Search for heavy resonances decaying into a $Z$ or $W$ boson and a Higgs boson
in final states with leptons and $b$-jets in $139\text{\,}{\mathrm{fb}}^{-1}$
of $pp$ collisions at $\sqrt{s}=$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$
with the ATLAS detector This article presents a search for new resonances
decaying into a $Z$ or $W$ boson and a $125$ GeV Higgs boson $h$, and it
targets the $\nu\bar{\nu}b\bar{b}$, $\ell^{+}\ell^{-}b\bar{b}$, or
$\ell^{\pm}{\nu}b\bar{b}$ final states, where $\ell=e$ or $\mu$, in
proton–proton collisions at $\sqrt{s}=$13 TeV. The data used correspond to a
total integrated luminosity of $139\text{\,}{\mathrm{fb}}^{-1}$ collected by
the ATLAS detector during Run 2 of the LHC at CERN. The search is conducted by
examining the reconstructed invariant or transverse mass distributions of $Z$h
or $Wh$ candidates for evidence of a localised excess in the mass range from
$220$ GeV to 5 TeV. No significant excess is observed and $95\%$ confidence-
level upper limits between 1.3 pb, and 0.3 fb are placed on the production
cross section times branching fraction of neutral and charged spin-1
resonances and CP-odd scalar bosons. These limits are converted into
constraints on the parameter space of the Heavy Vector Triplet model and the
two-Higgs-doublet model. HDBS-2020-19 CERN-EP-2022-115 JHEP

##  1 Introduction

Following the discovery of the Higgs boson [1, 2] by the ATLAS and CMS
collaborations at the Large Hadron Collider (LHC), measurements of its
properties so far indicate consistency with the Standard Model (SM)
predictions. Nevertheless, several questions related to electroweak symmetry
breaking remain open, in particular how the Higgs boson mass is protected
against large radiative corrections (the naturalness problem [3, 4, 5]) and
whether the Higgs boson is part of an extended scalar sector.

Various models with dynamical electroweak symmetry breaking scenarios attempt
to solve the naturalness problem by assuming new strong interactions at a
higher energy scale. These models generally predict the existence of new
vector resonances that have significant branching fractions for decays into a
vector boson and a Higgs boson. Examples are Minimal Walking Technicolour [6,
7, 8], Little Higgs [9], or composite Higgs models [10, 11]. Resonance
searches are typically not sensitive to all free parameters of the underlying
theory, so simplified models are generally used, such as the Heavy Vector
Triplet (HVT) parameterised Lagrangian [12, 13], which adds an additional
SU(2) field to the SM and provides a restricted number of new couplings.

A second class of models extend the scalar sector by including additional
Higgs singlets or doublets [14]. Examples of models with an extended scalar
sector are the minimal supersymmetric SM [15, 16, 17, 18, 19], axion models
[20], or baryogenesis models [21]. Again, instead of targeting any of these
specific theories, a generic two-Higgs-doublet model (2HDM) with a CP-
conserving Higgs potential is probed. The scalar sector of the theory consists
of five Higgs bosons: two charged ($H^{\pm}$), two neutral CP-even ($h$, $H$),
and one neutral CP-odd ($A$).

In this article, ATLAS presents a search for new heavy resonances decaying
into a SM Higgs boson $h$ and a $Z$ or $W$ boson in 139 $\mathrm{fb}^{-1}$ of
proton–proton ($pp$) collision data at $\sqrt{s}=$13 TeV. These resonances are
assumed to be either a new heavy vector boson, denoted hereafter by
$Z^{\prime}$ and $W^{\prime}$, or a heavy CP-odd scalar boson $A$. The article
targets leptonic decays of the $Z$ and $W$ bosons, and requires $b$-quark pair
decays for the Higgs boson. Therefore, the search is performed in the
following three final states: $\nu\bar{\nu}b\bar{b}$,
$\ell^{\pm}{\nu}b\bar{b}$ and $\ell^{+}\ell^{-}b\bar{b}$. In this article,
$\ell$ represents $e$ or $\mu$ unless otherwise stated.

Searches in the same final states have been performed at $\sqrt{s}=$13 TeV
with an integrated luminosity of 36.1 $\mathrm{fb}^{-1}$ by ATLAS [22] and of
137 $\mathrm{fb}^{-1}$ by CMS [23, 24]. Results in other final states have
also been published: ATLAS has performed a search in the fully hadronic final
state [25], based on an integrated luminosity of 139 $\mathrm{fb}^{-1}$, and
CMS has published results in the fully hadronic final state [26] and the
$\tau^{+}\tau^{-}\ell^{+}\ell^{-}$ final state [27], based on a 35.9
$\mathrm{fb}^{-1}$ dataset.

Apart from the significantly larger dataset used here, several improvements
have been implemented since the previous ATLAS publication [22], including
improved $b$-tagging, lepton isolation and jet reconstruction, a
reoptimisation of the lepton selection in the $\ell^{+}\ell^{-}b\bar{b}$
channel and a new selection based on the missing transverse momentum
($E_{\text{T}}^{\text{miss}}$) significance in the $\nu\bar{\nu}b\bar{b}$
channel (see Sections 4 and 5).

The search presented in this article is performed by seeking a localised
excess in the distribution of a channel-dependent proxy for the resonance mass
of the $\nu\bar{\nu}b\bar{b}$, $\ell^{\pm}{\nu}b\bar{b}$ and
$\ell^{+}\ell^{-}b\bar{b}$ systems. Signal mass hypotheses in a range from
$220\leavevmode\nobreak\ \text{GeV}$ to $5\leavevmode\nobreak\ \text{TeV}$ are
tested. Due to the wide range of probed Higgs boson transverse momenta, two
methods are used to reconstruct Higgs boson candidates. At low transverse
momenta, the decay products of the Higgs boson are reconstructed as individual
jets, while at high transverse momenta they are reconstructed as a single
large-radius jet.

The search selects events in signal regions (SR) and background-dominated
control regions (CR) based on requirements placed on kinematic properties of
final-state particles and on event-level quantities. The statistical
interpretation is performed with a binned maximum-likelihood fit to data based
on the reconstructed resonance mass, using the signal and control regions. The
major backgrounds are modelled using simulation and their normalisations are
determined by the fit. Different fit models are used depending on the targeted
signal hypothesis, production mode, or final state.

The results of the HVT $Z^{\prime}$ and $W^{\prime}$ searches are interpreted
in two benchmark models. In the first model, referred to as Model A, the
branching fractions to fermions and gauge bosons are comparable, as in some
models with an extended gauge symmetry [28]. For Model B, fermionic couplings
are suppressed, as in strong dynamical models such as the minimal composite
Higgs model [29]. Constraints are also set on the coupling-strength scale
factors $g_{\text{F}}$ and $g_{\text{H}}$, which modulate the coupling of the
$Z^{\prime}$ and $W^{\prime}$ bosons to fermions and the Higgs and gauge
bosons, respectively. The search focuses on high resonance masses, ranging
from 300 GeV to 5 TeV.

The search for $A\to Zh$ focuses on smaller masses, ranging from 220 GeV to 2
TeV, because the Higgs potential becomes unstable for sufficiently large $A$
boson masses in the class of models targeted by this article. The results of
the $A\to Zh$ search are interpreted as constraints on the ratio of the vacuum
expectation values of the two Higgs doublets, $\tan\beta$, and on
$\cos(\beta-\alpha)$, where $\alpha$ is the mixing angle between the two CP-
even Higgs bosons, for Type I, Type II, Lepton-specific, and Flipped 2HDMs.
These models differ with respect to which doublets couple to the up-type and
down-type quarks and to the charged leptons [14]. Production via gluon–gluon
fusion ($ggA$) and production with associated $b$-quarks ($b\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$)
are both considered in this search.

Representative lowest-order Feynman diagrams of the relevant production
modes111In the HVT benchmark Models A and B, the $Z^{\prime}$ and $W^{\prime}$
bosons are produced mainly via quark–antiquark annihilation, while production
via vector-boson fusion is relevant only for the fermiophobic scenario (i.e.
Model C). for $Z^{\prime}$, $W^{\prime}$, and $A$ bosons in $pp$ collisions
and the subsequent decays into the $\nu\bar{\nu}b\bar{b}$,
$\ell^{+}\ell^{-}b\bar{b}$, and $\ell^{\pm}{\nu}b\bar{b}$ final states are
depicted in Figure 1.

This article is structured as follows. Sections 2 and 3 provide a brief
description of the ATLAS experiment and of the data and simulated event
samples, respectively. The event reconstruction and selections are discussed
in Sections 4 and 5. The background estimation and systematic uncertainties
are described in Sections 6 and 7, respectively. Finally, Sections 8 and 9
detail the statistical analysis and then provide a discussion of the results
and concluding remarks.

((a))

((b))  ((c))

Figure 1: Representative lowest-order Feynman diagrams of the resonant
production of a $W$ or $Z$ boson and a Higgs boson via (a) quark–antiquark
annihilation, (b) gluon–gluon fusion and (c) $b$-associated production. The
subsequent decays into the $\nu\bar{\nu}b\bar{b}$, $\ell^{\pm}{\nu}b\bar{b}$
and $\ell^{+}\ell^{-}b\bar{b}$ final states are also depicted, where
$\ell=e,\mu,\mathrm{and}\,\tau$.

##  2 ATLAS detector

The ATLAS experiment [30] at the LHC is a multipurpose particle detector with
a forward–backward symmetric cylindrical geometry and a near $4\pi$ coverage
in solid angle.222ATLAS uses a right-handed coordinate system with its origin
at the nominal interaction point (IP) in the centre of the detector and the
$z$-axis along the beam pipe. The $x$-axis points from the IP to the centre of
the LHC ring, and the $y$-axis points upwards. Cylindrical coordinates
$(r,\phi)$ are used in the transverse plane, $\phi$ being the azimuthal angle
around the $z$-axis. The pseudorapidity is defined in terms of the polar angle
$\theta$ as $\eta=-\ln\tan(\theta/2)$. Angular distance is measured in units
of $\Delta R\equiv\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$.  It consists of
an inner tracking detector (ID) surrounded by a thin superconducting solenoid
providing a $2\text{\,}\mathrm{T}$ axial magnetic field, electromagnetic and
hadron calorimeters, and a muon spectrometer. The inner tracking detector
covers the pseudorapidity range $|\eta|<2.5$. It consists of silicon pixel,
silicon microstrip, and transition radiation tracking detectors. Lead/liquid-
argon (LAr) sampling calorimeters provide electromagnetic energy measurements
with high granularity. A steel/scintillator-tile hadron calorimeter covers the
central pseudorapidity range ($|\eta|<1.7$). The endcap and forward regions
are instrumented with LAr calorimeters for both the electromagnetic and
hadronic energy measurements up to $|\eta|=4.9$. The muon spectrometer
surrounds the calorimeters and is based on three large superconducting air-
core toroidal magnets with eight coils each. The field integral of the toroids
ranges between $2.0$ and $6.0\text{\,}\mathrm{T}\text{\,}\mathrm{m}$ across
most of the detector. The muon spectrometer includes a system of precision
tracking chambers and fast detectors for triggering. A two-level trigger
system is used to select events. The first-level trigger is implemented in
hardware and uses a subset of the detector information to accept events at a
rate below $100\text{\,}\mathrm{kHz}$. This is followed by a software-based
trigger that reduces the accepted event rate to $1\text{\,}\mathrm{kHz}$ on
average depending on the data-taking conditions. An extensive software suite
[31] is used in the reconstruction and analysis of real and simulated data, in
detector operations, and in the trigger and data acquisition systems of the
experiment.

##  3 Data and Monte Carlo samples

The data used in this analysis were recorded with the ATLAS detector between
2015 and 2018 in $pp$ collisions at $\sqrt{s}=13$ TeV and correspond to a
total integrated luminosity of 139 $\mathrm{fb}^{-1}$ [32]. The data are
required to satisfy criteria that ensure that the detector was in good
operating condition [33]. Monte Carlo (MC) simulation samples are used to
model all the major backgrounds and the signal processes.

For the HVT interpretations of the ATLAS data, the production of $Z^{\prime}$
and $W^{\prime}$ bosons via quark–antiquark annihilation was modelled at
leading-order (LO) accuracy by $\textsc{MadGraph}5$ 2.3.3 [34] interfaced with
Pythia 8.186 [35], which used the A14 set of tuned parameters [36] and the
NNPDF2.3lo parton distribution function (PDF) set [37]. Events were generated
for a range of resonance masses from 300 GeV to 5 TeV using the benchmark HVT
Model A. Separate generation of signal events for HVT Model A and Model B is
not necessary as both give rise to mass peaks in the $Zh$ and $Wh$ system with
a width that is dominated by the experimental resolution. Higgs boson decays
into $b\bar{b}$ or $c\bar{c}$ are considered, assuming the branching fractions
$B\left(h\to b\bar{b}/c\bar{c}\right)=0.569/0.0287$ and a Higgs boson mass of
$m_{h}=125\,\text{GeV}$. Even though this search does not target the decay
$h\rightarrow c\bar{c}$ directly, it is sensitive to such events. The $Z$ and
$W$ bosons were required to decay via $Z\rightarrow\nu\bar{\nu}$,
$Z\rightarrow\ell^{+}\ell^{-}$, or $W\rightarrow\ell^{\pm}\nu$, with
$\ell=e,\mu,\mathrm{and}\,\tau$. Contributions from $Z^{\prime}\rightarrow
WW\rightarrow\ell^{\pm}\nu qq$ or $W\textquoteright\rightarrow
WZ\rightarrow\ell^{\pm}\nu qq,\ell^{+}\ell^{-}qq$ decays are negligible to
these searches.

For the 2HDM interpretations of the ATLAS data, events for the gluon–gluon
fusion production of $A$ bosons ($ggA$) were generated in the narrow-width
approximation with $\textsc{MadGraph}5$ 2.3.3 at LO accuracy and interfaced to
Pythia 8.186, which used the A14 tune and the NNPDF2.3lo PDF set. Events for
$b$-quark associated production of $A$ bosons ($b\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$)
were simulated with MadGraph5_aMC@NLO 2.2.3 [34] using the four-flavour scheme
at next-to-leading order (NLO) with massive $b$-quarks and the CT10nlo nf4 PDF
set, and interfaced with Pythia 8.186, which used the A14 set of tuned
parameters. Events were generated for a range of $A$ boson masses from 220 GeV
to 2 TeV, assuming a narrow width. In the 2HDM, the Higgs boson branching
fractions vary strongly as a function of the relevant model parameters, and
therefore the $A$ boson signals were generated only for decays of the Higgs
boson $h$ into a $b\bar{b}$ pair. The $Z$ boson was required to decay either
via $Z\rightarrow\nu\bar{\nu}$ or via $Z\rightarrow\ell^{+}\ell^{-}$.

