Measurements of the Higgs boson width and anomalous HVV couplings from on-shell and off-shell production in the four-lepton final state

Studies of on-shell and off-shell Higgs boson production in the four-lepton final state are presented, using data from the CMS experiment at the LHC that correspond to an integrated luminosity of 80.2 fb−1 at a center-of-mass energy of 13 TeV. Joint constraints are set on the Higgs boson total width and parameters that express its anomalous couplings to two electroweak vector bosons. These results are combined with those obtained from the data collected at center-of-mass energies of 7 and 8 TeV, corresponding to integrated luminosities of 5.1 and 19.7 fb−1, respectively. Kinematic information from the decay particles and the associated jets are combined using matrix element techniques to identify the production mechanism and to increase sensitivity to the Higgs boson couplings in both production and decay. The constraints on anomalous HVV couplings are found to be consistent with the standard model expectation in both the on-shell and off-shell regions. Under the assumption of a coupling structure similar to that in the standard model, the Higgs boson width is constrained to be 3.2+2.8 −2.2 MeV while the expected constraint based on simulation is 4.1 +5.0 −4.0 MeV. The constraints on the width remain similar with the inclusion of the tested anomalous HVV interactions. ”Published in Physical Review D as doi:10.1103/PhysRevD.99.112003.” c © 2019 CERN for the benefit of the CMS Collaboration. CC-BY-4.0 license ∗See Appendix A for the list of collaboration members ar X iv :1 90 1. 00 17 4v 2 [ he pex ] 2 2 Ju n 20 19


Measurements of the Higgs boson width and anomalous
Studies of on-shell and off-shell Higgs boson production in the four-lepton final state are presented, using data from the CMS experiment at the LHC that correspond to an integrated luminosity of 80.2 fb −1 at a center-of-mass energy of 13 TeV. Joint constraints are set on the Higgs boson total width and parameters that express its anomalous couplings to two electroweak vector bosons. These results are combined with those obtained from the data collected at center-of-mass energies of 7 and 8 TeV, corresponding to integrated luminosities of 5.1 and 19.7 fb −1 , respectively. Kinematic information from the decay particles and the associated jets are combined using matrix element techniques to identify the production mechanism and to increase sensitivity to the Higgs boson couplings in both production and decay. The constraints on anomalous HVV couplings are found to be consistent with the standard model expectation in both the onshell and off-shell regions. Under the assumption of a coupling structure similar to that in the standard model, the Higgs boson width is constrained to be 3.2 þ2. 8 −2.2 MeV while the expected constraint based on simulation is 4.1 þ5.0 −4.0 MeV. The constraints on the width remain similar with the inclusion of the tested anomalous HVV interactions. DOI: 10.1103/PhysRevD.99.112003

I. INTRODUCTION
The standard model (SM) of particle physics postulates the existence of a Higgs field responsible for the generation of the masses of fundamental particles. The excitation of this field is known as the Higgs boson (H) [1][2][3][4][5][6][7]. The observation of an H boson with a mass of around 125 GeV by the ATLAS and CMS Collaborations [8][9][10] is consistent with the expectations of the SM, but further tests of the properties of this particle, such as its width and the structure of its couplings to the known SM particles, are needed to determine its nature.
The CMS and ATLAS experiments have set constraints of Γ H < 13 MeV at 95% confidence level (C.L.) on the H boson total width [11][12][13][14][15] using the off-shell production method [16][17][18], which relies on the relative measurement of off-shell and on-shell production. The upper bound on Γ H was set considering the gluon fusion and electroweak (EW) production mechanisms in the analysis. The precision on Γ H from on-shell measurements of the width of the resonance peak alone is approximately , which is significantly worse than the result from the off-shell method. The constraint on the H boson lifetime is equivalent to a lower bound on the width and was derived from the flight distance in the CMS detector as Γ H > 3.5 × 10 −9 MeV at 95% C.L. [13]. The SM expectation of the width of the H boson is around 4 MeV [22].
The CMS [13,[23][24][25][26][27] and ATLAS [28][29][30][31][32][33] experiments have set constraints on the spin-parity properties and anomalous couplings of the H boson, finding its quantum numbers to be consistent with J PC ¼ 0 þþ , but allowing small anomalous couplings to two EW gauge bosons (anomalous HVV couplings). Off-shell signal production may be enhanced in the presence of these anomalous HVV couplings [11,13,22,[34][35][36]. As a result, the measurement of Γ H using the off-shell technique may be affected by these deviations of the H boson couplings from the SM expectations. An attempt to measure Γ H using the off-shell technique while including anomalous HVV interactions has been made by the CMS experiment [13]. In that previous study, constraints are placed on Γ H and the on-shell crosssection fraction f ΛQ that expresses an anomalous coupling contribution sensitive to the invariant mass of the H boson, using a realistic treatment of interference between the H boson signal and the continuum background. Extending the application of the off-shell technique to a wider range of anomalous HVV contributions, studied previously using onshell H boson production [27], is the goal of this paper.
The presented investigation on the H boson width targets both gluon fusion and EW production mechanisms and tests the effects of possible anomalous HVV couplings in either production or decay. Nevertheless, it still relies on the knowledge of coupling ratios between the off-shell and onshell production, the dominance of the top quark loop in the gluon fusion production mechanism, and the absence of new particle contributions in the loop. A violation of the last assumption by itself would be a manifestation of physics beyond the SM (BSM), which may become evident if the measured width deviates from the SM expectation. The measured width may also deviate from the SM expectation if the H boson has new BSM decay channels or the known channels have non-SM rates. Therefore, the measurement of the width complements the search for H boson decay to invisible or undetected particles, and the measurement of the H boson couplings to the known SM particles.
The data sample used in this analysis corresponds to integrated luminosities of 35.9 fb −1 collected in 2016 and 41.5 fb −1 collected in 2017 during Run 2 of the CERN LHC at a center-of-mass energy of 13 TeV. These results are combined with results obtained earlier from the data collected at center-of-mass energies of 7 TeV (in 2011), 8 TeV (in 2012, and 13 TeV (in 2015), corresponding to integrated luminosities of 5.1, 19.7, and 2.7 fb −1 , respectively [25,27]. The increase in either energy and integrated luminosity leads to substantial improvement in the precision of the width measurement using the off-shell technique, either under the assumption of SM couplings or with BSM effects.
