Nexttoleadingorder electroweak corrections to the production of four charged leptons at the LHC
Received: November
Nexttoleadingorder electroweak corrections to the production of four charged leptons at the LHC
Benedikt Biedermann 0 1 2 4 5 6
Ansgar Denner 0 1 2 4 5 6
Stefan Dittmaier 0 1 2 4 5 6
Lars Hofer 0 1 2 3 4 5 6
0 08028 Barcelona , Spain
1 79104 Freiburg , Germany
2 97074 Wu ̈rzburg , Germany
3 Institut de Ci`encies del Cosmo (ICCUB)
4 Open Access , c The Authors
5 72076 Tu ̈bingen , Germany
6 [45] B. Biedermann, A. Denner , S. Dittmaier, L. Hofer and B. J ̈ager, Electroweak corrections to
and Barbara J¨agerd aInstitut fu¨r Theoretische Physik und Astrophysik, JuliusMaximiliansUniversita¨t Wu¨rzburg, Departament de F´ısica Qu`antica i Astrof´ısica (FQA), Universitat de Barcelona (UB), , ,
bPhysikalisches Institut; AlbertLudwigsUniversita¨t Freiburg

Abstract: We present a stateoftheart calculation of the nexttoleadingorder
electroweak corrections to ZZ production, including the leptonic decays of the Z bosons into
order matrix elements for fourlepton production, including contributions of virtual photons
and all offshell effects of Z bosons, where the finite Zboson width is taken into account
using the complexmass scheme. The matrix elements are implemented into Monte Carlo
programs allowing for the evaluation of arbitrary differential distributions.
integrated and differential cross sections for the LHC at 13 TeV both for an inclusive setup
where only lepton identification cuts are applied, and for a setup motivated by
Higgsboson analyses in the fourlepton decay channel. The electroweak corrections are divided
into photonic and purely weak contributions. The former show the wellknown pronounced
tails near kinematical thresholds and resonances; the latter are generically at the level of
leptons in the final state can reach up to 10% in offshellsensitive regions. Contributions
induced by incoming photons, i.e. photonphoton and quarkphoton channels, are included,
but turn out to be phenomenologically unimportant.
ArXiv ePrint: 1611.05338
1 Introduction
Details of the calculation
Partonic channels
Virtual corrections
Real corrections
Phenomenological results
3.1 Input parameters 2.1 2.2 2.3
Numerical implementation and independent checks of the calculation
Definition of observables and acceptance cuts
Results on integrated cross sections
Results on differential cross sections in the inclusive setup
Results on differential cross sections in the Higgsspecific setup
Introduction
The physics programme of the LHC at Run I was particularly successful in the
investigation of electroweak (EW) interactions and culminated in the discovery of a Higgs boson,
but no evidence for physics beyond the Standard Model (SM) was found. While the
community is looking forward to a major discovery at Run II, an important task is the precise
measurement of the properties of the Higgs boson and the other particles of the SM. Small
deviations from the predictions of the SM in the observed event rates or distributions might
reveal signs of new physics.
One class of processes particularly relevant for tests of the EW sector of the SM is
EW gaugeboson pair production. These reactions allow to measure the triple gaugeboson
couplings and to study the EW gauge bosons in more detail. Moreover, they constitute a
background to Higgsboson production with subsequent decay into weak gaugeboson pairs
and to searches for new physics such as heavy vector bosons. In the Higgssignal region
below the WW and ZZ production thresholds, offshell effects of the W and Z bosons are
of particular importance. In this paper we focus on the production of Zboson pairs with
subsequent decays to four charged leptons, covering all offshell domains in phase space.
While this channel has the smallest cross section among the vectorboson pair production
processes, it is the cleanest, as it leads to final states with four charged leptons that can
be well studied experimentally.
At Run I both ATLAS and CMS measured the cross section of Zboson pair
production [1–4] using final states with four charged leptons or two charged leptons and two
neutrinos. The results of these measurements are in agreement with the predictions of the
SM and permitted to derive improved limits on triple gaugeboson couplings between
neutral gauge bosons [5–7]. Run II allows to improve these measurements, and first analyses
have already been published [8, 9].
Precise measurements call for precise predictions. The nexttoleading order (NLO)
QCD corrections to Zboson pair production were calculated a long time ago for stable
Z bosons [10, 11] and including leptonic decays in the narrowwidth approximation [12].
Once the oneloop helicity amplitudes were available [13], complete calculations including
spin correlations and offshell effects became possible [14, 15]. Gluoninduced oneloop
contributions were evaluated for stable Z bosons [16, 17], including offshell effects [18, 19],
and studied as a background to Higgsboson searches [20]. NLO QCD corrections were
matched to parton showers in various frameworks without [21] and with [22–25] including
leptonic decays. In ref. [26], a comprehensive NLOQCDbased prediction was presented
for offshell weak diboson production as a background to Higgs production. Recently, the
nexttonexttoleading order (NNLO) QCD corrections to Zpair production have been
calculated for the total cross section [27] and including leptonic decays [28]. The NNLO QCD
calculation has been combined with nexttonexttoleadingorder resummation of multiple
softgluon emission [29]. Although formally being beyond NNLO in the pp cross section,
even the NLO corrections to the loopinduced gluonfusion channel were calculated [30–32]
because of their particular relevance in Higgsboson analyses.
Besides QCD corrections also EW NLO corrections are important for precise
predictions of vectorboson pair production at the LHC. EW corrections typically increase with
energy owing to the presence of large Sudakov and other subleading EW logarithms [33–
38] and reach several 10% in the highenergy tails of distributions. In addition, photonic
corrections lead to pronounced radiative tails near resonances or kinematical thresholds.
Logarithmic EW corrections to gaugeboson pair production at the LHC were studied
in ref. [39] and found to reach 30% for Zpair production for ZZ invariant masses in the
TeV range. Later, the complete NLO EW corrections were calculated for stable vector
bosons and all pair production processes including photoninduced contributions [40, 41].
The size and in particular the nonuniform effect on the shapes of distributions were
confirmed. Leptonic vectorboson decays were first included in NLO EW calculations in the
form of a consistent expansion about the resonances for Wpair production [42], and in an
approximate variant via the Herwig++ Monte Carlo generator for WW, WZ, and ZZ
production [43]. Most recently, NLO EW calculations based on full 2 → 4 particle amplitudes,
including all offshell effects, have been presented for Wpair [44] and Zpair production [45]
for fourlepton final states of different fermion generations (i.e. without identical particle
effects or WW/ZZ interferences). For Zpair production, the offshell effects include also the
contributions of virtual photons that cannot be separated from the Zpair signal, but only
suppressed by using appropriate invariantmass cuts. Note that these full offshell
calculations are essential to safely assess the EW corrections below the WW and ZZ thresholds,
¯
q
e−
¯
q
e−
e−
boson analyses. Moreover, a detailed comparison of the full fourlepton calculation [44] to
the doublepole approximation for Wboson pairs [42] revealed limitations of the latter
approach for transversemomentum distributions of the leptons in the highenergy domain
where newphysics signals are searched for.
