Boosting Higgs pair production in the \(b\bar{b}b\bar{b}\) final state with multivariate techniques
Abstract
The measurement of Higgs pair production will be a cornerstone of the LHC program in the coming years. Double Higgs production provides a crucial window upon the mechanism of electroweak symmetry breaking and has a unique sensitivity to the Higgs trilinear coupling. We study the feasibility of a measurement of Higgs pair production in the \(b\bar{b}b\bar{b}\) final state at the LHC. Our analysis is based on a combination of traditional cutbased methods with stateoftheart multivariate techniques. We account for all relevant backgrounds, including the contributions from light and charm jet misidentification, which are ultimately comparable in size to the irreducible 4b QCD background. We demonstrate the robustness of our analysis strategy in a high pileup environment. For an integrated luminosity of \({\mathcal {L}}=3\) ab\(^{1}\), a signal significance of \(S/\sqrt{B}\simeq 3\) is obtained, indicating that the \(b\bar{b}b\bar{b}\) final state alone could allow for the observation of double Higgs production at the High Luminosity LHC.
1 Introduction
The measurement of double Higgs production will be one of the central physics goals of the LHC program in its recently started highenergy phase, as well as for its future highluminosity upgrade (HLLHC) which aims to accumulate a total integrated luminosity of 3 ab\(^{1}\) [1, 2]. Higgs pair production [3] is directly sensitive to the Higgs trilinear coupling \(\lambda \) and provides crucial information on the electroweak symmetry breaking mechanism. It also probes the underlying strength of the Higgs interactions at high energies, and it can be used to test the composite nature of the Higgs boson [4, 5]. While Standard Model (SM) cross sections are small, many Beyond the SM (BSM) scenarios predict enhanced rates for double Higgs production; therefore searches have already been performed by ATLAS and CMS with Run I data [6, 7, 8, 9, 10] and will continue at Run II. The study of Higgs pair production will also be relevant to any future highenergy collider, either at a 100 TeV circular machine [11, 12, 13, 14] or at a linear or circular electron–positron collider [15].
Analogously to single Higgs production [16], in the SM the dominant mechanism for the production of a pair of Higgs bosons at the LHC is gluon fusion (see [3, 17] and references therein). For a centerofmass energy of \(\sqrt{s} = 14\) TeV, the nexttonexttoleading order (NNLO) total cross section is approximately 40 fb [18], which is increased by a further few percent once nexttonexttoleading logarithmic (NNLL) corrections are accounted for [19]. Feasibility studies in the case of a SMlike Higgs boson in the gluonfusion channel at the LHC have been performed for different final states, including \(b\bar{b}\gamma \gamma \) [20, 21, 22], \(b\bar{b}\tau ^+\tau ^\) [23, 24, 25, 26], \(b\bar{b}W^+W^\) [25, 27] and \(b\bar{b}b\bar{b}\) [21, 23, 25, 28, 29]. While these studies differ in their quantitative conclusions, the consistent picture emerges that the ultimate precision in the determination of the Higgs trilinear coupling \(\lambda \) requires the full integrated luminosity of the HLLHC, \({\mathcal {L}}=3\) ab\(^{1}\) and should rely on the combination of different final states. The interplay between kinematic distributions for the extraction of \(\lambda \) from the measured cross sections and the role of the associated theoretical uncertainties have been intensely scrutinised recently [17, 30, 31, 32, 33, 34, 35, 36, 37].
In addition to the gluonfusion channel, Higgs pairs can also be produced in the vectorboson fusion channel hhjj [5, 26, 38, 39], the associated production modes hhW and hhZ [3, 40, 41] (also known as HiggsStrahlung), and also in association with top quark pairs \(hht\bar{t}\) [42]. All these channels are challenging due to the small production rates: at 14 TeV, the inclusive total cross sections are 2.0 fb for VBF hhjj [43], 0.5 fb for W(Z)hh [3] and 1.0 for \(hht\bar{t}\) [42].
While the SM production rates for Higgs pairs are small, they are substantially enhanced in a variety of BSM scenarios. Feasibility studies of Higgs pair production in New Physics models have been performed in a number of different frameworks, including Effective Field Theories (EFTs) with higherdimensional operators and anomalous Higgs couplings [14, 44, 45, 46, 47, 48, 49, 50], resonant production in models such as extra dimensions [51, 52, 53, 54], and Supersymmetry and Two Higgs Doublet models (2HDMs) [55, 56, 57, 58, 59, 60, 61]. Since BSM dynamics modify the kinematic distributions of the Higgs decay products, for instance boosting the diHiggs system, different analysis strategies might be required for BSM Higgs pair searches as compared to SM measurements.
Searches for the production of Higgs pairs have already been performed with 8 TeV Run I data by ATLAS in the \(b\bar{b}b\bar{b}\) [7] and \(b\bar{b}\gamma \gamma \) [8] final states, and by CMS in the same \(b\bar{b}b\bar{b}\) [9] and \(b\bar{b}\gamma \gamma \) [10] final states. In addition, ATLAS has presented [6] a combination of its diHiggs searches in the \(bb\tau \tau ,\) \(\gamma \gamma WW^*\), \(\gamma \gamma bb\) and bbbb final states. Many other exotic searches involve Higgs pairs in the final state, such as the recent search for heavy Higgs bosons H [62].
In the context of SM production, the main advantage of the \(b\bar{b}b\bar{b}\) final state is the enhancement of the signal yield from the large branching fraction of Higgs bosons into \(b\bar{b}\) pairs, \(\mathrm{BR}\left( H\rightarrow b\bar{b}\right) \simeq 0.57\) [16]. However, a measurement in this channel needs to deal with an overwhelming QCD multijet background. Recent studies of Higgs pair production in this final state [28, 29] estimate that, for an integrated luminosity of \({{\mathcal {L}}}=3\) ab\(^{1}\), a signal significance of around \(S/\sqrt{B}\simeq 2.0\) can be obtained. In these analysis, irreducible backgrounds such as 4b and \(t\bar{t}\) are included, however, the reducible components, in particular bbjj and jjjj, are neglected. These can contribute to the signal yield when light and charm jets are misidentified as bjets. Indeed, due to both selection effects and bquark radiation in the parton shower, the contribution of the 2b2j process is as significant as the irreducible 4b component.
In this work, we revisit the feasibility of SM Higgs pair production by gluon fusion in the \(b\bar{b}b\bar{b}\) final state at the LHC. Our strategy is based upon a combination of traditional cutbased methods and multivariate analysis (MVA). We account for all relevant backgrounds, including the contribution from misidentified light and charm jets. We also assess the robustness of our analysis strategy in an environment with high pileup (PU). Our results indicate that the \(b\bar{b}b\bar{b}\) final state alone should allow for the observation of double Higgs production at the HLLHC.
The structure of this paper proceeds as follows. In Sect. 2 we present the modeling of the signal and background processes with Monte Carlo event generators. In Sect. 3 we introduce our analysis strategy, in particular the classification of individual events into different categories according to their topology. Results of the cutbased analysis are then presented in Sect. 4. In Sect. 5 we illustrate the enhancement of signal significance using multivariate techniques, and we assess the robustness of our results against the effects of PU. In Sect. 6 we conclude and outline future studies to estimate the accuracy in the determination of the trilinear coupling \(\lambda \) and to provide constraints in BSM scenarios.
2 Modeling of signal and background processes
In this section we discuss the Monte Carlo generation of the signal and background process samples used in this analysis. We shall also discuss the modeling of detector resolution effects.
2.1 Higgs pair production in gluon fusion
Details of the signal and background Monte Carlo samples used in this work. Also provided are the inclusive Kfactors which are applied to reproduce the known higherorder results
Process  Generator  \(N_{\mathrm {evt}}\)  \(\sigma _{\mathrm {LO}}\) (pb)  Kfactor 

