Constraints on anomalous Higgs boson couplings using production and decay information in the four-lepton final state

A search is performed for anomalous interactions of the recently discovered Higgs boson using matrix element techniques with the information from its decay to four leptons and from associated Higgs boson production with two quark jets in either vector boson fusion or associated production with a vector boson. The data were recorded by the CMS experiment at the LHC at a center-of-mass energy of 13 TeV and correspond to an integrated luminosity of 38.6 inverse femtobarns. These data are combined with the data collected at center-of-mass energies of 7 and 8 TeV, corresponding to integrated luminosities of 5.1 and 19.7 inverse femtobarns, respectively. All observations are consistent with the expectations for the standard model Higgs boson.

The CMS Collaboration analyzed the data collected at the CERN LHC at center-of-mass energies of 7 and 8 TeV (Run 1), corresponding to integrated luminosities of 5.1 and 19.7 fb −1 , measuring the spin-parity properties of the H boson and searching for anomalous HVV couplings using the H boson's decay modes to two electroweak gauge bosons [13]. That study focused on testing for the presence of anomalous effects in HZZ, HZγ, Hγγ, and HWW interactions under spin-zero, -one, and -two hypotheses. The spin-one hypotheses were excluded at greater than 99.999% confidence level (CL) in the ZZ and WW modes; they were also excluded via the Landau-Yang theorem [37,38] by the observation of the γγ decay mode with 5.7σ significance. The spin-two boson hypothesis with gravity-like minimal couplings was excluded at 99.87% CL, and nine other possible hypotheses of spin-two tensor structure of HVV interactions were excluded at 99% CL or higher. Given the exclusion of the spin-one and -two scenarios, constraints were set on the contribution of eleven anomalous couplings to the HZZ, HZγ, Hγγ, and HWW interactions under the hypothesis of a spin-zero state. Among others, these results constrained a CP-violation parameter f a3 , the fractional pseudoscalar cross section in the H → ZZ channel, which will be described in more detail in Section 2. The pure pseudoscalar hypothesis was excluded at 99.98% CL, and the limit f a3 < 0.43 was set at 95% CL. Similar results, for a smaller number of parameters and fewer exotic-spin models, were obtained by ATLAS [17].
All the above studies considered the decay of an on-shell H boson to two vector bosons. The accumulated data in Run 1 were not sufficient for precision tests of anomalous interactions in associated production, in off-shell production, or with fermions. Nonetheless, both CMS [14] and ATLAS [18] performed analyses of anomalous HVV interactions in VH and VBF production, respectively. Finally, the CMS experiment searched for anomalous HVV interactions in off-shell production of the H boson in pp → H → ZZ with Run 1 data [15]. Further measurements probing the tensor structure of the HVV and Hff interactions can test CP invariance and, more generally, any small anomalous contributions [39].
In this Letter, the analysis approach follows our previous Run 1 publication [13], expanded in two important ways. Information from the kinematic correlations of quark jets from VBF and VH production is used together with H → ZZ/Zγ * /γ * γ * → 4 decay information for the first time, applying the relevant techniques discussed in Ref. [33]. Moreover, data sets corresponding to integrated luminosities of 2.7 and 35.9 fb −1 collected at a center-of-mass energy of 13 TeV in Run 2 of the LHC during 2015 and 2016, respectively, are combined with the Run 1 data, increasing the data sample of H → 4 events by approximately a factor of four.
In what follows, the phenomenology of anomalous HVV interactions is discussed in Section 2. The CMS detector, reconstruction techniques, and Monte Carlo (MC) simulation are introduced in Section 3. Details of the analysis are discussed in Section 4, and results are presented in Section 5. We summarize in Section 6.

Phenomenology of anomalous H boson interactions
We assume that the H boson couples to two gauge bosons VV, such as ZZ, Zγ, γγ, WW, or gg, which in turn couple to quarks or leptons [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. Three general tensor structures that are allowed by Lorentz symmetry are tested. Each term includes a form factor F i (q 2 1 , q 2 2 ), where q 1 and q 2 are the four-momenta of the two difermion states, such as e + e − and µ + µ − in the H → e + e − µ + µ − decay. The H boson coupling to fermions is assumed not to be mediated by a new heavy state V , generating the so-called contact terms [35,36]. We therefore study the process H → VV → 4f and the equivalent processes in production, rather than H → VV → 4f or equivalent processes. Nonetheless, those contact terms are equivalent to the anomalous HVV couplings already tested using the f Λ1 and f Zγ Λ1 parameters, defined below. It is assumed that all lepton and quark couplings to vector bosons follow the SM predictions. Relaxing this requirement would be equivalent to allowing the contact terms to vary with flavor, which would result in too many unconstrained parameters to be tested with the present amount of data. Only the lowest order operators, or lowest order terms in the (q 2 j /Λ 2 ) form-factor expansion, are tested, where Λ is an energy scale of new physics.