In ref. [45] we have presented some selected results for the NLO EW corrections to
offshell ZZ production in a scenario relevant for Higgsboson studies. In this paper we provide
more detailed phenomenological studies in various phasespace regions relevant for LHC
including interference effects from identical finalstate leptons. We follow the same concepts
and strategies as in refs. [44, 45], i.e. finitewidth effects of the Z bosons are consistently
included using the complexmass scheme [46–48], so that we obtain NLO EW precision
everywhere in phase space. We also include photoninduced partonic processes originating
The paper is organized as follows: some details on the calculational methods are
presented in section 2. Phenomenological results for two different experimental setups are
discussed in section 3. Our conclusions are given in section 4.
Details of the calculation
Partonic channels
+X receive contributions from the quarkantiquark annihilation channels
q¯q/qq¯ → µ +µ −e+e−, µ +µ −µ +µ −,
channels in the following, are shown in figures 1(a) and 1(b). Note that all LO diagrams
involve Zboson and photon exchange only. There are LO channels with two photons in
the initial state as well,
γγ → µ +µ −e+e−, µ +µ −µ +µ −,
with corresponding diagrams shown in figure 1(c). Due to their small numerical impact,
LO cross section and do not include higherorder corrections to these processes.
The NLO EW corrections comprise virtual and real contributions of the q¯q channels,
q¯q/qq¯ → µ +µ −e+e− (+γ) , µ +µ −µ +µ −
and the real photoninduced contributions with one (anti)quark and one photon in the
γq¯/q¯γ → µ +µ −e+e− q¯, µ +µ −µ +µ −
Virtual corrections
The oneloop virtual corrections to the q¯q channels are computed including the full set
of Feynman diagrams. We employ the complexmass scheme for the proper handling of
unstable internal particles [46–48]. This approach allows the simultaneous treatment of
phasespace regions above, near, and below the Z resonances within a single framework,
leading to NLO accuracy both in resonant and nonresonant regions. Sample diagrams
for the virtual EW corrections are shown in figure 2. A first set of diagrams is obtained
by exchanging Z bosons or photons in all possible ways between the fermion lines of the
treelevel diagrams in figure 1: diagram types (a) and (b) of figure 2 would also be present
in narrowwidth or pole approximations for the Z bosons and contain separate corrections
to the production and the decay of the Z boson. Diagrams (c) and (d) feature correlations
between the initial and final states or between different Zboson decays and are only present
in a full offshell calculation. The sample diagrams (e)–(i) cannot be obtained by naive
vectorboson insertions between fermion lines. They involve, for example, closed fermion
loops (e) or the exchange of W or Higgs bosons.
In our calculation, we perform a gaugeinvariant decomposition of the full NLO EW
virtual photonic part is defined as the set of all diagrams with at least one photon in the
loop coupling to the fermion lines. The weak contribution is then the set of all remaining
oneloop diagrams, including also selfenergy insertions and vertex corrections induced by
closed fermion loops. The contributions to the renormalization constants have to be split
accordingly. This splitting is possible, because the LO process with four charged leptons
in the final state does not involve chargedcurrent interactions, i.e. there is no Wboson
exchange at tree level. The vectorboson insertions between fermion lines exemplarily
shown in the diagrams (a)–(d) of figure 2 thus exhaust all generic possibilities how a photon
appears in a loop propagator and can systematically be used to construct the virtual
photonic contribution. Figure 3 shows the decomposition of the eight diagrams represented
by figure 2(d) into the purely weak part with only Z bosons in the loop coupling to fermion
lines (upper row) and the photonic part with one or two photons in the loop (lower row).
¯
q
¯
q
¯
q
e−
e−
e−
¯
q
¯
q
¯
q
photon and Zboson insertions between the fermion lines of the treelevel diagrams in figure 1(a).
The remaining diagrams may involve couplings (f)–(i) or corrections to vertices (e) that are not
present at LO.
Note that the criterion for the splitting considers only the vector bosons in the loop,
while it does not refer to the treelevel part of the diagram. The contributions to the
field renormalization constants of the fermions are decomposed in an analogous manner.
Since only loops with internal photons lead to soft and collinear divergences, the purely
weak contribution is infrared (IR) finite. The full and finite photonic corrections to the q¯q
channels, on the other hand, comprise the virtual photonic corrections plus the real photon
emission described in the next section.
In general, the decomposition of an amplitude into gaugeinvariant parts requires great
caution. The gaugeinvariant isolation of photonic corrections in processes that proceed
in LO via neutralcurrent interactions is only possible because the LO amplitude can be
group with the photon as massless gauge boson and U(1)Z to an Abelian gauge theory with
the Z boson as gauge boson. The corrections within this theory, which is a consistent and
e−
e−
e−
¯
q
¯
q
¯
q
e−
e−
e−
¯
q
e−
¯
q
¯
q
contributions for the diagram type shown in figure 2(d).
renormalizable field theory on its own (see e.g. ref. [49] or ref. [50], chapter 12.9), form
a gaugeinvariant subset of the full SM corrections to our neutralcurrent process. Since
theory, the photonic contributions form a gaugeinvariant subset of the corrections (which
could even be further decomposed into subsets defined by the global charge factors of
the fermions linked by the photon). Note that the subset of closed fermion loops could be
isolated as another gaugeinvariant part of the EW correction, since each fermion generation
(in the absence of generation mixing) delivers a gaugeinvariant subset of diagrams. For
the process at hand, we, however, prefer to keep the closed fermion loops in the weak
Real corrections
The real corrections to the q¯q channels include all possible ways of photon radiation off the
initialstate quarks or off the finalstate leptons, schematically depicted in figure 4. The
phasespace integrations over the squared realemission amplitudes diverge if the radiated
photon becomes soft or collinear to one of the external fermions. For IRsafe (i.e.
softand collinearsafe) observables, however, the collinear finalstate singularities and the soft
singularities cancel exactly the corresponding soft and collinear divergences from the virtual
corrections after integration over the phase space. The collinear initialstate singularities
do not fully cancel between real and virtual corrections; the remnants are absorbed into the
parton distribution functions (PDFs) via factorization, in complete analogy to the usual
procedure applied in QCD.
We employ the dipole subtraction formalism for the numerical integration of the real
corrections. In detail, we apply two different variants based on dimensional
regularization [51] and mass regularization [52], respectively. The results obtained with the two
e−
e−
¯
q
e−
¯
q
e−
¯
q
e−
approaches are in perfect agreement. The underlying idea is to add and subtract
auxiliary terms Msub2 at the integrand level which pointwise mimic the universal singularity
structure of the squared real matrix elements Mreal2 and which, on the other hand, can
be integrated analytically in a processindependent way. In the dipole subtraction
approach, the subtraction function is constructed from “emitterspectator pairs”, where the
“emitter” is the particle whose collinear splitting leads to an IR singularity and the
“spectator” is the particle balancing momentum and charge conservation in the emission process.
+ 2 Re MBornMvirt
where 2 Re[MBornMvirt] denotes the interference of the Born amplitude MBorn with the
oneloop amplitude Mvirt. The last term represents the (IRdivergent) factorization
contribution from the PDF redefinition, which takes the form of a convolution over the
momentum fraction x quantifying the momentum loss via collinear parton/photon emission.
particle real phase space, which includes the bremsstrahlung photon, is decomposed into
integration over [dk] involves an integration over the variable x which controls the
momentum loss from initialstate radiation. Splitting off the soft singularities developing in this
integration of the subtraction terms over x, decomposes the integral R1[dk]Msub2 into
analytically combined with the virtual corrections, the latter with the factorization
contribution, to produce individually IRfinite terms which may be integrated separately in a
fully numerical way:
dΦ Mreal2Θ(p1, . . . , pn, pγ ) − Msub2Θ(p˜1, . . . , p˜n)
∗
dxhMsub(x)2fin + δ(1 − x)2 Re MBornMvirt finiΘ(p˜1, . . . , p˜n) .
