Combined NLO EW and QCD corrections to off-shell ttW production at the LHC
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Combined NLO EW and QCD corrections to off-shell ttW
production at the LHC
Ansgar Denner and Giovanni Pelliccioli
University of Würzburg, Institut für Theoretische Physik und Astrophysik, Emil-Hilb-Weg 22, 97074 Würzburg (Germany)
arXiv:2102.03246v2 [hep-ph] 21 Apr 2021
Abstract. The high luminosity that will be accumulated at the LHC will enable precise differential
measurements 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 we
combine for the first time the NLO QCD and electroweak corrections to the full off-shell ttW+ production
at the LHC in the three-charged-lepton channel, including all spin correlations, non-resonant effects, and
interferences. To this end, we have computed the NLO electroweak radiative corrections to the leading
QCD order as well as the NLO QCD corrections to both the QCD and the electroweak leading orders.
1 Introduction munity has invested a noticeable effort to address this
tension, so far no explanation emerged that is capable to
The hadronic production of top-antitop pairs in associa- fill the gap between the SM predictions and the data.
tion with a W boson is an interesting process to investi- An improved modelling of the tt̄W± process is re-
gate at the Large Hadron Collider (LHC), as it represents quired to allow for the comparison of SM predictions with
an important probe of the Standard Model (SM) as well future LHC data, particularly those that will be accu-
as a window to new physics. mulated during the high-luminosity run. The increased
This process is one of the heaviest signatures measur- statistics will enable not only more precise measurements
able at the LHC. It gives access to the top-quark cou- of tt̄W± cross-sections, but also measurements of differ-
pling to weak bosons and to possible deviations from its ential distributions and in different decay channels. This
SM value [1–3]. Due to the absence of a neutral initial target can only be achieved if the theoretical descrip-
state at a lower perturbative order than next-to-next- tion of realistic final states embedding the tt̄W resonance
to-leading order (NNLO) in QCD, it is also expected structure is available.
to improve substantially the sensitivity to the tt charge Many theoretical predictions for ttW± hadro-produc-
asymmetry [4]. Polarization observables and asymmetries tion are available in the literature. The first next-to-leading
in ttW± production are capable of enhancing the sensi- order (NLO) QCD calculation for TeV-scale colliders was
tivity to beyond-the-SM (BSM) interactions featuring a performed in a spin-correlated narrow-width approxima-
chiral structure different from the one of the SM [4, 5]. tion for the semi-leptonic decay channel [26]. The match-
The hadro-production of ttW± is in general well suited ing of NLO QCD predictions to a parton shower was first
to directly search for BSM physics, in particular super- tackled for the same decay channel in Ref. [27]. A number
symmetry [6, 7], supergravity [8], technicolour [9], vector- of calculations for ttW± inclusive production (on-shell
like quarks [10], Majorana neutrinos [11] and modified top/antitop quarks and W boson) have been carried out,
Higgs sectors [12–14]. Beyond its own importance in LHC targeting charge asymmetries [4], the impact of ttW± on
searches, the ttW production is a relevant background to the associated production of tt pairs with a Higgs boson
ttH production [15]. [15] at NLO QCD, and the effects of subleading NLO
The ATLAS and CMS collaborations have measured QCD and electroweak (EW) corrections [28–30]. Soft-
and investigated ttW± production at Run 1 [16, 17] and gluon resummation up to next-to-next-to-leading loga-
Run 2 [18–21] of the LHC. This signature has been in- rithmic accuracy [31–35] and multi-jet merging [36] have
cluded as a background in the recent experimental anal- also been investigated. NLO QCD corrections to the EW
yses for ttH production [22–25]. leading order (LO) have been computed in the narrow-
The most recent experimental results based on Run 2 width approximation (NWA), accounting for complete
show a tension between data and theory predictions in the spin correlations and including parton-shower effects [37,
tt̄W modelling both in direct measurements [19, 20] and 38]. Very recently a comparison of different fixed-order
in the context of the search for tt̄ associated production Monte Carlo generators matched to parton showers has
with a Higgs boson [24, 25]. While the theoretical com- been performed for inclusive ttW± , with a focus on the2 A. Denner, G. Pelliccioli: Combined NLO EW and QCD corrections to off-shell ttW production at the LHC
two-charged lepton signature (in the NWA) [38]. The The corrections that have the largest impact on the
first studies that aim at the off-shell modelling of tt̄W NLO cross-section are of order O(αs3 α6 ), which are pure
production concern the full NLO QCD corrections in the QCD corrections to LOQCD . Following the notation of
three-charged-lepton channel [5, 39, 40]. Refs. [29, 30], we label this perturbative order as NLO1 .
The calculation of subleading NLO corrections has These corrections have recently been computed for the
been performed for inclusive production [28–30, 37, 38], full off-shell process [39, 40]. This perturbative order shows
but is still missing for realistic final states. Our present a typical NLO QCD behaviour in the scale dependence,
work targets the complete fixed-order description of the and the NLO relative corrections to LOQCD are at the
off-shell tt̄W production, combining the NLO QCD and 10–20% level, depending on the choice of the renormal-
EW corrections that have a sizeable impact at the LHC@13TeV. ization and factorization scale [40].
This paper is organized as follows. In Section 2.1 we The NLO2 corrections are known only for on-shell
describe the process under investigation, providing de- top–antitop quarks and for an on-shell W boson [28–
tails on various NLO corrections that are presented. In 30]. They are expected to give a negative contribution
Sections 2.2 and 2.3 we provide the SM input parameters of about 4.5% of the inclusive LO cross-section.
and the selection cuts used for numerical simulations, re- In the off-shell calculation, as well as in the on-shell
spectively. The integrated results at LO and NLO are one, the NLO2 order receives contributions not only from
presented in Section 3.1, while in Section 3.2 a number the EW corrections to LOQCD , but also from the QCD
of differential distributions are described, focusing on the corrections to the LO interference, although at Born-level
relative impact of various NLO corrections to the LO pre- the O(αs α7 ) contribution vanishes.
