Constraints on the off-shell Higgs boson signal strength in the high-mass ZZ and WW final states with the ATLAS detector

Measurements of the ZZ and WW final states in the mass range above the 2mZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2m_Z$$\end{document} and 2mW\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2m_W$$\end{document} thresholds provide a unique opportunity to measure the off-shell coupling strength of the Higgs boson. This paper presents constraints on the off-shell Higgs boson event yields normalised to the Standard Model prediction (signal strength) in the ZZ→4ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ZZ \rightarrow 4\ell $$\end{document}, ZZ→2ℓ2ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ZZ\rightarrow 2\ell 2\nu $$\end{document} and WW→eνμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$WW\rightarrow e\nu \mu \nu $$\end{document} final states. The result is based on pp collision data collected by the ATLAS experiment at the LHC, corresponding to an integrated luminosity of 20.3 fb-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} at a collision energy of s=8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{s}=8$$\end{document} TeV. Using the CLs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CL_s$$\end{document} method, the observed 95 %\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\%$$\end{document} confidence level (CL) upper limit on the off-shell signal strength is in the range 5.1–8.6, with an expected range of 6.7–11.0. In each case the range is determined by varying the unknown gg→ZZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$gg\rightarrow ZZ$$\end{document} and gg→WW\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$gg\rightarrow WW$$\end{document} background K-factor from higher-order quantum chromodynamics corrections between half and twice the value of the known signal K-factor. Assuming the relevant Higgs boson couplings are independent of the energy scale of the Higgs boson production, a combination with the on-shell measurements yields an observed (expected) 95 %\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\%$$\end{document} CL upper limit on ΓH/ΓHSM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _H/\Gamma _H^{\mathrm {SM}}$$\end{document} in the range 4.5–7.5 (6.5–11.2) using the same variations of the background K-factor. Assuming that the unknown gg→VV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$gg\rightarrow VV$$\end{document} background K-factor is equal to the signal K-factor, this translates into an observed (expected) 95 %\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\%$$\end{document} CL upper limit on the Higgs boson total width of 22.7 (33.0) MeV.


Introduction
The observation of a new particle in the search for the Standard Model (SM) Higgs boson at the LHC, reported by the ATLAS [1] and CMS [2] Collaborations, is a milestone in the quest to understand electroweak symmetry breaking. Precision measurements of the properties of the new boson are of critical importance. Among its key properties are the couplings to each of the SM fermions and bosons, for which ATLAS and CMS presented results in Refs. [3,4], e-mail: atlas.publications@cern.ch and spin/CP properties, studied by ATLAS and CMS in Refs. [5,6].
The studies in Refs. [7][8][9][10] have shown that the high-mass off-peak regions beyond 2m V (V = Z , W ), well above the measured resonance mass of about 125 GeV [4,11], in the H → Z Z and H → W W channels are sensitive to Higgs boson production through off-shell and background interference effects. This presents a novel way of characterising the properties of the Higgs boson in terms of the off-shell event yields, normalised to the SM prediction (referred to as signal strength μ), and the associated off-shell Higgs boson couplings. Such studies provide sensitivity to new physics that alters the interactions between the Higgs boson and other fundamental particles in the high-mass region [12][13][14][15][16][17][18]. This approach was used by the CMS Collaboration [19] to set an indirect limit on the Higgs boson total width. The analysis presented in this paper is complementary to direct searches for Higgs boson to invisible decays [20,21] and to constraints coming from the Higgs boson coupling tests [3,4].
This paper presents an analysis of the off-shell signal strength in the Z Z → 4 , Z Z → 2 2ν and W W → eν μν final states ( = e, μ). It is structured as follows: Sect. 2 discusses the key theoretical considerations and the simulation of the main signal and background processes.  give details for the analysis in the Z Z → 4 , Z Z → 2 2ν and W W → eν μν final states, respectively. The dominant systematic uncertainties are discussed in Sect. 6. Finally the results of the individual analyses and their combination are presented in Sect. 7.
The ATLAS detector is described in Ref. [22]. The present analysis is performed on pp collision data corresponding to an integrated luminosity of 20.3 fb −1 at a collision energy of √ s = 8 TeV.

Theoretical predictions and simulated samples
The cross-section σ gg→H * →V V off-shell for the off-shell Higgs boson production through gluon fusion with subsequent decay into vector-boson pairs, 1 as illustrated by the Feynman diagram in Fig. 1a, is proportional to the product of the Higgs boson couplings squared for production and decay. However, unlike the on-shell Higgs boson production, σ gg→H * →V V off-shell is independent of the total Higgs boson decay width H [7,8]. Using the framework for Higgs boson coupling deviations as described in Ref. [23], the off-shell signal strength in the high-mass region selected by the analysis described in this paper at an energy scaleŝ, μ off-shell (ŝ), can be expressed as: where κ g,off-shell (ŝ) and κ V,off-shell (ŝ) are the off-shell coupling scale factors associated with the gg → H * production and the H * → V V decay. Due to the statistically limited sensitivity of the current analysis, the off-shell signal strength and coupling scale factors are assumed in the following to be independent ofŝ in the high-mass region selected by the analysis. The off-shell Higgs boson signal cannot be treated independently from the gg → V V background, as sizeable negative interference effects appear [7]. The interference term is proportional to √ μ off-shell = κ g,off-shell · κ V,off-shell . In contrast, the cross-section for on-shell Higgs boson production allows a measurement of the signal strength: which depends on the total width H . Assuming identical on-shell and off-shell Higgs boson coupling scale factors, the ratio of μ off-shell to μ on-shell provides a measurement of the total width of the Higgs boson. This assumption is particularly relevant to the running of the effective coupling κ g (ŝ) for the loop-induced gg → H production process, as it is 1 In the following the notation gg → (H * →)V V is used for the full signal + background process for V V = Z Z and W W production, including the Higgs boson signal (S) gg → H * → V V process, the continuum background (B) gg → V V process and their interference. For vector-boson fusion (VBF) production, the analogous notation VBF (H * →)V V is used for the full signal plus background process, with VBF H * → V V representing the Higgs boson signal and VBF V V denoting the background.
sensitive to new physics that enters at higher mass scales and could be probed in the high-mass m V V signal region of this analysis. More details are given in Refs. [12][13][14][15][16]. With the current sensitivity of the analysis, only an upper limit on the total width H can be determined, for which the weaker assumption that the on-shell couplings are no larger than the off-shell couplings, is sufficient. It is also assumed that any new physics which modifies the off-shell signal strength μ off-shell and the off-shell couplings κ i,off-shell does not modify the predictions for the backgrounds. Further, neither are there sizeable kinematic modifications to the off-shell signal nor new, sizeable signals in the search region of this analysis unrelated to an enhanced off-shell signal strength [18,24]. While higher-order quantum chromodynamics (QCD) and electroweak (EW) corrections are known for the off-shell signal process gg → H * → Z Z [25], which are also applicable to gg → H * → W W , no higher-order QCD calculations are available for the gg → V V background process, which is evaluated at leading order (LO). Therefore the results are given as a function of the unknown K-factor for the gg → V V background. QCD corrections for the off-shell signal processes have only been calculated inclusively in the jet multiplicity. The experimental analyses are therefore performed inclusively in jet observables, and the event selections are designed to minimise the dependence on the boost of the V V system, which is sensitive to the jet multiplicity.
The dominant processes contributing to the high-mass signal region in the Z Z → 4 , Z Z → 2 2ν and W W → eν μν final states are: the gg → H * → V V off-shell signal, the gg → V V continuum background, the interference between them, V V production in association with two jets through VBF and V H-like production modes pp → V V + 2 j (s-, t-and u-channel) and the qq → V V background. The LO Feynman diagrams for the gg → H * → V V signal, the continuum gg → V V background and the dominant irreducible qq → V V background are depicted in Fig. 1  125.36 GeV [11], is assumed for the off-shell signal processes. This small difference has a negligible impact on the predicted off-shell production yields. Figure 2 illustrates the size and kinematic properties of the gluon-induced signal and background processes by showing the four-lepton invariant mass (m 4 ) distribution for the gg → (H * →)Z Z → 2e2μ processes after applying the event selections in the Z Z → 4 channel (see Sect. 3) on generator-level quantities. The process gg → (H * →)Z Z → 2e2μ is shown for the SM μ off-shell = 1 case and for an increased off-shell signal with μ off-shell = 10. For low masses m Z Z < 2m Z the offshell signal is negligible, while it becomes comparable to the continuum gg → Z Z background for masses above the 2m t threshold. The interference between the gg → H * → Z Z signal and the gg → Z Z background is negative over the whole mass range. A very similar relation between the gg → H * → V V signal and the gg → V V background is also seen for the gg → (H * →)Z Z → 2 2ν and gg → (H * →)W W → eν μν processes.
The detector simulation for most generated Monte Carlo (MC) event samples is performed using Geant4 [26,27]. Some background MC samples in the W W → eν μν analysis for processes with large cross-sections are simulated with the fast detector simulation package Atlfast-II [27].

