Tools for estimating fake/non-prompt lepton backgrounds with the ATLAS
detector at the LHC Measurements and searches performed with the ATLAS
detector at the CERN Large Hadron Collider often involve signatures with one
or more prompt leptons. Such analyses are subject to ‘fake/non-prompt’ lepton
backgrounds, where either a hadron or a lepton from a hadron decay or an
electron from a photon conversion satisfies the prompt-lepton selection
criteria. These backgrounds often arise within a hadronic jet because of
particle decays in the showering process, particle misidentification or
particle interactions with the detector material. As it is challenging to
model these processes with high accuracy in simulation, their estimation
typically uses data-driven methods. Three methods for carrying out this
estimation are described, along with their implementation in ATLAS and their
performance. CERN-EP-2022-214 JINST

###### Contents

  1. 1 Introduction
  2. 2 Methods
    1. 2.1 Matrix method
      1. 2.1.1 Asymptotic matrix method
      2. 2.1.2 Poisson likelihood matrix method
    2. 2.2 Fake-factor method
    3. 2.3 Generalisation for multi-lepton final states
    4. 2.4 Use with weighted events
    5. 2.5 Performance studies
  3. 3 The ATLAS detector
  4. 4 Lepton selection criteria
    1. 4.1 Electron reconstruction and identification
    2. 4.2 Muon reconstruction and identification
    3. 4.3 Lepton isolation
    4. 4.4 Removing overlaps between jets and leptons
  5. 5 Monte Carlo simulation samples
  6. 6 Sources of fake/non-prompt leptons
  7. 7 Measurement of real and fake/non-prompt lepton efficiencies
    1. 7.1 Real-lepton efficiencies
    2. 7.2 Fake/non-prompt lepton efficiencies
  8. 8 Systematic uncertainties
    1. 8.1 Statistical uncertainties in the measured efficiencies
    2. 8.2 Systematic uncertainties in the measured efficiencies
    3. 8.3 Uncertainties in the modelling of real-lepton processes
    4. 8.4 Uncertainties due to biases in the likelihood matrix method
  9. 9 Examples of application in ATLAS analyses
    1. 9.1 Measurement of the $t\bar{t}Z$ cross-section in final states with three or four leptons
      1. 9.1.1 Real-lepton efficiencies
      2. 9.1.2 Fake/non-prompt lepton efficiencies
      3. 9.1.3 Results in the fake/non-prompt lepton validation regions
    2. 9.2 Model-independent search for new phenomena in multi-lepton final states
      1. 9.2.1 Fake/non-prompt lepton selection
      2. 9.2.2 Fake factors
      3. 9.2.3 Validation regions
  10. 10 Conclusions

##  1 Introduction

Many measurements and searches for new physics performed with the ATLAS
detector [1] at the Large Hadron Collider (LHC) at CERN require the presence
of one or more leptons (electrons or muons) to indicate that a high-energy
electroweak process occurred in the collision. Lepton candidates are
reconstructed from signals in the inner tracker, calorimeters, and muon
spectrometer [2, 3]. Identification criteria are then applied to suppress
candidates originating from physical objects other than leptons from the hard-
scattering process of the event. These background candidates fall into two
categories: $i$) ‘non-prompt leptons’ from the semileptonic decay of hadrons,
or from photon ($\gamma$) conversions in detector material, and $ii$) ‘fake
leptons’ where the reconstructed object is not, in fact, due to a lepton. In
contrast to the aforementioned categories, ‘real leptons’ are defined as
electrons or muons produced either directly in the hard-scattering process or
directly in the decay of a short-lived non-hadronic resonance (such as a
$W$/$Z$ boson).

The rates at which fake or non-prompt leptons are selected are difficult to
model accurately from simulation. They can depend strongly on details of the
physics simulation, including in non-perturbative regions where the simulation
would not be expected to be reliable. They also depend on the modelling of the
material composition and response of the detector. In addition, fake leptons
sometimes occur with low probability as a result of misidentifying a hadronic
jet in multi-jet events. The computing resources required to simulate these
processes with a sufficient sample size would be prohibitive. Therefore,
‘data-driven’ approaches are commonly used to estimate these backgrounds.

To simplify the adoption of such methods, and to ensure that they are applied
uniformly, a set of standard tools and prescriptions has been developed for
use in ATLAS physics analyses that are subject to fake/non-prompt lepton
backgrounds. The principles and performance of these tools are described in
this paper. The motivation and mathematical basis of the methods are explained
in Section 2; a description of the relevant features of the ATLAS detector is
given in Section 3; the criteria used to select leptons are given in Section
4; the simulated signal and background processes, as well as the different
processes that can lead to fake/non-prompt leptons, are discussed in Sections
5 and 6; the procedures used to measure the efficiencies for real and
fake/non-prompt leptons are described in Section 7; the systematic
uncertainties associated with the methods are described in Section 8; and
Section 9 provides examples of the application of the fake/non-prompt lepton
estimation methods in two published ATLAS physics analyses, with details that
are not included in the existing publications.

##  2 Methods

The fake/non-prompt lepton estimation methods considered in this paper depend
on defining two tiers of lepton selection criteria, called the ‘baseline’ and
‘tight’ criteria. The tight criteria are used to select the signal leptons in
a physics analysis, while the baseline criteria accept all of the tight lepton
candidates as well as an additional set of candidates with a higher rate of
fake/non-prompt contributions. Candidates that satisfy the baseline criteria
but not the tight criteria are called ‘loose’ leptons. If the two sets of
criteria are chosen well, the fraction of real leptons in the baseline sample
that satisfy the tight criteria will be substantially higher than the
corresponding fraction for fake/non-prompt leptons. These fractions are called
the ‘real efficiency’ ($\varepsilon_{\mathrm{r}}$) and ‘fake efficiency’
($\varepsilon_{\mathrm{f}}$), respectively.

The $\varepsilon_{\mathrm{r}}$ values are generally taken from Monte Carlo
(MC) simulated events that are corrected to account for differences between
data and the simulation, while the $\varepsilon_{\mathrm{f}}$ values are
typically measured in a data sample that is orthogonal to the one that is used
for the data analysis, as detailed in Section 7.

###  2.1 Matrix method

With the efficiencies known, a simple counting of the numbers of lepton
candidates that satisfy the tight and loose criteria provides an estimate for
the number of fake/non-prompt leptons. In the simplest case, where an analysis
selects signal events containing exactly one tight lepton candidate and no
loose lepton candidates, the relationship between the numbers of tight and
loose leptons observed in data and the composition of the sample in terms of
real and fake/non-prompt leptons is:

$\left({\begin{array}[]{*{20}{c}}{{N^{\mathrm{t}}}}\\\ {{N^{\mathrm{l}}}}\\\
\end{array}}\right)=\left({\begin{array}[]{*{20}{c}}\varepsilon_{\mathrm{r}}&\varepsilon_{\mathrm{f}}\\\
{1-\varepsilon_{\mathrm{r}}}&{1-\varepsilon_{\mathrm{f}}}\\\
\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{N_{\mathrm{r}}^{\mathrm{b}}}\\\
{N_{\mathrm{f}}^{\mathrm{b}}}\end{array}}\right),$ (1)

where $N^{\mathrm{t}}$ and $N^{\mathrm{l}}$ are the numbers of events with
tight and loose lepton candidates, and $N_{\mathrm{r}}^{\mathrm{b}}$ and
$N_{\mathrm{f}}^{\mathrm{b}}$ are the unknown numbers of real and fake/non-
prompt leptons in the baseline sample. In matrix notation, the relationship is
given by:

${\mathbf{N}}_{\mathrm{tl}}={\mathbf{M}}_{\varepsilon}{\mathbf{N}}_{\mathrm{rf}}^{\mathrm{b}}.$
(2)

The fact that the unknown values ($N_{\mathrm{r}}^{\mathrm{b}}$ and
$N_{\mathrm{f}}^{\mathrm{b}}$) and the observed yields ($N^{\mathrm{t}}$ and
$N^{\mathrm{l}}$) are related via the matrix ${\mathbf{M}}_{\varepsilon}$
gives rise to the name of this method: the ‘matrix method’. Inversion of the
matrix allows $N_{\mathrm{f}}^{\mathrm{b}}$ to be determined:

$N_{\mathrm{f}}^{\mathrm{b}}=\frac{1}{\varepsilon_{\mathrm{r}}-\varepsilon_{\mathrm{f}}}\left[(\varepsilon_{\mathrm{r}}-1)N^{\mathrm{t}}+\varepsilon_{\mathrm{r}}N^{\mathrm{l}}\right].$
(3)

In the typical use case, the quantity of interest is the number of events in
the tight sample where the lepton is fake/non-prompt,
$N_{\mathrm{f}}^{\mathrm{t}}$. This is related to the number of such events in
the baseline sample, $N_{\mathrm{f}}^{\mathrm{b}}$, by:

$N_{\mathrm{f}}^{\mathrm{t}}=\varepsilon_{\mathrm{f}}N_{\mathrm{f}}^{\mathrm{b}}.$
(4)

Similarly, the number of real leptons in the tight sample is

$N_{\mathrm{r}}^{\mathrm{t}}=\varepsilon_{\mathrm{r}}N_{\mathrm{r}}^{\mathrm{b}},$
(5)

and these can be treated as elements of a column matrix
${\mathbf{N}}_{\mathrm{rf}}^{\mathrm{t}}$.

The fact that $N^{\mathrm{t}}$ appears in Eq. (1) means that information about
the content of the analysis signal region is used in the estimate, and
therefore an analysis is not completely blinded when using this approach.

Generally, the values of $\varepsilon_{\mathrm{r}}$ and
$\varepsilon_{\mathrm{f}}$ depend on the lepton candidate’s momentum,
proximity to other objects, or other factors. Details of how these variations
are accounted for in the estimation are given below.

####  2.1.1 Asymptotic matrix method

In this method, events in the baseline sample are considered one at a time,
and a ‘fake weight’ $w_{i}$ is defined for each event, corresponding to the
two terms in Eq. (3) via Eq. (4):

$w_{i}=\left\\{{\begin{array}[]{*{20}{c}}{\frac{\varepsilon_{\mathrm{f},i}}{\varepsilon_{\mathrm{r},i}-\varepsilon_{\mathrm{f},i}}(\varepsilon_{\mathrm{r},i}-1)}&{\hbox{if
lepton candidate $i$ is tight (so $N^{\mathrm{t}}=1$ and
$N^{\mathrm{l}}=0$)}}\\\\[5.0pt]
{\frac{\varepsilon_{\mathrm{f},i}}{\varepsilon_{\mathrm{r},i}-\varepsilon_{\mathrm{f},i}}}\varepsilon_{\mathrm{r},i}&{\hbox{if
lepton candidate $i$ is loose (so $N^{\mathrm{t}}=0$ and
$N^{\mathrm{l}}=1$)}}\end{array}},\right.$

where $\varepsilon_{\mathrm{r},i}$ and $\varepsilon_{\mathrm{f},i}$ are the
values of $\varepsilon_{\mathrm{r}}$ and $\varepsilon_{\mathrm{f}}$ that are
appropriate for lepton $i$. Since $\varepsilon_{\mathrm{r},i}$ is always less
than one, the weight for an event with a tight lepton candidate is negative.
By extension, the total fake/non-prompt lepton background in the tight sample
is then estimated by

$N_{\mathrm{f}}^{\mathrm{t}}=\sum_{\mathrm{events}}w_{i}.$

This approach is convenient since the $w_{i}$ need only be calculated once and
then can be stored with the event, allowing the distribution of the fake-
lepton yield to be binned in any variable of interest, as well as a simple re-
computation of $N_{\mathrm{f}}^{\mathrm{t}}$ if the event selection is
modified. One drawback is that since the value of $w_{i}$ may be negative,
there is no guarantee that $N_{\mathrm{f}}^{\mathrm{t}}$ will be positive. The
value of $N_{\mathrm{f}}^{\mathrm{t}}$ is also sensitive to fluctuations in
the input $\varepsilon_{\mathrm{r},i}$ and $\varepsilon_{\mathrm{f},i}$
values.

The statistical uncertainty of $N_{\mathrm{f}}^{\mathrm{t}}$ is given by:

$\sigma_{N_{\mathrm{f}}^{\mathrm{t}}}=\sqrt{\sum_{\mathrm{events}}w^{2}_{i}}.$
(6)

The method that makes use of the $w_{i}$ is known as the ‘asymptotic matrix
method’, since Eq. (6) is only valid in the asymptotic limit with a large
number of events.