dictions. In Section 4 we draw our conclusions. Sample contributions to the virtual corrections at or-
der O(αs2 α7 ) are shown in Fig. 3. The diagram on the
left involves one-loop amplitudes of order O(gs4 g 6 ) inter-
2 Details of the calculation fered with tree-level EW amplitudes of order O(g 8 ) and is
obviously a QCD correction to the LO interference. The
2.1 Description of the process diagram on the right involves one-loop amplitudes of or-
der O(gs2 g 8 ) interfered with tree-level QCD amplitudes of
We consider the process order O(gs2 g 6 ) and could be naı̈vely classified as an EW
correction to LOQCD . However, it can also be regarded
pp → e+ νe τ + ντ µ− ν̄µ b b̄ + X , (1) as contributing to the QCD corrections to the LO inter-
ference. In fact, the IR singularities of this contribution
which receives contributions only from quark-induced par- are partially cancelled by the real photonic corrections
tonic channels at LO. Gluon–quark and photon–quark to LOQCD and partially by the real gluonic corrections
channels open up at NLO, while the pure gluonic chan- to the LO interference. Diagrams with weakly-interacting
nel enters only at NNLO in QCD. particles in the loops are more demanding from the com-
Although we consider the final state with three char- putational point of view, as the corresponding one-loop
ged leptons with different flavours, the corresponding re- amplitudes include up to 10-point functions, while the lat-
sults for the case of identical positively-charged leptons ter ones feature at most 7-point functions. A selection of
can be estimated by multiplying our results by a factor one-loop diagrams which contribute at this perturbative
1/2, up to interference contributions, which are expected order are shown in Fig. 4.
to be small. The real-radiation contributions to NLO2 corrections
In this work we focus on the production of tt̄ pairs in are computationally demanding due to the large multi-
association with a W+ boson, but the calculation of the plicity of electrically charged final-state particles. In con-
charge-conjugate process (tt̄W− ) can be performed with trast to the virtual ones, the real NLO2 corrections can be
the same techniques and no additional conceptual issues. uniquely classified into two types: the NLO EW correc-
At LO, the largest contribution is given by the QCD- tions to the LOQCD process, which involve a real photon
mediated process of order O(αs2 α6 ) (labelled LOQCD ), (see Fig. 5 left for an example), and the NLO QCD cor-
which always embeds a gluon s-channel propagator if no rections to the LO interference, which involve a real gluon
quark-family mixing is assumed (diagonal quark-mixing (see Fig. 5 right for an example). In the first class of con-
matrix with unit entries). The tree-level EW contribution tributions, the photon can be either in the final or in the
of order O(α8 ) (labelled LOEW ), despite being character- initial state. The processes with a photon in the final state
ized by many more diagram topologies, is expected to are characterized by many singular regions, as the photon
give a cross-section that is roughly 1% of the LOQCD one can become soft or collinear to any of the seven charged
owing to the ratio of EW and strong coupling constants. external particles. This results in a large number of sub-
The interference contribution, formally of order O(αs α7 ), traction counterterms that are required to ensure a stable
is identically zero due to colour algebra. In Fig. 1 we show calculation of the NLO cross-section. The real processes
sample diagrams for the QCD-mediated and purely-EW with a photon in the initial state possess a smaller num-
process. Note that the diagrams with a resonant top– ber of singular phase-space regions and are suppressed
antitop-quark pair are present also in EW tree-level con- due to the small luminosity of photons in the proton.
tributions. For on-shell production, the contribution of the photon-
At NLO, the ttW process receives contributions from induced channels to the leading-order cross-section is at
four different perturbative orders, as depicted in Fig. 2. the sub-percent level [28]. The QCD corrections to theA. Denner, G. Pelliccioli: Combined NLO EW and QCD corrections to off-shell ttW production at the LHC 3
e+ e+ e+
W+
u u u
νe b W + e+ νe b νe τ+
W+ W+
τ+ u τ+
νe W+ ντ
t τ+ t
ντ W+ ντ
g W+ Z/γ W+ H W−
Z ντ
d̄ d̄ Z ν̄µ
t̄ − µ− ν̄µ t̄ − µ− d̄
W W− W Z µ−
µ− b̄
ν̄µ g b̄ ν̄µ
d̄ b
b̄ b b̄
Fig. 1. Sample diagrams contributing to LOQCD (left) and to LOEW (right) cross-sections for off-shell ttW+ production in
the three-charged-lepton channel.
LOQCD LOEW
αs2 α6 αs α7 α8
QCD EW QCD QCD EW
αs3α6 αs2α7 αs α8 α9
NLO1 NLO2 NLO3 NLO4
Fig. 2. Contributing perturbative orders at LO and NLO for ttW hadro-production in the three-charged-lepton channel.
e+ e+ e+ e+
W+ νe b b W+ νe b b
t νe t νe
u τ+ τ+ W+ u τ+ τ+ W+
t u t u
g W+ W+ Z/γ W+ W+
ντ ντ Z/γ d ντ ντ g d
d d
t µ− µ− t µ− µ−
g g
t̄ W− W− t̄ t̄ W− W− t̄
d̄ ν̄µ ν̄µ d̄ d̄ ν̄µ ν̄µ d̄
b̄ b̄ b̄ b̄
O(gs4 g 6) O(g 8) O(gs2 g 8) O(gs2 g 6)
Fig. 3. Sample contributions to the virtual corrections at order O(αs2 α7 ) 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 either
the QCD corrections to the LO interference or the EW corrections to the LOQCD (right).
LO interference are non-vanishing only if the radiated LO colour structure. Therefore, the only NLO corrections
gluon is emitted by an initial-state light quark and ab- that contribute at order O(αs α8 ) are genuine QCD cor-
sorbed by a final-state b quark or top quark (or the other rections to the LOEW cross-section. This order is labelled
way around). A sample contribution is shown in Fig. 5 as NLO3 in Fig. 2. By simply counting the powers of αs
right. These corrections, although necessary to account the NLO3 corrections are expected to give a smaller con-
for all O(αs2 α7 ) contributions, turn out to be very small, tribution than the NLO2 ones. However, at the inclusive
as detailed in Section 3. level [29] and in the narrow-width approximation [37, 38],
To sum up, the full set of real partonic channels that they are noticeably larger than the NLO2 ones. This re-
contribute to the NLO2 corrections is sults from the fact that this perturbative order is domi-
nated by hard real radiation diagrams in the gluon-quark
u d¯ → e+ νe µ− ν̄µ τ + ντ b b̄ γ
partonic channel that embed the tW scattering process
γ u → e+ νe µ− ν̄µ τ + ντ b b̄ d EW corr. to LOQCD [1]. Sample diagrams are shown in Fig. 6. Thanks to the
γ d¯ → e+ νe µ− ν̄µ τ + ντ b b̄ ū
genuine QCD nature of the NLO3 corrections, it is pos-
sible to match them to a QCD parton shower with no
and
subtleties due to EW corrections, as it has been done in
u d¯ → e+ νe µ− ν̄µ τ + ντ b b̄ g
Refs. [37, 38].
