Vector-boson pair production at the LHC to O([alpha]^sup 3^) accuracy

Abstract

Abstract

Building on earlier work on electroweak corrections to W-pair production, the first calculation of the full electroweak one-loop corrections to on-shell ZZ, W^sup ±^Z and γγ production at hadron colliders is presented, explicitly taking into account the full vector-boson mass dependence. As a consequence, our results are valid in the whole energy range probed by LHC experiments. Until now, the electroweak corrections have only been known in dedicated high-energy approximations limited to a specific kinematic regime, in particular requiring high boson transverse momenta. Therefore, our results comprise an important and so far missing ingredient to improve on the theory predictions for these fundamental Standard-Model benchmark processes also at intermediate energies and small scattering angles, where actually the bulk of events is located. In case of Z-pair production we have also included the leptonic decays and the associated weak corrections in our analysis. For this particular channel, corrections of about -4% are observed even close to the production threshold. For hard scattering processes with momentum transfers of several hundred GeV one finds large negative corrections which may amount to several tens of percent and lead to significant distortions of transverse-momentum and rapidity distributions.

Published for SISSA by Springer

Received: June 4, 2013 Accepted: November 25, 2013 Published: December 16, 2013

Anastasiya Bierweiler, Tobias Kasprzik and Johann H. KhnKarlsruhe Institute of Technology (KIT), Institut fr Theoretische Teilchenphysik, D-76128 Karlsruhe, Germany

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: Building on earlier work on electroweak corrections to W-pair production, the rst calculation of the full electroweak one-loop corrections to on-shell ZZ, W[notdef]Z and

production at hadron colliders is presented, explicitly taking into account the full vector-boson mass dependence. As a consequence, our results are valid in the whole energy range probed by LHC experiments. Until now, the electroweak corrections have only been known in dedicated high-energy approximations limited to a specic kinematic regime, in particular requiring high boson transverse momenta. Therefore, our results comprise an important and so far missing ingredient to improve on the theory predictions for these fundamental Standard-Model benchmark processes also at intermediate energies and small scattering angles, where actually the bulk of events is located. In case of Z-pair production we have also included the leptonic decays and the associated weak corrections in our analysis. For this particular channel, corrections of about 4% are observed even close to the production

threshold. For hard scattering processes with momentum transfers of several hundred GeV one nds large negative corrections which may amount to several tens of percent and lead to signicant distortions of transverse-momentum and rapidity distributions.

Keywords: NLO Computations, Hadronic Colliders

ArXiv ePrint: 1305.5402

c

Vector-boson pair production at the LHC to O( 3) accuracy

JHEP12(2013)071

[circlecopyrt] SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP12(2013)071

Web End =10.1007/JHEP12(2013)071

Contents

1 Introduction 1

2 Details of the calculation 32.1 Contributions at leading order 32.2 Electroweak radiative corrections 42.3 Radiation of massive gauge bosons 5

3 Numerical results 53.1 Input and setup 63.2 Leading-order cross sections 73.3 Electroweak corrections 133.3.1 Integrated cross sections 133.3.2 Partially integrated cross sections 133.3.3 Di erential distributions 153.3.4 Comparison with existing results 203.4 Z-pair production: polarization and decays 223.4.1 Polarization e ects 223.4.2 Leptonic decays 243.5 Real-radiation contributions 25

4 Conclusions 26

1 Introduction

A profound understanding of vector-boson pair production processes at the LHC will provide deeper insight into the physics of weak interaction at the high-energy frontier, possibly even allowing for the discovery of physics beyond the Standard Model through the presence of anomalous couplings or the presence of resonances in the gauge-boson pair mode. Consequently, great e ort has been made during the last years to push the theory predictions for this process class to a new level, where, besides the dominating QCD corrections, also electroweak (EW) e ects have been studied extensively.

Next-to-leading (NLO) QCD corrections have been obtained long time ago and are implemented in Monte Carlo programs [19], and rst steps toward NNLO have been made in refs. [1012]. Gluon fusion as a source of loop-induced WW, ZZ or production has been studied extensively in refs. [1316], as of late also taking into account Higgs interference e ects [17, 18].

The importance of EW corrections at the LHC has been taken up signicantly later. During the last decade emphasis has been put on the large corrections in the TeV region

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which are a consequence of the Sudakov enhancement discussed in refs. [1931]. Considering the dominance of these large logarithmic corrections and the limited precision of LHC experiments (when compared to LEP analyses), it seemed su cient to limit the discussion for gauge-boson pair production to leading and partially subleading terms, rst in one-loop approximation [3234], including leptonic decays and related o -shell e ects, subsequently in two-loop [35] approximation for stable W bosons.

For WW and ZZ production in electron-positron annihilation at LEP a number of increasingly sophisticated NLO calculations [3645] have been performed, starting from the case of on-shell production to more rened situations including gauge-boson decays and the full set of resonant and non-resonant amplitudes for annihilation into the four-fermion nal state.

To arrive at predictions which satisfy the needs of increasingly precise LHC experiments and which are also valid for small invariant masses of the gauge-boson pair, the complete NLO predictions for W-pair production (without decay) have been obtained in ref. [46].

Extending these investigations, we present1 corresponding results for on-shell W[notdef]Z, ZZ and pair production in the SM. We will restrict ourselves to pair production through quark-antiquark annihilation. Photon-photon collisions do not contribute to WZ production (in contrast to the case of W pairs), are of higher order for Z-pair and -pair production and will not be discussed further. Also gluon fusion is, evidently, irrelevant for WZ production. For the case of W-pair production we have demonstrated that gluon fusion amounts to order of 5% relative to the quark-antiquark annihilation with decreasing importance for increasing transverse momenta. Gluon fusion, furthermore, does not lead to a strong modication of the angular and rapidity distributions of the W bosons. A qualitatively similar behaviour is expected for ZZ and production through gluon fusion, which therefore will not be discussed further. Also QCD corrections for WZ and ZZ/ production are expected to be similar to those for W pairs discussed in ref. [46] and will not be analyzed in the present paper, which, instead, will be entirely devoted to genuine EW corrections.

Earlier papers have emphasized the drastic inuence of the EW corrections on cross sections and distributions in the high-pT region. However, only pair-production channels with at least one massive gauge boson were considered [3234] while a phenomenological analysis of the channel is still missing. To close this gap, the rst computation of the full EW corrections to photon-pair hadroproduction is presented in this work. Moreover, the above papers neglect terms of order M2V /. Also this issue will be addressed in the present paper. All the processes under consideration are a ected by large corrections, amounting to several tens, sometimes up to fty percent. (This fact has even triggered studies of next-to-next-to-leading logarithmic corrections for W-pair production at the LHC [35], which, however, where not extended to Z-pair or WZ production.) Nevertheless, signicant di erences are observed between the di erent nal states, as far as the size of the corrections is concerned. We, furthermore, investigate rapidity distributions, both for small and for large invariant masses of the diboson system and observe signicant distortions that could be misinterpreted as anomalous triple-boson couplings. The full mass dependence, namely

1Preliminary results of this investigation have been presented in ref. [47].

