Comprehensive measurements of t-channel single top-quark production cross sections at $\mathbf{\sqrt{\mathrm{{\bf{\emph{s}}}}}=7}$ TeV with the ATLAS detector

The ATLAS detector ATLAS Collaboration 2008a is built from a set of cylindrical subdetectors, which cover almost the full solid angle around the interaction point ¹¹ 1 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point in the center of the detector and the $z$-axis along the beam direction. The $z$-axis is parallel to the anti-clockwise beam viewed from above. The pseudorapidity $\eta$ is defined as $\eta=-\ln[\tan(\theta/2)]$, where the polar angle $\theta$ is measured with respect to the $z$-axis. The azimuthal angle $\phi$ is measured with respect to the $x$-axis, which points toward the center of the LHC ring. Transverse momentum and energy are defined as $p_{\mathrm{T}}=p\sin\theta$ and $E_{\mathrm{T}}=E\sin\theta$, respectively. The $\Delta R$ distance in ($\eta$,$\phi$) space is defined as $\Delta R=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$.. ATLAS is composed of an inner tracking detector (ID) close to the interaction point, surrounded by a superconducting solenoid providing a $2\,$T axial magnetic field, electromagnetic and hadronic calorimeters, and a muon spectrometer (MS). The ID consists of a silicon pixel detector, a silicon microstrip detector (SCT), and a straw-tube transition radiation tracker (TRT). The electromagnetic calorimeter is a lead and liquid-argon (LAr) sampling calorimeter with high granularity. An iron/scintillator tile calorimeter provides hadronic energy measurements in the central pseudorapidity range. The endcap and forward regions are instrumented with LAr calorimeters for both the electromagnetic and hadronic energy measurements. The MS consists of three large superconducting toroids with eight coils each, a system of trigger chambers, and precision tracking chambers.

ATLAS employs a three-level trigger system. The first level (L1) is built from custom-made hardware, while the second and third levels are software based and collectively referred to as the High Level Trigger (HLT). The data sets used in this analysis are defined by high-$p_{\mathrm{T}}$ single electron or single muon triggers ATLAS Collaboration 2012c. During the data-taking period slightly different trigger conditions were used to cope with the increasing number of multiple $pp$ collisions per bunch crossing (pile-up).

At L1, electron candidate events are required to have an electromagnetic energy deposit of $E_{\rm T}>14{\mathrm{\ Ge\kern-1.00006ptV}}$; in the second part of the data-taking period the requirement was $E_{\rm T}>16{\mathrm{\ Ge\kern-1.00006ptV}}$. At the HLT level, the full granularity of the calorimeter and tracking information is available. The calorimeter cluster is matched to a track and the trigger electron object has to have $E_{\rm T}>20{\mathrm{\ Ge\kern-1.00006ptV}}$ or $E_{\rm T}>22{\mathrm{\ Ge\kern-1.00006ptV}}$, exceeding the corresponding L1 requirements by $6{\mathrm{\ Ge\kern-1.00006ptV}}$.

The single muon trigger is based on muon candidates reconstructed in the muon spectrometer. At L1, a threshold of $p_{\mathrm{T}}=10{\mathrm{\ Ge\kern-1.00006ptV}}$ is applied. At the HLT level, the requirement is tightened to $p_{\mathrm{T}}>18\,\rm{GeV}$.

Samples of simulated $t$-channel single top-quark events are produced with the NLO matrix-element generator POWHEG-BOX Nason 2004 interfaced to PYTHIA Sjostrand et al. 2006 (version 6.4.27) for showering and hadronization. In POWHEG-BOX the four-flavor scheme calculation is used to simulate $t$-channel single top-quark production. The events are generated using the fixed four-flavor NLO PDF set CT104f Lai et al. 2010 and the renormalization and factorization scales are calculated event-by-event Frederix et al. 2012 with $\mu_{R}=\mu_{F}=4\cdot\sqrt{m_{b}^{2}+p_{{\rm T},b}^{2}}$, where $m_{b}$ and $p_{{\rm T},b}$ are the mass and $p_{\mathrm{T}}$ of the $b$-quark from the initial gluon splitting.

Samples of $t\bar{t}$ events, $Wt$ events, and $s$-channel single top-quark events are generated with POWHEG-BOX interfaced to PYTHIA using the CT10 NLO PDF set Lai et al. 2010. All processes involving top quarks are produced assuming $m_{t}=172.5{\mathrm{\ Ge\kern-1.00006ptV}}$, and the parameters of the PYTHIA generator controlling the modeling of the parton shower and the underlying event are set to the values of the Perugia 2011 tune Skands 2010.

Vector-boson production in association with jets ($W/Z$+jets) is simulated using the multileg LO generator ALPGEN Mangano et al. 2003 (version 2.13) using the CTEQ6L1 PDF set Pumplin et al. 2002. The partonic events are showered with HERWIG Corcella et al. 2001 (version 6.5.20), and the underlying event is simulated with the JIMMY Butterworth et al. 1996 model (version 4.31) using values of the ATLAS Underlying Event Tune 2 ATLAS Collaboration 2011a. $W+$jets and $Z+$jets events with up to five additional partons are generated. The MLM matching scheme Alwall et al. 2008 is used to remove overlap between partonic configurations generated by the matrix element and by parton shower evolution. The double counting between the inclusive $W+n$-parton samples and samples with associated heavy-quark pair-production is removed utilizing an overlap removal based on a $\Delta R$ matching. The diboson processes $WW$, $WZ$ and $ZZ$ are generated using HERWIG and JIMMY.

