Toppair production at the LHC through NNLO QCD and NLO EW
Received: May
Toppair production at the LHC through NNLO QCD and NLO EW
Michal Czakon 0 1 3 6 7
David Heymes 0 1 3 5 7
Alexander Mitov 0 1 3 5 7
Davide Pagani 0 1 2 3 7
Ioannis Tsinikos 0 1 3 7
Marco Zaro 0 1 3 4 7
Cambridge 0 1 3 7
HE U.K. 0 1 3 7
0 JamesFranckStr. 1, Garching, D85748 Germany
1 Universite Catholique de Louvain , Chemin du Cyclotron 2, Louvain la Neuve, B1348 Belgium
2 Technische Universitat Munchen
3 Aachen , D52056 Germany
4 Sorbonne Universites, UPMC Universite Paris 06 , UMR 7589, LPTHE
5 Cavendish Laboratory, University of Cambridge
6 Institut fur Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University
7 Paris , F75005 France
In this work we present for the rst time predictions for topquark pair differential distributions at the LHC at NNLO QCD accuracy and including EW corrections. For the latter we include not only contributions of O( s2 ), but also those of order O( s 2 and O( 3). Besides providing phenomenological predictions for all main di erential distributions with stable top quarks, we also study the following issues. 1) The e ect of the
NLO Computations; QCD Phenomenology

EW
photon PDF on toppair spectra: we
nd it to be strongly dependent on the PDF set
used  especially for the top pT distribution. 2) The di erence between the additive and
multiplicative approaches for combining QCD and EW corrections: with our scale choice,
we nd relatively small di erences between the central predictions, but reduced scale
dependence within the multiplicative approach. 3) The potential e ect from the radiation of
heavy bosons on inclusive toppair spectra: we nd it to be, typically, negligible.
Phenomenological predictions for the LHC at 13 TeV
Comparison of two approaches for combining NNLO QCD predictions
Multiplicative combination and comparison with the additive one
Contributions from heavy boson radiation
1 Introduction
3.1
3.2
and EW corrections
Additive combination
4 Impact of the photon PDF
2
3
5
6
Conclusions
A Notation
1
Introduction
The availability of NNLO QCD predictions for stable toppair production at the LHC, both
for the total crosssection [1{4] with NNLL softgluon resummation [5, 6] and for all the
main di erential distributions [7, 8], has made it possible to compare Standard Model (SM)
theory with LHC data at the fewpercent level accuracy. Such a high precision has led,
among others, to further scrutiny of the di erences between LHC measurements [9] and
the ability of MonteCarlo event generators to describe hadronic tt production. As a result
of these ongoing studies, new MC developments are taking place, such as the incorporation
of nonresonant and interference e ects [10, 11], which builds upon previous works that
included NLO top decay corrections through xed order [12{16] and/or showered [17{19]
calculations.
One of the remaining ways for further improving SM theory predictions is by
consistently including the so called ElectroWeak (EW) corrections on top of the NNLO QCD
ones. Weak [20{28], QED [29] and EW (weak+QED) [30{34] corrections to topquark pair
production have been known for quite some time, and also EW corrections to the fully
o shell dilepton signature are nowadays available [35]. As it has been documented in the
literature, although EW e ects are rather small at the level of total crosssection, they
can have a sizeable impact on di erential distributions and also on the topquark charge
asymmetry.
The goal of this work is to consistently merge existing NNLO QCD predictions with
EW corrections into a single coherent prediction and to study its phenomenological impact.
{ 1 {
ing the photon content of the proton [
36, 37
]. As shown in ref. [34], depending on the
PDF set, photoninitiated contributions can be numerically signi cant in some regions of
phase space.1 If the photon density from the NNPDF3.0QED set [40, 41] is employed,
the photoninitiated contribution is large in size and of opposite sign with respect to the
Sudakov EW corrections, leading to the almost complete cancellation of the two e ects.
Nevertheless, large PDF uncertainties from the photon PDF are still present after this
cancellation. On the other hand, theoretical consensus about the correctness of the novel
approach introduced in ref. [
37
] appears to have emerged by now.2 The PDF set provided
with ref. [
37
], named LUXQED, includes a photon PDF whose central value and relative
uncertainty are both much smaller than in the case of NNPDF3.0QED. Thus, at
variance with the NNPDF3.0QED set, neither large cancellation between Sudakov e ects and
photoninduced contributions nor large photon PDF uncertainty is present in
LUXQEDbased predictions.
In order to document the ambiguity arising from the di erences between the
photon densities in the available PDF sets, with the exception of section 2, in this work we
always give predictions for toppair di erential distributions at the LHC based on the
LUXQED [
37
] and NNPDF3.0QED [40, 41] PDF sets. We believe that our
ndings will
provide a valuable input to future PDF determinations including EW e ects.