g u → e+ νe µ− ν̄µ τ + ντ b b̄ d QCD corr. to LO int.,
g d¯ → e+ νe µ− ν̄µ τ + ντ b b̄ ū
The last NLO perturbative order, O(α9 ), is furnished
where u and d stand for up-type and down-type quarks, by the EW corrections to the LOEW process. It has been
respectively (of the first and second generation). shown at the inclusive level that such contributions are
The vanishing LO interference implies that the cor- at the sub-permille level [30], as expected by naı̈ve power
responding EW corrections vanish as well, since addi- counting. Even with a substantially larger data set than
tional EW propagators (virtual contributions) and ra- the one of Run 2 (i.e. 3000 fb−1 at the high-luminosity
diated photons (real contributions) do not modify the LHC) these EW effects are out of reach in a realistic4 A. Denner, G. Pelliccioli: Combined NLO EW and QCD corrections to off-shell ttW production at the LHC
τ+ ντ
e+ νe
νe τ+ ντ ν̄µ
+
W
e+ u µ−
u W u νe u νe
ντ ντ ντ Z/γ Z/γ
W+ W Z/γ e+ νe W νµ
H W e+
τ+ e
d W d d W d W
νe
b W g b
t g t t g τ
g b b b Z/γ
d̄ d̄ µ− d̄ b ντ
W− µ− d̄ W−
t̄ µ−
ν̄µ τ+
ν̄µ ν̄µ
b̄ b̄ b̄ b̄ b
Fig. 4. One-loop diagrams of order O(gs2 g 8 )
contributing to the EW virtual corrections (NLO2 ) to off-shell ttW production
in the three-charged-lepton channel. From left to right: sample diagrams involving 7-, 8-, 9- and 10-point functions.
e+ e+ e+ e+
γ γ νe W+ g g νe W+
νe νe
u W+ b b u u W+ b u
b d
b τ +
τ + b b τ +
τ +
d d
t ντ ντ t d t ντ ντ t d
+ + +
g W W g g W W+ Z/γ
d̄ d̄ d̄ d̄
t̄ µ− µ− t̄ µ− µ−
W− W− W− W−
t̄ t̄
ν̄µ ν̄µ
ν̄µ ν̄µ
b̄ b̄ b̄ b̄
Fig. 5. Sample contribution to the real corrections at order O(αs2 α7 ) for ttW production in the three-charged-lepton channel:
photonic corrections to LOQCD (left) and gluonic corrections to the LO interference (right).
b̄ b̄ b̄
ν̄µ ν̄µ ν̄µ
W − µ− W − µ− W − µ−
b b b
g g g
t νe t νe t νe
t t t
W+ W+ W+
e+ e+ e+
Z/γ H b
+ + + + +
W W ντ W W ντ W+ W ντ
u u u
τ+ τ+ τ+
d d d
+ + − +
Fig. 6. Sample diagrams for the partonic channel gu → b b̄ e νe τ ντ µ ν̄µ d that contain tW scattering as a subprocess and
contribute to NLO3 corrections to ttW production in the three-charged-lepton channel.
fiducial region. Therefore, we are not providing results sector and use a unit quark-mixing matrix. The three
for this order. charged leptons that we consider in the final state are
In the following we focus on the first three NLO per- massless and characterized by three different flavours.
turbative orders. Tree-level and one-loop SM amplitudes The on-shell masses and widths of weak bosons are set to
are computed with the Recola matrix-element provider the following values [49],
[41, 42]. For the tensor reduction and evaluation of loop
OS OS
integrals we use the Collier library [43]. The multi- MW = 80.379 GeV, ΓW = 2.085 GeV ,
channel Monte Carlo integration is performed with Mo- MZOS = 91.1876 GeV, ΓZOS = 2.4952 GeV , (2)
CaNLO, a generator that has already been used to com-
pute the NLO QCD corrections to ttW [40] and the NLO and then translated into their pole values [50] that enter
EW corrections to several LHC processes involving top the Monte Carlo simulations. The Higgs-boson mass and
quarks [44, 45]. The subtraction of infrared and collinear width are fixed, following Ref. [49], to
singularities is carried out using the dipole formalism of
Refs. [46–48] both for QCD and for EW corrections. The MH = 125 GeV, ΓH = 0.00407 GeV . (3)
initial-state collinear singularities are absorbed in the par-
ton distribution functions (PDFs) in the MS factorization We have computed the LO top-quark width according to
scheme. Ref. [51], using the pole values for the W-boson mass and
width. The NLO top-quark width is obtained applying
the NLO QCD and EW correction factors of Ref. [52] to
2.2 Input parameters the LO width. The numerical values read
mt = 173.0 GeV ,
We consider proton–proton collisions at a centre-of-mass
energy of 13 TeV. We neglect flavour mixing in the quark ΓtLO = 1.4437 GeV , ΓtNLO = 1.3636 GeV . (4)A. Denner, G. Pelliccioli: Combined NLO EW and QCD corrections to off-shell ttW production at the LHC 5
The top-quark width is kept fixed when performing var- The second dynamical choice, already used in Ref. [40],
iations of the factorization and renormalization scales. is based on the transverse masses of the top and antitop
The EW coupling is treated in the Gµ scheme [53], quarks. Due to the ambiguity in choosing the `+ ν` pair
√ that results from the top quark, we pick the pair of lep-
M2
2 2 tons that when combined with the bottom quark forms
α= Gµ MW 1− W , (5) an invariant mass closest to the top-quark mass. We con-
π MZ2
sider two different central-scales based on this dynamical
where Gµ = 1.16638 · 10−5 GeV−2 is the Fermi constant. choice,
The masses of weak bosons and of the top quark, and rq (d)
therefore also the EW mixing angle, are treated in the (d)
q
(e) µ
µ0 = m2t + pT,t
2 m2t + pT,t
2 , µ0 = 0 .
complex-mass scheme [53–56]. 2
Both for LO and NLO predictions, we employ (10)
NNPDF 3.1 luxQED PDFs [57]. Using these PDFs, the The scale-dependence study performed in Ref. [40] shows
photon contribution is properly accounted for in the evo- (e)
that using µ0 as a central scale reduces the scale de-
lution. The evaluation of PDFs and the running of the pendence based on the conventional 7-point scale varia-
strong coupling are obtained via the LHAPDF6 inter- tion and gives smaller QCD corrections than using µ0 .
(d)
face [58]. The employed PDF set uses αs (MZ ) = 0.118 (e)
and one QCD loop in the calculation of the αs evolution. Therefore, the choice µ0 is preferable for the study of
the impact of NLO2 and NLO3 corrections, which is the
focus of this work.