After the event generation step, all samples are passed through the full simulation of the ATLAS detector ATLAS Collaboration 2010 based on GEANT4 Agostinelli et al. 2003 and are then reconstructed using the same procedure as for collision data. The simulation includes the effect of multiple $pp$ collisions per bunch crossing. The events are weighted such that the distribution of the number of collisions per bunch crossing is the same as in collision data.

Electron candidates are selected from energy deposits (clusters) in the LAr electromagnetic calorimeter matched to tracks ATLAS Collaboration 2014a and are required to have $E_{\mathrm{T}}>25{\mathrm{\ Ge\kern-1.00006ptV}}$ and $|\eta_{\mathrm{cl}}|<2.47$, where $\eta_{\mathrm{cl}}$ denotes the pseudorapidity of the cluster. Clusters falling in the calorimeter barrel/endcap transition region, corresponding to $1.37<|\eta_{\mathrm{cl}}|<1.52$, are ignored. The energy of an electron candidate is taken from the cluster, while its $\eta$ and $\phi$ are taken from the track. The $z$-position of the track has to be compatible with the hard-scatter primary vertex. Electron candidates are further required to fulfil stringent criteria regarding calorimeter shower shape, track quality, track–cluster matching, and fraction of high-threshold hits in the TRT to ensure high identification quality.

Hadronic jets mimicking the signature of an electron, electrons from $b$-hadron or $c$-hadron decays, and photon conversions constitute the major backgrounds for high-$p_{\mathrm{T}}$ electrons originating from the decay of a $W$ boson. Since signal electrons from $W$-boson decay are typically isolated from jet activity, these backgrounds can be suppressed via isolation criteria that require minimal calorimeter activity (calorimeter isolation) and only few tracks (track isolation) in an ($\eta$,$\phi$) region around the electron. Electron candidates are isolated by imposing thresholds on the scalar sum of the transverse momenta of calorimeter energy deposits $\Sigma p_{\rm T}^{\rm calo}$ within a surrounding cone of radius $\Delta R=0.2$, excluding the energy deposit associated with the candidate, and on the scalar sum of the transverse momenta of tracks $\Sigma p_{\rm T}^{\rm track}$ in a cone of radius $\Delta R=0.3$ around the candidate excluding the track associated with the electron candidate. The $\Sigma p_{\rm T}^{\rm calo}$ variable is corrected for pile-up effects as a function of the number of reconstructed vertices. The thresholds applied to $\Sigma p_{\rm T}^{\rm calo}$ and $\Sigma p_{\rm T}^{\rm track}$ vary as a function of the electron $p_{\rm T}$, the electron $\eta$, and the number of reconstructed primary vertices and are chosen such that the efficiency for electrons from $W$-boson or $Z$-boson decays to pass this isolation requirement is 90%.

Muon candidates are reconstructed by combining track segments found in the ID and in the MS ATLAS Collaboration 2014b. The momentum as measured using the ID is required to agree with the momentum measured using the MS after correcting for the predicted muon energy loss in the calorimeter. Only candidates that have $p_{\mathrm{T}}>25\;{\mathrm{\ Ge\kern-1.00006ptV}}$ and $|\eta|<2.5$ are considered. Selected muons must additionally satisfy a series of requirements on the number of track hits present in the various tracking subdetectors. Muon tracks are required to have at least two hits in the pixel detector, and six or more hits in the SCT. Tracks are rejected if they have more than two missing hits in the SCT and pixel detectors, or tracks with an excessive number of outlier hits in the TRT. Isolated muon candidates are selected by requiring $\Sigma p_{\rm T}^{\rm calo}<4\,{\mathrm{\ Ge\kern-1.00006ptV}}$ within a surrounding cone of radius $\Delta R=0.2$, and $\Sigma p_{\rm T}^{\rm track}<2.5\,{\mathrm{\ Ge\kern-1.00006ptV}}$ within a surrounding cone of radius $\Delta R=0.3$. The efficiency of this combined isolation requirement varies between 95% and 97%, depending on the data-taking period.

The reconstruction, identification and trigger efficiencies of electrons and muons are measured using tag-and-probe methods on samples enriched with $Z~\to\ell\ell$, $J/\psi~\to\ell\ell$, or $W^{\pm}~\to\ell\nu$ ($\ell=e,\mu$) events ATLAS Collaboration 2014a; ATLAS Collaboration 2014b.