The production of $W$ and $Z$ bosons in association with jets was modelled by
Sherpa 2.2.1 [38] with the NNPDF3.0nlo set of PDFs [39]. Diagrams with up to
two additional parton emissions were simulated at NLO precision, and those
with three or four additional parton emissions at LO accuracy, using the Comix
[40] and OpenLoops [41, 42, 43] libraries. Matrix elements (ME) were merged
with the Sherpa parton shower [44] (PS) using the MEPS@NLO prescription [45,
46, 47, 48] and the set of tuned parameters developed by the Sherpa authors.
The inclusive production cross section for these samples was normalised to
match a next-to-next-to-leading-order (NNLO) prediction [49].

The production of top-quark pair
($\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$)
events was modelled using the Powheg Box v2 [50, 51, 52, 53] generator. The
matrix elements were calculated at NLO precision in QCD using the five-flavour
scheme and the NNPDF3.0nlo PDF set, assuming a top-quark mass of
$172.5\,\text{GeV}$. The PS, hadronisation, and underlying event were modelled
with the Pythia 8.230 [54] generator using the A14 set of tuned parameters and
the NNPDF2.3lo set of PDFs. The $h_{\mathrm{damp}}$ parameter333The
$h_{\mathrm{damp}}$ parameter is a resummation damping factor and one of the
parameters that control the matching of Powheg matrix elements to the PS,
effectively regulating the high-$p_{\text{T}}$ radiation against which the
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$
system recoils. was set to 1.5 $m_{\mathrm{top}}$ [55]. The cross section was
calculated at NNLO precision, including the resummation of next-to-next-to-
leading logarithmic (NNLL) soft gluon terms with Top++ 2.0 [56, 57, 58, 59,
60, 61, 62].

The
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+h$
samples were generated at NLO accuracy using the MadGraph5_aMC@NLO 2.3.2
generator with the NNPDF3.0nlo PDF set and interfaced to Pythia 8.186, which
used the A14 set of tuned parameters and the NNPDF2.3lo PDF set.

The production of
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+V$
events (where $V=W,Z$) was modelled using the MadGraph5_aMC@NLO 2.3.3
generator at NLO with the NNPDF3.0nlo PDF set. The events were interfaced to
Pythia 8.230, which used the A14 set of tuned parameters and the NNPDF2.3lo
PDF set.

The associated production of top quarks with $W$ bosons ($Wt$) and the $s$\-
and $t$-channel production of single top quarks were modelled by the Powheg
Box v2 generator at NLO in QCD using the five-flavour scheme and the
NNPDF3.0nlo set of PDFs. The diagram removal (DR) scheme [63] was used to
remove interference and overlap with
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$
production. The events were interfaced to Pythia 8.230, which used the A14 set
of tuned parameters and the NNPDF2.3lo set of PDFs.

Diboson events ($WW$, $WZ$, $ZZ$) with semileptonic decays were simulated
using Sherpa 2.2.1 with the NNPDF3.0nnlo PDF set, including off-shell effects
and Higgs boson contributions where appropriate. Diagrams with up to one
additional emission were calculated at NLO accuracy in QCD, while diagrams
with two or three parton emissions were described at LO accuracy [64]. They
were merged and matched using the MEPS@NLO prescription. Loop-induced diboson
processes that are initiated via the $gg$ production mode were simulated at LO
in QCD for diagrams with up to one additional parton emission in the matrix
element using OpenLoops in Sherpa 2.2.2, which used the NNPDF3.0nnlo PDF set.
Diboson events with fully leptonic decays were modelled using Sherpa 2.2.2 and
the NNPDF3.0nnlo PDF set.

Finally, the production of a SM Higgs boson in association with a vector boson
was simulated using Powheg Box v2, interfaced with Pythia 8.212 for PS and
non-perturbative effects. The Powheg prediction is accurate to next-to-leading
order for production of $Vh$ plus one jet. The loop-induced $gg\to Zh$ process
was generated separately at LO. The PDF4LHC15 PDF set [65] and the AZNLO set
of tuned parameters [66] of Pythia 8.212 were used. The $gg\to Zh$ production
cross section was calculated at NLO precision, including the resummation of
next-to-leading logarithmic (NLL) soft gluon terms [67]. For the generation of
$Vh$ events, the Higgs boson mass was set to $125\,\text{GeV}$.

A summary of MC generators and programs used to model the underlying event and
parton shower (UEPS) for the simulation of background processes is provided in
Table 1.

Table 1: Summary of the Monte Carlo generators used to produce the various
background processes. The perturbative accuracy (in QCD and, if relevant, in
EW corrections) of the total production cross section $\sigma_{\mathrm{prod}}$
is stated for each process. The order at which the corresponding matrix
elements are calculated in the Monte Carlo simulation is not necessarily the
same as for the cross section. Details on the final state are also provided,
where $\ell=e,\mu,\mathrm{and}\,\tau$.

Process | Generator | Perturbative accuracy of $\sigma_{\mathrm{prod}}$
---|---|---
$q\bar{q}\to Z^{\prime}\to Zh\to\ell\ell/\nu\nu+bb/cc$ |  $\textsc{MadGraph}5$ 2.3.3 + Pythia 8.186 | LO
$q\bar{q}\to W^{\prime}\to Wh\to\ell\nu+bb/cc$ |  $\textsc{MadGraph}5$ 2.3.3 + Pythia 8.186 | LO
$gg\to A\to Zh\to\ell\ell/\nu\nu+bb$ |  $\textsc{MadGraph}5$ 2.3.3 + Pythia 8.186 | LO
$gg\to b\bar{b}A\to Zh\to\ell\ell/\nu\nu+bb$ |  MadGraph5_aMC@NLO 2.2.3 + Pythia 8.186 | NLO
$W\to\ell\nu$, $Z\to\ell\ell$, $\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\to\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\mathit{{\nu}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$ |  Sherpa 2.2.1 | NNLO
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern 0.0mu\overline{\mkern 0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$ |  Powheg Box v2 + Pythia 8.230 | NNLO+NNLL
single top quarks ($s$\- and $t$-channels) |  Powheg Box v2 + Pythia 8.230 | NLO
single top quarks ($Wt$-channel) |  Powheg Box v2 + Pythia 8.230 | approx. NNLO
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern 0.0mu\overline{\mkern 0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+h$ |  MadGraph5_aMC@NLO 2.3.2 + Pythia 8.186 | NLO (QCD) and NLO (EW)
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern 0.0mu\overline{\mkern 0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+V$ |  MadGraph5_aMC@NLO 2.3.3 + Pythia 8.230 | NLO
$qg/q\bar{q}\to VV\to\ell\ell/\ell\nu/\nu\nu+q\bar{q}$ |  Sherpa 2.2.1 | NLO
$gg\to VV\to\ell\ell/\ell\nu/\nu\nu+q\bar{q}$ |  Sherpa 2.2.2 | NLO
$qg/q\bar{q}\to\ell\ell\nu\nu$ |  Sherpa 2.2.2 | NLO
$qq\to Wh\to\ell\nu+b\bar{b}$ |  Powheg Box v2 + Pythia 8.212 | NNLO (QCD) and NLO (EW)
$qq\to Zh\to\ell\ell/\nu\nu+b\bar{b}$ |  Powheg Box v2 + Pythia 8.212 | NNLO (QCD) and NLO (EW)
$gg\to Zh\to\ell\ell/\nu\nu+b\bar{b}$ |  Powheg Box v2 + Pythia 8.212 | NLO+NLL

All simulated event samples include the effect of multiple
$\mathit{{p}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{p}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
interactions in the same and neighbouring bunch crossings (pile-up) by
overlaying simulated minimum-bias events on each generated signal or
background event. The minimum-bias events were simulated with the single-,
double- and non-diffractive
$\mathit{{p}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{p}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
processes of Pythia 8.186 using the A3 set of tuned parameters [68] and the
NNPDF2.3lo PDF. For all MadGraph and Powheg Box samples, the EvtGen 1.6.0
program [69] was used for the bottom and charm hadron decays. The generated
samples were processed using the Geant4-based ATLAS detector simulation [70,
71] and the simulated events were reconstructed using the same algorithms as
were used for the data events.

Simulated events were corrected to compensate for differences between data and
simulations regarding the energy (or momentum) scale and resolution of leptons
and jets, the efficiencies for the reconstruction, identification, isolation
and triggering of leptons, and the tagging efficiency for heavy-flavour jets.

##  4 Event reconstruction

Collision vertices are reconstructed from at least two ID tracks, each with a
transverse momentum $p_{\text{T}}>500$ MeV [72]. Among all vertices, the one
with the highest $p_{\text{T}}^{2}$ sum of all associated tracks is chosen to
be the primary vertex (PV) of the event.

Electrons are reconstructed from ID tracks that are matched to energy clusters
in the electromagnetic calorimeter and which originate from the PV. The latter
condition is satisfied by a requirement on the transverse impact parameter
significance, $|d_{0}|/\sigma(d_{0})<5.0$, and on the longitudinal impact
parameter, $|z_{0}\sin(\theta)|<0.5$ mm. Electrons must satisfy requirements
for the electromagnetic shower shape, track quality, and track–cluster
matching, using a likelihood-based approach. A Tight operating point is used
for the 1-lepton channel and a Loose operating point is used for the 2-lepton
channel as well as the electron veto in the 0-lepton channel [73]. Electrons
also have to be isolated from hadronic activity in the event: both the
transverse energy sum and the scalar sum of transverse momenta of all ID
tracks within a cone of variable size around the electron have to be smaller
than $0.06$ times the electron’s transverse energy $E_{\text{T}}$. The maximum
cone size is $\Delta R=0.2$, shrinking for larger $E_{\text{T}}$ [73].
Electron candidates are required to have a minimum $p_{\text{T}}$ of 7 GeV and
to lie within $|\eta|<2.47$.

Muons [74] are identified by matching tracks found in the ID to either energy
deposits in the calorimeter system consistent with a minimum-ionising particle
(calorimeter-tagged muons) or full tracks (combined muons) or track segments
(segment-tagged muons) reconstructed in the muon spectrometer. Muons
reconstructed as stand-alone tracks in the muon spectrometer are also
considered. Like electrons, they have to fulfil impact-parameter requirements:
$|d_{0}|/\sigma(d_{0})<3.0$ and $|z_{0}\sin(\theta)|<0.5$ mm. Muons are
required to pass the Tight identification operating point for the 1-lepton
channel or the Loose identification operation point for both the 2-lepton
channel and the muon veto in the 0-lepton channel. Muon candidates with
$p_{\text{T}}>300\leavevmode\nobreak\ \text{GeV}$ must satisfy tighter
identification requirements in the muon spectrometer in order to improve the
$p_{\text{T}}$ resolution [74]. Muons must also be isolated: in the ID system,
the $p_{\text{T}}$ sum within a variable-size cone around the combined track
has to be smaller than $0.06$ times the muon’s transverse momentum. The
maximum cone size is $\Delta R=0.3$, shrinking for larger muon $p_{\text{T}}$.
Muon candidates are required to have a minimum $p_{\text{T}}$ of 7 GeV and to
lie within $|\eta|<2.5$.

Three jet types are reconstructed, using the anti-$k_{t}$ [75] algorithm
implemented in the FastJet package [76]. $\text{Small-}R$ jets are
reconstructed from noise-suppressed topological clusters in the calorimeter
[77, 78] using a radius parameter of $R=0.4$. They are required to have
$p_{\text{T}}>20$ GeV for central jets ($|\eta|<2.5$) and $p_{\text{T}}>30$
GeV for forward jets ($2.5<|\eta|<4.5$). To suppress central jets from pile-up
interactions, they are required to pass the jet vertex tagger [79] selection
if they have $p_{\text{T}}<120$ GeV and $|\eta|<2.5$. $\text{Large-}R$ jets
are used to reconstruct high-momentum Higgs boson candidates, for which the
$b$-quarks are emitted with small angular separation. These jets are built
using a radius parameter of $R=1.0$ and a combination of ID tracks and
calibrated clusters of energy deposits in calorimeter cells, referred to as
Track-CaloClusters [80]. Trimming [81] is applied to remove the energy of
clusters that originate from initial-state radiation, pile-up interactions or
the underlying event. This is done by reclustering the constituents of the
initial jet, using the $k_{t}$ algorithm [82, 83], into smaller
$R_{\mathrm{sub}}=0.2$ subjets and then removing any subjet that has a
$p_{\text{T}}$ less than 5% of the $p_{\text{T}}$ of the parent jet [84]. The
$\text{large-}R$ jets are required to have $p_{\text{T}}>250\,\text{GeV}$ and
$|\eta|<2.0$. The momenta of both the $\text{large-}R$ and $\text{small-}R$
jets are corrected for energy losses in passive material and for the non-
compensating response of the calorimeter. $\text{Small-}R$ jets are also
corrected for the average additional energy due to pile-up interactions [85,
86]. The third type of jets are clustered from ID tracks using a variable
radius (VR) that shrinks with increasing $p_{\mathrm{T}}$ of the studied
proto-jet [87]. VR track-jets are used in this analysis to identify $b$-jets
from decays of boosted Higgs bosons. They must contain at least two ID tracks
compatible with the primary vertex and must have
$p_{\text{T}}>10\leavevmode\nobreak\ \text{GeV}$ as well as $|\eta|<2.5$. In
this analysis, only $\text{large-}R$ jets with at least one ghost-associated
[88] VR track-jet are retained.

$\text{Small-}R$ jets and VR track-jets containing $b$-hadrons are identified
using the MV2c10 $b$-tagging algorithm [89, 90, 91, 92], with an operating
point that corresponds to a selection efficiency for $b$-jets of $70\%$, as
measured in simulated
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$
events. Applying the $b$-tagging algorithm reduces the number of light-flavour
and gluon jets, jets containing hadronically decaying $\tau$-leptons, and
$c$-quark jets, by a factor of 300 (304), 36 (55) and 9 (9), respectively, for
$\text{small-}R$ jets (VR track-jets) [89].

Hadronically decaying $\tau$-lepton candidates [93, 94] are used in the
$\nu\bar{\nu}b\bar{b}$ channel to reject backgrounds with real hadronically
decaying $\tau$-leptons. They are reconstructed using calorimeter-based
$\text{small-}R$ jets. They are required to have either one or three
associated tracks, $p_{\text{T}}>20\leavevmode\nobreak\ \text{GeV}$ and
$|\eta|<1.37$ or $1.52<|\eta|<2.5$. They are identified using a multivariate
tagging algorithm, which is $55\%$ ($40\%$) efficient for one-track (three-
track) $\tau$-lepton candidates [94].

The missing transverse momentum, with magnitude $E_{\text{T}}^{\text{miss}}$,
is calculated as the negative vector sum of the transverse momenta of
calibrated leptons and jets, plus a track-based soft term, i.e. all tracks
compatible with the primary vertex and not associated with any lepton or jet
used in the $E_{\text{T}}^{\text{miss}}$ calculation [95, 96]. In addition, a
track-based missing transverse momentum estimator
$\vec{p}_{\mathrm{T}}^{\mathrm{\ miss}}$ is built as the negative vector sum
of the transverse momenta of all tracks from the primary vertex. In order to
reduce backgrounds from mismeasurements of jet and lepton energies, an object-
based $E_{\text{T}}^{\text{miss}}$ significance $\mathcal{S}$ [97] is
exploited. This observable takes the resolutions of objects entering the
$E_{\text{T}}^{\text{miss}}$ calculation into account and quantifies how
likely it is that there is significant $E_{\text{T}}^{\text{miss}}$ from
undetectable particles (such as neutrinos).