####  2.1.2 Poisson likelihood matrix method

In this method [4]111An earlier variant of the likelihood matrix method is
described in Ref. [5]. the elements of
${\mathbf{N}}_{\mathrm{rf}}^{\mathrm{t}}$ are treated as free parameters,
which are varied to maximise the likelihood of the observed
${\mathbf{N}}_{\mathrm{tl}}$ values. By doing so, the Poisson-distributed
nature of the ${\mathbf{N}}_{\mathrm{tl}}$ values is taken into account, so
there is no need to use the asymptotic approximation. In the fit, the
${\mathbf{N}}^{\mathrm{t}}_{\mathrm{rf}}$ values are converted to
${\mathbf{N}}^{\mathrm{b}}_{\mathrm{rf}}$ using Eqs. (4) and (5), where the
entries in the matrix ${\mathbf{M}}_{\varepsilon}$ are calculated using the
averages of the prompt and fake/non-prompt lepton efficiencies in the baseline
sample:

${\mathbf{M}}_{\varepsilon}=\left({\begin{array}[]{*{20}{c}}\langle\varepsilon_{\mathrm{r}}\rangle&\langle\varepsilon_{\mathrm{f}}\rangle\\\
{1-\langle\varepsilon_{\mathrm{r}}\rangle}&{1-\langle\varepsilon_{\mathrm{f}}\rangle}\\\
\end{array}}\right).$ (7)

The resulting ${\mathbf{N}}^{\mathrm{b}}_{\mathrm{rf}}$ values are used to
obtain the expectation values for ${\mathbf{N}}_{\mathrm{tl}}$ using Eq. (2).
These expectation values are denoted by
${\mathbf{N}}_{\mathrm{tl},\mathrm{exp}}$. The
${\mathbf{N}}_{\mathrm{rf}}^{\mathrm{t}}$ parameters are adjusted (subject to
the constraint that they must be non-negative) to maximise the joint Poisson
likelihood

$L\left({\mathbf{N}}^{\mathrm{t}}_{\mathrm{rf}}\right)\equiv\prod_{i}P\left[N_{\mathrm{tl}_{i}}|N_{\mathrm{tl},\mathrm{exp}_{i}}\left({\mathbf{N}}^{\mathrm{t}}_{\mathrm{rf}}\right)\right],$
(8)

where the product is over the elements of ${\mathbf{N}}_{\mathrm{tl}}$;
$N_{\mathrm{tl}_{i}}$ and $N_{\mathrm{tl},\mathrm{exp}_{i}}$ are the $i$th
elements of ${\mathbf{N}}_{\mathrm{tl}}$ and
${\mathbf{N}}_{\mathrm{tl},\mathrm{exp}}$, respectively; and
$P\left[n|\mu\right]$ is the Poisson probability for observing $n$ events when
$\mu$ are expected. The output of the fit consists of an estimate of the
number of fake/non-prompt leptons and the uncertainty in this quantity, which
is obtained by noting the value for which $-\ln L$ exceeds its minimum by
$0.5$.

The primary advantages of the Poisson likelihood approach are that the result
is constrained to be non-negative, and the uncertainty is a better
approximation to the range that gives 68% coverage, particularly in samples
with few events. In addition, in some scenarios it can provide smaller
statistical uncertainties than the asymptotic matrix method or the fake-factor
method (described in Section 2.2). The main drawback is that the estimated
yield must be calculated for the sample as a whole rather than from a sum of
individual event weights, which complicates the process of producing a
distribution of the fake-lepton yield, as required for any differential
measurement (to do so, the likelihood must be applied in every bin of the
distribution).

###  2.2 Fake-factor method

The fact that real-lepton kinematic distributions and efficiencies are
generally modelled well in simulation, and that scale factors can be applied
to account for any differences observed between the values in simulation and
in data control samples, leads to an alternative method that uses simulation,
rather than the data, to measure the real-lepton contribution to the loose
lepton sample.

The number of fake-lepton events in the loose sample is

$N_{\mathrm{f}}^{\mathrm{l}}=N^{\mathrm{l}}-N_{\mathrm{r}}^{\mathrm{l}}=(1-\varepsilon_{\mathrm{f}})N_{\mathrm{f}}^{\mathrm{b}}$

and the number of fake-lepton events in the tight sample is

$N_{\mathrm{f}}^{\mathrm{t}}=\varepsilon_{\mathrm{f}}N_{\mathrm{f}}^{\mathrm{b}}=F(N^{\mathrm{l}}-N_{\mathrm{r}}^{\mathrm{l}}),$

where the ‘fake factor’ $F$ is defined as
$F\equiv\varepsilon_{\mathrm{f}}/(1-\varepsilon_{\mathrm{f}})$. Thus, in the
fake-factor method, the number of tight fake/non-prompt leptons for a given
analysis can be computed using the fake factor $F$, the total number of loose
lepton candidates $N^{\mathrm{l}}$, and the number of real leptons in the
loose lepton sample $N_{\mathrm{r}}^{\mathrm{l}}$, where the latter quantity
can be estimated using MC simulated samples, and its contribution subsequently
subtracted from the quantity $FN^{\mathrm{l}}$ observed in the data. In
practice, the calculation is performed on an event-by-event basis to account
for potential variations in $F$ due to properties of the lepton:

$N_{\mathrm{f}}^{\mathrm{t}}=\sum_{\mathrm{data},i=1}^{N^{\mathrm{l}}}F_{i}-\sum_{\mathrm{MC},j=1}^{N^{\mathrm{l}}_{\mathrm{MC}}}w_{\mathrm{MC},j}F_{j},$

where $F_{i}$ is the fake factor appropriate for lepton $i$, all sources of
prompt leptons are considered in the sum over MC simulated events,
$N^{\mathrm{l}}_{\mathrm{MC}}$ is the number of MC events in the loose sample,
and $w_{\mathrm{MC},j}$ is the weight assigned to simulated event $j$, based
on the cross-section of the simulated process and any corrections to the
selection efficiency that may be needed to reflect the performance on data
events.

The main advantage of the fake-factor method is that this result does not
depend on $N^{\mathrm{t}}$, i.e. the yield in the analysis signal region. This
means that unlike the matrix method, the fake-factor method can be applied
while remaining ‘blind’ to the contents of the signal region. However, the
method does have some of the same drawbacks as the asymptotic matrix method,
namely the possibility of $N_{\mathrm{f}}^{\mathrm{t}}$ being negative, and
sensitivity to fluctuations in the $\varepsilon_{\mathrm{f}}$ values.

###  2.3 Generalisation for multi-lepton final states

The above methods can be generalised to cases where multiple baseline lepton
candidates are considered in each event. For the matrix methods, this is done
by increasing the dimensionality of ${\mathbf{M}_{\varepsilon}}$,
${\mathbf{N}}_{\mathrm{tl}}$, and
${\mathbf{N}}_{\mathrm{rf}}^{\mathrm{(t,b)}}$ to $2^{n_{\mathrm{b}}}$, where
$n_{\mathrm{b}}$ is the number of baseline lepton candidates in each event.
The estimated fake/non-prompt yield depends on the requirements of a
particular analysis in three ways: first, from the requirement placed on the
desired number $n_{\mathrm{t}}$ of tight lepton candidates per event; second,
whether or not events with additional loose lepton candidates are vetoed; and
third, on the minimum number of fake/non-prompt leptons $m_{\mathrm{f}}$
defining the background to be evaluated with one of the data-driven
methods.222An example of a case where $m_{\mathrm{f}}$ is greater than one
would be a dilepton analysis where there are backgrounds from both
$W(\rightarrow\ell\nu)+j$ events where the jet forms a fake/non-prompt lepton,
and dijet events where both jets form fake/non-prompt leptons. The analysers
may choose to estimate the first contribution from MC simulation, and use
data-driven methods for the second contribution, and therefore setting
$m_{\mathrm{f}}=2$ for the data-driven approach is required to avoid double-
counting. The consideration of $m_{\mathrm{f}}$ is reflected in the transition
from the number of fake/non-prompt lepton events in the baseline sample to the
number in the tight sample, in a generalisation of Eq. (4):

$N_{\mathrm{f}}^{\mathrm{t}}=\sum_{i}g_{i}\left(\varepsilon_{\mathrm{r}},\varepsilon_{\mathrm{f}}\right)N_{\mathrm{rf},i}^{\mathrm{b}}.$

Here, the sum is over all combinations of real and fake/non-prompt leptons
that include at least $m_{\mathrm{f}}$ fake/non-prompt leptons, $g_{i}$ is a
function of the real and fake/non-prompt efficiencies that will result in the
required number of tight lepton candidates for a given set of real and
fake/non-prompt leptons, and $N_{\mathrm{rf},i}^{\mathrm{b}}$ is the $i$th
element of $\mathbf{N_{\mathrm{rf}}^{\mathrm{b}}}$

To address the requirements on $n_{\mathrm{t}}$ and the possible presence of
additional loose lepton candidates, the analysis must consider all events in
the baseline sample with lepton multiplicities up to the sum of the allowed
numbers of tight and loose lepton candidates in the signal region.

As an example, consider the case where $n_{\mathrm{b}}=2$. If an analysis
selects signal events containing exactly two tight lepton candidates and no
loose lepton candidates, and the background with $m_{\mathrm{f}}\geq 1$ is
being evaluated, then

$N_{\mathrm{f}}^{\mathrm{t}}=\varepsilon_{\mathrm{r},1}\varepsilon_{\mathrm{f},2}N_{\mathrm{r_{1}f_{2}}}^{\mathrm{b}}+\varepsilon_{\mathrm{f},1}\varepsilon_{\mathrm{r},2}N_{\mathrm{f_{1}r_{2}}}^{\mathrm{b}}+\varepsilon_{\mathrm{f},1}\varepsilon_{\mathrm{f},2}N_{\mathrm{f_{1}f_{2}}}^{\mathrm{b}},$

where $N_{\mathrm{r_{1}f_{2}}}^{\mathrm{b}}$ is the number of events in the
baseline sample where the first lepton candidate is real and the second is
fake/non-prompt, and $N_{\mathrm{f_{1}r_{2}}}^{\mathrm{b}}$ and
$N_{\mathrm{f_{1}f_{2}}}^{\mathrm{b}}$ are defined correspondingly. The
ordering of the lepton candidates is typically according to $p_{\text{T}}$,
but the method does not depend on the ordering used.

For an analysis with $m_{\mathrm{f}}\geq 1$ that accepts events with two tight
lepton candidates or one tight and one loose lepton candidate, the expression
would change to

$N_{\mathrm{f}}^{\mathrm{t}}=(\varepsilon_{\mathrm{r},1}+\varepsilon_{\mathrm{f},2}-\varepsilon_{\mathrm{r},1}\varepsilon_{\mathrm{f},2})N_{\mathrm{r_{1}f_{2}}}^{\mathrm{b}}+(\varepsilon_{\mathrm{f},1}+\varepsilon_{\mathrm{r},2}-\varepsilon_{\mathrm{f},1}\varepsilon_{\mathrm{r},2})N_{\mathrm{f_{1}r_{2}}}^{\mathrm{b}}+(\varepsilon_{\mathrm{f},1}+\varepsilon_{\mathrm{f},2}-\varepsilon_{\mathrm{f},1}\varepsilon_{\mathrm{f},2})N_{\mathrm{f_{1}f_{2}}}^{\mathrm{b}},$

where the additional terms are needed to account for the additional ways that
fake/non-prompt leptons might satisfy the signal selection (and the terms
involving products of the efficiencies are subtracted to avoid double-
counting). For an analysis that imposes the same lepton candidate requirements
but only the background with $m_{\mathrm{f}}=2$ is being evaluated, the
expression is:

$N_{\mathrm{f}}^{\mathrm{t}}=(\varepsilon_{\mathrm{f},1}+\varepsilon_{\mathrm{f},2}-\varepsilon_{\mathrm{f},1}\varepsilon_{\mathrm{f},2})N_{\mathrm{f_{1}f_{2}}}^{\mathrm{b}}.$

The methods can also be extended to cases where there are more than two levels
of lepton selection criteria, or distinct categories of fake/non-prompt
leptons, as in Ref. [6].

As with the matrix method, the fake-factor method can be generalised to higher
lepton candidate multiplicities. In the dilepton final state, the number of
events with two tight lepton candidates, of which at least one is fake/non-
prompt, is:

$N_{\mathrm{f}}^{\mathrm{t}}=N^{\mathrm{t_{1}t_{2}}}-N_{\mathrm{r_{1}r_{2}}}^{\mathrm{t_{1}t_{2}}}=\\\
F_{1}(N^{\mathrm{l_{1}t_{2}}}-N_{\mathrm{r_{1}r_{1}}}^{\mathrm{l_{1}t_{2}}})+F_{2}(N^{\mathrm{t_{1}l_{2}}}-N_{\mathrm{r_{1}r_{2}}}^{\mathrm{t_{1}l_{2}}})-F_{1}F_{2}(N^{\mathrm{l_{1}l_{2}}}-N_{\mathrm{r_{1}r_{2}}}^{\mathrm{l_{1}l_{2}}}),$
(9)

where $N^{\mathrm{t(l)_{1}t(l)_{2}}}$ is the number of events where the first
lepton candidate is tight (loose) and the second lepton candidate is tight
(loose), $N_{\mathrm{r_{1}r_{2}}}^{\mathrm{t(l)_{1}t(l)_{2}}}$ is the
contribution to $N^{\mathrm{t(l)_{1}t(l)_{2}}}$ from events where both lepton
candidates are real leptons, and $F_{1}$ and $F_{2}$ correspond to the fake
factors associated with the first and second lepton candidate, respectively.

However, the algebraic simplification that leads to Eq. (9), where the result
depends simply on products of the fake factors and the observed tight and
loose lepton candidate yields, with a correction term that depends only on
events where all the leptons are real, does not hold for all possible event
selections nor all values of $m_{\mathrm{f}}$; such a simplification is
restricted to cases where the baseline and tight candidate lepton
multiplicities are the same in all events, and where $m_{\mathrm{f}}=1$.

