Experiment: The Higgs Boson Discovery

Tests Phenomena 104.59 and 104.62. Assuming Proposition 104.26, Proposition 104.61 and Definition 11.70.

On 4 July 2012 the ATLAS [Aad:2012] and CMS [Chatrchyan:2012] collaborations independently reported a new boson near \(125\,\mathrm{GeV}\)/\(c^{2}\), each at five standard deviations, in the diphoton and four-lepton channels. This chapter is the experimental closure of Electroweak Unification and the Higgs Boson: the Brout–Englert–Higgs mechanism [Englert:1964] [Higgs:1964] [Guralnik:1964] had been established indirectly for decades by the masses of the \(W\) and \(Z\) [Arnison:1983a] [Arnison:1983b] and by the precision electroweak fits [Schael:2006], but the scalar itself — the one particle of the Standard Model whose mass the theory does not predict — remained unobserved, its allowed mass window closed from below by LEP [Barate:2003] and eroded by the Tevatron [Aaltonen:2012]. The chapter sits at the end of the Standard Model sequence because it completes its particle content; what it does not complete is the theory, and the final sections say so.

Two features of this experiment demand more space than usual. The first is statistics: a discovery claim built on a few hundred events over a large smooth background needs its test statistic, its look-elsewhere correction and its blinding protocol stated as carefully as its detector [Read:2002] [Cowan:2011]. All of that machinery is derived in Probability and Statistics, and Section 113.5.1 applies it to the published numbers rather than gesturing at “five sigma”. The second is that the measurement did not stop in 2012 — spin-parity [Aad:2013] [Chatrchyan:2013], couplings against mass [Aad:2016] [Aad:2022] [Tumasyan:2022] and the total width [Aad:2023] have all been measured since, while the Higgs self-coupling, which is what would actually test the shape of the potential, has not.

Historical context and the prediction under test

The Brout–Englert–Higgs mechanism

The problem the mechanism solves is stated in Theorem 104.8: a mass term for a gauge field is not gauge invariant, so a gauge theory of the weak interaction predicts massless carriers, and the carriers are not massless. The obvious repair — write the mass term anyway — destroys renormalizability (Theorem 104.4). The mechanism is the non-obvious repair: leave the Lagrangian invariant and let the vacuum fail to be.

Three papers in 1964 said this independently. Englert and Brout [Englert:1964] computed the vector self-energy directly and found a pole at nonzero momentum; Higgs [Higgs:1964a] [Higgs:1964] gave the abelian model and, in the second paper, noted explicitly that a massive scalar excitation survives; Guralnik, Hagen and Kibble [Guralnik:1964] gave the operator argument for why Goldstone's theorem does not apply. Higgs later wrote the argument out in full [Higgs:1966], and Kibble extended it to a non-abelian group [Kibble:1967], which is the case the Standard Model needs. The physical precedent was already in the literature: Anderson [Anderson:1963] had pointed out that the plasmon in a superconductor is exactly a gauge field that has acquired a mass from a condensate, and that the would-be massless mode is absent there for the same reason. Section 104.3.1 develops the analogy through Nambu's treatment of superconductivity [Nambu:1960].

What has to be evaded is Goldstone's theorem [Goldstone:1961], proved as Theorem 104.17: a spontaneously broken continuous global symmetry produces one massless scalar for every broken generator. Massless scalars are not observed. The evasion, derived as Theorem 104.19 and in the non-abelian case as Theorem 104.22, is that when the broken symmetry is gauged the would-be Goldstone modes are not physical states at all: a gauge transformation removes them from the scalar sector, and they reappear as the longitudinal polarizations that a massless vector does not have and a massive one does.

The bookkeeping is worth doing explicitly, because it is what makes the prediction sharp.

The one free parameter

The scalar potential is

with \(\Phi\) of dimension \(\mathrm{J}^{1/2}\,\mathrm{m}^{-1/2}\), \(\mu\) of dimension \(/\mathrm{m}\) and \(\lambda\) of dimension \(/\mathrm{J}/\mathrm{m}\), so that \(V\) is an energy density (Definition 104.24). Two parameters appear, and one combination of them is already measured.

The consequence for the experiment is structural and it is what makes a null result meaningful. A search that scans \(m_{H}\) is testing a one-parameter family of fully specified theories, not fitting a shape to data. At each candidate mass the predicted rate in every channel is a number, so the data either accommodate it or exclude it; there is nothing to adjust. And when a signal does appear, the same rigidity makes the discovery over-determined: the mass measured in the diphoton channel must agree with the mass measured in the four-lepton channel, the ratio of the two rates must be the predicted one, and the production mechanisms must appear in the predicted proportion. Two experiments each measuring two channels give eight numbers where the theory has one.

The wider accounting — how many free parameters the Standard Model has in total, and which measurement fixes each — is tabulated in The Free Parameters of Physics. The scalar mass is one entry of that table, and Section 113.5.2 is the entry that this chapter fills in.

Indirect constraints

Before the scalar was produced it was constrained, because it circulates in loops. The relation between \(M_{W}\), \(M_{Z}\), \(\alpha\) and \(G_{F}\) receives a radiative correction \(\Delta r\), defined by Equation (104.43) and evaluated from the measured masses in Remark 104.33, which finds \(\Delta r=0.036\) and identifies the scalar-mass logarithm as one of the pieces making it up. What matters here is that dependence's form: it is logarithmic,

with \(\hat{g}\) the dimensionless \(\SU(2)\) coupling of Notation 104.1 and \(c_{H}\) a pure number of order unity that this book does not derive, whereas the dependence on the top mass is quadratic, \(\Delta r|_{t}\propto m_{t}^{2}\) by Equation (104.52). The two sensitivities have very different consequences and the difference is Veltman's screening theorem [Veltman:1977]: a heavy scalar decouples logarithmically, so precision data constrain it weakly, while a heavy top does not decouple at all.

Weak is not zero. The LEP and SLC programme measured \(M_{Z}\), the \(Z\) partial widths and a dozen asymmetries to a part in \(10^{3}\) or better [Schael:2006]; combined with \(M_{W}\), \(m_{t}\), \(\alpha\) and \(G_{F}\) the system is over-constrained, and the global fit before 2012 preferred a scalar mass of order \(100\,\mathrm{GeV}\)/\(c^{2}\) with an upper bound in the low hundreds [Baak:2014]. The particle was found at \(125\,\mathrm{GeV}\)/\(c^{2}\), inside that range.

What made the constraint credible in advance was that the same method had already worked once, on a harder case. The quadratic top sensitivity let the LEP data locate the top quark's mass before the Tevatron produced it, and the indirect value agreed with the direct measurement [Abachi:1995] [Abe:1995]. A method that has predicted one unseen particle correctly is evidence about the next one; a method that has not is an extrapolation.

