Next-to-leading-order electroweak corrections to the production of three charged leptons plus missing energy at the LHC
aa r X i v : . [ h e p - ph ] A ug ICCUB-17-016
Next-to-leading-order electroweak corrections to the productionof three charged leptons plus missing energy at the LHC
Benedikt Biedermann , Ansgar Denner , Lars Hofer Julius-Maximilians-Universit¨at W¨urzburg, Institut f¨ur Theoretische Physik und Astrophysik,Emil-Hilb-Weg 22, 97074 W¨urzburg, Germany Universitat de Barcelona (UB), Departament de F´ısica Qu`antica i Astrof´ısica (FQA),Institut de Ci`encies del Cosmo (ICCUB), 08028 Barcelona, Spain
Abstract:
The production of a neutral and a charged vector boson with subsequent decays into threecharged leptons and a neutrino is a very important process for precision tests of the StandardModel of elementary particles and in searches for anomalous triple-gauge-boson couplings. Inthis article, the first computation of next-to-leading-order electroweak corrections to the pro-duction of the four-lepton final states µ + µ − e + ν e , µ + µ − e − ¯ ν e , µ + µ − µ + ν µ , and µ + µ − µ − ¯ ν µ atthe Large Hadron Collider is presented. We use the complete matrix elements at leading andnext-to-leading order, including all off-shell effects of intermediate massive vector bosons andvirtual photons. The relative electroweak corrections to the fiducial cross sections from quark-induced partonic processes vary between −
3% and − −
30% in the high-energy tails of distributions originating from electroweak Sudakov loga-rithms. Photon-induced contributions at next-to-leading order raise the leading-order fiducialcross section by +2%. Interference effects in final states with equal-flavour leptons are at thepermille level for the fiducial cross section, but can lead to sizeable effects in off-shell sensitivephase-space regions. ontents
Vector-boson pair production belongs to the most important process classes at the Large HadronCollider (LHC). Owing to its sensitivity to the triple-gauge-boson couplings (TGC), it allowsfor fundamental precision tests of the Standard Model (SM) of elementary particles. In partic-ular, WZ production is considered as one of the key processes in searches for new physics viaanomalous TGC. Moreover, WZ production is an important SM background to many directsearches for new physics because the corresponding final state with three charged leptons plusmissing energy leads to a relatively clean signature.Both the ATLAS and CMS collaboration have measured WZ production at 7, 8 and 13 TeVcentre-of-mass energy [1–5]. Since the most recent determinations of anomalous TGC fromATLAS data of run II [6] are compatible with the SM prediction, possible new-physics effectsare severely constrained and expected to be found by looking for small deviations in high-energytails of differential distributions. It is thus of prime importance to have precise theoreticalpredictions for this process at hand.Most of the efforts for improving the theoretical accuracy of WZ production have beendedicated to perturbative higher-order calculations in the strong coupling α s . The first next-to-leading order (NLO) QCD computation treating the W and Z boson as on-shell externalparticles dates back more than two decades [7]. Systematic improvements followed, includingleptonic decays [8], off-shell effects and spin correlations [9, 10]. Fixed-order calculations for WZproduction have been matched to parton-shower generators at NLO QCD [11–13]. Recently, thefirst calculation of next-to-next-to-leading order (NNLO) QCD corrections for the integratedcross section has been completed [14] and extended to the level of differential distributions [15].At this level of accuracy, NLO electroweak (EW) corrections, which are proportional to theelectromagnetic coupling α , become relevant as well. On the one hand, naive power count-ing O ( α ) ≈ O ( α s ) suggests that they are of a similar order of magnitude as the NNLO QCDcorrections. On the other hand, EW corrections can be enhanced by logarithms of EW origin[16–21] and may distort differential distributions at large transverse momenta by several tensof percent. The latter property is of particular importance, since these phase-space regions aremost sensitive to effects of new physics. NLO EW corrections to WZ production have firstbeen studied in a logarithmic approximation [22]. A full NLO EW computation for on-shell Wand Z bosons has been presented later on [23, 24] including also photon-induced corrections.The NLO EW corrections for the complete four-lepton-production processes, i.e. including lep-tonic vector-boson decays and irreducible background diagrams, exist so far only for WW andZZ production [25–28], while corresponding results for WZ production are still missing in theliterature. 1 Z , γ ν e e + µ − µ + q i ¯ q j (a) Z , γ ν e e + µ − µ + q i ¯ q j W W (b)
W Z , γ ν e e + µ − µ + q i ¯ q j (c) W W µ + µ − ν e e + q i ¯ q j ν µ (d)Figure 1: Sample tree-level diagrams contributing at O ( α ) to q i ¯ q j → µ + µ − e + ν e .The aim of the present article is to fill this gap and to provide results for the NLO EWcorrections to the production of three charged leptons plus missing energy at the LHC. Weconsider the four different and experimentally well-defined final states µ + µ − e + ν e , µ + µ − e − ¯ ν e , µ + µ − µ + ν µ , and µ + µ − µ − ¯ ν µ . We use the complete matrix elements including besides diagramswith intermediate W and Z bosons also those with virtual photons as well as background dia-grams with only one possibly resonant vector boson. In addition, we include also photon-inducedcontributions at NLO. Using the complex-mass scheme [29–31] for a consistent treatment of res-onant propagators, our calculation provides NLO EW predictions for the entire fiducial volume.We apply acceptance cuts inspired by those of the experimental collaborations and study theimpact of the corrections on differential observables that are relevant in TGC searches.This article is organized as follows: In Section 2, some details of the computation are out-lined. The numerical setup and the phenomenological results are presented in Section 3. Finally,conclusions are given in Section 4. We consider the four independent processes pp → µ + µ − e + ν e + X , pp → µ + µ − e − ¯ ν e + X ,pp → µ + µ − µ + ν µ + X , and pp → µ + µ − µ − ¯ ν µ + X . At leading-order (LO), the correspondinghadronic cross sections at O ( α ) in the EW coupling receive contributions from the followingpartonic channels: q i ¯ q j / ¯ q j q i → µ + µ − e + ν e , q i ¯ q j ∈ { u¯d , c¯s , u¯s , c¯d } ,q i ¯ q j / ¯ q j q i → µ + µ − µ + ν µ , q i ¯ q j ∈ { u¯d , c¯s , u¯s , c¯d } ,q i ¯ q j / ¯ q j q i → µ + µ − e − ¯ ν e , q i ¯ q j ∈ { d¯u , s¯c , s¯u , d¯c } ,q i ¯ q j / ¯ q j q i → µ + µ − µ − ¯ ν µ , q i ¯ q j ∈ { d¯u , s¯c , s¯u , d¯c } . (2.1)We include quark-flavour mixing between the first two quark families as described by theCabibbo–Kobayashi–Maskawa (CKM) matrix defined in Eq. (3.5), i.e. we take into accountfirst-order mixing but neglect any higher-order quark-flavour mixings. The dominant channelsinvolving only quarks and antiquarks of the first generation contribute about 80% to the inte-grated LO cross section, the corresponding channels of the second family between 10% and 20%.Channels involving quarks of the first and anti-quarks of the second generation or vice versa, stayat the percent level. Sample tree-level diagrams contributing to the process q i ¯ q j → µ + µ − e + ν e are