2.3 Selection cuts The scale uncertainties shown in the following results
are based on 7-point scale variations, i.e. rescaling the
Coloured partons with |η| < 5 are clustered into jets by central factorization and renormalization scale by the fac-
means of the anti-kt algorithm [59] with resolution radius tors
R = 0.4. The same algorithm but with R = 0.1 is applied
to cluster photons into charged particles. (0.5, 0.5) , (1, 0.5) , (0.5, 1) , (1, 1) , (1, 2) , (2, 1) , (2, 2) ,
We choose cuts that mimic the fiducial selections ap-
while keeping the NLO QCD top-quark width fixed.
plied by ATLAS in Ref. [20] and that have already been
In the following we present the results combining NLO
used for the study of NLO QCD corrections to the same
corrections with the so-called additive approach,
final state [40]. We select events with exactly two b jets
(assuming perfect b-tagging efficiency) that are required σLO+NLO = σLOQCD + σNLO1 + σNLO2
to satisfy
+ σLOEW + σNLO3 . (11)
pT,b > 25 GeV , |ηb | < 2.5 . (6)
This approach is exact at the order of truncation of the
Furthermore, we ask for three charged leptons that fulfil perturbative expansion. Furthermore, it represents a nat-
standard acceptance and isolation cuts, ural choice for our process, as the combination involves
NLO corrections to two different leading orders that do
pT,` > 27 GeV , |η` | < 2.5 , ∆R`b > 0.4 , (7) not interfere.
where the R distance is defined as the sum in quadrature
of the azimuthal and rapidity separations, 3.1 Integrated cross-sections
q
∆Rij = ∆φij2 + ∆yij2 . (8) In Table 1, we present the integrated cross-sections in the
fiducial region defined in Section 2.3.
We do not constrain the missing transverse momentum The leading corrections to the LOQCD cross-section
and do not apply any veto to additional light jets. are expected to come from the corresponding pure QCD
radiative corrections (NLO1 ). Their inclusion has been
proved to decrease the theoretical uncertainty due to scale
3 Results variations and to stabilize the perturbative convergence
for this process. Nonetheless, their impact depends on
For the factorization and renormalization scale (µF = the choice of the scale, as already pointed out both in
µR ), we consider two different dynamical choices that inclusive [28, 29] and off-shell [39, 40] computations. In
(c)
have proved to behave better than a fixed scale [39, 40]. fact, with the resonance-blind dynamical choice µ0 , the
The first one, introduced in Ref. [39], depends on the NLO1 corrections give a 6.6% enhancement to the LOQCD
transverse-momentum content of the final-state particles, (d)
cross-section, with the resonance-aware choices, µ0 and
regardless of the top–antitop resonances,1 (e)
µ0 , they give a 18% and a 0.4% correction, respectively.
Note that, at variance with Ref. [40] in this paper we
P
(c) HT pT,miss + i=b,` pT,i
µ0 = = . (9) compute both LO and NLO predictions with the NLO
3 3
top-quark width, which gives roughly a 12% enhancement
1
We use the same notation for the scales as in Ref. [40]. to the LO cross-section.6 A. Denner, G. Pelliccioli: Combined NLO EW and QCD corrections to off-shell ttW production at the LHC
(c) (d) (e)
µ0 µ0 µ0
perturbative order σ (fb) ratio σ (fb) ratio σ (fb) ratio
LOQCD (αs2 α6 ) 0.2218(1)+25.3%
−18.8% 1 +23.9%
0.1948(1)−18.1% 1 0.2414(1)+26.2%
−19.3% 1
LOEW (α ) 8
0.002164(1)+3.7%
−3.6% 0.010 0.002122(1)+3.7%
−3.6% 0.011 0.002201(1)+3.7%
−3.6% 0.009
NLO1 (αs3 α6 ) 0.0147(6) 0.066 0.0349(6) 0.179 0.0009(7) 0.004
NLO2 (αs2 α7 ) -0.0122(3) -0.055 -0.0106(3) -0.054 -0.0134(4) -0.056
8
NLO3 (αs α ) 0.0293(1) 0.131 0.0263(1) 0.135 0.0320(1) 0.133
LOQCD +NLO1 0.2365(6)+2.9%
−6.0% 1.066 0.2297(6)+5.5%
−7.3% 1.179 0.2423(7)+3.5%
−5.2% 1.004
LOQCD +NLO2 0.2094(3)+25.0%
−18.7% 0.945 +23.8%
0.1840(3)−17.9% 0.946 0.2277(4)+25.9%
−19.2% 0.944
LOEW +NLO3 0.03142(4)+22.2%
−16.8% 0.141 0.02843(6)+20.5%
−15.6% 0.146 0.03425(7)+22.8%
−17.0% 0.142
LO+NLO 0.2554(7)+4.0%
−6.5% 1.151 0.2473(7)+6.3%
−7.6% 1.270 0.2628(9)+4.3%
−5.9% 1.089
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 LOQCD cross-section. The scale uncertainties from
7-point scale variations (in percentage) are listed for LO and NLO cross-sections. The result in the last row is the sum of all
LO cross-sections and NLO corrections, namely LOQCD + LOEW + NLO1 + NLO2 + NLO3 .
The NLO2 corrections are negative and amount to and we cluster photons into charged particles with
about −5.5% of the LOQCD cross-section for all scale isolation radius R = 0.1;
choices. Such supposedly subleading corrections have a – we employ the same dynamical scale as in Ref. [30],
sizeable impact on the fiducial NLO cross-section, and but using the kinematics after photon recombination
this is likely due to large EW Sudakov logarithms en- and choosing the top-quark candidate with the same
hancing the cross-section in the high-energy regime [60]. invariant-mass prescription as for the calculation of
This is supported by the fact that the average partonic (d) (e)
the scales µ0 and µ0 .
centre-of-mass energy and HT are quite high, 850 GeV
and 520 GeV, respectively. A crude estimate of the Su- The obtained inclusive cross-sections,
dakov logarithms gives a result which is of the same or-
der of magnitude as the full NLO2 corrections we have σLOQCD = 262+23.7%
−18.1% fb,
obtained for this process.
σLOQCD +NLO2 = 254+23.4%
−17.8% fb , (12)
The impact of QCD corrections that can be uniquely
attributed to the LO interference is very small, both for
real and for virtual corrections, accounting respectively exhibit NLO2 corrections of −3% of LOQCD , which is
for 5% and for less than 1% of the total NLO2 result. not far from the −4.5% of Refs. [28, 30]. The remaining
At O(αs2 α7 ), we have also included the contribution discrepancy should be due to both the additional cuts
from photon-initiated partonic channels, which are posi- we have applied and the non-resonant effects that are
tive and account for about 0.1% of the LO QCD cross- included in our full calculation, while being absent in on-
section. As already observed at the inclusive level [28], shell calculations. The comparison of results in the fidu-
this contribution is very small and its effect will be hardly cial and the inclusive setup reveals that the NLO2 cor-
visible even at the high-luminosity run of LHC (they will rections are more sizeable for realistic final states and in
√ the presence of reasonable fiducial cuts.
yield about 1 event for s = 13 TeV and L = 3000 fb−1 ).
