--- title: The Higgs Boson module: Electroweak Unification and the Higgs moduleNumber: 9 lessonNumber: 4 order: 904 summary: > The Higgs boson is produced at the LHC chiefly through gluon fusion, with vector-boson fusion and associated production as cleaner but rarer channels. It decays most often to $b\bar b$ and $WW^\ast$, but the discovery rested on two rare clean modes, $H \to \gamma\gamma$ and $H \to ZZ^\ast \to 4\ell$, whose narrow invariant-mass peaks emerged over smooth backgrounds. ATLAS and CMS announced a boson near 125 GeV in 2012; its measured spin-parity $0^+$ and its couplings, which scale with particle mass, identify it as the Standard Model Higgs. topics: [Electroweak Unification and the Higgs] sources: - book: Thomson ref: "Ch. 17 §17.8–17.10 (Higgs production, decay, and discovery at the LHC)" - book: Griffiths ref: "Ch. 11 (Higgs phenomenology)" - book: Tong ref: "The Standard Model (Cambridge Part III), §5.2 (the Higgs boson and its couplings)" draft: false --- The [Higgs mechanism](/particle-physics/electroweak-higgs/higgs-mechanism) predicts one physical scalar left over after the $W^\pm$ and $Z$ eat their Goldstone modes. Its mass $M_h = \sqrt{2\lambda}\,v$ 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 $\sqrt s = 13\ \text{TeV}$. - **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 $50\ \text{pb}$, 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 ($VH$).** 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 $H \to b\bar b$ decay. - **Top-associated production ($t\bar t H$).** 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. $$ % caption: 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. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=0.95] \definecolor{acc}{HTML}{4A6FA5} % --- ggF (left) --- \draw[black, thick, dashed] (-7.0,1.0) -- (-6.0,1.0); \draw[black, thick, dashed] (-7.0,-0.2) -- (-6.0,-0.2); \node[black, left, font=\scriptsize] at (-7.0,1.0) {g}; \node[black, left, font=\scriptsize] at (-7.0,-0.2) {g}; % top loop triangle \draw[thick] (-6.0,1.0) -- (-5.2,0.4) -- (-6.0,-0.2) -- cycle; \node[font=\scriptsize] at (-5.55,0.4) {top}; \draw[thick, dashed] (-5.2,0.4) -- (-4.2,0.4); \node[right, font=\scriptsize] at (-4.2,0.4) {H}; \node[black, below, font=\scriptsize] at (-5.6,-0.9) {gluon fusion}; % --- VBF (middle) --- \draw[thick, ->] (-3.0,1.4) -- (-2.2,1.1); \draw[thick] (-2.2,1.1) -- (-1.4,0.9); \draw[thick, ->] (-1.4,0.9) -- (-0.7,1.2); \draw[thick] (-0.7,1.2) -- (0.0,1.5); \draw[thick, ->] (-3.0,-0.7) -- (-2.2,-0.4); \draw[thick] (-2.2,-0.4) -- (-1.4,-0.2); \draw[thick, ->] (-1.4,-0.2) -- (-0.7,-0.5); \draw[thick] (-0.7,-0.5) -- (0.0,-0.8); \draw[black, thick, dashed] (-1.4,0.9) -- (-1.4,0.35); \draw[black, thick, dashed] (-1.4,-0.2) -- (-1.4,0.35); \node[black, right, font=\scriptsize] at (-1.3,0.6) {W, Z}; \draw[black, thick, dashed] (-1.4,0.35) -- (-0.4,0.35); \node[black, right, font=\scriptsize] at (-0.4,0.35) {H}; \node[black, below, font=\scriptsize] at (-1.5,-1.1) {vector-boson fusion}; % --- VH (right) --- \draw[thick, ->] (2.2,1.2) -- (2.9,0.85); \draw[thick] (2.9,0.85) -- (3.6,0.5); \draw[thick, ->] (2.2,-0.2) -- (2.9,0.15); \draw[thick] (2.9,0.15) -- (3.6,0.5); \node[left, font=\scriptsize] at (2.2,1.2) {q}; \node[left, font=\scriptsize] at (2.2,-0.2) {q-bar}; \draw[black, thick, dashed] (3.6,0.5) -- (4.7,0.5); \fill[black] (3.6,0.5) circle (1.6pt); \node[black, above, font=\scriptsize] at (4.15,0.55) {W, Z}; \fill[black] (4.7,0.5) circle (1.6pt); \draw[black, thick, dashed] (4.7,0.5) -- (5.5,1.1); \node[black, right, font=\scriptsize] at (5.5,1.1) {H}; \draw[black, thick] (4.7,0.5) -- (5.5,-0.1); \node[black, right, font=\scriptsize] at (5.5,-0.1) {W, Z}; \node[black, below, font=\scriptsize] at (3.9,-1.1) {associated}; \end{tikzpicture} $$ ## Decay modes A $125\ \text{GeV}$ 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 $b\bar b$ — the heaviest fermion pair kinematically open — followed by the off-shell $WW^\ast$. | Mode | Branching ratio | Role | | --- | --- | --- | | $b\bar b$ | $58\%$ | largest; huge QCD background | | $WW^\ast$ | $21\%$ | large; used with leptonic $W$ | | $gg$ | $8.2\%$ | via top loop; indistinct | | $\tau^+\tau^-$ | $6.3\%$ | accessible, moderate background | | $c\bar c$ | $2.9\%$ | hard to tag | | $ZZ^\ast$ | $2.6\%$ | golden $4\ell$ subchannel | | $\gamma\gamma$ | $0.23\%$ | rare, clean, discovery channel | The two channels that carried the discovery are not the largest but the cleanest. $H \to b\bar b$ has by far the highest rate, yet the LHC produces $b\bar b$ 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 **$H \to \gamma\gamma$.** 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. $$ % caption: The diphoton invariant-mass spectrum. Non-resonant photon-pair production % gives a smooth, steeply falling background; the Higgs adds a narrow excess at its % mass near 125 GeV. The peak width is the detector resolution, not the natural % width, which is only a few MeV. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.6,0) node[right, font=\scriptsize] {diphoton mass (GeV)}; \draw[->, black] (0,0) -- (0,3.6) node[above, font=\scriptsize] {events}; % smooth falling background: y = 3.0*exp(-0.28 x) approx via plot, overflow safe \draw[black, thick] plot[domain=0.3:6.2, samples=60] (\x, {3.0/(1 + 0.55*\x)}); \node[black, right, font=\scriptsize] at (4.0,0.85) {background}; % peak on top near x=3.3 (=125 GeV mark) \draw[acc, very thick] plot[domain=2.3:4.3, samples=60] (\x, {3.0/(1 + 0.55*\x) + 1.5*exp(-6*(\x-3.3)*(\x-3.3))}); \draw[black, densely dotted] (3.3,0) -- (3.3,2.6); \node[acc, above, font=\scriptsize] at (3.3,2.65) {125 GeV}; \node[acc, right, font=\scriptsize] at (4.3,2.0) {Higgs peak}; \end{tikzpicture} $$ **$H \to ZZ^\ast \to 4\ell$.** 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 $\mu$). 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 $ZZ^\ast$ production — is small and well modeled. A handful of clean four-lepton events clustered at one mass is a nearly background-free signal. $$ % caption: The golden four-lepton channel. The Higgs decays to two Z bosons, one % on-shell and one virtual; each Z decays to a charged-lepton pair. The four-lepton % invariant mass reconstructs the Higgs mass with high resolution over a small, % well-understood background. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[acc, thick, dashed] (-3.4,0) -- (-2.2,0); \node[acc, left, font=\scriptsize] at (-3.4,0) {H}; \fill[acc] (-2.2,0) circle (1.8pt); % two Z lines \draw[black, thick, dashed] (-2.2,0) -- (-0.6,1.1); \node[black, above, font=\scriptsize] at (-1.4,0.7) {$Z$}; \draw[black, thick, dashed] (-2.2,0) -- (-0.6,-1.1); \node[black, below, font=\scriptsize] at (-1.4,-0.7) {Z (virtual)}; % lepton pairs \fill[black] (-0.6,1.1) circle (1.6pt); \draw[thick, ->] (-0.6,1.1) -- (1.0,1.7); \draw[thick, ->] (-0.6,1.1) -- (1.0,0.7); \node[right, font=\scriptsize] at (1.0,1.7) {lepton}; \node[right, font=\scriptsize] at (1.0,0.7) {antilepton}; \fill[black] (-0.6,-1.1) circle (1.6pt); \draw[thick, ->] (-0.6,-1.1) -- (1.0,-0.5); \draw[thick, ->] (-0.6,-1.1) -- (1.0,-1.7); \node[right, font=\scriptsize] at (1.0,-0.5) {lepton}; \node[right, font=\scriptsize] at (1.0,-1.7) {antilepton}; \end{tikzpicture} $$ ## The 2012 discovery In July 2012 the ATLAS and CMS collaborations independently reported a new boson near $125\ \text{GeV}$, each combining the $\gamma\gamma$ and $ZZ^\ast \to 4\ell$ channels (with $WW^\ast$ contributing) to reach the $5\sigma$ threshold — a background fluctuation probability below $3 \times 10^{-7}$ — 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.