Measurement of lepton differential distributions and the top quark mass in
$t\bar{t}$ production in $pp$ collisions at $\sqrt{s}=8$ TeV with the ATLAS
detector TOPQ-2015-02 TOPQ-2015-02 Eur. Phys. J. C Eur. Phys. J. C 77 (2017)
804 10.1140/epjc/s10052-017-5349-9 CERN-EP-2017-200 This paper presents single
lepton and dilepton kinematic distributions measured in dileptonic $t\bar{t}$
events produced in 20.2 fb-1 of $\sqrt{s}=8$ TeV $pp$ collisions recorded by
the ATLAS experiment at the LHC. Both absolute and normalised differential
cross-sections are measured, using events with an opposite-charge $e\mu$ pair
and one or two $b$-tagged jets. The cross-sections are measured in a fiducial
region corresponding to the detector acceptance for leptons, and are compared
to the predictions from a variety of Monte Carlo event generators, as well as
fixed-order QCD calculations, exploring the sensitivity of the cross-sections
to the gluon parton distribution function. Some of the distributions are also
sensitive to the top quark pole mass; a combined fit of NLO fixed-order
predictions to all the measured distributions yields a top quark mass value of
$\mbox{$m_{t}^{\mathrm{pole}}$}=173.2\pm 0.9\pm 0.8\pm 1.2$ GeV, where the
three uncertainties arise from data statistics, experimental systematics, and
theoretical sources.

##  1 Introduction

The top quark is the heaviest known fundamental particle, with a mass
($m_{t}$) that is much larger than any of the other quarks, and close to the
scale of electroweak symmetry breaking. The study of its production and decay
properties in proton–proton ($pp$) collisions forms an important part of the
ATLAS physics program at the CERN Large Hadron Collider (LHC). Due to its
large mass and production cross-section, top quark production is also a
significant background to many searches for physics beyond the Standard Model,
making precise predictions of absolute rates and differential distributions
for top quark production a vital tool in fully exploiting the discovery
potential of the LHC.

At the LHC, top quarks are primarily produced as quark-antiquark pairs
($t\bar{t}$). The inclusive $t\bar{t}$ production cross-section
$\sigma_{t\bar{t}}$ has been calculated at full next-to-next-to-leading-order
(NNLO) accuracy in the strong coupling constant $\alpha_{\text{S}}$, including
the resummation of next-to-next-to-leading logarithmic (NNLL) soft gluon terms
[1, 2, 3, 4, 5]. The resulting prediction at a centre-of-mass energy
$\sqrt{s}=8$ TeV is $\mbox{$\sigma_{t\bar{t}}$}=252.9\pm 11.7^{+6,4}_{-8.6}$
pb for a top quark mass of 172.5 GeV, calculated using the top++ 2.0 program
[6]. The first uncertainty is due to parton distribution function (PDF) and
$\alpha_{\text{S}}$ uncertainties, calculated using the PDF4LHC prescription
[7] with the MSTW2008 68 % [8, 9], CT10 NNLO [10, 11] and NNPDF 2.3 5f FFN
[12] PDF sets, and the second to quantum chromodynamics (QCD) scale
variations. This prediction, which has a relative precision of 5.5 %, agrees
with measurements from ATLAS and CMS at $\sqrt{s}=8$ TeV [13, 14, 15] which
have reached a precision of 3–4 %. Measurements in LHC $pp$ collisions at
$\sqrt{s}=7$ TeV [13, 15] and more recently at $\sqrt{s}=13$ TeV [16, 17] are
also in good agreement with the corresponding NNLO + NNLL predictions.

Going beyond the inclusive production cross-section, measurements of
$t\bar{t}$ production as a function of the top quark and $t\bar{t}$ system
kinematics properties allow the predictions of QCD calculations and Monte
Carlo event-generator programs to be probed in more detail. These comparisons
are typically more sensitive at the level of normalised differential cross-
sections, i.e. shape comparisons, where both experimental and theoretical
uncertainties are reduced. Measurements by ATLAS [18, 19, 20, 21] and CMS [22,
23, 24] have generally demonstrated good agreement with the predictions of
leading-order (LO) multi-leg and next-to-leading-order (NLO) event generators
and calculations, though the top quark $p_{\text{T}}$ spectrum is measured to
be softer than the predictions by both experiments; this distribution appears
to be sensitive to the additional corrections contributing at NNLO [25].
Measurements of jet activity in $t\bar{t}$ events [26, 27, 28, 29] are also
sensitive to gluon radiation and hence the $t\bar{t}$ production dynamics,
without the need to fully reconstruct the kinematics of the $t\bar{t}$ system.
However, all these measurements require sophisticated unfolding procedures to
correct for the detector acceptance and resolution. This leads to significant
systematic uncertainties, especially due to modelling of the showers and
hadronisation of the quarks produced in the top quark decays, and the
measurement of the resulting jets in the detector.

In the Standard Model (SM), the top quark decays almost exclusively to a $W$
boson and a $b$ quark, and the final state topologies in $t\bar{t}$ production
are governed by the decay modes of the $W$ bosons. The channel where one $W$
boson decays to an electron ($W\rightarrow e\nu$) and the other to a muon
($W\rightarrow\mu\nu$), giving rise to the $e^{+}\mu^{-}\nu\bar{\nu}b\bar{b}$
final state111Charge-conjugate decay modes are implied unless otherwise
stated., is particularly clean and was exploited to make the most precise
ATLAS measurements of $\sigma_{t\bar{t}}$ [13, 17]. The leptons carry
information about the underlying top quark kinematics, are free of the
uncertainties related to the hadronic part of the final state, and are
precisely measured in the detector. Measurements of the $t\bar{t}$
differential cross-section as a function of the lepton kinematics therefore
have the potential to provide a complementary view of $t\bar{t}$ production
and decay dynamics to that provided by the complete reconstruction of the
$t\bar{t}$ final state.

This paper reports such a measurement of the absolute and normalised
differential cross-sections for $t\bar{t}\rightarrow e\mu\nu\bar{\nu}b\bar{b}$
produced in $pp$ collisions at $\sqrt{s}=8$ TeV, as a function of the
kinematics of the single leptons and of the dilepton system. Eight
differential cross-section distributions are measured: the transverse momentum
$p_{\mathrm{T}}^{\ell}$ and absolute pseudorapidity et al. of the single
leptons (identical for electrons and muons), the $p_{\text{T}}$, invariant
mass and absolute rapidity of the dilepton system ($p_{\mathrm{T}}^{e\mu}$,
$m^{e\mu}$ and $|y^{e\mu}|$), the azimuthal angle in the transverse plane
$\Delta\phi^{e\mu}$ between the two leptons, the scalar sum
$p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ of the $p_{\text{T}}$ of the two
leptons, and the sum $E^{e}+E^{\mu}$ of the energies of the two
leptons.222ATLAS uses a right-handed coordinate system with its origin at the
nominal interaction point in the centre of the detector, and the $z$ axis
along the beam line. Pseudorapidity is defined in terms of the polar angle
$\theta$ as $\eta=-\ln\tan{\theta/2}$, and transverse momentum and energy are
defined relative to the beamline as $p_{\text{T}}=p\sin\theta$ and
$E_{\text{T}}=E\sin\theta$. The azimuthal angle around the beam line is
denoted by $\phi$, and distances in $(\eta,\phi)$ space by $\Delta
R=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$. The rapidity is defined as
$y=\frac{1}{2}\ln\left(\frac{E+p_{z}}{E-p_{z}}\right)$, where $p_{z}$ is the
$z$-component of the momentum and $E$ is the energy of the relevant object or
system. The measurements are corrected to particle level and reported in a
fiducial volume where both leptons have $p_{\text{T}}>25$ GeV and
$|\eta|<2.5$, avoiding extrapolations into regions of leptonic phase space
which are not measured. The particle-level definition includes the
contribution of events where one or both $W$ bosons decay to electrons or
muons via leptonic decays of $\tau$-leptons ($t\rightarrow
W\rightarrow\tau\rightarrow e/\mu$), but an alternative set of results is
provided where the contributions of $\tau$-leptons are removed with a
correction derived from simulation. The definition of the fiducial volume does
not make any requirement on the presence of jets from the hadronic decay
products of the $t\bar{t}$ system. The measurements are made using events with
an opposite-charge $e\mu$ pair and one or two $b$-tagged jets, and
extrapolated to the fiducial volume (without jet requirements), using an
extension of the double-tagging technique used in the inclusive $t\bar{t}$
cross-section measurement [13]. This approach minimises the systematic
uncertainties due to the use of jets and $b$-tagging in the experimental event
selection. Since the lepton kinematics are precisely measured in the ATLAS
detector, a simple bin-by-bin correction technique is adequate to correct for
efficiency and resolution effects, without the need for a full unfolding
procedure.

The results are compared to the predictions of various NLO and LO multi-leg
$t\bar{t}$ event generators, and to fixed-order perturbative QCD predictions
from the MCFM [30] program, which is used to explore the sensitivity to PDFs
and QCD scale uncertainties. These comparisons are complementary to previous
ATLAS analyses exploring how well $t\bar{t}$ event generators can describe the
jet activity [27] and production of extra heavy-flavour jets [31] in the
$\sqrt{s}=8$ TeV $t\bar{t}$ dilepton sample.

Some of the cross-section distributions are sensitive to the top quark mass,
as suggested in Ref. [32], and mass measurements are made by comparing the
measured distributions to predictions from both NLO plus parton shower event
generators and fixed-order QCD calculations. The former are similar to
traditional measurements where the top quark mass is reconstructed from its
decay products [33, 34, 35, 36], but rely only on the leptonic decay products
of the $t\bar{t}$ system and are less sensitive to experimental uncertainties
related to the hadronic part of the final state. The measurements based on
fixed-order QCD predictions in a well-defined renormalisation scheme
correspond more directly to a measurement of the top quark pole mass
$m_{t}^{\mathrm{pole}}$, the mass definition corresponding to that of a free
particle, which may differ from that measured in direct reconstruction of the
decay products by $O(1\text{Ge\kern-1.00006ptV})$ [37, 38, 39]. Previous
determinations of $m_{t}^{\mathrm{pole}}$ from inclusive and differential
$t\bar{t}$ cross-section measurements are compatible with the top quark mass
measured from direct reconstruction, with uncertainties of 2–3 GeV [40, 13,
15, 41].

The data and Monte Carlo simulation samples used in this analysis are
described in Section 2, followed by the event reconstruction and selection in
Section 3, definition and determination of the fiducial differential cross-
sections in Section 4 and systematic uncertainties in Section 5. Results and
comparisons with predictions are given in Section 6. The ability of the data
to constrain the gluon PDF is investigated in Section 7 and the determination
of the top quark mass is discussed in Section 8. Finally, conclusions are
given in Section 9.

##  2 Data and simulated samples

The ATLAS detector [42] at the LHC covers nearly the entire solid angle around
the collision point, and consists of an inner tracking detector surrounded by
a thin superconducting solenoid magnet producing a 2 T axial magnetic field,
electromagnetic and hadronic calorimeters, and an external muon spectrometer
incorporating three large toroidal magnet assemblies. The analysis was
performed on a sample of proton–proton collision data at $\sqrt{s}=8$ TeV
recorded by the ATLAS detector in 2012, corresponding to an integrated
luminosity of 20.2 fb-1. Events were required to pass a single-electron or
single-muon trigger, with thresholds set to be fully efficient for leptons
with $p_{\text{T}}>25$ GeV passing offline selections. Each triggered event
also includes the signals from on average 20 additional inelastic $pp$
collisions in the same bunch crossing, referred to as pileup.

Monte Carlo simulated event samples were used to develop the analysis
procedures, to compare with data, and to evaluate signal efficiencies and
background contributions. An overview of the samples used for signal and
background modelling is shown in Table 1, and further details are given below.
Samples were processed using either the full ATLAS detector simulation [43]
based on GEANT4 [44], or a faster simulation making use of parameterised
showers in the calorimeters [45]. The effects of pileup were simulated by
generating additional inelastic $pp$ collisions with Pythia8 [46] using the A2
parameter set (tune) [47] and overlaying them on the primary simulated events.
These combined events were then processed using the same reconstruction and
analysis chain as the data. Small corrections were applied to the lepton
trigger and selection efficiencies better to model the performance measured in
data.

Process | Matrix-element | PDF | Parton shower | Tune | Comments
---|---|---|---|---|---
$t\bar{t}$ | Powheg | CT10 | Pythia6 | P2011C | $\mbox{$h_{\mathrm{damp}}$}=\mbox{$m_{t}$}$
| Powheg | CT10 | Herwig+Jimmy | AUET2 | $\mbox{$h_{\mathrm{damp}}$}=\infty$
| MC@NLO | CT10 | Herwig+Jimmy | AUET2 |
| Alpgen | CTEQ6L1 | Herwig+Jimmy | AUET2 | incl. $t\bar{t}$ $b\bar{b}$, $t\bar{t}$ $c\bar{c}$
| Powheg | CT10 | Pythia6 | P2012 radHi |  $\mbox{$h_{\mathrm{damp}}$}=2\mbox{$m_{t}$}$, $\frac{1}{2}\mu_{F,R}$
| Powheg | CT10 | Pythia6 | P2012 radLo |  $\mbox{$h_{\mathrm{damp}}$}=\mbox{$m_{t}$}$, $2\mu_{F,R}$
$Wt$ | Powheg | CT10 | Pythia6 | P2011C | diagram removal
$Z,W$+jets | Alpgen | CTEQ6L1 | Pythia6 | P2011C | incl. $Zb\bar{b}$
$WW$, $WZ$, $ZZ$ | Alpgen | CTEQ6L1 | Herwig | AUET2 |
$t\bar{t}$ +$W,Z$ | MadGraph | CTEQ6L1 | Pythia6 | P2011C |
$W\gamma$+jets | Sherpa | CT10 | Sherpa | default |
$t$-channel top | AcerMC | CTEQ6L1 | Pythia6 | AUET2B |
Table 1: Summary of simulated event samples used for $t\bar{t}$ signal and
background modelling, giving the matrix-element event generator, PDF set,
parton shower and associated tune parameter set. More details, including
generator version numbers and references, are given in the text.

The baseline simulated $t\bar{t}$ sample was produced using the NLO matrix
element event generator Powheg-Box v1.0 (referred to hereafter as Powheg) [48,
49, 50, 51] using the CT10 PDFs [10], interfaced to Pythia6 (version 6.426)
[52] with the CTEQ6L1 PDF set [53] and the Perugia 2011C (P2011C) tune [54]
for parton shower, hadronisation and underlying event modelling. This setup
provides an NLO QCD prediction of the $t\bar{t}$ production process, a
leading-order prediction for the top quark decays, and an approximate
treatment of the spin correlations between the quark and antiquark. The Powheg
parameter $h_{\mathrm{damp}}$, used in the damping function that limits the
resummation of higher-order effects incorporated into the Sudakov form factor,
was set to $m_{t}$. This value was found to give a better modelling of the
$t\bar{t}$ system $p_{\text{T}}$ at $\sqrt{s}=7$ TeV [55] than the setting of
$\mbox{$h_{\mathrm{damp}}$}=\infty$ used for the baseline $t\bar{t}$ sample in
Ref. [13], which corresponds to no damping.

Alternative $t\bar{t}$ simulation samples used to evaluate systematic
uncertainties were generated with Powheg interfaced to Herwig (version 6.520)
[56, 57] with the ATLAS AUET2 tune [58] and Jimmy (version 4.31) [59] for
underlying event modelling, with MC@NLO (version 4.01) [60, 61] interfaced to
Herwig + Jimmy, and with the leading-order ‘multi-leg’ event generator Alpgen
(version 2.13) [62], also interfaced to Herwig \+ Jimmy. The Alpgen samples
used leading-order matrix elements for $t\bar{t}$ production accompanied by up
to three additional light partons, and dedicated matrix elements for
$t\bar{t}$ plus $b\bar{b}$ or $c\bar{c}$ production, together with the MLM
parton-jet matching scheme [63] to account for double-counting of
configurations generated by both the parton shower and matrix-element
calculation. The effects of additional radiation in $t\bar{t}$ events were
further studied using two additional Powheg + Pythia6 samples, one using the
Perugia 2012 radHi tune [54], with $h_{\mathrm{damp}}$ set to
$2\mbox{$m_{t}$}$ and factorisation and renormalisation scales $\mu_{F}$ and
$\mu_{R}$ reduced from their event generator defaults by a factor of two,
giving more parton shower radiation; and one with the Perugia 2012 radLo tune
[54], $\mu_{F}$ and $\mu_{R}$ increased by a factor of two and
$\mbox{$h_{\mathrm{damp}}$}=\mbox{$m_{t}$}$, giving less parton shower
radiation. The parameters of these samples were chosen to span the
uncertainties in jet observables measured by ATLAS in $t\bar{t}$ events at
$\sqrt{s}=7$ TeV [55, 64, 26]. The top quark mass was set to 172.5 GeV in all
these samples, consistent with recent measurements by ATLAS [35] and CMS [36].
They were all normalised to the NNLO + NNLL cross-section prediction discussed
in Section 1 when comparing simulation with data. Further $t\bar{t}$
simulation samples with different event generator setups were used for
comparisons with the measured differential cross-sections as discussed in
Section 6.2, and in the extraction of the top quark mass as discussed in
Section 8.

Backgrounds to the $t\bar{t}$ event selection are classified into two types:
those with two real prompt leptons from $W$ or $Z$ boson decays (including
those produced via leptonic $\tau$ decays), and those where one of the
reconstructed lepton candidates is misidentified, i.e. a non-prompt lepton
from the decay of a bottom or charm hadron, an electron from a photon
conversion, hadronic jet activity misidentified as an electron, or a muon
produced from the decay in flight of a pion or kaon. The first category is
dominated by the associated production of a $W$ boson and a single top quark,
$Wt$, that is simulated using Powheg + Pythia6 with the CT10 PDFs and the
P2011C tune. The ‘diagram removal’ scheme was used to handle the interference
between the $t\bar{t}$ and $Wt$ final states that occurs at NLO [65, 66].
Smaller backgrounds result from $Z\rightarrow\tau\tau(\rightarrow e\mu)$+jets,
modelled using Alpgen + Pythia6 including leading-order matrix elements for
$Zb\bar{b}$ production, and diboson ($WW$, $WZ$ and $ZZ$) production in
association with jets, modelled with Alpgen + Herwig + Jimmy. The $Wt$
background was normalised to the approximate NNLO cross-section of $22.4\pm
1.5$ pb, determined as in Ref. [67]. The inclusive $Z$ cross-section was set
to the NNLO prediction from FEWZ [68], but the normalisation of the
$Z\rightarrow\tau\tau$ background with $b$-tagged jets was determined with the
help of data control samples as discussed in Section 4.2. The small diboson
background was normalised to the NLO QCD inclusive cross-section predictions
calculated with MCFM [69], using the Alpgen + Herwig prediction for the
fraction of diboson events with extra jets. Production of $t\bar{t}$ in
association with a $W$ or $Z$ boson, which contributes to the control sample
with two same-charge leptons, was simulated with MadGraph [70] interfaced to
Pythia6 with CTEQ6L1 PDFs, and normalised to NLO cross-section predictions
[71, 72].

