Measurement of the top quark mass using events with a single reconstructed top quark in pp collisions at $$ \sqrt{s} $$ = 13 TeV

Abstract
 A measurement of the top quark mass is performed using a data sample enriched with single top quark events produced in the t channel. The study is based on proton- proton collision data, corresponding to an integrated luminosity of 35.9 fb−1, recorded at $$ \sqrt{s} $$
 
 s
 
 = 13 TeV by the CMS experiment at the LHC in 2016. Candidate events are selected by requiring an isolated high-momentum lepton (muon or electron) and exactly two jets, of which one is identified as originating from a bottom quark. Multivariate discriminants are designed to separate the signal from the background. Optimized thresholds are placed on the discriminant outputs to obtain an event sample with high signal purity. The top quark mass is found to be $$ {172.13}_{-0.77}^{+0.76} $$
 
 172.13
 
 −
 0.77
 
 
 +
 0.76
 
 
 GeV, where the uncertainty includes both the statistical and systematic components, reaching sub-GeV precision for the first time in this event topology. The masses of the top quark and antiquark are also determined separately using the lepton charge in the final state, from which the mass ratio and difference are determined to be $$ {0.9952}_{-0.0104}^{+0.0079} $$
 
 0.9952
 
 −
 0.0104
 
 
 +
 0.0079
 
 
 and $$ {0.83}_{-1.35}^{+1.79} $$
 
 0.83
 
 −
 1.35
 
 
 +
 1.79
 
 
 GeV, respectively. The results are consistent with CPT invariance.


Introduction
The mass of the top quark, m t , is an important parameter of the standard model (SM) of particle physics. Its precise measurement is of profound importance, both for theory and experiment. On the theory side, it constitutes a major input to global electroweak (EW) fits [1,2], used to verify the self-consistency of the SM. It is also directly related to the stability of the EW vacuum, because among all known elementary particles it has the largest contribution in terms of radiative corrections to the Higgs boson mass [3,4]. From the experimental perspective, it also provides a benchmark for the identification and calibration of heavy-flavor jets [5] arising from bottom or charm quarks, both in resolved and boosted topologies [6].
The majority of LHC results on m t [7-14] have been obtained with top quark pair (tt) events. Such events are predominantly produced via gluon-gluon fusion (90%) along with a subdominant contribution from quark-antiquark annihilation (10%). The most precise m t measurements of 172.08 ± 0. 48  Top quarks can be singly produced at the LHC through charged-current EW interactions via the exchange of a W boson. At leading order (LO) in the SM, single top quark production can be realized in three modes, depending on the virtuality of the W boson involved in the process, namely the t channel (spacelike), the tW channel (on-shell), and the s channel (timelike). The t-channel diagram shown in Fig. 1 constitutes the dominant process for single top quark production in proton-proton (pp) collisions at the LHC, with a total cross section of 217 +9 −8 pb at √ s = 13 TeV calculated at next-to-LO (NLO) [17,18] in perturbative quantum chromodynamics (QCD). Within uncertainties, the measured cross section [19] agrees with this prediction. The t-channel single top quark production occurs at a lower energy scale (≥ 170 GeV) compared to tt (≥ 340 GeV), offering a partially independent event sample for measurement in a complementary region of phase space. Furthermore, it enhances the range of available measurements with systematic sources that are partially uncorrelated from those for tt events. The t-channel process involves CR only between the top quark and the proton from which the b quark or the gluon in the initial state arises, not to the whole event, unlike in the case of tt [16]. Such alternative measurements provide an important validation for the modeling of nonperturbative QCD processes in Monte Carlo (MC) simulations, which could be a source of large systematic effects. The unique production mode and event topology may also shed light on the interpretation of the m t parameter [20], since, as opposed to tt events, the single top quark process probes energy scales down to the production threshold of about 170 GeV.
The signature of t-channel single top quark production comprises an isolated high-momentum charged lepton, a neutrino, which results in an overall transverse momentum p T imbalance, a light-quark jet, often produced in the forward direction, and a jet arising from the hadronization of a bottom quark (b jet) from the top quark decay. The second b jet, arising in the production process via gluon splitting, as shown in Fig. 1 (left), has a softer p T spectrum and a broader pseudorapidity η distribution compared to the b jet originating from the top quark, and thus frequently escapes detection. Hence, candidate events are required to contain one muon or electron and exactly two jets, of which one is b tagged, in the final state.
In this paper, a measurement of m t is reported based on data recorded at √ s = 13 TeV by the CMS experiment in 2016 and corresponding to an integrated luminosity of 35.9 fb −1 [21]. The study uses a sample enriched in t-channel single top quark events selected via a multivariate analysis (MVA) technique, and supersedes the earlier CMS measurement at 8 TeV [22]. The masses of the top quark and antiquark are also measured separately using events with positively and negatively charged leptons in the final state, respectively. The mass ratio and difference from these measurements are used to test CPT invariance. Violation of CPT symmetry would imply a nonlocal field theory [23], signaling physics beyond the SM. Tabulated results are provided in HEPData [24].
The paper is organized as follows. The CMS detector is briefly discussed in Section 2. Section 3 describes the simulation samples used to model signal and background events. Section 4 provides an overview of the reconstruction and identification of physics objects, as well as of the selection of candidate events. A method relying on sideband (SB) data used to estimate the QCD multijet background is described in Section 5. Section 6 discusses the MVA technique designed to distinguish the t-channel single top quark signal from the tt, EW, and QCD backgrounds. The maximum-likelihood (ML) fit used to extract the value of m t from its reconstructed distribution is explained in Section 7. In Section 8 we describe various systematic sources affecting the measurement and their individual contributions. Results and overall impact are explained in Section 9, and Section 10 summarizes the main findings.

