Combined NLO EW and QCD corrections to off-shell \text{t}\overline{\text{t}}\text{W} production at the LHC
CCombined NLO EW and QCD corrections to off-shell ttWproduction at the LHC
Ansgar Denner and Giovanni Pelliccioli
University of W¨urzburg, Instit¨ut f¨ur Theoretische Physik und Astrophysik, Emil-Hilb-Weg 22, 97074 W¨urzburg (Germany)
Abstract.
The high luminosity that will be accumulated at the LHC will enable precise differentialmeasurements of the hadronic production of a top–antitop-quark pair in association with a W boson.Therefore, an accurate description of this process is needed for realistic final states. In this work wecombine for the first time the NLO QCD and electroweak corrections to the full off-shell ttW + productionat the LHC in the three-charged-lepton channel, including all spin correlations, non-resonant effects, andinterferences. To this end, we have computed the NLO electroweak radiative corrections to the leadingQCD order as well as the NLO QCD corrections to both the QCD and the electroweak leading orders. The hadronic production of top-antitop pairs in associa-tion with a W boson is an interesting process to investi-gate at the Large Hadron Collider (LHC), as it representsan important probe of the Standard Model (SM) as wellas a window to new physics.This process is one of the heaviest signatures measur-able at the LHC. It gives access to the top-quark cou-pling to weak bosons and to possible deviations from itsSM value [1–3]. Due to the absence of a neutral initialstate at a lower perturbative order than next-to-next-to-leading order (NNLO) in QCD, it is also expectedto improve substantially the sensitivity to the tt chargeasymmetry [4]. Polarization observables and asymmetriesin ttW ± production are capable of enhancing the sensi-tivity to beyond-the-SM (BSM) interactions featuring achiral structure different from the one of the SM [4, 5].The hadro-production of ttW ± is in general well suitedto directly search for BSM physics, in particular super-symmetry [6, 7], supergravity [8], technicolour [9], vector-like quarks [10], Majorana neutrinos [11] and modifiedHiggs sectors [12–14]. Beyond its own importance in LHCsearches, the ttW production is a relevant background tottH production [15].The ATLAS and CMS collaborations have measuredand investigated ttW ± production at Run 1 [16, 17] andRun 2 [18–21] of the LHC. This signature has been in-cluded as a background in the recent experimental anal-yses for ttH production [22–25].The most recent experimental results based on Run 2show a tension between data and theory predictions in thet¯tW modelling both in direct measurements [19, 20] andin the context of the search for t¯t associated productionwith a Higgs boson [24, 25]. While the theoretical com- munity has invested a noticeable effort to address thistension, so far no explanation emerged that is capable tofill the gap between the SM predictions and the data.An improved modelling of the t¯tW ± process is re-quired to allow for the comparison of SM predictions withfuture LHC data, particularly those that will be accu-mulated during the high-luminosity run. The increasedstatistics will enable not only more precise measurementsof t¯tW ± cross-sections, but also measurements of differ-ential distributions and in different decay channels. Thistarget can only be achieved if the theoretical descrip-tion of realistic final states embedding the t¯tW resonancestructure is available.Many theoretical predictions for ttW ± hadro-produc-tion are available in the literature. The first next-to-leadingorder (NLO) QCD calculation for TeV-scale colliders wasperformed in a spin-correlated narrow-width approxima-tion for the semi-leptonic decay channel [26]. The match-ing of NLO QCD predictions to a parton shower was firsttackled for the same decay channel in Ref. [27]. A numberof calculations for ttW ± inclusive production (on-shelltop/antitop quarks and W boson) have been carried out,targeting charge asymmetries [4], the impact of ttW ± onthe associated production of tt pairs with a Higgs boson[15] at NLO QCD, and the effects of subleading NLOQCD and electroweak (EW) corrections [28–30]. Soft-gluon resummation up to next-to-next-to-leading loga-rithmic accuracy [31–35] and multi-jet merging [36] havealso been investigated. NLO QCD corrections to the EWleading order (LO) have been computed in the narrow-width approximation (NWA), accounting for completespin correlations and including parton-shower effects [37].Very recently a comparison of different fixed-order MonteCarlo generators matched to parton showers has beenperformed for inclusive ttW ± , with a focus on the two- a r X i v : . [ h e p - ph ] F e b A. Denner, G. Pelliccioli : Combined NLO EW and QCD corrections to off-shell ttW production at the LHC charged lepton signature (in the NWA) [38]. The firststudies that aim at the off-shell modelling of t¯tW pro-duction concern the full NLO QCD corrections in thethree-charged-lepton channel [5, 39, 40].The calculation of subleading NLO corrections hasbeen performed for inclusive production [28–30, 37], but isstill missing for realistic final states. Our present work tar-gets the complete fixed-order description of the off-shellt¯tW production, combining the NLO QCD and EW cor-rections that have a sizeable impact at the [email protected] paper is organized as follows. In Section 2.1 wedescribe the process under investigation, providing de-tails on various NLO corrections that are presented. InSections 2.2 and 2.3 we provide the SM input parametersand the selection cuts used for numerical simulations, re-spectively. The integrated results at LO and NLO arepresented in Section 3.1, while in Section 3.2 a numberof differential distributions are described, focusing on therelative impact of various NLO corrections to the LO pre-dictions. In Section 4 we draw our conclusions.
We consider the processpp → e + ν e τ + ν τ µ − ¯ ν µ b ¯b + X , (1)which receives contributions only from quark-induced par-tonic channels at LO. Gluon–quark and photon–quarkchannels open up at NLO, while the pure gluonic chan-nel enters only at NNLO in QCD.Although we consider the final state with three char-ged leptons with different flavours, the corresponding re-sults for the case of identical positively-charged leptonscan be estimated by multiplying our results by a factor1 /
2, up to interference contributions, which are expectedto be small.In this work we focus on the production of t¯t pairs inassociation with a W + boson, but the calculation of thecharge-conjugate process (t¯tW − ) can be performed withthe same techniques and no additional conceptual issues.At LO, the largest contribution is given by the QCD-mediated process of order O ( α α ) (labelled LO QCD ),which always embeds a gluon s -channel propagator if noquark-family mixing is assumed (diagonal quark-mixingmatrix with unit entries). The tree-level EW contributionof order O ( α ) (labelled LO EW ), despite being character-ized by many more diagram topologies, is expected togive a cross-section that is roughly 1% of the LO QCD oneowing to the ratio of EW and strong coupling constants.The interference contribution, formally of order O ( α s α ),is identically zero due to colour algebra. In Fig. 1 we showsample diagrams for the QCD-mediated and purely-EWprocess. Note that the diagrams with a resonant top–antitop-quark pair are present also in EW tree-level con-tributions.At NLO, the ttW process receives contributions fromfour different perturbative orders, as depicted in Fig. 2. The corrections that have the largest impact on theNLO cross-section are of order O ( α α ), which are pureQCD corrections to LO QCD . Following the notation ofRefs. [29, 30], we label this perturbative order as NLO .These corrections have recently been computed for thefull off-shell process [39, 40]. This perturbative order showsa typical NLO QCD behaviour in the scale dependence,and the NLO relative corrections to LO QCD are at the10–20% level, depending on the choice of the renormal-ization and factorization scale [40].The NLO corrections are known only for on-shelltop–antitop quarks and for an on-shell W boson [28–30]. They are expected to give a negative contributionof about 4.5% of the inclusive LO cross-section.In the off-shell calculation, as well as in the on-shellone, the NLO order receives contributions not only fromthe EW corrections to LO QCD , but also from the QCDcorrections to the LO interference, although at Born-levelthe O ( α s α ) contribution vanishes.Sample contributions to the virtual corrections at or-der O ( α α ) are shown in Fig. 3. The diagram on theleft involves one-loop amplitudes of order O ( g g ) inter-fered with tree-level EW amplitudes of order O ( g ) and isobviously a QCD correction to the LO interference. Thediagram on the right involves one-loop amplitudes of or-der O ( g g ) interfered with tree-level QCD amplitudes oforder O ( g g ) and could be na¨ıvely classified as an EWcorrection to LO QCD . However, it can also be regardedas contributing to the QCD corrections to the LO inter-ference. In fact, the IR singularities of this contributionare partially cancelled by the real photonic correctionsto LO
QCD and partially by the real gluonic correctionsto the LO interference. Diagrams with weakly interactingparticles in the loops are more demanding from the com-putational point of view, as the corresponding one-loopamplitudes include up to 10-point functions, while the lat-ter ones feature at most 7-point functions. A selection ofone-loop diagrams which contribute at this perturbativeorder are shown in Fig. 4.The real-radiation contributions to NLO correctionsare computationally demanding due to the large multi-plicity of electrically charged final-state particles. In con-trast to the virtual ones, the real NLO corrections can beuniquely classified into two types: the NLO EW correc-tions to the LO QCD process, which involve a real photon(see Fig. 5 left for an example), and the NLO QCD cor-rections to the LO interference, which involve a real gluon(see Fig. 5 right for an example). In the first class of con-tributions, the photon can be either in the final or in theinitial state. The processes with a photon in the final stateare characterized by many singular regions, as the photoncan become soft or collinear to any of the seven chargedexternal particles. This results in a large number of sub-traction counterterms that are required to ensure a stablecalculation of the NLO cross-section. The real processeswith a photon in the initial state possess a smaller num-ber of singular phase-space regions and are suppresseddue to the small luminosity of photons in the proton.For on-shell production, the contribution of the photon-induced channels to the leading-order cross-section is atthe sub-percent level [28]. The QCD corrections to the . Denner, G. Pelliccioli : Combined NLO EW and QCD corrections to off-shell ttW production at the LHC u¯d e + ν e τ + ν τ b¯b µ − ¯ ν µ t ¯t W + W − W + g u¯d e + ν e τ + ν τ ¯bb µ − ¯ ν µ W + g W + W − Z u¯d e + ν e τ + ν τ b¯b µ − ¯ ν µ t ¯t W + W − W + Z /γ u¯d e + ν e µ − ¯ ν µ W + HZ Z τ + ν τ W + W − ¯bb Fig. 1.