\(pp \rightarrow hh\rightarrow 4b\)  MadGraph5_aMC@NLO  1M  \(6.2\times 10^{3}\)  
\(pp \rightarrow b\bar{b}b\bar{b}\)  SHERPA  3M  \(1.1\times 10^3\)  1.6 (NLO [63]) 
\(pp \rightarrow b\bar{b}jj\)  SHERPA  3M  \(2.7\times 10^5\)  1.3 (NLO [63]) 
\(pp \rightarrow jjjj\)  SHERPA  3M  \(9.7\times 10^6\)  0.6 (NLO [77]) 
\(pp \rightarrow t\bar{t}\rightarrow b\bar{b}jjjj\)  SHERPA  3M  \(2.5\times 10^3\)  1.4 (NNLO+NNLL [78]) 
In Fig. 1 we show representative Feynman diagrams for LO Higgs pair production in gluon fusion. The nontrivial interplay between the heavy quark box and the triangle loop diagrams can lead to either constructive or destructive interference and complicates the extraction of the trilinear coupling \(\lambda \) from the measurement of the Higgs pair production cross section. Higherorder corrections [17, 18] are dominated by gluon radiation from either the initialstate gluons or from the heavy quark loops.
The total inclusive cross section for this processes is known up to NNLO [18]. Resummed NNLO+NNLL calculations for Higgs pair production are also available [19], leading to a moderate enhancement of the order of few percent as compared to the fixedorder NNLO calculation. To achieve the correct higherorder value of the integrated cross section, we rescale our LO signal sample to match the NNLO+NNLL inclusive calculation. This corresponds to a Kfactor \(\sigma _\mathrm{NNLO+NNLL}/\sigma _\mathrm{LO}=2.4\), as indicated in Table 1.
Partonlevel signal events are then showered with the Pythia8 Monte Carlo [71, 72], version v8.201. We use the default settings for the modeling of the underlying event (UE), multiple parton interactions (MPI), and PU, by means of the Monash 2013 tune [73], based on the NNPDF2.3LO PDF set [74, 75].
2.2 Backgrounds
Background samples are generated at leading order with SHERPA [76] v2.1.1. As in the case of the signal generation, the NNPDF 3.0 \(n_f = 4\) LO set with strong coupling \(\alpha _s(m_Z^2)=0.118\) is used for all samples, and we use as factorisation and renormalisation scales \(\mu _F=\mu _R=H_T/2\). We account for all relevant background processes that can mimic the \(hh\rightarrow 4b\) signal process. This includes QCD 4b multijet production, as well as QCD 2b2j and 4j production, and topquark pair production. The latter is restricted to the fully hadronic final state, since leptonic decays of top quarks can be removed by requiring a lepton veto. Single Higgs production processes such as \(Z(\rightarrow b\bar{b})h(\rightarrow b\bar{b})\) and \(t\bar{t}h(\rightarrow b\bar{b})\) (see Appendix A) along with electroweak backgrounds e.g. \(Z(\rightarrow b\bar{b})b\bar{b}\), are much smaller than the QCD backgrounds [28, 29] and are therefore not included in the present analysis.
The LO cross sections for the background samples have been rescaled so that the integrated distributions reproduce known higherorder QCD results. For the 4j sample, we rescale the LO cross section using the BLACKHAT [77] calculation, resulting in an NLO/LO Kfactor of 0.6. For the 4b and 2b2j samples NLO/LO Kfactors of 1.6 and 1.3, respectively, have been determined using MadGraph5_aMC@NLO [63]. Finally, the LO cross section for \(t\bar{t}\) production has been rescaled to match the NNLO+NNLL calculation of Ref. [78], leading to a Kfactor of 1.4. The Kfactors that we use to rescale the signal and background samples are summarised in Table 1.
At the generation level, the following loose selection cuts are applied to background events. Each finalstate particle in the hard process must have \(p_T \ge 20\) GeV, and be located in the central rapidity region with \( \eta  \le 3.0\). At the matrixelement level all finalstate particles must also be separated by a minimum \(\Delta R_{\mathrm {min}} =0.1\). We have checked that these generatorlevel cuts are loose enough to have no influence over the analysis cuts. From Table 1 we see that the \(t\bar{t}\) and QCD 4b cross sections are of the same order of magnitude. However, the former can be efficiently reduced by using top quark reconstruction criteria. The bbjj cross section is more than two orders of magnitude larger than the 4b result, but it will be suppressed by the light and charm jet misidentification rates, required to contribute to the 4b final state.
As a crosscheck of the SHERPA background cross sections reported in Table 1, we have produced leadingorder multijet samples using MadGraph5_aMC@NLO, benchmarked with the results for the same processes reported in Ref. [63]. Using common settings, we find agreement, within scale uncertainties, between the MadGraph5_aMC@NLO and SHERPA calculations of the multijet backgrounds.
2.3 Modeling of detector resolution
To account for the finite angular resolution of the calorimeter, the \(\left( \eta ,\phi \right) \) plane is divided into regions of \(\Delta \eta \times \Delta \phi =0.1\times 0.1\), and each finalstate particle which falls in each of these cells is set to the same \(\eta \) and \(\phi \) values of the center of the corresponding cell. Finally, the energy of each finalstate particle is recalculated from the smeared \(p_T^\prime \), \(\eta ^\prime \) and \(\phi ^\prime \) values to ensure that the resulting fourmomentum is that of a lightlike particle, since we neglect all jet constituent masses in this analysis.
Our modeling of detector simulation has been tuned to lead to a mass resolution of the reconstructed Higgs candidates consistent with the hadronic mass resolutions of the ATLAS and CMS detectors [79, 80, 81], as discussed in Sect. 3.5.
3 Analysis strategy
In this section we describe our analysis strategy. First of all we discuss the settings for jet clustering and the strategy for jet btagging. Following this we discuss the categorisation of events into different topologies, and how the different topologies may be prioritised. We motivate our choice of analysis cuts by comparing signal and background distributions for representative kinematic variables. Finally, we describe the simulation of PU and validate the PUsubtraction strategy.
3.1 Jet reconstruction

SmallR jets. These are jets reconstructed with the anti\(k_T\) clustering algorithm [84] with \(R=0.4\) radius. These smallR jets are required to have transverse momentum \(p_T \ge 40\) GeV and pseudorapidity \(\eta <2.5\), within the central acceptance of ATLAS and CMS, and therefore within the region where btagging is possible.

LargeR jets. These jets are also constructed with the anti\(k_T\) clustering algorithm, now using a \(R=1.0\) radius. LargeR jets are required to have \(p_T \ge 200\) GeV and lie in a pseudorapidity region of \(\eta <2.0\). The more restrictive range in pseudorapidity as compared to the smallR jets is motivated by mimicking the experimental requirements in ATLAS and CMS related to the trackjet based calibration [85, 86]. In addition to the basic \(p_T\) and \(\eta \) acceptance requirements, largeR jets should also satisfy the BDRS massdrop tagger (MDT) [87] conditions, where the FastJet default parameters of \(\mu _\mathrm{mdt} = 0.67\) and \(y_\mathrm{mdt}=0.09\) are used. Before applying the BDRS tagger, the largeR jet constituents are reclustered with the Cambridge/Aachen (C/A) algorithm [88, 89] with \(R=1.0\). In the case of the analysis including PU, a trimming algorithm [106] is applied to all largeR jets to mitigate the effects of PU, especially on the jet mass. For further details, see Sect. 3.5.