This paper is organised as follows: section 2 is devoted to the phenomenological study
of our combined QCD and EW predictions for the LHC at 13 TeV. The reasons behind
some of the choices made in section 2  like the choice of PDF set and combination
approach  are revealed in section 3, where we compare in great detail two approaches
for combining NNLO QCD and EW corrections in toppair di erential distributions. The
socalled additive approach is discussed in section 3.1, while the multiplicative one in
section 3.2. Section 4 is dedicated to studying the impact of the photon PDF on
toppair spectra. In section 5 we provide an estimate of the impact of inclusive Heavy Boson
Radiation (HBR), namely the contribution from ttV nal states with V = H; W ; Z. While most of the notation is introduced in the main text some technical details are delegated to appendix A.
2
Phenomenological predictions for the LHC at 13 TeV
In this section we present predictions for tt distributions for the LHC at 13 TeV at NNLO
QCD accuracy including also EW corrections. We focus on the following distributions: the
toppair invariant mass m(tt), the top/antitop average transverse momentum (pT;avt) and
1This has been studied also in refs. [38, 39] for the case of neutralcurrent DrellYan production.
2The consensus has been also supported by a preliminary study in the determination of the photon PDF
including new LHC data [42].
{ 2 {
rapidity (yavt) and the rapidity y(tt) of the tt system. The pT;avt (yavt) distributions are
calculated not on an eventbyevent basis but by averaging the results of the histograms
for the transverse momentum (rapidity) of the top and the antitop.
Our calculation is performed using the following input parameters
mt = 173:3 GeV ;
mH = 125:09 GeV ;
mW = 80:385 GeV ;
mZ = 91:1876 GeV ;
while all other fermion masses are set to zero. All masses are renormalised onshell and
all decay widths are set to zero. The renormalisation of s is performed in the 5 avour
scheme while EW input parameters and the associated
renormalisation condition are in
refs. [44, 45], and in ref. [46] for the calculation of the complete NLO corrections.
We work with dynamical renormalisation ( r) and factorisation ( f ) scales. Their
common central value is de ned as
=
=
=
mT;t
mT;t
2
2
HT =
4
1
4
for the pT;t distribution;
for the pT;t distribution;
mT;t + mT;t
for all other distributions;
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
where mT;t
respectively.
qmt2 + p2T;t and mT;t
qmt2 + p2T;t are the transverse masses of the top
and antitop quarks. As already mentioned, pT;avt and yavt distributions are obtained by
averaging the top and antitop distributions for the transverse momentum and rapidity,
These scale choices have been motivated and studied at length in ref. [8]. In all cases
theoretical uncertainties due to missing higher orders are estimated via the 7point variation
of r and
f in the interval f =2; 2 g with 1=2
r= f
2.
We remark that the
combination of QCD and EW corrections is independently performed for each value of f;r.
For theoretical consistency, a set of PDFs including QED e ects in the DGLAP
evolution should always be preferred whenever NLO EW corrections are computed. At the
moment, the only two NNLO QCD accurate PDF sets that include them are NNPDF3.0QED
and LUXQED.3 Both sets have a photon density, which induces additional contributions
to tt production [29, 34].
As motivated and discussed at length in section 3, the phenomenological predictions
in this section are based on the LUXQED PDF set and on the multiplicative approach for
combining QCD and EW corrections, which we will denote as QCD
EW. We invite the
3The PDF sets MRST2004QED [47] and CT14QED [48] also include QED e ects in the DGLAP
evolution, but they are not NNLO QCD accurate. A PDF set including full SM LO evolution (not only
QCD and QED but also weak e ects) has also recently become available [49].
{ 3 {
1.2 total unc.
QCD×EW
PDF unc.
scale unc.
total unc.
QCD×EW
PDF unc.
scale unc.
impact is observed in the two rapidity distributions: the relative e ect for yavt is around
2 permil and is much smaller than the scale uncertainty. The y(tt) distribution is slightly
more sensitive, with a relative impact of slightly above 1% for large values of y(tt). This
correction is also well within the scalevariation uncertainty band. The impact of EW
corrections on the m(tt) distribution is larger. Relative to NNLO QCD it varies between
+2% at the absolute threshold and
6% at high energies. Still, this correction is well within
the scale variation uncertainty. The small sensitivity of the yavt and y(tt) distributions to
EW corrections supports the ndings of ref. [9] where these two distributions were used for
constraining the PDF of the gluon.
The largest correction due to EW e ects is observed in the pT;avt distribution. Relative
to NNLO QCD, the correction ranges from +2% at low pT;avt to
25% at pT;avt
3 TeV.
The correction is signi cant and is comparable to the scale variation band already for
pT;avt
500 GeV. Overall, the EW contribution to the pT;avt distribution is as large as
the total theory uncertainty band in the full kinematic range pT;avt
3 TeV considered in
this work.