In inclusive calculations with on-shell top–antitop Coming back to our default fiducial setup (see
quarks, the NLO2 corrections were found to give a −4.5% Section 2.3), the NLO3 perturbative order is dominated
contribution to the inclusive production cross-section by the real radiation in the ug partonic channel, owing to
[28, 30]. In order to compare our results with those of a PDF enhancement and the tW scattering embedded in
Ref. [30], we have performed a full off-shell calculation this channel. These real corrections account for 85% of the
in a very inclusive setup, and divided by the branching NLO3 corrections, and are one order of magnitude larger
ratios for the decays of the top and antitop quarks and than the corresponding leading order LOEW . The total
of the W boson. The setup is the same as the one of NLO3 corrections amount to 13% of the LOQCD cross-
Ref. [30], up to a few unavoidable differences: section, almost independently of the scale choice. This
confirms the 12% effect obtained in the case of on-shell
– we use finite top-quark and W widths (same values top and antitop quarks [30].
as those of Section 2.2) and we include Higgs-boson It is worth stressing that the inclusion of NLO2 and
contributions; NLO3 corrections gives a noticeable effect to the tt̄W
– we apply a minimum invariant-mass cut of 5 GeV to cross-section. Therefore such corrections must definitely
the bb̄ system to protect from infrared singularities be accounted for in experimental analyses. Furthermore,A. Denner, G. Pelliccioli: Combined NLO EW and QCD corrections to off-shell ttW production at the LHC 7
their relative contribution to the LO result is rather inde- but its analysis is useful to compare the full off-shell cal-
pendent of the scale choice, while the NLO1 corrections culation with the on-shell ones. The negative growth of
are much more scale dependent. the NLO2 corrections behaves very similarly in the inclu-
As a last comment of this section, we point out that sive calculations, as can be seen for example in Ref. [28]
the scale uncertainty of the combined LO+NLO cross- (figure 5 therein). This confirms that for sufficiently in-
section is driven by the NLO1 corrections which reduce clusive variables the NLO EW effects are dominated by
the LOQCD uncertainty roughly from 20% to 5%. Due to contributions with resonant top and antitop quarks. The
their EW nature, the NLO2 corrections do not diminish NLO1 corrections increase by roughly 25% in the consid-
the scale uncertainty of the corresponding leading order ered spectrum.
(LOQCD ). The NLO3 corrections exceed the correspond- In both transverse-momentum distributions of
ing pure EW LO process and, thus, imply a LO-like scale Fig. 7, the inclusion of subleading NLO corrections
dependence for LOEW +NLO3 . (NLO2 , NLO3 ) gives a decreasing effect towards large
So far, we have focused on the relative contributions of transverse momenta to the NLO QCD cross-section. In
various NLO corrections to the fiducial tt̄W cross-section. fact, the ratio between the combined LO + NLO cross-
However, the interplay among different corrections can be section and the LOQCD + NLO1 one ranges between 1.10
rather different for more exclusive observables. Therefore, and 1.15 for small transverse momenta and drops below
it is essential to study differential distributions. 1 already at moderate transverse momenta. We have
checked numerically that this conclusion can be drawn
(c) (d)
also for other scale choices (µ0 , µ0 ), confirming the
3.2 Distributions almost scale-independent impact of the NLO2 and NLO3
corrections.
In the following we present a number of relevant dis- In Fig. 8 we consider the distributions in the trans-
tributions focusing on the impact of the various NLO verse momentum of the tt̄ and the bb̄ system. The former
corrections relative to the LOQCD cross-sections. Since is not measurable at the LHC but can be reconstructed
in most of the LHC experimental analyses the theoret- from Monte Carlo truth, choosing the positively-charged
ical predictions are NLO QCD accurate, we also com- lepton–neutrino pair that best reconstructs the top-quark
ment on the distortion of NLO QCD distribution shapes mass when combined with the momentum of the bottom
(LOQCD + NLO1 in our notation) due to the inclusion quark. The latter variable is directly observable at the
of NLO2 and NLO3 corrections. We choose to present LHC.
(e)
all differential distributions with the µ0 scale.2 The The NLO1 corrections to the tt̄ transverse-momentum
shown scale uncertainties are based on the predictions for distribution [Fig. 8(a)] have already been investigated in
LOQCD + NLO1 , normalized to predictions at the central Ref. [40]: these QCD corrections grow monotonically and
scale in the relative plots. become dramatically large and positive at high pT,tt̄ . Note
We start by presenting transverse-momentum distri- that at LO this variable coincides with the pT of the re-
butions in Figs. 7–8. coiling W+ boson, and therefore is sensitive to the real
In Fig. 7(a) we consider the distribution in the trans- QCD radiation which is not clustered into b jets (and that
verse momentum of the positron, which is precisely mea- cannot be clustered to the W+ -boson decay products).
surable at the LHC. Since we also include EW corrections The NLO1 corrections receive a sizeable contribution by
(NLO2 ), the positron is understood as dressed (a radiated the gq/gq̄ partonic channels, which are enhanced by the
photon could be clustered into the positron). The distri- gluon PDF. In contrast, the NLO2 ones feature a typical
bution peaks around 50 GeV, where the relative impact of NLO EW behaviour in the tail of the distribution, giving
QCD and EW corrections follows straight the integrated a negative and monotonically decreasing correction to the
results. Relative to the LOQCD , all three radiative cor- LO cross-section. Differently from the QCD corrections,
rections drop in a monotonic manner. Nonetheless, the the additional photon can be clustered into any of the ex-
decrease of NLO3 corrections is very mild (14% below ternal charged particles, thus also into the decay products
50 GeV, 10% at 380 GeV), while the NLO1 and NLO2 of the recoiling W boson. Furthermore, the cross-section
corrections decrease steeper: the former become nega- is not enhanced by the γq/γ q̄ partonic channels due to the
tive around 80 GeV and give −9% at 380 GeV, the lat- very small photon luminosity in the proton. The NLO3
ter become lower than −10% already at moderate pT,e+ contribution is positive in the whole analyzed spectrum,
(200 GeV). The behaviour of EW corrections in the tails and increases from 6% (below 50 GeV) to a maximum of
of this distribution is likely driven by the impact of Su- 25% around pT,tt̄ = 2mt , then it slowly decreases in the
dakov logarithms, which become large at high pT . large-pT region. Relative to the LOQCD + NLO1 result,
The same behaviour of the NLO2 and NLO3 correc- the combination of all other corrections gives an effect
tions characterizes also the distribution in the transverse which is about 10% in the soft-pT region and diminishes
momentum of the antitop quark, shown in Fig. 7(b). The towards negative values at large pT .
antitop momentum is computed as the sum of the mo- The transverse momentum of the bb̄ system
menta of the muon, its corresponding antineutrino, and [Fig. 8(b)] is correlated to the one of tt̄ system. The NLO2
the antibottom quark. This is not observable at the LHC, corrections to this observable behave in the same manner
as those for the pT,tt̄ distributions, giving a −20% con-
2
In Ref. [40] the scale for the differential distributions is tribution around 400 GeV. The NLO3 corrections grow
(d)
µ0 , i.e. exactly twice the default scale used here. monotonically from +10% (at low transverse momentum)8 A. Denner, G. Pelliccioli: Combined NLO EW and QCD corrections to off-shell ttW production at the LHC
pp e+ e + bb, s = 13 TeV, 0 = (MT, t MT, t)1/2/2 pp e+ e + bb, s = 13 TeV, 0 = (MT, t MT, t)1/2/2
LOQCD 10 3 LOQCD
LOEW (×10) LOEW (×10)
10 3 LOQCD + NLO1 LOQCD + NLO1
LO+NLO LO+NLO
10 4
d /dpT, e + [fb/GeV]
d /dpT, t [fb/GeV]
10 4
10 5
10 5
10 6
0.3 LOEW NLO2 0.3
0.2 NLO1 NLO3 0.2
ratio [/LOQCD]
ratio [/LOQCD]
0.1 0.1
0.0 0.0
0.1 0.1
0.2 0.2 LOEW NLO2
0.3 0.3 NLO1 NLO3
ratio [/ LOQCD + NLO1]
ratio [/ LOQCD + NLO1]
1.15 LOQCD + NLO1 LO + NLO 1.15 LOQCD + NLO1 LO + NLO
1.10 1.10
1.05 1.05
1.00 1.00
0.95 0.95
50 100 150 200 250 300 350 100 200 300 400 500 600 700
pT, e + [GeV] pT, t [GeV]
(a) Transverse momentum of the positron. (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: differential
cross-sections (in fb) for LOQCD , LOEW (scaled by a factor 10), LOQCD + NLO1 and for the complete NLO, which is the sum
of all LO cross-sections and all NLO corrections. Middle panel: ratio of the LOEW , NLO1 , NLO2 , and NLO3 corrections over
the LOQCD cross-section. Bottom panel: ratio of the LO + NLO cross-section over the LOQCD + NLO1 one. Uncertainties from
7-point scale variations are shown in all panels for the LOQCD + NLO1 predictions.
to +30% (around 400 GeV). In the soft part of the spec- rections (unclustered photons). At values larger than the
trum (pT,bb̄ < 150 GeV), the NLO1 corrections are rather top-quark mass, the distribution receives an increasingly
flat, while in the large-pT region they grow positive and positive contribution from NLO1 corrections, while the
become very large, similarly to the pT,tt̄ distribution. The NLO2 ones give an almost flat correction of −10% to
overall NLO corrections are very small below 150 GeV due the LOQCD cross-section. The LOEW contribution shows
to mutual cancellations among the three contributions, a slightly wider distribution than the LOQCD one. This
while at larger transverse momenta the corrections are could be attributed to the relatively larger contribution
dominated by the NLO1 contribution. Furthermore, rela- of non-resonant background diagrams in the LOEW con-
tively to the LOQCD + NLO1 distribution, the combined tribution. Nonetheless, the impact of this difference on
NLO corrections give a flat and positive effect between the full distribution is almost invisible owing to the very
7% and 8% in the whole analyzed spectrum. small size of the LOEW contribution. The NLO3 correc-
In all analyzed transverse-momentum distributions of tions behave differently from the NLO1 ones, giving a
Figures 7–8, the LOEW contribution increases monotoni- rather flat enhancement of the fiducial LO cross-section,
cally (relatively to the LOQCD one) but never exceeds 3% which is minimal around the peak (+11% at mt ) and
of the LOQCD cross-section. mildly increases towards the tails (+20% at 200 GeV,
+30% at 150 GeV). This is due to the very large con-
In Fig. 9(a), we display the distribution in the invari- tribution of the u(c)g partonic channel, which has a light
ant mass of the antitop quark, which in our setup can be d(s) in the final state that cannot come from the radiative
reconstructed from the Monte Carlo truth. The lineshape decay of the top or of the antitop quark (differently from
is dominated by the Breit-Wigner distribution of the lep- final-state gluons). As can be seen in the bottom panel of
tonic decay of the antitop quark. The NLO1 corrections Fig. 9(a), the total LO + NLO result is 12% higher than
are negative at the peak while below the pole mass they the LOQCD + NLO1 one below the top-quark mass. This
give a very large enhancement to the LO result. Such a enhancement is smaller at (4%) and above (7%) the top
radiative tail, coming from unclustered real radiation, is mass.
also present, though less sizeable, in the NLO EW cor-A. Denner, G. Pelliccioli: Combined NLO EW and QCD corrections to off-shell ttW production at the LHC 9
pp e+ e + bb, s = 13 TeV, 0 = (MT, t MT, t)1/2/2 pp e+ e + bb, s = 13 TeV, 0 = (MT, t MT, t)1/2/2
10 3 LOQCD LOQCD
LOEW (×10) 10 3 LOEW (×10)
LOQCD + NLO1 LOQCD + NLO1
LO+NLO LO+NLO
d /dpT,+ttbest [fb/GeV]
d /dpT, bb [fb/GeV]
10 4
10 4
10 5
10 5
10 6
1.0 1.0
0.8 LOEW NLO2 0.8 LOEW NLO2
0.6 NLO1 NLO3 0.6 NLO1 NLO3
ratio [/LOQCD]
ratio [/LOQCD]
0.4 0.4
0.2 0.2
0.0 0.0
0.2 0.2
0.4 0.4
ratio [/ LOQCD + NLO1]
ratio [/ LOQCD + NLO1]
1.15 LOQCD + NLO1 LO + NLO 1.15 LOQCD + NLO1 LO + NLO
1.10 1.10
1.05 1.05
1.00 1.00
0.95 0.95
100 200 300 400 500 600 700 50 100 150 200 250 300 350 400
pT,+ttbest [GeV] pT, bb [GeV]
(a) Transverse momentum of the tt̄ system. (b) Transverse momentum of the bb̄ system.
Fig. 8. Distributions in the transverse momentum of the reconstructed tt̄ system (left) and of the bb̄ system (right). Same
structure as Fig. 7.
