The Standard Model (SM) of particle physics is one of the most successful scientific theories ever developed, with confirmed predictions spanning many orders of magnitude at great precision, and the Higgs boson plays a pivotal role in it. According to the SM, as particles interact with the Higgs field, they acquire mass through a mechanism called electroweak symmetry breaking (EWSB). The Higgs boson is the quantum excitation associated with fluctuations in the Higgs field, and its presence is a necessary consequence of this mechanism. The Higgs boson is, therefore, essential for explaining the origin of mass and maintaining the internal consistency of the theory, and its existence was one of the most important predictions of the theory.

Given its importance in the SM, one of the main goals of the CERN Large Hadron Collider (LHC) was indeed to produce, observe, and study the Higgs boson. The LHC started colliding protons in 2010, for the first period of data collection at a center of mass energy of s = 7 TeV until 2012, when the energy in the center of mass was increased to 8 TeV for 1 year. This period is known as Run 1. After a shutdown period dedicated to upgrades and maintenance, the LHC Run 2 ran from 2015 to 2018 at an energy of 13 TeV. After another shutdown period, Run 3 started in 2022 at s = 13.6 TeV. The task of discovery and studying the Higgs boson is performed by the ATLAS [1] and CMS [2] experiments and collaborations; two large multipurpose experiments developed mainly to study proton–proton collisions at the LHC. So far, the LHC has delivered to the CMS (ATLAS) experiment 29 fb⁻¹ (28 fb⁻¹), 163 fb⁻¹ (157 fb⁻¹), and 50 fb⁻¹ (48 fb⁻¹) in runs 1, 2, and 3, respectively, for a total of ∼ 243 fb⁻¹ (232 fb⁻¹). The experiments have been able to record more than 90% of the delivered luminosity [3–8]. It is important to mention that the increase in the center of mass energy and luminosity obtained throughout the years came along with an important increase in the amount of pile-up events for each collisions, from an average of 10 pile-up collision for event in 2010 up to an average of 46 (48) in Run 2 (Run 3). Much effort has been expended by the collaborations in order to keep or improve the detectors performances despite this larger background.

At hadron colliders, such as the LHC, the Higgs boson can be produced via several different mechanisms and can decay in several different final states. The cross sections (branching ratios) of the different production (decay) modes are shown in Figures 1A, B for a Higgs boson of about 125 GeV/c ². The most common production mode is the gluon fusion (g g H) production, occurring when two gluons from the colliding protons interact by exchanging quarks. The quarks themselves can then emit a Higgs boson. A Feynman diagram of this process is shown in Figure 2A [9]. This process has a cross section of 48.58 pb for a Higgs boson with a mass of 125 GeV/c ² [10]. At 3.78 pb, the vector boson fusion (VBF) is more than 10 times rarer than g g H, but it is extremely relevant for the study of the Higgs boson coupling. In this process, as shown in Figure 2B, a pair of quarks from the two incoming protons exchange a vector boson (W or Z), emitting a Higgs boson in the process. Even rarer production modes are VH (Figure 2C, 2.25 pb), t t ̄ H and b b ̄ H (Figure 2D, 1 pb), and t H (Figures 2E, F, 0.07 pb). Even if rarer, these latest production modes can provide additional handles in the final state to separate the signal from the background and are, therefore, especially suited to address final states affected by a large background component.