Simulation of gg → (H * →)V V
To generate the gg → H * → V V and gg → V V processes, including the interference, the LO MC generators gg2VV [7,28] and MCFM [9,10] together with PYTHIA8 [29] and SHERPA+OpenLoops [30][31][32][33] are used. The QCD renor-malisation and factorisation scales are set to m V V /2 [9]. The CT10 next-to-next-to-leading-order (NNLO) PDF set [34] is used, as the LO gg → V V process is part of the NNLO calculation for pp → V V . The default parton showering and hadronisation option for the events processed with the full detector simulation is PYTHIA8 with the "power shower" parton shower option [29].
For the gg → H * → V V signal, a NNLO/LO K-factor 2 including the next-to-leading-order (NLO) electroweak corrections, K H * (m V V ) = σ NNLO gg→H * →V V /σ LO gg→H * →V V , is applied. The K-factor and associated uncertainties are calculated in Ref. [25] as a function of the Higgs boson virtuality m V V for m H ∼125.5 GeV, using the MSTW2008 PDF set [35]. Additional corrections are used to re-weight the predictions to the CT10 NNLO PDF set used in the simulation.
For the gg → V V background and the interference with the gg → H * → V V signal, no higher-order QCD calculations are available. However, these corrections are studied for the W W final state in Ref. [36] in the soft-collinear approximation, which is considered suitable for high-mass Higgs boson production. In this approximation, the signal K-factor is found to provide a reliable estimate for the higher-order QCD corrections to the signal-background interference term.
The K-factor for the gg → V V background process, K(gg → V V ), remains unknown. Therefore, the results in this note are given as a function of the unknown Kfactor ratio between the gg → V V background and the gg → H * → V V signal, defined as where K B (m V V ) is the unknown mass-dependent K-factor for the gg → V V background, and K H * gg (m V V ) is the gluoninitiated K-factor [25] for the signal 3 as motivated by the soft-collinear approximation in Ref. [36]. Because the Kfactor K H * gg (m V V ) changes by less than 10 % as a function of m V V in the relevant region of phase space, no mass dependence on R B H * is assumed. The range 0.5-2 is chosen for the variation of the K-factor ratio R B H * in order to include the full correction from the signal K-factor K H * gg (m V V ) ∼ 2 in the variation range. With respect to the LO gg → V V process, this corresponds to an absolute variation in the approximate range 1-4. Using the K-factors discussed above, the crosssection for the gg → (H * →)V V process with any off-shell Higgs boson signal strength μ off-shell can be parameterised as: More details are given in Appendix A.1.
In addition, higher-order QCD corrections to the transverse momentum 4 p T and the rapidity y of the V V system are studied using SHERPA+OpenLoops, which includes matrix-element calculations for the first hard jet emission. A difference of order 20 % in the ratio of the p T of the V V system in the relevant kinematic region is observed when comparing the LO generators with parton shower to SHERPA+OpenLoops, while the difference in the rapidity y of the V V system is small. This difference in the p T of the V V system can modify the kinematic observables used in the analyses, leading to variations in both the kinematic shapes and acceptance which are not covered by the m V V dependent systematic uncertainties derived in Ref. [25]. To account for these effects, the LO generators are re-weighted by ∼2 % as the higher-order QCD contribution from qg and qq production is small. However, K H * gg (m Z Z ) has substantially larger uncertainties than K H * (m Z Z ). Therefore K H * (m Z Z ) is substituted here, ignoring the 2 % shift in central value, but taking the difference in the systematic uncertainty into account. 4 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector, and the z-axis along the beam line. The x-axis points from the IP to the centre of the LHC ring, and the y-axis points upwards. Cylindrical coordinates (r, φ) are used in the transverse plane, φ being the azimuthal angle around the beam line. Observables labelled "transverse" are projected into the x-y plane. The pseudorapidity is defined in terms of the polar angle θ as η = − ln tan(θ/2).
to SHERPA+OpenLoops in the p T of the V V system. Due to the different jet emission mechanisms in the signal and the background processes, different re-weighting functions are derived for the gg → H * → V V signal, the gg → V V background, and the total gg → (H * ) → V V , respectively. The impact of the re-weighting on the acceptance is below 1 % for the signal and at the level of 4-6 % for the background. In the Z Z → 4 channel, the re-weighting procedure is only used to account for the acceptance effects, as the matrix-element-based discriminant is insensitive to the p T of the Z Z system. For the Z Z → 2 2ν channel, the re-weighting is used in both the transverse mass shape and acceptance as the m T depends on the p T of the Z Z system. For the W W → eν μν channel, the re-weighting affects only the acceptance.