∗
Here, Msub(x)2fin and 2 Re[MBornMvirt]fin are the finite parts resulting from this splitting
of the x integration into continuum and endpoint parts. The momenta p˜i of the reduced
nparticle phase space in the first integral are related to the momenta pi of the (n+1)particle
The integration over x is a remainder from the factorized oneparticle phase space and
the phasespace cuts applied to the particle momenta {p1, . . . }, possibly after applying a
recombination procedure which is discussed in the next paragraphs.
Quarkinduced channels — the collinearsafe setup.
Observables that are collinear
safe with respect to finalstate radiation — as described in ref. [52] — are constructed by
applying an appropriate procedure for recombining radiated photons with nearly collinear
finalstate leptons. In collinear regions, a photonlepton pair with photon and lepton
moAny phasespace cut or any evaluation of an observable is performed for the recombined
momenta, while the matrix elements themselves are evaluated with the original kinematics.
Both the local and the integrated subtraction terms and the virtual corrections are cut in
edgeofphasespace effects which do not spoil integrability). Since collinear leptonphoton
configurations are treated fully inclusively within some collinear cone defined by the photon
recombination, the conditions for the KLN theorem are fulfilled, guaranteeing the
cancellation of the collinear mass singularity. The formation of such a quasiparticle is close to the
experimental concept of “dressed leptons”, as e.g. described by the ATLAS collaboration
in ref. [53].
Quarkinduced channels — the collinearunsafe setup. It is not a priori
necessary that observables sensitive to photon radiation off a finalstate lepton are defined in
a collinearsafe way. The reason is that photons and charged leptons may be detected
in geometrically separated places, i.e. the photons in the electromagnetic calorimeter and
the muons in the muon chamber. This allows the measurement of an arbitrarily collinear
photon emission off a muon. In the absence of photon recombination, the lepton masses
serve as a physical cutoff for collinear singularities. On the computational side, this simply
forbids the recombination of a muon with momentum pµ and a photon with momentum
do in a collinearsafe setup. In the case of photon emission off electrons, the detection of
the two particles takes place in the electromagnetic calorimeter. The finite resolution of
the detector then defines a natural “cone size” for the recombination of the leptonphoton
pair to a single quasiparticle. In our collinearunsafe setup, we exclude the muons from
recombination, while the electrons are recombined with photons like in the collinearsafe
In ref. [54], the dipole subtraction formalism was extended to collinearunsafe
observables. As in the collinearsafe case, the starting point of the formalism is eq. (2.5), the
fundamental difference being that without recombination of a leptonphoton pair, some
observables may now be sensitive to the individual lepton and photon momenta within
the collinear region. While this is obvious for the realemission matrix element, this is
also required from the subtraction terms in order to guarantee the local subtraction of the
singularities. To this end, the reduced npoint kinematics of the local subtraction terms
(which are integrated over the (n + 1)particle phase space) is a posteriori extended to an
effective (n + 1)particle configuration with a resolved collinear leptonphoton pair with
Here p˜i denotes the momentum of a particular finalstate emitter in the reduced kinematics,
leptonphoton pair is denoted by z; it is constructed from kinematical invariants of the
(n + 1)particle phase space. The local subtraction terms are evaluated in the same way
as in the collinearsafe case. However, any collinearunsafe contribution is now cut with
respect to the unrecombined (n + 1)particle phase space:
dΦ Mreal2Θ(p1, . . . , pn, pγ) − Msub2Θ(p˜1, . . . , p˜n; z)i
Z dΦ˜ Z 1dxZ 1dzhMsub(x, z)2fin +δ(1−x)δ(1−z)2 Re MBornMvirt finiΘ(p˜1, . . . , p˜n; z) .
∗
The schematic shorthand notation
emission. Note that the oneparticle phasespace integral [dk] in the integrated subtraction
terms in eq. (2.5) is modified, because the z dependence is required in the readded
subtraction contribution in order to allow for cuts on the bare lepton momentum also in the
collinear region. This is in contrast to the collinearsafe case where z could be integrated
in a processindependent way [52]. Splitting off the soft and collinearsingular
contributions in the zintegration, the subtraction terms can be separated into an inclusive part for
the collinearsafe case plus extra terms for the collinearunsafe case. The detailed form of
Msub(x, z)2fin can be found in ref. [54] and is not repeated here.
For collinearunsafe observables, the integration over z is not inclusive, so that the
conditions for the KLN theorem are not fulfilled. Hence, the collinear singularities from
the virtual corrections do not entirely cancel against those from the real corrections. Using
and modify the cross section, which often leads to significant shape distortions of differential
distributions. Since partonic scattering energies at the LHC are much larger than the muon
mass, all terms suppressed by factors of mµ can be safely neglected. From a practical
point of view this means that all kinematics is evaluated with exactly massless muons
and the relicts from collinearlysensitive observables remain in the finite but possibly large
¯
q
e−
¯
q
e−
Quarkphotoninduced channels.
nels with one photon and one (anti)quark in the initial state. Since an external soft quark
does not lead to a singularity and since there is no collinear divergence when the finalstate
quark becomes collinear to one of the finalstate leptons, the matrix elements exhibit only
collinear initialstate singularities. As illustrated in figure 5, they can be grouped into two
classes: first, the incoming photon splits into a quarkantiquark pair with the finalstate
(anti)quark becoming collinear to the incoming photon, or second, the incoming (anti)quark
splits into a photon(anti)quark pair with the final and initialstate (anti)quark
becoming collinear. With the dipole subtraction method each of the two collinear singularities
may be locally subtracted with a single dipole whose functional form is given in ref. [54].
The singularities in the two corresponding integrated subtraction terms cancel against the
collinear counterterm from the PDFs, which can, e.g., be found in ref. [55].
Numerical implementation and independent checks of the calculation
We have performed a complete calculation of all contributions using the publicly available
matrix element generator Recola [56] for the evaluation of the virtual corrections and for
all treelevel amplitudes at Born level and at the level of the real corrections. The
phasespace integration has been carried out with a multichannel Monte Carlo integrator with
an implementation of the dipolesubtraction formalism [51, 52, 54, 57] for collinearsafe
and collinearunsafe observables. The calculation has been crosschecked both at the level
of phasespace points and differential cross sections with two other independent
implementations. The oneloop matrix elements of the equalflavour case have been checked against
amplitudes from the Mathematica package Pole [58], which employs FeynArts [59, 60]
and FormCalc [61]. The oneloop matrix elements of the mixedflavour case have been
checked against a calculation based on diagrammatic methods like those developed for
fourfermion production in electronpositron collisions [47, 62], starting from a
generation of amplitudes with FeynArts [59, 60] and further algebraic processing with inhouse
Mathematica routines. In all three calculational approaches, the oneloop integrals are
evaluated with the tensorintegral library Collier [63] containing two independent
implementations of the tensor and scalar integrals. Collier employs the numerical reduction
schemes of refs. [64, 65] for oneloop tensor integrals and the explicit results of oneloop
scalar integrals of refs. [66–68] for complex masses. The phasespace integration in all three
approaches is carried out with independent multichannel Monte Carlo integrators which
are further developments of the ones described in refs. [69–71]. For all differential and total
cross sections obtained with the different implementations we find agreement within the
statistical uncertainty of the Monte Carlo integration.