The fraction of the theory uncertainty induced by PDFs is strongly dependent on
kinematics. For the yavt and y(tt) distributions, the PDF error is slightly smaller than the
scale uncertainty for central rapidities, but is larger in the peripheral region, especially for
the y(tt) distribution. The PDF uncertainty becomes the dominant source of theory error
in the pT;avt distribution for pT;avt as large as 500 GeV, while for the m(tt) distribution it
begins to dominate over the scale uncertainty for m(tt)
2:5 TeV.
There are many applications for the results derived in this work. Examples are:
inclusion of EW e ects in PDF determinations from LHC tt distributions, highmass LHC
searches, precision SM LHC measurements and benchmarking of LHC event generators.
A practical and su ciently accurate procedure for the utilisation of our results could be
as follows. One starts by deriving an analytic t for the QCD
EW=QCD Kfactor (all
Kfactors in
gure 1 are available in electronic form4); as evident from
gure 1 it is a
very smooth function for all four di erential distributions. Under the assumption that
this Kfactor is PDF independent, such an analytic t could then be used to rescale the
NNLOQCDaccurate di erential distributions derived with any PDF set from existing
NNLO QCD [
50
] fastNLO [
51, 52
] tables. Regarding the PDF error of NNLO QCD
differential distributions, it can be calculated very fast with any PDF set with the help of the
fastNLO tables of ref. [
50
]. As we show in the following the PDF error of the QCD and
combined QCD and EW predictions is almost the same, especially for the LUXQED PDF
set used for our phenomenological predictions.
3
Comparison of two approaches for combining NNLO QCD predictions
and EW corrections
In this work we compare two approaches for combining QCD and EW corrections. For
brevity, we will refer to them as additive and multiplicative approaches. As already
men4Repository with results and additional plots of NNLO QCD + EW tt di erential distributions:
{ 5 {
tioned, the results presented in section 2 have been calculated using the multiplicative
approach.
In the additive approach the NNLO QCD predictions (de ned as the complete set of
O( sn) terms up to n = 4) are combined with all possible remaining LO and NLO terms
arising from QCD and electroweak interactions in the Standard Model. In other words, at
LO we include not only the purely QCD O( s2) contribution, but also all O( s ) and O( 2)
terms. Similarly, at NLO we take into account not only the NLO QCD O( s3) contribution
but also the O( s2 ) one, the socalled NLO EW, as well as the subleading contributions
of O( s 2) and O( 3). For brevity, we will denote as \EW corrections" the sum of all LO
and NLO terms of the form O( sm n) with n > 0. Moreover, when we will refer to \QCD"
results, we will understand predictions at NNLO QCD accuracy. For a generic observable
in the additive approach we denote the prediction at this level of accuracy as
In the multiplicative approach one assumes complete factorisation of NLO QCD and
NLO EW e ects. This approach is presented in section 3.2 and is denoted as
The precise de nition of the various quantities mentioned in the text is given in appendix A
where an appropriate notation for the classi cation of the di erent contributions is
introduced. Here, we just state the most relevant de nitions for the following discussion,
QCD+EW.
QCD EW.
QCD
EW
QCD+EW
QCD EW
LO QCD +
denotes a generic observable in tt production and KQNCLOD is the standard NLO/LO
Kfactor in QCD. A variation of the multiplicative approach denoted as
QCD2 EW will
also be considered; it is de ned similarly to
QCD EW in eq. (3.1) but with NNLO/LO
QCD Kfactor.
As it has been discussed in ref. [34], the usage of di erent PDF sets leads to a very
di erent impact of the photoninduced contribution on tt distributions. While in the case
of NNPDF3.0QED the impact of photoninduced contributions is relatively large and with
very large uncertainties, in the case of LUXQED it is expected to be negligible. For this
reason in the rest of this work we always show predictions with both PDF sets.
3.1
Additive combination
Distributions for pT;avt and m(tt) are shown in gure 2, while the yavt and y(tt) distributions
are shown in
gure 3. The format of the plots for all distributions is as follows: for each
observable, we show two plots sidebyside, with the same layout. The plot on the
lefthand side shows predictions obtained using the LUXQED set, while for the one on the
right the NNPDF3.0QED set is employed. Results at NNLO QCD accuracy are labelled
as \QCD" while the combination of NNLO QCD predictions and EW corrections in the
additive approach are labelled as \QCD+EW".
{ 6 {
1 (QCD+EW)/QCD; scale unc.
QCD
EW
1 (QCD+EW)/QCD; scale unc.
QCD
EW
1 (QCD+EW)/QCD; scale unc. QCD
1 (QCD+EW)/QCD; scale unc. QCD
In each plot the three insets display ratios of di erent quantities5 over the centralscale
QCD result (i.e., in the case of LUXQED, the black line in the main panel of gure 1).