SmallR subjets. All finalstate particles are clustered using the anti\(k_T\) algorithm, but this time with a smaller radius parameter, namely \(R=0.3\). The resulting anti\(k_T\) \(R=0.3\) (AKT03) jets are then ghostassociated to each largeR jets in order to define its subjets [7]. These AKT03 subjets are required to satisfy \(p_T > 50\) GeV and \(\eta <2.5\), and they will be the main input for btagging in the boosted category.
 The \(k_T\)splitting scale [87, 92]. This variable is obtained by reclustering the constituents of a jet with the \(k_T\) algorithm [93], which usually clusters last the harder constituents, and then taking the \(k_T\) distance measure between the two subjets at the final stage of the recombination procedure,with \(p_{T,1}\) and \(p_{T,2}\) the transverse momenta of the two subjets merged in the final step of the clustering, and \(\Delta R_{12}\) the corresponding angular separation.$$\begin{aligned} \sqrt{d_{12}} \equiv \mathrm{min}\left( p_{T,1},p_{T,2}\right) \cdot \Delta R_{12}. \end{aligned}$$(3)
 The ratio of 2to1 subjettiness \(\tau _{21}\) [94, 95]. The Nsubjettiness variables \(\tau _N\) are defined by clustering the constituents of a jet with the exclusive \(k_t\) algorithm [96] and requiring that N subjets are found,where \(p_{T,k}\) is the \(p_T\) of the constituent particle k and \(\delta R_{ik}\) the distance from subjet i to constituent k. In this work we use as input to the MVA the ratio of 2subjettiness to 1subjettiness, namely$$\begin{aligned} \tau _N\equiv & {} \frac{1}{d_0} \sum _k p_{T,k}\cdot \mathrm{min}\left( \delta R_{1k}, \ldots , \delta R_{Nk}\right) ,\nonumber \\ d_0\equiv & {} \sum _k p_{T,k}\cdot R, \end{aligned}$$(4)which provides good discrimination between QCD jets and jets arising from the decay of a heavy resonance.$$\begin{aligned} \tau _{21} \equiv \frac{\tau _2}{\tau _1}, \end{aligned}$$(5)
 The ratios of energy correlation functions (ECFs) \(C^{(\beta )}_2\) [97] and \(D_2^{(\beta )}\) [98]. The ratio of energy correlation functions \(C_2^{(\beta )}\) is defined aswhile \(D_2^{(\beta )}\) is instead defined as a double ratio of ECFs, that is,$$\begin{aligned} C_2^{(\beta )} \equiv \frac{ \mathrm{ECF}(3,\beta ) \mathrm{ECF}(1,\beta )}{\left[ \mathrm{ECF}(2,\beta )\right] ^2}, \end{aligned}$$(6)The energy correlation functions \(\mathrm{ECF}(N,\beta )\) are defined in [97] with the motivation that \((N+1)\)point correlators are sensitive to Nprong substructure. The free parameter \(\beta \) is set to a value of \(\beta =2\), as recommended by Refs. [97, 98].$$\begin{aligned}&e_3^{(\beta )}\equiv \frac{ \mathrm{ECF}(3,\beta )}{\left[ \mathrm{ECF}(1,\beta )\right] ^3}, \quad e_2^{(\beta )}\equiv \frac{ \mathrm{ECF}(2,\beta )}{\left[ \mathrm{ECF}(1,\beta )\right] ^2},\nonumber \\&\quad D_2^{(\beta )} \equiv \frac{ e_3^{(\beta )}}{\left( e_2^{(\beta )} \right) ^3}. \end{aligned}$$(7)
3.2 Tagging of bjets

SmallR jets. If a smallR jet has at least one bquark among their constituents, it will be tagged as a bjet with probability \(f_b\). In order to be considered in the btagging algorithm, bquarks inside the smallR jet should satisfy \(p_T \ge 15\) GeV [99]. The probability of tagging a jet is not modified if more than one bquark is found among the jet constituents. If no bquarks are found among the constituents of this jet, it can be still be tagged as a bjet with a mistag rate of \(f_l\), unless a charm quark is present instead, and in this case the mistag rate is \(f_c\). Only jets that contain at least one (light or charm) constituent with \(p_T \ge 15\) GeV can induce a fake btag. We attempt to btag only the four (two) hardest smallR jets in the resolved (intermediate) category. Attempting to btag all of the smallR jets that satisfy the acceptance cuts worsens the overall performance as the rate of fake btags increases substantially.

LargeR jets. LargeR jets are btagged by ghostassociating anti\(k_T\) \(R=0.3\) (AKT03) subjets to the original largeR jets [7, 91, 102, 103]. A largeR jet is considered btagged if both the leading and the subleading AKT03 subjets, where the ordering is done in the subjet \(p_T\), are both individually btagged, with the same criteria as the smallR jets. Therefore, a largeR jet where the two leading subjets have at least one bquark will be tagged with probability \(f_b^2\). As in the case of smallR jets, we only attempt to btag the two leading subjets, else one finds a degradation of the signal significance. The treatment of the bjet misidentification from light and charm jets is the same as for the smallR jets.
3.3 Event categorisation
The present analysis follows a strategy similar to the scaleinvariant resonance tagging of Ref. [51]. Rather than restricting ourselves to a specific event topology, we aim to consistently combine the information from the three possible topologies: boosted, intermediate and resolved, with the optimal cuts for each category being determined separately. This approach is robust under variations of the underlying production model of Higgs pairs, for instance in the case of BSM dynamics, which can substantially increase the degree of boost in the final state.

Boosted category. An event which contains at least two largeR jets, with the two leading jets being btagged. Each of these two btagged, largeR jets are therefore candidates to contain the decay products of a Higgs boson.

Intermediate category. An event with exactly one btagged, largeR jet, which is assigned to be the leading Higgs candidate. In addition, we require at least two btagged, smallR jets, which must be separated with respect to the largeR jet by an angular distance of \(\Delta R\ge 1.2\). The subleading Higgs boson candidate is reconstructed by selecting the two btagged smallR jets that minimise the difference between the invariant mass of the largeR jet with that of the dijet obtained from the sum of the two smallR jets.

