Top-quark mass determination from t-channel single top production at the LHC

We study determination of top-quark mass using leptonic observables in t-channel single top-quark production at the LHC. We demonstrate sensitivity of transverse momentum of the charged lepton on the input top-quark mass. We present next-to-next-to-leading order (NNLO) QCD predictions with narrow width approximation and structure function approach. Further corrections due to parton showering and hadronizations, non-resonant and non-factorized contributions are discussed. To reduce impact of SM backgrounds we propose to use charge weighted distribution for the measurement, i.e., differences between distributions of charged lepton with positive and negative charges. Projections for (HL-)LHC are found to be very promissing with a total theoretical uncertainty on extracted top-quark mass of about 1 GeV, from modeling on both signal and background processes.


Introduction
The top quark (t) is the heaviest particle in the standard model (SM). The mass of top quark has been one of the most important input parameters of the SM and its experimental determination is crucial for precision test of the SM. For example, the recent global analysis of electroweak precision observables reveals a good agreement with top-quark mass from direct measurements [1]. Top quark also plays important role in renormalization group running of the SM especially due to the large Yukawa coupling and its precise mass value is responsible for the stabilities of the electroweak vacuums [2].
The top-quark mass can be measured directly at the CERN LHC in top-quark pair production with subsequent decays, e.g., through invariant mass distributions of various decay products, with which the CDF, D0, ATLAS and CMS collaborations have reported an unprecedented precision of about 0.5 GeV [3][4][5]. The above measurements are supposed to be affected by various non-perturbative QCD effects that are modeled by MC event generators.
The associated systematic uncertainties become more and more important as the improvement of the experimental precision, and receive a lot of attentions recently [6][7][8][9]. There are also discussions on intrinsic ambiguities of precise definition of the top-quark mass due to infrared renormalon effects [10][11][12]. Many alternative methods on determination of the top-quark mass have been proposed and carried out at the LHC in order to scrutinize the experimental precision. That includes utilizing kinematic variables other than conventional invariant masses in top-quark pair production, e.g., [13][14][15][16][17][18], and using measurements on total inclusive cross sections [19][20][21] or on distributions inclusive to decay products of top quark [22][23][24]. There are cases of using processes of associated production of top-quark pair with a jet [25] or single top-quark production [26][27][28]. Besides, at future electron-positron colliders the top-quark mass can be measured much precisely via a scan of energy at threshold of top-quark pair production [29]. A recent review on various topics in determination of top-quark mass can be found in [30].
In this work we perform a theoretical study on determination of the top-quark mass through t-channel single top-quark production at the LHC. Especially we study in details transverse momentum distribution of the charged lepton from decay of the top quark with which we determine the top-quark mass. Similar approaches have been adopted for mass measurement in top-quark pair production [14]. It has been argued that using methods with pure leptonic variables, i.e., not directly involving kinematics of jets, will be less affected by various non-perturbative effects, as well as by uncertainties due to jet energy scale. Measuring top-quark mass in single top-quark production [26][27][28] is complementary to those measurements in pair production because of different dynamics of production including QCD color flows, which may lead to different systematic uncertainties. It can provide independent and valuable inputs to the global determination of top-quark mass. Besides, significant efforts have been made to improve the theoretical description of t-channel single top-quark production. We notice that the QCD corrections in single top-quark productions are in general much smaller than those in pair production. The next-to-leading order (NLO) QCD corrections in the 5-flavor scheme (5FS) are calculated in Refs. [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. The NLO calculation in the 4-flavor scheme (4FS) is carried out in Ref. [47]. Full NLO corrections including top-quark leptonic decay are studied within the on-shell top-quark approximation [38,40,42] and beyond [43,44,48,49]. Code for fast numerical evaluation at NLO is provided in Ref. [45]. The NLO electroweak corrections are also calculated [50]. Soft gluon resummation is considered in Refs. [51][52][53][54][55][56][57]. Matching NLO calculations to parton showers is done in the framework of POWHEG and MC@NLO [48,[58][59][60]. Next-to-next-to-leading order (NNLO) QCD corrections with a stable top quark are calculated in Ref. [61]. The study here are based on the NNLO calculations including top-quark leptonic decay under narrow width approximation (NWA) as developed in Refs. [62][63][64][65] that provide a realistic parton-level simulation at NNLO. In our study, we have used the structure function approach, namely neglecting certain color suppressed contributions at NNLO. Progresses on calculation of those corrections have been made in [66,67].
However, we should point out several potential problems on determination of top-quark mass via single top-quark production. It suffers from large backgrounds due to top-quark pair production as well as W/Z boson plus jets production. A pure signal sample can only be obtained in a signal-enriched fiducial volume as shown in the ATLAS and CMS measurements [26,27]. That means the measurements are less inclusive and also have relatively low statistics as comparing to those in pair production. The former one is less concerned as far as precise theory predictions can be provided which will be the main topic of this paper. The shortcoming on statistics can also be overcome thanks to the high luminosity of LHC. The rest of our paper is organized as follows. In Sec. 2, we explain leptonic observables and the sensitivity to top-quark mass. In Sec. 3, we present our nominal predictions on leptonic distributions including their intrinsic and parametric uncertainties. Sec. 4 provides results of alternative theory predictions including from different heavy quark scheme and from MC generators. In Sec. 5 we address further questions on both theory and experimental sides related to the measurement and in Sec. 6 we show our projected precision of measurements at (HL-)LHC. Finally our summary and conclusions are presented in Sec. 7.

