HIG-17-031

$HeadURL:svn+ssh://svn.cern.ch/reps/tdr2/papers/HIG-17-031/trunk/HIG-17-031.tex$
$Id:HIG-17-031.tex4941392019-04-2811:39:13Zalverson$

HIG-17-031

# Combined measurements of Higgs boson couplings in proton-proton collisions
at $\sqrt{s}=13\TeV$

###### Abstract

Combined measurements of the production and decay rates of the Higgs boson, as
well as its couplings to vector bosons and fermions, are presented. The
analysis uses the LHC proton-proton collision data set recorded with the CMS
detector in 2016 at $\sqrt{s}=13\TeV$, corresponding to an integrated
luminosity of 35.9$\fbinv$. The combination is based on analyses targeting the
five main Higgs boson production mechanisms (gluon fusion, vector boson
fusion, and associated production with a $\PW$ or $\cPZ$ boson, or a top
quark-antiquark pair) and the following decay modes: $\PH\to\PGg\PGg$,
$\cPZ\cPZ$, $\PW\PW$, $\Pgt\Pgt$, $\PQb\PQb$, and $\Pgm\Pgm$. Searches for
invisible Higgs boson decays are also considered. The best-fit ratio of the
signal yield to the standard model expectation is measured to be $\mu=1.17\pm
0.10$, assuming a Higgs boson mass of $125.09\GeV$. Additional results are
given for various assumptions on the scaling behavior of the production and
decay modes, including generic parametrizations based on ratios of cross
sections and branching fractions or couplings. The results are compatible with
the standard model predictions in all parametrizations considered. In
addition, constraints are placed on various two Higgs doublet models.

##  0.1 Introduction

Understanding the mechanism behind electroweak symmetry breaking (EWSB)
remains one of the main objectives of the physics program at the CERN LHC. In
the standard model (SM) of particle physics [1, 2, 3, 4], EWSB is realized
through the addition of a complex scalar doublet field. A salient feature of
this is the prediction of one physical, neutral, scalar particle, the Higgs
boson () [5, 6, 7, 8, 9, 10]. The Higgs scalar field can also account for the
fermion masses through Yukawa interactions [2, 11]. The Higgs boson was
discovered by the ATLAS and CMS Collaborations [12, 13, 14], and is the
subject of much study. The Yukawa coupling strengths are free parameters in
the SM and do not explain the observed pattern of fermion masses. Furthermore,
it is not understood why the Higgs boson mass is near the electroweak scale,
since it is not protected in the SM from large quantum corrections [15, 16,
17, 18, 19]. This has led to the development of many beyond the SM (BSM)
theories that can alter the properties of the Higgs boson [20, 21, 22, 23,
24]. Precision measurements of the properties of the Higgs boson are therefore
an important test of the SM.

This paper describes combined measurements of the Higgs boson production
rates, decay rates, and couplings using analyses of $\sqrt{s}=13\TeV$
proton–proton collision data recorded with the CMS detector in 2016. The data
set corresponds to an integrated luminosity of 35.9. The following decay
channels are included in the combination: $\PH\to\PGg\PGg$, $\PH\to\cPZ\cPZ$,
$\PH\to\PW\PW$, $\PH\to\Pgt\Pgt$, $\PH\to\PQb\PQb$, and $\PH\to\Pgm\Pgm$, as
shown in Fig. 1. Here and in what follows, we do not distinguish between
particles and antiparticles in our notations of production and decay
processes. Searches for invisible decays of the Higgs boson, which are
predicted to be considerably enhanced by several BSM theories [25, 26, 27,
28], are also considered for selected measurements. The data samples
considered for each decay channel are ensured to have negligible overlap to
avoid introducing nontrivial correlations.

Figure 1: Examples of leading-order Feynman diagrams for Higgs boson decays in
the $\PH\to\PQb\PQb$, $\PH\to\Pgt\Pgt$, and $\PH\to\Pgm\Pgm$ (upper left);
$\PH\to\cPZ\cPZ$ and $\PH\to\PW\PW$ (upper right); and $\PH\to\PGg\PGg$
(lower) channels.

The analyses included in this combination target production via gluon fusion
($\Pg\Pg\PH$), vector boson fusion ($\mathrm{VBF}$), associated production
with a vector boson ($\mathrm{V}\PH$, V$=\PW$ or ), and associated production
with a pair of top quarks ($\cPqt\cPqt\PH$). The prediction for $\Pg\Pg\PH$
production has advanced to next-to-next-to-next-to-leading order (N3LO) in
perturbative quantum chromodynamics (QCD) [29, 30] and next-to-leading order
(NLO) for electroweak (EW) corrections, reducing its uncertainty from
${}^{+7.6\%}_{-8.1\%}$ (next-to-NLO) to ${}^{+4.6\%}_{-6.7\%}$. The
calculations of the $\mathrm{VBF}$ and $\mathrm{V}\PH$ cross sections are
performed at next-to-NLO QCD and NLO EW accuracy, while the calculation of the
$\cPqt\cPqt\PH$ cross section is performed at NLO QCD and NLO EW accuracy. The
updated theoretical predictions used for the various production and decay
modes in this paper can be found in Refs. [29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52] and are summarized
in Ref. [53]. Examples of leading-order (LO) Feynman diagrams for these
production processes can be seen in Figs. 2 and 3. In addition to the five
main production processes, the contributions due to Higgs boson production in
association with a single top quark ($\cPqt\PH$) and either a $\PW$ boson
($\cPqt\PH\PW$) or a quark ($\cPqt\PH\Pq$), as shown in Fig. 4, are included
in the analyses that have some sensitivity to them.

Figure 2: Examples of leading-order Feynman diagrams for the $\Pg\Pg\PH$
(upper left), $\mathrm{VBF}$ (upper right), $\mathrm{V}\PH$ (lower left), and
$\cPqt\cPqt\PH$ (lower right) production modes.

Figure 3: Examples of leading-order Feynman diagrams for the
$\Pg\Pg\to\cPZ\PH$ production mode.

Figure 4: Examples of leading-order Feynman diagrams for $\cPqt\PH$ production
via the $\cPqt\PH\PW$ (upper left and right) and $\cPqt\PH\Pq$ (lower) modes.

For certain measurements in this paper, such as $\Pg\Pg\PH$ production and
$\PH\to\PGg\PGg$ decay, the interference between the diagrams that contribute
to the process is considered. In addition, the $\cPqt\PH$ cross section is
small in the SM, being approximately 14% of the $\cPqt\cPqt\PH$ cross section,
due to the destructive interference between the diagrams shown in Fig. 4,
which involve the coupling of the Higgs boson to $\PW$ bosons ($\cPqt\PH\PW$
process) and top quarks ($\cPqt\PH\Pq$ process). This interference becomes
constructive, however, when the relative sign between these couplings is
negative, and so the $\cPqt\PH$ process is sensitive to the relative sign of
the $\PH\PW\PW$ and $\cPqt\cPqt\PH$ couplings.

The ATLAS and CMS Collaborations have published combined measurements of Higgs
boson production rates, decay rates, and couplings with the $\sqrt{s}=7$ and
$8\TeV$ LHC Run 1 data [54, 55]. A combination of the Run 1 ATLAS and CMS
analyses has also been performed [56]. All results were found to be in
agreement, within their uncertainties, with the predictions of the SM. In this
paper, due to the larger integrated luminosity and increased signal cross
section at $\sqrt{s}=13\TeV$, the measured precision for several parameters of
interest has significantly increased with respect to Ref. [56]. In particular,
the predicted cross sections for the dominant $\Pg\Pg\PH$ production mode and
the $\cPqt\cPqt\PH$ production mode increase by factors of approximately 2.3
and 3.8, respectively, between $\sqrt{s}=8$ and $13\TeV$. In addition, some of
the theoretical predictions have improved, as mentioned earlier.

This paper is organized as follows: A brief description of the CMS detector is
given in Section 0.2, Section 0.3 provides a summary of the various analyses
included in the combination, and Section 0.4 describes the modifications made
to these analyses to ensure a common signal and uncertainty model. Section 0.5
outlines the statistical procedure used to derive the results, and Section 0.6
outlines the treatment of the systematic uncertainties. Section 0.7 reports
the results of the signal parametrizations in terms of signal strength
modifiers and fiducial cross sections, while Section 0.8 describes the results
obtained from an alternative set of signal parametrizations in terms of Higgs
boson couplings. Section 0.9 details interpretations in terms of various two
Higgs doublet models. The paper is summarized in Section 0.10.

##  0.2 The CMS detector

The central feature of the CMS apparatus is a superconducting solenoid of 6m
internal diameter, providing a magnetic field of 3.8T. Within the solenoid
volume are a silicon pixel and strip tracker, a lead tungstate crystal
electromagnetic calorimeter, and a brass and scintillator hadron calorimeter,
each composed of a barrel and two endcap sections. Forward calorimeters extend
the pseudorapidity coverage provided by the barrel and endcap detectors. Muons
are detected in gas-ionization chambers embedded in the steel flux-return yoke
outside the solenoid. A more detailed description of the CMS detector,
together with a definition of the coordinate system used and the relevant
kinematic variables, can be found in Ref. [57].

##  0.3 Analyses included in the combination

In this section, the individual analyses included in the combination are
briefly described. More detailed information on each analysis can be found in
the corresponding references. Many of the analyses split their primary data
sample in multiple event categories with specific signatures that enhance the
discrimination power between different Higgs boson production processes. This
is achieved through selections that require the presence of additional leptons
or jets, as expected in the decay of a $\PW$ or $\cPZ$ boson in the $\PW\PH$
and $\cPZ\PH$ modes, or in top quark decays in the $\cPqt\cPqt\PH$ mode, and
that exploit the distinctive kinematic properties of the final state objects,
such as the presence of two jets with a large separation in pseudorapidity
$\Delta\eta_{\text{jj}}$, and a large invariant mass $m_{\text{jj}}$, in the
$\mathrm{VBF}$ topology. In some categories, the kinematic features of an
event as a whole are used to select particular production processes. For
example, requiring a large missing transverse momentum , defined as the
magnitude of the negative vector sum over the transverse momenta of all
particles reconstructed in an event, targets $\cPZ\PH$ production in which the
$\cPZ$ boson decays to neutrinos. The event categories within and amongst the
individual analyses are constructed to ensure a negligible level of overlap
(i.e. the same event entering more than one category). In many cases, this is
accomplished by synchronizing the object (e.g. electron, muon, tau, or jet)
identification definitions and imposing strict requirements on the number of
reconstructed objects. In other cases, the orthogonality is ensured by
imposing opposing requirements on higher level observables formed using
multiple objects. For rare cases where potential overlap is not explicitly
removed, the lists of selected data events were checked and found to contain a
negligible number of duplications. In total, up to 265 event categories are
considered, and there are over 5500 nuisance parameters corresponding to
various sources of experimental and theoretical systematic uncertainty. A
summary of the production and decay modes, which are described in more detail
in the following sections, is shown in Table 0.3.

[] Summary of the event categories in the analyses included in this
combination. The first column indicates the decay channel and the second
column indicates the production mechanism targeted by an analysis. The third
column provides the total number of categories per production tag, excluding
control regions. Notes on the expected fractions of different Higgs signal
production and decay modes with respect to the total signal yield in the given
category are given in the fourth column. Where the numbers do not sum to 100%,
the remaining contributions are from other signal production and decay
processes. Finally, where relevant, the fifth column specifies the approximate
expected relative mass resolution for the SM Higgs boson.

Decay tags | Production tags | Number of |  Expected signal fractions | Mass resolution
---|---|---|---|---
|  | categories
$\PH\to\PGg\PGg$, Section 0.3.1 |  |  |  |
| Untagged | 4 | 74–91% $\Pg\Pg\PH$ |
| $\mathrm{VBF}$ | 3 | 51–80% $\mathrm{VBF}$ |
|  $\mathrm{V}\PH$ hadronic | 1 | 25% $\PW\PH$, 15% $\cPZ\PH$ |
|  $\PW\PH$ leptonic | 2 | 64–83% $\PW\PH$ |
|  $\cPZ\PH$ leptonic | 1 | 98% $\cPZ\PH$ |
|  $\mathrm{V}\PH$ $\ptmiss$ | 1 | 59% $\mathrm{V}\PH$ |
$\PGg\PGg$ | $\cPqt\cPqt\PH$ | 2 | 80–89% $\cPqt\cPqt\PH$, $\approx$8% $\cPqt\PH$ | $\approx$1–2%
$\PH\to\cPZ\cPZ^{(\ast)}\to 4\ell$, Section 0.3.2
| Untagged | 3 |  $\approx$95% $\Pg\Pg\PH$ |
|  $\mathrm{VBF}$ 1, 2-jet | 6 |  $\approx$11–47% $\mathrm{VBF}$ |
|  $\mathrm{V}\PH$ hadronic | 3 |  $\approx$13% $\PW\PH$, $\approx$10% $\cPZ\PH$ |
|  $\mathrm{V}\PH$ leptonic | 3 |  $\approx$46% $\PW\PH$ |
|  $\mathrm{V}\PH$ $\ptmiss$ | 3 |  $\approx$56% $\cPZ\PH$ |
$4\Pgm$, $2\Pe 2\Pgm/2\Pgm 2\Pe$, $4\Pe$ | $\cPqt\cPqt\PH$ | 3 |  $\approx$71% $\cPqt\cPqt\PH$ | $\approx$1–2%
$\PH\to\PW\PW^{(\ast)}\to\ell\cPgn\ell\cPgn$, Section 0.3.3
|  $\Pg\Pg\PH$ 0, 1, 2-jet | 17 |  $\approx$55–92% $\Pg\Pg\PH$, up to $\approx$15% $\PH\to\Pgt\Pgt$ |
$\Pe\PGm/\PGm\Pe$ |  $\mathrm{VBF}$ 2-jet | 2 |  $\approx$47% $\mathrm{VBF}$, up to $\approx$25% $\PH\to\Pgt\Pgt$ |
$\Pe\Pe$+$\Pgm\Pgm$ |  $\Pg\Pg\PH$ 0, 1-jet | 6 |  $\approx$84–94% $\Pg\Pg\PH$ |
$\Pe\PGm$+jj |  $\mathrm{V}\PH$ 2-jet | 1 | 22% $\mathrm{V}\PH$, 21% $\PH\to\Pgt\Pgt$ |
$3\ell$ |  $\PW\PH$ leptonic | 2 |  $\approx$80% $\PW\PH$, up to 19% $\PH\to\Pgt\Pgt$ |
$4\ell$ |  $\cPZ\PH$ leptonic | 2 | 85–90% $\cPZ\PH$, up to 14% $\PH\to\Pgt\Pgt$ | $\approx$20%
$\PH\to\Pgt\Pgt$, Section 0.3.4 |  |  |  |
| 0-jet | 4 |  $\approx$70–98% $\Pg\Pg\PH$, 29% $\PH\to\PW\PW$ in $\Pe\Pgm$ |
| $\mathrm{VBF}$ | 4 |  $\approx$35–60% $\mathrm{VBF}$, 42% $\PH\to\PW\PW$ in $\Pe\Pgm$ |
$\Pe\Pgm,\Pe\tauh,\Pgm\tauh,\tauh\tauh$ | Boosted | 4 |  $\approx$48–83% $\Pg\Pg\PH$, 43% $\PH\to\PW\PW$ in $\Pe\Pgm$ | $\approx$10–20%
$\mathrm{V}\PH$ production with $\PH\to\PQb\PQb$, Section 0.3.5
$\cPZ(\cPgn\cPgn)\PH(\PQb\PQb)$ |  $\cPZ\PH$ leptonic | 1 |  $\approx$100% $\mathrm{V}\PH$, 85% $\cPZ\PH$ |
$\PW(\ell\cPgn)\PH(\PQb\PQb)$ |  $\PW\PH$ leptonic | 2 |  $\approx$100% $\mathrm{V}\PH$, $\approx$97% $\PW\PH$ |
| Low-$\pt(\mathrm{V})$ $\cPZ\PH$ leptonic | 2 |  $\approx$100% $\cPZ\PH$, of which $\approx$20% $\Pg\Pg\cPZ\PH$ |
$\cPZ(\ell\ell)\PH(\PQb\PQb)$ | High-$\pt(\mathrm{V})$ $\cPZ\PH$ leptonic | 2 |  $\approx$100% $\cPZ\PH$, of which $\approx$36% $\Pg\Pg\cPZ\PH$ | $\approx$10%
Boosted $\PH$ Production with $\PH\to\PQb\PQb$, Section 0.3.6
$\PQb\PQb$ |  $\pt(\PH)$ bins | 6 |  $\approx$72–79% $\Pg\Pg\PH$ |  $\approx$10%
$\cPqt\cPqt\PH$ production with $\PH\to\mathrm{leptons}$, Section 0.3.7
$2\ell$ss |  | 10 |  $\PW\PW/\Pgt\Pgt\approx 4.5$, $\approx$5% $\cPqt\PH$ |
$3\ell$ |  | 4 |  $\PW\PW:\Pgt\Pgt:\cPZ\cPZ\approx 15:4:1$, $\approx$5% $\cPqt\PH$ |
$4\ell$ |  | 1 |  $\PW\PW:\Pgt\Pgt:\cPZ\cPZ\approx 6:1:1$, $\approx$3% $\cPqt\PH$ |
$1\ell$$+2\tauh$ |  | 1 | 96% $\cPqt\cPqt\PH$ with $\PH\to\Pgt\Pgt$, $\approx$6% $\cPqt\PH$ |
$2\ell$ss$+1\tauh$ |  | 2 |  $\Pgt\Pgt:\PW\PW\approx 5:4$, $\approx$5% $\cPqt\PH$ |
$3\ell$$+1\tauh$ | $\cPqt\cPqt\PH$ | 1 |  $\Pgt\Pgt:\PW\PW:\cPZ\cPZ\approx 11:7:1$, $\approx$3% $\cPqt\PH$ |
$\cPqt\cPqt\PH$ production with $\PH\to\PQb\PQb$, Section 0.3.7
|  $\ttbar\to$ jets | 6 |  $\approx$83–97% $\cPqt\cPqt\PH$ with $\PH\to\PQb\PQb$ |
|  $\ttbar\to 1\ell$+jets | 18 |  $\approx$65–95% $\cPqt\cPqt\PH$ with $\PH\to\PQb\PQb$, up to 20% $\PH\to\PW\PW$ |
$\PQb\PQb$ |  $\ttbar\to 2\ell$+jets | 3 |  $\approx$84–96% $\cPqt\cPqt\PH$ with $\PH\to\PQb\PQb$ |
Search for $\PH\to\Pgm\Pgm$, Section 0.3.8
$\Pgm\Pgm$ | S/B bins | 15 | 56–96% $\Pg\Pg\PH$, 1–42% $\mathrm{VBF}$ |  $\approx$1–2%
Search for invisible $\PH$ decays, Section 0.3.9
| $\mathrm{VBF}$ | 1 | 52% $\mathrm{VBF}$, 48% $\Pg\Pg\PH$ |
|  $\Pg\Pg\PH$ \+ $\geq 1$ jet | 1 | 80% $\Pg\Pg\PH$, 9% $\mathrm{VBF}$ |
|  $\mathrm{V}\PH$ hadronic | 1 | 54% $\mathrm{V}\PH$, 39% $\Pg\Pg\PH$ |
Invisible |  $\cPZ\PH$ leptonic | 1 |  $\approx$100% $\cPZ\PH$, of which 21% $\Pg\Pg\cPZ\PH$ |

###  0.3.1 $\PH\to\PGg\PGg$

The $\PH\to\PGg\PGg$ analysis [58] provides good sensitivity to nearly all
Higgs boson production processes. Since the $\PH\to\PGg\PGg$ decay proceeds
mainly through - and top-loop processes, interference effects make its
branching fraction sensitive to the relative sign of the fermion and vector
boson couplings. The analysis measures a narrow signal peak in the diphoton
invariant mass ($m_{\PGg\PGg}$) spectrum over a smoothly falling continuum
background, originating mainly from prompt, nonresonant diphoton production,
or from events where at least one jet is misidentified as an isolated photon.

