Nexttoleadingorder QCD and electroweak corrections to WWW production at protonproton colliders
Received: May
Nexttoleadingorder QCD and electroweak corrections to W W W production at protonproton
Stefan Dittmaier 0 2
Alexander Huss 0 1
Gernot Knippen 0 2
Zurich 0
Switzerland 0
0 D79104 Freiburg , Germany
1 Institute for Theoretical Physics , ETH
2 AlbertLudwigsUniversitat Freiburg, Physikalisches Institut
TripleWboson production in protonproton collisions allows for a direct access to the triple and quartic gauge couplings and provides a window to the mechanism of electroweak symmetry breaking. It is an important process to test the Standard Model (SM) and might be background to physics beyond the SM. We present a calculation of the nexttoleading order (NLO) electroweak corrections to the production of WWW
NLO Computations

W
W
W
nal
states at protonproton colliders with onshell W bosons and combine the electroweak with
the NLO QCD corrections. We study the impact of the corrections to the integrated cross
sections and to kinematic distributions of the W bosons. The electroweak corrections are
generically of the size of 510% for integrated cross sections and become more pronounced
in speci c phasespace regions. The real corrections induced by quarkphoton scattering
turn out to be as important as electroweak loops and photon bremsstrahlung corrections,
but can be reduced by phasespace cuts. Considering that prior determinations of the
photon parton distribution function (PDF) involve rather large uncertainties, we compare
the results obtained with di erent photon PDFs and discuss the corresponding uncertainties
in the NLO predictions. Moreover, we determine the scale and total PDF uncertainties at
the LHC and a possible future 100 TeV pp collider.
1 Introduction
2
3 Input parameter scheme 4
Numerical results
4.1
4.2
4.3
4.4
Di erential distributions
NLO WWW cross sections with a jet veto
Discussion of PDF uncertainties in the photoninduced channel
Estimating the uncertainties of the total cross sections
5
Conclusion
A Results at a single phasespace point
lished in the narrow width approximation of the W bosons [6]. Our NLO calculation of
EW and QCD corrections, which is based on onshell W bosons, complements this
calculation by presenting additional results and carefully assessing the impact of the uncertainties
that arise from the PDFs. The issue of PDF uncertainties is particularly important for
WWW production, since quarkphoton induced channels have a large impact on the cross
section and previous determinations of the photon PDF su er from large uncertainties.
The recently released LUXqed photon PDF [7], however, is rather precise and stabilizes
the prediction considerably.
{ 1 {
This paper is structured as follows: section 2 describes the basic properties of
production at protonproton colliders and technical details of our NLO
calculation. It further covers the checks and validations we have performed on our calculation.
The setup of the calculation and the input parameters are summarized in section 3. We
present results on total and di erential cross sections, determine the scale dependence of
the NLO cross section, and assess the error induced by the uncertainty of the PDF in
section 4. We conclude with section 5.
2
Triple Wboson production at NLO
At leading order (LO), the production of three W bosons at protonproton colliders is
induced by the two chargeconjugated partonic subprocesses
uidj ! W
W+W+
and
uidj ! W+W
W ;
(2.1)
where i and j are the indices of the fermion generation. The di erent Feynman diagrams
contributing to W
W+W+ production at LO are shown in
gure 1. As can be seen from
the last diagram, the quartic WWWW coupling already enters the LO prediction. The
production of three W bosons further incorporates Higgs production in association with
a W boson, where the Higgs particle decays into two W bosons. However, owing to the
onshell requirement on the W bosons in our calculation, the Higgs boson is restricted to
be purely o shell.
At NLO, additional partons appear in the real emission contributions, namely photons
in the NLO EW real emission, gluons in the NLO QCD real emission, and quarks in the
gluoninduced and photoninduced channels. A selection of NLO real emission Feynman
diagrams is depicted in
gure 2. In our calculation, infrared (IR) singularities, which
arise due to soft and/or collinear emission, are dealt with using the dipole subtraction
formalism [8{12]. In the virtual contribution, which incorporates additional closed fermion
loops and virtual photon, gluon, or weakvectorboson exchange, we encounter oneloop
topologies up to pentagon diagrams. The tensor and scalar loop integrals are evaluated
using the Collier library [
13
], which is based on the results of refs. [14{16]. Examples of
NLO EW loop diagrams are shown in gure 3.