The Higgs boson generated in the collision can then decay in several final states, as shown in Figures 2G–J. Decays to heavier particles (up to a pair of b-quarks) are generally preferred according to the SM. Thus, the generation of each Higgs boson in a proton–proton collision at the LHC involves its coupling to different SM particles, whether in production (quarks and vector bosons) or in decay (leptons, quarks, and vector bosons). Thanks to the small predicted width of the Higgs boson, the production and decay mechanism can be considered independent and can be factorized. The number of Higgs bosons produced in any given production (ii) and decay (ff) combination can, therefore, be computed according to the following formula: N i i → H → f f ≈ σ i i → H ⋅ B H → f f = σ i × B R f ≈ σ i Γ f Γ tot , (1) where Γ _f represents the partial decay width of the Higgs boson to a pair of f particles, and Γ_tot represents the total Higgs boson decay width. To simplify the notation, one can introduce the signal strength modifiers as μ i f = σ i × BR f ( σ i × BR f ) SM , in order to parameterize eventual deviations from the SM. To disentangle effects in production and decay, for any particle j that couples in decay (production) with the Higgs boson, it is possible to define the coupling modifier κ such that κ j 2 = Γ j / Γ j SM for a Higgs boson decay and κ j 2 = σ j / σ j SM in production. With this definition, also known as the κ-framework, the total Higgs boson width is thus Γ H = κ H 2 Γ H SM 1 − B R H SM [11, 12]. In addition to the SM direct decays (vector bosons, leptons, and quarks), the κ _g and κ _γ couplings are also included to model the loop-mediated Higgs boson interaction with gluons and photons, respectively, without needing to resolve the loops. A convenient result of this notation is that, by definition, in the SM κ _j = 1 and μ i f = 1 for all the allowed decays. Any significant deviation from unity would, therefore, indicate the presence of physics beyond the standard model (BSM).

This paper will focus on the determination of the k parameters. It is nevertheless important to mention that the high-energy physics community is exploring frameworks that are not only able to evaluate the (dis)agreement between the current results and the SM, such as with the signal strengths and the k-framework, but also to explore possible new physics signals hiding in the data collected by the ATLAS and CMS experiments. To this scope, the STXS [13] framework has been developed to provide a common definition for the measurements’ phase space. This allows for an easier comparison between the experiment and the reinterpretation of the results within an effective field theory (EFT) framework.

Experimental signatures in the final states are exploited by the CMS and ATLAS collaborations in order to categorize the events according to their most likely production and decay mechanisms. Careful simulations of the experimental apparatus and precise predictions of the Higgs boson production cross sections [10] are used to predict the exact component, in terms of production processes, of each category, and with this knowledge, as we will discuss in the following sections, it is possible to perform a multidimensional statistical analysis to extract the couplings of the Higgs boson to each SM particle.

Each of the decay channels discussed so far is sensitive only to those couplings that enter the Higgs boson production and the decay mode of that channel. The best way to assess the Higgs boson couplings in a comprehensive way is, therefore, to perform a combined measurement of all the channels together. This is a challenging task since one must take into account not only correlations between the Higgs boson couplings across different channels, whose determination is the goal of this exercise, but also among different background processes and systematic uncertainties. To avoid the possible double-counting effects, the final states should be mutually exclusive whenever possible. Excluding channels for which the actual sensitivity is too low to provide a meaningful constraint, such as b b ̄ H, in the most general realization of the combined fit after the LHC Run 2, there are six production modes (g g H, VBF, WH, ZH, t t ̄ H, and t H) and seven decay channels (Z Z, W W, γ γ, τ τ, b b, μ μ, and Z γ). It is useful to merge the t t ̄ H and t H production modes as they are both dominated by κ _t , and the statistical significance of the t H production mode is still very low. Not all combinations of production and decay are actually accessible with the current datasets, so some of them are either neglected or merged in order to improve their significance. In total, CMS (ATLAS) identified 30 (25) production × decay unique categories. The information from these categories is combined to measure up to nine coupling modifiers: κ _Z , κ _W , κ _γ , κ _g , κ _t , κ _τ , κ _b , κ _μ , and κ _Zγ . Furthermore, extra couplings such as Higgs to invisible or to BSM particles can also be included in the fit. The measurement of κ _c is not included since it is not yet precise enough to contribute to the global fit but has been established by CMS to be in the range 1.1 < |κ _c| < 5.5 [45] and by ATLAS in the range |κ _c | < 8.5 [46].

