HIG-21-007

HIG-21-007

[cern]Olivier Davignon, David Anthony, Eshwen Bhal, Jim Brooke, Aaron Bundock,
Henning Flaecher, Maciej Glowacki, Ben Krikler, Sudarshan Paramesvaran, Rob
White

# A search for decays of the Higgs boson to invisible particles in events with
a top-antitop quark pair or a vector boson in proton-proton collisions at
$\sqrt{s}=13\TeV$

###### Abstract

A search for decays to invisible particles of Higgs bosons produced in
association with a top-antitop quark pair or a vector boson, which both decay
to a fully hadronic final state, has been performed using proton-proton
collision data collected at ${\sqrt{s}=13\TeV}$ by the CMS experiment at the
LHC, corresponding to an integrated luminosity of 138. The 95% confidence
level upper limit set on the branching fraction of the 125Higgs boson to
invisible particles, ${\mathcal{B}(\PH\to\text{inv})}$, is 0.47 (0.40
expected), assuming standard model production cross sections. The results of
this analysis are combined with previous ${\mathcal{B}(\PH\to\text{inv})}$
searches carried out at ${\sqrt{s}=7}$, 8, and 13in complementary production
modes. The combined upper limit at 95% confidence level on
${\mathcal{B}(\PH\to\text{inv})}$ is 0.15 (0.08 expected).

##  0.1 Introduction

The Higgs boson () [1, 2, 3, 4, 5, 6] of mass 125was discovered by the ATLAS
and CMS Collaborations in 2012 [7, 8, 9]. Since then its properties, including
its coupling to other standard model (SM) particles, have been extensively
studied using proton-proton (${\Pp\Pp}$) collision data from the CERN LHC
collected at ${\sqrt{s}=7}$, 8, and 13with the ATLAS [10] and CMS [11]
detectors. Properties of the Higgs boson can be exploited to probe for signs
of behaviour beyond the SM (BSM). In the SM, the decay of the Higgs boson to
an invisible final state ($\PH\to\text{inv}$) is only possible via
$\PH\to\PZ\PZ^{*}\to 4\PGn$, with a branching fraction of 0.1% [12]. Several
BSM theories predict a larger branching fraction to invisible final states,
${\mathcal{B}(\PH\to\text{inv})}$ [13, 14, 15, 16], namely in Ref. [17] and
references therein. For example, in a scenario where the Higgs boson connects
the SM and dark matter (DM) sectors [18, 19, 20, 21, 22, 23],
${\mathcal{B}(\PH\to\text{inv})}$ is enhanced as the Higgs boson can decay to
a pair of DM particles of mass $m_{\text{DM}}<m_{\PH}/2$.

Direct searches for $\PH\to\text{inv}$ have been performed by the ATLAS [24,
25, 26, 27, 28] and CMS [29, 30, 31, 32, 33, 34] Collaborations using data
collected during Run 1 (2011–2012) and Run 2 (2015–2018). These target
channels in which the Higgs boson is produced via vector boson fusion (VBF),
gluon-gluon fusion ($\cPg\cPg\PH$), and in association with either a vector
boson ($\PV\PH$, where stands for either a or boson) or with a quark pair
($\ttbar\PH$). The current most stringent constraint on
${\mathcal{B}(\PH\to\text{inv})}$ set by the CMS experiment is via the VBF
channel using Run 1 and Run 2 data, which reports a 95% confidence level ()
upper limit of 0.18 (0.10 expected) [34].

In this paper, a search for an invisibly decaying Higgs boson, produced in
association with a quark pair or a boson, where the associated particles decay
to a fully hadronic final state, is reported. Representative leading order
(LO) Feynman diagrams for $\ttbar\PH$ and $\PV\PH$ are presented in Fig. 1.
The search in the $\PV\PH$ channel looks only at topologies in which the
presence of the boson is inferred from well separated decay products,
complementing the previous $\PV\PH$ search with merged decay products arising
from boosted bosons [33]. The search uses LHC ${\Pp\Pp}$ collision data
collected during the years 2016–2018, corresponding to a total integrated
luminosity of 138at $\sqrt{s}=13\TeV$. This is the first search for hadronic
final states of $\ttbar\PH$ in a $\PH\to\text{inv}$ analysis using CMS data,
and the first for resolved $\PV\PH$ topologies using this data.

Figure 1: Representative LO Feynman diagrams for the SM Higgs boson production
channels $\ttbar\PH$ and $\PV\PH$.

Missing transverse momentum, , is the transverse component of the negative
vector sum of all reconstructed particle momenta in an event, and its
magnitude, , is used as the discriminating variable to separate the
$\PH\to\text{inv}$ signal from backgrounds. There are two main sources of
background affecting the measurement. The first is events with invisible boson
decays and visible jets ($\PZ\to\text{inv}$). The second, referred to as the
lost lepton background, $\Pell_{\text{lost}}$ where stands for either an or ,
includes events from $\ttbar+\text{jets}$ and $\PW+\text{jets}$ production
processes where one or more leptons are misreconstructed, excluded by the
phase space selection, or fall outside the detector acceptance. Control
regions (CRs) enriched in these background sources, requiring either one
lepton, one photon, or two same-flavour opposite-sign leptons, are used to
constrain these backgrounds from data. The 95% upper limit on
${\mathcal{B}(\PH\to\text{inv})}$ is extracted from a fit to the hadronic
recoil distribution of selected events, performed across the signal regions
(SRs) and CRs. The hadronic recoil is defined as the of an event excluding the
transverse momentum, , of any identified charged leptons or photons. In the
SRs, the hadronic recoil is equivalent to the while in the CRs it effectively
measures the of the boson or photon. The exclusion of leptons and photons
ensures good correspondence between SRs and CRs.

This paper is organised as follows: Section 0.2 is a brief description of the
CMS detector. The simulated samples used in this analysis are summarised in
Section 0.3. Section 0.4 describes the event reconstruction and object
definitions used in this analysis, while the event selection and event
categorisation are detailed in Section 0.5. The data CRs used for estimating
the SM backgrounds are introduced in Section 0.6. Section 0.7 describes the
statistical procedure used to constrain the backgrounds and extract the
signal. The results of the search are presented in Section 0.8. The results of
combining this search with other CMS searches for invisibly decaying Higgs
bosons are described in Section 0.9, and the results are summarised in Section
0.10.

##  0.2 The CMS detector

The CMS apparatus is a multipurpose, nearly hermetic detector, designed to
trigger on [35, 36] and identify electrons, muons, photons, and (charged and
neutral) hadrons [37, 38, 39]. The central feature of the CMS apparatus is a
superconducting solenoid, of $6~{}\mathrm{m}$ internal diameter. Within the
field volume are the silicon pixel and strip tracker, the crystal
electromagnetic calorimeter (ECAL), and the brass-scintillator hadron
calorimeter (HCAL). Muons are measured in gas-ionisation chambers embedded in
the steel flux-return yoke of the magnet. Besides the barrel and endcap
detectors, CMS has extensive forward calorimetry, performed on high $\eta$
objects in the HCAL forward calorimeter, which is located $11.2~{}\mathrm{m}$
from the interaction region along the beam axis. A global “particle-flow” (PF)
algorithm [40] aims to reconstruct all individual particles in an event,
combining information provided by all subdetectors. The reconstructed
particles are used to build leptons, jets, and [41, 42, 43].

The first level of the CMS trigger system, composed of custom hardware
processors, uses information from the calorimeters and muon detectors to
select the most interesting events, at a rate of roughly $100~{}\mathrm{kHz}$.
The high-level trigger (HLT) processor farm performs event reconstruction
similar to that of the full CMS reconstruction, but optimised for speed. This
decreases the event rate from around $100\unit{kHz}$ to around $1\unit{kHz}$,
before data storage [36].

The procedures for calculating the integrated luminosity recorded by the CMS
detector for each data-taking year are documented in Refs. [44, 45, 46] for
2016–2018, respectively.

A more detailed description of the CMS experiment can be found in Ref. [47].

##  0.3 Simulated samples

Monte Carlo (MC) simulated events are used to model signal and background
contributions in all analysis regions, except for quantum chromodynamics (QCD)
multijet production processes, which are estimated from data using a dedicated
control sample and simulation-based transfer factors. The method for
estimating QCD multijet production is detailed in Section 0.6.2. In all cases,
MC samples are produced using either version 1.0 or higher [48] or version
2.4.2 or higher [49] matrix element (ME) generators. The ME is encoded with
the maximum amount of information available for a hard scattering event. The
parton-level simulation provided by the ME generators is interfaced with
version 8 [50] to model the shower and hadronisation of partons in the initial
and final states, along with the underlying event description, using the tune
CUETP8M1 (CP5) when simulating events for the 2016 (2017 and 2018) data-taking
periods [51]. The propagation of all final state particles through the CMS
detector is simulated using the [52] toolkit. Samples for 2016 make use of the
NNPDF3.0 LO or next-to-LO (NLO) parton distribution functions (PDFs) [53],
whereas samples for the years 2017 and 2018 use the NNPDF3.1 next-to-NLO
(NNLO) PDFs.

Processes featuring $\PH\to\text{inv}$ occurring in $\ttbar\PH$, $\PV\PH$,
VBF, and $\cPg\cPg\PH$ channels are modelled by version 2.0 [54, 55, 56, 57]
at NLO in QCD. These samples require the SM Higgs boson to decay to four
neutrinos ($\PH\to\PZ\PZ^{*}\to 4\PGn$) resulting in
${\mathcal{B}(\PH\to\text{inv})}$ $=1$. The cross sections are appropriately
normalised to the corresponding SM predictions computed at NLO ($\ttbar\PH$),
NNLO ($\PV\PH$, VBF), and next-to-NNLO ($\cPg\cPg\PH$) accuracy in QCD, and to
NLO accuracy in electroweak (EW) corrections [58]. Background $\PZ\PH$
processes with the Higgs boson decaying to $\PQb\overline{\PQb}$ and the
associated boson decaying to $\PQb\overline{\PQb}$, ${\Pell\Pell}$, and (where
q represents a light or charm quark) are generated at NLO using with the FxFx
[59] matching scheme for 2016 samples, and with version 2.0 [60] for 2017 and
2018 samples.

The $\PV+\text{jets}$ processes are generated at LO in QCD using with up to
four partons in the final state using the MLM [61] matching scheme between
hard scatters and parton showers. These processes are generated in bins of
hadronic transverse energy, , which is the magnitude of the sum of all jets
reconstructed at generator level. The LO simulation of $\PV+\text{jets}$
processes is corrected to account for missing higher-order diagrams with
K-factors derived from -generated NLO QCD $\PV+\text{jets}$ processes with up
to two partons. These K-factors are extracted as a function of boson and the
of the leading jet in the event.

The $\PGg+\text{jets}$ processes are generated at NLO in QCD with , using a
binning based on the of the photon. The binning scheme for this sample is
defined at the ME level to increase the statistical precision in the phase
space regions probed by this analysis.

Processes including t-channel single quarks, and pairs with up to two
additional partons in ME computation are generated at NLO with version 2.0
[62, 63]. Single quarks produced in the s-channel are modelled using , and
also in the tW channel using version 1.0 [64]. The quark spectrum in processes
is corrected to match the spectrum obtained from the NNLO QCD $+$ NLO EW
simulation, following Ref. [65]. Rare ${\ttbar\text{X}}$ $+$ jets backgrounds
cover processes where is produced in association with a boson X (, , or a
visibly decaying ), generated at NLO. The ${\ttbar\PGg}$ $+$ jets,
${\ttbar\PW}$ $+$ jets, and ${\ttbar\PZ}$ $+$ jets samples are generated using
, with subsequent decays generated using MadSpin [66] to account for spin
correlations in the former two cases. The $\ttbar\PH$ $+$ jets sample, where
the decays to visible states, is generated using .

Diboson ${\PZ\PZ}$ and ${\PW\PZ}$ production processes are generated at LO
using , while the ${\PW\PW}$ process is simulated at NLO in QCD using the
version 2.0 [67]. The QCD multijet samples are generated at LO using in
exclusive ranges of in order to increase the statistical precision in the
phase-space probed by this analysis.

