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For an adjoint vacuum expectation value proportional to $\mathrm{diag}(1,1,-2)$, the centralizer is $(SU(2)\times U(1))/{\mathbb{Z}}_2$. Four gauge bosons remain massless and four acquire a common mass. This differs from Fundamental-Higgs breaking of SU(3) to SU(2).

For a Hermitian adjoint scalar field with kinetic term $\mathrm{tr}(D_{\mu }\Phi (D^{\mu }\Phi {)}^{†})$, use $D_{\mu }\Phi ={\partial }_{\mu }\Phi +ig[A_{\mu },\Phi ]$ and generators normalized by $\mathrm{tr}(t^a{t}^b)={\delta }^{ab}/2$. Expanding at a constant vacuum expectation value ${\Phi }_0$ and matching ${\mathcal{L}}_{\mathrm{mass}}=A_{\mu }^a(m^2{)}^{ab}{A}^{b\mu }/2$ gives the displayed expression. A real combination of generators $X=x_a{t}^a$ has quadratic form $2g^2\mathrm{tr}([X,{\Phi }_0{]}^{†}[X,{\Phi }_0])\ge 0$. Thus the massless directions are exactly the generators commuting with the vacuum.

For ${\Phi }_0=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)$, the two Hermitian generators associated with each matrix unit pair $E_{ij},E_{ji}$ have squared mass $g^2({\lambda }_i-{\lambda }_j{)}^2$. The surviving group is the centralizer of a Hermitian matrix in SU(n).

Let $b_k=\cosh {\theta }_k{a}_k+\sinh {\theta }_k{a}_{-k}^{†}$ for real even ${\theta }_k$, and let every $b_k$ annihilate a normalized vacuum state. Inverting the Bogoliubov transformation gives $a_k=\cosh {\theta }_k{b}_k-\sinh {\theta }_k{b}_{-k}^{†}$. Only $⟨b_k{b}_q^{†}⟩={\delta }_{kq}$ survives, proving the displayed vacuum expectation values. Taking the Hermitian adjoint gives the same real negative pair amplitude for two creation operators. The normal occupation number is quadratic at small ${\theta }_k$, whereas the pair amplitude is linear. For one opposite-momentum pair the two-mode squeezed vacuum is $\mathrm{sech}\theta {\sum }_{n\ge 0}(-\tanh \theta {)}^n|n,n⟩$, fixing the sign convention directly.

A theory of an $N\times N$ matrix of complex scalar fields with kinetic term $\mathrm{tr}(\partial {\Phi }^{†}\partial \Phi )$ and potential $V=\alpha \mathrm{tr}[({\Phi }^{†}\Phi {)}^2]/2+\beta [\mathrm{tr}({\Phi }^{†}\Phi ){]}^2/2+m^2\mathrm{tr}({\Phi }^{†}\Phi )$. It is invariant under global $SU(N{)}_L\times SU(N{)}_R\times U(1)$ transformations. For $\alpha ,\beta >0$ and $m^2<0$, minimizing the nonnegative eigenvalues of ${\Phi }^{†}\Phi$ gives a vacuum expectation value $C1$ with $C^2=-m^2/(\alpha +N\beta )$, up to a global symmetry transformation. The connected stabilizer is diagonal $SU(N)$, with $N^2$ broken generators. Expanding $\Phi =C1+(H+iP)/\sqrt{2}$ with Hermitian matrices gives $N^2$ massless Goldstone bosons in $P$ and squared masses $2C^2(\alpha +N\beta )$ for the singlet in $H$ and $2\alpha C^2$ for its traceless Adjoint representation. The faithful unbroken conjugation action factors through $SU(N)/{\mathbb{Z}}_N$.

A Dirac mass term pairs left- and right-handed Weyl fields in conjugate Lorentz representations. Gauge invariance requires the two chiral fields to carry matching internal quantum numbers, directly or after multiplication by a scalar field that acquires a vacuum expectation value.