The procedure used to perform the combined fit and to estimate the parameters of interest was established by a common agreement before the Higgs boson discovery [52], and it is still used today. It is based on a profile likelihood technique with asymptotic approximation, where the systematic uncertainties are the nuisance parameters. The full correlation of the nuisance parameters across years and channels is taken into account when performing the fit, and systematic uncertainties are considered correlated when they are related to the same underlying effect. The fit is an extremely onerous and complex task. It includes O (10⁴) nuisance parameters for each experiment, and in the end is a summary of most of our knowledge of the Higgs boson. The results of the combined likelihood fit by the CMS collaboration [9, 53] are shown in Figure 7. The same results from the ATLAS collaboration are very similar [54]. It can be observed that the measurements obtained at the LHC are in remarkable agreement with the SM predictions. To appreciate the extent of the work performed in these years by the experimental collaborations, it is possible, for example, to look at the improvements obtained on the precision on some of these couplings, as shown in Figure 7B. At the discovery of the Higgs boson, even the couplings accessible to the most precise channels contributing to the discovery, such as κ _Z , κ _γ , κ _g , and κ _W , were measured with less than 50% accuracy. By the end of Run 1, it was possible to measure κ _t and most couplings were known with an accuracy close to 10%. Moreover, in some cases such as κ _t , κ _b , and κ _g , the statistical uncertainty was already subdominant with respect to the systematic uncertainty. By the end of Run 2, which is our current knowledge, we passed the 10% threshold for most couplings. Systematic and statistical uncertainties are roughly the same size. Further reducing these uncertainties significantly will be very challenging at the LHC, and future machines will be better suited for this task, as discussed in Section 5. It is also possible to look for more specific BSM effects by imposing symmetries on the fit. For example, assuming that there is a universal coupling modifier for all couplings to vector bosons and a different one for all the couplings to fermions so that κ _V = κ _Z = κ _W and κ _f = κ _t = κ _τ = κ _b = κ _μ . The effective couplings corresponding to loops in the gluon fusion and H → γ γ vertices are resolved in terms of their fundamental SM couplings. Figure 8 from the ATLAS collaboration (a similar result can be seen from the CMS collaboration in [9]) shows the two-dimensional fit on the κ _V and κ _f parameters, along with the contribution of each decay channel to it.

Since a combination, even within a single experiment, is an extremely challenging task, only a pair of cross-experiment Higgs physics combinations between ATLAS and CMS have been performed so far, both of them based on Run 1 data. They were aimed at measuring as precisely as possible the Higgs boson mass [55] and its couplings [56]. Further efforts on Run 2 are being discussed between the collaborations, also targeting some rarer channels such as double-Higgs boson and Z γ productions. The STXS scheme was developed to implement a compatible binning between the experiment and ease future combinations.

For a given value of the Higgs boson mass, the relationship between the mass of a SM particle and its coupling to the Higgs boson is well known in the SM. For bosons, it is represented as κ m v e v , while for fermions, it is κ m v e v , where κ is the coupling modifier for that particular coupling and vev is the vacuum expectation value (∼246 GeV). It is possible to check whether this relationship holds at the LHC with the measurements described so far. The results are similar for the CMS [9] and ATLAS [54] experiments. The measurements from the ATLAS collaboration are shown in Figure 9 and show a truly impressive agreement with the SM across four orders of magnitude in the couplings strength and three orders of magnitude in masses of the particles, one of the most stringent test of the SM in the Higgs sector that it is possible to develop.

The potential of the Higgs field φ is given by V ( φ ) = 1 2 m H 2 φ 2 + λ / 2 m H φ 3 + 1 4 λ φ 4 , where the cubic term of strength λ represents the Higgs boson self-interaction. In the SM, λ = m H 2 / ( 2 vev 2 ) . The shape of the potential is, therefore, determined by the value of the Higgs boson self-coupling, also known as the Higgs boson trilinear coupling and could be related to models of the strong first-order phase transition necessary for baryogenesis [57]. Since the Higgs boson acquires its mass through the self-interaction and since a Higgs boson loop is present in the Higgs boson propagator, the value of λ can be measured indirectly from the measurements of the Higgs boson couplings described in Section 3. Indeed, both CMS and ATLAS reported measurements obtained in this method, with CMS constraining the trilinear coupling in the region − 3.54 < κ λ = λ λ SM < 12.62 [9] and ATLAS at −4.0 < κ _λ < 10.3 [58].