Simulation of electroweak V V production through VBF and V H-like processes
The electroweak 5 pp → V V + 2 j processes contain both VBF-like events and V H-like events, which are simulated using MadGraph5 [37] and cross-checked using PHANTOM [38]. The QCD renormalisation and factorisation scales are set to m W following the recommendation in Ref. [39] and the CTEQ6L1 PDF set [40] is used. PYTHIA6 [41] is used for parton showering and hadronisation. The high-mass range selected by this analysis includes Higgs boson signal events arising from: • the off-shell VBF H → V V process, which scales with κ 4 V,off-shell and is independent of H , • VBF-like V V processes with a t-channel Higgs boson exchange, which scale with κ 4 V,off-shell and are independent of H , • W H and Z H processes with an on-shell Higgs boson, with decays Z → 2 or W → ν and H → 2 2 j or H → ν2 j, which scale with κ 4 V,on-shell / H , where we assume the same coupling strength κ V,off-shell in the two VBF-like contributions, although the energy scale of the Higgs boson propagator is different between the two cases. Due to the different H dependence, the on-shell and off-shell Higgs boson production processes are separated in the analysis by requiring that the generated Higgs boson mass satisfies |m gen.
H − 125.5 GeV| < 1 GeV. This requirement is fully efficient in selecting the on-shell V H process. The NNLO QCD corrected cross-section in Ref. [23] is used for the on-shell V H production process. The cross-section σ pp→V V +2 j (μ off-shell ) for the electroweak pp → V V + 2 j process for any off-shell Higgs boson signal strength μ off-shell is parameterised in the same way as for the gg → (H * →) V V process. Details are given in Appendix A.2.

Simulation of qq → Z Z, W W and W Z backgrounds
The qq → Z Z, qq → W W , and qq → W Z backgrounds are simulated at NLO in QCD using POWHEG-BOX [42] with dynamic QCD renormalisation and factorisation scales of m V V ( ) and the CT10 NLO PDF set. In addition, SHERPA is used as a cross-check for the qq → W Z background. Parton showering and hadronisation are done with PYTHIA8 for qq → V Z and PYTHIA6 for qq → W W . The interference between the qq → W W and qq → Z Z processes for the 2 2ν final state is negligible [42] and thus not considered.
The cross-sections for the qq → Z Z and qq → W W processes are calculated in Refs. [43,44], respectively, for two on-shell Z or W bosons in the final state at NNLO QCD accuracy. As these calculations include the gg → V V processes as part of the NNLO calculation, a different K-factor is provided by the authors of the Refs. [43,44] excluding the gg → V V component and using a QCD renormalisation and factorisation scale μ QCD of m V V /2 in order to consistently match the simulation of the gg → (H * →)V V process: Electroweak higher-order corrections are not included in POWHEG-BOX. These corrections are calculated in Refs. [45,46] for on-shell outgoing vector bosons and found to be about −10 % in the high-mass V V region of this analysis. To account for these corrections, the POWHEG-BOX events are re-weighted using a procedure comparable to the one described in Ref. [47], based on the kinematics of the diboson system and the initial state quarks.

Simulation of top-quark backgrounds
In the W W → eν μν channel, the tt and single-top (schannel and W t) backgrounds are simulated with POWHEG-BOX [48,49] with parton showering and hadronisation done with PYTHIA6, using the CT10 NLO PDF set. The tchannel single-top background is simulated using AcerMC [50]+PYTHIA6 and uses the CTEQ6LI PDF set. The relative rates of tt and single-top production are evaluated with Top++2.0 [51] and the calculations in Refs. [52][53][54] respectively.

Analysis of the Z Z → 4 final state
The analysis for the Z Z → 4 final state closely follows the Higgs boson measurements in the same final state described in Ref. [55], with the same physics object definitions, trigger and event selections, and background estimation methods. A matrix-element-based discriminant (ME-based discriminant) is constructed to enhance the separation between the gg → H * → Z Z signal and the gg → Z Z and qq → Z Z backgrounds, and is subsequently used in a binned maximum-likelihood fit for the final result.

Event selection
To minimise the dependence of the gg → Z Z kinematics on higher-order QCD effects, the analysis is performed inclusively, ignoring the number of jets in the events.
The analysis is split into four lepton channels (2μ2e, 2e2μ, 4e, 4μ) as in Ref. [55]. Each electron (muon) must satisfy E T > 7 GeV (p T > 6 GeV) and be measured in the pseudorapidity range |η| < 2.47 (|η| < 2.7). The highestp T lepton in the quadruplet must satisfy p T > 20 GeV, and the second (third) lepton in p T order must satisfy p T > 15 GeV ( p T > 10 GeV). Lepton pairs are formed from same-flavour opposite-charge leptons. For each channel, the lepton pair with the mass closest to the Z boson mass is referred to as the leading dilepton pair and its invariant mass, m 12 , is required to be between 50 and 106 GeV. The second (subleading) pair is chosen from the remaining leptons (more than four leptons are allowed per event) as the pair closest in mass to the Z boson and in the range of 50 GeV < m 34 < 115 GeV. The off-peak region is defined to include the range from 220 GeV < m 4 < 1000 GeV. Figure 3a shows the observed and expected distributions of m 4 combining all lepton channels in the full off-peak region. The data are in agreement with the SM predictions, with a small deficit of the order of one standard deviation (1σ ). Table 1 shows the expected and observed number of events in the signal-enriched region, 400 GeV < m 4 < 1000 GeV, combining all lepton channels. This mass region was chosen since it is optimal for a m 4 cut-based analysis.

Matrix-element-based kinematic discriminant
The matrix-element kinematic discriminant fully exploits the event kinematics in the centre-of-mass frame of the 4 system, based on eight observables: {m 4 , m 12 , m 34 , cos θ 1 , cos θ 2 , φ, cos θ * , φ 1 }, defined in Refs. [5,55]. These observables are used to create the four-momenta of the leptons and incoming partons, which are then used to calculate matrix elements for different processes, provided by the MCFM pro-   Table 1 Expected and observed numbers of events in the signal region for all final states in the cut-based approaches. For the Z Z → 4 analysis a mass range of 400 < m 4 < 1000 GeV is used. The other backgrounds in the Z Z → 4 final state include contributions from Z +jets and top-quark processes. For the Z Z → 2 2ν analysis the range 380 GeV < m Z Z T < 1000 GeV is considered. For the W W → eν μν analysis, the region R 8 > 450 GeV is used and background event yields are quoted after the likelihood fit was performed. The expected events , including the Higgs boson signal, background and interference, are reported for both the SM predictions (in bold) and μ off-shell = 10. A relative gg → V V background K-factor of R B H * = 1 is assumed. The uncertainties in the number of expected events include the statistical uncertainties from MC samples and systematic uncertainties. The entries with a − are for processes with event yields < 0.1 tt, W t, and tb/tqb gram [9]. The following matrix elements are calculated for each event in the mass range 220 GeV < m 4 < 1000 GeV: • P qq : matrix element squared for the qq → Z Z → 4 process, • P gg : matrix element squared for the gg → (H * →)Z Z → 4 process including the Higgs boson (m H = 125.5 GeV) with SM couplings, continuum background and their interference, • P H : matrix element squared for the gg → H * → Z Z → 4 process (m H = 125.5 GeV).
The kinematic discriminant is defined as in Ref. [9]: where c = 0.1 is an empirical constant, to approximately balance the overall cross-sections of the qq → Z Z and gg → (H * →)Z Z processes. The value of c has a very small effect on the analysis sensitivity. Figure 3b shows the observed and expected distributions of the ME-based discriminant combining all lepton final states. Events with the ME-based discriminant value between −4.5 and 0.5 are selected with a signal efficiency of >99 %. In addition, an alternative multivariate discriminant based on a boosted decision tree (BDT) algorithm was studied to further separate the gg → H * → Z Z signal and the main qq → Z Z background, by exploiting additional kinematic information ( p T and η) of the Z Z system. The analysis sensitivity improves very little (∼2 %) compared to the ME-based discriminant alone. Due to the dependence on the p T of the Z Z system, the BDT-based discriminant introduces additional systematic uncertainties from the higher-order QCD corrections. For these reasons, the BDT-based discriminant is not used for the final result.