Leptonic observables
We demonstrate dependence of leptonic variables on the top quark mass in single top-quark production. We use on-shell renormalization scheme in perturbative calculations. Thus the top-quark mass we refer to in the remaining sections is always the pole mass. Specifically we focus on transverse momentum distribution of the charged lepton.
We start with a pedagogical discussion based on calculations at Born level. Kinematic distributions of the charged-lepton can be understood as below. For decay of an on-shell top quark in its rest frame, the triple differential decay width can be expressed as [68] where m b , m t , m W and Γ W are masses of the bottom quark, top quark and W boson, and width of the W boson, respectively. G F and V tb are the Fermi constant and the Cabibbo-Kobayashi-Maskawa (CKM) matrix element. The three kinematic variables are x = 2E l /m t , y = M 2 lν /m 2 t , and the cosine between direction of the charged lepton and the spin axis of the top quark. S = 1(0) corresponds to fully (un)polarized case. The Dalitz variables x and y fulfill kinematic constraints Distribution of transverse momentum of the charged lepton (p T,l ) defined with respect to axis z can be obtained from Eq. (2.1). For simplicity we first assume a zero width of W boson and massless bottom quark. Their effects will be added later. For unpolarized top quark average transverse momentum of the charged lepton can be derived as At hadron colliders kinematics of the charged lepton can be measured in both single top-quark production and top-quark pair production in which the top quarks are boosted in general. For a boost along the z axis it will not affect the transverse momentum distributions of the decay products. Now considering the top quark travels perpendicularly to the z axis with a velocity β, the average p T,l from decay of an unpolarized top quark is given by as derived from Eq. (2.1) by keeping up to O(β 2 ) terms. At LHC 13 TeV the top quark in t-channel single production has an average p T of around 40 GeV, while the average is about 120 GeV in pair production. They correspond to roughly a velocity of top quark of 0.2 and 0.6 respectively. From direct calculations of production with subsequent leptonic decay of the top quark at leading order (LO), we obtain the following results for LHC 13 TeV, p T,l t−ch = 38.38 GeV, p T,l tt = 56.37 GeV, (2.9) which are in agreement with estimations using Eq. (2.8) and the corresponding velocities at the LHC.
From above discussion we understand the precise distributions of the charged lepton will depend on modeling of the top-quark kinematics and polarizations in the production. They can be sensitive to QCD corrections in production and similarly to those in decay of the top quark. Besides, in experimental measurements various selection cuts applied can further change distributions of the charged lepton.

Theory predictions
In this section we present our major results on predictions of the leptonic observables. We first identify the signal regions used for LHC measurement and show sensitivity of the proposed observable to top-quark mass. Then we present our theory predictions based on a nextto-next-to-leading order calculation including decay of the top quark. Discussions on scale variations and parametric uncertainties are also included.