Phenomenological results
Input parameters
In the numerical analysis presented below, we consider the LHC at a centreofmass (CM)
energy of 13 TeV and choose the following input parameters. For the values of the onshell
masses and widths of the gauge bosons we use
Gµ = 1.16637 × 10−5 GeV−2
1 − MZ2
MZos = 91.1876 GeV,
M Wos = 80.385 GeV,
In the complexmass scheme, the onshell masses and widths need to be converted to pole
quantities according to the relations [72]
MV =
V = W, Z .
The complex weak mixing angle used in the complexmass scheme is derived from the ratio
EW parameters, we refer to ref. [47]. Since the Higgs boson and the top quark do not
appear as internal resonances in our calculation, their widths are set equal to zero. For the
corresponding masses we choose the values
MH = 125 GeV,
mt = 173 GeV.
particles with zero mass throughout the calculation. In the computation of collinearunsafe
observables, the physical muon mass appears as a regulator with numerical value
mµ = 105.6583715 MeV.
Note that this nonzero value for the muon mass is only kept in otherwise divergent
logarithms from photon radiation off muons, while everywhere else the muons are strictly
treated as massless particles.
scale into the LO cross section and thus avoids mass singularities in the charge
renormalization (see refs. [73, 74] and the discussion in the “EW dictionary” in ref. [75]). Moreover,
is only used as coupling parameter in the relative photonic corrections, i.e. the NLO
conwe do not consider QCD corrections in this paper and work in the onshell renormalization
scheme [47, 73], our crosssection predictions do not depend on the renormalization scale
µ ren. The dependence of the relative NLO EW corrections on the factorization scale µ fact is
need to investigate residual scale dependences or alternative scale choices. As PDFs we use
the NNPDF23 nlo as 0118 qed set [76].1 Throughout our calculation of EW corrections, we
employ the deepinelasticscattering (DIS) factorization scheme, following the arguments
given in ref. [78]. The corresponding finite terms for the EW corrections to be included in
the subtraction formalism can be found in ref. [55].
Definition of observables and acceptance cuts
In the following we define two different event selections: an “inclusive” and a
“Higgsspecific” setup. The former uses typical leptonidentification cuts without any further
selection criteria; the latter is motivated by specific criteria designed for Higgsboson
analyses by ATLAS [79] and CMS [80].
Inclusive setup. In the collinearsafe case, photons emerging in the realemission
contributions are recombined with the closest charged lepton (cf. section 2.3) if their separation
in the rapidityazimuthalangle plane obeys
ΔRℓi,γ = q(yℓi − yγ )2 + (Δφℓiγ )2 < 0.2 .
with larger rapidities as lost in the beam pipe. In the collinearunsafe case, photons are
recombined only with electrons/positrons, while no recombination with muons/antimuons
1In this calculation we take the photon density of NNPDF23 nlo as 0118 qed in spite of its larger
uncertainty compared to the LUXqed photon distribution [77], in order to rely on one consistent PDF set. This
leptons are thus labelled as ℓ1±, and subleading leptons as ℓ2±.
is performed. For observables of the equalflavourlepton final state, the leading lepton pair
As default setup, we consider a minimal set of selection cuts inspired by the ATLAS
analysis [1]. For each charged lepton ℓi, we restrict transverse momentum pT,ℓi and rapidity
yℓi according to
pT,ℓi > pT,min = 15 GeV,
Any pair of charged leptons (ℓi, ℓj ) is required to be well separated in the
rapidityazimuthalangle plane,
Higgsspecific setup. For the Higgsspecific setup, motivated by the ATLAS and CMS
analyses [79, 80], we replace the cuts of eq. (3.9) by the less restrictive criteria
pT,ℓi > pT,min = 6 GeV,
retain the cut (3.10), and complement them with additional invariantmass cuts on the
charged leptons. For the mixedflavour final state, we require for the two sameflavour
40 GeV < M + − < 120 GeV,
12 GeV < M + − < 120 GeV,
or further away from the nominal Zboson mass, respectively. For the sameflavour final
state, we apply the cuts of eq. (3.12) after selecting the leading and subleading lepton pairs
(ℓ1+, ℓ1 ) and (ℓ2+, ℓ2−) in the same way as described above. The invariant mass M4ℓ of the
−
fourlepton system is subjected to the cut
M4ℓ > 100 GeV,
which is independent of the flavour of the finalstate leptons.
In both setups we treat the additional (anti)quark in the final state of the
photoninduced contributions in a fully inclusive way, i.e. we do not apply any jet veto. Finally, we
note that the employed singleparticle lepton identification cuts are chosen to be equal for
all charged leptons. Since the lepton pairing in the equalflavour final state is flavour
indefinal state within the collinearsafe setup. In the following, [4µ ] denotes the equalflavour
final state, while [2µ 2e] denotes the mixedflavour final state.
Whenever possible, we have compared the results of our full offshell calculation with
the available results for onshell Zboson pair production [40, 41]. Since our calculation
sets phasespace cuts on the charged leptons, a direct comparison is not possible for most
observables. Moreover, the calculations for stable Z bosons do not take into account
corrections to the Zboson decays. Finally, there are differences in the setup: the values in
incl. [2µ 2e]
Higgs [2µ 2e] 13.8598(3)
−4.32
−4.32
−3.59
−3.42
−0.93
−0.94
−0.04
−0.09
−1.68
−2.43
−0.28
−0.66
−0.09
−0.14
corrections δi = σi/σqL¯qO for the LHC at √
(“incl.”) with the cuts of eqs. (3.9)–(3.10) and the Higgsspecific setup (“Higgs”) with the selection
cuts of eqs. (3.10)–(3.13), respectively.
the literature are given for a CM energy of √
PDFs. Nevertheless, the relative EW corrections to those observables that are less
sensitive to offshell effects and corrections to the Zboson decays can still be directly compared.
This holds in particular for the Sudakov enhancement at large transverse momentum of
the onshell Z boson or the corresponding charged finalstate lepton pair.
Results on integrated cross sections
given for the inclusive selection cuts of eqs. (3.9)–(3.10) and the Higgsspecific selection
cuts of eqs. (3.10)–(3.13). Together with the LO cross sections, the NLO EW corrections
. For the photonic corrections, we further distinguish between the collinearsafe
phot,safe, where the bremsstrahlung photon is recombined with any charged lepton,
phot,unsafe, where the muons are excluded from recombination,
EW/weak/phot to the q¯q channels are practically independent
with stateoftheart QCD predictions.
EW/weak/phot well suited for a combination
We first analyse the inclusive scenario. The major contribution to the corrections stems
The photoninduced contribution matters only at the permille level, justifying that we
neglect higherorder corrections to this channel.
Besides the small photon flux in the
proton, yet another reason for the strong suppression is that channels with two photons
in the initial state involve at most one resonant Z boson, as illustrated by the sample
diagram of figure 1(c). We count such kinematical topologies as background topologies to
the dominant contribution with two possibly resonant Zboson propagators as they appear
another order of magnitude and are thus entirely negligible in the integrated cross section.