Analysis of the Z Z → 2 2ν final state
The analysis of the Z Z → 2 2ν channel follows strategies similar to those used in the invisible Higgs boson search in the Z H channel [20]. The definitions of the reconstructed physics objects (electrons, muons, jets, and missing transverse momentum) are identical, but some of the kinematic cuts were optimised for the current analysis.

Event selection
As the neutrinos in the final state do not allow for a kinematic reconstruction of m Z Z , the transverse mass (m Z Z T ) reconstructed from the transverse momentum of the dilepton system ( p T ) and the magnitude of the missing transverse momentum (E miss T ): is chosen as the discriminating variable to enhance sensitivity to the gg → H * → Z Z signal. The selection criteria are optimised to maximise the signal significance with respect to the main backgrounds, which are Z Z, W Z, W W , top-quark, and W/Z +jets events, as described in Sect. 4.2. The impact of the background uncertainty is considered in the significance calculation.
First, events with two oppositely charged electron or muon candidates in the Z mass window 76 GeV < m < 106 GeV are selected. Events with a third lepton (e or μ) identified using looser identification criteria for the electrons and a lower p T threshold of 7 GeV are rejected. A series of selection requirements are necessary to suppress the Drell-Yan background, including: E miss T > 180 GeV; 380 GeV < m Z Z T < 1000 GeV; the azimuthal angle between the transverse momentum of the dilepton system and the missing transverse momentum φ( p T , E miss T ) > 2.5; and |E miss T + jet p jet T | − p T / p T < 0.3 Events with a b-jet with p T > 20 GeV and |η| < 2.5, identified by the MV1 algorithm [56,57] with 70 % tagging efficiency, are rejected to suppress the top-quark background. Finally, the selection on the azimuthal angle between the two leptons φ < 1.4 is applied to select events with boosted Z bosons to further discriminate the signal from the background.

Background estimation
The dominant background is qq → Z Z production, followed by qq → W Z production. Background contributions from events with a genuine isolated lepton pair, not originating from a Z → ee or Z → μμ decay, arise from the W W , tt, W t, and Z → τ τ processes. The remaining backgrounds are from Z → ee or Z → μμ decays with poorly reconstructed E miss T , and from events with at least one misidentified electron or muon coming from W +jets, semileptonic top decays (tt and single top), and multi-jet events.
The qq → Z Z background is estimated in the same way as for the Z Z → 4 analysis using the POWHEG-BOX simulation as described in Sect. 2.3. The W Z background is also estimated with the simulation (described in Sect. 2.3) and validated with data in a three-lepton control region. The observed number of events in the control region for E miss T > 180 GeV (300 GeV) is 30 (3), whereas the predicted event yield is 22.9 ± 0.8 (3.4 ± 0.3). No significant difference is observed between the data and simulation.
The W W , tt, W t, and Z → τ τ backgrounds are inclusively estimated with data assuming lepton flavour symmetry in an eμ control region using a relaxed selection. The following equations show how these backgrounds in the signal region can be estimated with eμ events: where N bkg ee and N bkg μμ are the number of dielectron and dimuon events in the signal region. N data,sub eμ is the num-ber of events in the eμ control region with W Z, Z Z, and other small backgrounds (W +jets, tt V , and triboson) subtracted using simulation. The different e and μ efficiencies are taken into account using the α variable, which is an efficiency correction factor determined from the ratio of dielectron to dimuon event yields after the inclusive Z mass requirement (76 GeV < m < 106 GeV). The measured value of α is 0.942 with a systematic uncertainty of 0.004 and a negligible statistical uncertainty. This scale factor is applied to the MC predictions. The other source of systematic uncertainty comes from the subtraction of W Z, Z Z, and other small backgrounds in the eμ control region using the simulation. As no data event remains after applying the full selection, a scale factor of 1.4 ± 0.3 is derived by comparing the event yields from the data-driven and MC predictions with a relaxed selection applying the E miss T and m Z Z T requirements but no further cuts. Experimental systematic uncertainties are considered for the MC predictions.
Imperfect modelling of detector non-uniformities and E miss T response could lead to an incorrect estimate of the Z boson background in the signal region. The Z boson background is estimated with data using the two-dimensional sideband regions constructed by reversing one or both of the φ( p T ,E miss T ) and φ selections [20]. The main uncertainty on the mis-measured Z boson background arises from the differences in shape of the E miss T and m Z Z T distributions in the signal and sideband regions and the small correlation between these two variables. Other systematic uncertainties originate from the subtraction of the non-Z boson backgrounds in the sideband regions.
The W +jets and multi-jet backgrounds are estimated from data using the fake-factor method [20]. The predicted background with a looser E miss T selection applied at 100 GeV, and without the m Z Z T selection, is 0.04 ± 0.01 events. No event remains after applying the full event selection for both the data-driven method and MC samples, and hence this background is estimated to be negligible.
The predicted signals and backgrounds with statistical and systematic uncertainties are summarised in Table 1. The observed event yields agree with the total predicted ones from the SM within the uncertainties. Figure 4 shows the distributions of m Z Z T for the ee and μμ channels in the signal region, compared to the predicted contributions from the SM as well as to a Higgs boson with μ off-shell = 10.

