A portrait of the Higgs boson by the CMS experiment ten years after the discovery – Nature

LHC project and the Higgs boson

The primary goals of the LHC and its two general-purpose experiments, ATLAS and CMS, are to: (1) elucidate the mechanism of electroweak symmetry breaking and find the associated particle, which in the SM of particle physics is the Higgs boson^4,5,6; and (2) search for BSM physics.

The necessity to study the wide range of processes in Fig. 1 largely drove the design of the ATLAS and CMS experiments. The production cross-sections and the decay branching fractions for a SM Higgs boson with a mass of 125.38 GeV are shown in Extended Data Table 1.

The LHC²⁰ is designed to accelerate protons to an energy up to 7 TeV by powerful electric fields generated in superconducting radio-frequency cavities and guided around their circular orbits by strong (8.3 T) superconducting dipole magnets in tubes under very high vacuum. The counterrotating LHC beams are organized in approximately 2,800 bunches comprising more than 10¹¹ protons per bunch, separated by 25 ns, leading to a bunch crossing rate of about 32 MHz. The two proton beams are brought into collision at the centre of the four LHC experiments. In Run 2, pp, interaction rates of 2 GHz were reached. Multiple pairs of protons interact in each bunch crossing, the average number ranging from 21 in 2012 to 32 in 2018. These are superposed on the triggered interaction and are labelled ‘pileup’.

The CMS experiment

Design criteria and the SM Higgs boson

In the early 1990s, during the design phase of the Compact Muon Solenoid (CMS) experiment, considerable emphasis was placed on the identification and measurement of high-energy electrons, photons and muons, as these particles were expected to play an important role in the search for the SM Higgs boson and in the search for BSM physics.

As the rate of production of energetic muons at high-luminosity hadron colliders is very large, the online selection of events using muons is a particularly formidable task. The muon momentum has to be measured in real time and a momentum threshold placed to limit the rate. This requires a high bending power (high magnetic field) and an adequately precise and robust measurement of the trajectory of muons. This consideration determined the starting point of the design of CMS, and by implication the choice, size and the power of the analysing magnet. The next design priority was driven by the search for the Higgs boson via its decay H → γγ, requiring an excellent electromagnetic calorimeter (ECAL). The muon system and the ECAL were to be complemented by a precision inner-tracking system, immersed in a high magnetic field, giving good momentum resolution, and a hadron calorimeter (HCAL) that provided an almost full calorimetric coverage (for example, for the search for the Higgs boson if its mass turned out to be larger than 500 GeV).

The CMS detector

The longitudinal cut-away view of the CMS detector is shown in Extended Data Fig. 1. The CMS detector comprises four principal layers: the inner tracker, the ECAL, the HCAL and the muon system. The various types of detecting element and their channel counts are also indicated. Physics objects (for example, electrons, photons, muons, quark or gluon jets, and so on) are identified by different combinations of the patterns of energy deposits and/or traces in these four layers.

The defining choice and the central element of the CMS detector is the long (13 m), large-inner-diameter (about 6 m), state-of-the-art high-field (3.8 T) superconducting solenoid, generating the magnetic field for both the inner tracker and the muon system. The large size of the solenoid allows the inner tracker and almost all the calorimetry to be installed inside the solenoid.

Inner tracking

Particles emerge from the interaction region into the inner tracker, housed in a cylindrical volume with a length of 5.8 m and a diameter of 2.5 m. The particles first encounter the pixel detector, configured in three (four) cylindrical layers of silicon sensors in the barrel region, and two (three) disks in the endcap region before (after) 2017. The pixel detector is surrounded by 10 concentric layers of silicon sensors in the barrel region, with 10-cm-long or 20-cm-long silicon microstrips, and 12 vertical planes in each endcap region. Points are measured with an accuracy of about 15 μm in the bending plane. The geometric coverage extends down to angles of 9° from the beamline.

