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ATL-PHYS-PUB-2017-018 24/11/2017 ATLAS PUB Note ATL-PHYS-PUB-2017-018 24th November 2017 Constraints on an effective Lagrangian from the combined H ZZ * 4and H γγ channels using 36. 1 fb -1 of s = 13 TeV pp collision data collected with the ATLAS detector The ATLAS Collaboration Six parameters of an effective field theory are constrained using the combined H γγ and H ZZ * 4channels, based on 36.1 fb -1 of proton-proton collision data collected by the ATLAS experiment at the LHC at s = 13TeV. The parameters are sensitive to modifications of the Higgs boson couplings to strong and electroweak gauge bosons, and to the top quark. The most stringent constraints are those on the effective couplings to photons and to gluons. With respect to the original version, Table 2 has been corrected to remove the H γγ vertex from the operators O HB and O B . © 2017 CERN for the benefit of the ATLAS Collaboration. Reproduction of this article or parts of it is allowed as specified in the CC-BY-4.0 license.

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ATL

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YS-

PUB-

2017

-018

24/1

1/20

17

ATLAS PUB NoteATL-PHYS-PUB-2017-018

24th November 2017

Constraints on an effective Lagrangian from thecombined H → ZZ∗→ 4` and H → γγ channelsusing 36.1 fb−1 of

√s = 13 TeV pp collision data

collected with the ATLAS detector

The ATLAS Collaboration

Six parameters of an effective field theory are constrained using the combined H → γγ andH → Z Z∗ → 4` channels, based on 36.1 fb−1 of proton-proton collision data collected by theATLAS experiment at the LHC at

√s = 13 TeV. The parameters are sensitive to modifications

of the Higgs boson couplings to strong and electroweak gauge bosons, and to the top quark.The most stringent constraints are those on the effective couplings to photons and to gluons.With respect to the original version, Table 2 has been corrected to remove the Hγγ vertexfrom the operators OHB and OB.

© 2017 CERN for the benefit of the ATLAS Collaboration.Reproduction of this article or parts of it is allowed as specified in the CC-BY-4.0 license.

1 Introduction

Following the discovery of the Higgs boson [1, 2], the Standard Model (SM) description of Higgs bosoncouplings was tested by constraining the κ values that parameterize non-SM effects and are defined asmultiplicative factors to the interaction terms in the SM Lagrangian. To extend the interpretation beyondthis framework to more general Lagrangian operators, an effective field theory (EFT) framework has beenconstructed [3]. This framework allows broader tests of Higgs boson interactions and kinematics througha systematic probe of new perturbative physics at a high energy scale.

This note describes measurements of EFT parameters sensitive to Higgs boson production and decay. Fitsto these parameters are performed using the full 2015-2016 H→ γγ [4] and H→ Z Z∗→ 4` [5] datasetsthat correspond to an integrated luminosity of 36.1 fb−1. Six parameters are chosen for the fit based onthe expected sensitivity of the measurements. The parameters are related to the data through simplifiedtemplate cross sections (STXS) based on Stage 1 of Ref. [3, 6].

The note is organized as follows: Section 2 reviews the dataset and input measurements, Section 3describes the fit parameters and their relationship to the simplified template cross sections, Section 4discusses the statistical model, Section 5 presents the results, and Section 6 provides a summary.

2 Data and input measurements

The proton–proton collision data were collected by the ATLAS experiment [7, 8] in 2015 and 2016,with the LHC operating at a centre-of-mass energy of 13 TeV. The dataset corresponds to an integratedluminosity of 36.1 fb−1, of which 3.2 fb−1 were collected in 2015 and the remainder in 2016, for both theH → γγ and H → Z Z∗ → 4` analyses.

The analyses of the individual decay channels separate the measured events into exclusive kinematicand topological categories, as summarized in Table 1 [9]. The category selection separates individualproduction modes and their kinematic properties based on SM predictions, in some cases aided by boosteddecision trees (BDTs). In categories with BDTs, the full binned output is utilized. The categories aredesigned to measure the simplified template cross sections, defined in Refs. [3, 6] and illustrated inFigure 1, using the SM kinematics of each STXS region as a template. The target STXS regions includegluon-fusion production in bins of jet multiplicity and pH

T , and in two bins with VBF-like kinematicsdefined by the presence of two jets with large dijet mass. Quark-initiated production processes qq→ Hqqare split into a bin with a high-pT jet, two VBF-like bins, a bin with two jets consistent with V(→ qq)Hproduction, and a bin for the remaining events. Finally, leptonic decays of the vector boson in VHproduction are split into H`ν and H`` final states, and further split by the Higgs-boson pT.