This analysis follows closely the general H → 4l (leptons l ¼ e or μ) selection and reconstruction documented in Ref. [21] using the data collected in 2016, and the on-shell study of anomalous HVV couplings with the combined 2015 and 2016 data set in Ref. [27]. Many of the technical details of the search for a scalar resonance X → ZZ at high mass in Run 2 data, documented in Ref. [37], are also shared in the analyses presented here. The rest of the paper is organized as follows. The phenomenology of anomalous HVV interactions is discussed in Sec. II. The CMS detector, reconstruction techniques, and Monte Carlo (MC) simulation methods are introduced in Sec. III. The addition of the 2017 data to that used in Refs. [21,27], and the relevant differences in the detector and reconstruction techniques are also discussed in this section. The details of the analysis are discussed in Secs. IV and V, and the results are presented in Sec. VI. We provide a summary of these results in Sec. VII.

II. PHENOMENOLOGY OF ANOMALOUS HVV INTERACTIONS
The constraints on Γ H are set using the off-shell production method, which considers the H boson production relationship between the on-shell (105 < m 4l < 140 GeV) and off-shell (m 4l > 220 GeV) regions. Denoting each production mechanism with vv → H → VV → 4l for the H boson coupling to either strong (vv ¼ gg) or EW (vv ¼ WW; ZZ; Zγ; γγ) vector bosons in its production, the on-shell and off-shell H boson signal yields are related by [16] σ on-shell vv→H→4l ∝ μ vvH and σ off-shell vv→H→4l ∝ μ vvH Γ H ; ð1Þ where μ vvH is defined as the on-shell signal strength, the ratio of the observed number of on-shell four-lepton events relative to the SM expectation. This ratio is interpreted as either μ F for H boson production via gluon fusion (ggH) or in association with a tt (ttH) or bb pair (bbH), or μ V for H boson production via vector boson fusion (VBF) or in association with an EW vector boson W or Z (VH). There is sizable interference between the H boson signal and the continuum background in the off-shell region [17], contrary to on-shell production, and this formalism scales the interference contribution with ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ vvH Γ H p . This analysis is based on a phenomenological framework [22, that describes the anomalous couplings of a Higgs-like boson to two gauge bosons, such as WW; ZZ; Zγ; γγ, and gg. These couplings appear in either the production of the H boson or its decay, regardless of the m 4l region in which the H boson is produced. The relationship in Eq. (1) is therefore meant to imply concurrent variations in vvH couplings in both on-shell and off-shell regions. The coupling of the H boson to two gluons is assumed to be as in the SM, via quark loops with Yukawa couplings to quarks, where the contribution from the top-quark is dominant. This assumption is valid as long as the production is dominated by the top-quark loop and no new particles contribute to this loop. The Yukawa couplings also appear in direct interactions with fermionantifermion pairs, such as in ttH and bbH productions. These interactions are of less importance in this study, since they are highly suppressed at high off-shell mass, but they are included in the analysis of the on-shell H boson production with similar assumptions as in the case of production via gluon fusion. Variation of the HVV couplings, in either the VBF or VH productions, or the H → 4l decay, are allowed to depend on anomalous coupling contributions.
In the following, we assume that the H boson couples to two gauge bosons VV, such as WW, ZZ, Zγ or γγ, which in turn couple to fermions, either four leptons in H boson decay, or quarks or leptons in its production or in the decay of associated EW bosons. It is assumed that the H boson does not couple to fermions through a new heavy state, generating a so-called contact interaction [57,58]. However, the inclusion of amplitude terms pertaining to contact interactions is equivalent to the anomalous HVV couplings already tested [25] under the assumption of flavor universality in Vff couplings. Both approaches test three general tensor structures allowed by Lorentz symmetry, with form factors F i ðq 2 1 ; q 2 2 Þ in front of each term, where q 1 and q 2 are the four-momenta of the two difermion states, such as ðe þ e − Þ and ðμ þ μ − Þ in the H → e þ e − μ þ μ − decay, and equivalent states in production. We also fix all lepton and quark couplings to vector bosons according to SM expectations. Relaxing this requirement would make it equivalent to flavor nonuniversal couplings of the contact terms, but would also introduce too many unconstrained parameters, which cannot be tested with the present data sample. Only the lowest order operators, or lowest order terms in the ðq 2 j =Λ 2 Þ form-factor expansion, are tested, where Λ is the energy scale of new physics.
The signal scattering amplitude describing the interaction between a spin-zero H boson and two spin-one gauge bosons VV is written as [54] A ∼ In this expression of the scattering amplitude, ϵ i is the polarization vector of gauge boson is a scalar tensor constructed from this polarization vector and the momentum of the gauge boson, andf ðiÞ μν ¼ 1 2 ϵ μνρσ f ðiÞρσ is the pseudoscalar tensor counterpart. When at least one of the gauge bosons V is massive, m V1 is the pole mass of that gauge boson. The scales of BSM physics are denoted with Λ 1 and Λ Q , so a i VV , or 1=Λ 1 and 1=Λ Q , become the coupling-strength modifiers of the relevant HVV amplitudes, where a i VV may in general be any complex number, and jκ VV 1;2;3 j ¼ 0 or 1 are complex numbers. Under the assumption that the couplings are constant and real, the above formulation is equivalent to an effective Lagrangian notation. Therefore, in this paper, the real coupling constants are tested. The above approach allows a sufficiently general test of the H → 4l kinematics in decay and equivalent kinematics in production, as discussed below, including production and decay of virtual intermediate photons. If deviations from the SM are detected, a more detailed study of F i ðq 2 1 ; q 2 2 Þ could be performed, eventually providing a measurement of the double-differential cross section for each tensor structure tested.
In the above, the only leading tree-level contributions are a ZZ 1 ≠ 0 and a WW 1 ≠ 0, and in the following we assume the custodial symmetry a ZZ 1 ¼ a WW 1 . The rest of the couplings are considered anomalous contributions, which are either tiny contributions arising in the SM due to loop effects or new BSM contributions. The SM loop contributions are not accessible experimentally with the available data. Among anomalous contributions, considerations of symmetry and gauge invariance require κ ZZ While not strictly required, the same symmetry is considered in the WW case κ WW Neither HZγ nor Hγγ couplings produce a sizable offshell enhancement, since there is no interplay between the vector bosons or the H boson going off-shell, and there is no off-shell threshold for these couplings. Therefore, offshell treatment for these couplings can be ignored. While the a Zγ 2;3 and a γγ 2;3 terms are tested in the Run 1 analysis [25], the precision of those constraints is still not competitive with the on-shell photon measurements in H → Zγ and γγ. Therefore, we omit those measurements in this paper. The Λ Zγ 1 coupling, on the other hand, can only be observed with off-shell photons decaying to a pair of fermions, so it is considered in the on-shell analysis. The Λ Q term depends only on the invariant mass of the H boson, so its contribution is not distinguishable from the SM in the on-shell region and is only testable through the off-shell region. Tight constraints are already set on this parameter in the Run 1 analysis [13], so it is also not considered in this paper.
