Higgs boson

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.