Electrons, muons, hadronically decaying $\tau$-leptons and jets are
reconstructed and identified independently. This can lead to ambiguous
identifications when these objects are spatially close to each other. An
overlap removal procedure is therefore applied to uniquely identify these
objects. $\text{Small-}R$ jets are discarded if they are within a cone of size
$\Delta R=0.2$ around an electron or $\tau$-lepton or if they have fewer than
three associated tracks and are within a cone of $\Delta R=0.2$ around a muon
candidate. A $\tau$-lepton is removed if it is within a cone of $\Delta R=0.2$
around an electron or a muon. Finally, electrons and muons are discarded if
they are within $\Delta R=\min(0.4,0.04+10\,\text{GeV}/p^{\ell}_{\mathrm{T}})$
of the axis of any surviving jet, where $p^{\ell}_{\mathrm{T}}$ is the
transverse momentum of the electron or muon.

Two dedicated correction procedures are applied to improve the mass resolution
of the reconstructed Higgs boson candidates. If any muons are found within a
cone of $p_{\text{T}}$-dependent size around the jet axis, the four-momentum
of the closest muon in $\Delta R$ with $p_{\text{T}}$ larger than
$5\leavevmode\nobreak\ \text{GeV}$ is added to the jet four-momentum after
subtracting the energy contribution deposited in the calorimeter by the muon
[22, 98] (refered to as the muon-in-jet correction). For this correction,
muons are not required to pass any isolation requirements. Additional flavour-
specific energy corrections are applied, after the standard jet-energy-scale
calibration [85], to both $b$-tagged $\text{small-}R$ and $\text{large-}R$
jets to account for biases in the response to jets with leptonic or hadronic
decays of heavy-flavour hadrons (refered to as the
$p_{\text{T}}$-reconstruction correction). The $p_{\text{T}}$-reconstruction
corrections are determined from simulated SM $Vh(h\rightarrow b\bar{b})$
events by calculating the ratio of the $p_{\text{T}}$ of the generator-level
$b$-jets from the Higgs boson decay to the $p_{\text{T}}$ of the reconstructed
$b$-tagged jets after the muon-in-jet correction is applied.

##  5 Analysis strategy and event selection

The search for the $Z^{\prime}$ and $A$ bosons in the
$Zh\rightarrow\nu\bar{\nu}b\bar{b}$ and
$Zh\rightarrow\ell^{+}\ell^{-}b\bar{b}$ decay modes uses event selections
wherein the number of reconstructed charged leptons is exactly zero or two
(0-lepton and 2-lepton channels). In the 0-lepton channel, large
$E_{\text{T}}^{\text{miss}}$ signals the presence of two neutrinos. For the
$W^{\prime}$ search in the $Wh\rightarrow\ell^{\pm}\nu b\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$
channel, events with exactly zero or one charged lepton are used (0-lepton or
1-lepton channels).

For the 0-lepton channel and the 1-muon channel, $E_{\text{T}}^{\text{miss}}$
triggers [99] with thresholds of 70–110 GeV are used for the various data-
taking periods, corresponding to the increasing instantaneous luminosity. The
$E_{\text{T}}^{\text{miss}}$ triggers are more efficient than single-muon
triggers for the relatively large momentum of the $W$ boson required in the
analysis444Since muons are not included in the online
$E_{\text{T}}^{\text{miss}}$ calculation, $E_{\text{T}}^{\text{miss}}$ and
$p_{\mathrm{T},\textit{W}}$ are equivalent at the trigger level.
($p_{\mathrm{T},\textit{W}}>150\leavevmode\nobreak\ \text{GeV}$) [100]. The
trigger efficiency is above 80% for events with
$E_{\text{T}}^{\text{miss}}>150\leavevmode\nobreak\ \text{GeV}$ and
essentially $100\%$ above $200\leavevmode\nobreak\ \text{GeV}$.

In the 2-lepton channel and the 1-electron channel, events were recorded using
a combination of single-lepton triggers [101, 102] with isolation
requirements. The lowest $p_{\text{T}}$ thresholds range from
$24\leavevmode\nobreak\ \text{GeV}$ to $26\leavevmode\nobreak\ \text{GeV}$ in
the electron channel and from $20\leavevmode\nobreak\ \text{GeV}$ to
$26\leavevmode\nobreak\ \text{GeV}$ in the muon channel. Additional triggers
without an isolation requirement are used to recover efficiency for leptons
with $p_{\text{T}}>60\leavevmode\nobreak\ (50)$ GeV in the electron (muon)
channel. The trigger-level electron (muon) is required to match a
reconstructed electron (muon) with $p_{\text{T}}>27$ GeV. The trigger
efficiencies for the combined single-electron and combined single-muon
triggers are larger than $95\%$.

The Higgs boson candidate is reconstructed from the four-vectors of its decay
products, the $b$-quarks. When the Higgs boson has relatively low
$p_{\text{T}}$, the $b$-quarks can be reconstructed as two $\text{small-}R$
jets (resolved category). As the momentum of the Higgs boson increases, the
two $b$-quarks become more collimated and a selection using a single
$\text{large-}R$ jet becomes more efficient (merged category).

For the resolved signal region, two central $\text{small-}R$ jets are required
to have an invariant mass $m_{jj}$ in the range 110–140 GeV for the 0-lepton
and 1-lepton channels and in the range 100–145 GeV for the 2-lepton channel.
The latter selection is looser in order to take advantage of the lower
background contribution in this channel. When two $b$-tagged jets are present
in the event, the dijet system representing the Higgs boson candidate is
defined by the two $b$-tagged $\text{small-}R$ jets. In events with more than
two $b$-tagged jets, the dijet system is defined by the two leading $b$-tagged
jets. In the case where only one $b$-tagged jet is present, the dijet pair is
defined by the $b$-tagged jet and the leading $\text{small-}R$ jet in the
remaining set. The leading jet in the pair must have $p_{\text{T}}>45$ GeV.
For the merged signal region, a $\text{large-}R$ jet is required with mass
$m_{\text{J}}$ in the range 75–145 GeV and at least one associated $b$-tagged
VR track-jet. Events which satisfy the selection requirements of both the
resolved and merged categories are assigned to the resolved category because
of its better dijet mass resolution. In resolved event topologies, the four-
momentum of the dijet system is scaled by 125 GeV / $m_{jj}$.

Higgs boson candidates with one or two $b$-tagged jets define the 1 $b$-tag
and 2 $b$-tags categories, respectively. For the merged selection, only one or
two leading VR track-jets associated with the $\text{large-}R$ jet are
considered in this counting. In the searches for $Z^{\prime}$, $W^{\prime}$
and $A$ boson production via the $ggA$ process, events with exactly one or
exactly two $b$-tagged jets are considered, while signal production via the
$b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
process is addressed by introducing further categories for events with
additional $b$-tagged jets. In the resolved region a category with at least
three $b$-tagged jets (3+ $b$-tags) is used, while in the merged regions,
events are assigned to additional categories based on the number of $b$-tagged
VR track-jets outside of the $\text{large-}R$ jet. The 0-lepton channel uses a
category for events in which two $b$-tagged VR track-jets are associated with
the $\text{large-}R$ jet and at least one $b$-tagged VR track-jet is outside
of the $\text{large-}R$ jet (2 $b$-tags & 1+ add).555The category for events
with exactly one $b$-tagged VR track-jet associated with the $\text{large-}R$
jet and at least one $b$-tagged VR track-jet outside of the $\text{large-}R$
jet (i.e. 1 $b$-tag & 1+ add) does not add sensitivity to the 0-lepton channel
and is therefore not used for the $A$ boson search. Furthermore, the 2-lepton
channel uses a category in which at least one $b$-tagged VR track-jet is
associated with the $\text{large-}R$ jet and at least one $b$-tagged VR track-
jet is outside of the $\text{large-}R$ jet (1+ $b$-tags & 1+ add).

The calculation of the reconstructed mass $m_{Vh}$ of the $Vh$ resonance
depends on the decay channel. In the 0-lepton channel, where it is not
possible to fully reconstruct the $Vh$ system, the transverse mass is used as
the final discriminant, defined as:
$m_{\mathrm{T},Vh}=\sqrt{\left(E_{\mathrm{T},h}+E^{\mathrm{miss}}_{\mathrm{T}}\right)^{2}-\left(\vec{p}_{\mathrm{T},h}+\vec{E}^{\mathrm{miss}}_{\mathrm{T}}\right)^{2}}$,
where $E_{\mathrm{T},h}$ is the transverse energy of the Higgs boson
candidate, while $\vec{E}^{\mathrm{miss}}_{\mathrm{T}}$ and
$\vec{p}_{\mathrm{T},h}$ are vectors of the missing transverse momentum and
Higgs boson transverse momentum, respectively. In the 1-lepton and 2-lepton
channels, $m_{Vh}$ is defined as the invariant mass calculated from the sum of
four-momenta of the vector boson and Higgs boson candidates. In the 1-lepton
channel, the vector boson is reconstructed from the sum of the four-momenta of
the charged lepton and the neutrino. The momentum of the neutrino in the
$z$-direction, $p_{z,\nu}$, is obtained by imposing a $W$ boson mass
constraint on the lepton–neutrino system, which results in a quadratic
equation. Here $p_{z,\nu}$ is taken as either the real component of the
complex solutions or the real solution with the smaller absolute magnitude.
The transverse mass of the $W$ boson, $m_{\mathrm{T},W}$, is calculated from
the transverse component of the lepton momentum and the
$\vec{E}^{\mathrm{miss}}_{\mathrm{T}}$. In the 2-lepton channel, the vector
boson is reconstructed from the sum of the four-momenta of the two charged
leptons. In events with two muons, the mass resolution in the signal region is
improved by scaling the four-momentum of the dimuon system by 91.2 GeV /
$m_{\mu\mu}$ to account for poorer dimuon mass resolution at high
$p_{\mathrm{T}}$.

An event-cleaning procedure is applied to all lepton channels. Events are
removed if they contain overlaps between one of the VR track-jets used for
$b$-tagging and at least one VR track-jet with a $p_{\text{T}}$ above
$5\leavevmode\nobreak\ \text{GeV}$ and with at least two associated tracks, in
order to prevent the selection of jets with ambiguously matched tracks from
heavy-flavour hadron decays. Additional selections are applied for each lepton
channel, as outlined below, to reduce the main backgrounds and enhance the
signal sensitivity.

The 0-lepton channel vetoes electrons and muons and, for the $Z^{\prime}$,
$ggA$ and $b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
searches, also hadronically decaying $\tau$-lepton candidates
($\mathit{{\tau}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}_{\text{had}}$).
In order to recover $W^{\prime}\rightarrow
Wh\rightarrow\mathit{{\tau}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}_{\text{had}}\nu
bb$ events via the 0-lepton channel, the veto on
$\mathit{{\tau}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}_{\text{had}}$
is relaxed for the $W^{\prime}$ search such that events with at most one
$\mathit{{\tau}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}_{\text{had}}$
are selected. The multijet background is reduced by requiring a missing
transverse momentum of $E_{\text{T}}^{\text{miss}}>150\leavevmode\nobreak\
(200)\leavevmode\nobreak\ \text{GeV}$ in the resolved (merged) category, an
$E_{\text{T}}^{\text{miss}}$ significance $\mathcal{S}$ of 9.0–13.6,
increasing linearly as a function of $m_{\mathrm{T},Vh}$, a track-based
missing transverse momentum of
$p_{\text{T}}^{\text{miss}}>60\leavevmode\nobreak\ \text{GeV}$, an angular
separation between $E_{\text{T}}^{\text{miss}}$ and
$p_{\text{T}}^{\text{miss}}$ of
$\Delta\phi(\vec{E}^{\mathrm{miss}}_{\mathrm{T}},\vec{p}^{\mathrm{miss}}_{\mathrm{T}})<\pi/2$,
an angular separation between $E_{\text{T}}^{\text{miss}}$ and the Higgs boson
candidate of $\Delta\phi(\vec{E}^{\mathrm{miss}}_{\mathrm{T}},h)>2\pi/3$, and
the minimum angular separation between $E_{\text{T}}^{\text{miss}}$ and the
jets that are used in the event to satisfy
$\min\left[\Delta\phi(\vec{E}^{\mathrm{miss}}_{\mathrm{T}},{\textrm{small-}R\leavevmode\nobreak\
\mathrm{jet}})\right]>\pi/9\leavevmode\nobreak\ (\pi/6)$ for events with up to
three (more than three) jets. In the resolved channel, the angular separation
between the two Higgs boson candidate jets is required to be
$\Delta\phi_{jj}<7\pi/9$, and the scalar sum of all jet $p_{\text{T}}$ values
must satisfy $S_{\mathrm{T}}>120\leavevmode\nobreak\ (150)\leavevmode\nobreak\
\text{GeV}$ if the event contains two (more than two) central jets. Finally,
the resolved channel requires at least two central $\text{small-}R$ jets, and
the merged channel at least one $\text{large-}R$ jet.

For the 1-lepton channel, selected events are required to have exactly one
electron or one muon with $p_{\text{T}}>27\leavevmode\nobreak\ \text{GeV}$.
The multijet background is reduced by requiring
$E_{\text{T}}^{\text{miss}}>80\,(40)\leavevmode\nobreak\ \text{GeV}$ for
events with an electron (muon) in the resolved category, and in the merged
category the requirement is increased to
$E_{\text{T}}^{\text{miss}}>100\leavevmode\nobreak\ \text{GeV}$ for both
electrons and muons. A $m_{Wh}$-dependent requirement is placed on
$p_{\mathrm{T},W}$: it has to be larger than both $150\,\text{GeV}$ and
$\left[710-(3.3\cdot 10^{5})/m_{Wh}\right]\,\text{GeV}$ for the resolved
channel, and larger than both $150\,\text{GeV}$ and
$\left[394\cdot\ln(m_{Wh})-2350\right]\,\text{GeV}$ for the merged channel,
where $m_{Wh}$ is expressed in GeV. The selection requirements on
$p_{\mathrm{T},W}$ exploit the fact that the $m_{Wh}$ and $p_{\mathrm{T},W}$
observables are more strongly correlated for the signal process than for the
various backgrounds. Finally, the transverse mass of the $W$ boson candidate
is required to be less than $300\leavevmode\nobreak\ \text{GeV}$ in order to
remove events in which the $E^{\mathrm{miss}}_{\mathrm{T}}$ or lepton momentum
is not well reconstructed. In addition, the angular separation between the
charged lepton and the Higgs boson candidate must fulfil $\Delta
R(\ell,h)>2.0$.