Backgrounds with one real and one misidentified lepton arise from $t\bar{t}$
events with one hadronically-decaying $W$; $W$+jets production, modelled as
described above for $Z$+jets; $W\gamma$+jets, modelled with Sherpa 1.4.1 [73]
with CT10 PDFs; and $t$-channel single top production, modelled with AcerMC
[74] with the AUET2B tune [75] and CTEQ6L1 PDFs interfaced to Pythia6. The
normalisations of these backgrounds in the opposite-charge $e\mu$ samples were
determined with the help of the corresponding same-charge $e\mu$ samples in
data. Other backgrounds, including processes with two misidentified leptons,
are negligible after the event selections used in this analysis.

##  3 Event reconstruction and selection

The analysis makes use of reconstructed electrons, muons, and $b$-tagged jets,
selected exactly as described in Ref. [13]. In brief, electron candidates [76]
were required to satisfy $E_{\text{T}}>25$ GeV and $|\eta|<2.47$, and to not
lie within the transition region $1.37<|\eta|<1.52$ between the barrel and
endcap electromagnetic calorimeters. Muon candidates [77] were required to
satisfy $p_{\text{T}}>25$ GeV and $|\eta|<2.5$. In order to reduce background
from non-prompt leptons, electrons were required to be isolated from nearby
hadronic activity using both calorimeter and tracking information, and muons
were required to be isolated using tracking information alone. Jets were
reconstructed using the anti-$k_{t}$ algorithm [78, 79] with radius parameter
$R=0.4$ using calorimeter energy clusters calibrated with the local cluster
weighting method [80]. Jets were further calibrated using information from
both simulation and data [81, 82], and required to satisfy $p_{\text{T}}>25$
GeV and $|\eta|<2.5$. Jets satisfying $p_{\text{T}}<50$ GeV and $|\eta|<2.4$
were additionally required to pass pileup rejection criteria based on their
associated tracks [82]. To further suppress non-isolated leptons likely to
originate from heavy-flavour decays within jets, electron and muon candidates
within $\Delta R<0.4$ of selected jets were discarded. Finally, jets likely to
contain $b$-hadrons were $b$-tagged using the MV1 algorithm [83], a
multivariate discriminant making use of track impact parameters and
reconstructed secondary vertices. A tagging working point corresponding to a
70 % efficiency for tagging $b$-quark jets from top decays in $t\bar{t}$
events was used, giving a rejection factor of about 140 against light-quark
and gluon jets, and about five against jets originating from charm quarks.

As in Ref. [13], events were required to have at least one reconstructed
primary vertex333The reconstructed vertex with the largest sum of
$p_{\text{T}}^{2}$ for the constituent tracks was selected as the primary
vertex. and to have no jets with $p_{\text{T}}>20$ GeV failing jet quality
requirements [81]. Events having muons compatible with cosmic-ray interactions
or losing substantial energy following bremsstrahlung in the calorimeter
material were rejected. A preselection requiring exactly one electron and one
muon selected as described above was then applied, requiring at least one
selected lepton to be matched to a corresponding electron or muon trigger
signature. Events with an opposite-charge-sign $e\mu$ pair formed the main
analysis sample, with events having a same-sign pair being used to estimate
the background from misidentified leptons.

A total of 66453 data events passed the opposite-sign $e\mu$ preselection.
Events were then further sub-divided according to the number of $b$-tagged
jets, irrespective of the number of untagged jets, and events having one or
two $b$-tagged jets were retained for further analysis. The numbers of one and
two $b$-tagged jet events selected in data are shown in Table 2, compared with
expected non-$t\bar{t}$ contributions from $Wt$ and dibosons evaluated from
simulation, and $Z(\rightarrow\tau\tau\rightarrow e\mu)$+jets and
misidentified leptons evaluated from data and simulation, as discussed in
detail in Section 4.2 and Section 5 below.444The background event counts and
uncertainties shown in Table 2 differ from those in Ref. [13] due to the use
of different simulation samples and the estimation of the background in bins
of lepton kinematic variables. In simulation, the one $b$-tagged sample is
about 88 % pure and the two $b$-tagged sample 96 % pure in $t\bar{t}$ events,
with the largest backgrounds coming from $Wt$ production in both cases. The
distribution of the number of $b$-tagged jets in preselected opposite-sign
$e\mu$ events is shown in Figure 1(a), compared to the predictions from
simulation using Powheg + Pythia6 (PY6), MC@NLO + Herwig (HW) and Alpgen +
Herwig $t\bar{t}$ samples, normalising the total simulation prediction in each
case using the integrated luminosity of the data sample. The distributions of
the $p_{\text{T}}$ of $b$-tagged jets, and the reconstructed electron and muon
$p_{\text{T}}$ and $|\eta|$ in events with at least one $b$-tagged jet are
shown in Figure 1(b–f), with the total simulation prediction normalised to the
same number of events as the data to facilitate shape comparisons. The
distributions of the reconstructed dilepton variables $p_{\mathrm{T}}^{e\mu}$,
$m^{e\mu}$, $|y^{e\mu}|$, $\Delta\phi^{e\mu}$,
$p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ and $E^{e}+E^{\mu}$ are shown in
Figure 2, with the simulation normalised as for Figure 1(b–f). In general the
data are well described by the predictions using the different $t\bar{t}$
models, but a few differences are visible. The lepton $p_{\text{T}}$ spectra
are softer in data than in simulation, the lepton et al. and dilepton
$|y^{e\mu}|$ distributions are more central than the Powheg + Pythia6 and
MC@NLO + Herwig predictions, and the $\Delta\phi^{e\mu}$ distribution is
slightly flatter in data than in all the predictions.

Event counts | $N_{1}$ | $N_{2}$
---|---|---
Data | 21666 | 11739
$Wt$ single top | $2080\pm 210$ | $350\pm 120$
$Z(\rightarrow\tau\tau\rightarrow e\mu)$+jets | $210\pm 40$ | $7\pm 2$
Diboson | $120\pm 30$ | $3\pm 1$
Misidentified leptons | $220\pm 80$ | $78\pm 50$
Total background | $2630\pm 230$ | $440\pm 130$
Table 2: Observed numbers of opposite-sign $e\mu$ events with one and two
$b$-tagged jets ($N_{1}$ and $N_{2}$) together with the estimates of
backgrounds and associated total uncertainties described in Section 5.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: Distributions of (a) the number of $b$-tagged jets in preselected
opposite-sign $e\mu$ events; and (b) the $p_{\text{T}}$ of $b$-tagged jets,
(c) the $p_{\text{T}}$ of the electron, (d) the $|\eta|$ of the electron, (e)
the $p_{\text{T}}$ of the muon and (f) the $|\eta|$ of the muon, in events
with an opposite-sign $e\mu$ pair and at least one $b$-tagged jet. The
reconstruction-level data are compared to the expectation from simulation,
broken down into contributions from $t\bar{t}$ (Powheg + Pythia6), single top,
$Z$+jets, dibosons, and events with misidentified electrons or muons. The
simulation prediction is normalised to the same integrated luminosity as the
data in (a) and to the same number of entries as the data in (b–f). The lower
parts of the figure show the ratios of simulation to data, using various
$t\bar{t}$ signal samples and with the cyan band indicating the data
statistical uncertainty. The last bin includes the overflow in panels (b), (c)
and (e).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Distributions of (a) the dilepton $p_{\mathrm{T}}^{e\mu}$, (b)
invariant mass $m^{e\mu}$, (c) rapidity $|y^{e\mu}|$, (d) azimuthal angle
difference $\Delta\phi^{e\mu}$, (e) lepton $p_{\text{T}}$ sum
$p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ and (f) lepton energy sum
$E^{e}+E^{\mu}$, in events with an opposite-sign $e\mu$ pair and at least one
$b$-tagged jet. The reconstruction-level data are compared to the expectation
from simulation, broken down into contributions from $t\bar{t}$ (Powheg +
Pythia6), single top, $Z$+jets, dibosons, and events with misidentified
electrons or muons, normalised to the same number of entries as the data. The
lower parts of the figure show the ratios of simulation to data, using various
$t\bar{t}$ signal samples and with the cyan band indicating the data
statistical uncertainty. The last bin includes the overflow in panels (a),
(b), (e) and (f).

##  4 Fiducial cross-section determination

The cross-section measurements were made for a fiducial region, where the
particle-level electron and muon were required to have opposite charge signs,
to each come from $W$ decays either directly or via
$W\rightarrow\tau\rightarrow e/\mu$ and to each satisfy $p_{\text{T}}>25$ GeV
and $|\eta|<2.5$. The lepton four-momenta were taken after final-state
radiation, and ‘dressed’ by including the four-momenta of any photons within a
cone of size $\Delta R=0.1$ around the lepton direction, excluding photons
produced from hadronic decays or interactions with the detector material. The
total cross-section within this fiducial volume corresponds to the fiducial
cross-section measured in Ref. [13]. According to the predictions of the
baseline Powheg + Pythia6 $t\bar{t}$ simulation, it is about 44 % of the total
$t\bar{t}\rightarrow e\mu\nu\bar{\nu}b\bar{b}$ cross-section without
restrictions on the lepton acceptance and including contributions via
$W\rightarrow\tau\rightarrow e/\mu$.

###  4.1 Cross-section extraction

The differential cross-sections were measured using an extension of the
technique used in Ref. [13], counting the number of leptons or events with one
($N^{i}_{1}$) or two ($N^{i}_{2}$) $b$-tagged jets where the lepton(s) fall in
bin $i$ of a differential distribution at reconstruction level. For the
single-lepton distributions $p_{\mathrm{T}}^{\ell}$ and et al., there are two
counts per event, in the two bins corresponding to the electron and muon. For
the dilepton distributions, each event contributes a single count
corresponding to the bin in which the appropriate dilepton variable falls. For
each measured distribution, these counts satisfy the tagging equations:

$\begin{array}[]{lll}\mbox{$N^{i}_{1}$}&=&L\mbox{$\sigma^{i}_{t\bar{t}}$}\
\mbox{$G^{i}_{e\mu}$}2\mbox{$\epsilon^{i}_{b}$}(1-\mbox{$C^{i}_{b}$}\mbox{$\epsilon^{i}_{b}$})+\mbox{$N_{1}^{i,\mathrm{bkg}}$},\\\\[5.69054pt]
\mbox{$N^{i}_{2}$}&=&L\mbox{$\sigma^{i}_{t\bar{t}}$}\
\mbox{$G^{i}_{e\mu}$}\mbox{$C^{i}_{b}$}(\mbox{$\epsilon^{i}_{b}$})^{2}+\mbox{$N_{2}^{i,\mathrm{bkg}}$},\end{array}$
(1)

where $\sigma^{i}_{t\bar{t}}$ is the absolute fiducial differential cross-
section in bin $i$, and $L$ is the integrated luminosity of the sample. The
reconstruction efficiency $G^{i}_{e\mu}$ represents the ratio of the number of
reconstructed $e\mu$ events (or leptons for $p_{\mathrm{T}}^{\ell}$ and et
al.) falling in bin $i$ at reconstruction level to the number of true $e\mu$
events (or leptons) falling in the same bin at particle level, evaluated using
$t\bar{t}$ simulation without making any requirements on reconstructed or
particle-level jets. It therefore corrects for both the lepton reconstruction
efficiency and bin migration, where events corresponding to bin $j$ at
particle level appear in a different bin $i\neq j$ at reconstruction level.
The values of $G^{i}_{e\mu}$ in simulation are typically in the range 0.5–0.6,
with some dependence on lepton kinematics due to the varying reconstruction
efficiencies with lepton $|\eta|$ and $p_{\text{T}}$, and the effect of
isolation requirements when the leptons are close together in the detector.

The efficiency $\epsilon^{i}_{b}$ represents the combined probability for a
jet from the quark $q$ in the $t\rightarrow Wq$ decay to fall within the
detector acceptance, be reconstructed as a jet with $p_{\text{T}}>25$ GeV and
be tagged as a $b$-jet. Although this quark is almost always a $b$-quark,
$\epsilon^{i}_{b}$ also accounts for the 0.2 % of top quarks that decay to
$Ws$ or $Wd$. If the kinematics of the two $b$ quarks produced in the top
quark decays are uncorrelated, the probability to tag both is given by
$\mbox{$\epsilon^{i}_{bb}$}=(\mbox{$\epsilon^{i}_{b}$})^{2}$. In practice,
small correlations are present, for example due to kinematic correlations
between the $b$-jets from the top quark decays, or extra $b\bar{b}$ or
$c\bar{c}$ pairs produced in association with the $t\bar{t}$ system [13].
Their effects are corrected via the tagging correlation coefficient
$\mbox{$C^{i}_{b}$}=\mbox{$\epsilon^{i}_{bb}$}/(\mbox{$\epsilon^{i}_{b}$})^{2}$,
whose values are taken from $t\bar{t}$ simulation. They depend slightly on the
bin $i$ of the dilepton system but are always within 1–2 % of unity, even for
the bins at the edges of the differential distributions. The correlation
$C^{i}_{b}$ also corrects for the small effects on $N^{i}_{1}$, $N^{i}_{2}$
and $\epsilon^{i}_{b}$ of the small fraction of $t\bar{t}$ events which have
additional $b$ quarks produced in association with the $t\bar{t}$ system, and
the even smaller effects from mistagged light quark, charm or gluon jets in
$t\bar{t}$ events. This formalism involving $\epsilon^{i}_{b}$ and $C^{i}_{b}$
allows the fraction of top quarks where the jet was not reconstructed to be
inferred from the counts $N^{i}_{1}$ and $N^{i}_{2}$, minimising the exposure
to systematic uncertainties from jet measurements and $b$-tagging, and
allowing the fiducial cross-sections $\sigma^{i}_{t\bar{t}}$ to be defined
with no requirements on the jets in the final state.

Backgrounds from sources other than $t\bar{t}\rightarrow
e\mu\nu\bar{\nu}b\bar{b}$ events also contribute to the counts $N^{i}_{1}$ and
$N^{i}_{2}$, and are represented by the terms $N_{1}^{i,\mathrm{bkg}}$ and
$N_{2}^{i,\mathrm{bkg}}$ in Eqs. (1). These contributions were evaluated using
a combination of simulation- and data-based methods as discussed in Section
4.2 below.

The tagging equations were solved numerically in each bin $i$ of each
differential distribution separately. The bin ranges for each distribution
were chosen according to the experimental resolution, minimising the bin-to-
bin migration by keeping the bin purities (the fractions of reconstructed
events in bin $i$ that originate from events which are also in bin $i$ at
particle level) above about 0.9. The resolution on the reconstructed kinematic
quantities is dominated by the electron energy and muon momentum measurements,
and the purities for the distributions which depend mainly on angular
variables are higher, around 0.96 for $|y^{e\mu}|$ and 0.99 for et al. and
$\Delta\phi^{e\mu}$. For these distributions, the bin ranges were chosen so as
to give about ten bins for each distribution. The bin range choices for all
distributions can be seen in Tables 3 to 6 in Section 6, and the last bin of
the $p_{\mathrm{T}}^{\ell}$, $p_{\mathrm{T}}^{e\mu}$, $m^{e\mu}$,
$p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ and $E^{e}+E^{\mu}$ distributions
includes overflow events falling above the last bin boundary, indicated by the
‘+’ sign after the upper bin limit.

The normalised fiducial differential cross-section distributions
$\varsigma^{i}_{t\bar{t}}$ were calculated from the absolute cross-sections
$\sigma^{i}_{t\bar{t}}$ determined from Eqs. (1) as follows:

$\displaystyle\mbox{$\varsigma^{i}_{t\bar{t}}$}=\frac{\mbox{$\sigma^{i}_{t\bar{t}}$}}{\Sigma_{j}\
\mbox{$\sigma^{j}_{t\bar{t}}$}}=\frac{\mbox{$\sigma^{i}_{t\bar{t}}$}}{\mbox{$\sigma^{t\bar{t}}_{\mathrm{fid}}$}},$
(2)

where $\sigma^{t\bar{t}}_{\mathrm{fid}}$ is the total cross-section summed
over all bins of the fiducial region. The $\varsigma^{i}_{t\bar{t}}$ values
are divided by the bin widths $W_{i}$, to produce the cross-sections
differential in the variable $x$ ($x=\mbox{$p_{\mathrm{T}}^{\ell}$}$, et al.,
etc.):

$\displaystyle\frac{1}{\sigma}\left(\frac{{\mathrm{d}}\sigma}{{\mathrm{d}}x}\right)_{i}=\frac{\mbox{$\varsigma^{i}_{t\bar{t}}$}}{W_{i}}\
.$

The normalisation condition in Eq. (2) induces a statistical correlation
between the normalised measurements in each bin. The absolute dilepton cross-
section measurements are not statistically correlated between bins, but
kinematic correlations between the electron and muon in each event induce
small statistical correlations between bins of the absolute single lepton
$p_{\mathrm{T}}^{\ell}$ and et al. distributions, as discussed in Section 4.3
below.

The measured cross-sections include contributions where one or both leptons
are produced via leptonic tau decays ($t\rightarrow
W\rightarrow\tau\rightarrow e/\mu$), but the fixed-order predictions discussed
in Section 6.3 only include the direct decays $t\rightarrow W\rightarrow
e/\mu$. To allow comparison with such predictions, a second set of cross-
section results were derived with a bin-by-bin multiplicative correction
$f^{i}_{\bar{\tau}}$ to remove the $\tau$ contributions:

$\displaystyle\mbox{$\sigma^{i}_{t\bar{t}}$}\,(\mbox{no--$\tau$})=\mbox{$f^{i}_{\bar{\tau}}$}\mbox{$\sigma^{i}_{t\bar{t}}$}\
,$ (3)

and similarly for the normalised cross-sections
$\mbox{$\varsigma^{i}_{t\bar{t}}$}\,(\mbox{no-$\tau$})$. The corrections
$f^{i}_{\bar{\tau}}$ were evaluated from the baseline Powheg + Pythia6
$t\bar{t}$ simulation and are typically close to 0.9, decreasing to 0.8–0.85
at low lepton $p_{\text{T}}$.

###  4.2 Background estimates

The $Wt$ single top and diboson backgrounds were estimated from simulation
using the samples discussed in Section 2, whilst the $Z$+jets background (with
$Z\rightarrow\tau\tau\rightarrow e\mu 4\nu$) and the contribution from events
with one real and one misidentified lepton were estimated using both
simulation and data as discussed below. The backgrounds in both the one and
two $b$-tagged samples are dominated by $Wt$ (see Table 2). The total
background fraction (i.e. the predicted fraction of events in each bin which
do not come from $t\bar{t}$ with two real prompt leptons) varies significantly
as a function of some of the differential variables, as shown in Figure 3.
This variation is taken into account by estimating the background
contributions $N_{1}^{i,\mathrm{bkg}}$ and $N_{2}^{i,\mathrm{bkg}}$ separately
in each bin of each differential distribution.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 3: Estimated background fractions in the one and two $b$-tagged samples
as functions of each lepton and dilepton differential variable, estimated from
simulation alone. The error bars correspond to the statistical uncertainties
of the simulation samples, and are often smaller than the marker size.

