HIG-22-008

HIG-22-008

# Constraints on anomalous Higgs boson couplings from its production and decay
using the $\PW\PW$ channel in proton-proton collisions at $\sqrt{s}=13\TeV$

###### Abstract

A study of the anomalous couplings of the Higgs boson to vector bosons,
including $CP$-violation effects, has been conducted using its production and
decay in the $\PW\PW$ channel. This analysis is performed on proton-proton
collision data collected with the CMS detector at the CERN LHC during
2016–2018 at a center-of-mass energy of 13, and corresponds to an integrated
luminosity of 138. The different-flavor dilepton ($\Pe\PGm$) final state is
analyzed, with dedicated categories targeting gluon fusion, electroweak vector
boson fusion, and associated production with a or boson. Kinematic information
from associated jets is combined using matrix element techniques to increase
the sensitivity to anomalous effects at the production vertex. A simultaneous
measurement of four Higgs boson couplings to electroweak vector bosons is
performed in the framework of a standard model effective field theory. All
measurements are consistent with the expectations for the standard model Higgs
boson and constraints are set on the fractional contribution of the anomalous
couplings to the Higgs boson production cross section.

##  0.1 Introduction

After the discovery of the Higgs boson () by the ATLAS and CMS Collaborations
in 2012 [1, 2, 3], the CMS [4, 5, 6, 7, 8, 9, 10, 11] and ATLAS [12, 13, 14,
15, 16, 17, 18] experiments set constraints on the spin-parity properties of
the Higgs boson and its couplings with gluons and electroweak (EW) gauge
bosons, denoted here as $\PH\Pg\Pg$ and $\PH\PV\PV$, respectively. The Higgs
boson quantum numbers are consistent with the standard model (SM) expectation
$J^{PC}=0^{++}$, but the possibility of small, anomalous couplings is not yet
ruled out. In beyond-the-SM (BSM) theories, interactions with the Higgs boson
may occur through several anomalous couplings, which lead to new tensor
structures in the interaction terms that can be both $CP$-even or $CP$-odd.
The $CP$-odd anomalous couplings between the Higgs boson and BSM particles may
generate $CP$ violation in the interactions of the Higgs boson.

In this paper, we study the tensor structure of the $\PH\Pg\Pg$ and
$\PH\PV\PV$ couplings, and we search for several anomalous effects, including
$CP$ violation, using the different-flavor dilepton ($\Pe\PGm$) final state
from $\PH\to\PW\PW$ decays. The Higgs boson production processes include gluon
fusion ($\Pg\Pg\PH$), EW vector boson fusion (VBF), and associated production
with a or boson ($\PV\PH$). Higgs boson production and decay processes are
sensitive to certain anomalous contributions, which can be described by
higher-dimensional operators in an effective field theory (EFT) [19] that can
modify the kinematic distributions of the Higgs boson decay products and the
particles from associated production.

Each production process of the Higgs boson is identified using its kinematic
features, and events are assigned to corresponding production categories. The
matrix element likelihood approach (MELA) [20, 21, 22, 23, 24] is employed to
construct observables that are optimal for the measurement of anomalous
couplings, or EFT operators, at the production vertex. These and other decay-
based variables are used to explore all kinematic features of the events,
giving the analysis sensitivity to simultaneous anomalous effects at the Higgs
boson production and decay vertices. Fully simulated signal samples that
include anomalous couplings incorporate the detector response into the
analysis.

The analysis is based on the proton-proton ($\Pp\Pp$) collision data collected
at the CERN LHC from 2016 to 2018, at a center-of-mass energy of 13,
corresponding to an integrated luminosity of 138. This paper builds on a
previous analysis conducted by the CMS Collaboration in the $\PH\to\PW\PW$
channel [25], which focused on measuring the Higgs boson production cross
sections and coupling parameters in the so-called $\kappa$ framework [26]. We
follow a formalism used in previous CMS analyses of anomalous couplings in Run
1 and Run 2 [4, 5, 6, 7, 8, 9, 10, 11, 27, 9, 28], focusing on the case where
the Higgs boson is produced on-shell. The coupling parameters are extracted
using the signal strength and the fractional contributions of the couplings to
the cross section. A general study of the $\PH\PV\PV$ interaction is performed
with four anomalous couplings analyzed individually. Through SU(2) x U(1)
symmetry considerations, the anomalous $\PH\PV\PV$ couplings are reduced in
number to three and analyzed simultaneously. The primary $\PH\PV\PV$ coupling
measurements are performed in terms of cross section fractions with additional
interpretations in terms of EFT couplings included. A study of the $\PH\Pg\Pg$
interaction is also performed in terms of a $CP$-odd anomalous coupling cross
section fraction.

This paper is organized as follows. The phenomenology of anomalous couplings
is discussed in Section 0.2. Section 0.3 gives a brief overview of the CMS
apparatus. Data sets and Monte Carlo (MC) simulation samples are discussed in
Section 0.4. The event reconstruction and selection are outlined in Sections
0.5 and 0.6, respectively. Methods to estimate backgrounds are given in
Section 0.7. In Section 0.8, we discuss the kinematic variables associated
with Higgs boson production and decay. Sources of systematic uncertainties are
presented in Section 0.9. The results are presented and discussed in Section
0.10. Finally, a summary is given in Section 0.11. Tabulated results are
provided in the HEPData record for this analysis [29].

##  0.2 Phenomenology

In this analysis, we investigate anomalous coupling effects in gluon fusion or
electroweak Higgs boson production, as well as in its decay to $\PW\PW$ pairs.
A detailed discussion of the theoretical considerations can be found in Refs.
[28, 22, 24]. The interaction of the spin-zero Higgs boson with two spin-one
gauge bosons $\PV_{1}\PV_{2}$, such as $\PW\PW$, $\PZ\PZ$, $\PZ\PGg$,
$\PGg\PGg$, or $\Pg\Pg$, can be parametrized by the scattering amplitude

$\displaystyle
A(\PH\PV_{1}\PV_{2})\sim\left[a_{1}^{\PV\PV}+\frac{\kappa_{1}^{\PV\PV}q_{\PV
1}^{2}+\kappa_{2}^{\PV\PV}q_{\PV
2}^{2}}{\left(\Lambda_{1}^{\PV\PV}\right)^{2}}\right]m_{\vspace{2
mm}\noindent\PV 1}^{2}\epsilon_{\vspace{2 mm}\noindent\PV
1}^{*}\epsilon_{\vspace{2 mm}\noindent\PV 2}^{*}$ (1)
$\displaystyle+\frac{1}{v}a_{2}^{\PV\PV}f_{\mu\nu}^{*(1)}f^{*(2),\mu\nu}+\frac{1}{v}a_{3}^{\PV\PV}f^{*(1)}_{\mu\nu}{\tilde{f}}^{*(2),\mu\nu},$

where $q_{\PV i}$ and $\epsilon_{\vspace{2 mm}\noindent\PV i}$ are the spin-
one gauge boson four-momentum and polarization vectors, $m_{\vspace{2
mm}\noindent\PV 1}$ is the pole mass of the boson,
$f^{(i),{\mu\nu}}=\epsilon_{{\vspace{2 mm}\noindent\PV}i}^{\mu}q_{{\vspace{2
mm}\noindent\PV}i}^{\nu}-\epsilon_{{\vspace{2
mm}\noindent\PV}i}^{\nu}q_{{\vspace{2 mm}\noindent\PV}i}^{\mu}$ and
${\tilde{f}}^{(i)}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}f^{(i),\rho\sigma}$
(with $\epsilon_{\mu\nu\rho\sigma}$ the Levi-Civita symbol),
$\Lambda_{1}^{\PV\PV}$ is the scale of BSM physics, and $v$ is the Higgs field
vacuum expectation value.

The only leading tree-level contributions in the scattering amplitude are
$a_{1}^{\PZ\PZ}\neq 0$ and $a_{1}^{\PW\PW}\neq 0$; other $a_{1}$ coupling
parameters ($\PZ\PGg$, $\PGg\PGg$, $\Pg\Pg$) do not contribute because the
pole mass vanishes. Additional $\PZ\PZ$ and $\PW\PW$ couplings are considered
anomalous contributions. Anomalous terms arising in the SM via loop effects
are typically small and are not yet accessible experimentally. The BSM
contributions, however, could yield larger coupling parameters. Among the
anomalous contributions, considerations of symmetry and gauge invariance
require $\kappa_{1}^{\PZ\PZ}=\kappa_{2}^{\PZ\PZ}$,
$\kappa_{1}^{\PGg\PGg}=\kappa_{2}^{\PGg\PGg}=0$,
$\kappa_{1}^{\Pg\Pg}=\kappa_{2}^{\Pg\Pg}=0$, and $\kappa_{1}^{\PZ\PGg}=0$
[24]. The presence of $CP$-odd $a^{\PV\PV}_{3}$ couplings together with any of
the other couplings (all of them $CP$-even), will result in $CP$ violation. We
reduce the number of independent parameters by assuming that
$a_{2}^{\PGg\PGg}$, $a_{3}^{\PGg\PGg}$, $a_{2}^{\PZ\PGg}$ and
$a_{3}^{\PZ\PGg}$ are constrained in direct decays of $\PH\to\PGg\PGg$ and
$\PZ\PGg$, therefore fixing them to be zero. The $a^{\Pg\Pg}_{2}$ term results
from loop effects in the SM.

The relationship between the $\PZ\PZ$ and $\PW\PW$ couplings is mostly
relevant for VBF production. There are no kinematic differences between the
$\PZ\PZ$ and $\PW\PW$ fusion processes; therefore, it is not possible to
disentangle the couplings. One possibility is to set the $\PZ\PZ$ and $\PW\PW$
couplings to be equal, $a_{i}=a_{i}^{\PZ\PZ}=a_{i}^{\PW\PW}$, leaving four
$\PH\PV\PV$ anomalous couplings to be measured: $a_{2}$, $a_{3}$,
$\kappa_{1}/(\Lambda_{1})^{2}$, and
$\kappa_{2}^{\PZ\PGg}/(\Lambda_{1}^{\PZ\PGg})^{2}$. The
$a_{1}^{\PZ\PZ}=a_{1}^{\PW\PW}$ relationship also appears under custodial
symmetry. This approach provides a general test of the Higgs boson Lagrangian
tensor structure and a search for $CP$ violation in $\PH\PV\PV$ interactions.
In an alternative approach, the SU(2) $\times$ U(1) symmetry reduces the
number of independent $\PH\PV\PV$ anomalous couplings to three ($a_{2}$,
$a_{3}$, and $\kappa_{1}/(\Lambda_{1})^{2}$) through the introduction of the
following coupling parameter relationships [19] :

$\displaystyle a_{1}^{\PW\PW}$ $\displaystyle=a_{1}^{\PZ\PZ},$ (2)
$\displaystyle a_{2}^{\PW\PW}$ $\displaystyle=c_{\text{w}}^{2}a_{2}^{\PZ\PZ},$
(3) $\displaystyle a_{3}^{\PW\PW}$
$\displaystyle=c_{\text{w}}^{2}a_{3}^{\PZ\PZ},$ (4)
$\displaystyle\frac{\kappa_{1}^{\PW\PW}}{(\Lambda_{1}^{\PW\PW})^{2}}$
$\displaystyle=\frac{1}{c_{\text{w}}^{2}-s_{\text{w}}^{2}}\left(\frac{\kappa_{1}^{\PZ\PZ}}{(\Lambda_{1}^{\PZ\PZ})^{2}}-2s_{\text{w}}^{2}\frac{a_{2}^{\PZ\PZ}}{m_{\PZ}^{2}}\right),$
(5) $\displaystyle\frac{\kappa_{2}^{\PZ\PGg}}{(\Lambda_{1}^{\PZ\PGg})^{2}}$
$\displaystyle=\frac{2s_{\text{w}}c_{\text{w}}}{c_{\text{w}}^{2}-s_{\text{w}}^{2}}\left(\frac{\kappa_{1}^{\PZ\PZ}}{(\Lambda_{1}^{\PZ\PZ})^{2}}-\frac{a_{2}^{\PZ\PZ}}{m_{\PZ}^{2}}\right),$
(6)

where $c_{\text{w}}$ and $s_{\text{w}}$ are the cosine and sine of the weak
mixing angle, respectively, and $m_{\PZ}$ is the $\PZ$ boson mass. With this
approach, there is a linear relationship between the scattering amplitude
couplings and the SM EFT (SMEFT) couplings in the Higgs basis [19] :

$\displaystyle\delta c_{\text{z}}$
$\displaystyle=\frac{1}{2}a_{1}^{\PZ\PZ}-1,$ (7) $\displaystyle c_{\text{zz}}$
$\displaystyle=-\frac{2s_{\text{w}}^{2}c_{\text{w}}^{2}}{e^{2}}a_{2}^{\PZ\PZ},$
(8) $\displaystyle\tilde{c}_{\text{zz}}$
$\displaystyle=-\frac{2s_{\text{w}}^{2}c_{\text{w}}^{2}}{e^{2}}a_{3}^{\PZ\PZ},$
(9) $\displaystyle c_{\text{z}\Box}$
$\displaystyle=\frac{m_{\PZ}^{2}s_{\text{w}}^{2}}{e^{2}}\,\frac{\kappa_{1}^{\PZ\PZ}}{(\Lambda_{1}^{\PZ\PZ})^{2}},$
(10)

where $e$ is the electron charge. The amplitude couplings may also be related
to the SMEFT Warsaw basis [30, 19] couplings through the following translation
[28, 31] :

$\displaystyle\delta a_{1}^{\PZ\PZ}$
$\displaystyle=\frac{v^{2}}{\Lambda^{2}}\left(2c_{\text{H}\Box}+\frac{6e^{2}}{s_{\text{w}}^{2}}c_{\text{HWB}}+(\frac{3c_{\text{w}}^{2}}{2s_{\text{w}}^{2}}-\frac{1}{2})c_{\text{HD}}\right),$
(11) $\displaystyle\kappa_{1}^{\PZ\PZ}$
$\displaystyle=\frac{v^{2}}{\Lambda^{2}}\left(-\frac{2e^{2}}{s_{\text{w}}^{2}}c_{\text{HWB}}+(1-\frac{1}{2s_{\text{w}}^{2}})c_{\text{HD}}\right),$
(12) $\displaystyle a_{2}^{\PZ\PZ}$
$\displaystyle=-2\frac{v^{2}}{\Lambda^{2}}\left(s_{\text{w}}^{2}c_{\text{HB}}+c_{\text{w}}^{2}c_{\text{HW}}+s_{\text{w}}c_{\text{w}}c_{\text{HWB}}\right),$
(13) $\displaystyle a_{3}^{\PZ\PZ}$
$\displaystyle=-2\frac{v^{2}}{\Lambda^{2}}\left(s_{\text{w}}^{2}c_{\text{H}\tilde{\text{B}}}+c_{\text{w}}^{2}c_{\text{H}\tilde{\text{W}}}+s_{\text{w}}c_{\text{w}}c_{\text{H}\tilde{\text{W}}\text{B}}\right),$
(14)

where $\Lambda$ is the UV cutoff of the theory (set to 1), and $\delta
a_{1}^{\PZ\PZ}$ is a correction to the SM value of $a_{1}^{\PZ\PZ}$. Further
discussion on the EFT operators corresponding to the couplings considered here
may be found in Chapter 2.2 of Ref. [19]. The assumed constraints on
$a_{2}^{\PGg\PGg}$, $a_{3}^{\PGg\PGg}$, $a_{2}^{\PZ\PGg}$ and
$a_{3}^{\PZ\PGg}$ imply that only one of the three coupling parameters
$c_{\text{HW}}$ , $c_{\text{HWB}}$, and $c_{\text{HB}}$ is independent; the
same is also true for their $CP$-odd counterparts
$c_{\text{H}\tilde{\text{W}}}$, $c_{\text{H}\tilde{\text{W}}\text{B}}$, and
$c_{\text{H}\tilde{\text{B}}}$. Therefore, we have four independent
$\PH\PV\PV$ couplings in both the Higgs and Warsaw basis. All the EFT
couplings are expected to be zero in the SM.