The CMS detector
The central feature of the CMS apparatus is a superconducting solenoid of 6 m diameter, providing an axial magnetic field of 3.8 T. Within the solenoid volume lie a silicon pixel and microstrip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and plastic scintillator hadron calorimeter, each composed of a barrel and two endcap sections. Forward calorimeters, based on steel absorbers with quartz fibers, extend the fiducial coverage provided by the barrel and endcap detectors up to |η| = 5. Muons are measured in the range |η| < 2.4, with detection planes made using three technologies: drift tubes, cathode strip chambers, and resistive plate chambers, embedded in the steel flux-return yoke outside the solenoid. A detailed description of the CMS detector, together with a definition of the coordinate system used and relevant kinematic variables, can be found in Ref. [25].
Events of interest are selected with a two-tiered trigger system [26]. The first level [27], composed of custom hardware processors, uses information from the calorimeters and muon detectors to select events at a rate of 100 kHz. The second level, known as the high-level trigger, comprises a farm of processors running a version of the full event reconstruction software optimized for fast processing at a rate of about 1 kHz.

Simulation of events
We simulate signal and background events to NLO QCD accuracy using either the POWHEG [28][29][30] or MADGRAPH5 aMC@NLO 2.2.2 [31] generator. The signal process [32] is simulated with POWHEG 2.0 in the four-flavor scheme (4FS), where b quarks are produced via gluon splitting as shown in Fig. 1 (left). This scheme is expected to yield a more accurate description of the kinematic distributions of t-channel events than the five-flavor scheme (5FS) [33][34][35] indicated in Fig. 1 (right). For the normalization of signal samples, we employ the 5FS predictions calculated using HATHOR 2.1 [17,18] at NLO in perturbative QCD. This is because with the inclusion of both gluon-and b-quark-initiated diagrams, the 5FS cross section calculation yields a more accurate value than that of 4FS, which is based on the gluon-initiated diagram only.
The tt background [36] is simulated using POWHEG 2.0 and is normalized to the cross section calculated with TOP++ 2.0 [37] at next-to-NLO (NNLO) in perturbative QCD, including soft gluon resummation at next-to-next-to-leading-log (NNLL) accuracy. We use POWHEG 1.0 to simulate the production of single top quarks in association with W bosons (tW) in the 5FS [38], and normalize the events to a prediction providing approximate NNLO accuracy [39,40]. The s-channel contribution is modeled using MADGRAPH5 aMC@NLO 2.2.2 in the 4FS with up to one additional parton and with the FxFx merging scheme [41] and normalized to a cross section calculated at NLO QCD accuracy in 5FS by HATHOR 2.1 [17,18]. While the m t value is set to 172.5 GeV in the nominal simulation samples of signal and top quark background events, a number of such samples generated with alternate m t hypotheses ranging between 169.5 and 178.5 GeV are considered for the purpose of mass calibration (Section 7). In all these samples the top quark width is set to its nominal value of 1.31 GeV.
Events with W and Z bosons produced in association with jets (referred to as V+jets) are simulated at NLO accuracy using MADGRAPH5 aMC@NLO 2.2.2 and the FxFx merging scheme. Predictions calculated with FEWZ 3.1 [42][43][44] are employed for the normalization of these two processes. The W+jets events are simulated with zero, one, and two additional partons exclusively, in order to retain a sufficient number of events surviving the selection criteria, while the Z+jets events are generated with up to two additional partons inclusively. Contributions from WW, WZ, and ZZ (collectively referred to as VV) processes are simulated at NLO with MADGRAPH5 aMC@NLO 2.2.2 with the FxFx merging scheme.
For all samples, PYTHIA 8.212 [45] is used to simulate parton showering and hadronization. We model the underlying event (UE) activities with the tune CUETP8M1 [46] for all samples except for tt, where we use the tune CUETP8M2T4 [47], as it provides a more accurate description of the kinematic distributions of the top quark pair and jets in tt events. The parton distribution functions (PDFs) predicted by NNPDF3.0 NLO [48] are used in all simulations. The cross sections of simulated signal and background processes are listed in Table 1. presence of at least one isolated electron with p T > 32 GeV and |η| < 2.1.
Events with at least one reconstructed pp interaction vertex are retained for further analysis. The vertex must be reconstructed from at least four tracks that have a longitudinal distance |d z | < 24 cm and a radial distance |d xy | < 2 cm from the nominal interaction point. If multiple vertices are found in an event, the one with the largest value of summed p 2 T of physics objects is taken as the primary pp interaction vertex. The objects are the jets, clustered with the tracks assigned to the vertex as inputs, and the associated missing transverse momentum p miss T , taken as the negative of the p T sum of those jets.