Summing up all contributions, we find for both the [2µ 2e] and the [4µ ] final state the
in a slightly different setup. The differences can be attributed to the offshell effects,
including also additional virtual photon exchange, and to differences in phasespace cuts
and in the employed numerical setup (cf. comments at the end of section 3.2). Note that
onshell approximation [41].
Comparing the collinearunsafe photonic corrections with the collinearsafe case, we
tively. In the collinearunsafe case, finalstate radiation off muons is enhanced through the
bined with the muons, there are systematically more events with a large energy loss in one
of the muon momenta induced by finalstate radiation. For this reason, less events pass
the eventselection in the collinearunsafe case, leading to more negative corrections. This
“acceptance correction” scales with the number of leptons treated in a collinearunsafe way,
The amplitudes for the equalflavour final state can be obtained from the
differentflavour amplitudes by antisymmetrization with respect to a pair of equal finalstate leptons.
two different vector bosons. These interference terms lead to a deviation from a naive
rescaling of the differentflavour cross section by a symmetry factor of two. From the
integrated LO cross section. Comparing this number with the total relative correction of
section in the inclusive setup is a “LO effect” in the sense that the relative corrections do
not modify this behaviour. The reason for the smallness of the interferences is that the LO
cross section is dominated by contributions with two resonant Zboson propagators. In the
interference terms, however, in at least one diagram both Zboson propagators are off shell.
with at most one resonant Z boson.
We now turn to the Higgsspecific setup. Despite the additional cuts of eqs. (3.12)–
(3.13), the cross sections for this scenario are larger than for the previously considered
inclusive setup. This feature is due to the less severe cut of 6 GeV imposed on the transverse
momenta of the charged leptons in the Higgsspecific setup, as compared to a cut of 15 GeV
resonances due to the coupling of the W boson to the photon.
in the inclusive setup. Moreover, we observe a percentlevel deviation at LO from the naive
0.973. This reflects, on the one hand, an expected enhancement of the interference terms
since, by construction, the whole scenario is more sensitive to the offshell effects of the
vector bosons. On the other hand, the additional invariantmass cuts in eq. (3.12) depend
in the equalflavour case on the chosen leptonpairing algorithm, while they do not in the
mixedflavour case. A quantitative statement on the size of the pure interference effects
would thus only be possible if the same lepton pairing was applied also in the mixedflavour
case. The same arguments apply also to the corrections in table 1, i.e. the differences
between the equalflavour and the mixedflavour final state are due to interferences and
event selection. As a general pattern, we observe that the bulk of corrections to the total
contribute only at the subpercent level.
Results on differential cross sections in the inclusive setup
Invariantmass and transversemomentum distributions. In order to illustrate the
impact of EW corrections on differential observables, we present several results on
distris = 13 TeV. We choose
the collinearsafe setup as default and provide selected results within the collinearunsafe
case subsequently.
Figure 6 shows the fourlepton invariantmass distribution for the unequalflavour
[2µ 2e] and the equalflavour [4µ ] final states.
The lefthand side resolves the offshell
region with its threshold and resonance structures with a fine histogram binning, while
the panels on the righthand side show the whole range from the offshell region over
the resonances and thresholds up to the TeV regime in coarsegrained resolution. The
absolute predictions of the LO and NLO distributions of both the [2µ 2e] and [4µ ]
final states in the upper panels follow the characteristic pattern of Zboson pair
producZ boson in the schannel according to the sample diagram in figure 1(b), the
threshical cut of eq. (3.9) on the lepton transverse momentum. For M4ℓ & MZ + 2pT,min, the
cross section is dominated by events with one resonant Z boson (→ ℓ1 ℓ1 ) and the ℓ2 ℓ2
+ −
+ −
plane. Since E + + E − > 2pT,min = 30 GeV is necessary for the event to pass the cuts,
ℓ ℓ
the transition rate, since no resonance enhancement is present anymore in the diagram
type illustrated in figure 1(a).
The panels directly below the absolute predictions for the cross sections show the
relative EW corrections to the q¯q channels in the collinearsafe setup, comparing the purely
shell region below the single Z resonance, we observe that the relative EW corrections
of the mixedflavour final state and the equalflavour final state are equal over the whole
√s = 13 TeV
10−4
[]% 10
−10
−20
] 2
[
δ 1
−1
100 200 300 400 500 600 700 800 900 1000
unequalflavour [2µ 2e] and the equalflavour [4µ ] final states in the inclusive setup. The panels at
the bottom show the ratio of the [2µ 2e] and [4µ ] final states.
invariantmass spectrum. This confirms at the level of differential distributions that the
interference effect is mainly a LO effect, in accordance with what we have already seen for
the integrated cross section.
The fourlepton invariant mass in the inclusive setup is well suited to study the relative
size of the interferences, as this observable does not depend on the lepton pairing. We show
LO and NLO curves are, as expected, almost equal. The size of the interference effect
equalflavour final state there. In the region MZ + 2pT,min . M4ℓ . 2MZ, where only one
√s = 13 TeV
lepton pair can be resonant, the interference effect amounts to 2%. Above the ZZ threshold,
the ratio is equal to one up to fractions of a percent, since in this region of phase space
the doublyresonant contribution dominates over any nonresonant interference effect. For
higher invariant masses M4ℓ, the overlap of the two resonance pairs becomes smaller and
smaller in phase space, so that the ratio asymptotically approaches one.
We inspect the EW corrections in more detail. In the highinvariantmass region,
the correction is entirely dominated by the purely weak contribution and reaches about
regime of ZZ production where all Mandelstam variables (s, t, u) of the 2 → 2 particle
process would have to be large. Instead, Zpair production at high energies is dominated
by forward/backwardproduced Z bosons, where t and u are small. At the ZZ production
threshold, the weak corrections change their sign and reach up to 5% below. Note that
this nontrivial sign change makes it impossible to approximate the full NLO EW results
by a global rescaling factor. At the Z peak, the weak corrections are extremely suppressed.
LO cross section falls off steeply. Finalstate radiation of a real photon, however, may shift
the value of the measured invariant mass to smaller values. Since the LO cross section is
small in this phasespace region, the relative correction due to the bremsstrahlung photon
becomes large. The structure of the radiative tails follows precisely the thresholds and
another one below the Z resonance.
For completeness, we also show the photoninduced corrections in a separate panel
nel has only a single Z resonance according to the diagram in figure 1(c), it is strongest
suppressed with respect to the q¯q LO cross section above the ZZ threshold and near the
schannel resonance at MZ. In the nonresonant region below MZ + 2pT,min it reaches
up to 1%. Since there the LO cross section is small anyway, the overall impact remains
small, in agreement with the result for the integrated cross section. Differences between
the equalflavour and the unequalflavour final states due to interferences are only visible
over the whole spectrum at most at the permille level. The large correction near the
phasespace boundary is phenomenologically irrelevant as the corresponding LO cross section in
this region is very small anyway. Due to the negligible impact of any photoninduced
corrections in the inclusive setup, we do not show them separately any more in the following
Up to the details in the event selection and the corrections from the Zboson decays,
the fourlepton invariant mass M4ℓ can be compared to the ZZinvariantmass distribution
obtained in the NLO calculations of refs. [40, 41] with onshell Z bosons. The relative
10−1
10−2
−µ 10−4
pd10−5
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10−7
−10
]−20
−40
−50
(2· 1.02
/
]
√s = 13 TeV
subleading µ +µ − pair
√s = 13 TeV
subleading µ +µ − pair
Mµ +µ − [GeV]
pTµ +µ − [GeV]
(upper panels), corresponding EW corrections (middle panels), and ratio of the [2µ 2e] and [4µ ] final
states (lower panels) in the inclusive setup. The equalflavour case is binned with respect to the
subleading lepton pair.