At the LHC, it is possible to obtain a more direct measurement of this coupling by studying events where a pair of Higgs boson is produced. The study of double-Higgs events, where a pair of Higgs bosons is produced, was first suggested in the late 80s [59]. At the LHC, the most common production mode for Higgs boson pairs is via gluon fusion to H H. This can happen either through the production of an off-shell Higgs boson that subsequently decays into a pair of Higgs boson at their nominal mass or through the emission of a pair of Higgs bosons via a quark loop, as shown in the Feynman diagrams in Figures 2K, L. The two diagrams of Figures 2K, L produce a destructive interference among them, so the production cross section of double-Higgs events at the LHC is very small. For a center of mass energy of s = 13 TeV, it is just 32.76 fb [60–62] or less than 1/1000th of the single Higgs boson production cross section for the same energy (48 pb). Despite these events being exceedingly rare, the measurement of double-Higgs production has become one of the most interesting topics in the study of Higgs physics at the LHC, and the experimental collaborations are obtaining more precise results with the ultimate goal of obtaining evidence of this process in the coming years. This effort is clear in Figure 10A, showing the limit obtained on the existence of double-Higgs event production for the three main channels studied by the CMS collaboration. The first results on H H production were obtained on data collected in 2016 [63]. By the end of Run 2, the limits have improved by a factor between 10 (b b b b) and 3 (b b γ γ), while over the same period, the luminosity grew only by a factor 4.

In order to obtain a significant number of events, the collaborations resort to explore as many final states as possible. The most sensitive channels, and the first ones to have been explored, are those where one of the Higgs bosons decays into a b b ̄ pair, i.e., HH → b b ̄ b b ̄ , HH → b b ̄ γ γ, and HH → b b ̄ τ τ . Since H → b b ̄ is the most common Higgs boson decay in the SM, these channels have a relatively high branching ratio among the double-Higgs boson decay at 33.92, 0.26, and 7.3%, respectively. Other less sensitive channels have also been explored with the latest results from ATLAS and CMS covering the b b b b, b b γ γ, b b τ τ, b b W W, and W W W W channels [58, 64–68]. Moreover, the CMS collaboration also reported measurements in the b b Z Z [69], W W τ τ and τ τ τ τ channels [70] and the ATLAS collaboration in the W W γ γ channel [68].

The combined results from the collaborations were able to put a constraint on the Higgs boson trilinear coupling at −0.6 < κ _λ < 6.6 (ATLAS) and −1.2 < κ _λ < 6.5 (CMS) using the double-Higgs events alone. It must be noted that since it is not possible to distinguish between events coming from the Higgs boson trilinear interaction (Figure 2K) or from the quark loop (Figure 2L), these measurements rely on the assumption that κ _t = 1 and that all possible deviations are only due to effects from κ _λ . To overcome this limitation and to increase the sensitivity on κ _λ , the collaborations and the theory community are working on the combination of double- and single-Higgs measurements, where it would be possible to constrain separately the two couplings. Such an effort was recently shown by the ATLAS collaboration [57]. The final result is still dominated by the double-Higgs boson production sensitivity, with a final constraint of −0.4 < κ _λ < 6.3. The importance of the single Higgs measurement contribution in breaking the degeneracy between κ _λ and κ _t can be observed in Figure 10B, showing the constraints obtained in the κ _λ − κ _t plane by single- and double-Higgs measurements and their combination.

