TOP-22-009

TOP-22-009

# Inclusive and differential cross section measurements of $\ttbar\bbbar$
production in the lepton+jets channel at $\sqrt{s}=13\TeV$

###### Abstract

Measurements of inclusive and normalized differential cross sections of the
associated production of top quark-antiquark and bottom quark-antiquark pairs,
$\ttbar\bbbar$, are presented. The results are based on data from proton-
proton collisions collected by the CMS detector at a centre-of-mass energy of
13, corresponding to an integrated luminosity of 138. The cross sections are
measured in the lepton+jets decay channel of the top quark pair, using events
containing exactly one isolated electron or muon and at least five jets.
Measurements are made in four fiducial phase space regions, targeting
different aspects of the $\ttbar\bbbar$ process. Distributions are unfolded to
the particle level through maximum likelihood fits, and compared with
predictions from several event generators. The inclusive cross section
measurements of this process in the fiducial phase space regions are the most
precise to date. In most cases, the measured inclusive cross sections exceed
the predictions with the chosen generator settings. The only exception is when
using a particular choice of dynamic renormalization scale,
$\smash{\mu_{\mathrm{R}}=\frac{1}{2}\prod_{i=\PQt,\PAQt,\PQb,\PAQb}m_{\mathrm{T},i}^{1/4}}$,
where $\smash{m_{\mathrm{T},i}^{2}=m_{i}^{2}+p_{\mathrm{T},i}^{2}}$ are the
transverse masses of top and bottom quarks. The differential cross sections
show varying degrees of compatibility with the theoretical predictions, and
none of the tested generators with the chosen settings simultaneously describe
all the measured distributions.

##  0.1 Introduction

The associated production of top and bottom quark-antiquark pairs,
$\ttbar\bbbar$, in proton-proton ($\Pp\Pp$) collisions at the CERN LHC is
notoriously challenging to model because of the nonnegligible mass of the
quark, and the difference in the typical energy scales of interactions
involving top and quarks [1, 2]. Comparing predictions for $\ttbar\bbbar$
production cross sections with inclusive and differential measurements is an
important test of perturbative quantum chromodynamics (QCD) calculations.
Furthermore, $\ttbar\bbbar$ production is a leading background for searches
and other measurements, such as the measurement of the associated production
of top quark pairs with Higgs bosons ($\ttbar\PH$), where the Higgs boson
decays to a pair of quarks ($\PH\to\bbbar$) [3, 4, 5, 6, 7, 8], and
measurements of the simultaneous production of four top quarks
($\ttbar\ttbar$) [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. These two
processes provide direct access to the top quark Yukawa coupling, a crucial
parameter of the standard model (SM) [20, 21]. An improved understanding of
$\ttbar\bbbar$ production will help reduce the uncertainties in these
important measurements.

Fixed-order calculations of inclusive and differential cross sections have
been obtained at next-to-leading order (NLO) in QCD for $\ttbar\bbbar$
production [22, 23, 24, 25, 26] and for the associated production of
$\ttbar\bbbar$ with one additional jet ($\ttbar\bbbar{\HepParticle{j}{}{}}$)
[1]. In addition, full NLO QCD corrections to off-shell $\ttbar\bbbar$
production are available [27, 28]. While these fixed-order calculations
provide important insights into the dynamics of the $\ttbar\bbbar$ process,
they cannot easily be compared with measurements because of the large
extrapolations to the full parton-level phase space that this would involve.
Instead, predictions obtained by matching matrix-element (ME) generators to
parton showers (PSs) can be more directly compared to the data and can be used
to model the $\ttbar\bbbar$ background in other measurements. Such predictions
have been obtained using different modelling approaches [29, 30, 31, 32, 33,
34]. The current state of the art in simulating $\ttbar\bbbar$ production uses
ME calculations at NLO in QCD with massive quarks and matching to a PS [35,
36, 2]. These calculations with massive quarks use parton distribution
functions (PDFs) of the proton in the four-flavour scheme (4FS), where quarks
are not part of the proton PDF. In particular, in final states with gluon
splittings, $\Pg\to\bbbar$, the quark mass is taken into account. These
calculations provide a description of $\ttbar\bbbar$ production based on the
NLO MEs in the entire phase space where the pair can be resolved as one or two
quark jets.

Measurements of inclusive and differential $\ttbar\bbbar$ cross sections have
previously been performed by the ATLAS and CMS Collaborations in $\Pp\Pp$
collisions at centre-of-mass energies of 7, 8, and 13in final states with
zero, one, or two charged leptons ($\Pell=\Pe,\PGm$), using samples
corresponding to integrated luminosities of up to 41.5 [37, 38, 39, 40, 41,
42, 43, 44, 45]. To date, $\ttbar\bbbar$ simulations have shown a tendency to
underpredict the inclusive $\ttbar\bbbar$ cross sections. Normalized
differential distributions are generally in agreement with the predictions
from event generators, within experimental and theoretical uncertainties,
although the size of these uncertainties has not yet made it possible to
definitively rule out or prefer specific modelling approaches. The dominant
uncertainties in previous studies were those related to tagging calibration,
jet energy scale (JES), and the limited precision of the NLO calculations, the
choice of renormalization ($\mu_{\mathrm{R}}$) and factorization scales
($\mu_{\mathrm{F}}$).

This paper reports the measurement of inclusive and normalized differential
cross sections of $\ttbar\bbbar$ production in four fiducial phase space
regions in the lepton+jets channel. The measurement uses $\Pp\Pp$ collision
data recorded with the CMS detector at the LHC from 2016 to 2018 at
$\sqrt{s}=13\TeV$, corresponding to an integrated luminosity of 138. Top
quarks almost always decay into a boson and a quark. We consider events in
which one boson decays into a pair of quarks, and the other boson decays into
a charged lepton (electron or muon) and a neutrino. Decays of the boson into a
tau lepton decaying into an electron or muon are also implicitly considered.
After hadronization of the quarks, and including the quarks not originating
from the top quarks, the considered $\ttbar\bbbar$ events contain one electron
or muon and at least five jets, of which three result from the hadronization
of quarks (jets).

We measure inclusive cross sections and normalized differential cross
sections, where the latter correspond to the ratios of the absolute
differential cross sections to the inclusive ones. Measurements are performed
in different phase space regions, targeting distinct aspects of $\ttbar\bbbar$
production. One phase space region targets fully resolved $\ttbar\bbbar$
events (labelled as $\ttbar\bbbar$). Another, more inclusive, phase space
region requires only three jets, so as to select $\ttbar\bbbar$ events in
which the pairs of additional quarks are reconstructed as a single jet, or in
which one jet is outside of the detector acceptance (labelled as
$\ttbar\PQb$). Finally, we measure cross sections aimed at the study of
additional QCD radiation in $\ttbar\PQb$ or $\ttbar\bbbar$ events (labelled as
$\ttbar\PQb{\HepParticle{j}{}{}}$ and $\ttbar\bbbar{\HepParticle{j}{}{}}$,
respectively), as these have been shown to be sensitive to the modelling of
$\ttbar\bbbar$ production [1].

The observed distributions are unfolded to the particle level using binned
maximum likelihood fits, whereby the data from muon and electron channels from
all data-taking years are fitted simultaneously, and uncertainties are
estimated using a nuisance parameter profiling procedure. Differential cross
sections are measured for several observables, using two complementary
approaches for the event interpretation given below.

In the first approach, no attempt is made at directly identifying the
additional jets in $\ttbar\bbbar$ events in either data or simulation. The
generator-level observables are defined using stable particles exclusively,
with no reference to the simulated event history or the origin of jets from
top quark decays or QCD radiation. This ensures that the observable
definitions at the generator and detector levels are highly consistent.
Furthermore, the distributions unfolded in this way can be compared with
predictions from event generators which do not provide any information about
the origin of quarks in the final state. In the phase space region with four
jets ($\ttbar\bbbar$), observables are defined via the pair of jets with the
smallest angular separation to target jets produced through the splitting of
gluons into .

In the second approach, the jets not originating from top quark decays are
identified at the generator level using the simulated event history, and a
multivariate algorithm is developed to identify the resulting reconstructed
jets among all observed jets. This approach is more accurate at identifying
additional jets and thus these observables can be more sensitive to the
modelling of additional heavy-quark production in events, at the cost of
restricting future reinterpretations of the results and introducing additional
modelling assumptions about the parton history.

The unfolded results are compared to different predictions based on various
NLO ME calculations, interfaced with different PS simulations. Tabulated
results are provided in the HEPData record for this analysis [46].

This paper is organized as follows. Section 0.2 describes the CMS detector.
The event generation and detector simulation are detailed in Section 0.3. The
reconstruction of electrons, muons, and jets and the identification of jets,
as well as the corresponding definition of particle-level objects, are
discussed in Section 0.4. Section 0.5 describes the event selection and the
definition of the measured observables at the generator and detector levels.
In Section 0.6, the extraction of the signal, the identification of additional
jets using a multivariate algorithm, and the unfolding of the observables are
presented. The treatment of systematic uncertainties is detailed in Section
0.7, and the results, compared to several theoretical predictions, are
presented in Section 0.8. Finally, a summary of the results is provided in
Section 0.9.

##  0.2 The CMS detector

The central feature of the Compact Muon Solenoid (CMS) apparatus is a
superconducting solenoid of 6m internal diameter, providing a magnetic field
of 3.8T. Within the solenoid volume are a silicon pixel and strip tracker, a
lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and
scintillator hadron calorimeter (HCAL), each composed of a barrel and two
endcap sections. Forward calorimeters extend the pseudorapidity ($\eta$)
coverage provided by the barrel and endcap detectors. Muons are measured in
gas-ionization detectors embedded in the steel flux-return yoke outside the
solenoid. A more detailed description of the CMS detector, together with a
definition of the coordinate system used and the relevant kinematic variables,
can be found in Refs. [47, 48].

Events of interest are selected using a two-tiered trigger system. The first
level (L1), composed of custom hardware processors, uses information from the
calorimeters and muon detectors to select events at a rate of around 100kHz
within a fixed latency of about 4 [49]. The second level, known as the high-
level trigger, consists of a farm of processors running a version of the full
event reconstruction software optimized for fast processing, and reduces the
event rate to around 1kHz before data storage [50].

##  0.3 Simulated samples

Samples of simulated events, produced with Monte Carlo (MC) event generators,
are used in this analysis to estimate the contributions from background
processes, model the correspondence between the observables at the generator
and detector levels for unfolding, and compare the unfolded results with
theoretical predictions.

For all simulated samples, additional $\Pp\Pp$ interactions in the same or
neighbouring bunch crossings (pileup) are generated with (v8.240) [51] and
overlapped with the simulated hard interactions to match the pileup
multiplicity measured in data. The detector response is modelled using a
detailed simulation of the CMS detector, based on [52].

The response of the detector and event reconstruction of the $\ttbar\bbbar$
signal is modelled using a sample of $\ttbar\bbbar$ events generated using
-box-res [53] and OpenLoops [54], referred to as the nominal $\ttbar\bbbar$
sample (or powheg+ol+p8 $\ttbar\bbbar$ 4FS), where the $\ttbar\bbbar$ MEs are
calculated at NLO in QCD with massive quarks [2], and matched with for parton
showering and hadronization. The 4FS NNPDF3.1 next-to-NLO (NNLO) PDF set is
used for the description of the proton structure. The quark mass is set to
4.75and the damping parameter that regulates the damping of real emissions in
the NLO calculation when matching to the PS is set to a value of
$h_{\mathrm{damp}}=1.379m_{\PQt}$. Dynamic $\mu_{\mathrm{F}}$ and
$\mu_{\mathrm{R}}$ scales were chosen as $\mu_{\mathrm{F}}=\HT/4$ and
$\mu_{\mathrm{R}}=\frac{1}{2}\prod_{i=\PQt,\PAQt,\PQb,\PAQb}m_{\mathrm{T},i}^{1/4}$,
respectively, where $\HT=\sum_{i=\PQt,\PAQt,\PQb,\PAQb,\Pg}m_{\mathrm{T},i}$
and the transverse mass
$m_{\mathrm{T},i}=\sqrt{\smash[b]{m_{i}^{2}+p_{\mathrm{T},i}^{2}}}$, following
Ref. [1]. The choice of renormalization scale is a geometric average of top
and bottom quark transverse masses and is a natural choice for taking into
account the widely separated energy scales of both particles. The
factorization scale is based on the maximum momentum of final state radiation
that is still resummed into the PDF and hence is parameterized as the scalar
sum of transverse top and bottom quark masses.

The sensitivity of the detector response to the modelling of the
$\ttbar\bbbar$ process is evaluated using an alternative sample of
$\ttbar\bbbar$ events, obtained from the inclusive simulation with (v2)
matched to (also referred to as powheg+p8 5FS). The quarks are assumed to be
massless, and accordingly, five flavour scheme (5FS) proton PDFs (also
NNPDF3.1 NNLO) are used in the calculation. In that sample, events with one
additional quark are modelled by the ME at leading order (LO) in QCD, while
any further quarks are generated by the PS. In particular, in $\Pg\to\bbbar$
splittings only the emission of the gluon off initial-state partons or the top
quarks is described at the ME-level. Hence, for a description of
$\ttbar\bbbar$, the modelling of the splitting itself is necessarily handled
by the PS. Since it has been shown that $\Pg\to\bbbar$ splittings are the
dominant mechanism for the production of additional jets with top quark pairs,
both in $\ttbar\PQb$ and $\ttbar\bbbar$ events [2], this results in large
uncertainties from the choice of the $\mu_{\mathrm{R}}$ scale used for the
strong coupling constant in the final-state PS. Conversely, in $\ttbar\bbbar$
samples generated in the 4FS, the dominant uncertainty comes from the choice
of $\mu_{\mathrm{R}}$ scale in the MEs, while uncertainties from the PS scale
are smaller. The $\mu_{\mathrm{F}}$ and $\mu_{\mathrm{R}}$ scales in the
powheg+p8 5FS sample were set to
$\mu_{\mathrm{F}}=\mu_{\mathrm{R}}=m_{\mathrm{T},\PQt}$.