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terms proportional to powers of M2V /, is consistently accounted for to obtain results valid in the whole energy range probed by LHC experiments, in particular for V -pairs produced near threshold2 or at low transverse momenta. Therefore, our results are complementary to those presented in refs. [32, 33], where only logarithmic corrections were considered, but the leptonic decays of the vector bosons and related o -shell e ects were included in a double-pole approximation. Comparing both approaches, we try to estimate the remaining theoretical uncertainties related to EW corrections in this important process class.

To investigate this aspect in more detail, also the combined production and decay process for Z pairs is calculated, including NLO corrections.3 To simplify the discussion, only the four-lepton state e+e+ is considered. Excellent agreement with the results of

Accomando et al. is observed in the region where the high-energy approximation employed in ref. [33] is valid. We, furthermore, investigate the contributions from the di erent helicity congurations including NLO corrections and observe that the transverse polarization not only dominates completely at large pT, it is also the most prominent conguration as far as the total cross section is concerned.

In addition, we also study the impact of real gauge-boson radiation leading to three-boson nal states. In principle, these congurations might compensate some of the negative contributions from virtual corrections, in practice, however, real radiation is signicantly smaller than virtual contributions.

2 Details of the calculation

2.1 Contributions at leading order

At the LHC, WW, W[notdef]Z, ZZ and production is, at lowest order O( 2), induced by the

partonic processes

qq ! WW+ (q = u, d, s, c, b) , (2.1a) uidj ! W+Z ,idj ! WZ (i = 1, 2; j = 1, 2, 3) , (2.1b)

qq ! ZZ , (2.1c)

qq ! . (2.1d) where the corresponding LO partonic cross sections are evaluated according to

qq[prime]!V1V2

LO =

[integraldisplay]

0 , the two-particle phase-space measure d (V1V2) and the averaging factor Nqq[prime] = 36. It is understood that for ZZ and production the cross sections receive an additional symmetry factor of 1/2. Note that one nds

[integraltext]

dV (dpp!V V /dV ) = 2pp!V V for = pT or = y for these particular channels.

2Although EW corrections are expected to be small at low energies, for Z-pair production a shift of about 4% is observed even close to threshold, whereas the corresponding corrections are below 1 percent

in the remaining cases.

3The corresponding analysis for W-pair and WZ production is more involved since photon emission from the initial state and from the W cannot be treated separately. See section 3.4.2.

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1

2Nqq[prime]

d (V1V2)

Xcol

Xspin

Xpol[notdef]Mqq[prime]!V1V20[notdef]2 , (2.2)

with the tree-level helicity amplitudes Mqq

[prime]!V1V2

2.2 Electroweak radiative corrections

To allow for consistent predictions with full O( 3) accuracy, virtual EW corrections as

well as real corrections due to photon radiation have to be considered. The evaluation of the radiative corrections is based on the well-established FeynArts/FormCalc/LoopTools setup [4853], and all processes have been independently cross-checked by a setup based on QGraf [54] and Form [55].

To considerably reduce the computational e ort, light quark masses are neglected whenever possible. However, soft and collinear singularities occurring in intermediate steps of the calculation are regularized by small quark masses mq and an innitesimal photon mass , generating unphysical ln mq and ln terms. To allow for a numerically stable evaluation of those infrared(IR)-divergent parts of the cross sections related to real radiation, the phase-space slicing method is adopted as detailed in ref. [56] for mass regularization. Finally, adding real and virtual contributions, the regulator-mass dependence drops out in any properly dened physical result. However, in complete analogy to QCD, residual collinear singularities attributed to initial-state (IS) radiation survive and have to be absorbed in renormalized PDFs in a proper factorization procedure adding the collinear counterterm dened by eq. (3.16) of ref. [57]. In the present computation, we apply the MS factorization scheme for the QED factorization.

The input parameters to be specied in section 3.1 are renormalized in a modied on-shell scheme [58], where the Fermi constant G is used instead of (0) to e ectively account for universal corrections induced by the running of () to the weak scale [56]. However, for the computation of the channel, (0) is used as input for the EW coupling, since the corresponding radiative corrections do not receive universal contributions related to the running of the coupling constant. According to the previous considerations, the partonic cross section at O( 3) accuracy may be written as

qq

[prime]!V1V2( )

NLO =

qq[prime]!V1V2

LO +

qq[prime]!V1V2

LO +

qq[prime]!V1V2

virt , (2.3)

where the di erent NLO contributions are given by

qq[prime]!V1V2

LO =

1

2Nqq[prime]

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[integraldisplay]

d (V1V2 )

Xcol

Xspin

Xpol[notdef]Mqq[prime]!V1V2 0[notdef]2 , (2.4)

qq[prime]!V1V2

virt =

1

2Nqq[prime]

[integraldisplay]

d (V1V2)

Xcol

n(Mqq[prime]!V1V20 ) Mqq[prime]!V1V21[bracerightBig], (2.5)

with the properly renormalized one-loop amplitudes Mqq

Xspin

Xpol2 Re

1 . The hadronic results at the LHC are then obtained by convoluting the partonic cross sections with appropriately chosen PDFs and summing incoherently over all contributing channels,

pp!V1V2( )NLO =

[prime]!V1V2

[integraldisplay]

1

0 d [integraldisplay]

1 dxb xb

Xq,q[prime]fq/p(xa, 2F)fq[prime]/p(xb, 2F) qq[prime]!V1V2( )

NLO (s, 2F) , (2.6)

where the hadronic CM energy s is related to via = s, with = xaxb. The kinematic production threshold of a vector-boson pair in the nal state is reected by the choice of

4

the lower integration boundary 0 = (MV1 + MV2)2/s, corresponding to a minimal partonic CM energy of0 = 0s. The factorization scale F enters the partonic cross section through the redenition of the PDFs in the QED factorization procedure described above.

2.3 Radiation of massive gauge bosons

As in ref. [46] we study the e ect of massive boson radiation on the massive vector-boson pair production cross section, where the additional massive boson is treated fully exclusively in the event selection to allow for a robust estimate of the corresponding phenomenological e ects. This is motivated by the possibility that the logarithmically enhanced positive contributions from the radiation of an additional soft or collinear massive gauge boson might compensate the negative virtual corrections, as has been argued recently for Z+jet production [59].