shown in Fig. 1. Besides diagrams with a resonant W boson and a resonant Z boson or aphoton these involve also diagrams with only one possibly resonant vector boson.The NLO EW corrections at O ( α ) comprise virtual corrections to the partonic channels22.1) as well as real photon emission via the quark-induced channels q i ¯ q j / ¯ q j q i → µ + µ − e + ν e (+ γ ) , q i ¯ q j ∈ { u¯d , c¯s , u¯s , c¯d } ,q i ¯ q j / ¯ q j q i → µ + µ − µ + ν µ (+ γ ) , q i ¯ q j ∈ { u¯d , c¯s , u¯s , c¯d } ,q i ¯ q j / ¯ q j q i → µ + µ − e − ¯ ν e (+ γ ) , q i ¯ q j ∈ { d¯u , s¯c , s¯u , d¯c } ,q i ¯ q j / ¯ q j q i → µ + µ − µ − ¯ ν µ (+ γ ) , q i ¯ q j ∈ { d¯u , s¯c , s¯u , d¯c } . (2.2)Moreover, we include the photon-induced contributions with one (anti)quark and one photonin the initial state, γq i /q i γ → µ + µ − e + ν e q j , q i q j ∈ { ud , cs } ,γ ¯ q i / ¯ q i γ → µ + µ − e + ν e ¯ q j , ¯ q i ¯ q j ∈ { ¯d¯u , ¯s¯c } ,γq i /q i γ → µ + µ − µ + ν µ q j , q i q j ∈ { ud , cs } ,γ ¯ q i / ¯ q i γ → µ + µ − µ + ν µ ¯ q j , ¯ q i ¯ q j ∈ { ¯d¯u , ¯s¯c } ,γq i /q i γ → µ + µ − e − ¯ ν e q j , q i q j ∈ { du , sc } ,γ ¯ q i / ¯ q i γ → µ + µ − e − ¯ ν e ¯ q j , ¯ q i ¯ q j ∈ { ¯u¯d , ¯c¯s } ,γq i /q i γ → µ + µ − µ − ¯ ν µ q j , q i q j ∈ { du , sc } ,γ ¯ q i / ¯ q i γ → µ + µ − µ − ¯ ν µ ¯ q j ¯ q i ¯ q j ∈ { ¯u¯d , ¯c¯s } , (2.3)generically referred to as qγ channels in the following. In all considered contributions, thebottom quark can neither appear as initial-state nor final-state particle since its weak isospinpartner, the top quark, is by construction excluded as external particle in all considered partonicchannels at LO and NLO.Since all considered processes involve exactly one quark-flavour-changing vertex at tree leveland since we treat all quarks except the top quark as massless, all tree-level amplitudes can beconstructed by multiplying the amplitudes for a unit CKM matrix with the corresponding non-vanishing CKM matrix elements. This treatment is also exact for the real and virtual correctionsin our setup. Since the CKM matrix can be eliminated for massless down-type quarks by aredefinition of flavour eigenstates, no renormalization of the CKM matrix is required in thisapproximation. Non-trivial CKM effects only enter because the flavour symmetry is brokenwhen amplitudes for different quark flavours are weighted with different parton distributionfunctions (PDFs). For the photon-induced channels the CKM matrix drops out exactly owingto its unitarity when the sum over the flavour of the final-state quark is performed. For thatreason, we can restrict the evaluation of the corrections in (2.3) to flavour-diagonal channelswith the CKM matrix set to unity.The one-loop virtual corrections comprise the full set of Feynman diagrams to the processes(2.1) at order O ( α ). Both at tree and one-loop level, we employ the complex-mass schemefor a consistent treatment of massive resonant particles [29–31] leading to NLO EW accuracyeverywhere in phase space. The integration of the real corrections is performed with help of thesubtraction methods of Refs. [32, 33] in order to deal with soft and collinear photon emissionoff fermions and with the collinear initial-state singularities in the photon-induced corrections.The employed formalism extends the dipole subtraction from QCD [34] to the case with QEDsplittings. The general idea of subtraction methods is to add and subtract auxiliary terms thatmimic the singularity structure of the real squared matrix elements point-wise such that theresulting differences can be integrated in four space-time dimensions. The re-added subtractionterms, on the other hand, can be integrated in a process-independent way allowing for an isola-tion of the divergences of the real corrections in analytical form. For infrared-safe observables,the extracted collinear final-state singularities and the soft singularities cancel with the corre-sponding divergences from the one-loop corrections according to the Kinoshita–Lee–Nauenbergtheorem. The left-over collinear initial-state singularities are absorbed in redefined parton-distribution functions. A more detailed description of real and virtual NLO EW corrections to3ector-boson pair production is given in Ref. [26] for the more general case of the productionof four charged leptons.The full computation with all possible final states has been performed with a private MonteCarlo program that has already successfully been used for the integration of the NLO EWcorrections to ZZ and WW production [25–27] and for the NLO QCD and EW corrections tovector-boson scattering [35, 36]. All the tree-level and one-loop matrix elements for LO, realand virtual contributions have been evaluated with the computer program Recola [37] whichinternally uses the
Collier library [38] for the one-loop scalar [39–42] and tensor integrals[43–45]. As a cross check, we have performed an independent calculation of the process pp → µ + µ − e + ν e + X and found perfect agreement at the level of phase-space points and at the levelof differential cross sections within the statistical uncertainty of the Monte Carlo integration.The matrix elements of the second implementation are generated with the Mathematica package
Pole [46] which is based on
FeynArts [47, 48] and
FormCalc [49]. The phase-spaceintegration is carried out with an independent multi-channel Monte Carlo integrator based onthe ones described in Refs. [50, 51].
For the numerical analysis we choose the following input parameters based on Ref. [52]. Theon-shell masses and widths of the gauge bosons read M osZ = 91 . , Γ osZ = 2 . ,M osW = 80 .
385 GeV , Γ osW = 2 .
085 GeV . (3.1)For the use within the complex-mass scheme, they need to be converted to pole masses andwidths according to Ref. [53]: M = M os p os /M os ) , Γ = Γ os p os /M os ) . (3.2)For the masses of the Higgs boson and the top quark, we use M H = 125 GeV , m t = 173 GeV , (3.3)while their widths can be set equal to zero as they do not appear as internal resonances in theconsidered processes. Throughout the calculation, all the charged leptons ℓ = { e ± , µ ± , τ ± } andthe five quarks q = { u , d , c , s , b } are considered as light particles with negligible masses.The electromagnetic coupling α is derived from the Fermi constant according to α G µ = √ π G µ M (cid:18) − M M (cid:19) with G µ = 1 . × − GeV − , (3.4)i.e. we work in the G µ scheme. In this scheme, the effects of the running of α from zero-momentum transfer to the electroweak scale are absorbed into the LO cross section, and masssingularities in the charge renormalization are avoided. Moreover, α G µ partially accounts forthe leading universal renormalization effects related to the ρ -parameter. We use the follow-ing approximation for the CKM matrix that includes transitions between the first two quarkgenerations: V CKM = V ud V us V ub V cd V cs V cb V td V ts V tb = cos θ c sin θ c − sin θ c cos θ c
00 0 1 , sin θ c = 0 . . (3.5)4ollowing Ref. [15], the renormalisation and factorisation scales, µ ren and µ fact , are set equalto the average of the Z-boson and W-boson mass, µ ren = µ fact = ( M Z + M W ) / . (3.6)As PDFs we choose the LUXqed plus PDF4LHC15 nnlo 100 parameterisation [54, 55]. Through-out our calculation, we employ the MS factorisation scheme. We have numerically verified thatthe difference between this scheme and the often used deep-inelastic-scattering scheme is belowone permille for the relative NLO EW corrections and, thus, phenomenologically irrelevant.
