NLO electroweak and QCD corrections to off-shell ttW production at the LHC

The foreseen luminosities for the next LHC runs will enable precise differential measurements of the associated production of top-antitop pairs with a W boson. Therefore, providing accurate theory predictions for this process is needed for realistic final states. We present the first combination of NLO electroweak and QCD corrections to off-shell ttW$^+$ production in the three-charged-lepton channel, including non-resonant effects, full spin-correlations and interferences. Such radiative corrections comprise the electroweak and QCD corrections to the leading QCD order, and the QCD corrections to the leading electroweak order.


Introduction
The associated production of top-antitop pairs with W bosons represents one of the heaviest signatures at the LHC and an important process to study, both as a probe of the Standard Model (SM) and as a window to new-physics effects. It gives direct access to the coupling of top quarks to electroweak (EW) bosons in or beyond the SM [1][2][3] and it is expected to enhance the sensitivity to the tt charge asymmetry [4]. This process is also a relevant background to the associated tt production with a Higgs boson [5]. Recent ATLAS and CMS public results [6,7] and improved analyses [8,9] for ttW point in the direction of a tension between data and SM predictions, which has not been addressed yet in spite of strong efforts in the theoretical community. Since the high-luminosity run of the LHC will allow for differential measurements in ttW, it is essential to provide precise SM predictions for this process in specific decay channels. The next-to-leading-order (NLO) QCD and EW corrections are known since several years for the inclusive production [4,5,[10][11][12]. Softgluon resummation [13][14][15][16][17] and multi-jet merging [18,19] have been performed for inclusive production. The decay modelling has been tackled in the narrow-width approximation (NWA) at NLO QCD [20] and with matching to parton shower [21]. The subleading NLO QCD corrections to the LO EW have been computed in the NWA, including spin correlations and parton-shower effects [22,23]. The first predictions for off-shell ttW production in the three-charged-lepton α 2 s α 6 α s α 7 α 8 α 3 s α 6 α 2 s α 7 α s α 8 α 9 Figure 1: Perturbative orders contributing at LO and NLO to ttW production in the threecharged-lepton decay channel. Figure 2: Sample diagrams contributing to LO QCD (left) and to LO EW (right) crosssections for off-shell ttW + production in the three-charged-lepton decay channel.
channel have appeared recently [24][25][26] and have been also compared to NWA results matched with parton-shower [27]. Based on Ref. [28], we present the first complete fixed-order description of the off-shell production of ttW + in the three-charged-lepton decay channel, combining NLO QCD and EW corrections which are sizeable at the LHC@13TeV.

Description of the calculation
We consider the process pp → e + ν e τ + ν τ µ −ν µ bb + X . At LO exclusively quark-induced partonic channels contribute, while at NLO the gluon-quark and photon-quark channels open up. The gluon-gluon channel only enters the calculation at NNLO QCD. This process, embedding as dominant resonant structure a tt pair in association with a W + boson, receives contributions from three different coupling orders at LO, as can be observed in Fig. 1: the largest contribution (labelled LO QCD ) is of order (α 2 s α 6 ) , while the (α 8 ) contribution (labelled LO EW ) is roughly 1% of LO QCD . Sample diagrams are shown in Fig. 2. The interference among QCD and EW diagrams, of order (α s α 7 ), vanishes owing to colour algebra (in the case of a diagonal CKM matrix with unit entries). Both at LO QCD and at LO EW , diagrams with two, one, or zero resonant top/antitop quarks are present in the off-shell calculation.
At NLO, four coupling orders contribute to ttW hadro-production (see Fig. 1). The pure QCD corrections to LO QCD (labelled NLO 1 ), of order (α 3 s α 6 ) , are the dominant ones at NLO. The corresponding virtual corrections involve up to 7-point functions, while the real corrections are challenging due to the high multiplicity of particle in the final state. The NLO 1 corrections to offshell ttW, calculated by two independent groups [24,25], are strongly dependent on the (renormalization and factorization) central-scale choice and range between 10% and 20% of the LO QCD cross-section. Figure 3: Sample (α 2 s α 7 ) virtual corrections to off-shell ttW production in the threecharged-lepton decay channel: QCD corrections to the LO interference (left) and mixed contribution (right). Figure 4: Sample (α 2 s α 7 ) real corrections to off-shell ttW production in the threecharged-lepton decay channel: EW corrections to LO QCD (left) and QCD corrections to the LO interference (right).
The corrections of order (α 2 s α 7 ), labelled NLO 2 , come from the EW radiative corrections to LO QCD as well as from the QCD corrections to the LO interference, even though the contribution of order (α s α 7 ) vanishes. The corresponding virtual corrections require up to 10-point functions to be evaluated, and can be divided in two classes (see Fig. 3): one-loop amplitudes of order (g 4 s g 6 ) contracted with tree-level amplitudes of order (g 8 ) and one-loop amplitudes of order (g 2 s g 8 ) contracted with tree-level amplitudes of order (g 2 s g 6 ). Analogously, the real corrections receive contributions from photon radiation off LO QCD squared amplitudes as well as from gluon radiation off the LO interference. Sample diagrams for real radiation at this perturbative order are shown in Fig. 4. It is crucial to include both EW corrections to LO QCD and QCD corrections to the LO interference to ensure the infrared (IR) finiteness of the NLO correction. In fact, the IR singularities in one-loop amplitudes of order (g 2 s g 8 ) (right side of Fig. 3) are cancelled by both classes of realradiation contributions (Fig. 4).
Since the LO interference vanishes, the corresponding EW corrections vanish as well. Therefore, the NLO 3 corrections, of order (α s α 8 ), are pure QCD corrections to LO EW . Such corrections, although expected to be sub-leading w.r.t. the NLO 1 and NLO 2 ones by α s -power counting arguments, give a larger contribution than the NLO 2 ones at the inclusive level [10,12], as they are dominated by real radiation contributions in the quark-gluon partonic channel which embed the tW + scattering as a sub-process [1].
The pure EW corrections to LO EW , formally of order (α 9 ), have been shown to be at the subper-mille level [11,12] and will be hardly relevant even at the high-luminosity LHC. Therefore they are not considered in this context.
One-loop and tree-level amplitudes are calculated with RECOLA [29,30], interfaced with COL-LIER [31] for the reduction and evaluation of loop integrals. The Monte Carlo integration is performed via a multi-channel approach with the MOCANLO code, which has already been utilized (1)   for several LHC processes with top quarks [32,33] and now for ttW [25,28]. The subtraction of IR singularities is performed in the dipole scheme [34][35][36]. Top-quark and EW-boson masses, as well as the weak mixing angle are treated in the complex-mass scheme [37][38][39][40][41]. Full off-shell matrix elements are considered at LO and NLO, including finite-width and non-resonant effects, as well as complete spin correlations. For more information on input parameters and details of the calculation, we refer to Sect. (2.2) of Ref. [28].

Phenomenological results
We now present phenomenological results for a fiducial LHC setup that mimics the signal region defined in a recent ATLAS measurement [7]. For more details on the selection cuts we refer to Sect. (2.3) of Ref. [28]. In Table 1 we show integrated results for LO cross-sections and NLO corrections, for three different choices of renormalization and factorization scale (labelled following the notation of Ref. [25]), where the ambiguity in identifying the top quark (for scale choices µ The impact of NLO 1 corrections relative to the LO QCD cross-section is scale dependent, and such corrections range between +0.5% and +18% depending on the central-scale choice. At variance with NLO 1 , the relative NLO 2 and NLO 3 corrections are rather scale independent and amount to −5% and +13% of the LO QCD result, respectively. As expected from power counting, the LO EW cross-section is about 1% of the LO QCD one, while the NLO 3 corrections are 10-times larger than LO EW , giving more sizeable corrections than the NLO 2 ones, in spite of one power of α s less. Also in the off-shell calculation, hard real corrections embedding the tW scattering dominate the (α 3 s α 6 ) perturbative order, confirming the inclusive results [10,12]  corrections, obtained combining NLO 1 , NLO 2 , and NLO 3 in an additive way, range between +9% and +27%, depending on the central-scale choice. The scale uncertainties at NLO are at the 5% level and are dominated by NLO 1 corrections. The NLO 2 corrections have been calculated also in a very inclusive setup, obtaining similar relative corrections to the LO QCD cross-section (−3%) as in on-shell calculations [12]. In Figs. 5-7 we present differential results for the scale choice µ (e) 0 . For exclusive observables, the interplay among the three NLO corrections can differ sizeably from the results obtained for total cross-sections.
In the left panel of Fig. 5 we show the distributions in the azimuthal separation between the positron and the muon. The NLO corrections increase the rate of events with small azimuthal separation. The NLO 1 corrections dinimish from +18% to +11% with constant slope over the distribution range, while the NLO 2 and NLO 3 ones give a rather constant shift to the LO QCD +NLO 1 cross-section.
In the right panel of Fig. 5 we consider distributions in the muon rapidity. This observable represents a good proxy for the rapidity of the antitop quark [28]. The muon is preferably produced with central rapidity. The NLO 2 corrections give an almost flat negative shift to the LO QCD differential cross-section, while the relative NLO 1 corrections feature a variation of about 35% in the rapidity range. The shape of NLO 3 corrections (relatively to LO QCD ) is similar to the one of NLO 1 corrections. Such corrections give a positive shift to the LO cross-section, ranging from +16% (central region) to +8% (forward regions).
In the left panel of Fig. 6 the distributions in the antitop-quark invariant mass are shown. The antitop-quark system (bµ −ν µ ) is only known from the Monte Carlo truth, owing to the presence of three neutrinos in the final state. The NLO 1 corrections are negative near the Breit-Wigner peak, while they give a huge radiative enhancement to LO QCD below the top-quark pole mass, coming from real gluon radiation that is not clustered into b jets. A similar radiative tail, though less sizeable, is found also for NLO 2 corrections. For an invariant mass larger than the pole mass, NLO 1 relative corrections increase towards positive values while the NLO 2 ones give an almost flat negative shift to LO QCD (−10%). At variance with NLO 1 and NLO 2 , the NLO 3 corrections are rather flat, ranging between +10% at the peak and +30% in the tail region, due to the large quark-gluon partonic channel, which features a light quark as final-state particle, which cannot result from the radiative decay of top/antitop quarks.
In the right panel of Fig. 6 we consider the invariant mass of the system formed by the three charged leptons. The QCD corrections (NLO 1 and NLO 3 ) are rather flat, while the NLO 2 corrections, dominated by virtual EW corrections, negatively increase towards large masses (−10% around 500 GeV). Such a behaviour is driven by Sudakov logarithms, which become large at high energy.
An analogous effect is found for transverse-momentum distributions, that are considered in Fig. 7. In fact, large negative NLO 2 corrections are found in the tail of the transverse-momentum distribution for the reconstructed antitop quark (−20% at 800 GeV) as well as for the two-b-jet system (−20% at 350 GeV). The NLO 3 corrections are pretty flat for both observables considered in Fig. 7, ranging between +10% and +30% in the considered transverse-momentum ranges. The NLO 1 corrections increase by 25% from small to large p T of the antitop quark, while they become much larger at moderate values of the transverse momentum of the two-b-jet system. The two-b-jet system is correlated to the tt system [25], which recoils against a W + boson and thus receives large contributions from unclustered real QCD radiation. The combined NLO corrections to the bb-system transverse momentum is almost vanishing due to cancellation among different

Conclusion
We have presented the first calculation including complete off-shell effects at NLO QCD [ (α 3 s α 6 )] and subleading NLO orders [ (α 2 s α 7 ) and (α s α 8 )] for the hadronic production of ttW + in the three-charged-lepton decay channel. The NLO 1 corrections range between +0.5% and +18%, depending on the central-scale choice. The scale uncertainties are reduced from 25% to 5% from LO to NLO, driven by the NLO 1 corrections. The NLO 2 corrections are negative (about −5%) and independent of the scale choice (relative to the LO cross-section), but they become large (up to −20%) for high-energy regimes in transverse-momentum and invariant-mass distributions. The NLO 3 corrections are also large, ranging between +10% and +30% in the considered distributions, and are rather scale independent. These corrections are dominated by gluon-quark-induced contributions embedding the tW-scattering process. The inclusion of NLO 2 and NLO 3 corrections is necessary for the modelling of ttW + production, as all NLO contributions change sizeably distribution shapes. Futhermore, the off-shell effects become important in the tails of several distributions.