In addition to the above, we consider several alternative predictions of
$\ttbar\bbbar$ cross sections, to be compared with the measurements. The
generator settings used for the powheg+ol+p8 $\ttbar\bbbar$ 4FS and inclusive
simulation, as well as a number of alternative generator setups described
below, which are used for comparison of results.

Generator settings for different modeling approaches of $\ttbar\bbbar$
production. The top quark mass value is set to $m_{\PQt}=172.5\GeV$ for all
generator setups, and for the generator setups using massive quarks, the quark
mass value is set to $m_{\PQb}=4.75\GeV$. In the scale settings, corresponds
to the scalar sum, $\HT=\sum_{i=\PQt,\PAQt,\PQb,\PAQb,\Pg}m_{\mathrm{T},i}$,
and $m_{\mathrm{T},i}=\sqrt{\smash[b]{m_{i}^{2}+p_{\mathrm{T},i}^{2}}}$ is the
transverse mass. For generators setups using the $h_{\mathrm{damp}}$ value is
specified. Other generator setups do not use this parameter and are marked
with ().  Generator setup Process/ME order Generator/Shower Tune PDF set
$h_{\mathrm{damp}}$ Scales powheg+p8 5FS / v2/ CP5 5FS NNPDF3.1
1.379$m_{\PQt}$ $\mu_{\mathrm{F}}=\mu_{\mathrm{R}}=m_{\mathrm{T},\PQt}$ NLO
8.240 NNLO powheg+h7 5FS / v2/ CH3 5FS NNPDF3.1 1.379$m_{\PQt}$
$\mu_{\mathrm{F}}=\mu_{\mathrm{R}}=m_{\mathrm{T},\PQt}$ NLO 7.13 NNLO
powheg+ol+p8 $\ttbar\bbbar$ 4FS $\ttbar\bbbar$/ -box-res/ CP5 4FS NNPDF3.1
1.379$m_{\PQt}$
$\smash{\mu_{\mathrm{R}}=\frac{1}{2}\prod_{i=\PQt,\PAQt,\PQb,\PAQb}m_{\mathrm{T},i}^{1/4}}$,
NLO 8.240 NNLO as 0118 $\mu_{\mathrm{F}}=\HT/4$ sherpa+ol $\ttbar\bbbar$ 4FS
$\ttbar\bbbar$/ 2.2.4 4FS NNPDF3.0
$\mu_{\mathrm{R}}=\prod_{i=\PQt,\PAQt,\PQb,\PAQb}m_{\mathrm{T},i}^{1/4}$, NLO
NNLO as 0118 $\mu_{\mathrm{F}}=\HT/2$ mg5_amc+p8 $\ttbar\bbbar$ 4FS
$\ttbar\bbbar$/ v2.4.2/ CP5 4FS NNPDF3.1
$\mu_{\mathrm{F}}=\mu_{\mathrm{R}}=\sum\mT$ NLO 8.230 NNLO as 0118 mg5_amc+p8
$\ttbar{+}\textrm{jets}$ FxFx 5FS $\ttbar{+}\textrm{jets}$ FxFx/ v2.6.1/ CP5
5FS NNPDF3.1 $\mu_{\mathrm{F}}=\mu_{\mathrm{R}}=\sum\mT$, NLO [$\leq$2 jets]
8.240 NNLO $\textrm{qCut}=40\GeV$, $\textrm{qCutME}=20\GeV$

The inclusive generator is also interfaced with (v7.13) for parton showering
and hadronization [55, 56], using the CH3 underlying event tune [57] (referred
to as powheg+h7 5FS). Two other sets of simulated $\ttbar\bbbar$ events in the
4FS at NLO in QCD were obtained using (referred to as mg5_amc+p8
$\ttbar\bbbar$ 4FS), and (v2.2.4) [58, 35] with OpenLoops (referred to as
sherpa+ol $\ttbar\bbbar$ 4FS). The mg5_amc+p8 $\ttbar\bbbar$ 4FS simulation is
matched with and uses MadSpin to decay the top quarks; the $\mu_{\mathrm{F}}$
and $\mu_{\mathrm{R}}$ scales are set to the sums of the transverse masses of
all partons in the final state ($\sum\mT$). In the sherpa+ol $\ttbar\bbbar$
4FS sample, the ME and PS are matched in the scheme, the PDF set used is
NNPDF3.0 NNLO in the 4FS, and the scales are set to $\mu_{\mathrm{F}}=\HT/2$
and $\mu_{\mathrm{R}}=\prod_{i=\PQt,\PAQt,\PQb,\PAQb}m_{\mathrm{T},i}^{1/4}$.
Finally, is used to generate a sample of $\ttbar{+}\textrm{jets}$ events with
up to two additional jets described at NLO QCD in the ME calculations, merged
in the FxFx scheme [33] and using massless quarks (referred to as mg5_amc+p8
$\ttbar{+}\textrm{jets}$ FxFx 5FS). The $\mu_{\mathrm{F}}$ and
$\mu_{\mathrm{R}}$ scales are set to
$\mu_{\mathrm{F}}=\mu_{\mathrm{R}}=\sum\mT$, while the jet cutoff in the ME
calculations is set to 20and the merging scale to 40. The generator settings
of all $\ttbar\bbbar$ simulation approaches are summarized in Table 0.3.

The predictions from the exclusive $\ttbar\bbbar$ samples are normalized using
the total $\ttbar\bbbar$ cross sections obtained at NLO in QCD from the
corresponding generators, whereas the inclusive and $\ttbar{+}\textrm{jets}$
samples are normalized using the total cross section for production,
$\sigma(\ttbar)=833.9\unit{pb}$, computed at NNLO in QCD using (v2.0) [59],
including soft-gluon resummations to next-to-next-to-leading logarithmic
accuracy [60], and assuming a top quark mass of $m_{\PQt}=172.5\GeV$.

The powheg+p8 5FS sample is used for the estimation of the inclusive
background. However, using the nominal $\ttbar\bbbar$ sample for the signal in
conjunction with the inclusive sample for the backgrounds would result in a
double counting of the $\ttbar\bbbar$ contribution. This overlap is avoided by
removing from the inclusive sample events containing at least one jet not
originating from a top quark (as defined below in Section 0.4.2), with
$\pt>20\GeV$ and $\abs{\eta}<2.4$. Conversely, in the nominal $\ttbar\bbbar$
sample only the events with at least one additional jet are kept. Henceforth
the $\ttbar\PQB$ process is defined as the production of events passing these
criteria. All simulated events in our measured fiducial phase spaces (see
Section 0.5.2) are found to pass the $\ttbar\PQB$ selection criteria. The
remaining events in the inclusive samples are categorized into a
$\ttbar{\HepParticle{C}{}{}}$ contribution, with events containing at least
one charm () jet (also with $\pt>20\GeV$ and $\abs{\eta}<2.4$) for which the
simulation histories of the matched hadrons do not include any top quark, and
a $\ttbar{+}\textrm{light}$ contribution with all remaining events.

Simulated samples of minor backgrounds include single top quark production in
the $t$ and $s$ channels, as well as $\PQt\PW$ production (collectively Single
); $\ttbar\PW$, $\ttbar\PH$, and $\ttbar\PZ$ production (collectively
$\ttbar\PX$); and the production of /$\PGg^{\ast}$ or in association with jets
(collectively $\PV{+}\text{jets}$). An overview of the simulation settings
used for these backgrounds is given in Table 0.3.

Generator settings for various minor background samples simulated with [61,
62, 63, 34, 64, 65, 66] or [32]. The “Group” column refers to the grouping of
processes in the maximum likelihood fits.  Process Group ME order Generator
Notes $\PQt\PW$ Single  NLO v2 $t$ channel Single  NLO v2 MadSpin for heavy
particle decays [67] $s$ channel Single  NLO v2.6.1 $\ttbar\PH$ $\ttbar\PH$
NLO v2 $\ttbar\PZ$ $\ttbar\PV$ NLO v2.6.1 MadSpin for heavy particle decays
[67] $\ttbar\PW$ $\ttbar\PV$ NLO v2.6.1 FxFx merging up to 1 additional jet
[33] MadSpin for heavy particle decays [67] $\PW{+}\text{jets}$
$\PV{+}\text{jets}$ LO v2.6.5 MLM merging up to 4 additional jets [68]
$\PZ{+}\text{jets}$ $\PV{+}\text{jets}$ LO v2.6.5 MLM merging up to 4
additional jets [68]

For all of these minor background samples, the proton structure is described
by the NNPDF3.1 set of NNLO PDFs [69], and parton showering and hadronization
are simulated with , using the CP5 tune [70] for the underlying event
description. The value of the Higgs boson mass is assumed to be 125, while the
top quark mass value is set to $m_{\PQt}=172.5\GeV$. In the samples, the
$h_{\mathrm{damp}}$ parameter is set to a value of
$h_{\mathrm{damp}}=1.379m_{\PQt}$ as a part of the CP5 tune.

The cross sections for , $\PZ{+}\text{jets}$, $\PW{+}\text{jets}$, and single
top quark production are obtained at NNLO in QCD [59, 60, 71]. Samples for
$\ttbar\PW$, $\ttbar\PZ$, and $\ttbar\PH$ production are normalized to
predictions at NLO in QCD [32, 66].

##  0.4 Event reconstruction

###  0.4.1 Detector-level object reconstruction and identification

A particle-flow (PF) algorithm [72] is applied to reconstruct and identify
each individual particle in an event, with an optimized combination of
information from the various elements of the CMS detector. The primary vertex
(PV) is taken to be the vertex corresponding to the hardest scattering in the
event, evaluated using tracking information alone, as described in Section
9.4.1 of Ref. [73].

The energy of electrons is determined using a multivariate algorithm from a
combination of the electron momentum at the PV as determined by the tracker,
the energy of the corresponding ECAL cluster, and the energy sum of all
bremsstrahlung photons, obtained from the ECAL, spatially compatible with
originating from the electron track [74]. Electrons are identified by placing
requirements on the cluster shape in the ECAL, the track quality, and the
compatibility between the track and the ECAL cluster. Electrons from photon
conversions are rejected. The electrons used in this analysis are required to
have $\abs{\eta}<2.5$ and $\pt>29\GeV$ for data collected in 2016. In data
collected in 2017–2018, the threshold is raised to 34, except for electrons
with $\abs{\eta}<2.1$, for which the threshold is 30. These requirements are
chosen to be as low as possible so as to maximize the signal selection
efficiency while staying above the trigger thresholds in the respective data-
taking periods. Electrons with a cluster pseudorapidity
$\abs{\eta_{\mathrm{SC}}}$ between 1.444 and 1.566 are not considered, in
order to avoid the gap between the barrel and endcap ECAL sections. The
average identification efficiency for the primary electrons is about 70%,
including the isolation requirements described below. Another looser set of
requirements with 95% selection efficiency is also considered, with
$\pt>15\GeV$ and $\abs{\eta}<2.5$, and relaxed identification criteria. These
define the “veto” electrons, which are used to reject events with more than
one lepton.

The of muons is obtained from the curvature of the corresponding track,
combining information from the inner tracker and the outer muon detector
system [75]. The muons are identified based on the quality of the combined
track fit and on the number of hits in the different tracking detectors, with
an efficiency of about 95%. Muons are required to have $\abs{\eta}<2.4$ and
$\pt>26$ (29)in 2016 (2017–2018), while “veto” muons are defined with
$\pt>15\GeV$, $\abs{\eta}<2.4$, and looser identification criteria.

Electron or muon tracks are required to have a longitudinal and transverse
distance to the PV smaller than 0.5–5mm, depending on the lepton flavour and
$\abs{\eta}$. In order to suppress backgrounds from hadrons misidentified as
leptons, or from leptons produced from hadrons decaying inside of jets, the
leptons are required to be isolated from hadronic activity. The lepton
isolation is defined as the ratio between the scalar sum of all PF candidates
in a cone around the lepton excluding the lepton itself, and the lepton . The
cone size is $\DR<0.4$ (0.3) for electrons (muons). The isolation is corrected
by removing contributions from pileup [76]. The maximum primary (veto)
electron isolation varies between 0.03–0.08 (0.20–0.27), decreasing with and
increasing with $\abs{\eta_{\mathrm{SC}}}$. For primary (veto) muons, the
isolation is required to be $<$0.15 ($<$0.25). Residual differences between
lepton reconstruction, identification, and isolation efficiencies in data and
simulation are corrected. The efficiencies are measured as a function of
lepton and $\eta$ in data samples enriched in $\PZ\to\Pell^{+}\Pell^{-}$
events using a “tag-and-probe” method. For electrons (muons), the corrections
are between 1–5% ($<$2%) with uncertainties of less than 2 (1)% [74, 75].

The energy of charged hadrons is determined from a combination of their
momentum measured in the tracker and the matching ECAL and HCAL energy
deposits, corrected for the response function of the calorimeters to hadronic
showers. The energy of neutral hadrons is obtained from the corresponding
corrected ECAL and HCAL energies. Hadronic jets are reconstructed by
clustering PF candidates using the anti-algorithm [77], as implemented in the
package [78], with a distance parameter of $R=0.4$. Jet momentum is determined
as the vectorial sum of all particle momenta in the jet, and is found from
simulation to be, on average, within 5–10% of the true momentum over the whole
spectrum and detector acceptance. The jet energy resolution (JER) amounts
typically to 10 (15–20)% at 100 (30) [79]. Pileup interactions can contribute
additional tracks and calorimetric energy depositions, increasing the apparent
jet momentum. To mitigate this effect, tracks identified to be originating
from pileup vertices are discarded and an offset correction is applied to
correct for remaining contributions [76]. Jet energy corrections are derived
from simulation studies so that the average measured energy of jets becomes
identical to that of jets at generator level. In-situ measurements of the
momentum balance in dijet, $\PGg{+}\text{jet}$, $\PZ{+}\text{jet}$, and
multijet events are used to determine any residual differences between the JES
in data and in simulation, and appropriate corrections are applied to data
[79]. Additional selection criteria are applied to each jet to remove jets
potentially falsely reconstructed, with dominant contributions from
instrumental effects or reconstruction failures. A multivariate algorithm is
used to identify and remove jets likely originating from pileup interactions
[76]. Jets used in the analysis are required to have $\pt>30\GeV$ and
$\abs{\eta}<2.4$ and to be separated from the selected electron or muon by
$\DR>0.4$.

The DeepJet algorithm is used to identify jets [80, 81, 82]. This algorithm
uses a deep neural network (DNN) discriminant to combine information from
charged and neutral PF candidates clustered in the jet with features of
secondary vertices within the jet. Jets are labelled as tagged if they pass a
“medium” working point of the discriminant, corresponding to an efficiency for
correctly identifying jets in events of 75–80% and to a misidentification
probability of about 15–17% for jets and about 1.5–2% for other jets. We will
also refer to jets failing the medium working point as light-tagged jets.
Another, more restrictive, (“tight”) tagging working point is also used,
yielding a misidentification rate of 2.5–3.5 for jets and 0.15–0.25% for other
jets. This improved mistagging rate comes at the cost of reducing the jet
identification efficiency to around 60%.

The jet (mis)identification probabilities in the simulation are corrected to
match the efficiencies measured in data at both the medium and tight working
points [80]. The tagging efficiency in data for jets is obtained from a
combination of five different measurements, three of which make use of a
sample of multijet events enriched in jets, by requiring jets to contain a
muon, while the remaining two use samples enriched in events and containing
respectively one or two isolated electrons or muons. These three samples are
statistically independent of each other and with the events used in this
analysis and yield compatible measurements of the tagging efficiency. The
tagging efficiency correction factors are measured as functions of jet and
vary between 0.9–1.0 depending on the tagger working point, the data period,
and the jet , and have uncertainties of up to 10%. For jets, the
misidentification efficiency in the simulation is corrected using the same
correction factors as for jets, but with double their uncertainty. This has
been shown to cover the true jet misidentification probability [80]. For other
jets, the misidentification probability is measured as a function of jet in an
inclusive sample of multijet events. The corresponding simulation-to-data
correction factors vary between 0.6–1.9, depending on the tagger working
point, the data period, and the jet . Their uncertainties vary from 10–30%.

###  0.4.2 Particle-level object definitions

The fiducial phase space regions and observables for the inclusive and
differential cross sections reported in this paper are defined based on the
properties of stable final-state particles, with proper lifetimes
$\tau_{0}>10\mm$. Particles with $\abs{\eta}>5$ are not considered. The
definition of objects at the particle level, which follow closely those at the
detector level introduced earlier, is described in the following.

Prompt particle-level electrons are selected with $\pt>29\GeV$ and
$\abs{\eta}<2.5$ and are “dressed” with photons from final-state radiation
(FSR) by adding to their four-momentum the momentum of any photon within
$\DR<0.1$. Prompt particle-level muons are required to have $\pt>26\GeV$ and
$\abs{\eta}<2.4$. A similar FSR dressing procedure as for the electrons is
applied to the muons. Furthermore, “veto” electrons and muons are defined by
relaxing the thresholds to $\pt>15\GeV$, and are then used to reject events
with more than one lepton.

Jets are obtained by clustering all particles, excluding any neutrinos, using
the anti-algorithm with $R=0.4$. They are required to have $\pt>25\GeV$ and
$\abs{\eta}<2.4$ and are not considered if they are within $\DR<0.4$ of a
prompt electron or muon. At the particle level, the flavour of jets is defined
unambiguously by rescaling the momenta of all generated hadrons to a
negligible value and including them in the jet clustering procedure [83]. Jets
thus matched to at least one hadron are considered as jets, while all
remaining jets are labelled as light jets.

For the purpose of defining observables for the second approach introduced in
Section 0.1, and the definition of the $\ttbar\PQB$ process, we also define as
“additional” jets those particle-level jets for which the simulation histories
of the matched hadrons do not include any top quark. This definition is
therefore not uniquely based on stable particles, since it refers to the
simulated parton-level event content, which is specific to each generator and
not provided by every generator.

##  0.5 Event selection and definition of fiducial phase space regions and
observables

###  0.5.1 Event selection

Data are collected using a set of triggers requiring the presence of one
isolated electron or muon. The lepton thresholds applied by these single-
lepton triggers vary between 27–32 (24–27)for electrons (muons), depending on
the data-taking period. In addition, triggers selecting events with one
isolated electron and a scalar jet sum () above 150are used, allowing the
electron threshold to be set to 28for $\abs{\eta}<2.1$. The trigger selections
used in collision data are also applied in the simulation, and residual
differences between data and simulation are corrected. For electron triggers,
efficiencies are measured in a control region enriched with events, collected
using triggers requiring the presence of missing transverse momentum. These
present a negligible correlation with the triggers used in this analysis. Muon
trigger efficiencies are measured using the tag-and-probe technique in a
control region enriched in $\PZ\to\Pell^{+}\Pell^{-}$ events.

Data from the different data-taking periods (2016, 2017, 2018) are analyzed
separately and are only combined at the likelihood level, as will be explained
in Section 0.6.3. Additionally, data collected in 2016 are split into two
groups, the “2016preVFP” and “2016postVFP” eras, due to substantial changes in
the detector conditions between them. At the beginning of the 2016 period,
saturation effects in the readout chips of the strip tracker led to lower
signal-to-noise ratios and fewer hits on tracks. This issue was mitigated by
changing the feedback pre-amplifier bias voltage (VFP) during the 2016 data-
taking run [84].

###  0.5.2 Fiducial phase space regions

The measured cross sections are derived from the fiducial phase space regions
and observables described below. The definitions at the particle and detector
levels follow each other as closely as possible in order to minimize
extrapolations outside of the detector acceptance. Both at the particle and
detector levels, events are first required to have exactly one primary
electron or muon, and no additional “veto” electrons or “veto” muons, as
defined in Section 0.4. At the particle level, events with electrons and muons
are combined, and no requirement is placed on the decay channels of the top
quarks, so that the fiducial phase space regions can also contain
$\ttbar\bbbar$ events in which both top quarks decay leptonically, but one
lepton is outside the detector acceptance and is not selected as a “veto”
lepton. Furthermore, electrons and muons produced indirectly through the decay
of a tau lepton are included. At the detector level, events with electrons or
muons are classified into two distinct channels in this measurement.

Four different and partially overlapping phase space regions are then
considered, each targeting different aspects of $\ttbar\bbbar$ production, as
introduced in Section 0.1. The most inclusive selection considered, labelled
as $5{\HepParticle{j}{}{}}3\PQb$ (or $\geq$5 jets: $\geq$3) and targeting
$\ttbar\PQb$, requires the presence of at least five jets, of which at least
three must be (-tagged) jets at particle (detector) level. In a second
selection, labelled as $6{\HepParticle{j}{}{}}4\PQb$ (or $\geq$6 jets:
$\geq$4) and targeting $\ttbar\bbbar$, at least six jets are required, of
which at least four are (-tagged) jets. Two additional selections are defined
which help target the properties of additional light jets produced in the
event, targeting $\ttbar\PQb{\HepParticle{j}{}{}}$ and
$\ttbar\bbbar{\HepParticle{j}{}{}}$. These are the
$6{\HepParticle{j}{}{}}3\PQb 3{\HepParticle{l}{}{}}$ phase space (also
labelled $\geq$6 jets: $\geq$3, $\geq$3 light), requiring at least six jets,
including at least three (-tagged) jets and at least three light(-tagged)
jets; and the $7{\HepParticle{j}{}{}}4\PQb 3{\HepParticle{l}{}{}}$ phase space
(also labelled $\geq$7 jets: $\geq$4, $\geq$3 light), requiring at least seven
jets, including at least four (-tagged) jets and at least three light(-tagged)
jets.

###  0.5.3 Observables

As described in Section 0.1, we consider two classes of observables for
unfolding. In the first class, all observables are defined using stable
particles exclusively, without reference to any simulated event history. This
implies that no strict distinction is made between jets originating from the
decay of top quarks, or from additional radiation. In this way, there is a
close correspondence between particle- and detector-level observables (see
Sections 0.4.1 and 0.4.2), and the definition of the observables is
independent of any specific event generator. In order to probe different
features of $\ttbar\bbbar$ production, and in particular the properties of
jets produced from QCD radiation, we distinguish different categories of
objects using simple kinematic criteria.

In the $5{\HepParticle{j}{}{}}3\PQb$ phase space, we focus the measurements on
the jet with the third-largest , which at the generator level is a true
additional jet (not coming from top quark decays) in the nominal signal
simulation in approximately 49% of $5{\HepParticle{j}{}{}}3\PQb$ events. This
identification allows for the study of the properties of the additional jets
even in the case where they cannot be individually resolved. In that phase
space, we measure, each independently, the number of jets in the event
($N_{\text{jets}}$), the number of jets in the event ($N_{\PQb}$), the and
$\abs{\eta}$ of the 3rd jet ($\pt(\PQb_{3})$, $\abs{\eta(\PQb_{3})}$), the
scalar sum of all jets (), and the scalar sum of all jets ($\HT^{\PQb}$).

In the $6{\HepParticle{j}{}{}}4\PQb$ phase space where there are at least four
jets in the event, in order to measure properties that are expected to be
sensitive to the modelling of gluon splittings to , we define the “extra” jets
as the pair of jets with the smallest angular separation, defined by

$\DR_{\PQb\PQb}=\sqrt{(\Delta\phi_{\PQb\PQb})^{2}+(\Delta\eta_{\PQb\PQb})^{2}},$
(1)

where $\Delta\phi_{\PQb\PQb}$ and $\Delta\eta_{\PQb\PQb}$ are the angular
separations of the pair of jets in azimuthal angle $\phi$ (in radians) and
$\eta$, respectively. We denote that pair by $\PQb\PQb^{\text{extra}}$ in the
following. With this strategy, in the fiducial generator-level phase space,
the two $\PQb\PQb^{\text{extra}}$ jets correspond to true additional jets in
$\sim$49% of events in the $6{\HepParticle{j}{}{}}4\PQb$ phase space. Other
choices, such as picking the pairs of jets with the 3rd and 4th largest ,
result in smaller probabilities of correctly identifying the additional jets
($\sim$28%).

In the $6{\HepParticle{j}{}{}}4\PQb$ phase space, we measure, each
independently, the number of jets in the event, the and $\abs{\eta}$ of the
3rd and 4th jets ($\pt(\PQb_{3})$, $\pt(\PQb_{4})$, $\abs{\eta(\PQb_{3})}$,
$\abs{\eta(\PQb_{4})}$), the scalar jet sum (), and the scalar -jet sum
($\HT^{\PQb}$). Considering all possible pairs of distinct jets, we
independently measure the average $\DR_{\PQb\PQb}$ over all pairings
($\DR_{\PQb\PQb}^{\text{avg}}$), and the largest invariant mass
($m_{\PQb\PQb}^{\text{max}}$). For the $\PQb\PQb^{\text{extra}}$ pair, we
measure, each independently, the distance in $\eta$ between the two jets of
the pair ($\DR(\smash[t]{\PQb\PQb^{\text{extra}}})$), the invariant mass of
the pair ($m(\smash[t]{\PQb\PQb^{\text{extra}}})$), the and $\abs{\eta}$ of
the pair ($\pt(\smash[t]{\PQb\PQb^{\text{extra}}})$,
$\abs{\eta(\smash[t]{\PQb\PQb^{\text{extra}}})}$), and the and $\abs{\eta}$ of
the leading ($\PQb^{\text{extra}}_{1}$) and sub-leading
($\PQb^{\text{extra}}_{2}$) jets in the pair ($\pt(\PQb^{\text{extra}}_{1})$,
$\pt(\PQb^{\text{extra}}_{2})$, $\abs{\eta(\PQb^{\text{extra}}_{1})}$,
$\abs{\eta(\PQb^{\text{extra}}_{2})}$).

In the $6{\HepParticle{j}{}{}}3\PQb 3{\HepParticle{l}{}{}}$ and
$7{\HepParticle{j}{}{}}4\PQb 3{\HepParticle{l}{}{}}$ phase space regions, we
target the properties of the additional light jets (see Sections 0.4.1 and
0.4.2). In each of these phase space regions, we measure the scalar sum of the
light jets in the event ($\HT^{\text{light}}$). We then remove from
consideration the pair of light jets with the invariant mass closest to the
boson mass obtained from a fit to the two detector- or particle-level jets
matched to the boson decay in simulation. We then measure, independently, the
of the leading remaining light jet ($\pt(\text{lj}^{\text{extra}}_{1})$), and
the $\Delta\phi$ between that light jet and the lowest jet
($\abs{\Delta\phi(\text{lj}^{\text{extra}}_{1},\PQb_{\text{soft}})}$). The
latter variable probes the amount of recoil against additional QCD radiation
in $\ttbar\bbbar$ events that is absorbed by the softest jet [1]. In the
$6{\HepParticle{j}{}{}}3\PQb 3{\HepParticle{l}{}{}}$ and
$7{\HepParticle{j}{}{}}4\PQb 3{\HepParticle{l}{}{}}$ phase space regions, the
leading remaining light jet corresponds to a light jet not from top quark
decays in $\sim$94% of cases. The softest jet is an additional jet in $\sim$50
(65)% of cases in the $6{\HepParticle{j}{}{}}3\PQb 3{\HepParticle{l}{}{}}$
($7{\HepParticle{j}{}{}}4\PQb 3{\HepParticle{l}{}{}}$) phase space.

