Electroweak interaction

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 local gauge structure of the Standard Model electroweak sector. Left-chiral fermions form weak-isospin doublets and hypercharge assignments determine the electromagnetic generator $Q=T_3+Y$ in one common normalization. A chosen Higgs field vacuum leaves the electromagnetic subgroup manifest. Gauge transformations remain a redundancy of the description; physical vector masses are gauge-invariant observables.

An interaction of three electroweak gauge bosons, especially $W^{+}{W}^{-}\gamma$ and $W^{+}{W}^{-}Z$. Their Standard Model coefficients and momentum dependence come from the cubic and quartic Yang-Mills self-interactions after neutral-field mixing. W-pair production tests their relative strengths through interference with neutrino exchange; effective anomalous couplings require a stated convention and consistency constraints.^[ref]

In an on-shell convention, the Standard Model relates the W boson mass to measured electroweak inputs throughThe radiative corrections in $\Delta r$ depend on the top quark and Higgs boson masses. The leading heavy-top contribution to the electroweak rho parameter is $\Delta \rho \simeq 3G_F{m}_t^2/(8\sqrt{2}{\pi }^2)$, reflecting the large mass splitting in the top-bottom weak doublet. The Higgs dependence in the precision corrections is mainly logarithmic over the relevant range. Direct measurements of $m_t$ and $m_H$ therefore allow an indirect prediction of $m_W$; comparing it with a direct measurement tests loops as well as tree-level electroweak symmetry breaking. Any tension must include input uncertainties, experimental systematics and higher-order corrections.^[ref]

Sheldon Lee Glashow's 1979 lecture, Towards a Unified Theory: Threads in a Tapestry, recalls his 1961 $SU(2)\times U(1)$ construction: a photon, a massive neutral intermediary and a mixing angle. It also stresses its unresolved mass problem: symmetry-breaking masses had been inserted, rather than generated by the Higgs mechanism. Establishing the gauge structure and explaining the gauge boson masses were complementary steps.^[ref]

Steven Weinberg's A Model of Leptons (1967) builds a spontaneously broken gauge theory for the electron-type leptons, using a scalar doublet to give masses while retaining a massless photon. It derives neutral and charged couplings and discusses neutrino-electron scattering. Its closing question about renormalizability is explicitly unresolved; the paper is neither a proof of that property nor a complete treatment of quarks.^[ref]

Abdus Salam's 1979 lecture, Gauge Unification of Fundamental Forces, describes his and Steven Weinberg's completion of the spontaneously broken $SU(2)\times U(1)$ formulation for leptons. He separates that construction from the subsequent establishment of renormalizability. Its role is to connect the mass mechanism with predictive neutral and charged couplings, not to identify a photographed event. The Gargamelle evidence subsequently tested the predicted neutral current.^[ref]

With the plus-sign convention $D_{\mu }={\partial }_{\mu }+ig{T}_3{W}_{\mu }^3+i{g}^{\prime }Y{B}_{\mu }$, let $\tan {\theta }_W=g^{\prime }/g$ and $A_{\mu }=\sin {\theta }_W{W}_{\mu }^3+\cos {\theta }_W{B}_{\mu }$. The coefficient of $A_{\mu }$ is $g\sin {\theta }_W{T}_3+g^{\prime }\cos {\theta }_{W}Y=e(T_3+Y)$, where $e=g\sin {\theta }_W=g^{\prime }\cos {\theta }_W$. The extra minus sign in the interaction Lagrangian comes from the $i$ in $\bar{\psi }i{\gamma }^{\mu }{D}_{\mu }\psi$. It reproduces the electric charges of every component of an Standard Model representation.

The Weinberg angle rotates the neutral gauge fields into the photon and Z boson:$g$ belongs to $SU(2{)}_L$, $g^{\prime }$ to $U(1{)}_Y$, and $e$ is the positive magnitude of the electron electric charge. These definitions use $Q=T_3+Y$; doubling the definition of hypercharge requires halving its coefficient in the covariant derivative. The electromagnetic coupling of a weak multiplet follows by substituting this rotation.