In the 2-lepton channel, same-flavour leptons ($ee$ or $\mu\mu$) are selected
with $p_{\text{T}}>27\leavevmode\nobreak\ \text{GeV}$ for the leading lepton
and $p_{\text{T}}>20\leavevmode\nobreak\ (25)\leavevmode\nobreak\ \text{GeV}$
for the subleading lepton in the resolved (merged) category. Three kinematic
selections are optimised as a function of the resonance mass $m_{Vh}$ to
reduce the
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$
and
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
backgrounds. The selection requirement on the mass of the dilepton system,
max[$40\leavevmode\nobreak\ \text{GeV},\ 87\leavevmode\nobreak\
\text{GeV}-0.030\cdot m_{Vh}$] $<m_{\ell\ell}<97\leavevmode\nobreak\
\text{GeV}+0.013\cdot m_{Vh}$, and the requirement
$E_{\text{T}}^{\text{miss}}/\sqrt{H_{{\mathrm{T}}}}<1.15+(8\times
10^{-3})\cdot m_{Vh}\,/\,(1\ \text{GeV})$ vary as a function of the $m_{Vh}$,
and these requirements become looser for larger resonance masses to account
for smaller backgrounds and a degradation of the dilepton invariant mass and
energy resolution with increasing boost of the final-state particles. The
$H_{{\mathrm{T}}}$ is calculated as the scalar sum of the $p_{\text{T}}$ of
the leptons and $\text{small-}R$ jets in the event. For the third kinematic
selection, the momentum of the dilepton system ($p_{\mathrm{T,}\ell\ell}$) is
required to be greater than
$\left[20+9\cdot\sqrt{m_{Vh}/(1\leavevmode\nobreak\
\text{GeV})-320}\right]\,\text{GeV}$ for $m_{Vh}$ greater than 320 GeV. In the
resolved dimuon category, an opposite-charge requirement is applied to further
reduce diboson backgrounds and because the probability of mismeasuring the
charge of individual muons is very low.

All signal-region selections are summarised in Table 2. The products of
kinematic acceptance and reconstruction efficiency for $Z^{\prime}\
\rightarrow Zh(h\rightarrow b\bar{b}/c\bar{c})$, $b\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A\rightarrow
Zh(h\rightarrow b\bar{b})$ and $W^{\prime}\ \rightarrow Wh(h\rightarrow
b\bar{b}/c\bar{c})$ are shown in Figure 2 as a function of the resonance mass.
The $m_{Vh}$ resolution is expected to be in the range 2%–5% (5%–9%) for the
resolved (merged) $\ell^{+}\ell^{-}b\bar{b}$ channel and in the range 7%–12%
(9%–12%) for the resolved (merged) $\ell^{\pm}{\nu}b\bar{b}$ channel, while
the $m_{{\mathrm{T}},Vh}$ resolution is expected to range from $8\%$ to about
$28\%$ (from $12\%$ to $17\%$) for the resolved (merged)
$\nu\bar{\nu}b\bar{b}$ channel. The resolution of the $m_{{\mathrm{T}},Vh}$
and $m_{Vh}$ deteriorates in the resolved (merged) category with increasing
(decreasing) resonance mass.

In addition to the signal regions, dedicated control regions are used to
improve the modelling of major backgrounds. These control regions are
constructed by inverting one of the selection requirements of the signal
region: in the 2-lepton resolved category,
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$
control regions are created by selecting events with different-flavour
oppositely charged leptons $e\mu$, and without the
$E_{\text{T}}^{\text{miss}}/\sqrt{H_{{\mathrm{T}}}}$ requirement. The
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$
purity of this selection is greater than 90%. These events are used to
constrain the
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$
normalisation in the final fit. Dedicated $m_{jj}$ ($m_{\text{J}}$) sideband
regions are defined in the resolved (merged) $0$\- and $1$-lepton regions by
requiring $m_{jj}$ ($m_{\text{J}}$) to be between $50\leavevmode\nobreak\
\text{GeV}$ and $200\leavevmode\nobreak\ \text{GeV}$ but outside the
respective signal-region windows defined above. They are used to improve the
modelling of the
$W/\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
kinematics and to constrain the normalisation of the
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$
and
$W/\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
backgrounds. As the
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
backgrounds are by far the largest in the 2-lepton signal region, yielding up
to $90\%$ of the total background, and due to the significant shape difference
between the $m_{Vh}$ templates for the hypothetical signal processes and the
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
backgrounds, no additional control regions are required to constrain the
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
contributions. Thus, the 2-lepton $m_{jj}$ and $m_{\text{J}}$ sideband regions
are not included into the likelihood fit. Instead, these sidebands are used to
correct the kinematics of the
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
background (see Section 6). The channels and regions used for the different
fits are detailed in Table 3.

Table 2:  Topological and kinematic selections for each channel and category
as described in the text. (∗) Applies in the case of only two central jets.
(∗∗) Applied only for the $W^{\prime}$ search. (†) A higher threshold (80 GeV)
is used for the single-electron channel. (††) Applied only for $m_{Vh}>320$
GeV. (‡‡) Only the two leading VR track-jets matched by ghost-association to
the large-$R$ jet are considered when classifying events into $b$-tag
categories. Events are further classified according to the number of
$b$-tagged jets in the events.

Variable | Resolved | Merged
---|---|---
| Common selection
Number of jets |  $\geq$2 ${\textrm{small-}R\leavevmode\nobreak\ \mathrm{jets}}$ (0, 2-lep.) |  $\geq$1 large-$R$ jet
2 or 3 ${\textrm{small-}R\leavevmode\nobreak\ \mathrm{jets}}$ (1-lep.) |  $\geq 1$ VR track-jets (matched to leading large-$R$ jet)‡‡
Leading jet $p_{\text{T}}$ [GeV] | $>45$ | $>250$
$m_{h}$ [GeV] | 110–140 (0,1-lep.), 100–145 (2-lep.) | 75–145
| 0-lepton selection
$E_{\text{T}}^{\text{miss}}$ [GeV] | $>150$ | $>200$
$S_{\mathrm{T}}$ [GeV] |  $>150$ $(120{}^{\ast})$ | –
$\Delta\phi_{jj}$ | $<7\pi/9$ | –
$p^{\mathrm{miss}}_{\mathrm{T}}$ [GeV] | $>60$
$\Delta\phi(\vec{E}^{\mathrm{miss}}_{\mathrm{T}},\vec{p}^{\mathrm{miss}}_{\mathrm{T}})$ | $<\pi/2$
$\Delta\phi(\vec{E}^{\mathrm{miss}}_{\mathrm{T}},h)$ | $>2\pi/3$
$\min\left[\Delta\phi(\vec{E}^{\mathrm{miss}}_{\mathrm{T}},{\textrm{small-}R\leavevmode\nobreak\ \mathrm{jet}})\right]$ |  $>\pi/9$ (2 or 3 jets), $>\pi/6$ ($\geq$ 4 jets)
$N_{\tau_{\mathrm{had}}}$ | 0 ($\leq$ 1∗∗)
$E_{\text{T}}^{\text{miss}}$ significance $\mathcal{S}$ |  $\begin{cases}>9&\text{if }m_{Vh}<240\,\text{GeV},\\\ >6.6+0.01\cdot m_{Vh}&\text{if }240\,\text{GeV}\leq m_{Vh}<700\,\text{GeV},\\\ >13.6&\text{if }m_{Vh}>700\,\text{GeV},\end{cases}$
| 1-lepton selection
Leading lepton $p_{\text{T}}$ [GeV] | $>27$ | $>27$
$E_{\text{T}}^{\text{miss}}$ [GeV] | $>40\ (80^{\dagger})$ | $>100$
$p_{\mathrm{T},W}$ [GeV] | $>\max\left[150,\ 710-(3.3\cdot 10^{5}\ \text{GeV})/m_{Vh}\right]$ | $>\max\left[150,\ 394\cdot\log(m_{Vh}/(1\ \text{GeV}))-2350\right]$
$m_{\mathrm{T},W}$ [GeV] | < 300
$\Delta R(\ell,h)$ | > 2.0
| 2-lepton selection
Leading lepton $p_{\text{T}}$ [GeV] | $>27$ | $>27$
Subleading lepton $p_{\text{T}}$ [GeV] | $>20$ | $>25$
$E_{\text{T}}^{\text{miss}}/\sqrt{H_{{\mathrm{T}}}}$ [$\sqrt{\mathrm{\text{GeV}}}$] | $<1.15\ +8\times 10^{-3}\cdot m_{Vh}/(1\ \text{GeV})$
$p_{\mathrm{T},\ell\ell}$ [GeV] | $>20+9\cdot\sqrt{m_{Vh}/(1\ \text{GeV})-320}^{\dagger\dagger}$
$m_{\ell\ell}$ [GeV] | $\in\Bigl{[}\max\left[40\ ,\ 87-0.030\cdot m_{Vh}/(1\ \text{GeV})\right],97+0.013\cdot m_{Vh}/(1\ \text{GeV})\Bigr{]}$

((a))  ((b))

((c))  ((d))

((e))  ((f))

Figure 2:  Product of acceptance and efficiency for (a) $Z^{\prime}\rightarrow
Zh\rightarrow\nu\bar{\nu}b\bar{b}/c\bar{c}$, (b) $Z^{\prime}\rightarrow
Zh\rightarrow\ell^{+}\ell^{-}b\bar{b}/c\bar{c}$, (c) $b\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A\left(\rightarrow
Zh\rightarrow\nu\bar{\nu}b\bar{b}\right)$, (d) $b\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A\left(\rightarrow
Zh\rightarrow\ell^{+}\ell^{-}b\bar{b}\right)$ and (e, f)
$W^{\prime}\rightarrow Wh\rightarrow\ell^{\pm}{\nu}b\bar{b}/c\bar{c}$ as a
function of the resonance mass for the 0-lepton signal regions (a, c, e), the
2-lepton signal regions (b, d), and the 1-lepton signal regions (f).

Table 3:  A list of the signal and control regions included in the statistical
analysis of (a) the $ggA$, $Z^{\prime}$ and $W^{\prime}$ signal model
hypotheses and (b) the $b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
signal hypothesis. The searches for the pseudoscalar boson and the
$Z^{\prime}$ boson are based on the 0- and 2-lepton channels, while the search
for the $W^{\prime}$ boson is based on the 0- and 1-lepton channels. The
notation 1+2 $b$-tag indicates that the 1 and 2 $b$-tag regions are combined.

Channel | Region | Signal regions | Control regions | Signal hypotheses
---|---|---|---|---
Resolved
0-lepton |  $b$-tags | 1, 2 $b$-tags | 1, 2 $b$-tags | $ggA$, $Z^{\prime}$, $W^{\prime}$
Mass window | $110<m_{jj}\leavevmode\nobreak\ (\text{GeV})<140$ | $50<m_{jj}\leavevmode\nobreak\ (\text{GeV})<110\leavevmode\nobreak\ ||\leavevmode\nobreak\ 140<m_{jj}\leavevmode\nobreak\ (\text{GeV})<200$
1-lepton |  $b$-tags | 1, 2 $b$-tags | 1, 2 $b$-tags | $W^{\prime}$
Mass window | $110<m_{jj}\leavevmode\nobreak\ (\text{GeV})<140$ | $50<m_{jj}\leavevmode\nobreak\ (\text{GeV})<110\leavevmode\nobreak\ ||\leavevmode\nobreak\ 140<m_{jj}\leavevmode\nobreak\ (\text{GeV})<200$
2-lepton |  $b$-tags | 1, 2 $b$-tags | 1 + 2 $b$-tags | $ggA$, $Z^{\prime}$
Mass window | $100<m_{jj}\leavevmode\nobreak\ (\text{GeV})<145$ | $100<m_{jj}\leavevmode\nobreak\ (\text{GeV})<145$
Lepton flavour |  $ee$, $\mu\mu$ | $e\mu$
Merged
0-lepton |  $b$-tags | 1, 2 $b$-tags | 1, 2 $b$-tags | $ggA$, $Z^{\prime}$, $W^{\prime}$
Mass window | $75<m_{\text{J}}\leavevmode\nobreak\ (\text{GeV})<145$ | $50<m_{\text{J}}\leavevmode\nobreak\ (\text{GeV})<75\leavevmode\nobreak\ ||\leavevmode\nobreak\ 145<m_{\text{J}}\leavevmode\nobreak\ (\text{GeV})<200$
1-lepton |  $b$-tags | 1, 2 $b$-tags | 1, 2 $b$-tags | $W^{\prime}$
Mass window | $75<m_{\text{J}}\leavevmode\nobreak\ (\text{GeV})<145$ | $50<m_{\text{J}}\leavevmode\nobreak\ (\text{GeV})<75\leavevmode\nobreak\ ||\leavevmode\nobreak\ 145<m_{\text{J}}\leavevmode\nobreak\ (\text{GeV})<200$
2-lepton |  $b$-tags | 1, 2 $b$-tags | – | $ggA$, $Z^{\prime}$
Mass window | $75<m_{\text{J}}\leavevmode\nobreak\ (\text{GeV})<145$
| Lepton flavour |  $ee$, $\mu\mu$

((a)) Signal and control regions for the $ggA$, $Z^{\prime}$and
$W^{\prime}$searches

Channel | Region | Signal regions | Control regions | Signal hypothesis
---|---|---|---|---
Resolved
0-lepton |  $b$-tags | 1, 2, 3+ $b$-tags | – | $b\mathit{{\mkern 0.0mu\overline{\mkern 0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
Mass window | $110<m_{jj}\leavevmode\nobreak\ (\text{GeV})<140$
2-lepton |  $b$-tags | 1, 2, 3+ $b$-tags | 1+2, 3+ $b$-tags | $b\mathit{{\mkern 0.0mu\overline{\mkern 0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
Mass window | $100<m_{jj}\leavevmode\nobreak\ (\text{GeV})<145$ | $100<m_{jj}\leavevmode\nobreak\ (\text{GeV})<145$
Lepton flavour |  $ee$, $\mu\mu$ | $e\mu$
Merged
0-lepton |  $b$-tags | 1, 2 $b$-tag, 2 $b$-tags & 1+ add | – | $b\mathit{{\mkern 0.0mu\overline{\mkern 0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
Mass window | $75<m_{\text{J}}\leavevmode\nobreak\ (\text{GeV})<145$
2-lepton |  $b$-tags | 1, 2 $b$-tag, 1+ $b$-tags & 1+ add | – | $b\mathit{{\mkern 0.0mu\overline{\mkern 0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
Mass window | $75<m_{\text{J}}\leavevmode\nobreak\ (\text{GeV})<145$
| Lepton flavour |  $ee$, $\mu\mu$
((b)) Signal and control regions for the $b\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
search

##  6 Background estimation

The background composition in the signal region depends on the charged-lepton
and $b$-jet multiplicities. In the 0-lepton channel, the dominant background
sources are
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
and
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$
events, with a significant contribution from
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{+\,jets}$.
In the 2-lepton channel,
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
production is the dominant background followed by
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$
production. The dominant background sources for the 1-lepton channel are
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{+\,jets}$
and
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$.
Contributions from diboson, SM $Vh$,
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+h$
and
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+V$
production are small in all channels.

