Combination of searches for non-resonant and and decay channels using - CERN Document Server

ATLAS CONF Note
 ATLAS-CONF-2021-052
 21st October 2021

 Combination of searches for non-resonant and
 ¯
 resonant Higgs boson pair production in the ,
 ¯ + − and 
 ¯ ¯ decay channels using 
 √
 collisions at = 13 TeV with the ATLAS detector

 The ATLAS Collaboration

 This note presents a combination of searches for Higgs boson pair production using
 126–139 fb−1 of proton-proton collision data recorded with the ATLAS detector at a center-
 √
 of-mass energy of = 13 TeV at the LHC. Three searches for pairs of Higgs bosons, in the
 ,
 ¯ ¯ + − , and 
 ¯ ¯ final states, are included in this combination. The non-resonant
ATLAS-CONF-2021-052

 interpretation uses results from the ¯ ¯ + − searches, while the resonant interpreta-
 and 
 tion uses results from all three searches. No statistically significant excess above the Standard
 Model expectation has been found. Upper limits are set on the production rate of non-resonant
 21 October 2021

 Higgs boson pairs, at the 95% confidence level, assuming Standard Model kinematics. The
 observed (expected) combined upper limit is found to be 3.1 (3.1) times the Standard Model
 prediction. The value of the Higgs boson trilinear self-coupling modifier ≡ / SM is
 excluded outside the observed (expected) range −1.0 ≤ ≤ 6.6 (−1.2 ≤ ≤ 7.2) at 95%
 confidence level. Upper limits on the production cross-section of a heavy scalar resonance
 decaying to two Standard Model Higgs bosons are set at 95% confidence level between 1.1
 and 595 fb (1.2 and 392 fb) in observation (expectation), depending on the resonance mass,
 , within the studied mass range 251 GeV ≤ ≤ 3 TeV.

 © 2021 CERN for the benefit of the ATLAS Collaboration.
 Reproduction of this article or parts of it is allowed as specified in the CC-BY-4.0 license.

1 Introduction

Since the discovery of the Higgs boson ( ) at the Large Hadron Collider (LHC) in 2012 [1, 2], a priority of
the ATLAS and CMS collaborations has been to better understand its properties and couplings. The Higgs
boson self-coupling is directly related to the shape of the Higgs scalar field potential, which is important
for understanding the mechanism of electroweak symmetry breaking, and serves as a precision test of
the electroweak theory. Higgs boson pair ( ) production provides a direct probe of the Higgs boson
self-coupling, which has a direct effect on the size of the production cross-section.
In the Standard Model (SM), the gluon–gluon fusion process (ggF) accounts for more than 90% of non-
resonant production. It proceeds mainly through two diagrams at leading order (LO) [3], known as the
triangle diagram and the box diagram, which are shown in Figure 1. The former is the LO diagram which is
sensitive to the Higgs boson trilinear self-coupling . These diagrams interfere destructively, leading to
a small SM cross-section, which is three orders of magnitude smaller than single Higgs boson production.
The SM cross-section is predicted to be ggF SM
 ( ) = 31.05+6% −23% (scale + top ) ± 3.0%(PDF + ) fb [4–
11] at next-to-next-to-leading order (NNLO) in and including an approximation of finite top-quark-mass
 √
effects, for the Higgs boson mass = 125 GeV and = 13 TeV. The “PDF + ” uncertainty accounts
for the effects of the strong coupling constant and parton distribution functions, the “scale” uncertainty
is due to the finite order of quantum chromodynamics (QCD) calculations, and the “ top ” uncertainty is
related to the top-quark mass scheme.
The next leading production mode is Vector Boson Fusion (VBF), contributing 5% of the SM 
production rate. The cross-section of the VBF process is evaluated at next-to-next-to-next-to-leading
order (N3LO) in QCD [12–16], with a value of VBF SM
 ( ) = 1.73+0.03%
 −0.04% (scale) ± 2.1% (PDF + ) fb,
 √
for = 125 GeV and = 13 TeV. The tree-level VBF diagrams are shown in Figure 2. Figure 2(a)
involves the Higgs boson self-coupling, while the other two involve solely the Higgs boson’s couplings to
vector bosons.
 
 (a) triangle diagram (b) box diagram

Figure 1: Leading order Feynman diagrams showing ggF non-resonant production of pairs of Higgs bosons in
the Standard Model: (a) the triangle diagram, featuring the Higgs boson trilinear self-coupling, labeled with the
self-coupling modifier ≡ / SM and (b) the box diagram, featuring only a loop of quarks.