For a fermion component, the corresponding neutral coefficient isFor a left-handed charged lepton, $T_3=-1/2$ and $Q=-1$, giving $g_L=-1/2+{\sin }^2{\theta }_W$. Its right-handed component has $T_3=0$, giving $g_R={\sin }^2{\theta }_W$. This algebra links the neutral current to the electromagnetic charge assignment. Together with the Higgs doublet, it also gives the tree-level electroweak gauge-boson masses with $m_W/m_Z=\cos {\theta }_W$; precise comparisons require a defined renormalization convention and radiative corrections.

Electroweak gauge invariance is invariance under local $SU(2{)}_L\times U(1{)}_Y$ transformations. It fixes the gauge couplings and organizes left-handed fermions into weak doublets while right-handed charged fermions are weak singlets.

A sphaleron is an unstable static electroweak field configuration at the top of the barrier between vacua of different topological number. Thermal transitions across this barrier violate $B+L$ while preserving $B-L$.

Hypercharge is the generator of the electroweak $U(1{)}_Y$ factor. With the convention $D_{\mu }={\partial }_{\mu }-ig{W}_{\mu }^a{T}^a-i{g}^{\prime }Y{B}_{\mu }$, electric charge is $Q=T^3+Y$.

Charge quantization is the restriction of electric charges to discrete multiples of a basic unit. In the Standard Model, Yukawa gauge invariance and gauge-anomaly cancellation relate the quark and lepton hypercharges and imply that the proton and electron have opposite charges.

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.

A nonzero Higgs doublet has a real radial magnitude and three real angular coordinates. For a normalized vector $z=(z_1,z_2{)}^T$, the SU(2) matrix $U=\left(\begin{array}{cc}{z}_2^{\ast }& z_1\\ -z_1^{\ast }& z_2\end{array}\right)$ satisfies $U(0,1{)}^T=z$. Near the vacuum, the three broken generators can equivalently be parameterized by $e^{-i\mathbf{n}\cdot \mathbf{\sigma }+i{n}_3}$. A local gauge transformation removes these angular fields in unitary gauge. This coordinate choice is local about a nonzero vacuum and need not extend through zeros or topological defects.

Set $\epsilon =i{\tau }_2$, whose Pauli matrix identity is $\epsilon {\tau }_i^{\ast }=-{\tau }_i\epsilon$. Exponentiation gives $\epsilon U^{\ast }=U\epsilon$ for $U\in SU(2)$. Hence complex conjugating a Higgs doublet and multiplying by $\epsilon$ restores the fundamental representation of $SU(2)$, while reversing hypercharge. A Higgs field of hypercharge $1/2$ yields a conjugate doublet of hypercharge $-1/2$. Its product ${\tilde{\varphi }}^{†}L$ with a lepton doublet $L$ of hypercharge $-1/2$ transforms in the trivial representation of the whole electroweak gauge group. At vacuum $(0,v{)}^T/\sqrt{2}$, this composite is $v{\nu }_L/\sqrt{2}$.

For ${\mu }_h^2>0$ and $\lambda >0$, the minimum has ${\varphi }^{†}\varphi =v^2/2$ with $v^2={\mu }_h^2/\lambda$. In unitary gauge, $\varphi =(0,v+H{)}^T/\sqrt{2}$ gives $V=\text{constant}+\lambda v^2{H}^2+\lambda v{H}^3+\lambda H^4/4$. The canonically normalized Higgs boson therefore has $m_H^2=2\lambda v^2$. The angular modes produce the Goldstone bosons absorbed in the Higgs mechanism.

Set $r=\sqrt{2{\varphi }^{†}\varphi }$. The same Higgs potential becomesThe origin has negative radial curvature, while $r=v$ minimizes the scalar potential. Changing the field's angular coordinates at fixed $r$ costs no potential energy; changing its radial magnitude does. This geometric distinction explains why the surviving physical scalar field has a mass even though the angular directions are flat in the scalar potential. Local angular variations still enter the gauge-covariant kinetic term.

