Modeling uncertainties of $t\bar{t}W^\pm$ multilepton signatures

In light of recent discrepancies between the modeling of $t\bar{t} W^\pm$ signatures and measurements reported by the Large Hadron Collider (LHC) experimental collaborations, we investigate in detail theoretical uncertainties for multi-lepton signatures. We compare results from the state-of-the-art full off-shell calculation and its Narrow Width Approximation to results obtained from the on-shell $t\bar{t} W^\pm$ calculation, with approximate spin-correlations in top-quark and $W$ decays, matched to parton showers. In the former case double-, single-, and non-resonant contributions together with interference effects are taken into account, while the latter two cases are only based on the double resonant top-quark contributions. The comparison is performed for the LHC at $\sqrt{s} = 13$ TeV for which we study separately the multi-lepton signatures as predicted from the dominant NLO contributions at the perturbative orders $\mathcal{O}(\alpha_s^3\alpha^6)$ and $\mathcal{O}(\alpha_s\alpha^8)$. Furthermore, we combine both contributions and propose a simple way to approximately incorporate the full off-shell effects in the NLO computation of on-shell $pp\to t\bar{t} W^\pm$ matched to parton showers.


I. INTRODUCTION
The hadronic production of top-quark pairs in association with a W boson is one of the most massive signatures currently accessible at the Large Hadron Collider (LHC) and allows to study possible deviations from the Standard Model (SM) dynamics of top quarks in the presence of charged electroweak (EW) gauge bosons.As the accompanying top quarks are rapidly decaying into a W boson and a b quark the signature of the pp → t tW ± process involves two b jets and the subsequent decay pattern of three decaying W bosons.This gives rise to the rare Standard Model production of same-sign lepton pairs and other multi-lepton signatures that are relevant to a multitude of searches [1][2][3][4] for physics beyond the Standard Model (BSM) as well as to the measurement of Higgs-boson production in association with a top-quark pair [5][6][7][8] and of four top-quarks [9][10][11][12].
Due to the importance of the pp → t tW ± process to validate the EW interactions of the top quark and as a dominant background to many ongoing measurements and searches at the LHC, it is crucial to have the process under excellent theoretical control.Interestingly, small tensions between the t tW measurements and the SM predictions have been reported since its discovery.The first measurements [13,14] during the LHC Run 1 at a center-of-mass energy of √ s = 8 TeV already reported tensions for the measured cross section slightly above the 1σ level.Further measurements [15][16][17] performed at √ s = 13 TeV confirmed the picture that a slight excess of t tW events is observed.Additional measurements of the t tW cross section [7,8,10,11], from analyses aimed at the measurement of t tH and t tt t production and utilizing data corresponding to an integrated luminosity of up to L = 139 fb −1 , still see a persisting tension with respect to the corresponding SM predictions depending on the considered final-state signature, where the largest deviation of up to a factor of 1.7 has been found in multi-lepton signatures [7].
Fueled by these deviations, tremendous progress has been made in the last decade in the theoretical description of the pp → t tW ± processes.The first next-to-leading (NLO) QCD corrections have been computed in Refs.[18,19], while EW contributions have been investigated in Refs.[20][21][22].Furthermore, the resummation of soft gluon effects at the next-to-next-to-leading logarithmic (NNLL) level has been achieved recently in Refs.[23][24][25][26][27]. Targeting a more realistic description of fiducial signatures, the t tW ± process has been matched to parton showers either using the MC@NLO matching scheme [28,29] in Refs. [30][31][32], or the POWHEG method [33,34] in Refs.[35,36].The impact of higherorder corrections via multi-jet merging has been studied as well in Refs.[37][38][39].In parallel also the inclusion of off-shell effects for top quarks and W bosons as well as single and nonresonant contributions together with interference effects have been studied including NLO QCD corrections [40][41][42] as well as complete NLO SM corrections [43].
The emergent picture from all the aforementioned studies is that NLO QCD corrections are of the order of 8%−20% depending on the chosen SM input parameters and cuts applied on the final states.The resummation of soft gluon effects in on-shell t tW production improves only marginally the perturbative convergence of NLO QCD predictions and increases the rate by less than 10%.The second largest contribution in the perturbative expansion of the cross section comprises the NLO QCD corrections to the pure EW born and amounts roughly to a +10% correction to the total NLO cross section.Including higher-order corrections via multi-jet merging also increases the inclusive cross section by a similar amount.
On the other hand QCD corrections to top-quark decays are negative and decrease the production rate by roughly 5%.Finally, even though off-shell effects and non-resonant contributions are small at the integrated level, they become sizable in the tails of dimensionful observables where they can make a difference even up to 70%.Overall, none of the effects described above can fully explain the tension between the measurements and the SM prediction.However, given the complexity of the signatures involved and the progress made to theoretically model them at both the integrated and differential fiducial level, a study aimed at comparing existing computations and understanding the origin of their differences as well as the individual residual uncertainties is of the utmost importance to eventually propose a theory recommendation for the comparison with data.
As existing computations are often quite different in nature and therefore inherently affected by distinctive theoretical uncertainties, it is not evident how to combine the various findings into a precise modeling of the t tW process.In this paper we attempt to do a first comparison between parton-shower matched predictions and fixed-order full off-shell calculations, together with their corresponding narrow-width-approximations (NWA), in order to understand the different approaches in more detail.In all cases, we include NLO QCD corrections to both the QCD and EW born level, in the following simply denoted by QCD and EW (see Sec. II).Incidentally, we also quantify for the first time the size of full off-shell effects for the EW contribution.Our comparison focuses on the multi-lepton signature as it is not only the cleanest signature on the theoretical side but also yields the strongest discrepancies when compared to measurements.
The paper is organized as follows.In section II we summarize the computational setup of the calculations employed in our study, while in section III we highlight in detail the differences between the various theoretical approaches.In section IV we study the modeling of top-quark production and decays separately for the QCD and EW contributions.In section V we show our phenomenological results and propose a simple method to approximately capture the full off-shell effects in parton-shower matched calculations for on-shell t tW ± production.Finally, we give our conclusions in section VI.