To be specic, we take into account the following partonic channels, where the corresponding partonic cross sections have to be computed according to

qq[prime]!V1V2V3

LO =

1

2Nqq[prime]

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[integraldisplay]

Xpol[notdef]Mqq[prime]!V1V2V30[notdef]2 . (2.7)

In case of ZZ pair production, the contributions from the processes

qq ! ZZZ ,

qq[prime] ! ZZW[notdef] (2.8) have to be considered, where the two Zs with highest pT have to fulll the LO cuts to be specied in section 3.1, while the third massive particle is treated inclusively. To assess the real-radiation e ects in W[notdef]Z production, we compute the cross sections for

qq ! W[notdef]ZW ,qq[prime] ! W[notdef]ZZ . (2.9)

In the rst case the W is treated inclusively, in the second case the Z with lowest pT. In section 3.5, numerical results will be presented for massive boson radiation normalized to the LO qq-induced pair-production channels,

V1V2V3 = pp!V1V2V3LO/pp!V1V2LO 1 , (2.10)

as well as for hard photon radiation (i.e. pT, > 15 GeV, [notdef]y [notdef] < 2.5),

V1V2 = pp!V1V2 LO/pp!V1V2LO 1 . (2.11)

d (V1V2V3)

Xcol

Xspin

3 Numerical results

In this section, for the rst time the full EW corrections to Z- and -pair production as well as W[notdef]Z production are presented. Most of the discussion is focussed on LHC8 and LHC14.

For completeness, we also show cross sections for these processes at the Tevatron with a

5

CM energy of ps = 1.96 TeV. The corresponding results for W-pair production originally presented in ref. [46] are given for comparison. In the following, the relative e ects EW of

the EW corrections are dened as

EW = pp!V1V2( )NLOpp!V1V2LO

3.1 Input and setup

For the computation presented here the same setup as specied in ref. [46] is applied. To be specic, we use the following SM input parameters for the numerical analysis,

G = 1.16637 [notdef] 105 GeV2,

MW = 80.398 GeV, MZ = 91.1876 GeV,MH = 125 GeV, Mt = 173.4 GeV . (3.2)

For the evaluation of all tree-level contributions we assume a block-diagonal CKM matrix with

|Vud[notdef] = [notdef]Vcs[notdef] = 0.974 , [notdef]Vus[notdef] = [notdef]Vcd[notdef] =

p1 [notdef]Vud[notdef]2 . (3.3)

Ignoring, furthermore, quark masses within the rst two families, both tree-level and one-loop predictions for ZZ and are equivalent to those without quark mixing. As a consequence of the smallness of the bottom-quark PDF the tree-level contribution from bb annihilation to ZZ or is small to start with. In addition, the non-diagonal CKM elements involving b quarks are small, and the ansatz (3.3) is well justied. As a consequence, bb ! ZZ or can safely be handled within the third family.4 The situation is di erent

for the WZ channel. In this case, the interplay between CKM angles and PDFs leads to a shift of the tree-level prediction of about one percent. For the radiative corrections the CKM matrix can, therefore, still be set to unity.

In the on-shell scheme applied in our computation, the weak mixing angle cos2 w =

M2W/M2Z is a derived quantity. For the computation of the processes (2.1) and the corresponding EW radiative corrections, we use the MSTW2008LO PDF set [60] in the LHAPDF setup [61]. In order to consistently include O( ) corrections, in particular real radiation

with the resulting collinear singularities, PDFs in principle should take these QED e ects into account. Such a PDF analysis has been performed in ref. [62], and the O( ) e ects

are known to be small, as far as their e ect on the quark distribution is concerned [63]. In addition, the currently available PDFs incorporating O( ) corrections [62] include QCD

e ects at NLO, whereas our EW analysis is LO with respect to perturbative QCD only. For these reasons, the MSTW2008LO set is used as our default choice for the quark-induced processes. Our default choice for the factorization scale is the average of the vector-boson transverse masses

F = mT = 12[parenleftBig][radicalBig]

. (3.4)

4We point out that a non-vanishing top-quark mass is consistently included in the computation of the one-loop contributions discussed in this paper.

6

1 . (3.1)

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M2V1 + p2T,V1 +

qM2V2 + p2T,V2

[parenrightBig]

WW+ ZZ W+Z WZ

default cuts LO (pb) EW(%) LO (pb) EW(%) LO (pb) EW(%) LO (pb) EW(%) LO (pb) EW(%)

LHC8 23.99 0.7 3.810 4.4 5.256 1.4 3.343 1.2 41.38 0.2 LHC14 42.39 0.9 7.066 4.5 8.677 1.5 6.463 1.3 62.69 0.2

Tevatron 7.054 0.5 0.8624 4.9 1.023 1.1 1.023 1.1 19.21 0.2

Table 1. Integrated leading-order cross sections and relative EW corrections for the LHC and the Tevatron evaluated with the default setup dened in section 3.1.

WW+ ZZ W+Z WZno cuts LO (pb) EW(%) LO (pb) EW(%) LO (pb) EW(%) LO (pb) EW(%) LHC8 35.51 0.4 5.064 4.1 8.273 1.4 4.643 1.3

LHC14 75.02 0.4 11.02 4.2 17.11 1.4 10.65 1.3

Tevatron 7.916 0.2 0.9466 4.7 1.123 1.1 1.123 1.1

Table 2. Total leading-order cross sections and relative EW corrections for the LHC and the Tevatron evaluated without any phase-space cuts.

A similar scale choice was taken in ref. [33] for the computation of the EW corrections to four-lepton production at the LHC. Yet we point out that the relative EW corrections, which are the main subject of this paper, only depend on the choice of F at the subpercent level even for large transverse momenta.

In our default setup, we require a minimum transverse momentum and a maximum rapidity for the nal-state vector bosons,

pT,Vi > 15 GeV , [notdef]yVi[notdef] < 2.5 , i = 1, 2 , (3.5) to dene a V -boson pair production event. Thereby we exclude events where the bosons are emitted collinear to the initial-state partons, which for the channel would inevitably lead to collinear singularities in the LO cross section. However, for nal states with massive gauge bosons selected numerical results will also be presented without applying the above cuts.

For the denition of a two-photon nal state we require at least two visible photons fullling the acceptance cuts (3.5). If additional photon bremsstrahlung is present, any further phase-space cuts will only be applied to the two visible photons with highest pT,

while the third is treated inclusively to ensure IR safety.

3.2 Leading-order cross sections

Before discussing the EW corrections to V -pair production at hadron colliders in detail let us rst recall the most striking features at leading order.