For a chosen model, not fitted physical constants, take $v=240\mathrm{GeV}$ and $\lambda =1/8$ in units $\hslash =c=1$. Then ${\mu }_h^2=7200{\mathrm{GeV}}^2$ and $m_H=120\mathrm{GeV}$. Direct expansion gives a quadratic coefficient $7200{\mathrm{GeV}}^2$, cubic coefficient $30\mathrm{GeV}$ and quartic coefficient $1/32$. The quadratic coefficient is $m_H^2/2$, not $m_H^2$. Keeping the same named $\lambda$ in the Higgs radial mass for a half-normalized quartic potential would instead give $m_H^2=7200{\mathrm{GeV}}^2$; these are different potentials. The shifted quartic Higgs potential normalization agrees with this section when $\kappa =\lambda$.

For $V=\frac{1}{2}\lambda ({\varphi }^{†}\varphi -v^2/2{)}^2$ and the canonically normalized radial field, the expansion is $V=\frac{1}{2}\lambda v^2{H}^2+\frac{1}{2}\lambda v{H}^3+\frac{1}{8}\lambda H^4$. Comparing the quadratic term with $m_H^2{H}^2/2$ gives the displayed mass. Replacing the prefactor $\lambda /2$ by $\lambda$ changes the answer to $2\lambda v^2$, so the potential's normalization must be retained.

In unitary gauge, ${\varphi }^{†}\varphi -v^2/2=vH+H^2/2$, so this Higgs potential gives $M_H^2=2\kappa v^2$, a cubic coefficient $\kappa v$ and a quartic coefficient $\kappa /4$. The corresponding Feynman rules are $-3i{M}_H^2/v$ and $-3i{M}_H^2/v^2$. Writing the potential with an additional factor $1/2$ changes the relation between its named coupling and the Higgs boson mass; it does not change these mass-based rules.

In the Higgs mechanism, gauge fields associated with broken generators acquire longitudinal polarizations and masses by absorbing Goldstone modes. Gauge fields of the unbroken subgroup remain massless.

The phrase “absorbing” describes a rearrangement of degrees of freedom, not a collision in which a gauge boson swallows another particle. The gauge symmetry remains a redundancy of the description. A chosen vacuum expectation value leaves a smaller subgroup manifest, while gauge-invariant observables describe the physical masses.

A one-field calculation isolates the mechanism. In an Abelian Higgs model, write a complex scalar field locally as $\Phi =(v+H)e^{i\theta }/\sqrt{2}$, use $D_{\mu }={\partial }_{\mu }-ie{A}_{\mu }$, and define $B_{\mu }=A_{\mu }-{\partial }_{\mu }\theta /e$. ThenThe second term contains a Proca field mass $m_B=ev$ and its interactions with the radial Higgs boson. A local gauge transformation removes $\theta$ from the scalar field but leaves its information in the longitudinal vector polarization. The electroweak symmetry breaking calculation applies this idea to three broken directions.

A real four-vector vacuum ${\Phi }_0=v{e}_4$ leaves the rotations of its perpendicular three-space unbroken. In a Hermitian generator basis split into two commuting SU(2) triples with $(T^a+T^{a+3})e_4=0$, define $A^{a,\pm }=(A^a\pm A^{a+3})/\sqrt{2}$. For scalar kinetic term $|D\Phi {|}^2$ and canonical vector kinetic terms, its vacuum value is $g^2{v}^2{\sum }_a(A^{a,-}{)}^2/2$. Comparison with $m^2{A}^2/2$ gives three massless and three mass-$|gv|$ gauge bosons.

The factor multiplying the scalar kinetic term matters: replacing it by $|D\Phi {|}^2/2$ at fixed numerical $v$ changes the massive value to $|gv|/\sqrt{2}$. The unbroken global subgroup is SO(3), rather than its simply connected SU(2) cover. The Spin(4) double cover accounts for that central identification.

In their 1964 paper, François Englert and Robert Brout showed that gauge bosons associated with generators that rotate the chosen vacuum acquire masses while gauge invariance is maintained. Their discussion included both scalar and fermionic models.^[ref]

Independently, Peter Higgs exhibited a massive vector and a surviving massive scalar field by linearizing a simple gauge theory. His paper explicitly distinguishes this classical calculation from a proof for the full quantized theory. These 1964 mechanisms supplied an ingredient for the later electroweak interaction, rather than the complete Standard Model.^[ref]

In four spacetime dimensions a massless vector has two physical polarizations, while a Proca field has three. Before rearrangement, the electroweak sector has four vector fields and four real Higgs doublet components, hence twelve propagating degrees of freedom. In the broken description, W bosons of both charges and the Z boson contribute nine, the photon contributes two, and the radial Higgs boson contributes one. The three angular scalar components supply the three additional longitudinal polarizations. Gauge components and constraint variables are excluded from both counts. The gluon sector is unaffected.