The LHC plans to continue to collect data at s = 13.6 TeV until 2025, when the end of Run 3 will conclude the LHC physics program. Run 3 is expected to deliver at least 250 fb⁻¹. In addition to improving on the precision of the couplings discussed so far, the additional data gathered in the next few years will provide enough integrated luminosity to reach a precision on most couplings close to 5%–10% (40% for κ _μ ), and should start gathering evidence for the presence of the Higgs boson trilinear coupling with κ _λ > 0.

After 2025, the LHC will undergo a major overhaul, with the goal of reopening in 2029 with a new machine capable of running at an instantaneous luminosity 3–4 times higher than the LHC. The resulting machine has been named high-luminosity LHC (HL-LHC), and it is planned to deliver between 3,000 and 4,000 fb⁻¹ at a center of mass energy of 14 TeV by 2041. Both the ATLAS and CMS experiments will need substantial upgrades in order to be able to collect data at the HL-LHC, with the major challenge being posed by the pile-up, which will rise up to an average of 200 pile-up collisions for each hard proton–proton scattering. Timing detectors, ultrafast triggers, and much more granular trackers will be among the strategies deployed in order to maintain, if not improve, the current experiment performances. This amount of luminosity will allow constraining the Higgs boson coupling to most SM particles at around 2%–4%, including the Higgs boson coupling to muons, as shown in Figure 11A. To appreciate the improvement with respect to the LHC measurement, it is possible to compare the current LHC measurements with the projected HL-LHC performances in Figure 7B, where the small inset on the right shows the HL-LHC projections. At these sensitivities, even relatively small deviations from the SM should start to be visible [71]. Among the most important targets for the Higgs boson couplings at the HL-LHC, there will be precise measurements of the Higgs boson decay into muons. On the double-Higgs side, it will be possible to obtain evidence of the existence of double-Higgs boson production, with an expected significance of approximately 4.0 and a projected precision on the double-Higgs boson production of roughly 50% [71]. It could be possible to measure the first hints of ttHH production [72]. Figure 10A shows the expected evolution for the limit on the double-Higgs boson production cross section, with the combined limit going below the threshold of 1, meaning that the sensitivity is sufficient to establish the existence of the Higgs boson pair production. It is expected that in the absence of new physics and by combining results from both CMS and ATLAS, the constraints on the Higgs boson trilinear coupling will be set in the region 0.52 < κ _λ < 1.5 [71], as shown in Figure 11B.

In order to reach even tighter constraints on the couplings and to observe the Higgs boson coupling to light particles such as the c-quark, completely new machines will be needed, which are currently in their exploratory phases. A couple of the most promising proposals that are being circulated will be briefly discussed here.

The Future Circular Collider (FCC) is a collider with a circumference of 90–100 km capable of running both as a lepton–lepton (FCC-ee) [73], lepton–hadron (FCC-eh), and hadron–hadron (FCC-hh) [74] collider, which should be built in the area of CERN. The FCC-hh will deliver 30 ab⁻¹ of proton–proton collisions at the center-of-mass energy of s = 100 TeV over an operating time of 25 years. Such a powerful machine will greatly enhance the precision achievable in the Higgs sector. When considering the improvement in the knowledge of the SM cross sections and the results obtained during the FCC-ee and FCC-eh data collection, FCC-hh is projected to measure the Higgs boson couplings well below the percent precision [75]. It is particularly impressive to see a projected sensitivity at the percentage level on the Higgs boson coupling to the c-quark (κ _c ). Figure 12 shows the projected FCC performances compared to some other proposed future colliders and the HL-LHC. One of the main targets of the FCC-hh program is the precise determination of the Higgs boson trilinear coupling. The employed techniques are similar to those established at the LHC with the same main channels considered, although the much larger available dataset will allow obtaining a greater precision on this measurement. Projections have shown that by combining all main channels together and depending on assumption on the detector performances and systematic uncertainties, FCC-hh could measure the Higgs boson trilinear coupling with a precision between 3.4% and 7.8% [76], as shown in Figure 13.