Another variable that is often investigated in LHC enhance the LOQCD cross-section by 11% to 14% every-
analyses is HT , whose definition is given in Eq. (9). The where in the analyzed invariant-mass range. The NLO2
LO and NLO distributions in this observable are shown in contribution has a similar behaviour as the one found for
Fig. 9(b). As in the transverse-momentum distributions the previous variables, growing negative towards the tail
studied above, the NLO2 radiative corrections decrease of the distribution.
monotonically towards large values of HT (about −20% The distribution in the invariant mass of the three-
for HT ≈ 1.2 TeV). The LOEW contribution grows to 5% charged-lepton system is considered in Fig. 10(b). The
of the LOQCD cross-section at 1.5 TeV, where the NLO behaviour of the NLO2 and NLO3 corrections follows
cross-section is two orders of magnitude lower than its closely the one for the bb̄ system, apart from a less
value at the maximum of the distribution. The NLO3 steep decrease of the EW corrections towards large in-
corrections are rather flat and enhance the LOQCD re- variant masses. These corrections are at the −10% level
sult between 10% and 15%. The NLO1 corrections are for masses larger than 500 GeV. The NLO1 corrections
characterized by a non-flat shape that is increasing for vary by hardly more than 10% in the studied range.
HT < 800 GeV from −10% to +25% and decreasing in As shown in the bottom panels of Fig. 10, both for the
the rest of the considered spectrum. We further observe bb̄ system and for the three-charged-lepton system, the
that the combination of the three NLO perturbative or- inclusion of NLO2 and NLO3 corrections (as well as of
ders yields an almost vanishing correction in the soft re- LOEW , though hardly visible) gives a non-flat correction
gion of the spectrum, while in the tail of the distribution to the NLO QCD invariant-mass distributions, decreas-
the overall correction is dominated by the NLO2 contri- ing monotonically from +12% to zero in the considered
bution for our scale choice. In a similar fashion as in other spectra.
transverse-momentum distributions, the ratio of the com-
After presenting transverse-momentum and invariant-
bined LO+NLO result over the LOQCD +NLO1 decreases
mass distributions, we switch to some relevant angular
monotonically from 1.15 to 0.95 in the analyzed range.
variables. In Fig. 11(a) and Fig. 11(b) we display the dis-
In Fig. 10 we study more invariant-mass distributions. tributions in the rapidity of the muon and of the antitop
The distribution in the invariant mass of the two-b-jet quark, respectively. Since these two variables are corre-
system [Fig. 10(a)] is characterized by rather flat QCD lated (the dominant resonant structure involves the decay
corrections (NLO1 and NLO3 ). The NLO3 corrections t̄ → b̄µ− ν̄µ ), the muon rapidity, which is precisely mea-10 A. Denner, G. Pelliccioli: Combined NLO EW and QCD corrections to off-shell ttW production at the LHC
pp e+ e + bb, s = 13 TeV, 0 = (MT, t MT, t)1/2/2 pp e+ e + bb, s = 13 TeV, 0 = (MT, t MT, t)1/2/2
10 1
LOQCD LOQCD
10 3
LOEW (×10) LOEW (×10)
LOQCD + NLO1 LOQCD + NLO1
10 2 LO+NLO LO+NLO
10 4
d /dHT [fb/GeV]
d /dMt [fb/GeV]
10 3
10 5
10 4
10 6
101.05
LOEW NLO2 0.3 LOEW NLO2
0.8
0.6 NLO1 NLO3 0.2 NLO1 NLO3
ratio [/LOQCD]
ratio [/LOQCD]
0.4 0.1
0.2 0.0
0.0 0.1
0.2 0.2
0.4 0.3
ratio [/ LOQCD + NLO1]
ratio [/ LOQCD + NLO1]
1.15 LOQCD + NLO1 LO + NLO 1.15 LOQCD + NLO1 LO + NLO
1.10 1.10
1.05 1.05
1.00 1.00
0.95 0.95
160 170 180 190 400 600 800 1000 1200 1400
Mt [GeV] HT [GeV]
(a) Invariant mass of the antitop quark. (b) HT variable.
Fig. 9. Distributions in the invariant mass of the antitop quark (left) and in the HT variable (right). Same structure as Fig. 7.
surable at the LHC, represents a suitable proxy for the with small azimuthal separations. The NLO2 corrections
rapidity of the corresponding antitop quark (which can are negative and roughly constant (−5% to −7%) over
only be reconstructed from Monte Carlo truth). Note that the full angular range, while the NLO3 contribution de-
the muon rapidity is sharply cut at ±2.5 by fiducial se- creases monotonically from +18% to +11% relatively to
lections, while this is not the case for the antitop quark. the LOQCD result. As already observed in Ref. [40], the
However, thanks to the rapidity cut applied to b jets, the NLO1 correction to the LO QCD cross-section decreases
cross-section is strongly suppressed for |yt̄ | > 2.5. Both with an almost constant negative slope over the full range.
the muon and the antitop quark are produced preferably The overall NLO corrections to the LOQCD cross-section
in the central region. The NLO2 corrections are rather are positive everywhere except in the vicinity of the peak
flat, giving between −4% and −8% decrease to the LO at ∆φe+ µ− = π. Relative to the LOQCD + NLO1 pre-
cross-section. The relative NLO1 corrections to the muon- diction, the combination of NLO2 and NLO3 corrections
rapidity distribution are characterized by a large vari- gives a pretty flat enhancement (1.11 at ∆φe+ µ− = 0,
ation (about 35%) in the available range. Relative to 1.06 at ∆φe+ µ− = π).
LOQCD , the differential NLO3 corrections have a simi-
lar shape as the NLO1 ones, giving in the whole rapidity As a last differential result, we present in Fig. 12(b)
range a positive correction (8% in the forward regions, the distribution in the R distance between the two b jets
16% in the central region). Almost identical results are [see Eq. (8) for its definition]. This distribution is charac-
found in the rapidity distribution of the antitop quark. terized by an absolute maximum around ∆Rbb̄ ≈ π. The
Owing to the NLO3 corrections, the ratio between the negative NLO2 corrections diminish monotonically over
complete NLO prediction and the LOQCD +NLO1 one has the analyzed spectrum. At large distance (∆Rbb̄ > 5)
a maximum of 1.11 in the central region and diminishes they give a contribution of −15%. The positive NLO3
towards forward regions (close to unity). This holds both corrections diminish from +15% at ∆Rbb̄ ≈ 0 to +12%
for the muon and for the antitop-quark rapidity spectra. at ∆Rbb̄ ≈ π and then increase again. The NLO1 ones
In Fig. 12(a) we consider the distribution in the az- show a similar behaviour, however, with larger slopes.