Anomalous interactions of a spin-zero H boson with two spin-one gauge bosons VV, such as ZZ, Zγ, γγ, WW, and gg, are parameterized with a scattering amplitude that includes three tensor structures with expansion of coefficients up to (q 2 /Λ 2 ): where q i , Vi , and m V1 are the four-momentum, polarization vector, and pole mass of a gauge boson, [13,33], and a VV i and κ VV i / Λ VV 1 2 are parameters to be determined from data.
In Eq. (1), the only leading tree-level contributions are a ZZ 1 = 0 and a WW 1 = 0, and we assume custodial symmetry, so that a ZZ 1 = a WW 1 . The rest of the couplings are considered anomalous contributions. Tiny anomalous terms arise in the SM due to loop effects, and new, beyond standard model (BSM) contributions could make them larger. The SM values of those couplings are not yet accessible experimentally. Considerations of gauge invariance and symmetry between two identical bosons require κ ZZ . The generic notation a i refers to all four of these couplings, as well as the SM coupling a 1 .
Equation (1) parameterizes both the H → VV decay and the production of the H boson via either VBF or VH. All three of these processes, which are illustrated in Fig. 1, are considered. While q 2 i in the H → VV process does not exceed (100 GeV) 2 due to the kinematic bound, in associated production no such bound exists. In the present analysis it is assumed that the q 2 i range is not restricted within the allowed phase space. The effective fractional cross sections f ai and phases φ ai are defined as follows: f ai = |a i | 2 σ i ∑ |a j | 2 σ j , and φ ai = arg (a i /a 1 ) .
(2) Figure 1: Illustration of H boson production and decay in three topologies: gluon fusion gg → H → VV → 4 (left); vector boson fusion qq → VV(qq) → H(qq) → VV(qq) (middle); and associated production qq → V → VH → (ff)H → (ff)VV (right). In the latter two cases, although the full H decay chain is not shown in the figure, the production and decay H → VV may be followed by the same four-lepton decay shown in the first case. The five angles shown in blue and the invariant masses of the two vector bosons shown in green fully characterize either the production or the decay chain. The angles are defined in either the H or V boson rest frames [26,33].
This definition of f ai is valid for both the SM coupling a 1 and the anomalous couplings, but there is no need for a separate measurement of f a1 because ∑ f ai = 1. The cross sections σ i in Eq. 2 are calculated for each corresponding coupling a i . They are evaluated for the H → ZZ/Zγ * /γ * γ * → 2e2µ process, where a i = 1 and all other a j = 0 in Eq. (1). The resulting ratios are σ 1 /σ 3 = 6.53, σ 1 /σ 2 = 2.77, σ 1 /σ Λ1 = 1.47 × 10 4 TeV −4 , and σ 1 /σ Zγ Λ1 = 5.80 × 10 3 TeV −4 . In the case of the HZγ coupling the requirement √ |q 2 i | ≥ 4 GeV is introduced in the cross section calculations to avoid infrared divergence. Equation (2) can be inverted to recover the coupling ratio, It is convenient to measure the effective cross-section ratios ( f ai ) rather than the anomalous couplings themselves (a i ). First of all, most systematic uncertainties cancel in the ratio. Moreover, the effective fractions are conveniently bounded by 0 and 1 and do not depend on the normalization convention in the definition of the couplings. Until the effects of interference become important, the statistical uncertainties in these measurements scale with the integrated luminosity as 1/ √ L, in the same way as cross section measurements. The f ai values have a simple interpretation as the fractional size of the BSM contribution for the H → ZZ/Zγ * /γ * γ * → 2e2µ decay. For example, f ai = 0 indicates a pure SM Higgs boson, f ai = 1 gives a pure BSM particle, and f ai = 0.5 means that the two couplings contribute equally to the H → ZZ/Zγ * /γ * γ * → 2e2µ process. In particular, f a3 is the fractional pseudoscalar cross section in the H → ZZ → 2e2µ channel. A value 0 < f a3 < 1 would indicate CP violation, with a possible mixture of scalar and pseudoscalar states, while f a3 = 1 would indicate that the H boson is a pure pseudoscalar resonance, which has been excluded at 99.98% CL [13].