FCC will be a monumental task to complete, and it is difficult to imagine even larger colliders to be built in the future. To reach energies higher than what FCC would be capable of, new technologies must be deployed. One such proposal is the Muon Collider. Collisions between leptons provide a much cleaner environment with respect to proton–proton collision, but it is difficult to reach high energies colliding particles as light as electrons. One solution is to operate a lepton collider, but colliding muons instead of electrons and thus exploiting the much larger mass of the muon to reach higher center-of-mass energies. The idea is not new. The possibility of building muon storage rings was first mentioned in the 60s [77], and the idea of using heavy leptons, such as muons instead of electrons, has been floating around since the 70s, although building one such machine was technically impractical until now. The main technical issues for a Muon Collider, which we are now managing to address, are that the muons must be accelerated very quickly in order to boost their lifetime in the laboratory frame before they decay since their proper lifetime is only 2.2 μs. Moreover, the decay of muons inside the beam will generate a large beam-induced background that must be controlled. Nevertheless, the first estimation for the physics reach of the Muon Collider is very promising. The current proposal suggests operating a Muon Collider at s = 3 TeV and s = 10 TeV, although other configurations are also under study. It must be noted that although these energies are nominally one order of magnitude lower than what will be available at FCC-hh, in the case of the Muon Collider, the whole center of mass energy is available in the hard scattering, while in hadron collisions, only the fraction of energy carried by the two heavily interacting partons participates in the hard scattering. The energy of FCC-hh roughly corresponds to a 14-TeV Muon Collider [78], which would require a collider of 14 km of circumference, instead of the 100 km of FCC. The first projections show that depending on the actual configuration, the Muon Collider can reach sensitivities close to those obtained at FCC-hh, with a precision of O (0.1%) on the coupling to vector boson and of 1% on κ _c [79]. Where the muon collider show a potential to outperform even the FCC is in the Higgs boson trilinear and quartic couplings, measured in events with two and three Higgs bosons in the final state, respectively. As shown in Figure 14A the Muon Collider, when operating at 10 TeV, can potentially reach a 3.7% uncertainty on κ _λ , at the lower edge of the FCC band, and can reach even higher sensitivities if larger machines are built. Studies have also been performed on the capabilities of Muon Collider machines to measure triple-Higgs boson production [80], showing that a 10 TeV machine could measure the quartic-Higgs boson self-coupling with roughly 50% accuracy, as shown in Figure 14B.

The discovery of the Higgs boson at the CERN LHC by the ATLAS and CMS collaborations unleashed a new era of measurements of the Higgs sector. Among the studies of the properties of the Higgs boson are its couplings with the particles of the SM, which are a direct test of the electroweak symmetry breaking.

In the 10 years since the discovery, the LHC made huge improvements in luminosity delivered and in energy available in the center of mass. The experiments on their side adapted to these evolving conditions by developing ever more refined analysis tools and strategies. This allowed progressing from testing the more common production modes such as g g H and VBF, and the decay modes with the best signal-over-background ratio such as H → γ γ, H → Z Z, and H → W W to ever more comprehensive measurements. The latest combined fits address all the main production and decay mechanisms of the Higgs boson, including the first exploration of the Higgs boson self-coupling. The results obtained by the experiments running at the LHC show a remarkable agreement with the SM prediction, spanning three orders in magnitude in mass and four in the coupling values, with no significant deviations observed so far.

Nevertheless, there still is plenty of phase space available for possible deviations from the SM, in what could be an indication of higher-energy BSM effects. To tackle these effects and to narrow the uncertainties on the couplings, the HL-LHC upgrade planned for 2029 will deliver 10 times the luminosity of the LHC. This will allow reducing the uncertainty on the couplings down to the 2% level and will make channels such as μ μ and Z γ accessible for precision studies, as well as providing a definite observation of HH production.

Even at the HL-LHC, coupling to first- and second-generation fermions will remain elusive. Constraints below the 1% level and measurement of processes such as H → c c can be obtained at future facilities, such as the FCC and the Muon Collider, which are also projected to obtain a few percentage-level precision on the H H production.