We have performed two completely independent calculations: one where the
amplitudes are generated and further processed with the packages FeynArts [17] and
FormCalc [18], and a second calculation using inhouse software based on Feynman diagrams
generated with FeynArts 1 [19]. The results of the two calculations agree within the
Monte Carlo integration errors. We have further checked that the results do not depend
on the regularization scheme, employing either mass or dimensional regularization for the
treatment of IR divergences.
To check for consistency, we also compare with NLO results on WWW production
available in the literature [4{6]: tables 1 and 2 show the comparison of our results to the
results in the literature and agreement is found within the Monte Carlo integration errors,
with the exception of the QCD result of ref. [6], where a phenomenologically insigni cant
deviation is found at the few per mill level.
{ 2 {
HJEP09(217)34
dj
W+
ui
dj
/Z
ui
dj
W
W+
W
ui
dj
/Z
W
W+
W
W+
W
W+
ui
dj
H
W
W
W+
emission, gluonic emission, quarkphoton, and quarkgluon induced channels) of the process pp !
LO [fb]
W+W+ production at the LHC and a centerofmass (CM) energy of p
our results
[6]
de nition of the NLO cross section and the relative corrections were chosen as reported in ref. [6]
with
R =
F = 32 MW.
where the superscript OS stands for the onshell scheme. We neglect all fermion masses
except for bottom and topquark masses, and further ignore the negligible mixing involving
the third generation quarks.1
As a result, the CKM matrix factorizes from all matrix
elements and can be absorbed into the parton luminosities.2 Furthermore, the SM behaves
like a CPconserving theory in our calculation. We calculate with a blockdiagonal CKM
matrix, where the mixing among the rst two generations is parametrized by the Cabibbo
angle
Cabibbo = 0:22731, so that the relevant CKM entries are given by
jVudj = jVcsj = 0:97428;
jVusj = jVcdj = 0:22536:
We work in the G scheme (see, e.g., ref. [21]), where the electromagnetic coupling
=
G is a derived quantity and given by
p
2
G
=
G
M W2
1
M W2
MZ2
:
(3.2)
(3.3)
1Note that due to the negligible mixing involving quarks of the third generation, the bottom quark never
2Owing to the mass degeneracy among the quarks of the rst two generations and the absence of mixing
with the third generation in our setup, the dependence on the CKM matrix drops out whenever a summation
over internal and external nalstate
avors is performed. The only case where the unitarity of the CKM
matrix cannot be exploited in this way is when the up and downtype quark are both in the initial state
and thus receive di erent weights from the PDFs. The calculation can still be performed using a diagonal
CKM by absorbing the CKM factors into the parton luminosities in this case.
{ 4 {
The G scheme accounts for universal corrections to the
parameter and the running of
from the Thomson limit to the electroweak scale. The running of the strong coupling
constant s is taken from the PDF set used.
We employ a dynamic renormalization and factorization scale ( R and F, respectively)
given by
r
where pT;i is the (vectorial) transverse momentum of particle i and S is the set of all
outgoing particles which carry no color. Note that this scale choice is equal to the
production threshold 3MW if there are no colorcharged particles in the nal state. In order
to estimate the residual theory uncertainties from missing higherorder corrections, we
examine the scale dependence in section 4 by varying the scales with respect to the central
choice (3.4) by factors of 12 and 2.