The above approach allows a general test of the kinematic distributions associated with the couplings of H to 4 fermions, whether in the decay or in the associated production channels, as shown in Fig. 1. If deviations from the SM are detected, a more detailed study of the (q 2 j /Λ 2 ) form-factor expansion can be performed, eventually providing a measurement of the doubledifferential cross section for each tested tensor structure. Under the assumption that the couplings are constant and real (i.e., φ ai = 0 or π), the above formulation is equivalent to an effec-

The CMS detector, simulation, and reconstruction
The H → 4 decays are reconstructed in the CMS detector, which is composed of a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter, each composed of a barrel and two endcap sections, all within a superconducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. Outside the solenoid are the gas-ionization detectors for muon measurements, which are embedded in the steel flux-return yoke. Extensive forward calorimetry complements the coverage provided by the barrel and endcap detectors. Events of interest are selected using a two-tiered trigger system. 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. [40].
A dedicated MC program, JHUGEN 7.0.2 [26,29,33,41], is used to simulate the effect of anomalous couplings in the production and decay H → ZZ / Zγ * / γ * γ * → 4 . The gluon fusion production of an H boson is simulated with the POWHEG 2.0 [42][43][44] event generator at nextto-leading order (NLO) in quantum chromodynamics (QCD). The associated gluon fusion production of an H boson with two jets is affected by anomalous Hgg interactions. These effects are modeled with JHUGEN. It is also found that the NLO QCD effects that are relevant for the analysis of a spin-zero state are well described by a combination of leading-order (LO) matrix elements and parton showering [33]. For the SM case, JHUGEN simulations at LO in QCD and POWHEG simulations at NLO in QCD, with parton showering applied in both cases, are explicitly compared, and no significant differences are found. Therefore, JHUGEN at LO in QCD is adopted for the simulation of VBF, VH, and ttH production with anomalous couplings. The MELA package [2,26,29,33,41] contains a library of matrix elements from JHUGEN for the H boson signal and MCFM 7.0 [45][46][47] for the SM background and is used to apply weights to events in any MC sample to model any other set of couplings.
The main background in this analysis, qq → ZZ/Zγ * → 4 , is estimated from simulation with POWHEG, with the next-to-NLO (NNLO) K-factor, which is approximately 1.1 at m 4 = 125 GeV [48], applied to the NLO cross section. The gg → ZZ/Zγ * → 4 background process is simulated with MCFM 7.0, where the Higgs boson production K-factor at NNLO in QCD, which is approximately 2.3 at m 4 = 125 GeV, is applied to the LO cross section [49]. The VBF and triple-gauge-boson (VVV) backgrounds are estimated at LO with PHANTOM 1.2.8 [50]. The parton distribution functions (PDFs) used for all of these samples are NNPDF3.0 [51]. All MC samples are interfaced with PYTHIA 8.212 [52] tune CUETP8M1 [53] for parton showering and further processed through a dedicated simulation of the CMS detector based on GEANT4 [54].
The selection of the H → 4 events and associated particles closely follows the methods used in the analyses of Run 1 [12,13] and Run 2 [48] data. The main triggers for this analysis select a pair of leptons passing loose identification and isolation requirements, with p T of the leading and subleading electron (muon) at least 23 (17) and 12 (8) GeV, respectively. 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 higher p T thresholds. Electrons (muons) are reconstructed within the geometrical acceptance defined by |η| < 2.5 (2.4) for transverse momentum p T > 7 (5) GeV with an algorithm that combines information from the ECAL (muon system) and the tracker. It is required that the ratio of each lepton track's impact parameter in three dimensions, computed with respect to the chosen primary vertex position, to its uncertainty be less than 4. The primary vertex is defined as the vertex with the highest sum of p 2 T of physics objects defined by a jet-finding algorithm. To discriminate prompt leptons from Z/γ * boson decays from those arising from hadron decays within jets, an isolation requirement for leptons is imposed. An algorithm is used to collect the final-state radiation (FSR) of leptons. An FSR photon is associated to the closest selected lepton in the event if its angular separation from the lepton is below the required threshold, as discussed in Ref. [48]. Three mutually exclusive channels are considered: 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, irrespective of flavor, are required to satisfy m + − > 4 GeV. The Z/γ * candidates are required to satisfy the condition 12 GeV < m < 120 GeV; the invariant mass of at least one of the Z/γ * candidates must be larger than 40 GeV. The four-lepton invariant mass m 4 must be between 105 and 140 GeV.