###  2.4 Use with weighted events

In some cases, such as for self-consistency tests using simulated events, it
may be advantageous to weight the events that are input to the fake/non-prompt
background estimate. This is straightforward for the fake-factor and
asymptotic matrix methods, since the weight returned by the method for each
event can be multiplied by the event weight $w_{\mathrm{evt}_{i}}$. For the
likelihood matrix method, this is handled by using the scaled Poisson
distribution [7], in which the values of $N_{\mathrm{tl},\mathrm{exp}_{i}}$
and $N_{\mathrm{tl},\mathrm{evt}_{i}}$ from Eq. (8) are scaled according to
the event weights $w_{\mathrm{evt}_{i}}$:

$s_{\mathrm{tl}}\equiv{\sum_{\mathrm{tl},\mathrm{obs}}w_{\mathrm{evt}_{i}}\over\sum_{\mathrm{tl},\mathrm{obs}}w_{\mathrm{evt}_{i}}^{2}},$

so that the likelihood becomes

$L\left({\mathbf{N}}^{\mathrm{t}}_{\mathrm{rf}}\right)=\prod_{i}P\left[N_{\mathrm{tl}_{i}}/s_{\mathrm{tl}}|N_{\mathrm{tl},\mathrm{exp}_{i}}\left({\mathbf{N}}^{\mathrm{t}}_{\mathrm{rf}}\right)/s_{\mathrm{tl}}\right]$

###  2.5 Performance studies

MC simulations of experiments are utilised to assess the statistical
performance of the methods. These simulations consist of pseudoexperiments
that mimic scenarios that might occur in an actual analysis. Pseudoexperiments
with sample sizes of $10$ or $1000$ dilepton events in the baseline sample are
considered. The fraction of fake/non-prompt leptons in the baseline lepton
sample is varied for each pseudoexperiment, with a uniform distribution
between $0$ and 100%. The values of $\varepsilon_{\mathrm{r}}$ and
$\varepsilon_{\mathrm{f}}$ for each lepton are drawn from Gaussian
distributions with specified means and widths (limits are imposed such that
the values are always between zero and one, and $\varepsilon_{\mathrm{f}}$ is
always at least 10% less than $\varepsilon_{\mathrm{r}}$). Each lepton is
randomly assigned as fake/non-prompt or real, in accord with the fraction of
fake/non-prompt leptons assumed for the pseudoexperiment. Then each lepton is
judged to either meet or fail to meet the tight selection criteria, based on
whether or not it is a real lepton and the values of
$\varepsilon_{\mathrm{r}}$ and $\varepsilon_{\mathrm{f}}$ assigned to it. The
set of simulated leptons is input to the data-driven algorithms, and the
estimated fake yield and its statistical uncertainty are determined for each
pseudoexperiment. These values are then compared with the expectation value
for the number of fake/non-prompt lepton events in each pseudoexperiment,
which is determined by the numbers of real and fake/non-prompt baseline
leptons, the values of $\varepsilon_{\mathrm{r}}$ and
$\varepsilon_{\mathrm{f}}$ for each lepton, and the value of $m_{\mathrm{f}}$
for the simulated analysis. For example, in a dilepton sample with
$m_{\mathrm{f}}\geq 1$, the expectation value is:

$\langle N_{\mathrm{f}}^{\mathrm{t}}\rangle=\sum_{\textrm{rf
events}}\varepsilon_{\mathrm{r}_{1,i}}\varepsilon_{\mathrm{f}_{2,i}}+\sum_{\textrm{fr
events}}\varepsilon_{\mathrm{f}_{1,i}}\varepsilon_{\mathrm{r}_{2,i}}+\sum_{\textrm{ff
events}}\varepsilon_{\mathrm{f}_{1,i}}\varepsilon_{\mathrm{f}_{2,i}}.$

For the likelihood matrix method, the likelihood maximisation is implemented
using the minuit function minimisation package [8] to minimise the negative
log likelihood. The interface to minuit is provided by the TMinuit class in
root [9].

The fake-factor method requires a two-step process, where the contribution
from real-lepton events is subtracted in the second step. In an actual physics
analysis, this subtraction is done using MC simulation of the prompt lepton
contribution. For the simple MC simulation used here, the second step is
modelled by running each pseudoexperiment a second time with the same
parameters but a statistically independent sample of events, and running the
fake-factor method only on the events that do not have fake/non-prompt
leptons. The second sample has ten times more events than the primary sample,
to be consistent with the usual case where the MC simulation sample has a
multiple of the number of events in the data sample. The result of this second
run is then scaled down by a factor of ten and subtracted from the result when
using the initial set of simulated events.

As an initial example, one can investigate the performance for dilepton events
under conditions that are favourable for estimating the fake/non-prompt lepton
background. This means that the samples are large ($1000$ events per
pseudoexperiment), and the values of $\varepsilon_{\mathrm{r}}$ and
$\varepsilon_{\mathrm{f}}$ are on average well-separated (here,
$\langle\varepsilon_{\mathrm{r}}\rangle=0.90$ and
$\langle\varepsilon_{\mathrm{f}}\rangle=0.10$). The ratios of the estimated to
true fake yields are shown versus $\langle N_{\mathrm{f}}^{\mathrm{t}}\rangle$
in Figure 1(a). The average statistical uncertainties in the estimates for
each method are shown in Figure 1(b), and the fraction of pseudoexperiments in
which the true fake yield lies within the uncertainty reported for each method
is shown in Figure 1(c). The performance of the methods for dilepton analyses
with low statistical precision ($10$ events per pseudoexperiment) are shown in
Figure 2. Finally, to represent a more challenging situation, the case where
there is less separation between the values of $\varepsilon_{\mathrm{r}}$ and
$\varepsilon_{\mathrm{f}}$ (due, for example, to the application of stricter
online lepton selection criteria that might be required when the LHC runs at
higher luminosities) is also explored. Figure 3 shows the results when
$\langle\varepsilon_{\mathrm{r}}\rangle=0.70$ and
$\langle\varepsilon_{\mathrm{f}}\rangle=0.30$, and there are $10$ events per
pseudoexperiment.

((a))

((b))

((c))

Figure 1: Performance of the three methods in pseudoexperiments with dilepton
events where the leptons had an average $\varepsilon_{\mathrm{r}}=0.9$ and
$\varepsilon_{\mathrm{f}}=0.1$, both values were varied according to a
Gaussian distribution of width $0.1$ when simulating each lepton, and there
were $1000$ events per pseudoexperiment. The quantity $\langle
N_{\mathrm{f}}^{\mathrm{\ t}}\rangle$ is the expectation value for the number
of fake/non-prompt lepton events in each pseudoexperiment. Plot (a) shows the
ratio of the estimated to expected fake-lepton yields, (b) shows the absolute
uncertainty estimate for each method (for the likelihood method the average of
the upward and downward uncertainties is taken), and (c) shows the fraction of
pseudoexperiments where the true fake yield lies within the reported one
standard deviation ($1\sigma$) range.

((a))

((b))

((c))

Figure 2: Performance of the three methods in pseudoexperiments with dilepton
events where the leptons had an average $\varepsilon_{\mathrm{r}}=0.9$ and
$\varepsilon_{\mathrm{f}}=0.1$, both values were varied according to a
Gaussian distribution of width $0.1$ when simulating each lepton, and there
were $10$ events per pseudoexperiment. The quantity $\langle
N_{\mathrm{f}}^{\mathrm{t}}\rangle$ is the expectation value for the number of
fake/non-prompt lepton events in each pseudoexperiment. Plot (a) shows the
ratio of the estimated to expected fake-lepton yields, (b) shows the absolute
uncertainty estimate for each method (for the likelihood method the average of
the upward and downward uncertainties is taken), and (c) shows the fraction of
pseudoexperiments where the true fake yield lies within the reported one
standard deviation ($1\sigma$) range.

((a))

((b))

((c))

Figure 3: Performance of the three methods in pseudoexperiments with dilepton
events where the leptons had an average $\varepsilon_{\mathrm{r}}=0.7$ and
$\varepsilon_{\mathrm{f}}=0.3$, both values were varied according to a
Gaussian distribution of width $0.1$ when simulating each lepton, and there
were $10$ events per pseudoexperiment. The quantity $\langle
N_{\mathrm{f}}^{\mathrm{t}}\rangle$ is the expectation value for the number of
fake/non-prompt lepton events in each pseudoexperiment. Plot (a) shows the
ratio of the estimated to expected fake-lepton yields, (b) shows the absolute
uncertainty estimate for each method (for the likelihood method the average of
the upward and downward uncertainties is taken), and (c) shows the fraction of
pseudoexperiments where the true fake yield lies within the reported one
standard deviation ($1\sigma$) range.

These studies show that all three methods give accurate estimates, with nearly
equivalent performance, in high-statistics samples with a large separation
between $\varepsilon_{\mathrm{r}}$ and $\varepsilon_{\mathrm{f}}$ (as shown in
Figure 1). One notable feature is the dip in the uncertainty values for all
three methods near $\langle N_{\mathrm{f}}^{\mathrm{t}}\rangle=10$ in Figure
1(b). This occurs because there are two ways for the model to produce an
expectation around $10$: either a very low fake fraction in the baseline to
start with, or a very large fake fraction so that there are few real leptons
and most of the background is from events with two fake/non-prompt leptons,
which gives a minimum value of $\varepsilon_{\mathrm{f}}^{2}\cdot 1000=10$
when $\varepsilon_{\mathrm{f}}=0.10$. These two processes will have very
different uncertainties. There is also a bias in the likelihood matrix method
toward low values when the true number of fakes is large. This bias arises due
to the averaging of efficiencies over the entire baseline sample (see Eq. (7))
when in fact the efficiencies are on average different for real and fake
leptons. Such differences in the averages occur randomly in the ‘toy’ MC
tests, but can occur systematically in a physics analysis if, for example, the
real and fake efficiencies have different kinematic distributions. The biases
can be mitigated by binning the baseline sample in the variables for which the
real-lepton and fake-lepton distributions may differ, and performing the
likelihood fit separately in each bin. As an example of the effect of such a
binning, the pseudoexperiments can be run with the results binned according to
the values of $\varepsilon_{\mathrm{f}}$. The effect of using two such bins in
the value of $\varepsilon_{\mathrm{f}}$ for each lepton is shown in Figure 4.

When the situation becomes more challenging, such as in Figures 2 and 3, the
characteristics of each method become more distinct. For low-statistics
samples, the likelihood approach tends to exhibit a bias toward high values
when the true number of fakes is small, a natural consequence of the fact that
it cannot return negative values. The coverage of the true value by the
estimated uncertainty is, however, still reasonable. A clear distinction
between the precision of the methods also appears in Figures 2 and 3, where
the likelihood approach has the smallest uncertainty, followed by the
asymptotic matrix method and then the fake-factor method. This is because the
likelihood approach considers lepton efficiencies averaged over the entire
sample and is therefore less susceptible to event-by-event fluctuations in the
values of $\varepsilon_{\mathrm{r}}$ and $\varepsilon_{\mathrm{f}}$.

Figure 4: Ratio of the estimated to expected fake-lepton yields for the
likelihood matrix method in pseudoexperiments with dilepton events where the
leptons had an average $\varepsilon_{\mathrm{r}}=0.9$ and
$\varepsilon_{\mathrm{f}}=0.1$, both values were varied according to a
Gaussian distribution of width $0.1$ when simulating each lepton, and there
were $1000$ events per pseudoexperiment. The triangles show the result with a
single bin, while the circles show the result where the input events are
binned according to the values of $\varepsilon_{\mathrm{f}}$ for each lepton.
Two bins are used for each of the two leptons, and the results from all four
bins are summed to give the estimated $N_{\mathrm{fake}}$ value.

Despite the differences between them, none of the approaches is incorrect, as
shown by the statistical coverage plots: except in extreme cases, the
confidence intervals built from the estimates and their statistical
uncertainty do contain the true fake yield in at least 68% of
pseudoexperiments, and the visible overcoverage is due mostly to the fact that
the uncertainties are computed under the assumption that all of the
${\mathbf{N}}_{\mathrm{tl}}$ values are independent, while the
pseudoexperiments were generated with a fixed number of baseline events for
each, which means there was some anticorrelation among the
${\mathbf{N}}_{\mathrm{tl}}$.

When selecting which method to use, the analyser needs to consider the
relative benefits and complexities of implementing the methods, along with the
size of the uncertainty in the fake/non-prompt lepton background yield
relative to other uncertainties in the analysis.

##  3 The ATLAS detector

While the above description of the matrix and fake-factor methods is general,
the remainder of this paper discusses the application of these methods to
ATLAS physics analyses, and therefore a brief description of the experimental
apparatus follows.

The ATLAS detector [1] at the LHC covers nearly the entire solid angle around
the collision point.333 ATLAS uses a right-handed coordinate system with its
origin at the nominal interaction point (IP) in the centre of the detector and
the $z$-axis along the beam pipe. The $x$-axis points from the IP to the
centre of the LHC ring, and the $y$-axis points upwards. Cylindrical
coordinates $(r,\phi)$ are used in the transverse plane, $\phi$ being the
azimuthal angle around the $z$-axis. The pseudorapidity is defined in terms of
the polar angle $\theta$ as $\eta=-\ln\tan(\theta/2)$. Angular distance is
measured in units of $\Delta R\equiv\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$.
It consists of an inner tracking detector surrounded by a thin superconducting
solenoid, electromagnetic (EM) and hadronic calorimeters, and a muon
spectrometer (MS) incorporating three large superconducting toroidal magnets.
The inner-detector system is immersed in a $2\text{\,}\mathrm{T}$ axial
magnetic field and provides charged-particle tracking in the range
$|\eta|<2.5$.

The high-granularity silicon pixel detector covers the vertex region and
typically provides four measurements per track, the first hit normally being
in the insertable B-layer (IBL) installed before Run 2 [10, 11]. It is
followed by the silicon microstrip tracker, which usually provides eight
measurements per track. These silicon detectors are complemented by the
transition radiation tracker (TRT), which enables radially extended track
reconstruction up to $|\eta|=2.0$. The TRT also provides electron
identification information based on the fraction of hits (typically $30$ in
total) above a higher energy-deposit threshold corresponding to transition
radiation.

The calorimeter system covers the pseudorapidity range $|\eta|<4.9$. Within
the region $|\eta|<3.2$, EM calorimetry is provided by barrel and endcap high-
granularity lead/liquid-argon (LAr) calorimeters, with an additional thin LAr
presampler covering $|\eta|<1.8$ to correct for energy loss in material
upstream of the calorimeters. Hadronic calorimetry is provided by the
steel/scintillator-tile calorimeter, segmented into three barrel structures
within $|\eta|<1.7$, and two copper/LAr hadronic endcap calorimeters. The
solid angle coverage is completed with forward copper/LAr and tungsten/LAr
calorimeter modules optimised for EM and hadronic measurements respectively.