##  0.4 Event reconstruction

During LHC runs, each beam crossing results in several ${\Pp\Pp}$ collisions
in the detector. Additional ${\Pp\Pp}$ interactions within the same or nearby
bunch crossing, known as pileup, make PF object reconstruction more
challenging. The reduction of the effect of pileup relies on mitigation
techniques [68] that filter energy deposits associated with pileup vertices
and remove objects not associated with the primary interaction vertex (PV).
The PV is the vertex associated with the hardest scattering in the event,
according to tracking information, as described in Ref. [69]. All simulated
samples from Section 0.3 are reweighted to match the pileup distribution
observed in data. In the SR, the final state is required to contain jets, a
sizeable hadronic recoil, and no isolated leptons or photons. Candidate
leptons and photons are selected with ${\pt>10\GeV}$ and pseudorapidity
${\abs{\eta}<2.4}$ for muons [38], ${\pt>10\GeV}$ and either
${\abs{\eta}<1.44}$ or ${1.57<\abs{\eta}<2.5}$ for electrons [37],
${\pt>20\GeV}$ and ${\abs{\eta}<2.3}$ for hadronically decaying tau leptons
[41], and ${\pt>15\GeV}$ and either ${\abs{\eta}<1.44}$ or
${1.57<\abs{\eta}<2.5}$ for photons [37]. These selection criteria are
optimised to reject reducible background contributions, and other criteria,
which depend on the isolation of a given object (lepton or photon) relative to
other PF particles within a cone of small (tight) or large (loose) radius, are
used to select leptons and photons. Loose identification and isolation
criteria are used to veto candidate events in the SR that contain leptons or
photons. The veto efficiencies are $>$99, $\simeq$95, and $\simeq$90% for
loose muons, electrons, and photons, respectively. The SR background
contributions are estimated using $\PGm+\text{jets}$, $\Pe+\text{jets}$,
$\PGm\PGm+\text{jets}$, $\Pe\Pe+\text{jets}$, and $\PGg+\text{jets}$. Tight
and loose identification and isolation criteria are used to select and count
muons, electrons, and photons in the CRs, enhancing the purity at little
expense to the efficiency. These achieve typical selection efficiencies of
$\simeq$95, 70, and 70 ($\simeq$98, 95 and 90)%, for tight (loose) muons,
electrons, and photons, respectively.

Jets are reconstructed by clustering all PF candidates originating from the PV
with the anti-jet clustering algorithm [70, 71], using a distance parameter
${R=0.4}$ (AK4). Jet momentum is determined as the vectorial sum of all
particle momenta in the jet, and is found from simulation to be, on average,
within 5 to 10% of the true momentum over the whole spectrum and detector
acceptance. Charged-hadron subtraction [42] is then applied to remove charged
particles from pileup vertices [72]. To ensure the measured jet energy matches
that of the particle level jets, jet energy corrections (JEC) derived from
simulation as functions of and $\eta$ are applied. Further corrections are
applied due to residual discrepancies in the jet energy scale (JES) between
data and simulated samples [42]. Additionally, each jet must pass selection
criteria to remove jets adversely affected by instrumentation or
reconstruction failure. The jet energy resolution (JER) in simulated samples
is smeared to match the measured resolution, which is typically 15–20% at 30,
10% at 100, and 5% at 1 [42]. The AK4 jets are required to have ${\pt>30\GeV}$
and ${\abs{\eta}<5.0}$, and those with loose leptons and photons located
within a cone of ${\Delta R<0.4}$ of the jet direction are removed.

The AK4 jets that originate from the hadronisation of a bottom quark (-tagged
jets) are identified using the DeepCSV algorithm, which correctly identifies
jets with ${\pt>20\GeV}$ with a probability of 80% and has a charm or light
jet mistag probability of 10% [73]. Simulated events containing jets are
corrected to be in agreement with the data by deriving corrections from data
control samples that contain jets.

Pileup effects are mitigated at the reconstructed particle level using the
pileup per particle identification algorithm (PUPPI) [74, 75] by defining a
local shape variable that can discriminate between particles originating from
the PV and from pileup. Charged particles originating from pileup are
discarded. For neutral particles, a local shape variable is computed based on
the information from charged particles in their vicinity that originate from
the PV within the tracker acceptance, and information from both charged and
neutral particles outside this acceptance. The momenta of neutral particles
are then rescaled based on the probability that they originated from the PV as
deduced from the local shape variable [74].

When a high quark or boson decays hadronically, a large set of collimated
particles cross the detector. These can be clustered within a single jet of
radius $R=0.8$ (AK8) using the anti-algorithm. In order to reduce pileup
effects, PUPPI PF candidates are used to seed the AK8 jet finder. The main
feature that distinguishes hadronically decaying quarks or bosons from the
quark or gluon fragmentation is the jet mass. To improve the resolution, the
modified mass-drop tagger algorithm [76, 77, 78] (also known as the soft-drop
algorithm, SD) with the angular exponent ${\beta=0}$, soft cutoff threshold
${z_{\text{cut}}<0.1}$, and characteristic radius ${R_{0}=0.8}$ [79] is
applied to each AK8 jet to remove soft and wide-angle radiation. In addition,
a deep neural network (DNN) classifier called the DeepAK8 [80] algorithm is
employed by assigning a set of numerical scores to each reconstructed AK8 jet
corresponding to the probabilities that it originates from particular final
states of V boson decays, for example ${\PZ\to\PQb\PAQb}$, ${\PZ\to\qqbar}$,
${\PW\to\PQc\PQs}$, rather than from QCD multijet processes. For this
analysis, reconstructed AK8 jets originating from quarks (bosons) are selected
by requiring ${\pt>400~{}(200)\GeV}$, SD mass $m_{\text{SD}}$ between 120 and
210 (65 and 120), and a DeepAK8 probability score for quarks (bosons) larger
than between 72.5 and 83.4 (91.8 and 92.5)% depending on the year of data-
taking. The resulting quark (boson) tagging efficiency at the
${\pt>400~{}(200)\GeV}$ threshold limit is estimated from simulation as 28
(25)% with a 1% mistag rate from QCD jets. Simulated events containing AK8
jets are corrected to agree with the data using data-derived correction
factors, and dedicated JEC are also applied [80].

The calculation of energy sums such as the hadronic recoil, , and are based on
AK4 jets, therefore JEC are propagated through the use of the -corrected jets.

##  0.5 Event selection and categorisation

In this analysis the signal is extracted from a fit to the hadronic recoil
distribution of events in the SR. The CRs are used to estimate the
contributions of different SM processes in the SR. Where possible, the CRs
have kinematic requirements identical to the SR, and leptons or a photon are
used in the CR definition, but otherwise ignored in the calculation of event
observables. The $\Pe+\text{jets}$ and $\PGm+\text{jets}$ CRs, enriched in
$\PW+\text{jets}$ and quark background processes, are used to derive
corrections to the irreducible $\Pell_{\text{lost}}$ backgrounds predicted by
simulation. The $\Pe\Pe+\text{jets}$, $\PGm\PGm+\text{jets}$, and, in the case
of the $\PV\PH$ category, $\PGg+\text{jets}$ CR samples are used to derive
corrections to the expected contribution from $\PZ+\text{jets}$ production,
where the boson decays to a pair of neutrinos. A QCD multijet enriched CR
(hadronic sideband) is also used to estimate reducible hadronic backgrounds in
the SR.

###  0.5.1 Trigger requirements

Events of interest are collected via a suite of triggers that are applied to
variables calculated using PF candidates reconstructed at the level of the
HLT. The trigger requirements vary amongst analysis regions and data-taking
periods. Events in the SR, hadronic sideband, and muon CRs are collected using
HLT selection criteria on and the missing , $H_{\mathrm{T}}^{\text{miss}}$,
which is the magnitude of the negative sum of jets reconstructed at the HLT
level with a threshold of 20applied. Muons are not considered in the
calculation of PF and PF $H_{\mathrm{T}}^{\text{miss}}$ to allow the same
trigger to be used in the SR and the muon CRs, with a typical efficiency of
$>$90% for $>250\GeV$. The use of the combined and
$H_{\mathrm{T}}^{\text{miss}}$ triggers in the muon CRs instead of single-muon
triggers corresponds more closely to the selection in the SR and minimises
selection biases. Trigger thresholds increase with time due to the increase in
instantaneous luminosity during Run 2. In 2016, the and
$H_{\mathrm{T}}^{\text{miss}}$ thresholds vary between 90 and 120. In 2017 and
2018, these thresholds are 120. During data-taking in 2017, additional
corrections were applied to account for the effect of ECAL endcap noise at
high $\abs{\eta}$ on PF measurements. Additionally, for 2016 and 2017 data-
taking periods, there was an inefficiency arising from a gradual shift in the
timing of the ECAL trigger inputs in the region ${\abs{\eta}>2.0}$ [35]. This
resulted in events containing an electron or photon (jet) with
${\pt>50~{}(100)\GeV}$ having an efficiency loss of up to 20%, depending on
and $\eta$. Correction factors for this trigger inefficiency are obtained from
2016 and 2017 data and applied to simulation samples as a function of $\eta$.

Events in the $\Pe+\text{jets}$ and $\Pe\Pe+\text{jets}$ CRs from the 2016,
2017, and 2018 data sets are required to pass a tight (loose) single-electron
trigger with thresholds of 27, 35, and 32 (105, 115, and 115), respectively.
The low-threshold single-electron triggers require the electron candidate to
pass a tight isolation condition, while the high-threshold trigger imposes a
looser selection on the isolation to improve the efficiency at high . Photon
events are required to pass a single-photon trigger with a threshold of 175
(200)without any isolation condition for the 2016 (2017 and 2018) data sets.
Simulated electron or photon events are accepted if they pass exactly one of
the above trigger requirements, and the efficiency of this selection is
corrected with data-derived efficiency correction factors.

###  0.5.2 Offline selection

In order to select events with a large amount of jet activity and sizeable
hadronic recoil, a further offline selection is applied to all regions. To
improve the purity of the signal, large missing energy is desirable, therefore
events require , $H_{\mathrm{T}}^{\text{miss}}$, and to be greater than 200.
Furthermore, the largest of an AK4 jet in an event,
$\vec{p}^{~{}\text{j}}_{\mathrm{T,1}}$, is required to be greater than 80.
Agreement between the hadronic recoil and $H_{\mathrm{T}}^{\text{miss}}$ is
ensured by requiring ${H_{\mathrm{T}}^{\text{miss}}/\ptmiss<1.2}$ and
azimuthal separation ${\abs{\Delta\phi(\ptvecmiss,\htvecmiss)}<0.5}$. To
ensure good correspondence between as measured including and excluding charged
lepton PF candidates in both the SR and the hadronic sideband, a selection is
made on $p_{\mathrm{T},\text{track}}^{\text{miss}}$, which is equivalent to
but calculated using only charged PF particles. Requirements of
${p_{\mathrm{T},\text{track}}^{\text{miss}}>60\GeV}$ and azimuthal separation
to the recoil direction
${\abs{\Delta\phi(\text{recoil},\vec{p}_{\mathrm{T},\text{track}}^{\text{miss}})}<1}$
are applied. The kinematic selection for all regions is optimised according to
the Asimov significance between signal (S) and background (B) yields assuming
a background systematic uncertainty $\Delta\text{B}$ of 5% or 10% [81]. The
peaks of the distribution for a given variable corresponds to its selection
threshold.