We use the LHAPDF6 library [22] to perform the convolution of the partonic cross
sections with the PDFs. We calculate the pure LO cross section, denoted by
LO, with
the LO NNPDF3.0 set [23]. All NLO contributions, including the LO contribution to the
NLO cross section
1LO, are calculated with the NLO NNPDF3.0QED [23, 24] set. The
photoninduced contributions are calculated with the LUXqed set [7]. Since all PDFs in
the LUXqed set, except for the photon PDF, are based on the PDF4LHC NNLO set [25],
the error introduced by using di erent PDFs for the quarkphotoninduced and every other
channel should be negligible in the overall PDF uncertainty. We additionally provide results
using the NNPDF3.0QED and the CT14QED (inclusive) [26] PDF sets in the
quarkphotoninduced channels to better assess the corresponding uncertainty. Throughout this
work we use PDF sets with ve active avors.
4
Numerical results
EW
qq0
qq0
EW correction qEqW0 at large energies, from the longrange QCD e ects and is preferable over
a purely additive approach [27{29]. Note that by normalizing the QCD correction to the LO
cross section evaluated with LO PDFs, the term 1+ QCD is identical to the usual de nition
of the Kfactor, up to small QED corrections in the PDFs. Moreover, the EW correction
factor qEqW0 becomes rather insensitive to the PDF choice and the factorization scale.
{ 5 {
qEqW0 [%]
, at di erent CM energies p
Carlo integration errors.
s of the collider. The indicated errors are estimates for the Monte
1:7 at LHC energies of 13 TeV and 14 TeV. The
photoninduced contributions are positive and overcompensate the negative EW corrections of the
quarkinduced channels, leading to total EW corrections of
6% and
7% at the current
LHC energy of 13 TeV for the two chargeconjugated
nal states, respectively. Note that
this partial cancellation is not systematic in the sense that the two compensating e ects
are not directly correlated. Due to the impact of the EW corrections it is important to
take the EW corrections into account, when comparing data to theory. We estimate the
missing higherorder EW corrections to be of the order of the squares of the individual
NLO corrections, i.e.
1% for LHC energies. We observe that the EW corrections in the
pure quarkinduced channels, which are generically of O(
5%), show only very little sen
sitivity to the collider energy. The quarkphotoninduced contribution, on the other hand,
rises with the pp scattering energy, reaching
40% for the scenario of a future 100 TeV
collider. This demonstrates the importance of determining the photon PDF precisely for
highenergy protonproton scattering. The QCD corrections increase with growing collider
CM energy owing to the higher gluon luminosity. The large Kfactors of
1:7 (
2:5)
{ 6 {
at 13 TeV (100 TeV), which are driven by the quarkgluoninduced channels, ask for
further improvements by higherorder QCD corrections. At least improvements by multijet
merging seem mandatory.
production mode among the two chargeconjugated processes. As both
nal states can
be easily separated, we will focus on the dominant, positively charged
nal state in the
following. A selection of di erential distributions including a breakdown of the corrections
into the relative factors de ned in eq. (4.1) is presented in
gures 4, 5, and 6.
Note
HJEP09(217)34
that the size of these corrections will be inherited by distributions based on decay leptons
when dropping the onshell requirement on the W bosons. As high transverse momenta
of the W
or high total invariant masses correspond to high partonic CM energies, the
unitarizing e ect of Higgs exchanges can be seen in the drop of the associated di erential
distributions shown in
gures 4 and 5. In this highenergy regime, Sudakov logarithms
from soft EW gaugeboson exchange are the leading contribution to the EW correction in
the quarkinduced channel, yielding corrections of several
10 % which can even overrule
the large quarkphotoninduced corrections at very high pT. At low invariant masses near
the production threshold, the e ect of the Coulomb singularity, which arises due to photon
exchange between W bosons, is visible. In this region the leading behavior of the NLO
EW correction qEqW0 is dominated by
Coul
CM frame [
30
]. Even though the QCD corrections grow with increasing pT of the W boson
they are rather independent on the total invariant mass. Figure 6 shows that the NLO
QCD correction changes the shape of the distribution in the di erence of the azimuthal
angle, preferring smaller angle di erences. This e ect is slightly enhanced by the total
EW correction.
4.2
NLO WWW cross sections with a jet veto
The large impact of the quarkphotoninduced channel on the total cross section can be
reduced by restricting the phase space of the additional jet in the nal state by a jet veto.