LO results for the processes pp(p) ! V1V2 + X (see eqs. (2.1)) are given in table 1

for integrated cross sections evaluated with our default setup, in table 2 for the total cross

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LO(fb)

EW(%)

100000 10000

1000 100

10 1 0.1 0.01

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ps = 14 TeV

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WZ

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pcutT(GeV)

pcutT(GeV)

Figure 1. Integrated LO cross sections (left) and relative EW corrections (right) evaluated with our default setup for di erent cuts on the transverse momenta of the nal-state bosons.

sections without any cuts. The relative EW corrections are also displayed and will be discussed in detail in section 3.3. The rates for ZZ and WZ production are in the same ballpark, both at the LHC and the Tevatron. For the LHC, being a proton collider, the

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Figure 2. Integrated LO cross sections (left) and relative EW corrections (right) evaluated with our default setup for di erent cuts on the invariant mass of the nal-state bosons. The respective results are presented for LHC14 (top), LHC8 (center) and the Tevatron (bottom).

cross section for W+Z production is larger than for WZ production, while at the Tevatron they trivially coincide. The WW production cross sections, however, are roughly a factor of 5 (7) larger at the LHC (Tevatron) than those for ZZ production. For our default cuts,

9

pp ! V1V2(+ ) + X at ps = 14 TeV

default cuts ZZ W+Z WZ

pcutT (GeV) LO (pb) EW(%) LO (pb) EW(%) LO (pb) EW(%) LO (pb) EW(%) 50 3.660 6.3 4.498 1.9 3.228 1.7 2.979 0.6

100 1.087 10.4 1.296 3.7 0.849 3.3 0.432 1.6

250 7.495 102 23.0 90.56 102 12.9 4.583 102 12.3 2.451 102 7.0

500 49.89 104 38.9 63.64 104 24.9 24.79 104 24.4 16.95 104 13.0

750 72.16 105 50.6 92.43 105 33.3 31.25 105 33.0 25.66 105 17.3

1000 14.60 105 60.1 18.35 105 39.8 57.32 106 39.6 54.91 106 20.6

1250 35.28 106 68.4 42.92 106 45.1 12.86 106 45.1 14.15 106 23.5

1500 94.73 107 75.7 10.99 106 49.6 32.48 107 49.6 40.71 107 25.9

Table 3. Integrated leading-order cross sections and relative EW corrections at LHC14 for di erent cuts on the minimal boson transverse momenta.

pp ! V1V2(+ ) + X at ps = 14 TeV

default cuts ZZ W+Z WZ

McutVV (GeV) LO (pb) EW(%) LO (pb) EW(%) LO (pb) EW(%) LO (pb) EW(%) 200 6.094 5.0 7.820 1.6 5.790 1.5 0.966 0.4

300 1.859 7.4 3.252 2.6 2.273 2.4 0.304 1.8

500 0.352 11.2 0.716 5.1 0.440 4.7 64.54 103 3.8 1000 23.10 103 22.0 52.70 103 13.1 24.47 103 12.4 58.63 104 8.1

1500 36.65 104 31.2 94.50 104 20.1 36.67 104 19.6 11.09 104 11.0

2000 84.75 105 37.8 23.73 104 25.6 82.87 105 25.0 28.25 105 13.4

2500 23.17 105 43.4 68.36 105 30.2 22.52 105 29.6 84.20 106 15.4

3000 69.60 106 48.2 16.72 105 35.0 67.72 106 33.5 27.49 106 17.0

Table 4. Integrated leading-order cross sections and relative EW corrections at LHC14 for di erent cuts on the minimal boson invariant mass.

pp ! V1V2(+ ) + X at ps = 8 TeV

default cuts ZZ W+Z WZ

pcutT (GeV) LO (pb) EW(%) LO (pb) EW(%) LO (pb) EW(%) LO (pb) EW(%) 50 1.913 6.2 2.651 1.8 1.610 1.5 1.843 0.6

100 0.523 10.2 0.706 3.4 0.384 3.0 0.242 1.3

150 0.164 14.5 0.219 6.1 0.106 5.7 64.40 103 3.5

250 26.08 103 22.5 34.70 103 11.9 14.10 103 11.5 95.15 104 6.7

350 60.94 104 29.3 80.73 104 17.0 29.08 104 16.6 22.17 104 9.3

500 10.11 104 38.1 13.11 104 23.2 42.29 105 22.9 38.01 105 12.3

600 35.40 105 43.2 44.99 105 26.6 13.87 105 26.5 13.77 105 14.2

750 83.51 106 50.1 10.18 105 31.1 30.31 106 31.1 34.62 106 16.5

Table 5. Integrated leading-order cross sections and relative EW corrections at LHC8 for di erent cuts on the minimal boson transverse momenta.

the cross section for production is even larger than the one for W-pair production, as a consequence of the singular behaviour for small pT, .

Requiring tighter cuts on the boson transverse momenta (tables 3 and 5 and gure 1, left) and on invariant masses (tables 4 and 6 and gure 2, left), one observes a rapid

10

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pp ! V1V2(+ ) + X at ps = 8 TeV

default cuts ZZ W+Z WZ

McutVV (GeV) LO (pb) EW(%) LO (pb) EW(%) LO (pb) EW(%) LO (pb) EW(%) 200 3.271 4.9 4.728 1.5 2.980 1.4 0.579 0.3

300 0.950 7.1 1.921 2.5 1.117 2.2 0.169 1.6

400 0.360 8.7 0.821 3.5 0.439 3.2 66.45 103 2.6 500 0.159 10.2 0.387 4.5 0.191 4.2 30.52 103 3.6 700 39.70 103 13.5 0.103 6.9 44.69 103 6.5 84.71 104 5.2

800 21.27 103 15.4 56.47 103 8.3 23.07 103 7.8 48.66 104 5.8

1000 67.00 104 19.5 18.35 104 11.5 68.07 104 11.0 17.91 104 7.1

1200 23.88 104 23.7 68.65 104 14.4 23.82 104 14.0 72.84 105 8.3

1500 61.33 105 28.5 18.45 104 18.2 60.60 105 17.6 21.25 105 9.8

Table 6. Integrated leading-order cross sections and relative EW corrections at LHC8 for di erent cuts on the minimal boson invariant mass.

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decrease of the cross sections over several orders of magnitude, corresponding to the fact that V -pair production is dominated by both small scattering angles and low-energetic

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events, as has already been pointed out in ref. [46] for the WW case. Consequently, a cut on the transverse momentum leads to a much stronger reduction than a comparable cut on the invariant mass, McutVV = 2pcutT, as is obvious from tables 3 and 4. In addition, also the relative rates of the di erent channels change dramatically. While comparable at low pT,

the cross section for WZ production is 3 times smaller than the one for W+Z production when going to large values of pT and MV V . For large pT, ZZ production behaves similar to W+Z production, at large invariant masses the event rates are comparable to WZ production. In contrast, the cross section, which dominates at low transverse momenta, drops rapidly for high invariant masses, at high transverse momenta it is comparable to WZ production.

For high invariant masses the forward-backward peaking of the cross sections is even more pronounced than for low energies as can be deduced from the left plots of gures 3 and 4, where the di erential cross sections are presented as a function of the boson rapidity gap yV V = yV1 yV2, evaluated with our default cuts (top) and with a minimal invariant

mass of 1000 GeV (500 GeV) at LHC14 (LHC8) (bottom). Note that yV V for xed MV V corresponds to the scattering angle in the diboson rest frame. At low energies the

12

distributions reach their maximum in the central region, i.e. at rather low values of [notdef] yV V [notdef], corresponding to central events, at high invariant masses they become maximal at [notdef] yV V [notdef] [similarequal]

3, corresponding to a drastic peaking in the forward and backward directions.