Electromagnetic and hadron calorimeters

The ECAL employs dense lead tungstate scintillating crystals. Each crystal has a length of about 23 cm that is sufficient to contain the full energy of high-energy electron and photon showers. The amount of generated or collected light is proportional to the energy of the incident particle. The fine transverse size of the crystals means that the energy of an electromagnetic shower is distributed over a cluster of crystals ranging from 9 (3 × 3) to 25 (5 × 5) crystals. The geometric coverage of the ECAL goes down to about 6° from the beamline.

The HCAL, comprising about 7,000 channels, is a sandwich of about 5-cm-thick brass absorber plates and about 4-mm-thick scintillator plates. The charged particles in the shower, generated in the absorber plates, traverse the scintillator plates and produce light that is collected and guided by fibres to the photodetectors. The geometric coverage of the HCAL goes down to about 6° from the beamline. This coverage is augmented by the very forward calorimeter, comprising an iron absorber with quartz fibres embedded in a matrix arrangement. The relativistic charged particles in the showers traverse the fibres and generate Cherenkov light, a part of which is guided by the fibres to the photodetectors. This calorimeter extends the calorimetric coverage down to an angle of about 0.75° from the beamline. The thickness of the hadron calorimetry is sufficient to absorb almost all of the energy of high-energy hadrons.

Muon system

Muons (and neutrinos) are the only particles that normally reach the muon system. All other particles deposit almost all of their energies in the calorimeters, and hence are said to have been absorbed. In addition to the measurements inside the inner tracker, the momentum of muons is measured a second time in gas-ionization chambers. These chambers are organized in four ‘stations’ that measure several points, to a precision of about 150 μm, and generate track segments whose direction is measured online with an angular precision of about 5 mrad. An independent set of gas-ionization chambers provide a signal timing resolution of about 3 ns, aiding the triggering process. The instrumented geometric coverage of the muon system goes down to an angle of 10° from the beamline.

Event selection

As the resources needed to record data for later use from all of the approximately 32 million beam crossings per second would be prohibitively costly, specific filters (known as triggers) are used to select the most interesting ones. An online two-tiered trigger system^26,27 is deployed, with the first tier (Level 1) being hardware-based and the second one (high-level or HLT) being software-based. The Level 1 uses custom hardware that processes coarse information from the calorimeters or the muon chambers to select around 100,000 crossings of interest per second, corresponding to a reduction of a factor of about 400. Crossings of interest are selected if the energy deposits in the calorimeters or the momentum of muons, are above predefined thresholds. Upon the issuance of a Level-1 trigger, and after a fixed latency of just under 4μs, all data from the ‘triggered’ crossing are off-loaded from the pipeline memories in the approximately 100 million on-detector electronics channels. These data, after suitable treatment in electronics housed in the underground ‘services’ cavern, are sent up 100 m to the surface as fragments on approximately 1,000 optical fibres and fed into a commercial telecommunication ‘switch’. The switch takes the individual fragments, puts them together, ‘builds’ the event, and feeds the event into the next available central processing unit (CPU) core, in a computer farm of some 50,000 CPU cores. There, in real time, full-event physics-grade software algorithms, optimized for fast processing, reconstruct physics objects and select for permanent storage some 1,000 events or crossings per second, based on topological and kinematic information (Extended Data Table 3).

Event reconstruction

The CMS experiment generates a large amount of collision and simulated data. To handle, store and analyse all these data required the development of the worldwide LHC distributed computing grid (wLCG), providing universal access to data for all CMS Collaboration members.

The data from the stored events are transferred to the Tier-0 centre housed on CERN’s main site, where a first processing stage is performed. The result of this stage is then distributed to seven other major centres worldwide, labelled Tier-1 centres, for offline analysis. The Tier-1s are designed to carry out tasks of further reconstruction of the collision data with improved calibration and alignment of the various CMS subdetectors, whereas the generation and reconstruction of Monte Carlo event samples is carried out both at the Tier-1 centres and smaller university-based locations, labelled Tier-2 centres.