Signal regions
In experimental measurements various kinematic cuts are applied due to finite coverage of detectors, for example on transverse momentums or rapidities of the reconstructed jets, electrons, muons and photons. For single top-quark production, additional cuts or selections are required in order to suppress SM backgrounds from top-quark pair production and associated production of vector bosons and jets. We follow closely fiducial regions used in the CMS analyses at 8 and 13 TeV [27,70].
We require one charged lepton in the final state with p T > 26 GeV and |η| < 2.4. We include only one family of leptons from decay of the top quark in results through the paper unless otherwise specified. We use anti-k T jet algorithm [71] with a distance parameter of D = 0.4. Jets are required to have p T > 40 GeV and |η| < 4.7. A clustered jet at parton level is defined as b-tagged if it has a non-zero net bottom-quark number in the constituents and further has |η| < 2.4. In addition a constant b-tagging efficiency of 50% has been applied. We refer light jets as jets that are not b-tagged. We consider two signal regions for t-channel production, CMS-SA and CMS-SB. Both are required to have exactly two jets in the final state with one being b-tagged jet and the other being light jet. We require the transverse mass of the charged lepton and the missing momentum from neutrinos to be greater than 50 GeV. In CMS-SB the light jet is required to stay in the forward region, namely |y| > 2.5, which can further increase the signal to background ratio.
We demonstrate the sensitivity of the leptonic distributions to top-quark mass in Fig. 1 for LHC 13 TeV. We show transverse momentum distribution of the charged lepton in the two signal regions with top-quark mass of 172.5 GeV or shifted by 5 GeV, calculated at NLO in QCD. The lower inset shows ratio of the distributions with different top-quark masses. Details of the calculation will be explained later. The increase of top-quark mass leads to a harder p T spectrum except at very large p T , where enhancements of the distribution are cancelled out because of the increasing importance of top-quark kinematics in production. The two signal regions show a very similar dependence on the top-quark mass.  We prefer to use a single variable when extracting the top-quark mass, rather than a template fit to the full leptonic distributions. We choose the variable as average p T of the charged lepton. We can select different windows of the p T spectrum to be included. We plot the relative change of the average p T when varying top-quark mass by 1 GeV, as a function of an upper limit placed on p T in Fig. 2. For both signal regions the sensitivity saturates to a value of about 0.3% when the upper limit reaches above 100 GeV. In later sections we will always present results for two representative values of the upper limit, 100 GeV and 200 GeV. Inclusion of high p T region usually leads to larger theory uncertainties.

NNLO predictions
The NNLO predictions for t-channel single top-quark production in the 5-flavor scheme are calculated using phase-space slicing with the N -jettiness variable [72][73][74][75] together with the method of "projection-to-Born" in Ref. [76]. Details for the NNLO calculation in the 5FS can be found in Ref. [62,63]. In the calculation, the QCD corrections can be further factored as from either fermion line with heavy quarks or light quarks neglecting certain color suppressed contributions [66,67]. We also include consistently the NNLO corrections in decay of the top quark as originally calculated in [77] using narrow width approximation. We focus on predictions for top-quark production at LHC 13 TeV. Results for top anti-quark production can be obtained through a CP transformation with substitutions of the initial parton distributions.
We use a PDF set of PDF4LHC15 nnlo 30 with α S (m Z ) = 0.118 [78][79][80][81][82][83], and a nominal value of the top-quark mass of 172.5 GeV. The central scales of QCD renormalization and factorization are set to be half of the top-quark mass which leads to better convergence of the 5FS predictions [65,84]. We evaluate the scale uncertainty by varying the two scales independently with a factor of two and taking envelope of results with 9 scale choices. Effects due to finite width of the W boson and finite mass of the bottom quark in top-quark decay, and finite width of the top quark, are included by adding their contributions as calculated at leading order.
We show transverse momentum distribution of the charged lepton in the two signal regions in Fig. 3 at various orders in QCD together with scale uncertainties for LHC 13 TeV. In the lower inset of each plot we show ratios of the predictions to a common reference at NNLO with nominal scale choice. The NNLO corrections lead to a softer spectrum due to both soften of the top-quark p T and the additional radiations in decay [77]. Size of the NNLO corrections range from -5% to -35% for the p T region shown. Moderate reduction of scale uncertainties are seen when including the NNLO corrections. However, the scale variations at NLO underestimate the size of NNLO corrections for signal region CMS-SB especially in the high-p T tail.
We present detailed results on average p T of the charged lepton in Table 1 for the two signal regions and with two choices of the upper limit on p T . The numbers in parenthesis correspond to predictions when excluding QCD corrections in decay of the top quark, i.e., only including corrections in production of the top quark. We see the leading order predictions show a rather small scale variation, which can be understood since the change of scales at LO only impact the overall normalization and longitudinal boost of the system, not much for shape of the transverse momentum distribution. The LO predictions can not model well distribution of transverse momentum of the top quark, especially at high p T , as explained in [65]. The NNLO corrections lead to a reduction of average p T by less than 0.2 GeV if the   The QCD corrections from top-quark decay are large comparing to our aimed precision on average p T . They reduce the average p T by about 0.3∼0.4 GeV at NLO. The NNLO corrections from top-quark decay further decrease the average p T by 0.1 GeV in case of signal region CMS-SA. We recall that the NNLO corrections due to top-quark decay consist of two parts, one from the pure two-loop corrections in top-quark decay and the other from oneloop corrections in decay multiplied by one-loop corrections in production. Both of the two pieces are important. Cancellation between them may occur depending on the observables and kinematic region considered. We also show predictions on total fiducial cross sections in Table 2. The NNLO corrections reduce the cross sections by about 6% for signal region CMS-SA, with the NNLO predictions at the lower boundary of the NLO scale variations. The reduction is about 10% for signal region CMS-SB.

Parametric uncertainties
We investigate dependence of our predictions on various inputs and the associated parametric uncertainties. That includes the parton distribution functions, QCD coupling constant, and bottom quark mass. Uncertainties due to parton distribution functions are estimated following the PDF4LHC recommendation [78] using PDF4LHC15 NNLO 30 PDF set. We calculate the dependence on QCD coupling constant by varying α S (m Z ) by 0.0015 from its nominal value of 0.118, accounting for induced changes of PDFs as well. We use MMHT2014 PDF set [85] with different bottom quark mass values to calculate the changes when varying pole mass of bottom quark by 0.5 GeV. The impact on average p T of the charged lepton and on the total fiducial cross section are summarized in Table 3. We also include corresponding numbers for varying mass of top quark by 1 GeV for comparison. We find in all cases the parametric uncertainties of average p T are small comparing to the dependence on top-quark mass, at the level of 0.01∼0.02 GeV. On another hand, the total fiducial cross sections show larger uncertainties. For example, the PDF uncertainties are about 4%∼2% and the uncertainties due to bottom quark mass are about 3%∼1% if taking error of bottom quark mass as 0.2 GeV [85]. We can also see that the total fiducial cross sections are insensitive to top-quark mass, unlike the average p T of the charged lepton.

Alternative theories
We present two alternative theory predictions concerning both the perturbative and nonperturbative components in t-channel production of single top quark. The comparison to our nominal predictions can lead a better understanding on the related theoretical uncertainties.  Table 3: PDF uncertainties on the average transverse momentum of the charged lepton and on the fiducial cross section within the two fiducial regions, followed by induced changes on the same quantities when varying α S (m Z ), m b , and m t by the amount in parenthesis.

Heavy-quark schemes
It is known that the t-channel production can also be calculated in a factorization scheme with a fixed 4 flavor numbers of quarks. The 5FS has the advantages of resumming potential large logarithms of bottom quark mass due to gluon splitting into bottom quarks from initial state. The 4FS maintains full bottom quark mass dependence through fixed order with current predictions available only at NLO in QCD. We notice that leading order calculations in 4FS already contain ingredients appearing in next-to-leading order calculations in 5FS. Critical questions arise on the use and agreement of the two heavy-quark schemes in tchannel single top-quark production, with efforts at understanding made in Refs. [47,84,86]. In a recent study by one of the authors [65], we compare predictions at NNLO in 5FS to those at NLO in 4FS without decaying of the top quark. We find the two schemes agree within a few percent in general for the shape of kinematic distributions of the top quark, and differ on the overall normalizations. We conclude the 5FS provides a better modeling on tchannel production when evaluating at a comparable perturbative order. Now we extend the comparison to include leptonic decay of the top quark, focusing on the leptonic observables discussed.
We use the program MCFM [87,88] to calculate t-channel single top-quark production with subsequent decays in the 4FS. The original calculation was detailed in Ref. [47]. We use CT14 NNLO PDFs [82] with 4 flavor numbers through the comparison and a bottom quark mass of 4.75 GeV. We set the nominal QCD renormalization scale and factorization scale to half of the top-quark mass. Scale variations are evaluated by the 9-scales envelope same as before.
We show 4FS predictions on transverse momentum distribution of the charged lepton in Fig. 4 for the two fiducial regions, comparing with the NNLO predictions in 5FS. In the comparison the NLO predictions in 4FS include the NLO corrections in top-quark decay, and the NNLO predictions in 5FS include further NNLO corrections in decay. We find the LO and NLO predictions in 4FS show less differences as comparing to the case of stable top quark in Ref. [65]. That is because of the jet veto conditions applied, namely requiring exactly two jets in the final state. For the same reason the NLO predictions in 4FS agree well with NNLO ones in 5FS even for the overall normalizations. Shape differences of the two predictions can be understood as due to both the harder p T spectrum of the top quark in production in 4FS and the inclusion of NNLO corrections in decay in 5FS. The scale variations are slightly larger in the high p T region for the 4FS predictions.    More detailed comparison can be found in Table 4 for average p T of the charged lepton. The numbers in parenthesis correspond to predictions without including QCD corrections in decay of top quark. We find scale variations of LO predictions in 4FS are small and underestimate the genuine NLO corrections especially when including high-p T regions, similar to the case seen earlier for LO predictions in 5FS. The difference on average p T of NLO predictions in 4FS and NNLO predictions in 5FS is about 0.3 GeV for fiducial region CMS-SA and 0.4 GeV for fiducial region CMS-SB. Half of the difference can be attributed to different treatment on corrections in decay of the top quark. Scale variations are slightly larger for average p T from NLO predictions in 4FS. Predictions from the two schemes overlap in general once including both scale variations. Similar results for total fiducial cross sections are shown in Table 5 where even better agreement are seen between the two schemes.

Parton showering and hadronizations
We compare our parton-level results with those from various Monte Carlo generators in the 5-flavor number scheme. We calculate the fiducial cross sections and distributions at NLO in QCD matched with parton showering using MG5 aMC@NLO program [89]. We generate matched events with stable top quarks that are further decayed with MadSpin [90]. In the following events are passed to various generators for parton showering and hadronization, including PYTHIA6 [91], PYTHIA8 [92], and HERWIG7 [93]. Finally the events are analysed with MadAnalysis5 [94] and FastJet [95]. We use same input parameters as in previous fixedorder calculations for PDFs, QCD scales and selection cuts. One difference with respect to fixed-order calculation is on definition of true b-jet for which we use the default method implemented in MadAnalysis5. Be specific, in each event after jet clustering, one searches for intermediate b quarks in the MC record. The clustered jets are considered as a true b-jet if it can be associated with a MC b quark inside the jet cone. Similarly a b-tagging efficiency of 50% is applied on the true b-jet. We show MC predictions on transverse momentum distribution of the charged lepton in Fig. 5   while the latter two display very good agreement. That can be due to different showering algorithms or possibly the way of different showering programs handling the decayed resonance, for example as studied in Refs. [48,96]. We can also compare the MC predictions with the NNLO predictions. It is interesting that the two show a very good agreement on shape of the distribution though the normalization is higher by about 10% for PYTHIA6 and PYTHIA8. Such an agreement is non-trivial since the matrix elements used in MC predictions even do not include NLO corrections in top-quark decay which has large impact on the p T distribution as shown in Table 1. It is the parton showering resummation that takes into account part of the missing NLO and NNLO corrections and brings the MC predictions closer to the NNLO predictions.
Comparison on average p T of the charged lepton are summarized in Table 6 along with the fiducial cross section shown in Table 7. In both tables we further include NLO predictions with and without corrections in decay of the top quark. For MC predictions, numbers in parenthesis are for cases of turning hadronizaiton off in the generator. It is clear that hadronization corrections are small for PYTHIA6 and PYTHIA8, within the statistical uncertainties which are about 0.1 GeV for average p T . Hadronization reduces the fiducial cross section by a few percents and increases the average p T slightly for HERWIG7. From Table 7 we find normalization of MC predictions generally lie between NLO predictions with and without corrections in top-quark decay. On the other hand, for average p T , MC predictions are closer to the NNLO predictions. We note that there exist MC generators including full NLO QCD corrections [48,50] which will be discussed in the following section.