3.3 Electroweak corrections

This section will be devoted to a detailed numerical analysis of the EW corrections to vector-boson pair production at the LHC and Tevatron. The corresponding e ects on the total cross sections will be discussed in section 3.3.1, those for partially integrated cross sections in di erent kinematic regimes in section 3.3.2. In section 3.3.3, we discuss the EW corrections to di erential distributions of observables relevant at the LHC, namely trans-verse momenta, rapidities and invariant masses. In particular, we focus on the structure of the respective corrections at highest energies, where high- as well as low-pT production of vector-boson pairs is addressed separately.

3.3.1 Integrated cross sections

In a rst step we present the EW corrections for the total production cross sections for WW, W[notdef]Z, ZZ and in table 1, using the default cuts described before. The EW corrections amount to roughly one percent, with the ZZ channel being the only exception with its sizable negative correction of about 4%. As discussed below, this feature is present

already close to production threshold, applies to Tevatron, LHC8 and LHC14 and is also present in rapidity distributions, as long as we do not enforce large through selected cuts.

For reference purpose we also list the results for the total cross section without any cuts for all nal states apart of , where the corresponding cross sections would diverge (table 2). The size of the radiative corrections remains practically unchanged.

3.3.2 Partially integrated cross sections

In order to enhance hard scattering events and explore the TeV region it is useful to consider partially integrated cross sections, where additional cuts are introduced on the transverse momenta of the gauge bosons or on the invariant mass of the gauge-boson pair. Numerical results are listed in tables 3, 4, 5 and 6 and in gures 1 and 2.

As expected, one observes increasingly negative corrections with increasing pcutT or McutVV. The corrections are largest for ZZ production, reaching -50% at LHC14 for pcutT =

800 GeV, and smallest for production, where they remain below 20% for all kinematic congurations. As already stated in our paper on WW production, the EW corrections at the LHC are considerably more important than at the Tevatron.

For illustration we also present the results up to pcutT = 1 TeV which might be accessible in a high-luminosity run. In this case corrections exceeding 40% are observed in W-pair production and the question of two-loop contributions necessarily arises. For W-pair production the logarithmically enhanced (NNLL) corrections have been discussed in ref. [35], for the other nal states they are not yet available and we will not dwell further on this subject.

The particularly large negative EW corrections in case of Z-pair production may be understood as a consequence of two completely independent physical e ects. On the one

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hand, the EW corrections to the ZZ cross section exhibit a constant o set of 4% irre

spective of cuts when compared to WW, WZ and production. This feature is most

14

obvious in gure 5, where the McutVV dependence of the cross sections is displayed without additional rapidity or transverse-momentum restrictions, i.e. without the default cuts described in section 3.1. Sudakov-enhanced logarithms cannot be made responsible, since a

4% correction is already present in the threshold region, and it is hardly dependent on the actual value of the collider energy. We additionally observe that real radiation of hard photons, which is included in our results together with radiation of soft and virtual photons, is particularly small for ZZ production (see gure 12, bottom). Taking both e ects into account, the EW corrections at high pT which can be attributed to large logarithms stemming from the weak corrections are similar for WW and ZZ production, while they are signicantly smaller in case of W[notdef]Z production.

Nevertheless, also corrections to WZ production are sizable, reaching 35% at LHC14

for pcutT = 800 GeV. Furthermore they are quite similar to those for W+Z production, since at parton level the corresponding unpolarized cross sections coincide. Small deviations solely arise from the di erent parton-luminosities multiplying the real-radiation and virtual contributions, respectively, leading to slightly di erent relative corrections, since the di erential partonic corrections

EW(, t), which evidently are the same for both channels, enter the hadronic results with di erent weights.

Going back to gure 5, we nd that the relative corrections are small (below 10%) even for large invariant masses (without additional cuts on rapidities and transverse momenta), since vector-boson pair production is dominated by low scattering angles, i.e. small [notdef]t[notdef],

where no logarithmically enhanced EW corrections are expected. Surprisingly enough, even moderate cuts on the boson transverse momenta and rapidities lead to considerable relative corrections as shown in gure 2.

Another interesting nding is that the EW corrections in case of production are again substantially smaller than in the massive channels, reaching 20% for pcutT =1000 GeV, which is consistent with corresponding results from ref. [64], where logarithmic corrections to the processes ! f

f were computed in the high-pT approximation.5 As already stated above, up to now no phenomenological study at all exists of the EW e ects in photon-pair production at hadron colliders.

3.3.3 Di erential distributions

In this section di erential cross sections for various kinematic scenarios are investigated. Specically, in addition to transverse-momentum, invariant-mass and rapidity distributions generated in our default setup, we also present results with explicit cuts on the invariant mass of the vector bosons. This allows to investigate the EW corrections in the high-energy regime where new-physics signatures might have a sizable impact. As will be shown, signicant distortions of the angular distributions are observed which might easily be misinterpreted as a signal of anomalous couplings and hence new physics.

In gures 6, 7 and 8 we display the di erential distributions and corresponding EW corrections for the boson transverse-momentum, invariant-mass and rapidity distributions, respectively, evaluated in the default setup dened in section 3.1. As already stated for the partially integrated cross sections in the previous section, the LO distributions rapidly

5The full one-loop EW corrections to ! tt were presented even earlier in ref. [65].

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decrease with increasing values of pT and MV V , reecting that vector-boson pair production is in general dominated by events with low, as a consequence of the rapidly-falling PDFs

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and cross sections. At the same time, the EW corrections increase with pT and MV V , ranging between 15% for and 45% for ZZ if we consider pT values of 800 GeV at

LHC14 as example.

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The relative corrections for the rapidity distributions (gure 8, right) are small for WW, WZ and production and resemble the corrections for the integrated cross sec-

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tions presented in table 1. Again, for ZZ production a constant o set of 4% in the EW

corrections is evident, reecting a remarkably constant K-factor. This behaviour is expected, since both the total cross section and the rapidity distributions are dominated by small MV V .

Let us now consider vector-boson pair production at highest energies accessible at the LHC. This is achieved by restricting MV V to values above 500 GeV (gure 9) and 1 TeV (gure 10). Looking at the right panels of gure 9 and gure 10, we observe that the relative EW corrections to the boson rapidity distributions are sizable at low rapidities, corresponding to central events with high transverse momenta, i.e. the Sudakov region. Large rapidities, in contrast, correspond to small scattering angles (small [notdef]t[notdef] or [notdef][notdef]) and thus

small pT. EW corrections per se are logarithmically enhanced only in the Sudakov region. For identical cuts on the invariant mass MV V the corrections are very similar at LHC14 and LHC8, respectively, reaching 15% (30%) for MV V > 500 GeV (MV V > 1000 GeV).

The corresponding corrections for the distributions with respect to yV V = yV1 yV2 were presented in gures 3 and 4 for MV V > 500 GeV (MV V > 1000 GeV) for LHC8 (LHC14).

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Large corrections were observed for small rapidity gaps, amounting to nearly 40% (25%)

for central ZZ production at LHC14 (LHC8).