The particle-flow (PF) algorithm³¹ reconstructs and identifies each individual particle in an event, with an optimized combination of information from the various elements of the CMS detector. The energy of photons is obtained from the measurements in the ECAL. The energy of electrons is determined from a combination of the electron momentum at the primary interaction vertex as determined by the tracker, and the energy in the corresponding cluster of crystals, including the energy sum of all bremsstrahlung photons spatially compatible with originating from the electron track. The momentum of muons is derived from the curvature of the corresponding track. The energy of charged hadrons is determined from a combination of their momentum measured in the tracker, and the matching ECAL and HCAL energy deposits. The energy of neutral hadrons is obtained from the corresponding corrected ECAL and HCAL energies.

Hadronic jets, arising from quarks or gluons, are created from all the particles reconstructed by the PF algorithm within a cone of half-angle of about 25°, centred on the axis determined by the vectorial sum of the momenta of all particles in the jet.

Improvements of the CMS detector

Several improvements have been introduced into the CMS experiment since the discovery of the Higgs boson in 2012. These include:

• The replacement, in late 2016, of the silicon pixel detector, with a new one comprising four concentric layers in the barrel region, at radii of 29 mm, 68 mm, 109 mm and 160 mm, and six endcap disks placed at ±34, ±41, and ±51 mm from the interaction point, along the beam line. The new configuration leads to an improvement in the reconstruction of the secondary vertices and in the quality of tagging of b quarks. The sensitivity of H → bb analysis is found to be improved by a factor of 2.

• The replacement of photodetectors in HCAL (hybrid photodiodes replaced by silicon photomultipliers) and implementation of more precise timing, allowing a reduction of accidental or instrumental backgrounds, for example, stray or out-of-time particles.

• The installation in 2013 and 2014 of chambers in the fourth endcap muon station that were left out for Run 1.

• The upgrade of the Level-1 trigger hardware before LHC Run 2 to improve the selection of physics events of interest. The trigger rate from background processes is reduced and the trigger efficiency improved for a wide variety of physics signals. In the muon system, new trigger processor boards deploy powerful commercial field-programmable gate arrays (FPGAs). A time-multiplexed architecture was introduced that enabled data from all the calorimetry in each crossing to be pushed into a single FPGA of the type used in the muon trigger system. The FPGAs allow sophisticated and innovative algorithms to be implemented and evolved as conditions change.

• In the data acquisition system, a new switch was installed and the CPU power of the computer farm increased. The whole fabric of the distributed computing systems was upgraded to allow more events to be stored (at least 1,000 events per second instead of the initially foreseen 100 events per second).

Offline event analysis

The principal physics objects are required to have transverse momenta or energies above a set threshold. The thresholds are lowered for the second, or any further, objects. Typical values of these thresholds are listed in Extended Data Table 3.

Leptons and photons resulting from the decays of Higgs bosons are expected to be unaccompanied by other particles; they are said to be ‘isolated’. Isolation criteria are imposed by requiring no additional energetic particles within a cone of about 20° opening angle around the object’s direction. Particles, other than from decays of b and c quarks or τ leptons, are expected to emerge directly from the primary interaction vertex, defined as the vertex corresponding to the pp collision identified by the online selection.

Increased use of regression and classification algorithms implemented using machine-learning methods, such as deep neural networks (DNNs) and boosted decision trees, led to a simultaneous increase in purity and in efficiencies of identification and reconstruction of physics objects (electrons, muons, photons, b quarks, τ leptons, jets and ({p}_{{rm{T}}}^{{rm{miss}}})), and improvements in the calibration of related kinematic observables.

All analyses make extensive use of Monte Carlo simulation of the signal and background processes. The CMS detector is precisely described in software code that is used to generate Monte Carlo event samples. Multiple interactions are included, which match the distribution of the number of pileup interactions observed in data. All the simulated event samples are then processed through the same chain of software programs and procedures as are collision data. Simulated samples are used to evaluate or determine geometric acceptances, energy, momentum and mass resolutions, as well as for online and offline particle identification and reconstruction efficiencies, and for training for the many boosted decision tree algorithms and DNNs.

Notes on Higgs boson decay channels

The distributions of the invariant mass of final-state particles in the individual decay channels are shown in Extended Data Figs. 3 and  4.