II. COMPUTATIONAL SETUP
In our study we consider the pp → + ν − ν ± ν b b process at O(α 3 s α 6 ) and O(α s α 8 ), where = e, µ and all possible lepton-flavor combinations are considered.In the following we will refer to the former perturbative order as t tW ± QCD and the latter as t tW ± EW.We will provide predictions for the LHC operating at a center-of-mass energy of √ s = 13 TeV and parametrize the proton content using the NNPDF 3.1 [44] PDF set as provided by the LHAPDF interface [45].We employ the necessary SM input parameters in the G F input scheme [46].For the electroweak bosons we choose the following masses and corresponding decay widths Together with the Fermi constant G F = 1.166378 • 10 −5 GeV −2 the electromagnetic coupling is given by Finally, we choose for the top-quark mass We postpone the discussion of the top-quark width as it depends on the approximation used in the corresponding calculation.Let us now turn to the description of the various theoretical predictions employed in our study.
Full off-shell: Using the Helac-NLO framework [47][48][49][50][51][52][53] we generate results at fixed-order NLO QCD accuracy employing the matrix elements for the fully decayed final state ) and O(α s α 8 ).This approach includes all double-, single-, and non-resonant top-quark and W -boson contributions as well as interference and spin-correlation effects at the matrix element level.Top quarks and electroweak gauge bosons are described in the complex-mass scheme [46,54,55] by Breit-Wigner propagators.Therefore, the top-quark decay width is calculated at NLO QCD accuracy including off-shell W bosons according to formulae given in Refs.[56][57][58], which for our study corresponds to: Bottom quarks are treated consistently as massless quarks throughout the computation.The renormalization and factorization scales are chosen as More details on the calculation can be found in Ref. [40].
NWA: Benefiting from the recent automation of the NWA in the Helac-NLO framework [59] we are able to provide results at NLO QCD accuracy including spincorrelated top-quark decays.In this approach, the full matrix elements for pp → + ν − ν ± ν b b are approximated by the double resonant t tW ± contributions via a factorization of the cross section into a production and a decay stage by applying the following relation to top-quark and W -boson propagators lim In this limit the cross section can be written as where B denotes the corresponding branching ratios and ⊗ indicates that full spin correlations are kept.The computation is tremendously simplified with respect to the full off-shell calculation but still allows to systematically include NLO QCD corrections separately in the production and decay stage.In this case the top-quark decay width is given by If QCD corrections to the top-quark decay are omitted, which we denote as "NWA LO decay", we use the leading-order (LO) prediction for the top-quark width given by For the renormalization and factorization scales we choose as before Powheg-Box: We employ parton-shower matched results for pp → t tW ± using the recent Powheg-Box implementation presented in Ref. [36], based on one-loop amplitudes provided via NLOX [60,61].In this calculation the on-shell production of t tW ± is matched to parton showers at NLO accuracy, meaning that NLO QCD corrections are only included in the production stage.The renormalization and factorization scales are chosen according to In the matching procedure, the Powheg-Box employs two damping parameters, h damp and h bornzero , to define a jet-function that classifies the soft and hard real contributions.Based on the results of Ref. [36], in this study we choose the following central values and estimate the dependence of our results on these parameters by varying their values independently in the ranges The decay of the top quarks and the W bosons are included at LO accuracy using the method introduced in Ref. [62] However, the top-quark decay width enters only the decay chain matrix elements which employ Breit-Wigner propagators, see Ref. [36].Thus, the chosen value of Γ t has generally a very small impact on our results.We employ NLO accurate branching ratios for the W boson decays to be consistent with the full off-shell calculation.Thus, the total leptonic branching ratio is given by MG5 aMC@NLO: We also use parton-shower matched computations for the on-shell pp → t tW ± process in the MC@NLO framework as provided by MG5 aMC@NLO [63].
This approach has the same formal accuracy as the aforementioned Powheg-Box computation.The renormalization and factorization scales are also chosen as where E T is defined in Eq. ( 11) 1 , while the initial shower scale µ Q is left to the default option We study the dependence of our results on the initial shower scale µ Q by varying it up and down by a factor of 2. LO spin-correlated top-quark and W -boson decays are included via MadSpin [64] that automates the approach of Ref. [62].Within MadSpin the necessary branching ratios are evaluated at leading-order accuracy by default.However, we rescale our results to be consistent with Eq. ( 14) as discussed in the next section. 1The default dynamical scale choice for the renormalization and factorization scales in the Powheg-Box and in MG5 aMC@NLO actually corresponds to the definition of Eq. ( 16).We noticed that our choice of Eq. ( 11) increases the total cross section by approximately 11%, which is within the estimated scale uncertainties as we will see later on.
By searching for the minimum and the maximum of the resulting cross sections one obtains an uncertainty band.
For the parton-shower matched predictions, we generate Les Houches event files [65,66] that are subsequently showered with Pythia8 [67,68].In the parton-shower evolution we do not take into account hadronization effects and multiple parton interactions.The analysis of showered events is then performed within the Rivet framework [69,70] that also provides an interface to FastJet [71,72].
Due to the complexity of the full off-shell computation we store our theoretical predictions obtained with the Helac-NLO framework in the form of events, either available in the format of modified Les Houches event files [66] or ROOT Ntuples [73].Based on ideas presented in Ref. [74] events are stored with additional information about matrix elements and PDFs.We improve the performance of our framework by storing partially unweighted events, see e.g.Ref. [75] for more information.The Ntuples allow for on-the-fly reweighting for different PDF sets or scale choices as well as studying different fiducial phase space cuts.
The reweighting and event analysis is performed within Heplot [76].
The shower input cards, the Rivet analysis code and all histogram data are made available in the ancillary material.