Analysis of the W W → eν μν final state
The analysis of the W W → eν μν channel closely follows the Higgs boson measurements in the oppositely charged electron-muon pair final state in Ref. [58]. This selection ensures orthogonality with the Z Z → 2 2ν final state. The same object identification and selection as in Ref. [58] is used The transverse mass is modified compared to the definition in Eq. (8) as the neutrinos do not come from the same parent particle, and there is no m Z constraint.   In order to isolate the off-shell Higgs boson production while minimising the impact of higher-order QCD effects on gg → W W kinematics, a new variable, R 8 , is introduced: Both the coefficient a = 0.8 and the requirement R 8 > 450 GeV are optimised for off-shell signal sensitivity while also rejecting on-shell Higgs boson events, which have relatively low values of m and m W W T . The predicted on-shell signal contamination is 0.04 ± 0.03(stat.) events. The MV1 algorithm, at 85 % efficiency, is used to reject b-jets with p T > 20 GeV and |η| < 2.4 in order to reject backgrounds containing top quarks. A more efficient working point for b-jet tagging is used compared to the Z Z → 2 2ν analysis because of the need to reject a substantially larger top-quark background. A requirement on the separation between leptons, η < 1.2, suppresses qq-initiated W W production relative to gg-initiated production. The b-jet veto and η requirement are found to have a minimal impact on the W W -system kinematics and jet multiplicity in the gg → (H * →)W W processes. Table 1 contains the predicted and observed event yields in the signal region, 90 ± 4 and 82 respectively, in agreement with the SM with a small deficit in data. The distribution of the R 8 variable in the signal region is shown in Fig. 5c for the SM expectation and for a Higgs boson with μ off-shell = 10.

Background estimation
The dominant backgrounds arise from processes with real W bosons in the final state. The two backgrounds with the largest expected event yield are top-quark and qq → W W production. Dedicated control regions (CRs) are constructed to normalise these two backgrounds in the signal region with a simultaneous fit. Uncertainties on the extrapolation from the CRs to the signal region are described in Sects. 6.2 and 6.3.
The top-quark background predictions in the signal and W W control region are both normalised from the same top CR. A sample of top-quark events is obtained by starting from the signal region and reversing the b-jet veto by requiring exactly one b-tagged jet. This is closer in phase space to the b-jet-vetoed signal region than requiring at least one b-tag and results in a smaller uncertainty. The statistical error on the top-quark background normalisation is reduced by expanding the top CR down to R 8 > 160 GeV and dropping the η requirement. The impact of these changes is discussed in Sect. 6.3. An event yield of 13498 events is observed in the top CR (Fig. 5a), resulting in a fit normalisation factor of 1.03 ± 0.04, where the uncertainty includes all systematic sources, including extrapolation uncertainties described in Sect. 6.3. The top CR is approximately 96 % pure in top-quark events.
The qq → W W background is normalised to data using an additional CR. The region 160 GeV < R 8 < 450 GeV without the η requirement is used because it has a large W W contribution with negligible on-shell Higgs boson contamination and is adjacent to the signal region. A b-jet veto is applied to reject part of the substantial top-quark contamination. An event yield of 8007 events is observed in the W W CR (Fig. 5b), resulting in a fit normalisation factor of 1.03 ± 0.11, including all of the uncertainties as above. This CR is approximately 46 % pure in qq → W W , while the leading background of top-quark events contributes 39 %. The gg-initiated W W background is estimated from MC simulation, as discussed in Sect. 2.1.
The remaining background predictions, except for W +jets and multi-jet production, are taken from MC simulation, as described in Ref. [58]. The predicted fraction of the total background in the signal region arising from gg → W W , W +jets, and W γ /W γ * /W Z/Z Z events is approximately 4 % each, while for Z +jets it is 2 %. The W +jets and multijet backgrounds are estimated by applying a data-driven extrapolation factor to CRs with lepton candidates failing the nominal lepton identification and isolation, while passing a loosened requirement [58].

Systematic uncertainties
The largest systematic uncertainties for this analysis arise from theoretical uncertainties on the gg → H * → V V signal process, the gg/qq → V V background processes and the interference between the gg → V V signal and background processes. The electroweak (H * → V V ) processes in association with two jets contribute about 10-30 % of the total signal. The associated theoretical uncertainties due to the missing higher-order corrections and PDF variations are small for V H-like and VBF-like processes pp → Z Z + 2 j, and are therefore not included in the analysis. Compared to the theoretical uncertainties, the experimental uncertainties are small in the Z Z → 2 2ν and W W → eν μν analyses and close to negligible in the Z Z → 4 analysis. In the Z Z → 2 2ν and W W → eν μν analyses, uncertainties on the extrapolations from the control regions to the signal regions are included.

Systematic uncertainties on
The uncertainty from missing higher-order QCD and EW corrections to the off-shell gg → H * → V V signal is estimated in Ref. [25] as a function of the Higgs boson virtuality, m V V , and adopted for this analysis. The uncertainty is 20-30 % for the high-mass region used in this analysis. The PDF uncertainty for the gg → (H * →)V V process as a function of m V V is found to be 10-20 % in the high-mass region used in this analysis. This is consistent with an earlier study at √ s = 7 TeV [39]. For the gg → V V background, higher-order QCD calculations are not available. As discussed in Sect. 2.1, the gluoninduced part of the signal K-factor K H * gg (m V V ) is applied to the background and results are then given as a function of the unknown K-factor ratio R B H * between background and signal. The uncertainty on K H * gg (m V V ) is larger than the uncertainty on K H * (m V V ) because some contributions to the full signal NNLO QCD K-factor are not present in K H * gg (m V V ). Therefore, the following correlation treatment of uncertainties is applied: the uncertainty on the signal K-factor K H * (m V V ) is applied as a correlated uncertainty to K H * gg (m V V ). The difference in quadrature between the uncertainty on K H * gg (m V V ) and K H * (m V V ) is added as an uncorrelated uncertainty component only to K H * gg (m V V ). The interference between gg → H * → V V and gg → V V is calculated at LO only. In Ref. [36], a soft-collinear approximation is used to calculate the cross-section for the sum of a heavy Higgs boson (gg → H → W W ) and its interference with the background. The uncertainty on this calculation is estimated to be about 10 %, which leads to about 30 % uncertainty on the interference alone. Within the ansatz of using an unknown K-factor ratio between background and signal (see Eq. (5)), this additional uncertainty of roughly 30 % on the interference term can be represented by an approximately 60 % variation of the K-factor ratio R B H * for the background around the nominal value of 1.0. Therefore the variation of R B H * from 0.5 to 2.0 should cover both the leading corrections and uncertainties for the interference and the background component taken individually.
However, there is a large cancellation between the background and the negative interference at the expected 95 % confidence level upper limit value of μ off-shell , shown in Tables 3 and 4. This leads to a large artificial cancellation in the uncertainties of the gg → Z Z background and the interference, when treated as correlated. To account for additional uncertainties on the interference component that are not covered by the soft-collinear approximation, the 30 % uncertainty on the interference derived in Ref. [36] is applied to the interference component in addition to, and uncorrelated with, other uncertainties.
The systematic uncertainties associated with SHERPAbased re-weighting in p T of the V V system are assessed by varying the renormalisation, factorisation and resummation scales in SHERPA. The larger in value between the scale variations in SHERPA and 50 % of the difference between SHERPA and gg2VV+ PYTHIA8 is assigned as the systematic uncertainty. This conservative approach is chosen to consider potential uncertainties not accounted for by the scale variations. The impact of the PDF uncertainties is found to be negligible.