Bosonic decay channels

For H → γγ, the signal is extracted by measuring the narrow signal peak over a smoothly falling background distribution⁴². Despite its small branching fraction (0.23%), this mode is a sensitive one owing to the excellent precision in the measurement of the energies of photons. The diphoton invariant mass resolution is ({sigma }_{{m}_{{rm{gamma }}{rm{gamma }}}}/{m}_{{rm{H}}}approx 1 % ). All the principal production modes can be studied (ggH, VBF, VH, ttH and tH). The background largely consists of an irreducible one from quantum chromodynamics (QCD) production of two photons. There is also a reducible background where one or more of the reconstructed photon candidates originate from misidentification of jet fragments, that is dominated by QCD Compton scattering from quarks.

The study of the H → ZZ → 4({ell }) decay channel uses the distinctive decay of the Z bosons to charged leptons (({ell })) leading to a final state with 4e, or 4μ, or 2e2μ (ref. ⁴³). The signal appears as a narrow peak on top of a smooth and small background. The momentum (energy) measurement of muons (electrons) is precise enough to give an invariant mass resolution with ({sigma }_{{m}_{4{ell }}}/{m}_{{rm{H}}}approx 1 % ). The background comprises an irreducible part arising from the non-resonant production of two Z bosons or Zγ*, and a reducible part from the production of Z+ jets and top pair events, where the jets originate from heavy quarks, and thus could contain charged leptons, or are misidentified as charged leptons. The event yield for this process is tiny owing to the small branching fractions of H → ZZ (2.71%) and subsequent Z → ({ell }{ell }) (3.37% per lepton type) decays. To enhance the signal over background and to categorize events, discriminants exploiting the production and decay kinematics expected for the signal and background events based on a matrix element likelihood approach are used together with the invariant mass of the particle.

Extended Data Fig. 2 (top) shows a display of a candidate H → ZZ → eeμμ event produced in pp collisions at a centre-of-mass energy (sqrt{s}=13,{rm{TeV}}) and recorded in the CMS detector.

For H → WW → ({ell })ν({ell })ν, two high-p_T({ell }) and large ({p}_{{rm{T}}}^{{rm{miss}}}) characterize this final state⁴⁴ and benefit from the H → WW decay having one of the largest branching fractions (about 22%). Owing to the presence of two neutrinos, the computation of the WW invariant mass is not possible. However, an associated variable, the transverse mass, m_T, can be computed from the ({{bf{p}}}_{{rm{T}}}) of the charged leptons and the ({{bf{p}}}_{{rm{T}}}^{{rm{miss}}}). The square of transverse mass for a collection of particles ([{P}_{i}]) is defined as ({m}_{{rm{T}}}^{2}([{P}_{i}])={(sum | {{bf{p}}}_{{rm{T}},i}| )}^{2}-{| sum {{bf{p}}}_{{rm{T}},i}| }^{2}). The dominant background arises from irreducible non-resonant WW production and is estimated from data. The channel has a good sensitivity to the ggH and VBF production processes. In the analysis, (3{ell }) and (4{ell }) categories are also included, which are sensitive to production of the Higgs boson in association with a leptonically decaying vector boson. The analysis does not target the ttH and tH production modes, which are covered by a dedicated analysis discussed in ‘ttH and tH with multileptons’.

The H → Zγ signal is sought as a peak over a smoothly falling background distribution⁴⁵. This analysis targets decays of the Z boson into 2e or 2μ. To increase the sensitivity to the signal, the events are divided into different categories on the basis of the production mode. Multivariate analysis (MVA) techniques are used to further categorize regions with high and low signal-to-background ratios. The dominant background arises from Drell–Yan dilepton production in association with an initial-state photon.

Fermionic decay channels

For H → ττ, four different ditau final states are studied⁴⁶: eμ, eτ_h, μτ_h and τ_hτ_h, where τ_h refers to a hadronically decaying τ lepton. The analysis of this decay channel targets the ggH, VBF and VH production modes. The identification of τ_h candidates uses DNN discriminants to reject quark and gluon jets misidentified as τ_h. To separate the H → ττ signal events from the sizeable contribution of irreducible Z → ττ events, the likelihood estimate of the reconstructed mass of the system is used. This analysis does not target ttH production, which is covered by the dedicated analysis discussed in ‘ttH and tH with multileptons’.