Resolved category. An event with at least four btagged smallR jets. The two Higgs candidates are reconstructed out of the leading four smallR jets in the event by considering all possible combinations of forming two pairs of jets and then choosing the configuration that minimises the relative difference of dijet masses.
These three categories are not exclusive: a given event can be assigned to more than one category, for example, satisfying the requirements of both the intermediate and the resolved categories at the same time. The exception is the boosted and intermediate categories, which have conflicting jet selection requirements.
This is achieved as follows. First of all we perform an inclusive analysis, and optimise the signal significance \(S/\sqrt{B}\) in each of the three categories separately, including the MVA. We find that the category with highest significance is the boosted one, followed by the intermediate and the resolved topologies, the latter two with similar significance. Therefore, when ascertaining in which category an event is to be exclusively placed: if the event satisfies the boosted requirements, it is assigned to this category, else we check if it suits the intermediate requirements. If the event also fails the intermediate category requirements, we then check if it passes the resolved selection criteria. The resulting exclusive event samples are then separately processed through the MVA, allowing for a consistent combination of the significance of the three event categories.
3.4 Motivation for basic kinematic cuts
We now motivate the kinematic cuts applied to the different categories, comparing representative kinematic distributions between signal and background events. First of all, we present results without PU, and then discuss the impact of PU on the description of the kinematic distributions. In the following, all distributions are normalised to their total integral.
In Fig. 2 we show the \(p_T\) distributions of the leading and subleading largeR jets in the boosted category. We observe that the background distribution falls off more rapidly as a function of \(p_T\) than the diHiggs signal. On the other hand, the cut in \(p_T\) cannot be too strong to avoid a substantial degradation of signal selection efficiency, specially taking into account the subleading largeR jet. This comparison justifies the cut of \(p_T \ge 200\) GeV for the largeR jets that we impose in the boosted category.
Turning to the resolved category, an important aspect to account for in the selection cuts is the fact that the \(p_T\) distribution of the four leading smallR jets of the event can be relatively soft, especially for the subleading jets. As noted in [29], this is due to the fact that the boost from the Higgs decay is moderate; therefore the \(p_T\) selection cuts for the smallR jets cannot be too large. In Fig. 4 we show the distribution in \(p_T\) of the four leading smallR jets in signal and background events: we observe that both distributions peak at \(p_T \le 50\) GeV, with the signal distribution falling off less steeply at large \(p_T\). The feasibility of triggering on four smallR jets with a relatively soft \(p_T\) distribution is one of the experimental challenges for exploiting the resolved category in this final state, and hence the requirement that \(p_T \ge 40\) GeV for the smallR jets. In Fig. 4 we also show the rapidity distribution of the smallR jets in the resolved category. As expected, the production is mostly central, and more so in the case of signal events, since backgrounds are dominated by QCD tchannel exchange; therefore the selection criteria on the jet rapidity are very efficient.
One of the most discriminating selection cuts is the requirement that the invariant mass of the Higgs candidate (di)jets must lie within a window around the nominal Higgs value, Eq. (8). In Fig. 5 we show the invariant mass of the leading reconstructed Higgs candidates, before the Higgs mass window selection is applied, for the resolved and boosted categories. While the signal distribution naturally peaks at the nominal Higgs mass, the background distributions show no particular structure. The width of the Higgs mass peak is driven both from QCD effects, such as initialstate radiation (ISR) and outofcone radiation, as well as from the fourmomentum smearing applied to finalstate particles as part of our minimal detector simulation.
In the resolved case, we see that the distribution in \(m_{hh}\) is rather harder for the signal as compared to the background, and therefore one expects that cutting in \(m_{hh}\) would help signal discrimination. For the boosted category the overall trend of the \(m_{hh}\) distribution is different because of the selection criteria, and the distribution now peaks at higher values of the invariant mass. In this case, signal and background distributions are not significantly differentiated. Note that at parton level the \(m_{hh}\) distribution for signal events has a kinematic cutoff at \(m_{hh}^\mathrm{min}=250\) GeV, which is smeared due to parton shower and detector resolution effects.
In Fig. 7 we show the transverse momentum of the diHiggs system, \(p_T^{hh}\), for the resolved and boosted categories. Once more we see that the background has a steeper falloff in \(p_T^{hh}\) than the signal, in both categories, therefore this variable should provide additional discrimination power, motivating its inclusion as one of the inputs for the MVA. In our LO simulation the \(p_T^{hh}\) distribution is generated by the parton shower, an improved theoretical description would require merging highermultiplicity matrix elements [35] or matching to the NLO calculation [17],
From Fig. 8 we observe how for these substructure variables the shapes of the signal and background distributions reflect the inherent differences in the internal structure of QCD jets and jets originating from Higgs decays. Signal and background distributions peak in rather different regions. For example, the \(k_t\) splitting scale \(\sqrt{d_{12}}\) peaks around 80 GeV (40 GeV) for signal (background) events, while the distribution of the ECF ratio \(C_2^{(\beta )}\) is concentrated at small values for signal and is much broader for background events. From Fig. 8 we also see the distributions of the subjettiness ratio \(\tau _{21}\) are reasonably similar for both the leading and the subleading jets.
3.5 Impact of pileup
Now we turn to discuss how the description of kinematic distributions for signal and background processes are modified in the presence of pileup. To study the impact of PU, Minimum Bias events have been generated with Pythia8, and then superimposed to the signal and background samples described in Sect. 2. We have explored two scenarios, one with a number of PU vertices per bunch crossing of \(n_\mathrm{PU}=80\), and another with \(n_\mathrm{PU}=150\). In the following we adopt \(n_\mathrm{PU}=80\) as our baseline, and denote this scenario by PU80. We have verified that the combined signal significance is similar if \(n_\mathrm{PU}=150\) is adopted instead.
From its definition in terms of the median, it follows that the value of \(p_T^{(\mathrm cut)}\) will be dynamically raised until half of the regions have \(p_{Ti}=0\). The size and number of these regions is a free parameter of the algorithm—here we will use square regions with length \(a=0.4\). We restrict ourselves to the central rapidity region, \(\eta  \le 2.5\), for the estimation of the \(p_T\) flow density \(\rho \). The SoftKiller subtraction is then applied to particles at the end of the parton shower, before jet clustering.
In Fig. 9 we show the invariant mass distributions of the Higgs candidates for signal events in the resolved and boosted categories. In the resolved category, we compare the results without PU with those with PU80, with and without SK subtraction. If PU is not subtracted, there is a large shift in the Higgs mass peak, by more than 30 GeV. Once SK subtraction is performed, we recover a distribution much closer to the no PU case, with only a small shift of a few GeV and a broadening of the mass distribution. In the boosted case, the comparison is performed between no PU, PU with only SK subtraction, and PU with both SK and trimming. We find that the mass distribution for jets to which no trimming is applied peaks at around 160 GeV, even after PU subtraction with SoftKiller. When trimming is applied in addition to SoftKiller, the distribution peaks close to the nominal Higgs mass, as in the case of the resolved category.
In Fig. 10 we compare the transverse momentum of the leading Higgs candidate, \(p_T^{h}\) and the invariant mass of the diHiggs system \(m_{hh}\), in both the boosted and the resolved categories, between the no PU and the PU+SK+Trim cases. In the case of the \(p_T^{h}\) distribution, the differences between the selection criteria for the resolved and boosted categories is reflected in the rightward shift of the latter. After subtraction, the effects of PU are small in the two categories. A similar behaviour is observed in the diHiggs invariant mass distribution.
We can also assess the impact of PU on the substructure variables that will be used as input to the MVA in the boosted and intermediate categories. In Fig. 11 we show the 2to1 subjettiness ratio \(\tau _{21}\), Eq. (5), and the ratio of energy correlation functions \(C_2^{(\beta )}\), Eq. (6), for the leading Higgs candidate. We observe that the shapes of both substructure variables are reasonably robust in an environment including significant PU. Therefore we can consider the PU subtraction strategy as validated for the purposes of this study, although further optimisation should still be possible, both in terms of the SoftKiller and of the trimming input settings.
It is illustrative to determine the mass resolution obtained for the reconstructed Higgs candidates in the various cases considered in the present study. In Table 2 we indicate the shift of the fitted invariant mass peak as compared to the nominal Higgs mass, \(\langle m_h^\mathrm{reco}\rangle m_h\), and the corresponding width of the distribution, \(\sigma _{m_h}\), obtained from fitting a Gaussian to the mass distributions of leading and subleading Higgs candidates in the resolved and boosted categories. We show results for three cases: without PU, with PU80 but without subtraction (only for the resolved category), and the same with SK+Trim subtraction.
In both categories, we find a mass resolution of around 9 GeV in the case without PU. In the case of PU with SK+Trim subtraction, in the resolved category the mass resolution worsens only slightly to around 11 GeV, while in the boosted category we find the same resolution as in the no PU case. We also note that after SK+Trim subtraction, the peak of the invariant mass distributions of Higgs candidates coincides with the nominal values of \(m_h\) within a few GeV for the two categories.
4 PreMVA loose cutbased analysis
In this section we present the results of the preMVA loose cutbased analysis described in the previous section, and provide cut flows for the different analysis steps. We study how the signal significance is affected if only the 4b component of the QCD multijet background is taken into account. This section presents the results in an environment without pileup; the following one contains those obtained including significant PU.
Resolution of the invariant mass distribution of reconstructed Higgs candidates in the resolved and boosted categories. We show three cases: no PU, with PU80 without subtraction (only for resolved), and the same with SK+Trim subtraction. We indicate the shift of the fitted invariant mass peak \(\left\langle m_h^\mathrm{reco}\right\rangle \) for the Higgs candidates as compared to the nominal Higgs mass \(m_h\), as well as the fitted Gaussian width \(\sigma _{m_h}\)
\(\left\langle m_h^\mathrm{reco}\right\rangle m_h\) (GeV)  \(\sigma _{m_h}\) (GeV)  

Resolved category  
No PU  
Leading h  −3.8  \(\left( 8.5\pm 0.2\right) \) 
Subleading h  −5.8  \(\left( 9.1\pm 0.3\right) \) 
PU80  
Leading h  \(+\)33  \(\left( 8.8\pm 1.5\right) \) 
Subleading h  \(+\)31  \(\left( 11.7\pm 3.3\right) \) 
PU80+SK  
Leading h  \(+\)3.9  \(\left( 10.7\pm 0.3\right) \) 
Subleading h  \(+\)2.1  \(\left( 10.5\pm 0.3\right) \) 
Boosted category  
No PU  
Leading h  \(+\)2.0  \(\left( 8.2\pm 0.5\right) \) 
Subleading h  \(+\)1.0  \(\left( 8.8\pm 0.5\right) \) 
PU80+SK+Trim  
Leading h  −2.2  \(\left( 8.7\pm 0.7\right) \) 
Subleading h  −4.9  \(\left( 9.0\pm 0.8 \right) \) 
Definition of the cuts imposed successively for the three selections
Boosted  Intermediate  Resolved  

C1a  \(N_\mathrm{jets}^{R10}\ge 2\)  \(N_\mathrm{jets}^{R04}\ge 2\), \(N_\mathrm{jets}^{R10}=1\)  \(N_\mathrm{jets}^{R04}\ge 4\) 
+\(p_T\) cuts and rapidity cuts  
C1b  +\(N_\mathrm{MDT}\ge 2\)  +\(N_\mathrm{jets}^{R10}=1\) with MDT  +Higgs reconstruction 
+Higgs reconstruction  
C1c  +\(m_h\) window cut  
C2  +btagging 
4.1 Cut flow and signal significance

C1a: check that we have at least two largeR jets (in the boosted case), one largeR jet and at least 2 smallR jets (in the intermediate case) and at least four smallR jets (in the resolved case). In addition, require that these jets satisfy the corresponding \(p_T\) thresholds; \(p_T \ge 200\) GeV for largeR jets and \(p_T \ge 40\) GeV for smallR jets, as well as the associated rapidity acceptance constraints.

C1b: the two leading largeR jets must be massdrop tagged in the boosted category. In the intermediate category, the largeR jet must also be massdrop tagged.

C1c: after the two Higgs candidates have been reconstructed, their invariant masses are required to lie within a window around \(m_H\), in particular between 85 and 165 GeV, Eq. (8).