In order to facilitate the combination of this analysis with the results from
other $\PH\to\text{inv}$ searches, additional selections are introduced to
reduce the potential event overlap. A veto is implemented to ensure
orthogonality with the VBF phase space, through a veto on events with leading
(subleading) AK4 jets with ${\abs{\eta_{1}}~{}(\abs{\eta_{2}})>2.4}$, and an
inversion of the kinematic selection employed by the VBF $\PH\to\text{inv}$
analysis [34]. This removes events containing two AK4 jets with
$\vec{p}^{~{}\text{j}}_{\mathrm{T,1}}>80\GeV$ and the subleading jet ,
$\vec{p}^{~{}\text{j}}_{\mathrm{T,2}}$, to be greater than 40, where the jets
are from opposite detector hemispheres (${\eta_{1}\eta_{2}<0}$), have a large
$m_{\text{jj}}$ ($>$200), small azimuthal separation
(${\Delta\phi_{\text{jj}}<1.5}$), and a large $\eta$ gap
(${\abs{\eta_{m_{\text{jj}}}}>1.0}$). Moreover, orthogonality to leptonic
$\ttbar\PH$ decays is ensured in the single-lepton CRs by requiring the
transverse mass of the combined single-lepton and system, defined as

$m_{\mathrm{T}}^{\Pell}=\sqrt{2p^{\Pell}_{\mathrm{T}}\ptmiss[1-\cos{(\phi(\vec{p}^{\Pell}_{\mathrm{T}})-\phi(\ptvecmiss))}]},$
(1)

to be lower than 110. Orthogonality between leptonic $\ttbar\PH$ decays in the
dilepton CRs is ensured by requiring the invariant mass of the charged lepton
pair, $m_{\Pell\Pell}$, to be lower than 120in these CRs. Selecting on the
invariant masses of lepton pairs also suppresses the $\ttbar\PH$ signal
contamination in the CRs. Overlap between the ggH/boosted $\PV\PH$
$\PH\to\text{inv}$ analysis and the resolved $\PV\PH$ category of this
analysis is rendered negligible by explicitly removing events from the low-
purity boosted $\PV\PH$ category defined in Ref. [33] if they contain exactly
two AK4 jets with an invariant mass, $m_{\text{jj}}$, forming a dijet
candidate with $65<m_{\text{jj}}<120\GeV$. No corresponding selection is
necessary for the resolved $\PV\PH$ category as a result, while there is
negligible change to the sensitivity of the boosted $\PV\PH$ category.

During significant periods of data-taking in 2018, the HCAL portion
corresponding to the region $-1.57<\phi<-0.87$, $-3.0<\eta<-1.39$ was not
functional. Events from 2018 with $-1.8<\phi(\ptvecmiss)<-0.6$ are vetoed if
they contain jets within the affected region, which removes $\approx$65% of
the total data from the affected region. To ensure good correspondence between
data and simulation, the simulation is reweighted to account for the
efficiency loss. A summary of the offline requirements are provided in Table
0.5.2.

Offline selection applied to all categories and regions in this analysis to
improve signal purity and reduce overlap with the phase space of other
$\PH\to\text{inv}$ searches.  Variable Selection Purpose $>200\GeV$ Signal
purity $H_{\mathrm{T}}^{\text{miss}}$ $>200\GeV$
$\vec{p}^{~{}\text{j}}_{\mathrm{T,1}}$ $>80\GeV$
$H_{\mathrm{T}}^{\text{miss}}/\ptmiss$ $<1.2$ Event quality
$\abs{\Delta\phi(\ptvecmiss,\htvecmiss)}$ $<0.5$ $\abs{\eta_{1}}$,
$\abs{\eta_{2}}$ $<2.4$ Analysis orthogonalisation VBF signal Veto (inversion
on signal selection) $m_{T}^{\Pell}$ $<110\GeV$ $m_{\Pell\Pell}$ $<120\GeV$

###  0.5.3 Signal regions

The search focuses on three types of hadronic final states: those with boosted
quarks and/or boosted bosons reconstructed with dedicated merged jet
algorithms; those with at least five AK4 jets and one or more jets and no
boosted quark or boson, targetting the bulk of hadronic $\ttbar\PH$ events;
and those with two resolved jets with the $m_{\text{jj}}$ compatible with that
of a or boson. The latter complements the boosted $\PV\PH$ channel analysed in
Ref. [33].

Events are categorised into boosted and resolved $\ttbar\PH$, and resolved
$\PV\PH$ topologies. The $\ttbar\PH$ category requires that at least five AK4
jets and one jet are present. The boosted $\ttbar\PH$ topology requires that
at least one AK8 jet is reconstructed and either - or -tagged, and is
subcategorised by the AK8 jet and jet multiplicities. Events without such - or
-tagged jets are categorised as belonging to a resolved $\ttbar\PH$ topology,
with further selections on the leading AK4 jet (leading or subleading jet) and
, $\abs{\Delta\phi(\ptvecmiss,\vec{p}^{~{}\text{j}}_{\mathrm{T,1}})}$
($\abs{\Delta\phi(\ptvecmiss,\vec{p}^{~{}\text{b}}_{\mathrm{T,1}})}$ or
$\abs{\Delta\phi(\ptvecmiss,\vec{p}^{~{}\text{b}}_{\mathrm{T,2}})}$) applied
to discriminate between $\ttbar\PH$ and $\ttbar+\text{jets}$ processes.
Finally, the remaining events are allocated to the resolved $\PV\PH$ topology
category if they have exactly two AK4 jets with $m_{\text{jj}}$ between 65 and
120, compatible with a or boson decay. The resolved $\PV\PH$ subcategories are
separated according to the jet multiplicity. Subcategories are also defined
based on $\vec{p}^{~{}\text{j}}_{\mathrm{T,2}}$ to suppress QCD multijet
background. The subcategory definitions are summarised in Table 0.5.3. The
intended outcome of this categorisation is a set of event samples with high
purity for a given production mode, and minimal background contamination or
signal cross-contamination.

Categorisation of the $\ttbar\PH$ and $\PV\PH$ production modes in the
analysis. No additional selections are applied to the boosted $\ttbar\PH$
subcategories.  Category Subcategory $n_{\text{j}}$ $n_{\PQb}$ $n_{\PQt}$
$n_{\PW}$ $\vec{p}^{~{}\text{j}}_{\mathrm{T,2}}$ () Other Boosted $\ttbar\PH$
2Boosted1b $\geq 5$ 1 2 $>80$ 2Boosted2b $\geq 5$ $\geq 2$ 2 1t1b $\geq 5$ 1 1
0 1t2b $\geq 5$ $\geq 2$ 1 0 1W1b $\geq 5$ 1 0 1 1W2b $\geq 5$ $\geq 2$ 0 1
Resolved $\ttbar\PH$ 5j1b 5 1 0 0 $>80$
$\abs{\Delta\phi(\ptvecmiss,\vec{p}^{~{}\text{b}}_{\mathrm{T,1}})}>1.0$, 6j1b
$\geq 6$ 1 0 0
$\abs{\Delta\phi(\ptvecmiss,\vec{p}^{~{}\text{j}}_{\mathrm{T,1}})}>\pi/2$ 5j2b
5 $\geq 2$ 0 0
$\abs{\Delta\phi(\ptvecmiss,\vec{p}^{~{}\text{b}}_{\mathrm{T,1}})}>1.0$, 6j2b
$\geq 6$ $\geq 2$ 0 0
$\abs{\Delta\phi(\ptvecmiss,\vec{p}^{~{}\text{b}}_{\mathrm{T,2}})}>\pi/2$
$\PV\PH$ 2j0b 2 0 0 0 $>30$ $65<m_{\text{jj}}<120\GeV$ 2j1b 2 1 0 0 2j2b 2 2 0
0

A requirement on
$\abs{\Delta\phi_{\text{min}}({\ptvecmiss,\vec{p}_{\mathrm{T,1234}})}}$,
defined as the minimum azimuthal separation between and the momentum direction
of any of the four highest jets, of $>0.5$ is applied to suppress QCD multijet
events where the is aligned with a jet. A parameter
$\tilde{\omega}_{\text{min}}$ is designed to suppress events where missing
energy is the result of a jet mismeasurement, and is especially effective in
categories with no jets. This is particularly relevant in the VH 2j0b
subcategory, where the QCD multijet contribution can be as much as 15% of the
total background after applying the
$\abs{\Delta\phi_{\text{min}}({\ptvecmiss,\vec{p}_{\mathrm{T,1234}})}}$
selection. For the $i$th jet in the event, $\omega_{i}$ is defined as
$\arctan{(H_{\mathrm{T},\text{min}}^{\text{miss}}/p_{\mathrm{T},i})}$, where
$p_{\mathrm{T},i}$ is the of jet $i$ and
$H_{\mathrm{T},\text{min}}^{\text{miss}}$ is the minimum value of
$H_{\mathrm{T}}^{\text{miss}}$ that can be obtained by scaling
$p_{\mathrm{T},i}$ by a constant factor. The value of $\omega_{i}$ minimised
over $i$ is $\tilde{\omega}_{\text{min}}$. A detailed derivation of this
variable is given in Ref. [82]. QCD multijet events in the SR are further
suppressed by requiring $\tilde{\omega}_{\text{min}}>0.3$. Requirements to
suppress QCD events are applied in the SR only for $\ttbar\PH$ categories, and
to both SR and CRs in the $\PV\PH$ categories in order to ensure good
correspondence amongst the regions. The selections applied to
$\tilde{\omega}_{\text{min}}$ and
$\abs{\Delta\phi_{\text{min}}({\ptvecmiss,\vec{p}_{\mathrm{T,1234}})}}$ are
not applied in the CRs used for background estimation of the $\ttbar\PH$
categories, where does not stem from jet mismeasurement. This is to increase
event counts in the CRs, particularly in the boosted $\ttbar\PH$ categories.

The in $\ttbar\PH$ production is closely aligned with the direction of the
Higgs boson typically. In events, the $p_{\mathrm{T}}^{\text{miss}}$ direction
is usually parallel or antiparallel to the direction of the leading jet, as
the quarks are produced back-to-back. Therefore, the angles between the and
the leading or subleading jet or jet directions provide additional features
for background suppression in the resolved $\ttbar\PH$ categories. The angular
variables $\abs{\Delta\phi(\ptvecmiss,\vec{p}^{~{}\text{j}}_{\mathrm{T,1}})}$,
$\abs{\Delta\phi(\ptvecmiss,\vec{p}^{~{}\text{b}}_{\mathrm{T,1}})}$, and
$\abs{\Delta\phi(\ptvecmiss,\vec{p}^{~{}\text{b}}_{\mathrm{T,2}})}$ are the
most sensitive discriminators between $\ttbar\PH$ and . The selection based on
these angular variables has been optimised by maximising the combined expected
sensitivity of the $\ttbar\PH$ analysis and is summarised in Table 0.5.3.

##  0.6 Control regions and background estimation

The analysis makes use of the $\PGm+\text{jets}$ and $\Pe+\text{jets}$ CRs to
estimate irreducible background contributions due to $\Pell_{\text{lost}}$,
which are mainly from $\ttbar+\text{jets}$, single quark, and
$\PW+\text{jets}$ events. The irreducible background contributions from
invisible boson decays, which include ${\PZ\PZ}$ and ${\ttbar\PZ}$ and Drell-
Yan (DY) contributions, are estimated from the $\PGm\PGm+\text{jets}$,
$\Pe\Pe+\text{jets}$, and $\PGg+\text{jets}$ CRs. Reducible hadronic
backgrounds in the SR such as QCD multijet contributions are estimated using a
transfer factor method applied to a QCD enriched sideband CR.

###  0.6.1 Irreducible background estimation

The $\PGm+\text{jets}$ ($\Pe+\text{jets}$) CR is defined by requiring exactly
one tightly-isolated muon (electron) with $\pt>20(40)\GeV$. Both CRs require
${50<m_{\mathrm{T}}^{\Pell}<110\GeV}$. The single-lepton CRs are used to
constrain the $\Pell_{\text{lost}}$ background, which is the main source of
background in the $\ttbar\PH$ and $\PV\PH$ 2j2categories. In the $\ttbar\PH$
category, the $\Pell_{\text{lost}}$ contribution arises mainly from , single
quark, and ${\ttbar\PV}$ processes, while in the $\PV\PH$ category it is from
$+$ jet events.

In the $\PGm\PGm+\text{jets}$ ($\Pe\Pe+\text{jets}$) CR, one tightly-isolated
muon (electron) with ${\pt>20~{}(40)\GeV}$, and one loose muon (electron) with
the opposite charge and $\pt>10(10)\GeV$ are required with invariant mass,
$m_{\Pgm\Pgm}$ ($m_{\Pe\Pe}$), compatible with a boson. For the $\ttbar\PH$
($\PV\PH$) category, the invariant mass is required to be between 75 and 105
(60 and 120). The processes $\PZ\to\PGn\PAGn$ and ${\PZ\to\Pell\Pell}$ are
kinematically nearly identical, largely due to lepton universality, hence the
dilepton regions can be used to constrain the $\PZ\to\text{inv}$ background
and minimise theoretical uncertainties. This is important for the
$\PZ\to\text{inv}$ background, which dominates the $\PV\PH$ category and
contributes to the $\ttbar\PH$ category especially at high hadronic recoil. In
the $\ttbar\PH$ category, events are selected for which
${\Delta\phi(\text{recoil},\vec{p}_{\mathrm{T},\text{track}}^{\text{miss}})>\pi/2}$,
which reduces the $\ttbar+\text{jets}$ background and favours DY production in
the dilepton CRs.

The $\PGg+\text{jets}$ CR is used for background estimation in the $\PV\PH$
category only, and requires exactly one loose photon with ${\pt>230\GeV}$.
This region is used to constrain the $\PZ\to\text{inv}$ background as the
event kinematics and topologies are similar for $\PZ+\text{jets}$ and
$\PGg+\text{jets}$ events, improving the sensitivity to the $\PV\PH$ signal
primarily at high hadronic recoil compared to the dilepton CRs because of the
larger number of events.

Photons can usually be discriminated from other sources of ECAL deposits using
the properties of the deposits themselves, such as isolation in ECAL and HCAL,
or the shape of the electromagnetic showers. However, occasionally other
particles will be incorrectly identified as photons, for example where a jet
is misidentified as a photon in QCD multijet events. In order to estimate the
contribution from misidentified photons in the $\PGg+\text{jets}$ CR, a purity
measurement is performed. The purity is defined as the fraction of
reconstructed photon candidates that correspond to genuine isolated photons
originating from the PV in the event. The photon purity is measured in data
based on the lateral width $\sigma_{\eta\eta}$ [83], which parametrises the
shape of the energy deposit associated with the photon in the ECAL. The
characteristic $\sigma_{\eta\eta}$ distribution from genuine photons peaks at
$\sigma_{\eta\eta}<1$, while the distribution due to misidentified photons
possesses a less pronounced peak with a much broader decline for
$\sigma_{\eta\eta}>1$. A template fit to the $\sigma_{\eta\eta}$ distribution
is performed, where for genuine photons simulated $\PGg+\text{jets}$ events
are used to build the signal templates, while for misidentified photons a data
sample enriched in misidentified photon events is obtained by inverting the
isolation requirements in the $\PGg+\text{jets}$ CR. The purity is defined as
the fraction of genuine photons extracted from the fit that pass the
$\sigma_{\eta\eta}$ selection. The photon purity is measured separately in
bins of $p_{\mathrm{T}}^{\PGg}$ and for each data-taking period. The
contamination is the fraction of misidentified photons in the
$\PGg+\text{jets}$ CR, and is estimated at around 4% for
${p_{\mathrm{T}}^{\PGg}>200\GeV}$. The QCD multijet contribution in the
$\PGg+\text{jets}$ CR is then estimated by weighting events in data for each
$p_{\mathrm{T}}^{\PGg}$ bin by the corresponding contamination. A 25%
systematic uncertainty is attributed to the QCD multijet background
normalisation, and is estimated by performing the procedure for different
$\sigma_{\eta\eta}$ binning in the template fit, which accounts for any
mismodelling of the simulated $\sigma_{\eta\eta}$ distribution. The
statistical uncertainty in the photon purity estimate in each
$p_{\mathrm{T}}^{\PGg}$ bin is found to be much smaller than the systematic
one. The full requirements for the analysis CRs are shown in Table 0.6.1.

Summary of all CR requirements, excluding selections suppressing the QCD
multijet background, and excluding the requirement of
${\Delta\phi(\text{recoil},\vec{p}_{\mathrm{T},\text{track}}^{\text{miss}})>\pi/2}$
applied to the $\ttbar\PH$ category in the dilepton CRs. No mass requirements
are imposed in the $\PGg+\text{jets}$.  Control region Category
$n_{\text{object}}$ reqs. Mass reqs. () reqs. () $\PGm+\text{jets}$
$\ttbar\PH$ $n_{\Pgm}=1$ $50<\mT^{\Pgm}<110$ $p^{~{}\PGm}_{\mathrm{T,1}}>20$
$\PV\PH$ $\Pe+\text{jets}$ $\ttbar\PH$ $n_{\Pe}=1$ $50<\mT^{\Pe}<110$
$p^{~{}\Pe}_{\mathrm{T,1}}>40$ $\PV\PH$ $\PGm\PGm+\text{jets}$ $\ttbar\PH$
$n_{\PGm}=2$ $75<m_{\Pgm\Pgm}<105$ $p^{~{}\PGm}_{\mathrm{T,1}}>20$,
$p^{~{}\PGm}_{\mathrm{T,2}}>10$ $\PV\PH$ $60<m_{\Pgm\Pgm}<120$
$\Pe\Pe+\text{jets}$ $\ttbar\PH$ $n_{\Pe}=2$ $75<m_{\Pe\Pe}<105$
$p^{\mathrm{~{}\Pe}}_{\mathrm{T,1}}>40$,
$p^{\mathrm{~{}\Pe}}_{\mathrm{T,2}}>10$ $\PV\PH$ $60<m_{\Pe\Pe}<120$
$\PGg+\text{jets}$ VH $n_{\PGg}=1$ $p_{\mathrm{T}}^{\PGg}>230$

###  0.6.2 Residual backgrounds from QCD multijet production

The event selection aims to reduce background contributions from QCD multijet
production as much as possible by requiring
${\abs{\Delta\phi_{\text{min}}({\ptvecmiss,\vec{p}_{\mathrm{T,1234}})}}>0.5}$
and ${\tilde{\omega}_{\text{min}}>0.3}$, although a QCD multijet background
enriched sideband is used to estimate any remaining background contribution
with the help of a transfer factor between sideband and SR, which is derived
from simulation. The sideband is defined with an identical selection to that
of the SR, but with an inversion on the requirements on
$\abs{\Delta\phi_{\text{min}}({\ptvecmiss,\vec{p}_{\mathrm{T,1234}})}}$ and
$\tilde{\omega}_{\text{min}}$, such that
$\abs{\Delta\phi_{\text{min}}({\ptvecmiss,\vec{p}_{\mathrm{T,1234}})}}<0.5$
and more stringently $\tilde{\omega}_{\text{min}}<0.2$. The criteria for
$\tilde{\omega}_{\text{min}}$ is determined by optimising the sideband to be
as QCD multijet-enriched as possible while ensuring the SR has negligible QCD
multijet background. For the $\PV\PH$ category, the $m_{\text{jj}}$
requirement is also inverted in order to have the sideband sufficiently
populated.

The SRs in both the $\ttbar\PH$ and $\PV\PH$ categories suffer from limited
simulated QCD multijet event counts, so it is not possible to reliably define
a transfer factor for each SR bin in individual subcategories. Within the
statistical precision of the simulated QCD multijet samples, the shape of the
hadronic recoil and relative population of the $\ttbar\PH$ subcategories are
observed not to depend on $\tilde{\omega}_{\text{min}}$ and
$\abs{\Delta\phi_{\text{min}}({\ptvecmiss,\vec{p}_{\mathrm{T,1234}})}}$.
Therefore, the expected QCD sideband yields are integrated over all
$\ttbar\PH$ subcategories and hadronic recoil intervals, and over hadronic
recoil intervals for each $\PV\PH$ category, in the sideband and SR, to
construct the transfer factors. The resulting hadronic recoil distributions
are used to predict the relative QCD multijet background in each subcategory
and hadronic recoil interval.

The estimated QCD multijet background yield in the $\ttbar\PH$ SR for
subcategory $i$ and hadronic recoil interval $j$, $N^{\text{QCD,
SR}_{\ttbar\PH}}_{i,j}$, is given by

$N^{\text{QCD, SR}_{\ttbar\PH}}_{i,j}=\sum_{p}\sum_{q}(N^{\text{data,
CR}_{\ttbar\PH}}_{p,q}-N^{\text{EW,
CR}_{\ttbar\PH}}_{p,q})\text{TF}^{\ttbar\PH}_{\text{QCD}}f^{\ttbar\PH}_{c_{i}}f^{\ttbar\PH}_{m_{j}},$
(2)

where EW refers to processes that are not QCD multijet, summation indices $p$
and $q$ are the subcategory and hadronic recoil bins, respectively,
$\text{TF}^{\ttbar\PH}_{\text{QCD}}$ is the QCD multijet simulation transfer
factor defined as the ratio between the expected QCD multijet background
contribution in the SR and the sideband, and $f^{\ttbar\PH}_{c_{i}}$ and
$f^{\ttbar\PH}_{m_{j}}$ are the fractions of simulated QCD multijet events in
each subcategory and hadronic recoil bin, respectively.

In the $\PV\PH$ category, the sideband regions are defined for each
subcategory, as the number of simulated QCD multijet events is sufficient to
derive the hadronic recoil fractions $f_{m_{j}}$ separately for each
subcategory. The method is otherwise analogous to that of $\ttbar\PH$, given
by Eq. 2).

The results of the QCD prediction aggregated over data sets from the 2016–2018
period are found to be small in comparison to background contributions from
$\Pell_{\text{lost}}$ and $\PZ\to\text{inv}$ processes. In addition to the
statistical uncertainties, a 100% systematic uncertainty is assigned to the
predicted background yields from QCD multijet production. The actual
uncertainty in the QCD prediction is measured at around 50%, derived by
calculating the QCD contribution in the entire $\ttbar\PH$ category for a
signal depleted validation region analogous to the SR but requiring
$0.2<\tilde{\omega}_{\text{min}}<0.3$ and
${\abs{\Delta\phi_{\text{min}}({\ptvecmiss,\vec{p}_{\mathrm{T,1234}})}}}$, and
comparing the estimate to data. It is inflated to 100% to be more conservative
when handling the individual $\ttbar\PH$ subcategories that are limited by
event counts at larger hadronic recoil, which was found to have negligible
impact on the final fit.

##  0.7 Statistical interpretation

A maximum likelihood fit method is used to obtain an upper limit on
${\mathcal{B}(\PH\to\text{inv})}$. The fit is performed simultaneously across
each year, region, category, and hadronic recoil interval, with systematic
uncertainties acting as nuisance parameters in the fit correlated to varying
degrees across year and category.