Systematic uncertainties on qq → V V
The missing-higher-order and PDF uncertainties for the qq → Z Z background, as a function of m Z Z , are taken from Ref. [39], based on NLO 7 TeV calculations using a fixed scale of m Z . Slightly smaller systematic uncertainties are found for 8 TeV using a dynamic scale of m Z Z /2, hence applying the uncertainties from Ref. [39] can be considered a conservative choice. Both the QCD scale uncertainty and the PDF uncertainty are 5-10 % for the highmass region used in this analysis. The NNLO calculation in Ref. [43] does not yield a significantly reduced QCD scale systematic uncertainty. An evaluation of the PDF uncertainty correlations shows that the qq → Z Z background PDF uncertainties are anti-correlated with the PDF uncertainties for the gg → (H * →)Z Z process, and this is taken into account in the analysis. Acceptance uncertainties on the qq → Z Z background are evaluated by comparing PYTHIA8 and HERWIG6 [59] samples and found to be negligible. The PDF, QCD scale, and EW correction uncertainties for the qq → W Z process are considered in the same way as for the qq → Z Z process. Both the QCD scale uncertainty and the PDF uncertainty are estimated to be ∼5-10 % for the high-mass region used in this analysis.
Extrapolation uncertainties on the qq → W W process in the W W → eν μν channel are evaluated using the method described in Ref. [58]. Uncertainties due to missing higher-order corrections are estimated by varying the renormalisation and factorisation scales independently by factors of one-half and two, keeping the ratio of the scales between one-half and two. Parton shower Table 2 Uncertainties on the extrapolation of top-quark processes and qq → W W from their respective CRs to the SR, and from the top CR to the W W CR, from the parton shower and underlying event (UE/PS), from matching the matrix element to the UE/PS model (Gen), from the QCD renormalisation and factorisation scale (scale), and from the PDFs. These uncertainties are used in the W W analysis and derived with the same methods as used in Ref. [ [62]). The extrapolation uncertainties from the W W control region to the signal region are summarised in Table 2. The EW corrections for the qq → V V process described in Sect. 2.3 are strictly valid only for the LO QCD qq → V V process above the diboson production threshold when both vector bosons are on shell. This is the case for all three analyses after final selections. The EW corrections are computed at LO QCD because the mixed QCD-EW corrections have not yet been calculated. In events with high QCD activity, an additional systematic uncertainty is considered by studying the variable ρ = | i i,T + E miss T |/( i | i,T | + | E miss T |) introduced in Eq. (4.4) of Ref. [47] (here T represents the transverse momentum of the lepton i from vector boson decays). A phase space region with ρ < 0.3 is selected, where the NLO QCD event kinematics resembles the LO event kinematics in being dominated by recoiling vector bosons and therefore the corrections are applicable without additional uncertainty. For events with ρ > 0.3 the correction is applied with a 100 % systematic uncertainty to account for the missing mixed QCD-EW corrections which are expected to be of the same order of magnitude. The applied corrections are partial in that they include only virtual corrections, and do not include polarisation effects. The sum of both of these effects is estimated to be O(1 %) [47] and is neglected in this analysis.
While the EW corrections and uncertainties directly affect the predicted size of the qq → Z Z and qq → W Z backgrounds in the Z Z → 4 and Z Z → 2 2ν analyses, only the extrapolation of the qq → W W background from the control region to the signal region is affected in the W W → eν μν analysis.

Systematic uncertainties on top-quark events
Theory uncertainties on extrapolating top-quark processes from the control region to the signal region in the W W → eν μν channel are also evaluated using methods similar to those of Ref. [58]. For the evaluation of the extrapolation uncertainties, the signal region requirements are relaxed to increase the sample size; the region is extended down to R 8 > 160 GeV and the η requirement is dropped. The extra uncertainty from this extension is checked in a separate sample with at least one b-tagged jet, again defined so as to reduce the statistical uncertainties, which is simultaneously re-weighted in η and R 8 to match the b-vetoed region. With this b-tagged sample, the extra uncertainty from the removal of the η requirement, and from extending the range in R 8 , is found to be 3.5 %.
The method described in Sect. 6.2 is used to evaluate the systematic uncertainties on top-quark processes. Since the extended signal region covers the W W CR, the same systematic uncertainties are valid for the extrapolation from the top CR to the W W CR. These uncertainties, summarised in Table 2, are applied to both tt and single-top processes, which make up approximately 22 % of the top background in the signal region. A 20 % uncertainty is assigned to the single-top processes in order to take into account the uncertainty on the single-top fraction; the impact on the result is negligible.

Experimental systematic uncertainties
For the Z Z → 4 analysis, the same sources of experimental uncertainty as in Ref. [55] are evaluated. In the off-shell Higgs boson region, the leptons come from the decay of onshell Z bosons; hence the lepton-related systematic uncertainties are small compared to those for the leptons from on-shell Higgs boson production. The leading, but still very small, experimental systematic uncertainties are due to the electron and muon reconstruction efficiency uncertainties.
Similarly, for the 2 2ν channel, the same sources of experimental uncertainty as in Ref. [20] are evaluated. The electron energy scale, electron identification efficiency, muon reconstruction efficiency, jet energy scale, and systematic uncertainties from the data-driven Z background estimates are the main sources of the experimental systematic uncertainties. These experimental uncertainties affect the expected sensitivity of the μ off-shell measurement only at the percent level.
Finally, for the W W → eν μν channel, the same sources of experimental uncertainty as in Ref. [58] are evaluated. The uncertainty on the electron energy scale, followed by the uncertainty on the rate for mis-tagged light-flavour jets as b-jets, and the uncertainty on the jet energy scale and resolution, are the dominant experimental sources of uncertainty.
The remaining experimental sources are significantly smaller than the theoretical uncertainties.
The uncertainty on the integrated luminosity is 2.8 %. It is derived, following the same methodology as that detailed in Ref. [63], from a preliminary calibration of the luminosity scale derived from beam-separation scans performed in November 2012.