Exclusive event categories are defined using dedicated selections based on
additional reconstructed objects to separate the different Higgs boson
production mechanisms. The presence of additional leptons, , or jets is used
to classify events into one of the following categories: $\cPqt\cPqt\PH$
leptonic, $\cPqt\cPqt\PH$ hadronic, $\cPZ\PH$ leptonic, $\PW\PH$ leptonic,
loose $\mathrm{V}\PH$ leptonic with low requirement, $\mathrm{VBF}$,
$\mathrm{V}\PH$ , and $\mathrm{V}\PH$ hadronic. The $\mathrm{VBF}$ category is
divided into three subcategories of increasing purity against $\Pg\Pg\PH$
production. Finally, the remaining events are divided into four untagged
categories with increasing signal purity.

In each event class, the background in the signal region (SR) is estimated
from a fit to the observed $m_{\PGg\PGg}$ distribution in data. The dominant
experimental uncertainties in the measurement of the rate of Higgs boson
production in the $\PH\to\PGg\PGg$ decay channel are related to the modeling
of the electromagnetic shower shape observables used in the photon
identification and the background shape parametrization.

###  0.3.2 $\PH\to\cPZ\cPZ$

Despite the $\PH\to\cPZ\cPZ^{(\ast)}\to 4\ell$ ($\ell=\Pe$ or $\Pgm$) decay
having the lowest branching fraction of the decay channels considered, it also
has the lowest background contamination, resulting in very good sensitivity to
production processes with large cross sections, such as $\Pg\Pg\PH$. It is
also the most important decay channel in constraining the $\PH\cPZ\cPZ$
coupling. The $\PH\to\cPZ\cPZ^{(\ast)}\to 4\ell$ [59] analysis measures a
narrow four-lepton invariant mass peak over a small continuum background. The
dominant irreducible background in this analysis is due to nonresonant
$\cPZ\cPZ$ production with both $\cPZ$ bosons decaying to a pair of charged
leptons, and is estimated from simulation. The $4\Pe$, $4\Pgm$, and $2\Pe
2\Pgm$/$2\Pgm 2\Pe$ decay channels are treated separately to better model the
different mass resolutions and background rates arising from jets
misidentified as leptons.

To separate the different Higgs boson production mechanisms, the following
categories are defined on the basis of the presence of jets, $\PQb$-tagged
jets, leptons, , and various matrix element discriminants that make use of the
information about the additional objects: $\mathrm{VBF}$ (1- and 2-jet),
$\mathrm{V}\PH$ hadronic, $\mathrm{V}\PH$ leptonic, $\cPqt\cPqt\PH$,
$\mathrm{V}\PH$ , and untagged categories.

In the $\PH\to\cPZ\cPZ^{(\ast)}\to 4\ell$ analysis, the dominant experimental
uncertainties are related to the lepton efficiencies and the determination of
the +jets background from data.

###  0.3.3 $\PH\to\PW\PW$

The $\PH\to\PW\PW^{(\ast)}\to\ell\cPgn\ell\cPgn$ analysis [60] profits from
the fact that the $\PH\to\PW\PW$ decay mode has one of the largest branching
fractions and has a relatively low-background final state. As a result, this
decay channel has very good sensitivity to most production processes, in
particular $\Pg\Pg\PH$ and $\mathrm{VBF}$. Imposing tight lepton
identification criteria and requiring the absence of $\PQb$-tagged jets helps
to reduce the misidentified lepton and top quark backgrounds, respectively.
Several event categories with varying signal-to-background ratios are defined
to improve the sensitivity to the signal. Events are selected that contain two
leptons, denoted $2\ell$, which may be of different or same flavor. The
different-flavor $\Pe\Pgm$ decay channel dominates the sensitivity since it
has the largest branching fraction and is the least contaminated by
backgrounds. The same-flavor $\Pe\Pe$ and $\Pgm\Pgm$ final states are also
considered, although their sensitivity is limited by the contamination from
Drell–Yan (DY) background events with misreconstructed . Given the large
background contribution from $\PQt\PQt$ production in both the different-
flavor and same-flavor final states, events are further categorized into
categories with 0, 1, and 2 associated jets, with the 0-jet category
dominating the overall sensitivity. In addition, events are further
categorized on the basis of the of the subleading lepton, since the background
from misidentified leptons is larger in the low-region. In the different-
flavor final state, dedicated 2-jet categories are included to enhance the
sensitivity to VBF and $\mathrm{V}\PH$ production mechanisms.

The analysis also includes categories that are sensitive to the associated
production of the Higgs boson with a vector boson that decays leptonically.
Two $3\ell$ categories that are sensitive to $\PW\PH$ production are defined
by requiring the presence of a total of three leptons (electrons or muons).
The two are distinguished by whether or not they contain a pair of leptons
with the same flavor and opposite sign. Events with four charged leptons, in
which one pair is consistent with a $\cPZ$ boson decay, are separated into two
categories depending on whether the remaining pair consists of same-flavor
leptons or not. These $4\ell$ categories are sensitive to the $\cPZ\PH$
production mode. The signal extraction method depends on the event category.

When measuring the rate of Higgs boson production in the $\PH\to\PW\PW$ decay
channel, the dominant experimental uncertainties arise from the determination
of the top quark pair, $\PW\PW$ and DY backgrounds from data, and the
uncertainties related to the and $\eta$ dependent lepton reconstruction and
identification efficiencies.

###  0.3.4 $\PH\to\Pgt\Pgt$

The $\PH\to\Pgt\Pgt$ analysis [61] benefits from a relatively large branching
fraction and a reasonable mass resolution of $\approx$10–20%, providing
competitive sensitivity to both the $\Pg\Pg\PH$ and $\mathrm{VBF}$ production
processes. It also provides the best sensitivity for the direct measurement of
a fermionic Higgs boson coupling. The analysis utilizes the four most
sensitive $\Pgt\Pgt$ final states: $\Pe\Pgm$, $\Pe\tauh$, $\Pgm\tauh$, and
$\tauh\tauh$, where denotes a hadronically decaying lepton. In the analysis of
each $\Pgt\Pgt$ decay channel, events are divided into three categories
labeled 0-jet, boosted, and VBF.

The VBF category requires the presence of two additional jets with large
$m_{\text{jj}}$ and $\Delta\eta_{\text{jj}}$, designed to increase the purity
of $\mathrm{VBF}$ events. The 0-jet category does not have much sensitivity to
the signal, but is useful to constrain systematic uncertainties in the
background model. The boosted category contains all remaining events, and is
binned as a function of of the $\Pgt\Pgt$ system to increase the sensitivity
to $\Pg\Pg\PH$ production. There is a nonnegligible contribution from the
$\PH\to\PW\PW$ process in some categories, and this is treated consistently as
an $\PH\to\PW\PW$ signal in this combined measurement.

The and energy scale uncertainties are the dominant experimental uncertainties
in the measurement of the Higgs boson production rate in the $\PH\to\Pgt\Pgt$
decay channel, followed by the uncertainties in the determination of the
$\cPZ(\Pgt\Pgt)+$jets background from data.

###  0.3.5 $\mathrm{V}\PH$ production with $\PH\to\PQb\PQb$

The $\PH\to\PQb\PQb$ decay has the largest expected branching fraction in the
SM (58.1% for $m_{\PH}=125.09\GeV$) and a reasonable mass resolution of 15%.
By requiring $\mathrm{V}\PH$ production it is possible to increase the signal
purity with respect to the inclusive case for which the background from QCD
multijet production is dominant. The analysis of the $\PH\to\PQb\PQb$ decay
targeting $\mathrm{V}\PH$ production ($\mathrm{V}\PH(\PQb\PQb)$) [62] provides
the best sensitivity to the $\PW\PH$ and $\cPZ\PH$ processes as well as to the
$\PQb\PQb\PH$ coupling. Selected events are categorized based on the presence
of two $\PQb$-tagged jets, and two ($\cPZ(\ell\ell)\PH(\PQb\PQb)$), one
($\PW(\ell\cPgn)\PH(\PQb\PQb)$) or no ($\cPZ(\cPgn\cPgn)\PH(\PQb\PQb)$)
electrons or muons in the final state. The $\cPZ(\ell\ell)\PH(\PQb\PQb)$
categories are subdivided into low-boost ($50<\pt(\cPZ)<150\GeV$) and high
boost ($\pt(\cPZ)>150\GeV$) regions. Events selected in the
$\cPZ(\cPgn\cPgn)\PH(\PQb\PQb)$ category are further required to have
$\ptmiss>170\GeV$.

The main backgrounds come from $\cPZ$ or $\PW$ boson production in association
with light- and heavy-flavor (LF and HF) jets, as well as from top quark pair
and diboson production. The dominant experimental uncertainties in in this
analysis are related to the determination of these backgrounds, and
uncertainties in the $\PQb$ tagging discriminator shapes and efficiencies.

###  0.3.6 Boosted production with $\PH\to\PQb\PQb$

The $\PH\to\PQb\PQb$ decay is also measured in an analysis that targets
inclusive production of the Higgs boson [63], exploiting the higher signal to
background ratio at high $\pt(\PH)$ (the transverse momentum of the Higgs
boson). The decay products of a high-$\PH\to\PQb\PQb$ system are reconstructed
using the anti-algorithm [64, 65] with a distance parameter of 0.8 (AK8 jet),
and the soft-drop algorithm [66, 67] is used to reconstruct the jet mass
$m_{\rm SD}$, which peaks at the Higgs boson mass for signal events. Events
containing substantial , or identified and isolated electrons, muons or
leptons are vetoed to reduce the background contributions from vector boson
production and top quark processes.

The main background component, QCD multijet production, is estimated from a
signal-depleted Control Region (CR). The selected events are divided according
to the jet into six bins of increasing width from 450 GeV to 1.

The dominant experimental uncertainties in this analysis are the uncertainties
related to the extrapolation of the QCD multijet and top quark pair
backgrounds from the CRs.

###  0.3.7 $\cPqt\cPqt\PH$ production

Measurements of the rate of the $\cPqt\cPqt\PH$ production process provide a
direct test of the Higgs boson’s coupling to top quarks. A recent measurement
by CMS combining the $\sqrt{s}=7$, $8$ and $13\TeV$ datasets was able to
establish the first $5\sigma$ observation of the $\cPqt\cPqt\PH$ production
process [68]. Dedicated analyses targeting the $\PH\to\text{leptons}$ [69] and
$\PH\to\PQb\PQb$ [70, 71] decay channels using $\sqrt{s}=13\TeV$ data are
described in this section.

####  $\cPqt\cPqt\PH$ production with $\PH\to\text{leptons}$

The analysis of $\cPqt\cPqt\PH$ production with $\PH\to\text{leptons}$ [69] is
mainly sensitive to the Higgs boson decaying to $\Pgt\Pgt$, $\PW\PW$ or
$\cPZ\cPZ$ with electrons, muons and/or in the final state. This analysis
provides the best sensitivity to the $\cPqt\cPqt\PH$ production process. The
main irreducible backgrounds come from $\PQt\PQt$V and diboson production.
Reducible backgrounds containing misidentified leptons or leptons with
misidentified charge are estimated from CRs in data. Events are categorized
according to their lepton content. The light-lepton ($\Pe$/$\Pgm$) categories
are defined as:

  * •

2$\ell$ss: Events with two leptons having the same sign and at least four
additional jets. A veto on the presence of hadronic tau decays is applied.
Further categories based on lepton charge, flavor and the number of
$\PQb$-tagged jets are defined within this class.

  * •

3$\ell$: Events containing three leptons, with the sum of lepton charges equal
to $\pm$1, and at least two additional jets of which one or two are $\PQb$
tagged.

  * •

4$\ell$: Events with four leptons, with an explicit veto on
$\PH\to\cPZ\cPZ^{(\ast)}\to 4\ell$ events as these are selected by the
analysis described in Section 0.3.2.

The categories, which require the presence of hadronically decaying leptons,
are defined as:

  * •

1$\ell$+2$\tauh$: Events with two oppositely charged $\tauh$ candidates and an
additional $\Pe$/$\Pgm$, at least three additional jets, and at least one
$\PQb$-tagged jet.

  * •

2$\ell$ss+1$\tauh$: Events containing three leptons, with sum of lepton
charges equal to $\pm$1, and at least two additional jets of which one or two
are $\PQb$ tagged. These events are further sorted into two subcategories
based on whether or not all of the jets expected in the $\cPqt\cPqt\PH$
process are reconstructed.

  * •

3$\ell$+1$\tauh$: Events with three light leptons, one and at least two
additional jets and one $\PQb$-tagged jet.

In the $\Pe$/$\Pgm$ and categories, the dominant experimental uncertainties on
the measurement of the rate of Higgs boson production in the $\cPqt\cPqt\PH$
mode are related to the lepton reconstruction efficiencies, and the estimation
of the reducible background contributions from data.

####  $\cPqt\cPqt\PH$ production with $\PH\to\PQb\PQb$

There are two analyses that target the associated production of the Higgs
boson with a pair of top quarks in the $\PH\to\PQb\PQb$ decay mode [70, 71].
The leptonic analysis requires at least one lepton to be present in the final
state, from the $\PQt\PQt$ decay system, while the hadronic analysis selects
events in the all-hadronic final state. These analyses provide good
sensitivity to the $\cPqt\cPqt\PH$ production process and improve the
precision in the measurement of the $\PQb\PQb\PH$ coupling.

In the leptonic analysis, events are sorted into the $1\ell$ or $2\ell$
classes, depending on the presence of one or two well-identified leptons.
Events are further categorized based on the number of reconstructed jets (Nj)
and the number of jets that are tagged as $\PQb$ jets (Nb) in each event. The
largest background is due to top quark pair production with additional jets
that contain heavy flavor hadrons. In the $1\ell$ class, three categories are
used: 4j $\geq$3$\PQb$, 5j $\geq$3$\PQb$, and 6j $\geq$3$\PQb$. In each
category a multi-classification deep neural network (DNN) [72] is used to
define six classes on the basis of the most probable event hypothesis for each
event, yielding a total of 18 categories. In the $2\ell$ class, there are two
jet categories: $\geq$4j 3$\PQb$ and $\geq$4j $\geq$4$\PQb$. The $\geq$4j
$\geq$4$\PQb$ category is further divided into two subcategories.

The all-hadronic final-state analysis selects events that contain at least
seven jets, at least three of which are tagged as $\PQb$ jets. These events
are divided into seven categories: 7j 3$\PQb$, 7j $\geq$4$\PQb$, 8j 3$\PQb$,
8j $\geq$4$\PQb$, $\geq$9j 3$\PQb$, and $\geq$9j $\geq$4$\PQb$. Events
containing electrons or muons are vetoed to maintain an orthogonal selection
to the leptonic final state analysis. The dominant background is QCD multijet
production, with other important backgrounds coming from $\PQt\PQt$+jets
processes.

The dominant experimental uncertainties in the measurement of the rate of
$\cPqt\cPqt\PH$ production with $\PH\to\PQb\PQb$ decay in the leptonic and
all-hadronic final states are due to uncertainties in the determination of the
$\PQt\PQt\PQb\PQb$ backgrounds and $\PQb$ tagging efficiencies. In the all-
hadronic final state, the uncertainty in the determination of the QCD multijet
background also has a significant contribution to the overall systematic
uncertainty.

###  0.3.8 Search for $\PH\to\Pgm\Pgm$

The $\PH\to\Pgm\Pgm$ search [73] is the only analysis included here that is
sensitive to the coupling of the Higgs boson to second-generation fermions.
The analysis searches for a narrow peak in the dimuon invariant mass
($m_{\Pgm\Pgm}$) spectrum above a large continuum background from DY
production of muon pairs. Events are categorized using variables that are
uncorrelated with $m_{\Pgm\Pgm}$, in order to avoid introducing an irregular
shape in the background spectrum. Variables that distinguish between the
$\Pg\Pg\PH$ and $\mathrm{VBF}$ signals, and the DY and $\PQt\PQt$ backgrounds,
are used to define event categories with varying signal-to-background ratios.
The categories are further divided based on the momentum of the muon with the
largest $\abs{\eta}$ in the dimuon pair, to exploit the differences in the
$m_{\Pgm\Pgm}$ resolution. Since there are more variables associated with
$\mathrm{VBF}$ production that can be used to separate signal and background,
the events with the highest BDT output value are most compatible with that
process.

In each event category, the background is estimated from a fit to the observed
$m_{\Pgm\Pgm}$ distribution. As in the $\PH\to\PGg\PGg$ analysis, the
parameters of the functions used to describe the background contribute to the
statistical uncertainty in the measurements. This is the dominant source of
uncertainty in constraining the rate of Higgs boson decay in the
$\PH\to\Pgm\Pgm$ decay channel. The observed upper limit on the cross section
times branching fraction of $\PH\to\Pgm\Pgm$ obtained in Ref. [73] is 2.93
times the SM value.

###  0.3.9 Search for $\PH\to$ invisible

The direct search for the Higgs boson decaying into particles that cannot be
detected provides a constraint on the invisible Higgs boson branching fraction
($\mathcal{B}_{\text{inv}}$), which is predicted to be enhanced in BSM
scenarios [25, 26, 27, 28, 74]. The search is performed using events with
large , and containing additional particles consistent with Higgs boson
production via the $\mathrm{VBF}$ [75], $\cPZ\PH$ with $\cPZ\to\ell\ell$ [76],
$\mathrm{V}\PH$ with the $\PW$ or $\cPZ$ boson decaying hadronically, or
$\Pg\Pg\PH$ modes [77].

