Anatomy of inclusive $t\overline{t}W$ production at hadron colliders

In LHC searches for new and rare phenomena the top-associated channel $pp \to t\overline{t}W^\pm +X$ is a challenging background that multilepton analyses must overcome. Motivated by sustained measurements of enhanced rates of same-sign and multi-lepton final states, we reexamine the importance of higher jet multiplicities in $pp \to t\overline{t}W^\pm +X$ that enter at $\mathcal{O}(\alpha_s^3\alpha)$ and $\mathcal{O}(\alpha_s^4\alpha)$, i.e., that contribute at NLO and NNLO in QCD in inclusive $t\overline{t}W^\pm$ production. Using fixed-order computations, we estimate that a mixture of real and virtual corrections at $\mathcal{O}(\alpha_s^4\alpha)$ in well-defined regions of phase space can arguably increase the total $t\overline{t}W^\pm$ rate at NLO by at least $10\%-14\%$. However, by using non-unitary, NLO multi-jet matching, we estimate that same these corrections are at most $10\%-12\%$, and at the same time exhibit the enhanced jet multiplicities that are slightly favored by data. This seeming incongruity suggests a need for the full NNLO result. We comment on implications for the $t\overline{t}Z$ process.


Introduction
The discovery [1,2] of the pp → ttW process, and likewise pp → ttZ, is an important milestone of the Large Hadron Collider's (LHC's) Standard Model (SM), Higgs, and New Physics programs. In its own right, ttW, which at lowest order proceeds at O(α 2 s α) through the diagrams in figure 1, is a multi-scale process with large quantum chromodynamic (QCD) and electroweak (EW) corrections. Hence, it is a laboratory for stress-testing the SM paradigm. At the same time, the ttW ± → W + W − W ± bb decay mode can give rise to the samesign dilepton ± i ± j and trilepton i j k signal categories, encumbering [3,4,5,6] searches for lepton number and lepton flavor violation as well as measurements of the Higgs's couplings.
The findings are telling: Known QCD corrections increase total LHC rates by 20% − 60%, depending on theoretical inputs, and reflect the 15%−85% scale ambiguity at leading order (LO). Even at this level, however, typical scale choices leave an 10% − 30% uncertainty, suggesting additional corrections are needed to ensure theoretical control. While EW corrections increase rates by a net 5%, uncertainties essentially stay the same. In comparison to data, a consistent picture has also emerged: Whereas first observations of the ttZ process by the ATLAS and CMS collaborations at √ s = 8 TeV were within SM expectations at NLO in QCD, both collaborations measured a ttW rate exceeding predictions at the 68% (1σ) confidence level (CL) [1,2]. Measurements at √ s = 13 TeV with up to L ≈ 36 fb −1 support a ttW rate that is 15% − 50% larger than predictions at NLO in QCD at about the same CL tension [31,32,33]; a modest ttZ rate increase of 15% is also preferred [31]. Improved measurements with L ≈ 80 fb −1 [34] and L ≈ 140 fb −1 [35,36] affirm a ttW rate that, depending on the signal category, is 25%−70% larger than predictions at NLO in QCD and EW, and corresponds to a 1.4σ − 2.4σ discrepancy. Given the sustained nature of these and other multilepton excesses, which include a diverse number of final states that are dominated by several SM processes, it is reasonable to contemplate seriously the possible role of new physics [37,38,39,40]. That said, it is also necessary to investigate each anomaly separately to understand the possible importance of missing higher order corrections.
To support such investigations and to exhaust possible SM explanations we have reexamined the role of the ttW ± j and ttW ± j j sub-processes at O(α 3 s α) and O(α 4 s α) in inclusive ttW production. While complementary works have studied the phenomenology of these channels [4,15,17], the impact on the inclusive cross section were not among their intents.
As a first step we use fixed order computations and find that a subset of well-defined real and virtual contributions at O(α 4 s α), i.e., finite elements to inclusive ttW production at next-to-nextto-leading order (NNLO) in QCD, are positive and reach at least 10% − 14% of the ttW ± rate at NLO. Interestingly, we find that these same contributions only increase the ttW rate by at most 10% − 12% when using then the non-unitary, NLO multi-jet matching procedure FxFx [41]. Despite this seeming discrepancy, we report that after imposing selection cuts and signal categorizations employed LHC experiments, the FxFx results exhibit enhanced light and heavy jet multiplicities that are slightly favored by data. To resolve this enigma, we argue a need for the full NNLO in QCD description of inclusive ttW production.
The report of our investigation continues as follows: After summarizing our computational setup in section 2, we build up the anatomy of inclusive ttW production at hadron colliders in section 3. There we estimate higher order corrections to ttW production and discuss theoretical uncertainties. In section 4, we show how rate increases propagate to differential observables and survive analysis cuts. In section 5, we present on outlook for the ttZ process. We conclude in section 6.

Computational Setup
To conduct our study we employ a state-of-the-art simulation tool chain based on Monte Carlo methods. For matrix element evaluation and parton-level event generation, we use MadGraph5 aMC@NLO (v2.6.7) [15] (mg5amc). Its conglomeration of packages [42,43,44,45,46,47] enables us to simulate high-p T hadron collisions in the SM up to NLO in QCD with PS-matching within the MC@NLO formalism [42]. We model decays of heavy resonances using the spin-correlated narrow width approximation [45,46]. Parton-level events are passed through Pythia8 (v244) [48] for QCD and QED parton showering, hadronization, and modeling the underlying event. We use the FxFx prescription [41] as implemented in mg5amc. Parton-level sequential clustering is handled according to k T -class algorithms [49,50,51] as implemented in FastJet [52,53]. To compare against ATLAS ttW results at √ s = 13 TeV [34], events are processed with DELPHES3 (v3. 4.2) [54] to model detector resolution. We assume most of the default settings for the ATLAS detector card. However, to better mimic ATLAS's analysis we employ updated lepton and b-tagging efficiencies [55,56,57].
Throughout this analysis, we work in the n f = 4 active quark flavor scheme with SM inputs as set by the mg5amc module loop sm. We do so for a more realistic description of massive B hadrons decays, and particularly charged lepton multiplicities. We tune the top and Higgs masses to m t (m t ) = 172.9 GeV and m H = 125.1 GeV.
3. Anatomy of inclusive ttW production at the LHC It may be that sustained measurements [31,32,33,36,34,35] of a ttW cross section that is larger than expectations at NLO in QCD and EW is due to new physics. As such, we find it compelling to exhaust SM explanations for these observations. In this context, we review in section 3.1 the modeling and uncertainties of ttW production at various orders in perturbation theory. Motivated by our findings, we turn our focus in section 3.2 to the ttW j and ttW j j subprocesses and their roles in the inclusive channel. We then present in section 3.3 our estimations for the ttW production rate at the level of NLO multi-jet matching. Differential results are presented in section 4.

State-of-the-art modeling for inclusive production
Categorically, the anatomy of inclusive ttW production consists of several pieces and nuances. To start: at lowest order, i.e., at O(α 2 s α), cross sections at the √ s = 13 TeV LHC span σ LO ttW ∼ 375 fb − 525 fb, depending on choices for PDF and µ f , µ r . The normalization of α s (µ r ) heavily influences the outcome. For static scale choices of µ f , µ r ∼ O(m t ), 3-or 9-point scale variation reveals an ambiguity of 25%−35% [11]. For typical dynamic choices, such as equation 2, one finds comparable uncertainties of 20%−30% but rates that are about 25% smaller [15,16,4]. The same holds for static choices of O(2m t + M W ), indicating that the threshold and kinematic scales are similar.
At NLO in QCD, contributions at O(α 3 s α) improve the picture dramatically. Due to the opening of (qg)-scattering, cross sections jump by 20% − 50%, again depending on inputs, to σ NLO ttW ∼ 485 fb − 645 fb [4,11,15,16,20]. Excluding scale choices that favor values far below O(2m t + M W ), rates more moderately span σ NLO ttW ∼ 485 fb − 595 fb. However, the range does not reflect scale variation, which spans only 10% − 15%, and suggests that further high-order corrections are needed to ensure theoretical control. This is partly due to the (qg)scattering channel, which at this level only is described at LO.
Beyond leading contributions at O(α 2 s α) and O(α 3 s α), it is now known [26,27,28] that "supposedly" sub-leading EW contributions at the Born level, i.e., at O(α 3 ), and at NLO, i.e., at O(α 2 s α 2 ), O(α s α 3 ), and O(α 4 ), are not negligible in comparison to the above uncertainty budget. Cancellations among virtual EW diagrams, interference between mixed EW-QCD and pure EW diagrams, real radiation, and the opening of tW → tW scattering, culminate in a positive contribution to ttW production that is about 6% of the rate at NLO in QCD [28]: Despite these improvements, dynamic scale variation at this order remains about the same as at NLO in QCD.
First attempts to extract two-loop predictions through resuming soft gluon radiation in the qq → ttW channel up to NNLL yield positive corrections [20,21,22,23,24,25]. Depending on scale inputs, these range 1% − 7% and reduce slightly both the scale uncertainty and range of predictions [20].
In one 1 detailed comparison to data [34], measurements of the ttW cross section by ATLAS with L ≈ 80 fb −1 at √ s = 13 TeV find category-based signal strengths that are 25%−70% larger than SM expectations. Relative to the SM prediction of σ ATLAS−TH. ttW = 727 fb +13% −13% , the measurements indicate a best-fit rate and signal strength of [34], which corresponds to a 2.4σ discrepancy. Importantly, the prediction is built from an established [5] reference point with leading QCD and EW corrections, σ ref.
NNLO−QCD = 1.11 for NNLO in QCD contributions. While seemingly innocuous, the estimate of K est. NNLO−QCD is based on the observation [15] that the pp → ttW j process exhibits a large, O(40%) correction at NLO in QCD for a specific set of inputs. However, neither Ref. [15] nor follow-up work [4] evaluate the ttW rate beyond O(α 3 s α). Therefore, owing to the uncertainty in K est.
NNLO−QCD , we turn our focus to the roles of the ttW j and ttW j j subprocesses in inclusive ttW production.

The ttWj and ttWjj processes
To investigate the pp → ttW j and pp → ttW j j processes, we argue first for definitions of these channels that ensure their matrix elements (MEs) are perturbative in the CSS sense [60,61], i.e., are absent of large collinear logarithms. Such logarithms originate from real radiation that goes soft or collinear and require sufficiently stringent transverse momentum cuts (p j min T ) to render MEs physical. For a (ttW)-system invariant mass of M ttW , these contributions cause cross sections to scale as For too small p j min T one enters the Sudakov regime where log factors exceed 1/α s factors and k T -resummation is needed.
To establish a sufficiently "safe" p j min T , we consider at √ s = 13 TeV the pp → ttW process at NLO in QCD, i.e., up to O(α 3 s α), and the LO decay to charged leptons, 1 Other measurements at √ s = 13 TeV [31,32,33,36,34,35] show similar disagreements but provide fewer modeling details.  Here and throughout we choose the dynamic scale scheme of equation 2 following studies [62,63] of the pp → W + n j process. There, schemes not reflective of kinematic scales were shown to lead to negative cross sections at NLO. For the inclusive ttW process at NLO (LO) in QCD, we obtain (7) as our baseline NLO (LO) cross sections. Uncertainties reflect scale and PDF dependence, respectively.
After parton showering, anti-k T clustering (R = 0.4), and overlap removal between leptons and jets, we plot in figure 2 the normalized p T distribution of the leading light jet (solid). As a check of our computational setup, we also plot the leading (light dash) and sub-leading (dark dash) b-jet. The b-jet distributions reflect the characteristic momentum p T ∼ m t (1 − M 2 W /m 2 t )/2 ∼ 65 GeV − 70 GeV, modulo recoils against W and light jets.
Since the ttW system is fully decayed to leptons, the leading light jet is an O(α 3 s α) contribution that originates in the MC@NLO formalism [42] from (i) the tree-level ttW j ME at "large" (p j T /M ttW ) or (ii) PS corrections to the one-loopimproved ttW ME at "small" (p j T /M ttW ). Using the procedure in Ref. [64], which generalizes an analogous procedure in Ref. [60], we estimate that the transition between the two domains occurs at around p safe T ∼ 30 GeV for M ttW ∼ 425 GeV − 475 GeV. Notably, this estimate neglects β 0 factors in α s running. Accounting for this we obtain instead p safe T ∼ 95 GeV − 100 GeV. In comparison to figure 2 one sees that the transition occurs somewhere 2 between the Sudakov peak at p T ∼ 15 GeV − 20 GeV and p T ∼ 75 GeV − 100 GeV, with an with scale and PDF uncertainties, the QCD K-factor and the difference between cross sections at NLO and LO which quantifies O(α 4 s α) contributions. For p j min T = 30 GeV − 150 GeV, NLO rates span σ NLO ttW j ∼ 100 fb − 350 fb, in agreement with Refs. [4,15]. We report that scale uncertainties are uniform across p j min T and reduce from 30%−40% at LO to 15% at NLO, suggesting stability for p j min Importantly, ∆σ encapsulates corrections to ttW j for a leading jet j 1 that is well-defined in the CSS sense; it does not reflect  corrections for when j 1 is in the soft-wide angle limit. Extrapolating from table 1 hints that such corrections remain positive and that the K est.
NNLO−QCD in the previous section underestimates NNLO in QCD corrections by at least a factor of two. Isolating this, however, is complicated by the phase space region where j 1 is hard but collinear to the beam line, which cancels against negative-valued PDF/collinear counter terms at NNLO.
The second observation is that QCD corrections to the ttW j process are large, with K-factors ranging K QCD ∼ 1.7 for p j min T = 75 GeV − 150 GeV, and with the largest (smallest) p j min T exhibiting the largest (smallest) corrections. In analogy to ttW, it is possible that new partonic channels associated with the ttW j j sub-process at O(α 4 s α) drive these increases. To check this we consider at LO, i.e., O(α 4 s α), the channel and list in table 2 the √ s = 13 TeV cross sections [fb] with uncertainties [%], for benchmark p j k min T on the leading ( j 1 ) and sub-leading ( j 2 ) jets, and with |η j | < 4.0. Also shown is the decomposition according to initial/final-state partons.
For p j k min T = 75 GeV, 100 GeV, we find that rates span σ LO ttW j j ∼ 20 fb − 35 fb, with up to 60% scale uncertainty and about a 1% PDF uncertainty. This translates to about 5% − 9% (3% − 6%) of the baseline ttW rate at (N)LO. For all cases, quark-gluon scattering (Q, g) accounts for about 70% of the total ttW j j rate, whereas quark-quark scattering (Q, Q) contributes 20%. This effectively rules out enhancements to ttW production at NNLO from valence-valence scattering. Instead, we find evidence of a large gluon-valence component, with 60% of the ttW j j rate being due to gluon-up/down scattering (g, q V ).
Returning to ∆σ in equation 10, we recall that it encapsulates the O(α 4 s α) parts of ttW j production at NLO in QCD. Given then the LO ttW j j rate, we can estimate the net impact of the soft-real, virtual, and counter-term corrections (SR+V+CT) to ttW j by taking the difference of the two: , which introduces logarithmic structures not captured in equation 5 [66,67].) Importantly, the interplay in equation 13 highlights that adding ∆σ ttW j at low p j min T to inclusive ttW should be accompanied by a reweighting / subtraction scheme that systematically protects against double counting of low-p T radiation.
In summary, we report the existence of partonic configurations at O(α 4 s α), i.e., pure O(α 2 s ) corrections to inclusive ttW production, with cross sections in well-defined phase space regions that greatly exceed estimates by standard scale variation at NLO in QCD. By one measure (table 1) a subset of these corrections span at least ∆σ ∼ 65 fb−85 fb, or about 10%−14% of the inclusive rate at NLO in QCD, and follows from a mixture of gluon-valence scattering in ttW j j at LO (table 2) and unresolved radiation in ttW j at NLO (equation 13). While alone accounting for 20%−30% of the discrepancy in measured the ttW rate at √ s = 13 TeV (see, for example, equation 4), other contributions at O(α 4 s α), such as two emissions of soft, wide-angle legs and one soft leg at one-loop, which are arguably positive, are not included. Importantly, resummed results at NNLL do not suggest large cancelations against pure two-loop diagrams. Guided by this, we turn to the impact of combining the ttW and ttW j processes using NLO multi-jet matching.

Inclusive production with NLO multi-jet matching
As a complementary estimate of the O(α 4 s α) contributions to ttW production that stem from the ttW + n j sub-channels, we employ the FxFx NLO multi-jet matching [41]. In short, FxFx is an established [68,69,70,71], non-unitarity [41,72] procedure within the MC@NLO formalism for promoting jet observables at LO+LL to NLO+LL through CKKW-like [73] reweighting. In particular, hard, wide-angle emissions are included through exact MEs at one-loop and double counting is avoided by Sudakov reweighting. As such, cross sections at NLO are augmented with terms that are O(α 2 s ) or higher. The cost of this improvement is the introduction of a merging scale (Q FxFx cut ) akin to those at LO. To set Q FxFx cut we follow Refs. [15,41,72], which call for Q FxFx cut > 2p j min T . In principle, FxFx merging is independent of p j min T and ultra-low p j min T choices simply lead to high event-veto rates and therefore poorer Monte Carlo efficiency. Formally, however, the Sudakov-reweighting in FxFx only cancels the collinear logarithms that are shared by MEs and the PS; for sufficiently small p j min T , mis-cancellations of soft logarithms can technically spoil perturbative convergence. Therefore, as a jet p T threshold is needed to regulate Born-level ttW + n j MEs, we also require that p j k min T is not too low in the CSS sense and that |η j | < 4.0. We match up to the first jet multiplicity (denoted as FxFx1j) and set as our baseline configuration At this order we obtain as the inclusive ttW cross section where the uncertainties reflect scale and PDF dependence, respectively. We report that corrections at this order increase the baseline NLO rate in equation 7 by about 10% with the size uncertainties remaining essentially the same. This is in agreement with the estimated [34] K est. NNLO−QCD = 1.11, and hence is at odds with table 1, which suggests that such O(α 4 s α) contributions are larger.
Accounting now for the EW corrections in equation 3 [28] and assuming the FxFx uncertainties above, we obtain We find that this rate is about 5% smaller than the prediction used in the ATLAS measurement of Ref. [34], and that the difference is mainly due to the scale and PDF choices in the baseline NLO in QCD rate. (A 1% difference follows from our FxFx correction being smaller than the estimated NNLO Kfactor.) In principle, this revised cross section worsens slightly the discrepancy reported in equation 4. For the pure FxFx and FxFx+EW cases, the corresponding best-fit signal strengths arê and are consistent with SM expectations at 2.7σ − 3.0σ. While NNLL threshold corrections can improve this picture, direct application of Ref. [20] is hindered by the different scale choices that we assume. That said, taking a comparable correction of K est.
NNLL−QCD = σ NLO+NNLL−QCD /σ NLO−QCD = 1.03, the associated SM rate and best-fit signal strength are: where we again assume the FxFx uncertainties. With these estimated corrections the discrepancy stays at 2.7σ.
To explore the uncertainty associated with our baseline (p j min T , Q FxFx cut ), we report in the upper panel of table 3 the inclusive ttW cross section at √ s = 13 TeV at LO, NLO in QCD, and with FxFx1j matching for various inputs. Also shown are the µ f , µ r scale and PDF uncertainties, and the QCD K-factor as defined in equation 9. We denote our benchmark rate by †.
For the inputs considered we observe that NLO multi-jet matching rates span about σ FxFxj1 ∼ 600 fb − 670 fb. This is about 0% − 12% larger than the baseline rate at NLO in QCD in equation 7. Notably, the range of ∆σ FxFxj1 ∼ 70 fb is much smaller than the differences in the ttW j rate at NLO, which span ∆σ ∼ 250 fb (see table 1). Naïvely, this indicates a sizable phase space overlap between the ttW and ttW j processes at NLO in QCD. However, this appears contrary to figure 2, which shows that the characteristic light jet scale in inclusive ttW is well below p j T = 50 GeV, and otherwise suggests a much milder phase space overlap for p j min T 50 GeV − 75 GeV. As expected [41,72], we find that the range of FxFx1j predictions is driven by the dependence on the merging scale more than the jet p T threshold. In particular, for the largest Q FxFx cut considered the FxFx1j rate reduces to the NLO rate and can be tied to an "over suppression" of the ttW j multiplicity [74]. To quantify an uncertainty associated with Q FxFx cut , we consider the envelope spanned by all FxFx predictions. For and vary p j min T . For all cases, we find that FxFx rates change only about ∆σ FxFxj1 [p j min T ] ∼ 1 fb − 10 fb, or 0.1% − 1.5%.