In the following, the ZZ labels for the ZZ interactions are omitted, and we use a generic notation a i to denote a 3 , a 2 , 1=Λ 1 , and 1=Λ Zγ 1 , which are the four couplings tested in this paper as listed in Table I. Furthermore, the WW measurements are integrated into the ZZ measurements assuming a ZZ i ¼ a WW i . The HWW contributions appear in the VBF and WH productions. This assumption does not affect the kinematic analysis of events because there is very little difference in kinematic distributions in events initiated by either WW or ZZ fusion. However, this assumption may affect the interpretation of the results should a different relationship between a ZZ i and a WW i be assumed. Therefore, such a scenario is discussed in more detail below by introducing the parameter r ai , following Ref.
[25], as Including the parameter r ai in the probability parametrization despite the lack of sensitivity of the data would introduce complexity without a comparable gain in physics content. We proceed with the analysis assuming r ai ¼ 1,  [47,50,54].

Anomalous coupling
Coupling phase Effective fraction Translation constant but point out below how results could be reinterpreted should a different value be assumed. Most systematic uncertainties cancel when taking ratios to the total cross section, so measurements of a i relative to the dominant SM-like contribution a 1 are the preferred approach. For this purpose, the effective fractional ZZ cross sections f ai and phases ϕ ai are defined as where σ i is the cross section for the process corresponding to a i ¼ 1, a j≠i ¼ 0, whileσ Λ1 is the effective cross section for the process corresponding to Λ 1 ¼ 1 TeV, given in units of fb TeV 4 . The cross-section ratios are quoted in Table I. The a i =a 1 ratios can be obtained from the ratio f ai =f a1 , the cross-section ratios, and the phase ϕ ai as The effective fractions f ai are bounded between 0 and 1 and do not depend on the coupling convention. In most cases, uncertainties on these measurements scale with integrated luminosity as 1= ffiffiffi ffi L p until effects of interference become important. Furthermore, the values of f ai have a simple interpretation as the fractional size of the BSM contribution for the H → 2e2μ decay. For example, f ai ¼ 0 indicates a pure SM-like H boson, f ai ¼ 1 gives a pure BSM particle, and f ai ¼ 0.5 means that the two couplings contribute equally to the H → 2e2μ process.
As mentioned above in application to Eq. (3), the measurement of f ai is performed under the r ai ¼ 1 assumption. Let us denote this to be an effective f eff ai . Without such an assumption, there is a certain dependence of f ai on r ai and f eff ai , such that f ai ¼ f eff ai for r ai ¼ 1. This dependence is different for different processes, such as VBF production or H → 4l decay, where the latter case is in fact independent of r ai because the HWW coupling does not affect this decay process. In the former case, let us consider the relative contributions of WW and ZZ fusion on-shell. For example, the ratio of VBF cross sections driven by WW and ZZ fusion is σ WW 1 =σ ZZ 1 ¼ 2.59 for the SM tree-level couplings under custodial symmetry a WW 1 ¼ a ZZ 1 at 13 TeV pp collision energy. The same ratio for the CP-odd couplings is σ WW The dependence of f ai on r ai and f eff ai , as measured in the VBF process, becomes where custodial symmetry a WW 1 ¼ a ZZ 1 is assumed and the effects of interference between WW and ZZ fusion are negligible and are therefore ignored.
All of the above discussion, including Eq. (2), describes the production of a resonance via gluon fusion, VBF with associated jets, or associated production with an EW vector boson, VH. These mechanisms, along with the ttH and bbH production, are considered in the analysis of the spinzero hypothesis of the H boson, where the gluon fusion production is expected to dominate. It is possible to study HVV interactions using the kinematics of particles produced in association with the H boson, such as VBF jets or vector boson daughters in VH production, as we show below. More details can be found in, e.g., Ref. [54] and the experimental application in Refs. [26,27]. While the q 2 i range in the HVV process does not exceed approximately 100 GeV because of the kinematic bound, no such bound exists in the associated production, so consideration of more restricted q 2 i ranges might be required [54]. However, we only consider that the q 2 i range is not restricted in the allowed phase space.

III. THE CMS DETECTOR, SIMULATION, AND RECONSTRUCTION
The H → 4l decay candidates are reconstructed in the CMS detector [60]. The CMS detector is comprised of a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass/scintillator hadron calorimeter, each composed of a barrel and two end cap sections, all within a superconducting solenoid of 6m internal diameter, providing a magnetic field of 3.8 T. Extensive forward calorimetry complements the coverage provided by the barrel and end cap detectors. Outside the solenoid are the gas-ionization detectors for muon measurements, which are embedded in the steel flux-return yoke. A detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [60]. The JHUGEN 7.0.2 [47,50,54,59] Monte Carlo (MC) program is used to simulate anomalous couplings in the H boson production and H → ZZ=Zγ Ã =γ Ã γ Ã → 4l decay. The gluon fusion production is simulated with the POWHEG 2 [61][62][63][64][65] event generator at next-to-leading order (NLO) in QCD, and simulation with the MINLO [66] program at NLO in QCD is used for evaluation of systematic uncertainties related to modeling of two associated jets. The kinematics of events produced in gluon fusion with two associated jets are also modified by anomalous Hgg couplings. These effects are studied using JHUGEN, and it is found that the kinematic distributions relevant for this analysis are not affected significantly.
The production of the H boson through VBF, in association with a W or Z boson, or with a tt pair, is simulated using both JHUGEN at LO in QCD and POWHEG A. M. SIRUNYAN et al. PHYS. REV. D 99, 112003 (2019) at NLO in QCD. Production in association with a bb pair is simulated only at LO in QCD via JHUGEN. In the VBF, VH, and ttH production modes, the JHUGEN and POWHEG simulations are explicitly compared after parton showering in the SM case, and no significant differences are found in kinematic observables. Therefore, the JHUGEN simulation is adopted to describe kinematics in the VBF, VH, and ttH production modes with anomalous couplings in the onshell region, with expected yields taken from the POWHEG simulation. The POWHEG program is used to simulate wide resonances at masses ranging from 115 GeV to 3 TeV, produced in gluon fusion, VBF, or VH. The events from the POWHEG simulation are later reweighted using the package for the matrix element likelihood approach (MELA) [9,47,50,54,59] to model off-shell H boson production distributions, as discussed below.
