Higgs field

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.

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.

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.

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.

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.

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.

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.

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.

An electroweak scalar triplet transforms in the three-dimensional adjoint representation of $SU(2{)}_L$. A complex triplet of hypercharge one has components of electric charge two, one and zero, and its neutral component can couple to two left-handed lepton doublets.