III. DIFFERENCES OF THE VARIOUS COMPUTATIONAL APPROACHES
Before presenting our phenomenological results we want to discuss the intrinsic differences between the various computational approaches introduced above.
Let us start with the purely technical issue of choosing renormalization and factorization scales µ 0 for the various approaches.In order to obtain a good perturbative convergence for dimensionful observables a dynamic-scale choice has to be employed.However, due to the nature of the different approaches we can not use a common scale choice.For instance, the off-shell and NWA predictions use scales evaluated on the momenta of the fully decayed final state, while parton-shower based predictions use the on-shell momenta of the intermediate unstable particles.We checked explicitly that our conclusions in the following sections do not depend significantly on the scale chosen.Specifically we performed a comparison using the fixed scale µ 0 = m t + m W /2 that can be employed in all predictions.Shape differences of differential distributions, induced by the dynamic scale choices, are common in magnitude and for all predictions they are independent of the specific definition of the scale.
Next we want to discuss differences that affect predominantly the overall normalization of the predictions.First, we focus on differences between MG5 aMC@NLO and the Powheg-Box.The inclusive cross section for the three-lepton signature, as predicted by parton-shower matched calculations, is given in terms of production cross section times branching ratios where the factor 8 accounts for all combinations of = e, µ.In parton-shower based predictions the top-quark decays always into a W boson and a b quark, thus the branching ratio is Br(t → W b) = 1 and drops out from any inclusive or fiducial cross section.In other words, the obtained cross section is completely independent of Γ t , even in fiducial phase space volumes.However, the cross section is very sensitive to the branching ratio for the leptonic decay width of the W boson as it depends on it cubicly.In MadSpin all branching ratios are computed at LO accuracy from the given input parameters.On the other hand in the Powheg-Box the total leptonic W boson branching ratio is an independent input parameter that we set to the NLO value.The difference between LO and NLO branching ratios is only of the order of 2.5%.Nonetheless, due to the strong dependence of the cross section on this parameter, we observe a O(10%) higher cross section if LO branching ratios are employed.In order to obtain a cleaner comparison we rescale the MG5 aMC@NLO predictions globally by a factor of Considering next the NWA approach, we observe that in this case the cross section depends globally on the top-quark decay width via 1/Γ 2 t .Therefore, contrary to partonshower based calculation the inclusive cross section depends crucially on the chosen value of Γ t .Moreover, to ensure consistency in the comparison between full off-shell results and predictions in the NWA the unexpanded NWA results are used.We have studied in Ref. [40] that the impact of these higher-order terms is at the level of 3% − 4%.Feynman diagrams were produced with the help of the FeynGame program [77].
With respect to the NWA, the full off-shell calculation includes additional contributions.
Besides the double-resonant pp → t tW ± production the off-shell matrix elements also receive contributions from single-resonant pp → tW W b and non-resonant pp → W W W b b production modes as well as their interferences.We show some representative Feynman diagrams for each contribution in Fig. 1.We want to stress, however, that the single and non-resonant production cross sections can not be computed separately in a gauge invariant way, as these processes start to mix once higher-order corrections are included.Therefore, the only unambiguous way to account for these contributions is to employ the full matrix elements for pp → + ν − ν ± ν b b.Notice, that the Γ t dependence in the full off-shell computation is due to Breit-Wigner propagators of the double and single-resonant contributions.
Finally, the last conceptual difference between the fixed-order computations and the parton-shower predictions concerns the definition of b jets.The fixed-order computations are performed in the 5-flavor scheme treating b quarks as massless.In this case, we define the b-jet flavor using the net bottomness in a jet, according to the recombination rules which renders the jet-flavor definition infrared safe at NLO.Beyond NLO the flavored k T -jet algorithm of Ref. [78] is the most adopted solution to define the flavor of jets for massless partons.Recently a flavored anti-k T jet algorithm has also been introduced [79].On the other hand, the b quark is treated as a massive quark in the parton-shower evolution.As the collinear divergence in the g → b b splitting is regulated by the bottom-quark mass we can define a b jet to be a jet that has at least one b quark among its constituents.Hadronization effects can have an impact on the definition of b-jet observables.However, in preliminary calculations where we included hadronization effects we only found marginal differences for the considered b-jet observables.We have also checked explicitly that a parton-shower evolution with massless b quarks but employing the jet algorithm of Ref. [78] 2 yields results that are very similar to the default approach of massive b quarks clustered via the anti-k T jet algorithm.