Jets are reconstructed using the anti-$k_{t}$ algorithm Cacciari et al. 2008 with a radius parameter of 0.4, using topological clusters ATLAS Collaboration 2008b identified in the calorimeter as inputs to the jet clustering. The jet energy is corrected for the effect of multiple $pp$ interactions, both in collision data and in simulated events. Further energy corrections apply factors depending on the jet energy and the jet $\eta$ to achieve a calibration that matches the energy of stable particle jets in simulated events ATLAS Collaboration 2013b. Differences between data and Monte Carlo simulation are evaluated using in situ techniques and are corrected for in an additional step ATLAS Collaboration 2014c. The in situ calibration exploits the $p_{\mathrm{T}}$ balance in $Z$+jet, $\gamma$+jet, and dijet events. $Z$+jet and $\gamma$+jet data are used to set the jet energy scale (JES) in the central detector region, while $p_{\mathrm{T}}$ balancing in dijet events is used to achieve an $\eta$ intercalibration of jets in the forward region with respect to central jets.

Jets with separation $\Delta R<0.2$ from selected electron candidates are removed, as in these cases the jet and the electron are very likely to correspond to the same physics object. In order to reject jets from pile-up events, a quantity called the jet-vertex fraction $\epsilon_{\mathrm{jvf}}$ is defined as the ratio of $\sum p_{\mathrm{T}}$ for all tracks within the jet that originate from the hard-scatter primary vertex to the $\sum p_{\mathrm{T}}$ of all tracks matched to the jet. It is required that $\epsilon_{\mathrm{jvf}}>0.75$ for those jets that have associated tracks. The $\epsilon_{\mathrm{jvf}}$ criterion is omitted for jets without matched tracks. An overlap removal between jets and muons is applied, removing any muon with separation $\Delta R<0.4$ from a jet with $p_{\rm{T}}>25\,\rm{GeV}$ and $\epsilon_{\mathrm{jvf}}>0.75$. In the same way an overlap removal is applied between jets and electrons, removing any electron separated from a jet by $0.2<\Delta R<0.4$.

Only jets having $p_{\mathrm{T}}>30$ ${\mathrm{\ Ge\kern-1.00006ptV}}$ and $|\eta|<4.5$ are considered. Jets in the endcap/forward-calorimeter transition region, corresponding to $2.75<|\eta|<3.5$, must have $p_{\mathrm{T}}>35$ ${\mathrm{\ Ge\kern-1.00006ptV}}$.

The $E_{\mathrm{T}}^{\mathrm{miss}}$ is a measure of the momentum of the escaping neutrinos, but is also affected by energy losses due to detector inefficiencies. The $E_{\mathrm{T}}^{\mathrm{miss}}$ is calculated based on the vector sum of energy deposits in the calorimeter projected onto the transverse plane and is corrected for the presence of electrons, muons, and jets ATLAS Collaboration 2012d.

The identification of jets originating from the fragmentation of $b$-quarks is one of the most important techniques for selecting top-quark events. Several properties can be used to distinguish $b$-quark jets from other jets: the long lifetime of $b$-hadrons, the large $b$-hadron mass, and the large branching ratio to leptons. The relatively long lifetime of $b$-flavored hadrons results in a significant flight path length, leading to reconstructable secondary vertices and tracks with large impact parameters relative to the primary vertex.

Jets containing $b$-hadrons are identified in the region $|\eta|<2.5$ by reconstructing secondary and tertiary vertices from the tracks associated with each jet and combining lifetime-related information in a neural network ATLAS Collaboration 2011b. Three different neural networks are trained corresponding to an optimal separation of $b$-quark jets, $c$-quark jets, and light-quark jets. The output of the networks is given in terms of probabilities $p_{b}$, $p_{c}$, and $p_{\mathrm{l}}$, which are then combined to form a final discriminant. In order to achieve excellent rejection of $c$-quark jets the ratio $p_{b}/p_{c}$ is calculated. The chosen working point corresponds to a $b$-tagging efficiency of about 54% for $b$-quark jets in $t\bar{t}$ events. The misidentification efficiency is 4.8% for $c$-quark jets and 0.48% for light-quark jets, as derived from simulated $t\bar{t}$ events. Jets passing the requirement on the identification discriminant are called $b$-tagged jets. Scale factors, determined from collision data, are applied to correct the $b$-tagging efficiency in simulated events to match the data.

The $W$+jets background is initially normalized to the theoretical prediction and then subsequently determined simultaneously both in the context of the multijet background estimation and as part of the extraction of the signal cross section. The estimated number of events of the much smaller $Z$+jets background is calculated using the theoretical prediction.

The cross sections for inclusive $W$-boson production and $Z$-boson production are predicted with NNLO precision using the FEWZ program Anastasiou et al. 2004, resulting in a LO-to-NNLO scale factor of 1.2 and an uncertainty of 4%. The uncertainty includes the uncertainty on the PDF and scale variations. The scale factor is applied to the prediction based on the LO ALPGEN calculation for the $W$+$b\bar{b}$, $W$+$c\bar{c}$, and $W$+light-jets samples. An uncertainty for associated jet production is estimated using variations of the factorization and renormalization scale and the ALPGEN matching parameter. These variations yield an uncertainty of 5% for the production of two additional light-quark jets and 15% for two additional heavy-quark jets. An additional relative uncertainty of 50% is assigned to the $W$+$b\bar{b}$ and $W$+$c\bar{c}$ production rates to take uncertainties on heavy-flavor production into account. This uncertainty is estimated using a tag-counting method in control regions ATLAS Collaboration 2012a.