The particle-flow algorithm [51], which combines information from various subdetectors, is used to reconstruct the individual particles. Muon candidates must have at least one hit in the muon detector and a minimum of five hits in the silicon microstrip tracker. They are then reconstructed by a global fit to the combined information from the tracker and muon detector. Selected muons must have p T > 26 GeV and |η| < 2.4. Electron candidates are reconstructed [52] from good quality tracks in the tracker, matched to clusters in the ECAL. They are identified by applying dedicated selection criteria on nine variables related to tracking and shower shape. Electrons are required to pass the tight identification criteria [53] corresponding to an average efficiency of approximately 70% and have p T > 35 GeV and |η| < 2.1, while those falling into the gap between the ECAL barrel and endcap regions (1.44 < |η| < 1.57) are rejected. The relative isolation (I rel ) for a muon (electron) candidate is calculated by summing the transverse energy deposited by photons and charged and neutral hadrons within a cone of size ∆R = √ (∆η) 2 + (∆φ) 2 < 0.4 (0.3) around its direction, corrected for contributions from pileup [54], divided by its p T . The transverse energy is defined as E sin θ, where E is the energy and θ is the polar angle of the energy deposit. The muon and electron candidates are required to pass the criterion I rel < 0.06.
Events containing additional muons (electrons) with p T > 10 (15) GeV and |η| < 2.4 (2.5) are rejected. In such cases, the criteria on lepton isolation are relaxed to I rel < 0.2 for muons and I rel < 0.18 (0.16) for electrons in the barrel (endcap) ECAL. We apply p T -and η-dependent scale factors in simulation for both selected and vetoed leptons to correct for observed differences in the lepton reconstruction efficiencies with data.
Jets are reconstructed using the anti-k T clustering algorithm [55] with a distance parameter of 0.4, as implemented in the FASTJET package [56]. The effect of additional tracks and calorimetric energy deposits from pileup on the jet momentum is mitigated by discarding tracks identified to be originating from pileup vertices, as well as by applying an offset correction to account for residual neutral pileup contributions [57,58]. Loose identification criteria [57] are applied to suppress jets arising from spurious sources, such as electronics noise in the calorime-ters. Energy corrections are derived from simulation to bring the measured average response of jets to that of particle-level jets. In-situ measurements of the momentum balance in dijet, γ+jets, Z+jets, and QCD multijet events are used to account for any residual differences in the jet energy scale (JES) between data and simulation [59]. In this analysis, jets are required to have p T > 40 GeV and |η| < 4.7.
A combined MVA tagging algorithm [5] is used to identify b jets, which are required to have p T > 40 GeV and |η| < 2.4. The efficiency to correctly identify b jets is about 55% at the chosen working point, while the misidentification probability is 0.1% for light-quark or gluon jets and 6% for charm jets. Simulated events are corrected using dedicated scale factors that account for the differences in the b tagging efficiencies and misidentification probabilities when compared to data. Candidate events must contain exactly two jets. For the signal sample selection, one of these jets is required to be b-tagged and the other must not satisfy the b tagging criterion (referred to as the "untagged" jet in the following discussion).
To suppress the background from QCD multijet processes, we require the transverse mass of the charged lepton plus neutrino system, defined as to exceed 50 GeV. Here, p miss T is the magnitude of p miss T , which is the negative of the p T sum of all reconstructed particle-flow objects in an event. The energy scale corrections applied to jets are propagated to p miss T [60]. The variables p miss x and p miss y denote the x and y components of p miss T , respectively. The symbol p T,l represents the magnitude of p T of the charged lepton and p x,l (p y,l ) is its x (y) component.

Event categories
The selected events are divided into two categories, depending on the number of jets, n, and number of b-tagged jets, m, (labeled nJmT). The 2J1T category has the largest contribution from signal events and is referred to as the signal category. Besides this, we use the 2J0T category, where both the jets do not satisfy the b tagging criterion, to validate the estimation of the QCD multijet background contribution in data. A similar approach was used in earlier CMS measurements of the inclusive and differential t-channel single top quark cross sections [19,35]. The event yields in the 2J1T category after applying all selection criteria are shown in Fig. 2 for the muon (left) and electron (right) final states. The yields are shown separately for events with positively and negatively charged leptons. The contribution from the QCD multijet background is determined from data as described in Section 5. For other processes, the event yields are obtained from simulation.