IV. PHENOMENOLOGICAL RESULTS
For our comparison we focus on the three-lepton signature, where we exclude τ leptons as their decays produce very distinct signatures and can be easily distinguished from electron or muon final states.Jets are formed using the anti-k T jet algorithm [80] with a separation parameter R = 0.4.We require at least 2 b jets that satisfy the conditions Furthermore, we require the presence of exactly three charged leptons with Note that in our case the three charged leptons always consist of a two-same-sign lepton pair ss and a single lepton with opposite charge os .
A. QCD production of t tW ± We start our discussion with the dominant QCD production mechanism for the pp → t tW ± process and its leptonic decay at O(α 3 s α 6 ).

Integrated fiducial cross sections
First we present a comparison of fiducial cross sections and their uncertainties.For the off-shell calculation we obtain which shows a very reduced sensitivity to the choice of the renormalization and factorization scales with changes only between +3% and −6% around the central prediction.We do not investigate PDF uncertainties here, since these have already been addressed in Ref. [40] and were estimated to be of the order of 2% − 3% depending on the PDF set used.For the NWA with and without NLO QCD corrections to the top-quark decays we obtain σ NLO NWA = 1.57 We observe that the full NWA is in perfect agreement with the off-shell prediction.However, if QCD corrections to top-quark decays are neglected we obtain a 5% larger cross section and the residual scale sensitivity increases up to ±10%.Finally, for the parton-shower matched computations we find The remarkable agreement between the two predictions is not a coincidence since we have corrected for the different values of branching ratios employed in the calculations (see Sec. III) and have aligned as much as possible Pythia8 parameters between MG5 aMC@NLO and the Powheg-Box.The parton-shower matched computations predict a 11% smaller cross section if compared to the off-shell result.The reduction of the cross section is due to multiple emissions in the top-quark decays that are not clustered back into the corresponding b jet and thus decrease the amount of events passing our selection cuts.We checked explicitly that if we turn off parton-shower emissions in the decays we obtain a larger cross section, closer to the result of the NWA with LO decays.Finally, the NLO + PS results show, similar to the NWA LO-decay result, a scale sensitivity of the order of ±11%, as expected since both computations include NLO QCD corrections only in the production part of the pp → t tW ± processes.

Differential distributions
At On the left of Fig. 2 the transverse momentum distribution of the hardest b jet is shown.
We observe shape differences between the various predictions over the whole plotted range.
Results in the full NWA have the same shape as those of the off-shell calculation for p T (b 1 ) 300 GeV.However, all of them differ by 27% − 35% from the full calculation at p T ∼ 600 GeV.This is not surprising since in this region the single-resonant contribution from pp → tW W b becomes sizable3 .These kinds of corrections can only be provided by an off-shell computation and the comparison can not be improved by including among others NNLO QCD corrections in the NWA for the t tW ± process or by incorporating multi-jet merging.The dominant uncertainties for all predictions are attributed to missing higherorder corrections.In the case of the full off-shell calculation the scale sensitivity is the smallest, starting from ±5% and increasing up to ±15% towards the end of the plotted spectrum.On the contrary, the scale uncertainties of parton-shower based results already For the transverse momentum of the system of the two hardest b jets shown on the left hand side of Fig. 3, we observe first of all the enlarged uncertainty band, of the order of ±30%, for the off-shell calculation, which indicates that the observable is very sensitive to extra jet radiation.Again starting from p T ∼ 300 GeV the NWA increasingly deviates from the off-shell calculation all the way up to 55% at the end of the spectrum.This can be understood from the fact that in the tail of the distribution the single-resonant contribution for pp → tW W b scattering, which is only included in the full off-shell computation, becomes quite large.On the other hand, the parton-shower based predictions lie somewhat in between the NWA and the off-shell calculation for p T 600 GeV.Within the estimated uncertainties the parton-shower results are fully compatible with the full off-shell calculation over the whole considered range.
On the right of Fig. 3 we show the invariant mass of the opposite-sign lepton os and the b jet that minimizes the invariant mass itself.This observable is especially sensitive to off-shell effects and additional radiation since it has a natural kinematical edge at M (b os ) ≤ m 2 t − m 2 W ≈ 152 GeV in the case of the t → W + b on-shell decay.Thus, this observable is indeed particularly interesting for top-quark mass measurements [81][82][83][84].The full NWA recovers the shape of the off-shell result for M (b os ) ≤ 140 GeV but deviates up to 60% in the tail of the distribution.If top-quark decays are treated at leading-order accuracy in the NWA also the kinematical boundary is not described well, even below the edge.Finally, the parton-shower based predictions that account for multiple emissions in the top-quark decays wash out the kinematic edge even further.A considerable fraction of events are pushed above the edge and populate the tail of the distribution.We find only a difference of the order of 30% with respect to the off-shell calculation.However, here as well, the tail of the distribution receives sizable contributions from the single-resonant tW W b scattering.Scale uncertainties for the off-shell calculation are of the order of ±5% below the kinematic edge and ±15% above.On the other hand, the Powheg-Box and MG5 aMC@NLO obtain roughly ±10% below and ±20% uncertainties above the edge, while matching uncertainties are negligible throughout the whole range.
Next we discuss the transverse momentum of the opposite-sign lepton os and the H lep T observable, which is defined as For the H lep T observable on the right of Fig. 4 we observe a similar trend.Even though predictions based on the full NWA have minor shape differences, they are fully compatible with the off-shell calculation within the uncertainties.For the off-shell results the scale uncertainties are estimated to be below 8% in the whole plotted range.The Powheg-Box and MG5 aMC@NLO predictions are hardly distinguishable and show a different shape, with deviations of up to 20% in the tail of the distribution, when compared to the off-shell results.Only because of the larger scale uncertainty of the parton-shower matched predictions, which are of the order of 15%, the curves are compatible with the off-shell prediction up to 700 GeV.Above that point, the uncertainty bands do not overlap anymore.

B. Electroweak production of t tW ±
Let us now turn to the discussion of the electroweak production of the pp → t tW ± process including decays at O(α s α 8 ).We start again by investigating the integrated fiducial cross sections and turn to differential distributions afterwards.