The ALPGEN prediction for the $W$+$c$ process is scaled by a factor of $1.52$ that is obtained from a study based on NLO calculations using MCFM Campbell and Ellis 2010. Normalization uncertainties on the factorization and renormalization scale and PDF uncertainties are 24%.

The processes $W$+$b\bar{b}$, $W$+$c\bar{c}$, and $W$+light-jets, being asymmetric in lepton charge, are combined and are used as a single process in the binned maximum-likelihood fit to determine the signal yield.

Multijet background events pass the signal selection if a jet is misidentified as an isolated lepton or if the event has a non-prompt lepton that appears to be isolated. Since it is neither possible to simulate a sufficient number of those events nor possible to calculate the rate precisely, different techniques are developed to model multijet events and to estimate the production rate. These techniques employ both, collision data as well as simulated events.

In the electron channel, misidentified jets are the main source of multijet background events. This motivates the jet-lepton method in which an electron-like jet is selected with special requirements and redefined as a lepton. This jet has to fulfil the same $p_{\rm T}$ and $\eta$ requirements as a signal electron, and contain at least four tracks to reduce the contribution from converted photons. In addition, the jet must deposit 80–95% of its energy in the electromagnetic calorimeter. Events are selected using the same criteria as for the signal selection except for the selection of the electron. The event is accepted if exactly one such ‘jet lepton’ is found. The jet-lepton selection is applied to a Pythia dijet sample and the resulting set of events is used to model the multijet background in the electron channel.

To determine the normalization of the multijet background in the electron channel, a binned maximum-likelihood fit to observed data in the $E_{\mathrm{T}}^{\mathrm{miss}}$ distribution is performed after applying all selection criteria except for the $E_{\mathrm{T}}^{\mathrm{miss}}$ requirement. In each channel two fits are performed separately; one for electrons in the central ($|\eta|<1.5$) region and one for the endcap ($|\eta|>1.5$) region of the electromagnetic calorimeter. The multijet template is fitted together with templates derived from Monte Carlo simulation for all other background processes whose rate uncertainties are accounted for in the fitting process in the form of additional constrained nuisance parameters. For the purpose of these fits the contributions from $W$+light-jets and $W$+$b\bar{b}$, $W$+$c\bar{c}$, $W$+$c$, the contributions from $t\bar{t}$ and single top-quark production, and the contributions from $Z$+jets and diboson production, are each combined into one template. Distributions normalized to the fit results in the 2-jet-$e^{+}$ and 2-jet-$e^{-}$ signal regions for central electrons are shown in Fig. 2.

In the muon channel, the matrix method ATLAS Collaboration 2012e is used to obtain both the normalization and shape of the multijet background. The method estimates the number of multijet background events in the signal region based on loose and tight lepton isolation definitions, the latter selection being a subset of the former. Hence, the loose selection is defined to contain leptons of similar kinematics, but results in much higher event yields and is, except for the muon isolation requirement, identical to the signal selection. The number of multijet events $N^{\mathrm{tight}}_{\mathrm{fake}}$ passing the tight (signal) isolation requirements can be expressed as,

where $\epsilon_{\mathrm{real}}$ and $\epsilon_{\mathrm{fake}}$ are the efficiencies for real and fake loose leptons being selected as tight leptons, $N^{\mathrm{loose}}$ is the number of selected events in the loose sample, and $N^{\mathrm{tight}}$ is the number of selected events in the signal sample. The fake efficiencies are determined from collision data in a sample of selected muon candidates with high impact parameter significance which is defined by the impact parameter divided by its uncertainty. The real efficiencies are also estimated from collision data using a “tag-and-probe” method, which is based on the identification of a tight lepton and a loose lepton in events originating from a leptonically decaying $Z$ boson.

An uncertainty of 50% is applied to the estimated yield of multijet background events based on comparisons of the rates obtained by using alternative methods, i.e. the matrix method in the electron channel and the jet-lepton method in the muon channel, and using an alternative variable, i.e. $m_{\mathrm{T}}(\ell E_{\mathrm{T}}^{\mathrm{miss}})$ instead of $E_{\mathrm{T}}^{\mathrm{miss}}$ for the binned maximum-likelihood fit.

The $t\bar{t}$ cross section is calculated at NNLO in QCD including resummation of next-to-next-to-leading logarithmic (NNLL) soft gluon terms Cacciari et al. 2012; Baernreuther et al. 2012; Czakon and Mitov 2012; Czakon and Mitov 2013; Czakon et al. 2013 with Top++2.0 Czakon and Mitov 2011. The PDF and $\alpha_{\mathrm{s}}$ uncertainties are calculated using the PDF4LHC prescription Botje et al. 2011 with the MSTW2008 NNLO Martin et al. 2009a; Martin et al. 2009b at 68% confidence level (CL), the CT10 NNLO Lai et al. 2010; Gao et al. 2014, and the NNPDF2.3 Ball et al. 2013 PDF sets, and are added in quadrature to the scale uncertainty, yielding a final uncertainty of 6%.