Reconstruction of the top quark
The four-momentum of the top quark (and hence its mass) is reconstructed from the momenta of its decay products: the charged lepton, the reconstructed neutrino, and the b-tagged jet. The momenta of the lepton and b-tagged jet are measured, while the transverse momentum of the neutrino p T,ν is inferred from p miss T . Assuming energy-momentum conservation at the W → lν vertex and setting the W boson mass m W to 80.4 GeV [61], the longitudinal momentum of the neutrino p z,ν can be calculated from the following constraint:  where E l is the lepton energy and p z,l is the z component of its momentum. The above equation generally leads to two possible solutions for p z,ν , given by where, Λ = (m 2 W /2) + p T,l · p miss T . If both solutions have real values, the one with the smaller magnitude is retained [62,63]. This choice yields a better consistency between the inferred and true values of p z,ν in simulation. In the case of complex solutions, the p T,ν values are modified so that the radical in Eq. (3) becomes zero, while still fulfilling Eq. (2). Setting the radical equal to zero, we get two pairs of possible solutions for p x,ν and p y,ν . Out of the two pairs, the one resulting in a p T,ν with a lower |∆φ| value with respect to p miss T in the event is chosen.
The reconstructed m t distribution after event selection for the signal and background is shown in Section 6. Potential inadequacy in the determination of p z,ν leads to a softer reconstructed p z,ν spectrum compared to the true spectrum in simulation. This in turn leads to a mismatch of the reconstructed m t with the true value used in simulation. A calibration is applied to the reconstructed m t value in order to compensate for this difference, and the related uncertainty is considered as a source of systematic uncertainty, as discussed in Sections 7 and 8, respectively.

Estimation of the QCD multijet background
The QCD multijet production has a large cross section in pp collisions but a tiny acceptance in the phase space used in this analysis. Therefore, a very large sample of simulated QCD multijet events would be needed in order to retain a sufficient event yield surviving our selection criteria to ensure a reliable description of this background. In the absence of simulated event samples of the required size, an alternative approach is followed by defining an SB in data enriched in QCD multijet events. The SB is obtained by requiring the selected muon to have 0.2 < I rel < 0.5 and the selected electron to fail the tight identification criteria. The underlying assumption here is that the description of kinematic variables for QCD multijet events in the SB is similar to that in the signal region. We have verified this assumption using simulated samples. Shapes for QCD multijet events are derived by subtracting the total non-QCD con-tribution from data in this SB. As such, the SB data contain 93 (70)% QCD multijet events for the muon (electron) final state. The QCD multijet contribution in the signal region is estimated by means of a binned ML fit to the m T distribution with two components: QCD and non-QCD. As the QCD multijet background has a larger contribution in the 2J0T category, this category is used to validate the above method. The procedure is then applied in the signal-enriched 2J1T category where the background estimation is performed separately for positively and negatively charged leptons, as well as inclusive of the lepton charge in the final state.  and 2J1T (lower) categories for the muon (left) and electron (right) final states. To account for possible differences between QCD multijet m T shapes obtained from the SB and that in the signal region, a separate 50% systematic uncertainty is assigned to that background rate and shape (see Section 8.1). This propagated uncertainty is twice the maximum difference between the data and the prediction found in the tail of the m T distributions.

Multivariate analysis
A number of variables are combined into an MVA discriminant to separate t-channel single top quark events as the signal from the rest as the background. The latter is the sum total of top quark (tt, tW, and s-channel), EW (V+jets and VV), and QCD multijet events selected in the 2J1T category. All background contributions are obtained with simulation samples except for the QCD multijet where the SB data are used, as described in Section 5. We develop two boosted decision trees (BDTs) using the TMVA package [64] with the variables listed in Table 2 as inputs for the muon and electron final states in the 2J1T category, respectively. Figure 4 shows a comparison of the data and simulation for the two highest-ranked variables in Table 2. The correlations among the variables are also taken into account by using the decorrelation method available in TMVA during evaluation of the BDT response. The modeling of the other input variables in data and simulation, along with their correlations in signal and background events before and after applying the decorrelation, are presented in Appendix A.
The input variables to the BDTs are chosen keeping in mind the following aspects: • good separation power to discriminate signal from background; • low correlation with the reconstructed m t .
The variables listed in Table 2 constitute a minimal set satisfying the above conditions in the muon and electron final states, and are selected as the BDT input variables. They are ranked in decreasing order of their separation power (SP) in the respective final states. The SP of a given variable x is defined as whereX S (X B ) denotes the probability density of x in the signal (background) event category.  During training, signal and background events are weighted according to their relative contribution in the 2J1T category. In total, 400 decision trees with a depth of three layers per tree are combined into a forest. The adaptive boosting algorithm [67] implemented in TMVA is used with the learning rate and minimum node size set to 40% and 1%, respectively. The BDT setup is checked for overtraining by dividing the MC samples into two independent subsamples of equal size, one for training and the other to validate its performance. Signal and background events are picked randomly to set up the training and validation subsamples in order to avoid bias. We perform Kolmogorov-Smirnov (KS) tests to compare the BDT output distribution between the two subsamples, and obtain KS probabilities ranging between 57 and 93% for the signal and background events. These results confirm no significant overtraining. Therefore, we combine the training and validation subsamples for further studies and evaluate the BDT responses for all signal and background events according to the probabilities obtained from training in the respective final states. The data-to-simulation comparisons of the BDT response distributions are shown in Fig. 4, separately for the muon (left) and electron (right) final states in the 2J1T category. The performance of the BDTs is quantified via a combined receiver-operator-characteristic (ROC) curve, as shown in Fig  between the signal and background. The correlation between the reconstructed m t and BDT response is about −13% in simulated signal events.
To obtain an event sample enriched in t-channel single top quark events, selection thresholds on the BDT responses are optimized by studying the signal and background efficiencies, together with the signal purity after selection, as shown in Fig. 5 (upper right). Ideally, high signal purity with high yield or efficiency is desirable for a precise m t measurement in t-channel single top quark events. In reality, however, the signal efficiency and purity have opposite trends with increasing selection thresholds. The optimal working point is determined by studying the combined contribution of the statistical and profiled systematic uncertainties (Section 8) along with that of the mass calibration, in the m t value as a function of the BDT thresholds applied to both muon and electron final states in simulated events. These uncertainties are evaluated by means of an ML fit discussed in Section 7, based on pseudo-experiments derived from simulated events. The result of this study is presented in Fig. 5 (lower left). The starting point for the optimization is chosen to be 0.6, where about 50% signal purity is observed for both final states. The mass calibration uncertainty reaches its minimum when the selection threshold is at 0.8 and thus offers an optimum point where the relative impact of background contamination is at its lowest. Beyond this point, both uncertainties start to increase rapidly with higher threshold values due to the depletion of the signal events. The criterion BDT response >0.8 is chosen, which yields about 65 (60)% signal purity in the muon (electron) final state.
Because of the low correlation between the BDT response and the reconstructed m t in the signal events, a selection on the former does not alter the m t distribution significantly, as shown in Fig. 5 (lower right). The differences in signal m t shapes, obtained with various selection criteria on the BDT responses, are mostly covered by the normalization uncertainty around the peak. This is evident in the lower panel of the plot where a comparison of shape ratios obtained with different selection thresholds relative to the one without any selection on the BDT response is shown. The differences, observed in the higher tails of the m t distributions with and without the optimized BDT selection, are covered by the signal shape variation discussed in Section 8.2. The m t distribution in data and the simulated signal and background before and after the application of the BDT selection criteria are compared in Fig. 6.