The H → bb decay channel has by far the largest branching fraction of all the decay channels considered, with around 60% of Higgs bosons decaying in this way. The background from QCD production of pairs of b jets is very large; hence, final states with special characteristics have been chosen to enhance the signal-to-background ratio^{47,48,49,50,51}.

To select jets most likely to originate from b quarks, a DNN algorithm is used^75,76. It provides a continuous discriminant score, which combines information typical of b-quark jets, such as the presence of tracks displaced from the primary vertex, identified secondary vertices and the presence of low  p_T leptons in the jet. The threshold on the discriminant score is set such that the misidentification rate for light (u, d and s) quarks or gluons is low. For example, setting this misidentification rate at 0.1% gives a 50% efficiency for b-quark jet identification when applied to jets in top quark–antiquark events.

The VH production mode uses the presence of one or more leptons from the decay of the vector boson, or large ({p}_{{rm{T}}}^{{rm{miss}}}). In the signal-sensitive region, DNNs are used to separate the signal from the background dominated by QCD multijet production.

The ttH and tH production modes are included in the combination and MVA techniques are used to separate the signal from the large multijet backgrounds. This analysis uses the 2016 dataset.

Lastly, an inclusive analysis is included that targets Higgs bosons produced with large p_T (ref. ⁵¹). In this kinematic region, the signal-to-background ratio is larger. The two b jets from decays of highly Lorentz-boosted Higgs bosons are close in space and appear in the detector as a single broad jet with distinctive internal structure.

Extended Data Fig. 2 (bottom) shows a candidate H → bb event produced in pp collisions at a centre-of-mass energy (sqrt{s}=13,{rm{TeV}}) and recorded in the CMS detector.

The H → μμ signal is searched for as a peak in the dimuon mass distribution, over a smoothly falling background⁵². The dimuon invariant mass resolution is ({sigma }_{{m}_{{rm{mu }}{rm{mu }}}}/{m}_{{rm{H}}}approx 1 % ). The analysis of this decay channel targets the ggH, VBF, VH and ttH production modes, and is most sensitive in the first two modes. The largest background in this decay channel comes from Drell–Yan dimuon production in which an off-shell Z* boson decays to a pair of muons. Events are split into production modes based on their kinematical properties. To improve the sensitivity of the analysis, MVA techniques are used in each of these different categories.

The analysis of the H → cc final state in the VH production mode (Fig. 1c) has recently been presented⁷⁴ but has not been included in the present combination. The analysis yields (sigma ({rm{V}}{rm{H}}){mathcal{B}}({rm{H}}to {rm{c}}{rm{c}}) < 0.94,) pb at the 95% CL. The observed 95% CL interval (expected upper limit) for κ is found to be (1.1 < |{kappa }_{{rm{c}}}| < 5.5) ((| {kappa }_{{rm{c}}}| < 3.4)), the most stringent so far. A search for Z → cc in VZ events is used to validate the analysis strategy and yields a first observation of this decay channel, at a hadron collider, with a significance of 5.7 s.d.

ttH and tH with multileptons

The ttH (Fig. 1d) and tH (Fig. 1e,f) production channels, which probe the coupling of the Higgs boson to the top quarks, are studied in the case where the Higgs boson and the top quarks subsequently decay into final states with several leptons⁵³, supplementing dedicated studies of the H → γγ, H → ZZ → 4({ell }) and H → bb decay modes.

This analysis uses a categorization based on the number of leptons and/or τ_h candidates to target both the different Higgs boson final states and the tt decay channels. Categories with at least two leptons, or one lepton and two τ_h candidates, target cases where at least one top quark decays via a leptonically decaying W boson. Categories with one lepton and one τ_h, or with no leptons and two τ_h candidates are used to target events in which both top quarks decay via hadronically decaying W bosons. This analysis is sensitive to the H → WW, H → ττ and H → ZZ decay channels. Several MVA techniques are employed to better separate the ttH and tH production modes.