Since $Wt$ production is charge symmetric with respect to top-quark and top-antiquark production, the combined cross section of $\sigma(Wt)=15.7\pm 1.1\;$pb Kidonakis 2010a is used in the analysis. The predicted cross sections for $s$-channel production are $\sigma(t\bar{b})=3.1\pm 0.1\;$pb and $\sigma(\bar{t}b)=1.4\pm 0.1\;$pb Kidonakis 2010b. The predictions of $\sigma(Wt)$, $\sigma(t\bar{b})$, and $\sigma(\bar{t}b)$ are given at approximate NNLO precision, applying soft-gluon resummation. Theoretical uncertainties including PDF and scale uncertainties are 4.4% Kidonakis 2010b for $s$-channel single top-quark production and 7.0% Kidonakis 2010a for $Wt$ production. The PDF uncertainties are evaluated using the 40 associated eigenvector PDF sets of MSTW 2008 at 90% CL. The cross sections given above are used to compute the number of expected single top-quark events by normalizing the samples of simulated events.

All top-quark background processes are shown combined in the figures and used as a single process in the analysis. The charge asymmetry in $s$-channel production is taken from the approximate NNLO prediction.

Diboson events ($WW$, $WZ$ and $ZZ$) are normalized to the NLO cross-section prediction calculated with MCFM Campbell and Ellis 2010. The cross-section uncertainty for these processes is 5%.

Table 1 provides the event yields after event selection. The yields are presented for the tagged channels, where exactly one $b$-tagged jet is required, separated according to the lepton charge and for the 3-jet-2-tag channel. Small contributions from the $tq$ process in the $\ell^{-}$ regions and the $\bar{t}q$ process in the $\ell^{+}$ regions originate from lepton charge misidentification.

When reconstructing the $W$ boson, the transverse momentum of the neutrino is given by the $x$- and $y$-components of the $E_{\mathrm{T}}^{\mathrm{miss}}$, while the unmeasured $z$-component of the neutrino momentum $p_{z}(\nu)$ is inferred by imposing a $W$-boson mass constraint on the lepton–neutrino system. Since the constraint leads to a quadratic equation for $p_{z}(\nu)$, a two-fold ambiguity arises. In the case of two real solutions, the one with the lower $|p_{z}(\nu)|$ is chosen. In case of complex solutions, which can occur due to the low $E_{\mathrm{T}}^{\mathrm{miss}}$ resolution, a kinematic fit is performed that rescales the neutrino $p_{x}$ and $p_{y}$ such that the imaginary part vanishes and at the same time the transverse components of the neutrino momentum are kept as close as possible to the $E_{\mathrm{T}}^{\mathrm{miss}}$. As a result of this algorithm, the four-momentum of the neutrino is reconstructed.

The top quark is reconstructed by adding the four-momenta of the reconstructed $W$ boson and the $b$-tagged jet. Several angular variables, invariant masses and differences in $p_{\mathrm{T}}$ are defined using the reconstructed physics objects.

The NeuroBayes Feindt and Kerzel 2006 tool is used for preprocessing the input variables and for the training of the NNs. The ranking of the variables in terms of their discrimination power is automatically determined as part of the preprocessing step and is independent of the training procedure ATLAS Collaboration 2012a. Only the highest-ranking variables are chosen for the training of the NNs. Separate NNs are trained in the $2$-jet channel and $3$-jet channel. In the training, no separation is made according to lepton charge or lepton flavour. Dedicated studies show that training in the channels separated by lepton charge does not lead to an improvement in sensitivity.

As a result of the optimization procedure in the $2$-jet channel, 13 kinematic variables are identified as inputs to the NN. In the $3$-jet channel, 11 variables are used. It was found that reducing the number of variables further would result in a considerable loss of sensitivity. The input variables to the NNs are listed in Table 2. The separation between signal and the two most important backgrounds, the top-quark background and the combined $W$+light-jets, $W$+$c\bar{c}$, and $W$+$b\bar{b}$ background, is shown in Fig. 3 for the two most important discriminating variables in the $2$-jet channel.

The modeling of the input variables is checked in a control region (see Sec. IV for the definition) that is enriched in $W+$jets events. Figures 4 and 5 show the three most discriminating variables in the 2-jet-$\ell^{\pm}$ and 3-jet-$\ell^{\pm}$-1-tag channels, respectively. Good modeling of the variables is observed.

After choosing a set of variables based on the criteria outlined above, the analysis proceeds with the training of the NNs using a three-layer feed-forward architecture. The number of hidden nodes was chosen to be 15 for both networks. Samples of simulated events are used for the training process, the size of the signal samples in the $2$-jet channel being about 37,000 events for top-quark and about 40,000 events for top-antiquark $t$-channel production. In the 3-jet channel the sizes of the training samples are 14,000 and 13,000 events, respectively. All background processes are used in the training, except for the multijet background whose modeling is associated with large uncertainties. The total number of simulated background events used in the training is about 89,000 in the 2-jet channel and about 57,000 in the 3-jet channel. The ratio of signal events to background events in the training is chosen to be 1:1, while the different background processes are weighted relative to each other according to the number of expected events.