Extraction of the top quark mass
The high skewness of the m t distribution, coupled with the low background rate after the BDT selection, poses a considerable challenge in obtaining appropriate analytic shapes for both the signal and background. Instead, a suitable alternative is found using the variable ζ = ln(m t / 1 GeV). The natural logarithm significantly reduces the skewness of the m t distribution since more extreme values to the right of the peak are pulled in, and those to the left are stretched further away [68]. The transformed probability density functions are well-behaved since this is a monotonic one-to-one mapping.
The ζ distributions obtained from the muon and electron final states are considered in a simultaneous ML fit [69]. The fit is separately performed on events with a positively charged lepton (l + ), negatively charged lepton (l − ), and inclusive of the lepton charge (l ± ) in the final state. The inclusion of the QCD multijet background as an additional component in the fit would require a reliable parameterization, which turns out to be challenging in our case. Instead, this background contribution is subtracted from data before the fit. The QCD-subtracted binned ζ distribution is described by a analytic model F(ζ) for each final state (l = µ or e). The total likelihood is given as where i is the bin index, ζ 0 is the peak position of the ζ distribution, P denotes the probability of the analytic model, F l (ζ; ζ 0 , f j ) to describe the observed ζ distribution, and Θ is the penalty term to account for the nuisance parameters f j . These parameters are defined for the rates of various event components j, namely signal, tt, and EW backgrounds, as  where N obs j (N exp j ) is the observed (expected) yield for the event component j. We express F l (ζ; ζ 0 , f j ) as where F sig , F tt , and F EW represent the analytic shapes for the signal, tt, and EW background, respectively. Small contributions (6%) from the tW and s-channel single top quark processes are absorbed into the significantly larger (94%) tt component in forming what we call the tt background above.
The F sig shape is parameterized with a sum of an asymmetric Gaussian (ζ 0 ) function as the core and a Landau [70] function to model the higher tail. The F tt shape is described by a Crystal Ball (ζ 0 ) function [71]. The F EW shape describes contributions from the V+jets and diboson processes. It is modeled with a Novosibirsk function [72].
The parameter ζ 0 of the combined signal and tt background shapes is the parameter of interest (POI), and is allowed to float in the fit. We extract m t from the best-fit ζ 0 value. Parameters, except for the POI, that alter the signal and background shapes are fixed to their estimated values during the fit. These are obtained by fitting individual models to the respective distributions of simulated signal and background events in the muon and electron final states. Out of the shape parameters, the ones having large correlations with the POI are varied up to three standard deviations about their estimated values. The resulting uncertainty is considered as a separate systematic uncertainty in the measured mass. The nuisance parameters f sig , f tt , and f EW are constrained in the fit using log-normal priors with 15, 6, and 10% widths, respectively. The constraint on f sig takes into account the uncertainty in the cross section of the inclusive t-channel single top quark production measured at √ s = 13 TeV [19]. The constraint on f tt is driven by the uncertainty in the predicted tt production cross section [73]. The constraint on f EW is relaxed to around three times the uncertainty in the measured V+jets cross sections [74,75] in order to account for mismodeling of heavy-flavor jet production in simulation, as well as to cover the uncertainties due to the renormalization (µ R ) and factorization (µ F ) scales and PDF in the EW background. A similar approach was used in the measurement of the inclusive t-channel single top quark production cross section [19].
Projections of the fit results onto the ζ distributions for the l + , l − , and l ± cases are shown in Fig. 7. The lower panels in Fig. 7 show the normalized residuals or pulls, which are defined as the difference between the distribution in data and the one predicted by the fit, divided by the uncertainty ∆ = Here, ∆ data is the Poisson uncertainty in the data and ∆ fit is the uncertainty in the fit that includes both the statistical and profiled systematic components. Most of the pull values lie within ±2 for all three cases, with the maximum deviations occurring in the first ζ bin because of significantly less underflow events in the data than in the simulation.
We validate the fit model given in Eq. (7) by applying it to a control sample defined by −0.2 < BDT < 0.8, which is dominated by tt events. The resulting best-fit values of m t and the nuisance parameters are found to agree within the uncertainties with that obtained from the signalenriched region (BDT > 0.8). The scan of the profile likelihood ratio with the POI, together with the correlation among the fit parameters in the signal-enriched region for the l ± case, are presented in Appendix B. The fit consistency is checked by performing pseudo-experiments based on the F(ζ) model with profiled systematic uncertainties only. We do not observe any bias in the fit parameters, and find the corresponding pulls follow a Gaussian distribution having a mean and width consistent with 0 and 1.0, respectively, within their uncertainties.
The linearity of the best-fit mass (m fit ) is checked against different true mass (m true ) hypotheses using dedicated simulation samples for signal and tt processes. A calibration is performed by applying a suitable offset to m fit to account for the differences relative to m true . The difference between m fit and m true can be attributed to the inadequacy in the determination of p z,ν in the signal process discussed in Section 4.3. Details about the associated systematic uncertainty due to the mass calibration are discussed in Section 8.1.

Systematic uncertainties
We consider several sources of systematic uncertainties. They can be broadly classified into two categories as follows.

Profiled (prof):
The signal and background normalizations are profiled by including them as nuisance parameters in the ML fit. The impacts of the profiled systematic sources are obtained directly from the fit via correlations between the POI and the nuisance parameters.
2. Externalized (ext): All other uncertainty sources are externalized, i.e., the ML fit is repeated with varied ζ shapes. Thus, full variations of the shapes are considered due to these sources. The impacts of these sources are calculated by taking the difference between the offset corrected best-fit values of m t corresponding to the nominal and varied shapes.
The largest shift relative to the nominal result is quoted as the uncertainty for a particular systematic source in a conservative approach, unless otherwise specified. The total uncertainty due to the externalized systematic sources in the central value is obtained by separately combining the positive and negative shifts.
The externalized systematic sources can be further divided into two subcategories, namely experimental and modeling.

Experimental uncertainties
• JES: The energies of all reconstructed jets in simulated events are simultaneously scaled up and down according to their p T -and η-dependent uncertainties [76], split into correlation groups, namely intercalibration, MPFInSitu, and uncorrelated according to the procedure described in Ref. [77]. A similar approach to subcategorize the JES uncertainties was used in Ref.
[9] for the m t measurement with tt events in the l+jets final state.
• Jet energy resolution (JER): To account for the difference in JER between data and simulation, a dedicated smearing is applied [76] in simulation that improves or worsens the resolutions within their uncertainties. • b tagging: The scale factors used to calculate the efficiency corrections for the b tagging algorithm are varied up and down within their uncertainties. The efficiency corrections found from these variations are then applied to the simulation to estimate the corresponding systematic uncertainty.
• QCD multijet background: The contribution of the QCD multijet background is estimated based on data as discussed in Section 5. Its contribution is first subtracted from data before the final fit using the parametric model given in Eq. (7). Each bin of the QCD multijet shape derived from SB data is varied independently by an uncertainty of 50% and a new set of shapes is obtained. The resulting new shapes are subtracted from data in the signal region one at a time and the fit is repeated. In this method, a maximal variation in the rate and shape of the QCD multijet background is considered. The resulting uncertainty is obtained from the difference in the mean value of the offset-corrected fit results relative to the nominal case.
• Mass calibration: The mass calibration (∆m cal ), i.e., the difference between m fit and m true , is obtained as a function of m fit , using dedicated MC samples with alternate m true hypotheses (Fig. 8). The linear behavior between m fit and m true dictates that ∆m cal also has a linear dependence on m fit . The band about the central line represents the ±1 standard deviation owing to statistical fluctuations of the signal and tt samples with different m true hypotheses. The mass calibration is obtained from the central value, while its uncertainty is determined from the band and considered as an independent source of uncertainty. This procedure has been implemented separately for the l + , l − , and l ± cases.
• Luminosity: The relative uncertainty in the integrated luminosity is ±2.5% [81]. This is propagated to the uncertainties in the expected rates of the signal and background processes except for the QCD multijet, which is determined from data.

Modeling uncertainties
• CR and early resonance decay (ERD): The uncertainties due to ambiguities in modeling CR effects are estimated by comparing the default model in PYTHIA 8 with two alternative CR models, one with string formation beyond leading color ("QCD inspired") [15], and the other in which the gluons can be moved to another string ("gluon move") [16]. In addition, CR effects due to the top quark decay products are assessed by switching off (default) and on ERD [16] in PYTHIA 8. In the first case, the lifetime of the top quark is assumed to be long enough to shield its decay products from color reconnecting with the rest of the event, whereas this restriction is lifted when we enable the ERD option. All models are tuned to the UE measurements in Ref.
[82], and simultaneous variations of different CR models in signal and tt simulations are considered. The largest observed shift is quoted as the systematic uncertainty.
• Flavor-dependent JES: The Lund string fragmentation model of PYTHIA 8 [45] is compared with the cluster fragmentation of HERWIG++ [83]. Each model relies on a large set of tuning parameters that allows one to modify the individual fragmentation of jets initiated from gluons and light, charm, and bottom quarks. Therefore, the difference in JES between PYTHIA and HERWIG is determined for each jet flavor [76]. • Signal modeling: To determine the influence of possible mismodeling of the t-channel single top quark process, several sources are considered as listed below.

Parton shower (PS) scale:
We compare the nominal signal shape with reweighted shapes obtained by using per-event weights corresponding to independent variations of initial-and final-state radiation (ISR and FSR) scales by a factor of 2 and 1/2, respectively. During estimation of the related uncertainties, the ISR scale is kept fixed at the nominal value while the FSR scale is varied and viceversa. The uncertainty is estimated from the difference in the fit results using reweighted shapes relative to the nominal one.
2. µ R and µ F scales: The impacts of varying the µ R and µ F scales up and down by a factor of 2 relative to their respective nominal values (both set to 172.5 GeV) are considered by applying per-event weights [89] on the ζ distributions. Two cases are considered for the evaluation of related uncertainties. In the first case, one scale is varied while the other is kept fixed to its nominal value; in the other case, both scales are varied together in the same direction with respect to their nominal values. The resulting uncertainties from each case are added in quadrature and quoted as the uncertainty due to the µ R and µ F scales.
3. PDF+α S : The impact due to the choice of PDFs is studied using reweighted shapes that are derived from replicas of the NNPDF 3.0 NLO (α S = 0.118) PDF set [48]. In addition, NNPDF3.0 sets with α S = 0.117 and 0.119 are evaluated and the observed difference is added in quadrature.
• tt modeling: The impacts due to variation of the ISR and FSR scales, the h damp parameter responsible for ME-PS matching [47] (where ME is the acronym for matrix element), the µ R and µ F scales, and PDF+α S in the tt process are considered. The uncertainties due to the ISR and FSR scales, µ R and µ F scales, and PDF+α S variations in tt events are estimated by following exactly the same method as for the signal events. Additionally, variation of α S in the UE tune for the tt simulation sample is considered in order to cover the difference between the UE models used for simulated tt and all other processes, as discussed in Section 3. We also take into account the mismodeling of the top quark p T spectrum, which is harder in simulated tt events than in the data [90]. The uncertainties due to the aforementioned sources are determined from the difference in fit results obtained from the varied or reweighted tt shapes corresponding to each source relative to the nominal one. The contributions from individual sources are then added in quadrature to obtain the total uncertainty due to tt modeling.
• Parametric shapes: The impact from varying the shape parameters of the signal and background models is considered as a separate systematic uncertainty. The shape parameters are varied by three standard deviations about their estimated values, derived using simulation. The impacts due to individual sources are summed in quadrature to obtain the total uncertainty due to the parametric modeling of the signal and backgrounds. Table 3 summarizes the aforementioned sources of systematic uncertainty and their contributions. The impacts due to alternative ME (MADGRAPH5 aMC@NLO), FS (5FS), PS (HERWIG++), and UE (CUETP8M2T4) modeling of the signal process are also evaluated for the l ± final state using dedicated simulated event samples. Their individual contributions range between −0.36 and +0.16 GeV. As these values are covered by the total systematic uncertainty listed in Table 3, no additional uncertainty is assigned to the measured m t value due to these sources.
In the case of the signal process, top quarks are produced more abundantly relative to their antiquark partners due to the charge asymmetry of the W boson radiated from the initial-state quark in pp collisions at the LHC. This leads to a higher relative background contamination in the l − final state arising from top antiquark decay compared to the l + final state from top quark decay, as shown in Fig. 7. As a result, the measurement in the l − final state is more sensitive to the sources that significantly alter the background contributions along with the signal, compared to the ones that impact the signal contribution only. This is reflected in Table 3 where the uncertainties from the signal modeling are lower for the l − case; whereas other sources, except for the ones listed under flavor-dependent JES, that alter the background contributions along with the signal have a larger impact on the total uncertainty. In the case of the flavor-dependent Table 3: Summary of the m t uncertainties in GeV for each final-state lepton charge configuration. The statistical uncertainties are obtained by performing the fits again in each case while fixing the nuisance parameters to their estimated values from data. With the exception of the flavor-dependent JES sources, the total systematic uncertainty is obtained from the quadrature sum of the individual systematic sources. The amount of statistical fluctuations in the systematic shifts are quoted within parentheses whenever alternative simulated samples with systematic variations have been used. These are determined from 1000 pseudo-experiments in each case. Entries with <0.01 denote that the magnitude of the systematic bias is less than 0.01.
Source JES uncertainty sources, the uncertainty is primarily dictated by the untagged jet. The presence of a light quark in the final state is a salient feature of the signal process. This light quark, or the FSR gluon radiated from it, is often detected as the untagged jet in the endcap region. The untagged-jet kinematic properties are heavily exploited by the BDTs in order to achieve a better separation between the signal and backgrounds. Hence, it has a large impact on the final acceptance, as well as on the ζ shapes obtained after the BDT selection. The energy calibration of the endcap detector is known to have larger uncertainties compared to the barrel and hence it has a larger impact on the untagged jet found in the endcap region. In the l − final state, the relative contribution of the flavor-dependent JES uncertainty is smaller owing to a lower signal-to-background ratio.