Integrated fiducial cross sections
In the case of the full off-shell computation the integrated fiducial cross section evaluates to σ NLO off-shell = 0.206 We observe large scale uncertainties of the order of ±20% at NLO, which reflects the fact that the α s dependence only starts at NLO.For the EW part the NLO QCD corrections are anomalously large and of the order of K = NLO/LO ≈ 18, due to the t-channel tW → tW scattering in the qg channels that opens up at NLO.Overall, the EW contribution amounts to 13% of the dominant NLO QCD contributions to t tW ± production, i.e.
If instead the NWA is employed we find the following cross sections First of all, we notice that NLO QCD corrections to top-quark decays increase the cross section by 17%, but do not affect the overall scale uncertainties, which are of the same order as in the case of the full off-shell calculation.Furthermore, besides the large corrections from the top-quark decays we also observe large off-shell effects of the order of 8% at the level of integrated cross sections.Such effects originate from W W → W W scattering contributions as we have checked explicitly at LO where the off-shell effects are even larger and of the order of 12%.
The event generators have the smallest cross section among all the approaches considered in our study.The parton-shower matched results predict a 34% − 35% smaller cross section if compared to the full off-shell result.Also in the case of the EW contribution the Powheg-Box and MG5 aMC@NLO results align very well and are 16% − 18% smaller than the predictions obtained with the NWA with LO top-quark decays.This is the same level of reduction of the cross section as observed in the QCD contribution and, as in that case, can be attributed to additional radiation in top decays.However, in the EW case parton-shower based results show the same level of uncertainties as all the other approaches.