Background contributions from multijet events were studied and found to be
negligible for the 0-lepton channels, the 2-lepton channels and the merged
1-lepton channel, while a small multijet background contribution (a few per
mille of the total background) is found in the resolved 1 $b$-tag category for
the 1-lepton channel. This contribution is estimated using a data-driven
method: a template is produced in a dedicated control region enriched with
multijet events, constructed by inverting the lepton isolation requirements
decribed in Section 4. The shape of the $m_{Vh}$ template is obtained by
subtracting the other backgrounds from data, assuming that the other
background processes are well modelled in Monte Carlo simulation. The
normalisation of the template is derived from a two-component fit (multijet
and ‘rest’) to data in the $m_{jj}$ sideband control region. Further details
of the template method can be found in Ref. [100].

The other background distributions are estimated from the samples of simulated
events, with normalisations of the main backgrounds estimated from the data.
The simulated
$W/\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
samples are reweighted as a function of the transverse momentum of the dijet
system, $p_{\mathrm{T},jj}$, based on fits to the data/simulation distribution
in the $m_{jj}$ sideband regions, defined in Section 5. In the 2-lepton
channel, different $p_{\mathrm{T},jj}$ reweighting are derived for the 1
$b$-tag and 2 $b$-tag regions. The $p_{\mathrm{T},jj}$ reweighting functions
are derived after accounting for differences in the overall normalisation
between data and the simulations.

The
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$,
single-top-quark,
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+V$
and
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+h$
processes are combined into one single component, which is referred to as the
top-quark backgrounds. In the search for a pseudoscalar produced via the
$b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
mode, the top-quark backgrounds are split into two components. The
categorisation is based on the flavour of the additional jets666Generator-
level particle jets are reconstructed from stable particles (i.e. particles
with a mean lifetime of $\tau>3\cdot 10^{-11}$ seconds) using the anti-$k_{t}$
algorithm with a radius parameter of $R=0.4$. These jets are required to have
$p_{\mathrm{T}}>15\,\text{GeV}$ and $|\eta|<4.5$. in the event, following the
same strategy as described in Ref. [103]. Events containing at least one $b$\-
or $c$-quark jet not originating from a top quark, $W$, $Z$ or Higgs boson
decay are classified as $\mathrm{top}+\mathrm{hf}$ (with $\mathrm{hf}$
denoting ‘heavy flavour’), while all other events are classified as top + lf
(with $\mathrm{lf}$ denoting ‘light flavour’).

Simulated samples of
$W/\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
events are split into different jet-flavour components to help improve the
grouping of the background components in the final fit because their
contributions vary between the 1 $b$-tag and 2 $b$-tag categories. In the
resolved category, the samples are split according to the generated flavour of
the two $\text{small-}R$ jets forming the Higgs boson candidate. In the merged
category, they are split according to the generated flavour of the one or two
leading track-jets associated with the $\text{large-}R$ jet. The generated jet
flavour is determined by counting generated heavy-flavour hadrons with
$p_{\text{T}}>5$ GeV that are matched by ghost-association to the
reconstructed jet. If a $b$-hadron is found, the jet is labelled as a $b$-jet,
otherwise if a $c$-hadron is found the jet is labelled as a $c$-jet. If
neither a $b$-hadron nor a $c$-hadron is associated with the reconstructed
jet, it is labelled as a light-flavour jet. Based on this association scheme,
the
$W/\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
simulated event samples are split into three components: $W/Z+\mathrm{hf}$
($W/Z\text{\,+\,}bb$ , $W/Z\text{\,+\,}bc$ and $W/Z\text{\,+\,}cc$),
$W/Z+\mathrm{hl}$ ($W/Z\text{\,+\,}b\text{l}$ and $W/Z\text{\,+\,}c\text{l}$)
and $W/Z+\mathrm{lf}$; in this notation ‘$\mathrm{l}$’ refers to a light-
flavour jet. In the statistical analysis described in Section 8, the global
normalisations of the
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$,
$Z+\mathrm{hl}$,
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$
and $W+\mathrm{hl}$ components are determined via a fit to data, while the
normalisation of the $W/Z+\mathrm{lf}$ component is taken from simulation. In
the $A$ boson interpretation, the
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$
background normalisation in the 3+ $b$-tag region is determined from this
region independently to account for decreased perturbative accuracy of the
background predictions for events with additional heavy-flavour jets in the
final state.

In the statistical analysis described in Section 8 the global normalisations
of the dominant background contributions, i.e. the top-quark,
$W/Z+\mathrm{hf}$ , and $W/Z+\mathrm{hl}$ backgrounds, are determined
separately for the 0-, 1- and 2-lepton channels via a fit to data. In the 0-
and 1-lepton channels, the $m_{jj}$ ($m_{\text{J}}$) sideband regions are used
to constrain the
$W/\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
and top-quark backgrounds in the resolved (merged) event categories. In the
2-lepton channel, an $e\mu$ control region is used for resolved event
topologies to constrain the normalisation of the combined top-quark
backgrounds.

##  7 Systematic uncertainties

The distributions of $m_{Vh}$ and $m_{\text{T,}Vh}$ are affected by both
experimental and modelling uncertainties, which enter the final fits as
nuisance parameters. Uncertainties in the modelling of physics objects are
correlated over signal and background processes, channels, kinematic regions
and distributions of observables.

The largest experimental systematic uncertainties are associated with the
calibration and resolution of the $\text{small-}R$ and $\text{large-}R$ jet
energy and of the $\text{large-}R$ jet mass. Further dominant uncertainties
are related to the determination of the $b$-tagging efficiency and
misidentification rate. The uncertainties in the $\text{small-}R$ jet energy
scale have contributions from in situ calibration studies, from the dependency
on the pile-up activity, and from the flavour composition of jets [85, 86]. An
additional uncertainty in the energy calibration of $b$\- and $c$-jets is also
used. The uncertainty in the scale and resolution of the $\text{large-}R$ jet
energy and mass is estimated by comparing the ratio of calorimeter-based to
track-based measurements in dijet data and simulation [84, 104]. Differences
in the $b$-tagging efficiency measured in data and simulation result in
correction factors for $b$-jets, $c$-jets and light-flavour jets.
Uncertainties in these correction factors are decomposed into uncorrelated
components [89, 91, 92]. An additional term is included to extrapolate the
measured uncertainties to the high-$p_{\mathrm{T}}$ region of interest. This
term is calculated from simulated events by considering variations of the
quantities affecting the $b$-tagging performance, such as the impact parameter
resolution, percentage of tracks from random combinations of measurements in
the ID, description of the detector material, and track multiplicity per jet.

Uncertainties in the reconstruction, identification, isolation and trigger
efficiencies of muons [105], electrons [106] and $\tau$-leptons [107] and in
their energy scale and resolution have only a small impact on the result.
Uncertainties in the $E_{\text{T}}^{\text{miss}}$ trigger efficiency are taken
into account by following the approach described in Ref. [98].

The uncertainties in the energy scale and resolution of the $\text{small-}R$
jets and leptons are propagated to the calculation of
$E_{\text{T}}^{\text{miss}}$, which also has additional uncertainties from the
modelling of the underlying event and the momentum scale, momentum resolution
and reconstruction efficiency of the tracks used to compute the soft-term [95,
96]. Finally, a global luminosity uncertainty of $1.7\%$ is applied to the
normalisation of all simulated events [108, 109].

Theoretical uncertainties are derived for all simulated signal and background
processes and lead to variations in the normalisation and shape of the
templates used for the statistical analysis. These uncertainties are assessed
by comparing nominal and alternative event generators and UEPS models. In
general, the perturbative precision and the PDF sets used in these alternative
configurations match those of the nominal generators (unless explicitly
stated). Uncertainties due to the PDF set parameterisation are evaluated using
replica sets, and uncertainties due to missing higher orders are evaluated by
varying appropriate scale parameters, as described below. Uncertainties due to
the modelling of PDFs are evaluated for the signal processes and all relevant
backgrounds by comparing the predictions of their nominal PDF set with those
of alternative PDF sets and then comparing the maximum difference with the
difference from the root-mean-square spread of the nominal replica sets. The
larger of the two is taken as the uncertainty.

For the signal processes, uncertainties in the acceptance were derived by
replacing the nominal PDF set by the MSTW2008lo [110] and CT10 [111] PDF sets,
by using the eigenvariations of the Pythia 8.186 A14 set of tuned parameter
values [36], and by replacing Pythia 8.186 by Herwig 7.0.4 [112].
Renormalisation and factorisation scale variations are also considered for the
signal processes. The effects of these variations are sizeable for both of the
two $A$ boson production modes and negligible for the $Z^{\prime}$ and
$W^{\prime}$ signals. For all the $A$, $Z^{\prime}$ and $W^{\prime}$ signals,
the total acceptance uncertainties are between 2% and 7%.

The modelling uncertainties in
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$
events are derived as follows. To assess potential uncertainties in the
matching of the matrix element to the parton shower, the nominal sample is
compared with a sample where Powheg Box is replaced by MadGraph5_aMC@NLO. The
parton shower modelling uncertainty is assessed by replacing Pythia 8.230 by
Herwig 7.04 together with the H7UE set of tuned parameters [112]. The
uncertainty in the modelling of initial-state radiation (ISR) is addressed by
changing the renormalisation and factorisation scales by a factor of two in
combination with the eigenvariations of the A14 set of tuned parameters. The
uncertainty in the modelling of final-state radiation (FSR) is addressed by
varying the FSR scale [36].

To account for the ambiguities in the interference between top-quark pair and
$Wt$ production, an alternative sample generated with Powheg Box and Pythia
8.230, based on the diagram subtraction (DS) scheme [63], is used. The
difference between the DS and DR schemes for $Wt$ production is considered as
an additional systematic uncertainty.

The relative contributions of the single-top-quark,
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+V$
and
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+h$
processes to the combined top-quark background are varied by $19\%$ for single
top quarks and by $50\%$ for
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+V$
and
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+h$
production.

For
$W/\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
backgrounds, the effects of scale uncertainties are estimated by varying the
renormalisation and factorisation scales by a factor of two. In order to
account for variations of both the matrix element and the parton shower, the
nominal samples were compared with samples with LO-accurate matrix elements
that contain up to four final-state partons generated with MadGraph5_aMC@NLO
2.2.2 using the NNPDF3.0nlo PDF set and interfaced to Pythia 8.186 [113] using
the CKKW-L merging procedure [114, 115]. The A14 set of tuned parameters of
Pythia is used with the NNPDF2.3lo PDF set. Decays of bottom and charm hadrons
are performed by EvtGen 1.2.0.

In the 1-lepton channel, additional JetBinMigration uncertainties in the
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{+\,jets}$
backgrounds, based on Ref. [116], are derived to account for the modelling of
the veto of additional jets in the selection. These uncertainties are
evaluated by individually varying the contributions from
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{+\,jets}$
events with either less than two jets or more than three jets at the generator
level. In both cases, the contributions to the signal and control regions are
varied upwards/downwards by $50\%$. Only the shape uncertainties are retained
because the normalisations of
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{+\,jets}$
backgrounds are allowed to float in the final fit.

For both the
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$
and
$W/\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
backgrounds, uncertainties in the PDF were determined by the standard
deviation of the 100 NNPDF3.0nlo replicas, by using NNPDF3.0nlo PDFs with
varied $\alpha_{\text{s}}$ [39], and by comparisons with the nominal
MMHT2014nnlo and the CT14nnlo PDF sets [65].

For the major backgrounds, the theory uncertainties listed above translate
into pre-fit probability distributions (priors) for nuisance parameters
affecting the global cross section and/or acceptance, the acceptance ratios
between different fit regions,777Extrapolation effects between the signal and
control regions, the resolved and merged categories, and the different lepton
channels are taken into account. and the shape of the distribution of the
final mass discriminant. For the diboson and SM $Vh$ backgrounds, only a
normalisation uncertainty is assigned, as these processes give only a
relatively small contribution to the studied phase-space regions. This
normalisation uncertainty covers effects on both the cross section and
acceptance [117]. The largest pre-fit impact on the $m_{Vh}$ distribution’s
slope for
$W/\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
comes from ME+PS variations (10%–20% over the mass range studied), from scale
uncertainties (5%–10%) and from the comparison with alternative PDF replicas
(5%). The largest pre-fit impacts on the $m_{Vh}$ distribution’s slope in
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$
simulation come from the parton shower modelling and the matching of the
matrix element to the parton shower (5%–20%), the ISR and FSR (2%–10%) and the
PDF uncertainties (up to 5%).

The multijet background gives only minor contributions (of a few per mille of
the total background) to the 1 $b$-tag signal and control regions of the
1-lepton channel. Various conservative uncertainties are assigned to the data-
driven multijet background estimate (see Section 6): an uncertainty in the
normalisation is set to $50\%$ and an uncertainty in the shape is derived from
the observed differences when varying the $E_{\text{T}}^{\text{miss}}$
selection requirement by $\pm 25\%$. This variation yields a shape uncertainty
of up to $40\%$ for the $m_{Vh}$ template. However, due to the small size of
the multijet background, its uncertainties do not have a significant impact on
the final results.

Additional uncertainties in the $p_{\mathrm{T},jj}$ reweighting of the
$W/\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
backgrounds in the 0-lepton and 2-lepton channels (see Section 6) are
estimated from the full difference between the original and reweighted
distributions. Residual mismodellings (i.e. non-closures in comparisons
between data and simulation) of the $p^{\mathrm{miss}}_{\mathrm{T}}$
distribution in the 0-lepton channel, the $p_{\mathrm{T},W}$ distribution in
the 1-lepton channel, and the transverse momentum of the leading
$\text{large-}R$ jet in all three channels, were observed in the respective
$m_{jj}$ and $m_{J}$ sideband regions. The magnitude of these mismodellings
depends on $m_{Vh}$ and ranges from $1\%$ to $10\%$. These are not corrected,
but the data-to-simulation ratios are used to define systematic uncertainties
in the relevant shapes and those are applied to all backgrounds as
uncorrelated nuisance parameters.

The relative MC generator modelling systematic uncertainties (at the pre-fit
stage), connected to the normalisation, cross-region extrapolation, and shape
of the signal and background processes are detailed in Tables 4 and 5.