Events selected in the $\mathrm{VBF}$ category are required to contain two
jets, with a large $m_{\text{jj}}$ and a large $\Delta\eta_{\text{jj}}$. The
$\mathrm{V}\PH$ hadronic and $\Pg\Pg\PH$ categories comprise events containing
either a high-AK8 jet, consistent with a boosted, hadronically decaying vector
boson, or a jet from initial-state radiation, reconstructed in the fiducial
volume of the tracker. The dominant backgrounds in these categories are due to
the $\cPZ(\nu\nu)$\+ jets and $\PW(\ell\nu)$\+ jets processes. These are
estimated from dedicated lepton and photon CRs in data. In all three
categories, the dominant uncertainties are related to the extrapolation of the
lepton and photon CRs to determine the $\cPZ(\nu\nu)$\+ jets and
$\PW(\ell\nu)$\+ jets backgrounds in the SR.

The $\cPZ\PH$ leptonic category is defined by selecting events that contain a
pair of oppositely charged electrons or muons consistent with the decay of a
$\cPZ$ boson. The dominant backgrounds arise from $\cPZ(\ell\ell)\cPZ(\nu\nu)$
and $\PW(\ell\nu)\cPZ(\nu\nu)$ diboson production and are estimated using a
combination of CRs in data containing additional leptons, and simulated
events. The dominant uncertainty in this category is related to theoretical
uncertainties in the higher-order corrections used in the simulation of these
backgrounds.

The observed upper limit on the branching fraction of $\PH\to\text{invisible}$
assuming SM Higgs production rates is 26% [75]. As described later in Section
0.8, the $\PH\to\text{invisible}$ analyses are included only in models for
which a nonzero invisible branching fraction of the Higgs boson is considered.

##  0.4 Modifications to the input analyses

This section describes the changes in each analysis, as implemented for this
combination, compared to their respective publications.

###  0.4.1 Gluon fusion modeling

In order to consistently combine the various analyses, it is necessary to use
the same theoretical predictions for the signal. The most significant
difference between the input analyses is the modeling of the dominant
$\Pg\Pg\PH$ production mode in the $\PH\to\cPZ\cPZ$, $\PH\to\Pgt\Pgt$,
$\PH\to\PGg\PGg$, and $\PH\to\PW\PW$ decay channels. The published results in
these analyses used different generators with next-to-leading order matrix
elements merged with parton showering (NLO+PS). In order to synchronize these
analyses and take advantage of the most accurate simulation of $\Pg\Pg\PH$
available, a reweighting is applied. Gluon fusion events are generated using
the 2.0 [78, 79, 80, 81], version 2.2.2 [82, 83], and nnlops [84, 85]
generators. The nnlops simulation, which is the highest order parton shower
matched $\Pg\Pg\PH$ simulation available, includes the effects of finite quark
masses. Events are separated into 0, 1, 2, and $\geq$3 jet bins, where the
jets used for counting are clustered from all stable particles, excluding the
decay products of the Higgs boson or associated vector bosons, and have
$\pt>30\GeV$. The sums of weights in each sample are first normalized to the
inclusive N3LO cross section. The ratio of the $\pt(\PH)$ distribution from
the nnlops generator to that from the or generators in each jet bin is applied
to the $\Pg\Pg\PH$ signal samples. The reweighting procedure has been checked
against fully simulated nnlops samples in the $\PH\to\PGg\PGg$ and
$\PH\to\Pgt\Pgt$ decay channels and was found to give results compatible
within the statistical uncertainty of the simulated samples. The
$\PH\to\Pgm\Pgm$ and boosted $\PH\to\PQb\PQb$ analyses, which are much less
sensitive to $\Pg\Pg\PH$ production than other decay channels, use the NLO+PS
simulation.

###  0.4.2 Theoretical uncertainties in gluon fusion

The $\Pg\Pg\PH$ cross section uncertainty scheme for the $\PH\to\cPZ\cPZ$ and
$\PH\to\Pgt\Pgt$ decay channels has been updated to the one proposed in Ref.
[53], as already used in the $\PH\to\PGg\PGg$ and $\PH\to\PW\PW$ analyses.
This uncertainty scheme includes 9 nuisance parameters accounting for
uncertainties in the cross section prediction for exclusive jet bins
(including the migration between the 0- and 1-jet, as well as between the 1-
and $\geq$2-jet bins), the 2-jet and $\geq$3-jet $\mathrm{VBF}$ phase spaces,
different $\pt(\PH)$ regions, and the uncertainty in the $\pt(\PH)$
distribution due to missing higher-order finite top quark mass corrections.
The boosted $\PH\to\PQb\PQb$ search, which is only sensitive to $\Pg\Pg\PH$ in
the high $\pt(\PH)$ tail, uses a dedicated prediction in this region, and
hence the theoretical uncertainties assigned are assumed to be uncorrelated
with the other analyses.

###  0.4.3 Statistical uncertainties in simulation

In the combination, many of the nuisance parameters originate from the use of
a limited number of Monte Carlo events to determine SM signal and background
expectations. Some of the input analyses have been modified to use the
“Barlow-Beeston lite” approach, which assigns a single nuisance parameter per
bin that scales the total bin yield [86, 87]. This differs from the previous
implementation, which utilized separate nuisance parameters for each process
per bin. With the Barlow-Beeston approach, the maximum likelihood estimator
for each of these nuisance parameters is independent from the others, and can
be solved for analytically. This has been found to provide a significant
reduction in the minimization time, while reproducing the results obtained
with the full treatment to within 1%.

##  0.5 Combination procedure

The overall statistical methodology used in this combination is the same as
the one developed by the ATLAS and CMS Collaborations, and described in Ref.
[56]. The procedures used in this paper are described in more detail in Refs.
[88, 89, 14] and are based on the standard LHC data modeling and handling
toolkits RooFit [90] and RooStats [91].

The parameters of interest (POI) $\vec{\alpha}$ for a particular model are
estimated with their corresponding confidence intervals using a profile
likelihood ratio test statistic $q(\vec{\alpha})$ [92], in which experimental
or theoretical uncertainties are incorporated via nuisance parameters (NP)
$\vec{\theta}$:

$q(\vec{\alpha})=-2\ln\left(\frac{L\big{(}\vec{\alpha}\,,\,{\hat{\vec{\theta}}}_{\vec{\alpha}}\big{)}}{L(\hat{\vec{\alpha}},\hat{\vec{\theta}})}\right).$
(1)

The likelihood functions in the numerator and denominator of Eq. (1) are
constructed using products of probability density functions of signal and
background for the various discriminating variables used in the input
analyses, as well as constraint terms for certain NPs. The probability density
functions are derived from simulation for the signal and from both data and
simulation for the background. The quantities $\hat{\vec{\alpha}}$ and
$\hat{\vec{\theta}}$ denote the unconditional maximum likelihood estimates of
the parameter values, while $\hat{\vec{\theta}}_{\vec{\alpha}}$ denotes the
conditional maximum likelihood estimate for fixed values of the parameters of
interest $\vec{\alpha}$. The choice of the POIs, e.g., signal strengths
($\mu$), couplings modifiers, production cross sections, branching fractions
or related ratios of the above quantities, depends on the specific model under
consideration, while the remaining parameters are treated as NPs. An
individual NP represents a single source of systematic uncertainty, and its
effect is therefore considered fully correlated between all of the input
analyses included in the fit.

For each model considered, the maximum likelihood estimates
$\hat{\vec{\alpha}}$ are identified as the best fit parameter values. The
$1\sigma$ and $2\sigma$ confidence level () intervals for one-dimensional
measurements of each POI are determined as the union of intervals for which
$q(\vec{\alpha})<1$ and $q(\vec{\alpha})<4$, respectively, unless otherwise
stated. In models with more than one POI, these intervals are determined
treating the other POIs as NPs. The differences between the boundaries of the
$1\sigma$ and $2\sigma$ intervals and the best fit value yield the $\pm
1\sigma$ and $\pm 2\sigma$ uncertainties on the measurement. In cases where a
physical boundary restricts the interval, we report a truncated interval and
determine the uncertainty from that interval. (See Fig. 6 and Table 0.7, for
example). In these cases, the intervals are not expected to maintain coverage.
In the case where the intervals are not contiguous, the interval that contains
the best fit point is used to determine these uncertainties. The 2D $1\sigma$
and $2\sigma$ regions are determined from the set of parameter values for
which $q(\vec{\alpha})<2.30$ and $q(\vec{\alpha})<6.18$, respectively, unless
otherwise stated.

The likelihood functions are constructed with respect to either the observed
data or an Asimov data set [92] constructed using the expected values of the
POIs for the SM, in order to obtain the observed or expected results,
respectively. Because of fluctuations in the observed data the observed
intervals may differ from the expected ones.

Finally, the SM predictions for the production and decay rates of the Higgs
boson depend on the mass of the Higgs boson, $m_{\PH}$. For all measurements
in this paper, the mass is taken to be $m_{\PH}=125.09\pm
0.21\mathrm{(stat)}\pm 0.11\mathrm{(syst)}\GeV$, determined from the ATLAS and
CMS combined measurement, from the LHC Run 1 data, using the high-resolution
$\PH\to\PGg\PGg$ and $\PH\to\cPZ\cPZ^{(\ast)}\to 4\ell$ decay channels [93].

##  0.6 Systematic uncertainties

For many of the POIs, the systematic uncertainties in their determination are
expected to be as large as, or larger than, the statistical uncertainties. The
theoretical uncertainties affecting the signal are among the most important
contributions to the systematic uncertainties. The uncertainties in the total
cross section prediction for the signal processes arising from the parton
distribution functions, the renormalization and factorization scales used in
the calculations and the branching fraction predictions are correlated between
all input analyses. Instead, theoretical uncertainties that affect kinematic
distributions and cause migrations between event categories are largely
uncorrelated between the input analyses. An exception is the set of
theoretical uncertainties for the $\Pg\Pg\PH$ production mode, where the
correlation scheme described in Section 0.4.2 is used to correlate both the
normalization and shape uncertainties between input analyses. The theoretical
uncertainties affecting the background predictions, including the parton
distribution function uncertainties, are assumed to be uncorrelated with those
affecting the signal predictions [88], with the exception of the uncertainties
from the underlying event and parton shower model.

The majority of the systematic uncertainties arising from experimental sources
are uncorrelated between the input analyses, with a few exceptions. The
uncertainties in the integrated luminosity measurement [94], and in the
modeling of additional collisions in the event (pileup), are correlated
between all of the input input analyses. Certain analyses, namely the
$\PH\to\Pgt\Pgt$, $\mathrm{V}\PH(\PQb\PQb)$, and $\cPqt\cPqt\PH(\PQb\PQb)$
analyses, are able to further constrain the jet energy scale uncertainties
determined in auxiliary measurements. The jet energy scale uncertainty in
these analyses is decomposed into several nuisance parameters corresponding to
different sources of uncertainty (for example, different flavor composition
and kinematic regions) that are correlated among these analyses but
uncorrelated with the other analyses. An independent jet energy scale
uncertainty is assumed to be correlated between the input analyses that are
not sensitive to the different sources of uncertainty. The uncertainties in
the $\PQb$ tagging efficiency are correlated between the $\cPqt\cPqt\PH$
analyses, but are uncorrelated from the $\mathrm{V}\PH(\PQb\PQb)$ analysis,
which is sensitive to different kinematic regions. A separate set of NPs is
used to describe the uncertainty in the $\PQb$ tagging efficiency in the
$\PH\to\PW\PW$, $\PH\to\PGg\PGg$, and $\PH\to\cPZ\cPZ$ analyses. The
uncertainty in the efficiency of the double-$\PQb$-tagger algorithm described
in Section 0.3.6 is taken to be uncorrelated from the single $\PQb$ tagging
uncertainties. Finally, the uncertainties in the lepton efficiency and
misidentification rate in the $\cPqt\cPqt\PH$-$\tauh$ and
$\cPqt\cPqt\PH$-$\Pe/\Pgm$ event classes are correlated, since the same
reconstruction and identification algorithms were used. In other input
analyses, different algorithms were used and therefore the uncertainties are
assumed to be uncorrelated.

The free parameters describing the shapes and normalizations of the background
models, and parameters that allow for the choice of the background
parametrization in each of the $\PH\to\PGg\PGg$ analysis categories are fully
determined by the data without any additional constraints, and are therefore
assigned to the statistical uncertainty of a measurement. The remaining
uncertainties are assigned to groups of systematic uncertainties.

##  0.7 Signal strength and cross section fits

The signal strength modifier $\mu$, defined as the ratio between the measured
Higgs boson yield and its SM expectation, has been used extensively to
characterize the Higgs boson yields. However, the specific meaning of $\mu$
varies depending on the analysis. For a specific production and decay channel
$i\to H\to f$, the signal strengths for the production, $\mu_{i}$, and for the
decay, $\mu^{f}$, are defined as:

$\mu_{i}=\frac{\sigma_{i}}{(\sigma_{i})_{\mathrm{SM}}}\hskip 14.22636pt{\rm
and}\hskip
14.22636pt\mu^{f}=\frac{\mathcal{B}^{f}}{(\mathcal{B}^{f})_{\mathrm{SM}}},\,\,$
(2)

respectively. Here
$\sigma_{i}\;(i=\Pg\Pg\PH,\,\mathrm{VBF},\,\PW\PH,\,\cPZ\PH,\,\cPqt\cPqt\PH)$
and
$\mathcal{B}^{f}\;(f=\cPZ\cPZ,\,\PW\PW,\,\PGg\PGg,\,\Pgt\Pgt,\,\PQb\PQb,\,\Pgm\Pgm)$
are, respectively, the production cross section for $i\to\PH$ and the
branching fraction for $\PH\to f$. The subscript ”SM” refers to their
respective SM predictions, so by definition, the SM corresponds to
$\mu_{i}=\mu^{f}=1$. Since $\sigma_{i}$ and $\mathcal{B}^{f}$ cannot be
separately measured without additional assumptions, only the product of
$\mu_{i}$ and $\mu^{f}$ can be extracted experimentally, leading to a signal
strength $\mu_{i}^{f}$ for the combined production and decay:

$\mu_{i}^{f}=\frac{\sigma_{i}\mathcal{B}^{f}}{(\sigma_{i})_{\mathrm{SM}}(\mathcal{B}^{f})_{\mathrm{SM}}}=\mu_{i}\mu^{f}.$
(3)

This parametrization makes use of the narrow width approximation, and the
reliability of this approximation was studied in Ref. [95] and found to be
adequate for global fits.

In this section, results are presented for several signal strength
parametrizations starting with a single global signal strength $\mu$, which is
the most restrictive in terms of the number of assumptions. Further
parametrizations are defined by relaxing the constraint that all production
and decay rates scale with a common signal strength modifier.

The combined measurement of the common signal strength modifier at
$m_{\PH}=125.09\,\GeV$ is,

$\begin{split}\mu&=1.17\pm{0.10}\\\
&=1.17\pm{0.06}\,\mathrm{(stat)}\,^{+0.06}_{-0.05}\,\mathrm{(sig\,theo)}\,\pm{0.06}\,\mathrm{(other\,syst)},\end{split}$
(4)

where the total uncertainty has been decomposed into statistical, signal
theoretical systematic, and other systematic components. The largest single
source of uncertainty apart from the signal theoretical systematic
uncertainties is the integrated luminosity ($\Delta\mu/\mu=2.5\%$), which is
correlated between all of the input analyses. In this measurement and others,
however, the other systematic uncertainty component is mostly dominated by
uncertainties that only affect a single input analysis.

Relaxing the assumption of a common production mode scaling, but still
assuming the relative SM branching fractions, leads to a parametrization with
five production signal strength modifiers: $\mu_{\Pg\Pg\PH}$,
$\mu_{\mathrm{VBF}}$, $\mu_{\PW\PH}$, $\mu_{\cPZ\PH}$, and
$\mu_{\cPqt\cPqt\PH}$. In this parametrization, as well as all subsequent
parametrizations involving signal strengths or cross sections, the $\cPqt\PH$
production is assumed to scale like $\cPqt\cPqt\PH$. Conversely, relaxing the
common decay mode scaling, but assuming the relative SM production cross
sections, leads to one with the modifiers: $\mu^{\PGg\PGg}$, $\mu^{\cPZ\cPZ}$,
$\mu^{\PW\PW}$, $\mu^{\Pgt\Pgt}$, $\mu^{\mu\mu}$, and $\mu^{\PQb\PQb}$.
Results of the fits in these two parametrizations are summarized in Fig. 5.
The numerical values, including the decomposition of the uncertainties into
statistical and systematic components, and the corresponding expected
uncertainties, are given in Table 0.7.

Figure 5: Summary plot of the fit to the per-production mode (left) and per-
decay mode (right) signal strength modifiers. The thick and thin horizontal
bars indicate the $\pm 1\sigma$ and $\pm 2\sigma$ uncertainties, respectively.
Also shown are the $\pm 1\sigma$ systematic components of the uncertainties.
The last point in the per-production mode summary plot is taken from a
separate fit and indicates the result of the combined overall signal strength
$\mu$.