We thus adopt two approaches to the $\PH\PV\PV$ coupling study. In Approach 1,
we use the $a_{i}^{\PZ\PZ}=a_{i}^{\PW\PW}$ relationship and individually
analyze each of the four anomalous couplings. In Approach 2, we enforce the
SU(2) x U(1) relationships from Eqs. (2–6) and analyze the three independent
anomalous couplings both individually and simultaneously. Approach 1 may be
considered to follow the relationships from Eqs. (2–5) in the limiting case
$c_{\text{w}}=1$.

It is convenient to measure the fractional contribution of the anomalous
couplings to the Higgs boson cross section rather than the anomalous couplings
themselves. For the anomalous $\PH\PV\PV$ couplings, the effective fractional
cross sections $f_{ai}$ are defined as

$f_{ai}=\frac{\abs{a_{i}}^{2}\sigma_{i}}{\sum_{j}{\abs{a_{j}}^{2}\sigma_{j}}}\,\text{sign}\left(\frac{a_{i}}{a_{1}}\right),$
(15)

where $\sum_{j}$ sums over all the coupling parameters considered, including
$a_{1}$, and $\sigma_{i}$ is the cross section for the process corresponding
to $a_{i}=1$ and $a_{j\neq i}=0$. Many systematic uncertainties cancel out in
the ratio, and the physical range is conveniently bounded between $-1$ and
$+1$. Our primary measurements are performed in terms of cross section
fractions, with additional interpretations in terms of the SMEFT Higgs and
Warsaw basis couplings also included. For consistency with previous CMS
measurements, the $\sigma_{i}$ coefficients used to define the fractional
cross sections correspond to the $\Pg\Pg\to\PH\to\PV\PV\to 2\Pe 2\PGm$ process
[28]. The numerical values are given in Table 0.2 as calculated using the
JHUGen simulation [20, 21, 22, 23]. Two sets of values are shown corresponding
to the different coupling relationships adopted in Approach 1 and 2.

It has been shown that the angular correlations of the associated jets in the
$\Pg\Pg\PH$ \+ 2 jets process are sensitive to anomalous $\PH\Pg\Pg$ coupling
effects at the production vertex [32]. The quark-quark initiated process,
$\PQq\PQq\to\PQq\PQq\PH$, corresponds to the gluon scattering topology
sensitive to anomalous effects. For the anomalous $\PH\Pg\Pg$ coupling, the
effective fractional cross section can be defined as

$f_{a3}^{\Pg\Pg\PH}=\frac{\abs{a_{3}^{\Pg\Pg}}^{2}\sigma_{3}^{\Pg\Pg}}{\abs{a_{2}^{\Pg\Pg}}^{2}\sigma_{2}^{\Pg\Pg}+\abs{a_{3}^{\Pg\Pg}}^{2}\sigma_{3}^{\Pg\Pg}}\,\text{sign}\left(\frac{a_{3}^{\Pg\Pg}}{a_{2}^{\Pg\Pg}}\right).\\\
$ (16)

The $\sigma_{3}^{\Pg\Pg}$ and $\sigma_{2}^{\Pg\Pg}$ cross sections correspond
to $a_{3}^{\Pg\Pg}=1,a_{2}^{\Pg\Pg}=0$ and
$a_{2}^{\Pg\Pg}=1,a_{3}^{\Pg\Pg}=0$, respectively, and are equal. With this
analysis it is not possible to distinguish the top quark, bottom quark, and
heavy BSM particle contributions in the gluon fusion loop. As such, the
$\PH\Pg\Pg$ coupling is treated as an effective coupling with heavy degrees of
freedom integrated out.

The cross sections ($\sigma_{i}$) of the anomalous contributions ($a_{i}$)
relative to the SM value ($\sigma_{1}$) used to define the fractional cross
sections $f_{ai}$ for the Approach 1 and 2 coupling relationships. For the
$\kappa_{1}$ and $\kappa_{2}^{\PZ\PGg}$ couplings, the numerical values
$\Lambda_{1}=\Lambda_{1}^{\PZ\PGg}=100\GeV$ are chosen to keep all
coefficients of similar order of magnitude.  $f_{ai}$ $a_{i}$ Approach 1
$\sigma_{i}/\sigma_{1}$ Approach 2 $\sigma_{i}/\sigma_{1}$ $f_{a2}$ $a_{2}$
0.361 6.376 $f_{a3}$ $a_{3}$ 0.153 0.153 $f_{\Lambda 1}$ $\kappa_{1}$ 0.682
5.241 $f_{\Lambda 1}^{\PZ\PGg}$ $\kappa_{2}^{\PZ\PGg}$ 1.746

##  0.3 The CMS detector

The CMS apparatus [33] is a multipurpose, nearly hermetic detector, designed
to identify electrons, muons, photons, and (charged and neutral) hadrons [34,
35, 36, 37]. A global reconstruction “particle-flow” (PF) algorithm [38]
combines the information provided by the all-silicon inner tracker and by the
crystal electromagnetic and brass-scintillator hadron calorimeters, operating
inside a 3.8T superconducting solenoid, with data from gas-ionization muon
detectors interleaved with the solenoid return yoke, to build leptons, jets,
missing transverse momentum, and other physics objects [39, 40, 41].

Events of interest are selected using a two-tiered trigger system [42, 43].
The first level (L1), composed of custom hardware processors, uses information
from the calorimeters and muon detectors to select events at a rate of around
100kHz within a fixed latency of about 4 [42]. The second level, known as the
high-level trigger (HLT), consists of a farm of processors running a version
of the full event reconstruction software optimized for fast processing, and
reduces the event rate to around 1kHz before data storage [43]. A more
detailed description of the CMS detector, together with a definition of the
coordinate system and kinematic variables, can be found in Ref. [33].

##  0.4 Data sets and simulation

The data sets included in this analysis were recorded with the CMS detector in
2016, 2017, and 2018, and correspond to integrated luminosities of 36.3, 41.5,
and 59.7, respectively [44, 45, 46]. The collision events must fulfill HLT
selection criteria that require the presence of one or two leptons satisfying
isolation and identification requirements. For the 2016 data set, the single-
electron trigger has a transverse momentum () threshold of 25for electrons
with pseudorapidity $\abs{\eta}<2.1$ and 27for $2.1<\abs{\eta}<2.5$, whereas
the single-muon trigger has a threshold of 24for $\abs{\eta}<2.4$. For the
2017 (2018) data set, the threshold is 35 (32)for the single-electron trigger
(covering $\abs{\eta}<2.5$) and 27 (24)for the single-muon trigger
($\abs{\eta}<2.4$). The dilepton $\Pe\PGm$ trigger has thresholds of 23 and
12for the leading and subleading leptons, respectively, with the same coverage
in pseudorapidity for electrons and muons as above. During the first part of
data taking in 2016, a lower threshold of 8for the subleading muon was used.

Monte Carlo event generators are used to model the signal and background
processes. For each process, three independent sets of simulated events,
corresponding to the three years of data taking, are used. This approach
includes year-dependent effects in the CMS detector, data taking, and event
reconstruction. All simulated events corresponding to a given data set share
the same set of parton distribution functions (PDFs), underlying event (UE)
tune, and parton shower (PS) configuration. The PDF sets used are NNPDF 3.0
[47, 48] for 2016 and NNPDF 3.1 [49] for 2017 and 2018. The CUETP8M1 [50] tune
is used to describe the UE in 2016 simulations, whereas the CP5 [51] tune is
adopted in 2017 and 2018 simulated events. The MC samples are interfaced with
8.226 [52] in 2016, and 8.230 in 2017 and 2018, for the modeling of UE, PS,
and hadronization. Standard Model Higgs boson production through $\Pg\Pg\PH$,
VBF, and $\PV\PH$ is simulated at next-to-leading order (NLO) accuracy in
quantum chromodynamics (QCD), including finite quark mass effects, using v2
[53, 54, 55, 56, 57, 58, 59]. The minlo hvj [58] extension of v2 is used for
the simulation of $\PW\PH$ and quark-induced $\PZ\PH$ production, providing
NLO accuracy for the $\PV\PH+0$\- and 1-jet processes. For $\Pg\Pg\PH$
production, the simulated events are weighted to match the NNLOPS [60, 61]
prediction in the hadronic jet multiplicity ($N_{\text{jet}}$) and Higgs boson
distributions. The weighting is based on and $N_{\text{jet}}$ as computed in
the simplified template cross section scheme 1.0 [62]. The minlo hjj [63]
generator, which provides NLO accuracy for $N_{\text{jet}}\geq 2$, is also
used for $\Pg\Pg\PH$ production. The associated production processes with top
quarks ($\ttbar{}\PH$) and bottom quarks () are simulated with v2 and v2.2.2
[64], respectively, and have a negligible contribution in the analysis phase
space. All SM Higgs boson samples are normalized to the cross sections
recommended in Ref. [19]. The Higgs boson mass in the event generation is
assumed to be 125, while a value of 125.38 [65] is used for the calculation of
cross sections and branching fractions. The decay to a pair of bosons and
subsequently to leptons or hadrons is performed using the JHUGen v5.2.5
generator in 2016, and v7.1.4 in 2017 and 2018, for $\Pg\Pg\PH$, VBF, and
quark-induced $\PZ\PH$ samples. The Higgs boson and boson decays are performed
using 8.212 for the other signal simulations.

The $\Pg\Pg\PH$, VBF, and $\PV\PH$ Higgs boson events with $\PH\PV\PV$
anomalous couplings are generated with JHUGen at LO accuracy. With respect to
the $\kappa_{2}^{\PZ\PGg}/(\Lambda_{1}^{\PZ\PGg})^{2}$ coupling parameter
discussed in Section 0.2, the sign convention of the photon field is
determined by the sign in front of the gauge fields in the covariant
derivative. In this analysis, we define the covariant derivative
$D_{\mu}=\partial_{\mu}-\mathrm{i}e\sigma^{i}W_{\mu}^{i}/(2s_{w})+\mathrm{i}eB_{\mu}/(2c_{w})$
following the convention in JHUGen [31]. The JHUGen and SM Higgs boson
simulations were compared after parton showering and no significant
differences in the distributions of kinematic observables were found. We adopt
the JHUGen simulation to describe the kinematic features in all production
modes with $\PH\PV\PV$ anomalous couplings. The expected yields are scaled to
match the SM theoretical predictions [19] for inclusive cross sections and the
SM prediction of relative event yields in the event categorization based on
associated particles. Simulation of the $\Pg\Pg\PH$ \+ 2 jets process with
$\PH\Pg\Pg$ anomalous couplings is done using minlo X0jj [66] at NLO in QCD. A
large number of signal samples with various anomalous couplings were
generated. The MELA package [20, 21, 22, 23, 24] contains a library of matrix
elements from JHUGen for different Higgs boson signal hypotheses. The ratio of
matrix elements allows reweighting of generated signal events to any coupling
signal hypothesis with the same production mechanism. This procedure is used
in the construction of the predictions for the different coupling components
and their interference, allowing us to cover all points in the signal model
phase space with sufficient statistical precision.

Background events are produced using several simulations. The quark-initiated
nonresonant $\PW\PW$ process is simulated with v2 [67] at NLO accuracy for
inclusive production. A reweighting is performed to match the diboson spectrum
computed at NNLO+NNLL QCD accuracy [68, 69]. The v7.0 [70, 71, 72] generator
is used to simulate gluon-induced $\PW\PW$ production at LO accuracy, with the
normalization chosen to match the NLO cross section [73]. Nonresonant EW
production of $\PW\PW$ pairs with two additional jets is simulated at LO
accuracy with v2.4.2 using the MLM matching and merging scheme [74]. Top quark
pair production () and single top quark processes, including $\PQt\PW$, s- and
t-channel contributions, are simulated with v2 [75, 76, 77]. A reweighting of
the top quark and antiquark spectrum at parton level is performed for the
simulation in order to match the NNLO and next-to-next-to-leading logarithm
(NNLL) QCD predictions, including also the NLO EW contribution [78].

The Drell–Yan (DY) production of a charged-lepton pair is simulated with
v2.4.2 at NLO accuracy with up to two additional partons, using the FxFx
matching and merging scheme [79]. Production of a boson associated with an
initial state radiation photon ($\PW{}\PGg$) is simulated with v2.4.2 at NLO
accuracy with up to 1 additional parton, using the FxFx jet merging. Diboson
processes containing at least one boson or a virtual photon ($\PGg^{*}$) with
a mass as low as 100are generated with v2 [67] at NLO accuracy. Production of
a boson in association with a $\PGg^{*}$ ($\PW{}{\PGg}^{*}$) for masses below
100is simulated by 8.212 in the parton showering of $\PW{}\PGg$ events.
Triboson processes with inclusive decays are also simulated at NLO accuracy
with v2.4.2.

For all processes, the detector response is simulated using a detailed
description of the CMS detector, based on the toolkit [80]. The distribution
of additional $\Pp\Pp$ interactions within the same or nearby bunch crossings
(pileup) in the simulation is reweighted to match that observed in data. The
efficiency of the trigger system is evaluated in data on a per lepton basis
using dilepton events consistent with the boson decay. The overall
efficiencies of the trigger selections used in the analysis are obtained as
the average of the per-lepton efficiencies weighted by their probability. The
resulting efficiencies are applied directly on simulated events.

##  0.5 Event reconstruction

The identification and measurement of the properties of individual particles
(PF candidates) in an event is achieved in the PF algorithm by combining
information from various subdetectors. Electrons are identified and their
momenta are measured in the pseudorapidity interval $\abs{\eta}<2.5$ by
combining tracks in the silicon tracker with spatially compatible energy
deposits in the electromagnetic calorimeter. Muons are identified and their
momenta are measured in the pseudorapidity range $\abs{\eta}<2.4$ by matching
tracks in the muon system and the silicon tracker. For better rejection of
nonprompt leptons, increasing the sensitivity of the analysis, leptons are
required to be isolated and well reconstructed using a set of criteria based
on the quality of the track reconstruction, shape of calorimetric deposits,
and energy flux in the vicinity of the particle’s trajectory [34, 35]. In
addition, a selection based on a dedicated multivariate analysis (MVA) tagger
developed for the CMS $\ttbar{}\PH$ analysis [81] is added in all channels for
muon candidates.