LEP and Tevatron searches

Two machines searched before the LHC, in two entirely different ways, and between them they set the window the LHC had to cover.

LEP. An \(e^{+}e^{-}\) collider produces the scalar through \(e^{+}e^{-}\to Z^{*}\to ZH\), the same \(HZZ\) vertex read as a production vertex. The event is fully constrained — the initial state has known energy — so the search is a bump hunt in the recoil mass against an identified \(Z\), with essentially no combinatorial background. What limits it is not statistics but kinematics.

The LEP combination of ALEPH, DELPHI, L3 and OPAL left a mild excess at the very edge of its sensitivity, near \(115\,\mathrm{GeV}\)/\(c^{2}\), of less than two standard deviations in the four-experiment combination and driven largely by one experiment [Barate:2003]. It did not survive, and the way it was reported is worth recording: the collaborations quoted the limit and the excess together, with the excess described as compatible with a background fluctuation, rather than either suppressing it or promoting it. The combination used the modified frequentist \(CL_{s}\) construction [Read:2002], whose purpose is exactly to prevent a downward background fluctuation from excluding a signal the experiment had no sensitivity to (Remark 11.95).

Tevatron. A \(p\bar{p}\) collider at \(\sqrt{s}\,c=1.96\,\mathrm{TeV}\) cannot use the diphoton channel — the rate is too low for its integrated luminosity — so CDF and D0 attacked the largest branching fraction instead, \(H\to b\bar{b}\), made usable by demanding an associated \(W\) or \(Z\) to suppress the enormous QCD production of \(b\) quarks. The combined analysis of the full Tevatron dataset reported a broad excess in that channel with a maximum local significance of about three standard deviations near \(125\,\mathrm{GeV}\)/\(c^{2}\), falling to roughly two and a half after correcting for the mass range searched [Aaltonen:2012]. The same combination excluded a Standard Model scalar between about \(147\,\mathrm{GeV}\)/\(c^{2}\) and \(179\,\mathrm{GeV}\)/\(c^{2}\).

The Tevatron result is the one place in this story where the associated-production channel with the largest branching ratio was competitive, and it is instructive that it produced evidence rather than a discovery. The excess was broad — the \(b\bar{b}\) mass resolution is of order \(15\,\mathrm{GeV}\)/\(c^{2}\), ten times worse than the diphoton channel — so it constrained a rate over a wide window rather than locating a resonance.

The window entering 2012. Combining the LEP lower bound, the Tevatron exclusion and the LHC's own 2011 exclusions, a Standard Model scalar was confined to roughly \(115\text{–}130\,\mathrm{GeV}\)/\(c^{2}\), with a smaller surviving region at high mass that the electroweak fit already disfavoured. Inside that window the branching fractions of Table 104.4 are changing rapidly: \(WW^{*}\) and \(ZZ^{*}\) are climbing as the thresholds approach, \(b\bar{b}\) is falling, and \(\gamma\gamma\) passes through its maximum. It is the one mass region in which the two high-resolution channels are simultaneously accessible, which is why the LHC was able to settle the question there and would have had a much harder time \(20\,\mathrm{GeV}\)/\(c^{2}\) higher.

Apparatus

Three instruments are involved: one accelerator and two detectors. The accelerator's job is to deliver, in a few years, enough proton–proton collisions that a process occurring once in \(3\times10^{9}\) of them yields a few hundred events. The detectors' job is to recognise those events in real time, discard the rest before they can be written to disk, and measure the survivors well enough that a resonance \(1\,\mathrm{GeV}/c^{2}\) wide can be seen on a continuum background a hundred times larger.

ATLAS and CMS were built to different designs on purpose. Their magnets, their calorimeter technologies and their reconstruction strategies differ in almost every respect, so their systematic errors are largely independent. That is not redundancy: it is the only protection available against a mistake that a single collaboration could make and not detect, and it is why the phrase “two experiments independently” carries weight in Section 113.4.3 that “two analyses of the same data” would not.

The Large Hadron Collider

The machine [Evans:2008] occupies the tunnel dug for LEP, a ring of circumference \(26659\,\mathrm{m}\) lying between \(45\,\mathrm{m}\) and \(170\,\mathrm{m}\) below the surface and tilted by \(1.4\,^\circ\). Reusing the tunnel fixed the circumference and therefore fixed everything else: to reach a beam energy of several \(\mathrm{TeV}\) in a ring of that size the bending field must be far beyond what iron can supply, so the LHC is a superconducting machine, and because two counter-rotating proton beams need opposite bending fields it uses twin apertures inside one cryostat and one yoke rather than two separate rings.

The beams are bunched by \(400\,\mathrm{MHz}\) superconducting radio-frequency cavities, eight per beam delivering about \(2\,\mathrm{MV}\) each. A proton completes the ring at

to within the \(2.8\times 10^{-8}\) by which \(v\) falls short of \(c\). In the 2011 and 2012 runs the bunches were spaced by \(50\,\mathrm{ns}\) — twice the design spacing — which puts consecutive bunches \(15.0\,\mathrm{m}\) apart along the beam and gives a crossing rate of \(20\,\mathrm{MHz}\) at each interaction point where bunches are present. With \(1374\) filled bunches in 2012 the actual crossing rate was \(1374\,f_{\text{rev}}=15.4\,\mathrm{MHz}\).

The last machine property that matters is not a machine property at all but a property of the proton. The scalar is produced dominantly by gluon fusion, \(gg\to H\), through a top-quark loop, and that dominance is a statement about parton densities rather than about couplings: at \(\sqrt{s}\,c=8\,\mathrm{TeV}\) a central resonance of mass \(125\,\mathrm{GeV}/c^{2}\) is made from partons carrying momentum fractions near \(x=m_{H}c^{2}/(\sqrt{s}\,c)=0.0156\), and at that \(x\) the gluon distribution is an order of magnitude above any single quark distribution (Section 116.6.5). The distributions themselves were measured in electron–proton scattering (Experiment: Deep Inelastic Scattering) and are usable here only because collinear factorization (Theorem 102.53) makes them universal — a theorem stated there at leading order, with its all-orders proof recorded as owed. Without it, and without the deep inelastic programme that measured its inputs, no LHC cross section could be predicted at all.

ATLAS

ATLAS [Aad:2008] is \(44\,\mathrm{m}\) long and \(25\,\mathrm{m}\) in diameter and weighs about \(7\times 10^{6}\,\mathrm{kg}\). Its distinguishing choices are a modest solenoid around a compact tracker, a sampling electromagnetic calorimeter with unusually fine longitudinal and lateral segmentation, and a very large air-core toroid outside the calorimeters that measures muons independently of the tracker.