The gg → ZZ=Zγ Ã → 4l background process is simulated with MCFM 7.0.1 [18,[67][68][69]. The vector boson scattering and triple-gauge-boson (VVV) backgrounds are obtained by reweighting the POWHEG simulation with the matrix elements provided by the MELA package using the MCFM and JHUGEN matrix elements, and the reweighted simulation is checked against the predictions of the PHANTOM 1.3 [70] simulation. Both the MCFM and PHANTOM generators allow one to model the H boson signal, background, and their interference in the off-shell production. However, they do not allow modeling of the anomalous interactions considered in this analysis. Therefore, a combined program has been developed for both gluon fusion and VBF with triple-gauge-boson production based on the modeling of signal and background scattering amplitudes from MCFM and anomalous contributions in the signal scattering amplitude from JHUGEN. This program is included within the JHUGEN and MELA packages, as detailed in Ref. [22]. A large number of MC events with anomalous couplings in the signal and their interference with background have been generated with these packages. The simulated events also include alternative weights to model various anomalous couplings in the signal.
In the gluon fusion process, the factorization and renormalization scales are chosen to be running as m 4l =2. In order to include higher-order QCD corrections, LO, NLO, and next-to-NLO (NNLO) signal cross-section calculations are performed using the MCFM and HNNLO 2 programs [71][72][73] for a wide range of masses using a narrow width approximation. The ratios between the NNLO and LO values (NNLO K factors) are used to reweight [22] the m 4l distributions from the MCFM and JHUGEN simulation at LO in QCD, and a uniform factor of 1.10 across all of the m 4l range is applied to normalize the cross section of the H boson production via gluon fusion to the predictions for m 4l ≈ 125 GeV at next-to-NNLO (N 3 LO) in QCD [22]. The simulated m 4l shapes or yields obtained from the POWHEG simulation of the gluon fusion process are corrected based on the above reweighted distributions. While the NNLO K factor calculation is directly applicable to the signal contribution, it is approximate for the background and its interference with the signal. The NLO calculation with some approximations [74][75][76][77] is available for the background and interference. Comparison with this calculation shows that while there is some increase of the NLO K factor for the interference close to the ZZ threshold, the NLO K factors for the background and interference are consistent with the signal within approximately 10% in the mass range m 4l > 220 GeV relevant for this analysis. We therefore multiply the background and interference contributions by the same NNLO K factor and uniform N 3 LO correction, both calculated for signal and including associated uncertainties, and introduce an additional unit factor with a 10% uncertainty for the background and the square root of this factor for the interference.
The MELA package contains a library of matrix elements from JHUGEN and MCFM for the signal, and MCFM for the background, and is used to apply weights to events in any MC sample to model any other set of anomalous or SM couplings in either on-shell or off-shell production. This matrix element library also allows reweighting of the signal POWHEG simulation of the wide resonances at NLO in QCD in either gluon fusion, VBF, or triple-gauge-boson production to model the signal, background, or their interference.
The main background in this analysis, qq → ZZ=Zγ Ã → 4l, is estimated from simulation with POWHEG. A fully differential cross section has been computed at NNLO in QCD [78], but it is not yet available in a partonic level event generator. Therefore the NNLO/NLO QCD correction is applied as a function of m 4l . Additional NLO EW corrections are also applied to this background process in the region m 4l > 2m Z [79,80]. The parton distribution functions (PDFs) used in this paper belong to the NNPDF 3.0 PDF sets [81]. All MC samples are interfaced to PYTHIA 8 [82] for parton showering, using version 8.212 for the simulation of the 2016 data period and 8.230 for the simulation of the 2017 data period. Simulated events include the contribution from additional pp interactions within the same or adjacent bunch crossings (pileup), and are weighted to reproduce the observed pileup distribution. The MC samples are further processed through a dedicated simulation of the CMS detector based on GEANT4 [83].
The selection of 4l events and associated particles closely follows the methods used in the analyses of the Run 1 [24] and Run 2 [21] data sets. The main triggers for the Run 2 analysis select either a pair of electrons or muons, or an electron and a muon. The minimal transverse momentum of the leading electron (muon) is 23 (17) GeV, while that of the subleading lepton is 12 (8) GeV. To maximize the signal acceptance, triggers requiring three leptons with lower p T thresholds and no isolation requirement are also used, as are isolated single-electron and single-muon triggers with thresholds of 27 and 22 GeV in 2016, or 35 and 27 GeV in 2017, respectively. The overall trigger efficiency for simulated signal events that pass the full selection chain of this analysis is larger than 99%. The trigger efficiency is measured in data using a sample of 4l events collected by the single-lepton triggers and is found to be consistent with the expectation from simulation.
Event reconstruction is based on the particle-flow (PF) algorithm [84], which exploits information from all the CMS subdetectors to identify and reconstruct individual particles in the event. The PF candidates are classified as charged hadrons, neutral hadrons, photons, electrons, or muons, and they are then used to build higher-level objects such as jets and lepton isolation quantities. Electrons (muons) are reconstructed within the geometrical acceptance defined by a requirement on the pseudorapidity jηj < 2.5ð2.4Þ for transverse momentum p T > 7ð5Þ GeV with an algorithm that combines information from the ECAL (muon system) and the tracker. A dedicated algorithm is used to collect the final-state radiation (FSR) of leptons [21].
The reconstructed vertex with the largest value of summed physics-object p 2 T is taken to be the primary pp interaction vertex. The physics objects are the jets and the associated missing transverse momentum, taken as the negative vector sum of the p T of those jets. The jets are clustered using the anti-k T jet finding algorithm [85,86] with a distance parameter of 0.4 and the associated tracks assigned to the vertex as inputs. Jets must satisfy p T > 30 GeV and jηj < 4.7 and must be separated from all selected lepton candidates and any selected FSR photons with a requirement on the distance parameter ΔRðl=γ; jetÞ > 0.4, where ðΔRÞ 2 ¼ ðΔϕÞ 2 þ ðΔηÞ 2 . For event categorization, jets are tagged as b-jets using the Combined Secondary Vertex algorithm [87,88], which combines information about impact parameter significance, the secondary vertex, and jet kinematics.
Each lepton track is required to have the ratio of the impact parameter in three dimensions, which is computed with respect to the chosen primary vertex position, and its uncertainty to be less than 4. To discriminate between leptons from prompt Z boson decays and those arising from hadron decays within jets, an isolation requirement for leptons is imposed in the analysis of the 2016 data [21]. For electrons, the isolation variable is included as part of the multivariate training inputs for electron identification in 2017.
We consider three mutually exclusive channels: H → 4e, 4μ, and 2e2μ. At least two leptons are required to have p T > 10 GeV, and at least one is required to have p T > 20 GeV. All four pairs of oppositely charged leptons that can be built with the four leptons are required to satisfy m l þ l 0− > 4 GeV regardless of lepton flavor. The Z candidates are required to satisfy the condition 12 < m l þ l − < 120 GeV, where the invariant mass of at least one of the Z candidates must be larger than 40 GeV. The region between 105 and 140 GeV in the four-lepton invariant mass m 4l is identified as the on-shell region, and the region above 220 GeV is identified as the off-shell region.