C2: the btagging conditions are imposed (see Sect. 3.2), and the event is categorised exclusively into one of the three topologies, according to the hierarchy determined in Sect. 3.3.
The cross sections for the signal and the background processes at different steps of the analysis (see Table 3), for the resolved (upper), intermediate (middle) and boosted (lower table) categories, for the analysis without PU. For each step, the signal over background ratio S / B, and the signal significance \(S/\sqrt{B}\) for \({\mathcal {L}}=3\) ab\(^{1}\) are also provided, considering either the total background, or only the 4b component
hh4b  Total bkg  Cross section [fb]  S / B  \(S/\sqrt{B}\)  

4b  2b2j  4j  \(t\bar{t}\)  Tot  4b  Tot  4b  
HLLHC, resolved category, no PU  
C1a  9  \(2.2\times 10^8\)  \(6.9\times 10^4\)  \(1.5\times 10^7\)  \(2.0\times 10^8\)  \(2.1\times 10^5\)  \(4.0\times 10^{8}\)  \(1.3\times 10^{4}\)  0.03  1.9 
C1b  9  \(2.2\times 10^8\)  \(6.9\times 10^4\)  \(1.5\times 10^7\)  \(2.0\times 10^8\)  \(2.1\times 10^5\)  \(4.0\times 10^{8}\)  \(1.3\times 10^{4}\)  0.03  1.9 
C1c  2.6  \(4.4\times 10^7\)  \(1.6\times 10^4\)  \(3.2\times 10^6\)  \(4.1\times 10^7\)  \(8.8\times 10^4\)  \(6.1\times 10^{8}\)  \(1.6\times 10^{4}\)  0.02  1.1 
C2  0.5  \(4.9\times 10^3\)  \(1.7\times 10^3\)  \(2.9\times 10^3\)  \(2.1\times 10^2\)  47  \( 1.1\times 10^{4}\)  \(2.9\times 10^{4}\)  0.4  0.6 
HLLHC, intermediate category, no PU  
C1a  2.8  \(8.4\times 10^7\)  \(2.1\times 10^4\)  \(5.3\times 10^6\)  \(7.9\times 10^7\)  \(3.3\times 10^4\)  \(3.4\times 10^{8}\)  \(1.3\times 10^{4}\)  0.02  1.1 
C1b  2.6  \(5.8\times 10^7\)  \(1.4\times 10^4\)  \(3.6\times 10^6\)  \(5.5\times 10^7\)  \(3.0\times 10^4\)  \(4.5\times 10^{8}\)  \(1.9\times 10^{4}\)  0.02  1.2 
C1c  0.5  \(3.5\times 10^6\)  \(8.7\times 10^2\)  \(2.1\times 10^5\)  \(4.3\times 10^7\)  \(8.8\times 10^3\)  \(1.6\times 10^{7}\)  \(6.1\times 10^{4}\)  0.02  1.0 
C2  0.09  \(1.8\times 10^2\)  56  96  22  3.1  \(5.3\times 10^{4}\)  \(1.6\times 10^{3}\)  0.4  0.6 
HLLHC, boosted category, no PU  
C1a  3.9  \(4.6\times 10^7\)  \(1.1\times 10^4\)  \(2.9\times 10^6\)  \(4.3\times 10^7\)  \(2.4\times 10^4\)  \(8.2\times 10^{8}\)  \(3.4\times 10^{4}\)  0.03  2.0 
C1b  2.7  \(3.7\times 10^7\)  \(7.5\times 10^3\)  \(2.1\times 10^6\)  \(3.5\times 10^7\)  \(2.2\times 10^4\)  \(7.4\times 10^{8}\)  \(3.7\times 10^{4}\)  0.03  1.7 
C1c  1.0  \(3.9\times 10^6\)  \(8.0\times 10^2\)  \(2.3\times 10^5\)  \(3.7\times 10^6\)  \(7.1\times 10^3\)  \(2.6\times 10^{7}\)  \(1.3\times 10^{3}\)  0.03  2.0 
C2  0.16  \(2.5\times 10^2\)  53  \(1.9\times 10^2\)  13  1.6  \(5.7\times 10^{4}\)  \(2.7\times 10^{3}\)  0.5  1.1 
In the boosted category, at the end of the loose cutbased analysis, we find that around 500 events are expected at the HLLHC, with a large number, \({\simeq } 10^6\), of background events. This leads to a preMVA signal significance of \(S/\sqrt{B}=0.5\) and a signal over background ratio of \(S/B=0.06~\%\). From Table 4 it is also possible to compute the corresponding preMVA expectations for the LHC Run II with \({\mathcal {L}}=300\) fb\(^{1}\): one expects in the boosted category around 50 signal events, with signal significance dropping down to \(S/\sqrt{B}\simeq 0.16\). Such signal significances could have been enhanced by applying tighter selection requirements, but our analysis cuts have been left deliberately loose so that such optimisation may be performed by the MVA.
The resolved category benefits from higher signal yields, but this enhancement is compensated for by the corresponding increase in the QCD multijet background. In both resolved and intermediate categories the signal significance is \(S/\sqrt{B}\simeq 0.4\), similar to that of the boosted category. A further drawback of the resolved case is that S / B is substantially reduced as compared to the boosted and intermediate cases.
Combining the results from the boosted, intermediate and resolved categories, we obtain an overall preMVA significance for the observation of the Higgs pair production in the \(b\bar{b}b\bar{b}\) final state at the HLLHC of \((S/\sqrt{B})_\mathrm{tot} \simeq 0.8\).
4.2 The role of light and charm jet misidentification
One of the main differences in the present study as compared to previous work is the inclusion of both irreducible and reducible background components, which allows us to quantify the impact of light and charm jet misidentification. Two recent studies that have also studied the feasibility of SM Higgs pair production in the \(b\bar{b}b\bar{b}\) final state are from the UCL group [28] and from the Durham group [29]. The UCL study is based on requiring at least four btagged \(R=0.4\) anti\(k_T\) jets in central acceptance with \(p_T \ge 40\) GeV, which are then used to construct dijets (Higgs candidates) with \(p_T \ge 150\) GeV, \(85 \le m_\mathrm{dijet} \le 140\) GeV and \(\Delta R \le 1.5\) between the two components of the dijet. In addition to the basic selection cuts, the constraints from additional kinematic variables are included by means of a Boosted Decision Tree (BDT) discriminant. The backgrounds included are the 4b and 2b2c QCD multijets, as well as \(t\bar{t}\), Zh, \(t\bar{t}h\) and \(hb\bar{b}\). For the HLLHC, a signal significance of \(S/\sqrt{B}\simeq 2.1\) is obtained.
From our results in Table 4, we observe that the signal significance for the boosted, intermediate, and resolved categories is increased to 1.1, 0.6 and 0.6, respectively, when only the QCD 4b background is included. Combining the signal significance in the three categories, we obtain \((S/\sqrt{B_\mathrm{4b}})_\mathrm{tot}\simeq 1.4\), twice as large as the result found when all background components are included. Note the importance of the combination of the three exclusive event topologies, as opposed the exploitation of a single specific category. Taking into account the loose selection cuts, we see that our preMVA results including only the 4b background are consistent with those reported in previous studies.
The relative fractions \(n^{(\text {bjet})}_j\) of events for the resolved selection for which out of the four leading smallR jets of the event, j jets contain at least one bquark with \(p_T^b\ge 15\) GeV. This information is provided for the diHiggs signal events and for the three QCD background samples. The last column indicates the overall selection efficiency as defined in Eq. (10)
\(n^{(\text {bjet})}_0\) (%)  \(n^{(\text {bjet})}_1\) (%)  \(n^{(\text {bjet})}_2\) (%)  \(n^{(\text { bjet})}_3\) (%)  \(n^{(\text {bjet})}_4\) (%)  \(\mathrm{EFF}_{\text {btag}}\) (%)  