Best fit values and $\pm 1\sigma$ uncertainties for the parametrizations with
per-production mode and per-decay mode signal strength modifiers. The expected
uncertainties are given in brackets.  Production process Best fit value
Uncertainty stat. syst. $\Pg\Pg\PH$ $1.22$ ${}^{+0.14}_{-0.12}$
${}^{+0.08}_{-0.08}$ ${}^{+0.12}_{-0.10}$ $({}^{+0.11}_{-0.11})$
$({}^{+0.07}_{-0.07})$ $({}^{+0.09}_{-0.08})$ $\mathrm{VBF}$ $0.73$
${}^{+0.30}_{-0.27}$ ${}^{+0.24}_{-0.23}$ ${}^{+0.17}_{-0.15}$
$({}^{+0.29}_{-0.27})$ $({}^{+0.24}_{-0.23})$ $({}^{+0.16}_{-0.15})$ $\PW\PH$
$2.18$ ${}^{+0.58}_{-0.55}$ ${}^{+0.46}_{-0.45}$ ${}^{+0.34}_{-0.32}$
$({}^{+0.53}_{-0.51})$ $({}^{+0.43}_{-0.42})$ $({}^{+0.30}_{-0.29})$ $\cPZ\PH$
$0.87$ ${}^{+0.44}_{-0.42}$ ${}^{+0.39}_{-0.38}$ ${}^{+0.20}_{-0.18}$
$({}^{+0.43}_{-0.41})$ $({}^{+0.38}_{-0.37})$ $({}^{+0.19}_{-0.17})$
$\cPqt\cPqt\PH$ $1.18$ ${}^{+0.30}_{-0.27}$ ${}^{+0.16}_{-0.16}$
${}^{+0.26}_{-0.21}$ $({}^{+0.28}_{-0.25})$ $({}^{+0.16}_{-0.15})$
$({}^{+0.23}_{-0.20})$ Decay mode Best fit value Uncertainty stat. syst.
$\PH\to\PQb\PQb$ $1.12$ ${}^{+0.29}_{-0.29}$ ${}^{+0.19}_{-0.18}$
${}^{+0.22}_{-0.22}$ $({}^{+0.28}_{-0.27})$ $({}^{+0.18}_{-0.18})$
$({}^{+0.21}_{-0.20})$ $\PH\to\Pgt\Pgt$ $1.02$ ${}^{+0.26}_{-0.24}$
${}^{+0.15}_{-0.15}$ ${}^{+0.21}_{-0.19}$ $({}^{+0.24}_{-0.22})$
$({}^{+0.15}_{-0.14})$ $({}^{+0.19}_{-0.17})$ $\PH\to\PW\PW$ $1.28$
${}^{+0.17}_{-0.16}$ ${}^{+0.09}_{-0.09}$ ${}^{+0.14}_{-0.13}$
$({}^{+0.14}_{-0.13})$ $({}^{+0.09}_{-0.09})$ $({}^{+0.11}_{-0.10})$
$\PH\to\cPZ\cPZ$ $1.06$ ${}^{+0.19}_{-0.17}$ ${}^{+0.16}_{-0.15}$
${}^{+0.11}_{-0.08}$ $({}^{+0.18}_{-0.16})$ $({}^{+0.15}_{-0.14})$
$({}^{+0.10}_{-0.08})$ $\PH\to\PGg\PGg$ $1.20$ ${}^{+0.18}_{-0.14}$
${}^{+0.13}_{-0.11}$ ${}^{+0.12}_{-0.09}$ $({}^{+0.14}_{-0.12})$
$({}^{+0.10}_{-0.10})$ $({}^{+0.09}_{-0.07})$ $\PH\to\Pgm\Pgm$ $0.68$
${}^{+1.25}_{-1.24}$ ${}^{+1.24}_{-1.24}$ ${}^{+0.13}_{-0.11}$
$({}^{+1.20}_{-1.17})$ $({}^{+1.18}_{-1.17})$ $({}^{+0.19}_{-0.03})$

The improvement in the precision of the measurement of the $\Pg\Pg\PH$
production rate of $\sim$50% (from $\sim$20% to $\sim$10%) compared to Ref.
[55] and $\sim$33% (from $\sim$15% to $\sim$10%) compared to Ref. [56], can be
attributed to the combined effects of an increased $\Pg\Pg\PH$ production
cross section, and a reduction in the associated theoretical uncertainties.
The improvements in the precision are up to $\sim$20% for the $\mathrm{VBF}$
and $\mathrm{V}\PH$ production rates compared to Ref. [55] . The uncertainty
in the measurement of the $\cPqt\cPqt\PH$ production rate is reduced by around
50% compared to Ref. [56]. This is in part due to the increase in the
$\cPqt\cPqt\PH$ cross section between 8 and 13, but also due to the inclusion
of additional exclusive event categories for this production process.

The most generic signal strength parametrization has one signal strength
parameter for each production and decay mode combination, $\mu_{i}^{f}$. Given
the five production and six decay modes listed above, this implies a model
with 30 parameters of interest. However not all can be experimentally
constrained in this combination. There is no dedicated analysis from CMS at
$\sqrt{s}=13\TeV$ targeting $\PW\PH$ and $\cPZ\PH$ production with
$\PH\to\Pgt\Pgt$ decay, or $\mathrm{VBF}$ production with $\PH\to\PQb\PQb$
decay, therefore these signal strength modifiers are fixed to the SM
expectation and are not included in the maximum likelihood fit. Likewise, the
$\PW\PH$, $\cPZ\PH$, and $\cPqt\cPqt\PH$ production rates with
$\PH\to\Pgm\Pgm$ decay are fixed to the SM expectation. In the case of
$\PW\PH$, $\cPZ\PH$, and $\cPqt\cPqt\PH$ production with $\PH\to\cPZ\cPZ$
decay, as well as $\cPZ\PH$ production with $\PH\to\PGg\PGg$ decay, the
background contamination is sufficiently low so that a negative signal
strength can result in an overall negative event yield. Therefore, these
signal strengths are restricted to nonnegative values. Figure 6 summarizes the
results in this model along with the $1\sigma$ intervals. The numerical
values, including the uncertainty decomposition into statistical and
systematic parts, and the corresponding expected uncertainties, are given in
Table 0.7.

Figure 6: Summary plot of the fit to the production–decay signal strength
products $\mu_{i}^{f}=\mu_{i}\mu^{f}$. The points indicate the best fit values
while the horizontal bars indicate the $1\sigma$ intervals. The hatched areas
indicate signal strengths that are restricted to nonnegative values as
described in the text.

Best fit values and $\pm 1\sigma$ uncertainties for the parameters of the
model with one signal strength parameter for each production and decay mode
combination. The entries in the table represent the parameter
$\mu_{i}^{f}=\mu_{i}\mu^{f}$, where $i$ is indicated by the column and $f$ by
the row. The expected uncertainties are given in brackets. Some of the signal
strengths are restricted to nonnegative values, as described in the text.
Decay mode Production process $\Pg\Pg\PH$ $\mathrm{VBF}$ $\PW\PH$ $\cPZ\PH$
$\cPqt\cPqt\PH$ Best fit Uncertainty Best fit Uncertainty Best fit Uncertainty
Best fit Uncertainty Best fit Uncertainty value stat syst value stat syst
value stat syst value stat syst value stat syst $\PH\to\PQb\PQb$ $2.51$
${}^{+2.43}_{-2.01}$ ${}^{+1.96}_{-1.92}$ ${}^{+1.44}_{-0.59}$ — $1.73$
${}^{+0.70}_{-0.68}$ ${}^{+0.53}_{-0.51}$ ${}^{+0.46}_{-0.45}$ $0.99$
${}^{+0.47}_{-0.45}$ ${}^{+0.41}_{-0.40}$ ${}^{+0.23}_{-0.20}$ $0.91$
${}^{+0.45}_{-0.43}$ ${}^{+0.24}_{-0.23}$ ${}^{+0.38}_{-0.36}$
$({}^{+2.06}_{-1.86})$ $({}^{+1.85}_{-1.83})$ $({}^{+0.89}_{-0.33})$ —
$({}^{+0.69}_{-0.67})$ $({}^{+0.52}_{-0.51})$ $({}^{+0.45}_{-0.44})$
$({}^{+0.46}_{-0.44})$ $({}^{+0.40}_{-0.39})$ $({}^{+0.23}_{-0.20})$
$({}^{+0.44}_{-0.42})$ $({}^{+0.23}_{-0.23})$ $({}^{+0.37}_{-0.35})$
$\PH\to\Pgt\Pgt$ $1.05$ ${}^{+0.53}_{-0.47}$ ${}^{+0.25}_{-0.25}$
${}^{+0.46}_{-0.40}$ $1.12$ ${}^{+0.45}_{-0.43}$ ${}^{+0.37}_{-0.35}$
${}^{+0.25}_{-0.25}$ — — $0.23$ ${}^{+1.03}_{-0.88}$ ${}^{+0.80}_{-0.71}$
${}^{+0.65}_{-0.52}$ $({}^{+0.45}_{-0.41})$ $({}^{+0.23}_{-0.23})$
$({}^{+0.38}_{-0.34})$ $({}^{+0.45}_{-0.43})$ $({}^{+0.37}_{-0.35})$
$({}^{+0.25}_{-0.24})$ — — $({}^{+0.98}_{-0.87})$ $({}^{+0.80}_{-0.73})$
$({}^{+0.56}_{-0.47})$ $\PH\to\PW\PW$ $1.35$ ${}^{+0.21}_{-0.19}$
${}^{+0.12}_{-0.12}$ ${}^{+0.17}_{-0.15}$ $0.28$ ${}^{+0.64}_{-0.60}$
${}^{+0.58}_{-0.53}$ ${}^{+0.28}_{-0.28}$ $3.91$ ${}^{+2.26}_{-2.01}$
${}^{+1.89}_{-1.72}$ ${}^{+1.24}_{-1.05}$ $0.96$ ${}^{+1.81}_{-1.46}$
${}^{+1.74}_{-1.44}$ ${}^{+0.50}_{-0.22}$ $1.60$ ${}^{+0.65}_{-0.59}$
${}^{+0.40}_{-0.39}$ ${}^{+0.52}_{-0.45}$ $({}^{+0.17}_{-0.16})$
$({}^{+0.10}_{-0.10})$ $({}^{+0.13}_{-0.12})$ $({}^{+0.62}_{-0.58})$
$({}^{+0.57}_{-0.53})$ $({}^{+0.26}_{-0.25})$ $({}^{+1.47}_{-1.19})$
$({}^{+1.32}_{-1.06})$ $({}^{+0.64}_{-0.54})$ $({}^{+1.67}_{-1.37})$
$({}^{+1.61}_{-1.35})$ $({}^{+0.45}_{-0.20})$ $({}^{+0.56}_{-0.53})$
$({}^{+0.38}_{-0.38})$ $({}^{+0.41}_{-0.37})$ $\PH\to\cPZ\cPZ$ $1.22$
${}^{+0.23}_{-0.21}$ ${}^{+0.20}_{-0.19}$ ${}^{+0.12}_{-0.10}$ $-0.09$
${}^{+1.02}_{-0.76}$ ${}^{+1.00}_{-0.72}$ ${}^{+0.21}_{-0.22}$ $0.00$
${}^{+2.33}_{+0.00}$ ${}^{+2.31}_{-0.00}$ ${}^{+0.30}_{-0.00}$ $0.00$
${}^{+4.26}_{+0.00}$ ${}^{+4.19}_{-0.00}$ ${}^{+0.80}_{-0.00}$ $0.00$
${}^{+1.50}_{+0.00}$ ${}^{+1.47}_{-0.00}$ ${}^{+0.30}_{-0.00}$
$({}^{+0.22}_{-0.20})$ $({}^{+0.20}_{-0.19})$ $({}^{+0.10}_{-0.07})$
$({}^{+1.27}_{-0.99})$ $({}^{+1.25}_{-0.97})$ $({}^{+0.23}_{-0.21})$
$({}^{+4.46}_{-0.99})$ $({}^{+4.42}_{-0.99})$ $({}^{+0.57}_{-0.00})$
$({}^{+7.57}_{-1.00})$ $({}^{+7.45}_{-1.00})$ $({}^{+1.33}_{-0.00})$
$({}^{+2.95}_{-0.99})$ $({}^{+2.89}_{-0.99})$ $({}^{+0.59}_{-0.00})$
$\PH\to\PGg\PGg$ $1.16$ ${}^{+0.21}_{-0.18}$ ${}^{+0.17}_{-0.15}$
${}^{+0.13}_{-0.10}$ $0.67$ ${}^{+0.59}_{-0.46}$ ${}^{+0.49}_{-0.42}$
${}^{+0.32}_{-0.18}$ $3.76$ ${}^{+1.48}_{-1.35}$ ${}^{+1.45}_{-1.33}$
${}^{+0.33}_{-0.24}$ $0.00$ ${}^{+1.14}_{+0.00}$ ${}^{+1.14}_{-0.00}$
${}^{+0.09}_{-0.00}$ $2.18$ ${}^{+0.88}_{-0.75}$ ${}^{+0.82}_{-0.74}$
${}^{+0.32}_{-0.14}$ $({}^{+0.17}_{-0.16})$ $({}^{+0.14}_{-0.14})$
$({}^{+0.11}_{-0.08})$ $({}^{+0.59}_{-0.48})$ $({}^{+0.48}_{-0.43})$
$({}^{+0.34}_{-0.21})$ $({}^{+1.28}_{-1.16})$ $({}^{+1.27}_{-1.16})$
$({}^{+0.13}_{-0.06})$ $({}^{+2.51}_{-1.04})$ $({}^{+2.50}_{-1.04})$
$({}^{+0.25}_{-0.00})$ $({}^{+0.74}_{-0.63})$ $({}^{+0.72}_{-0.63})$
$({}^{+0.16}_{-0.06})$ $\PH\to\Pgm\Pgm$ $0.31$ ${}^{+1.80}_{-1.79}$
${}^{+1.79}_{-1.78}$ ${}^{+0.19}_{-0.19}$ $2.72$ ${}^{+7.12}_{-7.03}$
${}^{+7.12}_{-7.04}$ ${}^{+0.26}_{-0.00}$ — — — $({}^{+1.69}_{-1.65})$
$({}^{+1.67}_{-1.67})$ $({}^{+0.28}_{-0.00})$ $({}^{+7.02}_{-6.94})$
$({}^{+7.01}_{-6.93})$ $({}^{+0.38}_{-0.50})$ — — —

###  0.7.1 Ratios of cross sections and branching fractions, relative to
$\Pg\Pg\PH\to\cPZ\cPZ$

Results are presented for a model based on the ratios of cross sections and
branching fractions. These are given relative to a well-measured reference
process, chosen to be $\Pg\Pg\PH\to\cPZ\cPZ$ ($\mu_{\Pg\Pg\PH}^{\cPZ\cPZ}$).
Using ratios has the advantage that some systematic or theoretical
uncertainties common to both the numerator and denominator cancel. The
following ratios are used: $\mu^{\PGg\PGg}/\mu^{\cPZ\cPZ}$,
$\mu^{\PW\PW}/\mu^{\cPZ\cPZ}$,
$\mu^{\Pgt\Pgt}/\mu^{\cPZ\cPZ}$,$\mu^{\mu\mu}/\mu^{\cPZ\cPZ}$,
$\mu^{\PQb\PQb}/\mu^{\cPZ\cPZ}$, $\mu_{\mathrm{VBF}}/\mu_{\Pg\Pg\PH}$,
$\mu_{\PW\PH}/\mu_{\Pg\Pg\PH}$, $\mu_{\cPZ\PH}/\mu_{\Pg\Pg\PH}$, and
$\mu_{\cPqt\cPqt\PH}/\mu_{\Pg\Pg\PH}$. These results are summarized in Fig. 7,
and the numerical values are given in Table 0.7.1. The uncertainties in the SM
predictions are included in the measurements.

Figure 7: Summary of the cross section and branching fraction ratio model. The
thick and thin horizontal bars indicate the $\pm 1\sigma$ and $\pm 2\sigma$
uncertainties, respectively. Also shown are the $\pm 1\sigma$ systematic
components of the uncertainties.

Best fit values and $\pm 1\sigma$ uncertainties for the parameters of the
cross section and branching fraction ratio model. The expected uncertainties
are given in brackets.  Uncertainty Uncertainty Parameter Best fit stat syst
Parameter Best fit stat syst $\mu_{\Pg\Pg\PH}^{\cPZ\cPZ}$ $1.07$
${}^{+0.20}_{-0.18}$ ${}^{+0.16}_{-0.15}$ ${}^{+0.11}_{-0.09}$
${\mathcal{B}}^{\PQb\PQb}/{\mathcal{B}}^{\cPZ\cPZ}$ $0.84$
${}^{+0.37}_{-0.27}$ ${}^{+0.27}_{-0.21}$ ${}^{+0.26}_{-0.17}$
$(^{+0.19}_{-0.16})$ $(^{+0.15}_{-0.14})$ $(^{+0.11}_{-0.08})$
$(^{+0.56}_{-0.37})$ $(^{+0.38}_{-0.28})$ $(^{+0.40}_{-0.25})$
$\mu_{\mathrm{VBF}}/\mu_{\Pg\Pg\PH}$ $0.60$ ${}^{+0.30}_{-0.24}$
${}^{+0.24}_{-0.21}$ ${}^{+0.17}_{-0.13}$
${\mathcal{B}}^{\Pgt\Pgt}/{\mathcal{B}}^{\cPZ\cPZ}$ $1.07$
${}^{+0.37}_{-0.30}$ ${}^{+0.25}_{-0.21}$ ${}^{+0.27}_{-0.22}$
$(^{+0.40}_{-0.32})$ $(^{+0.31}_{-0.27})$ $(^{+0.24}_{-0.17})$
$(^{+0.35}_{-0.28})$ $(^{+0.25}_{-0.20})$ $(^{+0.25}_{-0.19})$
$\mu_{\PW\PH}/\mu_{\Pg\Pg\PH}$ $2.19$ ${}^{+0.86}_{-0.69}$
${}^{+0.68}_{-0.56}$ ${}^{+0.52}_{-0.39}$
${\mathcal{B}}^{\PW\PW}/{\mathcal{B}}^{\cPZ\cPZ}$ $1.23$ ${}^{+0.27}_{-0.22}$
${}^{+0.22}_{-0.18}$ ${}^{+0.16}_{-0.13}$ $(^{+0.65}_{-0.52})$
$(^{+0.53}_{-0.44})$ $(^{+0.39}_{-0.29})$ $(^{+0.24}_{-0.19})$
$(^{+0.19}_{-0.16})$ $(^{+0.14}_{-0.11})$ $\mu_{\cPZ\PH}/\mu_{\Pg\Pg\PH}$
$0.88$ ${}^{+0.57}_{-0.44}$ ${}^{+0.49}_{-0.41}$ ${}^{+0.30}_{-0.17}$
${\mathcal{B}}^{\gamma\gamma}/{\mathcal{B}}^{\cPZ\cPZ}$ $1.14$
${}^{+0.28}_{-0.20}$ ${}^{+0.23}_{-0.18}$ ${}^{+0.16}_{-0.09}$
$(^{+0.68}_{-0.47})$ $(^{+0.53}_{-0.41})$ $(^{+0.43}_{-0.23})$
$(^{+0.23}_{-0.18})$ $(^{+0.20}_{-0.16})$ $(^{+0.11}_{-0.08})$
$\mu_{\cPqt\cPqt\PH}/\mu_{\Pg\Pg\PH}$ $1.06$ ${}^{+0.34}_{-0.27}$
${}^{+0.20}_{-0.18}$ ${}^{+0.27}_{-0.20}$
${\mathcal{B}}^{\mu\mu}/{\mathcal{B}}^{\cPZ\cPZ}$ $0.63$ ${}^{+1.24}_{-1.21}$
${}^{+1.24}_{-1.20}$ ${}^{+0.15}_{-0.11}$ $(^{+0.36}_{-0.30})$
$(^{+0.23}_{-0.21})$ $(^{+0.27}_{-0.21})$ $(^{+1.26}_{-1.19})$
$(^{+1.25}_{-1.19})$ $(^{+0.20}_{-0.03})$

###  0.7.2 Stage 0 simplified template cross sections

Measurements of production cross sections, which are complementary to the
signal strength parametrization, are made for seven processes defined
according to the simplified template cross sections proposed in Ref. [53]. The
results given here are for the stage 0 fiducial regions defined by the
rapidity of the Higgs boson $\abs{y_{\PH}}<2.5$. All input analyses have a
negligible acceptance for $\abs{y_{\PH}}>2.5$. Defining the fiducial region in
this way reduces the theoretical uncertainty that would otherwise apply while
extrapolating to the fully inclusive phase space. Subsequent stages involve
splitting the fiducial regions into a number of smaller ones, for example
based on ranges of the Higgs boson . The measured cross sections are defined
as:

  * •

$\sigma_{\Pg\Pg\PH+\PQb\PQb\PH}$: gluon fusion and $\PQb$-associated
production. While Ref. [53] proposes separate bins for these modes, they are
merged here because of the current lack of sensitivity to the associated
production with $\PQb$ quarks.