Differential distributions
Let us now turn again to the differential comparison of the various computational approaches in our study.First, we focus on b-jet observables like the transverse momentum of the hardest b jet and the pair of the two hardest b jets, as shown in Fig. 5.It is evident that, even though the scale uncertainties of the full off-shell prediction are of the order of ±20%, none of the other predictions can reproduce the shape of the spectrum.While the full NWA at least agrees with the off-shell prediction for low transverse momenta p T 200 GeV within scale uncertainties, the other predictions differ by 20% − 40% already in the bulk of the distribution.Towards the end of the plotted spectrum all predictions deviate by 50% from the full off-shell calculation.Missing higher-order corrections are the dominant source of uncertainties, while matching uncertainties stay below 5% for the Powheg-Box they can increase up to 20% − 40% for MG5 aMC@NLO.For the transverse momentum of the system of the two hardest b jets shown on the right hand side of Fig. 5 we observe even stronger deviations.As in the case of the QCD production mode the scale uncertainties for this observable are slightly larger and of the order of 20%−25%.Nonetheless, all predictions diverge from the off-shell result by 65% − 75% at p T ≈ 900 GeV.The estimated matching uncertainties are very different between the Powheg-Box and MG5 aMC@NLO.While the parton-shower starting scale µ Q has a big impact for MG5 aMC@NLO and can affect the shape the distribution by up to 90%, the dependence of the various damping parameters in the Powheg-Box induces differences up to ±10%.
Next, we investigate lepton observables such as the transverse momentum of the hardest same-sign lepton ss 1 and the opposite-sign lepton os as depicted in Fig. 6.These two observables behave very differently.In the case of p T ( ss 1 ) the lepton can either originate from the top-quark decay or from the W boson radiated in the initial state.For the offshell calculation even more topologies are possible as single-and non-resonant contributions are present.However, we find that all predictions generate a very similar shape of the spectrum and differences are mainly due to overall normalizations.To be precise, the full NWA is globally 10%, the NWA with LO decays 20%, and the parton-shower predictions 35% lower than the full off-shell calculation.Scale uncertainties are for all predictions at the level of ±20%.However due to the overall shift in the normalization the parton-shower results barely overlap with the full off-shell results.In addition, matching uncertainties are negligible for both parton-shower predictions.On the other hand, when we look at the transverse momentum of the opposite-sign lepton os we expect larger off-shell effects.This is due to fact that, in the case of the double-resonant t tW ± contribution, the lepton can only originate from a top-quark decay, which is not the case for the full off-shell computation.Indeed, we find that all approximations undershoot the tail of the distribution by 40%−60%.
Comparing the same observable to the QCD production mode as shown in Fig. 4 we notice that the single resonant contribution must be much larger in the EW contribution.As in the previous cases the residual dependence on the renormalization and factorization scales are the dominant uncertainties, which for all predictions are of the order of 22%.Nonetheless, the matching uncertainties become sizable in the tail of the spectrum and can reach up to 13% for the Powheg-Box and 20% for MG5 aMC@NLO.
At last we discuss the transverse momentum and the rapidity of the leading light jet as shown in Fig. 7. First of all we notice that for both observables the NWA reproduces the shape of the full off-shell prediction perfectly.The curves only differ by an overall shift of −10% in the normalization.Additionally, we can infer that NLO QCD corrections to the decay are not important for this observable and that both the full NWA and NWA with LO decays are compatible with the full off-shell prediction within the scale dependence.On the other hand, the parton-shower matched calculations receive large corrections.Especially in the beginning of the transverse momentum spectrum deviations up to 85% are visible.
The uncertainties of the various predictions turn out to be very different.However, the full off-shell calculation has a residual scale dependence at the level of 20%−25%, which is at the same level as for the parton shower based computations.In addition in the Powheg-Box the spectrum is very stable with respect to the various matching parameters with shape modifications below 4%.Thus, in the case of the Powheg-Box the dominant uncertainties are attributed to missing higher-order corrections.Contrary, the MG5 aMC@NLO curve depends crucially on the choice of the initial shower scale µ Q and can alter the tails of the distribution by more than 100%, making the matching uncertainties the dominant source of uncertainties.For the rapidity of the hardest light jet, shown on the right of Fig. 7 we also find large parton-shower corrections.While the leading jet in fixed-order perturbation theory is constrained to the very forward region, the inclusion of additional radiation in the parton-shower evolution fills the gap in the central rapidity region.Thus, parton-shower matched calculations predict significantly more events in the central regions with nearly 200% corrections.Still, the Powheg-Box and MG5 aMC@NLO predictions are very different.Similar effects for these two observables have been already observed in Ref. [36] in the case of an on-shell t tW ± signature.
V. COMBINED QCD AND ELECTROWEAK PRODUCTION OF t tW ± In the following we provide combined results including the QCD and EW production modes discussed in section IV and propose a way to improve theoretical predictions in the absence of fully exclusive NLO parton-shower event generators based on full off-shell In the previous sections we have seen that the theoretical predictions with parton-shower effects include many kinematical features that are not accessible to a fixed-order NLO computation.However, while comparing the various approaches to the full off-shell computation, we observed that the single-resonant contributions start to dominate in some phase-space regions and give rise to sizable corrections in the tail of dimensionful observables.Naturally the question arises how one can combine these two different theoretical predictions that describe the same observables but differ in the amount and kind of physical effects taken into account.In the case at hand we would like to supplement a NLO parton-shower matched computation with full off-shell effects.Our method of choice is the following additive combination where fixed-order full off-shell effects are simply added to a parton-shower based computation.Let us comment on the choice of subtraction used in defining the d∆σ off-shell /dX contribution.The subtraction of the full NWA removes approximately the double counting between the double-resonant contributions in the full off-shell and the NLO + PS computation.Strictly speaking one should subtract the NLO + PS result expanded to the same order in α s as the full off-shell computation.This would include parton-shower specific contributions due to parton-shower radiation in the top-quark decays.We approximate this additional parton-shower radiation by subtracting the full NWA with NLO top-quark decays instead.We note, that our differential corrections d∆σ off-shell /dX are independent of the parton-shower event generator employed.In addition, d∆σ off-shell /dX provides differential corrections for the single and non-resonant contributions as well as interference effects in an approximate way.To estimate the theoretical uncertainty of our improved predictions we compute the scale variations and matching uncertainties independently and combine them by adding them in quadrature where δ ∆σ scale is the estimated uncertainty of d∆σ off-shell /dX and computed from correlated scale variations.Finally, all contributions to δ th are symmetrized.
Finally, given the good agreement we found between the Powheg-Box and MG5 aMC@NLO in the course of our study, and considering the independence of the d∆σ off-shell /dX of the parton shower used, we will show in the following results for the Powheg-Box only.
Incidentally, if theoretical predictions including multi-jet merging are available then the aforementioned differential corrections can also be included.

Integrated fiducial cross sections
In Tab.I we present results for the integrated fiducial cross section.We observe that for the combined QCD+EW result including the ∆σ correction increases the NLOPS cross section by 2%.On the other hand, looking at the individual QCD and 10% − 15%, depending on the computational approach employed, at the beginning of the spectrum, while they decrease to 3% − 7% at the end of the spectrum.For the invariant mass spectrum of the same-sign lepton-pair the EW channels contribute a rather constant 10% − 12% correction over the whole spectrum.In this case we do not observe sizable improvements when NLOPS+∆σ is used instead of NLOPS.
At last, we study the differential distributions of the H T observable and p miss T as shown in Fig. 10.Let us remind the reader, that the H T observable is defined as where only the two hardest b jets are included.For the H T observable shown on the left of Fig. 10 we observe that the NLOPS+∆σ predictions are in better agreement with the offshell calculation.While the bulk of the distribution is dominated by parton-shower effects, the tail receives large single-resonant contributions of the order of 40%.As in all previous cases the full off-shell result obtains slightly larger EW contributions than in the case of NLOPS predictions.On the other hand, the p miss T observable only shows small corrections from single and non-resonant contributions below 12%.The NLOPS+∆σ prediction is fully compatible with the original NLOPS result within the estimated uncertainties.For the pure NLOPS prediction the EW production mode provides a nearly constant correction of 10%, while in the case of the full off-shell result the corrections are slightly larger with +14% at the beginning of the distribution.
calculation to parton showers is necessary.
Moving forward, we think that we could already learn a great deal by a detailed comparison of t tW ± unfolded data with the level of theoretical predictions presented in this paper.
Beyond that, a fixed-order calculation of NNLO QCD corrections to the on-shell t tW ± process could help removing some of the theoretical uncertainty in both scale-dependence and hard-radiation modeling.Equally important and equally challenging would be interfacing the full off-shell fixed-order calculation with parton-shower event generators.

FIG. 1 .
FIG. 1. Representative Feynman diagrams of the double-, single-, and non-resonant top-quark contributions that are part of the matrix elements employed in the full off-shell computation.
the differential level we start by comparing hadronic b-jet observables.All observables are shown as plots containing three panels.The upper panel always depicts the central differential distribution (i.e. the distribution obtained using the central values of both scale and matching parameters) for the various predictions employed in our study.The middle panel illustrates the scale uncertainty bands stemming from independent variations of factorization and renormalization scales.All curves are normalized to the central prediction of the off-shell calculation and for ease of readability we do not show uncertainty bands for the predictions based on the NWA.Finally, the bottom panel shows the matching uncertainties for the parton-shower based predictions which are estimated by a variation of the initial shower scale µ Q in the case of MG5 aMC@NLO and of the various damping parameters in the case of the Powheg-Box.

2 pFIG. 2 . 2 MFIG. 3 .
FIG. 2. Differential cross section distribution in the 3 fiducial region as a function of the transverse momentum of the hardest (l.h.s.) and the second hardest (r.h.s) b jet for the pp → t tW ± QCD process.The uncertainty bands correspond to independent variations of the renormalization and factorization scales (middle panel) and of the matching parameters (bottom panel).

Both observables are shown in Fig. 4 .FIG. 4 .
FIG.4.Differential cross section distribution in the 3 fiducial region as a function of the transverse momentum of the opposite-sign lepton os (l.h.s.) and the H lep T observable (r.h.s) for the pp → t tW ± QCD process.The uncertainty bands correspond to independent variations of the renormalization and factorization scales (middle panel) and of the matching parameters (bottom panel).

FIG. 5 .
FIG. 5. Differential cross section distribution in the 3 fiducial region as a function of the transverse momentum of the hardest b jet (l.h.s.) and of the system of the two hardest b jets (r.h.s) for the pp → t tW ± EW process.The uncertainty bands correspond to independent variations of the renormalization and factorization scales (middle panel) and of the matching parameters (bottom panel).

pFIG. 6 .
FIG.6.Differential cross section distribution in the 3 fiducial region as a function of the transverse momentum of the hardest same-sign lepton ss 1 (l.h.s.) and the opposite-sign lepton os (r.h.s) for the pp → t tW ± EW process.The uncertainty bands correspond to independent variations of the renormalization and factorization scales (middle panel) and of the matching parameters (bottom panel).

FIG. 7 .
FIG. 7. Differential cross section distribution in the 3 fiducial region as a function of the transverse momentum (l.h.s.) and the pseudorapidity (r.h.s) of the leading light jet for the pp → t tW ± EW process.The uncertainty bands correspond to independent variations of the renormalization and factorization scales (middle panel) and of the matching parameters (bottom panel).

FIG. 8 . 2 MFIG. 9 .
FIG. 8. Differential cross section distribution in the 3 fiducial region as a function of the transverse momentum of the hardest b jet (l.h.s.) and of the system of the two hardest b jets (r.h.s) for the pp → t tW ± QCD+EW process.The uncertainty bands correspond to scale variations (off-shell) and total uncertainties (NLOPS) (middle panel).The differential impact of the EW contribution is shown as well (bottom panel).

FIG. 10 .
FIG.10.Differential cross section distribution in the 3 fiducial region as a function of the total transverse momentum H T (l.h.s.) and the missing transverse momentum (r.h.s) for the pp → t tW ± QCD+EW process.The uncertainty bands correspond to scale variations (off-shell) and total uncertainties (NLOPS) (middle panel).The differential impact of the EW contribution is shown as well (bottom panel).

TABLE I .
Integrated fiducial cross sections for the pp → t tW ± process for various approaches.We use the Powheg-Box to obtain the NLOPS results.