Different sources of leptons such as the decays of heavy flavor jets or light mesons may produce additional background to the H boson signal in any of these decay channels, or the on-shell and off-shell regions. We denote this background collectively as the Z þ X background, and employ a data-driven method for its estimation and m 4l dependence. The lepton misidentification rates are first derived using Z þ 1l control regions with relaxed selection requirements on the third lepton, and the extracted rates are then applied on Z þ 2l control regions, where the two additional leptons with relaxed selection requirements have the same lepton flavor but may have opposite charge [21,24].

IV. ANALYSIS TECHNIQUES AND CATEGORIZATION OF EVENTS
The full kinematic information from each event using either the H boson decay or associated particles in its production is extracted using discriminants from matrix element calculations. These discriminants use a complete set of mass and angular input observables Ω [47,54,59] to describe kinematics at LO in QCD. The p T of either the combined H boson and two-jet system for the production discriminant (e.g., D VBF=VH ), or the H boson itself for the decay discriminants (e.g., D dec ), or for their combination (e.g., D VBF=VHþdec ) is not included in the input observables. This information is not used in the analysis of the H boson width and anomalous couplings, as the p T of the overall system is sensitive to QCD, parton shower, and underlying event uncertainties.
The kinematic discriminants used in this study are computed using the same MELA package that is utilized in simulation. The signal includes both the fourlepton decay kinematics in the processes H → ZZ=Zγ Ã = γ Ã γ Ã → 4l, and kinematics of associated particles in production H þ jet, H þ 2jets, VBF, WH, ZH, ttH, tqH, or bbH. The background includes gg or qq → ZZ=Zγ Ã =γ Ã γ Ã =Z → 4l processes, and VBF or associated production with a V boson of the ZZ system. Analytical algorithms are available for the cross-checks of the fourlepton kinematics in H decay and VH associated production within the MELA framework and were adopted in the previous CMS analyses [9,10,23].
Kinematic distributions of particles produced in the H boson decay or in association with H boson production are sensitive to the quantum numbers and anomalous couplings of the H boson. In the 1 → 4 process of the H → 4f decay, six observables fully characterize kinematics of the decay products Ω decay ¼ fθ 1 ; θ 2 ; Φ; m 1 ; m 2 ; m 4f g, while two other angles relate orientation of the decay frame with respect to the production axis, Ω prod ¼ fθ Ã ; Φ 1 g, as described in Ref. [47]. Moreover, two sets of observables, Ω assoc;VBF ¼ fθ VBF way to Ω decay for H boson associated production [54]. As a result, 13 kinematic observables, illustrated in Fig. 1, are defined for the 2 → 6 associated production process with subsequent H boson decay to a four-fermion final state.
With up to 13 observables, Ω, sensitive to the H boson anomalous couplings in Eq. (2), it is a challenging task to perform an optimal analysis in a multidimensional space of observables. The MELA approach introduced earlier is designed to reduce the number of observables to the minimum while retaining all essential information. Two types of discriminants were defined for either the production or decay process, and we also combine them into a joint discriminant for the full 2 → 6 process where relevant.
These types of discriminants are and where the probability of a certain process P is calculated using the full kinematics characterized by Ω for the processes denoted as "sig" for a signal model and "alt" for an alternative model, which could be an alternative H boson production mechanism (used to categorize events), background (to isolate signal), or an alternative H boson coupling model (to measure coupling parameters). The "int" label represents the interference between the two model contributions. The probabilities P are calculated from the matrix elements provided by the MELA package and are normalized to give the same integrated cross sections in the relevant phase space of each process. Such normalization leads to a balanced distribution of events in the range between 0 and 1 of the D alt discriminants, and between −1 and 1 of D int . One can apply the Neyman-Pearson lemma to prove that the two discriminants in Eqs. (7) and (8) become the minimal and complete set of optimal observables for the purpose of separating the two processes "sig" and "alt" while including their interference as well [54,59]. The selected events are split into three categories: VBFtagged, VH-tagged, and untagged. A set of discriminants D 2jet is constructed, following Eq. (7), where P sig corresponds to the signal probability for the VBF (WH or ZH) production hypothesis in the VBF-tagged (VH-tagged) category, and P alt corresponds to that of H boson production in association with two jets via gluon fusion. When more than two jets pass the selection criteria, the two jets with the highest p T are chosen for the matrix element calculations. Thereby, the D 2jet discriminants separate the target production mode of each category from gluon fusion production, in all cases using only the kinematics of the H boson and two associated jets. Figure 2 illustrates these discriminants, designed for the VBF or VH signal enhancement in the a 3 coupling analysis for a pseudoscalar contribution. A selection based on the D bkg observable, which utilizes information from the 4l decay kinematics and invariant mass, and which is discussed in more detail below, is applied in order to enhance the contribution of the signal over the background.
The three on-shell and off-shell categories are summarized in Tables II and III, and their sequential selection criteria are as follows: (i) VBF-tagged requires exactly four leptons, either two or three jets of which at most one is b-quark flavortagged, or more if none are b-tagged jets, and D VBF 2jet > 0.5 using either the SM or BSM signal hypothesis for the VBF production. In the first two cases the production and decay H → VV are followed by the same four-lepton decay shown in the third case. The angles are defined in either the H or V boson rest frames [47,54].
2jet Þ > 0.5 using either the SM or BSM signal hypothesis for the VH production. (iii) Untagged consists of the remaining events. The requirements on the number of b-tagged jets are applied to reduce crossfeed from ttH production. Even though VH cross sections are significantly lower with respect to VBF for m 4l > 220 GeV, the VH cross section becomes comparable to the VBF cross section in the presence of anomalous couplings. Therefore, the off-shell analysis also benefits from featuring the VH-tagged category with hadronic decays of the associated V. In either the on-shell or off-shell regions, events are not tagged for the smaller VH contribution with leptonic V decays explicitly, but this contribution is taken into account in the simulation and parametrization of the VH process in the three different categories. The expected and observed numbers of events are listed in Table IV for the on-shell region and Table V for the off-shell region.
In each category of events, typically three observables ⃗ x are defined following Eqs. (7) and (8), as summarized in Tables II and III. In the on-shell region, except for the SMlike analysis, these are ⃗ x ¼ fD bkg ; D ai ; D int g. The first observable, D bkg , is calculated differently in the three tagged categories. In the untagged category, P bkg is calculated for the dominant qq → 4l background process. The signal and background probabilities include both the Þ (right) in the on-shell region in the data from 2016 and 2017 from the analysis of the a 3 coupling for a pseudoscalar contribution. The requirement D bkg > 0.5 is applied in order to enhance the signal contribution over the background. The VBF signal under both the SM and pseudoscalar hypotheses is enhanced in the region above 0.5 for the former variable, and the WH and ZH signals are similarly enhanced in the region above 0.5 for the latter variable.