Multiple $\Pp\Pp$ interaction vertices are identified from tracking
information by use of the adaptive vertex fitting algorithm [82]. The primary
$\Pp\Pp$ interaction vertex is taken to be the vertex corresponding to the
hardest scattering in the event, evaluated using tracking information alone,
as described in Section 9.4.1 of Ref. [83]. Leptons are required to be
associated to the primary vertex using transverse and longitudinal impact
parameter criteria [34, 35].

Hadronic jets are clustered from PF candidates using the infrared- and
collinear-safe anti-algorithm with distance parameters of 0.4 (AK4) and 0.8
(AK8). The jet momentum is determined as the vectorial sum of all particle
momenta in the jet. The AK8 jets considered are required to be reconstructed
within the silicon tracker acceptance ($\abs{\eta}<2.4$), whereas AK4 jets are
reconstructed in the range $\abs{\eta}<4.7$. For AK4 jets, contamination from
pileup is suppressed using charged-hadron subtraction which removes charged PF
candidates originating from vertices other than the primary interaction
vertex. The residual contribution from neutral particles originating from
pileup vertices is removed by means of an event-by-event jet-area-based
correction to the jet four-momentum [84]. For AK8 jets, the pileup-per-
particle identification algorithm (PUPPI) [85] is used to mitigate the effect
of pileup at the reconstructed-particle level, making use of local shape
information, event pileup properties, and tracking information. Additional
selection criteria are applied to remove jets potentially dominated by
instrumental effects or reconstruction failures [84].

The AK8 jets are used to reconstruct hadronic boson decays in a single merged
jet when the decay products are highly collimated. This approach targets
boosted or bosons originating from the $\PV\PH$ production mode. Such Lorentz-
boosted decays are identified using the ratio of the 2- to 1-subjettiness
[86], $\tau_{2}/\tau_{1}$, and the groomed jet mass $m_{\text{J}}$. The
groomed mass is calculated after applying a modified mass drop algorithm [87,
88], known as the soft-drop algorithm [89], with parameters $\beta=0$,
$z_{\text{cut}}=0.1$, and $R_{0}=0.8$. The algorithm also identifies two hard
subjets within the AK8 jet.

We refer to the identification of jets likely originating from bottom quarks
as tagging [90, 91]. For each AK4 jet in the event, a score is calculated
through a multivariate combination of different jet properties, making use of
boosted decision trees and deep neural networks. A jet is considered -tagged
if its associated score exceeds a threshold, tuned to achieve a certain
tagging efficiency as measured in events. The chosen working point corresponds
to about 90% efficiency for bottom quark jets and to a mistagging rate of
about 10% for light-flavor or gluon jets and of about 50% for charm quark
jets.

The missing transverse momentum vector is computed as the negative vector sum
of the transverse momenta of all the PF candidates in an event, and its
magnitude is denoted as [41]. The PUPPI algorithm is applied to reduce the
pileup dependence of the observable by computing the from the PF candidates
weighted by their probability to originate from the primary interaction vertex
[41].

##  0.6 Event selection

The analysis is performed using $\PH\to\PW\PW$ candidate events in the
$\Pe\PGm$ final state. For an event to be selected, the transverse momenta of
the leading lepton $\pt^{\Pell 1}$ and the subleading lepton $\pt^{\Pell 2}$
must be greater than 25 and 13, respectively. The $\pt^{\Pell 2}$ threshold in
the case of a muon is lowered to 10for the 2016 data set because of the lower
threshold in the corresponding HLT algorithm. Events containing additional
leptons with $\pt>10\GeV$ are discarded. The dilepton system is required to
have an invariant mass $m_{\Pell\Pell}$ greater than 12and transverse momentum
$\pt^{\Pell\Pell}$ above 30. A requirement on the missing transverse momentum
of $\ptmiss>20\GeV$ is implemented. We define transverse mass discriminating
variables $\mT^{\PH}$ and $\mT^{\Pell 2}$ as

$\centering\mT^{\PH}=\sqrt{2\pt^{\Pell\Pell}\ptmiss[1-\cos{\Delta\Phi(\vec{p}_{\text{T}}^{\Pell\Pell},\vec{p}_{\text{T}}^{\text{miss}})}]},\>\@add@centering$
(17) $\centering\mT^{\Pell 2}=\sqrt{2\pt^{\Pell
2}\ptmiss[1-\cos{\Delta\Phi(\vec{p}_{\text{T}}^{\Pell
2},\vec{p}_{\text{T}}^{\text{miss}})}]},\>\@add@centering$ (18)

and select events with $\mT^{\PH}>60\GeV$ and $\mT^{\Pell 2}>30\GeV$. The
$\mT^{\PH}$ requirement suppresses the $\text{DY}\to\PGt{}\PGt$ background
process and avoids overlap with the $\PH\to\PGt\PGt$ analysis [11]. To ensure
orthogonality with a future off-shell $\PH\to\PW\PW$ analysis we require
$\mT^{\PH}<125\GeV$. In addition, the region $76.2<m_{\Pell\Pell}<106.2\GeV$
is excluded to avoid overlap with the off-shell $\PH\to\PZ\PZ\to 2\Pell 2\PGn$
analysis [10]. These requirements will simplify a future combination of Higgs
boson decay final states. Finally, events with any b-tagged jets with
$\pt>20\GeV$ are vetoed. These base selection criteria are summarized in Table
0.6.

Summary of the base selection criteria.  Variable Selection Number of leptons
2 ($\Pe\PGm$ of opposite charge) $\pt^{\Pell 1}$ $>$25 $\pt^{\Pell 2}$
$>$13(10for 2016 data) $m_{\Pell\Pell}$ 12–76.2or $>$106.2 $\pt^{\Pell\Pell}$
$>$30 $\ptmiss$ $>$20 $\mT^{\Pell 2}$ $>$30 $\mT^{\PH}$ 60–125
$N_{\text{jet}}$ (jets) 0

For the $\PH\PV\PV$ coupling analysis, exclusive selection criteria, which are
based on the associated jet activity in the event, are applied that target the
$\Pg\Pg\PH$, VBF, and $\PV\PH$ production processes. The AK4 (AK8) jets
considered are required to have $\pt>30~{}(200)\GeV$. In the $\Pg\Pg\PH$
channel, zero or one AK4 jet is required in the event. For the VBF and
Resolved $\PV\PH$ channels, we require two AK4 jets with dijet masses of
$m_{\text{jj}}>120\GeV$ and $60<m_{\text{jj}}<120\GeV$, respectively. The
Boosted $\PV\PH$ channel requires the presence of a -tagged AK8 jet (jet);
such jets have a groomed mass in the region $65<m_{\text{J}}<105\GeV$ and
satisfy the requirement $\tau_{2}/\tau_{1}<0.4$. In the other channels, a jet
veto is implemented to ensure orthogonality. These production channels for the
$\PH\PV\PV$ coupling study are summarized in Table 0.6.

Summary of the $\Pg\Pg\PH$, VBF, and $\PV\PH$ production channels used for the
$\PH\PV\PV$ coupling study.  Variable $\Pg\Pg\PH$ VBF Resolved $\PV\PH$
Boosted $\PV\PH$ $N_{\text{jet}}$ (jets) 0 0 0 $>$0 $N_{\text{jet}}$ (AK4
jets) 0 & 1 2 2 $m_{\text{jj}}$ $>$120 60–120

As the production vertex of the $\Pg\Pg\PH$ \+ 2 jets process is sensitive to
anomalous $\PH\Pg\Pg$ coupling effects, we use a 2-jet $\Pg\Pg\PH$ channel
that follows the VBF selection described above for the $\PH\Pg\Pg$ coupling
analysis. The $\PH\PW\PW$ decay vertex is not sensitive to anomalous
$\PH\Pg\Pg$ effects, and so decay-based variables are not studied in this
channel. This permits a relatively tight selection of $m_{\Pell\Pell}<55\GeV$
which is beneficial for background suppression. The 0- and 1-jet $\Pg\Pg\PH$
channels are also included to constrain the $\Pg\Pg\PH$ signal strength. All
channels included for the $\PH\Pg\Pg$ coupling study are summarized in Table
0.6.

Summary of $\Pg\Pg\PH$ channel selections used for the $\PH\Pg\Pg$ coupling
study.  Variable $\Pg\Pg\PH$ 2-jet $\Pg\Pg\PH$ $N_{\text{jet}}$ (AK4 jets) 0 &
1 2 $m_{\text{jj}}$ $>$120 $m_{\Pell\Pell}$ $<$55

Control regions (CRs) are defined using the base selection criteria together
with a set of alternative requirements summarized in Table 0.6. They are used
to validate the background description and to estimate the number of
background events in the signal region (SR). A dedicated $\PGt{}\PGt$ CR
targets events from the DY process $\PZ\to\PGt{}\PGt$ with $\PGt$ leptons
decaying leptonically to produce the $\Pe\PGm$ final state. Also a top quark
CR is defined to enhance events with one or more top quarks decaying to a
boson and bottom quark. Splitting events according to the number of associated
jets, separate $\PGt{}\PGt$ and top quark CRs are defined for the 0-, 1- and
2-jet SRs. An additional CR with an enhanced contribution from the nonresonant
$\PW\PW$ background is used in the 2-jet SR. All CRs are used in the final
data fit to constrain the DY, top quark, and $\PW\PW$ background
normalizations.

Summary of the $\PGt{}\PGt$, top quark, and $\PW\PW$ control region
requirements.  Variable $\PGt{}\PGt$ top quark $\PW\PW$ $m_{\Pell\Pell}$ 40–80
$>$50 $>$106.2 $\mT^{\PH}$ $<$60 60–125 $\mT^{\Pell 2}$ $>$30 $>$30
$N_{\text{jet}}$ (jets) 0 $>$0 0

Additional $\PGt{}\PGt$, top quark, and $\PW\PW$ CRs are defined requiring a
jet. These CRs are used to validate the background description in the Boosted
$\PV\PH$ channel. However, they generally do not have a sufficient number of
events to significantly constrain the background normalizations in the final
fit to the data. As such, we rely on the 2-jet CRs to determine the
normalizations to be used in the Boosted $\PV\PH$ channel. Agreement between
data and the background prediction in the jet CRs is observed when using
normalizations determined in the 2-jet CRs.

##  0.7 Background estimation

The nonprompt-lepton backgrounds originating from leptonic decays of heavy
quarks, hadrons misidentified as leptons, and electrons from photon
conversions are suppressed by identification and isolation requirements
imposed on electrons and muons. In this analysis, the nonprompt-lepton
background primarily originates from $\PW+$jets events and is estimated from
data, as described in detail in Ref. [92]. The procedure involves measuring
the rate at which a nonprompt lepton passing a loose selection further passes
a tight selection (misidentification rate) and the corresponding rate for a
prompt lepton to pass this selection (prompt rate). The misidentification rate
is measured in a data sample enriched in multijet events, whereas the prompt
rate is measured using a tag-and-probe method [93] in a data sample enriched
in DY events. The nonprompt-lepton background estimation is validated with
data in a CR enriched with $\PW+$jets events, in which a pair of same-sign
leptons is required.

The backgrounds from top quark processes and nonresonant $\PW\PW$ production
are estimated using a combination of MC simulations and the dedicated CRs
described in the previous section. The normalisations of these backgrounds are
left as free parameters in the fit, keeping different parameters for each jet
multiplicity region. The top quark background normalization is measured from
the observed data in the top quark enriched CRs. A separate normalization
parameter is included for the quark-induced and gluon-induced $\PW\PW$
backgrounds. For the 2-jet regions, the $\PW\PW$ enriched CR is used to
constrain the $\PW\PW$ background normalisation parameters. In the 0- and
1-jet channels, these parameters are constrained directly in the signal
regions, which span the high $m_{\Pell\Pell}$ phase space enriched in $\PW\PW$
events.

The $\text{DY}\to\PGt{}\PGt$ background process is estimated with a data-
embedding technique [94]. As for the top quark and $\PW\PW$ backgrounds, the
DY normalization is left unconstrained in the data fit. The
$\text{DY}\to\PGt{}\PGt$ enriched CR described in Section 0.6 is used to
constrain the free normalization parameters in the 0-, 1-, 2-jet regions. The
data-embedded samples cover the events that pass the $\Pe\PGm$ triggers, which
represent the vast majority of the selected events. The remaining
$\text{DY}\to\PGt{}\PGt$ events, which enter the analysis through the single-
lepton triggers (${\approx}5\%$ of the total), are estimated using MC
simulation.

The $\PW\PZ$ and $\PW{}{\PGg}^{*}$ background contributions are simulated as
described in Section 0.4, and a data-to-simulation scale factor is derived in
a three-lepton CR, as described in Ref. [92]. The contribution of the
$\PW{}\PGg$ process may also be a background because of photon conversions in
the detector material. This process is estimated using MC simulation and
validated using data in a CR requiring events with a leading and a trailing
with same sign and a separation in $\Delta
R=\sqrt{\smash[b]{\Delta\phi^{2}+\Delta\eta^{2}}}$ (where $\phi$ is the
azimuthal angle in radians) smaller than 0.5. Triple vector boson production
is a minor background in all channels and is estimated using MC simulation.

##  0.8 Observables and kinematic discriminants

In this paper, we search for anomalous $\PH\PV\PV$ and $\PH\Pg\Pg$ coupling
effects by studying:

  1. 1.

the two quark jets from VBF and $\PV\PH$ production ($\PH\PV\PV$ coupling);

  2. 2.

the $\PH\to\PW\PW$ decay products ($\PH\PV\PV$ coupling); and

  3. 3.

the two quark jets from $\Pg\Pg\PH$ \+ 2 jets production ($\PH\Pg\Pg$
coupling).

The VBF, $\PV\PH$, and $\Pg\Pg\PH$ production and decay topologies relevant
for the $\PH\PV\PV$ coupling are illustrated in Fig. 1.

Figure 1:  Topologies of the Higgs boson production and decay for vector boson
fusion $\PQq{\PQq^{\prime}}\to\PQq{\PQq^{\prime}}\PH$ (left),
$\PQq\bar{\PQq}^{\prime}\to\PV\PH$ (center), and gluon fusion with decay
$\Pg\Pg\to\PH\to 2\ell 2\nu$ (right). For the electroweak production
topologies, the intermediate vector bosons and their decays are shown in green
and the $\PH\to\PW\PW$ decay is marked in red. For the $\Pg\Pg\to\PH\to 2\ell
2\nu$ topology, the $\PW$ boson leptonic decays are shown in green. In all
cases, the incoming particles are depicted in brown and the angles
characterizing kinematic distributions are marked in blue. Five angles fully
characterize the orientation of the production and decay chain and are defined
in the suitable rest frames.