Tracking. A superconducting solenoid \(5.3\,\mathrm{m}\) long with a \(2.4\,\mathrm{m}\) bore produces an axial field of \(2\,\mathrm{T}\); it is deliberately thin, about \(0.66\) radiation lengths, because it sits in front of the electromagnetic calorimeter. Inside it are three subsystems covering \(\abs{\eta}<2.5\): a silicon pixel detector of three barrel layers at radii \(50.5\,\mathrm{mm}\), \(88.5\,\mathrm{mm}\) and \(122.5\,\mathrm{mm}\) with \(8.0\times 10^{7}\) channels of \(50\,\mu\mathrm{m}\times400\,\mu\mathrm{m}\); a silicon microstrip tracker of four double barrel layers out to \(514\,\mathrm{mm}\) with \(80\,\mu\mathrm{m}\) pitch; and a transition-radiation tracker of about \(3.5\times 10^{5}\) straw tubes of \(4\,\mathrm{mm}\) diameter, giving roughly \(36\) measurements per track and, from the transition radiation emitted in the interleaved radiator, an electron-versus-pion discriminant.

Electromagnetic calorimetry. Lead absorbers immersed in liquid argon at \(87.3\,\mathrm{K}\), folded into an accordion so that the readout can be brought out from the front and back with no cracks in azimuth. The barrel covers \(\abs{\eta}<1.475\) and the two endcaps \(1.375<\abs{\eta}<3.2\); the total depth is at least \(22\) radiation lengths in the barrel. The segmentation is the point of the design and is worth stating in full:

• a presampler layer for \(\abs{\eta}<1.8\), which measures the energy already lost upstream;

• a first sampling of about \(4.3\) radiation lengths divided into strips of \(\Delta\eta=0.0031\), eight times finer than the layers behind it;

• a second sampling of about \(16\) radiation lengths in cells of \(\Delta\eta\times\Delta\phi=0.025\times0.025\), which collects most of the shower;

• a third sampling of \(\Delta\eta=0.05\) to catch the tail.

The design energy resolution is

the first term being the sampling fluctuation of a lead–argon stack and the second the constant term set by calibration and mechanical uniformity.

The fine first sampling does two things that the discovery depended on. A neutral pion of \(50\,\mathrm{GeV}\) decays to two photons with a typical opening angle of \(2m_{\pi}c^{2}/E\approx5.4\times 10^{-3}\), which at the calorimeter face is a separation of a few millimetres — resolvable in strips of \(\Delta\eta=0.0031\) and invisible in cells of \(0.025\). And because the first and second samplings measure \(\eta\) at two different radii, the photon's direction can be reconstructed from the calorimeter alone, giving its origin along the beam to about \(15\,\mathrm{mm}\) with no help from the tracker. Proposition 113.14 explains why that number matters.

Hadronic calorimetry and muons. Steel and scintillating tile in the barrel, copper and liquid argon in the endcaps, and a copper–tungsten forward calorimeter reaching \(\abs{\eta}=4.9\), which is what makes a missing-transverse-momentum measurement possible. Outside everything sits the muon spectrometer: eight air-core superconducting coils forming a barrel toroid \(25.3\,\mathrm{m}\) long and \(20.1\,\mathrm{m}\) in outer diameter, plus two endcap toroids, giving a bending integral \(\int B\,\dd l\) between \(1.5\,\mathrm{T}\,\mathrm{m}\) and \(7.5\,\mathrm{T}\,\mathrm{m}\) depending on \(\eta\). Air-core means there is no iron to scatter the muon, so the spectrometer measures momentum on its own to about \(10\,\mathrm{\%}\) at \(1\,\mathrm{TeV}\); the price is a very large and mechanically demanding structure. Precision coordinates come from drift tubes of \(30\,\mathrm{mm}\) diameter filled with argon and carbon dioxide at \(3\times 10^{5}\,\mathrm{Pa}\) — three atmospheres — resolving about \(80\,\mu\mathrm{m}\) per tube.

Trigger. The crossing rate is \(20\,\mathrm{MHz}\) and the recording rate is a few hundred hertz, so the trigger must reject about \(5\times 10^{4}\) events for every one it keeps, in real time and without ever looking at an event twice. A hardware first level, using reduced-granularity calorimeter sums and dedicated muon chambers, made its decision within \(2.5\,\mu\mathrm{s}\) and reduced the rate to about \(70\,\mathrm{kHz}\); two software levels running on a farm of order \(2\times 10^{4}\) processor cores reduced it to about \(400\,\mathrm{Hz}\) recorded in 2012. The diphoton trigger required two electromagnetic clusters above roughly \(35\,\mathrm{GeV}\) and \(25\,\mathrm{GeV}\) of transverse energy. Any event not selected within \(2.5\,\mu\mathrm{s}\) is gone permanently, which is why the trigger menu is part of the physics analysis and not part of the plumbing.

CMS

CMS [Chatrchyan:2008] is \(21.6\,\mathrm{m}\) long, \(14.6\,\mathrm{m}\) in diameter and weighs about \(1.25\times 10^{7}\,\mathrm{kg}\) — half the length of ATLAS and nearly twice the mass. Every one of its choices is the opposite of the corresponding ATLAS choice, and the discovery is stronger for it.

The magnet. A single superconducting solenoid of \(6\,\mathrm{m}\) inner diameter and \(12.5\,\mathrm{m}\) length, operated at \(3.8\,\mathrm{T}\) with a stored energy of \(2.6\,\mathrm{GJ}\). Everything except the muon chambers sits inside it. The high field buys track momentum resolution in a small volume; the cost is that the coil is thick, so the electromagnetic calorimeter must be inside it, which in turn constrains its radius and therefore its cost per unit solid angle. The return flux is carried by an iron yoke of about \(1.8\,\mathrm{T}\) in which the muon chambers are embedded, so the muon system reuses the same magnetic circuit rather than needing one of its own.

Tracking. All silicon, about \(200\,\mathrm{m}^{2}\) of it, the largest such device built at the time. Three pixel barrel layers at radii \(4.4\,\mathrm{cm}\), \(7.3\,\mathrm{cm}\) and \(10.2\,\mathrm{cm}\) carry \(6.6\times 10^{7}\) pixels of \(100\,\mu\mathrm{m}\times150\,\mu\mathrm{m}\); ten microstrip barrel layers reach \(r=1.1\,\mathrm{m}\) with \(9.3\times 10^{6}\) strips. Coverage is \(\abs{\eta}<2.5\).