Discussions
In this section we further discuss several theory or experimental subjects which are relevant for extraction of the top-quark mass. That includes impact of experimental selections, for example, contributions from leptonic decay of τ lepton in top-quark decay, isolation of lepton from jets, and b-tagging efficiency. Theory topics include contributions of non-resonant diagrams, non-factorized and electroweak corrections. We also provide an estimation on various standard model backgrounds and propose a possible solution on reducing their impact. Results shown here are calculated with MG5 at leading order matched with parton showering and hadronization via PYTHIA6 unless otherwise specified.

Signal selection and corrections
We start with contributions from leptonic decay of τ lepton. They can be counted as either part of the signal events or as a background to be subtracted. The inclusive cross sections from τ decay are suppressed by a branching ratio of about 17%. In the fiducial regions selected, the τ contributions are further suppressed due to the p T threshold of charged lepton as well as the cut on transverse mass, since more neutrinos are presented. For the same reason it has a softer spectrum of the charged lepton comparing to those from direct production. As shown in Table 8, the τ contributions amount to about 2% of the direct contributions for fiducial cross sections, and reduce the average p T of charged lepton by 0.1 GeV.
[  Table 8: Changes of the average transverse momentum of the charged lepton and of the fiducial cross section within the two fiducial regions, when including contributions from τ decay, applying lepton isolation, varying b-tagging efficiency, and including non-resonant contributions.
In previous calculations we did not apply any isolation cuts on charged lepton from jets, which are usually imposed in experimental analyses. We repeat our NLO calculations by further requiring ∆R lj(b) > 0.4. The changes on fiducial cross section and average p T are summarized in Table 8. The isolation cut has less impact when requiring light jet in forward region, i.e., for signal region CMS-SB, since then the charged lepton is unlikely to be close to the light jet. We also vary the b-tagging efficiency from our nominal choice of 50% to a value of 40%. That leads to an overall rescaling of the cross section and distributions except when there exist more than one true b-jets in the final states, which is the case for beyond leading order. By repeating our NLO calculations we found the changes on average p T is negligible for signal region CMS-SB since it is unlikely the light jet is from a mistagged true b-jet. In reality nonuniformity of the experimental b-tagging efficiency may lead to further change of average p T due to correlations of kinematics of the b quark and the charged lepton.
Next we move to various theory aspects starting with non-resonant contributions, namely production of W + bj via electroweak interactions without a top-quark resonance. Those nonresonant diagrams can interfere with the resonant diagrams and induce non-negligible con-tributions as shown in Table 8. The effects are much smaller in fiducial region CMS-SB due to non-forward nature of the light jet in non-resonant production. We should mention that there are also non-resonant diagrams of W + bj production from QCD interactions. They do not interfere with the remaining at LO and we leave them to the W + JJ category that will be discussed later in the background section.  Figure 6: Transverse momentum distribution of the charged lepton within the two fiducial regions, comparing predictions at fixed-order and those from event generators tj and W bj (see text for details) in the 5FS for LHC 13 TeV.

Non-factorized and EW corrections
The NNLO predictions presented are based on a calculation using NWA and structure function approach. 1 There exist missing QCD corrections due to non-factorized diagrams starting at NLO, e.g., with a gluon connecting bottom quarks in production and decay of the top quark. Those non-factorized corrections have been studied in details in [44,49]. They are formally of the size α S Γ t /m t , namely suppressed by width of the top quark, but can be enhanced in certain kinematic region. We follow the strategy in Ref. [50] on identifying the corrections. We calculate the full NLO predictions for the production of final state W + bj and then subtract the contributions due to s-channel single top production and associated production of tW . All contributions are calculated using MG5 aMC@NLO with parton showering and hadronization applied from PYTHIA6, instead of calculated at fixed order. The resulting transverse momentum distribution of the charged lepton are presented in Fig. 6, comparing with fixed-order results and MC result with PYTHIA6 shown in early sections. The two MC predictions are denoted as tj and W + bj respectively. In comparison the latter includes exact NLO corrections in decay of the top quark and further the nonfactorized corrections just mentioned. For transverse momentum below 100 GeV we find the W + bj predictions locate well in between previous NLO and NNLO fixed-order predictions. We find a trend of large enhancement of the distribution beyond 100 GeV in W + bj predictions though accompanied with significant MC statistical errors. In Ref. [50] it also shows the NLO electroweak corrections can induce a significant change on shapes of various distributions. For comparison to experimental data, a recalculation of the full NLO QCD and EW corrections focusing on transverse momentum of the charged lepton will be desirable. That can be done following various techniques outlined in Ref. [50] and with a careful separation of backgrounds that are already accounted for in the experimental analyses.
Finally there are also non-factorized NNLO QCD corrections in production that are beyond the structure function approach, for example, from the double-box diagrams and also interferences of t-channel and s-channel at NNLO. They are color suppressed in principle, but it is not clear how they may change shape of various distributions. Progresses have been made recently on calculation of these non-factorized corrections for single top-quark production [66,67]. Similar non-factorized corrections also exist in production of the Higgs boson and are estimated using eikonal approximation [97,98].