In the rst inset we show the scale uncertainty due to EW corrections alone (red band),
without QCD contributions ( EW using the notation of appendix A). This quantity can
be compared to the scale uncertainty of the QCD prediction at NNLO accuracy (grey
band). In the second inset we present the scaleuncertainty band (red) for the combined
QCD+EW prediction. The grey band corresponds to the NNLO QCD scaleuncertainty
band already shown in the rst inset. The third inset is equivalent to the second one, but
it shows the PDF uncertainties. We combine, for each one of the PDF members, the QCD
prediction and the EW corrections into the QCD+EW result. The PDF uncertainty band
of the QCD+EW prediction is shown in red while the grey band corresponds to the PDF
uncertainty of the QCD prediction. For all insets, when the grey band is covered by the
red one, its borders are displayed as black dashed lines.
As can be seen in gures 2 and 3, the e ect of EW corrections is, in general, within the
NNLO QCD scale uncertainty. A notable exception is the case of the pT;avt distribution
with LUXQED. In the tail of this distribution the e ect of Sudakov logarithms is large and
5It is actually in all cases the QCD+EW= QCD ratio, but the bands refer to three di erent quantities,
as explained in the text.
{ 7 {
1.05 (QCD+EW)/QCD; scale unc.
QCD
EW
negative, of the order of (10{20%), and is not compensated by the photoninduced
contribution. On the contrary, in the case of NNPDF3.0QED, photoninduced contributions
mostly compensate the negative corrections due to Sudakov logarithms. As it has already
been noted in ref. [34], with this PDF set, the e ect of photoninduced contributions is not
negligible also for large values of m(tt), yavt and y(tt).
As it can be seen in the rst inset, in the large pT;avt regime the scale dependence of
the EW corrections alone is of the same size as, or even larger than, the scale variation
at NNLO QCD accuracy. For this reason, as evident from the second inset, the scale
uncertainty of the combined QCD+EW prediction is much larger than in the purely QCD
case, both with the LUXQED and NNPDF3.0QED PDF sets. This feature is present only
in the tail of the pT;avt distribution.
The PDF uncertainties (third inset) for all distributions do not exhibit large di
erences between QCD and QCD+EW predictions, despite the fact that the photoninduced
contribution in NNPDF3.0QED is large and has very large PDF uncertainty (relative to
LUXQED).
3.2
Multiplicative combination and comparison with the additive one
The additive approach
QCD+EW for combining QCD and EW corrections discussed in
section 3.1 is exact to the order at which the perturbative expansion of the production
cross{ 8 {
section is truncated. An alternative possibility for combining QCD and EW corrections is
what we already called the multiplicative approach,
QCD EW. This approach is designed
to approximate the leading EW corrections at higher orders. In the case of tt production
these are NNLO EW contributions of order O( s3 ).
The multiplicative approach is motivated by the fact that soft QCD and EW
Sudakov logarithms factorise, with the latter typically leading to large negative corrections
for boosted kinematics. Thus, when dominant NLO EW and NLO QCD corrections are
at the same time induced by these two e ects, the desired
xed order can be very well
approximated via rescaling NLO EW corrections with NLO QCD Kfactors.6 Otherwise,
if one is in a kinematical regime for which the dominant NLO EW or NLO QCD corrections
are of di erent origin (i.e. not Sudakov or soft), the di erence between the multiplicative
and additive approaches given by the term
mixed in eq. (A.8) can be considered as an
indication of theory uncertainty in that kinematics. It must be stressed that the
perturbative orders involved in the additive approach are included exactly also in the multiplicative
approach; the only addition the multiplicative approach introduces on top of the additive
one is the approximated O( s3 ) contribution.
One of the advantages of the multiplicative approach is the stabilisation of scale
dependence. As we saw in section 3.1, when QCD and EW corrections are combined in the
additive approach, the scale dependence at large pT;avt can exceed that of the NNLO QCD
prediction. On the other hand, the large pT;avt limit is precisely the kinematic regime where
the multiplicative approach is a good approximation and can be trusted: at large pT;avt
the NLO EW and NLO QCD corrections are mainly induced by Sudakov logarithms and
soft emissions, respectively, and as we just pointed out these two contributions factorise.
The presence of large Sudakov logarithms in the NLO EW result at large pT;avt is easy
to see since for Born kinematics large pT;t implies large pT;t which, in turn, implies large
s^; t^ and u^ Mandelstam variables. That NLO QCD corrections at large pT;avt are mainly of
soft origin can be shown with an explicit NLO calculation; by applying appropriate cuts
on the jet, the top and/or the antitop one can easily see that the di erential crosssection
is dominated by kinematic con gurations containing almost backtoback hard top and
antitop and a jet with small pT . Plots demonstrating this can be found in footnote 4.