Results
In this section the results for the Z Z → 4 , Z Z → 2 2ν and W W → eν μν analyses are presented and translated into limits on the off-shell signal strength μ off-shell for the individual analyses and for the combination of all three channels. In a second step, the off-shell analyses are combined with the on-shell Z Z * → 4 [55] and W W * → ν ν [58] analyses based on the 8 TeV data taken in 2012. In combining the Z Z and W W channels it is assumed that the ratio of the Z Z cross-section σ gg→H ( * ) →Z Z (ŝ) to the W W cross-section σ gg→H ( * ) →W W (ŝ) (and similarly for VBF) is as predicted in the SM for both the on-and off-shell processes.
Two different off-shell combinations are presented based on different assumptions. First, a single off-shell signal strength parameter is applied for all production modes. This is equivalent to assuming that the ratio of the off-shell production rates via the process gg → H to those via the VBF process are as predicted in the SM. In a second combination, only the off-shell signal strength of the gg → H * → V V production process is considered while the VBF production process is fixed to the SM prediction. In this case the combined signal strength μ gg→H * →V V off-shell can be interpreted as a constraint on the off-shell coupling strength κ g,off-shell associated with the gg → H * production mode.
The combination with the on-shell analyses is also performed under two assumptions that correspond to different interpretations of the results. The first is performed using different signal strengths for the gg → H ( * ) and the VBF production modes. 6 The parameter of interest is described by the ratio of the off-shell to the on-shell signal strengths, which can be interpreted as the Higgs boson total width normalised to its SM prediction: μ off-shell /μ on-shell = H / SM H . This interpretation requires that the off-and on-shell couplings are the same for both gg → H ( * ) and VBF production modes (i.e., κ g,on-shell = κ g,off-shell and κ V,on-shell = κ V,off-shell 7 ). In a second combination, the coupling scale factors κ V = κ V,on-shell = κ V,off-shell associated with the on-and off-shell VBF production and the H ( * ) → V V decay, are assumed to be the same and fitted to the data (profiled). In this case the parameter of interest, R gg = μ gg→H * off-shell /μ gg→H on-shell , can be interpreted as the ratio of the off-shell to the on-shell gluon couplings: R gg = κ 2 g,off-shell /κ 2 g,on-shell . This also assumes that the total width is equal to the SM prediction.
In the Z Z → 4 channel, a binned maximum-likelihood fit to the ME-based discriminant distribution is performed to extract the limits on the off-shell Higgs boson signal strength. The fit model accounts for signal and background processes, including gg → (H * →)Z Z, VBF(H * →)Z Z and qq → Z Z. The probability density functions (pdf) of the signal-related processes gg → (H * →)Z Z and VBF (H * →)Z Z are parameterised as a function of both the offshell Higgs boson signal strength μ off-shell and the unknown background K-factor ratio R B H * as given in Eqs. (15) and (19). Normalisation and shape systematic uncertainties on the signal and background processes are taken into account as described in Sect. 6.1, with correlations between different components and processes as indicated therein.
In the Z Z → 2 2ν channel, a similar maximumlikelihood fit to the transverse mass (m Z Z T ) is performed, comparing the event yield in the signal-enriched region in data with the predictions. The fit model accounts for the signal and all background processes mentioned in Table 1. The modelling of the dominant signal and background processes is the same as in the Z Z → 4 channel.
In the W W → eν μν channel, a maximum-likelihood fit is performed using the event yields in the signal region and the two control regions. As in the Z Z channels, the fit model accounts for the parameterised signal and all background processes mentioned in Sect. 5.2. Unconstrained strength parameters common among fit regions for the qq → W W and topquark processes allow the control regions to constrain the predicted event yields in the signal region.
The likelihood is a function of a parameter of interest μ and nuisance parameters θ . Hypothesis testing and confidence intervals are based on the profile likelihood ratio [64]. The parameters of interest are different in the various tests, while the remaining parameters are profiled. Hypothesised values for a parameter of interest μ are tested with a statistic where the single circumflex denotes the unconditional maximum-likelihood estimate of a parameter and the double circumflex [e.g.ˆ θ(μ)] denotes the conditional maximumlikelihood estimate (e.g. of θ ) for given fixed values of μ. This test statistic extracts the information on the parameters of interest from the full likelihood function.
All 95 % confidence level (CL) upper limits are derived using the C L s method [65], based on the following ratio of one-sided p-values: C L s (μ) = p μ /(1 − p 1 ) where p μ is the p-value for testing a given μ = μ off-shell or μ = H / SM H (the non-SM hypothesis) and p 1 is the p-value derived from the same test statistic under the SM hypothesis of μ off-shell = 1 in the first case and H / SM H = μ on-shell = 1 in the second case. 8 The 95 % C L s upper limit is found by solving for C L s (μ 95 % ) = 5 %. Values μ > μ 95 % are regarded as excluded at 95 % CL. A detailed description of the implementation of the C L s procedure can be found in Ref. [66].
The results presented in this paper rely on the asymptotic approximation [64] for the test statistic (μ). This approximation was cross-checked with Monte Carlo ensemble tests that confirm its validity in the range of the parameters for which the 95 % CL limits are derived. Deviations appear close to the boundary of μ off-shell ≥ 0 imposed by Eq. (5) and hence the 1σ uncertainties can only be seen as approximate.
While the final 95 % CL limits are given as a function of the unknown background K-factor ratio R B H * , comparisons between the data and the MC predictions, and values in other figures and tables, are given assuming R B H * = 1.

Results of the individual off-shell analyses
The scan of the negative log-likelihood, −2 ln , as a function of μ off-shell for data and the expected curve for an SM Higgs boson for the three individual off-shell analyses is illustrated in Fig. 6. The observed and expected 95 % CL upper limits on μ off-shell as a function of R B H * are shown in Fig. 7 and are summarised in Table 3. The Z Z → 4 and Z Z → 2 2ν analysis have a very similar expected sensitivity. The Z Z → 4 analysis is statistics limited, while the sensitivity in the Z Z → 2 2ν analysis is significantly reduced by the theoretical systematic uncertainties as can be seen in Fig. 6. The similar expected C L s limits for the two channels for R B H * = 0.5 and 1.0 in Table 3 is a coincidence, caused by the different statistical and systematic uncertainty components.
The typical off-shell mass scales tested by the analyses are in the range 400 GeV < m V V < 1000 GeV, with a small fraction of the expected H * → W W signal extending to substantially higher mass scales. 9 This is illustrated in Fig. 8, which shows the generated m V V mass for the gg → H * → V V and the VBF H * → V V signal processes weighted by the expected S/B ratio in each bin of the final discriminant

Combination of the off-shell Z Z and W W analyses
The analyses described in the previous sections are combined to obtain a limit on μ off-shell . In combining the off-shell results the main systematic uncertainties related to the theory uncertainties on the gg → (H * →)V V (including signal and interference contributions) and qq → V V processes are treated as correlated between the different channels. The same K-factor ratio R B H * is assumed for the gg → Z Z and gg → W W backgrounds. Where appropriate, the experimental systematic uncertainties are also treated as correlated. However, they are found to have a very small impact on the final combined limit.
The limits on μ off-shell are obtained under two different assumptions: • Determination of the signal strength μ off-shell when fixing the ratio of the signal strength in gg → H * and VBF to the SM prediction, namely μ The scan of the negative log-likelihood, −2 ln , as a function of μ off-shell for data and the expected curve for an SM Higgs boson for the two cases above are shown in Fig. 9.
The limits on μ off-shell and μ gg→H * off-shell are computed with the C L s method, assuming for the alternative hypothesis that all the off-shell rates are at their SM predictions. They are derived as a function of the gg → V V background K-factor ratio R B H * . These results are reported in Table 4 and shown in Fig. 10, assuming either one common scale factor for both the gg → H * and VBF processes or using a scale factor for the gg → H * process and fixing the VBF production to the SM prediction.  The impact of the various systematic uncertainties on the combined expected limit in the off-shell fit are listed in Table  5 when fixing the ratio of the signal strength in gg → H * and VBF to the SM prediction. The values in this table were derived by fixing all the nuisance parameters associated with the systematic uncertainties to the values derived from the SM-conditional fit to the data, with the exception of the one under study.