Backgrounds
The SM backgrounds mainly consist of the top-quark pair production, single top-quark production in s-channel and in associated with a W boson, QCD production of W JJ, and diboson productions. For the QCD production of W JJ, the jet J can arise from not only a bottom quark, but also a charm quark, even a gluon or a light quark. In the latter it mimics the signal due to mistagging which we choose a rate of 3% for charm quark and 0.1% for gluon and light quarks [70]. We summarize the fiducial cross sections of various backgrounds and the average transverse momentum in Table 9. For comparison we also show backgrounds for top anti-quark production, and include numbers for signal process as well which are calculated at NNLO. In Table 9 we do not repeat the rows if the background contributes equally to top quark and anti-quark processes.
In general the top-quark pair production is dominant among all backgrounds due to the large cross section, though that requires additional charged lepton or jets lie outside the acceptance region. We veto any event with more than one charged lepton with p T > 10 GeV. The primary charged lepton has larger transverse momentum in pair production due to the relatively large p T of the top-quark. Large contributions are also seen for QCD production of W JJ that has a harder p T spectrum of the charged lepton. The tW associated production can contribute at a level of tens percents of the signal cross sections. We note for top-quark production, both tW − andtW + production can contribute as backgrounds. In the later case the primary charged lepton comes directly from W + decay leading to a harder p T spectrum. Backgrounds due to s-channel production or diboson production are small. A typical feature can be seen from Table 9 is that in signal region CMS-SB where the light jet is required to be forward, almost all backgrounds are suppressed by a factor of ten at least. The signal from t-channel production are less affected due to the forward nature of the light jet.
[  Table 9: Average transverse momentum of the charged lepton and fiducial cross section within the two fiducial regions, for various background processes to t-channel top quark and anti-quark production. The top-quark pair production or top-quark associated production with W boson contribute equally to the two charge conjugate final states.
From Table 9 we see even in region CMS-SB, the rates of tt background can still reach the same level as the signal processes. That may easily spoil the precision on measurement of average p T of the signal processes due to uncertainties on modeling of the tt background. Further more, any backgrounds from top-quark production depend on the top-quark mass as well, which will complicate the extraction of top-quark mass. One important observation is that both the tt and tW backgrounds contribute equally to signal processes of charged lepton with positive and negative charges, with differences being negligible. The charge asymmetry first enters at NLO for tt production and is small at the LHC. In case of tW production the asymmetry vanishes even at NLO. Thus one possibility is to measure the difference of lepton p T spectrums with positive and negative charges. In this way dependence and associated uncertainties on modeling of the tt and tW backgrounds are minimized, though their statistical fluctuations remain. Sensitivity to top-quark mass and the theoretical uncertainties are almost unchanged for the signal when taking differences of processes with opposite charges. That is because the differences of t-channel single top quark and anti-quark production are only induced by different parton distributions at the light-quark line. Afterwards, the uncertainty due to modeling of QCD W JJ background will be dominant. However, as mentioned earlier, a large fraction of W JJ background arise from production of charm quark, gluon or light quarks which are misidentified as b-jets. One can further reduce their impact by either imposing a tighter b-tagging criteria or using data-driven methods.

