The Higgs Boson

The Higgs mechanism predicts one physical scalar left over after the $W^{\pm}$ and $Z$ eat their Goldstone modes. Its mass $M_{h} = 2 λ v$ is a free parameter, but every one of its couplings is fixed: to a gauge boson in proportion to the boson's mass squared, to a fermion in proportion to the fermion's mass. That rigidity is what made the search a sharp test rather than an open-ended hunt — a particle with the wrong couplings would not have been the Standard Model Higgs. This lesson covers how the boson is made at a proton collider, how it decays, the two channels that revealed it in 2012, and the coupling measurements that confirmed its identity.

Production at the LHC

The Higgs couples to mass, so it is produced most readily through the heaviest particles available. At a proton-proton collider the relevant partons are gluons and light quarks, and four production mechanisms dominate, in decreasing rate at $s = 13 TeV$ .

Gluon fusion (ggF). Two gluons fuse into the Higgs through a virtual quark loop, overwhelmingly the top quark since its Yukawa coupling is largest. This is the largest cross section, near $50 pb$ , but the final state (Higgs plus hadronic activity) is experimentally messy.
Vector-boson fusion (VBF). Each incoming quark radiates a $W$ or $Z$ , and the two bosons fuse into a Higgs; the quarks continue as two forward tag jets. The rate is about a tenth of ggF, but the two forward jets with little activity between them give a distinctive, low-background signature.
Associated production ( $V H$ ). A quark-antiquark pair annihilates into a virtual $W$ or $Z$ that radiates a Higgs, so the Higgs is produced alongside a leptonically decaying vector boson. The lepton tags the event, which makes this the channel of choice for the dominant but otherwise-invisible $H \to b \overset{ˉ}{b}$ decay.
Top-associated production ( $t \overset{ˉ}{t} H$ ). The Higgs is radiated from a top-quark pair. The smallest of the four, but it measures the top Yukawa coupling directly rather than through the ggF loop.

The three leading Higgs production channels at the LHC. In gluon fusion two gluons couple to the Higgs through a top-quark loop. In vector-boson fusion two incoming quarks each radiate a W or Z, which fuse into a Higgs, leaving two forward jets. In associated production a quark-antiquark pair makes a virtual W or Z that radiates the Higgs alongside a tagging boson.

Decay modes

A $125 GeV$ Higgs sits below the $2 M_{W}$ and $2 M_{Z}$ thresholds, so its decays to weak bosons proceed through one off-shell boson. The branching ratios are fixed by the mass-proportional couplings, weighted by phase space and by color and multiplicity factors. The dominant decay is to $b \overset{ˉ}{b}$ — the heaviest fermion pair kinematically open — followed by the off-shell $W W^{*}$ .

Mode	Branching ratio	Role
$b \overset{ˉ}{b}$	$58%$	largest; huge QCD background
$W W^{*}$	$21%$	large; used with leptonic $W$
$g g$	$8.2%$	via top loop; indistinct
$τ^{+} τ^{-}$	$6.3%$	accessible, moderate background
$c \overset{c}{ˉ}$	$2.9%$	hard to tag
$Z Z^{*}$	$2.6%$	golden $4 ℓ$ subchannel
$γ γ$	$0.23%$	rare, clean, discovery channel

The two channels that carried the discovery are not the largest but the cleanest. $H \to b \overset{ˉ}{b}$ has by far the highest rate, yet the LHC produces $b \overset{ˉ}{b}$ pairs by QCD at a rate millions of times higher, drowning the signal. The narrow-peak channels win instead: a fully reconstructed final state with small, smoothly varying background lets a sharp mass peak stand out even at a fraction of a percent branching.

The two discovery channels

$H \to γ γ$ . The Higgs does not couple to the massless photon directly; the decay proceeds through a loop of charged particles, dominated by the $W$ boson and the top quark. Despite the tiny $0.23%$ branching ratio, the two-photon final state is fully measured, and its invariant mass reconstructs a narrow peak — width set by detector resolution, a couple of GeV — sitting on a smooth falling background of non-resonant photon pairs. Fitting the background and looking for an excess is the whole analysis.

The diphoton invariant-mass spectrum. Non-resonant photon-pair production gives a smooth, steeply falling background; the Higgs adds a narrow excess at its mass near 125 GeV. The peak width is the detector resolution, not the natural width, which is only a few MeV.

$H \to Z Z^{*} \to 4 ℓ$ . The Higgs decays to one on-shell and one off-shell $Z$ , each of which decays to a charged-lepton pair, giving four leptons ( $e$ or $μ$ ). This golden channel has an even smaller rate than diphoton once the leptonic branchings are folded in, but the four-lepton invariant mass is measured with excellent resolution and the irreducible background — non-resonant $Z Z^{*}$ production — is small and well modeled. A handful of clean four-lepton events clustered at one mass is a nearly background-free signal.