Photons emitted in the Bremsstrahlung corrections are recombined with the closest chargedlepton if their separation ∆ R in the rapidity–azimuthal-angle plane fulfils∆ R ℓ i ,γ = q ( y ℓ i − y γ ) + (∆ φ ℓ i ,γ ) < . , (3.7)where y denotes the rapidity of the final-state particle and ∆ φ ℓ i ,γ the azimuthal-angle differencebetween a charged lepton ℓ i and the photon γ . Final-state photons with rapidity | y γ | > ℓ +Z ℓ − Z ) as the lepton pair associated with the Z-boson decay and to ℓ ± W as thecharged lepton from the W-boson decay. For the processes involving both muons and electrons,the equal-flavour lepton pair is associated with the Z-boson decay, while the other chargedlepton is associated with the W-boson decay. For the processes with three equal-flavour leptonsin the final state, the lepton pair emerging from the Z-boson decay is defined as the one whoseinvariant mass M ℓ + i ,ℓ − j is closer to the nominal Z-boson mass.We have investigated each process class in two different scenarios: first with a minimal set ofselection cuts, in the following referred to as “inclusive setup”, and second for a setup inspiredby the ATLAS measurements [2, 3] that is tailored to the investigation of TGC, referred to as“TGC setup”. The corresponding fiducial volumes are defined as follows: Inclusive setup:
We treat all charged final-state leptons on the same footing, requiring p T ,ℓ i >
15 GeV , | y ℓ i | < . , ∆ R ℓ i ,ℓ j > . , (3.8)where p T denotes the transverse momentum. Exclusive setup for TGC analysis:
For each charged lepton ℓ i , we demand a minimaltransverse momentum and a maximal rapidity: p T ,ℓ Z >
15 GeV , p T ,ℓ W >
20 GeV , | y ℓ i | < . . (3.9)Any pair of charged leptons ( ℓ i , ℓ j ) is required to be well separated in the rapidity–azimuthal-angle plane: ∆ R ℓ Z ,ℓ Z > . , ∆ R ℓ Z ,ℓ W > . . (3.10)The invariant mass of the ℓ +Z ℓ − Z pair is allowed to differ by at most 10 GeV from the nominalZ-boson mass: M Z −
10 GeV < M ℓ +Z ,ℓ − Z < M Z + 10 GeV . (3.11)The W-boson transverse mass M WT must obey M WT = q p missT p T ,ℓ W [1 − cos ∆ φ ( ℓ W , ~p missT )] >
30 GeV , (3.12)5ixed flavour [2 µ e ν ] σ LO [fb] δ ¯ qq ′ (%) δ qγ (%) δ NLO (%)inclusive pp → µ + µ − e + ν e + X − . . − . → µ + µ − e − ¯ ν e + X − . . − . → µ + µ − e + ν e + X − . . − . → µ + µ − e − ¯ ν e + X − . . − . µν ] σ LO [fb] δ ¯ qq ′ (%) δ qγ (%) δ NLO (%)inclusive pp → µ + µ − µ + ν µ + X − . . − . → µ + µ − µ − ¯ ν µ + X − . . − . → µ + µ − µ + ν µ + X − . . − . → µ + µ − µ − ¯ ν µ + X − . . − . φ ( ℓ W , ~p missT ) denotes the azimuthal angle between the momentum of the W-boson decaylepton ℓ W and the missing momentum in the transverse plane ~p missT , and p missT = | ~p missT | .We define the missing momentum as the negative vector sum of the momenta of all observedparticles. Final-state quarks from the photon-induced corrections are considered as observablejets if their transverse momentum satisfies p jetT > p jetT , min = 25 GeV . (3.13)Hence, quarks in the final state with transverse momentum below p jetT , min contribute to themissing momentum. All photons and jets from real radiation with rapidity | y γ/ jet | > The results for the fiducial cross sections at a centre-of-mass energy of 13 TeV are presented inTable 1 for all considered final states both in the inclusive and in the TGC setup. The secondcolumn shows the absolute prediction for the cross section at leading order, σ LO , followedby the relative EW corrections of the quark-induced contributions δ ¯ qq ′ , the relative photon-induced corrections δ qγ , and the total relative EW corrections δ NLO = δ ¯ qq ′ + δ qγ . According tothe total electric charge of the final-state leptons, we sometimes refer to the processes pp → µ + µ − e + ν e + X and pp → µ + µ − µ + ν µ + X as ZW + and to the processes pp → µ + µ − e − ¯ ν e + X andpp → µ + µ − µ − ¯ ν µ + X as ZW − . We stress, however, that we include all contributions leadingto the considered four-lepton final state, also those which do not proceed through intermediateZW ± production. The cross sections for the ZW + channels are about 50% larger than theones for the ZW − channels, both in the mixed-flavour case [2 µ e ν ] and in the equal-flavour case[3 µν ]. This can be attributed to the parton flux within the proton which is larger for the upquark than for the down quark. The fiducial cross section in the TGC setup is about 30%smaller than in the inclusive setup, as expected owing to the reduced fiducial volume. Forall setups and channels, the photon-induced contributions are of the order of +2% with onlyminor variations at the subpercent level. The quark-induced EW corrections are negative anddepend significantly on the phase-space cuts. The corrections in the TGC setup are with about −
6% almost twice as large as in the inclusive setup where they reach about − ∼ − . − −
4% in the TGC setup.The inclusion of first-order transitions in flavour-changing currents lowers the total crosssection with respect to a unit CKM matrix by 0 .
7% in the ZW + channels and by 0 .
9% in theZW − channels independently of the leptons in the final state. We have also performed a LOstudy including transitions between the second and third quark generation, which prooved thatthis effect is phenomenologically irrelevant.Since the cuts of the inclusive setup are by construction not sensitive to the lepton pairing,the scenario is well suited to study the size of interference effects present for equal-flavourleptons in the final state. In the absence of any interference, the equal-flavour and mixed-flavourcross sections would be equal. The deviation of the ratio σ [3 µν ] /σ [2 µ e ν ] from one thus gives ameasure of the impact of interferences. At LO, we find σ µ + µ − µ + ν µ /σ µ + µ − e + ν e = 0 . σ µ + µ − µ − ¯ ν µ /σ µ + µ − e − ¯ ν e = 0 . .
4% from the unit ratio [ σ µ + µ − µ + ν µ /σ µ + µ − e + ν e =1 . σ µ + µ − µ − ¯ ν µ /σ µ + µ − e − ¯ ν e = 1 . qγ and ¯ qq ′ channels for a detailed comparison. The computation in Ref. [23] does notinclude photon-induced corrections. For the LHC at 14 TeV, the authors state corrections of δ ¯ qq ′ = − .
5% for ZW + and δ ¯ qq ′ = − .
3% for ZW − , applying a minimal event selection that isroughly comparable with our inclusive setup. We attribute the difference of 2% to our resultsmainly to photon radiation off the µ + µ − pair. Inspection of the (unmeasurable) four-leptoninvariant-mass distribution reveals that above the pair-production threshold, where the crosssection receives the largest contribution, the NLO EW corrections are negative at the level of −
3% and dominated by real photon radiation. For an on-shell Z boson, the effect of final-stateradiation and thus of real corrections is reduced.