The MS comprises separate trigger and high-precision tracking chambers
measuring the deflection of muons in a magnetic field generated by the
superconducting air-core toroids. The field integral of the toroids ranges
between $2.0$ and $6.0\text{\,}\mathrm{T}\mathrm{m}$ across most of the
detector. A set of precision chambers covers the region $|\eta|<2.7$ with
three layers of monitored drift tubes, complemented by cathode-strip chambers
in the forward region, where the background is highest. The muon trigger
system covers the range $|\eta|<2.4$ with resistive-plate chambers in the
barrel, and thin-gap chambers in the endcap regions. Interesting events are
selected to be recorded by the first-level trigger system implemented in
custom hardware, followed by selections made by algorithms implemented in
software in the high-level trigger [12]. The first-level trigger accepts
events from the $40\text{\,}\mathrm{MHz}$ bunch crossings at a rate below
$100\text{\,}\mathrm{kHz}$, which the high-level trigger reduces in order to
record events to disk at about $1\text{\,}\mathrm{kHz}$.

An extensive software suite [13] is used in data simulation, in the
reconstruction and analysis of real and simulated data, in detector
operations, and in the trigger and data acquisition systems of the experiment.

##  4 Lepton selection criteria

Full descriptions of the electron and muon reconstruction algorithms and
available selection criteria used in ATLAS are provided in Refs. [2] and [3],
respectively. Here the features most relevant to the fake/non-prompt lepton
background estimation are summarised briefly.

###  4.1 Electron reconstruction and identification

Electron candidates are reconstructed within $|\eta|<2.47$ as tracks in the
inner detector matched to energy clusters in the EM calorimeter. In order to
separate true electrons from other unwanted reconstructed candidates, electron
identification (ID) algorithms are used. These rely upon a set of variables
that quantify the distribution of energy in the calorimeter, the quality of
the spatial match between the calorimeter deposit and the associated track,
and the transition radiation signal in the TRT (see Table 1 of Ref. [2] for a
complete list). Rather than place individual requirements on these variables,
they are combined into a likelihood discriminant based upon the probability
density functions of the variables measured for prompt electrons in
$Z\rightarrow e^{+}e^{-}$ events and for background candidates reconstructed
in inclusive collision events.

Since different analyses have different requirements for electron selection
efficiency and background rejection, several ‘working points’ (WPs) are
defined by different values of the likelihood. The likelihood threshold values
are varied according to the $p_{\text{T}}$ and $|\eta|$ of the electron
candidate, so that the selection efficiency varies smoothly with the electron
$p_{\text{T}}$. The most commonly used ID WPs (and their average
efficiencies444 Those efficiencies, as well as those quoted for muons in the
next section, do not include requirements for the leptons to be successfully
identified in the context of the trigger decision, which may include small
additional inefficiencies [14, 15].  measured for typical electroweak
processes) are ‘Loose’ (93%), ‘Medium’ (88%) and ‘Tight’ (80%). In addition to
the listed WPs, there is another (‘LooseAndBLayer’) WP that imposes the same
requirement on the likelihood as the ‘Loose’ WP, but also requires that the
electron track have a hit in the IBL to suppress candidates originating from
photon conversions.

Often, additional requirements are imposed on the impact parameter of the
electron’s track: the impact parameter in the transverse plane, $d_{0}$, with
respect to the centre of the beamspot must satisfy $|d_{0}|<5\sigma(d_{0})$,
where $\sigma(d_{0})$ is its estimated uncertainty, while the longitudinal
separation $z_{0}$ between the point where $d_{0}$ is measured and the chosen
primary vertex of the event, multiplied by a moderating factor $\sin(\theta)$
which accounts for reduced $z_{0}$ accuracy for more forward tracks, cannot
exceed $0.5\text{\,}\mathrm{mm}$ in absolute value.

###  4.2 Muon reconstruction and identification

Muon candidates are reconstructed in the region $|\eta|<2.5$ by combining MS
tracks with matching inner-detector tracks. The muon reconstruction efficiency
is approximately 98% per muon in simulated $Z\rightarrow\mu^{+}\mu^{-}$
events. After reconstruction, high-quality muon candidates used for physics
analyses are selected by a set of requirements on the number of hits in the
different inner subdetectors and different MS stations, on the track fit
properties, and on variables that test the compatibility of the individual
measurements in the two detector systems, as detailed in Ref. [3]. These
criteria reduce the background from in-flight decays of light-flavour hadrons,
which often result in kinked tracks. The most commonly used muon ID WPs (and
their efficiencies measured in $t\bar{t}$ MC events) are ‘Medium’ (98%) and
‘HighPt’ (80%), the latter optimised to offer the best momentum resolution for
$p_{\text{T}}>$100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$. The same impact
parameter requirements as defined for electrons are also often applied to muon
candidates, with a tighter condition in the transverse plane:
$|d_{0}|<3\sigma(d_{0})$.

###  4.3 Lepton isolation

In addition to the ID criteria mentioned above, most analyses place
requirements on the isolation of the lepton from other detector activity. This
is especially helpful in reducing the contribution from leptons produced in
heavy-flavour decays, or muons from $\pi^{\pm}$ or $K^{\pm}$ decays, within
jets555Jets are reconstructed from clusters of topologically connected
calorimeter cells (topo-clusters), as described in Ref. [16]. The anti-$k_{t}$
algorithm [17, 18] is used to form jets from the topo-clusters, with the
radius parameter $R$ usually set to 0.4; when reconstructing jets formed from
the merged decay products of boosted resonances, a larger $R$ value, typically
10, is used. Typically, jets with $p_{\text{T}}>$20\text{\,}\mathrm{GeV}$$ and
$|\eta|<4.5$ are considered in physics analyses. as there are often other
components of the jet that are near the lepton in these cases. In the context
of the matrix and fake-factor methods, the baseline lepton selection usually
does not require isolation, while the tight lepton selection usually does.
However, many of the single-lepton triggers [15, 14] used in ATLAS require
isolation, so analyses that depend on such triggers cannot avoid applying
isolation requirements for baseline leptons.

The calorimeter isolation [2, 3] is calculated from the sum of transverse
energies of calorimeter energy clusters within $\Delta
R\equiv\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}=\text{XX}/100$ of the lepton
candidate, not including the contribution expected from the candidate itself.
Typical values of XX are $20$, $30$ or $40$. Expected residual contributions
to the isolation from the lepton candidate, as well as expected contributions
from particles produced by additional proton–proton ($pp$) interactions, are
subtracted [19], resulting in the variable $E_{\mathrm{T}}^{\text{coneXX}}$.

The track-based isolation [2, 3], denoted by $p_{\text{T}}^{\text{coneXX}}$,
is based on tracks near the lepton candidate with either $p_{\text{T}}>0.5$ or
$1\text{\,}\mathrm{GeV}$ that satisfy basic track-quality requirements and are
spatially consistent with the primary vertex of the event. The scalar sum of
the transverse momenta of such tracks, excluding tracks associated with the
lepton candidate, is compared with the $p_{\text{T}}$ of the candidate to
assess the isolation. For muon candidates, only the single track associated
with the candidate is excluded; for electron candidates, additional tracks
consistent with pair-production from a bremsstrahlung photon are also
excluded. The track isolation can also be defined with a variable cone size
($p_{\text{T}}^{\text{varconeXX}}$). In this case, the size of the cone around
the lepton candidate within which tracks are considered is varied as a
function of the $p_{\text{T}}$ of the candidate:

$\Delta
R=\min\left(\frac{$10\text{\,}\mathrm{GeV}$}{p_{\text{T}}\,[\text{GeV}]},\,R_{\mathrm{max}}\right),$

where $R_{\mathrm{max}}$ is the maximum cone size (typically $0.2$ or $0.3$).

Combining selections on track-based and calorimeter-based isolation provides
even better fake/non-prompt lepton background rejection, as the two isolation
variables use complementary information. Track-based isolation was found to be
less sensitive to detector noise and pile-up effects than calorimeter-based
isolation, and the inner detector provides a better $p_{\text{T}}$ measurement
than the calorimeters for individual soft hadrons. On the other hand,
calorimeter-based isolation includes neutral particles as well as some
particles below the inner detector’s track-$p_{\text{T}}$ threshold, which are
ignored when computing track isolation. However, track and calorimeter
isolation variables measure hadronic activity in a redundant manner, since
charged particles are measured by both the calorimeters and the inner
detector, and simple selection cuts applied independently to those two
variables may not achieve optimal rejection power. To avoid this, an analysis
can use a ‘particle-flow’ algorithm, which allows removal of overlapping
contributions from the track-based and calorimeter-based isolation, decreasing
the correlation between the two variables. For the time being, particle-flow-
based isolation variables are defined only for muons, and discussed in Ref.
[3].

For analyses where the fake/non-prompt lepton background may be dominated by
non-prompt electrons and muons from the decays of $b$\- and $c$-hadrons [20,
21], isolation WPs using a boosted decision tree (BDT) discriminant based on
isolation and secondary vertex information, referred to as the non-prompt
lepton BDT, are also proposed.

Several isolation WPs based on tracking, a combination of calorimetry and
tracking, particle-flow, or a non-prompt lepton BDT are defined to allow for
consistency across analyses that require different levels of lepton isolation.
They are described in Refs. [2] and [3].

###  4.4 Removing overlaps between jets and leptons

In some cases the same object can result in multiple signatures in the
detector. For example, an electron will deposit energy in the calorimeter, and
that energy will generally also be clustered into a jet. In addition,
sometimes objects will be spatially correlated due to the underlying physics,
as when a muon is produced by heavy-flavour decay within a jet. To avoid
double-counting, and to select only the isolated leptons that are typically of
interest in physics analyses, an overlap removal (OR) procedure is applied to
resolve these ambiguities. The procedure used for most physics analyses is as
follows, where the lepton candidates are those that satisfy the baseline ID
criteria for the analysis:

  * •

All electron candidates that are within $\Delta R=0.01$ of a muon candidate
(or share a track with a muon candidate) are removed.

  * •

All jets that are within $\Delta R=0.2$ of any remaining electron candidates
are removed.

  * •

All electron candidates that are within $\Delta R=0.4$ of any remaining jet
are removed.

  * •

Cases where a remaining jet is within $\Delta R=0.4$ of a muon candidate are
examined to determine the number of tracks associated with the jet. If there
are more than two such tracks the muon candidate is removed; otherwise the jet
is removed.

Variations of this procedure are also supported, primarily for analyses that
focus on heavy-flavour jets or that select boosted massive particles using
large-$R$ jets.

##  5 Monte Carlo simulation samples

The studies of fake/non-prompt leptons that are presented in Sections 6 and 7
make use of large MC samples of simulated events. The production of $t\bar{t}$
events at next-to-leading order (NLO) in the quantum chromodynamics (QCD)
coupling constant $\alpha_{\text{s}}$ is described in Ref. [22] and relies on
the Powheg Box v2 event generator [23] interfaced with Pythia 8.230 [24] for
parton showering and subsequent steps, with the A14 set of tuned parameters
[25]. The parton distribution functions used for matrix element calculation
and parton showering are NNPDF3.0nlo [26] and NNPDF2.3lo [27] respectively.
Generated events were filtered such that at least one of the top quarks decays
semileptonically. The EvtGen 1.2.0 program [28] was used to model heavy-
flavour hadron decays.

The production of Drell–Yan $Z/\gamma^{*}\to\ell^{+}\ell^{-}$ events
($\ell=e,\mu,\tau$) at NLO in $\alpha_{\text{s}}$ is described in Ref. [29]
and relies on the Sherpa 2.2 event generator [30] with the dedicated set of
tuned parameters and the NNPDF3.0nnlo [26] parton distribution function.
Events were generated according to a partition of the phase space described in
Ref. [29], resulting in a set of orthogonal samples which were combined with
weights corresponding to the NLO cross-section calculated by the generator.
The $\ell^{+}\ell^{-}$ invariant mass was required to be at least
$40\text{\,}\mathrm{GeV}$.

A full ATLAS detector simulation [31] based on Geant4 [32] was then used to
faithfully reproduce particle interactions with the detector and its response.
Additional $pp$ interactions in the same or neighbouring bunch crossings were
also simulated in order to reproduce the conditions of data-taking during the
LHC Run 2 operation.

The source of a lepton candidate in simulation, discussed in Section 6, is
identified by using preserved generator-level information to match each
reconstructed lepton candidate to the closest particle produced by the event
generator, based primarily on the angular separation between the latter and
the reconstructed track of the former. If the matched particle is a charged
lepton (or, in the case of electron candidates, a photon), its origin is
checked so that the different sources of non-prompt leptons can be
distinguished.

In Section 9, the simulation of several other processes is involved, and is
performed with a similar workflow; complete information about the MC
generators employed, their configurations, and the cross-section calculations
used to normalise the simulated samples, is available in the references
provided in that section for those two analyses.

##  6 Sources of fake/non-prompt leptons

((a))  ((b))

Figure 5: Relative contributions of fake or non-prompt muons from different
sources as a function of $p_{\text{T}}$ in simulated (a) $t\bar{t}$ and (b)
Drell–Yan processes. Punch-through hadrons are charged hadrons reaching the MS
and leading to the reconstruction of a fake muon candidate. Some other minor
sources are not displayed; the sum of their contributions is less than 2% in
every bin of (a), and less than 4% in bins of (b) up to
$p_{\text{T}}=$100\text{\,}\mathrm{GeV}$$. The muons are required to satisfy
minimal track-quality criteria specified in the text. Error bars represent
statistical uncertainties of the simulation.