Sample diagrams contributing to LO
QCD (left) and to LO EW (right) cross-sections for off-shell ttW + production inthe three-charged-lepton channel. α α α s α α α α α α α s α α QCD QCD QCDEW EWLO
QCD
NLO NLO NLO NLO LO EW Fig. 2.
Contributing perturbative orders at LO and NLO for ttW hadro-production in the three-charged-lepton channel. u¯d e + ν e τ + ν τ b¯b µ − ¯ ν µ t ¯t W + W − W + g td g ¯du τ + ν τ µ − ¯ ν µ ¯bb e + ν e dW + W − ¯tt W + Z /γ O ( g g ) O ( g ) u¯d e + ν e τ + ν τ b¯b µ − ¯ ν µ t ¯t W + W − W + g td Z /γ ¯du τ + ν τ µ − ¯ ν µ ¯bb e + ν e dW + W − ¯tt W + g O ( g g ) O ( g g ) Fig. 3.
Sample contributions to the virtual corrections at order O ( α α ) for off-shell ttW production in the three-charged-lepton channel: QCD corrections to the LO interference (left) and a contribution that cannot be uniquely attributed to eitherthe QCD corrections to the LO interference or the EW corrections to the LO QCD (right).
LO interference are non-vanishing only if the radiatedgluon is emitted by an initial-state light quark and ab-sorbed by a final-state b quark or top quark (or the otherway around). A sample contribution is shown in Fig. 5right. These corrections, although necessary to accountfor all O ( α α ) contributions, turn out to be very small,as detailed in Section 3.To sum up, the full set of real partonic channels thatcontribute to the NLO corrections is u ¯ d → e + ν e µ − ¯ ν µ τ + ν τ b ¯b γγ u → e + ν e µ − ¯ ν µ τ + ν τ b ¯b dγ ¯ d → e + ν e µ − ¯ ν µ τ + ν τ b ¯b ¯ u EW corr. to LO
QCD and u ¯ d → e + ν e µ − ¯ ν µ τ + ν τ b ¯b gg u → e + ν e µ − ¯ ν µ τ + ν τ b ¯b d g ¯ d → e + ν e µ − ¯ ν µ τ + ν τ b ¯b ¯ u QCD corr. to LO int.,where u and d stand for up-type and down-type quarks,respectively (of the first and second generation).The vanishing LO interference implies that the cor-responding EW corrections vanish as well, since addi-tional EW propagators (virtual contributions) and ra-diated photons (real contributions) do not modify the LO colour structure. Therefore, the only NLO correctionsthat contribute at order O ( α s α ) are genuine QCD cor-rections to the LO EW cross-section. This order is labelledas NLO in Fig. 2. By simply counting the powers of α s the NLO corrections are expected to give a smaller con-tribution than the NLO ones. However, at the inclusivelevel [29] and in the narrow-width approximation [37],they are noticeably larger than the NLO ones. This re-sults from the fact that this perturbative order is domi-nated by hard real radiation diagrams in the gluon-quarkpartonic channel that embed the tW scattering process[1]. Sample diagrams are shown in Fig. 6. Thanks to thegenuine QCD nature of the NLO corrections, it is pos-sible to match them to a QCD parton shower with nosubtleties due to EW corrections, as it has been done inRefs. [37, 38].The last NLO perturbative order, O ( α ), is furnishedby the EW corrections to the LO EW process. It has beenshown at the inclusive level that such contributions areat the sub-permille level [30], as expected by na¨ıve powercounting. Even with a substantially larger data set thanthe one of Run 2 ( i.e. − at the high-luminosityLHC) these EW effects are out of reach in a realistic A. Denner, G. Pelliccioli : Combined NLO EW and QCD corrections to off-shell ttW production at the LHC u¯d e + ν e τ + ν τ µ − ¯ ν µ H W − Wgd W + W + t bb ¯bW u¯d e + ν e τ + ν τ µ − ¯ ν µ Wgd t b¯bZ /γ ¯t W e ν τ u¯d e + ν e τ + ν τ µ − ¯ ν µ gd W − t bb ¯b WW ν τ Z /γ ν e τ + ν τ µ − ¯ ν µ W b¯bb Z /γ WZ /γ u¯d d e + ν e ν µ τν e g Fig. 4.
One-loop diagrams of order O ( g g ) contributing to the EW virtual corrections (NLO ) to off-shell ttW productionin the three-charged-lepton channel. From left to right: sample diagrams involving 7-, 8-, 9- and 10-point functions. u¯d e + ν e τ + ν τ b ¯b µ − ¯ ν µ t ¯t W + W − W + g µ − ν τ τ + b e + ν e W + b ¯bd d u¯dgt¯t γ γ W + W − b¯ ν µ u¯d e + ν e τ + ν τ b ¯b µ − ¯ ν µ t ¯t W + W − W + g µ − ν τ τ + b e + ν e W + b ¯bd dd u¯dZ /γ t¯tg g W + W − ¯ ν µ Fig. 5.
Sample contribution to the real corrections at order O ( α α ) for ttW production in the three-charged-lepton channel:photonic corrections to LO QCD (left) and gluonic corrections to the LO interference (right). b¯bgu dW − µ − ¯ ν µ W + W + W + t t e + ν e τ + ν τ Z /γ b¯bgu dW − µ − ¯ ν µ W + W + W + t t e + ν e τ + ν τ H b¯bgu dW − µ − ¯ ν µ W + W + W + t t e + ν e τ + ν τ b Fig. 6.
Sample diagrams for the partonic channel gu → b ¯b e + ν e τ + ν τ µ − ¯ ν µ d that contain tW + scattering as a subprocess andcontribute to NLO corrections to ttW production in the three-charged-lepton channel. fiducial region. Therefore, we are not providing resultsfor this order.In the following we focus on the first three NLO per-turbative orders. Tree-level and one-loop SM amplitudesare computed with the Recola matrix-element provider[41, 42]. For the tensor reduction and evaluation of loopintegrals we use the
Collier library [43]. The multi-channel Monte Carlo integration is performed with
Mo-CaNLO , a generator that has already been used to com-pute the NLO QCD corrections to ttW [40] and the NLOEW corrections to several LHC processes involving topquarks [44, 45]. The subtraction of infrared and collinearsingularities is carried out using the dipole formalism ofRefs. [46–48] both for QCD and for EW corrections. Theinitial-state collinear singularities are absorbed in the par-ton distribution functions (PDFs) in the MS factorizationscheme.