To this end, we require the transverse momentum of the additional outgoing parton, which
can be experimentally identi ed with a jet, to be below a certain threshold value pT;cut.
This threshold should not be chosen too small in order to not a ect the e ective cancellation
of IR singularities. Otherwise, large logarithms of the jetveto cut would remain in the nal
result requiring resummation [
31
], which however is beyond the scope of this work. As we
cut on the transverse momentum of the jet alone, only the quarkphotoninduced channel,
the quarkgluoninduced channel and the QCD real emission contribution are a ected. The
integrated cross sections for di erent values of pT;cut are presented in table 4. In
gure 7
the impact of the pTcut on the relative corrections in W
W+W+ production is shown. A
{ 7 {
10− 4
10− 7
10− 8
50
0.1 × δ QCD
δ qEqW0
NLO
LO
HJEP09(217)34
0
200
800
1000
400
600
curve of the QCD correction is scaled down by a factor of 0:1.
NLO
LO
10
5
0
√pps→ = 13 TeV
0.1 × δ QCD
1000
1500
MWWW [GeV]
{ 8 {
0.1 × δ QCD
1
2
π
Δφ W+W+
NLO
corrections. The curve of the QCD correction is scaled down by a factor of 0:1.
relatively strong cut at a transverse momentum of 100 GeV reduces the total NLO cross
section by
23% at the current LHC CM energy of 13 TeV. In detail, the QCD correction
drops by a factor of
2 and the photoninduced channel decreases to approximately 40%
of its original value. With increasing CM energy the impact of the pTcut increases. In
combination with the strong growth of the quarkphotoninduced and the QCD corrections
(see above) this results in a reduction of the NLO cross section by
50% for a value of
pT;cut = 100 GeV.
4.3
Discussion of PDF uncertainties in the photoninduced channel
The inclusion of QED corrections into the determination of PDFs was rst considered by
the MRST collaboration [
32
], which imposed strong model assumptions on the
parametrization of the photon PDF and based the t mostly on DIS data. Later, the NNPDF [24]
and CT [26] collaborations provided photon PDFs as well, the former without any model
assumptions and using mostly LHC data to constrain the photon PDF, the latter with
similar, but less strict assumptions than the MRST collaboration and using ep ! e + X
data. Recently, a new approach was put forward which made it possible to derive the
photon PDF directly from the proton structure functions F2(x; Q2) and FL(x; Q2), which
are well determined from ep scattering data. This new LUXqed [7] PDF set exhibits a
very small uncertainty. Another approach, using the structure functions as well, was used
by HarlandLang et al. [33]. Not long ago, the xFitter collaboration published results on a
photon PDF t to highmass DrellYan data at the LHC [
34
].
{ 9 {
ps [TeV] pT(jet)<100 GeV
pT(jet)<150 GeV
pT(jet)<200 GeV
momentum at di erent CM energies p
resulting from the Monte Carlo integration.
s of the collider. The error is an estimated integration error
Until recently, the photon PDF was largely unconstrained due to the limited amount
of data. Therefore, the uncertainty on quarkphoton and photonphotoninduced
contributions could be easily as large as the contributions themselves. The quarkphotoninduced
channel of WWW production constitutes the largest contribution to the NLO EW
corrections. Thus, an uncertainty estimate is essential for a meaningful physical prediction.
We assess the uncertainty by analyzing our results with di erent available PDF sets
incorporating QED corrections: NNPDF3.0QED, CT14QEDinc, and LUXqed.