Jets are reconstructed using the particle-flow (PF) algorithm [55], with PF candidates clustered by the anti-k T algorithm [56,57] with a distance parameter of 0.4, and with the constraint that the charged particles be compatible with the primary vertex. The jet momentum is determined as the vectorial sum of all PF candidate momenta in the jet. Jets must satisfy p T > 30 GeV and |η| < 4.7 and be separated from all selected lepton candidates and any selected FSR photons by an angular distance ∆R( /γ, jet) > 0.4, where the angular distance between two particles i and j is ∆R(i, j) =

Analysis techniques
The full kinematic information from each event is extracted using the matrix element calculations in the MELA package. For either the H boson decay or associated production with two jets, up to seven kinematic observables, five angles and two invariant masses, are defined, as shown in Fig. 1 [26,33]. In the 2 → 6 process of associated H boson production via either VBF, ZH, or WH and its subsequent decay to a four-fermion final state, up to 13 independent observables Ω remain. In the following, we use either the production kinematics, the decay kinematics, or both, as appropriate. The p T of the system of the H boson and two jets, which would appear at NLO in QCD, is not included in the input observables in order to reduce associated QCD uncertainties. The MELA approach retains all relevant kinematic information in a minimal set of discriminants D, computed from ratios of probabilities P. We use two types of discriminants, and where "sig" stands for the SM signal; "alt" denotes an alternative hypothesis [29], which could be background ("bkg"), an alternative H boson production mechanism ("2jet"), or an alternative H boson coupling model ("a i "); and "int" represents the contribution to the probability from the interference between "sig" and "alt" [33]. By the Neyman-Pearson lemma [58], the D alt discriminant contains all the information available from the kinematics to separate the SM signal hypothesis from the alternative hypothesis. Because all intermediate hypotheses are a linear combination of the SM hypothesis and the alternative hypothesis, the combination of D alt with D int also contains all the information available to separate the interference component. The discriminants used in this analysis are summarized in Table 1 and described in more detail below. Table 1: Summary of the three production categories in the analysis of 2016 data. The discriminants D are calculated from Eqs. (4) and (5), as discussed in more detail in the text. For each analysis, the appropriate BSM model is considered in the definition of the categories: f a3 = 1,  The selected events in the 2016 data sample are split into three categories: VBF-jet, VH-jet, and untagged. The VBF-jet category requires exactly four leptons with either two or three jets of which at most one is b quark flavor-tagged, or at least four jets and no b-tagged jets. The VH-jet category requires exactly four leptons and two or more jets; if there are four or more jets, none of them should be b tagged. The requirements on the number of b-tagged jets are applied to reduce cross-feed from ttH production. In order to separate the target production mode for each category from gluon fusion production, the requirement D 2jet > 0.5 is applied following Eq. (4), where P sig corresponds to the signal probability for the VBF (ZH or WH) production hypothesis in the VBF-jet (VH-jet) category, and P alt corresponds to the gluon fusion production of the H boson in association with two jets. In each of the four f ai analyses, the requirement D 2jet > 0.5 is tested with both the f ai = 0 and f ai = 1 signal hypotheses in P sig . Thus, this categorization differs slightly in the four analyses. The two highest p T jets are used in the calculation of the matrix elements. All events not assigned to the VBF-jet or VHjet categories are assigned to the untagged category. The above requirements are summarized in Table 1. Due to the small size of the 2015 data sample, those events were not categorized and were all treated as untagged, as was done in the analysis of 2011 and 2012 data [13]. The expected and observed numbers of events are listed in Table 2.
We perform an unbinned extended maximum likelihood fit to the events split into the categories according to the lepton flavor and production topology. An independent fit is performed for each parameter defined in Table 3. In each category of events, three observables D = {D bkg , D ai , D int } are defined following Eqs. (4) and (5), as summarized in Table 1.