In the second class of observables, we focus on the
$6{\HepParticle{j}{}{}}4\PQb$ phase space and explicitly target the jets that
do not originate from decaying top quarks. At the particle level, these are
the additional jets as defined in Section 0.4.2, labelled as
$\PQb\PQb^{\text{add.}}$. In case more than two additional jets are present,
the two additional jets leading in are selected. At the detector level, the
pair of -tagged jets most consistent with the true additional jets is
identified using a DNN discriminant, described in Section 0.6.2. For the
$\PQb\PQb^{\text{add.}}$ pair defined in this way, we measure the opening
angle between the two jets of the pair ($\DR(\PQb\PQb^{\text{add.}})$), the
invariant mass of the pair ($m(\PQb\PQb^{\text{add.}})$), the and $\abs{\eta}$
of the pair ($\pt(\PQb\PQb^{\text{add.}})$,
$\abs{\eta(\PQb\PQb^{\text{add.}})}$), and the and $\abs{\eta}$ of the leading
and subleading jets in the pair ($\pt(\PQb^{\text{add.}}_{1})$,
$\pt(\PQb^{\text{add.}}_{2})$, $\abs{\eta(\PQb^{\text{add.}}_{1})}$,
$\abs{\eta(\PQb^{\text{add.}}_{2})}$).

A summary of all observables is given in Table 0.5.3. In the measurement of
each observable we also measure the inclusive cross section in the respective
phase space ($\sigma_{\text{fid}}$).

Description of all measured observables for each of the four fiducial phase
space regions. Observables marked as ($\checkmark^{\ast}$) rely on the
definition of additional jets, and do not fully correspond to the
$6{\HepParticle{j}{}{}}4\PQb$ fiducial phase space defined at the particle
level, but also require the presence of jets without top (anti)quarks in their
simulated history.  Observable $5{\HepParticle{j}{}{}}3\PQb$
$6{\HepParticle{j}{}{}}4\PQb$ $6{\HepParticle{j}{}{}}3\PQb
3{\HepParticle{l}{}{}}$ $7{\HepParticle{j}{}{}}4\PQb 3{\HepParticle{l}{}{}}$
$\sigma_{\text{fid}}$ Inclusive cross section ✓ ✓ ✓ ✓ Global observables
$N_{\text{jets}}$ Jet multiplicity ✓ ✓ $N_{\PQb}$ jet multiplicity ✓
$\HT^{{\HepParticle{j}{}{}}}$ Scalar sum of all jets ✓ ✓ $\HT^{\PQb}$ Scalar
sum of all jets ✓ ✓ $\HT^{\text{light}}$ Scalar sum of all light jets ✓ ✓
Observables related to jets $\pt(\PQb_{3})$ of third hardest jet ✓ ✓
$\abs{\eta(\PQb_{3})}$ $\abs{\eta}$ of third hardest jet ✓ ✓ $\pt(\PQb_{4})$
of fourth hardest jet ✓ $\abs{\eta(\PQb_{4})}$ $\abs{\eta}$ of fourth hardest
jet ✓ Observables considering all pairs of jets ($\PQb\PQb$)
$\DR_{\PQb\PQb}^{\text{avg}}$ Average $\DR$ of all $\PQb\PQb$ pairs ✓
$m_{\PQb\PQb}^{\text{max}}$ Highest invariant mass among all $\PQb\PQb$ pairs
✓ Observables related to the pair of jets closest in $\DR$
($\PQb\PQb^{\text{extra}}$) $\pt(\PQb^{\text{extra}}_{1})$ of leading extra
jet ✓ $\abs{\eta(\PQb^{\text{extra}}_{1})}$ $\abs{\eta}$ of leading extra jet
✓ $\pt(\PQb^{\text{extra}}_{2})$ of subleading extra jet ✓
$\abs{\eta(\PQb^{\text{extra}}_{2})}$ $\abs{\eta}$ of subleading extra jet ✓
$\DR(\smash[t]{\PQb\PQb^{\text{extra}}})$ $\DR$ of $\PQb\PQb^{\text{extra}}$
pair ✓ $\abs{\eta(\smash[t]{\PQb\PQb^{\text{extra}}})}$ $\abs{\eta}$ of
$\PQb\PQb^{\text{extra}}$ pair ✓ $m(\smash[t]{\PQb\PQb^{\text{extra}}})$
invariant mass of $\PQb\PQb^{\text{extra}}$ pair ✓
$\pt(\smash[t]{\PQb\PQb^{\text{extra}}})$ of $\PQb\PQb^{\text{extra}}$ pair ✓
Observables related to the pair of jets not from decay
($\PQb\PQb^{\text{add.}}$) $\pt(\PQb^{\text{add.}}_{1})$ of leading additional
jet $\checkmark^{\ast}$ $\abs{\eta(\PQb^{\text{add.}}_{1})}$ $\abs{\eta}$ of
leading additional jet $\checkmark^{\ast}$ $\pt(\PQb^{\text{add.}}_{2})$ of
subleading additional jet $\checkmark^{\ast}$
$\abs{\eta(\PQb^{\text{add.}}_{2})}$ $\abs{\eta}$ of subleading additional jet
$\checkmark^{\ast}$ $\DR(\PQb\PQb^{\text{add.}})$ $\DR$ of
$\PQb\PQb^{\text{add.}}$ pair $\checkmark^{\ast}$
$\abs{\eta(\PQb\PQb^{\text{add.}})}$ $\abs{\eta}$ of $\PQb\PQb^{\text{add.}}$
pair $\checkmark^{\ast}$ $m(\PQb\PQb^{\text{add.}})$ invariant mass of
$\PQb\PQb^{\text{add.}}$ pair $\checkmark^{\ast}$
$\pt(\PQb\PQb^{\text{add.}})$ of $\PQb\PQb^{\text{add.}}$ pair
$\checkmark^{\ast}$ Observables related to extra light jets
$\pt(\text{lj}^{\text{extra}}_{1})$ of leading extra light jet ✓ ✓
$\abs{\Delta\phi(\text{lj}^{\text{extra}}_{1},\PQb_{\text{soft}})}$
$\Delta\phi$ of leading extra light jet and softest jet ✓ ✓

##  0.6 Signal extraction and unfolding

In the selected regions with at least three -tagged jets, the data are highly
enriched in $\ttbar{+}\textrm{jets}$ events, which consist of about 30%
$\ttbar\PQB$, 20% $\ttbar{\HepParticle{C}{}{}}$, and 50%
$\ttbar{+}\textrm{light}$ events, with minor background contributions, in
descending order of importance, from single top quark production, $\ttbar\PH$,
$\PV{+}\text{jets}$, $\ttbar\PZ$, and $\ttbar\PW$ processes. Events from
$\ttbar\PH$, $\ttbar\PZ$, or $\ttbar\PW$ processes are referred to as a
combined $\ttbar\PX$ contribution if a distinction is not necessary. In the
other selected regions with at least four -tagged jets, the $\ttbar\PQB$
contribution is about two thirds of all events. Background contamination from
inclusive multijet production, where a jet or a nonisolated lepton is
misidentified as a primary lepton, has been verified with simulated multijet
events to be negligible at the level of the precision of this measurement
($\leq$1% in the $5{\HepParticle{j}{}{}}3\PQb$ phase space), and this
background is therefore not considered further. The prefit composition of the
events as a function of number of jets and number of -tagged jets at the
medium working point is shown in Figure 1, with the
$5{\HepParticle{j}{}{}}3\PQb$ selection applied. The events displayed in these
distributions are the superset of events used for the measurements presented
in Sec. 0.8.

Figure 1: Jet (left) and -tagged jet (right) multiplicity with the $\geq$5
jets: $\geq$3 selection prior to any fit, shown for both lepton channels and
all data periods combined. For the purpose of visualisation, the contributions
from simulation have been scaled by a common factor to match the yield in
data. The $\ttbar\PX$ contribution includes the $\ttbar\PH$, $\ttbar\PW$, and
$\ttbar\PZ$ processes. The shaded bands include all a priori uncertainties
described in Section 0.7, including the $\ttbar\PQB$ cross section uncertainty
estimated from the nominal $\ttbar\bbbar$ simulation. Only effects on the
shape of the distributions are considered. The last bins also contain the
overflow.

For each observable, the cross section is measured using a dedicated binned
maximum likelihood fit, which simultaneously yields the inclusive cross
section in the corresponding fiducial phase space and the unfolded normalized
differential cross section of the observable (Section 0.6.3). In order to
improve the separation between the signal and background components, thereby
better constraining the background contributions using the data, we use an
“ancillary” variable that divides the detector-level selections into signal-
and background-enriched categories, which are then fitted simultaneously.

###  0.6.1 Ancillary variable

We use the number of jets passing the tight tagging working point as an
ancillary variable. The distributions of the number of tight -tagged jets with
the $5{\HepParticle{j}{}{}}3\PQb$ and $6{\HepParticle{j}{}{}}4\PQb$ selections
applied are shown in Fig. 2. The regions with three or more tight -tagged jets
are highly enriched in $\ttbar\PQB$ signal events, as defined in Section 0.3,
since they typically feature three or four true jets after fiducial selection,
whereas the regions with 0–2 tight -tagged jets are dominated by
$\ttbar{\HepParticle{C}{}{}}$ or $\ttbar{+}\textrm{light}$ backgrounds where
or light jets have been misidentified as medium -tagged jets and, therefore,
serve as control regions for these backgrounds. In the fits of observables in
the $5{\HepParticle{j}{}{}}3\PQb$ and $6{\HepParticle{j}{}{}}3\PQb
3{\HepParticle{l}{}{}}$ phase space regions we use three ancillary regions
containing zero or one, two, or three or more tight -tagged jets. In the other
phase space regions we only use two ancillary regions: one containing events
with fewer than three tight -tagged jets, and another one containing events
with at least three tight -tagged jets. The use of ancillary variables is not
only useful for obtaining signal-enriched regions but also allows for the
constraint of tagging related uncertainties in the fit.

Figure 2: Number of jets tagged at the tight working point with the $\geq$5
jets: $\geq$3 (left) and $\geq$6 jets: $\geq$4 selections (right) prior to any
fit, shown for all lepton channels and years combined. For the purpose of
visualisation, the contributions from simulation have been scaled by a common
factor to match the yield in data. The shaded bands include all uncertainties
described in Section 0.7, including the $\ttbar\PQB$ cross section uncertainty
estimated from the nominal $\ttbar\bbbar$ simulation. Only effects on the
shape of the distributions are considered. The last bins also contain the
overflow. The vertical dashed lines indicate the ancillary regions.

###  0.6.2 Multivariate algorithm for jet assignment

For the eight observables in the $6{\HepParticle{j}{}{}}4\PQb$ phase space
with explicit identification of the additional jets, a multivariate algorithm
based on a DNN is used to identify the pair of -tagged jets most consistent
with the true additional jets as defined by the generator-level information.
The four -tagged jets in an event with the highest , in the following referred
to as candidate jets, are grouped into six permutations of candidate jet
pairs, depending on their ranking in . Permutations within a pair are not
treated separately. A detector-level jet is considered correctly identified as
an additional jet if it lies within an angular distance of $\DR<0.4$ to an
additional particle-level jet.

An illustration of the DNN architecture is shown in Fig. 3. The DNN makes use
of two sets of input variables, targeting jet-specific input information and
global event information separately.

Figure 3: Structural representation of the neural network used for the
assignment of the additional -jet pair. The neural network uses two sets of
input variables: global event information is connected to three dense network
layers, and jet-specific information is connected via convolutional network
layers (CNN) and a long short-term memory (LSTM) cell. The input sequences are
concatenated into one dense layer. The output layer consists of six nodes,
each representing one of the six possible candidate jet combinations.

For the input variables targeting jet-specific information, features of the
four candidate jets, an automated feature engineering is performed for each
jet using five convolutional neural network (CNN) layers [85] with filter
matrices of size 1$\times$1 [81]. The size of the filter matrices is chosen
such that they aggregate the different features, separately for each jet, into
higher-level features of the consecutive layers. These layers are followed by
a long short-term memory (LSTM) cell [86], where the independent jet features
are combined into a sequence, allowing the network to learn from the
correlations of the jet features. The five features used for each candidate
jet are the , $\eta$, a flag indicating whether it passes the tight tagging
working point, the angular separation ($\DR$) with the charged lepton, and the
invariant mass with the charged lepton. The 30 input variables targeting
global event information include the properties of the six dijet combinations
of the four candidate jets. These input variables are connected via three
dense network layers. The inputs consist of the scalar sum of the four
candidate jets, the , $\eta$, and $\phi$ of the charged lepton, the
$\Delta\phi$, $\Delta\eta$, and invariant mass of the dijet combinations, the
$\DR$ of the dijet combinations and the charged lepton, and the jet and
-tagged jet multiplicities. Both input sequences are concatenated into one
dense layer, which is connected to an output layer consisting of six nodes,
each representing one of the six possible candidate jet combinations. The pair
of -tagged jets with the highest DNN output value per event is chosen as the
additional -jet pair to be used further in the analysis.

The training of the DNN is performed with simulated events passing the
fiducial $6{\HepParticle{j}{}{}}4\PQb$ phase space definition using the keras
[87] package with a TensorFlow [88] backend and the adam optimizer [89]. In
the training, the categorical cross-entropy loss function is minimized. To
reduce biases from imbalances in the numbers of events for the different dijet
categories, events are weighted in the training such that each category has
the same number of weighted events. Potential overfitting is mitigated using a
dropout percentage of 10% [90] and an early-stopping procedure to stop the
training if no decrease in the loss minimization has been achieved for 20
epochs (iterations over the training data set) on a set of spectator events
not used for the training (validation data set). In order not to bias the
evaluation of the DNN, events from the powheg+p8 5FS simulation passing the
fiducial $6{\HepParticle{j}{}{}}4\PQb$ definition are used in the training,
while the evaluation for the measurement is performed with the nominal
$\ttbar\bbbar$ sample. It has been validated that the performance of the DNN
is independent of the choice of which of those samples of simulated
$\ttbar\bbbar$ events is used for the training or the evaluation. This implies
that the measurements of the DNN-based observables are not biased by the
choice of signal model used for training the DNN.

The performance of the DNN is evaluated in the simulation based on the
fraction of events in which it selects the correct pair of additional jets
(accuracy). The accuracy of the DNN is determined to be about 49%, which
represents a significant increase in identification accuracy compared to
choosing the two jets closest in $\DR$ (as in Section 0.5), which only yields
an accuracy of about 41%.

###  0.6.3 Unfolding methodology

The observed distributions are unfolded to the particle level by removing
estimated background contributions and correcting for resolution, acceptance
and efficiency effects of the detector and event reconstruction so that the
measured distributions can be directly compared to theoretical predictions
provided by MC event generators.