In the following, we present results for distributions for the LHC at 13 TeV. In each of thefigures, the upper panels show the absolute predictions for the LO and NLO differential crosssection while the lower panels display the relative EW corrections.We first discuss the mixed-flavour final state where the µ + µ − pair can be associated with thedecay of the neutral vector boson, distinguishing between the ZW + and ZW − case. Figure 2shows the invariant-mass distributions of the µ + µ − system. The absolute prediction in theinclusive setup (left panel) exhibits the characteristic pattern of this observable similar to thecorresponding µ + µ − invariant-mass distribution in ZZ production in Ref. [26]: 1) the resonancepeak at M µ + µ − = M Z , 2) the increase of the cross section towards M µ + µ − = 0 owing to thetail of the photon pole, and 3) a little bump between 30 GeV and 50 GeV from the s -channelresonance at M µ + µ − e ± ( − ) ν e = M W [c.f. diagrams (b), (c) and (d) in Fig. 1]. Turning to the EWcorrections, we observe in the quark-induced channels a typical radiative tail with corrections7 − δ qγ [ µ + µ − e − ¯ ν e ] δ ¯ qq ′ [ µ + µ − e − ¯ ν e ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] M µ + µ − [GeV] δ [ % ] − − µ + µ − e − ¯ ν e ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − e − ¯ ν e ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, inclusive setup d σ d M µ + µ − h f b G e V i − − δ qγ [ µ + µ − e − ¯ ν e ] δ ¯ qq ′ [ µ + µ − e − ¯ ν e ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] M µ + µ − [GeV] δ [ % ] − − µ + µ − e − ¯ ν e ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − e − ¯ ν e ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, TGC setup d σ d M µ + µ − h f b G e V i Figure 2: Distribution in the invariant mass of the µ + µ − pair in the inclusive setup (left panel)and in the TGC setup (right panel).of up to +75%: Photon radiation off the final-state charged leptons may shift the measuredvalue of the invariant mass to lower values. Since the LO cross section falls off steeply belowthe resonance, the relative real NLO corrections become large. At the resonance the correctionschange sign, and above they are of the order of −
10% (they reach −
25% at 1 . + and ZW − , the only visibledifference being in the radiative tail which is up to 6% larger for ZW + . This difference resultsfrom folding the partonic cross sections with the PDFs. The photon-induced corrections arepositive over the whole spectrum with variations between 1 .
8% and 5%. Owing to the cutaround the Z-boson resonance (3.11), the invariant µ + µ − mass in the TGC scenario (rightpanel) is restricted to [ M Z − , M Z + 10]. Evidently, this cut removes a substantial part of theradiative tail present in the inclusive setup.Figure 3 shows the distribution in the transverse mass M ℓν T of the four-lepton system inthe inclusive setup (left panel) and in the TGC setup (right panel) as defined in Ref. [2], M ℓν T = vuuut X ℓ i =1 p T ,ℓ i + | ~p missT | − X ℓ i =1 p ℓ i , x + p miss x + X ℓ i =1 p ℓ i , y + p miss y (3.14)with ℓ = ℓ +Z , ℓ = ℓ − Z , ℓ = ℓ ± W and the missing momentum ~p miss defined at the end ofSection 3.2. Note that for contributions with only leptons in the final state, like the virtualcorrections or the LO contribution, the transverse mass in Eq. (3.14) reduces to the scalar sumof the lepton transverse momenta. The absolute prediction has its maximum slightly below M ℓν T = M Z + M W . The observable does not show a sharp pair-production threshold (like theunmeasurable four-lepton invariant-mass distribution would exhibit at M ℓ = M Z + M W , c.f.the discussion in Ref. [26] for ZZ production) as the unmeasurable boost of the four-leptonsystem along the beam axis allows for on-shell production of the W and Z boson below M ℓν T = M Z + M W . The little peak directly below 80 GeV in the inclusive setup stems from a singleW-boson resonance with M = ( p ℓ +Z + p ℓ − Z + p ℓ ± W + p ν ) [c.f. diagrams (b), (c) and (d) in8 − − − − − δ qγ [ µ + µ − e − ¯ ν e ] δ ¯ qq ′ [ µ + µ − e − ¯ ν e ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] M ℓν T [GeV] δ [ % ] − − − − µ + µ − e − ¯ ν e ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − e − ¯ ν e ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, inclusive setup d σ d M ℓ ν T h f b G e V i − − − − − δ qγ [ µ + µ − e − ¯ ν e ] δ ¯ qq ′ [ µ + µ − e − ¯ ν e ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] M ℓν T [GeV] δ [ % ] − − − − µ + µ − e − ¯ ν e ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − e − ¯ ν e ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, TGC setup d σ d M ℓ ν T h f b G e V i Figure 3: Distribution in the transverse mass of the four-lepton system in the inclusive setup(left panel) and in the TGC setup (right panel).Fig. 1]. This resonance is removed in the TGC setup owing to the lower cut on the invariant µ + µ − mass in Eq. (3.11) and the minimal transverse momentum p minT ,ℓ W of the charged W-decay lepton candidate in Eq. (3.9) since ( M Z −
10 GeV) + p minT ,ℓ W > M W . The shape of thequark-induced EW corrections above the maximum of the distribution is very similar in bothsetups. We observe a plateau region from the maximum on up to about 300 GeV with − −
7% in the TGC setup) and then a constant decrease to −
20% ( − M ℓν T . The combination of the virtual and real corrections gives rise to the plateauin the distribution. The region below the maximum is entirely dominated by the subtractedreal corrections: the kink at the maximum followed by increasing corrections is due to theradiative return of the real photon at the relatively broad peak. The difference between theTGC and the inclusive setup in this region results from the enhanced radiative tail from thereconstructed Z-boson resonance, as we checked explicitly by switching off the invariant-masscut in Eq. (3.11). The photon-induced corrections have their minimum with about 1% wherethe LO quark-induced channels are largest, and constantly increase with growing M ℓν T up to5% to 8%, depending on the final state and the setup.Figure 4 compares the transverse mass M ℓν T of the four-lepton system for the equal-flavour[3 µν ] and mixed-flavour [2 µ e ν ] final states in ZW + production for the inclusive setup (left panel)and for the TGC setup (right panel). For both scenarios, the relative EW corrections of the µ + µ − e + ν e and µ + µ − µ + ν µ final states are almost equal (separately for the ¯ qq ′ and qγ channels).This is in agreement with the results for the fiducial cross section where only permille-leveldifferences between corrections of the mixed- and equal-flavour final states are observed. In9 . . . . . NLO [3 µν ] / [2 µeν ]LO [3 µν ] / [2 µeν ] M ℓν T [GeV] [ µ ν ] / [ µ e ν ] − δ qγ [ µ + µ − µ + ν µ ] δ ¯ qq ′ [ µ + µ − µ + ν µ ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] δ [ % ] − − − − µ + µ − µ + ν µ ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − µ + ν µ ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, inclusive setup d σ d M ℓ ν T h f b G e V i . . . . . NLO [3 µν ] / [2 µeν ]LO [3 µν ] / [2 µeν ] M ℓν T [GeV] [ µ ν ] / [ µ e ν ] − δ qγ [ µ + µ − µ + ν µ ] δ ¯ qq ′ [ µ + µ − µ + ν µ ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] δ [ % ] − − − − − µ + µ − µ + ν µ ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − µ + ν µ ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, TGC setup d σ d M ℓ ν T h f b G e V i Figure 4: Comparison between equal-flavour and mixed-flavour final state for the distributionin the four-lepton transverse mass M ℓν T for ZW + production. The left panel shows the inclusivesetup, the right panel the TGC setup.the lowest panel, the ratio (d σ [3 µν ](N)LO / d M ℓν T ) / (d σ [2 µ e ν ](N)LO / d M ℓν T ) / is shown. By construction, theobservable M ℓν T is not sensitive to the assignment of the decay leptons to the Z or the W boson.Since the inclusive setup is symmetric in the equal-flavour final-state leptons, the deviationfrom one of this ratio gives a direct measure of the impact of interferences. In the off-shell-sensitive region below 100 GeV, the interferences become indeed sizeable, lowering the [3 µν ]cross section by about one third with respect to the [2 µ e ν ] case. As expected, the interferencesare irrelevant in the on-shell region, where they are suppressed with respect to the doubly-resonant contributions. In the TGC setup we observe relative differences between the twofinal states of the order of 2 −
3% also in the on-shell region. Because of the smallness of theinterference effects in the inclusive setup we attribute the deviation from one in the on-shellregion in the TGC setup to the lepton pairing in the presence of asymmetric cuts on ℓ W and ℓ Z .Around the maximum, the ratio deviates from one at the percent level, in agreement with theratio for the fiducial cross section. Further below the maximum, a separation of interference andlepton-pairing effects is not possible. In both setups, the NLO EW corrections do not modifythe shape of the ratio.The left plot in Fig. 5 shows the distribution in the transverse mass of the reconstructedW boson, M WT = M e ν T , as defined in Eq. (3.12) in the TGC setup. The peak of the distributionis located below the W-boson mass (the reconstructed invariant W-boson mass is experimentallynot accessible owing to the undetected neutrino). The quark-induced corrections follow a similar10 − − − δ qγ [ µ + µ − e − ¯ ν e ] δ ¯ qq ′ [ µ + µ − e − ¯ ν e ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] M e ν T [GeV] δ [ % ] − − − − − µ + µ − e − ¯ ν e ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − e − ¯ ν e ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, TGC setup d σ d M e ν T h f b G e V i − − − δ qγ [ µ + µ − e − ¯ ν e ] δ ¯ qq ′ [ µ + µ − e − ¯ ν e ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] p T ,µ + µ − [GeV] δ [ % ] − − − − − µ + µ − e − ¯ ν e ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − e − ¯ ν e ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, TGC setup d σ d p T , µ + µ − h f b G e V i Figure 5: Distribution in the transverse mass of the reconstructed W boson (left panel) and inthe transverse momentum of the µ + µ − pair (right panel) in the TGC setup.pattern as already observed in M ℓν T : Below the maximum, the subtracted virtual correctionsare small (less than 1% in magnitude). Above, they constantly increase in size up to − .
5% withgrowing M e ν T due to logarithms of EW origin. The subtracted real corrections above 100 GeVgive again an off-set of the order of − . µ + µ − invariant mass in Eq. (3.11), does not directly influence M e ν T as this observable doesnot depend on the muon momenta. The photon-induced corrections are relatively flat and donot show any particularly interesting pattern.The right plot in Fig. 5 shows the transverse-momentum distribution of the µ + µ − systemin the TGC setup, i.e. the transverse momentum of the reconstructed Z boson. We observethe typical feature of large EW corrections in the quark-induced channels that reach −
25% at600 GeV due to EW Sudakov logarithms. We can compare this number with the correspondingresults for the distribution in the Z-boson transverse momentum of the on-shell calculationsof Refs. [23, 24]. From the plots in these references we extract a correction of about −
22% at p T , Z = 600 GeV. We attribute the difference of −
3% to the slightly different setup and to themissing final-state radiation off muons (radiative energy loss shifts events to smaller transversemomentum and thus leads to more negative corrections). Similarly to the previously consideredobservables that depend on transverse momenta, the subtracted virtual corrections are small(below 1%) in the low- p T region and start growing negative with constant slope above 100 GeV.The corrections in the low- p T region, where the bulk of the cross section stems from, areentirely dominated by the subtracted real radiation. The fact that the corrections are flat there11 − − − δ qγ [ µ + µ − e − ¯ ν e ] δ ¯ qq ′ [ µ + µ − e − ¯ ν e ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] p T , e ± [GeV] δ [ % ] − − − − − µ + µ − e − ¯ ν e ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − e − ¯ ν e ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, TGC setup d σ d p T , e ± h f b G e V i − − − δ qγ [ µ + µ − e − ¯ ν e ] δ ¯ qq ′ [ µ + µ − e − ¯ ν e ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] p missT [GeV] δ [ % ] − − − − − µ + µ − e − ¯ ν e ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − e − ¯ ν e ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, TGC setup d σ d p m i ss T h f b G e V i Figure 6: Distribution in the transverse-momentum of the charged W-decay lepton (left panel)and the missing-transverse momentum (right panel) in the TGC setup.is again due to the invariant-mass cut (3.11). In the inclusive setup (not shown) the correctionscontinuously decrease in size until approaching − .
5% at zero transverse momentum. The mostremarkable feature of this observable is the large increase of the photon-induced contributionsfor high transverse momenta. At 600 GeV, they reach +18% in the ZW + case and even +25%in the ZW − case, and, thus, almost compensate the large negative EW corrections from thequark-induced channels. The large difference between ZW + and ZW − is caused by the differentPDFs involved. In Ref. [24], it has been shown that the large increase of the photon-inducedcross section is mainly due to the coupling of the photon to the W boson. The photon-inducedcorrections presented in Ref. [24] show with +28% for ZW + and +41% for ZW − qualitativelya similar behaviour, though the numerical values differ. We attribute the difference mainlyto the different PDF set as the ratio between the photon-induced real corrections and thepurely quark-induced LO contribution is very sensitive to the employed photon PDF. The largephoton-induced corrections can be reduced by imposing a jet veto [27].The transverse-momentum distributions in Fig. 6 for the charged W-decay lepton (left panel)and the missing transverse momentum (right panel) show similar features as already observedin the transverse-momentum distribution of the µ + µ − pair in Fig. 5: Large negative EW correc-tions in the ¯ qq ′ channel and large positive corrections from the photon-induced contributions inthe high- p T regime that partially compensate each other. Among all transverse-momentum dis-tributions, those for the transverse momentum of the W-decay lepton show the largest differencein the photon-induced corrections between ZW + and ZW − .The distribution in the azimuthal-angle difference of the µ + µ − pair in the TGC setup isshown in the left panel of Fig. 7 for the ZW + and ZW − mixed-flavour case. The maximum at∆ φ → π has the same origin as for the corresponding observable in ZZ production describedin Ref. [26]: The whole distribution is dominated by events in the energy region just above thepair-production threshold with two resonant vector bosons. Owing to the t -channel nature of thedominant contributions [c.f. diagram (a) in Fig. 1], the vector bosons are preferably producedin forward direction with small momenta in the transverse plane. The Z-boson decay leptonsare thus mainly back-to-back in the transverse plane which explains the maximum at ∆ φ → π .12 . . . − − − − − δ qγ [ µ + µ − e − ¯ ν e ] δ ¯ qq ′ [ µ + µ − e − ¯ ν e ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] ∆ φ µ + µ − δ [ % ] NLO EW [ µ + µ − e − ¯ ν e ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − e − ¯ ν e ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, TGC setup d σ d ∆ φ µ + µ − [ f b ] . . . − − − − − δ qγ ∆ φ ( µ + e + ) [ µ + µ − e + ν e ] δ ¯ qq ′ ∆ φ ( µ + e + ) [ µ + µ − e + ν e ] δ qγ ∆ φ ( µ − e + ) [ µ + µ − e + ν e ] δ ¯ qq ′ ∆ φ ( µ − e + ) [ µ + µ − e + ν e ] ∆ φ µ ± e + δ [ % ] NLO EW ∆ φ ( µ + e + ) [ µ + µ − e + ν e ]NLO EW ∆ φ ( µ − e + ) [ µ + µ − e + ν e ]LO ∆ φ ( µ + e + ) [ µ + µ − e + ν e ]LO ∆ φ ( µ − e + ) [ µ + µ − e + ν e ] √ s = 13 TeV, TGC setup d σ d ∆ φ µ ± e + [ f b ] Figure 7: Distributions in the azimuthal-angle difference of the charged leptons in the TGCsetup for the mixed-flavour case. The left panel shows the correlation between the µ + µ − pairfor ZW + and ZW − . In the right panel, the correlations between the µ − e + pair and the µ + e + pair are plotted for the ZW + channel.The EW corrections are nearly equal for both presented final states. Around the maximum,the ¯ qq ′ channels receive corrections of −
6% as for the fiducial cross section. Towards ∆ φ → − φ = π of ∼ +1%, and a maximum at ∆ φ = 0 . ∼ +5%. Hence, both for the¯ qq ′ and qγ channels, the NLO EW corrections to ∆ φ µ + µ − reflect qualitatively the behaviour ofthe corrections in the transverse-momentum distribution of the µ + µ − pair in Fig. 5.The plot on the right-hand side in Fig. 7 compares the azimuthal-angle difference of the µ − e + pair with the one of the µ + e + pair in the mixed-flavour ZW + final state. In both caseswe observe a maximum at ∆ φ = π , and a minimum at ∆ φ = 0 resulting from boosts of back-to-back W and Z bosons. The kink at ∆ φ = 0 . φ µ + µ − , and smallestfor ∆ φ µ + e + . The photon-induced corrections show a rather flat behaviour and are practicallyindependent of the observable. The ¯ qq ′ -induced corrections, however, differ significantly for thetwo observables: The corrections in the µ − e + case decrease from − .
5% at ∆ φ µ − e + = 0 to − .
4% at ∆ φ µ − e + = π , while those in the µ + e + case increase from − .
2% to − .
2% within thesame range of ∆ φ µ + e + . The observed difference is mainly caused by the real corrections anddue to events close to the WZ production threshold.Figure 8 shows various distributions in the rapidities of the charged leptons for the mixed-flavour final states of ZW + and ZW − in the TGC setup. In the upper row, distributions in therapidities of the µ + and the µ − are presented. The corresponding photon-induced correctionsare rather flat and almost equal for y µ + and y µ − . The quark-induced corrections show a strikingdifference in the curvature of the relative corrections: those for y µ + are minimal in the centralregion and maximal in forward direction, while it is just the other way round for y µ − . In forward13 − − − − − − δ qγ [ µ + µ − e − ¯ ν e ] δ ¯ qq ′ [ µ + µ − e − ¯ ν e ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] y µ + δ [ % ] NLO EW [ µ + µ − e − ¯ ν e ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − e − ¯ ν e ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, TGC setup d σ d y µ + [ f b ] − − − − − − δ qγ [ µ + µ − e − ¯ ν e ] δ ¯ qq ′ [ µ + µ − e − ¯ ν e ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] y µ − δ [ % ] NLO EW [ µ + µ − e − ¯ ν e ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − e − ¯ ν e ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, TGC setup d σ d y µ − [ f b ] − − − − − − δ qγ [ µ + µ − e − ¯ ν e ] δ ¯ qq ′ [ µ + µ − e − ¯ ν e ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] y µ + µ − δ [ % ] NLO EW [ µ + µ − e − ¯ ν e ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − e − ¯ ν e ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, TGC setup d σ d y µ + µ − [ f b ] − − − − − − δ qγ [ µ + µ − e − ¯ ν e ] δ ¯ qq ′ [ µ + µ − e − ¯ ν e ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] y e ± δ [ % ] NLO EW [ µ + µ − e − ¯ ν e ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − e − ¯ ν e ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, TGC setup d σ d y e ± [ f b ] Figure 8: Distributions in the rapidities of the W and Z decay leptons in the TGC setup forZW + and ZW − : µ + (upper left panel), µ − (upper right panel), µ + µ − pair (lower left panel)and e ± (lower right panel). 14 − − − − − − − δ qγ [ µ + µ − e − ¯ ν e ] δ ¯ qq ′ [ µ + µ − e − ¯ ν e ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] ∆ y µ ± e ± δ [ % ] NLO EW [ µ + µ − e − ¯ ν e ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − e − ¯ ν e ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, TGC setup d σ d ∆ y µ ± e ± [ f b ] − − − − − − − δ qγ [ µ + µ − e − ¯ ν e ] δ ¯ qq ′ [ µ + µ − e − ¯ ν e ] δ qγ [ µ + µ − e + ν e ] δ ¯ qq ′ [ µ + µ − e + ν e ] ∆ y µ ∓ e ± δ [ % ] NLO EW [ µ + µ − e − ¯ ν e ]NLO EW [ µ + µ − e + ν e ]LO [ µ + µ − e − ¯ ν e ]LO [ µ + µ − e + ν e ] √ s = 13 TeV, TGC setup d σ d ∆ y µ ∓ e ± [ f b ] Figure 9: Distribution in the rapidity difference of the µ ± e ± pairs (left panel) and the µ ± e ∓ pairs (right panel) for ZW + and ZW − in the TGC setup.direction, the difference between the corrections to the two observables amounts to 2 .
5% forZW + and 3 .
