Searching for Pairs of Higgs Bosons in the LHC Run 2 Dataset

Abstract

The discovery of the Higgs boson confirms the existence of a scalar sector of the standard model, responsible for electroweak symmetry breaking, but the nature and properties of the potential at the origin of this mechanism are still unknown. By studying the production of pairs of Higgs bosons (HH), physicists can directly measure the coupling of the Higgs boson to itself and thus determine the shape of this potential, which has far-reaching implications on the origin and evolution of our Universe. Because of this deep connection to the foundations of electroweak symmetry breaking, HH production is also an ideal place to search for manifestations of yet-unknown physics, such as modifications of the strength of the self-coupling and of the interaction between pairs of vector bosons and Higgs bosons. In this review article, we summarize the current searches for HH production at ATLAS and CMS, using the LHC Run 2 dataset, discuss the implications of our current constraints on physics beyond the standard model, and briefly review prospect for future HH searches.

The study of the production of Higgs boson pairs (HH) is the focus of a broad program of research developed at the CERN Large Hadron Collider (LHC) by the ATLAS [1] and CMS Collaborations [2]. This review aims at providing an overview of the motivations and the status of these searches at the LHC and of the prospects at future colliders. We introduce the topic in Section 1 by discussing the role of HH and of the Higgs boson self-coupling in our understanding of fundamental physics, followed by an overview of the HH production modes and decay channels in Section 1.2. The latest LHC Run 2 results are discussed in Section 2, with a particular focus on the recent developments that have improved the experimental sensitivity. Finally, we discuss the prospects for the measurement of HH in Section 3, and conclude with a summary in Section 4. Throughout this review, we tried to streamline the discussion and avoid the details of individual searches, which can be found in the corresponding papers referenced in the text, and rather focus on the general philosophy of the analyses. We hope that this choice will be a useful introduction to readers that want to learn about HH searches, and a comprehensive overview of the current results for experts on the topic.

1. Introduction

Characterizing the properties of the standard model of particle physics (SM) to deepen our knowledge of the fundamental mechanisms of Nature is the scientific focus of the experiments at the LHC. After the discovery of the Higgs boson (H) by the ATLAS and CMS Collaborations in 2012 [3,4,5], the focus of the high energy physics community has shifted toward the precise characterization of the properties of this particle. As the observation of the Higgs boson confirms the existence of a scalar sector of the SM, testing the mechanism at the origin of the existence of the Higgs boson itself is of primary importance. It is in this context that the study of the production of HH should be regarded, with its potential to shed light on key properties of this fundamental mechanism.

In this section, we briefly discuss the theoretical motivations to study HH as a tool to verify the validity of the SM description, to search for the existence of new physics beyond it, and to consider the implications of these measurements in our understanding of nature.

1.1. The Higgs Boson Self-Coupling

The Brout–Englert–Higgs (BEH) mechanism [6,7,8], which explains the quantum–relativistic nature of the mass of fundamental particles, postulates the existence of a complex scalar doublet of fields, $\mathsf{\Phi }$, that is subject to a potential of the form

$$V\left({\mathsf{\Phi }}^{†}\mathsf{\Phi }\right)=-{\mu }^2{\mathsf{\Phi }}^{†}\mathsf{\Phi }+\lambda {\left({\mathsf{\Phi }}^{†}\mathsf{\Phi }\right)}^2\mathrm{with}{\mu }^2,\lambda >0$$

(1)

which is minimal for non-zero field values defined by the relation

$$|{\mathsf{\Phi }}^2|={\displaystyle \frac{{\mu }^2}{2\lambda }}\equiv {\displaystyle \frac{v^2}{2}}$$

(2)

We refer to the constant v as the vacuum expectation value. When the symmetry of this potential is spontaneously broken to a ground state, the degrees of freedom of the field $\mathsf{\Phi }$ give rise to the longitudinal polarizations of the W and Z bosons, and hence to their masses, and to a physical field H whose quantum corresponds to the Higgs boson, H. In particular, the resulting Lagrangian contains the following terms:

$$\mathcal{L}\supset {\displaystyle \frac{1}{2}}{m}_{\mathrm{H}}^2{H}^2+\lambda v{H}^3+{\displaystyle \frac{\lambda }{4}}{H}^4$$

(3)

meaning that the strengths of the trilinear ($H^3$) and quadrilinear ($H^4$) self-interactions are governed by the value of $\lambda$ and thus directly depend on the shape of the scalar potential of Equation (1).

The possibility to probe the shape of the scalar potential through the direct measure of $\lambda$ represents the key physics goal of the experimental study of HH production. The SM predicts the following relation for value of $\lambda$:

$$\lambda ={\displaystyle \frac{m_{\mathrm{H}}^2}{2v^2}}\approx 0.13$$

(4)

implying that once the measurement of the Higgs boson mass of $m_{\mathrm{H}}=125.25\pm 0.17\mathrm{GeV}$ [9] is taken into account, a precise prediction for the value of $\lambda$ exists. HH measurements can thus provide a test of the consistency of the SM and a verification that the BEH mechanism is realized in nature.

Studying this interaction is not a simple verification of an as-yet unmeasured property of the SM. The interaction of the Higgs boson with fermions connects to the Yukawa coupling structure, and the interactions with vector bosons to the breaking of electroweak symmetry and the local quantization around the minimum of the scalar potential. In contrast, the self-interaction of the Higgs boson depends on a global shape property of the potential that is deeply connected to the foundations of the BEH mechanism. Measuring the Higgs boson self-coupling would be a confirmation that the scalar potential of Equation (1) does exist in Nature and would reveal fundamental information on its properties.

Owing to this deep connection with the fundamentals of the SM, the study of the Higgs boson self-coupling has implications in our understanding of the evolution of our Universe. The value of $\lambda$ runs with the energy with a function that depends on the values of the top quark (t) and H masses, and an instability of the BEH potential minimum can occur if its value becomes negative at a certain scale [10], opening up the possibility of quantum tunneling from the current state to a new minimum. Figure 1 shows the running of $\lambda$ once the measured t and H masses are taken into account, and the tunneling probability as function of their values. These measurements thus imply that, in the SM, our Universe is not stable but is in a state of meta-stability, where the timescale associated with the tunneling probability is comparable to or longer than the currently estimated age of the Universe. While there is not a one-to-one correspondence between the vacuum instability and the trilinear self-coupling, the experimental determination of $\lambda$ could help shed light on whether the long-term existence of the electroweak vacuum is truly challenged in nature, or whether a new mechanism exists to ensure the stability of the Universe.

Other examples of the cosmological implications of the BEH potential are the role of the Higgs boson as the possible inflaton in the primordial Universe [11], or its role in primordial baryogenesis [12]. The study of HH production at the LHC and the determination of the Higgs self-coupling can deepen our understanding on these topics.

1.2. HH Production in Effective Field Theories

Many experimental and theoretical indications hint at the possibility that as-yet-undiscovered physics beyond the SM (BSM) exists in nature. Owing to its deep connection to the scalar potential and thus to the foundations of the scalar sector, the study of HH might reveal hints of that BSM physics.

Direct searches for new particles that decay to HH, responsible for resonant HH production, are an important area of direct searches for BSM physics that we will not cover in this review. For a discussion on the status of the LHC Run 2 resonant HH searches, see ref. [13].

Another possibility is that BSM physics manifests in the modification of the strength of the couplings involved in HH production. In particular, if such BSM physics is associated to an energy scale that is not directly reachable at the LHC, we could still observe its low-energy effects as modifications of the properties of HH production. The theoretical framework to study this case is provided by effective field theories (EFTs), where we expand the Lagrangian of the SM with higher-order operators that represent the low-energy effects induced by a complete theory at a higher energy scale $\mathsf{\Lambda }$. EFTs provide a convenient theoretical framework to interpret the experimental results, potentially serving as a bridge to reinterpret them in the context of specific UV-complete models. The choice of the criterion to perform the expansion of the SM Lagrangian, and the choice of the symmetries that are imposed on the new theory and its fields, result in the definition of different EFTs.