Higgs boson decays beyond the SM

In addition to the invisible Higgs boson decays discussed in ‘The κ framework for coupling modifiers’, other BSM decays are possible, into undetected particles. That is, these particles may or may not leave a trace in the CMS detector, but we do not have dedicated searches looking for these signatures. Nevertheless, the presence of undetected decays can be inferred indirectly from a reduction in the branching fraction for SM decays (or by an increase in the total Higgs boson width). In this interpretation, the total width becomes ({varGamma }_{{rm{H}}}=sum {varGamma }_{f}(kappa )/(1-{{mathcal{B}}}_{{rm{Inv}}.}-{{mathcal{B}}}_{{rm{Undet}}.})), where ({ {mathcal B} }_{{rm{Undet}}.}) is the branching fraction to undetected particles.

To probe invisible or undetected decays of the Higgs boson, another fit can be performed, including ({ {mathcal B} }_{{rm{Inv}}.}) and ({ {mathcal B} }_{{rm{Undet}}.}) as additional floating parameters, while imposing as an upper bound on κ_W and κ_Z their SM values, also valid in most proposed extensions of the SM^77,78. As can be seen from Extended Data Fig. 8 (right), ({ {mathcal B} }_{{rm{Inv}}.}) and ({ {mathcal B} }_{{rm{Undet}}.}) are found to be consistent with zero. The 95% CL upper limit on ({ {mathcal B} }_{{rm{Undet}}.}) is found to be <0.16, with only small changes to the other κ_i fitted values, as shown in Extended Data Fig. 8 (right). The measurement of the width⁶⁸ of the Higgs boson will be used in the future to constrain these quantities without imposing bounds on κ_W and κ_Z.

Statistical analysis

The statistical framework used to build the combination of all the channels is based on an established combined likelihood method (ref. ⁴⁰ and references therein), and briefly detailed in this section.

Given the enormous number of pp collisions produced at the LHC and the relatively small probability that one of those collisions will produce a signal-like event, the observations in data are described by Poisson probability functions, ({mathscr{P}}(k|lambda )={{rm{e}}}^{-lambda }{lambda }^{k}/k!), where k is the observed number of events, and the parameter λ is the expected number of events in a particular bin or region of one or more of the discriminating distributions used for each channel entering the combination. The combined likelihood is obtained from the product of the individual Poisson probability functions, accounting for the observed data and expected number of events across all channels.

The parameters λ are functions of the model parameters of interest: μ, which represent the Higgs boson couplings or signal strengths, and nuisance parameters θ, which model the effect of systematic uncertainties on the predicted signal and background contributions. Additional terms are included in the combined likelihood to represent constraints on the nuisance parameters owing to external measurements, such as energy- and momentum-scale calibrations or an integrated luminosity determination. The measurements reported in this paper are determined using the profile likelihood ratio (q(mu )=-,2,mathrm{ln}, {mathcal L} (mu ,{hat{theta }}_{mu })/ {mathcal L} (hat{mu },hat{theta })) where (hat{mu }) and (hat{theta }) are the values of the parameters of interest and nuisance parameters that maximize the likelihood ( {mathcal L} (mu ,theta )), and ({hat{theta }}_{mu }) are the values of the nuisance parameters that maximize the likelihood for a fixed value of (mu ). The compatibility between a given set of measurements and their corresponding SM predictions is reported as a P value, derived from the difference between q_SM and (q(hat{mu })). Expected intervals are derived from the Asimov dataset, in which the nuisance parameters are set to their maximum likelihood estimator values.

The modified likelihood ratio test statistic (tilde{q}(mu )=-,2,mathrm{ln},[ {mathcal L} (mu ,{hat{theta }}_{mu })/)( {mathcal L} (hat{mu },hat{theta })]) with a constraint (0le hat{mu }le mu ) is used to set 95% CL upper limits on signal strengths and production cross-sections using the  “CL_s criterion”⁴⁰.