  * •

$\sigma_{\mathrm{VBF}}$: $\mathrm{VBF}$ production.

  * •

$\sigma_{\PH+\mathrm{V}(\mathrm{qq})}$: Associated production with a $\cPZ$ or
$\PW$ boson, either quark or gluon initiated, in which the vector boson decays
hadronically.

  * •

$\sigma_{\PH+\PZ(\ell\ell/\nu\nu)}$: Associated production with a $\cPZ$
boson, in which the $\cPZ$ boson decays leptonically. While Ref. [53] proposes
separate bins for the quark- and gluon-initiated modes, they are merged here
because they cannot easily be distinguished experimentally, and therefore,
their measurements would be highly anticorrelated.

  * •

$\sigma_{\PH+\PW(\ell\nu)}$: Associated production with a $\PW$ boson, in
which the $\PW$ decays leptonically.

  * •

$\sigma_{\cPqt\cPqt\PH+\PQt\PH}$: Associated production with a pair of top
quarks or a single top quark. While Ref. [53] proposes separate bins for these
modes, they are merged here because of the lack of a dedicated analysis
targeting $\cPqt\PH$ production in this combination.

In addition to the cross sections, the Higgs boson branching fractions are
also included as POIs via ratios with respect to $\mathcal{B}^{\cPZ\cPZ}$. A
summary of the results in this model, normalized to the expected SM cross
sections, is given in Fig. 8 and Table 0.7.2. Since cross sections are
measured and not signal strength modifiers, the theoretical uncertainties in
these cross sections do not enter as sources of uncertainty. In Fig. 8, the
uncertainties in the SM predictions are indicated by gray bands.

Figure 8: Summary of the stage 0 model, ratios of cross sections and branching
fractions. The points indicate the best fit values, while the error bars show
the $\pm 1\sigma$ and $\pm 2\sigma$ uncertainties. The $\pm 1\sigma$
uncertainties on the measurements considering only the contributions from the
systematic uncertainties are also shown. The uncertainties in the SM
predictions are indicated.

Best fit values and $\pm 1\sigma$ uncertainties for the parameters of the
stage 0 simplified template cross section model. The values are all normalized
to the SM predictions. The expected uncertainties are given in brackets.
Uncertainty Uncertainty Parameter Best fit stat syst Parameter Best fit stat
syst $\sigma_{\Pg\Pg\PH}{\mathcal{B}}^{\cPZ\cPZ}$ $1.00$ ${}^{+0.19}_{-0.16}$
${}^{+0.16}_{-0.15}$ ${}^{+0.09}_{-0.07}$
${\mathcal{B}}^{\PQb\PQb}/{\mathcal{B}}^{\cPZ\cPZ}$ $0.96$
${}^{+0.44}_{-0.31}$ ${}^{+0.32}_{-0.24}$ ${}^{+0.30}_{-0.20}$
$(^{+0.18}_{-0.16})$ $(^{+0.16}_{-0.15})$ $(^{+0.09}_{-0.07})$
$(^{+0.57}_{-0.38})$ $(^{+0.40}_{-0.29})$ $(^{+0.41}_{-0.25})$
$\sigma_{\mathrm{VBF}}{\mathcal{B}}^{\cPZ\cPZ}$ $0.66$ ${}^{+0.32}_{-0.26}$
${}^{+0.27}_{-0.22}$ ${}^{+0.17}_{-0.13}$
${\mathcal{B}}^{\Pgt\Pgt}/{\mathcal{B}}^{\cPZ\cPZ}$ $0.98$
${}^{+0.35}_{-0.28}$ ${}^{+0.24}_{-0.20}$ ${}^{+0.25}_{-0.20}$
$(^{+0.40}_{-0.32})$ $(^{+0.33}_{-0.27})$ $(^{+0.22}_{-0.16})$
$(^{+0.36}_{-0.29})$ $(^{+0.26}_{-0.21})$ $(^{+0.25}_{-0.19})$
$\sigma_{\PH+\rm{V}(\Pq\Pq)}{\mathcal{B}}^{\cPZ\cPZ}$ $3.93$
${}^{+2.00}_{-1.71}$ ${}^{+1.77}_{-1.53}$ ${}^{+0.93}_{-0.75}$
${\mathcal{B}}^{\PW\PW}/{\mathcal{B}}^{\cPZ\cPZ}$ $1.30$ ${}^{+0.29}_{-0.24}$
${}^{+0.24}_{-0.20}$ ${}^{+0.17}_{-0.13}$ $(^{+1.66}_{-1.05})$
$(^{+1.49}_{-1.05})$ $(^{+0.72}_{-0.00})$ $(^{+0.24}_{-0.20})$
$(^{+0.20}_{-0.16})$ $(^{+0.14}_{-0.11})$
$\sigma_{\PH+\PW(\ell\nu)}{\mathcal{B}}^{\cPZ\cPZ}$ $1.95$
${}^{+0.88}_{-0.68}$ ${}^{+0.72}_{-0.57}$ ${}^{+0.51}_{-0.38}$
${\mathcal{B}}^{\gamma\gamma}/{\mathcal{B}}^{\cPZ\cPZ}$ $1.14$
${}^{+0.26}_{-0.20}$ ${}^{+0.23}_{-0.18}$ ${}^{+0.13}_{-0.09}$
$(^{+0.69}_{-0.52})$ $(^{+0.56}_{-0.44})$ $(^{+0.40}_{-0.29})$
$(^{+0.23}_{-0.19})$ $(^{+0.21}_{-0.17})$ $(^{+0.11}_{-0.08})$
$\sigma_{\PH+{\cPZ(\ell\ell/\nu\nu)}}{\mathcal{B}}^{\cPZ\cPZ}$ $0.84$
${}^{+0.57}_{-0.43}$ ${}^{+0.49}_{-0.40}$ ${}^{+0.29}_{-0.17}$
${\mathcal{B}}^{\mu\mu}/{\mathcal{B}}^{\cPZ\cPZ}$ $0.67$ ${}^{+1.40}_{-1.36}$
${}^{+1.39}_{-1.35}$ ${}^{+0.18}_{-0.13}$ $(^{+0.71}_{-0.46})$
$(^{+0.56}_{-0.41})$ $(^{+0.44}_{-0.22})$ $(^{+1.35}_{-1.28})$
$(^{+1.34}_{-1.28})$ $(^{+0.17}_{-0.05})$
$\sigma_{\cPqt\cPqt\PH}{\mathcal{B}}^{\cPZ\cPZ}$ $1.08$ ${}^{+0.37}_{-0.30}$
${}^{+0.26}_{-0.22}$ ${}^{+0.26}_{-0.19}$ $(^{+0.38}_{-0.31})$
$(^{+0.28}_{-0.23})$ $(^{+0.26}_{-0.20})$

##  0.8 Measurements of Higgs boson couplings

In the $\kappa$-framework [96], coupling modifiers are introduced in order to
test for deviations in the couplings of the Higgs boson to other particles. In
order to measure the individual Higgs couplings in this framework, some
assumption must be made to constrain the total Higgs boson width since it
cannot be directly measured at the LHC. Unless stated otherwise, it is assumed
that there are no BSM contributions to the total Higgs boson width. With this
assumption, the cross section times branching fraction for a production
process $i$ and decay $f$ can be expressed as,

$\sigma_{i}{\mathcal{B}}^{f}=\frac{\sigma_{\mathit{i}}(\vec{\kappa})\Gamma^{\mathit{f}}(\vec{\kappa})}{\Gamma_{\PH}(\vec{\kappa})},$
(5)

where $\Gamma_{\PH}(\vec{\kappa})$ is the total width of the Higgs boson and
$\Gamma^{\mathit{f}}(\vec{\kappa})$ is the partial width of the Higgs boson
decay to the final state $f$. A set of coupling modifiers, $\vec{\kappa}$, is
introduced to parameterize potential deviations in the bosonic and fermionic
couplings of the Higgs boson from the SM predictions. For a given production
process or decay mode $j$, a coupling modifier $\kappa_{j}$ is defined such
that,

$\kappa_{j}^{2}=\sigma_{j}/\sigma_{j}^{\mathrm{SM}}\ \ \ \mathrm{or}\ \ \
\kappa_{j}^{2}=\Gamma^{j}/\Gamma^{j}_{\mathrm{SM}}.$ (6)

In the SM, all $\kappa_{j}$ values are positive and equal to unity. In this
parametrization it is assumed that the higher-order accuracy of the QCD and
electroweak corrections to the SM cross sections and branching fractions is
preserved when the values of $\kappa_{j}$ deviate from unity. While this does
not hold in general, for the parameter ranges considered in this paper the
dominant higher-order QCD corrections largely factorize from the rescaling of
the couplings, therefore the approach is considered valid. Individual coupling
modifiers, corresponding to tree-level Higgs boson couplings to the different
particles, are introduced, as well as effective coupling modifiers
$\kappa_{\cPg}$ and $\kappa_{\PGg}$ that describe $\Pg\Pg\PH$ production and
$\PH\to\PGg\PGg$ decay. This is possible because the presence of any BSM
particles in these loops is not expected to significantly change the
corresponding kinematic properties of the processes. This approach is not
possible for $\cPg\cPg\to\cPZ\PH$ production, which occurs at leading order
through box and triangular loop diagrams, because a tree-level contact
interaction from BSM physics would likely exhibit a kinematic structure very
different from the SM, and is expected to be highly suppressed [97]. Other
possible BSM effects on the $\cPg\cPg\to\cPZ\PH$ process are related to
modifications of the $\PH\cPZ\cPZ$ and $\cPqt\cPqt\PH$ vertices, which are
best taken into account, within the limitation of the framework, by resolving
the loop in terms of the corresponding coupling modifiers, $\kappa_{\cPZ}$ and
$\kappa_{\PQt}$. More details on the development of this framework as well as
its theoretical and phenomenological foundations and extensions can be found,
for example, in Refs. [98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108,
109, 110, 111, 112].

The normalization scaling effects of each of the $\kappa$ parameters are given
in Table 0.8. Loop processes such as $\Pg\Pg\PH$ and $\PH\to\PGg\PGg$ can be
studied through either the effective coupling modifiers, thereby providing
sensitivity to potential BSM physics in the loops, or the modifiers of the SM
particles themselves. Interference between different diagrams, such as those
that contribute to $\cPg\cPg\to\cPZ\PH$, provides some sensitivity to relative
signs between Higgs boson couplings to different particles. Modifications to
the kinematic distributions of the $\cPqt\PH$ production are also expected
when the relative sign of $\kappa_{\PQt}$ and $\kappa_{\PW}$ is negative.
These effects were studied and the distributions of the final observables were
found to be insensitive with the present dataset to the relative sign of
$\kappa_{\PQt}$ and $\kappa_{\PW}$.

Normalization scaling factors for all relevant production cross sections and
decay partial widths. For the $\kappa$ parameters representing loop processes,
the resolved scaling in terms of the fundamental SM couplings is also given.
Effective Loops Interference scaling factor Resolved scaling factor Production
$\sigma(\Pg\Pg\PH)$ $\checkmark$ - $\kappa_{\Pg}^{2}$
$1.04\kappa_{\PQt}^{2}+0.002\kappa_{\PQb}^{2}-0.038\kappa_{\PQt}\kappa_{\PQb}$
$\sigma(\mathrm{VBF})$ — — $0.73\kappa_{\PW}^{2}+0.27\kappa_{\PZ}^{2}$
$\sigma(\PW\PH)$ — — $\kappa_{\PW}^{2}$ $\sigma(\Pq\Pq/\Pq\cPg\to\PZ\PH)$ — —
$\kappa_{\PZ}^{2}$ $\sigma(\cPg\cPg\to\PZ\PH)$ $\checkmark$ -
$2.46\kappa_{\PZ}^{2}+0.47\kappa_{\PQt}^{2}-1.94\kappa_{\PZ}\kappa_{\PQt}$
$\sigma(\cPqt\cPqt\PH)$ — — $\kappa_{\PQt}^{2}$
$\sigma(\cPg\PQb\to\PW\PQt\PH)$ — -
$2.91\kappa_{\PQt}^{2}+2.31\kappa_{\PW}^{2}-4.22\kappa_{\PQt}\kappa_{\PW}$
$\sigma(\Pq\PQb\to\PQt\PH\Pq)$ — -
$2.63\kappa_{\PQt}^{2}+3.58\kappa_{\PW}^{2}-5.21\kappa_{\PQt}\kappa_{\PW}$
$\sigma(\PQb\PQb\PH)$ — — $\kappa_{\PQb}^{2}$ Partial decay width
$\Gamma^{\PZ\PZ}$ — — $\kappa_{\PZ}^{2}$ $\Gamma^{\PW\PW}$ — —
$\kappa_{\PW}^{2}$ $\Gamma^{\gamma\gamma}$ $\checkmark$ - $\kappa_{\PGg}^{2}$
$1.59\kappa_{\PW}^{2}+0.07\kappa_{\PQt}^{2}-0.67\kappa_{\PW}\kappa_{\PQt}$
$\Gamma^{\tau\tau}$ — — $\kappa_{\tau}^{2}$ $\Gamma^{\PQb\PQb}$ — —
$\kappa_{\PQb}^{2}$ $\Gamma^{\mu\mu}$ — — $\kappa_{\mu}^{2}$ Total width for
$\mathcal{B}_{\mathrm{BSM}}=0$
$0.58\kappa_{\PQb}^{2}+0.22\kappa_{\PW}^{2}+0.08\kappa_{\Pg}^{2}+$
$\Gamma_{\PH}$ $\checkmark$ — $\kappa_{\PH}^{2}$
$+\,0.06\kappa_{\PGt}^{2}+0.026\kappa_{\PZ}^{2}+0.029\kappa_{\PQc}^{2}+$
$+\,0.0023\kappa_{\PGg}^{2}+\,0.0015\kappa_{\PZ\PGg}^{2}+$
$+\,0.00025\kappa_{\PQs}^{2}+0.00022\kappa_{\PGm}^{2}$

###  0.8.1 Generic model within $\kappa$-framework assuming resolved loops

Under the assumption that there are no BSM particles contributing to the
$\Pg\Pg\PH$ production or $\PH\to\PGg\PGg$ decay loops, these processes can be
expressed in terms of the coupling modifiers to the SM particles as described
previously. There are six free coupling parameters: $\kappa_{\PW}$,
$\kappa_{\cPZ}$, $\kappa_{\PQt}$, $\kappa_{\Pgt}$, $\kappa_{\PQb}$, and
$\kappa_{\Pgm}$. Without loss of generality, the value of $\kappa_{\PQt}$ is
restricted to be positive, while both negative and positive values of
$\kappa_{\PW}$, $\kappa_{\cPZ}$ and $\kappa_{\PQb}$ are allowed. In this
model, the rates of the $\Pg\Pg\PH$ and $\PH\to\PGg\PGg$ processes, which
occur through loop diagrams at leading order, are resolved, meaning that they
are described by the functions of $\kappa_{\PW}$, $\kappa_{\cPZ}$,
$\kappa_{\Pgt}$, and $\kappa_{\PQb}$ given in Table 0.8. The results of the
fits with this parametrization are given in Fig. 9 and Table 0.8.1.

Figure 9: Summary of the $\kappa$-framework model assuming resolved loops and
$\mathcal{B}_{\mathrm{BSM}}=0$. The points indicate the best fit values while
the thick and thin horizontal bars show the $1\sigma$ and $2\sigma$ intervals,
respectively. In this model, the $\Pg\Pg\PH$ and $\PH\to\PGg\PGg$ loops are
resolved in terms of the remaining coupling modifiers. For this model, both
positive and negative values of $\kappa_{\PW}$, $\kappa_{\cPZ}$, and
$\kappa_{\PQb}$ are considered. Negative values of $\kappa_{\PW}$ in this
model are disfavored by more than $2\sigma$.

The rate of the $\PH\to\cPZ\cPZ$ decay and $\cPZ\PH$ production depend only on
the absolute value of $\kappa_{\cPZ}$. The interference between the two
diagrams shown in Fig. 3, however, allows contributions from the
$\Pg\Pg\to\cPZ\PH$ production mode to break the degeneracy between the signs,
leading to a positive value of $\kappa_{\cPZ}$ being preferred. As these
contributions are typically small compared to other production modes, the
1$\sigma$ and 2$\sigma$ intervals also include negative values of
$\kappa_{\cPZ}$. Although a negative value of $\kappa_{\PQb}$ is preferred in
this model, the difference in $q$ between the best fit point and the minimum
in the region $\kappa_{\PQb}>0$ is smaller than 0.1.

An additional fit is performed using a phenomenological parametrization
relating the masses of the fermions and vector bosons to the corresponding
$\kappa$ modifiers using two parameters, denoted $M$ and $\epsilon$ [113,
114]. In such a model one can relate the coupling modifiers to $M$ and
$\epsilon$ as
$\kappa_{\mathrm{F}}=v\;m_{\mathrm{f}}^{\epsilon}/M^{1+\epsilon}$ for fermions
and $\kappa_{\mathrm{V}}=v\;m_{\mathrm{V}}^{2\epsilon}/M^{1+2\epsilon}$ for
vector bosons. Here, $v=246.22$, is the SM Higgs boson vacuum expectation
value [115]. The SM expectation, $\kappa_{i}=1$, is recovered when
$(M,\epsilon)=(v,0)$.

The lepton and vector boson mass values are taken from Ref. [115], while the
top quark mass is taken to be 172.5for consistency with theoretical
calculations used in setting the SM predictions. The bottom quark mass is
evaluated at the scale of the Higgs boson mass,
$m_{\PQb}(m_{\PH}=125\GeV)=2.76\GeV$.

The $1\sigma$ and $2\sigma$ regions in the $(M,\epsilon)$ fit are shown in
Fig. 10 (left). The results of the fit using the six parameter $\kappa$ model
are plotted versus the particle masses in Fig. 10 (right), and the result of
the $(M,\epsilon)$ fit is also shown for comparison. For the $\PQb$ quark,
since the best fit point for $\kappa_{\PQb}$ is negative, the absolute value
of this coupling modifier is shown. In order to show both the Yukawa and
vector boson couplings in the same plot, a “reduced” vector boson coupling
$\sqrt{\kappa_{\mathrm{V}}}m_{\mathrm{V}}/v$ is shown.

Figure 10: Likelihood scan in the $M$-$\epsilon$ plane (left). The best fit
point and the $1\sigma$ and $2\sigma$ regions are shown, along with the SM
prediction. Result of the phenomenological $(M,\epsilon)$ fit overlayed with
the resolved $\kappa$-framework model (right).