Table 4:  Relative MC generator modelling systematic uncertainties (at the pre-fit stage), connected to the normalisation, cross-region extrapolation, and shape of the $W/\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$ background processes included in the fits described in the text. An ‘S’ indicates that a shape variation is included for the sources listed, ‘$A\leftrightarrow B$’ indicates an extrapolation uncertainty between the regions $A$ and $B$, and ‘norm.’ is the sum of cross-section and acceptance variations. A value of ‘float’ indicates that the parameter is not constrained in the fit. The uncertainty in the normalisation ratios of two regions, determines to what degree the normalisations of two regions can diverge in the combined fit. A range of values means the value depends on the lepton channel. Parentheses indicate when the uncertainty applies only to a given fit or a given region.  | Process | Quantity/source | Value
---|---|---
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$ | 0/2 lep. norm. | float
1-lep. norm. | 50%
|  0/2-lep. resolved $\leftrightarrow$ merged | 10%–19%
|  1-lep. resolved $\leftrightarrow$ merged |  9%–18%
|  0-lep. SR $\leftrightarrow$ CR |  5%–12%
|  1-lep. SR $\leftrightarrow$ CR |  6%–20%
|  0-lep. $\leftrightarrow$ 1-lep. |  7%–27%
|  0-lep. $\leftrightarrow$ 2-lep. |  4%–16%
| generator, PDF, scale | S
|  $p_{\mathrm{T},jj}$ reweighting | S
|  $p^{\mathrm{miss}}_{\mathrm{T}}$ non-closure (0-lep.) | S
|  $p_{\mathrm{T},W}$ non-closure (1-lep.) | S
|  $\text{Large-}R$ jet $p_{\mathrm{T}}$ non-closure | S
$Z+\mathrm{hl}$ | 0/2 lep. norm. | float
1-lep. norm. | 50%
|  0/2 lep. resolved $\leftrightarrow$ merged | 15%–28%
|  1-lep. resolved $\leftrightarrow$ merged | 12%–13%
|  0-lep. SR $\leftrightarrow$ CR |  3%–20%
|  1-lep. SR $\leftrightarrow$ CR | 5%–7%
|  0-lep. $\leftrightarrow$ 1-lep. |  7%–27%
|  0-lep. $\leftrightarrow$ 2-lep. |  6%–17%
| generator, PDF, scale | S
|  $p_{\mathrm{T},jj}$ reweighting | S
|  $p^{\mathrm{miss}}_{\mathrm{T}}$ non-closure (0-lep.) | S
|  $p_{\mathrm{T},W}$ non-closure (1-lep.) | S
|  $\text{Large-}R$ jet $p_{\mathrm{T}}$ non-closure | S
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{lf}$ | norm. | 19%
0/2 lep. resolved $\leftrightarrow$ merged |  8%–50%
|  1-lep. resolved $\leftrightarrow$ merged | 10%–29%
|  0-lep. SR $\leftrightarrow$ CR |  5%–20%
|  1-lep. SR $\leftrightarrow$ CR | 29%–99%
|  0-lep. $\leftrightarrow$ 1-lep. |  8%–39%
|  0-lep. $\leftrightarrow$ 2-lep. |  4%–17%
| generator, PDF, scale | S
|  $p_{\mathrm{T},jj}$ reweighting | S
|  $p^{\mathrm{miss}}_{\mathrm{T}}$ non-closure (0-lep.) | S
|  $p_{\mathrm{T},W}$ non-closure (1-lep.) | S
|  $\text{Large-}R$ jet $p_{\mathrm{T}}$ non-closure | S
Process | Quantity/source | Value
---|---|---
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$ | 0/2 lep. norm. | 30%
1-lep. norm. | float
|  0/2 lep. resolved $\leftrightarrow$ merged | 13%–28%
|  1-lep. resolved $\leftrightarrow$ merged | 15%–22%
|  0-lep. SR $\leftrightarrow$ CR |  5%–28%
|  1-lep. SR $\leftrightarrow$ CR | 3%
|  0-lep. $\leftrightarrow$ 1-lep. | 4%
| generator, PDF, scale | S
|  $p_{\mathrm{T},jj}$ reweighting (0-lep.) | S
|  $p^{\mathrm{miss}}_{\mathrm{T}}$ non-closure (0-lep.) | S
|  $p_{\mathrm{T},W}$ non-closure (1-lep.) | S
|  $\text{Large-}R$ jet $p_{\mathrm{T}}$ non-closure | S
| JetBinMigration (1-lep.) | S
$W+\mathrm{hl}$ | 0/2 lep. norm. | 30%
1-lep. norm. | float
|  0/2 lep. resolved $\leftrightarrow$ merged |  2%–43%
|  1-lep. resolved $\leftrightarrow$ merged | 12%–13%
|  0-lep. SR $\leftrightarrow$ CR |  2%–20%
|  1-lep. SR $\leftrightarrow$ CR | 1%–2%
|  0-lep. $\leftrightarrow$ 1-lep. | 4%
| generator, PDF, scale | S
|  $p_{\mathrm{T},jj}$ reweighting (0-lep.) | S
|  $p^{\mathrm{miss}}_{\mathrm{T}}$ non-closure (0-lep.) | S
|  $p_{\mathrm{T},W}$ non-closure (1-lep.) | S
|  $\text{Large-}R$ jet $p_{\mathrm{T}}$ non-closure | S
| JetBinMigration (1-lep.) | S
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{lf}$ | 0/2 lep. norm. | 20%
1-lep. norm. | float
|  0/2 lep. resolved $\leftrightarrow$ merged | 14%–18%
|  1-lep. resolved $\leftrightarrow$ merged | 20%–21%
|  0-lep. SR $\leftrightarrow$ CR |  4%–20%
|  1-lep. SR $\leftrightarrow$ CR | 2%–4%
| generator, PDF, scale | S
|  $p_{\mathrm{T},jj}$ reweighting (0-lep.) | S
|  $p^{\mathrm{miss}}_{\mathrm{T}}$ non-closure (0-lep.) | S
|  $p_{\mathrm{T},W}$ non-closure (1-lep.) | S
|  $\text{Large-}R$ jet $p_{\mathrm{T}}$ non-closure | S
| JetBinMigration (1-lep.) | S

Table 5:  Relative MC generator modelling systematic uncertainties (at the pre-fit stage), connected to the normalisation, cross-region extrapolation, and shape of signal and the top, SM $Vh$ and diboson background processes included in the fits described in the text. An ‘S’ indicates that a shape variation is included for the sources listed, ‘$A\leftrightarrow B$’ indicates an extrapolation uncertainty between the regions $A$ and $B$, and ‘norm.’ is the sum of cross section and acceptance variations. A value of ‘float’ indicates that the parameter is not constrained in the fit. The uncertainty in the normalisation ratios of two regions, determines to what degree the normalisations of two regions can diverge in the combined fit. A range of values means the value depends on the lepton channel. Parentheses indicate when the uncertainty applies only to a given fit or a given region.  | Process | Quantity/source | Value
---|---|---
Signal | PS, ISR/FSR, PDF | 2%–7%, S
SM $V$$h$ | 0/1/2-lep norm. | 32%
|  $p^{\mathrm{miss}}_{\mathrm{T}}$ non-closure (0-lep.) | S
|  $p_{\mathrm{T},W}$ non-closure (1-lep.) | S
|  $\text{Large-}R$ jet $p_{\mathrm{T}}$ non-closure | S
Diboson | 0/1-lep norm. | 50%
| 2-lep norm. | 20%
|  $p^{\mathrm{miss}}_{\mathrm{T}}$ non-closure (0-lep.) | S
|  $p_{\mathrm{T},W}$ non-closure (1-lep.) | S
|  $\text{Large-}R$ jet $p_{\mathrm{T}}$ non-closure | S
Process | Quantity/source | Value
---|---|---
Top quark | 0/1/2-lep. norm. | float
($b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$:
separate
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$
+hf norm.)
| single-top-quark contribution | 19%
|  $\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern 0.0mu\overline{\mkern 0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+V$ contribution | 50%
|  $\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern 0.0mu\overline{\mkern 0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+h$ contribution | 50%
|  0-lep. resolved $\leftrightarrow$ merged |  9%–20%
|  1-lep. resolved $\leftrightarrow$ merged | 18%–20%
|  2-lep. resolved $\leftrightarrow$ merged | 18%
|  0-lep. SR $\leftrightarrow$ CR |  2%–12%
|  1-lep. SR $\leftrightarrow$ CR | 2%–3%
|  2-lep. SR $\leftrightarrow$ CR | 1.2%
| PS, ISR/FSR, ME, PDF | S
|  DS vs DR scheme ($Wt$) | S
|  $p^{\mathrm{miss}}_{\mathrm{T}}$ non-closure (0-lep.) | S
|  $p_{\mathrm{T},W}$ non-closure (1-lep.) | S
|  $\text{Large-}R$ jet $p_{\mathrm{T}}$ non-closure | S

##  8 Results

In order to test for the presence of a massive resonance, the
$m_{\text{T,}Vh}$ and $m_{Vh}$ templates obtained from the signal and
background simulated event samples are fitted to data using a binned maximum-
likelihood approach based on the RooFit/RooStats framework [118, 119, 120].
Fits are simultaneously performed on the SRs and CRs defined in Table 3 in
order to constrain the normalisation of backgrounds and the nuisance
parameters (NPs) describing the systematic uncertainties detailed in Section
7. Each systematic variation enters the fit as an individual NP using a log-
normal constraint so that the expected event yields remain positive for all
values of the corresponding NPs (even for bins with low event counts). The
correlations between systematic uncertainties are maintained across regions
and processes. Fits are performed on the $m_{\text{T,}Vh}$ distribution in the
0-lepton channel and the $m_{Vh}$ distribution in the 1-lepton and 2-lepton
channels using a binning of the distributions chosen to optimise the search
sensitivity while minimising statistical fluctuations. Asimov datasets [121]
are used to evaluate the expected performance of each fit.

The results are expressed as upper limits at the $95\%$ confidence level (CL)
on the production cross section times branching fraction of $Z^{\prime}$,
$W^{\prime}$, and $A$ bosons for a wide range of resonance masses. The limits
are evaluated using a modified frequentist method known as CL${}_{\text{s}}$
[122] and the profile-likelihood-ratio test statistic in an asymptotic
approximation [121].

Different fit configurations (i.e. fit models) are used depending on the
targeted signal hypothesis, production mode, or final state:

  * •

Searches for heavy vector bosons produced via quark–antiquark annihilation:

    * –

The search for a $Z^{\prime}$ boson is based on a combined fit in the 0-lepton
and 2-lepton channels. The signal cross section is a free parameter, as are
the global normalisations of the top-quark,
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$
and $Z+\mathrm{hl}$ backgrounds. Common normalisation factors are used for the
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$
and $Z+\mathrm{hl}$ backgrounds, while the normalisation of the top-quark
backgrounds is determined separately for the 0-lepton and 2-lepton event
categories. The results of these searches are presented as one-dimensional
upper limits at $95\%$ CL on the production cross section times branching
ratio, as well as constraints on the parameter space of the HVT model.

    * –

The search for a $W^{\prime}$ boson is based on a combined fit in the 0-lepton
and 1-lepton channels. The signal cross section is a free parameter, as are
the global normalisations of the top-quark,
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$
, $Z+\mathrm{hl}$ ,
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$
and $W+\mathrm{hl}$ backgrounds. Results are presented as one-dimensional
upper limits at $95\%$ confidence level on the production cross section times
branching ratio, as well as constraints on the parameter space of the HVT
model.

  * •

Searches for a pseudoscalar $A$ boson based on combined fits in the 0-lepton
and 2-lepton channels.

    * –

The search for the gluon–gluon fusion production mode follows the same
strategy as the search for the $Z^{\prime}$ boson.

    * –

In the search for the $b$-associated production mode, the signal cross section
is a free parameter, as are the global normalisations of the top + lf,
$\mathrm{top}+\mathrm{hf}$,
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$
and $Z+\mathrm{hl}$ backgrounds. An additional normalisation factor is used
for the
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{\,+\,jets}$
background contribution in the event categories with three or more $b$-tagged
jets.

    * –

A simultaneous fit of the $ggA$ and $b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
signal hypotheses is performed for a variety of different
$\sigma_{b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A}$
and $\sigma_{ggA}$ combinations. A negative log-likelihood (NLL) value is
calculated for each $(\sigma_{b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A}$,
$\sigma_{ggA})$ combination, and the best estimate is obtained at the point
where the NLL curve reaches its minimum. Contours at $68\%$ and $95\%$ CL are
determined relative to the minimum. The background model is the same as for
the $b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
search.

    * –

A combined fit of the $ggA$ and $b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
production modes with one free parameter is performed to determine the total
cross section of the two components. The fit is performed for different
admixtures of $ggA$ and $b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
contributions, $\sigma_{b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A}/(\sigma_{b\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A}+\sigma_{ggA})$,
as well as different $A$ boson widths $\Gamma_{A}$. The results of these fits
are used to set constraints on the parameter space of the 2HDM. The background
model is the same as for the $b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
search.

The search for a $Z^{\prime}$ and $W^{\prime}$ boson seeks resonances in the
mass range from $300\leavevmode\nobreak\ \text{GeV}$ to $5\leavevmode\nobreak\
\text{TeV}$, while the search for a pseudoscalar seeks resonances in a mass
range from $220\leavevmode\nobreak\ \text{GeV}$ to $2\leavevmode\nobreak\
\text{TeV}$. In total, 19 simulated signal mass hypotheses are tested. In
addition, parameter morphing [123] is applied to the shape and normalisation
of the $m_{\mathrm{T},Vh}$ and $m_{Vh}$ distributions for all four signal
hypotheses in order to interpolate between the simulated signal mass points.

The same normalisation factors are used across the various regions entering
the likelihood fit. However, dedicated nuisance parameters exist to account
for extrapolation effects between the signal and control regions, between the
resolved and merged categories as well as between the different lepton
channels. The corresponding extrapolation uncertainties are detailed in Table
4 and Table 5. The normalisation scale factors and their uncertainties
obtained from a background-only fit in the various signal and control regions
of the combined 0- and 2-lepton channels, as well as the combined 0- and
1-lepton channels, are summarised in Table 6. In the search for the $bbA$
process, separate normalisation scale factors are used for
$\mathrm{top}+\mathrm{hf}$ and top + lf production. Separate normalisation
scale factors for top + lf are used for the 0-lepton and 2-lepton channels,
while a common normalisation scale factor is used for
$\mathrm{top}+\mathrm{hf}$ due to statistical limitations in the 3+ $b$-tags
regions of the 2-lepton channel. The normalisation scale factors are
consistent with unity for top + lf and $1.27\pm 0.15$ for
$\mathrm{top}+\mathrm{hf}$. Since the $\mathrm{top}+\mathrm{hf}$ process gives
only small contributions to the 1 $b$-tag and 2 $b$-tag signal regions, no
bias was found in the $ggA$, $Z^{\prime}$ and $W^{\prime}$ fit models, in
which $\mathrm{top}+\mathrm{hf}$ and top + lf are described by a common
normalisation scale factor.