We
include numbers for the largely outdated MRSTQED04 set as well, but do not use it in
the uncertainty estimate. The uncertainty of the photon PDF of the NNPDF set is highly
nonGaussian and only loosely constrained by data. Following the procedure used by the
LHC Higgs Cross section Working Group [20] in the calculation of the Higgs production
cross sections via vectorboson fusion and Higgsstrahlung, we therefore take the lower limit
of the cross sections calculated separately with all NNPDF3.0QED replicas as the lower
bound, the median of the cross sections as the central value, and the maximum of the 68%
smallest cross sections as the upper bound. Note that obtaining an error estimate based
on the uncertainty of the photon PDF alone is di cult in the case of the NNPDF3.0QED
sets as there are no dedicated variations for the photon PDF. We have computed the
NNPDF3.0QED errors both using the full PDF error for the quarkphoton channel and
xing the incoming quark to the central member PDF and only varying the photon PDF
W− W+W+ + X
√pps →= 7 TeV
QCD correction (blue) for di erent cut values pT;cut and collider energies for the process pp !
W+W+ + X. The values at pT;cut = 1 are the relative corrections of the full integrated cross
section without any jet veto.
over the replicas. Both approaches result in similar error estimates. The CT14QEDinc
PDF set does not give any information on a central value, only a range of their free t
parameter, the initial photon momentum fraction at the t scale p0 . At the 68% con dence
level, p0 is restricted to be between 0% and 0.11%. This range yields the error bar used.
The error on the LUXqed PDF set was calculated as described in the corresponding
paper [7]. As the LUXqed PDF set uses the Hessian method with symmetric eigenvectors
to describe the uncertainties, the variance of a cross section
is given by
Var( ) =
2
0) ;
NEV
X ( j
j=1
(4.3)
where NEV is the number of eigenvector PDF sets, j the cross section evaluated with
eigenvector set j and
0 the central value. The MRSTQED04 set does not provide any
uncertainty information. Figure 8 shows the central values of the photoninduced
contribution, where the error bars represent the photonPDF uncertainty for the di erent PDF
sets. In gure 9 the impact of the photon PDF uncertainty on di erential distributions is
PDF set
MRSTQED04
NNPDF3.0QED
CT14QEDinc
LUXqed
6
8
10
12
13 TeV. A central value for the CT14QEDinc PDF set was calculated by taking the midpoint of the
range of cross sections calculated in the table on the left. The additional dashed gray error bar for
the LUXqed PDF set shows the total PDF uncertainty for the quarkphotoninduced correction.
pT,W− [GeV]
momentum of W
QED corrections.
with uncertainties due to the photon PDF for di erent PDF sets incorporating
shown. We observe that the results based on the recent PDF sets are consistent with each
other. Due to the limited amount of data and no model assumptions, the uncertainty of
the NNPDF3.0QED set is the largest. The photon PDF of the LUXqed PDF set shows
an outstanding small uncertainty, which is even less than the remaining PDF uncertainties
(cf. gure 8).
4.4
Estimating the uncertainties of the total cross sections
We investigate the two main sources to the uncertainty of the total cross sections: missing
higherorder corrections estimated through the residual dependence on the factorization
and renormalization scale, and the uncertainties of the PDFs.
At LO, the production of three W bosons at a protonproton collider is a purely
electroweak process. As a consequence, there is no dependence on the renormalization
scale
R at LO, so that no reduction of the scale dependence when going from LO to NLO
is expected. In order to estimate the scale uncertainties we vary our scale choice in eq. (3.4)
up and down by a factor of 2. The scale uncertainty of the total LO and NLO cross section
for di erent collider CM energies is shown in table 5.
Another signi cant contribution to the total uncertainty is the overall PDF uncertainty
which is given by the square root of the variance de ned by
Var
NLO
PDF = Var
This formula has to be handled with care as we chose to use di erent PDF sets for di
erent contributions. We can assume that the covariance between the quarkphotoninduced
channel and every other contribution is independent of the choice of the PDF. This is
a valid assumption as the used PDF sets agree reasonably well in their values for quark
and gluon PDFs (cf. ref. [25]) and the covariance is ruled by the quark PDFs. Therefore,
eq. (4.4) simpli es to
+ Var
As the NNPDF collaboration uses Monte Carlo replicas, the variance of a cross section
evaluated with the NNPDF3.0QED PDF set is given by
(4.5)
(4.6)
Var( )jPDF =
1
Nrep
1
Nrep
X ( j
j=1
2
0) ;
where Nrep is the number of replicas, j the cross section evaluated with replica j and
0 = h i the central value of the cross section. In contrast to the NNPDF collaboration,
the LUXqed PDF set uses the Hessian method where the variance is given by the sum
over the squared di erences between the central value and the contribution evaluated with
each eigenvector PDF set (cf. section 4.3). The results of the PDF uncertainty estimation
are shown in table 5. Note that using LUXqed the impact of the uncertainty of the photon
PDF is rather small and negligible in comparison to the scale and other PDF uncertainties.