In the following, for all observables
considered in this work, we present predictions in
the multiplicative approach denoted as
QCD EW. As a further check of the stability of the
multiplicative approach we display also the quantity
QCD2 EW, whose precise de nition
can be found in appendix A.
QCD2 EW is de ned analogously to
QCD EW, but by
rescaling NLO EW corrections via NNLO QCD Kfactors. By comparing
QCD EW and
QCD2 EW one can further estimate the uncertainty due to mixed QCDEW higher orders.
Figure 4 shows the pT;avt and m(tt) distributions, while gure 5 refers to yavt and y(tt).
As in section 3.1, the plots on the left are produced using the LUXQED PDF set, while
those on the right using the NNPDF3.0QED PDF set. We next describe the format of the
plots.
Each plot consists of ve insets, which all show ratios of di erent quantities over the
central value of
QCD. In the
rst inset we compare the centralscale results for the
6The precise de nitions of QCD EW is given in eq. (A.13).
{ 9 {
(QCD×EW)/QCD
(QCD2×EW)/QCD
1.2 (QCD+EW)/QCD
(QCD×EW)/QCD
(QCD2×EW)/QCD
di erential distributions at 13 TeV. The format of the plots is described in the text.
three alternative predictions:
QCD+EW= QCD (red line),
QCD EW= QCD (green line)
and
QCD2 EW= QCD (violet line). These quantities are further displayed in the second,
third and fourth inset, respectively, where not only the central value but also the scale
dependence of the numerator is shown. In all cases we calculate the scaleuncertainty
band as a scalebyscale combination and subsequent variation in the 7point approach.
Scale variation bands have the same colour as the corresponding centralvalue line. For
comparison we also display (grey band) the relative scale uncertainty of
QCD. Thus, the
second inset is exactly the same as the second inset in the corresponding plots in section 3.1.
The last inset shows a comparison of the ratio
QCD+EW= QCD including (red line) or not
(orange line) the contribution
res, where \res" stands for residual and denotes the fact
that
res are contributions to
EW that are expected to be small, regardless of the PDF
set used (see eq. (A.6)).
1.5 (QCD×EW)/QCD; scale unc.
QCD
(QCD+EW)/QCD
(QCD×EW)/QCD
(QCD2×EW)/QCD
(QCD+EW)/QCD
(QCD×EW)/QCD
(QCD2×EW)/QCD
As expected, the multiplicative approach shows much smaller dependence on the scale
variation. This is particularly relevant for the tail of the pT;avt distribution, where the scale
uncertainty of
EW alone is comparable in size with the one of
QCD; with this reduction
of the scale uncertainty the
QCD EW and
QCD uncertainty bands do not overlap when
LUXQED is used. In the case of m(tt) and yavt distributions, the
QCD EW centralvalue
predictions are typically larger in absolute value than those of
QCD+EW, while they are all
almost of the same size for the y(tt) distribution. In the case of yavt the di erence between
the additive and multiplicative approaches is completely negligible compared to their scale
uncertainty. Therefore, besides the kinematic region where Sudakov e ects are the
dominant contribution, the multiplicative and additive approaches are equivalent. Moreover,
the di erence between
QCD EW and
QCD2 EW is in general small; a sizeable di erence
between their scale dependences can be noted only in the tail of the pT;avt distribution.
For all the reasons mentioned above we believe that the multiplicative approach should
be preferred over the additive one and, indeed, it has been used for the calculation of our
best predictions in section 2. As can be seen from
gures 4 and 5 and their
thresholdzoomedin versions in footnote 4 the di erence between
QCD+EW and
QCD EW for
nonboosted kinematics is much smaller than the total theory uncertainty (scale+PDF) shown
in
gure 1. Thus, the di erence between the two approaches can be safely ignored in the
estimation of the theory uncertainty. One should bear in mind that this conclusion depends
on the choice of scale, which in our case, as explained in ref. [8], is based on the principle of
fastest convergence. A di erent scale choice with larger K factors would likely arti cially
enhance the di erence between
QCD+EW and
QCD EW.
In the last inset in
gures 4 and 5 we compare the quantities
EW and
EW
res,
where the
res contribution is exactly included in both the additive and multiplicative
approaches. As expected, one can see that the
res contribution is typically at and very
small. The only exception is the m(tt) distribution where a visible di erence between the
two curves ( EW and
EW
res) is present, especially in the tail. The
res contribution
includes the squared EW treelevel diagrams, the O( 2) contribution denoted as
in (A.2), and the two subleading NLO corrections of respectively O( s 2) and O( 3),
LO;3
denoted as
negligible, the O( 2) and O( s 2) ones both lead to positive nonnegligible contributions
of similar size to the m(tt) distribution. Indeed, the O( 2) contribution involves bb !
tt squared diagrams with a W boson in the tchannel, which at large m(tt) are not as
much suppressed as the contributions from the other initial states. Similarly, the O( s 2
)
contributions contain QCD corrections to them, featuring the same beahaviour. Relevant
plots displaying individually all the aforementioned contributions can be found at 4.
4
Impact of the photon PDF
In this section we quantify the impact on tt di erential distributions of the di erence
between the photon densities provided by the LUXQED and NNPDF3.0QED PDF sets.
In other words, we repeat the study performed in ref. [34] for these two PDF sets since
they were not considered in that work. We compare the size of the electroweak corrections
with and without the photon PDF for both PDF sets. In each plot of gure 6 we show
the relative impact induced by the electroweak corrections (the ratio
EW= QCD; see
de nitions in appendix A) for four cases: NNPDF3.0QED setting the photon PDF equal
to zero (red) or not (green), and LUXQED setting the photon PDF equal to zero (violet)
or not (blue). For the cases including the photon PDF, we also show the PDFuncertainty
band of
The impact of the photoninduced contribution can be evaluated via the di erence
between the green and red lines in the case of NNPDF3.0QED and the di erence between
the blue and violet lines in the case of LUXQED. As can be seen in gure 6, the impact of
the photon PDF on the pT;avt, m(tt), yavt and y(tt) distributions is negligible in the case of
LUXQED, while it is large and with very large uncertainties for the case of NNPDF3.0QED,
as already pointed out in ref. [34] for NNPDF2.3QED. At very large pT;avt and m(tt) also
HJEP10(27)86
yavt
at 13 TeV. The format of the plots is described in the text.
LUXQED show a nonnegligible relative PDF uncertainty, which is not induced by the
photon but from the PDFs of the coloured partons at large x. We checked that a similar
behaviour is exhibited also by NNPDF3.0QED when its photon PDF is set to zero.
5
Contributions from heavy boson radiation
In the calculation of EW corrections to QCD processes the inclusion of real emissions of
massive gauge bosons (heavy boson radiation or HBR) is not mandatory since, due to
the nite mass of the gauge bosons, real and virtual weak corrections are separately
nite
(albeit the virtual corrections are enhanced by large Sudakov logarithms). Furthermore,
such emissions are typically resolved in experimental analyses and are generally considered
as a di erent process ttV (+X) with V = H; W ; Z. For these reasons, the results in
section 2 do not include HBR contributions.
It is, nonetheless, interesting to estimate the contribution of HBR to inclusive tt
production. Our motivation is threefold: rst, resolved or not, HBR is a legitimate
contribution to the tt(+X) nal state considered in this work. Secondly, it is clear that one cannot
guarantee that HBR is resolved with 100% e ciency. Therefore, it is mandatory to have a
prior estimate for the size of the e ect. Finally, we are unaware of prior works where the
HBR contribution has been estimated in inclusive tt production. Recently, refs. [44, 53]
have provided estimates for HBR in the processes ttV (+X), with V = H; Z; W .
We have investigated the impact of HBR on all four distributions considered in this
work: pT;avt, m(tt), yavt and y(tt). Our results are shown in
gure 7, where we plot the
e ect of HBR on the central scale normalised to the QCD prediction. We show separately
the LO HBR e ect of order O( s2 ) as well as the NLO QCD HBR prediction which
includes terms of order O( s3 ). As a reference we also show the EW corrections for tt.
In our calculations we include HBR due to H; W and Z. We are fully inclusive in
HBR, i.e., no cuts on the emitted heavy bosons are applied. Clearly, any realistic
experi0.05
0
yavt
0
y(tt)
1
1
0.1 EW/QCD
HBR(αs2α)/QCD
HBR(αs2α+αs3α)/QCD
0.1 EW/QCD
HBR(αs2α)/QCD
HBR(αs2α+αs3α)/QCD
0
500
1000
1500
2000
2500
3000
0
500
1000
1500
2000
2500
3000
yavt
0
y(tt)
1
1
2
2
0
−0.1
−0.2
distributions at 13 TeV. The format of the plots is described in the text.
mental analysis will require an estimate of HBR subject to experimental cuts, but such an
investigation would be well outside the scope of the present work.
From
gure 7 we conclude that the e ect of HBR is generally much smaller than
the EW corrections. In particular, higherorder QCD corrections to HBR are completely
negligible, i.e. HBR is well described in LO for all the tt inclusive distributions and for the
full kinematic ranges considered here. The absolute e ect of HBR on the pT;avt distribution
is positive and small; it never exceeds 2{3% (relative to the tt prediction at NNLO QCD
accuracy) and is always much smaller than the EW correction. The only distribution
where the HBR contribution is not negligible compared to the EW one is m(tt) computed
with LUXQED. For this distribution the HBR correction is positive and only about half the
absolute size of the (negative) EW correction. Still, the absolute size of the HBR, relative to
the prediction at NNLO QCD accuracy, is within 1% and so its phenomenological relevance
is unclear. The impact of HBR on the two rapidity distributions is tiny, typically within
3 permil of the NNLO QCD prediction. { 14 {
In this work we derive for the rst time predictions for all main topquark pair di erential
distributions7 with stable top quarks at the LHC at NNLO QCD accuracy and including
the following EW corrections: the NLO EW e ects of O( s2 ), all subleading NLO terms
of order O( s 2) and O( 3) as well as the LO contributions of order O( s ) and O( 2).
We present a detailed analysis of toppair production at the LHC at 13 TeV and we
nd that the e ect of EW corrections on di erential distributions with stable top quarks
is in general within the current total (scale+PDF) theory uncertainty. A notable
exception is the pT;avt distribution in the boosted regime where the e ect of EW corrections is
signi cant with respect to the current total theory error. We have checked that similar
conclusions apply also for LHC at 8 TeV. All results derived in this work in the
multiplicative approach, for both 8 and 13 TeV, are available in electronic form 4 as well as with the
ArXiv submission of this paper.
Providing phenomenological predictions for the LHC is only one of the motivations for
the present study. In this work we also quantify the impact of the photon PDF on toppair
di erential distributions and study the di erence between the additive and multiplicative
approaches for combining QCD and EW corrections. Moreover, we analyse the contribution
from inclusive Heavy Boson Radiation on inclusive toppair di erential distributions.
In order to quantify the impact of the photon PDF, we use two recent PDF sets
whose photon components are constructed within very di erent approaches. The rst set,
LUXQED, is based on the PDF4LHC15 set [54] and adds to it a photon contribution that is
derived from the structure function approach of ref. [
37
]. The second set, NNPDF3.0QED,
is based on the NNPDF3.0 family of PDFs and adds a photon component that is extracted
from a t to collider data. NNPDF3.0QED photon density has both a much larger central
value and PDF uncertainty than those of LUXQED. On the other hand, the two sets are
compatible within PDF errors and they both include QED e ects in the DGLAP evolution
on top of the usual NNLO QCD evolution.
We con rm the observations already made in ref. [34], namely, the way the photon
PDF is included impacts all di erential distributions. The size of this impact is di erent
for the various distributions; the most signi cant impact can be observed in the pT;avt
distribution at moderate and large pT where the net e ect from EW corrections based on
NNPDF3.0QED is rather small and with large PDF uncertainties, while using LUXQED
it is negative, with small PDF uncertainties and comparable to the size of the NNLO QCD
scale error. The m(tt) distribution displays even larger e ects, but only at extremely high
m(tt). The y(tt) distribution is also a ected at large y(tt) values.8
It seems to us that a consensus is emerging around the structurefunction approach
of ref. [
37
]. Given its appealing predictiveness, this approach will likely be utilised in the
7One distribution we do not consider is pT;tt which is not known in NNLO QCD, and for which
resummation is mandatory in order to have reliable predictions.
are even more pronounced at 8 TeV.
8As it has been lengthly motivated and discussed in ref. [34], e ects due to the photon PDF a la NNPDF
future in other PDF sets. Therefore, at present, it seems to us that as far as the photon
PDF is concerned predictions based on the LUXQED set should be preferred.
Our best predictions in this work are based on the socalled multiplicative approach
for combining QCD and EW corrections. We have also presented predictions based on
the standard additive approach. In general, we nd that the di erence between the two
approaches is small and well within the scale uncertainty band. The di erence between the
two approaches is more pronounced for the m(tt) and pT;avt distributions. Nevertheless,
both approaches agree within the scale variation. The scale uncertainty is smaller within
the multiplicative approach and, especially in the case of the pT;avt distribution, does not
overlap with the NNLO QCD uncertainty band. We stress that these features may be
sensitive to the choice of factorisation and renormalisation scales.
Since we are unaware of a past study of Heavy Boson Radiation (i.e. H; W
and
Z) in inclusive tt production, for completeness, we have also presented the impact of
inclusive HBR on the inclusive toppair di erential spectrum. While it is often assumed
that additional HBR emissions can be removed in the measurements, it is nevertheless
instructive to consider the contribution of such
nal states. We nd that, typically, the
HBR contribution is negligible, except for the m(tt) distribution, where it tends to partially
o set the EW correction (when computed with LUXQED). We have also checked that
NLO QCD corrections to the LO HBR result are negligible for all inclusive tt distributions
considered by us.
Acknowledgments
We thank Stefano Frixione for his suggestions and interest at the early stage of this project.
D.P., I.T. and M.Z. acknowledge also Fabio Maltoni for his strong encouragement and
support in pursuing this study.
The work of M.C. is supported in part by grants of the DFG and BMBF. The work of
D.H. and A.M. is supported by the U.K. STFC grants ST/L002760/1 and ST/K004883/1.
A.M. is also supported by the European Research Council Consolidator Grant
\NNLOforLHC2". The work of D.P is partially supported by the ERC grant 291377 \LHCtheory:
Theoretical predictions and analyses of LHC physics: advancing the precision frontier" and
by the Alexander von Humboldt Foundation, in the framework of the Sofja Kovalevskaja
Award Project \Event Simulation for the Large Hadron Collider at High Precision". The
work of I.T. is supported by the F.R.S.FNRS \Fonds de la Recherche Scienti que"
(Belgium) and in part by the Belgian Federal Science Policy O ce through the Interuniversity
Attraction Pole P7/37. The work of M.Z. is supported by the European Union's Horizon
2020 research and innovation programme under the Marie SklodovskaCurie grant
agreement No. 660171 and in part by the ILP LABEX (ANR10LABX63), in turn supported by
French state funds managed by the ANR within the \Investissements d'Avenir" programme
under reference ANR11IDEX000402.
A
Notation
In this appendix we specify how EW corrections and NNLO QCD results are combined
in the additive and multiplicative approaches. The notation matches the one introduced
in [53]. The phenomenology of tt production within the additive approach is presented in
section 3.1. The multiplicative approach is studied in section 3.2 where it is also compared
to the additive one.
A generic observable tt in the process pp ! tt(+X) can be expanded simultaneously
in the QCD and EW coupling constants as:
X
In order to simplify the notation, we further de ne the following purely QCD quantities
and those involving EW corrections
res
EW
LO;3 +
NLO;3 +
NLO;4 ;
Throughout this work with the term \EW corrections" we refer to the quantity
while the term \NLO EW corrections" will only refer to
NLO EW. In the additive
approach, which is presented in section 3.1, QCD and electroweak corrections are combined
through the linear combination
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
EW,
(A.7)
(A.8)
The so called \multiplicative approach", which has been discussed in section 3.1, is
precisely de ned in the following. The purpose of the multiplicative approach is to
estimate the size of
NNLO;2, which for convenience we rename
mixed and assuming complete
factorisation of NLO QCD and NLO EW e ects we estimate as
mixed
NNLO;2
NLO QCD
NLO EW :
LO QCD
In the regime where NLO QCD corrections are dominated by soft interactions and NLO
EW by Sudakov logarithms, eq. (A.8) is a very good approximation, since the two e ects
factorise and are dominant. In other regimes
mixed can be used as an estimate of the
leading missing mixed QCD{EW higher orders. The advantage of the inclusion of
mixed
is the stabilisation of the scale dependence of the term
NLO EW, which in tt production
has almost9 the same functional form of LO QCD. To this end we de ne the multiplicative
approach as
res
res
res
1)
NLO EW
where we used the standard Kfactors
KQNCLOD
In order to test the stability of the multiplicative approach under even higher mixed
QCDEW orders, we combine NNLO QCD corrections and NLO EW corrections in order
to estimate, besides the
mixed term, also NNNLO contributions of order
s4 . For this
purpose we de ne the quantity
QCD2 EW
KENWLO
QCD +
res
= KQNCNDLO ( LO QCD +
=
=
QCD + KQNCNDLO
QCD+EW + (KQNCNDLO
NLO EW) +
res
res
where we introduced the Kfactor
KQNCNDLO
QCD
LO QCD
:
Finally, we brie y describe how the dependence on the photon PDF enters the di
erent perturbative orders. At LO and NLO accuracy, all contributions, with the exception
of
to
LO QCD and
NLO QCD, depend on the photon PDF. The dominant photoninduced
process is g
! tt, which contributes to
LO EW and, via QCD corrections to this order,
NLO EW. In addition,
NLO EW, but also
NLO;4, receive contributions
from the q
also the
! ttq and q
! ttq processes. Moreover, in the case of
LO;3 and
NLO;4,
initial state contributes. As already discussed in ref. [34], almost all of the
photoninduced contribution arises form
LO EW. In this work, at variance with ref. [34],
we also include the term
res in our calculations. However, since the size of res is in
general small, the previous argument still applies. The numerical impact of res is discussed
in section 3.2.
9We say \almost" because this order receives also QCD corrections to the
LO EW contributions from
the g and bb initial states. Besides these e ects NLO EW( 2) =
NLO EW( 1) LLOO QQCCDD(( 12)) .
(A.9)
(A.10)
(A.11)
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
(A.17)
(A.18)
(A.19)
Given the structure of the photoninduced contributions described before, it is also
important to note that, with LUXQED, the multiplicative approach is a better approximation
mixed than in the case of NNPDF3.0QED. Indeed, the order
NLO EW contains also
terms that can be seen as \QCD corrections" to the g contributions in
LO;2 (negligible
only with the LUXQED), but are not taken into account in the multiplicative approach.
Open Access.
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any medium, provided the original author(s) and source are credited.
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