###  0.7.1 Likelihood model

The limits on ${\mathcal{B}(\PH\to\text{inv})}$ are extracted via a
simultaneous binned maximum likelihood fit to the hadronic recoil
distributions obtained in the SR and CRs. The likelihood can be written as

$\displaystyle\mathcal{L}=\mathcal{L}_{\text{SR}}~{}\mathcal{L}_{\PGm}~{}\mathcal{L}_{\Pe}~{}\mathcal{L}_{\PGm\PGm}~{}\mathcal{L}_{\Pe\Pe}~{}\mathcal{L}_{\PGg},$
(3)

where $\mathcal{L}_{\text{SR}}$ is the likelihood function for the SR (boosted
$\ttbar\PH$, resolved $\ttbar\PH$, $\PV\PH$), and $\mathcal{L}_{\PGm}$,
$\mathcal{L}_{\Pe}$, $\mathcal{L}_{\PGm\PGm}$, $\mathcal{L}_{\Pe\Pe}$, and
$\mathcal{L}_{\PGg}$ designate the likelihood functions for the
$\PGm+\text{jets}$, $\Pe+\text{jets}$, $\PGm\PGm+\text{jets}$,
$\Pe\Pe+\text{jets}$, and $\PGg+\text{jets}$ CRs, respectively. The likelihood
function for the SR is defined as

$\mathcal{L}_{\text{SR}}=\prod_{\text{cat}=i}^{n_{\text{cat}}}\prod_{\text{recoil}=j(i)}^{n_{\xi(i)}}\text{Poisson}(n_{\text{obs}}^{i,j}\mid
n^{i,j}_{\text{pred}}),$ (4)

with

$\displaystyle n^{i,j}_{\text{pred}}$
$\displaystyle=\hat{\mu}s^{i,j}\rho_{s}^{i,j}$ (5)
$\displaystyle+b^{i,j}_{\Pell_{\text{lost}}}I^{i,j}\rho_{\Pell_{\text{lost}}}^{i,j}$
$\displaystyle+b^{i,j}_{\PZ\to\text{inv}}L^{i,j}\rho_{\PZ\to\text{inv}}^{i,j}$
$\displaystyle+b^{i,j}_{\text{QCD}}\rho^{i,j}_{\text{QCD}},$

where the symbols are defined in Table 0.7.1. The signal strength,
$\hat{\mu}$, is interpreted as the maximum likelihood estimator for
${\mathcal{B}(\PH\to\text{inv})}$, where the signal prediction assumes that
${\mathcal{B}(\PH\to\text{inv})}=1$. The fit also includes additional free
parameters $I^{i,j}$ and $L^{i,j}$, which depend on category $i$, hadronic
recoil bin $j$, and the number of recoil bins in each category $n_{\xi(i)}$.
The first of these parameters, $I^{i,j}$, simultaneously scales the
normalisation of the $\Pell_{\text{lost}}$ background in the SR and the sum of
the backgrounds in the $\PGm+\text{jets}$ and $\Pe+\text{jets}$ CRs. The
second of these parameters, $L^{i,j}$, simultaneously scales the normalisation
of the $\PZ\to\text{inv}$ background in the SR and the sum of the backgrounds,
except the QCD multijet background, in the $\PGm\PGm+\text{jets}$,
$\Pe\Pe+\text{jets}$, and $\PGg+\text{jets}$ CRs.

Meaning of the symbols used in Eqs. 4 and 5 that define the likelihood
function.  Symbol Meaning $\hat{\mu}$ Signal strength estimator of
${\mathcal{B}(\PH\to\text{inv})}$ $s^{i,j}$ Simulation predicted number of
signal events in bin $i,j$ of the SR $\rho_{s}^{i,j}$ Systematic uncertainties
affecting signal prediction in bin $i,j$ of the SR
$b^{i,j}_{\Pell_{\text{lost}}}$ Simulation predicted number of
$\Pell_{\text{lost}}$ events in bin $i,j$ of the SR $I^{i,j}$ Normalisation
parameter for the $\Pell_{\text{lost}}$ estimation in bin $i,j$
$\rho_{\Pell_{\text{lost}}}^{i,j}$ Systematic uncertainties affecting the
$\Pell_{\text{lost}}$ background in bin $i,j$ of the SR
$b^{i,j}_{\PZ\to\text{inv}}$ Simulation predicted number of $\PZ\to\text{inv}$
events in bin $i,j$ of the SR $L^{i,j}$ Normalisation parameter for the
$\PZ\to\text{inv}$ estimation in bin $i,j$ $\rho_{\PZ\to\text{inv}}^{i,j}$
Systematic uncertainties affecting the $\PZ\to\text{inv}$ background in bin
$i,j$ of the SR $b^{i,j}_{\text{QCD}}$ Predicted number of QCD events in bin
$i,j$ of the SR $\rho^{i,j}_{\text{QCD}}$ Systematic uncertainties of the QCD
component in bin $i,j$ of the SR

The likelihood for each CR is given by

$\mathcal{L}_{\text{CR~{}X}}=\prod_{\text{cat}=i}^{n_{cat}}\prod_{\text{recoil}=j(i)}^{n_{\xi(i)}}\text{Poisson}(n_{\text{obs}}^{i,j}\mid\text{X}^{i,j}_{\text{bkg}}L^{i,j}\rho_{\text{bkg}}^{i,j}),$
(6)

where X${}^{i,j}_{\text{bkg}}$ is the sum of background yields in CR X, and
$\rho_{\text{bkg}}^{i,j}$ refers to the associated systematic uncertainty.

Because of the low event counts in the dilepton CRs, the subcategory yields
are summed into the boosted and resolved $\ttbar\PH$ categories. For the
boosted $\ttbar\PH$ category, the $\PGm\PGm+\text{jets}$ and
$\Pe\Pe+\text{jets}$ CR yields are summed together to form a single
$\Pell\Pell+\text{jets}$ CR. Furthermore, in the boosted and resolved
$\ttbar\PH$ subcategories, $I^{i,j}$ are shared across subcategories,
therefore $i$ takes only two values corresponding to the boosted and resolved
$\ttbar\PH$ classes.

###  0.7.2 Systematic uncertainties

The model on which the maximum likelihood fit is based is inclusive of
experimental and theoretical uncertainties. Theoretical uncertainties related
to the PDF parameters and missing higher order corrections in the QCD and EW
perturbative expansions are estimated by following the procedure outlined in
Ref. [58] for $\ttbar\PH$ and $\PV\PH$ processes, and in Ref. [84] for
$\PV+\text{jets}$ and $\PGg+\text{jets}$ processes.

Data-derived correction factors are applied to simulated events containing , ,
and jets, and therefore the systematic uncertainties due to the limited
precision in these corrections are propagated to the simulated samples. These
are referred to as tagging uncertainties, and also account for the
uncertainties in the tagging efficiencies and misidentification probabilities.

The uncertainty in the combined and $H_{\mathrm{T}}^{\text{miss}}$ trigger
efficiency is computed using the $\PGm+\text{jets}$ and $\PGm\PGm+\text{jets}$
CRs. These are independent of the and $H_{\mathrm{T}}^{\text{miss}}$ data sets
in the SR, ensuring an unbiased measurement of the uncertainty. This
uncertainty is measured at 2%. An additional trigger inefficiency uncertainty
due to the mistiming of ECAL trigger inputs detailed in Section 0.5.1 is
applied to the data-taking years 2016 and 2017.

The uncertainty in the integrated luminosity varies between 1.2–2.5% depending
on the data-taking year, with an overall uncertainty of 1.6% for the 2016–2018
period [44, 45, 46].

The uncertainties considered in the analysis are presented in Table 0.7.2 with
the pre-fit ranges corresponding to the maximum and minimum deviations of the
event yields from their nominal values across each region, year of data-
taking, category, recoil bin, and all SM background processes, when the
respective systematic uncertainty is changed within $\pm$1 standard deviation.
Most uncertainties are assumed uncorrelated from year to year when performing
the fit. Those for which the source of the systematic uncertainty is identical
for each year are treated as correlated. All systematic uncertainties are
correlated across regions.

In addition to the uncertainties in Table 0.7.2, a photon normalisation
uncertainty of 40% is included in the $\PGg+\text{jets}$ CR, to cover
uncertainties in the translation between the $\PGg+\text{jets}$ and
${\PZ\to\Pell\Pell}$ yields. The 100% normalisation uncertainty, also omitted
from Table 0.7.2, is assigned to the predicted QCD multijet contribution to
the SR, as described in Section 0.6.2.

The overall experimental uncertainty is found to be dominated by tagging for
the $\ttbar\PH$ and tagging for the $\PV\PH$ categories in the SR. The lepton
and photon candidate efficiencies for identification, isolation, and
reconstruction, and uncertainties in the JER, JES, and trigger efficiencies
also make significant contributions. The theoretical uncertainty is dominated
by variations in the renormalisation scale, factorisation scale, and PDF for
$\PV+\text{jets}$ processes, although these are particularly sensitive to the
high exclusive jet multiplicity characterising the $\ttbar\PH$ and $\PV\PH$
categories.

The ranges corresponding to the maximum and minimum deviations of the event
yields from their nominal values, provided where applicable across each
region, year of data-taking, category, recoil bin, and all SM background
processes, when the respective systematic uncertainty is changed within $\pm$1
standard deviation.  Systematic uncertainties on background yields (pre-fit)
Signal region $\Pell+\text{jets}$ $\Pell\Pell+\text{jets}$ $\PGg+\text{jets}$
$\ttbar\PH$ cat. $\PV\PH$ cat. $\ttbar\PH$ cat. $\PV\PH$ cat. $\ttbar\PH$ cat.
$\PV\PH$ cat. $\PV\PH$ cat. Theoretical uncertainties Fact. scale
$\PV+\text{jets}$ (QCD) $<$1.0-7.7 % $<$1.0-19 % $<$1.0-2.6 % $<$1.0-11 %
$<$1.0-20 % $<$1.0-22 % 6.0 % Ren. scale $\PV+\text{jets}$ (QCD) $<$1.0-7.2 %
$<$1.0-8.6 % $<$1.0-3.6 % $<$1.0-10 % $<$1.0-14 % 2.0-11 % 12 % PDF
$\PV+\text{jets}$ $<$1.0-9.1 % 2.0-23 % $<$1.0-3.1 % $<$1.0-15 % $<$1.0-23 %
$<$1.0-26 % 8.0 % Ren. & Fact. scale $\ttbar\PH$ (QCD) $<$1.0-1.7 % $<$1.0 %
$<$1.0-1.4 % $<$1.0-1.4 % $<$1.0 % $<$1.0 % Ren. & Fact. scale (QCD) 7.8-15 %
2.5-9.3 % 6.4-17 % $<$1.0-6.3 % $<$1.0-5.8 % $<$1.0-5.8 % NNLO QCD & NLO EW
quark reweighting (inc. PDF) $<$1.0-3.1 % $<$1.0-1.2 % $<$1.0-4.0 % $<$1.0-3.9
% $<$1.0 % $<$1.0 % Ren. & Fact. scale VV (QCD) $<$1.0 % $<$1.0 % $<$1.0 %
$<$1.0 % $<$1.0 % $<$1.0 % $<$1.0 % $\ttbar\PH$ & $\PV\PH$ cat. cross section
(QCD) 5.8-9.2 % $<$1.0-3.8 % $\ttbar\PH$ & $\PV\PH$ cat. cross section (PDF &
$\alpha_{s}$) 3.6 % 1.6-1.8 % Initial-state radiation 2.0 % 3.0-6.0 % 2.0 %
$<$1.0-4.2 % 2.0 % 6.0 % $<$1.0-4.0 % Final-state radiation 5.0 % 3.0-5.0 %
2.0-2.2 % $<$1.0-3.1 % 4.6-5.0 % 5.0 % 2.0-3.0 % Experimental uncertainties
Integrated luminosity 1.2-2.5 % 1.2-2.5 % 1.2-2.5 % 1.2-2.5 % 1.2-2.5 %
1.2-2.5 % 1.2-2.5 % -tagging 3.2-6.5 % 2.1-5.7 % -tagging 7.8-18 % 7.1-18 %
-tagging 8.2-12 % 8.2-22 % 6.5-11 % 2.4-11 % 5.6-8.7 % 1.6-9.6 % 6.6-9.0 %
Electron identification & isolation 3.7-11 % 4.7-9.6 % $<$1.0-15 % $<$1.0-20 %
Electron reconstruction $<$1.0-1.8 % $<$1.0 % 1.0-1.5 % $<$1.0-1.4 % Muon
identification $<$1.0-1.0 % $<$1.0-1.0 % $<$1.0-1.8 % $<$1.0-1.9 % Muon
isolation $<$1.0 % $<$1.0 % $<$1.0 % $<$1.0 % Lepton veto $<$1.0 % $<$1.0 %
Photon identification & isolation 2.4-12 % Photon reconstruction $<$1.0 %
Pileup 1.4-8.8 % $<$1.0-4.5 % $<$1.0-4.8 % $<$1.0-4.7 % $<$1.0-2.1 %
$<$1.0-7.9 % $<$1.0-3.3 % Trigger inefficiency $<$1.0-12 % $<$1.0-1.4 %
$<$1.0-3.4 % $<$1.0-2.4 % $<$1.0-1.6 % $<$1.0-1.5 % $<$1.0-0.3 % Trigger 2.0 %
2.0 % 2.0 % 2.0 % 2.0 % 2.0 % 2.0 % Tau lepton veto $<$1.0 % $<$1.0 %
$<$1.0-1.0 % $<$1.0-2.4 % $<$1.0 % $<$1.0 % $<$1.0 % JER 2.4-3.6 % $<$1.0-1.1
% 1.7-3.0 % $<$1.0-1.5 % $<$1.0-3.5 % $<$1.0-1.4 % $<$1.0-2.9 % JES $<$1.0-6.3
% $<$1.0-2.9 % $<$1.0-5.0 % $<$1.0-2.2 % $<$1.0-6.7 % $<$1.0-2.8 % $<$1.0-3.8
%