\(hh\rightarrow 4b\)  0.1  3  25  53  20  8.5 
QCD 4b  1  8  27  44  20  8.4 
QCD 2b2j  9  42  49  1  0.1  0.04 
QCD 4j  96  3.5  0.5  0.01  \(3\times 10^{4}\)  \(2\times 10^{4}\) 
In Fig. 14 we show a comparison of the shapes of the 4b and 2b2j components of the QCD background for the transverse momentum \(p_T^h\) of the leading Higgs candidate and for invariant mass \(m_{hh}\) of the reconstructed diHiggs system in the resolved and boosted categories. The two components possess a rather similar shape for the two distributions, albeit with some differences. In the boosted category, the 4b component exhibits a less steep falloff of the \(p_T^h\) distribution at large \(p_T\), while in the resolved case the 2b2j component has a slightly harder distribution of the invariant mass \(m_{hh}\). We also observe that the 2b2j distributions are affected by somewhat larger Monte Carlo fluctuations as compared to 4b, despite the large size of the initial sample.
In the resolved category, the cross section before btagging is two orders of magnitude larger in the 2b2j sample as compared to the 4b sample. After btagging, a naive assessment would suggest a suppression of the 2b2j cross section by a factor \((f_l/f_b)^2 \simeq 1.5\times 10^{4}\), as compared to the 4b component, since a total of four btags are required to classify the event as a Higgs candidate. In this case the ratio of 2b2j over 4b would be around \({\simeq } 3~\%\), and therefore negligible. While we have checked that this expectation is borne out at the parton level, we find that when parton shower effects are accounted for the situation is different, due both to radiation of \(b\bar{b}\) pairs and from selection effects. Due to these, the number of b quarks in the final state is increased substantially in the 2b2j component as compared to the parton level, while at the same time the number of events in the 4b sample with 4 bjets passing selection cuts is reduced.
5 Multivariate analysis
At the end of the loose cutbased analysis, by combining the three event topologies, we obtain a signal significance of \(S/\sqrt{B}\simeq 0.8~(1.4)\) with all backgrounds (only QCD 4b) considered. This section describes how this signal significance can be enhanced when the cutbased analysis is complemented by multivariate techniques. These are by now a mature tool in highenergy physics data analysis, opening new avenues to improve the performance of many measurements and searches at highenergy colliders. In particular, the classification of events into signal and background processes by means of MVAs is commonly used in LHC applications [28, 46, 80, 118, 119, 120].
In this section, first we present the specific MVA that we use, based on feedforward multilayer neural networks. Then we introduce the input variables that are used in the MVA, including the jet substructure variables, and then present the signal significance obtained by applying the MVA. Then we assess the robustness of the MVA strategy in the case of significant contamination from pileup.
5.1 Deep artificial neural networks
The specific type of MVA that we use to disentangle signal and background events is a multilayer feedforward artificial neural network (ANN), known as a perceptron.^{2} This family of ANNs are also known as deep neural networks, due to their multilayered architecture. The MVA inputs are a set of kinematic variables describing the signal and background events which satisfy the requirements of the cutbased analysis. The output of the trained ANNs also allows for the identification, in a fully automated way, of the most relevant variables in the discrimination between signal and background.
The training of the neural networks therefore consists of the minimisation of the crossentropy error, Eq. (16), which in this work is achieved using a Genetic Algorithm (GA). GAs [125, 126, 127, 128] are nondeterministic minimisation strategies suitable for the solution of complex optimisation problems, for instance when a very large number of quasiequivalent minima are present. GAs are inspired on natural selection processes that emulate biological evolution. In our case, the GA training is performed for a very large number of generations, \(N_\mathrm{gen}=5\times 10^{4}\), to avoid the risk of undertraining. We have verified that if a much larger number of generations are used, the results are unchanged.
5.2 Input kinematic variables

The transverse momenta of the leading and subleading Higgs, \(p_{T,h_1}\) and \(p_{T,h_2}\).

The transverse momentum of the reconstructed Higgs pair, \(p_{T,hh}\).

The invariant masses of the leading and subleading Higgs candidates, \(m_{h,1}\) and \(m_{h,2}\).

The invariant mass of the reconstructed Higgs pair, \(m_{hh}\).

The separation in the \(\phi \)–\(\eta \) plane between the two Higgs candidates, \(\Delta R_{hh}\).

The separation in \(\eta \) between the two Higgs candidates, \(\Delta \eta _{hh}\).

The separation in \(\phi \) between the two Higgs candidates, \(\Delta \phi _{hh}\).
In the boosted and intermediate categories, we also include the jet substructure variables introduced in Sect. 3 for the largeR jets: the \(k_T\) splitting scales \(\sqrt{d_{12}}\), the ratio of 2to1 subjettiness \(\tau _{12}\), and the ratios of energy correlation functions \(C^{(\beta )}_2\) and \(D_2^{(\beta )}\). This leads to a total of \(N_{\mathrm {var}}=13,17\) and 21 variables for the resolved, intermediate and boosted categories, respectively.
Given that the MVA is able to identify the most discriminatory variables in an automated way, and to suppress those which have little effect, it is advantageous to include a wide array of input variables. This is one of the main advantages of ANNs in this context: their inherent redundancy means that adding additional information, even if carries very little weight, should not degrade the classification power of the MVA.
5.3 MVA results
We now present the results of the MVA, first without PU, and then later including the effects of PU. First of all, in Fig. 16 we show the distribution of the ANN output at the end of the GA minimisation, separately for the boosted, intermediate and resolved categories. All distributions are normalised so that their integral adds up to one. The separation between signal and background is achieved by introducing a cut, \(y_\mathrm{cut}\), on the ANN output, so that MC events with \(y_i\ge y_\mathrm{cut}\) are classified as signal events, and those with \(y_i < y_\mathrm{cut}\) as background events. Therefore, the more differentiated the distribution of the ANN output is for signal and background events, the more efficient the MVA discrimination will be.
From Fig. 16 we see that in the boosted category the MVA can produce a clear discrimination between signal and background, with the two distributions forming peaks at their respective optimal limits. This indicates that introducing a suitable cut \(y_\mathrm{cut}\) in the ANN output will substantially reduce the background, while keeping a reasonable signal efficiency. The performance of the MVA discrimination is similar, although slightly worse, in the intermediate and resolved categories.
The results for the signal selection efficiency and the background rejection rate as a function of the cut in the ANN output \(y_\mathrm{cut}\) define the socalled ReceiverOperating Characteristic (ROC) curve, shown in Fig. 17. It is clear that we can achieve high signal efficiency by using a small value of \(y_\mathrm{cut}\), but such a choice would be affected by poor background rejection. Conversely, using a higher value of the cut will increase background rejection at the cost of dropping signal efficiency. As could already be inferred from the distribution of neuralnetworks output in Fig. 16, we find that our MVA is reasonably efficient in discriminating signal over background. The performance is best in the case of the boosted category, and then slightly worse in the resolved and intermediate categories, consistent with the distributions of the ANN outputs in Fig. 16.
It is useful to estimate, for each value of the cut in the ANN output \(y_\mathrm{cut}\), how many signal and background events are expected at the HLLHC with \({\mathcal {L}}=3\) ab\(^{1}\). This comparison is shown in Fig. 17. We observe that in the boosted category, for a value \(y_\mathrm{cut}\simeq 0.9\) we end up with around 300 signal events and \(10^4\) background events. Similar results are obtained in the intermediate and resolved categories: in the former we find 130 (\(3\times 10^3\)) signal (background) events for \(y_\mathrm{cut}\simeq 0.85\) (0.60), and in the latter 630 (\(10^5\)) signal (background) events for \(y_\mathrm{cut}\simeq 0.6\). Therefore, the MVA achieves a substantial background suppression with only a moderate reduction of signal efficiency.
In Fig. 18 we show the distribution of the total associated weight, Eq. (17) for each of the \(N_\mathrm{var}\) input variables of the three categories, using the notation for the kinematic variables as in Sect. 5.2. In the resolved category, the variables that carry a higher discrimination power are the transverse momentum of the two reconstructed Higgs candidates and their invariant masses. In the case of the boosted category, the invariant mass distribution of the Higgs candidates is also the most discriminatory variable, followed by the subjet \(p_T\) distributions and substructure variables such as \(C_2^{(\beta )}\) and \(D_2^{(\beta )}\).
PostMVA results, for the optimal value of the ANN discriminant \(y_\mathrm{cut}\) in the three categories, compared with the corresponding preMVA results (\(y_\mathrm{cut}=0\)). We quote the number of signal and background events expected for \({\mathcal {L}}=3\) ab\(^{1}\), the signal significance \(S/\sqrt{B}\) and the signal over background ratio S / B. The preMVA results correspond to row C2 in Table 4
HLLHC, no PU  