Electromagnetic calorimetry. This is the subsystem CMS built for the diphoton channel specifically, and the reasoning is explicit in the design report: a homogeneous crystal calorimeter has no sampling fluctuation, so its stochastic term can be several times smaller than a sampling device's, and the diphoton mass resolution is dominated by that term. The calorimeter is \(75\,848\) lead-tungstate (\(\text{PbWO}_{4}\)) scintillating crystals — \(61\,200\) in the barrel for \(\abs{\eta}<1.479\) and \(7324\) in each endcap. The material was chosen for density: \(8280\,\mathrm{kg}/\mathrm{m}^{3}\), a radiation length of \(0.89\,\mathrm{cm}\) and a Molière radius of \(2.2\,\mathrm{cm}\), so a barrel crystal \(230\,\mathrm{mm}\) long is \(25.8\) radiation lengths in a \(22\,\mathrm{mm}\times22\,\mathrm{mm}\) front face. The design resolution measured in a test beam is

a stochastic term nearly four times smaller than Equation (113.13).

That advantage is bought at a price that had to be engineered away. Lead tungstate is a poor scintillator — about \(30\) photons per \(\mathrm{MeV}\) reach the photodetector — so the barrel is read out with avalanche photodiodes of gain \(50\) and the endcaps with vacuum phototriodes, both able to operate inside \(3.8\,\mathrm{T}\) where a photomultiplier cannot. Its light yield also falls by about \(2\,\mathrm{\%}\) per kelvin, so the whole calorimeter is held at \(18\,\mathrm{^\circ\mathrm{C}}\) stabilised to \(0.05\,\mathrm{K}\); and the crystals darken under irradiation and recover between fills, which forces a continuous transparency monitoring with injected laser light. A preshower of two lead converters and two silicon strip planes, three radiation lengths in all with a strip pitch of \(1.9\,\mathrm{mm}\), sits in front of the endcaps and does there what ATLAS's fine strips do in the barrel: separate a \(\pi^{0}\)'s two photons from one.

Reconstruction. CMS reconstructs events by particle flow: rather than reading each subdetector separately, the algorithm builds a list of individual particles by linking tracks to calorimeter clusters to muon segments, and then constructs jets, missing transverse momentum and isolation variables from that list. The method needs a tracker that measures nearly every charged particle and a calorimeter finely segmented enough to associate clusters with tracks, which is exactly what the \(3.8\,\mathrm{T}\) solenoid and the crystal array provide. Its relevance here is isolation: a photon candidate's isolation can be computed from the charged particles associated with the chosen vertex alone, which suppresses the pile-up dependence that a purely calorimetric isolation would carry.

Trigger. The same two-stage structure with different numbers: a hardware level with a \(3.2\,\mu\mathrm{s}\) latency reducing \(20\,\mathrm{MHz}\) to at most \(100\,\mathrm{kHz}\), and a software high-level trigger on a commercial farm reducing that to a few hundred hertz. CMS's diphoton triggers used lower transverse-energy thresholds than ATLAS's, near \(26\,\mathrm{GeV}\) and \(18\,\mathrm{GeV}\), compensated by isolation and shower-shape requirements applied online.

Procedure

Production and decay channels

Four production mechanisms contribute at the LHC, and all four are the vertices of Equation (104.69) read as production rather than decay. In descending order of rate at \(m_{H}c^{2}=125\,\mathrm{GeV}\) and \(\sqrt{s}\,c=8\,\mathrm{TeV}\):

• gluon fusion, \(gg\to H\), through a loop of virtual quarks dominated by the top because its Yukawa coupling is the largest, \(\sigma\approx1.93\times 10^{-39}\,\mathrm{m}^{2}\) (\(19.3\) picobarns), about \(87\,\mathrm{\%}\) of the total;

• vector-boson fusion, \(qq\to qqH\), in which each incoming quark radiates a \(W\) or \(Z\) that fuse, \(\sigma\approx1.58\times 10^{-40}\,\mathrm{m}^{2}\), and which is identified by two forward jets with a large rapidity gap;

• associated production, \(q\bar{q}\to VH\) with \(V=W\) or \(Z\), \(\sigma\approx1.12\times 10^{-40}\,\mathrm{m}^{2}\) summed over the two, the channel the Tevatron used;

• top-quark associated production, \(gg\to t\bar{t}H\), \(\sigma\approx1.29\times 10^{-41}\,\mathrm{m}^{2}\), which measures the top Yukawa coupling at tree level rather than through a loop.

The sum is \(2.21\times 10^{-39}\,\mathrm{m}^{2}\) at \(8\,\mathrm{TeV}\) and \(1.74\times 10^{-39}\,\mathrm{m}^{2}\) at \(7\,\mathrm{TeV}\) — in the customary units \(22.1\) and \(17.4\) picobarns [Aad:2012] [Chatrchyan:2012]. Set against the inelastic proton–proton cross section of \(7.3\times 10^{-30}\,\mathrm{m}^{2}\), the scalar is produced in \(3.0\times 10^{-10}\) of all collisions: one in three thousand million.

Event selection and background estimation

Photons. A photon candidate is an electromagnetic cluster with no matched track (or, for a converted photon, a matched conversion vertex) whose lateral and longitudinal shower shape is consistent with a single electromagnetic shower rather than with two overlapping ones or with a jet fluctuating to a leading \(\pi^{0}\). About a quarter of photons convert in the tracker material before reaching the calorimeter, which is why both experiments treat converted and unconverted candidates separately. Isolation — the transverse energy in a cone around the candidate, corrected for pile-up — rejects photons produced inside jets. ATLAS required two candidates with transverse energies above \(40\,\mathrm{GeV}\) and \(30\,\mathrm{GeV}\) within \(\abs{\eta}<2.37\) excluding the barrel to endcap transition; CMS scaled its thresholds with the candidate mass, requiring \(E_{T}>m_{\gamma\gamma}c^{2}/3\) and \(m_{\gamma\gamma}c^{2}/4\), which keeps the selection from sculpting a peak into the background.

Leptons. Electrons are clusters with a matched track; muons are tracks matched between the inner detector and the muon system. The four-lepton selection requires two same-flavour opposite-charge pairs, with the pair closer to the \(Z\) mass required to be near it and the second allowed to be far off shell — which is the whole point, since at \(m_{H}c^{2}=125\,\mathrm{GeV}\) the decay \(H\to ZZ\) is below threshold by \(57\,\mathrm{GeV}\) and one boson must be virtual. Lepton transverse-momentum thresholds are low, of order \(7\,\mathrm{GeV}/c\) for the softest, because the four leptons share \(125\,\mathrm{GeV}\) between them.

The measurement that everything else serves is the invariant mass, and its resolution is where the two detector designs cash out.