Best fit values and $\pm 1\sigma$ uncertainties for the parameters of the
$\kappa$ model in which the loop processes are resolved. The expected
uncertainties are given in brackets.  Parameter Best fit value Uncertainty
stat. syst. $\kappa_{\PW}$ $1.10$ ${}^{+0.12}_{-0.17}$ ${}^{+0.08}_{-0.16}$
${}^{+0.08}_{-0.06}$ $({}^{+0.11}_{-0.10})$ $({}^{+0.08}_{-0.08})$
$({}^{+0.06}_{-0.06})$ $\kappa_{\cPZ}$ $0.99$ ${}^{+0.11}_{-0.12}$
${}^{+0.09}_{-0.10}$ ${}^{+0.06}_{-0.07}$ $({}^{+0.11}_{-0.11})$
$({}^{+0.09}_{-0.09})$ $({}^{+0.06}_{-0.06})$ $\kappa_{\PQt}$ $1.11$
${}^{+0.12}_{-0.10}$ ${}^{+0.07}_{-0.07}$ ${}^{+0.09}_{-0.08}$
$({}^{+0.11}_{-0.12})$ $({}^{+0.07}_{-0.08})$ $({}^{+0.09}_{-0.09})$
$\kappa_{\PQb}$ $-1.10$ ${}^{+0.33}_{-0.23}$ ${}^{+0.29}_{-0.16}$
${}^{+0.15}_{-0.17}$ $({}^{+0.22}_{-0.22})$ $({}^{+0.15}_{-0.15})$
$({}^{+0.17}_{-0.16})$ $\kappa_{\Pgt}$ $1.01$ ${}^{+0.16}_{-0.20}$
${}^{+0.11}_{-0.17}$ ${}^{+0.12}_{-0.11}$ $({}^{+0.17}_{-0.15})$
$({}^{+0.12}_{-0.10})$ $({}^{+0.12}_{-0.11})$ $\kappa_{\mu}$ $0.79$
${}^{+0.58}_{-0.79}$ ${}^{+0.56}_{-0.80}$ ${}^{+0.14}_{-0.00}$
$({}^{+0.50}_{-1.01})$ $({}^{+0.50}_{-1.01})$ $({}^{+0.08}_{-0.10})$

###  0.8.2 Generic model within $\kappa$-framework with effective loops

The results of the fits to the generic $\kappa$ model where the $\Pg\Pg\PH$
and $\PH\to\PGg\PGg$ loops are scaled using the effective coupling modifiers
$\kappa_{\cPg}$ and $\kappa_{\PGg}$ are given in Fig. 11 and Table 0.8.2. In
this parametrization, additional contributions from BSM decays are allowed for
by rewriting the total width of the Higgs boson, relative to its SM value, as,

$\frac{\Gamma_{\PH}}{\Gamma_{\PH}^{\text{SM}}}=\frac{\kappa_{\PH}^{2}}{1-\left(\mathcal{B}_{\text{undet}}+\mathcal{B}_{\text{inv}}\right)},$
(7)

where $\kappa_{\PH}$ is defined in Table 0.8.

Figure 11: Summary plots for the $\kappa$-framework model in which the
$\Pg\Pg\PH$ and $\PH\to\PGg\PGg$ loops are scaled with effective couplings.
The points indicate the best fit values while the thick and thin horizontal
bars show the $1\sigma$ and $2\sigma$ intervals, respectively. In the left
figure the constraint $\mathcal{B}_{\mathrm{BSM}}=0$ is imposed, and both
positive and negative values of $\kappa_{\PW}$ and $\kappa_{\cPZ}$ are
considered. In the right figure a constraint
$\abs{\kappa_{\PW}},\,\abs{\kappa_{\cPZ}}\leq 1$ is imposed (same sign of
$\kappa_{\PW}$ and $\kappa_{\cPZ}$), while $\mathcal{B}_{\text{inv}}>0$ and
$\mathcal{B}_{\text{undet}}>0$ are free parameters.

Best fit values and $\pm 1\sigma$ uncertainties for the parameters of the
$\kappa$-framework model with effective loops. The expected uncertainties are
given in brackets.  $\mathcal{B}_{\mathrm{BSM}}=0$
$\mathcal{B}_{\mathrm{BSM}}>0$, $\abs{\kappa_{\mathrm{V}}}<1$ Uncertainty
Uncertainty Parameter Best fit stat syst Parameter Best fit stat syst
$\kappa_{\cPZ}$ $1.00$ ${}^{+0.11}_{-0.11}$ ${}^{+0.09}_{-0.09}$
${}^{+0.06}_{-0.07}$ $\kappa_{\cPZ}$ $-0.87$ ${}^{+0.08}_{-0.08}$
${}^{+0.07}_{-0.06}$ ${}^{+0.04}_{-0.04}$ $(^{+0.11}_{-0.11})$
$(^{+0.09}_{-0.09})$ $(^{+0.06}_{-0.06})$ $(^{+0.00}_{-0.12})$
$(^{+0.00}_{-0.10})$ $(^{+0.00}_{-0.06})$ $\kappa_{\PW}$ $-1.13$
${}^{+0.16}_{-0.13}$ ${}^{+0.15}_{-0.10}$ ${}^{+0.06}_{-0.08}$ $\kappa_{\PW}$
$-1.00$ ${}^{+0.09}_{-0.00}$ ${}^{+0.07}_{-0.00}$ ${}^{+0.05}_{-0.00}$
$(^{+0.12}_{-0.12})$ $(^{+0.09}_{-0.09})$ $(^{+0.07}_{-0.07})$
$(^{+0.00}_{-0.12})$ $(^{+0.00}_{-0.09})$ $(^{+0.00}_{-0.07})$ $\kappa_{\PQt}$
$0.98$ ${}^{+0.14}_{-0.14}$ ${}^{+0.08}_{-0.08}$ ${}^{+0.12}_{-0.11}$
$\kappa_{\PQt}$ $1.02$ ${}^{+0.19}_{-0.15}$ ${}^{+0.13}_{-0.09}$
${}^{+0.13}_{-0.13}$ $(^{+0.14}_{-0.15})$ $(^{+0.08}_{-0.09})$
$(^{+0.12}_{-0.12})$ $(^{+0.18}_{-0.15})$ $(^{+0.13}_{-0.09})$
$(^{+0.13}_{-0.12})$ $\kappa_{\Pgt}$ $1.02$ ${}^{+0.17}_{-0.17}$
${}^{+0.11}_{-0.13}$ ${}^{+0.12}_{-0.10}$ $\kappa_{\Pgt}$ $0.93$
${}^{+0.13}_{-0.13}$ ${}^{+0.08}_{-0.09}$ ${}^{+0.11}_{-0.10}$
$(^{+0.16}_{-0.15})$ $(^{+0.11}_{-0.11})$ $(^{+0.12}_{-0.11})$
$(^{+0.14}_{-0.15})$ $(^{+0.09}_{-0.10})$ $(^{+0.11}_{-0.11})$ $\kappa_{\PQb}$
$1.17$ ${}^{+0.27}_{-0.31}$ ${}^{+0.18}_{-0.29}$ ${}^{+0.20}_{-0.10}$
$\kappa_{\PQb}$ $0.91$ ${}^{+0.17}_{-0.16}$ ${}^{+0.11}_{-0.12}$
${}^{+0.13}_{-0.11}$ $(^{+0.25}_{-0.23})$ $(^{+0.18}_{-0.17})$
$(^{+0.17}_{-0.16})$ $(^{+0.19}_{-0.22})$ $(^{+0.14}_{-0.16})$
$(^{+0.13}_{-0.15})$ $\kappa_{\cPg}$ $1.18$ ${}^{+0.16}_{-0.14}$
${}^{+0.10}_{-0.09}$ ${}^{+0.12}_{-0.10}$ $\kappa_{\cPg}$ $1.16$
${}^{+0.18}_{-0.13}$ ${}^{+0.14}_{-0.09}$ ${}^{+0.12}_{-0.10}$
$(^{+0.14}_{-0.12})$ $(^{+0.10}_{-0.09})$ $(^{+0.10}_{-0.09})$
$(^{+0.17}_{-0.12})$ $(^{+0.13}_{-0.09})$ $(^{+0.11}_{-0.09})$ $\kappa_{\PGg}$
$1.07$ ${}^{+0.14}_{-0.15}$ ${}^{+0.10}_{-0.14}$ ${}^{+0.09}_{-0.05}$
$\kappa_{\PGg}$ $0.96$ ${}^{+0.09}_{-0.09}$ ${}^{+0.06}_{-0.08}$
${}^{+0.06}_{-0.06}$ $(^{+0.12}_{-0.12})$ $(^{+0.10}_{-0.09})$
$(^{+0.07}_{-0.07})$ $(^{+0.09}_{-0.11})$ $(^{+0.07}_{-0.09})$
$(^{+0.05}_{-0.07})$ $\kappa_{\Pgm}$ $0.80$ ${}^{+0.59}_{-0.80}$
${}^{+0.56}_{-0.81}$ ${}^{+0.17}_{-0.00}$ $\kappa_{\Pgm}$ $0.72$
${}^{+0.50}_{-0.72}$ ${}^{+0.50}_{-0.71}$ ${}^{+0.00}_{-0.07}$
$(^{+0.51}_{-1.01})$ $(^{+0.50}_{-1.01})$ $(^{+0.09}_{-0.09})$
$(^{+0.49}_{-1.01})$ $(^{+0.48}_{-1.00})$ $(^{+0.06}_{-0.08})$
$\mathcal{B}_{\text{inv}}$ $0.07$ ${}^{+0.08}_{-0.07}$ ${}^{+0.03}_{-0.03}$
${}^{+0.07}_{-0.06}$ $(^{+0.09}_{+0.00})$ $(^{+0.04}_{-0.00})$
$(^{+0.08}_{-0.00})$ $\mathcal{B}_{\text{undet}}$ $0.00$ ${}^{+0.17}_{+0.00}$
${}^{+0.14}_{-0.00}$ ${}^{+0.09}_{-0.00}$ $(^{+0.20}_{+0.00})$
$(^{+0.17}_{-0.00})$ $(^{+0.11}_{-0.00})$

Two different model assumptions are made concerning the BSM branching
fraction. In the first parametrization, it is assumed that
$\mathcal{B}_{\mathrm{BSM}}=\mathcal{B}_{\text{inv}}+\mathcal{B}_{\text{undet}}=0$,
whereas in the second, $\mathcal{B}_{\text{inv}}$ and
$\mathcal{B}_{\text{undet}}$ are allowed to vary as POIs, and instead the
constraint $\abs{\kappa_{\PW}},\,\abs{\kappa_{\cPZ}}\leq 1$ is imposed. This
avoids a complete degeneracy in the total width where all of the coupling
modifiers can be scaled equally to account for a non-zero
$\mathcal{B}_{\text{undet}}$. The parameter $\mathcal{B}_{\text{undet}}$
represents the total branching fraction to any final state that is not
detected by the analyses included in this combined analysis. The likelihood
scan for the $\mathcal{B}_{\text{inv}}$ parameter in this model, and the 2D
likelihood scan of $\mathcal{B}_{\text{inv}}$ vs. $\mathcal{B}_{\text{undet}}$
are given in Fig. 12. The 68 and 95% regions for Fig. 12 (right) are
determined as the regions for which
$q(\mathcal{B}_{\text{undet}},\mathcal{B}_{\text{inv}})<2.28$ and $5.99$,
respectively. The 95% upper limits of $\mathcal{B}_{\text{inv}}<0.22$ and
$\mathcal{B}_{\text{undet}}<0.38$ are determined, corresponding to the value
for which $q<3.84$ [92]. The uncertainty in the measurement of $\kappa_{\PQt}$
is reduced by nearly 40% compared to Ref. [56]. This improvement is because of
the improved sensitivity to the $\cPqt\cPqt\PH$ production mode as described
in Section 0.7.

Figure 12: Results within the generic $\kappa$-framework model with effective
loops and with the constraint $\abs{\kappa_{\PW}},\,\abs{\kappa_{\cPZ}}\leq 1$
(same sign of $\kappa_{\PW}$ and $\kappa_{\cPZ}$), and with
$\mathcal{B}_{\text{inv}}>0$ and $\mathcal{B}_{\text{undet}}>0$ as free
parameters. Scan of the test statistic $q$ as a function of
$\mathcal{B}_{\text{inv}}$ (left), and 68 and 95% regions for
$\mathcal{B}_{\text{inv}}$ vs. $\mathcal{B}_{\text{undet}}$ (right). The scan
of the test statistic $q$ as a function of $\mathcal{B}_{\text{inv}}$ expected
assuming the SM is also shown in the left figure.

In both of the generic $\kappa$ models, the best fit point for $\kappa_{\PW}$
is negative. The value of $q(\kappa_{\PW})$ as a function of $\kappa_{\PW}$ in
the two cases is shown in Fig. 13. While different combinations of signs for
$\kappa_{\PW}$ and $\kappa_{\cPZ}$ are shown, the minimum value of $q$ across
all combinations is used to determine the best fit point and the $1\sigma$ and
$2\sigma$ regions.

Figure 13: Scan of the test statistic $q$ as a function of $\kappa_{\PW}$ in
the generic $\kappa$ model assuming $\mathcal{B}_{\mathrm{BSM}}=0$ (left) and
allowing $\mathcal{B}_{\text{inv}}$ and $\mathcal{B}_{\text{undet}}$ to float
(right). The different colored lines indicate the value of $q$ for different
combinations of signs for $\kappa_{\PW}$ and $\kappa_{\cPZ}$. The solid black
line shows the minimum value of $q(\kappa_{\PW})$ in each case and is used to
determine the best fit point and the $1\sigma$ and $2\sigma$ regions. The scan
in the right figure is truncated because of the constraints of
$\abs{\kappa_{\PW}}\leq 1$ and $\abs{\kappa_{\cPZ}}\leq 1$, which are imposed
in this model.

The preferred negative value of $\kappa_{\PW}$ is due to the interference
between some of the diagrams describing $\cPqt\PH$ production, which
contributes in several analyses entering the combination. In particular, the
excess in the $\cPqt\cPqt\PH$ tagged categories of the $\PH\to\PGg\PGg$
analysis can be accounted for by a negative value of $\kappa_{\PW}$ as this
increases the contribution of $\cPqt\PH$ production. In these models, the
$\PH\to\PGg\PGg$ decay is treated as an effective coupling so that it has no
dependence on $\kappa_{\PW}$. This means that a negative value of
$\kappa_{\PW}$ will not result in excesses in the other categories of the
$\PH\to\PGg\PGg$ analysis.

Using Eq. (7), this model is also reinterpreted as a constraint on the total
Higgs boson width, and the corresponding likelihood scan is shown in Fig. 14.
Using this parametrization, the total Higgs boson width relative to the SM
expectation is determined to be
$\Gamma_{\PH}/\Gamma_{\PH}^{\mathrm{SM}}=0.98^{+0.31}_{-0.25}$. The different
behavior between the observed and expected likelihood scans for large
$\Gamma_{\PH}/\Gamma_{\PH}^{\mathrm{SM}}$ is due to the preference in data for
the $\kappa_{\PQt}\kappa_{\PW}<0$ relative sign combination.

An additional fit is performed assuming that the only BSM contributions to the
Higgs couplings appear in the loop-induced $\Pg\Pg\PH$ and $\PH\to\PGg\PGg$
processes. In this fit, $\kappa_{\cPg}$ and $\kappa_{\PGg}$ are the POIs,
$\mathcal{B}_{\text{inv}}$ and $\mathcal{B}_{\text{undet}}$ are floated, and
the other couplings are fixed to their SM predictions. The best fit point and
the $1\sigma$ and $2\sigma$ regions in the $\kappa_{\cPg}$-$\kappa_{\PGg}$
plane for this model are shown in Fig. 15.

Figure 14: The scan of the test statistic $q$ as a function of
$\Gamma_{\PH}/\Gamma_{\PH}^{\mathrm{SM}}$ obtained by reinterpreting the model
allowing for BSM decays of the Higgs boson. The expected scan of $q$ as a
function of $\Gamma_{\PH}/\Gamma_{\PH}^{\mathrm{SM}}$ assuming the SM is also
shown. Figure 15: The $1\sigma$ and $2\sigma$ regions in the $\kappa_{\Pg}$
vs. $\kappa_{\PGg}$ parameter space for the model assuming the only BSM
contributions to the Higgs boson couplings appear in the loop-induced
processes or in BSM Higgs decays.

###  0.8.3 Generic model with effective loops and coupling modifier ratios

An analogous parametrization to the ratios of cross sections and branching
fractions described in the previous section can be derived in terms of ratios
of the coupling modifiers ($\lambda_{ij}=\kappa_{i}/\kappa_{j}$). In this
parametrization a reference combined coupling modifier is defined that
accounts for modifications to the total event yield of a specific production
times decay process, thereby avoiding the need for assumptions on the total
Higgs boson width. The reference coupling modifier is taken to be
$\kappa_{\Pg\cPZ}=\kappa_{\Pg}\kappa_{\cPZ}/\kappa_{\PH}$. The remaining
parameters of interest are ratios of the form: $\lambda_{\cPZ\Pg}$,
$\lambda_{\PQt\Pg}$, $\lambda_{\PW\cPZ}$, $\lambda_{\PGg\cPZ}$,
$\lambda_{\Pgt\cPZ}$, $\lambda_{\PQb\cPZ}$. A summary of the results in this
model is given in Fig. 16, and the numerical values along with the $\pm
1\sigma$ uncertainties are shown in Table 0.8.3.

Figure 16: Summary of the model with coupling ratios and effective couplings
for the $\Pg\Pg\PH$ and $\PH\to\PGg\PGg$ loops. The points indicate the best
fit values while the thick and thin horizontal bars show the $1\sigma$ and
$2\sigma$ intervals, respectively. For this model, both positive and negative
values of $\lambda_{\PW\cPZ}$ and $\lambda_{\PQt\cPg}$ are considered.