((a))  ((b))

Figure 6: Relative contributions of fake or non-prompt electrons from
different sources as a function of $p_{\text{T}}$ in simulated (a) $t\bar{t}$
and (b) Drell–Yan processes. Some other minor sources are not displayed; the
sum of their contributions is less than 0.5% in every bin. These do not
include electron candidates corresponding to high-energy muon ionisation
signal in the calorimeters, or from conversions of final-state radiation (FSR)
photons with small angular separation from a genuine electron or muon. The
electrons are required to satisfy loose ID criteria specified in the text and
Ref. [33]. Error bars represent statistical uncertainties of the simulation.

The relative contributions of fake/non-prompt leptons from different sources
to the sample of selected lepton candidates depend on the energy and spatial
location of the candidate in the detector, the identification, impact
parameter and isolation criteria applied, the overlap removal procedure, and
the nature of the selected final state (e.g. the presence or absence of bottom
quarks).

Figures 5 and 6 present for illustration the relative contributions of the
different sources of fake/non-prompt muons and electrons respectively, as a
function of the transverse momentum of the candidate, as measured in MC
simulated events. They are shown for two different processes: $t\bar{t}$
production, leading to final states rich in heavy-flavour hadrons, and
Drell–Yan production of $e^{+}e^{-}$ or $\mu^{+}\mu^{-}$ pairs; for $t\bar{t}$
events at least one of the top quarks is required to decay leptonically. For
these particular figures, the reconstruction/ID of electron and muons and the
general event selection follow those described in Ref. [33] for baseline
lepton candidates, which correspond to rather loose criteria (in particular,
no isolation nor transverse impact parameter requirements are applied, and no
overlap removal is done). For both processes, events are considered only if
they contain a pair of reconstructed baseline leptons with identical charges,
a signature for which fake/non-prompt leptons usually represent a non-
negligible source of background.

The ‘prompt $\gamma$-conversion’ category only includes electron candidates
where the photon is separated by $\Delta R>0.1$ from any generator-level
high-$p_{\text{T}}$ electron from the hard-scatter interaction (photons
emitted at smaller separation are generally reconstructed as part of the
electron candidates).

Non-prompt leptons are those arising from electroweak decays of hadrons.
Heavy-flavour $b$\- and $c$-hadrons decay close to the interaction point and
the resulting leptons are distinguishable from real leptons mostly by
isolation and impact parameter.

Light-flavour hadrons can also be a major source of fake leptons via decay-in-
flight in the tracker volume. This happens mainly in final states for which
QCD multi-jet production is a significant contributor. A charged hadron
stopping early in the calorimeter, and generating a narrower-than-average
shower, can mimic the experimental signature of an electron. Electron ID
criteria are particularly powerful in rejecting these candidates rather than
those from other sources, based notably on three-dimensional profiles of the
shower, but since many orders of magnitude more hadrons than leptons are
produced in collisions at the LHC, a substantial number of such fake electrons
may be selected in a physics analysis. With regard to muons, the depth of the
ATLAS calorimeter is sufficient to stop most pions and kaons before they can
reach the MS. Muon candidates arising from kaons decaying semileptonically in
flight before reaching the MS are a more important contribution to the
fake/non-prompt lepton background.

Among hadrons faking electrons, one significant class is neutral pion decays
into photons ($\pi^{0}\to\gamma\gamma$); the collimated photons create a
single energy deposit in the EM calorimeter, while the associated track might
be provided by the conversion of one of the photons in the upstream detector
material. Due to the importance of this phenomenon, the electron ID criteria
specifically discriminate against these by attempting to identify a two-peak
structure in the distribution of the cell energies matched to the electron
cluster [2], which is not present for real electrons. Other contributions such
as Dalitz decay of pions [34] also create comparable experimental signatures.

The conversion of photons into electron–positron pairs represents the last
important class of non-prompt electrons. These $\gamma$-conversions must
typically occur early on (e.g. in the beam pipe) and be largely asymmetric in
the splitting of the momentum between the two electrons; otherwise, the
conversion vertex can be reconstructed, or the candidate electron’s track
lacks hits in the first layers of the inner detector, both leading to the
proper classification of the reconstructed object as a photon instead of an
electron [2]. The origin of the photon itself influences the characterisation
of the candidate as non-prompt or real: photons emitted close to a real
electron, either due to bremsstrahlung or as higher-order quantum
electrodynamic (QED) corrections to the production process, are typically
considered part of the electron candidate (the calorimeter energy deposits
tend to overlap to the extent that a single cluster is reconstructed);
furthermore, from the perspective of quantum field theory, well-defined
electrons must include extra radiation (‘dressed leptons’ [35]). The electron
reconstruction procedure [2] accounts for bremsstrahlung, in particular by
allowing kinks in the track consistent with bremsstrahlung emission in dense
material regions. In contrast, photons from other origins, such as initial-
state radiation, QED processes not involving leptons or where photons are
sufficiently separated from leptons, or hadronic jet fragmentation, may be
considered as sources of fake electrons.

It can be seen in Figure 5 that non-prompt muons constitute the only
substantial contribution to the fake/non-prompt muon background, while for
electrons Figure 6 shows more variety: in general, non-prompt electrons are
particularly represented in the lower $p_{\text{T}}$ range, especially in
processes involving the production of heavy-flavour hadrons, while hadron
fakes and converted photons populate the higher $p_{\text{T}}$
range.666However, for the range $p_{\text{T}}<$10\text{\,}\mathrm{GeV}$$, not
shown in the figure, hadron fakes are also the dominant contribution.

((a))  ((b))

Figure 7:  Rejection (defined as 1/$\varepsilon_{\mathrm{f}}$) as a function
of $p_{\text{T}}$ for different sources of fake/non-prompt (a) muons and (b)
electrons in simulated $t\bar{t}$ events. The efficiency
$\varepsilon_{\mathrm{f}}$ is the fraction of lepton candidates selected using
the same baseline criteria as in Figures 5 and 6 that also pass the tight
selection criteria of Ref. [33]. Error bars represent statistical
uncertainties of the simulation, and for purposes of clarity, values with
relative uncertainties greater than 30% are not shown.

The different sources of fake/non-prompt leptons have distinct probabilities
to satisfy the tight lepton selection criteria described in Section 4. Figure
7 illustrates those differences for the particular example of signal leptons
definitions used in Ref. [33] (including lepton–jet overlap removal) and
simulated $t\bar{t}$ events. Such variability between sources is unwelcome, as
the fake efficiencies required for the application of the methods described in
Section 2 then depend upon the relative contributions of each source to the
regions of interest, which may not be easy to assess. It is therefore
desirable to measure the efficiencies in regions similar in composition to the
regions where the background estimate is needed, otherwise large extrapolation
uncertainties may apply.

Precise simulation of these various sources of background, including their
relative contributions, is indeed very challenging, as it relies heavily on
the modelling of the soft-QCD regime by event generators, including modelling
of fragmentation and hadronisation processes, hadron decay modelling, and soft
emissions and detector modelling. Another issue is that only a small fraction
of fake or non-prompt lepton candidates survive the ID and isolation
requirements, so the simulation of a very large number of events is needed to
obtain a statistically accurate prediction. For inclusive processes with large
cross-sections (e.g. multi-jet production), this is often impractical.

For these reasons, many of the fake/non-prompt lepton background predictions
used in ATLAS publications are based on methods using the data, such as the
ones described in this paper. They rely on common properties shared to some
extent by the different sources of fake/non-prompt leptons that differentiate
them from real leptons, such as a high likelihood to not meet the combination
of ID and isolation criteria.

##  7 Measurement of real and fake/non-prompt lepton efficiencies

The matrix and fake-factor methods both rely on knowledge of the efficiency
for leptons that pass the baseline selection to also pass the tight selection.
For the fake-factor method, only the efficiency for fake/non-prompt leptons is
used explicitly in the calculation, while for the matrix method the efficiency
for real leptons must also be measured. In many cases these efficiencies
depend on the properties of the lepton (such as its $p_{\text{T}}$ or angular
distance from a jet) or on the event in which it is found (such as the overall
activity of the event, as measured for example by the number of reconstructed
primary vertices).

###  7.1 Real-lepton efficiencies

The efficiencies of specific working points of lepton selections are
calibrated precisely in ATLAS for general purposes, primarily with ‘tag-and-
probe methods’ based e.g. on $Z\rightarrow\ell\ell$ events. By performing
these measurements on both data and MC-simulated events, ‘scale factors’ (SFs)
that account for differences between the efficiencies observed in data and
simulation are derived. These SFs can then be applied to simulated events
using the selection criteria that are relevant to a given analysis to
determine the appropriate real-lepton efficiencies. This MC-based approach is
valid as long as both the baseline and tight lepton selection criteria are
taken from the set for which SFs have been measured; only in extraordinary
cases would an analysis utilise different selection criteria. Since the
efficiencies depend more strongly on the environment than the SFs do, the main
advantage of this approach over purely data-based measurements is to allow
efficiencies to be obtained directly in the desired environment (i.e. the
region in which the fake/non-prompt background estimate is needed), rather
than being extrapolated from a more distant region which would be needed for
reliable measurements in data.

Details of the real-electron and real-muon efficiency measurements can be
found in Refs. [2] and [3], respectively. The real efficiencies are often
parameterised with respect to the $p_{\text{T}}$ and $|\eta|$ of the leptons,
and measured separately for electrons and muons.

###  7.2 Fake/non-prompt lepton efficiencies

The fake/non-prompt lepton efficiencies are specific to each analysis,
primarily since there are several sources for such leptons (see Section 6),
which will contribute with different weights depending on the chosen selection
criteria. In general, though, the first step in the efficiency measurement is
to identify a region that has a large contribution from fake leptons. Two
approaches are commonly used. In the first, events with a pair of leptons with
the same electric charge are selected. Since such lepton pairs are only rarely
produced at the LHC (via processes such as $WZ\text{\,+\,jets}$,
$ZZ\text{\,+\,jets}$, and $t\bar{t}+X$ ($X=W/Z$) production) it is likely that
one of the two leptons is fake/non-prompt. By placing stringent quality
criteria on one lepton in these events, the probability that the remaining
lepton is fake/non-prompt is enhanced. The second approach is to use single-
lepton events, where criteria are imposed to suppress the contribution from
real leptons. Examples of such criteria are requiring the missing transverse
momentum777 $\mathbf{E_{\text{T}}^{\text{miss}}}$ is defined as the negative
vector sum of the $p_{\text{T}}$ of the reconstructed and calibrated objects
in the event, with a correction applied for inner detector tracks that
originate from the primary collision vertex and are not associated with any
other objects, and $E_{\text{T}}^{\text{miss}}$ is defined as the magnitude of
$\mathbf{E_{\text{T}}^{\text{miss}}}$ [36]. $E_{\text{T}}^{\text{miss}}$, or
transverse mass888Here
$m_{\text{T}}\equiv\sqrt{2\,p_{\text{T}}^{\ell}\,E_{\text{T}}^{\text{miss}}(1-\cos\Delta\phi_{\ell,E_{\text{T}}^{\text{miss}}})}$,
where $\Delta\phi_{\ell,E_{\text{T}}^{\text{miss}}}$ is the azimuthal angle
between the lepton and $\mathbf{E_{\text{T}}^{\text{miss}}}$ directions.
$m_{\text{T}}$, to be below specific thresholds, thereby reducing the
contribution from $W\text{+\,jets}$ or $t\bar{t}$ events, or requiring via the
track impact parameters that the lepton originate from a position inconsistent
with the primary event vertex, thereby enhancing the contribution of leptons
from heavy-flavour decay.

In either approach, there will be a residual contribution from events with
only real leptons in the selected sample. This contribution is typically
estimated using MC simulation and subtracted separately from both the tight
and baseline samples before the ratio of these samples is taken to measure the
efficiency.

As with the real-lepton efficiencies, the fake/non-prompt lepton efficiencies
depend on properties of the lepton candidates or of the event in which they
are found. Therefore, it is generally helpful to bin the efficiencies in the
lepton $p_{\text{T}}$ and $|\eta|$, and possibly in terms of other quantities
as well. The optimal binning to be used is chosen in the context of each
physics analysis, considering the given numbers of events and potential
changes in the relative contributions from the different sources of fake/non-
prompt leptons; illustrative examples are provided in Section 9.

((a))  ((b))

Figure 8: Probability for a muon produced in the decay of a $b$-hadron to
satisfy a track-based isolation requirement, in simulated $t\bar{t}$ events,
shown as a function of (a) the muon $p_{\text{T}}$; and (b) the sum
$p_{\text{T}}+E_{\text{T}}^{\text{cone40}}$ of the muon $p_{\text{T}}$ and
nearby calorimeter energy deposits. In each case, the universality of these
probabilities is gauged by changing the kinematic properties of the parent
hadron (markers of various colours), achieved indirectly by selecting events
in different ranges of global transverse energy $H_{\text{T}}$. Precise
definitions of the isolation criteria, the
$p_{\text{T}}+E_{\text{T}}^{\text{cone40}}$ and $H_{\text{T}}$ variables, as
well as descriptions of the muon and jets selections, are given in Section
7.2.