All the reported confidence intervals, confidence regions and P values are obtained assuming various asymptotic approximations for the distributions of the (modified) likelihood ratio test statistic⁷⁹. The validity of the asymptotic assumptions has been routinely checked in the context of individual analyses whenever the event yields are small or particular validity conditions are not met.

Signal strengths of production channels and decay modes

For a Higgs boson produced in mode i and decaying into a final state f, the signal event yields are proportional to ({sigma }_{i}{ {mathcal B} }^{f}), where σ_i is the production cross-section and ({ {mathcal B} }^{f}) is the decay branching fraction. The branching fraction is in turn given by ({{mathcal{B}}}^{f}={varGamma }^{f}/{varGamma }_{{rm{H}}},) where Γ^f is the partial decay width in the final state f and Γ_H the total natural width of the Higgs boson.

Fits are performed under different assumptions: per overall single signal strength, yielding μ = 1.002 ± 0.057; per production channel signal strengths (({mu }_{i}={sigma }_{i}/{sigma }_{i}^{{rm{SM}}}) with ({ {mathcal B} }^{f}={ {mathcal B} }_{{rm{SM}}}^{f})), Fig. 2 (left); per decay mode signal strengths (({mu }^{f}={ {mathcal B} }^{f}/{ {mathcal B} }_{{rm{SM}}}^{f}), with ({sigma }_{i}={sigma }_{i}^{{rm{SM}}})), Fig. 2 (right); and with a free parameter per individual combination of production modes and decay channels, as illustrated in Extended Data Fig. 6.

The covariance matrices for the fitted signal strengths per production mode ({mu }_{i}) and per decay channel μ^f are shown in Extended Data Fig. 5.

Notes on self-interaction strength

The potential energy of the BEH field (ϕ) is given by (V(varphi ),=)(frac{1}{2}{m}_{{rm{H}}}^{2}{varphi }^{2}+sqrt{lambda /2}{m}_{{rm{H}}}{varphi }^{3}+frac{1}{4}lambda {varphi }^{4}.) The first term accounts for the mass of the Higgs boson m_H. The second term represents the Higgs boson self-interaction, of strength λ. In the SM, (lambda ={m}_{{rm{H}}}^{2}/(2{upsilon }^{2})) (where the vacuum expectation value of the BEH field, corresponding to its minimum, is (upsilon =246,{rm{GeV}})) and it can be measured via the study of Higgs boson pair production. The third term represents the interaction of four Higgs bosons at a point, a process that is even rarer than its pair production. Knowledge of the exact shape of the potential V is crucial for understanding the phase transition that occurred in the early Universe and its consequences⁸⁰.

The search for Higgs boson pair production is performed by combining Higgs boson pairs, each with differing decay modes. The decay modes that have been used are bb, ττ and WW^57,58,59,60, benefitting from the large branching fractions, and γγ⁶¹ and ({rm{ZZ}}to 4{ell })⁶², benefitting from the presence of narrow mass peaks, thus improving the signal-to-background ratio. All final states analysed are defined to be mutually exclusive so that they could be properly combined as statistically independent observations.

Measurements of Higgs boson pair production are used to constrain the Higgs boson self-interaction strength λ. Several combinations of individual Higgs boson decay modes are used in this search. The highest rate for Higgs boson pair production and decays occurs when both Higgs bosons decay to b-quark pairs, HH → bbbb, corresponding to about 35% of all the possible HH decays in the SM.

The search in the 4b decay mode^57,58 is performed separately under the assumptions that ({m}_{{{rm{H}}}^{* }}gg 2{m}_{{rm{H}}}) or not. In the case ({m}_{{{rm{H}}}^{* }}gg 2{m}_{{rm{H}}}), each Higgs boson is energetic (and hence said to be boosted), such that its decay products, for example, b-quark jets, merge and appear as one broad jet, but with a distinctive internal structure. In the latter case, all four b-quark jets rarely overlap, and hence are said to be resolved.

Another group of analyses targets the HH final states where one H decays to b quarks and the other to ττ⁵⁹, γγ⁶¹ or ({rm{ZZ}}to 4{ell })⁶². Analyses targeting a set of multileptons final states with ({p}_{{rm{T}}}^{{rm{miss}}}) are HH → (WW)(WW), HH → (WW)(ττ) or HH → (ττ)(ττ)⁶⁰, where hadronic τ lepton decays are also included.