Regularization techniques are applied in the training process to dampen statistical fluctuations in the training sample and to avoid overtraining. At the preprocessing stage mentioned above (Sec. VI.2), the input variables are transformed in several steps to define new input variables that are optimally prepared to be fed into an NN. First, the variables are transformed, such that they populate a finite interval and are distributed according to a uniform distribution. The influence of outliers is thereby strongly reduced. The distributions of the transformed variables are discretized using 100 bins, and the distributions for signal events are divided by the sum of signal and background events bin-by-bin, yielding the purity distributions in each variable. Next, these purity curves are fitted with a regularized spline function, thereby yielding a continuous transformation from the original input variables to the purities. By means of the spline fit statistical fluctuations in the input variables are significantly reduced. Applying the continuous purity functions to the input variables yields purity distributions that are further transformed, such that the distributions of the resulting variables are centered at zero and have an RMS of one. These variables are input to the NNs. In the training process, the network structure is pruned to arrive at a minimal topology, i.e. statistically insignificant network connections and nodes are removed.

In Fig. 6, the probability densities of the resulting NN discriminants are shown for the signal, the top-quark backgrounds, and the combined $W$+light-jets, $W$+$c\bar{c}$, and $W$+$b\bar{b}$ background.

The separation between signal and backgrounds is equally good for the positive and the negative charge channels, which demonstrates that the choice of training the NNs with a charge-combined sample is appropriate.

The cross sections $\sigma(tq)$ and $\sigma(\bar{t}q)$ are extracted by performing a binned maximum-likelihood fit to the NN discriminant distributions in the 2-jet-$\ell^{+}$, 2-jet-$\ell^{-}$, 3-jet-$\ell^{+}$-1-tag, and 3-jet-$\ell^{-}$-1-tag channels and to the event yield in the 3-jet-2-tag channel, treating $t$-channel top-quark and $t$-channel top-antiquark production as independent processes. The signal rates, the rate of the combined top-quark background ($t\bar{t}$, $Wt$, $t\bar{b}$, and $\bar{t}b$), the rate of the combined $W$+light-jets, $W$+$c\bar{c}$, and $W$+$b\bar{b}$ background, and the $b$-tagging efficiency correction factor (discussed in Sec. III.4) are fitted in all channels simultaneously. The event yields of the multijet background and the $W$+$c$ background are not allowed to vary in the fit, but instead are fixed to the estimates given in Table 1. The cross-section ratio is subsequently computed as $R_{t}=\sigma(tq)/\sigma(\bar{t}q)$.

The maximum-likelihood function is given by the product of Poisson probability terms for the individual histogram bins (see Ref. ATLAS Collaboration 2012a). Gaussian priors are added multiplicatively to the maximum-likelihood function to constrain the background rates subject to the fit and the correction factor of the $b$-tagging efficiency to their predictions within the associated uncertainties.

The sensitivity to the background rates is mostly given by the background-dominated region close to zero in the NN discriminant distributions, while the sensitivity to the $b$-tagging efficiency stems from the event yield in the 3-jet-2-tag channel with respect to the event yields in the 1-tag channels.

In Fig. 7 the observed NN discriminant distributions are shown compared to the compound model of signal and background normalized to the fit results. Figures 8 and 9 show the three most discriminating variables normalized to the fit results in the 2-jet-$\ell^{\pm}$ and 3-jet-$\ell^{\pm}$-1-tag channels, respectively. Differences between data and prediction are covered by the normalization uncertainty of the different processes after the fit.

A high-purity region (HPR) is defined to measure the differential cross sections in the 2-jet-$\ell^{+}$ and 2-jet-$\ell^{-}$ channels, by requiring the NN discriminant to be larger than $0.8$. In the 2-jet-$\ell^{+}$ HPR the signal contribution is twice as large as the background contribution. The signal and background contributions in the 2-jet-$\ell^{-}$ HPR are of approximately the same size. The result of the fit described above is used to normalize the background in the HPR. Figure 10 shows the three most discriminating variables in the 2-jet-$\ell^{+}$ and 2-jet-$\ell^{-}$ high-purity channels, normalized to the fit results. The data are well described by the predicted compound model.

Systematic uncertainties on the reconstruction and energy calibration of jets, electrons and muons are propagated through the entire analysis. The main source of object modeling uncertainty comes from the jet energy scale (JES). The JES uncertainty has been evaluated for the in situ jet calibration ATLAS Collaboration 2013b; ATLAS Collaboration 2014c, which uses $Z$+jet, $\gamma$+jet, and dijet $p_{\mathrm{T}}$-balance measurements in data. The JES uncertainty is evaluated in several different categories:

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  Detector: The different $p_{\mathrm{T}}$-balance measurements have uncertainties due to the jet energy resolution, the electron and photon energy scale and the photon purity.

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  Physics modeling: The uncertainties in the in situ calibration techniques due to the choice of Monte Carlo generator, radiation modeling, and the extrapolation of $\Delta\phi$ between the jet and the $Z$ boson.