We consider proton–proton collisions at a centre-of-massenergy of 13 TeV. We neglect flavour mixing in the quark sector and use a unit quark-mixing matrix. The threecharged leptons that we consider in the final state aremassless and characterized by three different flavours.The on-shell masses and widths of weak bosons are set tothe following values [49], M OSW = 80 .
379 GeV , Γ
OSW = 2 .
085 GeV ,M OSZ = 91 . , Γ OSZ = 2 . , (2)and then translated into their pole values [50] that enterthe Monte Carlo simulations. The Higgs-boson mass andwidth are fixed, following Ref. [49], to M H = 125 GeV , Γ H = 0 . . (3)We have computed the LO top-quark width according toRef. [51], using the pole values for the W-boson mass andwidth. The NLO top-quark width is obtained applyingthe NLO QCD and EW correction factors of Ref. [52] tothe LO width. The numerical values read m t = 173 . ,Γ LOt = 1 . , Γ NLOt = 1 . . (4) . Denner, G. Pelliccioli : Combined NLO EW and QCD corrections to off-shell ttW production at the LHC The top-quark width is kept fixed when performing var-iations of the factorization and renormalization scales.The EW coupling is treated in the G µ scheme [53], α = √ π G µ M (cid:20) − M M (cid:21) , (5)where G µ = 1 . · − GeV − is the Fermi constant.The masses of weak bosons and of the top quark, andtherefore also the EW mixing angle, are treated in thecomplex-mass scheme [53–56].Both for LO and NLO predictions, we employNNPDF 3.1 luxQED PDFs [57], computed at NLO with α s = 0 . LHAPDF6 interface [58].
Coloured partons with | η | < k t algorithm [59] with resolution radius R = 0 .
4. The same algorithm but with R = 0 . p T , b >
25 GeV , | η b | < . . (6)Furthermore, we ask for three charged leptons that fulfilstandard acceptance and isolation cuts, p T ,(cid:96) >
27 GeV , | η (cid:96) | < . , ∆R (cid:96) b > . , (7)where the R distance is defined as the sum in quadratureof the azimuthal and rapidity separations, ∆R ij = (cid:113) ∆φ ij + ∆y ij . (8)We do not constrain the missing transverse momentumand do not apply any veto to additional light jets. For the factorization and renormalization scale ( µ F = µ R ), we consider three different dynamical choices thathave proved to behave better than a fixed scale [39, 40].The first one, introduced in Ref. [39], depends on thetransverse-momentum content of the final-state particles,regardless of the top–antitop resonances, µ (c)0 = H T p T , miss + (cid:80) i =b ,(cid:96) p T ,i . (9) We use the same notation for the scales as in Ref. [40].
The second and third choices, already used in Ref. [40],are based on the transverse masses of the top and antitopquarks. Due to the ambiguity in choosing the (cid:96) + ν (cid:96) pairthat results from the top quark, we pick the pair of leptonsthat when combined with the bottom quark forms aninvariant mass closest to the top-quark mass. We considertwo choices that differ by a factor of 1 / µ (d)0 = (cid:114)(cid:113) m + p , t (cid:113) m + p , t , µ (e)0 = µ (d)0 . (10)In the following we present the results combining NLOcorrections with the so-called additive approach, σ LO+NLO = σ LO QCD + σ NLO + σ NLO + σ LO EW + σ NLO . (11)This approach is exact at the order of truncation of theperturbative expansion. Furthermore, it represents a nat-ural choice for our process, as the combination involvesNLO corrections to two different leading orders that donot interfere. In Table 1, we present the integrated cross-sections in thefiducial region defined in Section 2.2.The leading corrections to the LO
QCD cross-sectionare expected to come from the corresponding pure QCDradiative corrections (NLO ). Their inclusion has beenproved to decrease the theoretical uncertainty due to scalevariations and to stabilize the perturbative convergencefor this process. Nonetheless, their impact depends onthe choice of the scale, as already pointed out both ininclusive [28, 29] and off-shell [39, 40] computations. Infact, with the resonance-blind dynamical choice µ (c)0 , theNLO corrections give a 6 .
6% enhancement to the LO
QCD cross-section, with the resonance-aware choices, µ (d)0 and µ (e)0 , they give a 18% and a 0 .
4% correction, respectively.Note that, at variance with Ref. [40] in this paper wecompute both LO and NLO predictions with the NLOtop-quark width, which gives roughly a 12% enhancementto the LO cross-section.The NLO corrections are negative and amount toabout − .
5% of the LO
QCD cross-section for all scalechoices. Such supposedly subleading corrections have asizeable impact on the fiducial NLO cross-section, andthis is likely due to large EW Sudakov logarithms en-hancing the cross-section in the high-energy regime [60].This is supported by the fact that the average partoniccentre-of-mass energy and H T are quite high, 850 GeVand 520 GeV, respectively. A crude estimate of the Su-dakov logarithms gives a result which is of the same or-der of magnitude as the full NLO corrections we haveobtained for this process.The impact of QCD corrections that can be uniquelyattributed to the LO interference is very small, both forreal and for virtual corrections, accounting respectivelyfor 5% and for less than 1% of the total NLO result. A. Denner, G. Pelliccioli : Combined NLO EW and QCD corrections to off-shell ttW production at the LHC µ (c)0 µ (d)0 µ (e)0 perturbative order σ (fb) ratio σ (fb) ratio σ (fb) ratioLO QCD ( α α ) 0.2218(1) +25 . − . +23 . − . +26 . − . EW ( α ) 0.002164(1) +3 . − . +3 . − . +3 . − . ( α α ) 0.0147(6) 0.066 0.0349(6) 0.179 0.0009(7) 0.004NLO ( α α ) -0.0122(3) -0.055 -0.0106(3) -0.054 -0.0134(4) -0.056NLO ( α s α ) 0.0293(1) 0.131 0.0263(1) 0.135 0.0320(1) 0.133LO QCD +NLO +2 . − . +5 . − . +3 . − . QCD +NLO +25 . − . +23 . − . +25 . − . EW +NLO +22 . − . +20 . − . +22 . − . +4 . − . +6 . − . +4 . − . Table 1.
LO cross-sections and NLO corrections (in fb) in the fiducial setup, for three different dynamical scale choices.Numerical errors (in parentheses) are shown. Ratios are relative to the LO
QCD cross-section. The scale uncertainties from7-point scale variations (in percentage) are listed for LO and NLO cross-sections. The result in the last row is the sum of allLO cross-sections and NLO corrections, namely LO
QCD + LO EW + NLO + NLO + NLO . At O ( α α ), we have also included the contributionfrom photon-initiated partonic channels, which are posi-tive and account for about 0 .
1% of the LO QCD cross-section. As already observed at the inclusive level [28],this contribution is very small and its effect will be hardlyvisible even at the high-luminosity run of LHC (they willyield about 1 event for √ s = 13 TeV and L = 3000 fb − ).In inclusive calculations with on-shell top–antitopquarks, the NLO corrections were found to give a − . – we use finite top-quark and W widths (same valuesas those of Section 2.2) and we include Higgs-bosoncontributions; – we apply a minimum invariant-mass cut of 5 GeV tothe b¯b system to protect from infrared singularitiesand we cluster photons into charged particles withisolation radius R = 0 . – we employ the same dynamical scale as in Ref. [30],but using the kinematics after photon recombinationand choosing the top-quark candidate with the sameinvariant-mass prescription as for the calculation ofthe scales µ (d)0 and µ (e)0 .The obtained inclusive cross-sections, σ LO QCD = 262 +23 . − . fb ,σ LO QCD +NLO = 254 +23 . − . fb , (12)exhibit NLO corrections of −
3% of LO
QCD , which isnot far from the − .