When combined with the momentum transfer of the vector bosons, the five angles
illustrated for VBF/$\PV\PH$ production provide complete kinematic information
for production and decay of the Higgs boson. The illustration for Higgs boson
production via $\Pg\Pg\PH$ in association with two jets is identical to the
VBF diagram, except for replacing the intermediate vector bosons by gluons.
Full production kinematic information is extracted for VBF, $\PV\PH$, and
$\Pg\Pg\PH$ \+ 2 jets candidate events using discriminants built from the
matrix element calculations of the MELA package. The MELA approach is designed
to reduce the number of observables to a minimum, while retaining all
essential information. To form the production-based MELA kinematic
discriminants, we use jets to reconstruct the four-momentum of the associated
production particles. The presence of two neutrinos in the final state means
it is not possible to reconstruct the four-momentum of all the Higgs boson
decay products. Therefore, decay-based kinematic discriminants built from
matrix elements are not used in this analysis. Instead, we rely on kinematic
variables related to the measured final state of the Higgs boson decay. The
strategies used for each of the topologies listed above are now discussed in
more detail.

###  0.8.1 Kinematic features of two quark jets in VBF and $\PV\PH$ channels

Kinematic distributions of associated particles in VBF and $\PV\PH$ production
are sensitive to the anomalous $\PH\PV\PV$ couplings of the Higgs boson. As
illustrated in Fig. 1, a set of seven observables can be defined for the VBF
and $\PV\PH$ production topologies :
$\Omega=\\{\theta_{1}^{(\prime)},\theta_{2}^{(\prime)},\theta^{*},\Phi,\Phi_{1},q_{1}^{2},q_{2}^{2}\\}$,
with $q_{1}^{2}$ and $q_{2}^{2}$ the squared four-momenta of the vector bosons
[22]. Three types of discriminants are defined using the full kinematic
description characterized by $\Omega$. The first type of discriminant is
designed to separate signal and background Higgs boson production processes:

$\mathcal{D}_{\text{sig}}=\frac{\mathcal{P}_{\text{sig}}(\Omega)}{\mathcal{P}_{\text{sig}}(\Omega)+\mathcal{P}_{\text{bkg}}(\Omega)},$
(19)

where the probability density $\mathcal{P}$ for a specific process is
calculated from the matrix elements provided by the MELA package. The second
type of discriminant separates the anomalous coupling BSM process from that of
the SM:

$\mathcal{D}_{\text{BSM}}=\frac{\mathcal{P}_{\text{BSM}}(\Omega)}{\mathcal{P}_{\text{BSM}}(\Omega)+\mathcal{P}_{\text{SM}}(\Omega)}.$
(20)

Throughout this document the generic BSM label is generally replaced by the
specific anomalous coupling state targeted. For the $a_{3}$ $CP$-odd and
$a_{2}$ $CP$-even coupling parameters, we use, respectively,
$\mathcal{D}_{0-}$ and $\mathcal{D}_{0+}$, whereas for the $\Lambda_{1}$
coupling parameters we use $\mathcal{D}_{\Lambda_{1}}$ and
$\mathcal{D}_{\Lambda_{1}}^{\PZ\gamma}$. The third type of discriminant
isolates the interference contribution:

$\mathcal{D}_{\text{int}}=\frac{\mathcal{P}_{\text{SM-
BSM}}^{\text{int}}(\Omega)}{\mathcal{P}_{\text{SM}}(\Omega)+\mathcal{P}_{\text{BSM}}(\Omega)},$
(21)

where $\mathcal{P}_{\text{SM-BSM}}^{\text{int}}$ is the interference part of
the probability distribution for a process with a mixture of the SM and BSM
contributions. The $CP$ label is generally used for the $a_{3}$ coupling
parameter, as the BSM signal in this case is a pseudoscalar and the
interference discriminant is a $CP$-sensitive observable. The $\mathcal{P}$
values are normalized to give the same integrated cross sections in the
relevant phase space of each process. Such normalization leads to a balanced
distribution of events in the range between 0 and 1 for
$\mathcal{D}_{\text{sig}}$ and $\mathcal{D}_{\text{BSM}}$, and between $-1$
and $+1$ for $\mathcal{D}_{\text{int}}$.

The selected events are split into three main production channels: VBF,
Resolved $\PV\PH$, and Boosted $\PV\PH$. In the first two channels, the four-
momenta of the two AK4 jets assigned as the associated particles are used in
the MELA probability calculation. For the Boosted $\PV\PH$ category, we use
the four-momentum of the two subjets of the -tagged AK8 jet. An estimate of
the Higgs boson four-momentum is also required for the probability
calculation. This can not be measured directly since the final state contains
two neutrinos. As such, we construct a proxy Higgs boson four-momentum in the
following manner. The $p_{\text{x}}$ and $p_{\text{y}}$ of the dineutrino
system are estimated from the in a given event. The corresponding
$p_{\text{z}}$ is then set to equal that of the dilepton system, which is
based on the observed correlation between these variables at the generator
level for simulated signals. Finally, the mass of the dineutrino system is set
equal to the mean value of the generator-level dineutrino mass. The resulting
four-momentum can then be combined with that of the measured dilepton system
to create a proxy Higgs boson four-momentum. We note that the MELA probability
calculation for the production vertices is largely based on the kinematic
features of the associated particles, so the reconstruction of the proxy Higgs
boson has a relatively small effect on the final discriminants. As an
illustrative example of the MELA based discriminants used in this analysis,
Fig. 2 shows the $\mathcal{D}_{0-}$ discriminant in the VBF and Resolved
$\PV\PH$ production channels for a number of different signal hypotheses. The
discriminants are designed to target the dominant signal production process in
a given channel.

Figure 2:  The $\mathcal{D}_{0-}$ discriminant in the VBF (left) and Resolved
$\PV\PH$ (right) production channels for a number of VBF (left) and $\PV\PH$
(right) signal hypotheses. Pure $a_{1}$ ($f_{a3}=0$) and $a_{3}$ ($f_{a3}=1$)
$\PH\PV\PV$ signal hypotheses are shown along with a mixed coupling hypothesis
($f_{a3}=0.5$). All distributions are normalized to unity.

In the VBF channel, a $\mathcal{D}_{\text{VBF}}$ discriminant is constructed,
following Eq. (19), where $\mathcal{P}_{\text{sig}}$ corresponds to the
probability for the VBF production hypothesis, and $\mathcal{P}_{\text{bkg}}$
corresponds to that of gluon fusion production in association with two jets.
The discriminant is also suitable for separating SM backgrounds from the VBF
signal process. In the Resolved and Boosted $\PV\PH$ channels, the
corresponding discriminants do not give a significant level of separation with
respect to $\Pg\Pg\PH$ production or SM backgrounds. This is due to the
relatively tight selection criteria, which limit the phase space to
$\PV\PH$-like events. Hence, these discriminants are not included in the
$\PV\PH$ channels.

The $\mathcal{D}_{CP}$ discriminant is sensitive to the sign of the
interference between the $CP$-even SM and $CP$-odd BSM states. An asymmetry
between the number of events detected with positive and negative
$\mathcal{D}_{CP}$ values is expected for mixed $CP$ states. Therefore, a
forward-backward categorization (forward defined as $\mathcal{D}_{CP}>0$ and
backward as $\mathcal{D}_{CP}<0$) is used to analyze the $CP$-odd couplings.
Similarly, $\mathcal{D}_{\text{int}}$ gives sensitivity to the sign of the
interference between the SM and $a_{2}$ $\PH\PV\PV$ BSM states. A forward-
backward $\mathcal{D}_{\text{int}}$ categorization is also included. The value
of $\mathcal{D}_{\text{int}}$ used to define the categories is chosen to
symmetrize the SM Higgs boson expectation. In the case of the $\Lambda_{1}$
measurements, the interference discriminants were shown to be highly
correlated with the $\mathcal{D}_{\text{BSM}}$ discriminants and so are not
considered.

We now discuss the categorization and construction of the final
multidimensional discriminants used for the two $\PH\PV\PV$ coupling
approaches defined in Section 0.2. The binning of the final discriminants was
optimized to ensure sufficient statistical precision in the predictions of all
bins, while retaining the kinematic information required to discriminate
between the SM and anomalous coupling signal hypotheses.

#### VBF/$\PV\PH$ analysis strategy for Approach 1

In Approach 1, each of the four anomalous $\PH\PV\PV$ coupling parameters
($a_{2}$, $a_{3}$, $\kappa_{1}/(\Lambda_{1})^{2}$, and
$\kappa_{2}^{\PZ\PGg}/(\Lambda_{1}^{\PZ\PGg})^{2}$) are analyzed separately.
For this purpose, we construct a multidimensional discriminant for each of the
four anomalous couplings in the VBF, Resolved $\PV\PH$, and Boosted $\PV\PH$
channels.

In the VBF channel, we use two bins of the production discriminant
$\mathcal{D}_{\text{VBF}}$, corresponding to low and high purity, using a bin
boundary of 0.75. The $m_{\Pell\Pell}$ variable, which is sensitive to
anomalous effects at the $\PH\to\PW\PW$ decay vertex, is included with two
bins in the range 12–76.2. A bin boundary of 45is chosen based on the expected
signal shape changes induced by anomalous effects. Finally, one of the
$\mathcal{D}_{\text{BSM}}$ discriminants is included with ten equally sized
bins. Depending on the anomalous coupling under study this discriminant may be
$\mathcal{D}_{0+}$, $\mathcal{D}_{0-}$, $\mathcal{D}_{\Lambda_{1}}$ or
$\mathcal{D}_{\Lambda_{1}}^{\PZ\gamma}$.

For the $\PV\PH$ channels, the $m_{\Pell\Pell}$ and $\mathcal{D}_{\text{BSM}}$
observables are used to build 2D kinematic discriminants. The $m_{\Pell\Pell}$
bins are the same as for the VBF channel. In the Resolved $\PV\PH$ channel, we
use four $\mathcal{D}_{\text{BSM}}$ bins of equal size. For the Boosted
$\PV\PH$ case, three variable bins with boundaries of 0.6 and 0.8 are used, a
large first bin is chosen because relatively little signal is expected at low
values of $\mathcal{D}_{\text{BSM}}$. A distinct multidimensional discriminant
is constructed for each anomalous coupling hypothesis in the $\PV\PH$
channels.

For the $a_{3}$ coupling parameter, a forward-backward categorization of
events based on $\mathcal{D}_{CP}$ is implemented. In the case of the $a_{2}$
coupling parameter, $\mathcal{D}_{\text{int}}$ is largely correlated with
$\mathcal{D}_{0+}$ in the $\PV\PH$ channels. Therefore, a forward-backward
$\mathcal{D}_{\text{int}}$ categorization is implemented only in the VBF
channel. Figures 3–5 show the discriminants used in the final fit to the data
for the $a_{2}$, $a_{3}$, $\kappa_{1}/(\Lambda_{1})^{2}$, and
$\kappa_{2}^{\PZ\PGg}/(\Lambda_{1}^{\PZ\PGg})^{2}$ Approach 1 coupling studies
in the VBF and $\PV\PH$ channels. A summary of the observables used in the
$\PH\PV\PV$ Approach 1 analysis may be found in Table 0.8.3.

#### VBF/$\PV\PH$ analysis strategy for Approach 2

In Approach 2, we use one categorization strategy and build one
multidimensional discriminant in each channel to target all the $\PH\PV\PV$
coupling parameters ($a_{2}$, $a_{3}$, $\kappa_{1}/(\Lambda_{1})^{2}$)
simultaneously. In the VBF channel, the $\mathcal{D}_{CP}$ and
$\mathcal{D}_{\text{int}}$ discriminants are used to create four interference
categories. Both $\mathcal{D}_{\text{VBF}}$ and $m_{\Pell\Pell}$ are used as
for Approach 1. All three $\mathcal{D}_{\text{BSM}}$ discriminants that target
the $a_{2},a_{3}$ and $\kappa_{1}/(\Lambda_{1})^{2}$ coupling parameters are
included. However, the number of bins we implement is limited by the number of
simulated events. Also the $\mathcal{D}_{\text{BSM}}$ discriminants are
significantly correlated and so have similar performance for all couplings.
Therefore, we use the $CP$-odd discriminant $\mathcal{D}_{0-}$ and just one of
the $CP$-even discriminants, $\mathcal{D}_{0+}$, both with three bins and bin
boundaries of 0.1 and 0.9. A dedicated rebinning strategy is applied to the
$[\mathcal{D}_{0-},\mathcal{D}_{0+}]$ distribution merging bins dominated by
the SM Higgs boson prediction or with low precision in the background
prediction. In the $\PV\PH$ channels, just two categories using
$\mathcal{D}_{CP}$ are defined and the discriminant is built using
$m_{\Pell\Pell}$ as for Approach 1. Again, both $\mathcal{D}_{0-}$ and
$\mathcal{D}_{0+}$ are chosen for the final discriminant. For the Resolved
$\PV\PH$ channel, we use three bins with boundaries of 0.25 and 0.75, whereas
for the Boosted $\PV\PH$ case we use two bins with a boundary of 0.8. The same
rebinning strategy described for the VBF channel is applied to both Resolved
and Boosted $\PV\PH$ multidimensional discriminants. Table 0.8.3 includes a
summary of the observables used in the $\PH\PV\PV$ Approach 2 analysis.

###  0.8.2 Kinematic features of $\PH\to\PW\PW$ decay products in 0- and 1-jet
$\Pg\Pg\PH$ channels

Similar to the SM $\PH\to\PW\PW$ analysis [25], we use $m_{\Pell\Pell}$ and
$\mT$ to build 2D discriminants in the 0- and 1-jet $\Pg\Pg\PH$ channels. The
distributions have nine bins for $m_{\Pell\Pell}$ in the range 12–200and six
bins for in the range 60–125. The bin widths vary and are optimized to achieve
good separation between the SM Higgs boson signal and backgrounds, as well as
between the different anomalous coupling signal hypotheses. In particular, a
finer binning with respect to the SM $\PH\to\PW\PW$ analysis is implemented in
regions where anomalous effects are most significant. Figure 6 shows the
$[\mT,m_{\Pell\Pell}]$ distributions in the 0- and 1-jet $\Pg\Pg\PH$ channels.
The same $[\mT,m_{\Pell\Pell}]$ discriminant is used to study all $\PH\PV\PV$
anomalous couplings for both Approach 1 and 2.