TABLE II. Summary of the three production categories in the on-shell m 4l region. The selection requirements on the D 2jet discriminants are quoted for each category, and further requirements can be found in the text. Two or three observables (abbreviated as obs.) are listed for each analysis and for each category. All discriminants are calculated with the JHUGEN signal matrix elements and MCFM background matrix elements. The discriminants D bkg in the tagged categories also include probabilities using associated jets and decay in addition to the m 4l probability. The VH interference discriminants in the hadronic VH-tagged categories are defined as the simple average of the ones corresponding to the WH and ZH processes. and VH-tagged categories, P bkg and P sig include fourlepton kinematics and the m 4l probability parametrization, but they also include kinematics of the two associated jets. The P bkg probability density represents the EW and QCD   background processes 4l þ 2 jets, while P sig represents EW processes VBF and VH. It was found that jet kinematics in the D bkg calculation improves separation of the targeted signal production both against background and against the H boson gluon fusion production. However, in the off-shell region and in the SM-like on-shell analysis, the four-lepton invariant mass m 4l is one of the most important observables, because the mass parametrization becomes an important feature of the analysis. Therefore, the m 4l parametrization is not used in the D bkg calculation in these cases, and this is reflected with the superscript denoting which information is used, either with decay only information in D kin bkg or with both decay and production in D VBFþdec bkg and D VHþdec bkg . The other observable, D ai , separates the SM hypothesis f ai ¼ 0 as P sig from the alternative hypothesis f ai ¼ 1 as P alt , following Eq. (7). In the untagged category, the   probabilities are calculated using only the decay information, and the D ai observable is called D 0− in the a 3 , D ohþ in the a 2 , D Λ1 in the Λ 1 , and D Zγ Λ1 in the Λ Zγ 1 analyses [25]. In the VBF-tagged and VH-tagged categories, both the production and decay probabilities are used, with the matrix elements calculated as the product of the decay component and the component from either VBF production or ðWH þ ZHÞ associated production, respectively [27]. The resultant set of D ai discriminants are called in a similar manner to their counterparts in the untagged category but indicating the production assumption in their upper index.   The last observable, D int defined in Eq. (8), separates the interference of the two amplitudes corresponding to the SM-like H boson coupling and the alternative H boson coupling model, or the SM-like H boson coupling and background as an alternative model in the case of D bsi for the signal-background interference in the off-shell region. In the case of the a 3 analysis, this observable is called D CP because if CP is violated it would exhibit a distinctive forward-backward asymmetry. In the untagged category, decay information is used in the calculation of D int . In the VBF-tagged and VH-tagged categories, production information with the two associated jets is used. The D bsi discriminant extends the idea of the D gg discriminant introduced in Ref. [11] for the H boson width measurement, but allows independent treatment of the interference component. It is used only in the SM-like analysis.
The distributions of events for several of the observables ⃗ x from Tables II and III are illustrated in Fig. 3 for the onshell and in Fig. 4 for the off-shell regions. In Figs. 3 and 4, cross sections of all background processes are fixed to the SM expectations, except for the Z þ X background estimated from the data control regions discussed above. Cross sections of all signal processes, including BSM, are normalized to the SM expectations in the on-shell region.

V. THE FIT IMPLEMENTATION
We perform an unbinned extended maximum likelihood fit [89] to the events split into several categories (enumerated with an index k below) according to the three lepton flavor combinations (4e, 4μ, and 2e2μ), three production categories (VBF-tagged, VH-tagged, and untagged), five data periods (2011, 2012, 2015, 2016, and 2017), and two mass ranges (on-shell and off-shell). Therefore, there could be up to 90 categories of events. However, not all categories are used in each independent measurement because of the simpler categorization approach applied to the earlier data. Here we focus on discussion of the 2016 and 2017 data analyses, while treatment of the earlier data can be found in Refs. [13,25,27].
An independent fit is performed for each of the four anomalous HVV coupling parameters f ai cosðϕ ai Þ using the on-shell region only. These fits avoid any assumptions on how the behavior of each process considered in the analysis changes from the on-shell region to the off-shell region. Four independent joint fits to the on-shell and offshell regions are performed in order to determine the width of the H boson under the SM-like assumption or in the presence of the three anomalous couplings a 3 , a 2 , and Λ 1 . These fits are also used to constrain the three corresponding anomalous coupling parameters f ai cosðϕ ai Þ. When a certain anomalous coupling is tested, all other anomalous couplings are assumed to be zero, and only real couplings in Eq. (2) are tested, that is with a 1 ≥ 0 and cosðϕ ai Þ ¼ AE1.
The on-shell analysis with the study of the a 3 , a 2 , Λ 1 , and Λ Zγ 1 couplings has been presented in Ref.
[27] using a partial data set. This part of the analysis remains essentially unchanged, except for a small change in the definition of the interference discriminant in Eq. (8) and the inclusion of information from the kinematics of the two associated jets in the D bkg calculation discussed in Sec. IV. The SM-like on-shell analysis is similar to the one presented in Ref. [21] in methodology, but it uses the observables ⃗ x and categorization k described in Table II and Sec. IV. The on-shell probability density is normalized to the total event yield in each process j and category k according to where ⃗ ζ ¼ ðμ F ; μ V ; Γ H ; f ai cosðϕ ai ÞÞ are the unconstrained parameters of interest, ⃗ ξ jk are the constrained nuisance parameters for a particular parametrization, and ⃗ x are the observables listed in Table II, specific to each a i . The onshell signal strength μ j in Eq. (9) is defined in references to Eq. (1) as either μ F or μ V according to the process type j (gg, VBF, WH, ZH, ttH, bbH, qq → 4l, and Z þ X). Each process includes both signal (sig) and background (bkg) components, but may contain only signal (ttH and bbH) or only background (qq → 4l and Z þ X) contributions in the particular cases. The interference between the signal and background components, when both are present, is negligible in the on-shell region because of the very small width Γ H compared to the mass range of interest. This also leads to the on-shell parametrization in Eq. (9) being independent from the width Γ H .