5% of Refs. [28, 30]. The remainingdiscrepancy should be due to both the additional cuts we have applied and the non-resonant effects that areincluded in our full calculation, while being absent in on-shell calculations. The comparison of results in the fidu-cial and the inclusive setup reveals that the NLO cor-rections are more sizeable for realistic final states and inthe presence of reasonable fiducial cuts.Coming back to our default fiducial setup (seeSection 2.2), the NLO perturbative order is dominatedby the real radiation in the ug partonic channel, owing toa PDF enhancement and the tW scattering embedded inthis channel. These real corrections account for 85% of theNLO corrections, and are one order of magnitude largerthan the corresponding leading order LO EW . The totalNLO corrections amount to 13% of the LO QCD cross-section, almost independently of the scale choice. Thisconfirms the 12% effect obtained in the case of on-shelltop and antitop quarks [30].It is worth stressing that the inclusion of NLO andNLO corrections gives a noticeable effect to the t¯tWcross-section. Therefore such corrections must definitelybe accounted for in experimental analyses. Furthermore,their relative contribution to the LO result is rather inde-pendent of the scale choice, while the NLO correctionsare much more scale dependent.As a last comment of this section, we point out thatthe scale uncertainty of the combined LO+NLO cross-section is driven by the NLO corrections which reducethe LO QCD uncertainty roughly from 20% to 5%. Due totheir EW nature, the NLO corrections do not diminishthe scale uncertainty of the corresponding leading order(LO QCD ). The NLO corrections exceed the correspond-ing pure EW LO process and, thus, imply a LO-like scaledependence for LO EW +NLO .So far, we have focused on the relative contributions ofvarious NLO corrections to the fiducial t¯tW cross-section.However, the interplay among different corrections can be . Denner, G. Pelliccioli : Combined NLO EW and QCD corrections to off-shell ttW production at the LHC rather different for more exclusive observables. Therefore,it is essential to study differential distributions. In the following we present a number of relevant dis-tributions focusing on the impact of the various NLOcorrections relative to the LO
QCD cross-sections. Sincein most of the LHC experimental analyses the theoret-ical predictions are NLO QCD accurate, we also com-ment on the distortion of NLO QCD distribution shapes(LO
QCD + NLO in our notation) due to the inclusionof NLO and NLO corrections. We choose to present alldifferential distributions with the µ (e)0 scale. We start by presenting transverse-momentum distri-butions in Figs. 7–8.In Fig. 7(a) we consider the distribution in the trans-verse momentum of the positron, which is precisely mea-surable at the LHC. Since we also include EW corrections(NLO ), the positron is understood as dressed (a radiatedphoton could be clustered into the positron). The distri-bution peaks around 50 GeV, where the relative impact ofQCD and EW corrections follows straight the integratedresults. Relative to the LO QCD , all three radiative cor-rections drop in a monotonic manner. Nonetheless, thedecrease of NLO corrections is very mild (14% below50 GeV, 10% at 380 GeV), while the NLO and NLO corrections decrease steeper: the former become nega-tive around 80 GeV and give −
9% at 380 GeV, the lat-ter become lower than −
10% already at moderate p T , e + (200 GeV). The behaviour of EW corrections in the tailsof this distribution is likely driven by the impact of Su-dakov logarithms, which become large at high p T .The same behaviour of the NLO and NLO correc-tions characterizes also the distribution in the transversemomentum of the antitop quark, shown in Fig. 7(b). Theantitop momentum is computed as the sum of the mo-menta of the muon, its corresponding antineutrino, andthe antibottom quark. This is not observable at the LHC,but its analysis is useful to compare the full off-shell cal-culation with the on-shell ones. The negative growth ofthe NLO corrections behaves very similarly in the inclu-sive calculations, as can be seen for example in Ref. [28](figure 5 therein). This confirms that for sufficiently in-clusive variables the NLO EW effects are dominated bycontributions with resonant top and antitop quarks. TheNLO corrections increase by roughly 25% in the consid-ered spectrum.In both transverse-momentum distributions ofFig. 7, the inclusion of subleading NLO corrections(NLO , NLO ) gives a decreasing effect towards largetransverse momenta to the NLO QCD cross-section. Infact, the ratio between the combined LO + NLO cross-section and the LO QCD + NLO one ranges between 1 . .
15 for small transverse momenta and drops below1 already at moderate transverse momenta. We havechecked numerically that this conclusion can be drawn In Ref. [40] the scale for the differential distributions is µ (d)0 , i.e. exactly twice the default scale used here. also for other scale choices ( µ (c)0 , µ (d)0 ), confirming thealmost scale-independent impact of the NLO and NLO corrections.In Fig. 8 we consider the distributions in the trans-verse momentum of the t¯t and the b¯b system. The formeris not measurable at the LHC but can be reconstructedfrom Monte Carlo truth, choosing the positively-chargedlepton–neutrino pair that best reconstructs the top-quarkmass when combined with the momentum of the bottomquark. The latter variable is directly observable at theLHC.The NLO corrections to the t¯t transverse-momentumdistribution [Fig. 8(a)] have already been investigated inRef. [40]: these QCD corrections grow monotonically andbecome dramatically large and positive at high p T , t¯t . Notethat at LO this variable coincides with the p T of the re-coiling W + boson, and therefore is sensitive to the realQCD radiation which is not clustered into b jets (and thatcannot be clustered to the W + -boson decay products).The NLO corrections receive a sizeable contribution bythe g q/ g¯ q partonic channels, which are enhanced by thegluon PDF. In contrast, the NLO ones feature a typicalNLO EW behaviour in the tail of the distribution, givinga negative and monotonically decreasing correction to theLO cross-section. Differently from the QCD corrections,the additional photon can be clustered into any of the ex-ternal charged particles, thus also into the decay productsof the recoiling W boson. Furthermore, the cross-sectionis not enhanced by the γq/γ ¯ q partonic channels due to thevery small photon luminosity in the proton. The NLO contribution is positive in the whole analyzed spectrum,and increases from 6% (below 50 GeV) to a maximum of25% around p T , t¯t = 2 m t , then it slowly decreases in thelarge- p T region. Relative to the LO QCD + NLO result,the combination of all other corrections gives an effectwhich is about 10% in the soft- p T region and diminishestowards negative values at large p T .The transverse momentum of the b¯b system[Fig. 8(b)] is correlated to the one of t¯t system. The NLO corrections to this observable behave in the same manneras those for the p T , t¯t distributions, giving a −
20% con-tribution around 400 GeV. The NLO corrections growmonotonically from +10% (at low transverse momentum)to +30% (around 400 GeV). In the soft part of the spec-trum ( p T , b¯b <
150 GeV), the NLO corrections are ratherflat, while in the large- p T region they grow positive andbecome very large, similarly to the p T , t¯t distribution. Theoverall NLO corrections are very small below 150 GeV dueto mutual cancellations among the three contributions,while at larger transverse momenta the corrections aredominated by the NLO contribution. Furthermore, rela-tively to the LO QCD + NLO distribution, the combinedNLO corrections give a flat and positive effect between7% and 8% in the whole analyzed spectrum.In all analyzed transverse-momentum distributions ofFigures 7–8, the LO EW contribution increases monotoni-cally (relatively to the LO QCD one) but never exceeds 3%of the LO
QCD cross-section.In Fig. 9(a), we display the distribution in the invari-ant mass of the antitop quark, which in our setup can bereconstructed from the Monte Carlo truth. The lineshape
A. Denner, G. Pelliccioli : Combined NLO EW and QCD corrections to off-shell ttW production at the LHC d / dp T , e + [ f b / G e V ] pp e + e + bb, s = 13TeV, = ( M T,t M T,t ) /2 LO QCD LO EW (×10)LO QCD +NLO LO+NLO r a t i o [ / L O Q C D ] LO EW NLO NLO NLO
50 100 150 200 250 300 350 p T,e + [GeV] r a t i o [ / L O Q C D + N L O ] LO QCD +NLO LO+NLO (a) Transverse momentum of the positron. d / dp T , t [ f b / G e V ] pp e + e + bb, s = 13TeV, = ( M T,t M T,t ) /2 LO QCD LO EW (×10)LO QCD +NLO LO+NLO r a t i o [ / L O Q C D ] LO EW NLO NLO NLO
100 200 300 400 500 600 700 p T,t [GeV] r a t i o [ / L O Q C D + N L O ] LO QCD +NLO LO+NLO (b) Transverse momentum of the antitop quark.