stood from the fact that in the vicinity of the Z resonance there are two different types
of contributions: corrections to the resonant part of the squared amplitude, and
corrections to the interference of the resonant and nonresonant parts of the amplitude. The
and thus change sign at the Z resonance. This qualitatively explains the observed sign
change of the purely weak corrections which is slightly shifted below the resonance due to
the negative offset mentioned above. The corrections are to a large extent equal for the
Including also the photonic corrections, we observe in both cases the typical radiative tail
due to finalstate radiation effects below the Z resonance, similar to what has been
observed in the fourlepton invariant mass. In the highenergy spectrum of the steeply falling
invariantmass distribution, shown in figure 8, both photonic and purely weak corrections
differ significantly from the mixedflavour case due to the large differences at LO. At the
peak around 45 GeV, the purely weak corrections basically vanish, which is consistent with
the fourlepton invariantmass distribution in figure 6 where the purely weak corrections
the [2µ 2e] and the [4µ ] case because the dominant contribution where the leading and the
subleading lepton pairs are both close to the resonance is not sensitive to the lepton pairing
panel and the discussion for the subleading case above the resonance). The weak
corrections stay always below 5% in absolute size. The photonic corrections exhibit an additional
radiative tail below the peak around 45 GeV. The radiative tail below the Z resonance is
less pronounced due to the missing resonance enhancement by the subleading lepton pair.
In figure 8 (righthand side) we show the distribution in the transverse momentum
boson transverse momentum in onshell calculations. Since the two lepton pairs are back
to back at LO, the transversemomentum distribution depends on the choice of the lepton
pair only very weakly. The interference effect of a few percent is only visible for small
ZZ production with onshell Z bosons discussed in refs. [40, 41]. However, as mentioned
above, it should be kept in mind that it cannot be expected to find perfect agreement
because of the differences in the event selection, which is based on finalstate leptons
in our calculation, and the absence of corrections to the Zboson decays in the onshell
s = 14 TeV,
which agrees with our result for √
Figure 9 shows the transversemomentum distribution of the µ +. The left panels
compare the leading µ + from the [4µ ] final state with the µ + from the [2µ 2e] final state, the
panels in the right column show the corresponding comparison with the subleading µ +.
Recall that our ordering of muons into “leading” and “subleading” corresponds to the
oras described in section 3.2, but not to the muon pT, which is frequently used as well. We
observe again that the observable is very sensitive to the event selection with characteristic
differences between leading and subleading leptons. Especially at high transverse momenta
pT, the spectrum of the leading muon in [4µ ] is suppressed with respect to the spectrum of
pT in [2µ 2e], while the spectrum of the subleading muon in [4µ ] is enhanced. The difference
can be traced back to the impact of the ZZ signal and background contributions at large
transverse momenta: the leading lepton belongs to the “more resonant” Z boson, and
therefore, the contribution is in general dominated by the doublyresonant signal contributions
(cf. figure 1(a)). The main effect of the background contribution in figure 1(b) for large
pT arises when the µ + is backtoback with the three other charged leptons. As already
observed for the related process of pp → WW → leptons [44], the impact of background
diagrams on the pT spectrum of a single lepton can be as large as the doublyresonant
contribution in the TeV range. Since there is no preselection of the µ + in the [2µ 2e] final
state with respect to the resonance, the pTµ + spectrum of [2µ 2e] lies between the spectra of
i 10−3
d10−5
10−6
10−7
−10
]−20
−40
(2· 1.5
/
]
√s = 13 TeV
rections (middle panels), and ratio of the [2µ 2e] and [4µ ] final states (lower panels) in the inclusive
setup. The left panels compare the leading µ + from the [4µ ] final state with the µ + from the
[2µ 2e] final state, while the panels in the right column show the corresponding comparison with the
subleading µ +.
the leading and subleading muons. This behaviour is also reflected in the size of the purely
weak corrections: since the Sudakov enhancement is larger in doublyresonant
contribucase of the subleading µ +. The photonic corrections give an almost constant negative offset
the equalflavour final state is very similar to the mixedflavour case. For the subleading
−1% and −0.5%.
Rapidity and angular distributions. The rapidity distributions in figure 10 do not
show any significant dependence on the lepton pairing except for a small effect at the
percent level in the forward direction with rapidities yµ +  > 2. The EW corrections are
independent of the lepton pairing as well, and their size is almost equal for both final states.
The purely photonic corrections give, in good approximation, a constant negative offset of
10−2
i 10−3
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]−20
−40
(2· 1.5
/
]
√s = 13 TeV
−1
−2
]−3
−5
−6
−7
√s = 13 TeV
s√usbl=ea1d3inTgeµV+
−1
−2
−5
−6
−7
−2
−1
−2
−1
panels), and ratio of the [2µ 2e] and [4µ ] final states (lower panels) in the inclusive setup. The left
panels compare the leading µ + from the [4µ ] final state with the µ + from the [2µ 2e] final state, the
panels in the right column show the corresponding comparison with the subleading µ +.
less negative in the forward direction with about −3%.
In figure 11, the distribution in the azimuthalangle distance between the muons in the
proximation independently of the final state and the lepton pairing. This can be explained
as follows: the azimuthalangledistance distribution is dominated in the whole range by
tion receives the largest contribution from doublyresonant contributions. Moreover, the
tchannel nature of the doublyresonant diagrams favours small transverse momenta of the
Z bosons. As can also be seen in figure 9, most of the leptons have pT . MZ/2 as a result
of the decay of the Z bosons that move slowly in the transverse plane, i.e. the Z bosons
decay almost isotropically in the transverse plane, without a large influence of boost effects
(2· 1.5
/
√s = 13 TeV
leading µ +µ − pair
s√usbl=ea1d3inTgeµV+µ − pair
sponding EW corrections (middle panels), and ratio of the [2µ 2e] and [4µ ] final states (lower panels)
comparison with the subleading lepton pair.
separation cut of eq. (3.10).
]fb 1.4
[
−2
−3
−4
−6
−7
−8
√s = 13 TeV
panels), corresponding EW corrections (middle panels), and ratio of the [2µ 2e] and [4µ ] final states
(lower panels) in the inclusive setup.
configuration preferred at low energies. The effect of increasing negative weak corrections
tion owing to the absence of collinear enhancements, because collinear finalstate radiation
does not change the directions of the radiating leptons significantly. The photonic
correccase, the leading lepton pair, and the subleading lepton pair, respectively. They decrease
for the unequalflavour [2µ 2e] final state and the corresponding angle defined by the leading
the CM system of the four finalstate leptons by
kℓ1+ℓ1− × kℓ+
kℓ1+ℓ1− × kℓ1+ kℓ1+ℓ1− × kℓ2+ 
kℓ1+ℓ1− × kℓ+ × kℓ1+ℓ1− × kℓ+
1 2
1 −
consequence of the leptonseparation cuts in eq. (3.10). These cuts remove collinear lepton
configurations where the decay planes tend to be coplanar. The local minima around
different lepton helicities: if the two equally charged finalstate leptons do have the same
the interference pattern, it is instructive to consider the limit of nearly coplanar Z decays
decay planes. In the vicinity of the coplanar configurations this line divides the event plane
the matrix elements are antisymmetrized with respect to the exchange of µ 1+ ↔ µ 2+ or
µ 1 ↔ µ 2−, a destructive interference is favoured for φ → 0 in [4µ ], leading to the observed
−
enhancement in the [2µ 2e]/(2[4µ ]) ratio. This effect is not changed by the EW corrections.
only uniformly contribute by −1%.