[^atlas-cms] The current world-average mass is $$ M_h = 125.25 \pm 0.17\ \text{GeV}, $$ now one of the best-measured parameters of the Standard Model.[^pdg-higgs] Because $M_h$ fixes $\lambda = M_h^2 / 2v^2 \approx 0.13$, 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 $125\ \text{GeV}$ 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 $ZZ^\ast \to 4\ell$ and $WW^\ast$ 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 $\tau$ 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. $$ % caption: Measured Higgs coupling strength versus particle mass, on logarithmic % axes. The Standard Model predicts a straight line: fermion couplings proportional % to mass, weak-boson couplings proportional to mass squared over the vacuum value. % The measured points for tau, b, W, Z, and top lie along that line. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.4,0) node[right, font=\scriptsize] {particle mass (log)}; \draw[->, black] (0,0) -- (0,4.2) node[above, font=\scriptsize] {coupling (log)}; % predicted straight line \draw[acc, very thick] (0.4,0.4) -- (5.8,3.9); \node[acc, right, font=\scriptsize, rotate=33] at (2.6,1.8) {SM prediction}; % measured points along the line \fill[black] (1.4,0.95) circle (2pt); \node[black, above left, font=\scriptsize] at (1.4,0.95) {tau}; \fill[black] (2.4,1.6) circle (2pt); \node[black, above left, font=\scriptsize] at (2.4,1.6) {b}; \fill[black] (3.9,2.55) circle (2pt); \node[black, above left, font=\scriptsize] at (3.9,2.55) {W}; \fill[black] (4.3,2.8) circle (2pt); \node[black, below right, font=\scriptsize] at (4.35,2.75) {Z}; \fill[black] (5.4,3.6) circle (2.4pt); \node[above left, font=\scriptsize] at (5.4,3.6) {top}; \end{tikzpicture} $$ ## 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 $\lambda$ 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](/particle-physics/beyond-standard-model/hierarchy-problem-naturalness) — 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 $t\bar tH$ as cleaner or more direct channels. It decays most often to $b\bar b$ and $WW^\ast$, but the discovery rested on the rare, clean $H \to \gamma\gamma$ and $H \to ZZ^\ast \to 4\ell$ modes, whose narrow invariant-mass peaks rise over smooth backgrounds. ATLAS and CMS announced a $5\sigma$ signal near $125\ \text{GeV}$ in 2012; the world-average mass is now $125.25\ \text{GeV}$. 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](/particle-physics/electroweak-higgs/standard-model). [^atlas-cms]: The 2012 discovery of a boson near $125\ \text{GeV}$ in the $\gamma\gamma$ and $ZZ^\ast \to 4\ell$ 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](https://arxiv.org/abs/1207.7214), and the CMS collaboration, "Observation of a new boson at a mass of 125 GeV," [arXiv:1207.7235](https://arxiv.org/abs/1207.7235). The production and decay phenomenology is reviewed in Thomson, §17.8–17.10. [^pdg-higgs]: The world-average mass $M_h = 125.25 \pm 0.17\ \text{GeV}$, the branching ratios, and the spin-parity and coupling measurements are from the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov). The mass-proportional couplings and $\lambda = M_h^2/2v^2$ are discussed in Tong, _The Standard Model_ (Cambridge Part III), §5.2, [damtp.cam.ac.uk/user/tong/standardmodel.html](http://www.damtp.cam.ac.uk/user/tong/standardmodel.html).