Fig. 7.
Distributions in the transverse momentum of the positron (left) and of the antitop quark (right). Top panel: differentialcross-sections (in fb) for LO
QCD , LO EW (scaled by a factor 10), LO QCD + NLO and for the complete NLO, which is the sumof all LO cross-sections and all NLO corrections. Middle panel: ratio of the LO EW , NLO , NLO , and NLO corrections overthe LO QCD cross-section. Bottom panel: ratio of the LO + NLO cross-section over the LO
QCD + NLO one. is dominated by the Breit-Wigner distribution of the lep-tonic decay of the antitop quark. The NLO correctionsare negative at the peak while below the pole mass theygive a very large enhancement to the LO result. Such aradiative tail, coming from unclustered real radiation, isalso present, though less sizeable, in the NLO EW cor-rections (unclustered photons). At values larger than thetop-quark mass, the distribution receives an increasinglypositive contribution from NLO corrections, while theNLO ones give an almost flat correction of −
10% tothe LO
QCD cross-section. The LO EW contribution showsa slightly wider distribution than the LO QCD one. Thiscould be attributed to the relatively larger contributionof non-resonant background diagrams in the LO EW con-tribution. Nonetheless, the impact of this difference onthe full distribution is almost invisible owing to the verysmall size of the LO EW contribution. The NLO correc-tions behave differently from the NLO ones, giving arather flat enhancement of the fiducial LO cross-section,which is minimal around the peak (+11% at m t ) andmildly increases towards the tails (+20% at 200 GeV,+30% at 150 GeV). This is due to the very large con-tribution of the u(c)g partonic channel, which has a lightd(s) in the final state that cannot come from the radiativedecay of the top or of the antitop quark (differently from final-state gluons). As can be seen in the bottom panel ofFig. 9(a), the total LO + NLO result is 12% higher thanthe LO QCD + NLO one below the top-quark mass. Thisenhancement is smaller at (4%) and above (7%) the topmass.Another variable that is often investigated in LHCanalyses is H T , whose definition is given in Eq. (9). TheLO and NLO distributions in this observable are shown inFig. 9(b). As in the transverse-momentum distributionsstudied above, the NLO radiative corrections decreasemonotonically towards large values of H T (about − H T ≈ . EW contribution grows to 5%of the LO QCD cross-section at 1 . corrections are rather flat and enhance the LO QCD re-sult between 10% and 15%. The NLO corrections arecharacterized by a non-flat shape that is increasing for H T <
800 GeV from −
10% to +25% and decreasing inthe rest of the considered spectrum. We further observethat the combination of the three NLO perturbative or-ders yields an almost vanishing correction in the soft re-gion of the spectrum, while in the tail of the distributionthe overall correction is dominated by the NLO contri-bution for our scale choice. In a similar fashion as in other . Denner, G. Pelliccioli : Combined NLO EW and QCD corrections to off-shell ttW production at the LHC d / dp + b e s t T , tt [ f b / G e V ] pp e + e + bb, s = 13TeV, = ( M T,t M T,t ) /2 LO QCD LO EW (×10)LO QCD +NLO LO+NLO r a t i o [ / L O Q C D ] LO EW NLO NLO NLO
100 200 300 400 500 600 700 p + bestT,tt [GeV] r a t i o [ / L O Q C D + N L O ] LO QCD +NLO LO+NLO (a) Transverse momentum of the t¯t system. d / dp T , bb [ f b / G e V ] pp e + e + bb, s = 13TeV, = ( M T,t M T,t ) /2 LO QCD LO EW (×10)LO QCD +NLO LO+NLO r a t i o [ / L O Q C D ] LO EW NLO NLO NLO
50 100 150 200 250 300 350 400 p T,bb [GeV] r a t i o [ / L O Q C D + N L O ] LO QCD +NLO LO+NLO (b) Transverse momentum of the b¯b system.
Fig. 8.
Distributions in the transverse momentum of the reconstructed t¯t system (left) and of the b¯b system (right). Samestructure as Fig. 7. transverse-momentum distributions, the ratio of the com-bined LO+NLO result over the LO
QCD +NLO decreasesmonotonically from 1 .
15 to 0 .
95 in the analyzed range.In Fig. 10 we study more invariant-mass distributions.The distribution in the invariant mass of the two-b-jetsystem [Fig. 10(a)] is characterized by rather flat QCDcorrections (NLO and NLO ). The NLO correctionsenhance the LO QCD cross-section by 11% to 14% every-where in the analyzed invariant-mass range. The NLO contribution has a similar behaviour as the one found forthe previous variables, growing negative towards the tailof the distribution.The distribution in the invariant mass of the three-charged-lepton system is considered in Fig. 10(b). Thebehaviour of the NLO and NLO corrections followsclosely the one for the b¯b system, apart from a lesssteep decrease of the EW corrections towards large in-variant masses. These corrections are at the −
10% levelfor masses larger than 500 GeV. The NLO correctionsvary by hardly more than 10% in the studied range.As shown in the bottom panels of Fig. 10, both for theb¯b system and for the three-charged-lepton system, theinclusion of NLO and NLO corrections (as well as ofLO EW , though hardly visible) gives a non-flat correctionto the NLO QCD invariant-mass distributions, decreas-ing monotonically from +12% to zero in the consideredspectra. After presenting transverse-momentum and invariant-mass distributions, we switch to some relevant angularvariables. In Fig. 11(a) and Fig. 11(b) we display the dis-tributions in the rapidity of the muon and of the antitopquark, respectively. Since these two variables are corre-lated (the dominant resonant structure involves the decay¯t → ¯b µ − ¯ ν µ ), the muon rapidity, which is precisely mea-surable at the LHC, represents a suitable proxy for therapidity of the corresponding antitop quark (which canonly be reconstructed from Monte Carlo truth). Note thatthe muon rapidity is sharply cut at ± . | y ¯t | > .
5. Boththe muon and the antitop quark are produced preferablyin the central region. The NLO corrections are ratherflat, giving between −
4% and −
8% decrease to the LOcross-section. The relative NLO corrections to the muon-rapidity distribution are characterized by a large vari-ation (about 35%) in the available range. Relative toLO QCD , the differential NLO corrections have a simi-lar shape as the NLO ones, giving in the whole rapidityrange a positive correction (8% in the forward regions,16% in the central region). Almost identical results arefound in the rapidity distribution of the antitop quark.Owing to the NLO corrections, the ratio between thecomplete NLO prediction and the LO QCD +NLO one has A. Denner, G. Pelliccioli : Combined NLO EW and QCD corrections to off-shell ttW production at the LHC d / d M t [ f b / G e V ] pp e + e + bb, s = 13TeV, = ( M T,t M T,t ) /2 LO QCD LO EW (×10)LO QCD +NLO LO+NLO r a t i o [ / L O Q C D ] LO EW NLO NLO NLO
160 170 180 190 M t [GeV] r a t i o [ / L O Q C D + N L O ] LO QCD +NLO LO+NLO (a) Invariant mass of the antitop quark. d / d H T [ f b / G e V ] pp e + e + bb, s = 13TeV, = ( M T,t M T,t ) /2 LO QCD LO EW (×10)LO QCD +NLO LO+NLO r a t i o [ / L O Q C D ] LO EW NLO NLO NLO
400 600 800 1000 1200 1400 H T [GeV] r a t i o [ / L O Q C D + N L O ] LO QCD +NLO LO+NLO (b) H T variable. Fig. 9.
Distributions in the invariant mass of the antitop quark (left) and in the H T variable (right). Same structure as Fig. 7. a maximum of 1 .