Suppose fields with integer U(1) charges $Q_i$ acquire nonzero vacuum expectation values, with at least one $Q_i\ne 0$. The stabilizer in the circle group is exactly ${\mathbb{Z}}_d$, where $d=\gcd (|Q_1|,\dots ,|Q_r|)$. Indeed, $e^{i\theta }$ preserves every condensate if and only if $e^{i{Q}_i\theta }=1$ for every $i$. The Bezout identity implies $e^{id\theta }=1$, and conversely $d\mid Q_i$ proves every $d$th root of unity preserves every condensate. Neutral fields impose no constraint; if all condensing fields are neutral, the whole circle group survives.

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.

The electroweak interaction is the $SU(2{)}_L\times U(1{)}_Y$ gauge theory that unifies the weak and electromagnetic interactions.

An instructive tension motivates the unification: the photon is massless, while the W boson and Z boson are massive; electromagnetic interaction and weak interaction also have different observed low-energy strengths and chiral couplings. A gauge theory can relate their interactions while the Higgs field gives different masses after electroweak symmetry breaking. The resulting theory predicts relationships between processes, not merely a common name for observed phenomena.

Its useful organizing principle is the combination of weak isospin and hypercharge. The surviving electromagnetic generator is $Q=T_3+Y$ in the convention used here. The photon and Z boson are different mixtures of neutral gauge fields, while the charged current uses the charged combinations. The electroweak doublet gauge-boson mass matrix explains why one neutral mixture stays massless. At low energies, a massive W boson propagator instead produces the approximate Fermi interaction, withHere $g$ is the $SU(2{)}_L$ gauge coupling and $v$ the Higgs doublet vacuum expectation value. The inverse squared mediator mass suppresses low-energy weak rates. Unification does not require setting $g$ equal to the electromagnetic coupling $e$: their relation is $e=g\sin {\theta }_W$.

A striking experimental consequence is a neutral current in addition to the familiar charged current. The Gargamelle experiment tests this extra channel through visible charged secondaries, with a background problem fundamentally different from directly identifying a Z boson. The muon-neutrino electron recoil spectrum shows how the same Weinberg angle enters another process quantitatively.

The electroweak scale is the characteristic energy associated with electroweak symmetry breaking. With the Standard Model Higgs doublet normalized by $⟨H⟩=(0,v/\sqrt{2}{)}^T$, its vacuum expectation value satisfies $v=(\sqrt{2}{G}_F{)}^{-1/2}\simeq 246\mathrm{GeV}$ and $m_W=gv/2$. Its smallness compared with possible ultraviolet scales motivates the hierarchy problem.

Electroweak symmetry breaking is the Higgs-field breaking $SU(2{)}_L\times U(1{)}_Y\to U(1{)}_Q$. It gives masses to the $W$ and $Z$ bosons while leaving the photon massless.

With the hypercharge convention $Q=T_3+Y$, the lower component of the Higgs doublet has $T_3=-1/2$ and $Y=1/2$. Thus $Q⟨\varphi ⟩=0$: the vacuum expectation value is electrically neutral. This is why one combination of the neutral gauge fields remains the massless photon, as the electroweak doublet gauge-boson mass matrix shows.

Continue the chosen Higgs field potential example with $g=3/5$ and $g^{\prime }=4/5$. The tree-level electroweak gauge-boson masses are $m_W=72\mathrm{GeV}$, $m_Z=120\mathrm{GeV}$ and $m_A=0$. In the $(W^3,B)$ basis,Its orthonormal eigenvectors $(4/5,3/5{)}^T$ and $(3/5,-4/5{)}^T$ have eigenvalues zero and $14400{\mathrm{GeV}}^2$. Thus $\cos {\theta }_W=3/5=m_W/m_Z$. This is an exact model check, not a fit to measured electroweak interaction parameters.

A diagonal charged-fermion Yukawa coupling expands asThe Yukawa coupling remains a free parameter: the mechanism relates a fermion mass to its scalar interaction but does not predict the numerical mass. This example concerns charged fermions; the minimal Standard Model has no right-handed neutrino and does not account for neutrino oscillations without an extension.

After a scalar vacuum expectation value, a Yukawa matrix produces a bilinear $-{\bar{f}}_L{M}_f{f}_R+h.c.$. Independent unitary left/right rotations diagonalize $M_f$ with nonnegative masses. Extra scalar-dependent operators can make the physical scalar coupling matrix fail to diagonalize with the mass matrix.