Adopting a parameterisation of the efficiency with respect to variables other
than $p_{\text{T}}$ and $\eta$ can sometimes be beneficial. For example, when
considering solely the probability that fake or non-prompt leptons satisfy the
isolation criteria, another relevant quantity is the momentum of the parent
jet: the fake efficiency corresponds in that case to the probability that most
of the jet’s visible momentum is carried by the lepton. A parameterisation as
a function of lepton $p_{\text{T}}$ thus assumes that for a particular
$p_{\text{T}}$, the distribution of the parent jet’s momentum is similar in
the regions where the efficiencies are measured and the regions where the
background estimates are needed. If this assumption does not hold, it can be
useful to adopt instead a parameterisation as a function of the parent jet’s
momentum. Since this quantity is not easily accessed experimentally (unlike
the lepton $p_{\text{T}}$), proxy observables are used in practice. An example
of successful application is the analysis in Ref. [37], which employed the sum
of the lepton’s $p_{\text{T}}$ and the transverse energy in a cone around the
lepton as a proxy.

The preceding discussion is also illustrated in Figure 8 for non-prompt muons
produced in the decay of $b$-hadrons in simulated $t\bar{t}$ events. The
probabilities for such muons to satisfy a track-based isolation requirement,
as defined in Section 4, are shown for two alternative parameterisations: one
based on the muon $p_{\text{T}}$, and the other on the scalar sum of the muon
$p_{\text{T}}$ and the transverse energy deposited in calorimeter-cell
clusters within a cone of size $\Delta R=0.4$ around the muon (referred to as
$E_{\text{T}}^{\text{cone40}}$ in Section 4). This scalar sum serves as a
proxy for the parent jet’s transverse momentum.

In the second parameterisation, the fake efficiency is the fraction of jets
with a non-prompt muon and visible momentum
$p_{\text{T}}+E_{\text{T}}^{\text{cone40}}$ in which the muon is mostly
isolated, i.e. $p_{\text{T}}\gg E_{\text{T}}^{\text{cone40}}$. While one might
consider jets with arbitrarily soft muons in the denominator of this fraction,
for practical reasons Figure 8(b) only includes events where muons satisfy
$p_{\text{T}}>$10\text{\,}\mathrm{GeV}$$. To study the dependence of these
efficiencies on the momentum distribution of the underlying jet, different
regions of jet momentum are emphasised by imposing different requirements on
the global transverse energy $H_{\text{T}}$ in the event, defined for this
purpose as the scalar $p_{\text{T}}$ sum of all jets with
$p_{\text{T}}>$25\text{\,}\mathrm{GeV}$$ and $|\eta|<2.8$ that are a distance
$\Delta R>0.6$ from the muon. This quantity is indeed partially correlated
with the kinematics of the muon’s parent jet, via the momentum of the top
quarks producing all these jets. The reconstruction, calibration and selection
of jets and muons for this figure are otherwise those detailed in Ref. [33].

It can be observed that for the case of a $p_{\text{T}}$-dependent
parameterisation, the efficiencies vary strongly with $H_{\text{T}}$, although
large differences occur mostly for $p_{\text{T}}>$40\text{\,}\mathrm{GeV}$$.
Since most non-prompt muons are produced at low $p_{\text{T}}$, the overall
impact of this non-captured dependency might be small, unless regions of
interest in the analysis specifically select high-$p_{\text{T}}$ leptons. In
contrast, the parameterisation as a function of
$p_{\text{T}}+E_{\text{T}}^{\text{cone40}}$ is much less influenced by
$H_{\text{T}}$, making the measured efficiencies less dependent on the event
topology. In practice, a compromise has to be found between this observation
and other elements evoked above justifying a $p_{\text{T}}$-dependent
parameterisation, especially for electrons because ID criteria are usually
employed in addition to isolation. The direct dependency of efficiencies on
other variables that are also correlated with the event topology (e.g.
$E_{\text{T}}^{\text{miss}}$) may also be reduced by such a parameterisation.

As detailed in the next section, suitable uncertainties must be assigned to
the use of the fake efficiencies in different regions than those in which they
are measured, in particular to account for potential differences in relative
contributions of the different sources of fake/non-prompt leptons. To minimize
these uncertainties, it has sometimes been found beneficial to use an approach
closer to that of section 7.1, in which efficiencies are evaluated in the
simulation and supplemented by data-driven correction factors that depend on
the source of the fake/non-prompt lepton. These correction factors are derived
using dedicated control regions that are enriched in a particular source of
fake/non-prompt leptons. The main assumptions are then the universality of the
correction factors across different processes, and the ability of the
simulation to adequately predict the relative contributions of each source in
the regions of interest. Such an approach has for example been used in Ref.
[38].

##  8 Systematic uncertainties

Systematic uncertainties in the fake/non-prompt lepton background estimates
from the matrix method and the fake-factor method arise from uncertainties in
the values of $\varepsilon_{\mathrm{r}}$ and $\varepsilon_{\mathrm{f}}$. These
uncertainties can be traced to statistical uncertainties from the samples used
to measure the efficiencies, to potential biases that may cause the
efficiencies in the signal region for a particular analysis to differ from the
values obtained from control samples (such as differences in the origin of
fake/non-prompt leptons between these regions), and to uncertainties in the
modelling of contamination from real-lepton processes in the samples used to
measure the fake efficiency. Details of these uncertainties and their
estimation are provided below. The overall impact of variations in
$\varepsilon_{\mathrm{r}}$ and $\varepsilon_{\mathrm{f}}$ depends on the
characteristics of the analysis. For the matrix method, from Eqs. (3) and (4)
one obtains:

$\displaystyle{\partial(N_{\mathrm{f}}^{\mathrm{t}})\over\partial\varepsilon_{\mathrm{r}}}$
$\displaystyle={\varepsilon_{\mathrm{f}}\over\varepsilon_{\mathrm{r}}(\varepsilon_{\mathrm{r}}-\varepsilon_{\mathrm{f}})}N_{\mathrm{r}}^{\mathrm{t}}$
$\displaystyle{\partial(N_{\mathrm{f}}^{\mathrm{t}})\over\partial\varepsilon_{\mathrm{f}}}$
$\displaystyle={\varepsilon_{\mathrm{r}}\over\varepsilon_{\mathrm{f}}(\varepsilon_{\mathrm{r}}-\varepsilon_{\mathrm{f}})}N_{\mathrm{f}}^{\mathrm{t}}.$

Therefore, the relative importance of uncertainties in the real and fake
efficiencies (${\Delta\varepsilon_{\mathrm{r}}}$ and
${\Delta\varepsilon_{\mathrm{f}}}$) depend on the typical values of
$\varepsilon_{\mathrm{r}}$ and $\varepsilon_{\mathrm{f}}$ and on the values of
$N_{\mathrm{r}}^{\mathrm{t}}$ and $N_{\mathrm{f}}^{\mathrm{t}}$. For analyses
with

$N_{\mathrm{r}}^{\mathrm{t}}\times\frac{\Delta\varepsilon_{\mathrm{r}}}{\varepsilon_{\mathrm{r}}^{2}}>N_{\mathrm{f}}^{\mathrm{t}}\times\frac{\Delta\varepsilon_{\mathrm{f}}}{\varepsilon_{\mathrm{f}}^{2}}$

the uncertainty in $\varepsilon_{\mathrm{r}}$ will dominate, and vice versa.

###  8.1 Statistical uncertainties in the measured efficiencies

Statistical uncertainties in the real-lepton and fake-lepton efficiencies can
be accounted for either by analytically propagating the uncertainties through
to the estimated event yields or by varying the efficiencies input to the
nominal yield calculation by their statistical uncertainties and using the
resulting difference in the estimated fake/non-prompt background yield as the
resulting uncertainty. The real- and fake-efficiency uncertainties are
generally uncorrelated since the efficiencies are measured in statistically
independent samples. In the usual case where the efficiencies are measured in
bins of one or more quantities, the bins are uncorrelated, so the variations
in each bin are applied separately. The total statistical uncertainty in the
fake/non-prompt lepton background estimate is given by the sum in quadrature
of the uncertainties from all variations. This source of systematic
uncertainty is usually not dominant.

###  8.2 Systematic uncertainties in the measured efficiencies

Systematic uncertainties in $\varepsilon_{\mathrm{f}}$ are generally larger
and more challenging to assess than for $\varepsilon_{\mathrm{r}}$:

  1. 1.

Real-lepton efficiencies have only slight variations due to event environment
effects since real leptons have small contributions to their measurements from
underlying event and jet activity. This also means there are a wide variety of
samples with which they can be calibrated in great detail.

  2. 2.

There are several sources of fake/non-prompt leptons, and the efficiencies may
differ between these sources. Therefore, any differences in the fake/non-
prompt lepton composition between the sample used to measure the efficiencies
and the signal region for an analysis may lead to a bias in the efficiencies.

Several methods are used to estimate systematic uncertainties in the fake-
lepton efficiencies. One is simply to vary the selection criteria for events
in the control region used to measure the fake-lepton efficiencies, since the
composition of fake/non-prompt leptons in the standard and alternative control
regions may differ. A more sophisticated approach is to use MC simulation to
estimate the fake/non-prompt lepton compositions in both the control and
analysis regions. This information, combined with the MC-estimated selection
efficiencies for each source of fake/non-prompt leptons, can be used to
provide an estimate of the uncertainty.

###  8.3 Uncertainties in the modelling of real-lepton processes

When measuring the fake-lepton efficiencies, a correction must be applied to
account for contamination from processes with real leptons in the control
sample used for the measurement:

$\varepsilon_{\mathrm{f}}=\frac{N^{\mathrm{t}}-N_{\mathrm{r}}^{\mathrm{t}}}{(N^{\mathrm{t}}+N^{\mathrm{l}})-(N_{\mathrm{r}}^{\mathrm{t}}+N_{\mathrm{r}}^{\mathrm{l}})}$

where $N_{\mathrm{r}}^{\mathrm{t}}$ and $N_{\mathrm{r}}^{\mathrm{l}}$ are the
numbers of real leptons in the selected tight and loose samples, respectively.
MC simulation of real-lepton processes is generally used to estimate
$N_{\mathrm{r}}^{\mathrm{t}}$ and $N_{\mathrm{r}}^{\mathrm{l}}$, with
corrections applied to account for known differences in object selection
efficiencies between simulation and data. Nonetheless, several sources of
systematic uncertainty in the real-lepton contamination remain: uncertainties
in the cross-sections of the real-lepton processes, uncertainties in the
correction factors, and uncertainties in the parameters, e.g. parton
distribution functions (PDFs) and factorisation/renormalisation scales
($\mu_{\mathrm{f}}$, $\mu_{\mathrm{r}}$), used in the simulation.

###  8.4 Uncertainties due to biases in the likelihood matrix method

The biases that occur in some situations for the likelihood matrix method (see
Section 2.5) may also be considered as a source of systematic uncertainty,
especially when the signal region contains many bins with few events in each.
The magnitude of the bias can be estimated either by repeating the analysis
with coarser bins, or by constraining the total fake/non-prompt lepton yield
estimate to be the value returned by applying the likelihood matrix method to
the entire unbinned sample. Any resulting differences in the binned estimates
can be taken as a systematic uncertainty.

##  9 Examples of application in ATLAS analyses

In this section the application of the fake/non-prompt lepton background
estimation methods is described using two example ATLAS analyses. The first is
a measurement of the $t\bar{t}Z$ differential cross-section using events that
contain three or four lepton candidates, and the second is a model-independent
search for ‘beyond the Standard Model’ (BSM) phenomena in events with three or
more lepton candidates. In both cases, the high lepton multiplicity suppresses
the Standard Model (SM) backgrounds, which makes the relative contribution of
the fake/non-prompt lepton background larger.

Data from $\sqrt{s}=$13\text{\,}\mathrm{TeV}$$ $pp$ collisions recorded by the
ATLAS detector between 2015 and 2018 are used to perform these analyses. In
this period, the LHC delivered colliding beams with a peak instantaneous
luminosity of
$L=$2.1\text{\times}{10}^{34}\text{\,}\mathrm{c}\mathrm{m}^{-2}\mathrm{s}^{-1}$$,
achieved in 2018, and an average number of $pp$ interactions per bunch
crossing of $33.7$. After applying beam, detector and data-quality criteria,
the total integrated luminosity of the dataset is
$139\text{\,}{\mathrm{fb}}^{-1}$ [39]. The uncertainty in the combined
2015–2018 integrated luminosity is 1.7% [40], obtained using the LUCID-2
detector [41] for the primary luminosity measurements.

###  9.1 Measurement of the $t\bar{t}Z$ cross-section in final states with
three or four leptons

This analysis measured the differential production cross-section of the
$t\bar{t}Z$ process in final states with a total of three or four electron and
muon candidates. The tools described above are used for the lepton efficiency
measurements and the application of the likelihood matrix method (described in
Section 2.1.2). In addition to the fake/non-prompt lepton estimation in the
signal regions of the analysis, the results are checked in several validation
regions enriched in fake/non-prompt leptons. The likelihood matrix method was
chosen for this analysis since the number of fake/non-prompt leptons in the
signal regions with three or four leptons is expected to be very low, and the
likelihood matrix method provides more stable results for binned estimations,
which are necessary for differential background predictions in these low-
statistics signal regions. The fraction of events with more than one fake/non-
prompt lepton in the signal or validation regions has been checked and found
to be negligible. More details about the analysis and the fake/non-prompt
lepton background estimation can be found in Ref. [42].

####  9.1.1 Real-lepton efficiencies

((a))

((b))

Figure 9: The two-dimensional real-lepton efficiencies obtained for (a)
electrons and (b) muons, in bins of $p_{\text{T}}$ and $|\eta|$ of the
leptons. The last $p_{\text{T}}$ bin is inclusive. The real-lepton
efficiencies are obtained using MC simulation, but corrected to match the
performance seen in data control samples.

The first step in the application of the matrix method is to measure the
efficiencies for real and fake/non-prompt leptons that satisfy the baseline
criteria to also satisfy the tight criteria. In the $t\bar{t}Z$ cross-section
measurement, baseline electrons are required to satisfy the ‘LooseAndBLayer’
ID WP, whereas tight electrons are required to satisfy the stricter ‘Medium’
ID criteria and to be isolated from nearby tracks and calorimeter energy
deposits. Both the baseline and tight muons are required to satisfy the
‘Medium’ WP, and tight muons are in addition required to be isolated from
nearby tracks.