11 in the central region and diminishestowards forward regions (close to unity). This holds bothfor the muon and for the antitop-quark rapidity spectra.In Fig. 12(a) we consider the distribution in the az-imuthal separation between the positron and the muon.The two charged leptons tend to be produced in oppo-site directions both at LO and at NLO, but the inclusionof radiative corrections enhances the fraction of eventswith small azimuthal separations. The NLO correctionsare negative and roughly constant ( −
5% to − contribution de-creases monotonically from +18% to +11% relatively tothe LO QCD result. As already observed in Ref. [40], theNLO correction to the LO QCD cross-section decreaseswith an almost constant negative slope over the full range.The overall NLO corrections to the LO QCD cross-sectionare positive everywhere except in the vicinity of the peakat ∆φ e + µ − = π . Relative to the LO QCD + NLO pre-diction, the combination of NLO and NLO correctionsgives a pretty flat enhancement (1 .
11 at ∆φ e + µ − = 0,1 .
06 at ∆φ e + µ − = π ).As a last differential result, we present in Fig. 12(b)the distribution in the R distance between the two b jets[see Eq. (8) for its definition]. This distribution is charac-terized by an absolute maximum around ∆R b¯b ≈ π . Thenegative NLO corrections diminish monotonically overthe analyzed spectrum. At large distance ( ∆R b¯b > − corrections diminish from +15% at ∆R b¯b ≈ ∆R b¯b ≈ π and then increase again. The NLO onesshow a similar behaviour, however, with larger slopes.The combined NLO and NLO corrections enhance theLO QCD +NLO prediction between 6% and 11%, similarlyto the case of the azimuthal distance shown in Fig. 12(a),but with a somewhat different shape. In this work we have presented the NLO correctionsto the off-shell production of ttW + at the LHC in thethree-charged-lepton channel. These include the next-to-leading-order (NLO) QCD corrections to the QCD(NLO ) and to the electroweak leading order (NLO ),as well as the NLO electroweak corrections to the QCDleading order (NLO ). It is the first time that the NLO and NLO radiative corrections are computed with fulloff-shell dependence for a physical final state, accountingfor all non-resonant, interference, and spin-correlation ef-fects.Both integrated and differential cross-sections havebeen presented and discussed in a realistic fiducial re-gion, keeping in mind the limited statistics of the LHCdata and relating the off-shell description of the processto the inclusive predictions that are available in the liter-ature. . Denner, G. Pelliccioli : Combined NLO EW and QCD corrections to off-shell ttW production at the LHC d / d M bb [ f b / G e V ] pp e + e + bb, s = 13TeV, = ( M T,t M T,t ) /2 LO QCD LO EW (×10)LO QCD +NLO LO+NLO r a t i o [ / L O Q C D ] LO EW NLO NLO NLO
100 200 300 400 500 600 700 M bb [GeV] r a t i o [ / L O Q C D + N L O ] LO QCD +NLO LO+NLO (a) Invariant mass of the b¯b system. d / d M [ f b / G e V ] pp e + e + bb, s = 13TeV, = ( M T,t M T,t ) /2 LO QCD LO EW (×10)LO QCD +NLO LO+NLO r a t i o [ / L O Q C D ] LO EW NLO NLO NLO
100 200 300 400 500 600 700 M [GeV] r a t i o [ / L O Q C D + N L O ] LO QCD +NLO LO+NLO (b) Invariant mass of the three-charged-lepton system.
Fig. 10.
Distributions in the invariant mass of the b¯b system (left) and of the three-charged-lepton system (right). Samestructure as Fig. 7.
The NLO and NLO corrections give a − .
5% and a+13% contribution, respectively, to the LO cross-section,almost independently of the choice of the factorizationand renormalization scales. The sizeable impact of NLO and NLO corrections makes it essential to combine themwith the NLO ones, in order to arrive at reliable predic-tions.The theory uncertainties from 7-point scale variationsare driven by the NLO corrections, which are the onlycorrections which feature a NLO-like scale dependence.Their inclusion reduces the scale uncertainty of the LOcross-section from 20% to 5%.The investigation of differential distributions revealsa more involved interplay among the various perturbativeorders compared with the integrated results. The NLO corrections drop by up to −
20% in most of the transverse-momentum and invariant-mass distributions, showing thetypical behaviour of EW corrections with large Sudakovlogarithms at high energies. They are rather flat for an-gular observables. The NLO corrections give a positiveenhancement between +10% and +20% (30% in somecases) to the LO cross-section in all analyzed distribu-tions, They are dominated by the ug partonic channel(formally belonging to QCD real corrections to LO EW)that embeds tW scattering. The NLO corrections, whichhave already been presented in the literature, show quitevariable patterns in the various differential K -factors. We stress that all three NLO contributions usuallygive non-flat corrections to the LO distributions, also tothe angular ones. This indicates that rescaling QCD re-sults (either LO or NLO accurate) by flat K -factors couldresult in a bad description of some LHC observables.In the light of an improved experimental descriptionof the ttW process, the inclusion of decay and off-shelleffects is mandatory. Although for sufficiently inclusiveobservables the full computation is well approximated byon-shell calculations, the inclusion of off-shell effects inthe modeling of ttW production is definitely needed whenstudying the tails of transverse-momentum and invariant-mass observables. Acknowledgements
We thank Mathieu Pellen for useful discussions onphoton-induced electroweak real corrections, TimoSchmidt for performing checks with
MoCaNLO , andJean-Nicolas Lang and Sandro Uccirati for maintaining
Recola . This work is supported by the German Fed-eral Ministry for Education and Research (BMBF) undercontract no. 05H18WWCA1. A. Denner, G. Pelliccioli : Combined NLO EW and QCD corrections to off-shell ttW production at the LHC d / d y [ f b ] pp e + e + bb, s = 13TeV, = ( M T,t M T,t ) /2 LO QCD LO EW (×10)LO QCD +NLO LO+NLO r a t i o [ / L O Q C D ] LO EW NLO NLO NLO y r a t i o [ / L O Q C D + N L O ] LO QCD +NLO LO+NLO (a) Rapidity of the muon. d / d y t [ f b ] pp e + e + bb, s = 13TeV, = ( M T,t M T,t ) /2 LO QCD LO EW (×10)LO QCD +NLO LO+NLO r a t i o [ / L O Q C D ] LO EW NLO NLO NLO y t r a t i o [ / L O Q C D + N L O ] LO QCD +NLO LO+NLO (b) Rapidity of the antitop quark.
Fig. 11.
Distributions in the rapidity of the muon (left) and of the antitop quark (right). Same structure as Fig. 7.