Even using a more conservative approach and taking the CT14QEDinc uncertainty would
insigni cantly change the total PDF uncertainty by only
5% at 13 TeV for W
W+W+
production.
We conclude that at past and present LHC energies, the dominant theoretical
uncertainty arises from the scale dependence of the NLO prediction, given that modern and
uptodate PDFs are employed. In this case, the scale uncertainties are almost twice as
large as the PDF errors and a further improvement on the prediction would require the
inclusion of QCD corrections beyond NLO.
(second) at di erent pp CM energies ps.
Owing to its sensitivity to the mechanism of electroweak symmetry breaking and to triple
and quartic gauge couplings, tripleW production is an important process to further test
the validity of the SM and search for physics beyond. As precise predictions are necessary
to analyze experimental data, we provide full NLO cross sections for the production of
three onshell W bosons at protonproton colliders. We observe that NLO corrections are
dominated by QCD with Kfactors of
1:5 1:7. The electroweak correction are of the
order of
5 10% at LHC energies. In special kinematic regimes, the electroweak
corrections grow large due to highenergy logarithms. The main contribution of the electroweak
corrections results from the quarkphotoninduced channel, yielding corrections of
11%
at 13 TeV. However, we observe large cancellations between the positive corrections from
the photoninduced and the negative EW corrections to the quarkinduced channels, so
that the net EW corrections are at the level of
7%. The impact of the quarkphoton
induced channel can be e ectively suppressed by applying a veto on hard jet emissions.
We estimate the impact of the uncertainty of the photon PDF on the NLO prediction by
considering di erent PDF sets incorporating QED corrections. Using the recently released
LUXqed PDF set, we observe a signi cant reduction in the uncertainties that originate
from the photon PDF and note that the total theory error to this process is now governed
by scale uncertainties. To further improve the cross section predictions, it would be
necessary to perform at least some multijet merging, which is beyond the scope of this work. To
particle
improve the predictive power in the distributions one should drop the onshell requirement
and include leptonic decays of the W bosons.
Acknowledgments
S.D. and G.K. are supported by the Research Training Group GRK 2044 of the German
Research Foundation (DFG). S.D. and G.K. acknowledge support by the state of
BadenWurttemberg through bwHPC and the German Research Foundation (DFG) through grant
no INST 39/9631 FUGG. A.H. is supported by the Swiss National Science Foundation
(SNF) under contract CRSII2160814.
A
Results at a single phasespace point
phasespace point with the fourmomenta given in table 6.3
In this appendix, we provide results for the partonic process ud ! W
The input parameter scheme has been de ned in section 3 and the dynamical scale
setting (3.4) reduces to the production threshold,
= 3MW, for the 2 ! 3 kinematics
considered here. In the following, we provide the squared amplitude averaged over
initialstate colors and helicities and summed over
nalstate helicities. The virtual corrections
are renormalized according to our inputparameter scheme with external legs renormalized
onshell. Infrared singularities are regularized using dimensional regularization (D = 4 2 )
and we further extract a factor of c = (4 )
(1 + ) from the coe cients of the poles.
W+W+ at a single
At Born level, we obtain
jM0j2 = 2:1306869301777854
and the virtual electroweak correction is given by
3The process ud ! W+W W is trivially obtained by a CP transformation.
(A.1)
(A.2)
For the virtual QCD correction we obtain 2 Re
QCD
M1loop M0
where we have further pulled out a global factor of s
.
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p
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