At tree level and energies much greater than a gauge-vector mass, an amplitude with longitudinal external gauge bosons equals the corresponding amplitude with their would-be Goldstone bosons, up to conventional external-state phases and relative $O(m_V/E)$ corrections for a nonzero leading amplitude. In a renormalizable gauge, the broken-symmetry identities relate $k_{\mu }{\mathcal{M}}^{\mu }$ to $m_V{\mathcal{M}}_{\chi }$; inserting ${\epsilon }_L^{\mu }=k^{\mu }/m_V+O(m_V/E)$ explains the result. Loop-level versions include field-normalization factors. This is different from an equivalence theorem for field redefinitions.

The Goldstone-boson equivalence theorem reduces high-energy electroweak longitudinal scattering to scalar scattering. Without the radial Higgs boson, a nonlinear sigma-model description gives an amplitude of order $s/v^2$. In the corresponding linear scalar model, the contact and Higgs-exchange terms give, for the charged-to-neutral Goldstone channel,The growing terms cancel, leaving $-m_H^2/v^2$ at high energy. Thus the Higgs mechanism supplies both vector masses and the interactions required for controlled longitudinal scattering. The tree expression is not valid as an unresummed approximation arbitrarily close to the Higgs pole.

The gauge-covariant kinetic term of the Higgs doublet gives $M_W^2(1+H/v{)}^2{W}_{\mu }^{+}{W}^{-\mu }+M_Z^2(1+H/v{)}^2{Z}_{\mu }{Z}^{\mu }/2$. Differentiation with respect to the external fields yields Feynman rules $2i{M}_V^2{g}_{\mu \nu }/v$ for one Higgs boson and two matching massive electroweak gauge bosons, and $2i{M}_V^2{g}_{\mu \nu }/v^2$ for two Higgs bosons and the same pair. Identical neutral fields supply combinatorial factors. The neutral radial Higgs boson has no corresponding elementary coupling to photons at tree level.

A Yang-Mills theory with gauge group $SO(3)$ and a real three-component scalar field in its defining vector representation. With a symmetry-breaking scalar potential, a nonzero vector vacuum leaves its $SO(2)$ stabilizer subgroup unbroken. The two transverse internal directions produce massive gauge bosons.

For an orthogonal group gauge transformation $R\in O(3)$, a scalar vector transforms as $\varphi \mapsto R\varphi$. The gauge connection belongs to the Adjoint representation, identified with axial vectors: the displayed cross-product identity makes its homogeneous transformation $A\mapsto (\det R)RA$. The matrix connection also has the usual inhomogeneous derivative term. Thus $D\varphi$ transforms as an ordinary vector, including under improper orthogonal transformations. Treating both fields as ordinary vectors would fail because a cross product changes sign under an orientation reversal. A nonzero vector vacuum has stabilizer subgroup $O(2)$; its connected part is $SO(2)$.

For a gauged real scalar triplet with vacuum $v{e}_3$ and generators $(t^a{)}_{bc}=-i{ϵ}^{abc}$, the covariant kinetic term yields gauge-boson squared masses $g^2{v}^2\mathrm{diag}(1,1,0)$. Two Goldstone bosons become longitudinal vector modes through the Higgs mechanism. One radial scalar, two massive three-polarization vector fields and one massless two-polarization field remain. The physical count $3+3\cdot 2=1+2\cdot 3+2=9$ is unchanged. In unitary gauge, the scalar kinetic term includes $\frac{1}{2}{g}^2(v+h{)}^2[(B^1{)}^2+(B^2{)}^2]$.

The scalar potential $V(\Phi )=\lambda ({\Phi }^T\Phi -v^2{)}^2/4$, with $\lambda >0$, is invariant under the SO(3) group. For $v>0$, its vacuum manifold is a sphere. Canonical scalar normalization gives radial mass squared $2\lambda v^2$.