Two main EFTs are currently discussed in the context of HH searches. The first is called the standard model EFT (SMEFT), where the Higgs field is considered to be a doublet in the theory, as in the usual SM description, and the theory is formulated in a $SU\left(2\right)\times U\left(1\right)$ invariant way, where the electroweak symmetry is broken by the scalar potential minimization. The SM Lagrangian is expanded with higher-order operators, whose contribution is suppressed by powers of $1/\mathsf{\Lambda }$, giving

$$\mathcal{L}={\mathcal{L}}_{\mathrm{SM}}+\sum _i{\displaystyle \frac{c_i}{\mathsf{\Lambda }}}{\mathcal{O}}_i^5+\sum _i{\displaystyle \frac{c_i}{{\mathsf{\Lambda }}^2}}{\mathcal{O}}_i^6+\cdots$$

(5)

The values $c_i$, that represent the “strength” of a specific operator, are referred to as Wilson coefficients. There is a single dimension-5 operator, and it violates the lepton number [14]. Consequently, it is neglected in the discussion of the Higgs (and HH) phenomenology, and the focus is on the dimension-6 operators, which provide the leading contribution in the $1/\mathsf{\Lambda }$ expansion.

The second EFT considered in HH is the Higgs EFT (HEFT), where the Higgs doublet is not present. In this case, the theory is invariant under $U\left(1\right)$ and the electroweak symmetry is assumed to have been already broken. For the HH case, the HEFT results in a more generic phenomenology at the cost of a larger number of operators to be studied and constrained. More information of HEFT in the context of HH can be found in refs. [15,16].

In general, EFTs are a broad and active area of research that can encompass many types of experimental measurements beyond HH. We present in Section 1.3.2 an overview of the EFT phenomenology that can be studied in HH, without giving further details on the theoretical construct. A theoretical overview of EFTs can be found in ref. [17].

1.3. HH Production Modes

In proton–proton collisions at the LHC, Higgs bosons pairs can be produced through the mechanisms that are listed, together with the corresponding cross section in the SM, in

Table 1. Summary of the cross section predicted by the SM for the various HH production modes at the LHC at $\sqrt{s}$ = $13\mathrm{TeV}$. For ggF, FTapprox refers to the approximate full top mass approximation. In all cases, a reference value of $m_{\mathrm{H}}=125\mathrm{GeV}$ is used.

Prod. Mode	$\mathit{\sigma }$ [$\mathbf{fb}$]	Scale Unc.	${\mathit{m}}_{\mathbf{t}}$ Unc.	PDF + ${\mathit{\alpha }}_{\mathbf{S}}$ Unc.	Precision
ggF	31.05	${}_{-5.0\%}^{+2.2\%}$	${}_{-23\%}^{+6\%}$	$\pm 3\%$	NNLO FTapprox [18,19,20,21,22,23,24,25]
VBF	1.726	${}_{-0.04\%}^{+0.03\%}$	-	$\pm 2.1\%$	${\mathrm{N}}^3\mathrm{LO}$ [26,27,28,29,30]
${\mathrm{W}}^{+}\mathrm{HH}$	0.329	${}_{-0.41\%}^{+0.32\%}$	-	$\pm 2.2\%$	NNLO [26]
${\mathrm{W}}^{-}\mathrm{HH}$	0.173	${}_{-1.3\%}^{+1.2\%}$	-	$\pm 2.8\%$	NNLO [26]
$\mathrm{Z}\mathrm{HH}$	0.363	${}_{-2.7\%}^{+3.4\%}$	-	$\pm 1.8\%$	NNLO [26]
ttHH	0.775	${}_{-4.3\%}^{+1.5\%}$	-	$\pm 3.2\%$	NLO [27]
$\mathrm{tj}\mathrm{HH}$	0.0289	${}_{-3.6\%}^{+5.5\%}$	-	$\pm 4.7\%$	NLO [27]

. The experimental investigation so far focuses on the two main production modes, the fusion of gluons (ggF) and of vector bosons (VBF), whose corresponding Feynman diagrams at the lowest order are shown in Figure 2.

1.3.1. ggF Production with Anomalous Self-Coupling

The simplest approach to model BSM physics effects in HH consists of studying the case of self-coupling values that differ from the SM prediction. Deviations are parametrized in terms of the strength of the coupling with respect to the SM prediction as ${\kappa }_{\lambda }=\lambda /{\lambda }^{\mathrm{SM}}$, such that the SM prediction corresponds to the case ${\kappa }_{\lambda }=1$. Similarly, deviations of the strength of the Yukawa interaction $y_{\mathrm{t}}$ between the top quark and the Higgs boson are parametrized as ${\kappa }_t=y_{\mathrm{t}}/y_{\mathrm{t}}^{\mathrm{SM}}$.

From the Feynman diagrams of Figure 2a, it can be seen that the amplitude of ggF HH production can be written as

$$\mathcal{A}\simeq {\kappa }_{\lambda }{\kappa }_t\mathfrak{T}+{\kappa }_t^2\mathfrak{B}$$

(6)

and the above relation holds in all orders in the QCD perturbative expansion. The symbols $\mathfrak{T}$ and $\mathfrak{B}$ are the amplitudes associated to the triangle and box diagrams at leading order (respectively on the left and right sides of Figure 2a), or the amplitudes associated to groups of diagrams with the same powers of ${\kappa }_{\lambda }$ and ${\kappa }_t$ at higher QCD orders. As a consequence, by computing the square of the amplitude of Equation (6), the HH cross-section dependence on the couplings takes the form

$$\sigma (\mathrm{g}\mathrm{g}\to \mathrm{HH})\simeq {\kappa }_{\lambda }^2{\kappa }_t^{2}T+{\kappa }_{\lambda }{\kappa }_t^{3}I+{\kappa }_t^{4}B$$

(7)

with $T={\left|\mathfrak{T}\right|}^2$, $B={\left|\mathfrak{B}\right|}^2$, and $I=|{\mathfrak{T}}^{*}\mathfrak{B}+{\mathfrak{B}}^{*}\mathfrak{T}|$. It should be noted that a global factor ${\kappa }_t^4$ can be taken out from the formula above, so that the cross-section shape (but not its magnitude) is fully described as a function of a single variable, ${\kappa }_{\lambda }/{\kappa }_t$. Therefore, it will be assumed ${\kappa }_t=1$ (as in the SM) in the following discussion.

The dependence of the total HH cross section on ${\kappa }_{\lambda }$ is shown, for ggF and the other production modes, in Figure 3 [27]. The ggF quadratic dependence has a minimum around ${\kappa }_{\lambda }\approx 2.5$. It should be noted that the figure reports the dependence at NLO FTapprox for ggF, but this cross section is now known at NNLO with finite top quark mass effects [24], and this value is used by the full Run 2 analyses discussed in this review.

The T, B, and I distributions as a function of the HH invariant mass, $m_{\mathrm{HH}}$, are shown in Figure 4a. We use $m_{\mathrm{HH}}$ since this is a key variable in experimental analyses, and the most important variable to represent the properties of the HH system—at leading order, the properties of the HH system can be fully described with only $m_{\mathrm{HH}}$, and the angle of one Higgs boson with respect to the beam line, which is approximately flat [31]. Higher values of $m_{\mathrm{HH}}$ are correlated with a higher $p_{\mathrm{T}}$ of the Higgs bosons and harder kinematics of the events. Anomalous values of ${\kappa }_{\lambda }$ alter the relative importance of these three components and thus the shape of the $m_{\mathrm{HH}}$ distribution. In particular, the distribution in $m_{\mathrm{HH}}$ is the softest for ${\kappa }_{\lambda }\approx 5$ and has a typical two-peak structure with a tail at high values for ${\kappa }_{\lambda }\approx 2.5$, with strong effects on the shape for ${\kappa }_{\lambda }$ values between 0 and 7. In the SM, the $m_{\mathrm{HH}}$ distribution is wide with a broad peak at $m_{\mathrm{HH}}\approx 400\mathrm{GeV}$. These effects are illustrated in Figure 4b.

Experimental analyses use both the total cross section and the differential distributions in $m_{\mathrm{HH}}$ to constrain the value of the self-coupling. At the current sensitivity, the capability to constrain the self-coupling largely stems from the enhanced HH cross section. However, the differential $m_{\mathrm{HH}}$ information is important to develop analyses that are optimal over a broad range of ${\kappa }_{\lambda }$ values, and to solve the degeneracy between ${\kappa }_{\lambda }$ values that result in the same total cross section values.

1.3.2. ggF Production in EFTs

In the previous section, we implicitly assumed that BSM physics might manifest solely through the modification of the Higgs boson self-coupling, and that the interpretations of experimental searches in terms of constraints on ${\kappa }_{\lambda }$ assume that all the other couplings are fixed to the SM predictions. While this is a valid operative assumption if we want to determine how precisely we can measure the Higgs boson self-coupling given the available data, a more generic study of BSM physics effects requires the framework of EFTs introduced in Section 1.2.