References
1. J. A. Dror, M. Farina, E. Salvioni, J. Serra,Strong tW Scattering at the LHC, JHEP 01(2016) 071. arXiv:1511.03674 , doi:10.1007/JHEP01(2016)071 .2. A. Buckley, C. Englert, J. Ferrando, D. J. Miller,L. Moore, M. Russell, C. D. White, Constraining topquark effective theory in the LHC Run II era, JHEP04 (2016) 015. arXiv:1512.03360 , doi:10.1007/JHEP04(2016)015 .3. O. Bessidskaia Bylund, F. Maltoni, I. Tsinikos,E. Vryonidou, C. Zhang, Probing top quark neutralcouplings in the Standard Model Effective Field The-ory at NLO in QCD, JHEP 05 (2016) 052. arXiv:1601.08193 , doi:10.1007/JHEP05(2016)052 .4. F. Maltoni, M. Mangano, I. Tsinikos, M. Zaro, Top-quark charge asymmetry and polarization in ttW ± production at the LHC, Phys. Lett. B 736 (2014) 252–260. arXiv:1406.3262 , doi:10.1016/j.physletb.2014.07.033 .5. G. Bevilacqua, H.-Y. Bi, H. B. Hartanto, M. Kraus,J. Nasufi, M. Worek, NLO QCD corrections to off-shell t ¯ tW ± production at the LHC: Correlations andAsymmetries (12 2020). arXiv:2012.01363 .6. R. Barnett, J. F. Gunion, H. E. Haber, Discoveringsupersymmetry with like sign dileptons, Phys. Lett.B 315 (1993) 349–354. arXiv:hep-ph/9306204 , doi: 10.1016/0370-2693(93)91623-U .7. M. Guchait, D. Roy, Like sign dilepton signaturefor gluino production at CERN LHC including topquark and Higgs boson effects, Phys. Rev. D 52(1995) 133–141. arXiv:hep-ph/9412329 , doi:10.1103/PhysRevD.52.133 .8. H. Baer, C.-h. Chen, F. Paige, X. Tata, Signals forminimal supergravity at the CERN large hadroncollider. 2: Multi-lepton channels, Phys. Rev. D 53(1996) 6241–6264. arXiv:hep-ph/9512383 , doi:10.1103/PhysRevD.53.6241 .9. R. Chivukula, E. H. Simmons, J. Terning, A heavytop quark and the Zb ¯ b vertex in non-commutingextended technicolor, Phys. Lett. B 331 (1994)383–389. arXiv:hep-ph/9404209 , doi:10.1016/0370-2693(94)91068-5 .10. J. Aguilar-Saavedra, R. Benbrik, S. Heinemeyer,M. P´erez-Victoria, Handbook of vectorlike quarks:Mixing and single production, Phys. Rev. D 88 (2013)094010. arXiv:1306.0572 , doi:10.1103/PhysRevD.88.094010 .11. F. M. L. Almeida Jr., Y. do Amaral Coutinho, J. A.Martins Sim˜oes, P. Queiroz Filho, C. Porto, Same-sign dileptons as a signature for heavy Majorananeutrinos in hadron hadron collisions, Phys. Lett. B400 (1997) 331–334. arXiv:hep-ph/9703441 , doi:10.1016/S0370-2693(97)00143-3 . . Denner, G. Pelliccioli : Combined NLO EW and QCD corrections to off-shell ttW production at the LHC d / d e + [ f b ] pp e + e + bb, s = 13TeV, = ( M T,t M T,t ) /2 LO QCD LO EW (×10)LO QCD +NLO LO+NLO r a t i o [ / L O Q C D ] LO EW NLO NLO NLO
20 40 60 80 100 120 140 160 e + r a t i o [ / L O Q C D + N L O ] LO QCD +NLO LO+NLO (a) Azimuthal angle between the positron and the muon. d / d R bb [ f b ] pp e + e + bb, s = 13TeV, = ( M T,t M T,t ) /2 LO QCD LO EW (×10)LO QCD +NLO LO+NLO r a t i o [ / L O Q C D ] LO EW NLO NLO NLO R bb r a t i o [ / L O Q C D + N L O ] LO QCD +NLO LO+NLO (b) R distance between the two b jets. Fig. 12.
Distributions in the azimuthal difference between the positron and the muon (left) and in the azimuthal-angle–rapiditydistance between the b jet and the ¯b jet (right). Same structure as Fig. 7.
12. J. Maalampi, N. Romanenko, Single production ofdoubly charged Higgs bosons at hadron colliders,Phys. Lett. B 532 (2002) 202–208. arXiv:hep-ph/0201196 , doi:10.1016/S0370-2693(02)01549-6 .13. M. Perelstein, Little Higgs models and their phe-nomenology, Prog. Part. Nucl. Phys. 58 (2007)247–291. arXiv:hep-ph/0512128 , doi:10.1016/j.ppnp.2006.04.001 .14. R. Contino, G. Servant, Discovering the top part-ners at the LHC using same-sign dilepton final states,JHEP 06 (2008) 026. arXiv:0801.1679 , doi:10.1088/1126-6708/2008/06/026 .15. F. Maltoni, D. Pagani, I. Tsinikos, Associated pro-duction of a top-quark pair with vector bosons atNLO in QCD: impact on ttH searches at the LHC,JHEP 02 (2016) 113. arXiv:1507.05640 , doi:10.1007/JHEP02(2016)113 .16. G. Aad, et al., Measurement of the ttW and ttZ production cross sections in pp collisions at √ s = 8 TeV with the ATLAS detector, JHEP11 (2015) 172. arXiv:1509.05276 , doi:10.1007/JHEP11(2015)172 .17. V. Khachatryan, et al., Observation of top quarkpairs produced in association with a vector bo-son in pp collisions at √ s = 8 TeV, JHEP 01(2016) 096. arXiv:1510.01131 , doi:10.1007/JHEP01(2016)096 . 18. M. Aaboud, et al., Measurement of the t ¯ tZ and t ¯ tW production cross sections in multilepton fi-nal states using 3.2 fb − of pp collisions at √ s =13 TeV with the ATLAS detector, Eur. Phys. J. C77(2017) 40. arXiv:1609.01599 , doi:10.1140/epjc/s10052-016-4574-y .19. A. M. Sirunyan, et al., Measurement of the cross sec-tion for top quark pair production in association witha W or Z boson in proton-proton collisions at √ s =13 TeV, JHEP 08 (2018) 011. arXiv:1711.02547 , doi:10.1007/JHEP08(2018)011 .20. M. Aaboud, et al., Measurement of the t ¯ tZ and t ¯ tW cross sections in proton-proton collisions at √ s =13 TeV with the ATLAS detector, Phys. Rev. D99(2019) 072009. arXiv:1901.03584 , doi:10.1103/PhysRevD.99.072009 .21. A. M. Sirunyan, et al., Measurement of top quarkpair production in association with a Z boson inproton-proton collisions at √ s = 13 TeV, JHEP03 (2020) 056. arXiv:1907.11270 , doi:10.1007/JHEP03(2020)056 .22. M. Aaboud, et al., Observation of Higgs boson pro-duction in association with a top quark pair at theLHC with the ATLAS detector, Phys. Lett. B 784(2018) 173–191. arXiv:1806.00425 , doi:10.1016/j.physletb.2018.07.035 . A. Denner, G. Pelliccioli : Combined NLO EW and QCD corrections to off-shell ttW production at the LHC
23. A. M. Sirunyan, et al., Observation of ttH pro-duction, Phys. Rev. Lett. 120 (2018) 231801. arXiv:1804.02610 , doi:10.1103/PhysRevLett.120.231801 .24. Analysis of t ¯ tH and t ¯ tW production in multileptonfinal states with the ATLAS detector, Tech. Rep.ATLAS-CONF-2019-045, CERN, Geneva (Oct 2019).URL http://cds.cern.ch/record/2693930
25. Search for Higgs boson production in association withtop quarks in multilepton final states at √ s = 13 TeV,Tech. Rep. CMS-PAS-HIG-17-004, CERN, Geneva(2017).URL https://cds.cern.ch/record/2256103