The off-shell probability density follows Eqs. (1) and (9) closely but with the additional contribution of interference (int) between the signal and background amplitudes as where the notation remains the same as for Eq. (9). The ⃗ x observables are listed in Table III and are specific to each coupling analysis. They include m 4l and two other discriminants. The process type j does not include ttH and bbH because of their negligible contribution in the off-shell region, while the VBF, WH, and ZH processes are combined into one EW process. The parametrization in Eq. (10) depends on the width Γ H explicitly and the reference value is taken to be Γ 0 ¼ 4.07 MeV, which determines the relative strength of P sig jk and P int jk with respect to P bkg jk in the parametrization. The EW H boson production (VBF and VH) or production via gluon fusion have different dependence on anomalous HVV couplings, equally in the on-shell or offshell regions. There are two HVV vertices in the former production mechanism with the subsequent H → VV → 4l decay while there is only one HVV decay vertex in the latter case. In addition, there is interference with the background in the off-shell region. This leads to the following general expressions for the signal (sig) or interference (int) contributions appearing in Eqs. (9) and (10): where the sum over the index m runs up to M ¼ 4 in the case of the EW signal process; M ¼ 2 in the case of the gluon fusion, ttH, and bbH signal processes, or the interference between the signal and background in the EW process; and M ¼ 1 in the case of the interference between the signal and background in the gluon fusion process. In this expression, the index m corresponds to the exponent of a i in the squared scattering amplitude from Eq. (2), which may contain contributions from production and decay, and the factor cosðϕ ai Þ ¼ AE1 affects only the sign of the terms that scale with an odd power of a i . The P sig=int jk;m and P bkg jk probability densities are normalized to the expected number of events, and are binned histograms (templates) of the observables ⃗ x listed in Tables II and III, except for the signal m 4l parametrization in the on-shell region as discussed below. These templates are obtained by reweighting the existing signal or background samples for different couplings and then finding their linear combination. Since m 4l is treated directly as an observable in the on-shell SM-like fit, the signal m 4l shape for each process j and category k is parametrized using a double-sided crystal-ball function [90], and the full signal probability density is parametrized as the product of the parametric m 4l shape and a template of other discriminants conditional in m 4l . In all cases, the H boson mass m H ¼ 125 GeV is assumed.
The final constraints on f ai cosðϕ ai Þ and Γ H are placed using the profile likelihood method using the RooFit toolkit [91] within the ROOT [92] framework. The extended likelihood function is constructed using the probability densities in Eqs. (9) and (10) with each event characterized by the discrete category k and typically three continuous observables ⃗ x. The likelihood L is maximized with respect to the nuisance parameters ⃗ ξ jk describing the systematic uncertainties discussed below and the yield parameters μ F and μ V . The allowed 68% and 95% C.L. intervals are defined using the profile likelihood function, −2Δ ln L ¼ 1.00 and 3.84, for which exact coverage is expected in the asymptotic limit [93].
Several systematic uncertainties are featured in the vectors of constrained parameters ⃗ ξ jk . The template shapes describing probability distributions in Eqs. (9), (10), and (11) are varied separately within either theoretical or experimental uncertainties. In the following, a range of uncertainties affecting the template distributions is given for the m 4l values from around 100 GeV (typical for the onshell range) to around 1 TeV (in the off-shell range), respectively. The factorization (or renormalization) scale uncertainties are evaluated by multiplying the central scale by 2 or 1=2, and the uncertainties range from AE0.7% in the processes with an associated EW boson, and from þ3.5 −5.5 % to AE1% (AE3%) in the qq → 4l background. PDF parametrization uncertainties are evaluated by taking the envelope of the 100 alternative NNPDF variations. Variations due to PDF parametrization uncertainties [or due to uncertainties in −4 % to about þ30 −25 % in the EW processes, and are approximately AE3% (from þ1.0 −1.8 % to AE0.5%) for the qq → 4l background. The signal processes, and the backgrounds that interfere with the signal, feature the uncertainties as a function of the multiplicity and kinematics of associated jets due to the hadronization scale used in PYTHIA and the underlying event variations, obtained with the variations of the PYTHIA tune. In the VBF-tagged categories, the correlated template variations for the hadronization scale (underlying event) range from AE11% (AE45%) to ∓8% (∓40%) in the gg process, from AE8% (AE24%) to ∓6% (∓8%) in the VBF process, and from AE13% (AE20%) to ∓10% (∓32%) in the processes with an associated EW boson. In the VH-tagged categories, these correlated template variations instead range from AE15% (AE50%) to ∓9% (∓45%) in the gg process, from AE8% (AE25%) to ∓7% (∓30%) in the VBF process, and from AE4% (AE19%) to ∓4% (∓13%) in the processes with an associated EW boson. Template shapes in the gg processes are also varied to account for a second jet in the hard process, and these correlated variations range from AE18% (AE32%) to ∓15% (∓14%) in the VBF-tagged (VHtagged) category. The qq → 4l background further features an uncertainty in the NLO EW corrections applied to the simulation [79,80], which are significant at higher m 4l values, reaching up to 20% at 1 TeV.
Experimental uncertainties involve jet energy calibration (JEC) uncertainties, which are only relevant when production categories are considered, and lepton efficiency and momentum uncertainties, which are similar for the different processes and categories. Systematic uncertainties in the JEC account for variations in the VBF-tagged (VH-tagged) category, and range from AE13% (AE4%) to AE8% (AE1%) in the gg process, from AE5% ( −10 þ2 %) to about AE11% (AE6%) in the VBF process, from AE9% (AE4%) to AE12% (AE1%) in processes with an associated EW boson, and from AE17% (AE8%) to AE15% ( þ2.0 −0.5 %) for the qq → 4l background. The cross-section uncertainties due to electron (muon) efficiency range from þ6 −7 % ( þ3.0 −4.5 %) to þ3.5 −4.5 % ( þ0.8 −2.0 %) to þ7 −8 % ( þ0.8 −2.0 %) in the 2e2μ channel, and roughly double for the 4e (4μ) channel, from m 4l ∼ 100 GeV to 230 GeV to around 1 TeV. In the estimation of the Z þ X background, the flavor composition of hadronic jets misidentified as leptons may be different in the Z þ 1l and Z þ 2l control regions, and together with the statistical uncertainty in the Z þ 2l region, this uncertainty accounts for about AE30% variation in the background estimate from the 2017 data set. The uncertainty on the modeling of this misidentification as a function of p T and η, combined with the Z þ 1l control region statistical uncertainty, leads to a þ20 −12 % to þ30 −27 % variation in the 4e channel, AEð10-20Þ% variation in the m 4l shape in the 2e2μ channel, and AE4% to þ14 −17 % variation in the 4μ channel. Uncertainties in the Z þ X background in the 2016 data set are only slightly larger. The normalization of the background processes derived from the MC simulation is affected by the uncertainties in the integrated luminosity of 2.5% [94] and 2.3% [95] in the 2016 and 2017 data sets, respectively. The integrated luminosity is measured using data from the CMS silicon pixel detector, drift tubes, and the forward hadron calorimeters, or from the fast beam conditions monitor and pixel luminosity telescope. All systematic uncertainties are treated as correlated between different time periods except for the luminosity and jet-related uncertainties which originate from statistically independent sources.