The addition of new operators in the SMEFT and HEFT formalisms results in the existence of new effective interactions, and the associated Feynman diagrams are shown in Figure 5. In the SMEFT formalism, the method employed to determine the expansion results in the existence of an additional chromomagnetic operator that implies a $\mathrm{t}\mathrm{t}\mathrm{g}\mathrm{H}$ vertex, but this is absent in the HEFT formalism. To date, this diagram is not simulated with the NLO-accurate Monte Carlo tools discussed below in Section 1.3.4, and therefore, it is not shown and further discussed here. Further information can be found in [32].

In addition to the $\mathrm{t}\mathrm{t}\mathrm{H}$ (${\kappa }_t$) and $\mathrm{H}\mathrm{H}\mathrm{H}$ (${\kappa }_{\lambda }$) interactions, we obtain in this formalism the $\mathrm{t}\mathrm{t}\mathrm{H}\mathrm{H}$ (${\mathrm{c}}_2$), $\mathrm{g}\mathrm{g}\mathrm{H}$ (${\mathrm{c}}_{\mathrm{g}}$) and $\mathrm{g}\mathrm{g}\mathrm{H}\mathrm{H}$ (${\mathrm{c}}_{2\mathrm{g}}$) interactions, where the symbols in parentheses denote the strength of the coupling.

In the SMEFT formalism, there is a relation that binds the value of ${\kappa }_t$ to ${\mathrm{c}}_2$, and the values of ${\mathrm{c}}_{\mathrm{g}}$ to ${\mathrm{c}}_{2\mathrm{g}}$, so that there are, in practice, three free parameters involved in HH production. Considering that much tighter constraints on ${\kappa }_t$ and ${\mathrm{c}}_{\mathrm{g}}$ are expected to be set in single Higgs boson measurements, this means essentially that the only parameter that HH measurements can help to constrain is ${\kappa }_{\lambda }$.

Conversely, in the HEFT formalism, all five interactions are independent, and their effects should be all considered simultaneously when interpreting the HH results. The five couplings ${\kappa }_{\lambda }$, ${\kappa }_t$, ${\mathrm{c}}_2$, ${\mathrm{c}}_{\mathrm{g}}$, and ${\mathrm{c}}_{2\mathrm{g}}$ correspond, respectively, to the coefficients of the relevant HEFT operators $c_{hhh}$, $c_{tth}$, $c_{tthh}$, $c_{ggh}$, and $c_{gghh}$, as they are denoted in ref. [33]. Since the exploration of a five-dimensional parameter space is computationally very challenging, experiments have thus far taken the practical approach of studying “shape benchmarks” that represent arbitrary combinations of $({\kappa }_{\lambda },{\kappa }_t,{\mathrm{c}}_2,{\mathrm{c}}_{\mathrm{g}},{\mathrm{c}}_{2\mathrm{g}})$ that are associated to characteristic distributions of $m_{\mathrm{HH}}$. While these do not represent, per se, points of specific theoretical interest, they capture the main kinematic features that result from combinations of the couplings, guiding the development of analyses and allowing for an evaluation of their sensitivity to generic EFT scenarios. Two sets of benchmarks have been defined to date, one based on leading-order (LO) modeling of the signal [31,34,35] and a more recent one derived from the next-to-leading-order (NLO) simulation [36]. The values of these benchmarks are shown in

Table 2. Definition of the LO [31,34,35] and NLO [36] EFT shape benchmarks. For the NLO benchmarks, please note that we used the conversion ${\mathrm{c}}_{\mathrm{g}}=\frac{3}{2}{c}_{ggh}$ and ${\mathrm{c}}_{2\mathrm{g}}=-3c_{gghh}$, where $c_{ggh}$ and $c_{gghh}$ are the coefficient of the associated HEFT operators. The conversion table can be found in Ref. [33].

	${\mathit{\kappa }}_{\mathit{\lambda }}$	${\mathit{\kappa }}_{\mathit{t}}$	${\mathbf{c}}_2$	${\mathbf{c}}_{\mathbf{g}}$	${\mathbf{c}}_{2\mathbf{g}}$
LO benchmarks
1	7.5	1.0	−1.0	0.0	0.0
2	1.0	1.0	0.5	−0.8	0.6
3	1.0	1.0	−1.5	0.0	−0.8
4	−3.5	1.5	−3.0	0.0	0.0
5	1.0	1.0	0.0	0.8	−1.0
6	2.4	1.0	0.0	0.2	−0.2
7	5.0	1.0	0.0	0.2	−0.2
8	15.0	1.0	0.0	−1.0	1.0
9	1.0	1.0	1.0	−0.6	0.6
10	10.0	1.5	−1.0	0.0	0.0
11	2.4	1.0	0.0	1.0	−1.0
12	15.0	1.0	1.0	0.0	0.0
NLO benchmarks
1	3.94	0.94	−1/3	3/4	−1
2	6.84	0.61	1/3	0	1
3	2.21	1.05	−1/3	0	−3/2
4	2.79	0.61	1/3	1/4	−1/2
5	3.95	1.17	−1/3	−3/4	3/2
6	5.68	0.83	1/3	1/2	−1
7	−0.1	0.94	1	−1/4	1/2

.

1.3.3. VBF Production with Anomalous Couplings

As illustrated by the Feynman diagrams of Figure 2a, VBF HH production involves the $\mathrm{VV}\mathrm{H}$ and $\mathrm{VV}\mathrm{H}\mathrm{H}$ couplings (with $\mathrm{V}=\mathrm{W},\mathrm{Z}$). The strengths of these two interactions with respect to their SM predictions are represented by the coupling modifiers ${\kappa }_{\mathrm{V}}$ and ${\kappa }_{2\mathrm{V}}$, respectively. In the SM, both interactions originate from the spontaneous breaking of electroweak symmetry, and their values are thus related to the same fundamental mechanism. As a consequence, a cancellation occurs in the amplitude for longitudinally polarized vector bosons. For generic ${\kappa }_{\mathrm{V}}$ and ${\kappa }_{2\mathrm{V}}$ couplings, at energies that are lower compared to the scale of the new physics, the amplitude depends on the coupling and on the energy $\sqrt{\widehat{s}}$ as [37]

$$\mathcal{A}({\mathrm{V}}_L{\mathrm{V}}_L\to \mathrm{HH})\simeq {\displaystyle \frac{\widehat{s}}{v^2}}\left({\kappa }_{2\mathrm{V}}-{\kappa }_{\mathrm{V}}^2\right)$$

(8)

In the SM, where ${\kappa }_{\mathrm{V}}={\kappa }_{2\mathrm{V}}=1$, this amplitude is suppressed. However, if the Higgs boson is the manifestation of new strong dynamics at the $\mathrm{TeV}$ scale, such as in models with a composite Higgs boson [38,39], anomalous values of ${\kappa }_{\mathrm{V}}$ or ${\kappa }_{2\mathrm{V}}$ can result in a large enhancements of the VBF HH production rate at high $m_{\mathrm{HH}}$, as illustrated in Figure 6. The dependence of the total VBF HH cross section on the value of the anomalous couplings is shown in Figure 7 for various combinations of parameters. The effects of the cancellation from Equation (8) are clearly visible in the dependence of the cross section on ${\kappa }_{\mathrm{V}}$ and ${\kappa }_{2\mathrm{V}}$. It should be noted that the independent modification of ${\kappa }_{\mathrm{V}}$ and ${\kappa }_{2\mathrm{V}}$ that we illustrate here is possible in a HEFT framework but not in SMEFT.

1.3.4. Monte Carlo Generators for HH

The most accurate simulation of ggF HH production is at the NLO precision with finite top quark mass effects [33,41,42], and includes the simulation of anomalous values of ${\kappa }_{\lambda }$ and of the EFT couplings in the EWChL framework. All the analyses based on the full Run 2 data set performed by the ATLAS and CMS Collaborations and discussed in this review use this simulation.

The VBF HH signal is simulated at the leading order precision with MadGraph, including effects from anomalous ${\kappa }_{\mathrm{V}}$, ${\kappa }_{2\mathrm{V}}$, and ${\kappa }_{\lambda }$ couplings from [37].