Because the resolution and the signal fraction both vary strongly across the detector and across event topologies, neither experiment performed a single inclusive fit. ATLAS divided its diphoton candidates into ten categories in 2011 and fourteen in 2012, by conversion status, by pseudorapidity, and by the component of the diphoton transverse momentum orthogonal to the thrust axis; CMS used a multivariate classifier trained on the expected resolution and predicted signal fraction, plus a separate class tagged by two forward jets for vector-boson fusion. The gain from doing so is not a technicality.

Background estimation. The two channels are treated in opposite ways, and the difference is instructive.

For \(\gamma\gamma\) the background is large, smooth and not calculable: it comes from continuum diphoton production, from photon plus jet with the jet misidentified, and from dijets with two misidentifications, in proportions that no simulation predicts to the accuracy required. It is therefore taken from the data. An analytic function — an exponential, or a polynomial in a Bernstein basis — is fitted to the observed mass spectrum over a wide range simultaneously with a signal peak of fixed shape, so the background normalization and shape are determined by the sidebands and the fit returns the signal yield. The danger of the method is that a wrong functional form can absorb a real signal or manufacture a spurious one, and it is controlled by the spurious-signal test: the chosen function is fitted to a background-only sample far larger than the data, and the fitted “signal” yield it returns is required to be a small fraction of the statistical uncertainty on the real measurement. The residual bias is then carried as a systematic uncertainty. The whole procedure is fixed, function and categories and all, before the signal region is examined.

For \(4\ell\) the background is small and partly calculable. The irreducible part is continuum \(ZZ^{*}\) production, taken from simulation and normalized where the simulation can be checked — on the \(Z\) peak itself, in the same final state, where the rate is thousands of times higher and the physics is the same. The reducible part, \(Z\) plus jets and \(t\bar{t}\) with jets faking leptons, is estimated from control regions in the data defined by inverting the lepton identification or isolation requirements. Both estimates are validated in sidebands before use.

Blinding and the statistical procedure

The statistical machinery is derived in Probability and Statistics and is used here as derived. This subsection states which pieces are in force; Section 113.5.1 applies them to the published numbers.

The test statistic is the profile-likelihood ratio of Definition 11.70. The parameter of interest is a single signal strength \(\mu\), defined so that \(\mu=0\) is the background-only hypothesis and \(\mu=1\) the Standard Model prediction at the mass being tested; every systematic uncertainty — energy scales, resolutions, luminosity, background shape, theoretical cross sections — enters as a nuisance parameter, constrained by an auxiliary measurement and profiled out. The one-sided discovery statistic is \(\tilde{q}_{0}\) of Equation (11.88), which refuses to count a deficit as evidence for a signal, and the significance is \(Z=\sqrt{\tilde{q}_{0}}\) by Proposition 11.93. That identity is what allows a significance to be quoted without simulating an ensemble of experiments, and it rests on Wilks' theorem (Theorem 11.88) applied to the profile statistic (Corollary 11.89); the asymptotic formulae in the form the collaborations use them are [Cowan:2011].

Exclusion uses the same machinery reversed, with the modified frequentist \(CL_{s}\) criterion [Read:2002] — the ratio of the two tail probabilities rather than the numerator alone — so that a downward background fluctuation cannot exclude a signal the experiment had no power to see (Remark 11.95). This is the construction behind the LEP limit of Proposition 113.7 and behind every exclusion plot in the 2012 papers.

Two conventions must be stated because mixing them shifts a significance by more than a tenth of a standard deviation (Remark 11.94). The \(p\)-values quoted below are one-sided, so that \(Z=5\) corresponds to \(p=2.87\times 10^{-7}\); and the significances quoted by the collaborations as “local” are computed at a mass fixed in advance, which no search does. The correction is Section 113.5.1.

Blinding. None of this is worth anything if the analysis was adjusted after the data were seen. Both collaborations therefore fixed the entire chain — trigger, selection, categorization, background parametrization, systematic model and statistical procedure — using simulation and the mass sidebands, with the signal region hidden, and unblinded only afterwards. The reason is not procedural hygiene but arithmetic: the \(p\)-value of Definition 11.75 is the probability, under the null, of a result at least as extreme as the one observed by a test specified in advance. A cut adjusted after seeing the excess makes the number it produces uninterpretable, because the effective number of hypotheses tested is then unknown and unboundable. Blinding is what makes the five-sigma threshold mean the thing it appears to mean, and it is the reason the 2012 result was believed immediately rather than after a decade of argument.

Observations and data

The diphoton channel

What the data show is a narrow peak sitting on a steeply falling smooth continuum. The continuum falls by roughly an order of magnitude between \(m_{\gamma\gamma}c^{2}=100\,\mathrm{GeV}\) and \(160\,\mathrm{GeV}\) and has no structure in it; the peak is one resolution width across, of order \(1.5\,\mathrm{GeV}/c^{2}\) by Proposition 113.14, and stands a few per cent above the continuum in the best categories and well under one per cent in the worst. Nothing about the picture is dramatic; what makes it a measurement is the arithmetic done to it.

ATLAS reported a local significance of \(4.5\) standard deviations near \(m_{H}c^{2}=126.5\,\mathrm{GeV}\) in this channel alone, against \(2.5\) expected for a Standard Model scalar at that mass [Aad:2012]; CMS reported \(4.1\) against \(2.8\) expected near \(125\,\mathrm{GeV}\) [Chatrchyan:2012]. Two things about those pairs of numbers deserve saying plainly, because they are usually skipped.

First, the expected significances — \(2.5\) and \(2.8\) — are close to the crude estimate \(s/\sqrt{b}\approx2\) obtained in Proposition 113.12 from cross section, branching fraction, efficiency and background. That agreement is a check on the whole chain: an experiment whose expected sensitivity disagreed by a factor of two with a back-of-envelope calculation from published inputs would have something wrong with it.

Second, both observed values exceed both expected values. The fitted diphoton signal strength was correspondingly high in both experiments, around \(1.8\) with an uncertainty near \(0.5\) in ATLAS and similarly above unity in CMS. That is an upward fluctuation of about \(1.5\) standard deviations, present in the channel that dominates the combination — exactly the configuration in which an unblinded, after-the-fact analysis would be untrustworthy, and exactly why the selection was frozen beforehand. The later, much larger datasets brought the diphoton signal strength back to unity, which is the expected behaviour of a fluctuation and confirms the 2012 diagnosis.

One structural fact is available from this channel with no further analysis: a resonance that decays to two real photons cannot have spin \(1\). That is the Landau–Yang argument [Landau:1948] [Yang:1950], derived in Theorem 104.60 and again, in the vector form, in the derivation accompanying Phenomenon 113.22. The discovery channel therefore excluded one of the four spin hypotheses on the day it was announced, without any angular measurement.

