of $SU(2)$ acts trivially on this field, so both descriptions have the same local field content and perturbative spectrum. Take $(t^a{)}_{bc}=-i{ϵ}^{abc}$, for which $[t^a,t^b]=i{ϵ}^{abc}{t}^c$. The covariant kinetic term and the gauge kinetic term are invariant under local transformations with the associated connection transformation. The scalar inversion $\Phi \mapsto -\Phi$, with the gauge field unchanged, is also a discrete invariance of the displayed action.

The vacuum leaves rotations about its third axis unbroken. The residual gauge group is $SO(2)$, or the corresponding $U(1)$ subgroup in the $SU(2)$ description. Expanding the covariant kinetic term about ${\Phi }_0$ givesThus the gauge-boson mass matrix is $g^2{v}^2\mathrm{diag}(1,1,0)$. The third generator annihilates the vacuum, while the first two do not. The masses are $m_{B^1}=m_{B^2}=gv$, $m_{B^3}=0$, and $m_h=\sqrt{2\lambda }v$.

The quadratic derivative mixing displays what happens to the Goldstone fields. With the generator convention just chosen,Local rotations can set the two angular fields to zero in unitary gauge. Then $\Phi =(0,0,v+h{)}^T$ andThe two Goldstone modes supply the longitudinal polarizations of the two massive vector bosons. They do not survive as additional physical massless scalars. The radial scalar is still physical, and $B^3$ retains only two transverse polarizations. This adjoint triplet Higgs spectrum realizes the Higgs mechanism; the gauge group is only partially broken, so one massless gauge boson remains.

The degree-of-freedom check isIn contrast, the global theory has just the three scalar modes, including its two physical Goldstone bosons. A vacuum expectation value in the local theory is a choice of gauge-fixed description: the direction can be changed by gauge transformations and is not itself an observable. Gauge-independent masses and the physical degree count express the actual effect. Both theories retain Lorentz symmetry in the constant scalar vacuum.

Take ${\tau }^a={\sigma }^a/2$, $Y_{\varphi }=1/2$, metric $g_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$, and a Higgs potential with ${\mu }_h^2>0$, $\lambda >0$. The gauge-scalar electroweak interaction isHere $B_{\mu \nu }={\partial }_{\mu }{B}_{\nu }-{\partial }_{\nu }{B}_{\mu }$. With the printed plus-sign convention for $D_{\mu }$, define $[D_{\mu },D_{\nu }]=ig{W}_{\mu \nu }^a{\tau }^a+i{g}^{\prime }{Y}_{\varphi }{B}_{\mu \nu }$; thus $W_{\mu \nu }^a={\partial }_{\mu }{W}_{\nu }^a-{\partial }_{\nu }{W}_{\mu }^a-g{ϵ}^{abc}{W}_{\mu }^b{W}_{\nu }^c$. The minus sign in the nonlinear gauge field strength follows from this convention.

The minima have ${\varphi }^{†}\varphi =v^2/2$, with $v^2={\mu }_h^2/\lambda$. A gauge transformation rotates the vacuum to $(0,v{)}^T/\sqrt{2}$. In unitary gauge, the three angular Goldstone bosons are removed, leaving $\varphi =(0,v+H{)}^T/\sqrt{2}$ and one real Higgs boson. The generator $Q=T^3+Y$ annihilates the vacuum, because its lower component has $T^3=-1/2$ and hypercharge $1/2$. This identifies the unbroken electromagnetic gauge group.

The quadratic terms from the Higgs field kinetic term areDefine the Weinberg angle and physical fields byThe neutral combinations areThe neutral gauge-boson mass matrix is $(v^2/4)\left(\begin{array}{cc}{g}^2& -g{g}^{\prime }\\ -g{g}^{\prime }& g^{\prime 2}\end{array}\right)$. It has eigenvalues zero and $(g^2+g^{\prime 2})v^2/4$, with the zero eigenvector giving $A_{\mu }$. Expanding the Higgs potential about its minimum gives $V=\text{constant}+\lambda v^2{H}^2+\lambda v{H}^3+\lambda H^4/4$. Therefore the tree-level electroweak gauge-boson masses and scalar mass areAlso $m_W=m_Z\cos {\theta }_W$ and $e=g\sin {\theta }_W=g^{\prime }\cos {\theta }_W$. The three removed Goldstone bosons supply the longitudinal polarization vector degrees of freedom of the massive electroweak gauge bosons.

Replacing $v^2$ by $(v+H{)}^2$ in the neutral mass term gives all the Higgs boson couplings to Z bosons in unitary gauge:There are exactly the cubic $HZZ$ and quartic $HHZZ$ elementary tree-level Feynman diagrams. Differentiating with respect to the identical fields gives Feynman rules $2i{m}_Z^2{g}_{\mu \nu }/v$ and $2i{m}_Z^2{g}_{\mu \nu }/v^2$, respectively. There is no elementary $ZHH$ vertex for the real neutral radial field.

For the Higgs doublet vacuum $(0,v{)}^T/\sqrt{2}$, the neutral gauge-boson mass matrix in $(W^3,B)$ is $(v^2/4)\left(\begin{array}{cc}{g}^2& -g{g}^{\prime }\\ -g{g}^{\prime }& g^{\prime 2}\end{array}\right)$. Its zero eigenvector is proportional to $(g^{\prime },g)$, the photon; the perpendicular vector is the Z boson. The charged fields $W^{\pm }=(W^1\mp i{W}^2)/\sqrt{2}$ have the stated W boson mass. Thus $m_W=m_Z\cos {\theta }_W$, where ${\theta }_W$ is the Weinberg angle. The unbroken charge generator annihilates the vacuum and explains the zero eigenvalue.

A real scalar triplet of weak hypercharge zero with vacuum expectation value $(0,0,v)$ breaks $T^1,T^2$ but preserves both $T^3$ and the independent weak hypercharge generator. With canonical kinetic terms its gauge-boson mass matrix in $(W^1,W^2,W^3,B)$ is $g^2{v}^2\mathrm{diag}(1,1,0,0)$. The two massless neutral gauge bosons prevent this field alone from reproducing Standard Model electroweak symmetry breaking.