The non-resonant production rate is very sensitive to small anomalous couplings in Beyond the Standard
Model (BSM) scenarios. For example, changing the sign of the Higgs boson trilinear self-coupling with
respect to the Standard Model expectation would quadruple the production rate [12]. It is estimated
that the Higgs boson trilinear self-coupling can be measured to a precision of 50% at the HL-LHC [17] by
combining results from both ATLAS and CMS.
Pairs of Higgs bosons can also be produced via the decay of a hypothetical heavy resonance, and many
BSM theories predict the existence of such heavy particles. Models that predict additional scalars include:
electroweak singlet models [18, 19], two-Higgs-doublet models [20], and models inspired by the Minimal

 2

 

 (a) (b) (c)

Figure 2: Leading order Feynman diagrams showing VBF non-resonant production of pairs of Higgs bosons in the
Standard Model, featuring: (a) the Higgs boson trilinear self-coupling, represented by the coupling modifier , (b)
the coupling, and (c) two Higgs-vector boson couplings.

Supersymmetric Standard Model (MSSM) such as the hMSSM [21–24]. The leading order Feynman
diagram corresponding to the production of a heavy scalar particle decaying to a pair of Higgs bosons is
shown in Figure 3. Only the ggF production mode is considered in this note.

 g H

 X

 g H

Figure 3: Leading order Feynman diagram for the production of a heavy resonance decaying to a pair of Higgs
bosons.

This note presents a combination of searches for non-resonant and resonant Higgs boson pair production
using the full Run-2 LHC dataset, corresponding to an integrated luminosity of up to 139 fb−1 of data
 √
collected at = 13 TeV. It includes a combination of non-resonant to bb and bb + − searches and
a combination of resonant to bbbb, bb + − and bb searches. A Higgs boson mass of 125.0 GeV is
assumed.
A previous ATLAS combination of searches for non-resonant and resonant pair production was
performed on a partial Run-2 dataset, using up to 36.1 fb−1 of data [25]. It included six channels: bbbb [26],
bb + − [27], bb [28], + − + − [29], ¯ ∗
 [30] and + − [31]. The combined observed
(expected) limit for non-resonant production at 95% confidence level was 6.9 (10) times the predicted
Standard Model cross-section. When varying the Higgs boson trilinear self-coupling from its Standard
Model value, the allowed range for the self-coupling modifier ≡ / SM was observed (expected) to
be −5.0 ≤ ≤ 12.0 (−5.8 ≤ ≤ 12.0). Limits were also presented on the production cross-section of a
narrow-width scalar resonance , in the mass range 260–3000 GeV, decaying to . CMS also published
a combination of searches using the partial Run-2 dataset, up to 35.9 fb−1 of data [32]. The CMS
combined observed (expected) limit for non-resonant production at 95% confidence level was 22.2
(12.8) times the predicted Standard Model cross-section, and the observed (expected) allowed range for the
self-coupling modifier was −11 ≤ ≤ 18 (−7.1 ≤ ≤ 13.6). Limits on resonant production, in
the mass range 250–3000 GeV, were also presented.

 3

2 ATLAS detector

The ATLAS experiment [33] at the LHC is a multipurpose particle detector with a forward–backward
symmetric cylindrical geometry and a near 4 coverage in solid angle.1 It consists of an inner tracking
detector surrounded by a thin superconducting solenoid providing a 2 T axial magnetic field, electromagnetic
and hadron calorimeters, and a muon spectrometer. The inner tracking detector covers the pseudorapidity
range | | < 2.5, and it consists of silicon pixel, silicon microstrip, and transition radiation tracking
detectors. Lead/liquid-argon (LAr) sampling calorimeters with high granularity provide electromagnetic
(EM) energy measurements. A steel/scintillator-tile hadron calorimeter covers the central pseudorapidity
range (| | < 1.7). The endcap and forward regions are instrumented with LAr calorimeters for both the EM
and hadronic energy measurements up to | | = 4.9. The muon spectrometer surrounds the calorimeters and
is based on three large superconducting air-core toroidal magnets with eight coils each. The field integral
of the toroids ranges between 2.0 and 6.0 T m across most of the detector. The muon spectrometer includes
a system of precision tracking chambers and fast detectors for triggering. A two-level trigger system [34] is
used to select events. The first-level trigger is implemented in hardware and uses a subset of the detector
information to accept events at a rate below 100 kHz. This is followed by a software-based trigger that
reduces the accepted event rate to 1 kHz on average depending on the data-taking conditions. An extensive
software suite [35] is used for real and simulated data reconstruction and analysis, for operation and in the
trigger and data acquisition systems of the experiment.

3 Data and simulation
 √
The searches included in this combination use up to 139 fb−1 of collision data at = 13 TeV
collected by ATLAS during LHC Run-2 from 2015–2018. The uncertainty in the combined 2015–2018
integrated luminosity is 1.7 % [36], obtained using the LUCID-2 detector [37] for the primary luminosity
measurements.
The ggF non-resonant Monte Carlo (MC) samples are generated at next-to-leading order (NLO)
accuracy in QCD with finite top-quark mass in both real and virtual corrections (NLO FT) [38, 39], using
the Powheg Box v2 [40] generator with the PDF4LHC15 parton distribution function (PDF) set [41]. The
Pythia 8.244 [42] generator is used for parton showering, hadronization and underlying event simulation,
with the A14 set of tuned parameters [43, 44]. Herwig v7.1.6 [45, 46] is used as an alternative generator
to estimate the uncertainty arising from the choice of parton shower. The ggF samples are generated with
 values of 1 and 10. The VBF non-resonant samples are generated at LO accuracy [47, 48] using
MadGraph5_aMC@NLO v2.6.0 [47]. The NNPDF3.0nlo PDF set [49] is used in the matrix element,
interfaced with Pythia 8.244 with the A14 set of tuned parameters. VBF samples are generated with
 values of 0, 1, 2, and 10. The ggF and VBF MC samples are normalized to the higher order
cross-sections mentioned in Section 1. Other subleading non-resonant production modes are not
considered.

1
 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector
 and the -axis along the beam pipe. The -axis points from the IP to the centre of the LHC ring, and the -axis points
 upwards. Cylindrical coordinates ( , ) are used in the transverse plane, being the azimuthal angle around the -axis.
 The pseudorapidity
 q is defined in terms of the polar angle as = − ln tan( /2). Angular distance is measured in units of
 Δ ≡ (Δ ) 2 + (Δ ) 2 .

 4

Table 1: Summary of the main characteristics of the non-resonant (bb and bb + − ) and resonant (bb , bb + −
and bbbb) analyses considered in the combination. The different rows report the pair branching fraction
B ( → ¯ ¯ ) for each decay channel, the integrated luminosity Lint considered by each analysis, the final
discriminant for the statistical interpretation, and the mass range for analyses targeting a resonant production.
References to the latest publications are also provided in the "Ref." row.

 bb bb + − bbbb resolved (boosted)

 B ( → 
 ¯ ¯ ) 2.6 · 10−3 0.073 0.339
 −1
 Lint [fb ] 139 139 126 (139)
 Discriminant MVA outputs 
 Resonance mass ( ) range [GeV] 251–1000 251–1600 251–1500 (900–3000)
 Ref. [51] [52] [53]

A reweighting method is used to determine the signal yield at each value, in addition to the explicitly
generated values for the ggF and VBF non-resonant MC samples. The reweighting method for the
ggF samples [50] derives scale factors as a function of in bins of truth by performing a linear
combination of generator-level samples at three different values. For the reweighted ggF signal, the
NNLO FTapprox cross-section as a function of is taken from Ref. [10]. A linear combination of three
reconstructed samples is used to produce the necessary weights for the VBF samples. For the reweighted
VBF signal, the N3LO cross-section as a function of is obtained by applying the SM N3LO-to-LO
 -factor to the LO generator cross-section. Uncertainties on the reweighting methods are evaluated and
included in the results.
MC samples with a heavy spin-0 resonance produced via ggF and decaying into a pair of Higgs
bosons, → → , are generated using MadGraph5_aMC@NLO v2.6.1 at LO accuracy with the
NNPDF2.3lo PDF set. The event generator is interfaced with Herwig v7.1.3 to model the parton shower,
hadronization and underlying event. The mass of the heavy scalar is varied from 251 GeV to up to 3 TeV,
and its width is set to 10 MeV. The interference with non-resonant production is neglected.
Depending on the search channel, Standard Model backgrounds are modeled using both simulation and
data-driven techniques. Further details about background estimation can be found in the notes describing
the input searches [51–53].

4 Description of searches

The analysis strategies for the individual searches included in this combination are described below. A
summary is presented in Table 1.

4.1 bb 

While the to bb decay mode has a very small branching ratio (0.26%), its experimentally clean
signature makes it one of the most sensitive searches. The bb search [51] considers the full Run-2
dataset of 139 fb−1 , and starts with a selection of exactly two photons and two -tagged jets, each consistent

 5

with a SM Higgs boson decay. The -jets are identified using the DL1r algorithm [54, 55] at the 77%
efficient working point. An electron and muon veto is applied for orthogonality with other searches
with final states including leptons. The main background processes in the bb search are continuum
production of photon pairs and jets, and processes where a single Higgs boson decays to a pair of photons.
For the resonant search, non-resonant SM production is considered a background process. For both the
non-resonant and resonant searches, multivariate techniques are used to separate signal from background,
and the statistical results are obtained from a fit of the diphoton invariant mass, . In the non-resonant
search, four categories based on the four-body mass, , and the multivariate analysis (MVA) outputs
are defined. In the resonant search, selection criteria are defined depending on the mass of the narrow-width
scalar particle , over the mass range 251 GeV≤ ≤ 1000 GeV.

4.2 bb + −

The to bb + − decay mode has one of the largest branching fractions (7.3%) among the investigated
 decay channels. The bb + − search [52] has different signal regions targeting fully-hadronic and
semi-leptonic di- final states, labeled had had and lep had respectively. The search exploits the complete
Run-2 dataset of 139 fb−1 . Events in the lep had channel are categorized depending on whether they pass
a single-lepton-trigger (SLT) or lepton-plus-tau trigger (LTT). Both categories of events are required to
have exactly one electron or muon and one hadronic tau with opposite charge. Events in the had had
channel are selected using a combination of single- and di-tau triggers, and are required to have exactly two
reconstructed hadronic taus with opposite charge. An electron and muon veto is applied for orthogonality
with the lep had channel. In both the lep had and had had channels, exactly two -jets passing the 77%
efficient working point of the DL1r tagger are required. The dominant backgrounds come from tt and
 + heavy flavor production.
After this event selection, both resonant and non-resonant analyses use the output MVAs as the main
observable. MVAs are trained and evaluated separately in the bb had had , bb lep had (SLT), and bb lep had
(LTT) signal regions using MC samples. The MVAs targeting non-resonant are trained using the ggF
SM → bb + − sample, while the MVAs targeting resonant are trained on all signal samples
simultaneously and evaluated separately for each of the considered signal models with masses between
251 GeV and 1600 GeV. Finally, a fit of the MVA outputs in the bb had had and bb lep had signal regions
is performed, using the data in the + heavy flavor control region to constrain the normalization of this
background. The distributions of the MVA scores obtained by the non-resonant bb + − search are shown
in Figure 4.
As described in Section 3, a reweighting of the simulated ggF and VBF samples is performed to
create a full set of values. Both the signal yields and the shape of the MVA distribution are taken
from this reweighting method. The allowed observed (expected) range in the bb + − search at 95%
confidence level is: ∈ [−2.4, 9.2] ( ∈ [−2.0, 9.0]). Constraints on in the full Run-2 bb + − search
are reported for this first time in this note.

4.3 bbbb

The to bbbb decay mode has the advantage of the largest SM decay branching fraction of 33.9%.
Nevertheless, among the searches targeted in this combination, the bbbb channel also has the largest
SM background due to the abundant production of QCD multijet processes.The resonant bbbb search

 6

Events / 0.07

 Events / 0.07
 107 ATLAS Preliminary Data SM HH 108 ATLAS Preliminary Data SM HH

 s = 13 TeV, 139 fb-1 κλ = 2 κλ = 10 s = 13 TeV, 139 fb-1 κλ = 2 κλ = 10
 106 107
 τlepτhad LTT Top-quark Jet → τhad fakes
 6 τlepτhad SLT Top-quark Jet → τhad fakes
 105 Other Z → ττ + (bb,bc,cc)
 10 Other Z → ττ + (bb,bc,cc)
 SM Higgs Uncertainty 105 SM Higgs Uncertainty
 104
 Pre-fit background Pre-fit background
 104
 103
 103
 102
 102
 10
 10
 1 1
 10− 1 10 −1

 10− 2 10− 2
 Data/Pred.

 Data/Pred.
 1.5
 1.2
 1 NN score 1 NN score
 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.80 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
 0.5
 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
 NN score NN score

 (a) bb lep had LTT (b) bb lep had SLT
 Events / 0.14

 107 ATLAS Preliminary Data SM HH
 6 s = 13 TeV, 139 fb-1 κλ = 2 κλ = 10
 10
 τhadτhad Top-quark Jet → τhad fakes
 5
 10 Other Z → ττ + (bb,bc,cc)
 SM Higgs Jet → τhad fakes (tt)
 4
 10 Uncertainty Pre-fit background

 103

 102
 10
 1
 −1
 10
 10− 2
 Data/Pred.

 1.2
 1 BDT score
 0.8−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
 BDT score

 (c) bb had had

Figure 4: MVA scores of the non-resonant bb + − analysis: (a) shows the Neural-Network (NN) output in the
bb lep had signal region exploiting lepton-plus- had triggers (LTT), (b) reports the NN score in the bb lep had region
exploiting single-lepton triggers (SLT), and (c) shows shape of the Boosted Decision Tree (BDT) output used in the
bb had had signal region. Data are shown as black points, backgrounds are shown as stacked, colored histograms
(the “Other” background includes +jets, +jets, and diboson processes), and signals are shown as colored
lines for = 1, 2 and 10. The signals are scaled to the predicted cross-section. The normalization, shape and the
uncertainty of the backgrounds are determined from a background only fit to data. The size of the total combined
statistical and systematic uncertainty of the background is indicated by the hatched band.

[53] is divided into two complementary channels: the resolved channel, in which Higgs boson candidates
are formed from four -initiated small-radius ( = 0.4) anti- jets, and the boosted channel, in which
the high- T Higgs boson candidates are reconstructed as two separate large-radius ( = 1.0) anti- jets.

 7

The resolved channel relies on a combination of small-radius -jet triggers, while the boosted channel
uses the data recorded by large-radius jet triggers without any -tagging requirement applied. The search
was performed using the Run-2 dataset. The resolved (boosted) channel uses 126 fb−1 (139 fb−1 ) of data
collected in 2016–2018 (2015–2018). After trigger selection, -initiated jets are identified using the DL1r
algorithm [54, 55] at the 77% efficient working point, and Higgs boson candidates are reconstructed from
these jets. The invariant masses of the Higgs boson candidates are then used to classify events into signal,
validation, and control regions. The final discriminant of the two channels is provided by the total invariant
mass of the two Higgs boson system, . The resolved channel provides the best sensitivity in the
lower range (from 251 GeV to 1500 GeV), while the boosted channel covers the higher range (from
900 GeV to 3000 GeV). The two channels are statistically combined in the overlapping mass region to
enhance the sensitivity of the search. The orthogonality of the two channels is guaranteed by explicitly
vetoing all resolved events in the boosted selection.

5 Statistical treatment

The combination of the input searches is achieved by multiplying separate likelihood functions into
a combined likelihood function L (D | , ), which accounts for the data D, the parameter of interest ,
the nuisance parameters , and the statistical model of each search. Through this procedure, the various
searches can be simultaneously fit to the data to constrain parameters of interest such as the SM signal
strength, defined as the ratio of the measured signal rate with respect to the SM theory prediction, or
the resonant production cross-section.
The construction of L (D | , ) from separate likelihood functions relies on the fact that all of the search
signal regions are kinematically orthogonal, or at least have a negligible number of shared events. In
this combination, the bb , bb + − , and bbbb signatures are expected to either be orthogonal due to the
different number and type of physics objects required in their final states, or to have a negligible number of
overlapping events. In order to check for the presence of overlapping events, the complete Run-2 dataset
was used to compare events passing the requirements of the analysis signal regions. No overlapping events
in data were found to be shared between the signal regions of the three searches.
The measurement of the parameter of interest is performed using a statistical test based on the profile
likelihood ratio [56]:
 L (D | , ˆˆ ( ))
 (D | ) = , (1)
 L (D | ,
 ˆ ) ˆ

where ˆˆ ( ) indicates that the nuisance parameters in the numerator are set to their profiled values which
maximize the likelihood for a given value of , while ˆ and ˆ maximize the likelihood unconditionally.
Exclusion upper limits are set following the CLs prescription [57] using the asymptotic formula [56]. The
considered test statistic ˜ is:

 L ( D | , ˆˆ ( ))
 ˆ < 0,
 
 
  −2 ln
 L ( D |0, ˆˆ (0))
 
 
 
 
 ˜ = −2 ln L ( D | , ˆˆ ( ))
 0 ≤ ˆ ≤ , (2)
 
 
  L ( D | ,
 ˆ ) ˆ
 ˆ > .
 
 0
 

 8

The test statistic 00 is used when computing the local 0 -value:

 L ( D |0, ˆˆ (0))
  +2 ln ˆ ≤ 0,
 
 
 L ( D | ,
 00
  ˆ ) ˆ
 = (3)
 L ( D |0, ˆˆ (0))
  −2 ln
 
 L ( D | ,ˆ )
 ˆ ˆ > 0.
 

Systematic uncertainties relating to the data-taking conditions, such as those associated to the integrated
luminosity and the pileup mis-modeling, are considered fully correlated among the input searches.
Uncertainties related to physics objects used by multiple searches, such as jets and flavor-tagging, are
treated as correlated where possible.2 Theoretical uncertainties on simulated signal and background
processes, such as the and single Higgs boson cross-sections, QCD scale, and proton PDFs are treated
as correlated where possible. The cross-section uncertainty is included in the limit on the signal
strength, but not in the cross-section limit.

6 Results

6.1 Limits on non-resonant production

Upper limits are set on the non-resonant ggF+VBF signal strength and production cross-section,
assuming Standard Model kinematics. The observed (expected) combined upper limit, at 95% confidence
level, on the signal strength is found to be 3.1 (3.1), where the predicted SM ggF+VBF cross-section
 SM
is +VBF ( ) = 32.78 fb. The observed (expected) combined 95% confidence level upper limit on
the non-resonant SM production cross-section is 92.0 fb (92.2 fb). The limits are shown in Table 2
and Figure 5. The upper limits on the SM signal strength use the theoretical prediction for the SM
 cross-section and its uncertainty, while the upper limits on the cross-section do not include these
uncertainties on the SM cross-section. This results in a slightly lower SM-normalized upper limit
value in the case of the cross-section.
These results are obtained assuming a Higgs boson mass of = 125 GeV. The bb + − -only limits and
the combined limits (observed and expected) change by less than 0.6% when assuming a Higgs boson
mass of 125.09 GeV [58] instead. A relatively larger difference can be seen in the bb search alone, since
it uses the diphoton mass spectrum in the extraction of the final result. The diphoton mass is more
sensitive to the value of than the reconstructed Higgs boson masses in the hadronic decay modes, due
to its better energy resolution. The Higgs boson mass was considered to be 125.09 GeV in the preliminary
bb result [51]. Using 125 GeV instead, the upper limits on the to bb observed (expected) signal
strength are 1.6% (0.2%) lower. In addition, the bb limits reported here include the most up-to-date
theory uncertainties [6], which are larger than those used in the preliminary bb result. The inclusion of
these uncertainties increases the observed (expected) upper limit on the signal strength in the bb 
search by 3.1% (4.4%).

2
 The bb + − limits presented in this note include some flavor-tagging systematic uncertainties that were neglected in Ref. [52].

 9

Table 2: Observed and expected 95% confidence level upper limits on the signal strength for SM production
derived from the bb + − and bb searches, and their statistical combination.

 Obs. −2 −1 Exp. 1 2 
 bb 4.3 3.1 4.1 5.7 8.8 14.3
 bb + − 4.6 2.1 2.8 3.9 5.9 9.4
 Combined 3.1 1.7 2.2 3.1 4.7 7.3

 ATLAS Preliminary Observed
 s = 13 TeV, 139 fb 1 Expected
 SM
 ggF + VBF = 32.78 fb
 Comb. exp. limit ± 1
 Comb. exp. limit ± 2

 Obs. Exp.

 bb + 4.6 3.9

 bb 4.3 5.7

 Combined 3.1 3.1

 1 10
 95% CL upper limit on signal strength

Figure 5: Observed and expected 95% confidence level upper limits on the signal strength for SM production in
the bb and bb + − searches, and their statistical combination. The expected limits assume no production.

6.2 Constraints on the Higgs boson self-coupling

Changes to the Higgs boson self-coupling modifier from its Standard Model value result in changes to
the cross-section and to the kinematics of the events. The product of acceptance and efficiency as
a function of for the individual searches included in this combination is shown in Figure 6.
Upper limits are set at 95% confidence level on the cross-section, ggF+VBF ( ), for each 
hypothesis. Combined limits including both the bb and bb + − searches are shown in Table 3 and
Figure 7, and the theoretical prediction for the cross-section ggF+VBF ( ) as a function of is overlaid
on the computed limits. The intersections between the theoretical prediction and the observed (expected)
limit determine the observed (expected) allowed range for . These are found to be −1.0 ≤ ≤ 6.6
(−1.2 ≤ ≤ 7.2). The expected limits are calculated assuming no production.

 10

16
 Acceptance x Efficiency [%]

 ATLAS Simulation Preliminary bb
 14 s = 13 TeV, 139 fb 1 bb +

 12
 10
 8
 6
 4
 2
 010 8 6 4 2 0 2 4 6 8 10

Figure 6: Acceptance times efficiency as a function of for the bb and bb + − searches. Both ggF and VBF 
signals are included.

Table 3: Observed and expected 95% confidence level allowed ranges for , for the bb and bb + − searches, and
their statistical combination. The expected limits assume no production.

 Obs. Exp.
 bb [−1.6, 6.7] [−2.4, 7.7]
 bb + − [−2.4, 9.2] [−2.0, 9.0]
 Combined [−1.0, 6.6] [−1.2, 7.2]

 11

(HH) [fb]

 104 ATLAS Preliminary
 Observed limit (95% CL)
 1 Expected limit (95% CL)
 s = 13 TeV, 139 fb Comb. exp. limit ±1
 Comb. exp. limit ±2
 ggF + VBF

 Theory prediction
 SM prediction
 103

 102
 Observed: [ 1.0, 6.6] bb
 bb +
 Expected: [ 1.2, 7.2] Combined
 10110 8 6 4 2 0 2 4 6 8 10

Figure 7: Observed and expected 95% confidence level upper limits on the non-resonant production cross-section
as a function of in the bb and bb + − searches, and their statistical combination. The expected limits assume
no production. The theory prediction curve represents the scenario where all parameters and couplings are set to
their SM values except for .

 12

6.3 Limits on resonant production

The resonant searches target a heavy, spin-0 scalar , which has a narrow-width compared to the
experimental mass resolution. Limits are set at 95% confidence level on the resonant production
cross-section, ( → ), and presented for the bb , bb + − , and bbbb 3 searches, and their statistical
combination. Figure 8 shows the combined limits on ( → ), ranging between 1.1 and 595 fb (1.2
and 392 fb) in observation (expectation), depending on the resonance mass. The bb search is the most
sensitive at low , the bb + − search is the most sensitive in the 400–800 GeV range, and the bbbb
search dominates for high , demonstrating the complementary of these three searches. The largest
deviation from the Standard Model expectation is observed at 1.1 TeV. This feature has been investigated,
and the local (global) significance for = 1.1 TeV using the asymptotic formula [59] is found to be 3.2 
(2.1 ), where the trial factor is evaluated based on the number of up-crossings in data.
 (X HH) [fb]

 104 ATLAS Preliminary
 s = 13 TeV, 126 139 fb 1
 Spin-0
 Observed limit (95% CL)
 103 Expected limit (95% CL)
 Comb. exp. limit ± 1
 Comb. exp. limit ± 2
 102

 101 bbbb
 bb +
 bb
 100 Combined

 200 300 500 1000 2000 3000
 mX [GeV]

Figure 8: Expected and observed 95% confidence level upper limits on ( → ) for a spin-0 resonance as a
function of its mass in the bb , bb + − and bbbb searches, and their statistical combination. The discontinuities
in the limit visible in the range < 400 GeV are caused by the partial availability of the different analysis limits on
a point-by-point basis, which are provided only for the bb search at the weakest limit points. Further details can be
found in Tables 4–7 in the appendix.

3
 The boosted bbbb search results were updated with respect to Ref. [53] by the recovery of some events in data and by imposing
 additional requirements, following orthogonality checks between resolved and boosted topologies.

 13

7 Conclusion

A statistical combination of three searches for Higgs boson pair production has been presented. These
searches were performed using the complete LHC Run-2 dataset (126–139 fb−1 ) collected with the ATLAS
detector. No statistically significant excess above the Standard Model expectation has been found.
Using a combination of the bb and bb + − searches, the observed (expected) combined upper limit on
the SM signal strength is found to be 3.1 (3.1) at 95% confidence level. The limit on the non-resonant
SM production cross-section is found to be 92.0 fb observed (92.2 fb expected) at 95% confidence
level. Values of the Higgs boson trilinear self-coupling modifier ≡ / SM are excluded outside the
observed (expected) range −1.0 ≤ ≤ 6.6 (−1.2 ≤ ≤ 7.2) at 95% confidence level. For the resonant
interpretation, using a combination of the bb , bb + − , and bbbb searches, observed (expected) 95%
confidence level upper limits have been set between 1.1 and 595 fb (1.2 and 392 fb) on the resonant Higgs
boson pair production cross-section, depending on the mass of the heavy resonance within the studied
mass range 251 GeV ≤ ≤ 3 TeV.

 14

Appendix

Additional material is included in this section for the combination results. Figure 9 shows the local 0 -value
as a function of the spin-0 resonance mass for the combination of the resonant bb , bb + − and bbbb
analyses. After combination, the largest excess is found to be at = 1.1 TeV and with a local (global)
significance of 3.2 (2.1 ). At = 1.1 TeV the level of compatibility between the best-fit cross-sections
in bb + − and bbbb channels corresponds to a -value of 34%. Tables 4 to 7 also show the observed
and expected 95% confidence level upper limits for the individual resonant analyses, as well as for
the combined resonant results. The bb channel is expected to dominate below 400 GeV; the bb + −
channel is expected to dominate in the region 400–800 GeV; the bbbb channel is expected to dominate
above 800 GeV. If for a given mass point only a single analysis limit is available, the corresponding limit
is included in the combination only if the analysis dominates in that mass range. Figure 10 shows the
observed and expected upper limits on the ggF+VBF ( ) cross-section obtained by the non-resonant
bb + − search.

 0
 Local p0-value

 10 0
 1
 1
 10
 2
 2
 10

 3
 3
 10 bbbb
 ATLAS Preliminary bb +
 s = 13 TeV, 126 139 fb 1
 10
 4 Spin-0 bb
 4 Combined
 200 300 500 1000 2000 3000
 mX [GeV]
Figure 9: Local 0 -value as a function of the resonance mass for the spin-0 resonance model. Each curve represents
the 0 -value corresponding to the single bb , bb + − , bbbb analyses, as well as the 0 -value resulting from
statistical combination of the different analyses. The largest excess in the combined limit is found at = 1.1 TeV
and it corresponds to a local (global) significance of 3.2 (2.1 ).

 15

Table 4: Individual and combined limits for a spin-0 resonance in fb, for masses below 350 GeV. Blank indicates no
result is calculated at the mass point.

 Mass
 251.0 260.0 270.0 280.0 290.0 300.0 312.5 325.0 337.5 350.0
 [GeV]
 Exp. 
 ¯ 218 372 392 374 350 373 365 333 293 273
 Exp. 
 ¯ 350 751 834 654 468 353
 Exp. 
 ¯ ¯ 2980 6050 4270 2540 653
 Exp. combined 176 325 392 325 350 308 365 257 293 192
 Obs. 
 ¯ 385 641 595 331 241 355 403 252 261 348
 Obs. 
 ¯ 650 941 490 530 340 230
 Obs. 
 ¯ ¯ 3730 8410 7740 3480 391
 Obs. combined 407 584 595 237 241 275 403 173 261 149

Table 5: Individual and combined limits for a spin-0 resonance in fb, for masses between 375 and 1000 GeV. Blank
indicates no result is calculated at the mass point.

 Mass
 375.0 400.0 450.0 500.0 550.0 600.0 700.0 800.0 900.0 1000.0
 [GeV]
 Exp. 
 ¯ 241 190 146 130 95.2 81.1 76.5 65.8 45.8 50.1
 Exp. 
 ¯ 222 147 69.4 43.6 33.6 26.5 19.1 15.6 13.6 12.3
 Exp. ¯
 ¯ 267 94.3 45.4 24.6 15.9 11.1 8.16
 Exp. combined 152 99.5 58.7 35.9 30 20.7 14.1 10.4 7.97 6.41
 Obs. 
 ¯ 415 205 134 173 87.6 74.1 49.3 72.2 77.1 51.8
 Obs. 
 ¯ 132 80.5 49.9 46.8 25.4 22.5 25.9 31.6 32.4 32.7
 Obs. ¯
 ¯ 163 85.3 26.4 22.1 12 12.2 6.45
 Obs. combined 144 58.2 42.8 40.5 22.9 13.3 14.9 14.1 16.4 10.5

Table 6: Individual and combined limits for a spin-0 resonance in fb, for masses between 1100 and 2000 GeV. Blank
indicates no result is calculated at the mass point.

 Mass
 1100.0 1200.0 1300.0 1400.0 1500.0 1600.0 1800.0 2000.0
 [GeV]
 Exp. 
 ¯
 Exp. 
 ¯ 13.5 13.8 20.1 29.4
 Exp. 
 ¯ ¯ 6.19 5.25 4.5 3.81 3.35 3.18 2.49 2.04
 Exp. combined 5.5 4.81 4.5 3.71 3.35 3.14 2.49 2.04
 Obs. 
 ¯
 Obs. 
 ¯ 28.9 25.1 26.6 33.9
 Obs. 
 ¯ ¯ 13.2 9.31 6.05 7.75 7.4 5.16 3.27 2.8
 Obs. combined 14.5 10.1 6.05 7.85 7.4 5.21 3.27 2.8

 16

Table 7: Individual and combined limits for a spin-0 resonance in fb, for masses between 2500 and 3000 GeV. Blank
indicates no result is calculated at the mass point.

 Mass
 2500.0 3000.0
 [GeV]
 Exp. 
 ¯
 Exp. 
 ¯
 Exp. 
 ¯ ¯ 1.48 1.21
 Exp. combined 1.48 1.21
 Obs. 
 ¯
 Obs. 
 ¯
 Obs. ¯
 ¯ 1.97 1.15
 Obs. combined 1.97 1.15

 105
 (HH) [fb]

 ATLAS Preliminary Observed limit (95% CL)
 s = 13 TeV, 139 fb 1 Expected limit (95% CL)

 104 HH bb + Expected limit ±1
 Expected limit ±2
 ggF + VBF

 Theory prediction
 SM prediction
 103

 102
 Observed: [ 2.4, 9.2]
 Expected: [ 2.0, 9.0]
 10110 8 6 4 2 0 2 4 6 8 10

Figure 10: Observed and expected 95% CL upper limits on non-resonant production cross-section as a function
of in the bb + − channel. The theory prediction curve represents the scenario where all parameters and couplings
are set to their SM values except for . The uncertainty band shows the cross-section uncertainty of this prediction.

 17

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