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  Statistics: The uncertainty due to the limited size of the data sets of the in situ jet calibration measurements.

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  Mixed detector and modeling: In this category the uncertainty due to the modeling of the underlying event and soft radiation as well as modeling of the jet fragmentation are considered.

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  $\eta$ intercalibration modeling: The uncertainty in the dijet-$p_{\mathrm{T}}$-balance technique due to the modeling of additional parton radiation is estimated by comparing dijet events simulated with PYTHIA and HERWIG. This JES category is the largest contribution from the jet energy scale to the cross-section measurements.

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  Close-by jets: The jet calibration can be affected by the presence of close-by jets, located at radii $\Delta R<1.0$.

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  Pile-up: Uncertainties due to the modeling of the large pile-up effects in data are included as a function of jet $p_{\mathrm{T}}$  and $\eta$.

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  Flavor composition: This uncertainty covers effects due to the difference in quark–gluon composition between the jets used in the calibration and the jets used in this analysis. Since the response to quark and gluon jets is different, the uncertainty on the quark–gluon composition in a given data sample leads to an uncertainty in the jet calibration.

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  Flavor response: In this category an uncertainty is considered due to imperfect knowledge of the calorimeter response to light-quark jets and gluon jets.

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  $b$-JES: An additional JES uncertainty is evaluated for $b$-quark jets by varying the modeling of $b$-quark fragmentation.

The uncertainty due to the jet energy resolution is modeled by varying the $p_{\mathrm{T}}$  of the jets according to the systematic uncertainties of the resolution measurement performed on data using the dijet-balance method ATLAS Collaboration 2013c. The effect of uncertainties associated with the jet vertex fraction is also considered for each jet.

The tagging efficiencies of $b$-jets, $c$-jets, and light jets are derived from data ATLAS Collaboration 2012f; ATLAS Collaboration 2012g; ATLAS Collaboration 2012h and parameterized as a function of $p_{\mathrm{T}}$  and $\eta$ of the jet. The corresponding efficiencies in simulated events are corrected to be the same as those observed in data, and the uncertainties in the calibration method are propagated to the analysis. The difference in the $b$-tagging efficiency between jets initiated by $b$-quark and $b$-antiquark is $\sim$1%, estimated from simulated $tq$ and $\bar{t}q$ events. To account for a possible uncertainty in the modeling of the detector response the full difference is taken as a systematic uncertainty. In Table 3 this uncertainty is called $b/\bar{b}$ acceptance.

The uncertainties due to lepton reconstruction, identification and trigger efficiencies are evaluated using tag-and-probe methods in $Z\rightarrow\ell\ell$ events. Uncertainties due to the energy scale and resolution are considered for electrons and muons. Additionally, the lepton charge misidentification is taken into account and was evaluated to be about 0.1%. All lepton uncertainties are summarized in Table 3 in one item.

Other minor uncertainties are assigned to the reconstruction of $E_{\mathrm{T}}^{\mathrm{miss}}$ and to account for the impact of pile-up collisions on the calculation of $E_{\mathrm{T}}^{\mathrm{miss}}$. The uncertainties on $E_{\mathrm{T}}^{\mathrm{miss}}$ are summarized under $E_{\mathrm{T}}^{\mathrm{miss}}$ modeling in Table 3.

Systematic uncertainties arising from the modeling of the single top-quark signal, the $t\bar{t}$ background, and the $W$+jets background are taken into account.

The uncertainty due to the choice of single top-quark $t$-channel generator and parton shower model is estimated by comparing events generated with POWHEG-BOX interfaced to PYTHIA and events generated with the NLO matrix-element generator MG5_aMC@NLO Alwall et al. 2014 and showered with HERWIG and JIMMY. Again the fixed four-flavor PDF set CT10f4 Lai et al. 2010 is used, and the renormalization and factorization scales are set to $\mu_{\mathrm{R}}=\mu_{\mathrm{F}}=4\cdot\sqrt{m_{b}^{2}+p_{\mathrm{T},b}^{2}}$, where $m_{b}=4.75{\mathrm{\ Ge\kern-1.00006ptV}}$ is the $b$-quark mass, and $p_{\mathrm{T},b}$ is the transverse momentum of the $b$-quark. The uncertainty on the choice of $\mu_{\mathrm{R}}$ and $\mu_{\mathrm{F}}$ is estimated using events generated with POWHEG-BOX interfaced to PYTHIA. Factorization and renormalization scales are varied independently by factors of $0.5$ and $2.0$, while the scale of the parton shower is varied consistently with the renormalization scale. The uncertainty related to scale variations is then given by the envelope of all variations.

The modeling uncertainty for the $t\bar{t}$ background is evaluated by comparing events simulated with the NLO generator POWHEG-BOX interfaced to PYTHIA and the multileg LO generator ALPGEN interfaced to HERWIG. An additional uncertainty for the top-quark background processes comes from the amount of initial-state and final-state radiation, estimated using dedicated AcerMC samples interfaced to PYTHIA where parameters controlling initial-state and final-state radiation (ISR/FSR) emission are varied. The variations of the parameters are constrained by a measurement of $t\bar{t}$ production with a veto on additional central jet activity ATLAS Collaboration 2012i.