As discussed in Section 7.1, the real-lepton efficiencies are obtained using
MC simulation, corrected to match the performance seen in data control
samples. Those efficiencies are shown in Figure 9, binned in lepton
$p_{\text{T}}$ and $|\eta|$. To check for potential dependencies on the number
of additional jets in the events used for the measurements, the efficiencies
are derived for different jet multiplicities. No significant differences
between the real-lepton efficiencies are observed.

####  9.1.2 Fake/non-prompt lepton efficiencies

The fake/non-prompt lepton efficiencies are measured with same-charge
electron–muon ($e\mu$) or muon–muon ($\mu\mu$) data events using a tag-and-
probe method, where one ‘tag lepton’ with very stringent requirements on
momentum and isolation ($p_{\text{T}}>$40\text{\,}\mathrm{GeV}$$,
$\textrm{max}(p_{\text{T}}^{\textrm{cone}20},\,E_{\text{T}}^{\textrm{cone}20})/p_{\text{T}}<0.01$)
is selected and the remaining ‘probe lepton’ is used for the efficiency
measurement. Events with more than two leptons are not considered. Only the
aforementioned baseline electron or muon requirements are used to select the
probe lepton. The samples are dominated by events containing at least one
fake/non-prompt lepton, and are orthogonal to the signal regions, which
require a minimum of three tight lepton candidates. In addition, $t\bar{t}Z$
events constitute negligible fractions of these samples.

Table 1:  Definition of the fake-lepton control regions used for the electron ($e$-fakes-CR) and muon ($\mu$-fakes-CR) fake-efficiency measurements. $N_{\ell}$ is the number of leptons ($\ell$), while $N_{\textrm{jets}}$ ($N_{b\textrm{-jets}}$) is the number of jets ($b$-tagged jets, see text), respectively. For additional details of the definitions of physics objects, see Ref. [42]. Region |  $N_{\ell}$ $(\ell=e,\mu)$ |  $p_{\text{T}}$ $(\ell)$ |  $p_{\text{T}}$ (jets) | $N_{\textrm{jets}}$ | $N_{b\textrm{-jets}}$
---|---|---|---|---|---
$e$-fakes-CR |  $=2$ ($e^{\pm}\mu^{\pm}$) | $>$7\text{\,}\mathrm{GeV}$$ | $>$25\text{\,}\mathrm{GeV}$$ | $\geq 1$ | $\geq 1$
$\mu$-fakes-CR |  $=2$ ($\mu^{\pm}\mu^{\pm}$) | $>$7\text{\,}\mathrm{GeV}$$ | $>$25\text{\,}\mathrm{GeV}$$ | $\geq 1$ | $\geq 1$

The definition of the regions used for the fake-efficiency measurements is
summarised in Table 1. For the electron fake efficiencies an $e\mu$ signature
is used, with the muon being used as the tag lepton. For the muon fake
efficiencies, the $\mu\mu$ region is used,999In $\mu\mu$ events with both
muons satisfying the tag-selection, the one with the higher $p_{\text{T}}$ is
chosen as the tag lepton. since the $e\mu$ region also contains unwanted
events where an electron with an incorrectly measured charge is selected as
the tag lepton and paired with a real muon.

((a))

((b))

Figure 10: The two-dimensional fake/non-prompt lepton efficiencies measured
for (a) electrons and (b) muons, in bins of $p_{\text{T}}$ and $|\eta|$ of the
leptons. The last $p_{\text{T}}$ bin is inclusive. The indicated uncertainties
show only the statistical errors in the given bins. Compared to the real-
lepton efficiencies shown in Figure 9, the fake/non-prompt lepton efficiencies
depend much more on the specifications of the analysis.

Real-lepton background processes leading to the same-charge dilepton signature
are estimated using MC-simulated events and are subtracted from data to obtain
an unbiased efficiency measurement. The contribution from electrons with
misassigned charge in the same-charge $e\mu$ region is also subtracted using
estimates from MC-simulated events. After this subtraction, the dominant
source of fake/non-prompt leptons is found to be heavy-flavour hadron decays.
The fake efficiencies, binned in lepton $p_{\text{T}}$ and $|\eta|$, are shown
in Figure 10.

It is assumed that since the loose and tight lepton selection criteria depend
on quantities related to the lepton itself or to its immediate surroundings,
the chosen parameterisation captures the main variations in the fake
efficiencies, and residual dependencies on the event environment can be
covered by systematic uncertainties (see Section 8). Indeed, in the
simulation, the dependence of the fake efficiencies on the number of light-
flavour jets or $b$-tagged101010Jets containing $b$-hadrons are identified
(tagged) by the MV2c10 $b$-tagging algorithm [43]. The algorithm uses a
multivariate discriminant with quantities such as the impact parameters of
associated tracks, and well-reconstructed secondary vertices. jets is mild.
The uncertainties are evaluated by comparing, in the simulation, the fake
efficiencies in event selections corresponding to either the measurement
regions in Table 1 or the regions of interest in the analysis for which the
background estimates are needed. These differences, evaluated as a function of
$p_{\text{T}}$ and $|\eta|$, are applied as a systematic uncertainty of the
fake-efficiency measurement (as discussed in Section 8), and are of the order
of 10%–20%, except for muons with $p_{\text{T}}>$50\text{\,}\mathrm{GeV}$$,
for which they reach 40%. Furthermore, normalisation uncertainties are
considered for the real-lepton background processes, which are subtracted in
the fake-efficiency measurement. They are evaluated by scaling the real-lepton
background processes upwards and downwards within their cross-section
uncertainties before the subtraction and using the differences between the
modified and nominal efficiencies as uncertainties, which are added in
quadrature to the aforementioned uncertainties.

####  9.1.3 Results in the fake/non-prompt lepton validation regions

To validate the performance of the method, predictions are obtained and
compared with data in two dedicated validation regions called
‘VR-$3\ell$-$1b3j$’ and ‘VR-$3\ell$-$1b3j$-no$Z$’, which have a larger
proportion of fake/non-prompt leptons than is expected in the signal regions.
The definitions of these two validation regions are summarised in Table 2. No
charge requirements are placed on the reconstructed lepton candidates in these
regions.

Table 2:  Definition of the fake/non-prompt lepton enriched validation regions, VR-$3\ell$-$1b3j$ and VR-$3\ell$-$1b3j$-no$Z$. $N_{\ell}$ stands for the number of leptons (${\ell}$), while $N_{\textrm{jets}}$ ($N_{b\textrm{-jets}}$) is the number of jets ($b$-tagged jets), respectively. The leptons are ordered by decreasing $p_{\text{T}}$ ($\ell_{1,2,3}$). The mass ($m$) variables are discussed in the text. For additional details of the object definitions, see Ref. [42]. Region |  $N_{\ell}$ $(\ell=e,\mu)$ |  $p_{\text{T}}$ $(\ell_{1,2,3})$ |  $p_{\text{T}}$ (jets) | $|m_{\ell\ell}^{\textrm{SF}}-m_{Z}|$ | $N_{\textrm{jets}}$ | $N_{b\textrm{-jets}}$
---|---|---|---|---|---|---
VR-$3\ell$-$1b3j$ | $=3$ | $>27,20,$20\text{\,}\mathrm{GeV}$$ | $>$25\text{\,}\mathrm{GeV}$$ | – | $=3$ | $=1$
| (2 tight, 1 loose) |  |  |  |  |
VR-$3\ell$-$1b3j$-no$Z$ | $=3$ | $>27,20,$20\text{\,}\mathrm{GeV}$$ | $>$25\text{\,}\mathrm{GeV}$$ | $>$10\text{\,}\mathrm{GeV}$$ | $=3$ | $=1$
| (all tight) |  |  |  |  |

The variable $m_{\ell\ell}^{\textrm{SF}}$ refers to the invariant mass of the
same-flavour opposite-charge (SFOC) lepton pair with the invariant mass
closest to the $Z$ boson mass. VR-$3\ell$-$1b3j$ is a region similar to the
actual signal regions defined in Ref. [42], but without a requirement on the
$Z$ mass for the (SFOC) lepton candidate pair. Therefore, it contains a higher
fraction of fake/non-prompt leptons after the selection. To further enhance
the fake/non-prompt lepton contribution, the third-highest-$p_{\text{T}}$
lepton that satisfies the baseline selection criteria must not satisfy the
tight criteria. An additional validation region, VR-$3\ell$-$1b3j$-no$Z$, is
defined by requiring all three leptons to satisfy the tight selection
criteria, but placing a veto on SFOC lepton pairs that have an invariant mass
consistent with the $Z$ boson, thereby enhancing the fake/non-prompt lepton
fraction in this region. Both regions are orthogonal to the analysis signal
regions and intended to validate the predictions of the matrix method for
different levels of fake/non-prompt lepton contamination.

Some example distributions are shown in Figure 11 for VR-$3\ell$-$1b3j$ and
Figure 12 for VR-$3\ell$-$1b3j$-no$Z$. The processes with three real leptons
(modelled with MC simulations) plus the prediction from the matrix method can
be compared with data in these regions.

The hatched bands in Figures 11 and 12 show only the statistical uncertainties
of the MC prediction and the uncertainties associated with the fake/non-prompt
lepton estimates (i.e. no theoretical or detector-related systematic
uncertainties are included). The total uncertainty associated with the
fake/non-prompt lepton estimate itself contains a systematic component, which
is evaluated from variations of the input fake/real efficiencies (described in
the previous section), and the statistical uncertainty of the data sample to
which the likelihood matrix method is applied.111111The total uncertainties of
the fake/non-prompt lepton estimates may be different from the uncertainties
reported in Ref. [42], as the number of loose leptons in these validation
regions is larger than in the signal region of the $t\bar{t}Z$ analysis and,
therefore, the statistical uncertainties are smaller.

There is generally good agreement between the data and the total background
estimate, except that the background is overestimated at low $\Delta
R(\ell_{1},\ell_{2})$ and low $E_{\text{T}}^{\text{miss}}$ in Figure 11. One
contribution to that difference is that the two leading (two
highest-$p_{\text{T}}$) lepton candidates are likely to be real, yet when they
are near each other they have a lower efficiency for satisfying the isolation
criteria, and thus are misinterpreted as fake/non-prompt leptons by the matrix
method. Analyses that are sensitive to such issues may benefit from imposing a
minimum $\Delta R$ requirement between leptons. A discrepancy is also observed
in the higher $p_{\text{T}}$ bins of Figure 11(d). Events in these bins have
three leptons with $p_{\text{T}}$ above $50\text{\,}\mathrm{GeV}$. As shown in
Figures 9 and 10, only a single $p_{\text{T}}$ bin above
$50\text{\,}\mathrm{GeV}$ is available for measuring
$\varepsilon_{\mathrm{r}}$ and $\varepsilon_{\mathrm{f}}$, due to the limited
number of events in the control regions, so variations above
$50\text{\,}\mathrm{GeV}$ may be missed. This point was not investigated
thoroughly since only a small fraction of events in the analysis were
impacted.

((a))  ((b))

((c))  ((d))

Figure 11: (a) Angular separation between the leading and second leading (in
$p_{\text{T}}$) lepton candidates, $\Delta R(\ell_{1},\ell_{2})$, (b) angular
separation between the second- and third-leading lepton candidates, $\Delta
R(\ell_{2},\ell_{3})$, (c) missing transverse momentum in the event,
$E_{\text{T}}^{\text{miss}}$, and (d) the $p_{\text{T}}$ of the third-leading
lepton in VR-$3\ell$-$1b3j$. The processes with three real leptons are
modelled with MC simulation, while the contribution from fake/non-prompt
leptons (dark red) comes from the likelihood matrix method as described above.
The hatched band shows the uncertainty from the MC statistics and the
fake/non-prompt background estimate. The rightmost bins are inclusive and
contain all events above the $x$-axis ranges. The lower panel shows the ratio
of data to the total SM prediction (sum of the real-lepton background
contributions estimated with MC samples and the fake/non-prompt lepton
contribution estimated with the matrix method). Further details are available
in Ref. [42].

((a))  ((b))

((c))  ((d))

Figure 12: Comparisons of the predicted and observed yields in
VR-$3\ell$-$1b3j$-no$Z$, with respect to the $p_{\text{T}}$ of the (a) leading
and (b) subleading lepton candidates, (c) the scalar sum of the lepton and jet
transverse momenta, $H_{\text{T}}$, and (d) the missing transverse momentum in
the event, $E_{\text{T}}^{\text{miss}}$. The processes with three real leptons
are modelled with MC simulation, while the contribution from fake/non-prompt
leptons (dark red) comes from the likelihood matrix method as described above.
The hatched band shows the uncertainty from MC statistics and the fake/non-
prompt background estimate. The rightmost (leftmost) bins are inclusive and
contain all events above (below) the $x$-axis ranges. The lower panel shows
the ratio of data to the total SM prediction (sum of the real-lepton
background contributions estimated with MC samples and the fake/non-prompt
lepton contribution estimated with the matrix method). Further details are
available in Ref. [42].

###  9.2 Model-independent search for new phenomena in multi-lepton final
states

Many interesting new models for BSM physics predict final states with three or
more leptons. The general multi-lepton search for new phenomena [44]
agnostically considers such final states. Its aim is to be sensitive to BSM
phenomena in often-overlooked corners of phase space. The background
estimation for the multi-lepton search uses MC predictions to account for
events that contain only real leptons, and the fake-factor method for events
containing at least one fake/non-prompt lepton. The dominant sources of
fake/non-prompt leptons are semileptonic heavy-flavour decays (primarily of
$b$-hadrons), light-hadron decays, and misidentification of light hadrons as
leptons. These mainly arise in $Z\text{\,+\,jets}$ and $t\bar{t}$ events.