26. J. M. Campbell, R. K. Ellis, t ¯ tW ± production anddecay at NLO, JHEP 07 (2012) 052. arXiv:1204.5678 , doi:10.1007/JHEP07(2012)052 .27. M. V. Garzelli, A. Kardos, C. G. Papadopoulos,Z. Tr´ocs´anyi, t ¯ tW ± and t ¯ tZ Hadroproduction atNLO accuracy in QCD with Parton Shower andHadronization effects, JHEP 11 (2012) 056. arXiv:1208.2665 , doi:10.1007/JHEP11(2012)056 .28. S. Frixione, V. Hirschi, D. Pagani, H. S. Shao,M. Zaro, Electroweak and QCD corrections to top-pair hadroproduction in association with heavybosons, JHEP 06 (2015) 184. arXiv:1504.03446 , doi:10.1007/JHEP06(2015)184 .29. R. Frederix, D. Pagani, M. Zaro, Large NLO cor-rections in t ¯ tW ± and t ¯ tt ¯ t hadroproduction fromsupposedly subleading EW contributions, JHEP02 (2018) 031. arXiv:1711.02116 , doi:10.1007/JHEP02(2018)031 .30. R. Frederix, S. Frixione, V. Hirschi, D. Pagani,H.-S. Shao, M. Zaro, The automation of next-to-leading order electroweak calculations, JHEP 07(2018) 185. arXiv:1804.10017 , doi:10.1007/JHEP07(2018)185 .31. H. T. Li, C. S. Li, S. A. Li, Renormalization groupimproved predictions for t ¯ tW ± production at hadroncolliders, Phys. Rev. D 90 (2014) 094009. arXiv:1409.1460 , doi:10.1103/PhysRevD.90.094009 .32. A. Broggio, A. Ferroglia, G. Ossola, B. D. Pecjak,Associated production of a top pair and a W boson atnext-to-next-to-leading logarithmic accuracy, JHEP09 (2016) 089. arXiv:1607.05303 , doi:10.1007/JHEP09(2016)089 .33. A. Kulesza, L. Motyka, D. Schwartl¨ander, T. Stebel,V. Theeuwes, Associated production of a topquark pair with a heavy electroweak gauge bo-son at NLO+NNLL accuracy, Eur. Phys. J. C 79(2019) 249. arXiv:1812.08622 , doi:10.1140/epjc/s10052-019-6746-z .34. A. Broggio, A. Ferroglia, R. Frederix, D. Pa-gani, B. D. Pecjak, I. Tsinikos, Top-quark pairhadroproduction in association with a heavy bosonat NLO+NNLL including EW corrections, JHEP08 (2019) 039. arXiv:1907.04343 , doi:10.1007/JHEP08(2019)039 .35. A. Kulesza, L. Motyka, D. Schwartl¨ander, T. Stebel,V. Theeuwes, Associated top quark pair produc-tion with a heavy boson: differential cross sec-tions at NLO+NNLL accuracy, Eur. Phys. J. C 80 (2020) 428. arXiv:2001.03031 , doi:10.1140/epjc/s10052-020-7987-6 .36. S. von Buddenbrock, R. Ruiz, B. Mellado, Anatomyof inclusive t ¯ tW production at hadron colliders, Phys.Lett. B 811 (2020) 135964. arXiv:2009.00032 , doi:10.1016/j.physletb.2020.135964 .37. R. Frederix, I. Tsinikos, Subleading EW correctionsand spin-correlation effects in t ¯ tW multi-lepton sig-natures, Eur. Phys. J. C 80 (2020) 803. arXiv:2004.09552 , doi:10.1140/epjc/s10052-020-8388-6 .38. F. F. Cordero, M. Kraus, L. Reina, Top-quark pairproduction in association with a W ± gauge boson inthe POWHEG-BOX (1 2021). arXiv:2101.11808 .39. G. Bevilacqua, H.-Y. Bi, H. B. Hartanto, M. Kraus,M. Worek, The simplest of them all: t ¯ tW ± at NLOaccuracy in QCD, JHEP 08 (2020) 043. arXiv:2005.09427 , doi:10.1007/JHEP08(2020)043 .40. A. Denner, G. Pelliccioli, NLO QCD correctionsto off-shell t¯tW + production at the LHC, JHEP11 (2020) 069. arXiv:2007.12089 , doi:10.1007/JHEP11(2020)069 .41. S. Actis, A. Denner, L. Hofer, A. Scharf, S. Uccirati,Recursive generation of one-loop amplitudes in theStandard Model, JHEP 04 (2013) 037. arXiv:1211.6316 , doi:10.1007/JHEP04(2013)037 .42. S. Actis, A. Denner, L. Hofer, J.-N. Lang, A. Scharf,S. Uccirati, RECOLA: REcursive Computation ofOne-Loop Amplitudes, Comput. Phys. Commun. 214(2017) 140–173. arXiv:1605.01090 , doi:10.1016/j.cpc.2017.01.004 .43. A. Denner, S. Dittmaier, L. Hofer, COLLIER: afortran-based Complex One-Loop LIbrary in Ex-tended Regularizations, Comput. Phys. Commun.212 (2017) 220–238. arXiv:1604.06792 , doi:10.1016/j.cpc.2016.10.013 .44. A. Denner, M. Pellen, NLO electroweak correctionsto off-shell top-antitop production with leptonic de-cays at the LHC, JHEP 08 (2016) 155. arXiv:1607.05571 , doi:10.1007/JHEP08(2016)155 .45. A. Denner, J.-N. Lang, M. Pellen, S. Uccirati, Higgsproduction in association with off-shell top-antitoppairs at NLO EW and QCD at the LHC, JHEP02 (2017) 053. arXiv:1612.07138 , doi:10.1007/JHEP02(2017)053 .46. S. Catani, M. Seymour, A general algorithm forcalculating jet cross-sections in NLO QCD, Nucl.Phys. B 485 (1997) 291–419, [Erratum: Nucl. Phys.B (1998) 503–504]. arXiv:hep-ph/9605323 , doi:10.1016/S0550-3213(96)00589-5 .47. S. Dittmaier, A general approach to photon ra-diation off fermions, Nucl. Phys. B 565 (2000)69–122. arXiv:hep-ph/9904440 , doi:10.1016/S0550-3213(99)00563-5 .48. S. Catani, S. Dittmaier, M. H. Seymour, Z. Tr´ocs´anyi,The dipole formalism for next-to-leading order QCDcalculations with massive partons, Nucl. Phys. B 627(2002) 189–265. arXiv:hep-ph/0201036 , doi:10.1016/S0550-3213(02)00098-6 .49. M. Tanabashi, et al., Review of Particle Physics,Phys. Rev. D 98 (2018) 030001. doi:10.1103/PhysRevD.98.030001 . . Denner, G. Pelliccioli : Combined NLO EW and QCD corrections to off-shell ttW production at the LHC
50. D. Yu. Bardin, A. Leike, T. Riemann, M. Sachwitz,Energy-dependent width effects in e + e − annihilationnear the Z-boson pole, Phys. Lett. B206 (1988) 539–542. doi:10.1016/0370-2693(88)91627-9 .51. M. Je˙zabek, J. H. K¨uhn, QCD Corrections toSemileptonic Decays of Heavy Quarks, Nucl. Phys.B314 (1989) 1–6. doi:10.1016/0550-3213(89)90108-9 .52. L. Basso, S. Dittmaier, A. Huss, L. Oggero, Tech-niques for the treatment of IR divergences in decayprocesses at NLO and application to the top-quarkdecay, Eur. Phys. J. C76 (2016) 56. arXiv:1507.04676 , doi:10.1140/epjc/s10052-016-3878-2 .53. A. Denner, S. Dittmaier, M. Roth, D. Wackeroth,Electroweak radiative corrections to e + e − → W W → arXiv:hep-ph/0006307 , doi:10.1016/S0550-3213(00)00511-3 .54. A. Denner, S. Dittmaier, M. Roth, D. Wackeroth,Predictions for all processes e + e − → γ ,Nucl. Phys. B560 (1999) 33–65. arXiv:hep-ph/9904472 , doi:10.1016/S0550-3213(99)00437-X .55. A. Denner, S. Dittmaier, M. Roth, L. H.Wieders, Electroweak corrections to charged-current e + e − → (2012) 504]. arXiv:hep-ph/0505042 , doi:10.1016/j.nuclphysb.2011.09.001,10.1016/j.nuclphysb.2005.06.033 .56. A. Denner, S. Dittmaier, The complex-mass schemefor perturbative calculations with unstable par-ticles, Nucl. Phys. B Proc. Suppl. 160 (2006)22–26. arXiv:hep-ph/0605312 , doi:10.1016/j.nuclphysbps.2006.09.025 .57. V. Bertone, S. Carrazza, N. P. Hartland, J. Rojo,Illuminating the photon content of the protonwithin a global PDF analysis, SciPost Phys. 5(2018) 008. arXiv:1712.07053 , doi:10.21468/SciPostPhys.5.1.008 .58. A. Buckley, J. Ferrando, S. Lloyd, K. Nordstr¨om,B. Page, M. R¨ufenacht, M. Sch¨onherr, G. Watt,LHAPDF6: parton density access in the LHC preci-sion era, Eur. Phys. J. C 75 (2015) 132. arXiv:1412.7420 , doi:10.1140/epjc/s10052-015-3318-8 .59. M. Cacciari, G. P. Salam, G. Soyez, The anti- k t jetclustering algorithm, JHEP 04 (2008) 063. arXiv:0802.1189 , doi:10.1088/1126-6708/2008/04/063 .60. A. Denner, S. Pozzorini, One loop leading loga-rithms in electroweak radiative corrections. 1. Re-sults, Eur. Phys. J. C 18 (2001) 461–480. arXiv:hep-ph/0010201 , doi:10.1007/s100520100551doi:10.1007/s100520100551