Category  \(N_\mathrm{ev}\) signal  \(N_\mathrm{ev}\) back  \(S/\sqrt{B}\)  S / B 
Boosted  
\(y_\mathrm{cut}=0\)  440  \(7.6\times 10^5\)  0.5  \(6\times 10^{4}\) 
\(y_\mathrm{cut}=0.90\)  290  \(1.2\times 10^4\)  2.7  0.03 
Intermediate  
\(y_\mathrm{cut}=0\)  280  \(5.3\times 10^5\)  0.4  \(5\times 10^{4}\) 
\(y_\mathrm{cut}=0.85\)  130  \(3.1\times 10^3\)  2.3  0.04 
Resolved  
\(y_\mathrm{cut}=0\)  1500  \(1.5\times 10^{7}\)  0.4  \(1\times 10^{4}\) 
\(y_\mathrm{cut}=0.60\)  630  \(1.1\times 10^{5}\)  1.9  0.01 
It should be emphasised that MVAs such as the ANNs used in this work can always be understood as a combined set of correlated cuts. Once the ANNs have been trained, it is possible to compare kinematical distributions after and before the ANN cut to verify its impact. This information would allow one in principle to perform a cutbased analysis, without the need of using ANNs, and finding similar results.
A particularly challenging aspect of our analysis is the modeling of the 2b2j and 4j background, especially for the latter, which require extremely large MC samples. In the analysis reported here, out of the original 3M 4j generated events, only around 100 survive the analysis cuts, and thus these low statistics have associated a potentially large uncertainty in the calculation of the postMVA 4j cross section. On the other hand, since the 4j cross sections are always quite smaller than the sum of the 4b of the 2b2j components, these low statistics should not modify qualitatively our conclusions above. To verify explicitly this expectation, and obtain a more robust estimate of the background cross section from misidentified jets, we have increased by a factor 10 the size of the 2b2j and 4j background samples, up to a total of 30M each. Processing these events though our analysis, including retraining the MVA, we find \((S/\sqrt{B})_\mathrm{tot}=3.9\), consistent with Eq. (18), indicating that the low statistics of the 4j background is not a limiting factor.
5.4 Impact of PU in the MVA
Same as Table 4, now for the case of PU80+SK+Trim
hh4b  Total bkg  Cross section [fb]  S / B  \(S/\sqrt{B}\)  

4b  2b2j  4j  \(t\bar{t}\)  Tot  4b  Tot  4b  
HLLHC, resolved category, PU+SK with \(n_\mathrm{PU}=80\)  
C1a  11  \(4.4 \times 10^8\)  \(1.5\times 10^5\)  \(3.0\times 10^7\)  \(4.1\times 10^8\)  \(2.6 \cdot 10^5\)  \( 2.4 \times 10^{8}\)  \(7.2 \times 10^{5}\)  0.03  1.5 
C1b  11  \(4.4 \times 10^8 \)  \(1.5\times 10^5\)  \(3.0\times 10^7\)  \(4.1\times 10^8\)  \(2.6 \cdot 10^5\)  \(2.4 \times 10^{8}\)  \(7.2 \times 10^{5}\)  0.03  1.5 
C1c  3  \(1.1 \times 10^8\)  \(4.2\times 10^4\)  \(7.7\times 10^6\)  \(9.9\times 10^7\)  \(1.1 \cdot 10^5\)  \(2.8 \times 10^{8}\)  \(7.4 \times 10^{5}\)  0.02  0.8 
C2  0.6  \(9.0 \times 10^3\)  \(3.5\times 10^3\)  \(5.1\times 10^3\)  \(3.1\times 10^2\)  50  \(6.5 \times 10^{5}\)  \(1.7 \times 10^{4}\)  0.4  0.5 
HLLHC, intermediate category, PU+SK+Trim with \(n_\mathrm{PU}=80\)  
C1b  2.7  \(8.1\times 10^7\)  \(2.1\times 10^4\)  \(5.2\times 10^6\)  \(7.6\times 10^7\)  \(3.0\times 10^4\)  \(3.4 \times 10^{8}\)  \(1.3 \times 10^{4}\)  0.02  1.0 
C1c  2.6  \(6.2\times 10^7 \)  \(1.5\times 10^4\)  \(3.9\times 10^6\)  \(5.8\times 10^7\)  \(2.8\times 10^4\)  \(4.1 \times 10^{8}\)  \(1.7 \times 10^{4}\)  0.02  1.1 
C1d  0.5  \(2.8\times 10^6\)  \(7.9\times 10^2\)  \(1.9\times 10^5\)  \(2.7\times 10^6\)  \(6.5\times 10^3\)  \(1.8 \times 10^{7}\)  \(6.2 \times 10^{4}\)  0.02  1.0 
C2  0.09  \(2.6\times 10^2\)  47  \(1.8\times 10^2\)  30  2.2  \(3.4 \times 10^{4}\)  \(1.8 \times 10^{3}\)  0.3  0.7 
HLLHC, boosted category, PU+SK+Trim with \(n_\mathrm{PU}=80\)  
C1a  3.5  \(4.1\times 10^7\)  \(1.0\times 10^4\)  \(2.7\times 10^6\)  \(3.8\times 10^7\)  \(2.0\times 10^4 \)  \(8.6 \times 10^{8}\)  \(3.4 \times 10^{4}\)  0.03  1.9 
C1b  2.5  \(3.2\times 10^7\)  \(6.8\times 10^3\)  \(1.9\times 10^6\)  \(3.0\times 10^7\)  \(1.9\times 10^4 \)  \(7.8 \times 10^{8}\)  \(3.6 \times 10^{4}\)  0.02  1.6 
C1c  0.8  \(2.2\times 10^6\)  \(5.4\times 10^2\)  \(1.4\times 10^5\)  \(2.0\times 10^6\)  \(4.8\times 10^3 \)  \(3.8 \times 10^{7}\)  \(1.6 \times 10^{3}\)  0.03  2.0 
C2  0.14  \(1.5\times 10^2\)  40  86  22  1.8  \( 9.0 \times 10^{4}\)  \(3.5 \times 10^{3}\)  0.6  1.2 
Same as Table 6, now for the case of PU80+SK+Trim
HLLHC, PU80+SK+Trim  

Category  \(N_\mathrm{ev}\) signal  \(N_\mathrm{ev}\) back  \(S/\sqrt{B}\)  S / B 
Boosted  
\(y_\mathrm{cut}=0\)  410  \(4.5\times 10^5\)  0.6  \( 10^{3}\) 
\(y_\mathrm{cut}=0.8\)  290  \(3.7\times 10^4\)  1.5  0.01 
Intermediate  
\(y_\mathrm{cut}=0\)  260  \(7.7\times 10^5\)  0.3  \(3\times 10^{4}\) 
\(y_\mathrm{cut}=0.75\)  140  \(5.6\times 10^3\)  1.9  0.03 
Resolved  
\(y_\mathrm{cut}=0\)  1800  \(2.7\times 10^7\)  0.4  \(7\times 10^{5}\) 
\(y_\mathrm{cut}=0.60\)  640  \(1.0\times 10^5\)  2.0  0.01 
In Table 8 we compare the results for the PU80+SK+Trim case between the preMVA loose cutbased analysis and the postMVA results for the optimal values of the ANN output cut \(y_\mathrm{cut}\). As in Table 6, we also quote the number of signal and total background events expected for \({\mathcal {L}}=3\) ab\(^{1}\) and the values of \(S/\sqrt{B}\) and S / B. We observe that the preMVA signal significance is close to the results of the simulations without PU for the three categories. We now find values for \(S/\sqrt{B}\) of 0.4, 0.3 and 0.6, in the resolved, intermediate and boosted categories, respectively, to be compared with the corresponding values without PU, namely 0.4, 0.4 and 0.5. The number of selected signal events in each category at the end of the cutbased analysis is only mildly affected by PU. The slight preMVA improvement in \(S/\sqrt{B}\) for the boosted case arises from a reduction in the number of background events that are classified in this category as compared to the case without PU.
In Fig. 21 we show the number of signal and background events that are expected for \({\mathcal {L}}=3\) ab\(^{1}\) as a function of \(y_\mathrm{cut}\), together with the corresponding ROC curve. The slight degradation of the boosted category in the case of PU can be seen by comparing with the corresponding results without PU in Fig. 17. In Fig. 22 we show the signal significance, \(S/\sqrt{B}\), and the signal over background ratio, S / B, accounting now for the effects of PU. The corresponding results in the case without PU were shown in Fig. 19. As can be seen, the MVAdriven enhancement remains robust in the presence of PU, with \(S/\sqrt{B}\) only moderately degraded. Therefore, the qualitative conclusions drawn in the case without PU also hold when the analysis is performed in a highPU environment. Since no specific effort has been made to optimise PU subtraction, for instance by tuning the values of the patch length a in SoftKiller or the \(p_T\) threshold during jet trimming, we believe that there should be still room for further improvement.
It is useful to quantify which of the MVA input variables carry the highest discrimination power in the case of PU, by means of Eq. (17), and compare this with the corresponding results without PU shown in Fig. 18. We have verified that the relative weight of the different input variables to the MVA is mostly unchanged in the case of PU. In the resolved category, the highest total associated weight is carried by the Higgs candidates \(p_T\) and invariant mass, as well as by the \(p_T\) of the individual smallR jets. For the boosted category, the highest weight is carried by the Higgs invariant mass, followed by the Higgs \(p_T\), \(m_{hh}\), the \(p_T\) of the AKT03 subjets and the substructure variables, with a similar weighting among them.
PostMVA number of signal and background events with \({\mathcal {L}}=3\) ab\(^{1}\). For the backgrounds, both the total number, \(N_\mathrm{ev}^\mathrm{tot}\), and the 4b component only, \(N_\mathrm{ev}^\mathrm{4b}\), are shown. Also provided are the values of the signal significance and the signal over background ratio, both separated in categories and for their combination. We quote the results without PU and for PU80+SK+Trim
Category  Signal  Background  \(S/\sqrt{B_\mathrm{tot}}\)  \(S/\sqrt{B_\mathrm{4b}}\)  \(S/B_\mathrm{tot}\)  \(S/B_\mathrm{4b}\)  