Combination of the off-shell and on-shell Z Z and W W analyses
In this section, the off-shell results reported above are combined with the on-shell H → Z Z * → 4 [55] and H → W W * → ν ν [58] analyses based on the 8 TeV data taken in 2012. In these analyses a Higgs boson mass value of 125.36 GeV [11] is assumed. For the on-shell Z Z and W W combination the main common sources of theoretical and experimental systematic uncertainties are treated as correlated [3].
[GeV] The uncertainties from the impact of higher-order QCD corrections on the gg → H ( * ) and qq → V V processes are considered correlated between the on-shell and off-shell measurements. The PDF uncertainties are treated as uncorrelated between on-shell and off-shell analyses. The correlations between the PDF uncertainties for the on-shell and off-shell analyses are expected to be small with the exception of the ones for the qq → V V process, which have negligible impact on the on-shell results.
In addition to the main theoretical uncertainties, the common experimental systematic uncertainties are treated as correlated.
The results reported in the following are based on two different assumptions: • Determination of H / SM H when profiling the coupling scale factors κ g and κ V associated with the on-and offshell gg → H ( * ) and VBF production and the H ( * ) → V V decay, assuming κ g = κ g,on-shell = κ g,off-shell and   Fig. 10 The observed and expected combined 95 % CL upper limit on μ off-shell as a function of R B H * for the combined Z Z and W W analyses. The upper limits are calculated using the C L s method, with the SM as the alternative hypothesis. a Limit on the common signal strength μ off-shell applied to both the gg → H * and VBF processes. The ratio of the gg → H * and VBF processes is assumed to be as in the SM. b Limit on the signal strength μ gg→H * →V V off-shell for the gg → H * → V V process. The production rate for the VBF off-shell process is fixed to the SM prediction. The green (yellow) bands represent the 68 % (95 %) confidence intervals for the C L s expected limit κ V,off-shell associated with the VBF production and the H ( * ) → V V decay. The ratio H / SM H = 1 is fixed to the SM prediction. The parameter R gg is sensitive to possible modifications of the gluon couplings in the highmass range with respect to the on-shell value.
The negative log-likelihood scans for the above-defined fitting configurations as well as the combined upper limit at 95 % CL on H / SM H and R gg are illustrated in Figs. 11 and 12 and the corresponding limits are listed in Table 6. The limits are all computed with the C L s method, taking the SM values as the alternative hypothesis. The limit on H / SM H can be translated into a limit on the total width of the Higgs boson under the assumptions reported above, out of which the most important is that the relevant Higgs boson coupling scale factors are independent of the energy scale of the Higgs boson production. Assuming a value of R B H * = 1, this translates into an observed (expected) 95 % CL upper limit on the Higgs boson total width of 22.7 (33.0) MeV. 10

Conclusion
The measurement of the Z Z and W W final states in the mass range above the 2m Z and 2m W thresholds provides a unique opportunity to measure the off-shell coupling strengths of the observed Higgs boson. In this paper constraints on the off-shell Higgs boson signal strengths in the Z Z → 4 , Z Z → 2 2ν and W W → eν μν final states and their combination are presented. The result is based on pp collision data collected by the ATLAS experiment at the LHC, corresponding to an integrated luminosity of 20.3 fb −1 at a collision energy of √ s = 8 TeV. Using the C L s method, the observed 95 % confidence level (CL) upper limit on the off-shell signal strength is in the range 5.1-8.6, with an expected range of 6.7-11.0. In each 10 The value of the SM Higgs boson width of 4.12 MeV at a mass of 125.4 GeV [23] is used to convert the limit H / SM H into the total width limit.   case the range is determined by varying the unknown gg → Z Z and gg → W W background K-factor from higher-order QCD corrections between half and twice the value of the known signal K-factor. Assuming the relevant Higgs boson couplings are independent of the energy scale of the Higgs boson production, a combination with the on-shell measurements of Z Z and W W in the same dataset yields an observed (expected) 95 % CL upper limit on H / SM H in the range 4.5-7.5 (6.5-11.2) under the same variations of the background K-factor. Assuming the value of R B H * = 1 and under the assumptions reported above, this translates into an observed (expected) 95 % CL upper limit on the Higgs boson total width of 22.7 (33.0) MeV.
Assuming that the total width of the Higgs boson is as expected in the SM, the same combination can be interpreted as a limit on the ratio of the off-shell to the on-shell couplings to gluons R gg = κ 2 g,off-shell /κ 2 g,on-shell . An observed (expected) 95 % CL upper limit on R gg in the range 4.7-8.6 (7.1-13.4) under the same variations of the background K-factor is found.
Acknowledgments We are very thankful to M. Bonvini, J. Campbell, S. Forte, F. Krauss, K. Melnikov, G. Passarino, and M. Spannowsky for their essential input in the estimation of uncertainties in the theoretical predictions for the signal and background processes and their interference. We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently. We acknowledge the support of

Appendix A: Monte Carlo and PDF scaling to arbitrary μ off-shell
The known dependence of the off-shell Higgs boson signal process, the background process and the interference term on the off-shell signal strength μ off-shell can be used to construct MC samples for arbitrary values of μ off-shell from three basic samples generated at different fixed values of μ off-shell .
A.1: Dependence of the gg → (H * →)V V off-shell cross-sections on the signal strength An event sample σ gg→(H * →)V V (μ off-shell ) for the gg → (H * →)V V process with an arbitrary value of the offshell Higgs boson signal strength μ off-shell can be constructed from the MC sample for the SM Higgs boson signal gg → H * → V V (σ SM gg→H * →V V ), the gg → V V continuum background MC sample (σ gg→V V, cont ) and a full SM Higgs boson signal plus background gg → (H * →)V V MC sample (σ SM gg→(H * →)V V ) using the following weighting function: where the K-factors are calculated inclusively without any selections. As a direct simulation of an interference MC sample is not possible, Eq. (13) and R B H * are used to obtain: A.2: Dependence of the V V + 2 j off-shell signal and background interference on the signal strength An MC event sample for the EW pp → (H * + 2 j →)V V + 2 j process with an arbitrary value of the off-shell Higgs boson signal strength μ off-shell can be constructed from a pure pp → V V + 2 j continuum background MC sample, a full SM Higgs boson signal plus background pp → (H * + 2 j →)V V + 2 j MC sample and a third Higgs boson signal plus background pp → (H * + 2 j →) V V +2 j MC sample with μ off-shell = κ 4 V = H / SM H = 10. Using H / SM H = 10 for the last sample ensures that the onshell V H events are generated with SM-like signal strength.
Solving for the generated MC samples yields: