W + W − production at the LHC: fiducial cross sections and distributions in NNLO QCD
Received: May
production at the LHC: ducial cross sections and distributions in NNLO QCD
Massimiliano Grazzini 0 1 3 6
Stefan Kallweit 0 1 3 4 5
Stefano Pozzorini 0 1 3 5 6
Dirk Rathlev 0 1 2 3
Marius Wiesemann 0 1 3 6
Zurich 0 1 3
Switzerland 0 1 3
0 D22607 Hamburg , Germany
1 University of California , Santa Barbara, CA 93106 , U.S.A
2 Theory Group, Deutsches ElektronenSynchrotron , DESY
3 Johannes Gutenberg University , D55099 Mainz , Germany
4 PRISMA Cluster of Excellence, Institute of Physics
5 Kavli Institute for Theoretical Physics
6 PhysikInstitut, Universitat Zurich
7 (Higgs cuts) @ 8 TeV
We consider QCD radiative corrections to W +W
QCD Phenomenology

W
W
and H ! W +W
e + X at p
production at the LHC
and present the rst fully di erential predictions for this process at nexttonexttoleading
order (NNLO) in perturbation theory. Our computation consistently includes the leptonic
decays of the W bosons, taking into account spin correlations, o shell e ects and
nonresonant contributions. Detailed predictions are presented for the di erent avour channel
s = 8 and 13 TeV. In particular, we discuss ducial cross sections
and distributions in the presence of standard selection cuts used in experimental W +W
analyses at the LHC. The inclusive W +W
cross section receives large
NNLO corrections, and, due to the presence of a jet veto, typical ducial cuts have a sizeable
in uence on the behaviour of the perturbative expansion. The availability of di erential
NNLO predictions, both for inclusive and
ducial observables, will play an important role
in the rich physics programme that is based on precision studies of W +W
signatures at
the LHC.
1 Introduction 2
Description of the calculation 3
Results 4
Summary
1
Introduction
2.1
2.2
2.3
2.4
3.1
3.2
3.3
3.4
contamination through singletop and tt production
The production of W boson pairs is one of the most important electroweak (EW) processes
at hadron colliders. Experimental studies of W +W
production play a central role in
precision tests of the gauge symmetry structure of EW interactions and of the mechanism of
EW symmetry breaking. The W +W
cross section has been measured at the Tevatron [
1,
2
] and at the LHC, both at 7 TeV [3, 4] and 8 TeV [5{8]. The dynamics of W pair production
is of great interest, not only in the context of precision tests of the Standard Model, but
also in searches of physics beyond the Standard Model (BSM). Any small anomaly in
the production rate or in the shape of distributions could be a signal of new physics. In
particular, due to the high sensitivity to modi cations of the Standard Model trilinear
gauge couplings, W +W
measurements are a powerful tool for indirect BSM searches
via anomalous couplings [3, 4, 6, 8, 9].
Thanks to the increasing reach in transverse
momentum, Run 2 of the LHC will considerably tighten the present bounds on anomalous
couplings. Final states with W boson pairs are widely studied also in the context of direct
BSM searches [10].
In Higgsboson studies [11{16], W +W
production plays an important role as
irreducible background in the H ! W +W
channel. Such measurements are mostly based
on
nal states with two leptons and two neutrinos, which provide a clean experimental
a consequence, it is not possible to extract the irreducible W +W
signature, but do not allow for a full reconstruction of the H ! W +W
resonance. As
background from data
with a simple sideband approach. Thus, the availability of precise theory predictions
{ 1 {
for the W +W
background is essential for the sensitivity to H
! W +W
and to any
BSM particle that decays into W boson pairs. In the context of Higgs studies, the o
shell treatment of W boson decays is of great relevance, both for the description of the
signal region below the W +W
threshold, and for indirect determinations
of the Higgsboson width through signalbackground interference e ects at high invariant
The accurate description of the jet activity is another critical aspect of Higgs
measurements, and of W +W
measurements in general. Such analyses typically rely on a rather
strict jet veto, which suppresses the severe signal contamination due to the tt background,
but induces potentially large logarithms that challenge the reliability of xedorder
predictions in perturbation theory. All these requirements, combined with the ever increasing
accuracy of experimental measurements, call for continuous improvements in the theoretical
description of W +W
production.
Nexttoleading order (NLO) QCD predictions for W +W
production at hadron
colliders have been available for a long time, both for the case of stable W bosons [20, 21]
and with spincorrelated decays of vector bosons into leptons [22{25]. Recently, also the
NLO EW corrections have been computed [26{28]. Their impact on inclusive cross sections
hardly exceeds a few percent, but can be strongly enhanced up to several tens of percent
at transverse momenta of about 1 TeV.
Given the sizeable impact of O( S) corrections, the calculation of higherorder QCD
e ects is indispensable in order to reach high precision. The simplest ingredient of pp !
+ X at O( S2) is given by the loopinduced gluonfusion contribution. Due to the
strong enhancement of the gluon luminosity, the gg channel was generally regarded as the
H ! W +W
masses [17{19].
dominant source of NNLO QCD corrections to pp ! W +W
tions for gg ! W +W
contributions at LO are known also for gg ! W +W
at LO have been widely studied [25, 29{32], and squared quarkloop
+ X in the literature.
Predicg [33, 34]. Twoloop helicity
amplithe NLO QCD corrections to gg !
tudes for gg ! V V 0 became available in refs. [35, 36], and have been used to compute
[37], including all partonic processes with
external gluons, while the ones with external quarks are still unknown to date.
Calculations at NLO QCD for W +W
production in association with one [38{41] and two [42, 43]
jets are also important ingredients of inclusive W +W
production at NNLO QCD and
beyond. The merging of NLO QCD predictions for pp ! W +W
+ 0; 1 jets1 has been
presented in ref. [45]. This merged calculation also consistently includes squared quarkloop
+ 0; 1 jets in all gluon and quarkinduced channels.
contributions to pp ! W +W
First NNLO QCD predictions for the inclusive W +W
cross section became available
in ref. [46]. This calculation was based on twoloop scattering amplitudes for onshell
production, while twoloop helicity amplitudes are now available for all
vectorboson pair production processes, including o shell leptonic decays [47, 48]. In the energy
range from 7 to 14 TeV, NNLO corrections shift the NLO predictions for the total cross
section by about 9% to 12% [46], which is around three times as large as the gg ! W +W
contribution alone. Thus, contrary to what was widely expected, gluongluon fusion is
not the dominant source of radiative corrections beyond NLO. Moreover, the relatively
1See also [44] for a combination of xedorder NLO predictions for W +W +0,1 jet production.
{ 2 {
large size of NNLO e ects turned out to alleviate the tension that was observed between
earlier experimental measurements [5, 7] and NLO QCD predictions supplemented with
the loopinduced gluon fusion contribution [25]. In fact, NNLO QCD predictions are in
good agreement with the latest measurements of the W +W
cross section [6, 8].
Besides perturbative calculations for the inclusive cross section, the modelling of the
jetveto e ciency is another theoretical ingredient that plays a critical role in the
comparison of data with Standard Model predictions. In particular, it was pointed out that
a possible underestimate of the jetveto e ciency through the Powheg Monte Carlo [49],
which is used to extrapolate the measured cross section from the
ducial region to the
full phase space, would lead to an arti cial excess in the total cross section [50, 51]. The
+e
relatively large size of higherorder e ects and the large intrinsic uncertainties of NLO+PS
Monte Carlo simulations call for improved theoretical predictions for the jetveto e ciency.
The resummation of logarithms of the jetveto scale at nexttonexttoleading logarithmic
(NNLL) accuracy was presented in refs. [
52, 53
]. Being matched to the pp ! W +W
+ X
cross sections at NLO, these predictions cannot describe the vetoing of hard jets beyond
LO accuracy. In order to reach higher theoretical accuracy, NNLL resummation needs to
be matched to di erential NNLO calculations. Such NNLL+NNLO predictions have been
presented in ref. [
54
] for the distribution in the transverse momentum of the W +W
system, and could be used to obtain accurate predictions for the jetveto e ciency through a
reweighting of Monte Carlo samples, along the lines of refs. [50, 55].
In this paper we present, for the rst time, fully di erential predictions for W +W
production with leptonic decays at NNLO. More precisely, the full process that leads to
a nal state with two leptons and two neutrinos is considered, including all relevant o
shell and interference e ects in the complexmass scheme [56]. The calculation is carried
out with Matrix,2 a new tool that is based on the Munich Monte Carlo program3
interfaced with the OpenLoops generator of oneloop scattering amplitudes [57, 58], and
includes an automated implementation of the qT subtraction [59] and resummation [60]
formalisms. This widely automated framework has already been used, in combination with
the twoloop scattering amplitudes of refs. [48, 61], for the calculations of Z
tons and two neutrinos, but in this paper we will focus on the di erent avour signature
e. The impact of QCD corrections on cross sections and distributions will be
studied both at inclusive level and in presence of typical experimental selection cuts for W +W
measurements and H ! W +W
studies. The presented NNLO results for ducial cross
sections and for the e ciencies of the corresponding acceptance cuts provide rst insights
into acceptance e ciencies and jetveto e ects at NNLO.
2Matrix is the abbreviation of \Munich Automates qT subtraction and Resummation to Integrate
Xsections", by M. Grazzini, S. Kallweit, D. Rathlev and M. Wiesemann. In preparation.
3Munich is the abbreviation of \MUltichaNnel Integrator at Swiss (CH) precision" an automated
parton level NLO generator by S. Kallweit. In preparation.
{ 3 {
As pointed out in ref. [46], radiative QCD corrections resulting from real
bottomquark emissions lead to a severe contamination of W pair production through topquark
resonances in the W +W
b and W +W
bb channels. The enhancement of the W +W
cross
section that results from the opening of the tt channel at NNLO can exceed a factor of ve.
It is thus clear that a careful subtraction of tt and singletop contributions is indispensable
in order to ensure a decent convergence of the perturbative series. To this end, we adopt a
topfree de nition of the W +W
cross section based on a complete bottomquark veto in
the four avour scheme. The uncertainty related with this prescription will be assessed by
means of an alternative topsubtraction approach based on the topquarkwidth dependence
of the W +W
cross section in the ve avour scheme [46].
The manuscript is organized as follows. In section 2 we describe technical aspects of the
computation, including the subtraction of resonant topquark contributions (section 2.1),
qT subtraction (section 2.2), the Matrix framework (section 2.3), and the stability of
(N)NLO predictions based on qT subtraction (section 2.4). Section 3 describes our
numere + X: we present the input parameters (section 3.1), cross
sections and distributions without acceptance cuts (section 3.2) and with cuts
correspondsignal (section 3.3) and Higgs analyses (section 3.4). The main results are
ical results for pp !
+e
ing to W +W
summarized in section 4.
2
Description of the calculation
We study the process
pp ! l+l0
l l0 + X;
(2.1)
including all resonant and nonresonant Feynman diagrams that contribute to the
production of two charged leptons and two neutrinos.
Depending on the
avour of the
nalstate leptons, the generic reaction in eq. (2.1)
can involve di erent combinations of vectorboson resonances. The di erent avour nal
state l+l0
l l0 is generated, as shown in gure 1 for the qq process at LO,
(a) via resonant tchannel W +W
production with subsequent W + ! l+ l and
W
! l0
l0 decays;
(b) via schannel production in Z( )=
! W W ( ) topologies through a
triplegaugeboson vertex with subsequent W + ! l+ l and W
! l0
l0 decays, where either
both W bosons, or the Z boson and one of the W bosons can become simultaneously
(c) via Z=
production with a subsequent decay Z=
! l lW ! ll0 l l0 . Note that
kinematics again allows for a resonant W boson in the decay chain of a resonant Z
resonant;
boson.
Additionally, in the case of equal lepton avours, l = l0, o shell ZZ production diagrams
are involved, as shown in gure 2, where the l+l l l nal state is generated
(d) via resonant tchannel ZZ production with Z ! l+l and Z ! l l decays;
{ 4 {
avour case (l 6= l0) and in the same avour case (l = l0).
(e) via further Z ! 4 leptons topologies, Z=
! llZ ! ll l l or Z ! l lZ ! ll l l.
Any doubleresonant con gurations are kinematically suppressed or excluded by
phasespace cuts.
process as W +W
avour channel pp !
+e
production though.
Note that the appearance of infrared (IR) divergent
equal lepton
avours would prevent a fully inclusive phasespace integration.
! l+l splittings in the case of
Our calculation is performed in the complexmass scheme [56], and besides resonances,
it includes also contributions from o shell EW bosons and all relevant interferences; no
resonance approximation is applied. Our implementation can deal with any combination
of leptonic avours, l; l0 2 fe; ; g. However, in this paper we will focus on the di
erente + X. For the sake of brevity, we will often denote this
The NNLO computation requires the following scattering amplitudes at O( S2):
tree amplitudes for qq ! l+l0 l l0 gg, qq(0) ! l+l0 l l0 q(00)q(000), and crossingrelated
processes;
oneloop amplitudes for qq ! l+l0 l l0 g, and crossingrelated processes;
squared oneloop amplitudes for qq ! l+l0 l l0 and gg ! l+l0 l l0 ;
twoloop amplitudes for qq ! l+l0 l l0 .
All required treelevel and oneloop amplitudes are obtained from the OpenLoops
generator [57, 58], which implements a fast numerical recursion for the calculation of NLO
scattering amplitudes within the Standard Model. For the numerically stable evaluation of tensor
integrals we employ the Collier library [67{69], which is based on the DennerDittmaier
reduction techniques [70, 71] and the scalar integrals of ref. [72]. For the twoloop helicity
{ 5 {
amplitudes we rely on a public C++ library [73] that implements the results of ref. [48],
and for the numerical evaluation of the relevant multiple polylogarithms we use the
implementation [74] in the GiNaC [75] library. The contribution of the massivequark loops is
neglected in the twoloop amplitudes, but accounted for anywhere else, in particular in the
loopinduced gg channel. Based on the size of twoloop contributions with a masslessquark
loop, we estimate that the impact of the neglected diagrams with massivequark loops will
be well below the per mille level.
2.1
contamination through singletop and tt production
The theoretical description of W +W
production at higher orders in QCD is complicated
by a subtle interplay with topproduction processes, which originates from realemission
channels with
nalstate bottom quarks [38, 45, 46]. In the
ve avour scheme (5FS),
where bottom quarks are included in the partondistribution functions and the
bottomquark mass is set to zero, the presence of real bottomquark emission is essential to cancel
collinear singularities that arise from g ! bb splittings in the virtual corrections.
At
the same time, the occurrence of W b pairs in the realemission matrix elements induces
W b resonances that lead to a severe contamination of W +W
production.
The
problem starts with the NLO cross section, which receives a singleresonant tW ! W +W b
contribution of about 30% (60%) at 7 (14) TeV. At NNLO, the appearance of
doubleresonant tt ! W +W
bb production channels enhances the W +W
cross section by about
a factor of four (eight) [46]. Such singletop and tt contributions arise through the couplings
of W bosons to external bottom quarks and enter at the same orders in
and S as (N)NLO
QCD contributions from light quarks. Their huge impact jeopardises the convergence of
the perturbative expansion. Thus, precise theoretical predictions for W +W
production
require a consistent prescription to subtract the top contamination.
In principle, resonant top contributions can be suppressed by imposing a bjet veto,
similarly as in experimental analyses. However, for a bjet veto with typical pT values of
20 30 GeV, the top contamination remains as large as about 10% [46], while in the limit of
a vanishing bjet veto pT 's the NLO and NNLO W +W
singularities associated with massless bottom quarks in the 5FS.
cross sections su er from collinear
To circumvent this problem, throughout this paper we use the four avour scheme
(4FS), where the bottom mass renders all partonic subprocesses with bottom quarks in
the nal state separately
nite. In this scheme, the contamination from tt and singletop
production is easily avoided by omitting bottomquark emission subprocesses. However,
this prescription generates logarithms of the bottom mass that could have a nonnegligible
impact on the W +W
cross section. In order to assess the related uncertainty, results
in the 4FS are compared against a second calculation in the 5FS. In that case, the
contributions that are free from top resonances are isolated with a gaugeinvariant approach
that exploits the scaling behaviour of the cross sections in the limit of a vanishing
topquark width [46]. The idea is that doubleresonant (singleresonant) contributions depend
quadratically (linearly) on 1= t, while topfree W +W
contributions are not enhanced
at small t
. Exploiting this scaling property, the tt, tW and (topfree) W +W
components in the 5FS are separated from each other through a numerical t based on multiple
{ 6 {
highstatistics evaluations of the cross section for increasingly small values of t. The
subtracted result in the 5FS can then be understood as a theoretical prediction of the genuine
cross section and directly compared to the 4FS result. The di erence should be
regarded as an ambiguity in the de nition of a topfree W +W
cross section and includes,
among other contributions, the quantum interference between W +W
production (plus
unresolved bottom quarks) and tt or singletop production. This ambiguity was shown to
be around 1%
2% for the inclusive W +W
cross section at NNLO [46], and turns out to
be of the same size or even smaller in presence of a jet veto (see section 3).
2.2
The qT subtraction formalism
The implementation of the various IRdivergent amplitudes into a numerical code that
provides
nite NNLO predictions for physical observables is a highly nontrivial task. In
particular, the numerical computations need to be arranged in a way that guarantees the
cancellation of IR singularities across subprocesses with di erent parton multiplicities. To
this end various methods have been developed. They can be classi ed in two broad
categories. In the rst one, the NNLO calculation is organized so as to cancel IR singularities of
both NLO and NNLO type at the same time. The formalisms of antenna subtraction [76{
79], colourful subtraction [80{82] and Stripper [83{85] belong to this category. Antenna
subtraction and colourful subtraction can be considered as extensions of the NLO
subtraction methods of refs. [86{89] to NNLO. Stripper, instead, is a combination of the FKS
subtraction method [86] with numerical techniques based on sector decomposition [90, 91].
The methods in the second category start from an NLO calculation with one additional
parton (jet) in the
nal state and devise suitable subtractions to make the cross section
nite in the region in which the additional parton (jet) leads to further divergences. The qT
subtraction method [59] as well as N jettiness subtraction [92{94], and the Bornprojection
method of ref. [95] belong to this class.
The qT subtraction formalism [59] has been conceived in order to deal with the
production of any colourless4 highmass system F at hadron colliders. This method has already
been applied in several NNLO calculations [46, 59, 62{66, 97{100], and we have employed
it also to obtain the results presented in this paper. In the qT subtraction framework, the
pp ! F + X cross section at (N)NLO can be written as
d (FN)NLO = H(N)NLO
F
h
d LFO + d (FN+)jLeOt
d (CNT)NLO :
i
(2.2)
The term d (FN+)jLeOt represents the cross section for the production of the system F plus one
jet at (N)LO accuracy and can be evaluated with any available NLO subtraction formalism.
The counterterm d (CNT)NLO guarantees the cancellation of the remaining IR divergences of
the F +jet cross section. It is obtained via
xedorder expansion from the resummation
formula for logarithmically enhanced contributions at small transverse momenta [60]. The
practical implementation of the contributions in the square bracket in eq. (2.2) is described
in more detail in section 2.3.
4The extension to heavyquark production has been discussed in ref. [96].
{ 7 {
process and compensates5 for the subtraction of d (CNT)NLO. It is obtained from the (N)NLO
truncation of the processdependent perturbative function
H
F = 1 +
S
H
F(1) +
S 2
H
F(2) + : : : :
(2.3)
The NLO calculation of d F requires the knowledge of HF(1), and the NNLO calculation
also requires HF(2). The general structure of HF(1) has been known for a long time [101].
Exploiting the explicit results of HF(2) for Higgs [102] and vectorboson [103]
production, the result of ref. [101] has been extended to the calculation of the NNLO coe cient
HF(2) [104]. These results have been con rmed through an independent calculation in the
framework of SoftCollinear E ective Theory [105, 106]. The counterterm d (CNT)NLO only
depends on H(N)LO, i.e. for an NNLO computation it requires only HF(1) as input, which
F
can be derived from the oneloop amplitudes for the Born subprocesses.
2.3
Organization of the calculation in MATRIX
Our calculation of W +W
production is based on Matrix, a widely automated program
for NNLO calculations at hadron colliders. This new tool is based on qT subtraction, and
is thus applicable to any process with a colourless highmass nal state, provided that the
twoloop amplitudes for the Born subprocess are available. Moreover, besides xedorder
calculations, it supports also the resummation of logarithmically enhanced terms at NNLL
accuracy (see ref. [
54
], and ref. [107] for more details).
Matrix is based on Munich, a generalpurpose Monte Carlo program that includes a
fully automated implementation of the CataniSeymour dipole subtraction method [88, 89],
an e cient phasespace integration, as well as an interface to the oneloop generator
OpenLoops [57, 58] to obtain all required (spin and colourcorrelated) treelevel and oneloop
amplitudes. Munich takes care of the bookkeeping of all relevant partonic subprocesses.
For each subprocess it automatically generates adequate phasespace parameterizations
based on the resonance structure of the underlying (squared) treelevel Feynman diagrams.
These parameterizations are combined using a multichannel approach to simultaneously
atten the resonance structure of the amplitudes, and thus guarantee a fast convergence
of the numerical integration. Several improvements like an adaptive weightoptimization
procedure are implemented as well.
Supplementing the fully automated NLO framework of Munich with a generic
implementation of the qT subtraction and resummation techniques, Matrix achieves
NNLL+NNLO accuracy in a way that limits the additionally introduced dependence on
F
the process to the twoloop amplitudes that enter HNNLO in eq. (2.2). All other
processdependent information entering the various ingredients in eq. (2.2) are expressed in terms
of NLO quantities already available within Munich+OpenLoops.
5More precisely, while the behaviour of d (CNT)NLO for qT ! 0 is dictated by the singular structure of
d (FN+)jLeOt, its nondivergent part in the same limit is to some extent arbitrary, and its choice determines the
F
explicit form of H(N)NLO.
{ 8 {
All NNLO contributions with vanishing total transverse momentum qT of the
nalF
state system F are collected in the coe cient HNNLO. The remaining part of the NNLO
cross section, namely the di erence in the square bracket in eq. (2.2), is formally
nite in
the limit qT ! 0, but each term separately exhibits logarithmic divergences in this limit.
Since the subtraction is nonlocal, a technical cut on qT is introduced in order to render
both terms separately
nite. In this way, the qT subtraction method works very similarly
to a phasespace slicing method. In practice, it turns out to be more convenient to use a
cut, rcut, on the dimensionless quantity r = qT =M , where M denotes the invariant mass of
the nalstate system F .
The counterterm d (CNT)NLO cancels all divergent terms from the realemission
contributions at small qT , implying that the rcut dependence of their di erence should become
numerically negligible for su ciently small values of rcut. In practice, as both the
counterterm and the realemission contribution grow arbitrarily large for rcut ! 0, the statistical
accuracy of the Monte Carlo integration degrades, preventing one from pushing rcut too
low. In general, the absence of any strong residual rcut dependence provides a stringent
check on the correctness of the computation since any signi cant mismatch between the
contributions would result in a divergent cross section in the limit rcut ! 0. To monitor
the rcut dependence without the need of repeated CPUintensive runs, Matrix allows for
simultaneous crosssection evaluations at variable rcut values. The numerical information
on the rcut dependence of the cross section can be used to quantify the uncertainty due to
nite rcut values (see section 2.4).
2.4
Stability of qT subtraction for
+e
e production
+e
In the following we investigate the stability of the qT subtraction approach for pp !
e + X. To this end, in
gure 3 we plot the NLO and NNLO cross sections
as functions of the qT subtraction cut, rcut, which acts on the dimensionless variable
r = pT; +e
e =m +e
e
. Validation plots are presented at 8 TeV both for the fully
inclusive cross section (see section 3.2) and for the most exclusive case we have
investigated, i.e. the cross section in presence of standard
ducial cuts for Higgs background
analyses (see section 3.4). All considered scenarios at 8 and 13 TeV lead essentially to the
same conclusions.
At NLO the rcutindependent cross section obtained with CataniSeymour subtraction
perturbative correction at NnLO is
is used as a reference for the validation of the qT subtraction result. The comparison of the
NLO cross sections in the left panels of gure 3 demonstrates that qT subtraction reaches
about halfpermille accuracy already at the moderate value of rcut = 1%, where we can,
however, still resolve a di erence, which is slightly larger than the respective numerical
uncertainties, with respect to the rcutindependent result achieved using CataniSeymour
subtraction. This di erence is due to the powersuppressed contributions that are left
after the cancellation of the logarithmic singularity at small rcut. Going to even smaller
values of rcut, we observe a perfect convergence within statistical uncertainties towards the
CataniSeymoursubtracted result in the limit rcut ! 0. The expected behaviour of the
(n)(rcut) =
(n) + f (n)(rcut) n = 1; 2; : : :
(2.4)
{ 9 {
+0.70 σ/σNNLO − 1[%]
+0.70 σ/σNNLO − 1[%]
HJEP08(216)4
+0.08
+0.06
+0.04
+0.02
0
−0.02
−0.04
−0.06
−0.08
−0.10
+0.08
+0.06
+0.04
+0.02
0
−0.02
−0.04
−0.06
−0.08
−0.10
σNNLO
σNqTNLO(r)
σNNLO
σNqTNLO(r)
σNCLSO
σNqTLO(r)
σNCLSO
σNqTLO(r)
+0.60
+0.50
+0.40
+0.30
+0.20
+0.10
−0.10
−0.20
−0.30
−0.40
−0.50
−0.60
−0.70
0
+0.60
+0.50
+0.40
+0.30
+0.20
+0.10
−0.10
−0.20
−0.30
−0.40
−0.50
−0.60
−0.70
0
rcut, for both NLO (left plots) and NNLO (right plots) results in the inclusive phase space (upper
plots) and with Higgs cuts (lower plots). NLO results are normalized to the rcutindependent NLO
cross section computed with CataniSeymour subtraction, and the NNLO results are normalized to
their values at rcut ! 0, with a conservative extrapolationerror indicated by the blue bands.
where
(n) is the rcutindependent result and the function f (n)(rcut) has the general form
2n 1
k=0
f (n)(rcut) = rc2ut
X ak;n lnk(rcut) + : : :
(2.5)
At NNLO, where an rcutindependent control result is not available, we observe no
signi cant, i.e. beyond the numerical uncertainties, rcut dependence below about rcut = 1%;
we thus use the
nitercut results to extrapolate to rcut = 0, taking into account the
breakdown of predictivity for very low rcut values, and conservatively assign an additional
numerical error to our results due to this extrapolation.6 This procedure allows us to control
all NNLO predictions to inclusive and
ducial cross sections presented in section 3 well
below the level of two per mille. The increasing error bars indicate that arbitrarily low rcut
values cannot be tested as the contributions cancelling in the limit are separately divergent.
Based on the observation that no signi cant rcut dependence is found below rcut = 1%,
the value rcut = 0:25% was adopted for the calculation of the di erential observables
presented in section 3.
We have checked that the total rates for that value are fully
consistent within numerical uncertainties with our extrapolated results and that a smaller
value rcut = 0:1% leads to distributions in full statistical agreement, thus con rming the
robustness of our results also at the di erential level.
6In the NNLO calculation the O( S) contributions are evaluated by using CataniSeymour subtraction.
We present numerical results for the di erent avour process pp !
+e
e + X at
s = 8 TeV and 13 TeV. Cross sections and distributions are studied both in the inclusive
phase space and in presence of typical selection cuts for W +W
Di erent avour nal states provide the highest sensitivity both in W +W
ments and Higgs studies. We note that, due to the charge asymmetry of W +W
in protonproton collisions and the di erences in the muon and electron acceptance cuts (in
particular regarding the rapidity cuts), the two di erent avour channels,
analyses.
measureproduction
+e
e and
e , do not yield identical cross sections. However, we have checked that the absolute
di erences are not resolved on the level of our statistical errors. Thus (N)NLO predictions
and Kfactors for +e
e production can be safely applied also to pp ! e+
e
+ X.
Input parameters, PDFs and selection cuts
Results in this paper are based on the EW input parameters G
= 1:1663787 10 5 GeV 2
mW = 80:385 GeV and mZ = 91:1876 GeV. The other couplings in the EW sector are
derived in the G scheme, where cos w = mW =mZ and
complexmass scheme, the physical gaugeboson masses and the weak mixing angle are
= p
2G m2W sin2 w= . In the
replaced by V =
q
m2V
i V mV and cos ^w =
W = Z , while for
the above realvalued
expression is used. For the vectorboson widths we employ
W = 2:085 GeV and
Z =
2:4952 GeV [108], and for the heavy quarks we set mb = 4:92 GeV and mt = 172:5 GeV.
These input parameters result in a branching fraction BR(W
! l l) = 0:1090040 for
each massless lepton generation, i.e. l = e; . Contributions from resonant Higgs bosons and
continuum are fully supported in our implementation.
production as EW signal or as background
their interference with the W +W
However, since this study is focused on W +W
to H ! W +W , Higgs contributions have been decoupled by taking the mH ! 1 limit.
To compute hadronic cross sections, we use NNPDF3.0 partondistribution functions
(PDFs) [109], and, unless stated otherwise, we work in the 4FS, while removing all
contributions with nalstate bottom quarks in order to avoid any contamination from topquark
resonances. In the NNPDF framework, 4FS PDFs are derived from the standard
variableavournumber PDF set with
(5F)(MZ ) = 0:118 via appropriate backward and forward
s
evolution with ve and four active avours, respectively. The resulting values of the strong
coupling
(4F)(MZ ) at LO, NLO and NNLO are 0.1136, 0.1123 and 0.1123, respectively.
s
Predictions at NnLO are obtained using PDFs at the corresponding perturbative order
and the evolution of
S at (n + 1)loop order, as provided by the PDF set. The central
values of the factorization and renormalization scales are set to
F =
R = mW . Scale
uncertainties are estimated by varying F and
R in the range 0:5 mW
F ; R
2 mW
with the restriction 0:5
F = R
In the following subsections we investigate +e
e production in the inclusive phase
space (section 3.2) and in presence of typical selection cuts that are designed for
measurements of W +W
production (section 3.3) and for H ! W +W
studies (section 3.4) at
the LHC. The detailed list of cuts is speci ed in table 1. Besides the requirement of two
charged leptons within a certain transversemomentum and rapidity region, they involve
cut variable
W +W
cuts
pT;l1
pT;l2
jy j
jyej
pmiss
T
T
pmiss,rel
pT;ll
mll
Rll
ll
ll;
Njets
lepton de nition
leptonic cuts
> 25 GeV
> 20 GeV
antikT jets with R = 0:4, pT;j > 25 GeV, jyj j < 4:5
momentum of the
pmiss
T
closest lepton;
sin j
j, where
ll;
momenta, pT;ll, and pTmiss.
signal measurements (central column) and H !
W +W
studies (right column). The hardest and second hardest lepton are denoted as l1 and
l2, respectively. The missing transverse momentum, pmiss, is identi ed with the total transverse
T
pair, while the relative missing transverse momentum pmiss,rel is de ned as
T
is the azimuthal separation between pTmiss and the momentum of the
is the azimuthal angle between the vectorial sum of the leptons' transverse
additional restrictions on the missing transverse momentum (pTmiss = pT; ), the
transverse momentum (pT;ll) and invariant mass (mll) of the dilepton system, the combined
rapidityazimuth ( Rll) and azimuthal (
ll) separation of the charged leptons, as well as
on the relative missing transverse momentum (pTmiss,rel) and the azimuthal angle between
pT;ll, and pTmiss (
ll; ), as de ned in table 1. Moreover, the W +W
and Higgs selection
criteria involve a veto against antikT jets [110] with R = 0:4, pT > 25 GeV and jyj < 4:5.
3.2
Analysis of inclusive
+e
e production in absence of acceptance cuts. Predictions
for the total inclusive cross section at LO, NLO and NNLO are listed in table 2. The NLO
cross section computed with NNLO PDFs, denoted by NLO0, and NLO0 supplemented with
the loopinduced gluonfusion contribution (NLO0+gg) are provided as well.
At p
s = 8 (13) TeV the NLO corrections increase the LO cross section by 47% (55%),
and the NNLO corrections result in a further sizeable shift of +11% (+14%) with respect
to NLO. The total NNLO correction can be understood as the sum of three contributions
that can be read o table 2: evaluating the cross section up to O( S) with NNLO PDFs
increases the NLO result by about 2% (3%). The loopinduced gluonfusion channel, which
used to be considered the dominant part of the NNLO corrections, further raises the cross
inclusive [fb]
= NLO
p
s
LO
respect to NLO. The quoted uncertainties correspond to scale variations as described in the text,
and the numerical integration errors on the previous digit(s) are stated in brackets; for the NNLO
results, the latter include the uncertainty due the rcut extrapolation (see section 2.4).
section by only 3% (4%), while the genuine O( S2) corrections to the qq channel7 amount to
about +6% (+7%). Neglecting PDF e ects, we nd that the loopinduced gg contribution
corresponds to only 37% (38%) of the total O( S2) e ect, i.e. of NNLO
remaining 63% (62%) being due to genuine NNLO corrections.
NLO0 , with the
These results are in line with the inclusive onshell predictions of ref. [46], where the
relative weight of the gg contribution was found to be 35% (36%), and the small di erence
is due to the chosen PDFs. We also
nd by up to about 2% larger NNLO corrections
than stated in ref. [46], which can also be attributed to the chosen PDF sets. Indeed,
repeating the onshell calculation of ref. [46] using the input parameters of section 3.1
(with
W =
Z = 0), we
nd that the relative corrections agree on the level of the
statistical error when the same PDF sets are applied. Moreover, comparing the results of
table 2 with this onshell calculation allows us to quantify the size of o shell e ects, which
turn out to reduce the onshell result by about 2% with a very mild dependence (at the
permille level) on the perturbative order and the collider energy. The results for the two
considered collider energies con rm that the size of relative corrections slightly increases
with the centreofmass energy, as in the onshell case.
We add a few comments on the theoretical uncertainties of the above results. As is
well known, scale variations do not give a reliable estimate of the size of missing
higherorder contributions at the
rst orders of the perturbative expansion. In fact, LO and
NLO predictions are not consistent within scale uncertainties, and the same conclusion
can be drawn by comparing NLO or NLO0+gg predictions with their respective scale
uncertainties to the central NNLO result. This can be explained by the fact that the qg
(as well as qg) and gg (as well as qq(0), qq(0) and qq0) channels open up only at NLO and
NNLO, respectively. Since the NNLO is the rst order where all the partonic channels
are contributing, the NNLO scale dependence should provide a realistic estimate of the
7Here and in what follows, all NNLO corrections that do not stem from the loopinduced gg ! W +W
channel are denoted as genuine O( S2
) corrections or NNLO corrections to the qq channel. Besides
qqinduced partonic processes, they actually contain also gq and gq channels with one extra
nalstate parton
as well as gg, qq(0), qq(0) and qq0 channels with two extra
nalstate partons.
dσ/dmWW [fb/GeV] µ+e νµν‾ e(inclusive)@LHC 8 TeV
dσ/dmWW [fb/GeV] µ+e νµν‾ e(inclusive)@LHC 13 TeV
acceptance cuts are applied. Absolute LO (black, dotted), NLO (red, dashed) and NNLO (blue,
solid) predictions at p
s = 8 TeV (left) and p
s = 13 TeV (right) are plotted in the upper frames.
The lower frames display NLO0+gg (green, dotdashed) and NNLO predictions normalized to NLO.
The bands illustrate the scale dependence of the NLO and NNLO predictions. In the case of ratios,
scale variations are applied only to the numerator, while the NLO prediction in the denominator
corresponds to the central scale.
uncertainty from missing higherorder corrections. The loopinduced gluongluon channel,
which contributes only at its leading order at O( S2) and thus could receive large relative
corrections, was not expected to break this picture due to its overall smallness already in
ref. [46]. That conclusion is supported by the recent calculation of the NLO corrections to
the loopinduced gg channel [37].
In gures 4{7 we present distributions that characterize the kinematics of the
reconstructed W bosons.8 Absolute predictions at the various perturbative orders are
complemented by ratio plots that illustrate the relative di erences with respect to NLO. In order
to assess the importance of genuine NNLO corrections, full NNLO results are compared to
NLO0+gg predictions in the ratio plots.
In gure 4 we show the distribution in the total invariant mass, mW +W = m +e
e
.
This observable features the characteristic threshold behaviour around 2 mW , with a rather
long tail and a steeply falling cross section in the o shell region below threshold. Although
suppressed by two orders of magnitude, the Zboson resonance that originates from
topologies of type (b) and (c) in
gure 1 is clearly visible at m +e
e = mZ . Radiative QCD
e ects turn out to be largely insensitive to the EW dynamics that governs o shell W boson
8The various kinematic variables are de ned in terms of the o shell W boson momenta, pW + = p + +p
and pW
= pe + p e .
R
T
T
A
A
i
i
w
w
u
u
d
d
o
o
pT,WW [GeV]
(b)
applied. Absolute predictions and relative corrections as in gure 4.
decays and dictates the shape of the m +e
e distribution. In fact, the NNLO= NLO
ratio is rather at, and shape distortions do not exceed about 5%, apart from the strongly
suppressed region far below the 2 mW threshold.
The distribution in the transverse momentum of the W +W
pair, shown in gure 5,
vanishes at LO. Thus, at nonzero transverse momenta NLO (NNLO) results are formally
only LO (NLO) accurate.
Moreover, the loopinduced gg channel contributes only at
pT;W W = 0. The relative NNLO corrections are consistent with the results discussed in
ref. [
54
]: they are large and exceed the estimated scale uncertainties in the small and
intermediate transversemomentum regions, while the NLO and NNLO uncertainty bands
overlap at large transverse momenta. At very low pT , the xedorder NNLO calculation
diverges, but NNLL+NNLO resummation [
54
] can provide accurate predictions also in
that region.
In
gures 6 and 7 the transversemomentum distributions of the harder W boson,
pT;W1 , and the softer W boson, pT;W2 , are depicted. The rst eyecatching feature is the
large NLO/LO correction in case of the harder W boson, which grows with pT and leads
to an enhancement by a factor of ve at pT
500 GeV, whereas such large corrections
are absent for the softer W boson. This feature is due to the fact that the phasespace
region with at least one hard W boson is dominantly populated by events with the NLO
jet recoiling against this W boson, while the other W boson is relatively soft. The LOlike
nature of this dominant contribution for moderate and large values of pT;W1 is re ected by
the large NLO scale band. The phasespace region where the softer W boson has moderate
or high transverse momentum as well is naturally dominated by topologies with the two W
bosons recoiling against each other. Such topologies are present already at LO, and thus
do not result in exceptionally large corrections. Both for the leading and subleading W
101
102
103
1.4
1.3
1.2
1.1
1
0.9
0.8
0
101
100
101
102
103
1.4
1.3
1.2
1.1
1
0.9
0.8
0
pT,W2 [GeV]
NLO
NNLO
NLO'+gg
LO
NLO
NNLO
NLO'+gg
101
X
RIT 100
dσ/dpT,W1 [fb/GeV] µ+e νµν‾ e(inclusive)@LHC 8 TeV
dσ/dpT,W1 [fb/GeV] µ+e νµν‾ e(inclusive)@LHC 13 TeV
100
200
300
400
500
0
100
200
300
400
500
LO
NLO
NNLO
NLO'+gg
LO
NLO
NNLO
X
I
R
T
h
t
d
e
c
o
r
p
I
R
t
d
e
c
o
r
p
acceptance cuts are applied. Absolute predictions and relative corrections as in gure 4.
dσ/dpT,W2 [fb/GeV] µ+e νµν‾ e(inclusive)@LHC 8 TeV
dσ/dpT,W2 [fb/GeV] µ+e νµν‾ e(inclusive)@LHC 13 TeV
100
200
300
400
500
0
100
200
300
400
500
acceptance cuts are applied. Absolute predictions and relative corrections as in gure 4.
pT,W2 [GeV]
(b)
A
NLO'+gg
boson, the NNLO corrections tend to exceed the NLO scale band at moderate
transversemomentum values.
For all distributions discussed so far, we nd qualitatively the same e ects at 8 and
13 TeV, essentially only di ering by the larger overall size of the NNLO corrections at the
higher collider energy. Contributing only about one third of the total NNLO correction, the
NLO0+gg approximation does not provide a reliable description of the full NNLO result.
Moreover, in general the loopinduced gluongluon channel alone cannot reproduce the
correct shapes of the full NNLO correction.
In this section we investigate the behaviour of radiative corrections in presence of
acceptance cuts used in W +W
measurements. The full set of cuts is summarized in table 1
and is inspired by the W +W
analysis of ref. [6].9 Besides various restrictions on the
leptonic degrees of freedom and the missing transverse momentum, this analysis implements
a jet veto.
Predictions for ducial cross sections at di erent perturbative orders are reported in
very di erently as compared to the inclusive case. The NLO corrections with respect to
LO amount to only about +4% (+1%) at 8 (13) TeV. Neglecting the +2% (+3%) shift due
to the PDFs, the NNLO corrections amount to +5% (+7%). Their positive impact is,
however, entirely due to the loopinduced gluonfusion contribution, which is not a ected
by the jet veto. In fact, comparing the NNLO and NLO0+gg predictions we see that the
genuine O( S2) corrections are negative and amount to roughly
1% ( 2%).
The reduction of the impact of radiative corrections when a jet veto is applied is
a wellknown feature in perturbative QCD calculations [111]. A stringent veto on the
radiation recoiling against the W +W
system tends to unbalance the cancellation between
positive real and negative virtual contributions, possibly leading to large logarithmic terms.
The resummation of such logarithms has been the subject of intense theoretical studies,
especially in the important case of Higgsboson production [112{115], and it has been
recently addressed also for W +W
production [
52, 53
]. In the case at hand, the moderate
size of radiative e ects beyond NLO suggests that, similarly as for Higgs production,
xedorder NNLO predictions should provide a fairly reliable description of jetvetoed ducial
cross sections and distributions.
The reduced impact of radiative e ects in the presence of a jet veto is often
accompanied by a reduction of scale uncertainties in xedorder perturbative calculations.
Comparing the results in table 3 with those in table 2 we indeed see that the size of the NNLO
scale uncertainty is reduced when cuts, particularly the jet veto, are applied. Such a small
scale dependence should be interpreted with caution as it tends to underestimate the true
uncertainty due to missing higherorder perturbative contributions.
The e ect of radiative corrections on the e ciency of W +W
ducial cuts,
=
ducial= inclusive ;
(3.1)
9We do not apply any leptonisolation criteria with respect to hadronic activity.
1:3%
p
s
LO
NLO
NLO'
di erences with respect to NLO. Scale uncertainties and errors as in table 2.
acceptance cuts at di erent perturbative orders and relative di
erences with respect to NLO. Scale uncertainties and errors as in table 2.
is illustrated in table 4, where numerator and denominator are evaluated at the same
perturbative order and both with
R =
F = mW . Due to the negative impact of the
newly computed NNLO corrections on the
ducial cross section (see table 3) and their
positive impact on the inclusive cross section (see table 2), the NNLO corrections on
the cut e ciency are quite signi cant. In particular, at p
s = 8 (13) TeV the full NNLO
prediction lies about 6% (9%) below the NLO0+gg result. The uncertainties quoted in
table 4 are obtained by varying
R and
F in a fully correlated way in the numerator and
denominator of eq. (3.1). Clearly, there is a large correlation at LO, which results in a
particularly small uncertainty. At NNLO the uncertainties are comparable to those of the
ducial cross sections.
As discussed in section 2.1, our default 4FS predictions are compared to an alternative
obtain
NLO = 165:7(3) fb and
NNLO = 181:9(4) fb at p
topsubtracted computation in the 5FS, in order to assess the uncertainty related to the
prescription for the subtraction of the top contamination. Without top subtraction we
s = 8 TeV in the 5FS. Due to the
jet veto, these ducial cross sections feature a moderate top contamination of about 8% at
NLO and 12% at NNLO. Removing the top contributions, we nd NLO = 153:4(4) fb and
NNLO = 162:5(3) fb, which agree with the 4FS results within 1%. At p
s = 13 TeV, the top
contamination in the 5FS is somewhat larger and amounts to 12% (17%) at NLO (NNLO).
The topsubtracted
ducial cross sections,
NLO = 238:3(6) fb and
NNLO = 265(2) fb, on
the other hand, are again in agreement with the 4FS results at the 1%
LO
NLO
NNLO
200 dσ/dΔϕll [fb]
0.5
1
2
2.5
3
0.5
1
2
2.5
3
R
R
T
i
w
w
d
d
e
e
c
c
u
u
d
d
o
o
r
r
p
p
X
I
M
h
1.5
Δϕll
(b)
applied. Absolute predictions and relative corrections as in gure 4.
Di erential distributions in presence of W +W
ducial cuts are presented
in gures 8{15. We rst consider, in
gure 8, the distribution in the azimuthal
separation of the charged leptons,
full NNLO result at small
ll. The NLO0+gg approximation is in good agreement with
ll, but in the peak region the di erence exceeds 5%, and
the NLO0+gg result lies outside the NNLO uncertainty band. The di erence signi cantly
increases in the large
ll region, where the cross section is strongly suppressed though.
The uncertainty bands of the NLO and NNLO predictions do not overlap. This feature
is common to all distributions that are considered in the following. It is primarily caused
by the loopinduced gg contribution, which enters only at NNLO and is not accounted for
by the NLO scale variations. Ignoring the gluoninduced component, we observe a good
perturbative convergence, apart from some peculiar phasespace corners.
In gure 9 we study the cross section as a function of the azimuthal separation
ll;
between the transverse momentum of the dilepton pair (pT;ll) and the missing transverse
ll;
=
at LO, the (N)NLO calculation is only (N)LO
< . The NNLO corrections have a dramatic impact on the shape of
momentum (pTmiss). Since
accurate at
ll;
to O(10) in the region
the distribution: the NNLO= NLO Kfactor grows with decreasing
ll;
and reaches up
ll;
. 1, where the cross section is suppressed by more than
three orders of magnitude. This huge e ect results from the interplay of the jet veto with
the cuts on the pT 's of the individual leptons and on pTmiss. At small
momenta pT;ll and pTmiss must be balanced by recoiling QCD partons. However, at NLO
the emitted parton can deliver a sizeable recoil only in the region that is not subject to the
ll;
the transverse
jet veto, i.e. in the strongly suppressed rapidity range jyj j > 4:5. At NNLO, the presence
of a second parton relaxes this restriction to some extent, thereby reducing the suppression
1.5
Δϕll,νν
(b)
ton system and the missing transverse momentum. W +W
cuts are applied. Absolute predictions
and relative corrections as in gure 4.
dσ/dmll [fb/GeV]
µ+e νµν‾ e(WWcuts )@LHC 8 TeV
dσ/dmll [fb/GeV]
µ+e νµν‾ e(WWcuts )@LHC 13 TeV
LO
NLO
NNLO
X
I
R
R
T
LO
NLO
NNLO
LO
(b)
104 dσ/dΔϕll,νν [fb]
101
100
101
102
101
100
100
101
102
1.4
1.3
1.2
1.1
1
0.9
0
I
R
T
u
d
r
p
M
h
c
u
d
predictions and relative corrections as in gure 4.
HJEP08(216)4
103
RIT 102
1.4
1.3
1.2
1.1
1
0.9
d
d
o
o
LO
NLO
NNLO
T
t
i
d
e
c
u
r
p
NLO'+gg
102
103
104
1.5
1.4
mTATLAS [GeV]
NLO0+gg predictions at to the PDFs.
transverse mass. W +W
cuts are applied. Absolute
predictions and relative corrections as in gure 4.
by about one order of magnitude. The loopinduced gg contribution does not involve any
QCD radiation and contributes only at
ll;
=
. As a consequence, the NLO and
ll;
are almost identical, apart from minor di erences due
The invariantmass distribution of the dilepton pair is presented in gure 10. On the
one hand, if one takes into account NNLO scale variations, the NLO0+gg result is by and
large consistent with the NNLO prediction. On the other hand, the shapes of the NLO0+gg
and NNLO distributions feature nonnegligible di erences, which range from +5% at low
masses to
5% in the highmass tail. Nevertheless, NLO0+gg provides a reasonable
approximation of the full NNLO result, in particular regarding the normalization.
The distribution in the W +W
transverse mass,
mTATLAS =
q
ET;l1 + ET;l2 + pTmiss 2
pT;l1 + pT;l2 + pTmiss 2 ;
(3.2)
is displayed in
gure 11. Also in this case, apart from the strongly suppressed region
of small mTATLAS, the NLO0+gg approximation is in quite good agreement with the full
NNLO prediction.
In gures 12 and 13 we show results for the pT distributions of the leading and
subleading lepton, respectively. In both cases the impact of NNLO corrections grows with pT .
This is driven by the gluoninduced contribution, which overshoots the complete NNLO
result in the smallpT region and behaves in the opposite way as pT becomes large. In
the case of the subleading lepton, the genuine NNLO corrections are as large as O(10%)
around pT;l2 = 200 GeV. Overall, there is a visible di erence in shape between NLO0+gg
and NNLO for both the leading and subleading lepton transversemomentum distributions.
100
101
102
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0
101
100
101
102
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0
LO
NLO
NNLO
I
R
T
i
w
d
r
p
I
R
t
i
c
u
d
101 dσ/dpT,l1 [fb/GeV] µ+e νµν‾ e(WWcuts )@LHC 13 TeV
pT,l2 [GeV]
100
(a)
LO
NLO
NNLO
NLO'+gg
LO
NLO
NNLO
RT 100
I
i
dw 101
(b)
cuts are applied. Absolute
w
d
d
e
e
o
o
r
r
p
p
predictions and relative corrections as in gure 4.
dσ/dpT,l2 [fb/GeV]
µ+e νµν‾ e(WWcuts )@LHC 8 TeV
dσ/dpT,l2 [fb/GeV] µ+e νµν‾ e(WWcuts )@LHC 13 TeV
50
150
200
0
50
150
200
(b)
cuts are applied. Absolute
predictions and relative corrections as in gure 4.
100
101
102
103
101
102
103
1.8
1.6
1.4
1.2
1
0.8
0
LO
NLO
NNLO
NLO'+gg
LO
NLO
NNLO
X 100
I
R
R
T
predictions and relative corrections as in gure 4.
101 dσ/dpTmiss [fb/GeV]
101 dσ/dpTmiss [fb/GeV] µ+e νµν‾ e(WWcuts )@LHC 13 TeV
101 dσ/dpT,ll [fb/GeV] µ+e νµν‾ e(WWcuts )@LHC 13 TeV
(b)
cuts are applied. Absolute
R
R
t
i
i
w
w
u
u
d
d
o
o
LO
NLO
NNLO
NLO'+gg
LO
NLO
NNLO
X
I
d
e
c
r
p
X
I
d
e
c
r
p
50
150
200
0
50
150
200
(b)
cuts are applied. Absolute
predictions and relative corrections as in gure 4.
s = 13 TeV in presence of W +W
The pT distribution of the dilepton pair is displayed in gure 14. This observable has
a kinematical boundary at LO, where the requirement pTmiss > 20 GeV implies that pT;ll >
20 GeV. The region pT;ll < 20 GeV starts to be populated at NLO, but each perturbative
higherorder contribution (beyond LO) produces integrable logarithmic singularities leading
to perturbative instabilities at the boundary [116]. This becomes particularly evident in
the d NNLO=d NLO ratio. The loopinduced gg contribution, having Bornlike kinematics,
does not contribute to the region pT;ll < 20 GeV. In contrast, NNLO corrections are huge,
and the formal accuracy of NNLO predictions is only NLO in that region. In the region of
high pT;ll we observe signi cant NNLO corrections, and the NLO0+gg approximation works
rather well. Similar features are observed in the pmiss distribution, displayed in
gure 15,
but without the perturbative instability at pTmiss = 20 GeV, as the cut on pmiss is explicit.
T
In general, radiative corrections behave in a rather similar way at p
s = 8 TeV and
T
cuts. Comparing the NLO0+gg approximation with
the full NNLO prediction, we nd that the overall normalization is typically reproduced
quite well, while genuine NNLO corrections can lead to signi cant shape di erences of up
to 10%. It does not come as a surprise that in kinematic regions that imply the presence of
QCD radiation, loopinduced gg contributions cannot provide a reasonable approximation
of the full NNLO correction.
e production with Higgs selection cuts
In this section we repeat our study of radiative corrections in presence of cuts that are
designed for H ! W +W
studies at the LHC. In this case, W +W
production plays the
role of irreducible background, and more stringent cuts are applied in order to minimize
its impact on the H ! W +W
corresponds to the H ! W +W
of cuts similar to the ones used in W +W
suppression of onshell W +W
pT;ll, mll,
ll and
ll; .
signal. The precise list of cuts is speci ed in table 1 and
analysis of ref. [12].10 This selection implements a series
signal measurements, including a jet veto. The
production is achieved through additional restrictions on
In table 5 we report predictions for ducial cross sections at di erent perturbative
orders. The corresponding acceptance e ciencies, computed as in section 3.3, are presented
in table 6. It turns out that Higgs cuts suppress the impact of QCD radiative e ects in
a similar way as W +W
cuts. At 8 (13) TeV the NLO and NNLO corrections amount to
+5% (+3%) and to +9% (+13%), respectively. The latter consist of a positive +3% shift
due to NNLO PDFs, a sizeable loopinduced gg component of +9% (+13%), and a rather
small genuine O( S2) contribution of
2% ( 4%).
at p
We compare the 4FS predictions against the topsubtracted calculation in the 5FS:
s = 8 (13) TeV the latter yields
NLO = 48:7 (3) fb ( NLO = 73:4 (2) fb) and
NNLO =
53:0 (5) fb ( NNLO = 83:1 (5) fb), which corresponds to a 1%
2% agreement with the 4FS
results. The size of the subtracted top contamination in the 5FS is slightly smaller than
what was found for W +W
cuts. It amounts to 5% (9%) at NLO and 6% (11%) at NNLO.
10In our analysis, we require jy j < 2:4 as for W +W
signal cuts, in contrast to jy j < 2:5 in the ATLAS
analysis. Moreover, we do not apply any leptonisolation criteria with respect to hadronic activity.
HJEP08(216)4
s
LO
NLO
NLO'
ducial cuts at di erent perturbative orders and relative
di erences with respect to NLO. Scale uncertainties and errors as in table 2.
=
ducial(H cuts)= inclusive
= NLO
1
s
LO
8 TeV
with respect to NLO. Scale uncertainties and errors as in table 2.
Similarly to the case of W +W
impact on the acceptance e ciency: at p
cuts, genuine O( S2) corrections have a signi cant
s = 8 (13) TeV the NNLO prediction lies roughly
8% (10%) below the NLO0+gg result, which exceeds the respective scale uncertainties.
While the relative size of higherorder e ects on the Higgscut e ciency is almost identical
to the one found for W +W
selection cuts, the absolute size of the acceptance e ciencies
is much smaller. In the case of Higgs cuts it is almost a factor of three lower, primarily
due to the stringent cut on the invariant mass of the dilepton system.
Di erential distributions with Higgs cuts applied are presented in gures 16{23. In
general, they behave in a similar way as for the case of W +W
cuts discussed in section 3.3.
However, a few observables are quite sensitive to the additional cuts that are applied in the
Higgs analysis. Most notably, the distribution in the azimuthal separation of the charged
leptons in
gure 16 exhibits a completely di erent shape as compared to
gure 8. In
corrections with respect to the NLO distribution at p
particular, it features an approximate plateau in the region 0:4
ll
1:2. The NNLO
s = 8 (13) TeV range from about
+13% (+18%) at small
ll to roughly +2% (+5%) at separations close to the ducial cut.
The loopinduced gg component provides a good approximation of the complete NNLO
result for small separations, but in the large
ll region it overshoots the complete NNLO
result by about 5% (7%).
In the
case of W +W
ll;
distribution, displayed in
gure 17, we observe that, similarly to the
gure 9), also Higgs cuts lead to huge NNLO corrections at
40 dσ/dΔϕll [fb]
I
I
R
R
T
A
M
M
h
t
t
i
i
w
w
d
e
e
c
u
u
d
d
o
r
r
Absolute predictions and relative corrections as in gure 4.
dσ/dΔϕll,νν [fb]
µ+e νµν‾ e(Hcuts )@LHC 8 TeV
dσ/dΔϕll,νν [fb]
µ+e νµν‾ e(Hcuts )@LHC 13 TeV
LO
NLO
NNLO
NLO'+gg
60
55
50
45
40
35
30
25
1.2
I
TRA 101
M
h 100
LO
NLO
NNLO
NLO'+gg
103 dσ/dσNLO
102
101
100
Δϕll
Δϕll,νν
(b)
35
30
25
20
15
1.2
101
102
103
104
105
102
101
100
t
t
NLO'+gg
R
T
i
w
d
e
c
u
d
o
r
p
1.6
dilepton system and the missing transverse momentum. Higgs cuts are applied. Absolute
predictions and relative corrections as in gure 4.
1.6 dσ/dmll [fb/GeV]
2.5 dσ/dmll [fb/GeV]
AT 1.5
M
h
t
i
i
w
w
d
d
d
o
r
r
p
LO
NLO
NNLO
T
A
c
u
NLO'+gg
(b)
tions and relative corrections as in gure 4.
T
small
ll; . As discussed in section 3.3, this behaviour is due to the fact that at small
the leptonic and pmiss cuts require the presence of a sizeable QCD recoil, which is,
however, strongly suppressed by the jet veto at NLO. In the Higgs analysis, this suppression
mechanism becomes even more powerful due to the additional cut pT;ll > 30 GeV, which
forbids the two leptons to recoil against each other. This leads to the kink at
ll;
= 2:2
in the NLO distribution and to the explosion of NNLO corrections below and slightly above
this threshold.
The invariant mass of the dilepton system, shown in gure 18, is restricted to the region 10 GeV
mll
55 GeV. The peak of the distribution is around mll = 38 GeV, and
the
NNLO= NLO Kfactor is essentially
at. Also the NLO0+gg curve has a very similar
shape so that the radiative corrections precisely match those on the ducial rates.
The distribution in mTATLAS is presented in
gure 19. As compared to the W +W
analysis (see gure 11), we observe that the tail of the distribution drops signi cantly faster
when Higgs cuts are applied. Moreover, in the highmTATLAS region the size of the
loopinduced gg corrections relative to NLO and, hence, the size of the full NNLO correction,
is much larger than in the W +W
40% (60%) of the NLO cross section at p
when W +W
cuts are applied.
analysis. The NNLO corrections amount up to about
s = 8 (13) TeV, while they hardly exceed 15%
The distributions in the lepton pT 's, depicted in gures 20 and 21, behave in a similar
way as in
gures 12 and 13, apart from a steeper dropo in the tail and slightly larger
corrections. The shape of the pT;l2 distribution can be qualitatively explained as follows.
As pT;l2 becomes large, since the dilepton invariant mass is constrained to be smaller than
55 GeV (see table 1), the total transverse momentum of the dilepton system increases.
Such large dilepton pT has to be balanced by the missing transverse momentum and by the
i
w
w
d
d
o
o
X
I
R
t
d
c
u
r
HJEP08(216)4
100
101
102
103
1.8
mTATLAS [GeV]
mTATLAS [GeV]
tions and relative corrections as in gure 4.
transverse mass. Higgs cuts are applied. Absolute
predicrecoiling QCD radiation, whose pT must increase accordingly. At pT;l2
50 GeV the jet
of the impact of radiative corrections. This e ect is particularly visible at p
veto starts to suppress QCD radiation harder than 25 GeV, thereby leading to a reduction
s = 13 TeV,
where the available energy is larger.
W +W
T
of W +W
For the distributions in the pT of the dilepton pair (see gure 22) and in pmiss (see
T
gure 23), we also nd a similar behaviour as in the case where W +W
cuts are applied.
We note, however, that the perturbative instability observed in the pT;ll distribution with
gure 14) is removed by the explicit cut pT;ll > 30 GeV in the Higgs
analysis. Accordingly, the pT;ll cut implicitly vetoes events with pmiss < 30 GeV at Born
T
level, which leads to a perturbative instability in the pTmiss distribution, particularly visible
in the NNLO= NLO ratio. In fact, it is evident from
gure 23 that the phasespace region
pmiss < 30 GeV is lled only upon inclusion of higherorder corrections. Similarly to the case
cuts, the behaviour of radiative e ects is rather insensitive to the collider energy.
Comparing NLO0+gg and full NNLO predictions, in spite of the fairly good agreement at
the level of ducial cross section, we observe that the genuine O( S2) corrections lead to
signi cant shape distortions at the 10% level.
4
We have presented the rst fully di erential calculation of the NNLO QCD corrections to
W +W
production with decays at the LHC. O shell e ects and spin correlations, as well
as all possible topologies that lead to a nal state with two charged leptons and two
neutrinos are consistently taken into account in the complexmass scheme. At higher orders
(b)
tions and relative corrections as in gure 4.
101 dσ/dpT,l2 [fb/GeV]
µ+e νµν‾ e(Hcuts )@LHC 8 TeV
101 dσ/dpT,l2 [fb/GeV]
µ+e νµν‾ e(Hcuts )@LHC 13 TeV
100
101
102
103
104
1.8
101
102
103
104
1.8
1.6
1.4
1.2
1
LO
NLO
NNLO
LO
NLO
NNLO
R
T
A
i
e
c
p
i
w
e
c
u
r
p
I
I
h
h
d
d
d
d
HJEP08(216)4
I
I
h
h
d
d
d
d
o
o
LO
NLO
NNLO
LO
NLO
NNLO
R
MAT 101
wit 102
uce 103
R
MAT 101
wit 102
euc 103
predictions and relative corrections as in gure 4.
NLO'+gg
pT,l2 [GeV]
40
(b)
pT,ll [GeV]
(b)
predictions and relative corrections as in gure 4.
dσ/dpTmiss [fb/GeV]
µ+e νµν‾ e(Hcuts )@LHC 13 TeV
I
R
R
h
h
t
t
i
i
c
c
u
u
d
d
predictions and relative corrections as in gure 4.
101
102
103
1.6
1.4
1.2
1
100
101
102
103
1.8
1.6
1.4
1.2
1
0.8
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
pmiss [GeV]
T
(a)
(b)
I
I
R
R
h
h
t
t
i
i
c
c
u
u
d
d
HJEP08(216)4
1.8 dσ/dσNLO
LO
NLO
NNLO
T
A
M
w
r
p
T
A
M
w
d
e
o
r
p
LO
NLO
NNLO
LO
NLO
NNLO
100
in QCD perturbation theory, the inclusive W +W
cross section is plagued by a huge
contamination from topquark production processes, and the subtraction of top contributions
is mandatory for a perturbatively stable de nition of the W +W
rate. In our calculation,
any top contamination is avoided by excluding partonic channels with
nalstate bottom
quarks in the 4FS, where the bottomquark mass renders such contributions separately
nite. In order to quantify the sensitivity of the topfree W +W
cross section on the details
of the topsubtraction prescription, our default predictions in the 4FS have been compared
to an alternative calculation in the 5FS. In the latter case a numerical extrapolation in
the narrow topwidth limit is used to separate contributions that involve top resonances
from genuine W +W
production and its interference with tW and tt production. The
comparison of 4FS and 5FS predictions for inclusive and
ducial cross sections indicates
that the dependence on the topsubtraction prescription is at the 1%
Numerical predictions at p
di erent avour channel pp !
s = 8 and 13 TeV have been discussed in detail for the
e + X. As compared to the case of onshell W +W
production [46], the inclusion of leptonic decays leads to a reduction of the total cross
section that corresponds to the e ect of leptonic branching ratios plus an additional correction
of about
2% due to o shell e ects. The in uence of o shell W boson decays on the
behaviour of (N)NLO QCD corrections is negligible. In fact, apart from minor di erences
due to the employed PDFs, we
nd that the relative impact of QCD corrections on the
total cross sections is the same as for onshell W +W
production [46]. At p
s = 8 (13) TeV,
ignoring the shift of +2% (+3%) due to the di erence between NNLO and NLO PDFs, the
overall NNLO correction is as large as +9% (+11%), while the loopinduced gluongluon
contribution amounts to only +3% (+4%); i.e., contrary to what was generally expected
in the literature, the NNLO corrections are dominated by genuine NNLO contributions to
the qq channel, and the loopinduced gg contribution plays only a subdominant role.
The complete calculation of NNLO QCD corrections allows us to provide a rst
realistic estimate of theoretical uncertainties through scale variations: as is wellknown,
uncertainties from missing higherorder contributions obtained through scale variations are
completely unreliable at LO and still largely underestimated at NLO. This is due to the
fact that the qg (as well as qg) and gg (as well as qq(0), qq(0) and qq0) partonic channels
do not contribute at LO and NLO, respectively. In fact, NNLO is the rst order at which
all partonic channels contribute. Thus NNLO scale variations, which are at the level of
2%
3% for the inclusive cross sections, can be regarded as a reasonable estimate of the
theoretical uncertainty due to the truncation of the perturbative series. This is supported
by the moderate impact of the recently computed NLO corrections to the loopinduced gg
contribution [37].
Imposing a jet veto has a strong in uence on the size of NNLO corrections and on the
relative importance of NNLO contributions from the qq channel and the loopinduced gg
channel. This was studied in detail for the case of standard
ducial cuts used in W +W
and H ! W +W
analyses by the LHC experiments. As a result of the jet veto, such
cuts signi cantly suppress all (N)NLO contributions that involve QCD radiation, thereby
enhancing the relative importance of the loopinduced gg channel at NNLO. More precisely,
depending on the analysis and the collider energy, ducial cuts lift the loopinduced gg
contribution up to 6%
13% with respect to NLO, whereas the genuine NNLO corrections
to the qq channel are negative and range between
1% and
4%, while the NLO corrections
vary between +1% and +5%. The reduction of the impact of radiative corrections is
accompanied by a reduction of scale uncertainties, which, for the NNLO
sections, are at the 1%
2% level. This is a typical sidee ect of jet vetoes, and scale
uncertainties are likely to underestimate unknown higherorder e ects in this situation.
As a result of the di erent behaviour of radiative corrections to the inclusive and
ducial cross sections, their ratios, which determine the e ciencies of acceptance cuts, turn out
to be quite sensitive to higherorder e ects. More explicitly, the overall NNLO corrections
to the cut e ciency are small and range between
3% and
1%. However, they arise from
a positive shift between +3% and +9% due to the loopinduced gg channel, and a negative
shift between
6% and
10% from genuine NNLO corrections to the qq channel. The NLO
prediction supplemented by the loopinduced gg channel, i.e. the \best" prediction before
the complete NNLO corrections were known, would thus lead to a signi cant overestimation
of the e ciency, by up to about 10%. Similarly to the case of ducial cross sections, the
scale uncertainties of cut e ciencies are at the 1% level, and further studies are needed in
order to estimate unknown higherorder e ects in a fully realistic way. This, in particular,
involves a more accurate modelling of the jet veto, which is left for future work.
Our analysis of di erential distributions demonstrates that, in absence of ducial cuts,
genuine NNLO corrections to the qq channel can lead to signi cant modi cations in the
shapes of observables that are sensitive to QCD radiation, such as the transverse
momentum of the leading W boson or of the W +W
system. On the other hand, in presence of
ducial cuts, NLO predictions supplemented with the loopinduced gg contribution yield
a reasonably good description of the shape of di erential observables, such as dilepton
invariant masses and singlelepton transverse momenta. We nd, however, that even for
standard W +W
and Higgs selection cuts, which include a jet veto, genuine NNLO
corrections tend to distort such distributions by up to about 10%. In phasespace regions that
imply the presence of QCD radiation, loopinduced gg contributions cannot approximate
the shapes of full NNLO corrections.
The predictions presented in this paper have been obtained with Matrix, a widely
automated and
exible framework that supports NNLO calculations for all processes of
the class pp ! l+l0
l l0 + X, including in particular also the channels with equal lepton
avours, l = l0. More generally, Matrix is able to address fully exclusive NNLO
computations for all diboson production processes at hadron colliders.
Acknowledgments
We thank A. Denner, S. Dittmaier and L. Hofer for providing us with the oneloop
tensorintegral library Collier well before publication, and we are grateful to P. Maierhofer and
J. Lindert for advice on technical aspects of OpenLoops. This research was supported
in part by the Swiss National Science Foundation (SNF) under contracts 200020141360,
200021156585, CRSII2141847, BSCGI0157722 and PP00P2153027, and by the Kavli
Institute for Theoretical Physics through the National Science Foundation's Grant No.
NSF PHY1125915.
p
p
Open Access.
This article is distributed under the terms of the Creative Commons Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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