\(N_\mathrm{ev}\)  \(N_\mathrm{ev}^\mathrm{tot}\)  \(N_\mathrm{ev}^\mathrm{4b}\)  
Boosted  
No PU  290  \(1.2\times 10^4\)  \(8.0\times 10^3\)  2.7  3.2  0.03  0.04 
PU80+SK+Trim  290  \(3.7\times 10^4\)  \(1.2\times 10^4\)  1.5  2.7  0.01  0.02 
Intermediate  
No PU  130  \(3.1\times 10^3\)  \(1.5\times 10^3\)  2.3  3.3  0.04  0.08 
PU80+SK+Trim  140  \(5.6\times 10^3\)  \(2.4\times 10^3\)  1.9  2.9  0.03  0.06 
Resolved  
No PU  630  \(1.1\times 10^5\)  \(5.8\times 10^4\)  1.9  2.7  0.01  0.01 
PU80+SK  640  \(1.0\times 10^5\)  \(7.0\times 10^4\)  2.0  2.6  0.01  0.01 
Combined  
No PU  4.0  5.3  
PU80+SK+Trim  3.1  4.7 
6 Conclusions and outlook
In this work we have presented a feasibility study for the measurement of Higgs pair production in the \(b\bar{b}b\bar{b}\) final state at the LHC. Our strategy is based on the combination of traditional cutbased analysis with stateoftheart multivariate techniques. We take into account all relevant backgrounds, in particular the irreducible 4b and the reducible 2b2j and 4j QCD multijets. We have illustrated how the 2b2j component leads to a contribution comparable to that of QCD 4b production, due to a combination of parton shower effects, bquark pair radiation, and selection requirements. We have also demonstrated the robustness of our analysis strategy under the addition of significant PU. In particular, we have explored two scenarios, \(n_\mathrm{PU}=80\) and \(n_\mathrm{PU}=150\), and we found a comparable overall signal significance in the two cases.
Combining the contributions from the resolved, intermediate and boosted categories, we find that, for \({\mathcal {L}}=3\) ab\(^{1}\), the signal significance for the production of Higgs pairs turns out to be \(S/\sqrt{B}\simeq 3\). This indicates that, already from the \(b\bar{b}b\bar{b}\) final state alone, it should be possible to claim observation of Higgs pair production at the HLLHC. Our study also suggests possible avenues that the LHC experiments could explore to further improve this signal significance. One handle would be to reduce the contribution from light and charm jet misidentification, ensuring that the irreducible 4b background dominates over the 2b2j component. This would allow one to enhance \(S/\sqrt{B}\) almost to the discovery level; see Table 9. It would also be advantageous to improve the btagging efficiency, allowing to achieve higher signal yields. Another possibility would be to improve the mass resolution of the Higgs reconstruction in highPU environments, and, more generally, to optimise the PU subtraction strategy in order to reduce the impact of PU in the modeling of kinematic variables and the associated degradation in the MVA discrimination.
Another challenging aspect of the measurement of Higgs pairs in the \(b\bar{b}b\bar{b}\) final state is achieving an efficient triggering strategy. In order to reduce the rate from background QCD processes sufficiently, while being able to access the relevant \(p_T\) regimes, (multi)jet triggers using bquark tagging information online for one or more jets are likely to be necessary. The additional rejection provided by these triggers could enable events to be selected efficiently, with four jets down to \(p_T=40\) GeV in the resolved category, and boosted Higgs decays in largeR jets down to jet transverse momenta of \(p_T=200\) GeV. In addition, good control of the multijet backgrounds and the experimental systematics of the MVA inputs will be important to achieve these sensitivities.
Our strategy relies on the modeling of the kinematic distributions of signal and background events, since these provide the inputs to the MVA discriminant. In this respect, it would be important, having established the key relevance of the \(b\bar{b}b\bar{b}\) channel for the study of Higgs pair production, to revisit and improve the theoretical modeling of our signal and background simulation, in particular using NLO calculations matched to parton showers both for signal [17, 35] and for backgrounds [63, 76].
One important implication of this work is that it should be possible to significantly improve the accuracy on the extraction of the Higgs trilinear coupling \(\lambda \) from a measurement of the \(\sigma \left( hh\rightarrow b\bar{b}b\bar{b}\right) \) cross section, as compared to existing estimates. A determination of \(\lambda \) in our approach is however rather nontrivial, involving not only generating signal samples for a wide range of values of \(\lambda \), but also repeating the analysis optimisation, including the MVA training, for each of these values. This study is left to a future publication, where we will also compare the precision from the \(b\bar{b}b\bar{b}\) final state with the corresponding precision that has been reported from other final states such as \(b\bar{b}\gamma \gamma \) and \(b\bar{b}\tau \tau \). It will also be interesting to perform this exercise for a 100 TeV hadron collider [11, 12, 13, 14]. While at 100 TeV the signal yields would be increased, also the (gluondriven) QCD multijet background would grow strongly. Revisiting the present analysis, including the MVA optimisation, at 100 TeV would also allow us to assess the accuracy of an extraction of the trilinear coupling \(\lambda \) from the \(b\bar{b}b\bar{b}\) final state at 100 TeV.
In this work we have considered only the SM production mechanism, but many BSM scenarios predict deviations in Higgs pair production, both at the level of total rates and of differential distributions. In the absence of new explicit degrees of freedom, deviations from the SM can be parametrised in the EFT framework using higherorder operators [14, 48]. Therefore, we plan to study the constraints on the coefficients of these effective operators that can be obtained from measurements of various kinematic distributions in the \(hh\rightarrow b\bar{b}b\bar{b}\) process. Note that the higher rates of the \(b\bar{b}b\bar{b}\) final state as compared to other final states, such as \(b\bar{b}\gamma \gamma \), allow for better constraints upon operators that modify the highenergy behaviour of the theory, for instance, it would become possible to access the tail of the \(m_{hh}\) distribution.
As in the case of the extraction of the Higgs trilinear coupling \(\lambda \), such a study would be a computationally intensive task, since BSM dynamics will modify the shapes of the kinematic distributions and thus in principle each point in the EFT parameter space would require a reoptimisation with a newly trained MVA. In order to explore efficiently the BSM parameters without having to repeat the full analysis for each point, modern statistical techniques such as the Cluster Analysis method proposed in Ref. [46] might be helpful.
Footnotes
 1.
These techniques have also important applications in the subtraction of the UE/MPI contamination for jet reconstruction in heavy ion collisions [115].
 2.
 3.
The impact of PU on the separate significance of the three categories exhibits some dependence on the specific choice for \(n_\mathrm{PU}\) and on the settings of the PU subtraction strategy. We find, however, that the overall signal significance from combining the three categories is similar in the \(n_\mathrm{PU}=80\) and \(n_\mathrm{PU}=150\) cases.
Notes
Acknowledgments
We thank F. Bishara, R. Contino, A. Papaefstathiou and G. Salam for useful discussions on the topic of Higgs pair production. We thank E. Vryonidou and M. Zaro for assistance with diHiggs production in MadGraph5_aMC@NLO. The work of K. B. is supported by a Rhodes Scholarship. D. B., J. F. and C. I. are supported by the STFC. J. R. and N. H. are supported by an European Research Council Starting Grant “PDF4BSM”. J. R. is supported by an STFC Rutherford Fellowship and Grant ST/K005227/1 and ST/M003787/1.
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