This behaviour can be pushed even further by going to MV V > 1.5 TeV, a region potentially accessible at a high luminosity LHC. Results for a minimal invariant mass of MV V > 750 GeV (MV V > 1500 GeV) at LHC8 (LHC14) are shown in gure 11, and for

Z-pair production at LHC14 in this kinematic region the corresponding corrections reach

50%. In this regime, also weak two-loop e ects might be required to reliably predict the cross sections.

3.3.4 Comparison with existing results

To additionally validate our numerical analysis and to assess the remaining theoretical uncertainties, we compare our predictions for vector-boson pair production at hadron colliders with older results obtained in a high-energy approximation [33]. Although a tuned comparison in general is not possible since the approach taken in the present work does not allow to apply event-selection cuts to the leptonic decay products of the vector bosons,

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as is done in ref. [33], (See, however, the discussion in section 3.4.), qualitative statements can be made already now.

Comparing our results for Z-pair production ( ZZEW) to those given in table 3 of ref. [33] ( ZZ,ADKEW), we observe very good agreement in the whole energy range considered if we employ the additional constraint [notdef] yZZ[notdef] < 3 on the rapidity gap of the Z bosons to explicitly

enforce Sudakov kinematics (i.e., [notdef]t[notdef], [notdef][notdef] M2Z). For instance, we nd ZZEW = 28.5%, to

be compared with ZZ,ADKEW = 28.1% for MZZ > 1000 GeV. As expected, EW corrections to ZZ production in the Sudakov regime are exhaustively described by logarithmic weak corrections, and mass e ects do not play a signicant role. Surprisingly, also o -shell e ects and nal-state photon radiation, both included in ref. [33], as well as LHC acceptance cuts, do not seem to noticeably a ect the relative EW corrections after the Z reconstruction has been performed.

Turning to WZ production, the comparison is also straightforward. According to scenario (7.3) of ref. [33] we apply-in addition to our default cuts-a cut on the transverse

21

momentum of the Z boson and obtain W+ZEW = 22.4% for pT,Z > 500 GeV, which is in

reasonable agreement with WZ,ADKEW = 21.2% given in table 1 of ref. [33].

After submitting our paper, two preprints [66, 67] appeared which investigate gauge-boson pair production at the LHC. Ref. [66] considers all three channels WW, ZZ and WZ, includes the quark-photon-induced reaction, but does not consider gauge-boson decay and the four-fermion states. Ref. [67] is focussed on W-pair production, includes quark-photon-induced reactions and W decays, taking into account also o -shell e ects. Wherever the results of ref. [66] can be compared to our analysis they are in good agreement. A more detailed comparison of the results is performed in ref. [67] and again very good agreement is observed with our ndings.

3.4 Z-pair production: polarization and decays

Once su ciently large samples of gauge-boson pairs have been produced, it will be important to investigate the angular distributions of their decay products, which evidently carry the information about the Z (or W) polarization.

Up to the order considered in the present paper, real plus virtual photon radiation can be separated in a gauge-invariant manner, as far as ZZ production is concerned. It is quite remarkable that this purely electromagnetic subset of the corrections to ZZ production is tiny, in general below 1% in all cases discussed in this paper. In a rst step we thus evaluate the cross sections for the production of polarized Z pairs and the impact of purely weak corrections on these cross sections. In a second step, we consider combined production and decay for the mode pp ! Z(! e+e) Z(! +) + X, including weak corrections.

3.4.1 Polarization e ects

Let us start with polarized Z-pair production at LHC8 and LHC14. Longitudinal, right-and left-circular production are denoted by (L), (+) and (). The cross sections for the

production of one Z with polarization (i) and one with polarization (j) is represented by (ij). The unpolarized cross section is thus composed of the following combination

tot = (LL) + (++) + () + (+) + (L+) + (L) . (3.6) As a consequence of CP symmetry (L+) = (L) and (++) = (). The remaining

four independent combinations are listed in table 7. The results are presented for the default cuts, for events with pT,Z > 500 GeV and for pT,Z > 1000 GeV.

Let us rst discuss the Born cross section, which is the upper entry for each partial or summed cross section. Already for the default cuts it receives its major contribution (70%) from the (+) conguration, for larger transverse momenta the remaining congurations

die out quickly. This behaviour can be deduced directly from the equivalence theorem: neutral scalar pair production is strictly forbidden, which in the present case leads to the M2Z/ suppression of (LL). Also the (++) conguration is strictly forbidden for massless gauge bosons. The MZ/ps behaviour of (L[notdef]) is similar to the one for W pairs

e.g. discussed in ref. [68]. Qualitatively, this behaviour can indeed be read o from a comparison of the ratios (LL)/(+) and (L[notdef])/(+) for the pcutT values of 500 GeV

and 1000 GeV.

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LO/pb 7.067 0.402 0.734 0.100 4.997 weak/pb 0.338(0.292) 0.015(0.014) 0.029(0.025) 0.004(0.003) 0.257(0.223)

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LO/pb 102 [0.499 107 [0.921 104 [0.334 107 [0.230 102 [0.492

weak/pb 0.195(0.148)] 4.70(+5.577)] 0.087(0.067)] 0.426(0.185)] 0.192(0.147)] pT,Z > 1000 GeV

LO/pb 103 [0.146 109 [0.189 106 [0.306 1010 [0.475 103 [0.146

weak/pb 0.088(0.062)] 4.319(+30.04)] 0.126(0.090)] 2.953(+2.295)] 0.088(0.062)]

LHC8LO/pb 3.810 0.223 0.396 101 [0.559 2.676

weak/pb 0.179(0.155) 0.009(0.008) 0.016(0.014) 0.002(0.002)] 0.134(0.117) pT,Z > 500 GeV

LO/pb 102 [0.101 107 [0.202 105 [0.779 108 [0.504 103 [0.996

weak/pb 0.039(0.030)] 0.975(+0.748)] 0.204(0.157)] 0.895(0.425)] 0.383(0.293)] pT,Z > 1000 GeV

LO/pb 105 [0.919 1010 [0.121 107 [0.231 1011 [0.303 105 [0.915

weak/pb 0.557(0.387)] 2.599(+14.909)] 0.098(0.070)] 1.742(+1.043)] 0.555(0.387)]

Table 7. Polarized LO cross sections and corresponding weak corrections to ZZ production at the LHC for di erent cuts on the boson transverse momenta. The rst entry for the corrections represents the interference between Born and one-loop amplitude, the second entry (in brackets) includes the squared one-loop amplitude.

Also shown in table 7 as lower entry are the O( ) weak corrections weak. Again we observe the overall reduction by about 5%, arising mainly from small, which is fairly similar for the di erent polarizations, as far as low are concerned. For larger pT values a di erent pattern emerges. For the suppressed diagonal congurations (LL), (++), () the negative corrections increase with pT, and quickly exceed the Born contribution.