Unfolding is performed through a maximum likelihood fit by constructing, for
each observable, a statistical model that links the distributions at the
particle and detector levels. The values of the particle-level cross sections
which maximize the agreement between the predicted detector-level
distributions and the observed data are determined from the fit. In these
models, freely floating parameters of interest determine the total cross
section of the signal process in the corresponding phase space, as well as the
normalized differential cross section of the signal process in discrete bins
of the considered observable. The models are constructed using the simulated
signal and background samples described in Section 0.3 and include nuisance
parameters that model the effect of systematic uncertainties in the signal and
background predictions. The fit is performed by minimizing the negative-log-
likelihood (NLL) of the data with respect to the parameters of the model. The
calculation of the NLL and its minimization are performed using smoofit [91],
itself relying on the jax package [92] for automatic differentiation of the
NLL function, providing fast and numerically stable evaluations of the NLL
gradients and of the second derivative (Hessian) matrix. Both the central
values and confidence intervals for the absolute and normalized differential
cross sections are extracted from the fit.

The electron and muon channels as well as the four data-taking eras
(2016preVFP, 2016postVFP, 2017, and 2018) and the ancillary variable regions
are combined at the likelihood level. The likelihood used to measure the
unfolded distributions can be written as

$L(\vec{\mu},\vec{\alpha})=\left[\prod_{e,i}\text{Poi}\left(D_{e,i}\bigg{\rvert}S_{e,i}(\vec{\mu},\vec{\alpha})+\sum_{p\in\text{bkg.}}N^{p}_{e,i}(\vec{\alpha})\right)\right]\mathcal{N}(\vec{\alpha}),$
(2)

where the parameters $\vec{\mu}$ are freely-floating parameters of interest,
$\vec{\alpha}$ are profiled nuisance parameters used to model systematic
uncertainties, $D_{e,i}$ are the observed yields in data-taking era $e$ and
detector-level bin $i$ (including the lepton channels and ancillary regions),
$N^{p}_{e,i}$ are the predicted yields of background process $p$ in era $e$
and bin $i$, $S_{e,i}$ are the predicted signal yields in era $e$ and bin $i$,
$\text{Poi}(d|\nu)$ is the Poisson probability mass function for counts $d$
with mean $\nu$, and $\mathcal{N}(\vec{\alpha})$ is the Gaussian constraint
term (with mean of zero and width of one) of the nuisance parameters.

We denote by $M^{e}_{ij}$ the expected number of signal events in the fiducial
phase space in the detector-level bin $i$ and generator-level bin $j$ for the
data-taking era $e$. For every generator-level bin there are, therefore, up to

$(\text{4 eras})\times(\text{2 channels})\times(\text{2 or 3 anc.
regions})=\text{16 or 24 detector-level bins},$ (3)

depending on the phase space. In addition, the detector-level binning is
chosen to be finer than the generator-level binning, with every bin at the
detector level being about half the width of the generator-level bins (except
for discrete observables such as the number of jets or jets, and the
observables in the $7{\HepParticle{j}{}{}}4\PQb 3{\HepParticle{l}{}{}}$ region
where the data yields are the lowest). This improves the separation between
signal and background. Including other control regions in the fit or using a
finer binning at the detector level has not been found to provide any
significant further improvements in the sensitivity of the measurements. The
expected event yields in era $e$ can be written as
$\smash[b]{M^{e}_{ij}=\lumi_{e}\sigma^{0}_{j}K^{e}_{ij}}$, where
$\smash[b]{\lumi_{e}}$ is the integrated luminosity in era $e$,
$\sigma^{0}_{j}$ is the prefit cross section in bin $j$ estimated using the
nominal $\ttbar\bbbar$ sample, and $K^{e}_{ij}$ are response matrices, the
probability for a simulated event in era $e$ and generator-level bin $j$ to be
reconstructed and selected in detector-level bin $i$. The total expected
signal yields $S$ in Eq. (2) are computed as functions of the parameters of
interest as

$S_{e,i}(\vec{\mu},\vec{\alpha})=\mu_{\text{fid}}\sum_{j=1}^{n}\mu_{j}M^{e}_{ij}(\vec{\alpha}),$
(4)

where $\mu_{\text{fid}}=\sigma_{\text{fid}}/\sigma^{0}_{\text{fid}}$ is the
signal-strength modifier for the inclusive cross section and $\mu_{j}$ are the
parameters varying the fraction of signal events in each generator-level bin
$j$. To preserve unity, the yields in the last generator-level bin $n$ are not
scaled independently, but as a function of the other bins as

$\mu_{n}(\mu_{1}\dots\mu_{n-1})=\frac{1}{F_{n}}\left(1-\sum_{i=1}^{n-1}\mu_{i}F_{i}\right).$
(5)

where
$\smash[b]{F_{j}=\sigma^{0}_{j}/\sigma^{0}_{\text{fid}}=\sigma^{0}_{j}/\sum_{i=1}^{n}\sigma^{0}_{i}}$
is the a priori fractional cross section in bin $j$. In this way, the measured
inclusive cross section is directly obtained as
$\hat{\sigma}_{\text{fid}}=\hat{\mu}_{\text{fid}}\sigma^{0}_{\text{fid}}$.
Furthermore, the measured normalized differential cross section in bin $j$ is
extracted as
$1/\hat{\sigma}_{\text{fid}}\,\ddinline{\hat{\sigma}_{j}}{X}=\hat{\mu}_{j}F_{j}/w_{j}$,
where $w_{j}$ is the width of generator-level bin $j$. Thus, the yields in the
last bin $n$ are constrained through Eq. (5) to conserve the total signal
normalization for a given value of $\mu_{\text{fid}}$. The covariance matrix
of all fit parameters is obtained from the inverse Hessian of the NLL at the
minimum and is used to compute confidence intervals on the measured cross
sections. We have verified that the resulting intervals are equivalent to
those obtained by finding the level crossings of the profiled NLL, and we have
validated the frequentist coverage properties of the confidence intervals
using pseudo-experiments.

Figure 4 shows the response matrix for an example observable, both with and
without ancillary variables. For the purpose of visualization, the response
matrices shown are averaged across eras and lepton channels and are normalized
so that the values sum to 100% across each column (generator-level bin). Most
response matrices are highly diagonal, which can be explained by the close
correspondence between the particle- and detector-level definitions of the
observables. Compared to these purely particle-level based observables, there
is considerably more migration for the observables based on the additional
jets, which are defined via the simulated history at the particle level but
only identified with limited accuracy by the DNN at the detector level.

Figure 4: Response matrix for $\DR(\smash[t]{\PQb\PQb^{\text{extra}}})$ in the
$\geq$6 jets: $\geq$4 phase space. The $x$ ($y$) axes show the generator-
(detector-)level observables. The upper figure includes the ancillary
variable, unrolled on the same axis as the detector-level observable, so that
the binning of the detector-level observable, stacked vertically, is repeated
twice. For the lower figure, the ancillary variables are projected out to more
easily visualize the correspondence between true and reconstructed values. The
coloured bins show the finer binning used at reconstructed level (bins split
in two), while the numbers show the values one would obtain when using the
same binning at the generator and detector level.

Despite a careful definition of the fiducial phase space at the generator and
detector level, a fraction of the events selected at the detector level
consists of simulated $\ttbar\PQB$ events that do not pass the corresponding
generator-level fiducial phase space definitions. There is an inherent
ambiguity on how to treat this contribution, labelled as “out of acceptance”
(OOA). While technically part of the $\ttbar\bbbar$ process being measured,
these events are outside of the fiducial phase space and should not be
included in the measured cross sections. Therefore, we treat this OOA
contribution as a background and assign to it theoretical systematic
uncertainties estimated using the nominal $\ttbar\bbbar$ simulation. This OOA
contribution constitutes about 40% of selected $\ttbar\PQB$ events in the
$6{\HepParticle{j}{}{}}4\PQb$ and $7{\HepParticle{j}{}{}}4\PQb
3{\HepParticle{l}{}{}}$ regions and reduces to about 20 and 12% in the more
inclusive $6{\HepParticle{j}{}{}}3\PQb 3{\HepParticle{l}{}{}}$ and
$5{\HepParticle{j}{}{}}3\PQb$ regions, respectively.

Two distinct mechanisms explain the presence of OOA events in the detector-
level selection. First, mismeasurements of the energy or direction of leptons
or jets can cause events that are outside the generator-level phase space to
pass the detector-level selection. This class of events features a similar
topology to the fiducial signal events. Second, the misidentification of light
or jets as jets can lead to events with only two or three jets within the
fiducial acceptance to be selected in the regions requiring the presence of
three or four -tagged jets, respectively. This latter mechanism is dominant in
the two regions with four jets, $6{\HepParticle{j}{}{}}4\PQb$ and
$7{\HepParticle{j}{}{}}4\PQb 3{\HepParticle{l}{}{}}$, and explains why the OOA
contribution is larger in these regions, while the fractions of OOA events are
reduced in the signal-enriched ancillary regions. However, $\ttbar\PQB$ events
with only three jets selected within the generator-level fiducial volume
feature a distinct topology from those with four -tagged jets, since they
might either contain a soft (low-) or forward (high-$\abs{\eta}$) jet, or a
jet resulting from the hadronization of a collinear quark pair. Due to this
difference, the two OOA configurations are treated as separate background
sources in the $6{\HepParticle{j}{}{}}4\PQb$ and $7{\HepParticle{j}{}{}}4\PQb
3{\HepParticle{l}{}{}}$ regions and are labelled as $\ttbar\PQb$ OOA and
$\ttbar\bbbar$ OOA, respectively, based on the presence of one or two
“additional” jets (in the sense of Section 0.4.2).

For some of the unfolded distributions, there is no well-defined upper bound
on the measured observable. In order to include all observed events in the
measurement, while keeping the upper edge of the unfolded distributions at a
reasonable value, we normalize the differential cross sections in these bins
by the width of the bin as it appears in the histogram, while at the same time
including any overflow events, both in the data and in the simulation, into
these last bins.

In order to assess the presence of biases due to the choice of the nominal
signal model (powheg+ol+p8 $\ttbar\bbbar$ 4FS), bias tests were performed with
the powheg+p8 5FS simulation. These tests consisted of unfolding pseudo-data
generated using signal predictions from the powheg+p8 5FS simulation, while
using response matrices constructed from the powheg+ol+p8 $\ttbar\bbbar$ 4FS
samples, thereby verifying whether the differences in signal modelling and
response matrices resulted in any biases in the measurements. These studies
showed that the unfolded results were compatible with the expectations of the
powheg+p8 5FS signal prediction within systematic uncertainties, supporting
the validity of using the response matrices of the powheg+ol+p8 $\ttbar\bbbar$
4FS model for unbiased unfolding.

##  0.7 Systematic uncertainties

Systematic uncertainties are evaluated by appropriate variations of the signal
and background simulations. The uncertainty sources may affect background
yields and distributions, as well as the selection efficiency and the
kinematic distributions of the signals. These uncertainties are taken into
account via nuisance parameters in the likelihood fits.

In certain cases where the statistical uncertainty of a systematic variation
is comparable to or larger than the size of the systematic variation or there
is significant bin-to-bin variation, the systematic templates need to be
smoothed. This is done via a lowess-based smoothing algorithm, which
constrains the shape differences between the upward and downward fluctuations
if they have a shape effect (smoothing) or, if not, converts them to rate-only
effects. If, on the other hand, the systematic uncertainty in question is
determined not to have a shape or rate effect, then it is removed altogether.

A summary of all systematic uncertainties is given in Table 0.7 and in the
following sections, grouped by experimental uncertainties in Section 0.7.1 and
modelling uncertainties in Section 0.7.2.

Summary of the systematic uncertainty sources in the inclusive and
differential $\ttbar\bbbar$ cross section measurements. The first column lists
the source of the uncertainty. The second (third) column indicates the
treatment of correlations of the uncertainties between different data-taking
periods (processes), where $\checkmark$ means fully correlated, $\sim$ means
partially correlated (contains sub-sources that are either fully correlated or
uncorrelated), $\times$ means uncorrelated, and means not applicable.  Source
Corr. (period) Corr. (process) Experimental Integrated luminosity $\sim$
$\checkmark$ Pileup reweighting $\checkmark$ $\checkmark$ Electron
reconstruction and identification $\checkmark$ $\checkmark$ Muon
reconstruction and identification $\checkmark$ $\checkmark$ Trigger
efficiencies $\times$ $\checkmark$ L1 prefiring $\checkmark$ $\checkmark$ JES
$\sim$ $\checkmark$ JER $\times$ $\checkmark$ tagging $\sim$ $\checkmark$
Theoretical $\mu_{\mathrm{R}}$ scale $\checkmark$ $\sim$ $\mu_{\mathrm{F}}$
scale $\checkmark$ $\sim$ Top quark modelling $\checkmark$ $\checkmark$ PDF
$\checkmark$ $\checkmark$ PS scales: ISR $\checkmark$ $\times$ PS scales: FSR
$\checkmark$ $\times$ ME-PS matching ($h_{\mathrm{damp}}$) $\checkmark$ $\sim$
Underlying-event tune $\checkmark$ $\checkmark$ Colour reconnection
$\checkmark$ $\checkmark$ quark fragmentation $\checkmark$ $\checkmark$
Inclusive $\ttbar{\HepParticle{C}{}{}}$ cross section $\checkmark$

###  0.7.1 Experimental uncertainties

Integrated luminosity:

The integrated luminosities of the data-taking periods are individually
measured with uncertainties of 1.2, 2.3, and 2.5% for 2016, 2017, and 2018
data-taking periods, respectively [93, 94, 95]. The uncertainty in the
integrated luminosity of the combined data set is 1.6% when taking into
account the correlations between the periods.

Pileup reweighting:

The prediction of the number of pileup interactions in simulation is performed
by assuming a total inelastic $\Pp\Pp$ cross section of 69.2mb. Changes in the
assumed pileup multiplicity are estimated by varying the total inelastic cross
section by $\pm$4.6% [76]. This uncertainty is treated as fully correlated
between the data-taking periods.