0% for ZW − . This behaviour originates from the difference in the PDFs of upand down quarks in combination with the fact that the matrix element is not symmetric underexchange of the µ + and µ − momenta. The lower left plot shows the distribution in the rapidityof the µ + µ − pair, i.e. the rapidity of the reconstructed Z boson. Like in the upper plots, thephoton-induced corrections are flat and almost equal for ZW + and ZW − . This holds also for thequark-induced corrections in the central region. In forward direction, we observe a differencebetween the corrections to ZW + and ZW − of about one percent which can be attributed to theinterplay of PDFs and subtracted virtual corrections. The lower right plot shows the distributionin the rapidity of the charged lepton from W decay. The photon-induced corrections stay atthe level of 2% with sub-percent deviations between ZW + and ZW − . Differences of similar sizeare also observed in the quark-induced corrections. Like for y µ + µ − , we could show that thedifference is induced by the PDFs and largest for the virtual corrections.Figure 9 displays the distribution in the rapidity difference of the µ ± e ± pairs (left panel)and in the rapidity difference of the µ ∓ e ± pairs (right panel) for ZW + and ZW − in the TGCsetup. The NLO EW corrections of the quark-induced channels show characteristic percent-level differences between the ZW + and ZW − case. For the same-sign pair µ + e + ( µ − e − ), thecorrections in the ZW + case (ZW − case) have a maximum at ∆ y = 0 with −
4% ( − −
8% ( − | ∆ y | = 5. The maximal difference of the corrections is, thus, largest forthe ZW + case. In the opposite-sign case ( µ ∓ e ± pairs), the behaviour is the other way round.The corrections for ZW + vary between their extrema by only about 2%, while the variation forZW − amounts to almost 5%. The photon-induced corrections are basically equal for both finalstates and rather flat. 15 Conclusions
The production of a pair of a neutral and a charged vector boson with subsequent leptonic decaysis a very important process for precision tests of the Standard Model of elementary particlesand in searches for anomalous triple-gauge-boson couplings (TGC). In this article, the firstcomputation of next-to-leading order (NLO) electroweak (EW) corrections to the productionof three charged leptons plus missing energy at the Large Hadron Collider has been presented.We have analysed the four independent final states µ + µ − e + ν e , µ + µ − e − ¯ ν e , µ + µ − µ + ν µ , and µ + µ − µ − ¯ ν µ applying realistic experimental phase-space cuts, first in a rather inclusive setupwith minimal event selection and second in a scenario that is tailored to TGC searches. Weuse the complete matrix elements including all off-shell effects of intermediate massive vectorbosons and virtual photons as well as irreducible background diagrams.We have computed the NLO EW corrections resulting from quark–antiquark initial states aswell as from initial states with photons. The photon-induced corrections raise the leading-order(LO) cross sections by about +2% in both scenarios. The quark-induced corrections dependsignificantly on the fiducial volume. They lower the cross section by about −
3% in the inclusivecase and by about −
6% in the TGC setup. For the fiducial cross section, the corrections to finalstates with positive total charge (ZW + ) differ from those with negative total charge (ZW − ) atthe sub-percent level.At the level of differential distributions, we observe quark-induced corrections of up to − + and ZW − ), weobserve that the photon-induced corrections exhibit significant differences of more than 10% forcertain observables. The different behaviour between the ZW + and ZW − channels results fromthe differences in the parton distribution functions for the respective initial states. Concerningquark-induced corrections, percent-level differences between final states with opposite chargeare found in rapidity distributions. Differential distributions that are sensitive to kinematicthresholds or resonances show typical radiative tails induced by photon-radiation off final-stateleptons and lead to characteristic differences between the considered inclusive and TGC setups.We have studied the impact of interference effects arising in final states with equal-flavourleptons. In the inclusive setup, they lower the fiducial cross section with respect to the mixed-flavour case at the permille level only. While this holds true also at the level of differentialdistributions in phase-space regions dominated by on-shell vector-boson-pair production, in-terference effects become sizeable and lower the cross section by up to one third in off-shellsensitive phase-space regions. If the observables or the phase-space cuts depend on the selec-tion of equal-flavour final-state leptons (like in the considered TGC scenario), the differencesbetween equal- and mixed-flavour processes are, in general, more pronounced and cannot ex-clusively be attributed to interferences. Both at LO and NLO, the shape distortions owing tolepton selection and interferences are almost equal in size.The NLO EW corrections for WZ production presented in this article are important forprecision tests of the Standard Model and its possible extensions. Taking into account that thisprocess is meanwhile known at next-to-next-to-leading order in the strong coupling, the NLOEW corrections further reduce the theoretical uncertainty and can help to improve the exclusionlimits on anomalous TGC. We advocate for a systematic inclusion of the EW corrections infuture experimental analyses. 16 cknowledgements We would like to thank Stefan Dittmaier for useful discussions. The work of B.B. and A.D. wassupported by the German Federal Ministry for Education and Research (BMBF) under contractno. 05H15WWCA1 and by the German Science Foundation (DFG) under reference number DE623/6-1. The work of L.H. was supported by the Spanish MINECO grants FPA2013-46570-C2-1-P and FPA2016-76005-C2-1-P, by the grant 2014-SGR-104, and partially under the projectMDM-2014-0369 of ICCUB (Unidad de Excelencia “Mar´ıa de Maeztu”).
References [1]
ATLAS
Collaboration, G. Aad et al.,
Measurement of
W Z production in proton-protoncollisions at √ s = 7 TeV with the ATLAS detector , Eur. Phys. J.
C72 (2012) 2173,[ arXiv:1208.1390 ].[2]
ATLAS
Collaboration, G. Aad et al.,
Measurements of W ± Z production cross sectionsin pp collisions at √ s = 8 TeV with the ATLAS detector and limits on anomalous gaugeboson self-couplings , Phys. Rev.
D93 (2016) 092004, [ arXiv:1603.02151 ].[3]
ATLAS
Collaboration, M. Aaboud et al.,
Measurement of the W ± Z bosonpair-production cross section in pp collisions at √ s = 13 TeV with the ATLAS Detector , Phys. Lett.
B762 (2016) 1–22, [ arXiv:1606.04017 ].[4]
CMS
Collaboration, V. Khachatryan et al.,
Measurement of the WZ production crosssection in pp collisions at √ s =
13 TeV , Phys. Lett.
B766 (2017) 268–290,[ arXiv:1607.06943 ].[5]
CMS
Collaboration, V. Khachatryan et al.,
Measurement of the WZ production crosssection in pp collisions at √ s = 7 and 8 TeV and search for anomalous triple gaugecouplings at √ s = 8 TeV , Eur. Phys. J.
C77 (2017) 236, [ arXiv:1609.05721 ].[6]
ATLAS
Collaboration,
Measurement of W ± Z boson pair-production in pp collisions at √ s = 13 TeV with the ATLAS Detector and confidence intervals for anomalous triplegauge boson couplings , Tech. Rep. ATLAS-CONF-2016-043, CERN, Geneva, Aug, 2016.[7] J. Ohnemus,
An order α s calculation of hadronic W ± Z production , Phys. Rev.
D44 (1991) 3477–3489.[8] J. Ohnemus,
Hadronic ZZ , W − W + , and W ± Z production with QCD corrections andleptonic decays , Phys. Rev.
D50 (1994) 1931–1945, [ hep-ph/9403331 ].[9] L. J. Dixon, Z. Kunszt, and A. Signer,
Vector boson pair production in hadronic collisionsat order α s : Lepton correlations and anomalous couplings , Phys. Rev.
D60 (1999)114037, [ hep-ph/9907305 ].[10] J. M. Campbell and R. K. Ellis,
An update on vector boson pair production at hadroncolliders , Phys. Rev.
D60 (1999) 113006, [ hep-ph/9905386 ].[11] K. Hamilton,
A positive-weight next-to-leading order simulation of weak boson pairproduction , JHEP (2011) 009, [ arXiv:1009.5391 ].[12] T. Melia, P. Nason, R. R¨ontsch, and G. Zanderighi, W + W − , W Z and ZZ production inthe POWHEG BOX , JHEP (2011) 078, [ arXiv:1107.5051 ].[13] P. Nason and G. Zanderighi, W + W − , W Z and ZZ production in thePOWHEG-BOX-V2 , Eur. Phys. J.
C74 (2014) 2702, [ arXiv:1311.1365 ].1714] M. Grazzini, S. Kallweit, D. Rathlev, and M. Wiesemann, W ± Z production at hadroncolliders in NNLO QCD , Phys. Lett.
B761 (2016) 179–183, [ arXiv:1604.08576 ].[15] M. Grazzini, S. Kallweit, D. Rathlev, and M. Wiesemann, W ± Z production at the LHC:fiducial cross sections and distributions in NNLO QCD , JHEP (2017) 139,[ arXiv:1703.09065 ].[16] W. Beenakker, A. Denner, S. Dittmaier, R. Mertig, and T. Sack, High-energyapproximation for on-shell W pair production , Nucl.Phys.
B410 (1993) 245–279.[17] M. Beccaria, G. Montagna, F. Piccinini, F. Renard, and C. Verzegnassi,
Rising bosonicelectroweak virtual effects at high-energy e + e − colliders , Phys.Rev.
D58 (1998) 093014,[ hep-ph/9805250 ].[18] P. Ciafaloni and D. Comelli,
Sudakov enhancement of electroweak corrections , Phys.Lett.
B446 (1999) 278–284, [ hep-ph/9809321 ].[19] J. H. K¨uhn and A. Penin,
Sudakov logarithms in electroweak processes , hep-ph/9906545 .[20] V. S. Fadin, L. N. Lipatov, A. D. Martin, and M. Melles, Resummation of doublelogarithms in electroweak high-energy processes , Phys. Rev.