A shape uncertainty is assigned to the $W$+jets background, based on variation of the choices of the matching scale and of the functional form of the factorization scale in ALPGEN.

The impact of using simulation samples of limited size is also taken into account.

The systematic uncertainties related to the PDFs are taken into account for the acceptance of all single top-quark processes and $t\bar{t}$ production. The simulated events are reweighted according to each of the PDF uncertainty eigenvectors. The uncertainty is calculated following the recommendation of the respective PDF group. The final PDF uncertainty is the envelope of the estimated uncertainties for the CT10 PDF set, the MSTW2008nlo Martin et al. 2009b PDF set and the NNPDF2.3 Ball et al. 2013 PDF set. For all PDFs the variable flavor number scheme Thorne 2010 is used.

In Sec. V the theoretical cross sections and their uncertainties are quoted for each background process. Since the $t\bar{t}$, single top-quark $Wt$ and $s$-channel processes are grouped together in the statistical analysis, their uncertainties are added in proportion to their relative fractions, leading to a combined uncertainty of 6.7%. The uncertainty on the combined $Z$+jets and diboson background is 60% including a conservative estimate of the uncertainty of the heavy-flavor fraction of 50%, while the uncertainties of the $W$+jets backgrounds are 24% for $W$+$c$ and 36% for the combined $W$+$b\bar{b}$, $c\bar{c}$ and light jets including the same heavy-flavor-fraction uncertainty on the $b\bar{b}$ and $c\bar{c}$ contributions. Additionally, an uncertainty on the relative fraction of 2-jet to 3-jet events of 5% for events with light-flavor jets and 7% for events with heavy-flavor jets is applied for the $W$+jets estimation. This uncertainty was estimated by varying the following input parameters of the generation with ALPGEN by a factor of two: the hard scattering scale, the coupling of the hard interaction, and the minimum $p_{\mathrm{T}}$ and $\Delta R$ separation of the partons.

The luminosity measurement is calibrated using dedicated beam-separation scans, referred to as van der Meer scans, where the absolute luminosity can be inferred from the measurement of the beam parameters ATLAS Collaboration 2013a. The resulting uncertainty is 1.8%.

The systematic uncertainties on the individual top-quark and top-antiquark cross-section measurements and their ratio are determined using pseudo-experiments that account for variations of the signal acceptance, the background rates, and the shape of the NN discriminant due to all sources of uncertainty described above. As an example, Fig. 11 shows the shape variation of the NN discriminant for $t$-channel single top-quark signal events due to the variation of the JES because of the uncertainty on the $\eta$ intercalibration. The correlations between the different channels and the physics processes are fully accounted for.

The probability densities of all possible outcomes of the measurements of $\sigma(tq)$, $\sigma(\bar{t}q)$ and $R_{t}$ are obtained by performing the measurements on the pseudo-data. The values measured in data are used as central values when generating the pseudo-experiments. The root mean squares of the estimator distributions of the measured quantities are estimators of the measurement uncertainties.

Table 3 summarizes the contributions of the various sources of systematic uncertainty to the uncertainties on the measured values of $\sigma(tq)$, $\sigma(\bar{t}q)$, $R_{t}$ and $\sigma(tq+\bar{t}q)$. The dominant systematic uncertainty on the cross sections is the JES $\eta$-intercalibration uncertainty since one of the prominent features of $tq$ production is a jet in the forward region.

The inclusive $t$-channel cross section $\sigma(tq+\bar{t}q)$ is extracted by using only one scale factor $\beta(tq+\bar{t}q)$ in the likelihood function, scaling the top-quark and the top-antiquark contributions simultaneously. The top-quark-to-antiquark ratio is taken from the approximate NNLO prediction Kidonakis 2011 (see Sec. I). The systematic uncertainties on the measured value of inclusive cross-section are determined as described in Sec. VII. A detailed list of the uncertainties is given in Table 3.

The binned maximum-likelihood fit yields a cross section of

assuming $m_{t}=172.5{\mathrm{\ Ge\kern-1.00006ptV}}$. Figure 12 compares the measured value for $\sigma(tq+\bar{t}q)$ to NLO predictions Campbell et al. 2009; Kant et al. 2014 obtained with different PDF sets. All predictions are in agreement with the measurement.

The $t$-channel single top-quark cross sections are measured using a signal model with $m_{t}=172.5{\mathrm{\ Ge\kern-1.00006ptV}}$. The dependence of the cross-section measurements on $m_{t}$ is mainly due to acceptance effects and is expressed by the function:

with $\Delta m_{t}=m_{t}-172.5{\mathrm{\ Ge\kern-1.00006ptV}}$. The parameters $p_{1}$ and $p_{2}$ are determined using dedicated signal samples with different $m_{t}$ and are given in Table 4 for $\sigma(tq)$, $\sigma(\bar{t}q)$ and $\sigma(tq+\bar{t}q)$. The cross-section ratio $R_{t}$ is largely independent of the top-quark mass.