In the SO(3) vector Higgs model, a real scalar field $\Phi \in {\mathbb{R}}^3$ transforms as $\Phi \mapsto R\Phi$. Its gauge covariant derivative is $D_{\mu }\Phi ={\partial }_{\mu }\Phi +g{\mathbf{A}}_{\mu }\times \Phi$ for generators acting by the cross product.

If $D_{\mu }\Phi =0$ in the SO(3) vector Higgs model, antisymmetry of the gauge potential implies ${\partial }_{\mu }|\Phi {|}^2=0$. On a local patch with nonzero constant norm, unitary gauge aligns the field with a fixed axis. The connection and gauge field strength then take values in the axis stabilizer; curvature can remain nonzero. Global alignment can require several bundle patches.

An $SU(2)$ gauge theory with a real adjoint scalar and a vacuum $\varphi =v{e}_3$ breaks to $U(1)$. The Higgs mechanism gives two charged vectors mass $gv$, leaves one massless neutral vector, and gives the radial scalar mass $\sqrt{2\lambda }v$ for the quartic potential. Unlike the Standard Model electroweak interaction, this model has no additional hypercharge field and no massive neutral Z boson.

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 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.

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.

An electroweak doublet with $Y=-1/2$ has component charges $(0,-1)$ under $Q=T^3+Y$. Its upper-component vacuum therefore preserves electromagnetism. The canonical covariant-derivative term gives $m_W=g_{2}v/2$, $m_Z=v\sqrt{g_1^2+g_2^2}/2$ and a massless photon. Neutral eigenvectors are proportional to $(g_2,-g_1)$ for $Z$ and $(g_1,g_2)$ for the photon. This is the conjugate-doublet convention to the usual positive-hypercharge Higgs doublet and gives the same mass ratio and electric coupling.

A thermal change in the electroweak Higgs sector as the early Universe cools. Its order and strength depend on the particle content, couplings and effective description. A polynomial landau free energy is a simplified model and does not, by itself, establish a model-independent Higgs-mass bound.

If $\lambda /a=x=m_H^2/m_0^2$ and $c/a=A+x$, then $\gamma =[72{\pi }^{2}x(A+x){]}^{-1}$ and ${\varphi }^{\ast }/T^{\ast }=(6\pi x{)}^{-1}$. Requiring both $\gamma <1$ and ${\varphi }^{\ast }>T^{\ast }$ gives $x_{-}<x<1/(6\pi )$, where $x_{-}=[-A+\sqrt{A^2+1/(18{\pi }^2)}]/2$. This is an open interval for the specified model. Failing the strength condition does not alone eliminate its first-order jump.

Renormalization-group evolution can change a Higgs squared-mass parameter from positive at a high input scale to negative at lower scales, making a symmetry-breaking scalar expectation energetically favorable. In the Minimal supersymmetric Standard Model, the large top Yukawa coupling can drive the up-type Higgs soft squared mass downwards as the scale is lowered. Actual breaking also requires the other Higgs-potential parameters to give a stable suitable minimum; the running of one mass alone is not sufficient.

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.

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.

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.

Unitary gauge uses a gauge transformation to remove the Goldstone fields from the Higgs multiplet, leaving the physical radial Higgs excitation and massive vector fields.

For a real orthogonal gauge group representation, the tangent vectors to the vacuum orbit are $v_a=T_a{\varphi }_0$. Choose an independent basis of these vectors. Under an infinitesimal gauge transformation, the derivative of the displayed slice conditions with respect to the broken gauge parameters is the Gram matrix $v_a^T{v}_b$. It is invertible on the independent tangent directions, so the implicit function theorem selects this unitary gauge locally near the vacuum. The remaining scalar fluctuations are orthogonal to the gauge orbit, and their derivative has no quadratic mixing with the massive gauge fields. The statement is local and does not assert a global gauge slice through zeros of the scalar field.

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.

For a charged Dirac field excitation of mass $m$, multiplicity $N$, electric coupling $eQ$ and scalar Yukawa interaction $-yh\bar{\psi }\psi$, two orientations of a fermion loop induce a Higgs boson coupling to two photons. With metric $(+---)$ and the usual $S=1+iT$ convention, their physical on-shell scattering amplitude isThe Ward identity holds on both photon legs. At low external energy, the integral tends to $1/(3m^2)$. Thus fixed $y$ gives an inverse-mass coefficient, whereas $y=m/v$ gives a constant coefficient as $m$ increases at fixed $v$. The distinction is essential when applying heavy-field decoupling.