imuthal separation between the positron and the muon. The combined NLO2 and NLO3 corrections enhance the
The two charged leptons tend to be produced in oppo- LOQCD +NLO1 prediction between 6% and 11%, similarly
site directions both at LO and at NLO, but the inclusion to the case of the azimuthal distance shown in Fig. 12(a),
of radiative corrections enhances the fraction of events but with a somewhat different shape.A. Denner, G. Pelliccioli: Combined NLO EW and QCD corrections to off-shell ttW production at the LHC 11
pp e+ e + bb, s = 13 TeV, 0 = (MT, t MT, t)1/2/2 pp e+ e + bb, s = 13 TeV, 0 = (MT, t MT, t)1/2/2
LOQCD 10 3 LOQCD
10 3
LOEW (×10) LOEW (×10)
LOQCD + NLO1 LOQCD + NLO1
LO+NLO LO+NLO
d /dMbb [fb/GeV]
d /dM3 [fb/GeV]
10 4 10 4
10 5
10 5
10 6 10 6
0.3 0.3 LOEW NLO2
0.2 0.2 NLO1 NLO3
ratio [/LOQCD]
ratio [/LOQCD]
0.1 0.1
0.0 0.0
0.1 0.1
0.2 LOEW NLO2 0.2
0.3 NLO1 NLO3 0.3
ratio [/ LOQCD + NLO1]
ratio [/ LOQCD + NLO1]
1.15 LOQCD + NLO1 LO + NLO 1.15 LOQCD + NLO1 LO + NLO
1.10 1.10
1.05 1.05
1.00 1.00
0.95 0.95
100 200 300 400 500 600 700 100 200 300 400 500 600 700
Mbb [GeV] M3 [GeV]
(a) Invariant mass of the bb̄ system. (b) Invariant mass of the three-charged-lepton system.
Fig. 10. Distributions in the invariant mass of the bb̄ system (left) and of the three-charged-lepton system (right). Same
structure as Fig. 7.
The results for the differential distributions presented Both integrated and differential cross-sections have
in Figs. 7–12 show that in many kinematic regions the been presented and discussed in a realistic fiducial re-
NLO2 and NLO3 corrections give an enhancement at the gion, keeping in mind the limited statistics of the LHC
level of 10% to the LOQCD + NLO1 result that is larger data and relating the off-shell description of the process
than the QCD scale uncertainties at the same pertur- to the inclusive predictions that are available in the liter-
bative order. This concerns in particular the soft- and ature.
moderate-pT , the low-mass and the central-rapidity re- The NLO2 and NLO3 corrections give a −5.5% and a
gions, which are also the statistically most-populated +13% contribution, respectively, to the LO cross-section,
ones. This reinforces that including formally subleading almost independently of the choice of the factorization
corrections (NLO2 , NLO3 ) is necessary to give a more and renormalization scales. The sizeable impact of NLO2
realistic description of total and differential tt̄W cross- and NLO3 corrections makes it essential to combine them
sections. with the NLO1 ones, in order to arrive at reliable predic-
tions.
The theory uncertainties from 7-point scale variations
4 Conclusions are driven by the NLO1 corrections, which are the only
corrections which feature a NLO-like scale dependence.
In this work we have presented the NLO corrections Their inclusion reduces the scale uncertainty of the LO
to the off-shell production of ttW+ at the LHC in the cross-section from 20% to 5%.
three-charged-lepton channel. These include the next- The investigation of differential distributions reveals
to-leading-order (NLO) QCD corrections to the QCD a more involved interplay among the various perturbative
(NLO1 ) and to the electroweak leading order (NLO3 ), orders compared with the integrated results. The NLO2
as well as the NLO electroweak corrections to the QCD corrections drop by up to −20% in most of the transverse-
leading order (NLO2 ). It is the first time that the NLO2 momentum and invariant-mass distributions, showing the
and NLO3 radiative corrections are computed with full typical behaviour of EW corrections with large Sudakov
off-shell dependence for a physical final state, accounting logarithms at high energies. They are rather flat for an-
for all non-resonant, interference, and spin-correlation ef- gular observables. The NLO3 corrections give a positive
fects. enhancement between +10% and +20% (30% in some12 A. Denner, G. Pelliccioli: Combined NLO EW and QCD corrections to off-shell ttW production at the LHC
pp e+ e + bb, s = 13 TeV, 0 = (MT, t MT, t)1/2/2 pp e+ e + bb, s = 13 TeV, 0 = (MT, t MT, t)1/2/2
0.08
0.07 0.08
0.06
0.06
0.05 LOQCD LOQCD
d /dy [fb]
d /dyt [fb]
LOEW (×10) LOEW (×10)
0.04 LOQCD + NLO1 LOQCD + NLO1
0.04
0.03 LO+NLO LO+NLO
0.02 0.02
0.01
0.00 0.00
0.3 0.3
0.2 0.2
ratio [/LOQCD]
ratio [/LOQCD]
0.1 0.1
0.0 0.0
0.1 0.1
0.2 LOEW NLO2 0.2 LOEW NLO2
0.3 NLO1 NLO3 0.3 NLO1 NLO3
ratio [/ LOQCD + NLO1]
ratio [/ LOQCD + NLO1]
1.15 LOQCD + NLO1 LO + NLO 1.15 LOQCD + NLO1 LO + NLO
1.10 1.10
1.05 1.05
1.00 1.00
0.95 0.95
2 1 0 1 2 2 1 0 1 2
y yt
(a) Rapidity of the muon. (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.
cases) to the LO cross-section in all analyzed distribu- eral Ministry for Education and Research (BMBF) under
tions, They are dominated by the ug partonic channel contract no. 05H18WWCA1.
(formally belonging to QCD real corrections to LO EW)
that embeds tW scattering. The NLO1 corrections, which
have already been presented in the literature, show quite
variable patterns in the various differential K-factors. References
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pp e+ e + bb, s = 13 TeV, 0 = (MT, t MT, t)1/2/2 pp e+ e + bb, s = 13 TeV, 0 = (MT, t MT, t)1/2/2
LOQCD LOQCD
LOEW (×10) 0.12 LOEW (×10)
0.0020 LOQCD + NLO1 LOQCD + NLO1
LO+NLO 0.10 LO+NLO
[fb]
0.0015 0.08
d /dRbb [fb]
e+
0.06
0.0010
d /d
0.04
0.0005 0.02
0.00
0.0000 0.5
0.3 LOEW NLO2 0.4 LOEW NLO2
0.2 NLO1 NLO3 0.3 NLO1 NLO3
ratio [/LOQCD]
ratio [/LOQCD]
0.1 0.2
0.0 0.1
0.1 0.0
0.2 0.1
0.3 0.2
0.3
ratio [/ LOQCD + NLO1]
ratio [/ LOQCD + NLO1]
1.15 LOQCD + NLO1 LO + NLO 1.15 LOQCD + NLO1 LO + NLO
1.10 1.10
1.05 1.05
1.00 1.00
0.95 0.95
20 40 60 80 100 120 140 160 1 2 3 4 5
e+ Rbb
(a) Azimuthal angle between the positron and the muon. (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–rapidity
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