Results
The m t value is measured with events dominated by t-channel single top quark process, inclusive of the lepton charge in the final state, as The masses of the top quark and antiquark are determined separately by requiring positively and negatively charged leptons in the final state, respectively. We find The first uncertainty is the combination of the statistical and profiled systematic uncertainties, whereas the second denotes the uncertainty due to the externalized systematic sources. The total uncertainty is obtained by adding the two values in quadrature. The measured masses of the top quark and antiquark are consistent with each other, as well as with the l ± result, within uncertainties. The measured value corresponding to the charge-inclusive final states agrees with previous measurements by the ATLAS [7] and CMS [8-10, 14, 22] Collaborations at different center-of-mass energies using various final states, within the uncertainties. Comparisons of the previous measurements and the result from this analysis are shown in Fig. 9.
The mass ratio of the top antiquark to quark is determined to be and the mass difference between the top quark and antiquark is The uncertainties in the mass ratio and difference are obtained by combining the uncertainties from the individual measurements, as indicated in Eqs.    The precision of the m t measurement presented here shows about a 30% improvement over the previous CMS result [22] from single top quark events. The inclusion of the electron final state improves the overall signal yield, thus reducing the statistical component of the total uncertainty. The MVA discriminant and the nuisance parameters in the ML fit constrain the background contamination to a level where the impact of dominant systematic uncertainty sources including JES can be brought under control. These improved strategies are responsible for reducing the overall uncertainty in the measured mass. The statistical uncertainty plays a minor role in the achieved precision, which is limited by the systematic uncertainties due to JES, CR, and FSR modeling in the signal process. A deeper understanding of these effects would be needed to further improve the precision.

Summary
Measurements of the top quark and antiquark masses, as well as their ratio and difference, are performed using a data sample enriched with single top quark events produced in protonproton collisions at √ s = 13 TeV. The analyzed data correspond to an integrated luminosity of 35.9 fb −1 recorded by the CMS experiment at the LHC. Events containing an isolated muon or electron and two jets, of which one is b tagged, in the final state are used in the study. From the inclusive measurement the top quark mass is found to be 172.13 +0. 76 −0.77 GeV, where the uncertainty includes both the statistical and systematic components. The masses of the top quark and antiquark are separately determined as 172.62 +1.04 −0.75 and 171.79 +1.44 −1.51 GeV, respectively. These quantities are used to determine the mass ratio of the top antiquark to top quark of 0.9952 +0.0079 −0.0104 , along with the difference between the top quark and antiquark masses of 0.83 +1.79 −1.35 GeV, both for the first time in single top quark production. The obtained mass ratio and difference agree with unity and zero, respectively, within the uncertainties, and are consistent with the conservation of CPT symmetry. This is the first measurement of the top quark mass in this particular final state to achieve a sub-GeV precision.

Acknowledgments
We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centers and personnel of the Worldwide LHC Computing Grid and other centers for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC, the CMS detector, and the supporting computing infrastructure provided by the following funding agencies: [6] ATLAS Collaboration, "Performance of top-quark and W-boson tagging with ATLAS in Run 2 of the LHC", Eur. Phys. J. C 79 (2019) 375, doi:10.1140/epjc/s10052-019-6847-8, arXiv:1808.07858.
[15] J. R. Christiansen and P. Z. Skands, "String formation beyond leading colour", JHEP 08  Figures A.1 and A.2 show the distributions of the other six BDT input variables in data and simulation for the muon and electron final states. Figure A.3 presents the correlations among all BDT input variables for these final states before and after applying the decorrelation method available in TMVA. As expected, the correlations are significantly reduced after decorrelation.      Figure B.1 (left) shows the scan of the profile likelihood ratio as a function of the POI for the ML fit model used to determine the m t value in the charge-inclusive muon and electron final states. The scan results are presented both for data and simulation. Figure B.1 (right) presents correlations among the POI and three nuisance parameters corresponding to the ML fit applied to data in the same final states. Correlations in % among the POI and nuisance parameters corresponding to the fit to data for the l ± final state of the 2J1T category (right).