D61 (2000) 094002,[ hep-ph/9910338 ].[21] A. Denner and S. Pozzorini,
One-loop leading logarithms in electroweak radiativecorrections. 1. Results , Eur.Phys.J.
C18 (2001) 461–480, [ hep-ph/0010201 ].[22] E. Accomando, A. Denner, and A. Kaiser,
Logarithmic electroweak corrections togauge-boson pair production at the LHC , Nucl. Phys.
B706 (2005) 325–371,[ hep-ph/0409247 ].[23] A. Bierweiler, T. Kasprzik, and J. H. K¨uhn,
Vector-boson pair production at the LHC to O ( α ) accuracy , JHEP (2013) 071, [ arXiv:1305.5402 ].[24] J. Baglio, L. D. Ninh, and M. M. Weber, Massive gauge boson pair production at theLHC: a next-to-leading order story , Phys. Rev.
D88 (2013) 113005, [ arXiv:1307.4331 ].[25] B. Biedermann, A. Denner, S. Dittmaier, L. Hofer, and B. J¨ager,
Electroweak correctionsto pp → µ + µ − e + e − + X at the LHC: a Higgs background study , Phys. Rev. Lett. (2016) 161803, [ arXiv:1601.07787 ].[26] B. Biedermann, A. Denner, S. Dittmaier, L. Hofer, and B. J¨ager,
Next-to-leading-orderelectroweak corrections to the production of four charged leptons at the LHC , JHEP (2017) 033, [ arXiv:1611.05338 ].[27] B. Biedermann, M. Billoni, A. Denner, S. Dittmaier, L. Hofer, B. J¨ager, and L. Salfelder, Next-to-leading-order electroweak corrections to pp → W + W − → , JHEP (2016) 065, [ arXiv:1605.03419 ].[28] S. Kallweit, J. M. Lindert, S. Pozzorini, and M. Sch¨onherr, NLO QCD+EW predictionsfor ℓ ν diboson signatures at the LHC , arXiv:1705.00598 .[29] A. Denner, S. Dittmaier, M. Roth, and D. Wackeroth, Predictions for all processes e + e − → fermions + γ , Nucl.Phys.
B560 (1999) 33–65, [ hep-ph/9904472 ].[30] A. Denner, S. Dittmaier, M. Roth, and L. Wieders,
Electroweak corrections tocharged-current e + e − → fermion processes: Technical details and further results , Nucl.Phys.
B724 (2005) 247–294, [ hep-ph/0505042 ].1831] A. Denner and S. Dittmaier,
The complex-mass scheme for perturbative calculations withunstable particles , Nucl.Phys.Proc.Suppl. (2006) 22–26, [ hep-ph/0605312 ].[32] S. Dittmaier,
A general approach to photon radiation off fermions , Nucl.Phys.
B565 (2000) 69–122, [ hep-ph/9904440 ].[33] S. Dittmaier, A. Kabelschacht, and T. Kasprzik,
Polarized QED splittings of massivefermions and dipole subtraction for non-collinear-safe observables , Nucl.Phys.
B800 (2008) 146–189, [ arXiv:0802.1405 ].[34] S. Catani and M. Seymour,
A general algorithm for calculating jet cross-sections in NLOQCD , Nucl.Phys.
B485 (1997) 291–419, [ hep-ph/9605323 ].[35] B. Biedermann, A. Denner, and M. Pellen,
Large electroweak corrections to vector-bosonscattering at the Large Hadron Collider , Phys. Rev. Lett. (2017) 261801,[ arXiv:1611.02951 ].[36] B. Biedermann, A. Denner, and M. Pellen,
Complete NLO corrections to W + W + scattering and its irreducible background at the LHC , arXiv:1708.00268 .[37] S. Actis, A. Denner, L. Hofer, J.-N. Lang, A. Scharf, and S. Uccirati, RECOLA:REcursive Computation of One-Loop Amplitudes , Comput. Phys. Commun. (2017)140–173, [ arXiv:1605.01090 ].[38] A. Denner, S. Dittmaier, and L. Hofer,
Collier: a fortran-based Complex One-LoopLIbrary in Extended Regularizations , Comput. Phys. Commun. (2017) 220–238,[ arXiv:1604.06792 ].[39] G. ’t Hooft and M. J. G. Veltman,
Scalar one-loop integrals , Nucl. Phys.
B153 (1979)365–401.[40] W. Beenakker and A. Denner,
Infrared Divergent Scalar Box Integrals With Applicationsin the Electroweak Standard Model , Nucl. Phys.
B338 (1990) 349–370.[41] S. Dittmaier,
Separation of soft and collinear singularities from one-loop N-pointintegrals , Nucl. Phys.
B675 (2003) 447–466, [ hep-ph/0308246 ].[42] A. Denner and S. Dittmaier,
Scalar one-loop 4-point integrals , Nucl.Phys.
B844 (2011)199–242, [ arXiv:1005.2076 ].[43] G. Passarino and M. J. G. Veltman,
One-loop corrections for e + e − annihilation into µ + µ − in the Weinberg model , Nucl. Phys.
B160 (1979) 151–207.[44] A. Denner and S. Dittmaier,
Reduction of one-loop tensor five-point integrals , Nucl.Phys.
B658 (2003) 175–202, [ hep-ph/0212259 ].[45] A. Denner and S. Dittmaier,
Reduction schemes for one-loop tensor integrals , Nucl.Phys.
B734 (2006) 62–115, [ hep-ph/0509141 ].[46] E. Accomando, A. Denner, and C. Meier,
Electroweak corrections to
W γ and Zγ production at the LHC , Eur. Phys. J.
C47 (2006) 125–146, [ hep-ph/0509234 ].[47] J. K¨ublbeck, M. B¨ohm, and A. Denner,
Feyn Arts: Computer algebraic generation ofFeynman graphs and amplitudes , Comput. Phys. Commun. (1990) 165–180.[48] T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3 , Comput. Phys.Commun. (2001) 418–431, [ hep-ph/0012260 ].1949] T. Hahn and M. Perez-Victoria,
Automatized one loop calculations in four-dimensionsand D-dimensions , Comput. Phys. Commun. (1999) 153–165, [ hep-ph/9807565 ].[50] F. A. Berends, R. Pittau, and R. Kleiss,
All electroweak four fermion processes inelectron-positron collisions , Nucl. Phys.
B424 (1994) 308–342, [ hep-ph/9404313 ].[51] S. Dittmaier and M. Roth,
LUSIFER: A LUcid approach to six FERmion production , Nucl. Phys.
B642 (2002) 307–343, [ hep-ph/0206070 ].[52]
Particle Data Group
Collaboration, C. Patrignani et al.,
Review of Particle Physics , Chin. Phys.
C40 (2016) 100001.[53] D. Yu. Bardin, A. Leike, T. Riemann, and M. Sachwitz,
Energy-dependent width effects in e + e − annihilation near the Z-boson pole , Phys. Lett.
B206 (1988) 539–542.[54] A. Manohar, P. Nason, G. P. Salam, and G. Zanderighi,
How bright is the proton? Aprecise determination of the photon parton distribution function , Phys. Rev. Lett. (2016) 242002, [ arXiv:1607.04266 ].[55] J. Butterworth et al.,
PDF4LHC recommendations for LHC Run II , J. Phys.
G43 (2016)023001, [ arXiv:1510.03865arXiv:1510.03865