The production cross-sections for $Z$ bosons accompanied by heavy-flavour jets
are subject to large theoretical uncertainties. The background predictions
from Alpgen + Pythia6 in each bin of each distribution were therefore
normalised from data, by multiplying them by constant scale factors of $1.4\pm
0.2$ for the one $b$-tagged jet sample and $1.1\pm 0.3$ for the two $b$-tagged
jet sample. These scale factors were derived from the comparison of data and
simulated event yields for $Z\rightarrow ee$ and $Z\rightarrow\mu\mu$ plus one
or two $b$-tagged jets, inclusively for all lepton pairs passing the kinematic
selections for electrons and muons [13]. The uncertainties are dominated by
the dependence of the scale factors on lepton kinematics, investigated by
studying their variation with $Z$-boson $p_{\text{T}}$, reconstructed from the
$ee$ or $\mu\mu$ system.

The background from events with one real and one misidentified lepton was
estimated using a combination of data and simulation in control regions with
an electron and muon of the same charge [13]. Simulation studies showed that
the samples with a same-sign $e\mu$ pair and one or two $b$-tagged jets are
dominated by events with a misidentified lepton, with rates and kinematic
distributions similar to those in the opposite-sign sample. The distributions
of the dilepton kinematic variables for same-sign events with at least one
$b$-tagged jet in data are shown in Figure 4, and compared with the
predictions from simulation. The expected contributions are shown separately
for events with two prompt leptons, events where the electron candidate
originates from a converted photon radiated from an electron produced in a top
quark decay, events with a converted photon from other sources, and events
where the electron or muon originates from the decay of a bottom or charm
hadron. The analogous distributions for the electron and muon $p_{\text{T}}$
and $|\eta|$ are shown in Ref. [13]. In general, the simulation models the
rates and kinematic distributions of the same-sign events well. The modelling
of misidentified leptons was further tested in control samples where either
the electron or muon isolation requirements were relaxed in order to enhance
the contributions from heavy-flavour decays, and similar levels of agreement
were observed.

The contributions $N_{j}^{i,\mathrm{mis-id}}$ of events with misidentified
leptons to the opposite-sign samples with $j=1$, 2 $b$-tagged jets were
estimated in each bin $i$ of each distribution using

$\begin{array}[]{rll}\mbox{$N_{j}^{i,\mathrm{mis-
id}}$}&=&R^{i}_{j}(\mbox{$N_{j}^{i,\mathrm{data,SS}}$}-\mbox{$N_{j}^{i,\mathrm{prompt,SS}}$}),\\\\[2.84526pt]
R^{i}_{j}&=&\frac{\mbox{$N_{j}^{i,\mathrm{mis-
id,OS}}$}}{\mbox{$N_{j}^{i,\mathrm{mis-id,SS}}$}},\end{array}$ (4)

where $N_{j}^{i,\mathrm{data,SS}}$ is the number of observed same-sign events
in bin $i$ with $j$ $b$-tagged jets, $N_{j}^{i,\mathrm{prompt,SS}}$ is the
estimated number of events in this bin with two prompt leptons, and
$R^{i}_{j}$ is the ratio of the number of opposite- to same-sign events with
misidentified leptons in bin $i$ with $j$ $b$-tagged jets. This formalism uses
the observed data same-sign event rate in each bin to predict the
corresponding opposite-sign contribution from misidentified leptons. It relies
on simulation to predict the ratios of opposite- to same-sign rates and the
prompt same-sign contribution, but not the absolute normalisation of
misidentified leptons. The prompt-lepton contribution in Eq. (4) comes mainly
from semileptonic $t\bar{t}$ events with an additional $W$ or $Z$ boson,
diboson events with two same-sign leptons, and $t\bar{t}\rightarrow
e\mu\nu\bar{\nu}b\bar{b}$ events where the electron charge was
misreconstructed. These components were evaluated directly from simulation in
each bin $(i,j)$, and an uncertainty of $\pm 50$ % was assigned [13]. The
values of $R^{i}_{j}$ were taken from simulation, separately for each
differential distribution and $j=1$ and 2 $b$-tagged jets, and averaged over
several consecutive bins $i$ in order to reduce statistical fluctuations. The
values of $R^{i}_{1}$ range from 0.8 to 1.5, and $R^{i}_{2}$ from 1.2 to 2.0,
as the predicted background composition changes across the kinematic
distributions. As in Ref. [13], uncertainties of $\pm 0.25$ and $\pm 0.5$ were
assigned to $R^{i}_{1}$ and $R^{i}_{2}$, based on the variation of $R^{i}_{j}$
for different components of the misidentified lepton background, and taken to
be correlated across all bins $(i,j)$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: Distributions of (a) the dilepton $p_{\mathrm{T}}^{e\mu}$, (b)
invariant mass $m^{e\mu}$, (c) rapidity $|y^{e\mu}|$, (d) azimuthal angle
difference $\Delta\phi^{e\mu}$, (e) lepton $p_{\text{T}}$ sum
$p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ and (f) lepton energy sum
$E^{e}+E^{\mu}$, in events with a same-sign $e\mu$ pair and at least one
$b$-tagged jet. The simulation prediction is normalised to the same integrated
luminosity as the data, and broken down into contributions where both leptons
are prompt, or one is a misidentified lepton from a photon conversion
originating from a top quark decay or from background, or from heavy-flavour
decay. In the $p_{\mathrm{T}}^{e\mu}$, $m^{e\mu}$,
$p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ and $E^{e}+E^{\mu}$ distributions,
the last bin includes the overflows.

###  4.3 Validation of the analysis procedure

The method for the differential cross-section determination was tested on
simulated events in order to check for biases and determine the expected
statistical uncertainties. Pseudo-data samples corresponding to the data
integrated luminosity were produced by varying the event counts $N^{i}_{1}$
and $N^{i}_{2}$ in each bin $i$ independently, according to Poisson
distributions with mean values predicted from a chosen $t\bar{t}$ simulation
sample plus non-$t\bar{t}$ backgrounds. The tagging equations Eqs. (1) were
then solved for each pseudo-experiment using the values of $G^{i}_{e\mu}$,
$C^{i}_{b}$, $N_{1}^{i,\mathrm{bkg}}$ and $N_{2}^{i,\mathrm{bkg}}$ calculated
with the baseline simulation samples. An initial set of 1000 pseudo-
experiments was performed using the baseline simulation sample as a reference,
and the mean and RMS width of the deviations of the result in each bin from
the reference values were used to validate the analysis procedure. The black
points in Figure 5 show the mean deviation of the results (averaged over all
pseudo-experiments) for four of the measured normalised distributions, with
error bars corresponding to the uncertainty in the mean due to the finite size
of the simulation samples (about 17 times the data integrated luminosity). The
residual biases of the mean deviations away from the reference are compatible
with zero and in all cases much smaller than the expected statistical
uncertainties in data, measured by the RMS widths and shown by the cyan bands.
Similar results were obtained for the other normalised differential cross-
section distributions, and for the absolute distributions. The pull
distributions (i.e. the distributions of deviations divided by the estimated
statistical uncertainty from each pseudo-experiment) were also found to have
widths within a few percent of unity. The $\chi^{2}$ values for the
compatibility of each measured distribution with the reference were also
calculated for each pseudo-experiment and the distribution of the
corresponding $p$-values across all pseudo-experiments was found to be uniform
between zero and one. These tests confirm that the analysis procedure is
unbiased and correctly estimates the statistical uncertainties in each bin of
each distribution.

(a)

(b)

(c)

(d)

Figure 5: Results of pseudo-experiment studies on simulated events for the
extraction of the normalised differential cross-section distributions for (a)
$p_{\mathrm{T}}^{\ell}$, (b) $p_{\mathrm{T}}^{e\mu}$, (c) et al. and (d)
$|y^{e\mu}|$, shown as relative deviations
$(\sigma-\sigma_{\mathrm{ref}})/\sigma_{\mathrm{ref}}$ from the reference
cross-section values in the baseline Powheg+Pythia6 CT10 sample with
$\mbox{$m_{t}$}=172.5$ GeV. The black points show the mean deviations from the
reference when fitting pseudo-data samples generated with the baseline
simulation sample, with error bars indicating the uncertainties due to the
limited number of simulated events. The cyan bands indicate the expected
statistical uncertainties for a single sample corresponding to the data
integrated luminosity. The open red points show the mean deviations from the
reference values when fitting pseudo-experiments generated from alternative
simulation samples with $\mbox{$m_{t}$}=165$ GeV (a, b) or with the HERAPDF
1.5 PDF (c, d), with error bars due to the limited size of these alternative
samples. The red dotted lines show the true deviations from the reference in
the alternative samples.

Additional pseudo-experiments were performed to test the ability of the
analysis procedure to reconstruct distributions different from the reference,
taking the values of $G^{i}_{e\mu}$, $C^{i}_{b}$, $N_{1}^{i,\mathrm{bkg}}$ and
$N_{2}^{i,\mathrm{bkg}}$ from the baseline samples. Tests were conducted using
simulated Powheg + Pythia6 and MC@NLO + Herwig $t\bar{t}$ samples with
different top mass values, a Powheg + Pythia6 sample generated using the
HERAPDF 1.5 [84, 85] PDF set instead of CT10, and a Powheg + Pythia6 sample
reweighted to reproduce the top quark $p_{\text{T}}$ distribution calculated
at NNLO from Ref. [25]. In all cases, the analysis procedure recovered the
true distributions from the alternative samples within the statistical
precision of the test, demonstrating the adequacy of the bin-by-bin correction
procedure without the need for iteration or a more sophisticated matrix-based
unfolding technique. Some examples are shown by the red points and dotted
lines in Figure 5, for an alternative sample with $\mbox{$m_{t}$}=165$ GeV for
$p_{\mathrm{T}}^{\ell}$ and $p_{\mathrm{T}}^{e\mu}$, and for HERAPDF 1.5 for
et al. and $|y^{e\mu}|$, both simulation samples having about twice the
statistics of the data. These figures also demonstrate the sensitivities of
some of the measured distributions to $m_{t}$ and different PDFs.

For the single-lepton distributions $p_{\mathrm{T}}^{\ell}$ and et al., which
have two entries per event, the formalism of Eqs. (1) and the pseudo-
experiments generated by fluctuating each bin independently do not take into
account correlations between the kinematics of the electron and muon in each
event. This effect was checked by generating pseudo-data samples corresponding
to the data integrated luminosity from individual simulated events, taken at
random from a large $t\bar{t}$ sample combining both full and fast simulation
and corresponding to about 70 times the data integrated luminosity. The effect
of neglecting the electron-muon correlations within an event was found to
correspond to at most a 2 % fractional overestimate of the absolute and 2 %
fractional underestimate of the normalised cross-section uncertainties. Hence,
no corresponding corrections to the statistical uncertainties were made.

##  5 Systematic uncertainties

Systematic uncertainties in the measured cross-sections arise from
uncertainties in the values of the input quantities $G^{i}_{e\mu}$,
$C^{i}_{b}$, $N_{1}^{i,\mathrm{bkg}}$, $N_{2}^{i,\mathrm{bkg}}$ and $L$ used
in Eqs. (1). Each source of systematic uncertainty was evaluated by coherently
changing the values of all relevant input quantities and re-solving Eqs. (1),
thus taking into account correlations of the uncertainties in e.g.
$G^{i}_{e\mu}$ and $C^{i}_{b}$. The uncertainties are divided into five groups
($t\bar{t}$ modelling, leptons, jets/$b$-tagging, background and
luminosity/beam energy uncertainties) and are discussed in Sections 5.1 to
5.5. The resulting relative uncertainties in each measured differential cross-
section value are shown in the results Tables 3 to 6, and the grouped
systematic uncertainties for the normalised differential cross-sections are
shown in Figure 6, together with the statistical and total uncertainties.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 6: Relative uncertainties on the measured normalised differential
cross-sections coming from data statistics, $t\bar{t}$ modelling, leptons,
jets and background, as a function of each lepton or dilepton differential
variable. The total uncertainty is shown by the black lines, and also includes
small contributions from the integrated luminosity and LHC beam energy
uncertainties.

###  5.1 $t\bar{t}$ modelling

The uncertainties in $G^{i}_{e\mu}$ and $C^{i}_{b}$ (and $f^{i}_{\bar{\tau}}$
for the $\tau$-corrected cross-sections) were evaluated using the various
alternative $t\bar{t}$ simulation samples detailed in Section 2.

$t\bar{t}$ generator:

Event generator uncertainties were evaluated by comparing the baseline Powheg
+ Pythia6 $t\bar{t}$ sample (with $\mbox{$h_{\mathrm{damp}}$}=\mbox{$m_{t}$}$)
with alternative samples generated with MC@NLO interfaced to Herwig (thus
changing both the NLO hard-scattering event generator and the parton shower,
hadronisation and underlying event model), and with the LO multi-leg event
generator Alpgen, also interfaced to Herwig. The bin-by-bin shifts in
$G^{i}_{e\mu}$ and $C^{i}_{b}$ were fitted with polynomial functions in order
to reduce statistical fluctuations caused by the limited size of the simulated
samples, and the larger of the differences between the baseline and the two
alternative samples was taken in each bin to define the generator uncertainty.
As also found in the inclusive cross-section analysis [13], a substantial part
of the differences in $G^{i}_{e\mu}$ in the various samples arises from
differences in the hadronic activity close to the leptons, which affects the
efficiency of the lepton isolation requirements. These efficiencies were
therefore measured in situ in $t\bar{t}$ events selected in data as discussed
in Section 5.2 below, and the simulation uncertainties on $G^{i}_{e\mu}$
evaluated by considering the lepton reconstruction, identification and lepton-
jet overlap requirements only. The resulting uncertainties on $G^{i}_{e\mu}$
are typically 0.5–1 % in most regions of the phase space, varying only
slightly as a function of the lepton and dilepton kinematics. The same
procedure was used to evaluate uncertainties in $C^{i}_{b}$, and the
predictions of the three simulation samples were found to agree at the 0.5–1 %
level, giving similar predictions for the variations of $C^{i}_{b}$ across the
bins of the various measured distributions. Alternative $t\bar{t}$ samples
generated with Powheg + Pythia6 and Powheg + Herwig (both with
$\mbox{$h_{\mathrm{damp}}$}=\infty$) were also considered, but the resulting
differences in $G^{i}_{e\mu}$ and $C^{i}_{b}$ were found to be significantly
less than those from the comparisons with MC@NLO + Herwig and thus no
additional uncertainty was assigned. Variations in the predictions of
$f^{i}_{\bar{\tau}}$ from the three $t\bar{t}$ samples were found to be at the
0.2 % level, and were also taken into account for the $\tau$-corrected cross-
section results.

Initial/final-state radiation:

The effects on $G^{i}_{e\mu}$ $C^{i}_{b}$ and $f^{i}_{\bar{\tau}}$ of
uncertainties in the modelling of additional radiation in $t\bar{t}$ events
were assessed as half the difference between Powheg + Pythia6 samples tuned to
span the uncertainties in jet activity measured in $\sqrt{s}=7$ TeV ATLAS data
[55, 64, 26], as discussed in Section 2. The uncertainties were taken as half
the difference between the upward and downward variations, and were
substantially reduced by measuring the lepton isolation efficiencies from
data, in the same way as for the $t\bar{t}$ generator uncertainties discussed
above.

Parton distribution functions:

The uncertainties in $G^{i}_{e\mu}$ due to limited knowledge of the proton
PDFs were evaluated using the error sets of the CT10 [10], MSTW 2008 68 % CL
[8] and NNPDF 2.3 [12] NLO PDF sets, by reweighting the MC@NLO + Herwig
$t\bar{t}$ sample based on the $x$ and $Q^{2}$ values of the partons
participating in the hard scattering in each event. The final uncertainty in
each bin was calculated as half the envelope encompassing the predictions from
all three PDF sets and their associated uncertainties, following the PDF4LHC
prescription [7]. The resulting uncertainties on $G^{i}_{e\mu}$ are typically
around 0.3 % except at the high ends of the distributions, and were taken to
be fully correlated across all bins.

Top quark mass:

The values of $G^{i}_{e\mu}$ and the predicted levels of $Wt$ background
depend weakly on the assumed value of $m_{t}$. These effects were evaluated
with $t\bar{t}$ and $Wt$ samples simulated with $m_{t}$ values of 170 and 175
GeV, and scaled to a nominal $\pm 1$ GeV mass variation. The resulting effects
are at the level of 0.1-0.2 % on $G^{i}_{e\mu}$, and are partially cancelled
by the variations in the $Wt$ background, whose cross-section decreases with
increasing $m_{t}$. The residual uncertainties are typically around 0.1 % for
the absolute cross-sections except at the extreme ends of the distributions,
and smaller for the normalised cross-sections.

The total $t\bar{t}$ modelling uncertainties in the normalised differential
cross-sections also include the small uncertainties on $G^{i}_{e\mu}$ and
$C^{i}_{b}$ from the limited size of the simulated $t\bar{t}$ samples, and are
shown by the green lines in Figure 6. They are typically dominated by the
$t\bar{t}$ event generator comparisons.

###  5.2 Lepton identification and measurement

Uncertainties in the modelling of the detector response to electrons and muons
affect both $G^{i}_{e\mu}$ and the background estimates, with the largest
uncertainties in the cross-section measurements coming via the former.

Lepton identification:

The modelling of the electron and muon identification efficiencies, and the
rate of electron charge misidentification, were studied using $Z\rightarrow
ee/\mu\mu$, $J/\psi\rightarrow ee/\mu\mu$ and $W\rightarrow e\nu$ events in
data and simulation [76, 77], taking into account the systematic correlations
across different regions of the lepton $p_{\text{T}}$ and $\eta$ spectrum. The
uncertainties in $G^{i}_{e\mu}$ are typically below 0.5 % for electron and
below 0.3 % for muon efficiencies, with significant cancellations in the
normalised differential cross-sections.

Lepton scales and resolution:

The electron and muon energy/momentum scales and resolutions were determined
using $Z\rightarrow ee/\mu\mu$, $Z\rightarrow(ee/\mu\mu)\gamma$,
$J/\psi\rightarrow ee/\mu\mu$ and $\Upsilon\rightarrow\mu\mu$ decays [86, 77].
The largest uncertainty comes from the limited knowledge of the electron
energy scale, which gives uncertainties varying from 0.2 % to over 2% for the
bins involving the highest energy electrons. The muon momentum scale
uncertainties are small in comparison.