Collinearsafe versus collinearunsafe observables. In figure 13, the different
recombination schemes for the muon are illustrated for the fourlepton and twolepton
invariant masses. The recombination procedure only affects the photonic corrections. As
a general pattern, all the radiative tails induced by finalstate radiation off the charged
leptons are strongly enhanced if collinear photons are not recombined with muons. The
enhancement is due to the fact that the collinear logarithms are regularized by the muon
mass rather than the size of the recombination cone. The effects can be best isolated in the
M4ℓ invariantmass distribution (left panels of figure 13) which is not sensitive to the lepton
i 10−2
10−3
10−4
−20
NLO EW collinear unsafe [4µ ]
NLO EW collinear unsafe [2µ 2e]
10−1
10−3
10−4
−20
NLO EW collinear safe [2µ 2e]
NLO EW collinear unsafe [2µ 2e]
Mµ +µ − [GeV]
lepton invariantmass distributions. The upper panels show the absolute distributions and the lower
panels the relative EW corrections.
pairing. While the absolute prediction is only shown for the collinearunsafe case for the
mixed and equalflavour final states, the relative EW corrections are plotted both for the
collinearsafe and unsafe cases. The results illustrate the impact of the number of muons
excluded from recombination to the distribution: the maximum of the radiative tail below
the ZZ threshold increases from about +30% with full recombination to more than +50%
for excluding one muon pair ([2µ 2e]) up to about +70% by excluding both muon pairs
([4µ ]). For [4µ ], the increase is twice as large as for [2µ 2e], since the recombination effect
scales with the number of collinear cones that are subject to the changes in the
recombination. A similar behaviour is found for the other radiative tails at smaller values of M4ℓ.
panels) below the Z resonance where the relative correction increases from almost +60%
to +140%. Note that above the resonance the effect from the collinearunsafe treatment
pushes the negative collinearsafe corrections even more negative.
Results on differential cross sections in the Higgsspecific setup
Invariantmass and transversemomentum distributions. The production of
Zboson pairs at the LHC is interesting not only per se, as a signal process, but also
conmode. In order to assess the impact of this background on Higgs analyses, we impose the
Higgsspecific cuts of eqs. (3.12)–(3.11) in addition to the inclusive cut of eq. (3.10). In
ref. [45] we already presented some important results of this study, however, restricted to
the unequalflavour finalstate [2µ 2e] and ignoring photoninduced channels. In the
follow10−1
10−2
−5
] 2
√s = 13 TeV
ℓ
dσ dM410−3
10−4
−10
−20
−30
−0.5
−1
4
[2· 0.8
(
/
µ]e2 0.7
EW corrections (2nd panels from above), photonic contributions (third panels from above) for the
unequalflavour [2µ 2e] and the equalflavour [4µ ] final states in the Higgsspecific setup. The lower
panels show the ratio of the [2µ 2e] and [4µ ] final states.
ing we continue the discussion started there by comparing results for the [2µ 2e] and [4µ ]
final states and considering further observables.
Figure 14 illustrates the invariantmass distribution of the fourlepton system at LO
and the corresponding NLO EW corrections for both the [2µ 2e] and the [4µ ] final states. In
each case, we observe a steep shoulder at the Zboson pair production threshold at about
at smaller invariant masses. Though smaller in magnitude, a similar effect can be observed
mass cuts we impose on the charged leptons. Like in the inclusive setup, both the purely
weak and the photonic corrections exhibit a sign change at the pair production threshold
√s = 13 TeV
similar to the inclusive setup with at most permille level differences between the [4µ ] and
the [2µ 2e] case. The photonic corrections decrease in absolute size from approximately
region below the pair production threshold, the difference between the [4µ ] and the [2µ 2e]
cases in the purely weak corrections is below the percent level. The radiative tails in the
photonic corrections are up to 5% larger in the mixedflavour case. In contrast to the
inclusive setup, the phasespace cuts of the Higgsspecific setup introduce a dependence on
the lepton pairing even in otherwise symmetric observables like the fourlepton invariant
mass. The difference seen in the photonic corrections is thus due to both the lepton pairing
corrections with respect to the final states are, however, entirely negligible. The significant
differences between the [4µ ] and the [2µ 2e] case in the offshellsensitive region are, like
in the inclusive case, a priori a LO effect. Note that the nontrivial sign change of the
photonic corrections leads to significant cancellations between oppositesign contributions
below and above the ZZ threshold resulting in subpermille effects in the total cross section
(cf. table 1), although the individual photonic corrections can be sizable in distributions.
We also show the photoninduced contribution to the fourlepton invariantmass
distribution in the third panels from above in figure 14. Above the ZZ production threshold
large extent. The overall impact remains at the subpercent level. We do not show the
photoninduced corrections separately in the following plots.
final state. Due to the cuts of eq. (3.12), the invariant mass of the leading muon pair in
the equalflavour final state is restricted to the range of 40–120 GeV. This cut leads in the
[2µ 2e] final state to a little bump at 40 GeV. Moreover, the local maximum near MZ/2
because the invariantmass cut M4ℓ > 100 GeV in eq. (3.13) entirely removes the schannel
similar to the results in the inclusive setup (cf. figure 7). The distribution peaks at the
finalstate radiation effects. The weak corrections, on the other hand, are of the order of
5% and give rise to a change in sign near the Zboson resonance. Above the resonance
the EW corrections are qualitatively similar to the ones in the inclusive setup for both
expected, most prominent in the leading lepton pair where the local minimum of the weak
corrections at 45 GeV and the entire additional radiative tail of the photonic corrections
are removed. While the EW corrections show sizeable deviations between the mixed and
equalflavour final states, the main differences are LO effects that can be attributed to the
cuts and the lepton pairing.
√s = 13 TeV
leading µ +µ − pair
s√usbl=ea1d3inTgeµV+µ − pair
Mµ +µ − [GeV]
Mµ +µ − [GeV]
dle panels), and ratio of the [2µ 2e] and [4µ ] final states (lower panels) in the Higgsspecific setup.
In the left column the equalflavour case is binned with respect to the leading lepton pair, while
the right column shows results for the subleading one.
Figure 16 depicts the transversemomentum distribution of the µ + in the [2µ 2e] final
state together with the leading and the subleading µ + of the [4µ ] final state, respectively.
We once again remind the reader that the classification of leptons as “leading” or
“subleading” refers to the criteria of eq. (3.12), i.e. the leading muon is not necessarily the muon
closest to the mass of the Z boson. We find that, in contrast to the inclusive setup
illustrated in figure 9, the weak corrections to the transverse momenta are very similar in size
and shape for the equal and the unequalflavour cases. They become large and negative
suppression of background diagrams of the type shown in figure 1(b), which can already
be seen from the suppression of the absolute LO cross section at large transverse momenta
(cf. figure 9 and related discussion there). The impact of photonic corrections is at the
level of one percent for small transverse momenta and even smaller for large ones for both
leptonic final states.