The first observable, D bkg (shown in Fig. 2 (a)), is common to all events and is designed to separate the signal from the dominant qq → 4 background, for which P bkg is calculated. The signal and background probabilities include both the matrix element probability based on lepton kinematics and the m 4 probability parameterization extracted from simulation of detector effects. The signal m 4 parameterization assumes that m H = 125 GeV.
The second observable, D ai , separates the SM hypothesis f ai = 0 from the alternative hypothesis f ai = 1. It is defined following Eq. (4), with P sig calculated for f ai = 0 and P alt for the alternative H boson coupling hypothesis with f ai = 1. In the untagged category the probabilities are calculated using only the decay information, but in the VBF-jet and VH-jet categories both the production and decay probabilities are used, with the matrix elements calculated for either VBF × decay or (ZH + WH) × decay, respectively. This observable is called D 0− in the f a3 , D 0h+ in the f a2 , D Λ1 in the f Λ1 , and D Zγ Λ1 in the f Zγ Λ1 analyses [13]. Superscripts are added to the discriminant name to indicate the processes used to calculate the matrix elements: either dec, VBF+dec, or VH+dec to denote decay, VBF × decay, or (ZH + WH) × decay, respectively. Distributions of D 0− in the three categories are shown in Fig. 2 (e), (f), (g). Figure 2 (b), (c), (d) also shows the distributions of D 0h+ , D Λ1 , and D Zγ Λ1 , respectively, for the untagged events. The third observable, D int from Eq. (5), separates the interference of the two amplitudes corresponding to the SM coupling and the alternative H boson coupling model. In the case of the f a3 analysis, this observable is called D CP because if CP is violated it would exhibit a distinctive forward-backward asymmetry between D CP > 0 and D CP < 0, as illustrated in Fig. 2 (h) for the untagged category of events. In the untagged category, decay information is used in the calculation of D int . In the VBF-jet and VH-jet categories, production information is used. As in the case of D ai , superscripts indicate which processes were used to calculate the matrix elements. In the f Λ1 and f Zγ Λ1 analyses, the interference discriminant does not provide additional separation, and D 0h+ is used as the third observable.
In the likelihood fit, the signal probability density function (pdf) is parameterized for each production mode and in each category as where T n is the three-dimensional template probability obtained from MC simulation, |a i /a 1 | is calculated from f ai through Eq. and decay sides, and over three contributions n = 0, 1, and 2 for the other signal modes. The background pdf is also parameterized with templates extracted from simulation, except for the reducible background, Z+X, which is dominated by the Z+jets process but also includes the tt+jets, Zγ+jets, WZ+jets, and WW+jets processes. The Z+X background is estimated using independent control regions in data with loose identification requirements on two leptons.
The yields of signal events in 2016 data are expressed with two unconstrained parameters µ V and µ F , which are the ratios of the observed yields to the expectation in the SM for the production mechanisms driven by the HVV couplings (VBF and VH) and for the other modes (gluon fusion and ttH), respectively. The signal yield in 2015 data is expressed with a single parameter µ 13 TeV , which is a linear combination of µ V and µ F . The fit is also performed simultaneously with the 2011 and 2012 data from Ref. [13], where the two signal strength parameters µ 7 TeV and µ 8 TeV are also linear combinations of µ V and µ F including the effects of the cross section scaling for each value of f ai .
Most uncertainties in the signal yields cancel in this analysis because measurements of anomalous couplings are expressed as relative cross sections. Statistical uncertainties dominate over any systematic uncertainties in this analysis. In the decay-only observables the main effects come from lepton momentum uncertainties and are propagated into the template uncertainties as in the previous analyses [13], where the main effect is on the m 4 resolution affecting the D bkg parameterization.
The primary new feature in this analysis, compared to Run 1 [13], is the categorization based on jets and the kinematic discriminants using jet information. Both the shapes and the yields are varied according to uncertainties obtained from the jet energy variations. In addition, uncertainties in renormalization and factorization scales, PDFs, and the modeling of hadronization and the underlying event in MC simulation are propagated to the template and relative yield uncertainties. As part of these studies, comparisons were made between QCD production at NLO and LO, with matched PYTHIA hadronization in each case, for the VBF, VH, and ttH processes. In all cases, only small differences were observed. The uncertainties in the migration of signal and background events between categories amount to 3-13% for the signal and 4-25% for the background, depending on the category. Among the signal processes, the largest uncertainties arise from the prediction of the gg → H yield in the VBF-jet category. In ttH and gluon fusion production, anomalous couplings on the production side are not generally related to the HVV anomalous couplings considered here. There is a negligible effect on the observed distributions with large variations in the couplings.