Lepton reconstruction and identification:

Separate uncertainties are assigned to the corrections applied to the
reconstruction and identification efficiencies for electrons in the
simulation. Similarly, the muon tracking, identification, and isolation
efficiencies are estimated and the associated uncertainties are propagated to
signal and background distributions used in the fits. These uncertainties are
taken to be fully correlated across the data-taking periods.

Trigger efficiencies:

The trigger efficiency scale factors are varied within their uncertainties,
separately for electron and muon triggers and for each data-taking period.

L1 prefiring:

During the 2016–2017 data-taking periods, a gradual shift in the timing of the
inputs of the ECAL L1 trigger in the forward endcap region ($\abs{\eta}>2.4$)
led to a specific inefficiency, known as “prefiring”. A similar effect is
present for the muon system due to its limited time resolution, most
pronounced in 2016, but also impacting data collected in 2017–2018.
Corrections of this effect are applied to simulated events, and 20% of the
corrections are assigned as the associated uncertainties.

Jet energy scale and resolution:

Uncertainties in the determination of the JES are taken into account by
shifting the jet momenta in the simulation up and down, separately for several
sources of uncertainty such as the overall energy scale, differences in
flavour response, and residual differences between energy scale measurements.
Some of these sources are treated separately per data-taking period, while
some are correlated for all periods. Uncertainties in the JER are evaluated by
increasing or decreasing the variation of jet energies between the
reconstructed and particle levels, or by smearing the measured jet energy in
case no matching particle-level jet could be found [79]. This uncertainty is
uncorrelated between data-taking periods.

tagging:

Differences in the tagging efficiency between data and simulation are
corrected by applying correction factors to simulated events, derived as a
function of jet and $\abs{\eta}$. Systematic uncertainties are considered
separately for light jets and /jets. For and jets, nine different sources of
uncertainties from the measurements of the correction factors are considered
[80]. Included amongst these uncertainties are effects from variations in
fragmentation and gluon to splitting. Scale factors and their uncertainties on
the tagging efficiency and mistag rates are split based on the flavour of the
intiating quark, but do not discriminate based on the origin of the quark as
coming, for example, from a gluon splitting or top quark decay. Statistical
uncertainties in the scale factor measurements are treated independently for
the medium and tight tagging working points, and for the four data-taking
periods. All other sources of uncertainties are correlated between the data-
taking periods and the tagging working points.

###  0.7.2 Modelling uncertainties

This analysis is affected by uncertainties in the modelling of the background
processes and the migration matrices linking the measured particle-level
observables with the detector-level observables. Variations applied to the
signal process are defined such that the predicted yields at generator level
remain constant for all values of the corresponding nuisance parameters,
independently for each bin of the generator-level distributions. This
procedure ensures that the modelling uncertainties only have an effect on the
signal selection efficiency, due to shape variations within each generator-
level bin. All modelling uncertainties are correlated between the data-taking
eras. Some uncertainties are not correlated between processes. In the phase
space regions where the OOA contribution is separated into $\ttbar\PQb$ OOA
and $\ttbar\bbbar$ OOA contributions, the two components can have different
correlations with the signal. However, by repeating the fits with different
correlation assumptions for the signal and the OOA processes, it has been
verified that the results were not sensitive to this choice.

Renormalization and factorization scales:

Uncertainties covering the choice of $\mu_{\mathrm{R}}$ and $\mu_{\mathrm{F}}$
scales in the ME generators are considered by shifting the scales
independently up and down by a factor of two. These uncertainties are treated
separately for each process but correlated for $\ttbar{+}\textrm{light}$ and
$\ttbar{\HepParticle{C}{}{}}$, as these contributions are estimated using the
same powheg+p8 5FS simulation. Where separate $\ttbar\bbbar$ OOA and
$\ttbar\PQb$ OOA contributions are considered, the uncertainty in the former
contribution is correlated with the signal while the latter taken as
uncorrelated. For the $\ttbar\PQB$ OOA processes, shape and normalization
effects of the $\mu_{\mathrm{R}}$ scale are decorrelated, and both are
uncorrelated with the signal.

Top quark modelling:

Because of discrepancies between the observed and simulated spectrum of top
quarks in events, simulated events are reweighted to match the top quark
distribution predicted at NNLO [96]. This procedure improves the agreement of
the simulated predictions with the data. We consider as uncertainty in this
reweighting the full effect of the reweighting itself. The uncertainty is
applied to all subprocesses (including the signal and all OOA processes), but
not to single top quark or $\ttbar\PX$ production.

PDF:

The uncertainties in the PDFs are evaluated by using replicas of the NNPDF set
[69]. For the 4FS set used for the $\ttbar\bbbar$ signal simulation,
uncertainties are estimated using the root-mean-square of all residuals in the
predictions obtained using the PDF replicas, as these are defined by sampling
from the covariance matrix of the PDF fit. For the 5FS set used in all the
other simulated samples, the replicas correspond to the leading eigenvectors
of the PDF fit covariance matrix, and the uncertainties are obtained as the
quadratic sum of the residuals in the predictions across the replicas. This
uncertainty is treated as correlated for all processes. An additional
uncertainty from the value of the strong coupling constant, , used in the PDF,
is also included.

PS scales:

The uncertainty from the choice of scale at which the strong coupling constant
is evaluated in the PS is estimated by varying the scale up and down by a
factor of two, independently for initial-state radiation (ISR) and final state
radiation (FSR). These uncertainties are considered as uncorrelated for all
processes. Accordingly, the $\ttbar\PQB$ OOA processes is considered
uncorrelated with the signal. Where separate $\ttbar\bbbar$ OOA and
$\ttbar\PQb$ OOA contributions are considered, the $\ttbar\bbbar$ OOA
contribution is correlated with the signal, and $\ttbar\PQb$ OOA is treated
separately.

ME-PS matching ($h_{\mathrm{damp}}$):

In the generator, the scale that separates the phase space of the first QCD
emission into soft and hard parts is controlled by the $h_{\mathrm{damp}}$
parameter. The nominal value of the $h_{\mathrm{damp}}$ parameter in the CP5
tune [70] is $\smash[b]{h_{\mathrm{damp}}=1.379m_{\PQt}}$ and the
uncertainties are estimated with varied values of
$\smash[b]{h_{\mathrm{damp}}=2.305m_{\PQt}}$ and 0.874$m_{\PQt}$ for the and
$\ttbar\bbbar$ samples. This uncertainty is treated as correlated for
$\ttbar{+}\textrm{light}$ and $\ttbar{\HepParticle{C}{}{}}$ processes but
decorrelated from the $\ttbar\PQB$ contributions. This uncertainty is only
applied to subprocesses, which does not include single top or $\ttbar\PX$
production.

Underlying event tune:

The simulation of the underlying event is based on the CP5 tune of the event
generator, and uncertainties are estimated via varied tune settings [70],
applied to the samples. This uncertainty is only applied to $\ttbar\PQB$,
$\ttbar{+}\textrm{light}$, and $\ttbar{\HepParticle{C}{}{}}$ processes and is
treated as correlated.

Colour reconnection:

The default colour reconnection model in the PS simulation is replaced by
three alternative models [97, 98]. These uncertainties are treated as
correlated for $\ttbar\PQB$, $\ttbar{+}\textrm{light}$, and
$\ttbar{\HepParticle{C}{}{}}$ processes. This uncertainty is only applied to
subprocesses, which does not include single top or $\ttbar\PX$ production.

quark fragmentation:

We include a theoretical uncertainty for the fragmentation of the quarks into
hadrons. The fragmentation of the initiating parton into observable hadrons is
subject to modelling uncertainties. We estimate this effect by varying
simultaneously the generator- and detector-level momenta of jets up and down
by 1% in the simulated samples. This is consistent with the effect from
reasonable variations of the fragmentation functions on the jet distribution
in Ref. [99]. Residual effects on the efficiency to identify jets, due to a
possible mismodelling of the fragmentation of quarks, are already accounted
for by the calibration of the tagging efficiency in the simulation.

Inclusive $\ttbar{\HepParticle{C}{}{}}$ cross section:

For the $\ttbar{\HepParticle{C}{}{}}$ process an additional 20% normalization
uncertainty is applied, corresponding to the precision of the inclusive
$\ttbar{\HepParticle{C}{}{}}$ cross section measurement by CMS [45]. This
measurement found that the $\ttbar{\HepParticle{C}{}{}}$ cross section agreed
with the powheg+p8 5FS prediction.

No additional cross section uncertainties are considered for the backgrounds
since the uncertainties listed above already result in variations in the
predicted event yields that cover the uncertainties in the theoretical cross
sections used to normalize the background contributions.

The finite size of the simulated MC samples is taken into account as a
systematic uncertainty, following a method similar to the one proposed by
Barlow and Beeston [100, 101]. For every bin of the detector-level
distributions, a single Gaussian-constrained nuisance parameter varies the
predicted yields, summed over all processes including the signal, within their
statistical uncertainty.

The impact of a group of $k$ nuisance parameters
$\alpha_{n_{1}}\dots\alpha_{n_{k}}$ on the parameter of interest $p$ is
computed as

$I_{p}=\sqrt{\sum_{i,j=1}^{k}C_{pn_{i}}\widetilde{C}^{-1}_{n_{i}n_{j}}C_{n_{j}p}},$
(6)

where $C$ is the covariance matrix between all parameters, and $\widetilde{C}$
is the covariance matrix restricted to the parameters $n_{1}\dots n_{k}$. In
this way, the effects of a set of nuisance parameters are combined while
taking into account their correlation in the fit. The total systematic
uncertainties are calculated by considering all nuisance parameters in the sum
in Eq. (6), and the statistical uncertainties as the difference in quadrature
between the total uncertainties and the total systematic uncertainties.

Contributions of the considered sources of uncertainty to the total
uncertainty in the inclusive cross sections. For each group of uncertainty
sources, the impacts of the corresponding nuisance parameters on the total
cross section are combined, taking into account their correlation in the fit.
The numbers show relative uncertainties (in %). The statistical uncertainty is
obtained as the difference, in quadrature, between the total uncertainty and
the sum of all systematic uncertainties.  Relative uncertainty (%) Uncertainty
source $5{\HepParticle{j}{}{}}3\PQb$ $6{\HepParticle{j}{}{}}3\PQb
3{\HepParticle{l}{}{}}$ $6{\HepParticle{j}{}{}}4\PQb$
$7{\HepParticle{j}{}{}}4\PQb 3{\HepParticle{l}{}{}}$ Integrated luminosity 1.6
1.6 2.0 1.8 Pileup reweighting 0.2 0.8 0.4 0.5 Lepton and trigger 1.1 0.9 1.9
1.8 JES, JER 2.1 1.6 3.5 5.7 tagging 4.5 3.9 7.0 9.1 $\mu_{\mathrm{R}}$ and
$\mu_{\mathrm{F}}$ scales 2.8 6.8 8.2 12 Top quark modelling 0.3 1.0 0.6 1.3
PDF 0.2 0.7 1.0 1.9 PS scales 2.8 2.7 2.4 1.5 ME-PS matching
($h_{\mathrm{damp}}$) 0.4 0.9 1.3 2.8 Underlying event 0.4 $<$0.1 0.4 0.4
Colour reconnection 1.1 1.5 1.9 4.5 quark fragmentation 0.3 0.4 0.4 0.4
Inclusive $\ttbar{\HepParticle{C}{}{}}$ cross section 0.5 0.3 1.9 2.6 MC
statistical 0.8 1.6 2.4 2.8 Total systematic uncertainty 6.0 8.7 13 17
Statistical uncertainty 0.6 1.2 2.2 3.3 Total uncertainty 6.0 8.8 13 17

Table 0.7.2 shows the contributions of various sources of uncertainties to the
total uncertainties in the inclusive cross sections, obtained from the
combined impacts $I_{\mu_{\text{fid}}}$ on the parameters scaling the
inclusive cross sections in the fits of the representative observables
discussed in Section 0.8.1. For these fits, the contributions of the 20 single
nuisance parameters with the largest contributions to the uncertainty in the
inclusive cross section can be found in Appendix .10. Figure 5 shows the
effect of the sources of uncertainty on the normalized differential cross
section measurements for the of jets in the $5{\HepParticle{j}{}{}}3\PQb$
phase space. For the four representative observables, corresponding figures
are shown in Appendix .11. The uncertainties in the inclusive cross sections
are dominated by systematic sources, while the precision in the differential
measurements is mainly limited by the statistical uncertainty in the data
since the rate-based effects of many systematic uncertainties cancel out in
the ratio between the absolute differential cross sections and the inclusive
cross sections. The leading systematic uncertainties originate from the
calibration of the tagging and of the JES, the choice of $\mu_{\mathrm{R}}$
scale in the signal $\ttbar\bbbar$ and background processes, and, for the
differential measurements only, the finite number of simulated events.
Previous measurements of the inclusive $\ttbar\bbbar$ cross section by the
ATLAS [39] and CMS [42, 43, 44] Collaborations had the same leading sources of
systematic uncertainty.

Figure 5: Effect of the considered sources of uncertainties on the measurement
of the normalized differential cross section of the of jets in the $\geq$5
jets: $\geq$3 phase space, obtained by combining the impacts of associated
nuisance parameters according to Eq. (2). The ordering of the various sources
is similar for other observables and in the other phase space regions. The
last bin of the distribution is not shown, since it has no associated
parameter of interest but is constrained by the other bins as described in
Section 0.6.3. The category “other theory” includes quark fragmentation, top
quark modelling, PDF, $h_{\mathrm{damp}}$, colour reconnection, and underlying
event uncertainties. The category “other experimental” includes pileup and the
integrated luminosity uncertainties.

The nuisance parameter associated with the normalization of the
$\ttbar{\HepParticle{C}{}{}}$ process is not found to be constrained
significantly beyond its prefit expectation and also shows no significant
deviation from its expected value, which is consistent with the results of
Ref. [45]. The correlation of that nuisance parameter with the inclusive
$\ttbar\bbbar$ cross section in the fits to different observables is below 5
(20)% in the phase space regions with at least three (four) jets.