###  0.8.3 Kinematic features of two quark jets in 2-jet $\Pg\Pg\PH$ channel

For the $\PH\Pg\Pg$ coupling, we adopt a similar approach to the VBF $CP$
study, where the $CP$-odd $a_{3}$ $\PH\PV\PV$ coupling parameter is included.
In this case, the optimal observables are $\mathcal{D}^{\Pg\Pg\PH}_{0-}$ and
$\mathcal{D}^{\Pg\Pg\PH}_{CP}$, targeting the $CP$-odd $a_{3}$ $\PH\Pg\Pg$
coupling parameter. A forward-backward categorization is implemented using
$\mathcal{D}^{\Pg\Pg\PH}_{CP}$, and the $\mathcal{D}_{\text{VBF}}$ and
$\mathcal{D}^{\Pg\Pg\PH}_{0-}$ observables are used to build 2D discriminants.
The $m_{\Pell\Pell}$ variable is not considered in this case because it is not
sensitive to anomalous $\PH\Pg\Pg$ effects. For $\mathcal{D}_{\text{VBF}}$,
the bin boundary is relaxed to 0.5 to ensure sufficient $\Pg\Pg\PH$ events are
accepted in the more VBF-like bin. For $\mathcal{D}_{0-}$, eight (five) bins
are used in the more (less) VBF-like bin with larger bin sizes at the extremes
of the distribution to ensure sufficient precision in the background and
signal predictions. The 0- and 1-jet channels discussed previously are also
included in this study to constrain the $\Pg\Pg\PH$ signal strength. The
$[\mathcal{D}_{\text{VBF}}$, $\mathcal{D}^{\Pg\Pg\PH}_{0-}]$ distributions
used to analyze the $\PH\Pg\Pg$ $a_{3}$ anomalous coupling in the 2-jet
$\Pg\Pg\PH$ channel are shown in Fig. 7. A summary of the observables used in
the $\PH\Pg\Pg$ analysis is given in Table 0.8.3.

The kinematic observables used for the interference based categorization and
for the final discriminants used in the fits to data to study the $\PH\PV\PV$
and $\PH\Pg\Pg$ couplings. For each of the anomalous $\PH\PV\PV$ couplings in
Approach 1, we have a dedicated analysis in the VBF and $\PV\PH$ channels. In
Approach 2, we use one analysis to target all anomalous $\PH\PV\PV$ couplings
simultaneously.  Analysis Channel Categorization Final discriminant
$\PH\PV\PV$ VBF ($a_{2}$) $\mathcal{D}_{\text{int}}$
$[\mathcal{D}_{\text{VBF}}$, $m_{\Pell\Pell}$, $\mathcal{D}_{0+}]$ Approach 1
VBF ($a_{3}$) $\mathcal{D}_{CP}$ $[\mathcal{D}_{\text{VBF}}$,
$m_{\Pell\Pell}$, $\mathcal{D}_{0-}]$ VBF ($\kappa_{1}$)
$[\mathcal{D}_{\text{VBF}}$, $m_{\Pell\Pell}$, $\mathcal{D}_{\Lambda 1}]$ VBF
($\kappa_{2}^{\PZ\PGg}$) $[\mathcal{D}_{\text{VBF}}$, $m_{\Pell\Pell}$,
$\mathcal{D}_{\Lambda_{1}}^{\PZ\gamma}]$ $\PV\PH$ ($a_{2}$) $[m_{\Pell\Pell}$,
$\mathcal{D}_{0+}]$ $\PV\PH$ ($a_{3}$) $\mathcal{D}_{CP}$ $[m_{\Pell\Pell}$,
$\mathcal{D}_{0-}]$ $\PV\PH$ ($\kappa_{1}$) $[m_{\Pell\Pell}$,
$\mathcal{D}_{\Lambda 1}]$ $\PV\PH$ ($\kappa_{2}^{\PZ\PGg}$)
$[m_{\Pell\Pell}$, $\mathcal{D}_{\Lambda_{1}}^{\PZ\gamma}]$ 0- $\&$ 1-jet
$\Pg\Pg\PH$ $[\mT$, $m_{\Pell\Pell}]$ $\PH\PV\PV$ VBF $\mathcal{D}_{CP}$,
$\mathcal{D}_{\text{int}}$ $[\mathcal{D}_{\text{VBF}}$, $m_{\Pell\Pell}$,
$\mathcal{D}_{0-}$, $\mathcal{D}_{0+}]$ Approach 2 VH $\mathcal{D}_{CP}$
$[m_{\Pell\Pell}$, $\mathcal{D}_{0-}$, $\mathcal{D}_{0+}]$ 0- $\&$ 1-jet
$\Pg\Pg\PH$ $[\mT$, $m_{\Pell\Pell}]$ $\PH\Pg\Pg$ 2-jet $\Pg\Pg\PH$
$\mathcal{D}^{\Pg\Pg\PH}_{CP}$ $[\mathcal{D}_{\text{VBF}}$,
$\mathcal{D}^{\Pg\Pg\PH}_{0-}]$ 0- $\&$ 1-jet $\Pg\Pg\PH$ $[\mT$,
$m_{\Pell\Pell}]$

Figure 3:  Observed and predicted distributions after fitting the data for
$[\mathcal{D}_{\text{VBF}},m_{\Pell\Pell},\mathcal{D}_{0+}]$ in the VBF
channel (upper), and for $[m_{\Pell\Pell},\mathcal{D}_{0+}]$ in the Resolved
$\PV\PH$ (lower left) and Boosted $\PV\PH$ (lower right) channels. For the VBF
channel, the $\mathcal{D}_{\text{int}}<0.4$ (left) and
$\mathcal{D}_{\text{int}}>0.4$ (right) categories are shown. The predicted
Higgs boson signal is shown stacked on top of the background distributions.
For the fit, the $a_{1}$ and $a_{2}$ $\PH\PV\PV$ coupling contributions are
included. The corresponding pure $a_{1}$ ($f_{a2}=0$) and $a_{2}$ ($f_{a2}=1$)
signal hypotheses are also shown superimposed, their yields correspond to the
predicted number of SM signal events scaled by an arbitrary factor to improve
visibility. The uncertainty band corresponds to the total systematic
uncertainty. The lower panel in each figure shows the ratio of the number of
events observed to the total prediction.

Figure 4:  Observed and predicted distributions after fitting the data for
$[\mathcal{D}_{\text{VBF}},m_{\Pell\Pell},\mathcal{D}_{0-}]$ in the VBF
channel (upper), and for $[m_{\Pell\Pell},\mathcal{D}_{0-}]$ in the Resolved
$\PV\PH$ (middle) and Boosted $\PV\PH$ (lower) channels. For each channel, the
$\mathcal{D}_{CP}<0$ (left) and $\mathcal{D}_{CP}>0$ (right) categories are
shown. For the fit, the $a_{1}$ and $a_{3}$ $\PH\PV\PV$ coupling contributions
are included. More details are given in the caption of Fig. 3.

Figure 5:  Observed and predicted distributions after fitting the data for
$[\mathcal{D}_{\text{VBF}},m_{\Pell\Pell},\mathcal{D}_{\Lambda 1}]$ (upper
left) and $[\mathcal{D}_{\text{VBF}},m_{\Pell\Pell},\mathcal{D}_{\Lambda
1}^{\PZ\gamma}]$ (upper right) in the VBF channel, and for
$[m_{\Pell\Pell},\mathcal{D}_{\Lambda 1}]$ (left) and
$[m_{\Pell\Pell},\mathcal{D}_{\Lambda 1}^{\PZ\gamma}]$ (right) in the Resolved
$\PV\PH$ (middle) and Boosted $\PV\PH$ (lower) channels. For the fits, the
$a_{1}$ and $\kappa_{1}/(\Lambda_{1})^{2}$ (left) or $a_{1}$ and
$\kappa_{2}^{\PZ\PGg}/(\Lambda_{1}^{\PZ\PGg})^{2}$ (right) $\PH\PV\PV$
coupling contributions are included. More details are given in the caption of
Fig. 3.

Figure 6:  Observed and predicted distributions after fitting the data for
$[\mT,m_{\Pell\Pell}]$ in the 0- (left) and 1-jet (right) $\Pg\Pg\PH$
channels. For the fit, the $a_{1}$ and $a_{3}$ $\PH\PV\PV$ coupling
contributions are included. More details are given in the caption of Fig. 3.

Figure 7:  Observed and predicted distributions after fitting the data for
$[\mathcal{D}_{\text{VBF}}$, $\mathcal{D}^{\Pg\Pg\PH}_{0-}]$ in the 2-jet
$\Pg\Pg\PH$ channel. Both the $\mathcal{D}^{\Pg\Pg\PH}_{\text{CP}}<0$ (left)
and $\mathcal{D}^{\Pg\Pg\PH}_{\text{CP}}>0$ (right) categories are shown. In
this case, the VBF and $\Pg\Pg\PH$ signals are shown separately. For the fit,
the $a_{2}^{\Pg\Pg}$ and $a_{3}^{\Pg\Pg}$ coupling contributions are included.
The corresponding pure $a_{2}^{\Pg\Pg}$ ($f_{a3}^{\Pg\Pg\PH}=0$) and
$a_{3}^{\Pg\Pg}$ ($f_{a3}^{\Pg\Pg\PH}=1$) signal hypotheses are also shown
superimposed, their yields correspond to the predicted number of SM signal
events. More details are given in the caption of Fig. 3.

##  0.9 Systematic uncertainties

The signal extraction is performed using binned templates to describe the
various signal and background processes. Systematic uncertainties that change
the normalization or shape of the templates are included. All the
uncertainties are modeled as nuisance parameters that are profiled in the
maximum likelihood fit described in Section 0.10. The systematic uncertainties
arise from both experimental or theoretical sources.

###  0.9.1 Experimental uncertainties

The following experimental systematic uncertainties are included in the final
fit to data:

  * •

The total uncertainty associated with the measurement of the integrated
luminosity for 2016, 2017, and 2018 is
$1.2\%\text{~{}\cite[cite]{[\@@bibref{Number}{CMS:LUM-17-003}{}{}]},
}2.3\%\text{~{}\cite[cite]{[\@@bibref{Number}{CMS-PAS-LUM-17-004}{}{}]}, and
}2.5\%$ [46], respectively. This uncertainty is partially correlated among the
three data sets, resulting in an overall uncertainty of 1.6%.

  * •

The systematic uncertainty in the trigger efficiency is determined by varying
the tag lepton selection criteria and the boson mass window used in the tag-
and-probe method. It affects both the normalization and the shape of the
signal and background distributions, and is kept uncorrelated among data sets.
The total normalization uncertainty is less than 1%.

  * •

The tag-and-probe method is also used to determine the lepton identification
and isolation efficiency. Corrections are applied to account for any
discrepancy in the efficiencies measured in data and simulation. The
corresponding systematic uncertainty is about 1% for electrons and 2% for
muons.

  * •

The uncertainties in the determination of the lepton momentum scale mainly
arise from the limited data sample used for their estimation. The impact on
the normalization of the signal and background templates ranges between
0.6–1.0% for the electron momentum scale and is about 0.2% for the muon
momentum scale. They are treated as uncorrelated among the three data-taking
years.

  * •

The jet energy scale uncertainty is modeled by implementing eleven independent
nuisance parameters corresponding to different jet energy correction sources,
six of which are correlated among the three data sets. Their effects vary in
the range of 1–10%, mainly depending on the jet multiplicity in the analysis
phase space. Another source of uncertainty arises from the jet energy
resolution smearing applied to simulated samples to match the resolution
measured in data. The effect varies in a range of 1–5%, depending on the jet
multiplicity and is uncorrelated among the data sets. These uncertainties are
included for both AK4 and AK8 jets. In addition, the $m_{\text{J}}$ scale and
resolution, and tagging corrections with their corresponding uncertainties are
included for -tagged AK8 jets. These variables are calibrated in a top quark-
antiquark sample enriched in hadronically decaying bosons [95].

  * •

The effects of the unclustered energy scale, jet energy scale, and lepton
scales are included for the calculation of the missing transverse momentum.
The resulting normalization systematic uncertainty is 1–10% and is treated as
uncorrelated among the years.

  * •

Both the normalization and shape of the signal and background templates are
affected by the jet pileup identification uncertainty. The effect is below 1%.

  * •

The uncertainty associated with the tagging efficiency is modeled by seventeen
nuisance parameters out of which five are of a theoretical origin and are
correlated among the three data sets. The remaining set of four parameters per
data set are treated as uncorrelated as they arise from the statistical
accuracy of the efficiency measurement [90].

  * •

Estimation of the nonprompt-lepton background is affected by the limited size
of the data sets used for the misidentification rate measurements. It is also
affected by the difference in the flavor composition of jets misidentified as
leptons between the misidentification rate measurement region (enriched in
multijet events) and the signal phase space. The effects on the nonprompt-
lepton background estimation range between a few percent to about 10%
depending on the SR and are treated as nuisance parameters uncorrelated
between electrons and muons and among the three data sets. A normalization
uncertainty of 30% [92] is assigned to fully cover for any discrepancies with
respect to data in a $\PW+$jets CR and is treated as uncorrelated among data
sets.

  * •

The statistical uncertainties due to the limited number of simulated events
are also included for all bins of the background distributions used to extract
the results [96].

###  0.9.2 Theoretical uncertainties

Multiple theoretical uncertainties are considered and are correlated among
data sets, unless stated otherwise:

  * •

The uncertainties related to the choice of PDF and have a minor effect on the
shape of the distributions. Therefore, only normalization effects related to
the event acceptance and to the cross section are included. However, these
uncertainties are not considered for the backgrounds that have their
normalization constrained through data in dedicated CRs. For the Higgs boson
signal processes, these uncertainties are calculated by the LHC Higgs cross
section working group [19].

  * •

The theoretical uncertainties arising from missing higher-order corrections in
the cross section calculations are also included. Background simulations are
reweighted to the alternative scenarios corresponding to renormalization
$\mu_{\text{R}}$ and factorization $\mu_{\text{F}}$ scales varied by factors
0.5 or 2 and the envelopes of the varied templates are taken as the one
standard deviations. For background processes that have their normalization
constrained through data in dedicated CRs, we consider only the shape effect
of the uncertainties coming from the missing higher-order corrections. The
$\PW\PW$ nonresonant background has the uncertainties derived by varying
$\mu_{\text{R}}$, $\mu_{\text{F}}$, and the resummation scale. For the
$\Pg\Pg\PH$ and VBF signal processes, the effects of the missing higher-order
corrections on the overall cross section are decoupled into multiple sources
according to the recipes described in Ref. [19].

  * •

The uncertainty due to the pileup modeling was included for the main simulated
background processes (DY, $\PW\PW$, top quark) as well as the $\Pg\Pg\PH$ and
VBF signals. The effect is determined by varying the total inelastic $\Pp\Pp$
cross section (69.2mb [97, 98]) within the assigned 5% uncertainty.

  * •

The PS modeling mainly affects the jet multiplicity, causing migration of
events between categories that results in template shape changes. Associated
uncertainties are evaluated by reweighting events with varied PS weights
computed with 8.212. The effect on the signal strength is found to be below
1%.