VI. RESULTS
Four f ai cosðϕ ai Þ parameters sensitive to anomalous HVV interactions, as defined in Eqs. (2) and (6), are tested in the on-shell data sample using the probability densities defined in Eq. (9). Since only the real couplings are tested, cosðϕ ai Þ ¼ AE1. Figure 5 shows the results of the likelihood scans of these parameters for the 2016 and 2017 periods of the 13 TeV run and for the full combined data set from collisions at 7, 8, and 13 TeV. The analysis of the 2016 and 2017 data uses the approach presented here with the observables sensitive to anomalous couplings in both production and decay. Because of the smaller numbers of events, the data from the 2015 period of the 13 TeV run and from the 2011 and 2012 periods of the 7 and 8 TeV runs are analyzed using only the decay information as in Refs. [25,27], which is equivalent to having all events in the untagged category of this analysis. The results from onshell events in the combined data set are listed in Table VI. These results supersede our previous measurements of these parameters in Refs. [25,27].
The observed and expected 68% C.L. constraints are significantly tighter than in the Run 1 analysis [25] as it is evident from the narrow minima at f ai ¼ 0 in Fig. 5. This effect comes from utilizing production information because the cross section in VBF and VH production increases quickly with f ai . Moreover, the minima of the −2 ln L distributions appear rather sharp because of the higher order polynomial of the f ai parameters appearing in Eq. (11)  The combination of on-shell and off-shell regions allows the setting of tighter constraints on f ai cosðϕ ai Þ using the probability densities defined in Eqs. (9) and (10). As discussed above, the on-shell region is analyzed using the 2015, 2016, and 2017 data, and the earlier Run 1 data. The off-shell region is analyzed using only 2016 and 2017 data because no such analysis of the three anomalous couplings has been performed with the Run 1 or 2015 data in this region. The one-parameter likelihood scans of f ai cosðϕ ai Þ combining all such available on-shell and off-shell events is shown for two cases in Fig. 6, either with Γ H unconstrained in the fit or with the constraint Γ H ¼ Γ SM H . The corresponding 68% and 95% C.L. constraints are summarized in Table VI. The full two-parameter likelihood scans of f ai cosðϕ ai Þ and Γ H are likewise shown in Fig. 6. Using the transformation in Eq. (5), the f ai cosðϕ ai Þ results can be interpreted for the coupling parameters used in Eq. (2), as shown in Table VII. Limits on Γ H are set by combining events from the onshell and off-shell regions. The left-hand panel of Fig. 7 shows the results of the likelihood scans of Γ H for the 2016 and 2017 period of the 13 TeV run and for the combined data set from collisions at 7, 8 and 13 TeV under the assumption of SM-like couplings. The small contribution from the 2015 data set is not considered in this case, but the Run 1 analysis includes both the on-shell and off-shell regions in the analysis of the H → ZZ → 4l decay [11,13]. VI. Summary of allowed 68% C.L. (central values with uncertainties) and 95% C.L. (in square brackets) intervals for the anomalous coupling parameters f ai cosðϕ ai Þ obtained from the analysis of the combination of Run 1 (only on-shell) and Run 2 (on-shell and off-shell) data sets. Three constraint scenarios are shown: using only on-shell events, using both on-shell and off-shell events with the Γ H left unconstrained, or with the constraint Γ H ¼ Γ SM H .  Table IX. These results are obtained with the same fit configurations as for the study of anomalous couplings in the combination of the on-shell and off-shell regions.
The systematic uncertainties mostly cancel in the ratios of cross sections in the measurement of fractional parameters f ai cosðϕ ai Þ, and are therefore negligible. The width constraints are also dominated by the statistical uncertainties, but because of the nontrivial dependence of systematic uncertainties on m 4l , their dominant contributions may be worth examination. The two leading theoretical and two leading experimental uncertainties affecting the width constraints (observed and expected at 68% C.L.) are the uncertainty on the NLO EW corrections for the qq → 4l background (AE0.5 and AE1.9 MeV), the variation of renormalization scale in gluon fusion (AE0.2 and AE0.4 MeV), the muon efficiency uncertainty (AE0.1 and AE0.4 MeV), and the electron efficiency uncertainty (AE0.1 and AE0.3 MeV).
The width constraints could also be reinterpreted as an off-shell signal strength with a change of parameters. For this interpretation, we perform an SM-like analysis of only the off-shell events, where the signal strength is modified by the parameter μ off-shell common to all production mechanisms in Eqs. (1) and (10), with Γ H ¼ Γ 0 ¼ Γ SM H and the SM expectation corresponding to μ off-shell ¼ 1. In addition, we also perform a fit of the off-shell events with two unconstrained parameters μ off-shell F and μ off-shell V , which express the signal strengths in the gluon fusion and EW processes, respectively. These constraints are summarized in Table X.

VII. SUMMARY
Studies of on-shell and off-shell H boson production in the four-lepton final state are presented, using data from the CMS experiment at the LHC that correspond to an integrated luminosity of 80.2 fb −1 at a center-of-mass energy of 13 TeV. Joint constraints are set on the H boson total width and parameters that express its anomalous couplings to two electroweak vector bosons. These results are combined with those obtained from the data collected at center-of-mass energies of 7 and 8 TeV, corresponding to integrated luminosities of 5.1 and 19.7 fb −1 , respectively. Kinematic information from the decay particles and the associated jets are combined using matrix element techniques to identify the production mechanism and increase sensitivity to the H boson couplings in both production and decay. The constraints on anomalous HVV couplings are found to be consistent with the standard model expectation in both on-shell and off-shell regions, as presented in Tables VI and VII. Under the assumption of a coupling structure similar to that in the standard model, the H boson width is constrained to be 3.2 þ2.8 −2.2 MeV while the expected constraint based on simulation is 4.1 þ5.0 −4.0 MeV, as shown in Table VIII. The constraints on the width remain similar with the inclusion of the tested anomalous HVV interactions and are summarized in Table IX. The width results are also interpreted in terms of the H boson signal strength in the off-shell region in Table X. The observed off-shell signal strength, or equivalently a nonzero value of the width, is more than 2 standard deviations away from a background-only hypothesis, which provides a new direction to measure H boson properties when more data are available.

ACKNOWLEDGMENTS
We thank Markus Schulze for optimizing the JHUGEN Monte Carlo simulation program and matrix element library for this analysis. We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centers and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. We also acknowledge the Maryland Advanced Research Computing Center (MARCC) for providing computing resources essential for this analysis. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: BMBWF and FWF (Austria); FNRS and FWO