The four-lepton channel

\(H\to ZZ^{*}\to4\ell\) is the opposite kind of measurement. The final state is fully reconstructed — four charged leptons, all their momenta measured in the tracker — so the mass resolution is set by lepton momentum resolution alone and is of order \(2\,\mathrm{GeV}/c^{2}\), and the background is small enough that the signal fraction exceeds unity in a narrow window. What it does not have is events: the arithmetic of Proposition 113.12 gives about twenty-five produced decays in the ATLAS dataset and fewer than ten surviving selection.

Both experiments observed roughly a dozen candidates in a window \(10\,\mathrm{GeV}/c^{2}\) wide around \(125\,\mathrm{GeV}/c^{2}\), against an expected background of about five, and reported local significances of \(3.6\) standard deviations (ATLAS, against \(2.7\) expected) and \(3.2\) (CMS, against \(3.8\) expected) [Aad:2012] [Chatrchyan:2012]. With numbers this small the Gaussian approximation is doing real work, and the collaborations used the asymptotic formulae [Cowan:2011] whose accuracy at such yields is itself part of what Probability and Statistics establishes; the CMS value fell below its own expectation, which is an ordinary downward fluctuation of a Poisson count.

The complementarity is the point of quoting both channels. A high-rate, high-background channel and a low-rate, low-background one fail in different ways: a mismodelled background shape would move the diphoton result and not the four-lepton one, and a mismeasured lepton efficiency would do the reverse. Their masses agreed to within the resolutions, and their signal strengths agreed with each other and with unity. Two supporting channels, \(H\to WW^{*}\to\ell\nu\ell\nu\) and \(H\to\tau^{+}\tau^{-}\), contributed sensitivity but no mass information at all: the neutrinos in the first and in the tau decays of the second make full reconstruction impossible, so those channels report an excess of events with the expected kinematics and not a resonance. They are evidence for a signal and not evidence for a particle, and both papers treat them as such.

Combined significance, 4 July 2012

Combining channels, ATLAS reported a local significance of \(5.9\) standard deviations at \(m_{H}c^{2}=126.0\,\mathrm{GeV}\) against \(4.9\) expected, and CMS \(5.0\) at \(125.5\,\mathrm{GeV}\) against \(5.8\) expected [Aad:2012] [Chatrchyan:2012]. The masses were

the first uncertainty statistical and the second systematic, and the signal strengths relative to the Standard Model prediction were \(\mu=1.4(3)\) and \(\mu=0.87(23)\) respectively.

The two collaborations were blind to each other's results until the seminar of 4 July 2012. That is not a detail of etiquette. Two measurements that agree are evidence in proportion to how independent they are, and these two shared only the accelerator: different magnets, different calorimeter technologies, different reconstruction software, different people, and no exchange of intermediate results. An error common to both would have to originate in the beam or in the theory of the Standard Model, not in either apparatus.

The measured numbers

Table 113.3 collects the quantities of the discovery itself and Table 113.4 what has been measured since; Tables 113.5 and 113.6 collect the coupling test, which is the measurement that distinguishes this scalar from any other. All four sit here, under this rubric, rather than beside the sections that use them, so that a reader looking for the chapter's numbers finds them in one place. Rows that record a run setting carry no uncertainty because they define the conditions rather than measure anything; every row that reports a measured or derived quantity carries one, and a derived row names the equation that produced it.

Interpretation

The look-elsewhere effect

Every significance in Table 113.3 labelled local is the answer to a question nobody asked: what is the probability of a fluctuation this large at a mass named in advance? No search names a mass in advance. Both experiments scanned a range and reported the largest excess found in it, and the probability that some point of a range fluctuates upward is larger — here by one to two orders of magnitude — than the probability that a nominated point does. The correction is derived in Section 11.6 and is applied here to the published numbers.

The mass

The measured mass is now

a fractional precision of \(8.8\times 10^{-4}\) [Navas:2024]. Almost all of it comes from the two high-resolution channels, and the systematic error is dominated by one thing: the absolute energy scale of the electromagnetic calorimeters.

That scale is not known a priori. A crystal's light yield, a liquid-argon gap's response and the material in front of both are known to a per cent at best, which would give a mass uncertainty of over \(1\,\mathrm{GeV}/c^{2}\) — ten times the quoted error. What fixes it is a calibration source inside the same detector: the \(Z\to e^{+}e^{-}\) peak, tens of millions of events at \(91.1880(20)\,\mathrm{GeV}/c^{2}\) whose mass is known from LEP to two parts in \(10^{5}\) [Schael:2006] [Navas:2024]. Reconstructing it with the experiment's own electrons and demanding that the reconstructed peak sit at the LEP value calibrates the energy scale in situ, cell by cell and category by category. The residual uncertainty is then the extrapolation from the electron scale at \(45\,\mathrm{GeV}\) to the photon scale at \(60\,\mathrm{GeV}\) — a small step, and it is small because the calibration source and the measurement are in the same instrument and differ by less than a factor of two in energy.

Spin and parity

Four hypotheses were on the table in 2012: \(J^{P}=0^{+}\), the prediction; \(0^{-}\), a pseudoscalar; \(1^{\pm}\); and \(2^{+}\), a graviton-like tensor. One of them was excluded by the discovery channel itself.

That separation uses the four-lepton final state, which is fully reconstructed and therefore carries its complete angular information: five angles — the production polar angle, the two polar angles of the lepton pairs in their parent frames, and two azimuths, one of them the angle \(\Phi\) between the planes of the two lepton pairs — plus the two lepton-pair masses. The discriminating variable is \(\Phi\). A scalar couples to the two \(Z\) bosons through the structure \(\varepsilon_{1}^{*}\cdot\varepsilon_{2}^{*}\) of Equation (104.69), which produces a distribution containing a \(\cos\Phi\) term; a pseudoscalar can only couple through the antisymmetric structure \(\epsilon_{\mu\nu\rho\sigma}\varepsilon_{1}^{*\mu} \varepsilon_{2}^{*\nu}k_{1}^{\rho}k_{2}^{\sigma}\), which has no \(\cos\Phi\) term and a \(\cos2\Phi\) term of the opposite sign; and a spin-\(2\) state produces polar distributions that neither can. Both collaborations built a likelihood in the seven variables and reported the alternatives disfavoured at high confidence, CMS excluding the pseudoscalar at above \(99\,\mathrm{\%}\) and ATLAS reaching a comparable conclusion [Aad:2013] [Chatrchyan:2013]. The scalar hypothesis was not merely preferred: the alternatives were rejected.

The three structures named in the previous paragraph are quoted, not derived. Landau–Yang is derived above and disposes of \(J=1\) from the diphoton channel alone; the separation of \(0^{+}\) from \(0^{-}\) and from spin \(2\) is the load-bearing half of Phenomenon 113.22 and this chapter does not carry it.

Couplings against mass

Everything so far would be satisfied by a scalar that had nothing to do with mass generation. The measurement that closes the argument is the pattern of couplings.

The total width

What the discovery does and does not establish

It is worth separating the four claims, because they are established to very different degrees and are routinely run together.

A neutral scalar exists at \(125.20(11)\,\mathrm{GeV}/c^{2}\). Established, by two independent instruments, in two independent channels each, with the mass measured consistently and the rate as predicted. This is as secure as any result in particle physics.

Its quantum numbers are \(J^{P}=0^{+}\). Established: spin \(1\) by Landau–Yang from the existence of the diphoton decay, spin \(0\) over spin \(2\) and even over odd parity by the four-lepton angular analysis [Aad:2013] [Chatrchyan:2013]. A small \(CP\)-odd admixture is bounded, not excluded (Remark 113.23).

Its couplings are proportional to mass. Established over the range from the muon to the top — a factor of \(1630\) — to a precision of roughly ten per cent, with the first generation and the light quarks untested (Remark 113.25).

The potential is the quartic one of Equation (113.1). Not established. Everything above probes the theory at the minimum of the potential. Its shape away from the minimum enters only through the self-couplings, and those are bounded to within a factor of several of the predicted value and not measured (Section 113.6.1). The central postulate of the mechanism — that the vacuum sits at the minimum of a quartic — remains an assumption supported by the mass and the couplings, and not a measurement.

What is not yet measured

The self-coupling and the shape of the potential

Expanding Equation (113.1) in unitary gauge gives, by Proposition 104.63,

so the trilinear and quartic self-couplings are fixed by \(m_{H}\) and \(E_{v}\), both measured. Writing the trilinear coupling as \(\kappa_{\lambda}\) times its Standard Model value, the mechanism predicts \(\kappa_{\lambda}=1\) with nothing to adjust. Nobody has measured it.

The reason is the size of the only channel that carries it. Producing two scalars at once, \(gg\to HH\), proceeds through two interfering amplitudes: a triangle in which a top loop makes one off-shell scalar that then splits, carrying \(\kappa_{\lambda}\) at the splitting vertex, and a box in which two scalars are emitted from the same top loop, carrying no self-coupling at all. The two interfere destructively — a low-energy theorem forces them to cancel at threshold in the limit of a heavy top — so the cross section is a quadratic in \(\kappa_{\lambda}\) with a minimum inside the range of interest rather than a monotone function of it. Three consequences follow, and all three are unhelpful.

First, the rate is tiny: some three orders of magnitude below single production, tens of femtobarns at LHC energies against the tens of picobarns of Section 113.3.1. Second, because the dependence is quadratic with a minimum inside the interesting range, a measured rate maps onto two values of \(\kappa_{\lambda}\), so the constraint is naturally an interval with a possible second solution rather than a value. Third, the destructive interference means the observable loses sensitivity in the neighbourhood of the predicted value rather than gaining it there.

The present constraints [Aad:2022] [Tumasyan:2022] allow \(\kappa_{\lambda}\) over a range of order \(-1\) to \(+7\): consistent with the predicted \(1\), and far too loose to be called a test.

The high-luminosity programme is expected to reduce that to a band of order tens of per cent around the prediction, which would be the first real measurement of the shape of the potential; that is a projection, not a result, and it is recorded here as one.

First-generation couplings and rare decays

The scalar has been seen to couple to the top, the bottom, the tau and — at three standard deviations — the muon [Sirunyan:2021]. It has not been seen to couple to the electron, the up quark, the down quark or the strange quark, which between them account for essentially all the mass of ordinary matter that is not binding energy. The claim that one mechanism gives mass to all the fermions is therefore verified for the third generation, for one second-generation lepton, and nowhere else.

Three other classes of measurement are open and are being pursued.

Loop-level rare decays. \(H\to Z\gamma\) has a predicted branching fraction of \(1.5\times 10^{-3}\) (Table 104.4) and, like \(H\to\gamma\gamma\), no tree-level vertex: it proceeds through \(W\) and top loops in a combination different from the diphoton one, so measuring both constrains the loop content in two independent ways. Evidence for it has been reported by the two collaborations combined, at a significance near the evidence threshold; the treatise's bibliography does not currently hold that publication, and the claim is therefore stated here without a citation rather than attached to the wrong one.

Lepton-flavour-violating decays. \(H\to\mu\tau\) and \(H\to e\tau\) are forbidden in the Standard Model, where the Yukawa matrix and the mass matrix are diagonalized by the same transformation (Theorem 104.38) so that no flavour-changing neutral scalar current exists at tree level. Searches have found nothing and bound the branching fractions at the level of a part in a thousand. This is a null result and it is a meaningful one: it says the scalar's couplings are flavour-diagonal to that accuracy, which the mechanism requires and a general two-doublet sector would not.

Invisible decays. The Standard Model predicts \(\mathcal{B}(H\to ZZ^{*}\to4\nu)\approx1.1\times 10^{-3}\), which is one line of Table 104.4: the branching fraction to \(ZZ^{*}\) is \(0.026\) and each \(Z\) goes to neutrinos with probability \(0.20\), so \(0.026\times0.20^{2}=1.04\times 10^{-3}\), the small remainder coming from the off-shell boson's distorted phase space. Any additional invisible width would signal decays to particles the detector cannot see, so the bound on it — currently of order ten per cent of the total width — constrains a light unseen sector coupling to the scalar. Collecting those searches, together with the direct and indirect ones, belongs to The Dark Sector: Evidence Without Explanation, whose collider subsection is reserved for exactly this material and does not yet contain it.

Vacuum stability

The measured mass has one consequence that reaches far beyond the collider, and it is the most-quoted and least-secure statement in this chapter.

\(\lambda\) is a coupling, so it runs (The Renormalization Group). Its one-loop beta function Equation (104.72) is dominated by the top Yukawa through the term \(-6\hat{y}_{t}^{4}\), negative because it descends from a closed fermion loop, and numerically fifty times larger than the bosonic terms. Proposition 104.66 evaluates it at a renormalization scale \(\mu_{R}=m_{t}c^{2}\) — the scale of Notation 113.1, not the potential's mass parameter and not a signal strength — and finds \(\dd\hat{\lambda}/\dd\ln\mu_{R}=-0.0206\): the quartic coupling falls with scale. Integrating the coupled system upward, with the measured \(m_{H}c^{2}=125.20(11)\,\mathrm{GeV}\) and \(m_{t}c^{2}=172.57(29)\,\mathrm{GeV}\) as inputs, \(\hat{\lambda}\) crosses zero somewhere near \(10^{10}\text{–}10^{11}\,\mathrm{GeV}\) [Degrassi:2012] [Buttazzo:2013]. Above that scale the potential turns over and a deeper minimum exists at large field values: the electroweak vacuum is not the global minimum but a metastable one, with a computed tunnelling lifetime exceeding the age of the universe by hundreds of orders of magnitude.

This chapter records the result and, more importantly, records what it is worth. Remark 104.67 states the three caveats in full and none may be dropped: the conclusion sits on a knife edge in the top mass, where a shift of about \(1\,\mathrm{GeV}\) moves the answer between absolute stability, metastability and instability, and the relation between what a hadron collider reconstructs and the short-distance mass entering the beta function carries a theoretical uncertainty of that size; the extrapolation assumes that nothing couples to the scalar anywhere between \(10^{3}\,\mathrm{GeV}\) and the \(10^{10}\,\mathrm{GeV}\) at which the quartic changes sign — seven decades of energy in which nothing has been looked at, and the extrapolation is only worth as much as that assumption; and the running is a statement about a coupling in a renormalization scheme, not about an observable.

That interval is seven decades wide and not sixteen: \(\log_{10}(10^{10}/10^{3})=7\). Remark 104.67 names the same two endpoints and calls them sixteen, which is an arithmetic slip there; the count used here is the one the endpoints give.

The experimental content of this subsection is therefore narrow and should be stated as such: this chapter measured \(m_{H}\), and Experiment: Deep Inelastic Scattering and its successors measured \(m_{t}\) and \(\alpha_{s}\), and those measurements are the inputs to a calculation whose output is interesting and whose error bar is dominated by an ambiguity in one of its inputs. It is a striking near-coincidence that the Standard Model, extrapolated with no new physics, lands so close to the boundary. It is not evidence for anything.

The related question — why \(E_{v}=246.220\,\mathrm{GeV}\) at all, when nothing in the theory protects a fundamental scalar's mass from corrections scaling with whatever completes it at high energy — is not a question this chapter can address, and it is recorded among the open problems in Section 133.2. The discovery made it sharper rather than softer: before 2012 one could hope the electroweak sector was not built from an elementary scalar; it is.

Primary references

The experiment is two papers, submitted on the same day to the same journal and published back to back. ATLAS [Aad:2012] and CMS [Chatrchyan:2012] each report the diphoton and four-lepton analyses in full, with the supporting \(WW^{*}\), \(\tau\tau\) and \(b\bar{b}\) channels, the categorization, the background models and the statistical treatment. Both are readable end to end and both should be read in preference to any account of them, including this one; the numbers in Table 113.3 attributed to 2012 come from them and are not recomputed here.

The instruments have their own literature and it is where the apparatus section comes from. The ATLAS [Aad:2008] and CMS [Chatrchyan:2008] technical papers in the 2008 LHC machine special issue describe the detectors as built, subsystem by subsystem, with the design resolutions quoted in Equation (113.13) and Equation (113.14); Evans and Bryant [Evans:2008] in the same issue describe the accelerator, and it is the source for every machine parameter in Table 113.2 other than the two derived rows.

The statistical machinery has a small canonical literature and this chapter uses all of it. Cowan, Cranmer, Gross and Vitells [Cowan:2011] fix the asymptotic formulae, the one-sided discovery statistic and the Asimov construction that the collaborations use to quote expected sensitivities; Read [Read:2002] is the \(CL_{s}\) construction behind every exclusion limit quoted here, including LEP's; Wilks [Wilks:1938] is the theorem that makes \(Z=\sqrt{q}\) possible and Gross and Vitells [Gross:2010] the trials-factor practice applied in Proposition 113.19, resting in turn on Davies' bound [Davies:1977] [Davies:1987] and on Rice's level-crossing formula [Rice:1944]. All of it is derived in Probability and Statistics rather than quoted.

The theory the experiment tested was in print half a century before the data. Englert and Brout [Englert:1964], Higgs [Higgs:1964a] [Higgs:1964] [Higgs:1966] and Guralnik, Hagen and Kibble [Guralnik:1964] give the mechanism in 1964, with Kibble's non-abelian extension [Kibble:1967] supplying the case the Standard Model needs and Anderson's plasmon argument [Anderson:1963] the physical precedent; Goldstone [Goldstone:1961] and Nambu [Nambu:1960] give the theorem that has to be evaded and the superconducting analogy that shows how. Glashow [Glashow:1961], Weinberg [Weinberg:1967] and Salam [Salam:1968] assemble the electroweak theory around it, and Electroweak Unification and the Higgs Boson derives the whole of it; Weinberg's second volume [Weinberg:1996] is the standard reference treatment.

The searches that preceded the discovery are the LEP combination of ALEPH, DELPHI, L3 and OPAL [Barate:2003], which set the lower limit and reported the excess at the edge of its sensitivity, and the Tevatron combination of CDF and D0 [Aaltonen:2012], which reported the associated-production excess in \(b\bar{b}\). The indirect constraints are the LEP and SLD precision electroweak measurements [Schael:2006] and the global fits built on them [Baak:2014], together with Veltman's screening theorem [Veltman:1977], which is why the constraint on the scalar mass was logarithmic and therefore weak.

The measurements made since 2012 divide by what they test. The spin-parity analyses are ATLAS [Aad:2013] and CMS [Chatrchyan:2013]; the first joint mass combination is [Aad:2015] and the first joint coupling fit [Aad:2016]; the ten-year coupling maps of the full Run-2 dataset are ATLAS [Aad:2022] and CMS [Tumasyan:2022], which are the source for every \(\kappa\) statement in Tables 113.5 and 113.6 and for the self-coupling range in Section 113.6.1. Note that [Tumasyan:2022] is the CMS ten-year portrait and not a discovery paper, and it is cited here only for couplings. Individual channel observations are \(t\bar{t}H\) production [Sirunyan:2018] and the \(H\to\mu^{+}\mu^{-}\) evidence [Sirunyan:2021]; the off-shell width constraint of Section 113.5.5 is [Aad:2023]. Vacuum stability is Degrassi et al. [Degrassi:2012] and its sequel [Buttazzo:2013].

Two sources are used for numbers rather than for claims. The Particle Data Group's 2024 review [Navas:2024] supplies every world-average mass and width in this chapter, taken from its own machine-readable release and stored with the treatise; CODATA 2022 [Mohr:2025] supplies the exact defining constants used in every conversion between electronvolts, joules and kilograms.

Finally, one gap should be named rather than papered over. The Standard Model production cross sections quoted in Section 113.3.1 are the compilations of the LHC Higgs Cross Section Working Group, which the collaborations cite and which this treatise's bibliography does not hold; they are attributed above to the two discovery papers, which quote and use them. The same applies to the ATLAS and CMS combination reporting evidence for \(H\to Z\gamma\), referred to in Section 113.6.2 without a citation for the same reason.