##  0.8 Results

The results are obtained with the statistical procedures described in Section
0.6.3. Inclusive and differential cross section results are presented in
Sections 0.8.1 and 0.8.2, respectively.

###  0.8.1 Inclusive cross sections

The inclusive $\ttbar\bbbar$ cross sections, measured in each of the fiducial
phase space regions, are shown in Fig. 6 and listed in Table 0.8.1, along with
the predictions obtained from the $\ttbar\bbbar$ generator setups described in
Section 0.3. In each phase space, the representative observable for which we
report the measured inclusive cross section is that for which the measured
value is closest to the mean of all measured inclusive cross sections in order
to provide a value representative of the ensemble of measurements. These
observables are $\abs{\eta(\PQb_{3})}$ in the $5{\HepParticle{j}{}{}}3\PQb$
phase space, $\HT^{\text{light}}$ in the $6{\HepParticle{j}{}{}}3\PQb
3{\HepParticle{l}{}{}}$ phase space, $\abs{\eta(\PQb^{\text{extra}}_{2})}$ in
the $6{\HepParticle{j}{}{}}4\PQb$ phase space, and
$\abs{\Delta\phi(\text{lj}^{\text{extra}}_{1},\PQb_{\text{soft}})}$ in the
$7{\HepParticle{j}{}{}}4\PQb 3{\HepParticle{l}{}{}}$ phase space. Their
observed distributions at the detector level, combined for both lepton
channels and all data-taking periods, are shown in Figs. 7 and 8 after the
fits to data. The correlations between the parameters $\vec{\mu}$ in the fit
of $\abs{\eta(\PQb_{3})}$ in the $5{\HepParticle{j}{}{}}3\PQb$ phase space are
shown in Fig. 9. For the other representative observables, corresponding
figures are shown in Appendix .12. The inclusive cross section results for all
observables are provided in the HEPData record for this analysis [46].

Figure 6: Measured inclusive cross sections for each considered phase space,
compared to predictions from different $\ttbar\bbbar$ simulation approaches
shown as coloured symbols. The predictions include uncertainties (horizontal
bars) due to the limited number of simulated events. The blue colour is
reserved for models using massive quarks and NLO QCD $\ttbar\bbbar$ MEs, while
red is used for the inclusive generators at NLO in QCD with massless quarks.
The right panel shows the ratios between the predicted and measured cross
sections, with the black bars showing the relative uncertainties in the
measurements.

Measured and predicted inclusive cross sections in the four considered phase
space regions (in fb).  Fiducial phase space $5{\HepParticle{j}{}{}}3\PQb$
$6{\HepParticle{j}{}{}}3\PQb 3{\HepParticle{l}{}{}}$
$6{\HepParticle{j}{}{}}4\PQb$ $7{\HepParticle{j}{}{}}4\PQb
3{\HepParticle{l}{}{}}$ Measured cross section 2367 1037 291 147 ${}\pm 142$
${}\pm 90$ ${}\pm 36$ ${}\pm 24$ ${}\pm 14$ ${}\pm 12$ ${}\pm 6$ ${}\pm 5$
powheg+ol+p8 $\ttbar\bbbar$ 4FS 2361 1183 361 197 $\mu_{\mathrm{R}}$ variation
$+1161$ /$-737$ $+826$ /$-433$ $+183$ /$-113$ $+121$ /$-67$ $\mu_{\mathrm{F}}$
variation $+126$ /$-100$ $+97$ /$-78$ $+23$ /$-18$ $+16$ /$-13$ powheg+p8 5FS
1791 899 240 129 powheg+h7 5FS 1665 762 197 95 sherpa+ol $\ttbar\bbbar$ 4FS
1391 677 216 116 mg5_amc+p8 $\ttbar\bbbar$ 4FS 1024 524 187 101 mg5_amc+p8
$\ttbar{+}\textrm{jets}$ FxFx 5FS 1560 712 203 101

Figure 7: The $\abs{\eta}$ of the third-hardest jet in
($\abs{\eta(\PQb_{3})}$) in the $\geq$5 jets: $\geq$3 phase space (upper) and
the $\abs{\eta}$ of the subleading additional jet
($\abs{\eta(\PQb^{\text{extra}}_{2})}$) in the $\geq$6 jets: $\geq$4 phase
space (lower) after the fit to data, shown for both lepton channels and all
data periods combined. The distributions are shown separately for each
ancillary region, as defined in Section 0.6.1. In the $\geq$5 jets: $\geq$3
($\geq$6 jets: $\geq$4) phase space the ancillary regions are defined as
$\leq$2, 2, and $\geq$3 ($\leq$3 and $\geq$3) tight -tagged jets. The shaded
bands include all uncertainties described in Section 0.7 after profiling the
nuisance parameters in the fit, estimated by sampling the predicted yields
from the fit covariance matrix. The blue line shows the sum of the predicted
yields for all processes before the fit to data, using the nominal
$\ttbar\bbbar$ samples and its corresponding cross section for the signal. In
the ratio panel the expected yields before the fit to data are shown relative
to the predicted yields after the fit to data. The last bins contain the
overflow.

Figure 8: The of all light jets in the $\geq$6 jets: $\geq$3, $\geq$3 light
phase space (upper) and the azimuthal angle between the hardest remaining
light jet and the softest jet
($\abs{\Delta\phi(\text{lj}^{\text{extra}}_{1},\PQb_{\text{soft}})}$) in the
$\geq$7 jets: $\geq$4, $\geq$3 light phase space (lower) after the fit to
data, shown for both lepton channels and all data periods combined. The
distributions are shown separately for each ancillary region, as defined in
Section 0.6.1. In the $\geq$6 jets: $\geq$3, $\geq$3 light ($\geq$7 jets:
$\geq$4, $\geq$3 light) phase space the ancillary regions are defined as
$\leq$2, 2, and $\geq$3 ($\leq$3 and $\geq$3) tight -tagged jets. The shaded
bands include all uncertainties described in Section 0.7 after profiling the
nuisance parameters in the fit, estimated by sampling the predicted yields
from the fit covariance matrix. The blue line shows the sum of the predicted
yields for all processes before the fit to data, using the nominal
$\ttbar\bbbar$ samples and its corresponding cross section for the signal. In
the ratio panel the expected yields before the fit to data are shown relative
to the predicted yields after the fit to data. The last bins contain the
overflow.  Figure 9: Correlations between the parameters of interest
$\vec{\mu}$ in the fit for $\abs{\eta(\PQb_{3})}$ in the $\geq$5 jets: $\geq$3
phase space.

The measured cross sections in all phase space regions are larger than the
theoretical predictions, as was also observed in previous measurements of the
inclusive $\ttbar\bbbar$ cross section (with somewhat different fiducial
definitions) by the CMS [42, 43, 44] and ATLAS [39] Collaborations, with the
notable exception of the powheg+ol+p8 $\ttbar\bbbar$ 4FS generator setup. The
choice of reduced central $\mu_{\mathrm{R}}$ and $\mu_{\mathrm{F}}$ scales in
the powheg+ol+p8 $\ttbar\bbbar$ 4FS sample, justified by the previous
measurements and by studies of fixed-order NLO QCD corrections to the
$\ttbar\bbbar{\HepParticle{j}{}{}}$ process [1], results in significantly
larger cross sections in all phase space regions. This prediction agrees well
with the measurement in the $5{\HepParticle{j}{}{}}3\PQb$ phase space, but
overestimates the cross section by 10–35% in the other phase space regions.
Nevertheless, the powheg+ol+p8 $\ttbar\bbbar$ 4FS predictions agree with the
measurements when considering their $\mu_{\mathrm{R}}$ scale uncertainties of
about 50%, estimated by varying the $\mu_{\mathrm{R}}$ scale by a factor of
two. When considering $\mu_{\mathrm{R}}$ scale uncertainties in both the ME
and in the PS, the powheg+p8 5FS predictions agree with the measurements in
the phase space regions targeting additional light radiation in $\ttbar\PQb$
and $\ttbar\bbbar$ events.

###  0.8.2 Normalized differential cross sections

The normalized differential cross section is measured in four different
fiducial phase space regions for 29 observables that use exclusively stable-
particle level information without reference to any simulated event history,
and eight observables targeting explicitly the jets that do not originate from
decaying top quarks. For each observable, customized bin sizes are chosen,
depending on the resolution of the observables and the statistical uncertainty
in the measured event yields.

The resulting normalized differential cross sections are shown in Fig. 10 for
the observables of the $5{\HepParticle{j}{}{}}3\PQb$ phase space, in Figs.
11–13 for the observables of the $6{\HepParticle{j}{}{}}4\PQb$ phase space,
for the observables targeting the jets that do not originate from decaying top
quarks in Figs. 14 and 15, and finally in Fig. 16 for the observables of the
$6{\HepParticle{j}{}{}}3\PQb 3{\HepParticle{l}{}{}}$ and
$7{\HepParticle{j}{}{}}4\PQb 3{\HepParticle{l}{}{}}$ phase space regions. The
measurements are compared to six cross section predictions of the
$\ttbar\bbbar$ process, obtained at the particle level, produced with the
different combinations of event generators and PSs introduced in Section 0.3.
The various predictions are shown as symbols distinguished by colour and
shape. It should be noted that predictions from sherpa+ol $\ttbar\bbbar$ 4FS
cannot be compared to the observables related to the additional jets, since
that generator does not provide the necessary information (parton-level top
quarks) to assign jets to the decaying top quarks.

The compatibility of the predictions with the unfolded data in each of the
phase space regions is quantified using $\chi^{2}$ tests, which are converted
to $z$ scores [102], which quantify the $p$-value in terms of the equivalent
number of standard deviations by which each theoretical prediction differs
from the mean of a normal distribution centered at the normalized cross-
section measurements. These are shown in Fig. 17 and as tables in Appendix
.14. The $\chi^{2}$ test statistics are computed using the predicted and
measured normalized cross sections, as well as the estimated covariance
between the bins of the unfolded distributions, but do not include the
inclusive cross sections. No uncertainties in the predictions are considered
in these tests. Varying levels of agreement between the predicted and measured
distributions are observed, whereby no event generator setup describes all
observables well since they each result in $z\geq 2$ for several observables.

Figure 10: Predicted and observed normalized differential cross sections in
the $\geq$5 jets: $\geq$3 fiducial phase space, for the inclusive jet
multiplicity (upper left), the jet multiplicity (upper right), the inclusive
jet (middle left, $\HT^{{\HepParticle{j}{}{}}}$), the of jets (middle right,
$\HT^{\PQb}$), the $\abs{\eta}$ of the third jet (lower left), and the of the
third jet (lower right). The data are represented by points, with inner
(outer) vertical bars indicating the systematic (total) uncertainties, also
represented as blue (grey) bands. Cross section prediction obtained at the
particle level from different simulation approaches are shown, including their
statistical uncertainties, as coloured symbols with different shapes. For
$N_{\text{jets}}$, $N_{\PQb}$, $\HT^{{\HepParticle{j}{}{}}}$, $\HT^{\PQb}$,
and $\pt(\PQb_{3})$, the last bins contain the overflow.

Figure 11: Predicted and observed normalized differential cross sections in
the $\geq$6 jets: $\geq$4 fiducial phase space, for the inclusive jet (upper
left), the of jets (upper right), the $\abs{\eta}$ of the third jet (middle
left), the of the third jet (middle right), the $\abs{\eta}$ of the fourth jet
(lower left), and the of the fourth jet (lower right). The data are
represented by points, with inner (outer) vertical bars indicating the
systematic (total) uncertainties, also represented as blue (grey) bands. Cross
section predictions obtained at the particle level from different simulation
approaches are shown, including their statistical uncertainties, as coloured
symbols with different shapes. For and , the last bins contain the overflow.

Figure 12: Predicted and observed normalized differential cross sections in
the $\geq$6 jets: $\geq$4 fiducial phase space, for the average $\DR$ of all
possible $\PQb\PQb$ pairs (upper left), the largest invariant mass of any
$\PQb\PQb$ pair (upper right), the invariant mass (middle left), $\DR$ (middle
right), (lower left), and $\abs{\eta}$ (lower right) of the extra -jet pair.
The data are represented by points, with inner (outer) vertical bars
indicating the systematic (total) uncertainties, also represented as blue
(grey) bands. Cross section predictions obtained at the particle level from
different simulation approaches are shown, including their statistical
uncertainties, as coloured symbols. For $m_{\PQb\PQb}$ and , the last bins
contain the overflow.

Figure 13: Predicted and observed normalized differential cross sections in
the $\geq$6 jets: $\geq$4 fiducial phase space, for the $\abs{\eta}$ (upper
left) and (upper right) of the first extra jet, the $\abs{\eta}$ (middle left)
and (middle right) of the second extra jet, and the inclusive jet multiplicity
(lower left). The data are represented by points, with inner (outer) vertical
bars indicating the systematic (total) uncertainties, also represented as blue
(grey) bands. Cross section predictions obtained at the particle level from
different simulation approaches are shown, including their statistical
uncertainties, as coloured symbols. For $N_{\text{jets}}$ and , the last bins
contain the overflow.

Figure 14: Predicted and observed normalized differential cross sections in
the $\geq$6 jets: $\geq$4 fiducial phase space, for the invariant mass (upper
left), $\DR$ (upper right), (lower left), and $\abs{\eta}$ (lower right) of
the additional -jet pair not originating from decaying top quarks. The data
are represented by points, with inner (outer) vertical bars indicating the
systematic (total) uncertainties, also represented as blue (grey) bands. Cross
section predictions obtained at the particle level from different simulation
approaches are shown, including their statistical uncertainties, as coloured
symbols. For $m_{\PQb\PQb}$ and , the last bins contain the overflow.

Figure 15: Predicted and observed normalized differential cross sections in
the $\geq$6 jets: $\geq$4 fiducial phase space, for the $\abs{\eta}$ (left)
and (right) of the first (upper row) and second (lower row) additional of the
-jet pair not originating from decaying top quarks. The data are represented
by points, with inner (outer) vertical bars indicating the systematic (total)
uncertainties, also represented as blue (grey) bands. Cross section
predictions obtained at the particle level from different simulation
approaches are shown, including their statistical uncertainties, as coloured
symbols. For , the last bins contain the overflow.