  * •

Uncertainties associated with UE modeling are evaluated by varying the UE tune
parameters used in the MC sample generation. Systematic uncertainties are
correlated between the 2017 and 2018 data sets since they share the same UE
tunes, whereas for 2016 the uncertainty is considered uncorrelated. The UE
uncertainty has a minimal effect on the template shapes and affects the
normalization by about 1.5%.

  * •

A 15% uncertainty is applied to the relative fraction of the $\Pg\Pg$-induced
component in nonresonant $\PW\PW$ production [99]. The relative fraction
between single top quark and processes is assigned a systematic uncertainty of
8% [100]. Additional process-specific (DY, $\PV\PZ$, $\PV\PGg$, $\PV\PGg^{*}$)
uncertainties, related to corrections to account for possible discrepancies
between data and simulation, are assigned and are correlated among data sets.

##  0.10 Results

The optimization and validation of the analysis were performed using
simulation and data in CRs. The data in the SRs were examined once all details
of the analysis were finalized. For the final results, we perform a binned
maximum likelihood fit to the data combining all channels and data-taking
periods. The statistical approach was developed by the ATLAS and CMS
Collaborations in the context of the LHC Higgs Combination Group [101]. The
likelihood function is defined for candidate events as:

$\mathcal{L}(\text{data}|\mu_{\Pg\Pg\PH},\mu_{\text{EW}},f_{ai},\theta)=\prod_{\mathrm{j}}\text{Poisson}(n_{\mathrm{j}}|s_{\mathrm{j}}(\mu_{\Pg\Pg\PH},\mu_{\text{EW}},f_{ai},\theta)+b_{\mathrm{j}}(\theta))p(\tilde{\theta}|\theta)\text{,}$
(22)

where j runs over all bins and $n_{\mathrm{j}}$ is the observed number of data
events in each bin. Total signal and background expectations in each bin are
represented by $s_{\mathrm{j}}$ and $b_{\mathrm{j}}$, respectively. The
individual signal and background processes considered in each category are
described using binned templates of multidimensional discriminants as
described in Section 0.8. Each signal process is parametrized as a linear
combination of terms originating from the SM, and anomalous couplings and
their interference. The signal expectation depends on the parameters
$\mu_{\Pg\Pg\PH}$, $\mu_{\text{EW}}$, and $f_{ai}$, and is constrained by the
data fit. Both the signal and background expectations are functions of
$\theta$, which represents the full set of nuisance parameters corresponding
to the systematic uncertainties. The CRs described in Section 0.6 are included
in the fit in the form of single bins, representing the number of events in
each CR.

The $\mu_{\Pg\Pg\PH}$ and $\mu_{\text{EW}}$ parameters correspond to the Higgs
boson signal strength modifiers for the $\Pg\Pg\PH$ and VBF/$\PV\PH$ signals,
respectively. Signal yields for the VBF and $\PV\PH$ processes are related to
each other because the same $\PH\PV\PV$ couplings enter both in production and
decay of the Higgs boson. The $\Pg\Pg\PH$ signal is initiated predominantly by
the top fermion couplings and is unrelated to the VBF and $\PV\PH$ production
mechanisms. As the signal strength modifiers are free parameters in the fit,
the overall signal event yield is not used to discriminate between alternative
signal hypotheses. The $f_{ai}$ parameter corresponds to the anomalous
coupling cross section fraction and determines the shape of the signal
expectation. The cross section fraction for the SM coupling is simply taken as
$1-\abs{f_{ai}}$. In Approach 1, the SM and just one anomalous $\PH\PV\PV$
coupling are included, and each $f_{ai}$ is thus studied independently.
Depending on the particular anomalous coupling under investigation, $f_{ai}$
may represent $f_{a2}$, $f_{a3}$, $f_{\Lambda 1}$, or $f_{\Lambda
1}^{\PZ\PGg}$. For Approach 2, the SM and three anomalous $\PH\PV\PV$
couplings are included. In this case, $f_{ai}$ represents $f_{a2}$, $f_{a3}$
and $f_{\Lambda 1}$, which are studied simultaneously. It is explicitly
required that $\abs{f_{a2}}+\abs{f_{a3}}+\abs{f_{\Lambda 1}}\leq 1$ to avoid
probing an unphysical parameter space. Finally, there is just one anomalous
coupling corresponding to $f_{a3}^{\Pg\Pg\PH}$ to consider for the $\PH\Pg\Pg$
vertex. For this study, we also include the effect of the $CP$-odd $\PH\PV\PV$
anomalous coupling on the VBF process. This is achieved by including $f_{a3}$
as a free parameter in the fit. The $p(\tilde{\theta}|\theta)$ are the
probability density functions (PDFs) for the observed values of the nuisance
parameters, $\tilde{\theta}$, obtained from calibration measurements. The
systematic uncertainties that affect only the normalizations of the signal and
background processes are treated as PDFs following a log-normal distribution,
whereas shape-altering systematic uncertainties are treated as Gaussian PDFs
[101].

Additional interpretations in terms of the SMEFT Higgs and Warsaw basis
coupling parameters are also considered using Eqs. (7–10) and Eqs. (11–14),
respectively. In each case, four independent couplings are studied
simultaneously and the effect of the couplings on the total width of the Higgs
boson is taken into account. For the $f_{ai}$ measurements, this effect is
absorbed by the signal strength modifiers. A parameterization of the partial
widths of the main Higgs boson decay modes as a function of the couplings is
used to determine the effect on the Higgs boson width [28, 24].

The likelihood is maximized with respect to the signal modifier parameters and
with respect to the nuisance parameters. Confidence level () intervals are
determined from profile likelihood scans of the respective parameters. The
allowed 68% and 95% intervals are defined using the set of parameter values at
which the profile likelihood function $-2\Delta\ln{\mathcal{L}}$ = 1.00 and
3.84 [102], respectively, for which exact coverage is expected in the
asymptotic limit [103]. The likelihood value at a given $f_{ai}$ is determined
by the shape of the signal hypothesis and the relative signal event yields
between categories. Expected results are obtained using the Asimov data set
[104] constructed using the SM values of the signal modifier parameters.

For Approach 1, where we assume $a_{i}^{\PZ\PZ}=a_{i}^{\PW\PW}$, the expected
and observed $f_{a2}$, $f_{a3}$, $f_{\Lambda 1}$, and $f_{\Lambda
1}^{\PZ\PGg}$ likelihood scans are shown in Fig. 8. Significant interference
effects for negative values of $f_{a2}$, around $-0.25$, and positive values
of $f_{\Lambda 1}$, around 0.5, are evident. Relatively large changes in the
signal shape with respect to the SM are predicted at these values. Also
evident are narrow minima around $f_{ai}$ = 0. The anomalous coupling terms in
Eq. (1) have a $q_{i}^{2}$ dependence, which can be larger at the VBF/$\PV\PH$
production vertex than at the Higgs decay vertex. This causes the shape of the
VBF/$\PV\PH$ signal hypothesis to change quickly with $f_{ai}$. The axis
scales are varied to improve the visibility of important features for $f_{a2}$
and $f_{\Lambda 1}$. For Approach 2, where the SU(2) x U(1) coupling
relationships from Eqs. (2–6) are adopted, the expected and observed $f_{a2}$,
$f_{a3}$ and $f_{\Lambda 1}$ likelihood scans are shown in Fig. 9. The results
are shown for each $f_{ai}$ separately with the other two $f_{ai}$ either
fixed to zero or left floating in the fit. The measured values of the signal
strength parameters correspond to $\mu_{\text{EW}}=0.9^{+0.19}_{-0.24}$ and
$\mu_{\Pg\Pg\PH}=0.9^{+0.38}_{-0.20}$ when all parameters float
simultaneously. It is notable that the observed $-2\Delta\ln{\mathcal{L}}$
profile values are generally lower than expected. This is consistent with a
downward statistical fluctuation in the number of VBF and $\PV\PH$ events. The
lowest $\mu_{\text{EW}}$ value measured is 0.82 for the Approach 1 $f_{a3}$
fit which can be compared with the highest value of 0.97 for the corresponding
$f_{\Lambda 1}$ fit. In each case, the uncertainty in $\mu_{\text{EW}}$ is
about 20% and as such all fitted values are consistent with both the SM and
each other. More generally, all anomalous $\PH\PV\PV$ coupling parameter
measurements are consistent with the expectations for the SM Higgs boson. The
p-value compatibility of the full Approach 2 fit, where all signal parameters
float simultaneously, with the SM is 91%. A summary of constraints on the
anomalous $\PH\PV\PV$ coupling parameters with the best fit values and allowed
68% and 95% intervals are shown in Table 0.10. The most stringent constraints
on the $\PH\PV\PV$ anomalous coupling cross section fractions are at the per
mille level. Some constraints are less stringent than expected due to the
fitted values of $\mu_{\text{EW}}$ being lower than the SM expectation. The
observed correlation coefficients between $\PH\PV\PV$ anomalous coupling cross
section fractions and signal strength modifiers are displayed in Fig. 10.

For the SMEFT Higgs basis interpretation, the expected and observed
constraints on the $\delta c_{\text{z}}$, $c_{\text{z}\Box}$, $c_{\text{zz}}$,
and $\tilde{c}_{\text{zz}}$ coupling parameters are shown in Fig. 11. Table
0.10 presents a summary of the constraints on the couplings whereas Fig. 10
reports the observed correlation coefficients between them. For the Warsaw
basis interpretation, the expected and observed constraints on the
$c_{\text{H}\Box}$, $c_{\text{HD}}$, $c_{\text{HW}}$, $c_{\text{HWB}}$,
$c_{\text{HB}}$, $c_{\text{H}\tilde{\text{W}}}$,
$c_{\text{H}\tilde{\text{W}}\text{B}}$, and $c_{\text{H}\tilde{\text{B}}}$
coupling parameters are presented in Table 0.10. To cover all the Warsaw basis
coupling parameters, three independent fits to the data were performed with a
different choice of four independent couplings in each. A summary of the
constraints on the SMEFT Higgs and Warsaw basis coupling parameters is
presented in Fig. 12.

Finally, the expected and observed $f_{a3}^{\Pg\Pg\PH}$ likelihood scans are
shown in Fig. 13. The result is consistent with the expectation for a SM Higgs
boson. Excluding the effect of the $CP$-odd $\PH\PV\PV$ anomalous coupling, by
fixing $f_{a3}$ to zero, has a negligible effect. For $f_{a3}^{\Pg\Pg\PH}$
approaching unity, the observed $-2\Delta\ln{\mathcal{L}}$ profile values are
larger than expected. This is consistent with downward statistical
fluctuations in the data for a couple of bins where sensitivity to the $a_{3}$
$\PH\Pg\Pg$ coupling contribution is enhanced (Fig. 7 left). The constraint on
the anomalous $\PH\Pg\Pg$ coupling parameter with the best fit value and
allowed 68% interval is shown in Table 0.10.

Figure 8:  Expected (dashed) and observed (solid) profiled likelihood on
$f_{a2}$ (upper left), $f_{\Lambda 1}$ (upper right), $f_{a3}$ (lower left),
and $f_{\Lambda 1}^{\PZ\PGg}$ (lower right) using Approach 1. In each case,
the signal strength modifiers are treated as free parameters. The dashed
horizontal lines show the 68 and 95% regions. Axis scales are varied for
$f_{a2}$ and $f_{\Lambda 1}$ to improve the visibility of important features.

Figure 9:  Expected (dashed) and observed (solid) profiled likelihood on
$f_{a2}$ (upper left), $f_{\Lambda 1}$ (upper right) and $f_{a3}$ (bottom)
using Approach 2. The other two anomalous coupling cross section fractions are
either fixed to zero (blue) or left floating in the fit (red). In each case,
the signal strength modifiers are treated as free parameters. The dashed
horizontal lines show the 68 and 95% regions. Axis scales are varied for
$f_{a2}$ and $f_{\Lambda 1}$ to improve the visibility of important features.

Summary of constraints on the anomalous $\PH\PV\PV$ and $\PH\Pg\Pg$ coupling
parameters with the best fit values and allowed 68 and 95% (in square
brackets) intervals. For Approach 1, each $f_{ai}$ is studied independently.
For Approach 2, each $f_{ai}$ is shown separately with the other two cross
section fractions either fixed to zero or left floating in the fit. In each
case, the signal strength modifiers are treated as free parameters.  Analysis
$f_{ai}$ Observed ($\times 10^{-3}$) Expected ($\times 10^{-3}$) $\PH\PV\PV$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\begin{cases}\\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\
\\\ \\\ \end{cases}$ $f_{a2}$ best fit 0.5 0.0 Approach 1 68%  [$-$0.8, 3.5]
[$-$1.4, 1.3] 95%  [$-$5.7, 12.0] [$-$5.2, 6.1] $f_{a3}$ best fit 0.9 0.0 68%
[$-$2.7, 4.1] [$-$0.7, 0.7] 95%  [$-$553.0, 561.0] [$-$2.8, 2.9] $f_{\Lambda
1}$ best fit $-$0.2 0.0 68%  [$-$0.5, 0.0] [$-$0.2, 0.5] 95%  [$-$1.4, 0.7]
[$-$0.6,1.4] $f_{\Lambda 1}^{\PZ\PGg}$ best fit 3.0 0.0 68%  [$-$11.0, 9.1]
[$-$5.0, 3.8] 95%  [$-$55.0, 42.0] [$-$14.0, 11.0] $\PH\PV\PV$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\begin{cases}\\\ \\\ \\\ \\\ \\\ \\\ \\\
\end{cases}$ $f_{a2}$ best fit 38.0 0.0 Approach 2 68%  [$-$112.2, 129.3]
[$-$30.9,37.5] (Fix others) 95%  [$-$376.6, 430.0]$\cup$[$-$989.2, $-$826.3]
[$-$126.1,136.8] $f_{a3}$ best fit 0.8 0.0 68%  [$-$0.8, 3.5] [$-$0.8,1.1] 95%
[$-$7.6, 58.8] [$-$3.4,4.3] $f_{\Lambda 1}$ best fit $-$0.15 0.0 68%
[$-$1.21, 0.16] [$-$0.4,0.4] 95%  [$-$19.5, 118.5]$\cup$[909.9, 964.1]
[$-$1.7,18.9] $\PH\PV\PV$ $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\begin{cases}\\\
\\\ \\\ \\\ \\\ \\\ \\\ \end{cases}$ $f_{a2}$ best fit $-$1.0 0.0 Approach 2
68%  [$-$104.1, 139.9] [$-$31.1,39.8] (Float others) 95%  [$-$986.4, 981.2]
[$-$127.5,148.7] $f_{a3}$ best fit 0.34 0.0 68%  [$-$0.69, 3.4] [$-$1.0,1.2]
95%  [$-$8.0, 361.5] [$-$4.3,5.3] $f_{\Lambda 1}$ best fit $-$0.1 0.0 68%
[$-$1.08, 3.78]$\cup$[7.2, 20.7] [$-$0.4,0.9] 95%  [$-$994.8, 993.9]
[$-$1.9,21.4] $\PH\Pg\Pg$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\begin{cases}\\\ \\\ \end{cases}$
$f_{a3}^{\Pg\Pg\PH}$ best fit $-$34 0 68%  [$-$721, 383] [$-$1000, 1000] 95%
[$-$1000, 1000] [$-$1000, 1000]

Figure 10:  The observed correlation coefficients between $\PH\PV\PV$
anomalous coupling cross section fractions and signal strength modifiers
(left) and between SMEFT Higgs basis coupling parameters (right).