Lepton isolation:

Building on the studies described in Ref. [13], the efficiencies of the lepton
isolation requirements were measured in data, using the fractions of selected
opposite-sign $e\mu$ events with at least one $b$-tagged jet where either the
electron or the muon fails the isolation requirement. After correcting for the
contamination from events with a misidentified lepton, these fractions give
the inefficiency of the isolation requirements on signal $t\bar{t}$ events.
The misidentified lepton backgrounds were measured both by using the same-sign
$e\mu$ control samples discussed in Section 4.2 above, and by using the
distributions of lepton impact parameter significance
$|d_{0}|/\sigma_{d_{0}}$, where $d_{0}$ is the distance of closest approach of
the lepton track to the event primary vertex in the transverse plane, and
$\sigma_{d_{0}}$ its uncertainty. The isolation inefficiencies were measured
as functions of lepton $p_{\text{T}}$ separately for the barrel ($|\eta|<1.5$)
and endcap regions of the detector. Consistent results were obtained using
both misidentified lepton estimation methods, and showed that the baseline
Powheg + Pythia6 $t\bar{t}$ simulation sample overestimates the efficiencies
of the lepton isolation requirements by up to 1 % for electrons with
$p_{\text{T}}$ in the range 40–80 GeV, and by up to 2 % for muons at low
$p_{\text{T}}$, decreasing rapidly to less than 0.5 % for 40 GeV. The values
of $G^{i}_{e\mu}$ from the baseline simulation were corrected for these
$p_{\text{T}}$-dependent shifts using a reweighting technique. The
corresponding uncertainties are dominated by those on the misidentified lepton
subtraction (including a comparison of the same-sign and
$|d_{0}|/\sigma_{d_{0}}$-based methods) and amount to typically 0.5–1 % for
electrons and 0.2–0.5 % for muons. The effect on the normalised cross-sections
is about half that on the absolute measurements, taking into account
systematic correlations across lepton $p_{\text{T}}$ and $|\eta|$ bins.

Lepton trigger:

The efficiencies of the single-lepton triggers were measured in data using
$Z\rightarrow ee/\mu\mu$ events [87]. Since only one lepton trigger was
required to accept the $e\mu$ event, the trigger efficiency with respect to
the offline event selection is about 99 %, with a residual uncertainty of less
than 0.2 %.

The lepton-related uncertainties are shown by the blue dot-dashed lines in
Figure 6, and the largest uncertainties typically come from the electron
energy scale and electron isolation uncertainties.

###  5.3 Jet measurement and $b$-tagging

Uncertainties in the selection and $b$-tagging of jets affect the background
estimates $N_{1}^{i,\mathrm{bkg}}$ and $N_{2}^{i,\mathrm{bkg}}$, and to a
lesser extent, the correlation $C^{i}_{b}$. The jet uncertainties also have a
very small effect on $G^{i}_{e\mu}$, through the requirement that leptons be
separated from selected jets by $\Delta R>0.4$.

Jet-related uncertainties:

The jet energy scale was varied according to the uncertainties derived from
simulation and in situ calibration measurements [81], using a model with 22
orthogonal uncertainty components describing the evolution with jet
$p_{\text{T}}$ and $|\eta|$. The effects of residual uncertainties in the
modelling of the jet energy resolution [88] were assessed by smearing jet
energies in simulation. The jet reconstruction efficiency was measured in data
using track-based jets, and the effect of residual uncertainties assessed in
simulation by randomly discarding jets. The modelling of the pileup rejection
requirement applied to jets was studied using $Z\rightarrow ee/\mu\mu$+jets
events [82].

$b$-tagging uncertainties:

The efficiencies for $b$-tagging jets in $t\bar{t}$ signal events were
extracted from the data, but simulation was used to predict the numbers of
$b$-tagged jets in the $Wt$ single top and diboson backgrounds. The
corresponding uncertainties were assessed using studies of $b$-jets containing
muons, charm jets containing $D^{*+}$ mesons and inclusive jet events [83].

The jet- and $b$-tagging-related uncertainties are shown by the purple lines
on Figure 6, and are typically dominated by the effect of the jet energy scale
on the level of $Wt$ background.

###  5.4 Background modelling

As well as the detector-related uncertainties discussed above, the background
estimates depend on uncertainties in modelling the $Wt$ and diboson processes
taken from simulation, and uncertainties in the procedures used for estimating
the $Z$+jets and misidentified lepton backgrounds from data.

Single top modelling:

Uncertainties in the modelling of the $Wt$ background were assessed by
comparing the predictions from the baseline Powheg + Pythia6 sample with those
from MC@NLO + Herwig, and from two samples generated with AcerMC + Pythia6
utilising different tunes to vary the amount of additional radiation, in all
cases normalising the total production cross-section to the approximate NNLO
prediction based on Ref. [67]. The uncertainty in this prediction was
evaluated to be 6.8 %. The $Wt$ background with two $b$-tagged jets is
sensitive to the production of $Wt$ with an additional $b$-jet, an NLO
contribution which interferes with the $t\bar{t}$ final state. The
corresponding uncertainty was assessed by comparing the predictions of Powheg
+ Pythia6 with the diagram removal and diagram subtraction schemes for
handling this interference [65, 66]. The latter predicts up to 25 % less $Wt$
background in the one $b$-tagged and 60 % less in the two $b$-tagged channels
at the extreme high ends of the lepton $p_{\text{T}}$ and dilepton
$p_{\mathrm{T}}^{e\mu}$, $m^{e\mu}$, $p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$
and $E^{e}+E^{\mu}$ distributions, but only 1–2 % and 20 % differences for one
and two $b$-tagged $Wt$ events across the et al., $|y^{e\mu}|$ and
$\Delta\phi^{e\mu}$ distributions, similar to the differences seen for the
inclusive analysis [13]. The uncertainties due to the limited size of the $Wt$
simulation samples are negligible in comparison to the modelling
uncertainties.

Diboson modelling:

The uncertainties in modelling the diboson background events (mainly $WW$)
with one and two additional $b$-tagged jets were assessed by comparing the
predictions from Alpgen \+ Herwig with those of Sherpa 1.4.3 [73] including
the effects of massive $b$ and $c$ quarks. The resulting uncertainties in the
diboson background are typically in the range 20–30 %, substantially larger
than the differences between recent predictions for the inclusive diboson
cross-sections at NNLO in QCD [89] and the NLO predictions from MCFM used to
normalise the simulated samples. The background from SM Higgs production with
$H\rightarrow WW$ and $H\rightarrow\tau\tau$ is smaller than the uncertainties
assigned for diboson modelling, and was neglected.

Z+jets extrapolation:

The backgrounds from $Z\rightarrow\tau\tau\rightarrow e\mu$ accompanied by one
or two $b$-tagged jets were extrapolated from the analogous $Z\rightarrow
ee/\mu\mu$ event rates, with uncertainties of 20 % for one and 30 % for two
additional $b$-tagged jets, as discussed in Section 4.2.

Misidentified leptons:

Uncertainties in the numbers of events with misidentified leptons arise from
the statistical uncertainties in the corresponding same-sign samples, together
with systematic uncertainties in the opposite-to-same-sign ratios $R^{i}_{j}$
and the estimated contributions of prompt same-sign events. The total
uncertainties in the measured cross-sections are typically 0.2–0.5 %, except
at the extreme ends of distributions where the same-sign data statistical
uncertainties are larger.

The background uncertainties are shown by the solid red lines on Figure 6, and
are dominated by $Wt$ modelling uncertainties, in particular from the
$Wt$-$t\bar{t}$ interference at the high ends of some distributions.

###  5.5 Luminosity and beam energy

Uncertainties in the integrated luminosity and LHC beam energy give rise to
additional uncertainties in the differential cross-section results.

Luminosity:

The uncertainty in the integrated luminosity is 1.9 %, derived from beam-
separation scans performed in November 2012 [90]. The corresponding
uncertainty in the absolute cross-section measurements is slightly larger,
typically about 2.1 %, as the $Wt$ and diboson backgrounds were evaluated from
simulation, thus becoming sensitive to the assumed integrated luminosity. The
sensitivity varies with the background fractions, leaving a residual
uncertainty of typically less than 0.1 % in the normalised cross-section
results.

Beam energy:

The LHC beam energy during the 2012 $pp$ run was determined to be within 0.1 %
of the nominal value of 4 TeV per beam, based on the LHC magnetic model
together with measurements of the revolution frequency difference of proton
and lead-ion beams [91]. Following the approach used in Ref. [13] with an
earlier less precise determination of the LHC beam energy [92], an additional
uncertainty corresponding to the change in cross-sections for a 0.1 % change
in $\sqrt{s}$ was applied to the final results, allowing them to be
interpreted as measurements at exactly $\sqrt{s}=8$ TeV. The changes in each
differential cross-section bin were calculated by scaling the differences seen
in Powheg + Pythia6 samples generated at $\sqrt{s}=8$ TeV and $\sqrt{s}=7$
TeV. The resulting values were cross-checked with an explicit NLO fixed-order
calculation using Sherpa 2.1 [73], making use of the Applgrid framework [93]
to reweight an $\sqrt{s}=8$ TeV prediction so as to change the $\sqrt{s}$
value by $\pm 0.66$ % which was then rescaled to correspond to a $\sqrt{s}$
change of 0.1 %. The changes in the absolute cross-sections are in the range
0.2–0.4 %, and largely cancel in the normalised cross-sections.

These uncertainties are not shown separately in Figure 6, but are included in
the total uncertainties shown by the black lines, and given in Tables 3 to 6.

##  6 Results

The absolute differential cross-sections were determined by solving Eqs. (1)
separately for each bin $i$ of each lepton and dilepton differential
distribution, taking the effects of systematic uncertainties into account as
discussed in Section 5. The normalised differential cross-sections were
determined from the absolute results using Eq. (2). The values of
$\epsilon^{i}_{b}$, i.e. the product of jet acceptance, reconstruction and
$b$-tagging probabilities in each bin, were determined to be in the range
0.5–0.6, in agreement with the simulation prediction for each bin. The results
were found to be stable when changing the minimum jet $p_{\text{T}}$
requirement from 25 GeV up to 55 GeV, and when using $b$-tagging working
points corresponding to $b$-jet efficiencies of 60–80 %. The electron and muon
$p_{\text{T}}$ and $|\eta|$ distributions were also measured separately,
instead of combining them into lepton distributions with two entries per
event, and found to be compatible. The bin-by-bin comparison of the electron
and muon $p_{\text{T}}$ ($|\eta|$) distributions has a $\chi^{2}$ per degree
of freedom of 10.9/9 (12.5/8), in both cases taking into account statistical
and uncorrelated systematic uncertainties.

###  6.1 Fiducial cross-section measurements

The measured absolute and normalised fiducial differential cross-sections are
shown in Table 3 ($p_{\mathrm{T}}^{\ell}$ and et al.), Table 4
($p_{\mathrm{T}}^{e\mu}$ and $m^{e\mu}$), Table 5 ($|y^{e\mu}|$ and
$\Delta\phi^{e\mu}$) and Table 6 ($p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$
and $E^{e}+E^{\mu}$). Each table shows the measured cross-section values and
uncertainties, together with a breakdown of the total uncertainties into
components due to data statistics (‘Stat.’), $t\bar{t}$ modelling
uncertainties (‘$t\bar{t}$ mod.’), lepton-related uncertainties (‘Lept’), jet
and $b$-tagging uncertainties (‘Jet/$b$’), background uncertainties (‘Bkg.’)
and luminosity/beam energy uncertainties (‘$L/E_{\mathrm{b}}$’), corresponding
to the breakdown in Sections 5.1 to 5.5. The rightmost columns show the cross-
sections corrected to remove the contributions where one or both leptons
result from $W\rightarrow\tau\rightarrow e/\mu$ decays using Eq. (3). As can
also be seen from Figure 6, the total uncertainties on the normalised
differential cross-sections range from 1.2 % to around 10 %, typically smaller
than those for the measurements as a function of the $t\bar{t}$ system
kinematics in Ref. [21]. The largest uncertainties are generally statistical
(from 1.1 % to about 10 %), with the background uncertainties also becoming
large at high values of some kinematic variables. Other systematic
uncertainties due to $t\bar{t}$ modelling, leptons and jets are significantly
smaller than the statistical uncertainties, benefiting from cancellations
between bins. The cancellations are particularly important when leptons with
similar $p_{\text{T}}$ contribute to all bins, as is the case for
$\Delta\phi^{e\mu}$ and the bulk of the et al. and $|y^{e\mu}|$ distributions.
The uncertainties in the absolute cross-sections are substantially larger,
with the systematic uncertainties due to $t\bar{t}$ modelling and leptons
becoming comparable to the statistical uncertainties. The absolute cross-
sections also have an uncertainty of 2.1–2.5 % from the integrated luminosity
measurement, depending on the background level in each bin.

The integrals of the differential cross-sections across all bins of a given
distribution ($\sigma^{t\bar{t}}_{\mathrm{fid}}$ in Eq. (2)) agree in all
cases within 0.4 % of the integrated fiducial cross-sections of $3.455\pm
0.025$ pb (or $3.043\pm 0.022$ pb excluding $\tau$ contributions) measured
within the same fiducial region in Ref. [13, 14]. The quoted uncertainties are
statistical.555The integrals of the $p_{\mathrm{T}}^{\ell}$ and et al.
distributions correspond to twice these values, as the definitions include two
leptons per event.

The normalised differential cross-sections are shown graphically in Figures 7
and 8; in these and later Figures, the data points are plotted at the centre
of each bin. The measured cross-sections are compared to the particle-level
predictions from the Powheg + Pythia6, MC@NLO + Herwig and Alpgen + Herwig
$t\bar{t}$ samples within the fiducial volume of the measurement, including
the contributions from $W\rightarrow\tau\rightarrow e/\mu$ decays. Similar
trends in the description of the measured distributions by the predictions can
be seen as for the reconstructed distributions for events with at least one
$b$-tagged jet in Figures 1 and 2.

Absolute | $\mathrm{d}\sigma/\mathrm{d}\mbox{$p_{\mathrm{T}}^{\ell}$}$ | Stat. |  $t\bar{t}$ mod. | Lept. |  Jet/$b$ | Bkg. | $L/E_{\mathrm{b}}$ | Total |  $\mathrm{d}\sigma/\mathrm{d}\mbox{$p_{\mathrm{T}}^{\ell}$}$ (no $\tau$)
---|---|---|---|---|---|---|---|---|---
Bin [GeV] |  [fb/GeV] | (%) | (%) | (%) | (%) | (%) | (%) | (%) |  [fb/GeV]
25–30 | $154.8\pm 5.7$ | 1.6 | 1.3 | 1.8 | 0.8 | 1.2 | 2.1 | 3.7 | $127.2\pm 4.8$
30–40 | $146.1\pm 4.9$ | 1.1 | 1.2 | 1.5 | 0.8 | 1.0 | 2.1 | 3.3 | $124.6\pm 4.2$
40–50 | $118.8\pm 3.7$ | 1.2 | 1.1 | 1.0 | 1.0 | 0.9 | 2.1 | 3.1 | $104.3\pm 3.3$
50–60 | $93.5\pm 2.9$ | 1.4 | 1.0 | 1.0 | 0.8 | 0.9 | 2.1 | 3.1 | $83.4\pm 2.6$
60–80 | $60.0\pm 1.8$ | 1.2 | 0.9 | 0.9 | 0.6 | 0.9 | 2.1 | 3.0 | $54.1\pm 1.6$
80–100 | $32.4\pm 1.1$ | 1.6 | 0.8 | 1.1 | 1.4 | 1.1 | 2.1 | 3.5 | $29.4\pm 1.0$
100–120 | $16.23\pm 0.64$ | 2.3 | 0.9 | 1.1 | 1.1 | 1.5 | 2.2 | 3.9 | $14.75\pm 0.58$
120–150 | $7.61\pm 0.35$ | 2.7 | 1.1 | 1.4 | 1.5 | 1.9 | 2.2 | 4.6 | $6.91\pm 0.32$
150–200 | $2.41\pm 0.15$ | 3.9 | 1.6 | 1.7 | 1.6 | 3.2 | 2.2 | 6.2 | $2.17\pm 0.13$
200–300+ | $0.49\pm 0.06$ | 6.7 | 3.5 | 2.3 | 2.9 | 7.5 | 2.4 | 11.5 | $0.44\pm 0.05$
Normalised | $\frac{1}{\sigma}\mathrm{d}\sigma/\mathrm{d}\mbox{$p_{\mathrm{T}}^{\ell}$}$ | Stat. |  $t\bar{t}$ mod. | Lept. |  Jet/$b$ | Bkg. | $L/E_{\mathrm{b}}$ | Total |  $\frac{1}{\sigma}\mathrm{d}\sigma/\mathrm{d}\mbox{$p_{\mathrm{T}}^{\ell}$}$ (no $\tau$)
Bin [GeV] |  [$10^{-2}/$GeV] | (%) | (%) | (%) | (%) | (%) | (%) | (%) |  [$10^{-2}/$GeV]
25–30 | $2.235\pm 0.045$ | 1.5 | 0.4 | 0.9 | 0.6 | 0.8 | 0.0 | 2.0 | $2.090\pm 0.042$
30–40 | $2.108\pm 0.029$ | 1.0 | 0.3 | 0.7 | 0.3 | 0.5 | 0.0 | 1.4 | $2.048\pm 0.029$
40–50 | $1.714\pm 0.023$ | 1.1 | 0.2 | 0.5 | 0.4 | 0.4 | 0.0 | 1.3 | $1.714\pm 0.023$
50–60 | $1.350\pm 0.019$ | 1.3 | 0.2 | 0.4 | 0.3 | 0.3 | 0.0 | 1.4 | $1.370\pm 0.020$
60–80 | $0.866\pm 0.011$ | 1.1 | 0.3 | 0.4 | 0.4 | 0.3 | 0.0 | 1.3 | $0.890\pm 0.011$
80–100 | $0.4673\pm 0.0093$ | 1.5 | 0.4 | 0.7 | 0.8 | 0.5 | 0.0 | 2.0 | $0.4831\pm 0.0096$
100–120 | $0.2343\pm 0.0063$ | 2.3 | 0.5 | 0.8 | 0.6 | 1.0 | 0.0 | 2.7 | $0.2424\pm 0.0065$
120–150 | $0.1098\pm 0.0040$ | 2.7 | 0.6 | 1.2 | 1.3 | 1.5 | 0.1 | 3.6 | $0.1135\pm 0.0041$
150–200 | $0.0348\pm 0.0018$ | 3.9 | 0.9 | 1.6 | 1.2 | 2.8 | 0.1 | 5.3 | $0.0357\pm 0.0019$
200–300+ | $0.0070\pm 0.0007$ | 6.7 | 2.7 | 2.3 | 2.3 | 7.2 | 0.3 | 10.7 | $0.0072\pm 0.0008$