Table 6:  Post-fit normalisation scale factors (NF) and their uncertainties obtained from a combined background-only fit to the various signal and control regions of the 0- and 2-lepton channels and from a combined background-only fit of the 0- and 1-lepton channels. Numbers are presented for the background components that are allowed to float in the likelihood fit. The top-quark background is normalised with separate NFs for the 0-, 1- and 2-lepton regions, while the NFs for the $\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$ and $Z+\mathrm{hl}$ backgrounds are correlated between the 0- and 2-lepton channels. Background | 0-lepton | 2-lepton | 1-lepton
---|---|---|---
Combined $Z^{\prime}$ /$ggA$ fit of 0- and 2-lepton channels
top | $0.92\pm 0.05$ | $1.02\pm 0.02$ | –
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$ | $1.18\pm 0.07$ | –
$Z+\mathrm{hl}$ | $1.16\pm 0.06$ | –
Combined $W^{\prime}$ fit of 0- and 1-lepton channels
top | $0.92\pm 0.04$ | – | $0.93\pm 0.04$
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$ | $1.30\pm 0.16$ | – | –
$Z+\mathrm{hl}$ | $1.30\pm 0.16$ | – | –
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$ | – | – | $1.35\pm 0.18$
$W+\mathrm{hl}$ | – | – | $1.34\pm 0.12$
Combined $b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
fit of 0- and 2-lepton channels
$\mathrm{top}+\mathrm{hf}$ | $1.27\pm 0.15$ | –
top + lf | $0.92\pm 0.11$ | $1.03\pm 0.08$ | –
$Z$ \+ jets ($N^{b\textrm{-tags}}\geq 3$) | $1.19\pm 0.15$ | –
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$ | $1.21\pm 0.07$ | –
$Z+\mathrm{hl}$ | $1.12\pm 0.06$ | –

The $m_{\text{T,}Vh}$ and $m_{Vh}$ distributions after a fit of the background
templates to data are shown in Figures 3 and 4 for the signal regions used in
the search for a $Z^{\prime}$ boson. Similar distributions are obtained from
the 2HDM $ggA$ fits, with background yields consistent within the
uncertainties. Figure 5 shows the signal region distributions in the 1-lepton
channel after the $W^{\prime}$ fit. The distributions of the $m_{\text{T,}Vh}$
and $m_{Vh}$ observables are presented in Figure 6 for events with three or
more $b$-tags (i.e. regions used in the search for the $b\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
process) after a fit of the background templates to data. Control region
distributions are presented in Figure 7.

The expected and observed event yields after fits of the backgrounds as well
as the $b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
and $W^{\prime}$ signals to data are shown in Table 7.

((a))  ((b))

((c))  ((d))

Figure 3:  Event distributions of $m_{\text{T,}Vh}$ for the 0-lepton SR (a, b)
and of $m_{Vh}$ for the 2-lepton SR (c, d) in the resolved categories of the
$Z^{\prime}$ fit. The quantity on the vertical axis is the number of data
events divided by the bin width in GeV. The term ‘Top’ summarises events from
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$,
single-top-quark,
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+h$
and
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+V$
contributions. In each plot, the last bin contains the overflow. The
background prediction is shown after a background-only maximum-likelihood fit
to the data. The lower panels show the ratio of the observed to the estimated
SM background. The background uncertainty band shows the post-fit statistical
and systematic components added in quadrature. The signal for the benchmark
HVT Model A with $m_{Z^{\prime}}=1.4$ TeV, normalised to $\sigma\times
B(Zh)\times B(h\rightarrow b\bar{b})=0.1\leavevmode\nobreak\ \mathrm{pb}$, is
shown as a dashed line.

((a))  ((b))

((c))  ((d))

Figure 4:  Event distributions of $m_{\text{T,}Vh}$ for the 0-lepton channel
(a, b) and of $m_{Vh}$ for the 2-lepton channels (c, d) in the merged
categories of the $Z^{\prime}$ fit. The quantity on the vertical axis is the
number of data events divided by the bin width in GeV. The term ‘Top’
summarises events from
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$,
single-top-quark,
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+h$
and
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+V$
contributions. In each plot, the last bin contains the overflow. The
background prediction is shown after a background-only maximum-likelihood fit
to the data. The signal for the benchmark HVT Model A with
$m_{Z^{\prime}}=1.4$ TeV, normalised to $\sigma\times B(Zh)\times
B(h\rightarrow b\bar{b})=0.1\leavevmode\nobreak\ \mathrm{pb}$, is shown as a
dashed line. The lower panels show the ratio of the observed to the estimated
SM background. The background uncertainty band shows the post-fit statistical
and systematic components added in quadrature.

((a))  ((b))

((c))  ((d))

Figure 5:  Event distributions of $m_{Vh}$ for the 1-lepton SR in the resolved
(a, b) and merged (c, d) categories of the $W^{\prime}$ fit. The quantity on
the vertical axis is the number of data events divided by the bin width in
GeV. The term ‘Top’ summarises events from
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$,
single-top-quark,
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+h$
and
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+V$
contributions. In each plot, the last bin contains the overflow. The
background prediction is shown after a background-only maximum-likelihood fit
to the data. The lower panels show the ratio of the observed to the estimated
SM background. The background uncertainty band shows the post-fit statistical
and systematic uncertainties added in quadrature. The signal for the benchmark
HVT Model A with $m_{W^{\prime}}=1.4$ TeV is shown as a dashed line and
normalised to a cross section times branching fraction of 0.1 pb.

((a))  ((b))

((c))  ((d))

Figure 6:  Event distributions of $m_{\text{T,}Vh}$ for the 0-lepton channel
(a, c) and of $m_{Vh}$ for the 2-lepton channels (b, d) in the merged and
resolved 3+ tag SR categories of the $b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
fit. The term ‘Top’ summarises events from
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$,
single-top-quark,
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+h$
and
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+V$
contributions. The quantity on the vertical axis is the number of data events
divided by the bin width in GeV. In each plot, the last bin contains the
overflow. The background prediction is shown after a background-only maximum-
likelihood fit to the data. The signal for the benchmark 2HDM model with
$m_{A}=0.7$ TeV, normalised to $\sigma\times B(Zh)\times B(h\rightarrow
b\bar{b})=0.1\leavevmode\nobreak\ \mathrm{pb}$, is shown as a dashed line. The
lower panels show the ratio of the observed to the estimated SM background.
The background uncertainty band shows the post-fit statistical and systematic
components added in quadrature.

((a))

((b))

((c))

Figure 7: Distributions of the $m_{Vh}$ and $m_{\text{T,}Vh}$ observables in
the control regions of the (a) resolved and (b) merged event categories used
for the $W^{\prime}$ search as well as (c) the control regions of the resolved
and merged event categories used in the $Z^{\prime}$ search. The term ‘Top’
summarises events from
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$,
single-top-quark,
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+h$
and
$\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}+V$
contributions. The distributions are presented after a background-only
maximum-likelihood to data. The lower panels show the ratio of the observed to
the estimated SM background. The background uncertainty band shows the post-
fit statistical and systematic uncertainties added in quadrature.

Table 7:  The observed event yields and background predictions in the various signal regions after fitting a signal and the various background templates to the data. The yields in the 0- and 2-lepton channels correspond to the $b\mathit{{\mkern 0.0mu\overline{\mkern 0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$ fit for a signal of mass 420 GeV. The yields in the 1-lepton channel are derived from the $W^{\prime}$ fit for a signal of mass $1400\leavevmode\nobreak\ \text{GeV}$. The term Top summarises events from $\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern 0.0mu\overline{\mkern 0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$, single-top-quark, $\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern 0.0mu\overline{\mkern 0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$ \+ $h$ and $\mathit{{t}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern 0.0mu\overline{\mkern 0.0mu{t}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$ +$V$ contributions. The quoted uncertainties are the statistical and systematic uncertainties combined in quadrature after the fit. The uncertainties in the individual background predictions are larger than the total background uncertainty due to correlations between the normalisation parameters in the fit.  | Resolved | Merged
---|---|---
0-lepton | 1 $b$-tag | 2 $b$-tags | 3+ $b$-tags | 1 $b$-tag | 2 $b$-tags | add. $b$-tags
Top |  404000$\pm$ | 600 |  102000$\pm$ | 200 |  11400$\pm$ | 31 |  20000$\pm$ | 100 |  71.90$\pm$ | 3.4 |  770$\pm$ | 5
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$ |  37000$\pm$ | 100 |  30000$\pm$ | 100 |  82.10$\pm$ | 3.3 |  6510$\pm$ | 32 |  2980$\pm$ | 12 |  19.20$\pm$ | 1.4
$Z+\mathrm{hl}$ |  207000$\pm$ | 400 |  1060$\pm$ | 13 |  5.50$\pm$ | 0.8 |  23000$\pm$ | 100 |  12.90$\pm$ | 0.8 |  2.50$\pm$ | 0.2
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{lf}$ |  16000$\pm$ | 200 |  5.00$\pm$ | 2.1 |  0.10$\pm$ | 0.1 |  3960$\pm$ | 34 |  0.80$\pm$ | 0.3 | < 0.1
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$ |  11400$\pm$ | 80 |  8600$\pm$ | 60 |  30.90$\pm$ | 2.1 |  2430$\pm$ | 24 |  1060$\pm$ | 11 |  3.90$\pm$ | 0.6
$W+\mathrm{hl}$ |  100000$\pm$ | 400 |  900$\pm$ | 10 |  3.60$\pm$ | 0.5 |  11000$\pm$ | 100 |  5.80$\pm$ | 0.6 |  0.40$\pm$ | 0.1
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{lf}$ |  19150$\pm$ | 270 |  130$\pm$ | 4.4 | < 0.1 |  2340$\pm$ | 31 |  0.50$\pm$ | 0.2 | < 0.1
Diboson |  4700$\pm$ | 60 |  680$\pm$ | 8 |  3.80$\pm$ | 0.4 |  2800$\pm$ | 40 |  780$\pm$ | 11 |  2.60$\pm$ | 0.3
SM $Vh$ |  2230$\pm$ | 13 |  2920$\pm$ | 16 |  3.70$\pm$ | 0.2 |  24.20$\pm$ | 1.7 |  15.20$\pm$ | 1.0 |  0.50$\pm$ | 0.1
Total |  801000$\pm$ | 220 |  146500$\pm$ | 90 |  12660$\pm$ | 21 |  72600$\pm$ | 60 |  5870$\pm$ | 13 |  1060$\pm$ | 5
Data | 80110 | 14681 | 1265 | 7260 | 584 | 105
2-lepton | 1 $b$-tag | 2 $b$-tags | 3+ $b$-tags | 1 $b$-tag | 2 $b$-tags | add. $b$-tags
Top |  81000$\pm$ | 200 |  60000$\pm$ | 200 |  1350$\pm$ | 4 |  44.80$\pm$ | 3.0 |  2.80$\pm$ | 0.2 |  79.40$\pm$ | 4.8
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$ |  232000$\pm$ | 500 |  171000$\pm$ | 100 |  3270$\pm$ | 9 |  2080$\pm$ | 7 |  980$\pm$ | 4 |  33.20$\pm$ | 2.2
$Z+\mathrm{hl}$ |  1025000$\pm$ | 1100 |  4500$\pm$ | 40 |  16.90$\pm$ | 1.9 |  7090$\pm$ | 15 |  4.60$\pm$ | 0.2 |  1570$\pm$ | 7
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{lf}$ |  68000$\pm$ | 1000 |  290$\pm$ | 11 | < 0.1 |  1540$\pm$ | 11 |  0.30$\pm$ | 0.1 |  18.10$\pm$ | 1.7
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$ |  4.90$\pm$ | 1.2 |  1.50$\pm$ | 0.1 | < 0.1 |  0.20$\pm$ | 0.0 | < 0.1 | < 0.1
$W+\mathrm{hl}$ |  250$\pm$ | 4 |  0.20$\pm$ | 0.1 | < 0.1 |  1.30$\pm$ | 0.0 | < 0.1 | < 0.1
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{lf}$ |  0.70$\pm$ | 0.1 | < 0.1 | < 0.1 |  0.30$\pm$ | 0.0 | < 0.1 | < 0.1
Diboson |  16600$\pm$ | 80 |  4800$\pm$ | 20 |  13.10$\pm$ | 0.6 |  85.50$\pm$ | 3.0 |  20.50$\pm$ | 0.9 |  10.10$\pm$ | 0.4
SM $Vh$ |  3330$\pm$ | 13 |  4020$\pm$ | 11 |  5.10$\pm$ | 0.2 |  5.10$\pm$ | 0.2 |  2.60$\pm$ | 0.1 |  0.90$\pm$ | 0.1
Total |  1427000$\pm$ | 300 |  244000$\pm$ | 100 |  4960$\pm$ | 11 |  12080$\pm$ | 16 |  1290$\pm$ | 4 |  2980$\pm$ | 7
Data | 142672 | 24371 | 505 | 1220 | 133 | 311
1-lepton | 1 $b$-tag | 2 $b$-tags | 3+ $b$-tags | 1 $b$-tag | 2 $b$-tags | add. $b$-tags
Top |  865400$\pm$ | 470 |  181740$\pm$ | 88 | – |  260700$\pm$ | 200 |  10730$\pm$ | 23 | –
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{lf}$ |  1080$\pm$ | 20 |  0.40$\pm$ | 0.1 | – |  22.60$\pm$ | 5.9 | < 0.1 | –
$Z+\mathrm{hl}$ |  5610$\pm$ | 83 |  5.20$\pm$ | 0.9 | – |  1110$\pm$ | 14 |  0.90$\pm$ | 0.1 | –
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$ |  840$\pm$ | 20 |  590$\pm$ | 12 | – |  20.90$\pm$ | 4.5 |  10.90$\pm$ | 2.4 | –
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{lf}$ |  20800$\pm$ | 230 |  0.70$\pm$ | 0.3 | – |  10110$\pm$ | 81 |  1.80$\pm$ | 0.5 | –
$W+\mathrm{hl}$ |  223300$\pm$ | 530 |  92.80$\pm$ | 6.6 | – |  56700$\pm$ | 180 |  21.10$\pm$ | 1.7 | –
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$ |  29100$\pm$ | 110 |  14570$\pm$ | 54 | – |  12190$\pm$ | 38 |  600$\pm$ | 20 | –
Diboson |  1600$\pm$ | 15 |  16.30$\pm$ | 1.4 | – |  2510$\pm$ | 19 |  32.90$\pm$ | 2.7 | –
SM $Vh$ |  1500$\pm$ | 14 |  1900$\pm$ | 17 | – |  32.60$\pm$ | 2.7 |  25.90$\pm$ | 2.2 | –
Multijet |  2980$\pm$ | 58 | < 0.1 | – | < 0.1 | < 0.1 | –
Total |  1152100$\pm$ | 280 |  199950$\pm$ | 80 | – |  344100$\pm$ | 120 |  17690$\pm$ | 27 | –
Data | 115145 | 20017 | – | 34403 | 1771 | –

The largest deviation from the Standard Model expectations is found in the
$Z^{\prime}$ and $ggA$ searches for a resonance mass of $500\;\text{GeV}$ and
corresponds to a local significance of about 2.1 standard deviations (and a
$p$-value, i.e. the probability that the background can produce a fluctuation
greater than the excess observed in data, of 0.017). The global significance
of this excess corresponds to 1.1 standard deviations. The $b\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
search displays a similar excess of 1.6 standard deviations ($p$-value of
0.059) around the same resonance mass. For all three fit models, the excess
originates mainly from the 2 $b$-tag category of the 2-lepton channel, where
small differences between the observations and SM predictions are found for
$m_{Vh}$ values around $500\;\text{GeV}$. The local significance in the
$b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
search is lower than in the $Z^{\prime}$and $ggA$ searches because the
additional signal regions used in the search for the $b\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
process do not display any excess. Another excess is found in the $Z^{\prime}$
search at a resonance mass of $2.2\;\text{TeV}$ and corresponds to a local
significance of about 2.0 standard deviations ($p$-value of 0.02). The
$W^{\prime}$ search finds mild excesses at resonance masses of
$400\;\text{GeV}$ (about 2.0 standard deviations, $p$-value of 0.02) and
$3.0\;\text{TeV}$ (about 1.7 standard deviations, $p$-value of 0.05).