Production through exchange of weak gauge bosons emitted by incoming fermions. A characteristic Higgs boson process at a hadron collider has two outgoing quark jets and tests the Higgs coupling to weak vectors. Predictions for a specified measured final state must include the consistent set of interfering amplitudes rather than treating a diagram label as a separately measurable event.

Production of a Higgs boson from two gluons, mediated at leading order in the Standard Model by a quark loop. The large top quark Yukawa coupling makes the top loop important. There is no direct renormalizable tree-level coupling of the colorless Higgs field to two gluons.

A muon and antimuon can form a Higgs boson directly in the annihilation channel. Their Higgs Yukawa coupling is proportional to the muon mass, making the corresponding partial width much larger than for an electron pair. The observable resonant cross section also depends on the collider's energy distribution: a beam spread broader than the resonance smears its ideal peak.

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.

With the conventional electroweak vacuum value $v\simeq 246\mathrm{GeV}$, the electron Yukawa coupling is $y_e\simeq 2.9\times 10^{-6}$ and the physical Higgs-electron vertex coefficient is $m_e/v\simeq 2.1\times 10^{-6}$. Direct s-channel Higgs production in electron-positron annihilation is consequently very small away from a compensating resonance enhancement. This suppression belongs to the coupling at the initial vertex, not to a prohibition on the Higgs boson coupling to W bosons.

The ATLAS experiment and CMS experiment searched proton-proton collisions at the Large Hadron Collider at collision energies of $7$ and $8\mathrm{TeV}$. The ATLAS experiment used silicon tracking, a $2\mathrm{T}$ solenoid, liquid-argon electromagnetic calorimeters and an external muon spectrometer. Its July 2012 paper reported a combined local statistical significance of $6.0$ standard deviations near $126.5\mathrm{GeV}$; this paper result is distinct from the initial 4 July announcement.^[ref]

The CMS experiment used silicon tracking in a $3.8\mathrm{T}$ solenoid, lead-tungstate electromagnetic calorimeters and external muon chambers. Electrons combine tracks and particle showers; muons combine tracking and chamber hits; photons supply shower energies and directions. Selected $H\to \gamma \gamma$ and $H\to Z{Z}^{\ast }\to 4\ell$ candidates, with $\ell =e,\mu$, yielded invariant mass peaks. It checked lepton efficiencies with Z boson decays, estimated fake-lepton backgrounds from control samples with relaxed identification, and fitted the smooth diphoton background from data. Its combined local statistical significance was $5.0$ standard deviations near $125\mathrm{GeV}$.^[ref]

The ATLAS experiment compared signal-plus-background and background-only hypotheses with a profile likelihood, accounting for systematic uncertainties and correlations between decay channels. Diphoton and four-lepton candidates gave precise invariant masses; $W{W}^{\ast }$ decays supplied additional evidence with poorer mass reconstruction because of escaping neutrinos. A local p-value concerns one specified mass; scanning masses requires a look-elsewhere effect correction.^[ref]

Two-photon decay excludes spin one under the usual assumptions, but does not prove spin zero. A mass peak does not determine every Higgs boson coupling. Production rates, branching fractions and angular correlations require additional tests.^[ref]

The bridge from detector readout to a particle resonance is four-momentum conservation. In units $c=1$, sum the measured final-state four-momenta:For two photons separated by angle $\alpha$, $|{\mathbf{p}}_i|=E_i$ gives the displayed diphoton formula. Two back-to-back photons of $60\mathrm{GeV}$ each have $m_{\gamma \gamma }=120\mathrm{GeV}$. A four-lepton candidate instead uses four four-momenta in the same sum. Collision debris outside the selected decay is not included. Since the partons carry unknown fractions of the incoming proton momenta, reconstructing the decay products is more informative than assigning the full beam energy to each candidate.