Figure 11:  Expected (dashed) and observed (solid) profiled likelihood on the
$\delta c_{\text{z}}$ (upper left), $c_{\text{z}\Box}$ (upper right),
$c_{\text{zz}}$ (lower left), and $\tilde{c}_{\text{zz}}$ (lower right)
couplings of the SMEFT Higgs basis. All four couplings are studied
simultaneously. The dashed horizontal lines show the 68 and 95% regions.

Summary of constraints on the SMEFT Higgs basis coupling parameters with the
best fit values and 68% uncertainties. All four couplings are studied
simultaneously.  Coupling Observed Expected $\delta c_{\text{z}}$
$-0.06^{+0.09}_{-0.16}$ $0.00^{+0.08}_{-0.10}$ $c_{\text{z}\Box}$
$0.01^{+0.02}_{-0.06}$ $0.00^{+0.02}_{-0.02}$ $c_{\text{zz}}$
$0.03^{+0.30}_{-0.52}$ $0.00^{+0.23}_{-0.29}$ $\tilde{c}_{\text{zz}}$
$-0.17^{+0.42}_{-0.30}$ $0.00^{+0.29}_{-0.32}$

Summary of constraints on the SMEFT Warsaw basis coupling parameters with the
best fit values and 68% uncertainties. Only one of $c_{\text{HW}}$ ,
$c_{\text{HWB}}$, and $c_{\text{HB}}$ is independent, the same is also true
for $c_{\text{H}\tilde{\text{W}}}$, $c_{\text{H}\tilde{\text{W}}\text{B}}$,
and $c_{\text{H}\tilde{\text{B}}}$. Three independent fits to the data were
performed with a different choice of four independent couplings in each.
Coupling Observed Expected $c_{\text{H}\Box}$ $-0.76^{+1.43}_{-3.43}$
$0.00^{+1.37}_{-1.84}$ $c_{\text{HD}}$ $-0.12^{+0.93}_{-0.32}$
$0.00^{+0.43}_{-0.30}$ $c_{\text{HW}}$ $0.08^{+0.43}_{-0.87}$
$0.00^{+0.37}_{-0.48}$ $c_{\text{HWB}}$ $0.17^{+0.88}_{-1.79}$
$0.00^{+0.77}_{-0.96}$ $c_{\text{HB}}$ $0.03^{+0.13}_{-0.26}$
$0.00^{+0.11}_{-0.14}$ $c_{\text{H}\tilde{\text{W}}}$ $-0.26^{+0.67}_{-0.50}$
$0.00^{+0.48}_{-0.52}$ $c_{\text{H}\tilde{\text{W}}\text{B}}$
$-0.54^{+1.37}_{-1.03}$ $0.00^{+0.99}_{-1.07}$ $c_{\text{H}\tilde{\text{B}}}$
$-0.08^{+0.20}_{-0.15}$ $0.00^{+0.15}_{-0.16}$

Figure 12:  Summary of constraints on the SMEFT Higgs (left) and Warsaw
(right) basis coupling parameters with the best fit values and 68%
uncertainties. For the Warsaw basis, only one of $c_{\text{HW}}$ ,
$c_{\text{HWB}}$, and $c_{\text{HB}}$ is independent, the same is also true
for $c_{\text{H}\tilde{\text{W}}}$, $c_{\text{H}\tilde{\text{W}}\text{B}}$,
and $c_{\text{H}\tilde{\text{B}}}$.  Figure 13:  Expected (dashed) and
observed (solid) profiled likelihood on $f_{a3}^{\Pg\Pg\PH}$. The signal
strength modifiers and the $CP$-odd $\PH\PV\PV$ anomalous coupling cross
section fraction are treated as free parameters. The crossing of the observed
likelihood with the dashed horizontal line shows the observed 68% region.

##  0.11 Summary

This paper presents a study of the anomalous couplings of the Higgs boson ()
with vector bosons, including $CP$ violating effects, using its associated
production with hadronic jets in gluon fusion, electroweak vector boson
fusion, and associated production with a or boson, and its subsequent decay to
a pair of bosons. The results are based on the proton-proton collision data
set collected by the CMS detector at the LHC during 2016–2018, corresponding
to an integrated luminosity of 138at a center-of-mass energy of 13. The
analysis targets the different-flavor dilepton ($\Pe\PGm$) final state, with
kinematic information from associated jets combined using matrix element
techniques to increase sensitivity to anomalous effects at the production
vertex. Dedicated Monte Carlo simulation and matrix element reweighting
provide modeling of all kinematic features in the production and decay of the
Higgs boson with full simulation of detector effects. A simultaneous
measurement of four Higgs boson couplings to electroweak vector bosons has
been performed in the framework of a standard model effective field theory.
All measurements are consistent with the expectations for the standard model
Higgs boson and constraints are set on the fractional contribution of the
anomalous couplings to the Higgs boson cross section. The most stringent
constraints on the $\PH\PV\PV$ anomalous coupling cross section fractions are
at the per mille level. These results are in agreement with those obtained in
the $\PH\to\PZ\PZ$ and $\PH\to\PGt\PGt$ channels, and also significantly
surpass those of the previous $\PH\to\PW\PW$ anomalous coupling analysis from
the CMS experiment in both scope and precision.

###### 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 centers and personnel of the Worldwide LHC Computing Grid and other
centers 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, the CMS detector, and the supporting
computing infrastructure provided by the following funding agencies: SC
(Armenia), BMBWF and FWF (Austria); FNRS and FWO (Belgium); CNPq, CAPES,
FAPERJ, FAPERGS, and FAPESP (Brazil); MES and BNSF (Bulgaria); CERN; CAS,
MoST, and NSFC (China); MINCIENCIAS (Colombia); MSES and CSF (Croatia); RIF
(Cyprus); SENESCYT (Ecuador); ERC PRG, RVTT3 and MoER TK202 (Estonia); Academy
of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); SRNSF
(Georgia); BMBF, DFG, and HGF (Germany); GSRI (Greece); NKFIH (Hungary); DAE
and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); MSIP and NRF
(Republic of Korea); MES (Latvia); LMTLT (Lithuania); MOE and UM (Malaysia);
BUAP, CINVESTAV, CONACYT, LNS, SEP, and UASLP-FAI (Mexico); MOS (Montenegro);
MBIE (New Zealand); PAEC (Pakistan); MES and NSC (Poland); FCT (Portugal);
MESTD (Serbia); MCIN/AEI and PCTI (Spain); MOSTR (Sri Lanka); Swiss Funding
Agencies (Switzerland); MST (Taipei); MHESI and NSTDA (Thailand); TUBITAK and
TENMAK (Turkey); NASU (Ukraine); STFC (United Kingdom); DOE and NSF (USA).
Individuals have received support from the Marie-Curie program and the
European Research Council and Horizon 2020 Grant, contract Nos. 675440,
724704, 752730, 758316, 765710, 824093, 101115353, and COST Action CA16108
(European Union); the Leventis Foundation; the Alfred P. Sloan Foundation; the
Alexander von Humboldt Foundation; the Science Committee, project no. 22rl-037
(Armenia); 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 Beijing Municipal Science & Technology
Commission, No. Z191100007219010 and Fundamental Research Funds for the
Central Universities (China); the Ministry of Education, Youth and Sports
(MEYS) of the Czech Republic; the Shota Rustaveli National Science Foundation,
grant FR-22-985 (Georgia); the Deutsche Forschungsgemeinschaft (DFG), under
Germany’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306, and
under project number 400140256 - GRK2497; the Hellenic Foundation for Research
and Innovation (HFRI), Project Number 2288 (Greece); the Hungarian Academy of
Sciences, the New National Excellence Program - ÚNKP, the NKFIH research
grants K 124845, K 124850, K 128713, K 128786, K 129058, K 131991, K 133046, K
138136, K 143460, K 143477, 2020-2.2.1-ED-2021-00181, and TKP2021-NKTA-64
(Hungary); the Council of Science and Industrial Research, India; ICSC –
National Research Center for High Performance Computing, Big Data and Quantum
Computing, funded by the EU NexGeneration program (Italy); the Latvian Council
of Science; the Ministry of Education and Science, project no. 2022/WK/14, and
the National Science Center, contracts Opus 2021/41/B/ST2/01369 and
2021/43/B/ST2/01552 (Poland); the Fundação para a Ciência e a Tecnologia,
grant CEECIND/01334/2018 (Portugal); the National Priorities Research Program
by Qatar National Research Fund; MCIN/AEI/10.13039/501100011033, ERDF “a way
of making Europe”, and the Programa Estatal de Fomento de la Investigación
Científica y Técnica de Excelencia María de Maeztu, grant MDM-2017-0765 and
Programa Severo Ochoa del Principado de Asturias (Spain); the Chulalongkorn
Academic into Its 2nd Century Project Advancement Project, and the National
Science, Research and Innovation Fund via the Program Management Unit for
Human Resources & Institutional Development, Research and Innovation, grant
B37G660013 (Thailand); the Kavli Foundation; the Nvidia Corporation; the
SuperMicro Corporation; the Welch Foundation, contract C-1845; and the Weston
Havens Foundation (USA).

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##  .12 The CMS Collaboration

Yerevan Physics Institute, Yerevan, Armenia A. Hayrapetyan, A.
Tumasyan10009-0000-0684-6742

Institut für Hochenergiephysik, Vienna, Austria W. Adam0000-0001-9099-4341,
J.W. Andrejkovic, T. Bergauer0000-0002-5786-0293, S.
Chatterjee0000-0003-2660-0349, K. Damanakis0000-0001-5389-2872, M.
Dragicevic0000-0003-1967-6783, P.S. Hussain0000-0002-4825-5278, M.
Jeitler20000-0002-5141-9560, N. Krammer0000-0002-0548-0985, A.
Li0000-0002-4547-116X, D. Liko0000-0002-3380-473X, I.
Mikulec0000-0003-0385-2746, J. Schieck20000-0002-1058-8093, R.
Schöfbeck0000-0002-2332-8784, D. Schwarz0000-0002-3821-7331, M.
Sonawane0000-0003-0510-7010, S. Templ0000-0003-3137-5692, W.
Waltenberger0000-0002-6215-7228, C.-E. Wulz20000-0001-9226-5812

Universiteit Antwerpen, Antwerpen, Belgium M.R. Darwish30000-0003-2894-2377,
T. Janssen0000-0002-3998-4081, P. Van Mechelen0000-0002-8731-9051

Vrije Universiteit Brussel, Brussel, Belgium E.S. Bols0000-0002-8564-8732, J.
D’Hondt0000-0002-9598-6241, S. Dansana0000-0002-7752-7471, A. De
Moor0000-0001-5964-1935, M. Delcourt0000-0001-8206-1787, H. El
Faham0000-0001-8894-2390, S. Lowette0000-0003-3984-9987, I.
Makarenko0000-0002-8553-4508, D. Müller0000-0002-1752-4527, A.R.
Sahasransu0000-0003-1505-1743, S. Tavernier0000-0002-6792-9522, M.
Tytgat40000-0002-3990-2074, G.P. Van Onsem0000-0002-1664-2337, S. Van
Putte0000-0003-1559-3606, D. Vannerom0000-0002-2747-5095

Université Libre de Bruxelles, Bruxelles, Belgium B.
Clerbaux0000-0001-8547-8211, A.K. Das, G. De Lentdecker0000-0001-5124-7693, L.
Favart0000-0003-1645-7454, P. Gianneios0009-0003-7233-0738, D.
Hohov0000-0002-4760-1597, J. Jaramillo0000-0003-3885-6608, A. Khalilzadeh, K.
Lee0000-0003-0808-4184, M. Mahdavikhorrami0000-0002-8265-3595, A.
Malara0000-0001-8645-9282, S. Paredes0000-0001-8487-9603, L.
Pétré0009-0000-7979-5771, N. Postiau, L. Thomas0000-0002-2756-3853, M. Vanden
Bemden0009-0000-7725-7945, C. Vander Velde0000-0003-3392-7294, P.
Vanlaer0000-0002-7931-4496

Ghent University, Ghent, Belgium M. De Coen0000-0002-5854-7442, D.
Dobur0000-0003-0012-4866, Y. Hong0000-0003-4752-2458, J.
Knolle0000-0002-4781-5704, L. Lambrecht0000-0001-9108-1560, G. Mestdach, K.
Mota Amarilo0000-0003-1707-3348, C. Rendón, A. Samalan, K.
Skovpen0000-0002-1160-0621, N. Van Den Bossche0000-0003-2973-4991, J. van der
Linden0000-0002-7174-781X, L. Wezenbeek0000-0001-6952-891X

Université Catholique de Louvain, Louvain-la-Neuve, Belgium A.
Benecke0000-0003-0252-3609, A. Bethani0000-0002-8150-7043, G.
Bruno0000-0001-8857-8197, C. Caputo0000-0001-7522-4808, C.
Delaere0000-0001-8707-6021, I.S. Donertas0000-0001-7485-412X, A.
Giammanco0000-0001-9640-8294, K. Jaffel0000-0001-7419-4248, Sa.
Jain0000-0001-5078-3689, V. Lemaitre, J. Lidrych0000-0003-1439-0196, P.
Mastrapasqua0000-0002-2043-2367, K. Mondal0000-0001-5967-1245, T.T.
Tran0000-0003-3060-350X, S. Wertz0000-0002-8645-3670

Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, Brazil G.A.
Alves0000-0002-8369-1446, E. Coelho0000-0001-6114-9907, C.
Hensel0000-0001-8874-7624, T. Menezes De Oliveira0009-0009-4729-8354, A.
Moraes0000-0002-5157-5686, P. Rebello Teles0000-0001-9029-8506, M. Soeiro

Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil W.L. Aldá
Júnior0000-0001-5855-9817, M. Alves Gallo Pereira0000-0003-4296-7028, M.
Barroso Ferreira Filho0000-0003-3904-0571, H. Brandao
Malbouisson0000-0002-1326-318X, W. Carvalho0000-0003-0738-6615, J.
Chinellato5, E.M. Da Costa0000-0002-5016-6434, G.G. Da
Silveira60000-0003-3514-7056, D. De Jesus Damiao0000-0002-3769-1680, S.
Fonseca De Souza0000-0001-7830-0837, R. Gomes De Souza, J.
Martins70000-0002-2120-2782, C. Mora Herrera0000-0003-3915-3170, L.
Mundim0000-0001-9964-7805, H. Nogima0000-0001-7705-1066, J.P.
Pinheiro0000-0002-3233-8247, A. Santoro0000-0002-0568-665X, A.
Sznajder0000-0001-6998-1108, M. Thiel0000-0001-7139-7963, A. Vilela
Pereira0000-0003-3177-4626