####  9.2.1 Fake/non-prompt lepton selection

The fake factors are measured using events with a single lepton candidate. In
order to prevent a bias in the fake factor due to trigger selection criteria
[14, 15], the selected events are required to have fired a loose single-lepton
trigger where isolation requirements are not imposed. However, due to the high
rate of events that pass such triggers, a prescale factor is applied, which
reduces the number of events available for measuring the fake factors.
Baseline lepton candidates must pass a common object selection, as detailed in
Ref. [44]. Electron candidates are required to pass either the ‘Loose’ ID WP
with calorimeter- and track-based isolation requirements, or the ‘Tight’ ID WP
with no isolation requirement. Baseline muon candidates are required to pass
the ‘Medium’ ID WP (‘HighPt’ for $p_{\text{T}}>$300\text{\,}\mathrm{GeV}$$).
Only leptons with $p_{\text{T}}>$25\text{\,}\mathrm{GeV}$$ and satisfying the
longitudinal and transverse track impact parameter requirements (see Sections
4.1 and 4.2) are considered.

More stringent selection criteria are imposed for tight lepton candidates. For
muons, where fake/non-prompt candidates are mainly muons from semileptonic
heavy-flavour decays, only the isolation criteria is modified: the tight
selection requires that muons satisfy track-based isolation criteria. For
electrons, where both light- and heavy-flavour hadrons are a non-negligible
source of fake/non-prompt candidates, both the ID and isolation criteria are
modified. Tight electrons must satisfy both the ‘Tight’ ID WP and calorimeter-
and track-based isolation requirements.

Additional selection requirements are imposed on the single-lepton sample to
ensure a high purity of fake/non-prompt leptons, which reduces the statistical
uncertainty of the computed fake factor: the $E_{\text{T}}^{\text{miss}}$ is
required to be $<$25\text{\,}\mathrm{GeV}$$ ($<$40\text{\,}\mathrm{GeV}$$) for
events with electron (muon) candidates, and the number of jets in the event is
required to be $\geq 1\leavevmode\nobreak\ (2)$ for events with electron
(muon) candidates. For muon-candidate events, there must also be at least one
jet with $p_{\text{T}}>$35\text{\,}\mathrm{GeV}$$ (the ‘tag jet’), and the
azimuthal angle between the muon candidate and this jet is required to be
$>2.7$ radians.

The fake factor is binned in both the $p_{\text{T}}$ and $|\eta|$ of the
lepton candidate. The bins are defined to have tolerable statistical
uncertainties while preventing sizeable differences between the fake-factor
values in adjacent bins. The highest muon-$p_{\text{T}}$ bin includes all muon
candidates with $p_{\text{T}}>$80\text{\,}\mathrm{GeV}$$, as there are very
few fake/non-prompt muons at higher transverse momenta.

####  9.2.2 Fake factors

The fake factors calculated for the multi-lepton analysis are shown in Figures
13 and 14 for electrons and muons. The uncertainties in the fake factors are
categorised and evaluated as described below.

There are statistical uncertainties due to the data and MC sample sizes in
each fake-factor bin, which due to their small size are summed in quadrature
into a single uncertainty.

In the regions enriched in fake/non-prompt leptons, MC predictions are used to
subtract the real-lepton contribution from the data. The uncertainties in the
MC contributions are propagated to the fake factors. The main contributions
from real leptons in the single-lepton regions are from $Z\text{\,+\,jets}$,
$W\text{+\,jets}$ and $t\bar{t}$ events. For the $Z/W\text{+\,jets}$
processes, an uncertainty of 5% is applied to the cross-section. For the
$t\bar{t}$ process, the assumed uncertainties are ${}^{+2.4\%}_{-3.3\%}$
($\mu_{\mathrm{r}}$, $\mu_{\mathrm{f}}$) and $\pm 4.2\%$ (PDF).

Extrapolating the fake factors from the single-lepton sample to the multi-
lepton samples used in the analysis introduces an uncertainty because these
samples differ in the kinematic distributions of fake/non-prompt leptons, and
possibly also in the fake/non-prompt lepton composition. Two uncertainties are
included to address the bias caused by imposing a $E_{\text{T}}^{\text{miss}}$
upper bound in the fake/non-prompt lepton estimation sample, and by imposing a
$p_{\text{T}}$ requirement on the tag jet in the fake/non-prompt muon
estimation sample. These uncertainties are estimated by varying the
requirements on these variables upwards and downwards by
$10\text{\,}\mathrm{GeV}$.121212Only the $E_{\text{T}}^{\text{miss}}$
variation is displayed separately in the plots, as the variation of the jet
$p_{\text{T}}$ results in an uncertainty that is too small to be visible.
Plots showing the impact of these systematic effects on the fake factors are
given in Figures 13 and 14.

((a))  ((b))

((c))  ((d))

Figure 13: Measured fake factors for electrons as a function of $p_{\text{T}}$
for different $|\eta|$ ranges, and their dependence on individual variations
of parameters of the measurement, used to determine systematic uncertainties.
The ‘MC syst’ uncertainty covers variations in the model used to subtract the
real-lepton contribution in the control regions used to measure
$\varepsilon_{\text{f}}$. The combined impact of statistical and systematic
uncertainties added in quadrature is indicated by the shaded yellow area (the
grey area represents only the statistical uncertainty).

((a))  ((b))

((c))

Figure 14: Measured fake factors for muons as a function of $p_{\text{T}}$ for
different $|\eta|$ ranges, and their dependence on individual variations of
parameters of the measurement, used to determine systematic uncertainties. The
combined impact of statistical and systematic uncertainties added in
quadrature is indicated by the shaded yellow area (the grey area represents
only the statistical uncertainty).

Finally, a direct assessment of the uncertainty in the composition of the
fake/non-prompt lepton background in the multi-lepton sample is made. Since
fake/non-prompt leptons can come from both light- and heavy-flavour sources,
it is possible that the relative abundances from these sources can vary
between samples. This possibility is addressed through an additional
uncertainty which leverages the different event signatures produced by light-
and heavy-flavour hadrons, the latter consisting primarily of $b$-hadrons. To
evaluate this uncertainty, an alternative set of fake factors is computed
which, in addition to the binning in $p_{\text{T}}$ and $|\eta|$, are binned
according to the presence or absence of $b$-tagged jets in the event. This
alternative set of fake factors is shown in Figure 15. The composition
uncertainty in the total event yields is then derived from the difference
between estimates calculated with either the nominal fake factors oblivious to
the presence of $b$-tagged jets in the events, or the alternative set that
requires such a jet. This uncertainty is found to have a negligible impact on
the analysis, since most of the jets are not $b$-tagged.

((a))  ((b))

((c))  ((d))

Figure 15: Fake factors measured in bins of the lepton candidate’s
$p_{\text{T}}$ and $|\eta|$ for events with (a,c) electron candidates and
(b,d) muon candidates in events (a,b) without and (c,d) with $b$-tagged jets.

####  9.2.3 Validation regions

Validation regions are defined using appropriate sub-selections of $ee\mu$ and
$e\mu\mu$ events. These are used to check that the computed fake factors
extrapolate correctly from the regions where they are calculated to the
regions in which they are applied. The ‘on-$Z$’ validation region requires an
SFOC lepton pair with a dilepton mass within $10\text{\,}\mathrm{GeV}$ of the
$Z$ boson mass. The ‘off-$Z$’ validation region also requires a SFOC pair of
leptons, but requires the dilepton mass to fall outside of the $Z$-mass
window. Only mixed-flavour final states are selected for these validation
regions so that which lepton to choose as the third one, assumed to be the
fake/non-prompt lepton, is unambiguous. The sources of fake/non-prompt leptons
contribute in different ratios to the on-$Z$ and off-$Z$ validation regions:
the on-$Z$ validation region is more sensitive to $Z\text{\,+\,jets}$ events
than the off-$Z$ region, while the inverse is true for $t\bar{t}$ events,
although in absolute terms, $Z\text{\,+\,jets}$ events are more numerous than
$t\bar{t}$ events in both cases. Both validation regions target, through a
$m_{\text{T}}$ requirement of
$m_{\text{T}}(\ell,E_{\text{T}}^{\text{miss}})<$40\text{\,}\mathrm{GeV}$$, a
third lepton that is likely to be fake/non-prompt. The union of the on-$Z$ and
off-$Z$ validation regions is called the ‘fakes validation region’.

The variables of primary importance for this analysis are the invariant mass
of all lepton candidates in the event ($m_{\text{inv}}$) and the
$E_{\text{T}}^{\text{miss}}$. The signal regions are separated according to
the values of these quantities, as discussed in Ref. [44]. The
$m_{\text{inv}}$ distributions in the two validation regions are shown in
Figure 16, while the $E_{\text{T}}^{\text{miss}}$ distributions are shown in
Figure 17. Lastly, the electron and muon candidate $p_{\text{T}}$
distributions are shown in the fakes validation region in Figure 18. The
comparisons in Figure 16 were also presented in Ref. [44] (albeit in
logarithmic scale) and thus include fitted normalisation factors for the
$WZ\text{\,+\,jets}$ and $ZZ\text{\,+\,jets}$ backgrounds from their
respective control regions (‘post-fit’). On the other hand, the complementary
distributions shown in Figures 17 and 18 were obtained independently of this
statistical analysis, and thus employ the unconstrained SM background
normalisations and uncertainties (‘pre-fit’ distributions). The background
estimate is consistent with the data within the statistical and systematic
uncertainties.

((a))  ((b))

Figure 16: Comparison between data and prediction for the $m_{\text{inv}}$
distribution in the (a) on-$Z$ and (b) off-$Z$ validation regions, after
fitting the normalisation factors for the $WZ\text{\,+\,jets}$ and
$ZZ\text{\,+\,jets}$ backgrounds and systematic uncertainties [44]. All
uncertainties, systematic and statistical, are included. The leftmost
(rightmost) bin is inclusive and contains all events with
$m_{\text{inv}}<$200\text{\,}\mathrm{GeV}$\hbox{
}(>$500\text{\,}\mathrm{GeV}$)$. The hatched grey area in these figures shows
the total uncertainty. Further details are available in Ref. [44].

((a))  ((b))

Figure 17: Comparison between data and prediction for the
$E_{\text{T}}^{\text{miss}}$ distribution in the (a) on-$Z$ and (b) off-$Z$
validation regions. All uncertainties, systematic and statistical, are
included. The rightmost bin is inclusive and contains all events with
$E_{\text{T}}^{\text{miss}}>$475\text{\,}\mathrm{GeV}$$. The hatched grey area
in these figures shows the total uncertainty. Further details are available in
Ref. [44].

((a))  ((b))

Figure 18: Comparison between data and prediction for the $p_{\text{T}}$
distribution in the fakes validation region (which is the union of the on-$Z$
and off-$Z$ validation regions). Shown are (a) the electron $p_{\text{T}}$ in
$e\mu\mu$ and (b) the muon $p_{\text{T}}$ in $ee\mu$ events. All
uncertainties, systematic and statistical, are included. The rightmost bin is
inclusive and contains all events with
$p_{\text{T}}>$275\text{\,}\mathrm{GeV}$$. The hatched grey area in these
figures shows the total uncertainty. Further details are available in Ref.
[44].

##  10 Conclusions

For physics analyses exploring signatures with one or more prompt leptons,
background contributions due to fake/non-prompt leptons are often difficult to
estimate in simulations. Therefore, data-driven methods are commonly used.
Three related methods have been adopted by the ATLAS Collaboration as
recommended tools: the asymptotic matrix method, the likelihood matrix method,
and the fake-factor method. All three approaches depend on defining two
categories of leptons, one of which (‘tight’) is subject to the same
identification and selection criteria as are used in the analysis. The other
category (‘loose’) adds additional lepton candidates with less stringent
selection requirements. The union of the two sets is called the ‘baseline’
sample. The criteria are typically defined such that the probability for a
real baseline lepton to satisfy the tight criteria is substantially higher
than the corresponding probability for a fake/non-prompt lepton. Then, the
relative numbers of loose and tight leptons in the analysis sample can be used
to estimate the contribution of fake/non-prompt leptons, either inclusively or
differentially in any variables of interest.

Despite their similarities, the methods each have their own strengths and
drawbacks. The asymptotic matrix method and fake-factor method provide a
fake/non-prompt lepton weight for each event, which is convenient for
analyses. However, these methods are subject to large uncertainties if the
efficiency for loose fake/non-prompt leptons to satisfy the tight criteria is
large in parts of the analysis phase space. The likelihood matrix method
returns a smaller uncertainty in such cases, and avoids any possibility of
producing a negative estimate for the event yield, but does not provide a per-
event weight, introducing difficulties e.g. for differential estimations. The
fake-factor method uses simulation rather than data to incorporate the
contribution from events where all leptons are tight. This method can
therefore be employed while the signal region for an analysis is fully
blinded, although it may induce additional simulation-related uncertainties in
the background estimate.

The systematic uncertainties for all three methods arise from similar sources,
with the largest contributions related to the extrapolation of the
efficiencies measured in the control samples to events in the analysis sample.
Differences in the fake/non-prompt lepton composition in the samples must be
accounted for and appropriate uncertainties must be assigned to this
extrapolation.

The performance of the likelihood matrix method and the fake-factor method has
been demonstrated in a differential $t\bar{t}Z$ cross section measurement and
in a model-independent search for BSM phenomena in multi-lepton final states,
respectively. In both cases, the chosen method was shown to provide a reliable
estimate of the fake/non-prompt lepton background, as measured in validation
regions.

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