Differential Production
Assuming that the enlarged ttW cross section measurements are solely due to missing QCD corrections, then NLO multi-jet matching cannot be the full picture. At the same time, differences in initial/finite-state radiation, associated production of heavy flavors, and relative enhancements by virtual radiation all impact particle kinematics. Hence, complementary to the total rate itself, kinematic distributions provide a means to test and understand the modeling of inclusive ttW production.
A comprehensive investigation into the impact of NLO multijet merging on particle kinematics is beyond our present scope and left to future work. That said, to at least build a qualitative picture, we take as a benchmark the ATLAS analysis [34] associated with the signal strength in equation 4 and consider the ttW decay mode, Here the two same-sign W bosons decay leptonically and the odd-sign W decays inclusively. Following closely the selection criteria of table 3 in Ref. [34], then after selection cuts and vetoes, three signal categories are defined: • The "inclusive selection category" is identified as two same-sign, high-p T charged leptons , two reconstructed light jets, and at least one b-tagged jet.  • The "two same-sign leptons" (2lSS) category assumes category (i) but vetos events with three or more .
• The "three leptons" (3l) category again assumes (i) but requires exactly three with a net charge of ±1.
In figure 3, we plot the differential cross sections at √ s = 13 TeV for representative observables and signal categories at NLO in QCD with PS-matching (black) and FxFx-matching (blue). In the insets we plot the ratio of the FxFx and NLO+PS rates. Uncertainty bands are built from the envelope encapsulating the 27-point µ f , µ f , µ s variation and 1σ PDF uncertainty. Starting with figure 3(a), we show the b-jet multiplicity (N b− jets ) in the inclusive selection category. As anticipated from its larger cross section, we find that the normalization of the FxFx distribution is systematically larger than the NLO+PS one by at least 10%. More specifically, the bin-by-bin normalization grows to about 13% for N b− jets = 2 and 30% for N b− jets = 3, and stems from the opening of g * → bb splitting in qq →ttW g * production. While not shown, we report a slight suppression (enhancement) of low (high) light jet multiplicities relative to NLO+PS. Notably, an enhanced rate at high multiplicities for both heavy and light jets is slightly favored by data [34,35].
For the same signal category, we show in figure 3(b) the p T of the same-sign dilepton system (p T ). Over the range p T = 0 GeV − 200 GeV, we observe a slowly increasing binby-bin shift in the FxFx normalization relative to the NLO+PS normalization, ranging from about a 10% enhancement to about  20%. We attribute the growing FxFx rate with increasing p T to the larger hadronic activity that the dilepton system recoils against, i.e., the positive contributions from the O(α s ) corrections to the ttW j sub-channel.
Focusing now on more exclusive signal regions, we plot in figures 3(c) and 3(d), respectively, the invariant mass (m ) of the same-sign dilepton system for the 2lSS and 3l categories. For both categories we observe a qualitatively similar but quantitatively stronger trend than found for p T . Numerically, enhancements grow from about 10% to just over 30%, with the importance of NLO multi-jet matching increasing for larger m . We argue that the increases at larger m is due to additional final-state radiation off top quarks in the ttW j subchannel at NLO. Such radiation imbue top quarks with recoil momentum that propagate to charged leptons. This in turn leads to larger lepton momenta and thus larger invariant masses.

Outlook
Due the large impact of QCD radiation at O(α 4 s α) on the ttW process we are compelled to consider implications for other  Figure 4: Central cross section predictions at √ s = 13 TeV with 1σ uncertainty bands for the ttW and ttZ processes at NLO in QCD+EW (light) as well with FxFx matching (dark). Also shown are best-fit measurements from the ATLAS [32,34] and CMS [36] experiments.
scenarios. This includes, for example, inclusive ttZ production at the LHC, a situation that we now explore.

Inclusive ttZ production at the LHC
The SM's SU(2) L gauge symmetry dictates that ttW and ttZ production are intimately related. However, due to differences in gauge quantum numbers, inclusive ttZ production occurs via different partonic channels, especially at lowest order. Hence, ttZ possesses a different sensitivity to QCD corrections.
To investigate these differences we repeat the work of section 3 and report in the lower panel of table 3 the cross sections for the ttZ rate at O(α 2 s α) (LO), up to O(α 3 s α) (NLO), and FxFx matching up to the first jet multiplicity (FxFx1j) for various inputs. Also reported are the scale and PDF uncertainties, as well as the QCD K-factor. We observe that QCD corrections generally impact the ttZ process in a comparable manner to the ttW process. More specifically, the NLO and nominal ( †) FxFx1j ttZ rates possess the K-factors K ttZ QCD = 1.48 and 1.12, respectively. In comparison, for ttW one finds respectively K ttW QCD = 1.57 and 1.10. For other FxFx inputs we find comparable differences as in ttW but note that the ttZ FxFx1j rate suffers a slightly milder dependence on p j min T than ttW. To summarize our findings, we plot in figure 4 the cross section predictions with uncertainties at √ s = 13 TeV for the ttW and ttZ processes at NLO in QCD+EW (light) and FxFx+EW (dark). (For ttZ, we use the EW K-factor K NLO−EW = 0.98 [27].) For theoretical uncertainties, we combine scale and PDF uncertainties in quadrature. Also shown is the error-weighted combination of best-fit results from ATLAS [32,34] and CMS [36]. Unlike the ttW case, we see appreciable improvement in the agreement between the predicted and measured ttZ rate. However, like the ttW case, there remains a sizable theory uncertainty that prevents a more significant statement.

Summary and Conclusion
In light of sustained measurements of an enhanced ttW ± cross section at the LHC, we report a systematic investigation into the role of the ttW j and ttW j j sub-processes in inclusive ttW production. We focus particularly on their impact on total and differential rate normalizations.
To conduct this study we revisited (see section 3.1) the stateof-the-art modeling for inclusive ttW production and took special note of estimated NNLO in QCD corrections that are employed in LHC analyses. Using LO and NLO in QCD computations, we then examined (see section 3.2) the ttW j and ttW j j processes. We report that a subset of real and virtual contributions at O(α 4 s α) in well-defined regions of phase space are positive and can arguably increase the inclusive ttW rate at NLO by at least 10% − 14%. Resummed results at NNLL do not suggest large cancelations against pure two-loop diagrams.
Interestingly, using instead the non-unitary, NLO multi-jet matching scheme FxFx, we find (see section 3.3) that these same QCD corrections at O(α 4 s α) increase the inclusive ttW cross section at NLO by at most 10% − 12%. We obtain a slightly smaller central normalization for the inclusive ttW production rate than used in LHC analyses, which in turn worsens slightly reported discrepancies. At the same time, after selection cuts and signal categorization, the FxFx description of ttW exhibits enhanced jet multiplicities that are slightly favored by data (see section 4). Our main results are summarized in figure  4. There we compare FxFx-improved cross sections for the ttW and ttZ processes to measurements at √ s = 13 TeV. In conclusion, while production rates at the FxFx1j level do not obviously resolve existing tension between SM predictions and LHC measurements of the pp → ttW + X process, the magnitude residual scale uncertainties, the moderate dependence on matching inputs, and the discrepancy with numerical estimates of O(α 2 s ) corrections motivate the need for a full description of inclusive ttW and ttZ production at NNLO in QCD. that is supported by the National Research Foundation and the Department of Science and Innovation, and the Research office of the University of the Witwatersrand. RR is supported under the UCLouvain fund MOVE-IN Louvain and acknowledge the contribution of the VBSCan COST Action CA16108.
Computational resources have been provided by the supercomputing facilities of the Université catholique de Louvain (CISM/UCL) and the Consortium desÉquipements de Calcul Intensif en Fédération Wallonie Bruxelles (CÉCI) funded by the Fond de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under convention 2.5020.11 and by the Walloon Region.