For a complex fundamental scalar with vacuum expectation value proportional to $(0,0,v)$, the stabilizer is the $SU(2)$ acting on the first two components. Five broken generators produce five massive vector bosons, while three gauge bosons remain massless and one radial scalar remains physical.

With canonically normalized gauge field kinetic terms, a gauge-boson mass matrix is the real symmetric matrix of coefficients in the displayed quadratic mass term. Its eigenvalues are squared physical gauge boson masses. For a complex scalar field vacuum ${\Phi }_0$ in the Higgs mechanism, $(M^2{)}_{ab}=2\mathrm{Re}[(g_a{T}_a{\Phi }_0{)}^{†}(g_b{T}_b{\Phi }_0)]$. Thus for real $x_a$, $x^T{M}^{2}x=2\Vert {\sum }_a{x}_a{g}_a{T}_a{\Phi }_0{\Vert }^2\ge 0$: it is a Gram matrix and a positive semidefinite matrix. An unbroken generator annihilates the vacuum expectation value and lies in its null space. Diagonalizing the matrix identifies the massive and massless field combinations.

For generators normalized by $\mathrm{Tr}(t^a{t}^b)={\delta }^{ab}/2$, a constant adjoint vacuum expectation value ${\Phi }_0$ gives this matrix. A generator commuting with ${\Phi }_0$ corresponds to a massless gauge boson.

The Standard Model lepton doublet $L=({\nu }_L,e_L{)}^T$ has hypercharge $-1/2$, the electron singlet $e_R$ has $-1$, and the Higgs doublet has $1/2$. Therefore $-y_e\bar{L}\varphi e_R+h.c.$ is gauge-invariant even though a bare ${\bar{e}}_L{e}_R$ mass is not. The vacuum expectation value generates a Dirac mass $|y_e|v/\sqrt{2}$. In unitary gauge, the Higgs boson coupling is $-(m_e/v)h\bar{e}e$ after rephasing the electron.

An unbroken gauge symmetry forbids an explicit local Proca mass for its gauge vector. A Higgs mechanism can generate that mass through a charged vacuum expectation value, but the stability of the resulting vector scale then depends on the scalar sector that fixes the expectation. Gauge invariance alone does not protect a fundamental scalar mass.

The Higgs boson is the neutral radial excitation of the Higgs field about its vacuum expectation value.

In the single-doublet Standard Model this physical excitation is a spin zero excitation of a scalar field. Its mass depends on the Higgs potential, so the theory's gauge symmetry alone does not specify where to search. The Higgs boson discovery in 2012 tested the existence of a new particle resonance; establishing its detailed identity requires testing its interactions and spin as well.

Expanding the Higgs doublet kinetic term about its vacuum expectation value gives $\mathcal{L}\supset (2m_W^2/v)h{W}_{\mu }^{+}{W}^{-\mu }$. Its vertex is proportional to $g_{\mu \nu }$, so scalar exchange contracts the external particle polarization vectors.

At leading order, neglecting phase-space suppression and radiative corrections, Higgs decay to a fermion pair has partial decay width proportional to $N_c^f{m}_f^2$, with $N_c=3$ for a quark and one for a lepton. A known sum of branching fractions therefore fixes the proportionality between these weights and individual branching fractions without requiring the Higgs vacuum expectation value. The same normalization estimates a small additional fermionic channel if its weight is not included in the known sum.

The Standard Model Higgs field is a complex $SU(2{)}_L$ doublet of hypercharge $1/2$. Its nonzero vacuum expectation value triggers the Higgs mechanism.

The Higgs field is a scalar field: its components carry internal weak isospin and hypercharge, but no spacetime spin. A complex Higgs doublet contains four real field components. The vacuum expectation value is a property of the lowest-energy state, not a gas of Higgs bosons. A Higgs boson is an excitation about that state.

Why introduce this field? A bare mass term for a gauge field generally fails to respect the local gauge symmetry. A bare Dirac mass term for a charged Standard Model fermion also joins left- and right-handed fields with different Standard Model representations. A Higgs doublet permits gauge-invariant interactions whose expansion about a nonzero vacuum expectation value supplies both masses and definite Higgs boson interactions. The field therefore links the mass problem to experimentally testable coupling relations, as illustrated by the Higgs boson coupling to Z bosons.