1.4. HH Decay Channels

Owing to the large number of decays of the Higgs boson, HH production gives rise to a phenomenologically rich set of final states, which are illustrated in Figure 8. When choosing a final state to be investigated in experimental searches, one Higgs boson is usually required to decay to either $\mathrm{b}\mathrm{b}$ or $\mathrm{W}\mathrm{W}$ to ensure a sizable branching fraction, while the choice of the decay of the second Higgs boson represents a trade-off between the abundance of the final state and the background contamination. The HH decay channels that have been chosen for searches in the LHC Run 2 data set by the ATLAS and CMS Collaborations are listed below with a very short overview of the background processes and the main challenges. The references to the published analyses of each channel, a more detailed discussion of the analysis methods, and an overview of the results are presented in Section 2.

• $\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$ ($\mathcal{B}\approx 34\%$) is characterized by the largest branching fraction. The experimental challenge is the large multijet background from QCD and $t\overline{t}$ production, that requires a high performance in the identification of b jets, efficient online identification at trigger level, and dedicated strategies for the modeling of the background and its separation from the HH signal. The current searches at the LHC cover the topology, where all four b jets are separately reconstructed (“resolved” topology), and the topology where the H decays at high Lorentz boost resulting in collimated decay products and the HH system is reconstructed with two large-radius jets (“boosted” topology).
• $\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$ ($\mathcal{B}\approx 7.3\%$) is usually approached in $\tau \tau$ final states with at least one decay to hadrons (${\tau }_{\mathrm{h}}$) and a neutrino, ${\tau }_{\mathrm{h}}{\tau }_{\mathrm{h}}$, ${\tau }_{\mu }{\tau }_{\mathrm{h}}$, and ${\tau }_{\mathrm{e}}{\tau }_{\mathrm{h}}$, that together collect more than 88% of the $\tau \tau$ decays. The main irreducible backgrounds are $t\overline{t}$ and Z plus b jets production, plus instrumental backgrounds where hadronic jets are misidentified as $\tau$ and b jets. The kinematic information of the reconstructed objects is combined in a multivariate discriminant to identify the presence of a signal.
• $\mathrm{b}\overline{\mathrm{b}}{W}^{\pm }{W}^{\mp }$ ($\mathcal{B}\approx 25\%$) is examined in the decays of the $\mathrm{W}\mathrm{W}$ system containing either one (≈30%) or two (≈5%) light leptons (electrons and/or muons, while decays to taus are only indirectly accounted for in their subsequent decays to an electron or muon plus neutrinos). Presently, the results based on the complete Run 2 LHC data set only cover the two-lepton final state. The main background is $t\overline{t}$, that is suppressed with the usage of multivariate discriminants.
• $\mathrm{b}\overline{\mathrm{b}}\mathrm{Z}{\mathrm{Z}}^{*}$ ($\mathcal{B}\approx 3.1\%$) is currently investigated in the $\mathrm{Z}\mathrm{Z}\to 4\ell$ decay channel ($\ell =\mathrm{e},\mu$, about 0.45% of $\mathrm{Z}\mathrm{Z}$ decays) with the full Run 2 data set, although previous searches also investigate the $4\mathrm{q}$ and $2\mathrm{q}2\nu$ decay channels of the $\mathrm{Z}\mathrm{Z}$ system. The main backgrounds of the $4\ell$ search are Z, $\mathrm{Z}\mathrm{Z}$, and H production in association with jets. The analysis relies on the excellent $m_{4\ell }$ resolution to suppress the Z and $\mathrm{Z}\mathrm{Z}$ backgrounds and uses a boosted decision tree to reject single Higgs boson events.
• $\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$ ($\mathcal{B}\approx 0.26\%$) is characterized by a tiny branching fraction, but benefits from the low background contamination and the percent-level experimental resolution on the photon pair invariant mass to identify the signal. The main backgrounds are the continuum production of photons and jets, and single Higgs boson production in association with jets (ggF, $\mathrm{t}\mathrm{t}\mathrm{H}$). The latter can have an important impact on the sensitivity of the analysis since it is characterized by the same $\mathrm{H}\to \gamma \gamma$ signature, and is suppressed with dedicated multivariate techniques. The distribution of the diphoton invariant mass, $m_{\gamma \gamma }$, is used to search for the presence of the signal, optionally with the simultaneous fit of the broader $m_{\mathrm{b}\mathrm{b}}$ distribution.
• Decays of the HH system to $\mathrm{W}\mathrm{W}\mathrm{W}\mathrm{W}$ ($\mathcal{B}\approx 4.6\%$), $\mathrm{W}\mathrm{W}\tau \tau$ ($\mathcal{B}\approx 2.7\%$), and $\tau \tau \tau \tau$ ($\mathcal{B}\approx 0.39\%$), where W and $\tau$ decays contain two, three, or four reconstructed leptons (e, $\mu$, and ${\tau }_{\mathrm{h}}$), are collectively referred to as “multilepton” final state. Categories are defined based on the number, flavor, and electrical charge (same or opposite sign) of the leptons, and the main backgrounds are diboson production and processes with misidentified leptons. Boosted decision trees are used to enhance the presence of the signal in the selected data samples.

1.5. Overview of Earlier HH Results

The investigation of HH production at the LHC has been marked by spectacular progress over the past years. If the study of HH was initially regarded as difficult, even at the HL-LHC, because of its low cross section, the efforts of the ATLAS and CMS analysis teams have resulted in a large improvement of the experimental sensitivities. Channels such as $\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$ that were originally thought to be impossible because of the large backgrounds, provide some of the most stringent constraints on HH production today. Several analyses investigate both the ggF and VBF production modes to probe the strength of both the $\mathrm{H}\mathrm{H}\mathrm{H}$ and the $\mathrm{VV}\mathrm{H}\mathrm{H}$ vertices. As this review was prepared, the results based on the full LHC Run 2 data set (about $140{\mathrm{fb}}^{-1}$) in many channels have been made public, and the experiments are working toward their combination. We will cover these latest results and discuss the main drivers of analysis improvements in Section 2.

It is instructive to quickly recap the previous set of results that correspond to the combination of analyses based on the partial Run 2 data set collected in 2015 and 2016 (about $36{\mathrm{fb}}^{-1}$). A summary of the 95% confidence level (CL) limit on the signal strength, defined as the ratio of the excluded cross section with respect to the SM prediction, is shown in Figure 9 for the ATLAS and CMS HH combinations [43,44].

These results show that there is not a single “golden” channel for the investigation of HH, and the final result benefits from the combination of several analyses—in particular, from the combination of $\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$, $\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$, and $\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$. A summary these results is presented in

Table 3. 95% CL upper limits on $\sigma /{\sigma }_{SM}$ for HH searches in the partial Run 2 data set.

		$\mathit{\sigma }/{\mathit{\sigma }}_{\mathbf{SM}}$ Obs. (Exp.) Limit	Ref.
$\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$	ATLAS	12.9 (20.7)	[45]
	CMS	74.6 (36.9)	[46]
$\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$	ATLAS	12.6 (14.6)	[47]
	CMS	31.4 (25.1)	[48]
$\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$	ATLAS	20.4 (26.3)	[49]
	CMS	23.6 (18.8)	[50]
$\mathrm{b}\overline{\mathrm{b}}{\mathrm{VV}}^{*}$	ATLAS	300 (300)	[51]
	CMS	79 (89)	[52]
${\mathrm{W}}^{\pm }{\mathrm{W}}^{\mp }\gamma \gamma$	ATLAS	230 (160)	[53]
${\mathrm{W}}^{\pm }{\mathrm{W}}^{\mp }{\mathrm{W}}^{\pm }{\mathrm{W}}^{\mp }$	ATLAS	160 (120)	[54]

. The different hierarchy between the ATLAS and CMS results per channel was traced to different analysis methodologies, and the choice to focus more effort on the SM or BSM optimizations.

The upper limits on the HH cross section as function of ${\kappa }_{\lambda }$ are illustrated in Figure 10 for the two experiments. The shape of the limit curves directly relates to the modification of the interference pattern as discussed in Section 1.3.1. For very large $|{\kappa }_{\lambda }|$ values, the squared amplitude of the triangle diagram dominates, while in the region between ${\kappa }_{\lambda }$∼0 and 7, the changes induced in the shape of $m_{\mathrm{HH}}$ result in the large changes of the experimental sensitivity. The sensitivity is maximal when the spectrum is hardest (${\kappa }_{\lambda }\approx 2.5$) and minimal when the spectrum is softer (${\kappa }_{\lambda }\approx 5$). Since most of the cross section enhancement effects appear in the region close to the kinematic production threshold of $m_{\mathrm{HH}}=250\mathrm{GeV}$, channels with a good acceptance at low $m_{\mathrm{HH}}$, such as $\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$, have a smaller gap in the sensitivities between these two extreme regimes when compared to channels such as $\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$, where the higher thresholds imposed on the final state objects result in a lower acceptance at low $m_{\mathrm{HH}}$.

When comparing these early results with the most recent ones illustrated in Section 2, it should be noted that neither the contribution from the VBF HH signal nor the latest uncertainty related to the top quark mass scheme are included here.

2. Summary of LHC Run 2 HH Results

2.1. Updated HH Searches with Full LHC Run 2 Data Set

ATLAS and CMS are currently pursuing the full Run 2 HH analysis program. Several results have already been updated using the full Run 2 integrated luminosity. Updates on previous channels, including the ‘big three” $\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$, $\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$, and $\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$ that lead the sensitivity, were accompanied by the analysis of new HH channels, where branching ratios are smaller or where the event topology requires additional creativity to benefit of the Run 2 data set. We present in what follows an overview of the analysis techniques and discuss the results at the end of the Section.

2.1.1. HH→$\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$

The $\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$ channel benefits from the largest branching ratio, but must contend with the large multijet backgrounds and combinatorial challenges when reconstructing the two Higgs bosons. The precise modeling and the efficient rejection of the background are the key elements to maximize the sensitivity of the analyses. The CMS Collaboration has performed analyses of the full Run 2 data set in the $\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$ channel, investigating both the ggF and the VBF production modes, in resolved and boosted regimes. The resolved analysis covers the bulk of the HH signal, while the boosted focuses on a specific corner of the phase space, where the background contamination is minimized, and a powerful identification algorithm for $\mathrm{b}\mathrm{b}$ large radius jets can be deployed. The ATLAS Collaboration has published to date the study of the VBF production mode in the resolved topology with the full Run 2 data set.

Resolved HH →$\mathbf{b}\overline{\mathbf{b}}\mathbf{b}\overline{\mathbf{b}}$

Trigger represents a fundamental challenge in $\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$ searches, and both the ATLAS and CMS analyses [55,56] rely on online algorithms to trigger on events containing at least two (ATLAS) or three (CMS) b jets, achieving thresholds as low as 30 or 40$\mathrm{GeV}$ on the softest jet. Additional forward jets in opposite $\eta$ regions of the ATLAS or CMS detector are required to tag the VBF topology. In the ATLAS analysis, the VBF candidate jets are chosen by requiring |$\Delta {\eta }_{jj}$| > 5.0 and $m_{jj}$> 1000 $\mathrm{GeV}$, while in the CMS analysis, no selection on $m_{jj}$ is applied, but a boosted decision tree (BDT) discriminant is trained to separate the ggF and the VBF production modes.

The reconstruction of the two H candidates is performed by both analyses by choosing the pairing that minimizes the distance from a line in the $(m_{bb}^1$, $m_{bb}^2)$ plane that connects the origin to the expected peak position of the HH signal. In the CMS case, the choice that maximizes the $p_{\mathrm{T}}$ of the H candidates is additionally performed when the jet resolution limits the performance of this decision. As this procedure effectively minimizes the invariant mass difference between the two pairings but avoids directly imposing the $m_{\mathrm{H}}=125\mathrm{GeV}$ hypothesis, the combinatoric challenge is solved without sculpting the multijet background to look like the HH signal. Optimal resolution on b jets is ensured through the use of a regression algorithm to estimate the neutrino energy loss from B hadronization and decays.

Signal regions are defined by the presence of 4 b jets and by requiring the masses of the two H candidates to be compatible with the Higgs boson. The CMS analysis performs an additional categorization of the events depending on their properties (low and high $m_{\mathrm{HH}}$ for ggF, SM and BSM kinematics for VBF), providing maximal sensitivity to both the SM production and anomalous couplings.

The ATLAS analysis uses the $m_{\mathrm{HH}}$ distribution to search for the presence of the signal, as shown in Figure 11. In the CMS analysis, either $m_{\mathrm{HH}}$ or a counting experiment are used in the VBF categories, depending on the expected total number of events. For the ggF case, the output of a BDT that combines several kinematic variables is used, looking for the signal as an enhancement at high scores of this discriminant. An example of the distributions of these variables are shown in Figure 12.

The accurate modeling of the background calls for data-driven methods, and the multijet processes are estimated from background-enriched regions where the b tagging requirements and the H mass requirements are inverted. In the ATLAS VBF analysis, an iterative procedure is implemented to progressively correct the background template, following the method developed for the partial Run 2 data set analysis [45]. In the CMS analysis, a new method was deployed. It uses a BDT with a dedicated metric to learn the differences between $3\mathrm{b}$ and $4\mathrm{b}$ data in a mass sideband region and to correct the control region data, enabling the simultaneous modeling of the variables that are used in the final discriminant.

Boosted HH →$\mathbf{b}\overline{\mathbf{b}}\mathbf{b}\overline{\mathbf{b}}$

The boosted search performed by the CMS Collaboration [57,58] studies both the ggF and VBF signal. For the VBF case, the focus of the analysis is on anomalous ${\kappa }_{2\mathrm{V}}$ couplings, which results in the hard $m_{\mathrm{HH}}$ distributions shown in Figure 6. The two Higgs bosons are reconstructed as large-radius jets with $p_{\mathrm{T}}$ above 400 and 500 $\mathrm{GeV}$, and two standard radius jets are additionally required to reconstruct the outgoing VBF partons. For the ggF search, where the signal is expected to be softer, lower thresholds of 300$\mathrm{GeV}$ are used for the jets. The ParticleNet algorithm [59,60], based on a powerful graph neural network, is used to identify the $\mathrm{H}\to b\overline{b}$ decays and to improve the resolution on the jet mass, benefiting the definition of signal regions based on $m_{\mathrm{H}}$. Categories for different signal purity are defined based on the ParticleNet output to maximize the sensitivity. A combination of simulation for hadronic $t\overline{t}$ and data-driven methods (for QCD multijet) are employed to estimate the background from sidebands defined by inverted ParticleNet and Higgs boson mass requirements. A summary of the expected number of events in the boosted searches is shown in Figure 13.

Although the orthogonality with the resolved $\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$ CMS analysis is not yet enforced in these results, the overlap is expected to be small since the boosted selections define a small portion of the total phase space, which could be removed from the events selected in the resolved analysis with a small impact on the sensitivity. The two topologies are thus expected to contribute equally to the global sensitivity.

2.1.2. HH→$\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$

The $\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$ channel trades off between the large branching ratio for Higgs bosons decaying to bottom quarks, and the stronger background rejection when selecting two tau leptons. Both Higgs bosons can be reconstructed, and their masses can be used to discriminate between the signal and background.

Both ATLAS [61] and CMS [62] have updated their $\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$ searches to the full Run 2 data set. Events are categorized by the tau decays (${\tau }_{\mathrm{h}}{\tau }_{\mathrm{h}}$, ${\tau }_{\mu }{\tau }_{\mathrm{h}}$, and ${\tau }_{\mathrm{e}}{\tau }_{\mathrm{h}}$), and by the triggers used to select events (ATLAS), or by the number of b-tagged jets and by resolved, boosted, or VBF topology (CMS). Machine learning plays a big role in these analyses, from the b jet and tau tagging, to $\mathrm{H}\to \mathrm{b}\mathrm{b}$ candidate tagging (CMS), to signal and background discrimination. The latter combines the kinematic properties of the reconstructed final state objects to enhance the presence of signal at high values of the discriminant. Background modeling is a key challenge, and MC simulation is supplemented with dedicated control regions to constrain the normalization of key backgrounds involving Z bosons and top quarks. Signal extraction is performed in the MVA output in each category. Figure 14 shows the BDT output score in the ATLAS ${\tau }_{had}{\tau }_{had}$ category and aggregated DNN output score, sorted by the expected signal-to-background ratio, for all the CMS categories combined.

2.1.3. HH→$\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$

The $\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$ decay channel benefits from the excellent $m_{\gamma \gamma }$ resolution for the extraction of results, as well as the efficient and relatively low-threshold diphoton triggers used by both experiments. A large challenge in the last analysis round was the low statistics in data, once a simple signal selection was made. The size of the full Run 2 data set allowed for more creativity in analysis design for this round.

Both ATLAS [63] and CMS [64] updated their $\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$ searches to the full Run 2 data set. Events with at least two photons with ${p_{\mathrm{T}}}^1>0.35\left(0.33\right)\times m_{\gamma \gamma }$ and ${p_{\mathrm{T}}}^2>0.25\times m_{\gamma \gamma }$ and exactly two (at least two) b-tagged jets are selected with $p_{\mathrm{T}}>25\mathrm{GeV}$ by ATLAS (CMS). Both searches use advanced machine learning techniques to separate the tiny HH→$\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$ signal from the large background and from the contamination of single Higgs processes (mostly $\mathrm{t}\mathrm{t}\mathrm{H}$) that are characterized by the same $\mathrm{H}\to \gamma \gamma$ signature. A modified four-body-mass variable, M*, defined as M* = $m_{\gamma \gamma jj}-(m_{\gamma \gamma }-m_H)-(m_{jj}-m_H)$ is used to reduce dependence on the jet and photon energy resolutions. This is true in the case that the photons and jets originate from the signal process. Both searches take advantage of the good acceptance for low-p${}_T$ photons by dividing the M* spectrum into a number of categories (ATLAS divides the M* spectrum in half and then creates two additional categories based on BDT score, CMS creates two M* categories for the VBF selection, and four M* categories for the ggF selection, before splitting into 3 further categories based on MVA score), for a total of 4 ATLAS categories and 14 CMS categories. Then, the $m_{\gamma \gamma }$ spectrum is used to extract the final results: ATLAS fits the $m_{\gamma \gamma }$ spectrum in each category, while CMS performs a simultaneous fit in $m_{\gamma \gamma }$ × $m_{jj}$ in each category. The fit variables are shown in Figure 15, and the results are presented at the end of this section.

2.1.4. Final States with Leptons

The decay channels involving light leptons in the final state, including $\mathrm{b}\overline{\mathrm{b}}{\mathrm{VV}}^{*}$ ($\ell \nu \ell \nu$), $\mathrm{b}\overline{\mathrm{b}}\mathrm{Z}{\mathrm{Z}}^{*}$ ($\ell \ell \ell \ell$), ${\mathrm{W}}^{\pm }{\mathrm{W}}^{\mp }{\mathrm{W}}^{\pm }{\mathrm{W}}^{\mp }$, ${\mathrm{W}}^{\pm }{\mathrm{W}}^{\mp }{\tau }^{-}{\tau }^{+}$, and ${\tau }^{-}{\tau }^{+}{\tau }^{-}{\tau }^{+}$ bring unique challenges for signal and background separation, as well as HH event reconstruction. These channels often benefit from low light lepton ($e,\mu$) trigger thresholds.

Multilepton: The CMS multilepton search [65] targets HH→${\mathrm{W}}^{\pm }{\mathrm{W}}^{\mp }{\mathrm{W}}^{\pm }{\mathrm{W}}^{\mp }$, ${\mathrm{W}}^{\pm }{\mathrm{W}}^{\mp }{\tau }^{-}{\tau }^{+}$, and ${\tau }^{-}{\tau }^{+}{\tau }^{-}{\tau }^{+}$ decays. Together, these channels make up 7.7% of all HH decays, so this could be an important place to gain sensitivity to SM HH production. This analysis divides the signal region into seven categories based on the number of light leptons and/or hadronic taus. The categories and their sensitivity to SM HH production can be seen in Figure 16a. This plot demonstrates the power of applying a common analysis technique to selections that are not individually very powerful in the overall HH combination, but together, they can set an impressive limit.

$\mathrm{b}\overline{\mathrm{b}}Z{Z}^{*}$ (4ℓ): The CMS bbZZ(4ℓ) search [66] is performed in events with at least four leptons, forming two pairs of opposite sign, same flavor leptons, and at least two jets with p${}_T$> 20 $\mathrm{GeV}$. The leading lepton is required to have p${}_T$> 20 $\mathrm{GeV}$, and the subleading lepton is required to have p${}_T$> 12 (10) $\mathrm{GeV}$ for electrons (muons). No other requirements are placed on the jets, and the two jets with highest b-tagging score are chosen as the $\mathrm{H}\to \mathrm{b}\mathrm{b}$ candidate jets. A BDT is trained separately in nine categories based on lepton flavor ($4e$, $2e2\mu$, $4\mu$) and data-taking year (2016–2018). The combined BDT output score for all categories is shown in Figure 16b.

bb$\ell \nu \ell \nu$: The ATLAS bb$\ell \nu \ell \nu$ search [67] starts by selecting events with exactly two oppositely charged light leptons (e,$\mu$) and exactly two b-tagged jets. A DNN classifier is constructed using kinematic information from each of the two Higgs candidates and their relative position in the detector, and trained on the dominant HH→$\mathrm{b}\overline{\mathrm{b}}{W}^{\pm }{W}^{\mp }$ process. The signal region is split into events with same flavor lepton pairs and those with different flavor, which allows for better separate between the signal and the background from Z boson production. Figure 16c shows the comparison between the data and the background modeling, for the DNN discriminant variable, $d_{HH}$.

2.1.5. Results and Summary

Table 4. The 95% CL upper limits on $\sigma /{\sigma }_{SM}$ in the Run 2 HH searches presented in this section, improvement compared to previous searches (where applicable), and 95% CL exclusion range for the self-coupling modifer, ${\kappa }_{\lambda }$.

		$\mathit{\sigma }/{\mathit{\sigma }}_{\mathbf{SM}}$ Obs. (Exp.)	Improvement w.r.t $36{\mathbf{fb}}^{-1}$	${\mathit{\kappa }}_{\mathit{\lambda }}$ Exclusion Obs. (Exp.)	Ref.
$\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$ (boosted)	CMS	3.9 (7.8)	4.7×	[−2.3,9.4] ([−5.0,12.0])	[55]
	CMS	9.9 (5.1)	30×	[−9.9,16.9] ([−5.1,12.1])	[58]
$\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$	ATLAS	4.7 (3.9)	3.8×	[−2.4,9.2] ([−2.0,9.0])	[61]
	CMS	3.3 (5.2)	4.8×	[−1.8,8.8] ([−3.0,9.9])	[62]
$\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$	ATLAS	4.2 (5.7)	4.6×	[−1.5,6.7] ([−2.4,7.7])	[63]
	CMS	7.7 (5.2)	3.6×	[−3.3,8.5] ([−2.5,8.2])	[64]
multi-ℓ	CMS	21.8 (19.6)	-	[−7.0,11.7] ([−7.0,11.2])	[65]
bbZZ(4ℓ)	CMS	30 (37)	-	[−9.0,14.0] ([−10.5,15.5])	[66]
bb$\ell \nu \ell \nu$	ATLAS	40 (29)	-	-	[67]

presents the limits set on $\sigma /{\sigma }_{SM}$, a comparison to the earlier Run 2 limits, and the limits set on the self-coupling modifier, ${\kappa }_{\lambda }$, in each analysis. Figure 17 summarizes the limits on the HH production cross section compared to its SM value, for each of the analyses presented in this section.

Limits on the VBF SM cross section for the various channels are summarized in

Table 5. The 95% CL upper limits on ${\sigma }^{VBF}/{\sigma }_{SM}^{VBF}$ in the Run 2 VBF HH searches presented in this section and 95% CL exclusion range for the $HHVV$ coupling modifier, ${\kappa }_{2\mathrm{V}}$.

		${\mathit{\sigma }}^{\mathbf{VBF}}/{\mathit{\sigma }}_{\mathbf{SM}}^{\mathbf{VBF}}$ Obs. (Exp.)	${\mathit{\kappa }}_{2\mathbf{V}}$ Observed Exclusion	${\mathit{\kappa }}_{2\mathbf{V}}$ Expected Exclusion	Ref.
$\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$ (boosted)	CMS	-	0.62 < ${\kappa }_{2\mathrm{V}}$ < 1.41	0.66 < ${\kappa }_{2\mathrm{V}}$ < 1.37	[58]
$\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$	ATLAS	840 (550)	−0.43 <${\kappa }_{2\mathrm{V}}$< 2.56	−0.55 <${\kappa }_{2\mathrm{V}}$< 2.72	[56]
$\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$	CMS	226 (412)	−0.1 <${\kappa }_{2\mathrm{V}}$< 2.2	−0.4 <${\kappa }_{2\mathrm{V}}$< 2.5	[55]
$\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$	CMS	124 (154)	−0.4 <${\kappa }_{2\mathrm{V}}$< 2.6	−0.6 <${\kappa }_{2\mathrm{V}}$< 2.8	[62]
$\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$	CMS	225 (208)	−1.3 <${\kappa }_{2\mathrm{V}}$< 3.5	−0.9 <${\kappa }_{2\mathrm{V}}$< 3.0	[64]

. The excluded cross section as a function of ${\kappa }_{2\mathrm{V}}$ is shown in Figure 18 for the boosted $\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$ CMS analysis, that provides the strongest constraint. Results are shown under the variation of ${\kappa }_{2\mathrm{V}}$ only, and with the simultaneous variation of ${\kappa }_{\mathrm{V}}$ as well. The excluded contours connect to the dependence of the theoretical cross section on these couplings of Figure 7c.

Limits on the shape benchmarks discussed in Section 1.3.2 are illustrated in Figure 19 for the CMS $\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$ search (LO benchmarks) and the ATLAS combination of $\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$ and $\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$ (NLO benchmarks). The scatter of the excluded cross sections, that vary by more than one order of magnitude, illustrates the sensitivity of the analyses to the kinematic effects, and shows that constraints on the SM cannot be simply ported to generic EFT sensitivities. As we are approaching the SM sensitivity for HH production, a systematic study of the EFT effects and of the contributions of the operators beyond simple benchmarks will help in extracting the most information from the experimental measurements.

2.2. Preliminary Run 2 Combination

The ATLAS Collaboration performed a preliminary combination [69] of the recent $\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$ [63] and $\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$ [61] HH results. The observed (expected) combined 95% CL upper limit is found to be 3.1 (3.1) times the standard model prediction, which represents a factor of 3.2 improvement over the previous ATLAS HH combination, even though this preliminary combination only uses two channels. The observed (expected) allowed range for the self-coupling modifier, ${\kappa }_{\lambda }$, is found to be −1.0 <${\kappa }_{\lambda }$< 6.6 (−1.2 <${\kappa }_{\lambda }$< 7.2), at 95% CL.

Both the $\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$ and the $\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$ channels are limited by statistical uncertainties, and their dominant uncertainties are mostly independent between the two analyses. As a consequence, the combined result is close to a simple sum in the quadrature of the individual sensitivities, as shown in Figure 20. Since these assumptions hold for all the HH channels summarized in Table 4 and Figure 17, we can perform the same sum for all the channels and estimate that a sensitivity of the order of 2–3 times the SM prediction will be achieved by each experiment at the LHC Run 2.

2.3. Lessons Learned from the Run 2 HH Programme

Huge progress has already been made toward the eventual measurement of the Higgs pair production cross section. This progress can be attributed to the improvements in online and offline event selection, object reconstruction, and creative analysis techniques. These improvements have allowed the ATLAS and CMS collaborations to produce results that outpace the improvements expected simply from the addition of more data. Two elements that proved to be central to achieve these results are as follows:

HH event kinematics: understanding and exploring the unique kinematics of HH events with varied anomalous couplings has allowed this round of analyses to achieve unprecedented limits. As shown in Figure 4, varying ${\kappa }_{\lambda }$ from its SM value results in softer event kinematics. Following early Run 2 analyses, we used this information systematically in all channels, using the $m_{\mathrm{HH}}$ information in the analyses and designing the selections to also target low-$m_{\mathrm{HH}}$ events for ${\kappa }_{\lambda }$ extraction. This in turn highlights the importance of being able to select low-$p_{\mathrm{T}}$ objects in the trigger, which has traditionally been a challenge for hadronic signatures because of the large multi-jet background at the LHC. The latter point is one that we should keep in mind when preparing for Run 3 and HL-LHC operations.

Machine learning was used to great effect in all of the HH analyses presented in this review. It contributes to these HH searches at all levels, from object identification and selection to separating the small HH signal process from large backgrounds. The importance of high-performance object identification is illustrated by the $\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$ and $\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$ analyses, where improved neural network-based algorithms have boosted the sensitivities.

3. Prospects for Future Measurements

3.1. LHC Run 3

The LHC plans to increase the center-of-mass energy from $\sqrt{s}=13\mathrm{TeV}$ to $\sqrt{s}$ = $13.6\mathrm{TeV}$ for the Run 3, which will start in mid-2022. The larger center-of-mass energy will result in a ≈20% larger HH production cross section for gluon–gluon fusion (ggF). LHC Run 3 will produce approximately 400${\mathrm{fb}}^{-1}$ of data per experiment, which will be especially beneficial to the HH searches, which are currently statistically limited.

Both ATLAS and CMS have finished “Phase-I” upgrades to the detectors, which will improve the quality of data that can be recorded during Run 3. The ATLAS Phase-I upgrade involves the installation of new muon detectors [70], which will improve muon identification and trigger capabilities, an upgrade of the liquid argon calorimeter readout electronics, which will provide higher granularity information to the trigger [71], and and upgrade of the trigger and data acquisition system to make use of the upgraded detector capabilities [72]. The CMS Phase-I upgrades, partly already operating during Run 2, included an improved pixel detector [73], an upgraded hadron calorimeter to cope with the higher pileup environment [74], new endcap muon detectors [75] and a replacement of the calorimeter, muon, and global trigger electronics [76], that will continue to assure efficient online event selection during Run 3.

Following the considerations highlighted in Section 2.2, we can expect that the Run 2 combined limit will improve by at least $\sqrt{400/140}\approx 1.7$. A combination of the results of ATLAS and CMS should bring roughly a factor $\sqrt{2}$ of improvement, meaning that a Run 3 LHC sensitivity at the level of the SM predictions is within reach. If the impressive pace of improvements discussed in Section 2 is maintained, a hint or an evidence of HH production is in reach during LHC Run 3.

3.2. HH Prospects at the High-Luminosity LHC

The full LHC physics program, expected to extend until ∼2040, will culminate in the construction and operation of the high-luminosity LHC (HL-LHC), which is an ambitious upgrade to the accelerator complex [77]. The HL-LHC will collide protons at $\sqrt{s}=14\mathrm{TeV}$, and plans to provide 3000 ${\mathrm{fb}}^{-1}$ of data to each of the ATLAS and CMS experiments. The data-taking conditions at the HL-LHC will be challenging: the instantaneous luminosity will increase to 5–$7\times {10}^{34}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$, and the number of simultaneous collisions will increase to 140–200 per event.

The Phase-II upgrades of the ATLAS and CMS detectors [78,79] will facilitate data taking and event reconstruction in the much higher pileup environment of the HL-LHC. In order to maintain or lower the trigger thresholds with respect to those used in Run 2, the trigger and data acquisition systems will be substantially upgraded or replaced. New tracker systems will be installed, extending the tracking coverage up to about |$\eta$| = 4. The calorimeters will be upgraded with new electronics or replaced with new technologies in the challenging endcap regions. Detectors to precisely measure the time of arrival of particles will help to face the challenges of pileup. Finally, the muon systems will be upgraded with new electronics and additional muon chambers.

Early HL-LHC projections [80], based on extrapolations from early Run 2 HH analyses or on dedicated analyses of HL-LHC conditions with a parametric simulation, suggest that ATLAS and CMS would achieve together a combined significance of 4$\sigma$, including the $\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$, $\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$, $\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$, $\mathrm{b}\overline{\mathrm{b}}\mathrm{Z}{\mathrm{Z}}^{*}$ (4ℓ), and $\mathrm{b}\overline{\mathrm{b}}{\mathrm{VV}}^{*}$ ($\ell \nu \ell \nu$) channels. The projected precision on the value of the self-coupling is 50%. Figure 21 presents the negative log-likelihood scan as a function of ${\kappa }_{\lambda }$, for each of the individual HH channels considered and their combination. This corresponds to an exclusion range of $0.1<{\kappa }_{\lambda }<2.3$ at 95% CL.

A small number of individual channel HL-LHC projections was recently updated, and they are summarized here:

• $\mathrm{HH}\to$$\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$ + $\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$: The recent preliminary combination of the $\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$ and $\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$ channels by ATLAS (described in Section 2 is projected to the HL-LHC luminosity, and results in a combined significance of $3.2\sigma$, assuming the baseline systematic uncertainty scenario described in the reference [81]. This result also excludes values of ${\kappa }_{\lambda }$ outside the range $0.0<{\kappa }_{\lambda }<2.7$.
• HH in final states with photons: CMS prepared HH projection in three channels, in final states including photons. These studies were performed using dedicated simulations of the HL-LHC conditions (so they are not projections of earlier published work), and using the generic detector delphes [82]. In the $\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$ channel [83], a boosted decision tree is used to separate the signal and background, and the extraction of the signal significance is performed using a simultaneous fit to the $m_{\gamma \gamma }$ and $m_{bb}$ distributions. The significance is found to be $2.16\sigma$ (to be compared to $1.8\sigma$ in a previous projection). In the ${\mathrm{W}}^{\pm }{\mathrm{W}}^{\mp }\gamma \gamma$ and ${\tau }^{-}{\tau }^{+}\gamma \gamma$ channels [84], events are categorized by the number of light leptons or taus, and then the signal extraction is performed in each category using the $m_{\gamma \gamma }$ distribution. The resulting significance is $0.22\sigma$, when combining all categories.
• CMS ttHH (4b): The cross section for ttHH production in the standard model is incredibly small, at just 0.948 fb [85]. In this search [86], events with exactly one light lepton and at least four jets are fed into a 2-step deep neural network to first separate ttHH events from events containing top quarks and Z bosons, and then to separate the SM ttHH signal from all background processes, for the various jet and b-jet multiplicities. The statistical analysis is performed using the simultaneously fitting the DNN discriminants, and an upper limit is set on SM ttHH production of 3.14 times the SM cross section.

We expect the exciting trend of improvements due to creative analysis techniques to continue, and look forward to seeing further HH projections for the HL-LHC as they are prepared.

3.3. Future Colliders

While we expect that improvements to data-collection, reconstruction, and analysis techniques will improve enough such that observation of SM Higgs pair production will be possible at the HL-LHC, it is unlikely that the Higgs self-coupling will be measured at comparable precision to other Higgs couplings that have been studied at the LHC. A precision of 5–10% would give us sensitivity to a broad class of new particles that could couple to the Higgs boson, and a precision of 1% would be sensitive to quantum corrections to the value of the self-coupling [87].

Projections of the sensitivity to the Higgs self-coupling were performed for several proposed future colliders:

• Low-energy $e^{+}{e}^{-}$ colliders such as the FCC-ee, CEPC, or the proposed low-energy ILC runs, are able to make indirect measurements of the Higgs self-coupling through higher-order corrections to single-Higgs processes. These machines will be able to measure the self-coupling with a precision of 40% [88].
• Higher-energy $e^{+}{e}^{-}$ colliders such as CLIC or higher-energy ILC runs would be able to produce pairs of Higgs bosons, and thus have direct access to the self-coupling, which would allow an O(20%) determination of the Higgs self-coupling [88].
• A ${\mu }^{+}{\mu }^{-}$ collider operating at $\sqrt{s}=10\left(30\right)\mathrm{TeV}$ could achieve 5.6 (2.0)% precision on the self-coupling [89], studying pairs of Higgs bosons produced via vector boson fusion.
• Very high energy $pp$ colliders, such as the proposed 100 $\mathrm{TeV}$ FCC-hh, would also have the ability to produce pairs of Higgs bosons. A study [90] combining the $\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$, $\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$, and $\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$ decay modes projects a precision of 3.4–7.8% on the value of the self-coupling.

4. Conclusions

The study of HH production connects deeply to the foundations of the SM. This process gives direct access to the determination of the Higgs boson self-coupling, and thus to the shape of the scalar potential postulated in the Brout–Englert–Higgs mechanism, with deep implications on our understanding on the foundations of particle physics and on the cosmological history of our Universe. The measurement of HH provides at the same time a test of the consistency of the SM and a probe of the possible existence of as-yet-undiscovered physics beyond it.

The ATLAS and CMS Collaborations at the CERN LHC are pursuing a broad program of investigation of HH production. As this manuscript is being prepared, most of the analyses based on the full Run 2 $\mathrm{p}\mathrm{p}$ data set, recorded in 2016–2018 at a center-of-mass energy of 13 $\mathrm{TeV}$ and corresponding to an integrated luminosity $\mathcal{L}\approx 140{\mathrm{fb}}^{-1}$, have been completed, and their statistical combination is being prepared. The most sensitive individual channels achieve a sensitivity in the Run 2 data set ranging between 4 and 8 times the SM prediction, meaning that their combination is expected to probe signals that are 2–3 times the SM prediction. The analyses currently study both the ggF and the VBF HH production modes, and set constraints on the strength of the self-coupling, ${\kappa }_{\lambda }$, and on the $\mathrm{VV}\mathrm{HH}$ interaction, ${\kappa }_{2\mathrm{V}}$. The current best constraint of ${\kappa }_{\lambda }$ is between −1 and 6.6 from the combination of the $\mathrm{b}\overline{\mathrm{b}}\gamma \gamma$ and $\mathrm{b}\overline{\mathrm{b}}{\tau }^{-}{\tau }^{+}$ analyses, and will further improve, as the full set of channels will be combined. The best constraints of ${\kappa }_{2\mathrm{V}}$ from the $\mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}}$ channel in a search for high Lorentz boost Higgs bosons and corresponds to the range $0.6<{\kappa }_{2\mathrm{V}}<1.4$, and excludes the hypothesis that the $\mathrm{VV}\mathrm{HH}$ interaction does not exist when we assume that only the SM diagrams participate in VBF HH production.

When compared to the previous set of results, based on the partial Run 2 data set with $\mathcal{L}\approx 36{\mathrm{fb}}^{-1}$, the latest analyses show impressive improvements that are factors larger than the simple improvement expected from the increase in the size of the data set that is roughly proportional to $1/\sqrt{\mathcal{L}}$, i.e., $\sqrt{140/36}\approx 2$ times better. These results were enabled by the improvements in the analysis techniques and in the identification of physics objects, often leveraging on new machine learning techniques. The larger data set and the growing interest for HH also enabled the study of new channels and of new topologies of the final states that contribute to boost the total sensitivity.

Such considerations set the ground for excellent prospects for the LHC Run 3, where a further increase in the total data set size to about $400{\mathrm{fb}}^{-1}$ is expected after 3 years of $\mathrm{p}\mathrm{p}$ run in 2023–2025 at a center-of-mass energy of $13.6\mathrm{TeV}$. With the sole increase in the luminosity and the combination of the results of ATLAS and CMS, a global LHC sensitivity at the level of the SM prediction is within reach. Further progress in the object identification methods and analysis techniques, and a careful preparation of the Run 3 data taking with more efficient triggers may bring a first hint or an evidence of the existence of HH production in only a few years from now.

At the HL-LHC, where a data set of about $3000{\mathrm{fb}}^{-1}$ of $\sqrt{s}=14\mathrm{TeV}$ proton–proton collisions is expected to be collected by ATLAS and CMS, the ultimate LHC sensitivity to HH and to the self-coupling will be achieved. The prospects for these measurements were comprehensively studied in the context of the update of the European Strategy for Particle Physics submitted in 2018. These results show that a combination of the ATLAS and CMS HH analyses is expected to yield a significance of $4\sigma$ for the SM signal and to constrain the value of ${\kappa }_{\lambda }$ to a precision of about 50% at the 68% confidence level. It should be noted, however, that these projections are based on the extrapolation or on the reproduction with a simplified simulation of the early Run 2 analyses based on the 2015 and 2016 data sets, and therefore mostly account for the increase in the integrated luminosity and the performance of the upgraded detectors. The improvements already observed in the timescale of less than five years with the most recent results show that larger data sets enable more sophisticated and powerful analyses, and suggest that these projections may be conservative, as supported by recent projections in selected final states.

The study of HH at future $e^{+}{e}^{-}$ colliders will require high center-of-mass energies that are achievable only in the latest stages of the linear machines (CLIC, ILC), while at lower energies and at circular colliders (FCC-ee, CEPC), indirect constraints on the self-coupling can be set from the study of single Higgs bosons. New machines, such as a ${\mu }^{+}{\mu }^{-}$ collider, might achieve the energy for the study of HH. Finally, the 100 $\mathrm{TeV}$ $\mathrm{p}\mathrm{p}$ collider, FCC-hh, that is being discussed as the long-term CERN collider project has the capability to measure ${\kappa }_{\lambda }$ with a precision of ∼5%, providing an accurate map of the shape of the scalar potential.

There is no doubt that the study of HH is an essential part of the short-, mid-, and long-term programs of the research in particle physics, and that its study might hold the key to answer many open questions of high energy physics. We are currently in a very favorable moment of this program: we are about to fully exploit the Run 2 data set, getting ready to unlock the potential of LHC Run 3, and preparing for the challenges and opportunities of the HL-LHC. We are looking forward to everything that the study of HH production will reveal.