10−2
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/
µ]e2 0.9
i 10−3
d10−5
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]−20
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−50
(2· 1.0
/
√s = 13 TeV
√s = 13 TeV
Transversemomentum distribution of the µ + (upper panels), corresponding EW
corrections (middle panels), and ratio of the [2µ 2e] and [4µ ] final states (lower panels) in the
Higgsspecific setup. The left panels compare the leading µ + from the [4µ ] final state with the µ + from
the [2µ 2e] final state, while the panels in the right column show the corresponding comparison with
the subleading µ +.
Rapidity and angular distributions. The rapidity distributions of the µ + and the
corresponding EW corrections, shown in figure 17, do not change very much when going
from the inclusive to the Higgsspecific setup. The only visible changes are the constant
offsets in the relative corrections that can already be observed for the integrated cross
sections given in table 1.
Figure 18 illustrates the distribution in the azimuthalangle difference of the leading
the analogous distributions in the inclusive setup shown in figure 11 is absent in the
HiggsHiggsspecific selection cuts applied in the current setup remove such contributions, leaving
−1
]−2
−4
−5
√s = 13 TeV
√s = 13 TeV
−2
−1
−2
−1
panels), and ratio of the [2µ 2e] and [4µ ] final states (lower panels) in the Higgsspecific setup. The
left panels compare the leading µ + from the [4µ ] final state with the µ + from the [2µ 2e] final state,
the panels in the right column show the corresponding comparison with the subleading µ +.
weak corrections on normalization and shape of the azimuthalangle differences in the Higgs
setup is similar in size as in the inclusive setup. Purely photonic corrections are even more
suppressed than in the inclusive case. In the Higgsspecific scenario the fraction of events
with leading muon pairs close to the Z resonance is enhanced, while the one for subleading
muon pairs is reduced (compare figures 7 and 15). As a consequence, the distribution of
while the one of the subleading lepton pair is reduced. For small azimuthalangle differences
the situation is reversed.
We show in figure 19 the distribution in the angle between the two Zboson decay planes
in the fourlepton CM system in the Higgsspecific setup.4 The distribution as well as the
EW corrections closely resemble those of the inclusive setup shown in figure 12. The ratio
4The distribution in the angle between the two Zboson decay planes shown in figure 3 of ref. [45] is not
the lepton momenta in the laboratory system.
√s = 13 TeV
leading µ +µ − pair
√s = 13 TeV
subleading µ +µ − pair
corresponding EW corrections (middle panels), and ratio of the [2µ 2e] and [4µ ] final states (lower
corresponding comparison with the subleading lepton pair.
the Higgs setup, we cannot attribute the deviations of the ratio from one to interference
ZZ⋆
→ 4 leptons looks qualitatively similar to the distribution of direct ZZ production
shown in figure 19, but the distortions by EW corrections are quite different [82, 83].
Conclusions
The production of four charged leptons in hadronic collisions at the LHC is an important
process class both for the investigation of the interactions between the neutral Standard
Model gauge bosons and as background process to searches for new physics and to precision
studies of the Higgs boson. In the confrontation of experimental data with theory
predic−1
−2
−4
−5
−6
√s = 13 TeV
panels), corresponding EW corrections (middle panels), and ratio of the [2µ 2e] and [4µ ] final states
(lower panels) in the Higgssearch setup.
tions precision plays a key role. In this paper we have further improved the theory
prediction by calculating the nexttoleadingorder electroweak corrections to the production of
mediate states. Our results are thus accurate to nexttoleading order in all phasespace
regions, no matter whether they are dominated by two, one, or zero resonant Z bosons. Our
numerical discussion of the corrections focuses on two different eventselection scenarios,
one based on typical leptonidentification criteria only and another one that is specifically
designed for Higgsboson analyses. Since the Higgsboson mass of about 125 GeV lies below
the Zpair threshold, the flexibility of our calculation, allowing intermediate Z bosons to
be far off shell, is essential for the study of fourlepton production as background to the
Higgsboson decay H → ZZ⋆.
vestigated further observables and channels with photons in the initial state and included
corrections consist of photonic and purely weak contributions displaying rather different
features. Photonic corrections can grow very large, to several tens of percent, in
particular in distributions where resonances and kinematic shoulders lead to radiative tails.
These effects are significantly enhanced when observables within a collinearunsafe setup
are considered. While photonic corrections might be well approximated with QED parton
showers, this is not the case for the weak corrections, which are typically of the size of
invariantmass and transversemomentum distributions. Moreover, the weak corrections
below the ZZ threshold distort distributions that are important in Higgsboson analyses.
On the other hand, contributions induced by incoming photons, i.e. photonphoton and
quarkphoton channels, turn out to be phenomenologically unimportant. Comparing the
state to intermediate Z bosons. Interferences in equalflavourlepton final states lead to
deviations of up to 10% from the mixedflavour case in offshellsensitive phasespace
regions. Their effect is, however, in general hidden in the effects of the selection criteria for
the lepton pairing. The relative electroweak corrections are widely insensitive to details
nexttoleading order roughly in the same way.
The full calculation is available in the form of a Monte Carlo program allowing for
the evaluation of arbitrary differential cross sections. The best possible predictions for
ZZ production processes can be achieved by combining the electroweak corrections of our
calculation with the most accurate QCD predictions available to date. Practically, this
could be achieved, e.g., by reweighting differential distributions including QCD corrections
with electroweak correction factors. In this way, an overall accuracy at the percent level
can be achieved for integrated cross sections that are dominated by energy scales up to
a few 100 GeV, where the theoretical uncertainty is completely dominated by QCD. We
estimate the contribution of missing higherorder electroweak corrections on the integrated
cross section to 0.5%. The impact of missing higherorder electroweak corrections grows in
the highenergy tails of transversemomentum and invariantmass distributions where weak
Sudakov (and subleading highenergy) logarithms are known to be large. In this kinematic
domain, the size of this uncertainty may be estimated by the square of the relative
electroweak correction. The inclusion of the known leading twoloop effects or a resummation
of logarithmically enhanced contributions could reduce these theoretical uncertainties. At
the same time, multiphoton emission effects could be systematically taken into account
by structure functions or parton showers. Such improvements are, however, left to future
studies. For upcoming analyses of LHC data, nexttoleading order precision in electroweak
corrections is certainly sufficient, and the remaining electroweak uncertainties are
negligible compared to the larger uncertainties from missing QCD corrections and from parton
distribution functions.
Acknowledgments
We would like to thank Jochen Meyer for helpful discussions. The work of B.B. and
A.D. was supported by the German Federal Ministry for Education and Research (BMBF)
under contract no. 05H15WWCA1 and by the German Science Foundation (DFG) under
reference number DE 623/21. S.D. gratefully acknowledges support from the DFG research
training group RTG 2044.
The work of L.H. was supported by the grants
FPA201346570C21P and 2014SGR104, and partially by the Spanish MINECO under the project
MDM20140369 of ICCUB (Unidad de Excelencia “Mar´ıa de Maeztu”). The work of
B.J. was supported in part by the Institutional Strategy of the University of Tu¨bingen
(DFG, ZUK 63) and in part by the BMBF under contract number 05H2015. The authors
acknowledge support by the state of BadenWu¨rttemberg through bwHPC and the German
Research Foundation (DFG) through grant no. INST 39/9631 FUGG.
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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