Hence, if one would try to analyze the di erent polarizations separately, one should include the squared 1-loop correction. The corresponding values for weak representing the interference between Born and one-loop term plus one-loop squared contribution are given in round brackets in the same line. However, since all these contributions are below one permille, they are irrelevant for all practical considerations. Note, in addition, that the residual uncertainties on the integrated cross sections due to missing higher-order weak corrections are at the level of 10% and 20% for pT,Z > 500 GeV and pT,Z > 1000 GeV, respectively.

In total a fairly simple pattern emerges: for small pT electroweak corrections are very similar for all polarizations and can be taken as one global factor, for large pT only one combination survives and corrections are again trivially represented by one factor. Electroweak corrections can therefore be represented for the bulk of events at each and t by a correction factor which does not modify the relative importance of the di erent polarizations.

23

JHEP12(2013)071

3.4.2 Leptonic decays

Let us, in a next step, evaluate the complete production and decay process for the e+e+

nal state at the LHC. For the acceptance cuts on muon and electron transverse momenta and rapidities we adopt the prescriptions of ref. [33], namely

pT,l > 15 GeV , [notdef]yl[notdef] < 3 . (3.7)

The intermediate Z bosons are reconstructed from the nal-state leptons requiring

|Mll MZ[notdef] < 20 GeV (3.8)

to suppress the admixture of virtual photons and to improve the validity of the approximations used to compute the weak corrections.

The cuts employed in table 8 are chosen to mimic on the one hand the experimental acceptance, and on the other hand, select events with increasing ps, corresponding to the invariant mass Minv(4l) of the four-lepton system. The rst four columns represent predictions for LHC14 and LHC8 using four variants of the Born approximation. In the rst column we give the full LO cross section, including all o -shell e ects and non-resonant contributions, in a naive xed-width implementation using non-vanishing constant vector-boson widths6

Z = 2.505044 GeV , W = 2.099360 GeV (3.9)

in propagators with time-like 4-momenta,

1p2 M2V !

1p2 M2V + (p2)iMV V

JHEP12(2013)071

. (3.10)

The second column shows the corresponding results evaluated in the complex-mass scheme (CMS) [69, 70] which involves complex couplings. In addition, we present results for the Born cross section in the double-pole approximation (DPA) as described in ref. [33] and in the narrow-width approximation (NWA), where resonant propagators are replaced according to

1

(p2 M2V )2 + M2V 2V !

MV V (p2 M2V ) , (3.11)

corresponding to the limit V /MV ! 0. Both approximations only include doubly-resonant

contributions. Note that both the and Z-induced amplitudes are included in the full LO predictions, while the -mediated diagrams are absent in the approximate results. The relative weak corrections, which are identical to a level of 1[permil]

between DPA and NWA,

are listed in column 5.

With the cuts employed until now a sizable fraction of events corresponds to large but small [notdef]t[notdef], and the Sudakov approximation is not applicable. Imposing, however, a cut

on the rapidity di erence between the reconstructed Z bosons, [notdef] yZZ[notdef] < 3, removes events

with small scattering angle and decreases the total rates by roughly 30% at high Minv(4l).

Moreover, it leads to enhanced EW corrections, as shown in table 9. We point out that the

6For the numerical evaluation we adopt the values from ref. [33] for Z and W.

24

pp ! (Z/ )(Z/ ) + X ! e+e+ + X

Mcutinv(4l)/GeV naiveLO/pb CMSLO/pb DPALO/pb NWALO/pb DPAweak/% LHC14200 0.835 [notdef] 102 0.835 [notdef] 102 0.815 [notdef] 102 0.875 [notdef] 102 5.4

300 0.239 [notdef] 102 0.239 [notdef] 102 0.233 [notdef] 102 0.249 [notdef] 102 8.1

400 0.987 [notdef] 103 0.987 [notdef] 103 0.966 [notdef] 103 1.035 [notdef] 103 10.2

500 0.484 [notdef] 103 0.484 [notdef] 103 0.473 [notdef] 103 0.508 [notdef] 103 12.4

600 0.261 [notdef] 103 0.261 [notdef] 103 0.256 [notdef] 103 0.275 [notdef] 103 14.5

700 0.151 [notdef] 103 0.151 [notdef] 103 0.148 [notdef] 103 0.159 [notdef] 103 16.6

800 0.920 [notdef] 104 0.920 [notdef] 104 0.901 [notdef] 104 0.971 [notdef] 104 18.8

900 0.584 [notdef] 104 0.584 [notdef] 104 0.572 [notdef] 104 0.617 [notdef] 104 20.8

1000 0.384 [notdef] 104 0.384 [notdef] 104 0.376 [notdef] 104 0.406 [notdef] 104 22.9

LHC8

200 0.445 [notdef] 102 0.445 [notdef] 102 0.435 [notdef] 102 0.466 [notdef] 102 5.3

300 0.120 [notdef] 103 0.120 [notdef] 103 0.117 [notdef] 102 0.125 [notdef] 103 7.7

400 0.463 [notdef] 103 0.463 [notdef] 103 0.452 [notdef] 103 0.484 [notdef] 103 9.4

500 0.210 [notdef] 103 0.210 [notdef] 103 0.206 [notdef] 103 0.221 [notdef] 103 11.2

600 0.105 [notdef] 103 0.105 [notdef] 103 0.103 [notdef] 103 0.110 [notdef] 103 12.9

700 0.557 [notdef] 103 0.557 [notdef] 103 0.544 [notdef] 103 0.586 [notdef] 103 14.8

800 0.309 [notdef] 104 0.309 [notdef] 104 0.302 [notdef] 104 0.326 [notdef] 104 16.6

900 0.178 [notdef] 104 0.178 [notdef] 104 0.174 [notdef] 104 0.186 [notdef] 104 18.5

1000 0.106 [notdef] 104 0.106 [notdef] 104 0.103 [notdef] 104 0.111 [notdef] 104 20.4

Table 8. LO cross section for Z-boson pair production with 4-lepton nal states and corresponding weak corrections at the LHC for di erent cut values of the 4-lepton invariant mass.

results of table 9 are in good agreement (better than 2%) with those presented in table 3 of ref. [33], despite the fact that we do not include QED corrections and corresponding non-factorizable contributions which arise in the DPA.

We do not include leptonic decays in our analysis of EW corrections to WW and WZ production in this work. This would additionally require the inclusion of real-photon radiation in the production and decay processes, respectively, resulting in a slightly more involved calculation. However, we are currently working on this issue, combining the EW corrections to resonant four-lepton production with state-of-the-art Monte Carlo predictions [71], and a detailed discussion of the results will soon be available [72].