Best fit values and $\pm 1\sigma$ uncertainties for the parameters of the
coupling modifier ratio model. The expected uncertainties are given in
brackets.  Uncertainty Uncertainty Parameter Best fit stat syst Parameter Best
fit stat syst $\kappa_{\cPg\cPZ}$ $1.03$ ${}^{+0.09}_{-0.09}$
${}^{+0.07}_{-0.07}$ ${}^{+0.05}_{-0.05}$ $\lambda_{\PGg\cPZ}$ $1.07$
${}^{+0.12}_{-0.10}$ ${}^{+0.10}_{-0.08}$ ${}^{+0.06}_{-0.05}$
$(^{+0.09}_{-0.09})$ $(^{+0.07}_{-0.07})$ $(^{+0.05}_{-0.05})$
$(^{+0.11}_{-0.09})$ $(^{+0.09}_{-0.08})$ $(^{+0.05}_{-0.04})$
$\lambda_{\PW\cPZ}$ $-1.13$ ${}^{+0.10}_{-0.11}$ ${}^{+0.08}_{-0.09}$
${}^{+0.06}_{-0.06}$ $\lambda_{\PQb\cPZ}$ $1.17$ ${}^{+0.23}_{-0.20}$
${}^{+0.16}_{-0.14}$ ${}^{+0.16}_{-0.14}$ $(^{+0.11}_{-0.09})$
$(^{+0.09}_{-0.08})$ $(^{+0.06}_{-0.05})$ $(^{+0.22}_{-0.19})$
$(^{+0.16}_{-0.14})$ $(^{+0.15}_{-0.13})$ $\lambda_{\PQt\cPg}$ $0.83$
${}^{+0.14}_{-0.13}$ ${}^{+0.08}_{-0.08}$ ${}^{+0.11}_{-0.10}$
$\lambda_{\Pgt\cPZ}$ $1.02$ ${}^{+0.16}_{-0.15}$ ${}^{+0.11}_{-0.10}$
${}^{+0.12}_{-0.11}$ $(^{+0.17}_{-0.16})$ $(^{+0.11}_{-0.11})$
$(^{+0.12}_{-0.12})$ $(^{+0.16}_{-0.14})$ $(^{+0.11}_{-0.10})$
$(^{+0.11}_{-0.10})$ $\lambda_{\cPZ\cPg}$ $0.85$ ${}^{+0.14}_{-0.13}$
${}^{+0.10}_{-0.12}$ ${}^{+0.09}_{-0.05}$ $\lambda_{\mu\cPZ}$ $0.81$
${}^{+0.57}_{-0.81}$ ${}^{+0.56}_{-0.82}$ ${}^{+0.11}_{-0.00}$
$(^{+0.17}_{-0.16})$ $(^{+0.13}_{-0.13})$ $(^{+0.11}_{-0.09})$
$(^{+0.50}_{-1.01})$ $(^{+0.49}_{-1.01})$ $(^{+0.07}_{-0.07})$

###  0.8.4 Fits of vector boson and fermion coupling modifiers

A more constrained version of the loop-resolved $\kappa$ model is defined by
assuming a common scaling of all vector boson and fermion couplings,
respectively. Two models are defined: one in which all signal processes are
scaled according to these two $\kappa_{\mathrm{V}}$ and $\kappa_{\mathrm{F}}$
parameters, and one in which separate $\kappa_{\mathrm{V}}^{f}$ and
$\kappa_{\mathrm{F}}^{f}$ parameters are defined for each of the five decay
processes. The best fit points and the $1\sigma$ and $2\sigma$ regions in the
$\kappa_{\mathrm{V}}$-$\kappa_{\mathrm{F}}$ plane for both models are shown in
Fig. 17, and the results are summarized in Table 0.8.4. For large values of
$\kappa_{\mathrm{F}}^{\cPZ\cPZ}$ the likelihood becomes essentially flat,
resulting in the best fit point for this parameter being beyond the scale of
the axis shown. The 1D 68% region for $\kappa_{\mathrm{F}}^{\cPZ\cPZ}$ can be
expressed as $[1.22,\infty]$.

Figure 17: The $1\sigma$ and $2\sigma$ regions in the $\kappa_{\mathrm{F}}$
vs. $\kappa_{\mathrm{V}}$ parameter space for the model assuming a common
scaling of all the vector boson and fermion couplings.

Best fit values and $\pm 1\sigma$ uncertainties for the parameters of the
$\kappa_{\mathrm{V}},\kappa_{\mathrm{F}}$ model. The expected uncertainties
are given in brackets.  Uncertainty Uncertainty Parameter Best fit stat syst
Parameter Best fit stat syst $\kappa_{\mathrm{V}}^{\PW\PW}$ $1.10$
${}^{+0.08}_{-0.08}$ ${}^{+0.06}_{-0.06}$ ${}^{+0.06}_{-0.06}$
$\kappa_{\mathrm{F}}^{\PW\PW}$ $1.49$ ${}^{+1.55}_{-0.38}$
${}^{+1.04}_{-0.33}$ ${}^{+1.15}_{-0.18}$ $(^{+0.08}_{-0.08})$
$(^{+0.06}_{-0.06})$ $(^{+0.05}_{-0.05})$ $(^{+0.38}_{-0.20})$
$(^{+0.31}_{-0.17})$ $(^{+0.21}_{-0.11})$ $\kappa_{\mathrm{V}}^{\cPZ\cPZ}$
$0.96$ ${}^{+0.09}_{-0.08}$ ${}^{+0.08}_{-0.07}$ ${}^{+0.05}_{-0.04}$
$\kappa_{\mathrm{F}}^{\cPZ\cPZ}$ — — — $(^{+0.12}_{-0.11})$
$(^{+0.11}_{-0.10})$ $(^{+0.05}_{-0.05})$ $(^{+1.79}_{-0.31})$
$(^{+1.55}_{-0.30})$ $(^{+0.91}_{-0.06})$ $\kappa_{\mathrm{V}}^{\PQb\PQb}$
$1.15$ ${}^{+0.23}_{-0.22}$ ${}^{+0.18}_{-0.18}$ ${}^{+0.15}_{-0.13}$
$\kappa_{\mathrm{F}}^{\PQb\PQb}$ $1.01$ ${}^{+0.16}_{-0.18}$
${}^{+0.09}_{-0.10}$ ${}^{+0.13}_{-0.15}$ $(^{+0.22}_{-0.22})$
$(^{+0.17}_{-0.18})$ $(^{+0.13}_{-0.12})$ $(^{+0.16}_{-0.18})$
$(^{+0.09}_{-0.10})$ $(^{+0.13}_{-0.15})$ $\kappa_{\mathrm{V}}^{\Pgt\Pgt}$
$1.10$ ${}^{+0.38}_{-0.29}$ ${}^{+0.26}_{-0.24}$ ${}^{+0.27}_{-0.18}$
$\kappa_{\mathrm{F}}^{\Pgt\Pgt}$ $0.98$ ${}^{+0.15}_{-0.16}$
${}^{+0.08}_{-0.09}$ ${}^{+0.13}_{-0.14}$ $(^{+0.32}_{-0.29})$
$(^{+0.24}_{-0.23})$ $(^{+0.20}_{-0.17})$ $(^{+0.14}_{-0.14})$
$(^{+0.07}_{-0.08})$ $(^{+0.11}_{-0.12})$ $\kappa_{\mathrm{V}}^{\PGg\PGg}$
$1.10$ ${}^{+0.14}_{-0.08}$ ${}^{+0.11}_{-0.07}$ ${}^{+0.09}_{-0.05}$
$\kappa_{\mathrm{F}}^{\PGg\PGg}$ $1.14$ ${}^{+0.67}_{-0.29}$
${}^{+0.53}_{-0.26}$ ${}^{+0.41}_{-0.14}$ $(^{+0.10}_{-0.08})$
$(^{+0.08}_{-0.06})$ $(^{+0.05}_{-0.04})$ $(^{+0.47}_{-0.25})$
$(^{+0.41}_{-0.23})$ $(^{+0.23}_{-0.10})$

###  0.8.5 Benchmark models with resolved loops to test the symmetry of
fermion couplings

Several BSM models predict the existence of an extended Higgs sector. In such
scenarios, the couplings to up-and down-type fermions, or to leptons and
quarks, can be separately modified. In order to probe such models,
parametrizations are introduced in which the couplings of the Higgs boson to
fermions are scaled either by separate common modifiers for up-type
($\kappa_{\PQu}$) and down-type ($\kappa_{\PQd}$) fermions or by separate
common modifiers for quarks $\kappa_{\Pq}$ and leptons $\kappa_{\text{l}}$
($\text{l}=\Pe,\Pgm,\Pgt$).

Figure 18 shows the results of the fits where the ratio of the couplings to
up- and down-type fermions $\lambda_{\PQd\PQu}=\kappa_{\PQd}/\kappa_{\PQu}$ is
determined along with the ratio
$\lambda_{\mathrm{V}\PQu}=\kappa_{\mathrm{V}}/\kappa_{\PQu}$ and
$\kappa_{\PQu\PQu}=\kappa_{\PQu}^{2}/\Gamma_{\PH}$. Also shown are the results
of the fit where the ratio of the coupling to leptons and to quarks
$\lambda_{\text{l}\Pq}=\kappa_{\text{l}}/\kappa_{\Pq}$ is determined along
with the ratio $\lambda_{\mathrm{V}\Pq}=\kappa_{\mathrm{V}}/\kappa_{\Pq}$ and
$\kappa_{\Pq\Pq}=\kappa_{\Pq}^{2}/\Gamma_{\PH}$. The results of these two
parametrizations are summarized in Table 0.8.5.

Figure 18: Summary plots of the 3-parameter models comparing up- and down-type
fermions, and floating the ratio of the vector coupling to the up-type
coupling (left) and comparing lepton and quark couplings (right). The points
indicate the best fit values while the thick and thin horizontal bars show the
$1\sigma$ and $2\sigma$ intervals, respectively. Both positive and negative
values of $\lambda_{\PQd\PQu}$, $\lambda_{\mathrm{V}\PQu}$,
$\lambda_{\text{l}\Pq}$, and $\lambda_{\mathrm{V}\Pq}$ are considered.

Best fit values and $\pm 1\sigma$ uncertainties for the parameters of the two
benchmark models with resolved loops to test the symmetry of fermion
couplings. The expected uncertainties are given in brackets.
$\lambda_{\mathrm{V}\PQu}$ $\lambda_{\PQd\PQu}$ $\kappa_{\PQu\PQu}$ Best fit
Uncertainty Best fit Uncertainty Best fit Uncertainty value stat syst value
stat syst value stat syst $0.97$ ${}^{+0.14}_{-0.10}$ ${}^{+0.11}_{-0.08}$
${}^{+0.09}_{-0.06}$ $0.92$ ${}^{+0.12}_{-0.12}$ ${}^{+0.09}_{-0.09}$
${}^{+0.08}_{-0.08}$ $1.14$ ${}^{+0.20}_{-0.20}$ ${}^{+0.13}_{-0.16}$
${}^{+0.15}_{-0.12}$ $({}^{+0.15}_{-0.11})$ $({}^{+0.12}_{-0.09})$
$({}^{+0.09}_{-0.07})$ $({}^{+0.13}_{-0.13})$ $({}^{+0.10}_{-0.10})$
$({}^{+0.08}_{-0.08})$ $({}^{+0.16}_{-0.19})$ $({}^{+0.12}_{-0.16})$
$({}^{+0.11}_{-0.11})$ $\lambda_{\mathrm{V}\Pq}$ $\lambda_{\text{l}\Pq}$
$\kappa_{\Pq\Pq}$ Best fit Uncertainty Best fit Uncertainty Best fit
Uncertainty value stat syst value stat syst value stat syst $0.99$
${}^{+0.10}_{-0.08}$ ${}^{+0.07}_{-0.06}$ ${}^{+0.07}_{-0.06}$ $0.92$
${}^{+0.13}_{-0.13}$ ${}^{+0.09}_{-0.08}$ ${}^{+0.10}_{-0.10}$ $1.10$
${}^{+0.13}_{-0.12}$ ${}^{+0.08}_{-0.08}$ ${}^{+0.10}_{-0.09}$
$({}^{+0.12}_{-0.09})$ $({}^{+0.08}_{-0.07})$ $({}^{+0.09}_{-0.07})$
$({}^{+0.15}_{-0.14})$ $({}^{+0.10}_{-0.09})$ $({}^{+0.11}_{-0.10})$
$({}^{+0.13}_{-0.13})$ $({}^{+0.09}_{-0.09})$ $({}^{+0.10}_{-0.10})$

###  0.8.6 Compatibility of measurements with the SM

Table 0.8.6 shows a summary of the compatibility of the different models
considered, as described in Sections 0.7 and 0.8, with the SM predictions. For
each model, the value of $q$ at the values of the POIs for the SM expectation
($q_{\mathrm{SM}}$) is converted to a $p$-value with respect to the SM. This
is done assuming $q$ is distributed according to a $\chi^{2}$ function with
the number of degrees of freedom equal to the number of POIs. This $p$-value
is found to be greater than 5% for all parametrizations.

Compatibility of the fit results with the SM prediction under various signal
parametrizations. The value of $q$ at the values of the POIs for which the SM
expectation is obtained ($q_{\text{SM}}$) is shown along with the
corresponding $p$-value, with respect to the SM, assuming $q$ is distributed
according to a $\chi^{2}$ function with the specified number of degrees of
freedom (DOF).  Parameterization $p$-value ($q_{\text{SM}}$) DOF Parameters of
interest Global signal strength 6.28% (3.46) 1 $\mu$ Production processes
9.87% (9.27) 5 $\mu_{\Pg\Pg\PH}$, $\mu_{\mathrm{VBF}}$, $\mu_{\PW\PH}$,
$\mu_{\cPZ\PH}$, $\mu_{\cPqt\cPqt\PH}$ Decay modes 53.8% (5.05) 6
$\mu^{\PGg\PGg}$, $\mu^{\cPZ\cPZ}$, $\mu^{\PW\PW}$, $\mu^{\Pgt\Pgt}$,
$\mu^{\PQb\PQb}$, $\mu^{\Pgm\Pgm}$ $\sigma_{i}\mathcal{B}^{f}$ products 61.2%
(21.5) 24 $\sigma_{\Pg\Pg\PH}\mathcal{B}^{\PQb\PQb}$,
$\sigma_{\Pg\Pg\PH}\mathcal{B}^{\Pgt\Pgt}$,
$\sigma_{\Pg\Pg\PH}\mathcal{B}^{\mu\mu}$,
$\sigma_{\Pg\Pg\PH}\mathcal{B}^{\PW\PW}$,
$\sigma_{\Pg\Pg\PH}\mathcal{B}^{\cPZ\cPZ}$,
$\sigma_{\Pg\Pg\PH}\mathcal{B}^{\PGg\PGg}$,
$\sigma_{\mathrm{VBF}}\mathcal{B}^{\Pgt\Pgt}$,
$\sigma_{\mathrm{VBF}}\mathcal{B}^{\mu\mu}$,
$\sigma_{\mathrm{VBF}}\mathcal{B}^{\PW\PW}$,
$\sigma_{\mathrm{VBF}}\mathcal{B}^{\cPZ\cPZ}$,
$\sigma_{\mathrm{VBF}}\mathcal{B}^{\PGg\PGg}$,
$\sigma_{\PW\PH}\mathcal{B}^{\PQb\PQb}$,
$\sigma_{\PW\PH}\mathcal{B}^{\PW\PW}$,
$\sigma_{\PW\PH}\mathcal{B}^{\cPZ\cPZ}$,
$\sigma_{\PW\PH}\mathcal{B}^{\PGg\PGg}$,
$\sigma_{\cPZ\PH}\mathcal{B}^{\PQb\PQb}$,
$\sigma_{\cPZ\PH}\mathcal{B}^{\PW\PW}$,
$\sigma_{\cPZ\PH}\mathcal{B}^{\cPZ\cPZ}$,
$\sigma_{\cPZ\PH}\mathcal{B}^{\PGg\PGg}$,
$\sigma_{\cPqt\cPqt\PH}\mathcal{B}^{\Pgt\Pgt}$,
$\sigma_{\cPqt\cPqt\PH}\mathcal{B}^{\PW\PW}$,
$\sigma_{\cPqt\cPqt\PH}\mathcal{B}^{\cPZ\cPZ}$,
$\sigma_{\cPqt\cPqt\PH}\mathcal{B}^{\PGg\PGg}$,
$\sigma_{\cPqt\cPqt\PH}\mathcal{B}^{\PQb\PQb}$ Ratios of $\sigma$ and
$\mathcal{B}$ relative to $\Pg\Pg\to\PH\to\cPZ\cPZ$ 32.3% (11.5) 10
$\mu_{\Pg\Pg\PH}^{\cPZ\cPZ}$, $\mu_{\mathrm{VBF}}/\mu_{\Pg\Pg\PH}$,
$\mu_{\PW\PH}/\mu_{\Pg\Pg\PH}$, $\mu_{\cPZ\PH}/\mu_{\Pg\Pg\PH}$,
$\mu_{\cPqt\cPqt\PH}/\mu_{\Pg\Pg\PH}$, $\mu^{\PW\PW}/\mu^{\cPZ\cPZ}$,
$\mu^{\PGg\PGg}/\mu^{\cPZ\cPZ}$, $\mu^{\Pgt\Pgt}/\mu^{\cPZ\cPZ}$,
$\mu^{\PQb\PQb}/\mu^{\cPZ\cPZ}$, $\mu^{\PQb\PQb}/\mu^{\mu\mu}$ Simplified
template cross sections with branching fractions relative to
$\mathcal{B}^{\cPZ\cPZ}$ 21.2% (14.4) 11
$\sigma_{\Pg\Pg\PH}\mathcal{B}^{\cPZ\cPZ}$,
$\sigma_{\mathrm{VBF}}\mathcal{B}^{\cPZ\cPZ}$,
$\sigma_{\PH+\mathrm{V}(\Pq\Pq)}\mathcal{B}^{\cPZ\cPZ}$,
$\sigma_{\PH+\PW(\ell\nu)}\mathcal{B}^{\cPZ\cPZ}$,
$\sigma_{\PH+\cPZ(\ell\ell/\nu\nu)}\mathcal{B}^{\cPZ\cPZ}$,
$\sigma_{\cPqt\cPqt\PH}\mathcal{B}^{\cPZ\cPZ}$,
$\mathcal{B}^{\PQb\PQb}/\mathcal{B}^{\cPZ\cPZ}$,
$\mathcal{B}^{\Pgt\Pgt}/\mathcal{B}^{\cPZ\cPZ}$,
$\mathcal{B}^{\mu\mu}/\mathcal{B}^{\cPZ\cPZ}$,
$\mathcal{B}^{\PW\PW}/\mathcal{B}^{\cPZ\cPZ}$,
$\mathcal{B}^{\PGg\PGg}/\mathcal{B}^{\cPZ\cPZ}$ Couplings, SM loops 45.6%
(5.71) 6 $\kappa_{\cPZ}$, $\kappa_{\PW}$, $\kappa_{\PQt}$, $\kappa_{\Pgt}$,
$\kappa_{\PQb}$, $\kappa_{\Pgm}$ Couplings vs. mass 16.8% (3.57) 2 $M$,
$\epsilon$ Couplings, BSM loops 18.5% (11.3) 8 $\kappa_{\cPZ}$,
$\kappa_{\PW}$, $\kappa_{\PQt}$, $\kappa_{\Pgt}$, $\kappa_{\PQb}$,
$\kappa_{\Pgm}$, $\kappa_{\PGg}$, $\kappa_{\Pg}$ Couplings, BSM loops and
decays including $\PH\to\text{invisible}$ analyses 32.4% (11.5) 10
$\kappa_{\cPZ}$, $\kappa_{\PW}$, $\kappa_{\PQt}$, $\kappa_{\Pgt}$,
$\kappa_{\PQb}$, $\kappa_{\Pgm}$, $\kappa_{\PGg}$, $\kappa_{\Pg}$,
$\mathcal{B}_{\text{inv}}$, $\mathcal{B}_{\text{undet}}$ Ratios of coupling
modifiers 18.1% (11.4) 8 $\kappa_{\Pg\cPZ}$, $\lambda_{\PW\cPZ}$,
$\lambda_{\PGg\cPZ}$, $\lambda_{\PQt\Pg}$, $\lambda_{\PQb\cPZ}$,
$\lambda_{\Pgt\cPZ}$, $\lambda_{\mu\cPZ}$, $\lambda_{\cPZ\Pg}$ Fermion and
vector couplings 16.9% (3.55) 2 $\kappa_{\mathrm{F}}$, $\kappa_{\mathrm{V}}$
Fermion and vector couplings, per decay mode 76.7% (8.2) 12
$\kappa^{\PQb\PQb}_{\mathrm{F}}$, $\kappa^{\Pgt\Pgt}_{\mathrm{F}}$,
$\kappa^{\mu\mu}_{\mathrm{F}}$, $\kappa^{\PW\PW}_{\mathrm{F}}$,
$\kappa^{\cPZ\cPZ}_{\mathrm{F}}$, $\kappa^{\PGg\PGg}_{\mathrm{F}}$,
$\kappa^{\PQb\PQb}_{\mathrm{V}}$, $\kappa^{\Pgt\Pgt}_{\mathrm{V}}$,
$\kappa^{\mu\mu}_{\mathrm{V}}$, $\kappa^{\PW\PW}_{\mathrm{V}}$,
$\kappa^{\cPZ\cPZ}_{\mathrm{V}}$, $\kappa^{\PGg\PGg}_{\mathrm{V}}$ Up vs.
down-type couplings 25.5% (4.06) 3 $\lambda_{\mathrm{V}\PQu}$,
$\lambda_{\PQd\PQu}$, $\kappa_{\PQu\PQu}$ Lepton vs. quark couplings 27.2%
(3.91) 3 $\lambda_{\text{l}\Pq}$, $\lambda_{\mathrm{V}\Pq}$, $\kappa_{\Pq\Pq}$