##  0.8 Results

The hadronic recoil distributions across all $\ttbar\PH$ and $\PV\PH$
subcategories are shown in Figs. 2 to 5. The predicted background yield from
the fit to the CRs only is shown with the result of a fit including the data
in the SR. Figure 2 (3) shows the $\PGm+\text{jets}$ ($\Pe+\text{jets}$) CR
yields for the $\ttbar\PH$ and $\PV\PH$ categories, respectively, aggregated
over 2016–2018. In these CRs, $\Pell_{\text{lost}}$ background from ,
$\PW\to\Pell\PGn$, and single quark production dominates, with smaller
contributions from multiboson and processes. The $\PGm\PGm+\text{jets}$,
$\Pe\Pe+\text{jets}$, $\Pell\Pell+\text{jets}$ (only for $\ttbar\PH$), and
$\PGg+\text{jets}$ (only for $\PV\PH$) CR distributions used for the
prediction of backgrounds stemming from $\PZ\to\text{inv}$ decays are shown in
Fig. 4 for 2016–2018. In addition, the total SM background prediction in the
SR, consisting of $\Pell_{\text{lost}}$, $\PZ\to\text{inv}$, and QCD
backgrounds, is shown for the $\ttbar\PH$ and $\PV\PH$ category in Fig. 5. The
SR distributions contain all the Higgs boson production modes in the fitted
${\mathcal{B}(\PH\to\text{inv})}$ signal, including the $\cPg\cPg\PH$ and VBF
contamination in the $\ttbar\PH$ and $\PV\PH$ categories, with the prevalence
of the $\cPg\cPg\PH$ process due to its high production cross section. The
post-fit event yields for each subcategory and recoil bin in the SR are
tabulated in Table 0.8. For these results, a fit assuming
${\mathcal{B}(\PH\to\text{inv})}$ $=0$ such that only SM background
contributions are considered (B-only) is performed simultaneously using only
the CRs (CR only), which are independent of the SR, or across both SR and CRs
(CR$+$SR). A fit across all regions, including signal and background
contributions (S$+$B fit), is also performed, in which the signal contribution
is weighted by the best fit signal strength,
${\mathcal{B}(\PH\to\text{inv})}$. In all cases, uncertainties are inclusive
of statistical and systematic contributions.

Total post-fit yields in the SRs in each recoil bin and analysis category
obtained by summing the contributions from the individual data-taking periods.
B-only fits are performed for either CR$+$SR or CR only cases. The extracted
signal yields from an S$+$B fit are also reported, where the signal strength
is weighted by ${{\mathcal{B}(\PH\to\text{inv})}=0.10}$.
$\Pell_{\text{lost}}$ $\PZ\to\text{inv}$ QCD Total background Signal CR only
CR only CR only CR only CR + SR Data ${\mathcal{B}(\PH\to\text{inv})}=0.10$
Subcategory Hadronic recoil B-only fit B-only fit B-only fit B-only fit B-only
fit S$+$B fit $\ttbar\PH$ 1t1b [200, 300) 251.1 $\pm$ 9.5 35.2 $\pm$ 4.1 23.1
$\pm$ 16.8 309.4 $\pm$ 19.8 295.5 $\pm$ 11.6 288.0 $\pm$ 17.0 1.0 $\pm$ 0.8
[300, 400) 235.2 $\pm$ 9.5 35.7 $\pm$ 5.0 5.2 $\pm$ 4.2 276.1 $\pm$ 11.5 268.1
$\pm$ 9.1 257.0 $\pm$ 16.0 1.3 $\pm$ 1.0 [400, 500) 97.5 $\pm$ 5.3 27.6 $\pm$
4.9 0.9 $\pm$ 0.6 126.1 $\pm$ 7.2 135.5 $\pm$ 6.7 145.0 $\pm$ 12.0 1.0 $\pm$
0.8 [500, $\infty$) 37.5 $\pm$ 2.9 26.1 $\pm$ 4.9 0.3 $\pm$ 0.3 63.9 $\pm$ 5.7
70.1 $\pm$ 5.1 66.0 $\pm$ 8.1 0.9 $\pm$ 0.7 $\ttbar\PH$ 1t2b [200, 300) 312.5
$\pm$ 12.0 19.0 $\pm$ 2.2 10.9 $\pm$ 8.6 342.4 $\pm$ 14.9 328.1 $\pm$ 10.5
298.0 $\pm$ 17.3 1.4 $\pm$ 1.2 [300, 400) 265.9 $\pm$ 10.7 20.2 $\pm$ 2.7 2.5
$\pm$ 1.7 288.6 $\pm$ 11.2 287.1 $\pm$ 9.3 299.0 $\pm$ 17.3 1.6 $\pm$ 1.3
[400, 500) 93.6 $\pm$ 5.1 15.4 $\pm$ 2.6 0.4 $\pm$ 0.3 109.5 $\pm$ 5.7 116.5
$\pm$ 5.2 136.0 $\pm$ 11.7 1.2 $\pm$ 0.9 [500, $\infty$) 35.4 $\pm$ 2.9 13.8
$\pm$ 2.5 0.2 $\pm$ $<$0.1 49.4 $\pm$ 3.9 52.8 $\pm$ 3.5 53.0 $\pm$ 7.3 1.0
$\pm$ 0.8 $\ttbar\PH$ 1W1b [200, 300) 1704.6 $\pm$ 49.9 190.7 $\pm$ 21.2 18.8
$\pm$ 16.8 1914.1 $\pm$ 56.8 1855.7 $\pm$ 41.2 1819.0 $\pm$ 42.6 5.7 $\pm$ 4.0
[300, 400) 395.6 $\pm$ 15.1 90.2 $\pm$ 12.7 4.3 $\pm$ 2.9 490.0 $\pm$ 19.9
485.0 $\pm$ 16.2 486.0 $\pm$ 22.0 2.9 $\pm$ 1.9 [400, 500) 56.2 $\pm$ 3.9 35.8
$\pm$ 6.5 0.8 $\pm$ 0.5 92.7 $\pm$ 7.7 103.7 $\pm$ 7.1 111.0 $\pm$ 10.5 0.9
$\pm$ 0.6 [500, $\infty$) 9.9 $\pm$ 1.3 13.9 $\pm$ 2.9 0.3 $\pm$ $<$0.1 24.1
$\pm$ 3.2 29.5 $\pm$ 3.0 37.0 $\pm$ 6.1 0.4 $\pm$ 0.3 $\ttbar\PH$ 1W2b [200,
300) 1295.8 $\pm$ 40.7 53.1 $\pm$ 5.7 5.6 $\pm$ 3.8 1354.5 $\pm$ 41.3 1311.6
$\pm$ 29.4 1276.0 $\pm$ 35.7 3.9 $\pm$ 3.2 [300, 400) 266.2 $\pm$ 11.8 27.2
$\pm$ 3.8 1.3 $\pm$ 0.9 294.7 $\pm$ 12.4 291.3 $\pm$ 9.9 298.0 $\pm$ 17.3 1.9
$\pm$ 1.6 [400, 500) 38.3 $\pm$ 3.3 8.1 $\pm$ 1.5 0.2 $\pm$ $<$0.1 46.6 $\pm$
3.7 47.6 $\pm$ 3.1 47.0 $\pm$ 6.9 0.6 $\pm$ 0.4 [500, $\infty$) 6.0 $\pm$ 1.0
3.7 $\pm$ 0.7 0.1 $\pm$ $<$0.1 9.9 $\pm$ 1.2 11.3 $\pm$ 1.1 17.0 $\pm$ 4.1 0.2
$\pm$ $<$0.1 $\ttbar\PH$ 2Boosted1b [200, 300) 20.2 $\pm$ 3.6 3.8 $\pm$ 0.4
0.3 $\pm$ 0.3 24.3 $\pm$ 3.6 20.4 $\pm$ 2.6 14.0 $\pm$ 3.7 0.5 $\pm$ 0.3 [300,
$\infty$) 6.3 $\pm$ 1.4 6.1 $\pm$ 0.9 0.1 $\pm$ $<$0.1 12.5 $\pm$ 1.7 12.9
$\pm$ 1.6 15.0 $\pm$ 3.9 0.5 $\pm$ 0.4 $\ttbar\PH$ 2Boosted2b [200, 300) 15.8
$\pm$ 2.9 3.9 $\pm$ 0.9 0.3 $\pm$ $<$0.1 20.0 $\pm$ 3.1 18.0 $\pm$ 2.4 15.0
$\pm$ 3.9 0.4 $\pm$ 0.3 [300, $\infty$) 5.4 $\pm$ 1.3 3.8 $\pm$ 0.5 0.1 $\pm$
$<$0.1 9.3 $\pm$ 1.4 8.6 $\pm$ 1.1 6.0 $\pm$ 2.4 0.5 $\pm$ 0.4 $\ttbar\PH$
5j1b [200, 300) 5279.7 $\pm$ 114.4 1703.7 $\pm$ 82.8 99.1 $\pm$ 78.5 7082.4
$\pm$ 161.6 7122.6 $\pm$ 127.6 7207.0 $\pm$ 84.9 14.4 $\pm$ 7.7 [300, 400)
1135.0 $\pm$ 31.8 836.4 $\pm$ 50.0 22.5 $\pm$ 17.3 1994.0 $\pm$ 61.7 1960.9
$\pm$ 43.2 1907.0 $\pm$ 43.7 7.4 $\pm$ 3.8 [400, 500) 182.2 $\pm$ 9.0 267.5
$\pm$ 24.9 4.0 $\pm$ 2.8 453.6 $\pm$ 26.6 438.8 $\pm$ 16.2 427.0 $\pm$ 20.7
2.7 $\pm$ 1.4 [500, $\infty$) 54.2 $\pm$ 3.7 146.0 $\pm$ 20.3 1.5 $\pm$ 1.0
201.7 $\pm$ 20.6 226.2 $\pm$ 11.5 221.0 $\pm$ 14.9 1.5 $\pm$ 0.8 $\ttbar\PH$
5j2b [200, 300) 1317.8 $\pm$ 47.3 350.0 $\pm$ 16.6 11.8 $\pm$ 8.5 1679.6 $\pm$
50.9 1635.4 $\pm$ 33.9 1602.0 $\pm$ 40.0 6.3 $\pm$ 4.2 [300, 400) 188.7 $\pm$
9.2 174.1 $\pm$ 10.4 2.7 $\pm$ 2.0 365.5 $\pm$ 14.1 363.3 $\pm$ 10.7 367.0
$\pm$ 19.2 2.9 $\pm$ 1.8 [400, 500) 33.6 $\pm$ 3.5 53.8 $\pm$ 5.1 0.5 $\pm$
0.3 87.9 $\pm$ 6.2 86.3 $\pm$ 4.5 91.0 $\pm$ 9.5 0.9 $\pm$ 0.5 [500, $\infty$)
8.2 $\pm$ 1.4 24.6 $\pm$ 3.5 0.2 $\pm$ $<$0.1 33.0 $\pm$ 3.8 36.8 $\pm$ 2.5
36.0 $\pm$ 6.0 0.5 $\pm$ 0.3 $\ttbar\PH$ 6j1b [200, 300) 3851.5 $\pm$ 87.9
805.5 $\pm$ 38.8 85.9 $\pm$ 66.3 4742.9 $\pm$ 116.7 4672.6 $\pm$ 87.1 4632.0
$\pm$ 68.1 12.3 $\pm$ 8.1 [300, 400) 876.0 $\pm$ 27.5 438.8 $\pm$ 26.1 19.5
$\pm$ 13.4 1334.2 $\pm$ 40.2 1332.5 $\pm$ 30.4 1371.0 $\pm$ 37.0 6.7 $\pm$ 4.0
[400, 500) 179.6 $\pm$ 8.5 162.8 $\pm$ 15.4 3.4 $\pm$ 2.5 345.9 $\pm$ 17.8
330.9 $\pm$ 11.4 312.0 $\pm$ 17.7 2.4 $\pm$ 1.4 [500, $\infty$) 61.0 $\pm$ 4.0
98.2 $\pm$ 13.6 1.3 $\pm$ 1.0 160.5 $\pm$ 14.3 179.1 $\pm$ 8.4 197.0 $\pm$
14.0 1.6 $\pm$ 0.8 $\ttbar\PH$ 6j2b [200, 300) 1214.0 $\pm$ 38.7 237.2 $\pm$
11.4 15.6 $\pm$ 12.0 1466.8 $\pm$ 42.1 1433.1 $\pm$ 29.9 1404.0 $\pm$ 37.5 7.8
$\pm$ 6.1 [300, 400) 237.9 $\pm$ 12.0 118.8 $\pm$ 7.1 3.6 $\pm$ 2.9 360.3
$\pm$ 14.2 351.9 $\pm$ 10.8 341.0 $\pm$ 18.5 3.8 $\pm$ 2.9 [400, 500) 38.8
$\pm$ 3.8 40.9 $\pm$ 4.0 0.6 $\pm$ 0.4 80.3 $\pm$ 5.6 79.9 $\pm$ 4.3 91.0
$\pm$ 9.5 1.4 $\pm$ 1.0 [500, $\infty$) 12.9 $\pm$ 1.7 21.6 $\pm$ 3.0 0.2
$\pm$ $<$0.1 34.7 $\pm$ 3.5 38.1 $\pm$ 2.4 41.0 $\pm$ 6.4 0.7 $\pm$ 0.4
$\PV\PH$ 2j0b [200, 300) 17753.9 $\pm$ 373.6 29102.3 $\pm$ 655.5 105.8 $\pm$
68.3 46962.1 $\pm$ 757.6 47499.1 $\pm$ 460.7 47559.0 $\pm$ 218.1 185.6 $\pm$
92.5 [300, 400) 2535.2 $\pm$ 69.4 5505.3 $\pm$ 155.0 16.8 $\pm$ 12.0 8057.3
$\pm$ 170.3 8075.7 $\pm$ 106.8 8106.0 $\pm$ 90.0 44.3 $\pm$ 23.0 [400, 500)
278.9 $\pm$ 16.1 684.1 $\pm$ 34.7 2.8 $\pm$ 1.8 965.8 $\pm$ 38.3 944.5 $\pm$
26.7 938.0 $\pm$ 30.6 6.6 $\pm$ 3.4 [500, $\infty$) 19.2 $\pm$ 3.1 76.9 $\pm$
8.1 0.9 $\pm$ 0.5 97.1 $\pm$ 8.7 95.7 $\pm$ 6.6 98.0 $\pm$ 9.9 0.6 $\pm$ 0.3
$\PV\PH$ 2j1b [200, 300) 3020.1 $\pm$ 84.0 2490.4 $\pm$ 114.7 26.2 $\pm$ 24.5
5536.8 $\pm$ 144.3 5808.6 $\pm$ 111.1 5883.0 $\pm$ 76.7 20.3 $\pm$ 10.0 [300,
400) 360.1 $\pm$ 17.3 609.0 $\pm$ 44.1 3.6 $\pm$ 3.0 972.7 $\pm$ 47.5 962.3
$\pm$ 30.1 949.0 $\pm$ 30.8 5.2 $\pm$ 2.8 [400, $\infty$) 36.3 $\pm$ 4.5 66.7
$\pm$ 7.3 0.6 $\pm$ 0.5 103.7 $\pm$ 8.6 111.3 $\pm$ 7.7 120.0 $\pm$ 11.0 0.7
$\pm$ 0.4 $\PV\PH$ 2j2b [200, 300) 209.4 $\pm$ 14.0 422.3 $\pm$ 46.6 2.0 $\pm$
1.2 633.7 $\pm$ 48.6 620.1 $\pm$ 26.8 617.0 $\pm$ 24.8 10.8 $\pm$ 7.9 [300,
$\infty$) 30.7 $\pm$ 3.5 102.6 $\pm$ 15.4 0.2 $\pm$ $<$0.1 133.6 $\pm$ 15.8
131.1 $\pm$ 9.8 128.0 $\pm$ 11.3 3.5 $\pm$ 2.5

The best fit value for $\hat{\mu}$ and corresponding 68 and 95% confidence
intervals are extracted following the procedure outlined in Ref. [85] and Ref.
[86]. The computing of upper limits adheres to the CLs criterion [87, 88]
under the asymptotic approximation [81]. The upper limits on
${\mathcal{B}(\PH\to\text{inv})}$ as extracted from the likelihood fit
presented in Section 0.7.1 are found to be 0.51 (0.53 expected) and 0.68 (0.53
expected) at 95% for the $\ttbar\PH$ and $\PV\PH$ categories, respectively,
with a combined upper limit of 0.47 (0.40 expected). These results are shown
in Fig. 6 together with the observed and expected profile likelihood
distribution. The expected distribution assumes
${\mathcal{B}(\PH\to\text{inv})}=0$. The results are compatible with the
background expectation. The best fit ${\mathcal{B}(\PH\to\text{inv})}$ for the
$\ttbar\PH$ and $\PV\PH$ categories is
$\hat{\mu}=0.10_{-0.18}^{+0.20}(\text{syst.})$
${}_{-0.08}^{+0.08}(\text{stat.})$ $(0.00_{-0.17}^{+0.17}(\text{syst.})$
${}_{-0.08}^{+0.08}(\text{stat.}))$, where the pre-fit normalisation assumes
${\mathcal{B}(\PH\to\text{inv})}$ $=1$. The systematic uncertainty with the
largest impact on the ${\mathcal{B}(\PH\to\text{inv})}$ measurement for the
$\ttbar\PH$ and $\PV\PH$ categories using 2016–2018 data are those associated
with the JES, while the statistical uncertainty contributes significantly to
the overall uncertainty on ${\mathcal{B}(\PH\to\text{inv})}$. The breakdown of
the impacts into uncertainty groups are presented in Table 0.8, together with
the expectation assuming ${\mathcal{B}(\PH\to\text{inv})}$ $=0$. The best fit
estimate for the $\ttbar\PH$ ($\PV\PH$) category is
$\hat{\mu}=-0.04_{-0.27}^{+0.27}$ $(0.00_{-0.25}^{+0.26})$
($\hat{\mu}=0.19_{-0.27}^{+0.26}$ $(0.00_{-0.26}^{+0.27})$).

Figure 2: Distributions of hadronic recoil in the $\ttbar\PH$ (upper plot) and
$\PV\PH$ (lower plot) categories for the $\PGm+\text{jets}$ CR. The black
histogram shows the total background (bkg.) prediction from a CR only, B-only
fit, while the red histogram shows the yields from a CR$+$SR S$+$B fit.

Figure 3: Distributions of hadronic recoil in the $\ttbar\PH$ (upper plot) and
$\PV\PH$ (lower plot) categories for the $\Pe+\text{jets}$ CR. The black
histogram shows the total background (bkg.) prediction from a CR only, B-only
fit, while the red histogram shows the yields from a CR$+$SR S$+$B fit.

Figure 4: Distributions of hadronic recoil in the $\ttbar\PH$ category for the
$\PGm\PGm+\text{jets}$, $\Pe\Pe+\text{jets}$, and $\Pell\Pell+\text{jets}$ CRs
(upper plot), and the $\PV\PH$ category for the $\PGm\PGm+\text{jets}$,
$\Pe\Pe+\text{jets}$, and $\PGg+\text{jets}$ CRs (lower plot). The black
histogram shows the total background (bkg.) prediction from a CR only, B-only
fit, while the red histogram shows the yields from a CR$+$SR S$+$B fit.

Figure 5: Distributions of hadronic recoil in the $\ttbar\PH$ (upper plot) and
$\PV\PH$ (lower plot) categories for the SR, showing the signal contributions
from $\ttbar\PH$, $\PV\PH$, $\cPg\cPg\PH$, and VBF weighted by
${\mathcal{B}(\PH\to\text{inv})}$ $=$0.10. The black histogram shows the total
background (bkg.) prediction from a CR only, B-only fit, while the red
histogram shows the yields from a CR$+$SR S$+$B fit.

Figure 6: Left: Observed and expected limits at 95% for the $\ttbar\PH$ and
$\PV\PH$ categories using 2016–2018 data. Right: The profile likelihood scan
corresponding to observed and expected (where
${\mathcal{B}(\PH\to\text{inv})}=0$) limits in the fit to the $\ttbar\PH$ and
$\PV\PH$ categories.

The observed and expected impacts on ${\mathcal{B}(\PH\to\text{inv})}$ for
different groups of uncertainties, where the expected results are produced
with ${\mathcal{B}(\PH\to\text{inv})}$ $=0$.  Uncertainty group Impact on
${\mathcal{B}(\PH\to\text{inv})}$ Observed Expected Jet calibration
${}^{+0.12}_{-0.11}$ ${}^{+0.11}_{-0.10}$ Lepton veto $\pm 0.05$ $\pm 0.05$
Lepton/photon identification ${}^{+0.07}_{-0.06}$ ${}^{+0.06}_{-0.05}$ Theory
${}^{+0.06}_{-0.05}$ ${}^{+0.04}_{-0.03}$ Integrated luminosity/pileup $\pm
0.03$ $\pm 0.03$ QCD prediction $\pm 0.02$ $\pm 0.02$ Boosted object/jet
tagging $\pm 0.02$ $\pm 0.02$ Triggers $\pm 0.04$ $\pm 0.03$ Stat. uncertainty
of simulation $\pm 0.08$ $\pm 0.08$ Other uncertainties $\pm 0.10$ $\pm 0.10$

##  0.9 Combined $\PH\to\text{inv}$ limits

A variety of production modes of the Higgs boson can be used for searches for
$\PH\to\text{inv}$ decays. A combination of the results of this analysis,
analyses covering the years 2016–2018, and earlier published CMS combination
results using Run 1 (years 2011–2012) and 2015 data [89] at
$\sqrt{s}=7,~{}8,~{}$and 13, detailed in Table 0.9, is performed by means of a
combined likelihood fit in which systematic uncertainties are correlated
across search regions where appropriate. Unless explicitly specified below,
parameters of the individual likelihood functions are treated as independent.

Data sets and their respective integrated luminosities used for each
production mode across Run 1 and Run 2. For some data-taking periods, no
$\PH\to\text{inv}$ search have been performed for the given production mode,
and are not included in the combination.  Analysis tag Production mode
Integrated luminosity (fb-1) 7 8 13(Run 2) VBF-tagged VBF 19.2 [90] 140
[89][34] $\PV\PH$-tagged $\PZ(\Pell\Pell)\PH$ 4.9 [90] 19.7 [90] 140 [89][32]
${\PZ(\bbbar)\PH}$ 18.9 [90] ${\PV(\text{jj})\PH}$ 19.7 [91] 140 [89][this
paper] Boosted $\PV\PH$ 138 [33] $\ttbar\PH$-tagged $\ttbar\PH$ (hadronic) 138
[this paper] $\ttbar\PH$ (leptonic) 138 [29, 30] $\cPg\cPg\PH$-tagged
$\cPg\cPg\PH$ 19.7 [91] 140 [89][33]

For the $\ttbar\PH$ analysis with fully leptonic final states, a
reinterpretation of the supersymmetry searches in the semileptonic and
dileptonic decay channels in Ref. [29, 30] in the context of the $+$ DM model
studied in Ref. [31] has been performed. Another leptonic channel included in
this combination is from the ${\PZ(\Pell\Pell)\PH}$ analysis [32] using
2016–2018 data.

Analyses with hadronic final states partially overlap in their phase space
selection, and this must be accounted for in the statistical combination.
Those affected by overlap are the VBF analysis [34], the analysis targetting
hadronic $\cPg\cPg\PH$ and boosted $\PV\PH$ final states [33], and the
resolved $\PV\PH$ channel described in this paper.

To remove the overlap between the VBF analysis and $\cPg\cPg\PH$/boosted
$\PV\PH$ analysis, events are considered for rejection in the
$\cPg\cPg\PH$/boosted $\PV\PH$ analysis if they have at least two AK4 jets
each with $\abs{\eta}<4.7$. Specifically, an inversion of the VBF kinematic
selection is applied similarly to the $\ttbar\PH$ and resolved $\PV\PH$
analysis as described in Section 0.5.2. These requirements mirror the
selection used to enhance the characteristic VBF phase space in Ref. [34],
with negligible effect on the sensitivity of the $\cPg\cPg\PH$/boosted
$\PV\PH$ analysis to ${\mathcal{B}(\PH\to\text{inv})}$.

The overlap between the $\cPg\cPg\PH$/boosted $\PV\PH$ analysis and the
$\PV\PH$ 2j0b category of this analysis is driven by the low-purity $\PV\PH$
category of the boosted analysis. By removing events from the low-purity
boosted $\PV\PH$ category that contain exactly two AK4 jets forming a dijet
candidate with ${65<m_{\text{jj}}<120\GeV}$, there is negligible reduction in
the exclusion sensitivity of that analysis. The overlap meanwhile is reduced
from 30-40% in the CR phase spaces to about 1%.

The uncertainties in the overall cross section for the signal processes are
treated as correlated amongst analysis channels, and amongst data sets with
the same centre-of-mass energy. The uncertainties related to missing higher-
order corrections, as well as PDF variations, are obtained from Ref. [58]. In
some of the channels, additional uncertainty contributions relating to signal
acceptance modelling are considered. These are treated as uncorrelated amongst
the different analysis channels.

The main sources of theoretical modelling uncertainties in the background
estimate vary for the different analysis channels. The analyses preferentially
select different phase space regions, and employ different assumptions for the
modelling of theoretical uncertainties in transfer factors amongst different
analysis regions. The resulting uncertainties are therefore treated as
uncorrelated.

Significant correlations appear in the treatment of experimental
uncertainties. The determination of the integrated luminosity estimate is
affected by a number of sources of uncertainty, which are assumed to be
correlated amongst all channels, and partially correlated amongst data sets.
Some of the analysis channels share trigger requirements, and the
uncertainties in the efficiencies of these common triggers are assumed to be
correlated amongst channels and uncorrelated amongst data sets. Furthermore,
analysis channels often share criteria used for identifying -tagged jets, as
well as the hadronic decay products of tau leptons. The uncertainties in the
efficiencies of these identification criteria are assumed to be correlated
amongst channels using the same criteria in the same data set. Finally,
uncertainties in the calibration of the JER and JES are treated as correlated
amongst this analysis, the VBF, and the $\cPg\cPg\PH$/boosted $\PV\PH$
channels. All other experimental uncertainties are assumed to be uncorrelated
amongst channels. For earlier analyses using Run 1 and 2015 data, the
correlation scheme established in Ref. [89] is used.

Exclusion limits on ${\mathcal{B}(\PH\to\text{inv})}$ are calculated assuming
SM production cross sections. The 2016–2018 data yields an overall limit of
0.16 (0.09 expected). If the Run 1 and 2015 data-taking periods are included,
values larger than 0.15 (0.08 expected) are excluded at 95% . This value is
dominated by the VBF channel, which yields a limit for
${\mathcal{B}(\PH\to\text{inv})}$ of 0.18 (0.10 expected). The limits for Run
1 and Run 2, separated by the Higgs boson production mode as tagged by the
input analyses, are presented in Fig. 7. The integrated luminosities of the
Run 1 and Run 2 data sets [34, 33, 32, 31, 89] are described in Table 0.9. The
final combination represents an improvement in sensitivity of approximately
20% relative to the most sensitive single channel (VBF).

Maximum likelihood fits to the individual production channels are performed,
as well as to the combination of all channels. The dependence of the profile
negative log-likelihood functions on the signal strength parameter $\hat{\mu}$
is shown in Fig. 7 (right). The best fit values of $\hat{\mu}$ for the
individual production channels are compatible with one another and with the
combined value of ${0.08^{+0.04}_{-0.04}}$, and the observed signal strength
is compatible with the absence of a $\PH\to\text{inv}$ signal within two
standard deviations. A breakdown of the best fit values of $\hat{\mu}$ for
each channel are presented in Table 0.9. A saturated goodness-of-fit test is
performed using the final combined likelihood function [92], yielding a
probability of 12% that the S$+$B model is consistent with the observed
results from the CMS experiment. Tabulated yields and fit results are provided
in HEPData [93].

The observed best fit estimates of ${\mathcal{B}(\PH\to\text{inv})}$, for each
analysis channel in the combination, and the 95% observed and expected (exp)
upper limits on ${\mathcal{B}(\PH\to\text{inv})}$.  Channel Best fit
${\mathcal{B}(\PH\to\text{inv})}$ ${\mathcal{B}(\PH\to\text{inv})}$ Combined
$0.08^{+0.04}_{-0.04}$ 0.15 (0.08 exp) VBF-tag $0.09^{+0.05}_{-0.05}$ 0.18
(0.10 exp) $\PV\PH$-tag $0.04^{+0.09}_{-0.09}$ 0.21 (0.18 exp) $\ttbar\PH$-tag
$-0.08^{+0.16}_{-0.16}$ 0.26 (0.30 exp) $\cPg\cPg\PH$-tag
$0.22^{+0.16}_{-0.16}$ 0.49 (0.32 exp)

Figure 7: Left: Exclusion limits at 95% on ${\mathcal{B}(\PH\to\text{inv})}$.
The results are shown separately for each Higgs boson production mode as
tagged by the input analyses for Run 1 and Run 2, as well as combined across
modes. Right: Scan of the profile negative log-likelihood as a function of
${\mathcal{B}(\PH\to\text{inv})}$ broken down by the Higgs boson production
mode as tagged by the input analyses for Run 1 and Run 2.

The upper limit on ${\mathcal{B}(\PH\to\text{inv})}$ is interpreted in the
context of a set of Higgs portal models of DM interactions, where a stable
weakly interacting massive particle (WIMP), such as a singlet scalar, fermion,
or vector, has a substantial coupling to a Higgs boson of mass 125 [19, 20].
The interaction of a WIMP with an atomic nucleus can occur via the exchange of
a Higgs boson, and the resulting nuclear recoil is measured to obtain an upper
bound on the spin-independent DM-nucleon scattering cross section,
$\sigma^{\text{SI}}_{\text{{DM\mbox{-}nucleon}}}$. An effective field theory
(EFT) approach is considered for scalar and fermionic WIMPs, while in the
vectorial case two UV-complete DM models are considered, given the EFT
appraoch violates unitarity [23, 94]. The vector-spin WIMP model (Vector
DM${}^{\text{UV-comp}}$) described in Ref. [20], and its radiative portal
analogue (Vector DM${}^{\text{radiative}}_{m_{2}}$) introduced in Ref. [23]
for dark Higgs boson masses $m_{2}=65$ and $100\GeV$, and with a mixing angle
between the SM and dark Higgs bosons $\theta=0.2$, are presented. The results
are compared to direct-detection searches, where in these experiments it is
assumed DM particles interact with atomic nuclei. Direct-detection limits are
reported by the CRESST-III [95], DarkSide-50 [96], Panda-X 4T [97], and LUX-
ZEPLIN [98] experiments. For the CRESST-III limits, a range of masses from
$m_{\text{DM}}=150\MeV$ and above are reported, however for this paper only
values above $m_{\text{DM}}=1\GeV$ are used. Upper limits on
$\sigma^{\text{SI}}_{\text{{DM\mbox{-}nucleon}}}$ for a range of DM mass
points are presented in Fig. 8 at the 90% using the full CMS data set. The
uncertainties in $\sigma^{\text{SI}}_{\text{{DM\mbox{-}nucleon}}}$ are
obtained from the extrema of a coupling parametrisation factor as derived from
lattice theory [99, 100, 19]. Results of the Higgs portal interpretation and
direct-detection comparison are also provided in HEPData [93].

Figure 8: Upper limits on $\sigma^{\text{SI}}_{\text{{DM\mbox{-}nucleon}}}$ as
a function of DM candidate mass $m_{\text{DM}}$. Results are presented for a
fermion (red) and scalar (yellow) DM candidate. In addition, a vector DM
candidate is studied using two UV-comp approaches, the first denoted Vector
DM${}^{\text{UV-comp}}$ [20] (burgundy), and the second a radiative portal
version denoted Vector DM${}^{\text{radiative}}_{m_{2}}$ [23] (orange) with a
dark Higgs boson mass of $m_{2}=65$ and $100\GeV$. Uncertainties are derived
from Refs. [99, 100, 19]. Results are compared to direct-detection searches
from CRESST-III [95] (truncated at $m_{\text{DM}}>1\GeV$), DarkSide-50 [96],
PandaX-4T [97] and LUX-ZEPLIN [98].

The sensitivity of the Run 1 and Run 2 combination depends on the cross
sections assumed for the different Higgs boson production modes: VBF,
$\PV\PH$, $\cPg\cPg\PH$, and $\ttbar\PH$. Cross sections can be parameterised
by the coupling strength of the Higgs boson to bosons and fermions. The cross
sections can be directly scaled by coupling strength modifiers
$\kappa_{\text{V}}$ and $\kappa_{\text{F}}$ to investigate BSM scenarios
[101]. In this context, the observed 95% upper limits on
${\mathcal{B}(\PH\to\text{inv})}$ are evaluated as a function of
$\kappa_{\text{V}}$ and $\kappa_{\text{F}}$ and shown in Fig. 9. Best
estimates of $\kappa_{\text{V}}$ and $\kappa_{\text{F}}$ from CMS [11] are
presented with the 68 and 95% contours. For the best estimate of
$\kappa_{\text{V}}$ and $\kappa_{\text{F}}$ by CMS, the 95% limit on
${\mathcal{B}(\PH\to\text{inv})}$ is found to be 0.16, and varies between 0.14
and 0.17 inside the 95% contour.

Figure 9: Observed 95% upper limit on ${\mathcal{B}(\PH\to\text{inv})}$ as a
function of coupling strength modifiers, $\kappa_{\text{V}}$ and
$\kappa_{\text{F}}$, for a Higgs boson of mass 125. Best estimates for
$\kappa_{\text{V}}$ and $\kappa_{\text{F}}$ from Ref. [11] are shown as a
black cross, together with 68 and 95% contours.

##  0.10 Summary

The results of a search for invisible decays of the Higgs boson produced in
association with a top-antitop quark pair ($\ttbar\PH$) or a vector boson
($\PV\PH$, where stands for either a or boson), which decays to a fully
hadronic final state, are presented. The analysis is based on proton-proton
collision data collected at $\sqrt{s}=13\TeV$ during the 2016–2018 data-taking
period by the CMS experiment at the LHC, corresponding to an integrated
luminosity of 138. The $\ttbar\PH$ production mechanism is investigated using
final states containing jets, or boosted quarks or bosons. The $\PV\PH$
production channel focuses on resolving a dijet pair with an invariant mass
that is compatible with that of a or boson. No significant excess of events is
observed above the predicted SM background. A 95% confidence level upper limit
of 0.47 (0.40 expected) is set on the branching fraction of the decay of the
Higgs boson to an invisible final state, ${\mathcal{B}(\PH\to\text{inv})}$,
assuming SM production cross sections.

The results are combined with previous ${\mathcal{B}(\PH\to\text{inv})}$
searches carried out at $\sqrt{s}=7$, 8, and 13in complementary production
modes. The combined 95% confidence level upper limit on
${\mathcal{B}(\PH\to\text{inv})}$ of 0.15 (0.08 expected) is obtained using
Run 1 (2011–2012) and Run 2 (2015–2018) data. The combination represents an
improvement in sensitivity of 20% relative to the most sensitive single
channel. The results are interpreted in the context of a set of Higgs portal
models of dark matter interactions to produce model-dependent exclusion limits
that complement direct-detection experiments for light mass dark matter
candidates.

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