Figure 16: Predicted and observed normalized differential cross sections in
the $\geq$6 jets: $\geq$3, $\geq$3 light (left) and $\geq$7 jets: $\geq$4,
$\geq$3 light (right) fiducial phase space regions, for the of light jets
(upper row), the of the extra light jet (middle row), and the $\Delta\phi$
between the extra light jet and the softest jet (lower row). The data are
represented by points, with inner (outer) vertical bars indicating the
systematic (total) uncertainties, also represented as blue (grey) bands. Cross
section predictions obtained at the particle level from different simulation
approaches are shown, including their statistical uncertainties, as coloured
symbols with different shapes. For and , the last bins contain the overflow.
Figure 17: Observed $z$ score for each of the theoretical predictions, given
the unfolded normalized differential cross sections and their covariances. A
lower value indicates a better agreement between prediction and measurement.
The dashed line at $z=2$ indicates a $p$-value of 5%. Predictions for which
the $z$ score exceeds the visible range of the figure are marked with arrows
($\to$).

In the $5{\HepParticle{j}{}{}}3\PQb$ phase space (Fig. 10), with a few
exceptions, none of the distributions are described well by any of the
considered generator setups, quantified by the $z$ scores without any
consideration of the uncertainties on the predictions. Most generator setups
predict a higher number of inclusive jets than what is observed, with the
exception of powheg+h7 5FS and mg5_amc+p8 $\ttbar{+}\textrm{jets}$ FxFx 5FS
showing better agreement. The jet multiplicity distribution, which can be
interpreted as a measure of the ratio between the cross sections of
$\ttbar\bbbar$ and $\ttbar\PQb$, is only well described by the inclusive
simulations matched to or . The NLO $\ttbar\bbbar$ simulations all predict a
higher ratio of events with at least four jets over exactly three jets. The
$\HT^{{\HepParticle{j}{}{}}}$ distribution shows a significant trend towards
higher values than what is observed in data for mg5_amc+p8 $\ttbar\bbbar$ 4FS,
and towards lower values for powheg+h7 5FS, while the other generator setups
better describe the distribution. The $\HT^{\PQb}$ distribution shows a trend
towards lower values in data compared to most predictions, with the exception
of powheg+h7 5FS and sherpa+ol $\ttbar\bbbar$ 4FS. The predictions of the
different simulations for $\abs{\eta(\PQb_{3})}$ are all very similar to each
other and have $z$ scores around 2. All generator setups predict somewhat
larger values of the of the third jet.

Observables which are not fully described by the NLO $\ttbar\bbbar$ ME, such
as observables related to the radiation of light jets, are expected to have
strong dependencies on the $\mu_{\mathrm{R}}$ and $\mu_{\mathrm{F}}$ scale
choices when described with a NLO $\ttbar\bbbar$ ME simulation setup [2].
Hence, in Appendix .13, the measured normalized differential cross sections of
the $N_{\text{jets}}$ and $\HT^{{\HepParticle{j}{}{}}}$ observables are
compared with alternative $\mu_{\mathrm{R}}$ and $\mu_{\mathrm{F}}$ scale
choices of the powheg+ol+p8 $\ttbar\bbbar$ 4FS simulation approach. These
comparisons show that increased $\mu_{\mathrm{R}}$ and $\mu_{\mathrm{F}}$
scales relative to the nominal scale choices (see Table 0.3) tend to better
describe these observables.

The agreement between data and predictions is generally better in the
$6{\HepParticle{j}{}{}}4\PQb$ phase space (Figs. 11–13), at least in part due
to the larger uncertainties in the measurements. With a few exceptions, the
predictions between the various generator setups are also closer to each other
than in the $5{\HepParticle{j}{}{}}3\PQb$ phase space. $\HT^{\PQb}$ and
$\HT^{{\HepParticle{j}{}{}}}$ are generally modelled well, while for the
powheg+h7 5FS simulation a trend toward lower values is observed, similar to
the $5{\HepParticle{j}{}{}}3\PQb$ phase space. However, most generator setups
underpredict the threshold region at low values of . The predictions for
$\pt(\PQb_{3})$, $\abs{\eta(\PQb_{3})}$, $\pt(\PQb_{4})$, and
$\abs{\eta(\PQb_{4})}$ are generally compatible with the data, with only a
slight trend visible in data towards more central (lower $\abs{\eta}$) and
softer (lower ) jets. Similarly, predictions for the and $\abs{\eta}$ for the
extra jets all describe the data well within the measurement uncertainties.
The total number of jets is reasonably described by all predictions and does
not show any clear trend.

The measured $\DR_{\PQb\PQb}^{\text{avg}}$ shows a small trend towards lower
values for all simulation approaches, but is compatible with the data within
the uncertainties of the measurement, as is the maximum invariant mass of any
pair.

The observables related to the $\PQb\PQb^{\text{extra}}$ pair are modelled
well by most of the generator setups, except for powheg+h7 5FS, which predicts
a softer $m(\smash[t]{\PQb\PQb^{\text{extra}}})$, and a smaller
$\DR(\smash[t]{\PQb\PQb^{\text{extra}}})$. The variables related to the
$\PQb\PQb^{\text{add.}}$ pair (Figs. 14–15), the jets not part of the top
quark decay chain, show similar behaviour, whereby
$\DR(\PQb\PQb^{\text{add.}})$, $m(\PQb\PQb^{\text{add.}})$ and
$\pt(\PQb\PQb^{\text{add.}})$ are not described by powheg+h7 5FS. Predictions
for the and $\abs{\eta}$ of the individual additional jets all tend to
describe the data well.

In the $6{\HepParticle{j}{}{}}3\PQb 3{\HepParticle{l}{}{}}$ and
$7{\HepParticle{j}{}{}}4\PQb 3{\HepParticle{l}{}{}}$ phase space regions (Fig.
16), the $\HT^{\text{light}}$ and
$\abs{\Delta\phi(\text{lj}^{\text{extra}}_{1},\PQb_{\text{soft}})}$
observables are modelled well by most generator setups, with only powheg+ol+p8
$\ttbar\bbbar$ 4FS predicting a higher $\HT^{\text{light}}$. With the
exception of powheg+h7 5FS, $\pt(\text{lj}^{\text{extra}}_{1})$ is observed to
be softer than what is predicted by most generator setups. In Appendix .13,
the measured normalized differential cross sections of the
$\pt(\text{lj}^{\text{extra}}_{1})$ and $\HT^{\text{light}}$ observables in
the $6{\HepParticle{j}{}{}}3\PQb 3{\HepParticle{l}{}{}}$ phase space are
compared with alternative $\mu_{\mathrm{R}}$ and $\mu_{\mathrm{F}}$ scale
choices of the powheg+ol+p8 $\ttbar\bbbar$ 4FS simulation approach. These
comparisons show that increased $\mu_{\mathrm{R}}$ and $\mu_{\mathrm{F}}$
scales relative to the nominal scale choices (see Table 0.3) tend to better
describe these observables.

##  0.9 Summary

Measurements of inclusive and normalized differential cross sections of the
associated production of top quark-antiquark and bottom quark-antiquark pairs,
$\ttbar\bbbar$, for events containing an electron or a muon, have been
presented. These measurements use proton-proton collision data recorded by the
CMS detector at $\sqrt{s}=13\TeV$ and correspond to an integrated luminosity
of 138.

The inclusive cross sections are measured in four fiducial phase space regions
requiring different jet, jet, and light jet multiplicities. With total
uncertainties of 6–17%, depending on the phase space, these are the most
precise measurements of the $\ttbar\bbbar$ cross section to date. The
uncertainties are dominated by systematic sources, with the leading
uncertainties originating from the calibration of the tagging and of the jet
energy scale, and from the choice of renormalization scale in the signal
$\ttbar\bbbar$ and background processes. In most cases, the measured inclusive
cross sections exceed the predictions with the chosen generator settings. The
only exception is when using a particular choice of dynamic renormalization
scale,
$\smash[b]{\mu_{\mathrm{R}}=\frac{1}{2}\prod_{i=\PQt,\PAQt,\PQb,\PAQb}m_{\mathrm{T},i}^{1/4}}$,
where $\smash{m_{\mathrm{T},i}^{2}=m_{i}^{2}+p_{\mathrm{T},i}^{2}}$ are the
transverse masses of top and bottom quarks.

Differential cross section measurements are performed as a function of several
observables in the aforementioned phase space regions. These observables
mainly target jets as well as additional light jets produced in association
with the top quark pairs. In the phase space containing events with at least
six jets, of which at least four are tagged, the additional -jet radiation is
probed with two different approaches. The first approach uses observables
defined purely at the particle level, without any reference to the top quark
decay chains, by selecting the two jets with the smallest angular separation.
The second approach uses explicitly the jets at the generator level that do
not originate from top quark decays and identifies those jets at the detector
level with a neural network discriminant. The differential measurements have
relative uncertainties in the range of 2–50%, depending on the phase space and
the observable.

The results are compared to the predictions of several event generator setups,
and it is found that none of them simultaneously describe all measured
distributions in the various phase space regions. In the more inclusive phase
space with five jets and three jets, the agreement between data and
predictions is generally poor, while in the phase space with six jets and four
jets, corresponding to the case in which the two additional jets in
$\ttbar\bbbar$ production are resolved, most predictions are compatible with
the data within the larger experimental uncertainties. These measurements will
help to further tune and refine the theoretical predictions and better assess
the validity of the theoretical uncertainties estimated from the various
$\ttbar\bbbar$ event generators.

###### Acknowledgements.

We congratulate our colleagues in the CERN accelerator departments for the
excellent performance of the LHC and thank the technical and administrative
staffs at CERN and at other CMS institutes for their contributions to the
success of the CMS effort. In addition, we gratefully acknowledge the
computing centres and personnel of the Worldwide LHC Computing Grid and other
centres for delivering so effectively the computing infrastructure essential
to our analyses. Finally, we acknowledge the enduring support for the
construction and operation of the LHC, the CMS detector, and the supporting
computing infrastructure provided by the following funding agencies: SC
(Armenia), BMBWF and FWF (Austria); FNRS and FWO (Belgium); CNPq, CAPES,
FAPERJ, FAPERGS, and FAPESP (Brazil); MES and BNSF (Bulgaria); CERN; CAS,
MoST, and NSFC (China); MINCIENCIAS (Colombia); MSES and CSF (Croatia); RIF
(Cyprus); SENESCYT (Ecuador); MoER, ERC PUT and ERDF (Estonia); Academy of
Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); SRNSF (Georgia);
BMBF, DFG, and HGF (Germany); GSRI (Greece); NKFIH (Hungary); DAE and DST
(India); IPM (Iran); SFI (Ireland); INFN (Italy); MSIP and NRF (Republic of
Korea); MES (Latvia); LAS (Lithuania); MOE and UM (Malaysia); BUAP, CINVESTAV,
CONACYT, LNS, SEP, and UASLP-FAI (Mexico); MOS (Montenegro); MBIE (New
Zealand); PAEC (Pakistan); MES and NSC (Poland); FCT (Portugal); MESTD
(Serbia); MCIN/AEI and PCTI (Spain); MOSTR (Sri Lanka); Swiss Funding Agencies
(Switzerland); MST (Taipei); MHESI and NSTDA (Thailand); TUBITAK and TENMAK
(Turkey); NASU (Ukraine); STFC (United Kingdom); DOE and NSF (USA).
Individuals have received support from the Marie-Curie programme and the
European Research Council and Horizon 2020 Grant, contract Nos. 675440,
724704, 752730, 758316, 765710, 824093, and COST Action CA16108 (European
Union); the Leventis Foundation; the Alfred P. Sloan Foundation; the Alexander
von Humboldt Foundation; the Science Committee, project no. 22rl-037
(Armenia); the Belgian Federal Science Policy Office; the Fonds pour la
Formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA-
Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-
Belgium); the F.R.S.-FNRS and FWO (Belgium) under the “Excellence of Science –
EOS” – be.h project n. 30820817; the Beijing Municipal Science & Technology
Commission, No. Z191100007219010 and Fundamental Research Funds for the
Central Universities (China); the Ministry of Education, Youth and Sports
(MEYS) of the Czech Republic; the Shota Rustaveli National Science Foundation,
grant FR-22-985 (Georgia); the Deutsche Forschungsgemeinschaft (DFG), under
Germany’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306, and
under project number 400140256 - GRK2497; the Hellenic Foundation for Research
and Innovation (HFRI), Project Number 2288 (Greece); the Hungarian Academy of
Sciences, the New National Excellence Program - ÚNKP, the NKFIH research
grants K 124845, K 124850, K 128713, K 128786, K 129058, K 131991, K 133046, K
138136, K 143460, K 143477, 2020-2.2.1-ED-2021-00181, and TKP2021-NKTA-64
(Hungary); the Council of Science and Industrial Research, India; the Latvian
Council of Science; the Ministry of Education and Science, project no.
2022/WK/14, and the National Science Center, contracts Opus
2021/41/B/ST2/01369 and 2021/43/B/ST2/01552 (Poland); the Fundação para a
Ciência e a Tecnologia, grant CEECIND/01334/2018 (Portugal); the National
Priorities Research Program by Qatar National Research Fund;
MCIN/AEI/10.13039/501100011033, ERDF “a way of making Europe”, and the
Programa Estatal de Fomento de la Investigación Científica y Técnica de
Excelencia María de Maeztu, grant MDM-2017-0765 and Programa Severo Ochoa del
Principado de Asturias (Spain); the Chulalongkorn Academic into Its 2nd
Century Project Advancement Project, and the National Science, Research and
Innovation Fund via the Program Management Unit for Human Resources &
Institutional Development, Research and Innovation, grant B05F650021
(Thailand); the Kavli Foundation; the Nvidia Corporation; the SuperMicro
Corporation; the Welch Foundation, contract C-1845; and the Weston Havens
Foundation (USA).

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