Absolute | $\mathrm{d}\sigma/\mathrm{d}\textit{et al.}$ | Stat. |  $t\bar{t}$ mod. | Lept. |  Jet/$b$ | Bkg. | $L/E_{\mathrm{b}}$ | Total |  $\mathrm{d}\sigma/\mathrm{d}\textit{et al.}$ (no $\tau$)
---|---|---|---|---|---|---|---|---|---
Bin [unit $\eta$] |  [fb/unit $\eta$] | (%) | (%) | (%) | (%) | (%) | (%) | (%) |  [fb/unit $\eta$]
0.00–0.25 | $4590\pm 140$ | 1.2 | 1.0 | 1.0 | 0.9 | 0.9 | 2.1 | 3.1 | $4030\pm 130$
0.25–0.50 | $4440\pm 140$ | 1.2 | 1.0 | 1.0 | 0.9 | 0.9 | 2.1 | 3.1 | $3900\pm 120$
0.50–0.75 | $4230\pm 130$ | 1.2 | 1.0 | 1.0 | 0.9 | 0.9 | 2.1 | 3.1 | $3710\pm 120$
0.75–1.00 | $3660\pm 110$ | 1.3 | 1.0 | 1.0 | 0.8 | 1.0 | 2.1 | 3.1 | $3210\pm 100$
1.00–1.25 | $3100\pm 100$ | 1.5 | 1.0 | 1.0 | 0.9 | 1.0 | 2.1 | 3.3 | $2722\pm 89$
1.25–1.50 | $2470\pm 87$ | 1.9 | 1.1 | 1.0 | 0.9 | 1.0 | 2.1 | 3.5 | $2173\pm 77$
1.50–1.75 | $2035\pm 73$ | 1.9 | 1.1 | 1.4 | 0.7 | 1.0 | 2.1 | 3.6 | $1793\pm 65$
1.75–2.00 | $1431\pm 57$ | 2.4 | 1.2 | 1.4 | 0.6 | 1.4 | 2.1 | 4.0 | $1263\pm 50$
2.00–2.50 | $844\pm 34$ | 2.2 | 1.3 | 1.4 | 0.7 | 1.4 | 2.1 | 4.0 | $749\pm 30$
Normalised | $\frac{1}{\sigma}\mathrm{d}\sigma/\mathrm{d}\textit{et al.}$ | Stat. |  $t\bar{t}$ mod. | Lept. |  Jet/$b$ | Bkg. | $L/E_{\mathrm{b}}$ | Total |  $\frac{1}{\sigma}\mathrm{d}\sigma/\mathrm{d}\textit{et al.}$ (no $\tau$)
Bin [unit $\eta$] |  [$10^{-1}/$unit $\eta$] | (%) | (%) | (%) | (%) | (%) | (%) | (%) |  [$10^{-1}/$unit $\eta$]
0.00–0.25 | $6.646\pm 0.083$ | 1.1 | 0.4 | 0.2 | 0.1 | 0.3 | 0.0 | 1.2 | $6.632\pm 0.083$
0.25–0.50 | $6.428\pm 0.076$ | 1.1 | 0.3 | 0.2 | 0.1 | 0.2 | 0.0 | 1.2 | $6.416\pm 0.076$
0.50–0.75 | $6.117\pm 0.074$ | 1.1 | 0.2 | 0.2 | 0.1 | 0.2 | 0.0 | 1.2 | $6.103\pm 0.074$
0.75–1.00 | $5.297\pm 0.070$ | 1.3 | 0.2 | 0.2 | 0.2 | 0.2 | 0.0 | 1.3 | $5.286\pm 0.070$
1.00–1.25 | $4.482\pm 0.066$ | 1.4 | 0.3 | 0.2 | 0.2 | 0.3 | 0.0 | 1.5 | $4.484\pm 0.066$
1.25–1.50 | $3.574\pm 0.068$ | 1.8 | 0.4 | 0.2 | 0.2 | 0.3 | 0.0 | 1.9 | $3.579\pm 0.068$
1.50–1.75 | $2.944\pm 0.060$ | 1.9 | 0.5 | 0.6 | 0.2 | 0.4 | 0.0 | 2.0 | $2.954\pm 0.061$
1.75–2.00 | $2.070\pm 0.054$ | 2.3 | 0.6 | 0.6 | 0.4 | 0.8 | 0.0 | 2.6 | $2.080\pm 0.054$
2.00–2.50 | $1.221\pm 0.031$ | 2.1 | 0.7 | 0.6 | 0.4 | 1.0 | 0.0 | 2.5 | $1.233\pm 0.031$

Table 3: Absolute and normalised differential cross-sections as functions of
$p_{\mathrm{T}}^{\ell}$ (top) and et al. (bottom). The columns show the bin
ranges, measured cross-section and total uncertainty, relative statistical
uncertainty, relative systematic uncertainties in various categories (see
text), total relative uncertainty, and differential cross-section corrected to
remove contributions via $W\rightarrow\tau\rightarrow e/\mu$ decays. Relative
uncertainties smaller than 0.05 % are indicated by ‘0.0’. The last bin
includes overflows where indicated by the ‘+’ sign.

Absolute | $\mathrm{d}\sigma/\mathrm{d}\mbox{$p_{\mathrm{T}}^{e\mu}$}$ | Stat. |  $t\bar{t}$ mod. | Lept. |  Jet/$b$ | Bkg. | $L/E_{\mathrm{b}}$ | Total |  $\mathrm{d}\sigma/\mathrm{d}\mbox{$p_{\mathrm{T}}^{e\mu}$}$ (no $\tau$)
---|---|---|---|---|---|---|---|---|---
Bin [GeV] |  [fb/GeV] | (%) | (%) | (%) | (%) | (%) | (%) | (%) |  [fb/GeV]
0–20 | $11.50\pm 0.49$ | 2.8 | 1.2 | 1.2 | 1.1 | 1.2 | 2.1 | 4.2 | $9.62\pm 0.41$
20–40 | $26.72\pm 0.94$ | 1.9 | 1.0 | 1.2 | 1.0 | 1.0 | 2.1 | 3.5 | $22.62\pm 0.81$
40–60 | $35.9\pm 1.2$ | 1.6 | 0.9 | 1.2 | 1.1 | 1.0 | 2.1 | 3.4 | $30.6\pm 1.0$
60–80 | $39.0\pm 1.3$ | 1.5 | 0.9 | 1.1 | 0.9 | 0.9 | 2.1 | 3.2 | $34.4\pm 1.1$
80–100 | $29.19\pm 0.96$ | 1.7 | 1.0 | 1.0 | 0.8 | 1.0 | 2.1 | 3.3 | $26.48\pm 0.88$
100–120 | $16.38\pm 0.65$ | 2.3 | 1.2 | 1.3 | 1.3 | 1.1 | 2.1 | 3.9 | $15.11\pm 0.60$
120–150 | $6.53\pm 0.30$ | 2.9 | 1.4 | 1.5 | 1.2 | 1.6 | 2.2 | 4.6 | $6.06\pm 0.28$
150–200 | $1.39\pm 0.11$ | 5.0 | 2.2 | 2.1 | 2.2 | 4.2 | 2.3 | 7.9 | $1.27\pm 0.10$
200–300+ | $0.23\pm 0.04$ | 9.3 | 4.2 | 2.9 | 4.1 | 13.2 | 2.5 | 17.6 | $0.20\pm 0.04$
Normalised | $\frac{1}{\sigma}\mathrm{d}\sigma/\mathrm{d}\mbox{$p_{\mathrm{T}}^{e\mu}$}$ | Stat. |  $t\bar{t}$ mod. | Lept. |  Jet/$b$ | Bkg. | $L/E_{\mathrm{b}}$ | Total |  $\frac{1}{\sigma}\mathrm{d}\sigma/\mathrm{d}\mbox{$p_{\mathrm{T}}^{e\mu}$}$ (no $\tau$)
Bin [GeV] |  [$10^{-2}/$GeV] | (%) | (%) | (%) | (%) | (%) | (%) | (%) |  [$10^{-2}/$GeV]
0–20 | $0.332\pm 0.010$ | 2.7 | 0.5 | 0.5 | 0.7 | 0.8 | 0.0 | 3.0 | $0.316\pm 0.010$
20–40 | $0.772\pm 0.015$ | 1.7 | 0.4 | 0.5 | 0.5 | 0.6 | 0.0 | 2.0 | $0.743\pm 0.015$
40–60 | $1.036\pm 0.017$ | 1.4 | 0.3 | 0.4 | 0.5 | 0.5 | 0.0 | 1.6 | $1.006\pm 0.017$
60–80 | $1.127\pm 0.016$ | 1.3 | 0.3 | 0.3 | 0.4 | 0.3 | 0.0 | 1.5 | $1.130\pm 0.016$
80–100 | $0.843\pm 0.015$ | 1.5 | 0.3 | 0.3 | 0.6 | 0.4 | 0.0 | 1.8 | $0.870\pm 0.015$
100–120 | $0.473\pm 0.012$ | 2.2 | 0.4 | 0.8 | 0.8 | 0.5 | 0.0 | 2.5 | $0.497\pm 0.013$
120–150 | $0.1886\pm 0.0066$ | 2.8 | 0.6 | 1.2 | 1.0 | 1.2 | 0.0 | 3.5 | $0.1993\pm 0.0069$
150–200 | $0.0402\pm 0.0028$ | 4.9 | 1.4 | 1.9 | 1.9 | 3.9 | 0.2 | 7.0 | $0.0419\pm 0.0029$
200–300+ | $0.0066\pm 0.0011$ | 9.2 | 3.3 | 2.7 | 3.6 | 13.0 | 0.4 | 16.9 | $0.0067\pm 0.0011$

Absolute | $\mathrm{d}\sigma/\mathrm{d}\mbox{$m^{e\mu}$}$ | Stat. |  $t\bar{t}$ mod. | Lept. |  Jet/$b$ | Bkg. | $L/E_{\mathrm{b}}$ | Total |  $\mathrm{d}\sigma/\mathrm{d}\mbox{$m^{e\mu}$}$ (no $\tau$)
---|---|---|---|---|---|---|---|---|---
Bin [GeV] |  [fb/GeV] | (%) | (%) | (%) | (%) | (%) | (%) | (%) |  [fb/GeV]
0–20 | $3.37\pm 0.25$ | 6.3 | 2.0 | 1.9 | 1.3 | 1.3 | 2.1 | 7.4 | $2.97\pm 0.22$
20–40 | $10.94\pm 0.47$ | 2.8 | 1.5 | 1.4 | 0.9 | 1.2 | 2.1 | 4.3 | $9.61\pm 0.41$
40–60 | $17.66\pm 0.70$ | 2.2 | 1.4 | 1.5 | 0.7 | 1.3 | 2.1 | 4.0 | $15.29\pm 0.61$
60–80 | $23.98\pm 0.89$ | 1.9 | 1.3 | 1.4 | 0.8 | 1.2 | 2.1 | 3.7 | $20.54\pm 0.76$
80–100 | $26.00\pm 0.90$ | 1.8 | 1.2 | 1.1 | 0.8 | 0.9 | 2.1 | 3.4 | $22.42\pm 0.78$
100–120 | $23.03\pm 0.83$ | 2.0 | 1.1 | 1.0 | 1.2 | 0.9 | 2.1 | 3.6 | $20.07\pm 0.73$
120–150 | $16.71\pm 0.57$ | 1.9 | 1.0 | 0.9 | 1.0 | 1.0 | 2.1 | 3.4 | $14.72\pm 0.51$
150–200 | $9.38\pm 0.34$ | 1.9 | 0.8 | 0.9 | 1.3 | 1.2 | 2.1 | 3.6 | $8.41\pm 0.30$
200–250 | $4.09\pm 0.18$ | 3.0 | 0.8 | 1.1 | 1.1 | 1.6 | 2.1 | 4.4 | $3.73\pm 0.16$
250–300 | $1.95\pm 0.11$ | 4.4 | 1.0 | 1.2 | 1.4 | 2.1 | 2.2 | 5.7 | $1.80\pm 0.10$
300–400 | $0.66\pm 0.05$ | 5.2 | 1.3 | 1.5 | 2.0 | 3.1 | 2.2 | 7.1 | $0.62\pm 0.04$
400–500+ | $0.26\pm 0.03$ | 8.0 | 2.2 | 2.1 | 2.1 | 4.8 | 2.2 | 10.3 | $0.25\pm 0.03$
Normalised | $\frac{1}{\sigma}\mathrm{d}\sigma/\mathrm{d}\mbox{$m^{e\mu}$}$ | Stat. |  $t\bar{t}$ mod. | Lept. |  Jet/$b$ | Bkg. | $L/E_{\mathrm{b}}$ | Total |  $\frac{1}{\sigma}\mathrm{d}\sigma/\mathrm{d}\mbox{$m^{e\mu}$}$ (no $\tau$)
Bin [GeV] |  [$10^{-3}/$GeV] | (%) | (%) | (%) | (%) | (%) | (%) | (%) |  [$10^{-3}/$GeV]
0–20 | $0.973\pm 0.066$ | 6.2 | 1.4 | 1.2 | 1.4 | 1.3 | 0.0 | 6.7 | $0.977\pm 0.066$
20–40 | $3.157\pm 0.095$ | 2.7 | 0.7 | 0.5 | 0.4 | 0.9 | 0.0 | 3.0 | $3.156\pm 0.095$
40–60 | $5.10\pm 0.13$ | 2.1 | 0.5 | 0.7 | 0.4 | 1.0 | 0.0 | 2.5 | $5.02\pm 0.13$
60–80 | $6.92\pm 0.14$ | 1.8 | 0.4 | 0.6 | 0.3 | 0.7 | 0.0 | 2.1 | $6.75\pm 0.14$
80–100 | $7.51\pm 0.14$ | 1.7 | 0.3 | 0.3 | 0.6 | 0.4 | 0.0 | 1.8 | $7.37\pm 0.14$
100–120 | $6.65\pm 0.14$ | 1.8 | 0.3 | 0.3 | 0.7 | 0.3 | 0.0 | 2.0 | $6.60\pm 0.14$
120–150 | $4.823\pm 0.092$ | 1.7 | 0.3 | 0.3 | 0.5 | 0.3 | 0.0 | 1.9 | $4.839\pm 0.092$
150–200 | $2.707\pm 0.058$ | 1.8 | 0.4 | 0.5 | 0.7 | 0.6 | 0.0 | 2.1 | $2.763\pm 0.059$
200–250 | $1.180\pm 0.041$ | 2.9 | 0.7 | 0.8 | 1.0 | 1.2 | 0.0 | 3.4 | $1.224\pm 0.042$
250–300 | $0.563\pm 0.029$ | 4.4 | 1.0 | 1.0 | 1.4 | 1.7 | 0.0 | 5.1 | $0.590\pm 0.030$
300–400 | $0.191\pm 0.012$ | 5.2 | 1.8 | 1.4 | 1.6 | 2.8 | 0.1 | 6.5 | $0.203\pm 0.013$
400–500+ | $0.0763\pm 0.0076$ | 8.0 | 2.6 | 1.9 | 2.0 | 4.5 | 0.1 | 9.9 | $0.0820\pm 0.0081$

Table 4: Absolute and normalised differential cross-sections as functions of
$p_{\mathrm{T}}^{e\mu}$ (top) and $m^{e\mu}$ (bottom). The columns show the
bin ranges, measured cross-section and total uncertainty, relative statistical
uncertainty, relative systematic uncertainties in various categories (see
text), total relative uncertainty, and differential cross-section corrected to
remove contributions via $W\rightarrow\tau\rightarrow e/\mu$ decays. Relative
uncertainties smaller than 0.05 % are indicated by ‘0.0’. The last bin
includes overflows where indicated by the ‘+’ sign.

Absolute | $\mathrm{d}\sigma/\mathrm{d}\mbox{$|y^{e\mu}|$}$ | Stat. |  $t\bar{t}$ mod. | Lept. |  Jet/$b$ | Bkg. | $L/E_{\mathrm{b}}$ | Total |  $\mathrm{d}\sigma/\mathrm{d}\mbox{$|y^{e\mu}|$}$ (no $\tau$)
---|---|---|---|---|---|---|---|---|---
Bin [unit $y$] |  [fb/unit $y$] | (%) | (%) | (%) | (%) | (%) | (%) | (%) |  [fb/unit $y$]
0.00–0.25 | $3007\pm 95$ | 1.5 | 0.9 | 1.0 | 1.0 | 0.8 | 2.1 | 3.2 | $2639\pm 84$
0.25–0.50 | $2681\pm 86$ | 1.5 | 0.9 | 1.0 | 0.8 | 0.9 | 2.1 | 3.2 | $2353\pm 76$
0.50–0.75 | $2419\pm 80$ | 1.6 | 1.0 | 1.0 | 1.0 | 1.0 | 2.1 | 3.3 | $2123\pm 71$
0.75–1.00 | $2026\pm 71$ | 1.9 | 1.1 | 1.1 | 0.8 | 1.1 | 2.1 | 3.5 | $1780\pm 63$
1.00–1.25 | $1536\pm 57$ | 2.2 | 1.2 | 1.1 | 0.8 | 1.0 | 2.1 | 3.7 | $1351\pm 50$
1.25–1.50 | $1038\pm 43$ | 2.7 | 1.3 | 1.3 | 0.7 | 1.2 | 2.1 | 4.2 | $912\pm 38$
1.50–1.75 | $637\pm 33$ | 3.7 | 1.6 | 1.6 | 1.3 | 1.3 | 2.2 | 5.2 | $561\pm 29$
1.75–2.00 | $321\pm 23$ | 5.7 | 2.0 | 1.8 | 1.2 | 2.1 | 2.2 | 7.1 | $283\pm 20$
2.00–2.50 | $69.1\pm 7.7$ | 8.8 | 2.7 | 2.3 | 2.3 | 4.7 | 2.2 | 11.1 | $61.3\pm 6.8$
Normalised | $\frac{1}{\sigma}\mathrm{d}\sigma/\mathrm{d}\mbox{$|y^{e\mu}|$}$ | Stat. |  $t\bar{t}$ mod. | Lept. |  Jet/$b$ | Bkg. | $L/E_{\mathrm{b}}$ | Total |  $\frac{1}{\sigma}\mathrm{d}\sigma/\mathrm{d}\mbox{$|y^{e\mu}|$}$ (no $\tau$)
Bin [unit $y$] |  [$10^{-1}/$unit $y$] | (%) | (%) | (%) | (%) | (%) | (%) | (%) |  [$10^{-1}/$unit $y$]
0.00–0.25 | $8.71\pm 0.13$ | 1.3 | 0.3 | 0.2 | 0.3 | 0.3 | 0.0 | 1.4 | $8.71\pm 0.13$
0.25–0.50 | $7.77\pm 0.12$ | 1.4 | 0.3 | 0.2 | 0.3 | 0.3 | 0.0 | 1.5 | $7.76\pm 0.12$
0.50–0.75 | $7.01\pm 0.11$ | 1.5 | 0.3 | 0.2 | 0.3 | 0.3 | 0.0 | 1.6 | $7.00\pm 0.11$
0.75–1.00 | $5.87\pm 0.11$ | 1.7 | 0.4 | 0.2 | 0.3 | 0.4 | 0.0 | 1.9 | $5.87\pm 0.11$
1.00–1.25 | $4.451\pm 0.099$ | 2.1 | 0.5 | 0.2 | 0.4 | 0.3 | 0.0 | 2.2 | $4.458\pm 0.099$
1.25–1.50 | $3.009\pm 0.085$ | 2.6 | 0.6 | 0.4 | 0.6 | 0.6 | 0.0 | 2.8 | $3.009\pm 0.085$
1.50–1.75 | $1.846\pm 0.073$ | 3.6 | 0.8 | 0.7 | 0.8 | 0.8 | 0.0 | 4.0 | $1.850\pm 0.073$
1.75–2.00 | $0.930\pm 0.057$ | 5.6 | 1.3 | 1.0 | 0.9 | 1.8 | 0.1 | 6.2 | $0.935\pm 0.058$
2.00–2.50 | $0.200\pm 0.021$ | 8.8 | 2.1 | 1.7 | 2.5 | 4.4 | 0.1 | 10.5 | $0.202\pm 0.021$

Absolute | $\mathrm{d}\sigma/\mathrm{d}\mbox{$\Delta\phi^{e\mu}$}$ | Stat. |  $t\bar{t}$ mod. | Lept. |  Jet/$b$ | Bkg. | $L/E_{\mathrm{b}}$ | Total |  $\mathrm{d}\sigma/\mathrm{d}\mbox{$\Delta\phi^{e\mu}$}$ (no $\tau$)
---|---|---|---|---|---|---|---|---|---
Bin [rad] | [fb/rad] | (%) | (%) | (%) | (%) | (%) | (%) | (%) | [fb/rad]
0.00–0.31 | $696\pm 30$ | 2.9 | 1.4 | 1.3 | 0.7 | 0.9 | 2.1 | 4.2 | $630\pm 27$
0.31–0.63 | $735\pm 29$ | 2.7 | 1.3 | 1.3 | 0.6 | 0.9 | 2.1 | 4.0 | $664\pm 26$
0.63–0.94 | $780\pm 31$ | 2.6 | 1.3 | 1.2 | 0.7 | 0.9 | 2.1 | 3.9 | $704\pm 28$
0.94–1.26 | $850\pm 33$ | 2.5 | 1.2 | 1.2 | 0.7 | 0.9 | 2.1 | 3.9 | $763\pm 30$
1.26–1.57 | $947\pm 36$ | 2.3 | 1.1 | 1.2 | 0.8 | 1.0 | 2.1 | 3.8 | $844\pm 32$
1.57–1.88 | $1103\pm 41$ | 2.2 | 1.1 | 1.1 | 1.1 | 0.9 | 2.1 | 3.7 | $977\pm 37$
1.88–2.20 | $1235\pm 43$ | 2.1 | 1.0 | 1.0 | 0.7 | 0.9 | 2.1 | 3.5 | $1084\pm 38$
2.20–2.51 | $1410\pm 50$ | 2.0 | 1.0 | 1.0 | 1.1 | 1.0 | 2.1 | 3.5 | $1226\pm 44$
2.51–2.83 | $1575\pm 56$ | 2.0 | 0.9 | 0.9 | 1.1 | 1.1 | 2.1 | 3.6 | $1353\pm 49$
2.83–3.14 | $1696\pm 58$ | 1.9 | 0.9 | 0.9 | 0.9 | 1.2 | 2.1 | 3.4 | $1449\pm 51$
Normalised | $\frac{1}{\sigma}\mathrm{d}\sigma/\mathrm{d}\mbox{$\Delta\phi^{e\mu}$}$ | Stat. |  $t\bar{t}$ mod. | Lept. |  Jet/$b$ | Bkg. | $L/E_{\mathrm{b}}$ | Total |  $\frac{1}{\sigma}\mathrm{d}\sigma/\mathrm{d}\mbox{$\Delta\phi^{e\mu}$}$ (no $\tau$)
Bin [rad] |  [$10^{-1}/$rad] | (%) | (%) | (%) | (%) | (%) | (%) | (%) |  [$10^{-1}/$rad]
0.00–0.31 | $2.010\pm 0.060$ | 2.8 | 0.6 | 0.4 | 0.5 | 0.4 | 0.0 | 3.0 | $2.068\pm 0.062$
0.31–0.63 | $2.121\pm 0.057$ | 2.6 | 0.5 | 0.3 | 0.4 | 0.4 | 0.0 | 2.7 | $2.179\pm 0.058$
0.63–0.94 | $2.252\pm 0.059$ | 2.5 | 0.5 | 0.3 | 0.2 | 0.4 | 0.0 | 2.6 | $2.311\pm 0.060$
0.94–1.26 | $2.454\pm 0.064$ | 2.4 | 0.5 | 0.2 | 0.6 | 0.4 | 0.0 | 2.6 | $2.506\pm 0.065$
1.26–1.57 | $2.732\pm 0.064$ | 2.2 | 0.4 | 0.2 | 0.2 | 0.4 | 0.0 | 2.3 | $2.773\pm 0.065$
1.57–1.88 | $3.185\pm 0.070$ | 2.1 | 0.4 | 0.1 | 0.3 | 0.3 | 0.0 | 2.2 | $3.207\pm 0.071$
1.88–2.20 | $3.566\pm 0.073$ | 2.0 | 0.4 | 0.1 | 0.2 | 0.3 | 0.0 | 2.0 | $3.559\pm 0.072$
2.20–2.51 | $4.069\pm 0.079$ | 1.9 | 0.4 | 0.1 | 0.3 | 0.3 | 0.0 | 1.9 | $4.028\pm 0.078$
2.51–2.83 | $4.546\pm 0.088$ | 1.8 | 0.4 | 0.2 | 0.3 | 0.4 | 0.0 | 1.9 | $4.443\pm 0.086$
2.83–3.14 | $4.897\pm 0.090$ | 1.7 | 0.4 | 0.3 | 0.1 | 0.5 | 0.0 | 1.8 | $4.757\pm 0.088$

Table 5: Absolute and normalised differential cross-sections as functions of
$|y^{e\mu}|$ (top) and $\Delta\phi^{e\mu}$ (bottom). The columns show the bin
ranges, measured cross-section and total uncertainty, relative statistical
uncertainty, relative systematic uncertainties in various categories (see
text), total relative uncertainty, and differential cross-section corrected to
remove contributions via $W\rightarrow\tau\rightarrow e/\mu$ decays. Relative
uncertainties smaller than 0.05 % are indicated by ‘0.0’. The bin boundaries
for $\Delta\phi^{e\mu}$ correspond to exact multiples of $\pi/10$ but are
quoted to two decimal places.

Absolute | $\mathrm{d}\sigma/\mathrm{d}(\mbox{$p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$})$ | Stat. |  $t\bar{t}$ mod. | Lept. |  Jet/$b$ | Bkg. | $L/E_{\mathrm{b}}$ | Total |  $\mathrm{d}\sigma/\mathrm{d}(\mbox{$p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$})$ (no $\tau$)
---|---|---|---|---|---|---|---|---|---
Bin [GeV] |  [fb/GeV] | (%) | (%) | (%) | (%) | (%) | (%) | (%) |  [fb/GeV]
50–80 | $23.02\pm 0.89$ | 1.7 | 1.2 | 1.8 | 0.7 | 1.5 | 2.1 | 3.8 | $18.90\pm 0.73$
80–100 | $38.0\pm 1.2$ | 1.5 | 1.0 | 1.2 | 0.8 | 0.9 | 2.1 | 3.3 | $33.0\pm 1.1$
100–120 | $34.3\pm 1.1$ | 1.6 | 1.0 | 1.0 | 0.9 | 0.9 | 2.1 | 3.2 | $30.5\pm 1.0$
120–150 | $21.00\pm 0.69$ | 1.7 | 0.9 | 1.0 | 0.8 | 1.0 | 2.1 | 3.3 | $18.95\pm 0.63$
150–200 | $9.11\pm 0.34$ | 1.9 | 1.0 | 1.1 | 1.2 | 1.5 | 2.2 | 3.8 | $8.27\pm 0.31$
200–250 | $3.03\pm 0.16$ | 3.5 | 1.3 | 1.5 | 1.4 | 2.4 | 2.2 | 5.3 | $2.78\pm 0.15$
250–300 | $1.08\pm 0.09$ | 5.7 | 1.9 | 2.0 | 2.9 | 3.5 | 2.2 | 8.1 | $0.99\pm 0.08$
300–400+ | $0.38\pm 0.04$ | 7.0 | 3.3 | 2.3 | 3.3 | 6.5 | 2.3 | 11.1 | $0.35\pm 0.04$
Normalised | $\frac{1}{\sigma}\mathrm{d}\sigma/\mathrm{d}(\mbox{$p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$})$ | Stat. |  $t\bar{t}$ mod. | Lept. |  Jet/$b$ | Bkg. | $L/E_{\mathrm{b}}$ | Total |  $\frac{1}{\sigma}\mathrm{d}\sigma/\mathrm{d}(\mbox{$p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$})$ (no $\tau$)
Bin [GeV] |  [$10^{-2}/$GeV] | (%) | (%) | (%) | (%) | (%) | (%) | (%) |  [$10^{-2}/$GeV]
50–80 | $0.664\pm 0.015$ | 1.5 | 0.4 | 1.0 | 0.3 | 1.2 | 0.0 | 2.2 | $0.621\pm 0.014$
80–100 | $1.097\pm 0.017$ | 1.3 | 0.3 | 0.5 | 0.4 | 0.6 | 0.0 | 1.6 | $1.085\pm 0.018$
100–120 | $0.990\pm 0.016$ | 1.4 | 0.3 | 0.3 | 0.4 | 0.4 | 0.0 | 1.6 | $1.003\pm 0.016$
120–150 | $0.606\pm 0.010$ | 1.5 | 0.3 | 0.5 | 0.3 | 0.4 | 0.0 | 1.7 | $0.623\pm 0.010$
150–200 | $0.2627\pm 0.0062$ | 1.8 | 0.4 | 0.9 | 0.6 | 1.0 | 0.0 | 2.4 | $0.2716\pm 0.0063$
200–250 | $0.0875\pm 0.0039$ | 3.4 | 0.7 | 1.4 | 1.4 | 2.1 | 0.1 | 4.5 | $0.0912\pm 0.0041$
250–300 | $0.0311\pm 0.0023$ | 5.6 | 1.3 | 2.0 | 2.5 | 3.2 | 0.1 | 7.3 | $0.0326\pm 0.0024$
300–400+ | $0.0110\pm 0.0011$ | 7.0 | 2.5 | 2.4 | 2.9 | 6.2 | 0.2 | 10.4 | $0.0116\pm 0.0012$

Absolute | $\mathrm{d}\sigma/\mathrm{d}(\mbox{$E^{e}+E^{\mu}$})$ | Stat. |  $t\bar{t}$ mod. | Lept. |  Jet/$b$ | Bkg. | $L/E_{\mathrm{b}}$ | Total |  $\mathrm{d}\sigma/\mathrm{d}(\mbox{$E^{e}+E^{\mu}$})$ (no $\tau$)
---|---|---|---|---|---|---|---|---|---
Bin [GeV] |  [fb/GeV] | (%) | (%) | (%) | (%) | (%) | (%) | (%) |  [fb/GeV]
50–80 | $4.05\pm 0.22$ | 3.8 | 1.3 | 2.2 | 1.1 | 1.2 | 2.1 | 5.3 | $3.21\pm 0.17$
80–100 | $13.68\pm 0.57$ | 2.5 | 1.2 | 1.6 | 1.2 | 1.1 | 2.1 | 4.2 | $11.38\pm 0.48$
100–120 | $18.36\pm 0.67$ | 2.1 | 1.1 | 1.3 | 0.8 | 0.9 | 2.1 | 3.6 | $15.67\pm 0.57$
120–150 | $19.10\pm 0.64$ | 1.7 | 1.1 | 1.1 | 0.8 | 0.9 | 2.1 | 3.3 | $16.63\pm 0.56$
150–200 | $15.79\pm 0.51$ | 1.5 | 1.0 | 0.9 | 1.0 | 0.9 | 2.1 | 3.2 | $13.92\pm 0.45$
200–250 | $10.04\pm 0.35$ | 1.8 | 1.0 | 0.9 | 1.1 | 1.1 | 2.1 | 3.5 | $8.97\pm 0.31$
250–300 | $6.24\pm 0.25$ | 2.4 | 1.0 | 1.0 | 1.2 | 1.4 | 2.1 | 4.0 | $5.61\pm 0.22$
300–400 | $3.04\pm 0.13$ | 2.5 | 1.1 | 1.3 | 0.9 | 1.7 | 2.2 | 4.2 | $2.75\pm 0.12$
400–500 | $1.20\pm 0.07$ | 4.3 | 1.5 | 2.0 | 1.3 | 2.4 | 2.2 | 6.1 | $1.10\pm 0.07$
500–700+ | $0.48\pm 0.04$ | 4.8 | 2.3 | 2.3 | 1.6 | 3.8 | 2.2 | 7.4 | $0.44\pm 0.03$
Normalised | $\frac{1}{\sigma}\mathrm{d}\sigma/\mathrm{d}(\mbox{$E^{e}+E^{\mu}$})$ | Stat. |  $t\bar{t}$ mod. | Lept. |  Jet/$b$ | Bkg. | $L/E_{\mathrm{b}}$ | Total |  $\frac{1}{\sigma}\mathrm{d}\sigma/\mathrm{d}(\mbox{$E^{e}+E^{\mu}$})$ (no $\tau$)
Bin [GeV] |  [$10^{-3}/$GeV] | (%) | (%) | (%) | (%) | (%) | (%) | (%) |  [$10^{-3}/$GeV]
50–80 | $1.172\pm 0.050$ | 3.8 | 0.8 | 1.4 | 0.7 | 1.1 | 0.1 | 4.3 | $1.058\pm 0.046$
80–100 | $3.95\pm 0.11$ | 2.4 | 0.5 | 0.8 | 0.7 | 0.9 | 0.0 | 2.8 | $3.75\pm 0.11$
100–120 | $5.31\pm 0.12$ | 2.0 | 0.4 | 0.6 | 0.6 | 0.7 | 0.0 | 2.3 | $5.16\pm 0.12$
120–150 | $5.521\pm 0.099$ | 1.6 | 0.3 | 0.4 | 0.4 | 0.5 | 0.0 | 1.8 | $5.478\pm 0.099$
150–200 | $4.564\pm 0.067$ | 1.3 | 0.3 | 0.3 | 0.4 | 0.3 | 0.0 | 1.5 | $4.585\pm 0.067$
200–250 | $2.904\pm 0.055$ | 1.7 | 0.4 | 0.4 | 0.5 | 0.4 | 0.0 | 1.9 | $2.955\pm 0.056$
250–300 | $1.803\pm 0.048$ | 2.3 | 0.5 | 0.5 | 0.9 | 0.7 | 0.0 | 2.6 | $1.849\pm 0.049$
300–400 | $0.878\pm 0.026$ | 2.4 | 0.5 | 0.8 | 0.7 | 1.1 | 0.0 | 2.9 | $0.907\pm 0.026$
400–500 | $0.348\pm 0.018$ | 4.2 | 1.0 | 1.7 | 1.2 | 2.0 | 0.1 | 5.2 | $0.362\pm 0.019$
500–700+ | $0.1393\pm 0.0091$ | 4.7 | 1.8 | 1.9 | 1.2 | 3.5 | 0.1 | 6.5 | $0.1463\pm 0.0095$

Table 6: Absolute and normalised differential cross-sections as functions of
$p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ (top) and $E^{e}+E^{\mu}$ (bottom).
The columns show the bin ranges, measured cross-section and total uncertainty,
relative statistical uncertainty, relative systematic uncertainties in various
categories (see text), total relative uncertainty, and differential cross-
section corrected to remove contributions via $W\rightarrow\tau\rightarrow
e/\mu$ decays. Relative uncertainties smaller than 0.05 % are indicated by
‘0.0’. The last bin includes overflows where indicated by the ‘+’ sign.

(a)

(b)

(c)

(d)

Figure 7: Normalised differential cross-sections as a function of (a)
$p_{\mathrm{T}}^{\ell}$, (b) et al., (c) $p_{\mathrm{T}}^{e\mu}$ and (d)
$m^{e\mu}$. The measured values are shown by the black points with error bars
corresponding to the data statistical uncertainties and cyan bands
corresponding to the total uncertainties in each bin, and include the
contributions via $W\rightarrow\tau\rightarrow e/\mu$ decays. The results are
compared to the predictions from the Powheg + Pythia6, MC@NLO + Herwig and
Alpgen + Herwig $t\bar{t}$ simulation samples. The lower plots show the ratios
of predictions to data, with the error bars indicating the data statistical
uncertainties and the cyan bands indicating the total uncertainties in the
measurements.

(a)

(b)

(c)

(d)

Figure 8: Normalised differential cross-sections as a function of (a)
$|y^{e\mu}|$, (b) $\Delta\phi^{e\mu}$, (c)
$p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ and (d) $E^{e}+E^{\mu}$. The
measured values are shown by the black points with error bars corresponding to
the data statistical uncertainties and cyan bands corresponding to the total
uncertainties in each bin, and include the contributions via
$W\rightarrow\tau\rightarrow e/\mu$ decays. The results are compared to the
predictions from the Powheg + Pythia6, MC@NLO + Herwig and Alpgen + Herwig
$t\bar{t}$ simulation samples. The lower plots show the ratios of predictions
to data, with the error bars indicating the data statistical uncertainties and
the cyan bands indicating the total uncertainties in the measurements.

###  6.2 Comparison with event generator predictions

The measured normalised differential cross-sections are compared to a larger
set of predictions from different $t\bar{t}$ Monte Carlo event generator
configurations in Figures 9 to 12. The figures show the ratios of each
prediction to the data as a function of the differential variables, organised
into four groups of samples as summarised in Table 7. These event generator
setups and tunes were used in ATLAS top physics analyses at $\sqrt{s}=7$ TeV
and $\sqrt{s}=8$ TeV, or have been studied in preparation for analyses at
$\sqrt{s}=13$ TeV [55, 94, 95].

| Matrix-element | PDF | Parton shower | Tune | Comments
---|---|---|---|---|---
1 | Powheg | CT10 | Pythia6 | P2011C | $\mbox{$h_{\mathrm{damp}}$}=\mbox{$m_{t}$}$
| Powheg | CT10 | Pythia6 | P2012 radHi |  $\mbox{$h_{\mathrm{damp}}$}=2\mbox{$m_{t}$}$, $\frac{1}{2}\mu_{F,R}$
| Powheg | CT10 | Pythia6 | P2012 radLo |  $\mbox{$h_{\mathrm{damp}}$}=\mbox{$m_{t}$}$, $2\mu_{F,R}$
2 | Powheg | CT10 | Pythia6 | P2011C | $\mbox{$h_{\mathrm{damp}}$}=\infty$
| Powheg | CT10 | Pythia6 | P2011C |  $\mbox{$h_{\mathrm{damp}}$}=\mbox{$m_{t}$}$, NNLO top $p_{\text{T}}$
| Powheg | CT10 | Pythia8 | A14 | $\mbox{$h_{\mathrm{damp}}$}=\mbox{$m_{t}$}$
3 | Powheg | HERAPDF 1.5 | Pythia6 | P2011C | $\mbox{$h_{\mathrm{damp}}$}=\mbox{$m_{t}$}$
| Powheg | CT10 | Pythia6 | P2011C |  $\mbox{$h_{\mathrm{damp}}$}=\infty$, no spin corl.
4 | Alpgen | CTEQ6L1 | Herwig+Jimmy | AUET2 | incl. $t\bar{t}$ $b\bar{b}$, $t\bar{t}$ $c\bar{c}$
| MC@NLO | CT10 | Herwig+Jimmy | AUET2 |
| MG5_aMC@NLO | CT10 | Herwig++ | UE-EE-5 |
Table 7: Summary of particle-level simulation samples used in the comparison
to the corrected data distributions in Section 6.2, giving the matrix-element
event generator, PDF set, parton shower and associated tune parameter set. The
four groups shown correspond to the four panels for each measured distribution
shown in Figures 9 to 12.

The first group

shows the baseline Powheg + Pythia6 $t\bar{t}$ sample with
$h_{\mathrm{damp}}$=$m_{t}$ (which is also shown in Figures 7–8), together
with the two tunes giving more or less parton shower radiation—the Perugia
2012 radHi and radLo tunes [54] coupled to scale and $h_{\mathrm{damp}}$
parameter variations as discussed in Section 2.

The second group

shows a Powheg + Pythia6 sample with $\mbox{$h_{\mathrm{damp}}$}=\infty$ (i.e.
no damping of the first emission), the baseline Powheg + Pythia6 sample with
the top quark $p_{\text{T}}$ spectrum reweighted to the NNLO prediction of
Ref. [25], and a sample generated with Powheg and
$\mbox{$h_{\mathrm{damp}}$}=\mbox{$m_{t}$}$ interfaced to Pythia8 (version
8.186) [46] with the A14 tune [96] and the CTEQ6L1 PDF set for the parton
shower, hadronisation and underlying event modelling as described in Ref.
[94].

The third group

shows a Powheg + Pythia6 sample with
$\mbox{$h_{\mathrm{damp}}$}=\mbox{$m_{t}$}$ generated with the HERAPDF 1.5 PDF
set [84, 85] instead of CT10666Although HERAPDF 1.5 has been superseded by
HERAPDF 2.0 [97], which uses the final combined DIS data from the H1 and ZEUS
experiments, HERAPDF 1.5 is used here due to availability of the corresponding
simulation sample., and a Powheg + Pythia6 sample with
$\mbox{$h_{\mathrm{damp}}$}=\infty$ and no simulation of spin correlations
between the top and antitop quarks.

The fourth group

shows alternative matrix-element event generators—the Alpgen + Herwig and
MC@NLO + Herwig samples described in Section 2 and shown in Figures 7–8,
together with a sample generated using MadGraph5_aMC@NLO 2.2.1 [98] (referred
to as aMC@NLO below) and CT10 PDFs, interfaced to Herwig++ [99] with the UE-
EE-5 Herwig++ author tune.

(a)

(b)

Figure 9: Ratios of predictions of normalised differential cross-sections to
data as a function of (a) $p_{\mathrm{T}}^{\ell}$ and (b) et al.. The data
statistical uncertainties are shown by the black error bars around a ratio of
unity, and the total uncertainties are shown by the cyan bands. The $t\bar{t}$
predictions are shown in four groups from top to bottom, with error bars
indicating the uncertainties due to the limited size of the simulated samples.

(a)

(b)

Figure 10: Ratios of predictions of normalised differential cross-sections to
data as a function of (a) $p_{\mathrm{T}}^{e\mu}$ and (b) $m^{e\mu}$. The data
statistical uncertainties are shown by the black error bars around a ratio of
unity, and the total uncertainties are shown by the cyan bands. The $t\bar{t}$
predictions are shown in four groups from top to bottom, with error bars
indicating the uncertainties due to the limited size of the simulated samples.

(a)

(b)

Figure 11: Ratios of predictions of normalised differential cross-sections to
data as a function of (a) $|y^{e\mu}|$ and (b) $\Delta\phi^{e\mu}$. The data
statistical uncertainties are shown by the black error bars around a ratio of
unity, and the total uncertainties are shown by the cyan bands. The $t\bar{t}$
predictions are shown in four groups from top to bottom, with error bars
indicating the uncertainties due to the limited size of the simulated samples.

(a)

(b)

Figure 12: Ratios of predictions of normalised differential cross-sections to
data as a function of (a) $p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ and (b)
$E^{e}+E^{\mu}$. The data statistical uncertainties are shown by the black
error bars around a ratio of unity, and the total uncertainties are shown by
the cyan bands. The $t\bar{t}$ predictions are shown in four groups from top
to bottom, with error bars indicating the uncertainties due to the limited
size of the simulated samples.

The compatibility of each prediction with each measured normalised
distribution was assessed quantitatively using a $\chi^{2}$ test, calculated
as:

$\displaystyle\chi^{2}={\mathbf{\Delta}}_{(n-1)}^{T}{\mathbf{S}}^{-1}_{(n-1)}{\mathbf{\Delta}}_{(n-1)}\,,$
(5)

where ${\mathbf{\Delta}}_{(n-1)}$ is the vector of differences between the
measured and predicted normalised differential cross-section in each of the
$n$ bins, excluding the last one, and ${\mathbf{S}}_{(n-1)}$ is the
corresponding covariance matrix, including both the experimental uncertainties
in the measurement and the statistical uncertainties in the predictions. Bin-
to-bin correlations in both the statistical (from the normalisation condition)
and systematic uncertainties were taken into account via off-diagonal entries.
The last bin of each distribution was excluded due to the normalisation
condition, rendering the covariance matrix ${\mathbf{S}}_{(n-1)}$
invertible.777The $\chi^{2}$ value does not depend on the choice of which bin
is removed. The resulting $\chi^{2}$ values, number of degrees of freedom
$(n-1)$ and corresponding $\chi^{2}$ probability $p$-values are shown for each
distribution and prediction in Table 8.

Generator | $p_{\mathrm{T}}^{\ell}$ | et al. | $p_{\mathrm{T}}^{e\mu}$ | $m^{e\mu}$ | $|y^{e\mu}|$ | $\Delta\phi^{e\mu}$ | $p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ | $E^{e}+E^{\mu}$
---|---|---|---|---|---|---|---|---
$N_{\mathrm{d}of}$ | 9 | 8 | 8 | 11 | 8 | 9 | 7 | 9
Powheg + PY6 | 13.6 | 26.3 | 7.3 | 14.6 | 46.6 | 14.0 | 11.3 | 22.7
Powheg + PY6 radLo | 15.9 | 22.9 | 7.6 | 14.6 | 45.6 | 25.9 | 14.0 | 22.0
Powheg + PY6 radHi | 10.0 | 28.2 | 11.0 | 12.6 | 42.0 | 4.5 | 9.1 | 21.4
Powheg + PY6 $\mbox{$h_{\mathrm{damp}}$}=\infty$ | 17.2 | 22.5 | 14.5 | 12.9 | 42.8 | 5.0 | 15.6 | 23.4
Powheg + PY6 $p_{\text{T}}$ NNLO | 8.3 | 28.5 | 6.3 | 12.1 | 49.2 | 7.6 | 7.6 | 17.4
Powheg + PY8 $\mbox{$h_{\mathrm{damp}}$}=\infty$ | 15.1 | 28.9 | 8.3 | 14.4 | 44.3 | 13.0 | 12.7 | 25.8
Powheg + PY6 HERAPDF 1.5 | 11.4 | 11.8 | 3.6 | 11.1 | 6.7 | 10.3 | 7.0 | 1.9
Powheg + PY6 no spin corl. | 21.8 | 23.2 | 152 | 100 | 45.3 | 279 | 22.4 | 27.6
Alpgen + HW | 31.2 | 11.6 | 15.5 | 13.7 | 15.3 | 36.0 | 27.4 | 12.7
MC@NLO + HW | 15.7 | 18.8 | 9.4 | 9.3 | 39.4 | 7.1 | 11.8 | 16.2
aMC@NLO + HW++ | 7.8 | 29.2 | 7.6 | 24.5 | 46.6 | 8.2 | 12.0 | 13.8
Powheg + PY6 | 0.14 | 9 $10^{-4}$ | 0.51 | 0.20 | 2 $10^{-7}$ | 0.12 | 0.13 | 7 $10^{-3}$
Powheg + PY6 radLo | 0.070 | 3 $10^{-3}$ | 0.48 | 0.20 | 3 $10^{-7}$ | 2 $10^{-3}$ | 0.052 | 9 $10^{-3}$
Powheg + PY6 radHi | 0.35 | 4 $10^{-4}$ | 0.20 | 0.32 | 1 $10^{-6}$ | 0.87 | 0.24 | 0.011
Powheg + PY6 $\mbox{$h_{\mathrm{damp}}$}=\infty$ | 0.045 | 4 $10^{-3}$ | 0.069 | 0.30 | 1 $10^{-6}$ | 0.83 | 0.029 | 5 $10^{-3}$
Powheg + PY6 $p_{\text{T}}$ NNLO | 0.51 | 4 $10^{-4}$ | 0.62 | 0.36 | 6 $10^{-8}$ | 0.57 | 0.36 | 0.043
Powheg + PY8 $\mbox{$h_{\mathrm{damp}}$}=\infty$ | 0.089 | 3 $10^{-4}$ | 0.41 | 0.21 | 5 $10^{-7}$ | 0.16 | 0.080 | 2 $10^{-3}$
Powheg + PY6 HERAPDF 1.5 | 0.25 | 0.16 | 0.89 | 0.44 | 0.57 | 0.32 | 0.43 | 0.99
Powheg + PY6 no spin corl. | 0.010 | 3 $10^{-3}$ | 0 | 0 | 3 $10^{-7}$ | 0 | 2 $10^{-3}$ | 1 $10^{-3}$
Alpgen + HW | 3 $10^{-4}$ | 0.17 | 0.051 | 0.25 | 0.054 | 4 $10^{-5}$ | 3 $10^{-4}$ | 0.17
MC@NLO + HW | 0.073 | 0.016 | 0.31 | 0.60 | 4 $10^{-6}$ | 0.62 | 0.11 | 0.063
aMC@NLO + HW++ | 0.56 | 3 $10^{-4}$ | 0.47 | 0.011 | 2 $10^{-7}$ | 0.52 | 0.10 | 0.13
Table 8: The $\chi^{2}$ values (top) and associated probabilities (bottom) for
comparison of measured normalised differential fiducial cross-sections with
various $t\bar{t}$ simulation samples. Probabilities smaller than $10^{-10}$
are shown as zero.

As can be seen from Figure 9, in the single-lepton $p_{\mathrm{T}}^{\ell}$
distribution, the data are softer than the predictions from Powheg with CT10
PDFs, interfaced to either Pythia6 or Pythia8. The Powheg-based predictions do
not depend strongly on the choice of parton shower/hadronisation model or tune
parameters controlling the amount of radiation. However, the agreement with
data is improved when using HERAPDF 1.5 or reweighting to the NNLO top quark
$p_{\text{T}}$ prediction from Ref. [25]. The predictions from the samples
with alternative matrix-element event generators, i.e. MC@NLO + Herwig and
Alpgen + Herwig, are also harder than the data, though aMC@NLO + Herwig++
describes the data well. The $p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ and
$E^{e}+E^{\mu}$ distributions (Figure 12) show some similar features to
$p_{\mathrm{T}}^{\ell}$, being softer than the predictions from the Powheg +
Pythia6 samples with CT10, and better described with HERAPDF 1.5, and by
aMC@NLO + Herwig++.

The predictions for the single lepton et al. and dilepton $|y^{e\mu}|$
distributions (Figures 9 and 11) are insensitive to the choice of parton
shower/hadronisation model and tune, and are also insensitive to the top quark
$p_{\text{T}}$ reweighting. The data distributions are more central than the
predictions of all the NLO event generators (Powheg, MC@NLO and aMC@NLO) with
CT10 PDFs, but are better described by Powheg with HERAPDF 1.5, and to a
lesser extent also by Alpgen + Herwig, which uses the leading-order CTEQ6L1
PDF. These distributions, whose experimental measurements are limited by
statistical uncertainties over the full kinematic range, are thus particularly
suitable for constraining PDFs, as explored further in Section 7.

The dilepton $p_{\mathrm{T}}^{e\mu}$ and $m^{e\mu}$ distributions (Figure 10)
are generally well described by all the NLO event generators, except for
aMC@NLO which does not model the data well at low $m^{e\mu}$. The
$p_{\mathrm{T}}^{e\mu}$ distribution is sensitive to the amount of parton
radiation, and is better described by the radLo than the radHi Powheg +
Pythia6 sample, and by $\mbox{$h_{\mathrm{damp}}$}=\mbox{$m_{t}$}$ than
$\mbox{$h_{\mathrm{damp}}$}=\infty$. Both distributions are sensitive to the
modelling of $t\bar{t}$ spin correlations, and are not well-modelled by the
Powheg + Pythia6 sample without spin correlations.

The $\Delta\phi^{e\mu}$ distribution (Figure 11) is particularly sensitive to
spin correlations, and has been previously used to exclude $t\bar{t}$
simulation models without spin correlation and the pair-production of
supersymmetric top squarks with masses close to $m_{t}$, via template fits to
reconstruction-level distributions [100, 101]. The particle-level
$\Delta\phi^{e\mu}$ measurements shown here also exclude the prediction
without spin correlations and the LO implementation of spin correlations in
the Alpgen + Herwig sample. The $\Delta\phi^{e\mu}$ distribution is also
sensitive to radiation, this time favouring the radHi Powheg + Pythia6 sample.

The $\chi^{2}$ formalism of Eq. (5) was extended to consider several
normalised distributions simultaneously, by forming vectors $\Delta_{i}$ where
the index runs over the bins of several distributions, excluding the last bin
in each one to account for the normalisation condition. The covariance matrix
S was extended with off-block-diagonal components encoding the correlations
between bins of different measured distributions. The statistical correlations
between distributions were evaluated using pseudo-experiments generated by
sampling from the large simulated $t\bar{t}$ sample discussed in Section 4.3.
The individual sources of systematic uncertainty were assumed to be fully
correlated across the different distributions. Five sets of combined
distributions were considered: the combination of $p_{\mathrm{T}}^{\ell}$ and
$p_{\mathrm{T}}^{e\mu}$, combining all the information from single and
dilepton $p_{\text{T}}$; the combination of $p_{\mathrm{T}}^{e\mu}$,
$m^{e\mu}$ and $p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$, including all the
dilepton kinematic distributions except rapidity; the combination of et al.
and $|y^{e\mu}|$, combining the single and dilepton rapidity information; the
combination of et al., $|y^{e\mu}|$ and $E^{e}+E^{\mu}$, combining all the
distributions with longitudinal information; and the combination of all eight
measured distributions, denoted ‘All’. The resulting $\chi^{2}$ values,
numbers of degrees of freedom and $p$-values are shown for each combination
and prediction in Table 9.

The results for the combinations of distributions reflect the observations for
the individual distributions. The best modelling of the first two combinations
(involving $p_{\mathrm{T}}^{\ell}$, $p_{\mathrm{T}}^{e\mu}$, $m^{e\mu}$ and
$p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$) is given by Powheg + Pythia6 with
either HERAPDF 1.5 or with CT10 plus reweighting of the top quark
$p_{\text{T}}$ distribution to the NNLO prediction; the radHi variation of
Powheg + Pythia6 also does well. The combinations involving et al. and
$|y^{e\mu}|$ and the combination of all eight distributions are only well-
described by Powheg +Pythia6 with HERAPDF 1.5, and marginally well described
by the radHi variation. All other event generator setups (in particular the LO
multileg event generator Alpgen) fail to describe some of the distributions,
but this could potentially be improved by appropriate parameter tuning and
switching to a different PDF set. These results highlight the sensitivity of
the differential distributions to the choice of PDF, in particular that of the
gluon, as discussed further in Section 7. They also indicate that NNLO
corrections may be important in describing the kinematics of the decay
leptons, as well as for the prediction of the top quark $p_{\text{T}}$
spectrum as discussed in Ref. [25].

Generator |  $p_{\mathrm{T}}^{\ell}$, $p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ |  $p_{\mathrm{T}}^{e\mu}$, $m^{e\mu}$, |  et al., $|y^{e\mu}|$ |  et al., $|y^{e\mu}|$, | All
---|---|---|---|---|---
|  | $p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ |  | $E^{e}+E^{\mu}$ |
$N_{\mathrm{d}of}$ | 16 | 26 | 16 | 25 | 69
Powheg + PY6 | 20.7 | 38.2 | 57.6 | 70.0 | 120
Powheg + PY6 radLo | 24.6 | 50.6 | 57.6 | 70.6 | 138
Powheg + PY6 radHi | 16.4 | 29.7 | 52.3 | 62.8 | 98.7
Powheg + PY6 $\mbox{$h_{\mathrm{damp}}$}=\infty$ | 25.0 | 40.1 | 54.2 | 68.7 | 113
Powheg + PY6 $p_{\text{T}}$ NNLO | 15.1 | 30.0 | 60.0 | 68.2 | 109
Powheg + PY8 $\mbox{$h_{\mathrm{damp}}$}=\infty$ | 23.6 | 37.3 | 56.8 | 71.3 | 121
Powheg + PY6 HERAPDF 1.5 | 20.1 | 29.6 | 22.5 | 24.5 | 68.6
Powheg + PY6 no spin corl. | 30.2 | 284 | 58.3 | 77.4 | 462
Alpgen + HW | 38.9 | 79.3 | 49.3 | 67.2 | 154
MC@NLO + HW | 23.1 | 35.2 | 54.8 | 65.7 | 110
aMC@NLO + HW++ | 19.1 | 45.2 | 63.1 | 70.2 | 128
Powheg + PY6 | 0.19 | 0.058 | 1 $10^{-6}$ | 4 $10^{-6}$ | 1 $10^{-4}$
Powheg + PY6 radLo | 0.077 | 3 $10^{-3}$ | 1 $10^{-6}$ | 3 $10^{-6}$ | 2 $10^{-6}$
Powheg + PY6 radHi | 0.43 | 0.28 | 1 $10^{-5}$ | 4 $10^{-5}$ | 0.011
Powheg + PY6 $\mbox{$h_{\mathrm{damp}}$}=\infty$ | 0.069 | 0.038 | 5 $10^{-6}$ | 6 $10^{-6}$ | 6 $10^{-4}$
Powheg + PY6 $p_{\text{T}}$ NNLO | 0.51 | 0.27 | 5 $10^{-7}$ | 7 $10^{-6}$ | 2 $10^{-3}$
Powheg + PY8 $\mbox{$h_{\mathrm{damp}}$}=\infty$ | 0.100 | 0.071 | 2 $10^{-6}$ | 2 $10^{-6}$ | 1 $10^{-4}$
Powheg + PY6 HERAPDF 1.5 | 0.21 | 0.29 | 0.13 | 0.49 | 0.49
Powheg + PY6 no spin corl. | 0.017 | 0 | 1 $10^{-6}$ | 3 $10^{-7}$ | 0
Alpgen + HW | 1 $10^{-3}$ | 3 $10^{-7}$ | 3 $10^{-5}$ | 1 $10^{-5}$ | 2 $10^{-8}$
MC@NLO + HW | 0.11 | 0.11 | 4 $10^{-6}$ | 2 $10^{-5}$ | 1 $10^{-3}$
aMC@NLO + HW++ | 0.26 | 0.011 | 2 $10^{-7}$ | 4 $10^{-6}$ | 2 $10^{-5}$
Table 9: The $\chi^{2}$ values (top) and associated probabilities (bottom) for
comparison of combinations of measured normalised differential fiducial cross-
sections with various $t\bar{t}$ simulation samples. Probabilities smaller
than $10^{-10}$ are shown as zero.