Suppose the intrinsic decay width is negligible compared with the stipulated detector spread. Let the reconstructed signal mass follow a normal distribution with mean $m_0=120\mathrm{GeV}$ and standard deviation ${\sigma }_m$. For a total expected selected signal $S=60$ and flat background density $b=8$ events per $10^9$ electronvolts, the expected count density isThis is a deliberately chosen response model, not a fit to either experiment. In the fixed window $m_0-a<m<m_0+a$ with $a=3\mathrm{GeV}$, direct integration givesThe expected signal counts are $57.2700$ for ${\sigma }_m=1.5\mathrm{GeV}$ and $40.9614$ for ${\sigma }_m=3\mathrm{GeV}$. Doubling the spread lowers the fraction in the same window without changing the total signal. Detector resolution can therefore change the visibility of a peak even with identical particle production.

As a separate counting example, stipulate an observed integer count $n=70$ and an exactly known background mean $B_w=48$. The Poisson distribution gives a likelihood $L(s)=e^{-(48+s)}(48+s{)}^{70}/70!$ for nonnegative signal mean $s$. Its maximum-likelihood estimator is $\hat{s}=22$, and the likelihood ratio statistic against $s=0$ isThis tests a specified window with a specified background, and is not either experiment's discovery significance. If the background mean is allowed to vary freely, background alone can choose $70$ and fit that count exactly. Control samples, a defensible mass-shape model, nuisance parameters and corrections for searching many masses are therefore essential to the experimental inference.

Higgsstrahlung produces a Higgs boson together with a massive electroweak gauge boson. At an electron-positron collider the process $e^{+}{e}^{-}\to Z^{\ast }\to ZH$ uses the Higgs boson coupling to Z bosons. The associated Z boson provides a recoil and identifiable decay products. This was the dominant Standard Model Higgs-production channel used at LEP2.

A scalar Yukawa coupling $-m_{f}H\bar{f}f/v$ givesabove threshold. The fermion spin sum provides the additional factor of momentum squared that turns the phase-space threshold factor into its third power. The color multiplicity in a decay width is $N_c=3$ for a quark pair and one for a lepton pair.

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 tree-level two-body decay width from $2m_W^{2}H{W}_{\mu }^{+}{W}^{-\mu }/v$ isabove the on-shell threshold. The charged pair is distinguishable, so no identical-particle factor divides this result. Below threshold, physical multi-particle decays through virtual W bosons require a different Lorentz-invariant phase-space measure.

For $x=m_W^2/m_H^2<1/4$, the on-shell longitudinal pair contributes the displayed tree-level partial decay width. In the Higgs boson rest frame, its polarization vectors have contraction $(m_H^2/2-m_W^2)/m_W^2$. Multiplying by the Higgs boson coupling to W bosons gives $|{\mathcal{M}}_{LL}{|}^2=m_H^4(1-2x{)}^2/v^2$; the two-body Lorentz-invariant phase space then gives the width. The sum over transverse polarizations adds $8x^2$ to the longitudinal polynomial. Below threshold, physical decays through virtual W bosons require a different multiparticle calculation.

The neutral vector mass term is $\frac{1}{2}{m}_Z^2{Z}_{\mu }{Z}^{\mu }(1+H/v{)}^2$. Expanding it gives the cubic $HZZ$ and quartic $HHZZ$ interactions in unitary gauge. Differentiating the action with respect to identical fields gives Feynman rules $2i{m}_Z^2{g}_{\mu \nu }/v$ and $2i{m}_Z^2{g}_{\mu \nu }/v^2$. A single $Z$ does not couple to two copies of the real neutral radial Higgs boson through an elementary vertex.

For $m_H>2m_Z$, the Higgs boson coupling to Z bosons gives $\mathcal{M}=(2m_Z^2/v){ϵ}_1^{\ast }\cdot {ϵ}_2^{\ast }$. The massive vector polarization sum gives $\sum |\mathcal{M}{|}^2=(m_H^4/v^2)(1-4x+12x^2)$, with $x=m_Z^2/m_H^2$. The two-body decay phase space and identical final-state symmetry factor $1/2$ giveThe contraction uses $k_1\cdot k_2=(m_H^2-2m_Z^2)/2$. The on-shell rate vanishes at threshold; below threshold an off-shell calculation is required.