The 95% CL upper limits on the production cross section for
$Z^{\prime}\rightarrow Zh$ are shown in Figure 8(a) as a function of the
resonance mass. The observed limits range from $0.9\leavevmode\nobreak\
\mathrm{pb}$ for $m_{Z^{\prime}}=300\leavevmode\nobreak\ \text{GeV}$ to
$0.3\leavevmode\nobreak\ \mathrm{fb}$ for
$m_{Z^{\prime}}=5\leavevmode\nobreak\ \text{TeV}$. These limits exclude
$Z^{\prime}$ masses below $2.8\leavevmode\nobreak\ \text{TeV}$ for HVT
benchmark Model A with coupling constant $g_{\text{V}}=1$ [12]. For Model B
with coupling constant $g_{\text{V}}=3$ [12], $Z^{\prime}$ masses below
$3.2\leavevmode\nobreak\ \text{TeV}$ are excluded. The combined limits are
dominated by the 0-lepton (2-lepton) channel for resonance masses above
(below) $800\;\text{GeV}$. The limits on the
$m_{Z^{\prime}}=5\leavevmode\nobreak\ \text{TeV}$ signal hypothesis were also
calculated using pseudo-experiments to validate the asymptotic approximation
approach in a phase-space region strongly limited by a low number of data
events. It was found that the two statistical methods agree within $20\%$.

The dominant uncertainties for low resonance masses (i.e.
$m_{Z^{\prime}}=500\;\text{GeV}$) are due to the modelling of the parton
shower in $t\bar{t}$ background events and to the QCD scale uncertainties in
the
$\mathit{{Z}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}+\mathrm{hf}$
background. At intermediate resonance masses (i.e.
$m_{Z^{\prime}}=800\;\text{GeV}$), the $\text{large-}R$ jet mass resolution
uncertainties become increasingly important and so do the uncertainties
related to the matching of the matrix element to the parton shower for the
various
$\mathit{{W}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\text{+\,jets}$
background contributions. The latter uncertainties are also the dominant
systematic uncertainties for $Z^{\prime}$ resonance masses above 2 TeV. The
statistical uncertainties of the data become dominant for resonance masses
above $600\;\text{GeV}$.

Figure 8(b) shows the 95% CL upper limits on the production cross section for
$W^{\prime}\rightarrow Wh$ as a function of the resonance mass. The limits
range from $1.3\leavevmode\nobreak\ \mathrm{pb}$ for
$m_{W^{\prime}}=400\leavevmode\nobreak\ \text{GeV}$ to
$0.4\leavevmode\nobreak\ \mathrm{fb}$ for
$m_{W^{\prime}}=5\leavevmode\nobreak\ \text{TeV}$. These limits exclude
$W^{\prime}$ masses below $2.95\leavevmode\nobreak\ \text{TeV}$ for HVT
benchmark Model A with coupling constant $g_{\text{V}}=1$ [12]. For Model B
with coupling constant $g_{\text{V}}=3$, $W^{\prime}$ masses below
$3.3\leavevmode\nobreak\ \text{TeV}$ are excluded. The combined limits are
dominated by the 1-lepton channel over the full range of tested mass
hypotheses. The limits on the $m_{W^{\prime}}=5\leavevmode\nobreak\
\text{TeV}$ signal hypothesis were also calculated using pseudo-experiments to
validate the asymptotic approximation approach in a phase-space region
strongly limited by a low number of data events. It was found that the two
statistical methods agree within $10\%$.

Figure 9(a) shows the 95% CL upper limits on the production cross section of
$ggA$ multiplied by its branching fraction to $Zh$ as a function of the
resonance mass. The observed limits range from $0.6\leavevmode\nobreak\
\mathrm{pb}$ for ${m_{A}=220\leavevmode\nobreak\ \text{GeV}}$ to
$3\leavevmode\nobreak\ \mathrm{fb}$ for $m_{A}=2\leavevmode\nobreak\
\text{TeV}$. Figure 9(b) shows the 95% CL upper limits on the production cross
section of $b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
multiplied by its branching fraction to $Zh$ as a function of the resonance
mass. The observed limits range from $0.5\leavevmode\nobreak\ \mathrm{pb}$ for
${m_{A}=220\leavevmode\nobreak\ \text{GeV}}$ to $2.5\leavevmode\nobreak\
\mathrm{fb}$ for $m_{A}=2\leavevmode\nobreak\ \text{TeV}$.

((a))  ((b))

Figure 8:  Upper limits at the 95% CL on the product of the cross section for
$\mathit{{p}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{p}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\to\
$$Z^{\prime}$ and the branching fraction to $Zh$ from the combination of the
0-lepton (0L) and 2-lepton (2L) channels (a) and on the product of the cross
section for
$\mathit{{p}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{p}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\to\
$$W^{\prime}$ and the branching fraction to $Wh$ from the combination of the
0-lepton and 1-lepton (1L) channels (b). For the $Z^{\prime}$ and $W^{\prime}$
search, the branching fraction of $h\rightarrow b\bar{b},c\bar{c}$ is assumed
to be 0.598.

((a))  ((b))

Figure 9:  Upper limits at the 95% CL on the product of the cross section for
(a) $gg\to A$ and (b) $b\bar{b}\to A$ and their respective branching fraction
to $Zh$ from the combination of the 0-lepton (0L) and 2-lepton (2L) channels.
For the $gg\to A$ search, the possible signal components of the data are
interpreted assuming pure gluon–gluon fusion production. For both searches, a
branching fraction of 0.569 for $h\rightarrow b\bar{b}$ [124] is assumed.

The upper limits on the production cross section times branching ratio
obtained from the $Z^{\prime}$ and $W^{\prime}$ searches are used to set
$95\%$ CL exclusion contours in the HVT parameter plane {$g_{\mathrm{F}}$,
$g_{\mathrm{H}}$}888The coupling constants $g_{\mathrm{H}}$ and
$g_{\mathrm{F}}$ are related to those in Ref. [12] as follows: the Higgs boson
coupling is $g_{\mathrm{H}}=g_{\mathrm{V}}c_{\mathrm{H}}$ and the universal
fermion coupling is $g_{\mathrm{F}}=g^{2}c_{\mathrm{F}}/g_{\mathrm{V}}$, where
$g$ is the SM SU(2)L gauge coupling. [125]. Exclusion contours are shown in
Figure 10 for resonance masses of $2$, $3$ and $4\,\text{TeV}$. The
constraints on $g_{\mathrm{F}}$ and $g_{\mathrm{H}}$ are stronger for large
coupling parameter values and become weaker as these coupling parameters
approach zero. This is because the resonance couplings to $Vh$ vanish as the
$g_{\mathrm{H}}$ parameter reaches zero, while for $g_{\mathrm{F}}=0$ the
$Z^{\prime}$ and $W^{\prime}$ production cross sections in the quark–antiquark
annihilation mode become zero.

Figure 10:  Observed limits on the HVT model at $95\%$ CL in the
$g_{\mathrm{F}}$ vs $g_{\mathrm{H}}$ plane for resonance masses of $2$, $3$
and $4$ TeV in the $Wh$ channel (left) and in the $Zh$ channel (right). The
circles indicate the coupling values for models A and B and the grey region
corresponds to the area of phase space where the decay width of the resonance
is no longer negligible and the signal $m_{Vh}$ shape is no longer expected to
be dominated by the experimental resolution.

Figure 11 shows the expected and observed two-dimensional likelihood scans of
the $b$-associated production cross section times branching fraction
$B(A\rightarrow Zh)$ versus the gluon–gluon fusion production cross section
times branching fraction $B(A\rightarrow Zh)$ for various pseudoscalar masses
$m_{A}$. For each mass hypothesis, the best-fit value is compatible with the
absence of a signal. The largest difference between the observed and expected
best-fit values is found for a resonance mass of
${m_{A}=500\leavevmode\nobreak\ \text{GeV}}$.

To obtain the 95% CL upper limits on the 2HDM parameters, 95% CL upper limits
on the production cross section for $A\rightarrow Zh$ are calculated for
admixtures of $ggA$ and $b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
production modes and variations of the $A$ boson natural width up to
$\Gamma_{A}/m_{A}=20\%$. For this interpretation, $m_{\text{T,}Vh}$ and
$m_{Vh}$ distributions of the simulated signal events are smeared according to
a Breit–Wigner function with the width predicted by the parameters of the
model. It was verified that this procedure produces line-shapes that are the
same as those predicted by simulation.

Figure 12 shows the interpretation of the limits on $ggA$ and
$b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A$
production in the Type-I, Type-II, Lepton-specific and Flipped 2HDM scenarios
as a function of the parameters $\tan\beta$ and $\cos(\beta-\alpha)$ for
$m_{A}=700\leavevmode\nobreak\ \text{GeV}$. In the Type-I and Lepton-specific
2HDM models, a pseudoscalar with mass $m_{A}=700\leavevmode\nobreak\
\text{GeV}$ is excluded at the 95% CL for $\tan\beta$ values as large as 10.
Greater sensitivity is observed at high $\tan\beta$ for the Type-II and
Flipped models, due to a larger cross section for $b$-quark associated
production. In the alignment limit $\cos(\beta-\alpha)\rightarrow 0$, the
$A\rightarrow Zh$ branching fraction vanishes, thus the relatively low
sensitivity around $\cos(\beta-\alpha)\approx 0$. Nevertheless,
$\cos(\beta-\alpha)$ values down to around 0.07 can be excluded for all tested
model scenarios. The narrow inclined regions with no sensitivity at low
$\tan\beta$ are caused by a vanishing $h\rightarrow bb$ branching fraction.

Figure 13 shows the interpretation of the cross-section limits as a function
of the parameters $\tan\beta$ and $m_{A}$ for $\cos(\beta-\alpha)=0.1$. The
shape of the expected exclusions is determined by the interplay of the
expected cross-section limit, which decreases as a function of $m_{A}$, and
the signal production cross section times the $A\rightarrow Zh$ branching
fraction at given $m_{A}$ and $\tan\beta$ values. This branching fraction
decreases significantly at $m_{A}=350\,\text{GeV}$ because the $A\rightarrow
t\bar{t}$ decay becomes possible, but increases again at higher $m_{A}$
values. The parameter $\tan\beta$ controls the admixture of the gluon–gluon
fusion and $b$-quark associated production, and thus affects the rate at which
the signal cross section falls as a function of $m_{A}$, leading to a varying
sensitivity as a function of $\tan\beta$. The excesses or deficits in the data
visible in Figure 9 are also reflected in Figure 13.

Figure 11: Expected and observed two-dimensional likelihood scans of the
$\mathit{{b}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$-associated
production cross section times branching fraction, $\sigma_{b\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A}\times
B(A\to Zh)$, vs the gluon–gluon fusion cross section times branching fraction,
$\sigma_{ggA}\times B(A\to Zh)$, for a given
$\mathit{{A}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
boson mass
($m_{\mathit{{A}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$).
For each mass, a variety of different cross-section hypotheses are scanned. At
each point, $2\Delta(\text{NLL})$ is calculated, defined as the negative-log-
likelihood (NLL) of the conditional fit to the Asimov and observed datasets
with $\sigma_{b\mathit{{\mkern 0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}A}$
and $\sigma_{ggA}$ fixed to their values at that point and with the minimum
NLL value at any point subtracted. The best-fit point and the preferred
$68\text{\,}\mathrm{\char 37\relax}$ and $95\text{\,}\mathrm{\char 37\relax}$
CL boundaries are found at $2\Delta(\text{NLL})$ values of 0.0, 2.30 and 5.99,
respectively.

((a))

((b))

((c))

((d))

Figure 12:  The interpretation of the cross-section limits in the context of
the various 2HDM types as a function of the parameters $\tan\beta$ and
$\cos(\beta-\alpha)$ for $m_{A}=700$ GeV: (a) Type-I , (b) Type-II, (c)
Lepton-specific, and (d) Flipped. Variations of the natural width up to
$\Gamma_{A}/m_{A}=20\%$ have been taken into account. For the interpretation
in Type-II and Flipped 2HDM, the $b$-quark associated production is included
in addition to the gluon–gluon fusion production. The exclusion limits are
only evaluated for $\tan\beta<50$ because larger $\tan\beta$ values are
disfavoured by most models (and cross-section predictions become less
reliable).

((a))

((b))

((c))

((d))

Figure 13:  The interpretation of the cross-section limits in the context of
the various 2HDM types as a function of the parameters $\tan\beta$ and $m_{A}$
for $\cos(\beta-\alpha)=0.1$: (a) Type-I , (b) Type-II, (c) Lepton-specific,
and (d) Flipped. Variations of the natural width up to $\Gamma_{A}/m_{A}=20\%$
have been taken into account. The exclusion limits are only evaluated for
$\tan\beta<50$ because larger $\tan\beta$ values are disfavoured by most
models (and cross-section predictions become less reliable).

##  9 Conclusion

A search for $Z^{\prime}$ and $W^{\prime}$ bosons and for a CP-odd Higgs boson
$A$ in the $\nu\bar{\nu}b\bar{b}$, $\ell^{+}\ell^{-}b\bar{b}$ and
$\ell^{\pm}{\nu}b\bar{b}$ final states is performed using 139 fb-1 of 13 TeV
$pp$ collision data collected with the ATLAS detector at the LHC. No
significant excess of events is observed above the SM predictions in the three
channels, and upper limits are set on the respective production cross
sections.

The 95% CL upper limits on the production cross section for
$Z^{\prime}\rightarrow Zh$ range from $0.9\leavevmode\nobreak\ \mathrm{pb}$
for $m_{Z^{\prime}}=300\leavevmode\nobreak\ \text{GeV}$ to
$0.3\leavevmode\nobreak\ \mathrm{fb}$ for
$m_{Z^{\prime}}=5\leavevmode\nobreak\ \text{TeV}$. These limits exclude
$Z^{\prime}$ masses below $2.8\leavevmode\nobreak\ \text{TeV}$ for HVT
benchmark Model A with coupling constant $g_{\text{V}}=1$. For Model B with
coupling constant $g_{\text{V}}=3$, $Z^{\prime}$ masses below
$3.2\leavevmode\nobreak\ \text{TeV}$ are excluded. The $Z^{\prime}$ cross-
section limits are also converted into constraints on the couplings in the
$g_{\mathrm{F}}$ vs $g_{\mathrm{H}}$ plane for resonance mass hypotheses of
$2$, $3$ and $4$ TeV.

The 95% CL upper limits on the production cross section for
$W^{\prime}\rightarrow Wh$ range from around $1.3\leavevmode\nobreak\
\mathrm{pb}$ for ${m_{W^{\prime}}=400\leavevmode\nobreak\ \text{GeV}}$ to
$0.4\leavevmode\nobreak\ \mathrm{fb}$ for
$m_{W^{\prime}}=5\leavevmode\nobreak\ \text{TeV}$. These limits exclude
$W^{\prime}$ masses below $2.95\leavevmode\nobreak\ \text{TeV}$ for HVT