The golden four-lepton channel. The Higgs decays to two Z bosons, one on-shell and one virtual; each Z decays to a charged-lepton pair. The four-lepton invariant mass reconstructs the Higgs mass with high resolution over a small, well-understood background.

The 2012 discovery

In July 2012 the ATLAS and CMS collaborations independently reported a new boson near $125 GeV$ , each combining the $γ γ$ and $Z Z^{*} \to 4 ℓ$ channels (with $W W^{*}$ contributing) to reach the $5 σ$ threshold — a background fluctuation probability below $3 \times 1 0^{- 7}$ — that particle physics takes as the standard for a discovery. The two experiments' masses agreed, and the signal appeared at the same place in the same channels in both, which made a statistical fluke or a detector artifact untenable.¹ The current world-average mass is

M_{h} = 125.25 \pm 0.17 GeV,

now one of the best-measured parameters of the Standard Model.² Because $M_{h}$ fixes $λ = M_{h}^{2} /2 v^{2} \approx 0.13$ , the last free parameter of the Higgs potential is pinned, and every Higgs coupling becomes an absolute prediction to be checked.

Confirming it is the Standard Model Higgs

Finding a boson at $125 GeV$ is not the same as finding the Standard Model Higgs. Two properties had to match.

Spin and parity. A particle decaying to two photons cannot have spin 1 (the Landau-Yang theorem forbids it), leaving spin 0 or 2. Angular distributions of the decay products in $Z Z^{*} \to 4 ℓ$ and $W W^{*}$ discriminate among the hypotheses, and the data strongly favor $J^{P} = 0^{+}$ — a scalar, as the Higgs must be — over $0^{-}$ , $1^{\pm}$ , and $2^{+}$ .
Couplings proportional to mass. The defining prediction is that each coupling scales with the particle's mass: linearly for fermions ( $\propto m_{f}$ ), and set by $m_{V}^{2} / v$ for the weak bosons. Plotting the measured coupling strength of each particle against its mass, the points fall on the predicted line across three orders of magnitude — from the $τ$ and $b$ through the $W$ , $Z$ , and top. This mass-proportional pattern is the fingerprint of the Higgs mechanism and is not shared by any generic new scalar.

Measured Higgs coupling strength versus particle mass, on logarithmic axes. The Standard Model predicts a straight line: fermion couplings proportional to mass, weak-boson couplings proportional to mass squared over the vacuum value. The measured points for tau, b, W, Z, and top lie along that line.

Open questions

The discovery closed the Standard Model but opened its own questions. The Higgs self-coupling — the $h^{3}$ and $h^{4}$ terms fixed by $λ$ in the potential — has not been measured directly; it requires observing Higgs-pair production, a target for the high-luminosity LHC and future colliders, and it is the only direct probe of the shape of the potential that drives symmetry breaking. The couplings to the first-generation fermions and to the muon are only beginning to be tested. And the Higgs mass itself is unstable under quantum corrections — the hierarchy problem — which is the sharpest hint that the scalar sector is not the whole story. Whether the Higgs is truly elementary or a composite bound state remains an experimental question.

Summary

The Higgs boson is produced at the LHC mainly by gluon fusion through a top loop, with vector-boson fusion, associated production, and $t \overset{ˉ}{t} H$ as cleaner or more direct channels. It decays most often to $b \overset{ˉ}{b}$ and $W W^{*}$ , but the discovery rested on the rare, clean $H \to γ γ$ and $H \to Z Z^{*} \to 4 ℓ$ modes, whose narrow invariant-mass peaks rise over smooth backgrounds. ATLAS and CMS announced a $5 σ$ signal near $125 GeV$ in 2012; the world-average mass is now $125.25 GeV$ . Its spin-parity $0^{+}$ and its mass-proportional couplings identify it as the Standard Model Higgs. With its mass measured, the last parameter of the electroweak Lagrangian is fixed, completing the Standard Model.

The 2012 discovery of a boson near $125 GeV$ in the $γ γ$ and $Z Z^{*} \to 4 ℓ$ channels was reported independently by the ATLAS collaboration, Observation of a new particle in the search for the Standard Model Higgs boson, arXiv:1207.7214, and the CMS collaboration, Observation of a new boson at a mass of 125 GeV, arXiv:1207.7235. The production and decay phenomenology is reviewed in Thomson, §17.8–17.10. ↩
The world-average mass $M_{h} = 125.25 \pm 0.17 GeV$ , the branching ratios, and the spin-parity and coupling measurements are from the Particle Data Group, pdg.lbl.gov. The mass-proportional couplings and $λ = M_{h}^{2} /2 v^{2}$ are discussed in Tong, The Standard Model (Cambridge Part III), §5.2, damtp.cam.ac.uk/user/tong/standardmodel.html. ↩