3.5 Real-radiation contributions

Let us nally investigate the phenomenological e ect of additional massive-boson radiation as dened in section 2.3. In gures 12 and 13 we show the corresponding relative corrections for LHC14 and LHC8, respectively. As far as WW, ZZ and W+Z production at LHC14 is concerned, the contributions, which are always below 10%, are found to be of minor importance for the phenomenological analysis, and the e ects are even smaller

25

JHEP12(2013)071

pp ! (Z/ )(Z/ ) + X ! e+e+ + X, [notdef] yZZ[notdef] < 3

Mcutinv(4l)/GeV naiveLO/pb CMSLO/pb DPALO/pb NWALO/pb DPAweak/% LHC14200 0.815 [notdef] 102 0.815 [notdef] 102 0.795 [notdef] 102 0.855 [notdef] 102 5.4

300 0.219 [notdef] 102 0.219 [notdef] 102 0.214 [notdef] 102 0.229 [notdef] 102 8.4

400 0.791 [notdef] 103 0.791 [notdef] 103 0.770 [notdef] 103 0.828 [notdef] 103 11.6

500 0.326 [notdef] 103 0.326 [notdef] 103 0.319 [notdef] 103 0.343 [notdef] 103 15.9

600 0.168 [notdef] 103 0.168 [notdef] 103 0.164 [notdef] 103 0.177 [notdef] 103 19.3

700 0.962 [notdef] 104 0.962 [notdef] 104 0.941 [notdef] 104 1.017 [notdef] 104 22.3

800 0.587 [notdef] 104 0.587 [notdef] 104 0.575 [notdef] 104 0.621 [notdef] 104 24.9

900 0.374 [notdef] 104 0.374 [notdef] 104 0.367 [notdef] 104 0.397 [notdef] 104 27.4

1000 0.247 [notdef] 104 0.247 [notdef] 104 0.242 [notdef] 104 0.262 [notdef] 104 29.7

Table 9. LO cross section for Z-boson pair production with 4-lepton nal states and corresponding weak corrections at LHC14 for di erent cut values of the 4-lepton invariant mass in the Sudakov regime with [notdef] yZZ[notdef] < 3.

at LHC8, as expected. In contrast to this, in the case of WZ production we observe remarkably large corrections reaching +25% (+15%) at LHC14 (LHC8). However, we point out that this e ect, though sizable, cannot be attributed to large logarithms arising from the infrared structure of the corresponding squared matrix elements. In fact, it can be easily understood recalling that the corresponding real-radiation contributions as dened in section 2.3 include WZW+ production at LO which, compared to WZ production is from the beginning enhanced by a factor of 2 due to one u-quark PDF factor. To verify this line of argumentation, we have checked that for the Tevatron the relative corrections for W[notdef]Z production indeed coincide, not exceeding the level of +5% for pcutT = 300 GeV. Nevertheless, the numerical e ects discussed here can easily (and denitely should be) taken into account in the experimental analysis of the background.

Turning to the e ects of hard photon radiation displayed in the lower two plots of gures 12 and 13, respectively, we nd that the corresponding contributions are moderate and by far largest for W-pair production, exceeding +5%, while they are completely irrelevant for Z-pair production.

4 Conclusions

We have computed the full one-loop electroweak corrections to on-shell ZZ, W[notdef]Z, ZZ and production at hadron colliders, for the rst time consistently taking into account all mass e ects. Furthermore, the results presented are not limited to a particular kinematic regime, allowing for exible predictions valid in all regions of phase space. In case of ZZ production we have also included the leptonic decays and the corresponding weak corrections, nding good agreement with former computations restricted to Sudakov kinematics. The relative corrections are negative and grow with increasing center-of-mass energy. They are largest for ZZ production, reaching 50% at energies accessible at LHC14, and smallest for

26

JHEP12(2013)071

LO (pb)

LO (pb)

100 10

1 0.1 0.01 0.001 0.0001

100 10

1 0.1 0.01 0.001 0.0001

pp ! V1V2(+V3) + X

ps = 14 TeV MV V > McutVV

WW+

ZZ

W+Z

WZ

pp ! V1V2(+V3) + X

ps = 14 TeVpT,V1 and pT,V2 > pcutT,V

JHEP12(2013)071

200

400

600

800

1000

1200

1400

1600

1800

2000

100

200

400

500

600

700

800

McutVV (GeV)

300 pcutT,V (GeV)

V V V (%)

V V V (%)

30 25 20 15 10

5 0

30 25 20 15 10

5 0

W+W V

ZZV

W+ZV

WZV

200

400

600

800

1000

1200

1400

1600

1800

2000

100

200

400

500

600

700

800

McutVV (GeV)

300 pcutT,V (GeV)

V V (%)

V V (%)

6 5 4 3 2 1 0

6 5 4 3 2 1 0

W+W

ZZ

W+Z

WZ

200

400

600

800

1000

1200

1400

1600

1800

2000

100

200

400

500

600

700

800

McutVV (GeV)

300 pcutT,V (GeV)

Figure 12. Integrated LO cross sections (top) and relative corrections at the LHC14 due to radiation of one additional massive vector boson (center) and hard-photon radiation (bottom) evaluated with our default setup for di erent cuts on the invariant mass (left)/transverse momenta (right) of the nal-state bosons.

production. As an interesting new nding we observe that the relative corrections are not only sizable in the Sudakov regime as has been shown before, but may also in the ZZ

27

LO (pb)

LO (pb)

100 10

1 0.1 0.01 0.001 0.0001

100 10

1 0.1 0.01 0.001 0.0001

pp ! V1V2(+V3) + X

ps = 8 TeV MV V > McutVV

WW+

ZZ

W+Z

WZ

pp ! V1V2(+V3) + X

ps = 8 TeVpT,V1 and pT,V2 > pcutT,V

JHEP12(2013)071

200

300

400

500

600

700

800

900

1000

50

100

200

250

300

350

400

McutVV (GeV)

150 pcutT,V (GeV)

V V V (%)

V V V (%)

30 25 20 15 10

5 0

30 25 20 15 10

5 0

W+W V

ZZV

W+ZV

WZV

200

300

400

500

600

700

800

900

1000

50

100

200

250

300

350

400

McutVV (GeV)

150 pcutT,V (GeV)

V V (%)

V V (%)

6 5 4 3 2 1 0

6 5 4 3 2 1 0

W+W

ZZ

W+Z

WZ

200

300

400

500

600

700

800

900

1000

50

100

200

250

300

350

400

McutVV (GeV)

150 pcutT,V (GeV)

Figure 13. Integrated LO cross sections (top) and relative corrections at the LHC8 due to radiation of one additional massive vector boson (center) and hard-photon radiation (bottom) evaluated with our default setup for di erent cuts on the invariant mass (left)/transverse momenta (right) of the nal-state bosons.

case give signicant contributions at rather low transverse momenta. We have also investigated the e ect of massive boson radiation processes which may be considered as

28

background to vector-boson pair production depending on the details of the experimental setup. The e ects are smaller than expected from naive partonic considerations and may easily be included in the experimental studies. Our predictions rely on the experimental reconstruction of the intermediate bosons. In the future leptonic decays of the vector bosons will be included also in the analysis of WW and WZ production together with the corresponding O( ) corrections to allow for a more realistic event denition.

Acknowledgments

This work has been supported by Strukturiertes Promotionskolleg Elementarteilchen- und Astroteilchenphysik, SFB TR9 Computational and Particle Physics and BMBF Contract 05HT4VKATI3.

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