Backgrounds from the qq → 4 , gg → 4 , VBF, and V+(4 ) processes are estimated using MC simulation. Theoretical uncertainties in the background estimation include uncertainties from the renormalization and factorization scales, the PDFs, and the K-factors described above. An additional 10% uncertainty is assigned to the gg → 4 background K-factor to cover potential differences between signal and background.

Results and discussion
Four f ai parameters sensitive to anomalous H boson interactions, as defined in Eqs. (2) and (3), are tested in the observed data using the pdf in Eq. (6). The results of the likelihood scans of the f ai parameters on 13 TeV data only and on the full, combined data set from collisions at 13, 8, and 7 TeV are shown in Fig. 3. The combined results are listed in Table 3 and supersede our previous measurement in Ref. [13].
The expected 68% CL constraints improve by nearly an order of magnitude compared to the Run 1 analysis [13], as is evident from the narrow minima at f ai = 0 in the expectations in Fig. 3. This effect comes from utilizing production information, because the cross section in VBF and VH production increases quickly with f ai due to larger q 2 values contributing in Eq. (1) [33]. The narrow minima are shallower than expected, which may be understood by examining the best fitted (µ V , µ F ) values in the four analyses under the assumption that f ai = 0: 19 ) at f Zγ Λ1 = 0. The values obtained for the different analyses vary due to the different categorization and observables. The overall behavior with µ V less than 1 is consistent with a downward statistical fluctuation in the small number of VBF and VH events, with the total observed number of untagged events similar to the expectation. Because fewer VBF and VH events are observed than expected, the narrow minima of −2 ln(L) at f ai = 0, which come from the production information in these events, are observed to be less pronounced than expected. The minimum is most pronounced in the f a3 analysis in Fig. 3 (a) due to the largest observed µ V value.  The improvement in the 95% CL constraints with respect to Run 1 is mostly due to the increase in the number of events with H → 4 decay information by about a factor of four. Another factor of four increase in the data sample size is expected by the end of 2018, under similar running conditions. At that time, the inclusion of production information is expected to result in improvements to the 95% CL constraints in line with the improvements already seen in the 68% CL constraints.
Other features in Fig. 3 can be explained by examining the kinematic distributions in Fig. 2. The D dec 0− distribution in Fig. 2 (e) favors a mixture of the f a3 = 0 and f a3 = 1 models, resulting in the best fit value of f a3 = 0.30 ± 0.21 in Run 2. The D dec CP distribution in Fig. 2 (h) has a small forward-backward asymmetry, with more events at D dec CP > 0 than D dec CP < 0, which gives preference to the f a3 cos(φ a3 ) = +0.30 value as opposed to −0.30. The narrow local minimum at f a3 = 0 corresponds to the distribution of events in the tagged categories in Fig. 2 (f), (g), which favors the SM hypothesis. The Run 1 result [13] favors the SM strongly, and therefore combining the two data sets results in a global minimum at f a3 = 0.
Certain values of anomalous couplings, such as f a2 cos(φ a2 ) ∼ −0.5 and f Λ1 cos(φ Λ1 ) ∼ +0.5, lead to strong interference effects between the SM and anomalous amplitudes in Eq. (1). Therefore, kinematic distributions of such models are easily distinguished from SM distributions, and they are excluded at high CL in Fig. 3. Such anomalous models are shown in Fig. 2 (b), (c). The f a3 = 1 and f Zγ Λ1 = 1 models are shown in other cases in Fig. 2, as the most distinct from SM, except for (h), where maximal forward-backward asymmetry in D CP is shown for f a3 = 0.5. In all cases, the observed distributions in Fig. 2 are consistent with the SM expectations.

Summary
We study anomalous interactions of the H boson using novel techniques with a matrix element likelihood approach to simultaneously analyze the H → 4 decay and associated production with two quark jets. Three categories of events are analyzed, targeting events produced in vector boson fusion, with an associated vector boson, and in gluon fusion, respectively. The data collected at a center-of-mass energy of 13 TeV in Run 2 of the LHC are combined with the Run 1 data, collected at 7 and 8 TeV. No deviations from the standard model are observed and constraints are set on the four anomalous HVV contributions, including the CP-violation parameter f a3 , summarized in Table 3.

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. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following fund-