A fit to Higgs boson pair production data can be used to simultaneously constrain κ_λ and κ_2V, as shown in Extended Data Fig. 9 (left).

Measurements of single Higgs boson production and decay can also be used to constrain κ_λ as quantum corrections to the SM Higgs boson production modes and decay channels depend on κ_λ (refs. ^81,82). These corrections have been derived⁸³ for the different production and decay modes entering the combination, as shown in Extended Data Table 2.

The values of κ_λ extracted from single and pair Higgs boson production are shown in Extended Data Fig. 9 (right).

Upgrade of the CMS experiment for HL-LHC running

To exploit the full potential of the LHC, the accelerator and its experiments will be upgraded. The HL-LHC will operate at an instantaneous luminosity of 5 × 10³⁴ cm⁻² s⁻¹. The intention is to collect ten times more data than the 300 fb⁻¹ foreseen in the initial LHC phase. This means that the integrated radiation levels will be correspondingly larger.

The physics to be studied drives the technical choices for the upgrade. The physics goals are:

• precise measurements of the properties of the Higgs boson and its self-coupling, to elucidate further the physics of electroweak symmetry breaking;

• search for BSM physics; and

• selected precision SM measurements.

The translation of these physics goals into experimental design goals requires:

• The construction of a new higher-granularity, more radiation-hard silicon tracker. The design of the new front-end electronics will allow information from the inner tracker to participate in the Level-1 trigger. The size of the individual detecting elements will be decreased leading to about ten times larger number of electronics channels. All components inside the tracker (silicon sensors, front-end electronics, 10 Gb s⁻¹ data links and so on) will have to withstand integrated doses of up to 500 Mrad and fluences of 10¹⁶ (1 MeV equivalent neutrons) per cm². The geometric coverage of the inner tracker will be increased, extending it down an angle of 2° from the beamline.

• The replacement of other components affected by radiation. Principally, these are the endcap calorimeters and the ECAL front-end electronics. The endcap calorimeters will be replaced with a new high-granularity ‘imaging’ calorimeter with precision timing. It will be based on 600 m² of silicon sensors with detecting cells of sizes of 0.5 cm² to 1.0 cm². Regions in this calorimeter will reach integrated doses of up to 500 Mrad and fluences of 10¹⁶ (1 MeV equivalent neutrons) per cm². The new front-end electronics for the ECAL barrel will allow data from each crystal to be sent to the calorimeter Level-1 trigger processor, instead of the sum of 25 crystals today, and which will allow better measurement of the timing of the impact of electrons or photons.

• Higher-bandwidth Level-1 and high-level triggers. Information from the inner trackers will be used at Level 1. The Level-1 trigger latency will be increased from 4 μs to over 12 μs, requiring corresponding changes in the front-end electronics, allowing more processing time leading to a purer selection of events. The output rate from the Level-1 processors will be increased from 100 kHz to 750 kHz and correspondingly the number of events stored for later analysis will be increased from 1 kHz to 10 kHz.

• The introduction of precision timing detectors. A new set of detectors will be installed in the barrel and endcap regions, covering a region down to an angle of 9° from the beamline. The precision timing of photons (in the barrel region) and charged tracks will greatly improve the localization of the correct interaction vertex. At HL-LHC, on average, some 140 pairs of protons are expected to interact in each crossing, spread over a time characterized by σ ≈ 200 ps. Furthermore, suppression of energy can be carried out that is not consistent in time with the interaction of interest.

The upgraded CMS experiment at HL-LHC will be more powerful than the current one. Uncertainties in many measurements of the properties of the Higgs boson are expected to approach the percent level, benefitting from the anticipated larger event samples, reduced experimental systematic uncertainties and more accurate theoretical calculations.

Theoretical references

The theoretical works used in our analyses can be found in the LHC Higgs Cross Section Working Group reports^36,37,38,39 and in refs. ^{54,56,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108}.