Projection for (HL-)LHC
We provide an estimation on precision of the top-quark mass measurement can be achieved in the coming run of LHC and HL-LHC. As explained earlier, the observable used is the average transverse momentum of the charged lepton in the charge weighted distribution, where in the second line we have rewritten the average p T in terms of signal and background contributions, and r is the ratio of background to signal rate. We neglect backgrounds other than from top-quark pair production and QCD production of W JJ, which are small according to Table 9. From Eq. (6.1) we can extract average transverse momentum of the signal p T S using measurement on p T obs and inputs of r and p T B . From our theory calculation we can arrive at a linear model on dependence of average p T on the top-quark mass, where p T,0 is the signal average p T for a top-quark mass of 172.5 GeV. p T,0 and λ can be calculated from NNLO predictions shown in Tables 1-3 together with the counterpart for top anti-quark production. By combining Eqs. (6.1) and (6.2) we can extract the measurement on top-quark mass.
In the following we focus on signal region CMS-SB with p T,l < 100 GeV. It benefits from both smaller backgrounds and controlled theoretical uncertainty. We estimate several contributions to the final uncertainty of measured top-quark mass. The statistical uncertainty on p T obs due to both signal and background fluctuations, including tt contributions, are computed with pseudo experiment assuming an integrated luminosity of 300 and 3000 fb −1 respectively and top-quark decays into two charged lepton families. Theoretical uncertainties on p T,0 are estimated with scale variations of NNLO predictions shown in Table 1 Further uncertainties are related to modeling of the backgrounds. We only need to consider the W JJ background in this case since the systematic uncertainty for tt production cancels in the charge weighted p T distribution. A precise study on QCD W JJ background is beyond the scope of current paper and can be carried out with dedicated MC simulations. We simply assign some empirical numbers on systematic uncertainties of p T B and r from W JJ background. On one side we assume they are 0.5 GeV and 10% respectively, and reduced by a factor of two in the optimistic case. The results are shown in Fig. 8 with the horizontal bands representing uncertainty of p T S as propagated from background systematic errors. The induced uncertainty on measured top-quark mass is 0.8(0.4) GeV for the two scenarios as shown by the vertical lines in Fig. 8, comparing to the theoretical uncertainty shown earlier. Thus we expect the full error budget on the extracted top-quark mass consists of the theoretical uncertainty of about 1 GeV from signal modeling, 0.4 GeV due to background modeling, and even smaller for the statistical uncertainty.

Summary
In summary we have studied determination of top-quark mass using leptonic observables in t-channel single top-quark production at the LHC. Extraction of top-quark mass from single top-quark production benefits from partial uncorrelated systematic uncertainties to those from top-quark pair production on both experimental and theory sides. We demonstrate sensitivity of average transverse momentum of the charged lepton to the input top-quark mass. Leptonic observables are generally believed to be less affected by various non-perturbative QCD effects and the jet energy scale uncertainties. We identify the appropriate signal region for such a measurement at the LHC that has enhanced signal to background ratio as well as stable theory predictions. We present our NNLO QCD predictions with narrow width approximation and structure function approach. We show that QCD corrections in top-quark decay play important role for such leptonic observables. We find a good convergence on predictions of the average transverse momentum of charged lepton with scale variations well under control. By comparing fixedorder predictions to those from MC generators we confirm the parton showering resummation mimic part of the NLO and NNLO corrections, and the hadronization effects are in general small for leptonic observables. Besides, we discuss on several corrections that need to be applied when comparing our NNLO predictions with data, including non-resonant corrections, non-factorized QCD corrections, EW corrections, and so on.
Moreover, we estimate various SM backgrounds for the signal regions considered. We propose to use charge weighted distribution in the measurement, i.e., difference between distributions of charged lepton with positive and negative electric charges. That can reduce uncertainties due to modeling of SM backgrounds which contribute equally to final states with different charges, for example, from QCD jets production, top-quark pair production, and topquark associated production with a W boson. We construct a simple model on dependence of the observed average transverse momentum of charged lepton to the top quark mass, and present projections for future (HL-)LHC measurements. The statistical uncertainties are found to be small. Scale variations in our signal modeling transfer into an uncertainty of about 1 GeV of the extracted top-quark mass. Theoretical uncertainty due to modeling of remaining SM backgrounds is estimated to be 0.4∼0.8 GeV.