###  6.3 Comparison with fixed-order predictions

The comparisons described in Section 6.2 show that the predictions are
strongly sensitive to the choice of PDF, and also to the QCD scale (whose
variation approximates the effects of missing higher-order corrections) and
other parameters related to the amount of radiation. In this Section, these
aspects are further explored using a set of predictions from the MCFM program
(version 6.8) [30], combined with Applgrid (version 1.4.73) [93] to interface
to various PDF sets available in LHAPDF (version 6.1.5) [102]. Four recent NLO
PDF sets were considered, namely CT14 [103], MMHT14 [104], NNPDF 3.0 [105] and
HERAPDF 2.0 [97]. The data were also compared to HERAPDF 1.5 [85] for
comparison with the results of Section 6.2; the results from these two PDF
sets are similar.

MCFM provides an NLO fixed-order prediction of the $t\bar{t}$ process in the
dilepton channel, including NLO QCD corrections in both production and decay
in the on-shell approximation, and full NLO spin correlations [106]. Only the
direct decays of $W\rightarrow e/\mu$ are included, so these predictions were
compared to the measurements corrected to remove the leptonic $\tau$ decay
contributions. The top quark mass $m_{t}$ was set to 172.5 GeV. Informed by
the discussion in Ref. [107], the central values for the QCD renormalisation
and factorisation scales were set to $\mbox{$m_{t}$}/2$, the lower than
typical ($m_{t}$) scale choice being intended to account for the impact of
resummed soft-gluon contributions not included in the fixed-order
calculations. The MCFM predictions do not include quantum electrodynamics
(QED) final state photon radiation, unlike the experimental measurements where
the leptons are dressed with nearby photons as discussed in Section 4.
Therefore, the MCFM predictions were corrected bin-by-bin using corrections
derived from two $t\bar{t}$ samples generated with Pythia8 (version 8.205)
[108] and the ATTBAR tune [109] with QED final-state radiation enabled and
disabled. These corrections are typically 1–2 % on the absolute and always
smaller than 1 % on the normalised differential cross-sections. No corrections
were applied to the normalised et al. and $|y^{e\mu}|$ distributions, as the
determined corrections were always smaller than 0.3 % and consistent with
unity within the simulation statistical uncertainties.

The ratios of the MCFM normalised differential cross-section predictions with
HERAPDF 1.5 (the PDF set found to best fit the data when comparing with Powheg
+ Pythia6 samples in Section 6.2) to data are shown in Figure 13. The
uncertainties in the predictions include effects from PDFs, QCD scales and the
value of the strong coupling constant $\alpha_{\text{S}}$. For each individual
component variation, the prediction was renormalised to unity before
calculating the shift for each bin; the effects on the normalised cross-
section predictions are typically significantly smaller than those on the
absolute cross-sections. The PDF uncertainties for CT14 and MMHT were
evaluated from the sum in quadrature of the symmetrised up/down variations
from each individual eigenvector pair from the PDF error set. For the HERAPDF
sets, each pair of eigenvector or model parameter variations was treated as an
independent variation. For NNPDF 3.0, the 100 replica sets which represent the
NNPDF uncertainty were used to define a full covariance matrix taking into
account correlations between the bins of each distribution. The QCD scale
uncertainties were evaluated by varying the renormalisation and factorisation
scales $\mu_{R}$ and $\mu_{F}$ separately, and adding the variations in
quadrature. Each scale was varied by factors of one-half and two from its
central value ($\mbox{$m_{t}$}/2$), and the resulting variations symmetrised.
This procedure was used instead of taking an envelope including simultaneous
variations of $\mu_{F}$ and $\mu_{R}$ in order to properly account for the
correlations between bins of the normalised differential cross-section
predictions. Finally, the $\alpha_{\text{S}}$ uncertainty was evaluated using
the HERAPDF 1.5 PDF sets with $\alpha_{\text{S}}$ set to 0.116 and 0.120,
rescaling the resulting uncertainty to $\Delta\alpha_{\text{S}}=\pm 0.0015$,
in line with the corresponding PDF4LHC recommendation [110].

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 13: Ratios of MCFM + HERAPDF 1.5 fixed-order predictions of normalised
differential cross-sections to data as a function of lepton and dilepton
variables. Contributions via $W\rightarrow\tau\rightarrow e/\mu$ decays are
not included, and the MCFM predictions have been corrected to include QED
final-state radiation effects. The total data uncertainties are shown by the
error bars around unity. The separate uncertainties in the predictions from
QCD scales, PDFs and the strong coupling constant $\alpha_{\text{S}}$ are
shown by the hatched bands, and the total uncertainties in the predictions are
shown by the yellow band.

The compatibility of the predictions with the normalised cross-section data
was tested quantitatively using the $\chi^{2}$ of Eq. (5), updating the
covariance matrix S to also include the theoretical uncertainties discussed
above, including their bin-to-bin correlations via the off-diagonal terms. The
resulting $\chi^{2}$ and $p$-values are shown as the ‘MCFM + HERAPDF 1.5’
entries in Table 10 for individual distributions, and in Table 11 for the
combinations of distributions. As can be seen from these tables and from
Figure 13, MCFM with the HERAPDF 1.5 PDF describes the data well, once all the
theoretical uncertainties are taken into account. The predictions for
$p_{\mathrm{T}}^{\ell}$, $p_{\mathrm{T}}^{e\mu}$, $m^{e\mu}$,
$\Delta\phi^{e\mu}$ and $p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ have large
scale uncertainties, which largely cover any differences between the
measurements and central predictions with scales
$\mu_{R}=\mu_{F}=\mbox{$m_{t}$}/2$. The et al. and $|y^{e\mu}|$ distributions
have little scale dependence and are more sensitive to PDF variations, but are
again well-described within the uncertainties of the HERAPDF 1.5 set. The
$\alpha_{\text{S}}$-related uncertainties are small compared to the other two
classes.

Generator | $p_{\mathrm{T}}^{\ell}$ | et al. | $p_{\mathrm{T}}^{e\mu}$ | $m^{e\mu}$ | $|y^{e\mu}|$ | $\Delta\phi^{e\mu}$ | $p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ | $E^{e}+E^{\mu}$
---|---|---|---|---|---|---|---|---
$N_{\mathrm{d}of}$ | 9 | 8 | 8 | 11 | 8 | 9 | 7 | 9
MCFM + CT14 | 11.5 | 14.1 | 7.2 | 11.2 | 13.0 | 7.2 | 11.4 | 11.2
MCFM + MMHT | 11.3 | 12.8 | 7.2 | 11.2 | 12.6 | 7.1 | 11.2 | 9.6
MCFM + NNPDF 3.0 | 11.7 | 11.3 | 7.2 | 11.4 | 9.4 | 7.3 | 11.5 | 8.5
MCFM + HERAPDF 1.5 | 9.1 | 10.9 | 6.4 | 12.1 | 8.0 | 6.9 | 8.5 | 2.6
MCFM + HERAPDF 2.0 | 8.4 | 12.0 | 6.2 | 12.4 | 8.0 | 6.8 | 8.0 | 2.7
MCFM + CT14 | 0.24 | 0.080 | 0.51 | 0.43 | 0.11 | 0.62 | 0.12 | 0.27
MCFM + MMHT | 0.26 | 0.12 | 0.51 | 0.42 | 0.13 | 0.62 | 0.13 | 0.38
MCFM + NNPDF 3.0 | 0.23 | 0.18 | 0.52 | 0.41 | 0.31 | 0.61 | 0.12 | 0.49
MCFM + HERAPDF 1.5 | 0.43 | 0.21 | 0.61 | 0.36 | 0.44 | 0.65 | 0.29 | 0.98
MCFM + HERAPDF 2.0 | 0.49 | 0.15 | 0.63 | 0.33 | 0.44 | 0.66 | 0.34 | 0.97
Table 10: The $\chi^{2}$ values (top) and associated probabilities (bottom) for comparison of measured normalised differential fiducial cross-sections with the predictions of MCFM with various PDF sets. Contributions via $W\rightarrow\tau\rightarrow e/\mu$ decays are not included, and the MCFM predictions have been corrected to include QED final-state radiation effects. The results take into account the uncertainties in both the measurements and predictions. Generator |  $p_{\mathrm{T}}^{\ell}$, $p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ |  $p_{\mathrm{T}}^{e\mu}$, $m^{e\mu}$, |  et al., $|y^{e\mu}|$ |  et al., $|y^{e\mu}|$, | All
---|---|---|---|---|---
|  | $p_{\mathrm{T}}^{e}+p_{\mathrm{T}}^{\mu}$ |  | $E^{e}+E^{\mu}$ |
$N_{\mathrm{d}of}$ | 16 | 26 | 16 | 25 | 69
MCFM + CT14 | 19.5 | 29.6 | 24.2 | 32.4 | 73.0
MCFM + MMHT | 19.3 | 29.6 | 23.4 | 30.7 | 72.0
MCFM + NNPDF 3.0 | 19.9 | 29.7 | 20.1 | 27.4 | 69.3
MCFM + HERAPDF 1.5 | 16.1 | 28.8 | 21.5 | 26.1 | 68.8
MCFM + HERAPDF 2.0 | 15.3 | 30.0 | 22.7 | 27.4 | 69.0
MCFM + CT14 | 0.24 | 0.28 | 0.086 | 0.15 | 0.35
MCFM + MMHT | 0.25 | 0.28 | 0.10 | 0.20 | 0.38
MCFM + NNPDF 3.0 | 0.23 | 0.28 | 0.22 | 0.34 | 0.47
MCFM + HERAPDF 1.5 | 0.45 | 0.32 | 0.16 | 0.40 | 0.48
MCFM + HERAPDF 2.0 | 0.51 | 0.27 | 0.12 | 0.34 | 0.48
Table 11: The $\chi^{2}$ values (top) and associated probabilities (bottom)
for comparison of combinations of measured normalised differential fiducial
cross-sections with the predictions of MCFM with various PDF sets.
Contributions via $W\rightarrow\tau\rightarrow e/\mu$ decays are not included,
and the MCFM predictions have been corrected to include QED final-state
radiation effects. The results take into account the uncertainties in both the
measurements and predictions.

The predictions for all five PDF sets (including PDF uncertainties, scaled to
68 % CL for CT14, as well as scale and $\alpha_{\text{S}}$ uncertainties) are
compared to the data in Figure 14. The corresponding $\chi^{2}$ and
$p$-values, including the PDF, scale and $\alpha_{\text{S}}$ uncertanities on
the predictions, are shown in Tables 10 and 11. The results for HERAPDF 1.5
and HERAPDF 2.0 are close to the data, whereas the CT14, MMHT and NNPDF 3.0
PDF sets describe the data slightly less well, particularly for
$p_{\mathrm{T}}^{\ell}$, et al., $|y^{e\mu}|$ and $E^{e}+E^{\mu}$. These
conclusions are similar to those found for HERAPDF 1.5 and CT10 with the
Powheg + Pythia6 setup discussed in Section 6.2 above. However, the difference
in $\chi^{2}$ between the PDF sets is smaller for the fixed-order predictions,
as the explicit inclusion of PDF and scale uncertainties in the predictions
renders the differences between the central predictions of each PDF less
significant. The PDF comparisons would benefit from the availability of
predictions including NNLO QCD effects in both the top quark production and
decay, which should substantially reduce the scale uncertainties.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 14: Ratios of MCFM fixed-order predictions of normalised differential
cross-sections to data as a function of lepton and dilepton variables, using
the CT14, MMHT, NNPDF 3.0, HERAPDF 1.5 and HERAPDF 2.0 PDF sets for the
predictions. Contributions via $W\rightarrow\tau\rightarrow e/\mu$ decays are
not included, and the MCFM predictions have been corrected to include QED
final-state radiation effects. The total data uncertainties are shown by the
cyan bands around unity, and the total uncertainty for each prediction
(including QCD scales, PDFs, and the strong coupling constant
$\alpha_{\text{S}}$) are shown by the vertical bars.

##  7 Constraints on the gluon parton distribution function

As a demonstration of the ability of the normalised differential cross-section
measurements to constrain the gluon PDF, fits were performed to deep inelastic
scattering (DIS) data from HERA I+II [97], with and without the addition of
the constraints from $t\bar{t}$ dilepton et al., $|y^{e\mu}|$ and
$E^{e}+E^{\mu}$ distributions. As shown in Figure 13, these distributions are
the most sensitive to PDF variations, whilst being less sensitive to QCD scale
variations and the value of $m_{t}$. The fits are based on the predictions
from MCFM and ApplGrid discussed in Section 6.3, allowing predictions for
arbitrary PDF variations to be obtained much faster than if a full NLO plus
parton shower event generator setup were to be used. The QCD scales were set
to fixed values of $\mu_{F}=\mu_{R}=\mbox{$m_{t}$}/2$. The fits were performed
using the xFitter package [111, 112], which allows the PDF and other
theoretical uncertainties to be included via asymmetric error propagation. In
this formalism, the $\chi^{2}$ for the compatibility of the measurements with
the prediction is expressed by:

$\displaystyle\chi^{2}=\sum_{i,j}\left(\mbox{$\varsigma_{i}^{\mathrm{exp}}$}-\mbox{$\varsigma_{i}^{\mathrm{th}}$}\right)\
S^{-1}_{\mathrm{exp},ij}(\mbox{$\varsigma_{i}^{\mathrm{th}}$},\mbox{$\varsigma_{j}^{\mathrm{th}}$})\
\left(\mbox{$\varsigma_{j}^{\mathrm{exp}}$}-\mbox{$\varsigma_{j}^{\mathrm{th}}$}\right)\
,$ (6)

where $\varsigma_{i}^{\mathrm{exp}}$ is the measured normalised differential
cross-section in bin $i$ (equivalent to $\varsigma^{i}_{t\bar{t}}$ in Eq.
(2)), $\varsigma_{i}^{\mathrm{th}}$ is the corresponding theoretical
prediction, $S_{\mathrm{exp},ij}$ is the covariance matrix of experimental
uncertainties including both statistical and systematic contributions, and
correlations between bins, and the sums for $i$ and $j$ run over $n-1$ bins to
account for the normalisation condition. Unlike in the formulation of Eq. (5),
the covariance matrix is a function of the theoretical predictions, with the
statistical uncertainties being rescaled according to the difference between
the measured values and the predictions using a Poisson distribution, and the
systematic uncertainties being scaled in proportion to the predictions.

Following the formalism outlined in Ref. [113], the covariance matrix was
decomposed into a diagonal matrix $\mathbf{D}$ representing the uncorrelated
parts of the uncertainties, and a set of coefficients
$\gamma_{ij}^{\mathrm{exp}}$ giving the one standard deviation shift in the
measurement $i$ for source $j$, where $j$ runs over the correlated part of the
statistical uncertainties and each source of systematic uncertainty. Each
source of experimental uncertainty was then associated with a ‘nuisance
parameter’ $b_{j,{\mathrm{exp}}}$ parameterising the associated shift in units
of standard deviation. The $\chi^{2}$ becomes a function of the set of PDF
parameters ${\mathbf{p}}$ defining the theoretical prediction
$\varsigma_{i}^{\mathrm{th}}$ and the vector of experimental nuisance
parameters ${\mathbf{b}}_{\mathrm{exp}}$, and is given by:

$\displaystyle\chi^{2}(\mbox{${\mathbf{p}}$},\mbox{${\mathbf{b}}_{\mathrm{exp}}$})=\sum_{i}\frac{\left(\mbox{$\varsigma_{i}^{\mathrm{exp}}$}+\sum_{j}\mbox{$\gamma_{ij}^{\mathrm{exp}}$}\mbox{$b_{j,{\mathrm{exp}}}$}-\mbox{$\varsigma_{i}^{\mathrm{th}}$}(\mbox{${\mathbf{p}}$})\right)^{2}}{d_{ii}^{2}}+\sum_{j}\mbox{$b_{j,{\mathrm{exp}}}$}^{2}+L\
,$ (7)

where $d_{ii}$ are the non-zero elements of the diagonal matrix $\mathbf{D}$,
and the rescaling of the uncertainties leads to the logarithmic term $L$,
arising from the likelihood transition to $\chi^{2}$ as discussed in Refs.
[113, 114]. The $\chi^{2}$ was minimised as a function of the PDF parameters
${\mathbf{p}}$ and the nuisance parameters ${\mathbf{b}}_{\mathrm{exp}}$, and
the value at the minimum provides a compatibility test of the data and
prediction.

For the PDF fits, the perturbative order of the DGLAP evolution [115, 116,
117] was set to NLO, to match the order of the MCFM predictions. The gluon PDF
$g(x)$ was parameterised as a function of Bjorken-$x$ as:

$\displaystyle xg(x)=Ax^{B}(1-x)^{C}(1+Ex^{2})\,e^{Fx},$ (8)

which, compared to the standard parameterisation given in Eq. (27) of Ref.
[97], removes the negative $A^{\prime}$ term at low $x$ and adds more
flexibility at medium and high $x$ through the additional terms with the
parameters $E$ and $F$. The standard parameterisations were used for the quark
PDFs, giving a total of 14 free PDF parameters in the vector ${\mathbf{p}}$,
after imposing momentum and valance sum rules, and the constraint that the
$\bar{u}$ and $\bar{d}$ contributions are equal at low $x$. Other parameters
in the PDF fit were set as described in Ref. [113].

The minimised $\chi^{2}$ values from the fits without and with the $t\bar{t}$