##  0.9 Constraints on benchmark two Higgs doublet models

The generic models described in Section 0.8.5 can also be interpreted in the
context of explicit benchmark BSM models that contain a second Higgs doublet
(2HDM) [116, 117, 118]. Only models with CP conservation and a discrete
$\mathbb{Z}_{2}$ symmetry to prevent tree-level flavor changing neutral
currents are considered. Under these assumptions, four 2HDM types are
possible, referred to as Types I, II, III, and IV. Each of these 2HDMs contain
seven free parameters. Under the additional assumption that the Higgs boson
with a mass of 125.09is the lightest CP-even, neutral Higgs boson in the
extended Higgs sector, the predicted rates for its production and decay are
sensitive at leading order to only two 2HDM parameters: the angles $\alpha$
and $\beta$ that diagonalize the mass-squared matrices of the scalars and
pseudoscalars. These two parameters are conventionally substituted by
$\cos(\beta-\alpha)$ and $\tan\beta$, without loss of generality. In all of
the 2HDMs, the coupling of the Higgs boson to vector bosons is modified by a
factor $\sin(\beta-\alpha)$. The 2HDM types differ in how the fermions couple
to the Higgs doublets. In the Type I model, all fermions couple to just one of
the Higgs doublets. In Type II, the up-type fermions couple to one of the
Higgs doublets, while the down-type fermions and the right-handed leptons
couple to the second. In Type III 2HDM, also referred to as “lepton-specific”,
the quarks couple to one of the Higgs doublets and the right-handed leptons
couple to the other. In the Type IV 2HDM, also referred to as “flipped”, the
up-type fermions and right-handed leptons couple to one of the Higgs doublets,
while the down-type quarks couple to the other. Table 0.9 shows the relation
between the coupling modifiers to vector bosons, quarks and leptons and the
2HDM model parameters.

The minimal supersymmetric standard model (MSSM) [119, 120, 121, 122] is a
specific example of a 2HDM of Type II that includes additional particle
content compared to the SM. The additional strong constraints given by the
nontrivial fermion-boson symmetry fix all mass relations between the Higgs
bosons and the angle $\alpha$, at tree-level, leaving only two free parameters
to fully constrain the MSSM Higgs sector, usually chosen to be $m_{\PSA}$ and
$\tan\beta$. The hMSSM scenario [123, 124], in particular, is an effective
MSSM model, trading the precise knowledge of $m_{\PH}$ against unknown higher-
order corrections such that $m_{\PH}=125.09\GeV$ across the $m_{\PSA}$,
$\tan\beta$ parameter space. Another requirement of the scenario is that $\PH$
be identified as the lightest of the two neutral scalar Higgs bosons.
Furthermore, one also obtains relatively simple relations between $m_{\PSA}$,
$\tan\beta$, and the Higgs boson coupling modifiers [125], which are shown in
Table 0.9 and completed by Eqs. (8) and (9). Although many other MSSM
benchmark models have also been defined [126], the lack of analytic
expressions for the Higgs boson couplings renders these models technically
more challenging to consider and they are therefore beyond the scope of this
paper.

Modifications to the couplings of the Higgs bosons to up-type
($\kappa_{\PQu}$) and down-type ($\kappa_{\PQd}$) fermions, and vector bosons
($\kappa_{\mathrm{V}}$), with respect to the SM expectation, in 2HDM and for
the hMSSM. The coupling modifications for the hMSSM are completed by the
expressions for $s_{\PQu}$ and $s_{\PQd}$, as given by Eqs. (8) and (9).  2HDM
hMSSM Type I Type II Type III Type IV $\kappa_{\mathrm{V}}$
$\sin(\beta-\alpha)$ $\sin(\beta-\alpha)$ $\sin(\beta-\alpha)$
$\sin(\beta-\alpha)$
$\frac{s_{\PQd}+s_{\PQu}\tan\beta}{\sqrt{1+\tan^{2}\beta}}$ $\kappa_{\PQu}$
$\cos(\alpha)/\sin(\beta)$ $\cos(\alpha)/\sin(\beta)$
$\cos(\alpha)/\sin(\beta)$ $\cos(\alpha)/\sin(\beta)$
$s_{\PQu}\frac{\sqrt{1+\tan^{2}\beta}}{\tan\beta}$ $\kappa_{\PQd}$
$\cos(\alpha)/\sin(\beta)$ $-\sin(\alpha)/\cos(\beta)$
$\cos(\alpha)/\sin(\beta)$ $-\sin(\alpha)/\cos(\beta)$
$s_{\PQd}\sqrt{1+\tan^{2}\beta}$ $\kappa_{\text{l}}$
$\cos(\alpha)/\sin(\beta)$ $-\sin(\alpha)/\cos(\beta)$
$-\sin(\alpha)/\cos(\beta)$ $\cos(\alpha)/\sin(\beta)$
$s_{\PQd}\sqrt{1+\tan^{2}\beta}$

$\displaystyle
s_{\PQu}=\frac{1}{\sqrt{1+\frac{(m_{\PSA}^{2}+m_{\cPZ}^{2})^{2}\tan^{2}\beta}{(m_{\cPZ}^{2}+m_{\PSA}^{2}\tan^{2}\beta-
m_{\PH}^{2}(1+\tan^{2}\beta))^{2}}}}\,$ (8) $\displaystyle
s_{\PQd}=s_{\PQu}\frac{(m_{\PSA}^{2}+m_{\cPZ}^{2})\tan\beta}{m_{\cPZ}^{2}+m_{\PSA}^{2}\tan^{2}\beta-
m_{\PH}^{2}(1+\tan^{2}\beta)}\,$ (9)

To set constraints on the 2HDM model parameters, 3-dimensional likelihood
scans of the parametrizations described in Section 0.8.5 (with necessary
modifications to the lepton coupling modifiers to describe the Type IV 2HDM)
are performed. A test-statistic is then defined, for example in the Types I,
II and hMSSM scenarios,

$q(\lambda_{\PQd\PQu},\lambda_{\mathrm{V}\PQu},\kappa_{\PQu\PQu})=-2\ln\left(\frac{{L}(\lambda_{\PQd\PQu},\lambda_{\mathrm{V}\PQu},\kappa_{\PQu\PQu})}{{L}(\hat{\lambda}_{\PQd\PQu},\hat{\lambda}_{\mathrm{V}\PQu},\hat{\kappa}_{\PQu\PQu})}\right),$
(10)

where
$\hat{\lambda}_{\PQd\PQu},\hat{\lambda}_{\mathrm{V}\PQu},\hat{\kappa}_{\PQu\PQu}$
are the values of the POIs that maximize the likelihood. An interpolation
scheme is used to determine the value of $q$ as a function of
$\cos(\beta-\alpha)$ and $\tan\beta$, or $m_{\PSA}$ and $\tan\beta$, for the
Types I and II, or hMSSM scenarios, respectively, using the relations in Table
0.9.

A second quantity $q\prime$ is defined as,

$q\prime=-2\ln\left(\frac{L(\hat{\lambda}_{\PQd\PQu},\hat{\lambda}_{\mathrm{V}\PQu},\hat{\kappa}_{\PQu\PQu})}{L_{\mathrm{max}}}\right),$
(11)

where $L_{\mathrm{max}}$ is the maximum likelihood value attained in the
planes of $\cos(\beta-\alpha)$–$\tan\beta$, or $m_{\PSA}$–$\tan\beta$. The
allowed regions are determined as the points in each plane for which the
difference between $q$ and $q\prime$ ($\Delta q$) is less than 5.99. This
value corresponds to the 95% confidence region assuming $\Delta q$ is
distributed as a $\chi^{2}$ function with 2 degrees of freedom. A similar
procedure is performed using the model with the parameters
$\lambda_{\text{l}\Pq}$, $\lambda_{\mathrm{V}\Pq}$ and $\kappa_{\Pq\Pq}$, to
determine the allowed region for the Type III scenario.

Figure 19 shows the results of the fits for the different 2HDM benchmark
scenarios. The lobe features that can be seen in the Types II, III, and IV
constraints for $\cos(\beta-\alpha)>0$ are due to negative values of
$\kappa_{\PQd}$, $\kappa_{\Pgt}$, and $\kappa_{\PQb}$, which are not excluded
with the current sensitivity. In all of these 2HDM models, the Higgs boson
couplings are the same as those predicted in the SM for
$\cos(\beta-\alpha)=0$.

The results for the hMSSM scenario are also shown in Fig. 19. The constraints
observed are more stringent than those expected under the SM. This is due to
the best fit value of $\lambda_{\PQd\PQu}$ being smaller than 1, while in the
hMSSM for $\tan\beta>1$, $\lambda_{\PQd\PQu}$ is strictly greater than 1 and
asymptotically approaches unity only at large $m_{\PSA}$. Therefore the
observed data disfavors small values of $m_{\PSA}$, leading to the stronger
constraint.

Figure 19: Constraints in the $\cos({\beta-\alpha})$ vs. $\tan\beta$ plane for
the Types I, II, III, and IV 2HDM, and constraints in the $m_{\PSA}$ vs.
$\tan\beta$ plane for the hMSSM. The white regions, bounded by the solid black
lines, in each plane represent the regions of the parameter space that are
allowed at the 95% , given the data observed. The dashed lines indicate the
boundaries of the allowed regions expected for the SM Higgs boson.

The constraints in the 2HDM and hMSSM scenarios are complementary to those
obtained from direct searches for additional Higgs bosons [127, 128, 129, 130,
131].

##  0.10 Summary

A set of combined measurements of Higgs boson production and decay rates has
been presented, along with the consequential constraints placed on its
couplings to standard model (SM) particles, and on the parameter spaces of
several beyond the standard model (BSM) scenarios. The combination is based on
analyses targeting the gluon fusion and vector boson fusion production modes,
and associated production with a vector boson or a pair of top quarks. The
analyses included in the combination target Higgs boson production in the
$\PH\to\cPZ\cPZ,\,\PW\PW,\,\PGg\PGg,\,\Pgt\Pgt$, $\PQb\PQb$, and $\Pgm\Pgm$
decay channels, using $13\TeV$ proton-proton collision data collected in 2016
and corresponding to an integrated luminosity of 35.9. Additionally, searches
for invisible Higgs boson decays are included to increase the sensitivity to
potential interactions with BSM particles.

Measurements of the Higgs boson production cross section times branching
fractions are presented, along with a generic parametrization in terms of
ratios of production cross sections and branching fractions, which makes no
assumptions about the Higgs boson total width. The combined signal yield
relative to the SM prediction has been measured as $1.17\pm 0.10$ at
$m_{\PH}=125.09\GeV$. An improvement in the measured precision of the gluon
fusion production rate of around $\sim$50% is achieved compared to previous
ATLAS and CMS measurements. Additionally, a set of fiducial Higgs boson cross
sections, in the context of the simplified template cross section framework,
is presented for the first time from a combination of six decay channels.
Furthermore, interpretations are provided in the context of a leading-order
coupling modifier framework, including variants for which effective couplings
to the photon and gluon are introduced. All of the results presented are
compatible with the SM prediction. The invisible (undetected) branching
fraction of the Higgs boson is constrained to be less than 22 (38%) at 95%
Confidence Level. The results are additionally interpreted in two BSM models,
the minimal supersymmetric model and the generic two Higgs doublet model. The
constraints placed on the parameter spaces of these models are complementary
to those that can be obtained from direct searches for additional Higgs
bosons.

###### Acknowledgements.

We congratulate our colleagues in the CERN accelerator departments for the
excellent performance of the LHC and thank the technical and administrative
staffs at CERN and at other CMS institutes for their contributions to the
success of the CMS effort. In addition, we gratefully acknowledge the
computing centres and personnel of the Worldwide LHC Computing Grid for
delivering so effectively the computing infrastructure essential to our
analyses. Finally, we acknowledge the enduring support for the construction
and operation of the LHC and the CMS detector provided by the following
funding agencies: the Austrian Federal Ministry of Science, Research and
Economy and the Austrian Science Fund; the Belgian Fonds de la Recherche
Scientifique, and Fonds voor Wetenschappelijk Onderzoek; the Brazilian Funding
Agencies (CNPq, CAPES, FAPERJ, and FAPESP); the Bulgarian Ministry of
Education and Science; CERN; the Chinese Academy of Sciences, Ministry of
Science and Technology, and National Natural Science Foundation of China; the
Colombian Funding Agency (COLCIENCIAS); the Croatian Ministry of Science,
Education and Sport, and the Croatian Science Foundation; the Research
Promotion Foundation, Cyprus; the Secretariat for Higher Education, Science,
Technology and Innovation, Ecuador; the Ministry of Education and Research,
Estonian Research Council via IUT23-4 and IUT23-6 and European Regional
Development Fund, Estonia; the Academy of Finland, Finnish Ministry of
Education and Culture, and Helsinki Institute of Physics; the Institut
National de Physique Nucléaire et de Physique des Particules / CNRS, and
Commissariat à l’Énergie Atomique et aux Énergies Alternatives / CEA, France;
the Bundesministerium für Bildung und Forschung, Deutsche
Forschungsgemeinschaft, and Helmholtz-Gemeinschaft Deutscher
Forschungszentren, Germany; the General Secretariat for Research and
Technology, Greece; the National Research, Development and Innovation Fund,
Hungary; the Department of Atomic Energy and the Department of Science and
Technology, India; the Institute for Studies in Theoretical Physics and
Mathematics, Iran; the Science Foundation, Ireland; the Istituto Nazionale di
Fisica Nucleare, Italy; the Ministry of Science, ICT and Future Planning, and
National Research Foundation (NRF), Republic of Korea; the Lithuanian Academy
of Sciences; the Ministry of Education, and University of Malaya (Malaysia);
the Mexican Funding Agencies (BUAP, CINVESTAV, CONACYT, LNS, SEP, and UASLP-
FAI); the Ministry of Business, Innovation and Employment, New Zealand; the
Pakistan Atomic Energy Commission; the Ministry of Science and Higher
Education and the National Science Centre, Poland; the Fundação para a Ciência
e a Tecnologia, Portugal; JINR, Dubna; the Ministry of Education and Science
of the Russian Federation, the Federal Agency of Atomic Energy of the Russian
Federation, Russian Academy of Sciences, the Russian Foundation for Basic
Research and the Russian Competitiveness Program of NRNU “MEPhI”; the Ministry
of Education, Science and Technological Development of Serbia; the Secretaría
de Estado de Investigación, Desarrollo e Innovación, Programa Consolider-
Ingenio 2010, Plan Estatal de Investigación Científica y Técnica y de
Innovación 2013-2016, Plan de Ciencia, Tecnología e Innovación 2013-2017 del
Principado de Asturias and Fondo Europeo de Desarrollo Regional, Spain; the
Swiss Funding Agencies (ETH Board, ETH Zurich, PSI, SNF, UniZH, Canton Zurich,
and SER); the Ministry of Science and Technology, Taipei; the Thailand Center
of Excellence in Physics, the Institute for the Promotion of Teaching Science
and Technology of Thailand, Special Task Force for Activating Research and the
National Science and Technology Development Agency of Thailand; the Scientific
and Technical Research Council of Turkey, and Turkish Atomic Energy Authority;
the National Academy of Sciences of Ukraine, and State Fund for Fundamental
Researches, Ukraine; the Science and Technology Facilities Council, UK; the US
Department of Energy, and the US National Science Foundation. Individuals have
received support from the Marie-Curie programme and the European Research
Council and Horizon 2020 Grant, contract No. 675440 (European Union); the
Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt
Foundation; the Belgian Federal Science Policy Office; the Fonds pour la
Formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA-
Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-
Belgium); the F.R.S.-FNRS and FWO (Belgium) under the “Excellence of Science -
EOS” - be.h project n. 30820817; the Ministry of Education, Youth and Sports
(MEYS) of the Czech Republic; the Lendület (“Momentum”) Programme and the
János Bolyai Research Scholarship of the Hungarian Academy of Sciences, the
New National Excellence Program ÚNKP, the NKFIA research grants 123842,
123959, 124845, 124850 and 125105 (Hungary); the Council of Scientific and
Industrial Research, India; the HOMING PLUS programme of the Foundation for
Polish Science, cofinanced from European Union, Regional Development Fund, the
Mobility Plus programme of the Ministry of Science and Higher Education, the
National Science Center (Poland), contracts Harmonia 2014/14/M/ST2/00428, Opus
2014/13/B/ST2/02543, 2014/15/B/ST2/03998, and 2015/19/B/ST2/02861, Sonata-bis
2012/07/E/ST2/01406; the National Priorities Research Program by Qatar
National Research Fund; the Programa de Excelencia María de Maeztu and the
Programa Severo Ochoa del Principado de Asturias; the Thalis and Aristeia
programmes cofinanced by EU-ESF and the Greek NSRF; the Rachadapisek Sompot
Fund for Postdoctoral Fellowship, Chulalongkorn University and the
Chulalongkorn Academic into Its 2nd Century Project Advancement Project
(Thailand); the Welch Foundation, contract C-1845; and the Weston Havens
Foundation (USA).

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