Universidade Estadual Paulista, Universidade Federal do ABC, São Paulo, Brazil
C.A. Bernardes60000-0001-5790-9563, L. Calligaris0000-0002-9951-9448, T.R.
Fernandez Perez Tomei0000-0002-1809-5226, E.M. Gregores0000-0003-0205-1672,
P.G. Mercadante0000-0001-8333-4302, S.F. Novaes0000-0003-0471-8549, B.
Orzari0000-0003-4232-4743, Sandra S. Padula0000-0003-3071-0559

Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of
Sciences, Sofia, Bulgaria A. Aleksandrov0000-0001-6934-2541, G.
Antchev0000-0003-3210-5037, R. Hadjiiska0000-0003-1824-1737, P.
Iaydjiev0000-0001-6330-0607, M. Misheva0000-0003-4854-5301, M.
Shopova0000-0001-6664-2493, G. Sultanov0000-0002-8030-3866

University of Sofia, Sofia, Bulgaria A. Dimitrov0000-0003-2899-701X, L.
Litov0000-0002-8511-6883, B. Pavlov0000-0003-3635-0646, P.
Petkov0000-0002-0420-9480, A. Petrov0009-0003-8899-1514, E.
Shumka0000-0002-0104-2574

Instituto De Alta Investigación, Universidad de Tarapacá, Casilla 7 D, Arica,
Chile S. Keshri0000-0003-3280-2350, S. Thakur0000-0002-1647-0360

Beihang University, Beijing, China T. Cheng0000-0003-2954-9315, T.
Javaid0009-0007-2757-4054, L. Yuan0000-0002-6719-5397

Department of Physics, Tsinghua University, Beijing, China Z.
Hu0000-0001-8209-4343, J. Liu, K. Yi8,90000-0002-2459-1824

Institute of High Energy Physics, Beijing, China G.M.
Chen100000-0002-2629-5420, H.S. Chen100000-0001-8672-8227, M.
Chen100000-0003-0489-9669, F. Iemmi0000-0001-5911-4051, C.H. Jiang, A.
Kapoor110000-0002-1844-1504, H. Liao0000-0002-0124-6999, Z.-A.
Liu120000-0002-2896-1386, R. Sharma130000-0003-1181-1426, J.N. Song12, J.
Tao0000-0003-2006-3490, C. Wang10, J. Wang0000-0002-3103-1083, Z. Wang10, H.
Zhang0000-0001-8843-5209

State Key Laboratory of Nuclear Physics and Technology, Peking University,
Beijing, China A. Agapitos0000-0002-8953-1232, Y. Ban0000-0002-1912-0374, A.
Levin0000-0001-9565-4186, C. Li0000-0002-6339-8154, Q. Li0000-0002-8290-0517,
Y. Mao, S.J. Qian0000-0002-0630-481X, X. Sun0000-0003-4409-4574, D.
Wang0000-0002-9013-1199, H. Yang, L. Zhang0000-0001-7947-9007, C.
Zhou0000-0001-5904-7258

Sun Yat-Sen University, Guangzhou, China Z. You0000-0001-8324-3291

University of Science and Technology of China, Hefei, China N.
Lu0000-0002-2631-6770

Nanjing Normal University, Nanjing, China G. Bauer14

Institute of Modern Physics and Key Laboratory of Nuclear Physics and Ion-beam
Application (MOE) - Fudan University, Shanghai, China X.
Gao150000-0001-7205-2318, D. Leggat, H. Okawa0000-0002-2548-6567

Zhejiang University, Hangzhou, Zhejiang, China Z. Lin0000-0003-1812-3474, C.
Lu0000-0002-7421-0313, M. Xiao0000-0001-9628-9336

Universidad de Los Andes, Bogota, Colombia C. Avila0000-0002-5610-2693, D.A.
Barbosa Trujillo, A. Cabrera0000-0002-0486-6296, C. Florez0000-0002-3222-0249,
J. Fraga0000-0002-5137-8543, J.A. Reyes Vega

Universidad de Antioquia, Medellin, Colombia J. Mejia
Guisao0000-0002-1153-816X, F. Ramirez0000-0002-7178-0484, M.
Rodriguez0000-0002-9480-213X, J.D. Ruiz Alvarez0000-0002-3306-0363

University of Split, Faculty of Electrical Engineering, Mechanical Engineering
and Naval Architecture, Split, Croatia D. Giljanovic0009-0005-6792-6881, N.
Godinovic0000-0002-4674-9450, D. Lelas0000-0002-8269-5760, A.
Sculac0000-0001-7938-7559

University of Split, Faculty of Science, Split, Croatia M.
Kovac0000-0002-2391-4599, T. Sculac0000-0002-9578-4105

Institute Rudjer Boskovic, Zagreb, Croatia P. Bargassa0000-0001-8612-3332, V.
Brigljevic0000-0001-5847-0062, B.K. Chitroda0000-0002-0220-8441, D.
Ferencek0000-0001-9116-1202, S. Mishra0000-0002-3510-4833, A.
Starodumov160000-0001-9570-9255, T. Susa0000-0001-7430-2552

University of Cyprus, Nicosia, Cyprus A. Attikis0000-0002-4443-3794, K.
Christoforou0000-0003-2205-1100, S. Konstantinou0000-0003-0408-7636, J.
Mousa0000-0002-2978-2718, C. Nicolaou, F. Ptochos0000-0002-3432-3452, P.A.
Razis0000-0002-4855-0162, H. Rykaczewski, H. Saka0000-0001-7616-2573, A.
Stepennov0000-0001-7747-6582

Charles University, Prague, Czech Republic M. Finger0000-0002-7828-9970, M.
Finger Jr.0000-0003-3155-2484, A. Kveton0000-0001-8197-1914

Escuela Politecnica Nacional, Quito, Ecuador E. Ayala0000-0002-0363-9198

Universidad San Francisco de Quito, Quito, Ecuador E. Carrera
Jarrin0000-0002-0857-8507

Academy of Scientific Research and Technology of the Arab Republic of Egypt,
Egyptian Network of High Energy Physics, Cairo, Egypt S. Elgammal17, A.
Ellithi Kamel18

Center for High Energy Physics (CHEP-FU), Fayoum University, El-Fayoum, Egypt
M.A. Mahmoud0000-0001-8692-5458, Y. Mohammed0000-0001-8399-3017

National Institute of Chemical Physics and Biophysics, Tallinn, Estonia K.
Ehataht0000-0002-2387-4777, M. Kadastik, T. Lange0000-0001-6242-7331, S.
Nandan0000-0002-9380-8919, C. Nielsen0000-0002-3532-8132, J.
Pata0000-0002-5191-5759, M. Raidal0000-0001-7040-9491, L.
Tani0000-0002-6552-7255, C. Veelken0000-0002-3364-916X

Department of Physics, University of Helsinki, Helsinki, Finland H.
Kirschenmann0000-0001-7369-2536, K. Osterberg0000-0003-4807-0414, M.
Voutilainen0000-0002-5200-6477

Helsinki Institute of Physics, Helsinki, Finland S.
Bharthuar0000-0001-5871-9622, E. Brücken0000-0001-6066-8756, F.
Garcia0000-0002-4023-7964, K.T.S. Kallonen0000-0001-9769-7163, R. Kinnunen, T.
Lampén0000-0002-8398-4249, K. Lassila-Perini0000-0002-5502-1795, S.
Lehti0000-0003-1370-5598, T. Lindén0009-0002-4847-8882, L.
Martikainen0000-0003-1609-3515, M. Myllymäki0000-0003-0510-3810, M.m.
Rantanen0000-0002-6764-0016, H. Siikonen0000-0003-2039-5874, E.
Tuominen0000-0002-7073-7767, J. Tuominiemi0000-0003-0386-8633

Lappeenranta-Lahti University of Technology, Lappeenranta, Finland P.
Luukka0000-0003-2340-4641, H. Petrow0000-0002-1133-5485

IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France M.
Besancon0000-0003-3278-3671, F. Couderc0000-0003-2040-4099, M.
Dejardin0009-0008-2784-615X, D. Denegri, J.L. Faure, F.
Ferri0000-0002-9860-101X, S. Ganjour0000-0003-3090-9744, P.
Gras0000-0002-3932-5967, G. Hamel de Monchenault0000-0002-3872-3592, V.
Lohezic0009-0008-7976-851X, J. Malcles0000-0002-5388-5565, J. Rander, A.
Rosowsky0000-0001-7803-6650, M.Ö. Sahin0000-0001-6402-4050, A. Savoy-
Navarro190000-0002-9481-5168, P. Simkina0000-0002-9813-372X, M.
Titov0000-0002-1119-6614, M. Tornago0000-0001-6768-1056

Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, Institut
Polytechnique de Paris, Palaiseau, France C. Baldenegro
Barrera0000-0002-6033-8885, F. Beaudette0000-0002-1194-8556, A. Buchot
Perraguin0000-0002-8597-647X, P. Busson0000-0001-6027-4511, A.
Cappati0000-0003-4386-0564, C. Charlot0000-0002-4087-8155, M.
Chiusi0000-0002-1097-7304, F. Damas0000-0001-6793-4359, O.
Davignon0000-0001-8710-992X, A. De Wit0000-0002-5291-1661, B.A. Fontana Santos
Alves0000-0001-9752-0624, S. Ghosh0009-0006-5692-5688, A.
Gilbert0000-0001-7560-5790, R. Granier de Cassagnac0000-0002-1275-7292, A.
Hakimi0009-0008-2093-8131, B. Harikrishnan0000-0003-0174-4020, L.
Kalipoliti0000-0002-5705-5059, G. Liu0000-0001-7002-0937, J.
Motta0000-0003-0985-913X, M. Nguyen0000-0001-7305-7102, C.
Ochando0000-0002-3836-1173, L. Portales0000-0002-9860-9185, R.
Salerno0000-0003-3735-2707, J.B. Sauvan0000-0001-5187-3571, Y.
Sirois0000-0001-5381-4807, A. Tarabini0000-0001-7098-5317, E.
Vernazza0000-0003-4957-2782, A. Zabi0000-0002-7214-0673, A.
Zghiche0000-0002-1178-1450

Université de Strasbourg, CNRS, IPHC UMR 7178, Strasbourg, France J.-L.
Agram200000-0001-7476-0158, J. Andrea0000-0002-8298-7560, D.
Apparu0009-0004-1837-0496, D. Bloch0000-0002-4535-5273, J.-M.
Brom0000-0003-0249-3622, E.C. Chabert0000-0003-2797-7690, C.
Collard0000-0002-5230-8387, S. Falke0000-0002-0264-1632, U.
Goerlach0000-0001-8955-1666, C. Grimault, R. Haeberle0009-0007-5007-6723,
A.-C. Le Bihan0000-0002-8545-0187, M. Meena0000-0003-4536-3967, G.
Saha0000-0002-6125-1941, M.A. Sessini0000-0003-2097-7065, P. Van
Hove0000-0002-2431-3381

Institut de Physique des 2 Infinis de Lyon (IP2I ), Villeurbanne, France S.
Beauceron0000-0002-8036-9267, B. Blancon0000-0001-9022-1509, G.
Boudoul0009-0002-9897-8439, N. Chanon0000-0002-2939-5646, J.
Choi0000-0002-6024-0992, D. Contardo0000-0001-6768-7466, P.
Depasse0000-0001-7556-2743, C. Dozen210000-0002-4301-634X, H. El Mamouni, J.
Fay0000-0001-5790-1780, S. Gascon0000-0002-7204-1624, M.
Gouzevitch0000-0002-5524-880X, C. Greenberg, G. Grenier0000-0002-1976-5877, B.
Ille0000-0002-8679-3878, I.B. Laktineh, M. Lethuillier0000-0001-6185-2045, L.
Mirabito, S. Perries, A. Purohit0000-0003-0881-612X, M. Vander
Donckt0000-0002-9253-8611, P. Verdier0000-0003-3090-2948, J.
Xiao0000-0002-7860-3958

Georgian Technical University, Tbilisi, Georgia G. Adamov, I.
Lomidze0009-0002-3901-2765, Z. Tsamalaidze160000-0001-5377-3558

RWTH Aachen University, I. Physikalisches Institut, Aachen, Germany V.
Botta0000-0003-1661-9513, L. Feld0000-0001-9813-8646, K.
Klein0000-0002-1546-7880, M. Lipinski0000-0002-6839-0063, D.
Meuser0000-0002-2722-7526, A. Pauls0000-0002-8117-5376, N.
Röwert0000-0002-4745-5470, M. Teroerde0000-0002-5892-1377

RWTH Aachen University, III. Physikalisches Institut A, Aachen, Germany S.
Diekmann0009-0004-8867-0881, A. Dodonova0000-0002-5115-8487, N.
Eich0000-0001-9494-4317, D. Eliseev0000-0001-5844-8156, F.
Engelke0000-0002-9288-8144, J. Erdmann0000-0002-8073-2740, M.
Erdmann0000-0002-1653-1303, P. Fackeldey0000-0003-4932-7162, B.
Fischer0000-0002-3900-3482, T. Hebbeker0000-0002-9736-266X, K.
Hoepfner0000-0002-2008-8148, F. Ivone0000-0002-2388-5548, A.
Jung0000-0002-2511-1490, M.y. Lee0000-0002-4430-1695, L. Mastrolorenzo, F.
Mausolf0000-0003-2479-8419, M. Merschmeyer0000-0003-2081-7141, A.
Meyer0000-0001-9598-6623, S. Mukherjee0000-0001-6341-9982, D.
Noll0000-0002-0176-2360, F. Nowotny, A. Pozdnyakov0000-0003-3478-9081, Y.
Rath, W. Redjeb0000-0001-9794-8292, F. Rehm, H. Reithler0000-0003-4409-702X,
U. Sarkar0000-0002-9892-4601, V. Sarkisovi0000-0001-9430-5419, A.
Schmidt0000-0003-2711-8984, A. Sharma0000-0002-5295-1460, J.L.
Spah0000-0002-5215-3258, A. Stein0000-0003-0713-811X, F. Torres Da Silva De
Araujo220000-0002-4785-3057, L. Vigilante, S. Wiedenbeck0000-0002-4692-9304,
S. Zaleski

RWTH Aachen University, III. Physikalisches Institut B, Aachen, Germany C.
Dziwok0000-0001-9806-0244, G. Flügge0000-0003-3681-9272, W. Haj
Ahmad230000-0003-1491-0446, T. Kress0000-0002-2702-8201, A.
Nowack0000-0002-3522-5926, O. Pooth0000-0001-6445-6160, A.
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Deutsches Elektronen-Synchrotron, Hamburg, Germany H. Aarup
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Yang0009-0009-3430-0558, A. Zimermmane Castro Santos0000-0001-9302-3102

University of Hamburg, Hamburg, Germany A. Albrecht0000-0001-6004-6180, S.
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