Details of the experimental analyses are given in Refs. [4] and [5].

3 Effective Lagrangian parameters

Effective field theories can be used to describe new short-distance physics not included in the StandardModel. Taking the leading terms in an expansion in inverse distance (or equivalently, energy), the firstterms that conserve baryon and lepton number have coefficients that are quadratic in distance (making

2

Table 1: Data categories entering the combined measurements for the H → γγ and H → Z Z∗ → 4` decay modes,as described in Refs. [4] and [5], respectively. The categories are listed in order of prioritization such that eventsassigned to a given category are not considered for subsequent categories. The purity of the targeted productionmode varies from category to category.

H → γγ

ttH+tH leptonic (two tHX and one ttH categories)ttH+tH hadronic (two tHX and four BDT ttH categories)VH dileptonVH one-lepton, p

`+EmissT

T ≥ 150 GeVVH one-lepton, p

`+EmissT

T <150 GeVVH Emiss

T , EmissT ≥ 150 GeV

VH EmissT , Emiss

T <150 GeVVH+VBFpj1

T ≥ 200 GeVVH hadronic (BDT tight and loose categories)VBF, pγγ j jT ≥ 25 GeV(BDT tight and loose categories)VBF, pγγ j jT <25 GeV(BDT tight and loose categories)ggF 2-jet, pγγT ≥ 200 GeVggF 2-jet, 120 GeV≤ pγγT <200 GeVggF 2-jet, 60 GeV≤ pγγT <120 GeVggF 2-jet, pγγT < 60 GeVggF 1-jet, pγγT ≥ 200 GeVggF 1-jet, 120 GeV≤ pγγT <200 GeVggF 1-jet, 60 GeV≤ pγγT <120 GeVggF 1-jet, pγγT < 60 GeVggF 0-jet (central and forward categories)

H → Z Z∗ → 4`ttHVH leptonic2-jet VH2-jet VBF, pj1

T ≥ 200 GeV2-jet VBF, pj1

T <200 GeV1-jet ggF, p4`

T ≥ 120 GeV1-jet ggF, 60 GeV<p4`

T <120 GeV1-jet ggF, p4`

T <60 GeV0-jet ggF

the corresponding field operators dimension-6 in energy). The general form of the Lagrangian includingdimension-6 operators is [3]:

L = LSM +∑i

c(6)i O(6)i /Λ

2, (1)

whereΛ is the energy scale of new processes; in the following the parameters are simplified to ci = c(6)i /Λ2.

Several bases of these operators are available for gauge-invariant products of SM fields; of these, thestrongly-interacting light Higgs (SILH) [10] and Warsaw [11] bases have the most complete publicimplementations. The fit described here focusses on the dominant operator coefficients in the SILH basis,based on leading-order predictions and taking into account precision electroweak constraints [12].

There are 59 operators in the dimension-6 basis assuming flavour-universal couplings, with an additionalseventeen operators for the hermitian conjugates. The majority of these operators do not affect Higgsphysics or have coefficients that are tightly constrained by precision electroweak data at leading order.Constraints on the coefficients of operators of the SILH implementation in Madgraph (the Higgs EffectiveLagrangian, or HEL [13]) have been tabulated in an LHC Higgs working group document [14]. Of thefifteen operators whose coefficients are constrained by Higgs boson interactions, four are CP-odd and areneglected because they do not enter any STXS observable at leading order in 1/Λ2 and are degenerate withcorresponding CP-even operators at 1/Λ4. Other operators that do not directly affect the H → γγ andH → Z Z∗ measurements are those that affect the Higgs boson self-couplings and the Yukawa couplings

3

= 0-jet

ggF

≥ 2-jet

pHT [200, ∞]

BSM

pHT [0, 60]

pHT [60, 120]

pHT [120, 200]

= 1-jet

pHT [200, ∞]

BSM

pHT [0, 60]

pHT [60, 120]

pHT [120, 200]

(+)

(+)

(+) (+)

(+)

≥ 2-jet

pHjjT [0, 25]

pHjjT [25, ∞]

≃ 2j

! 3j

pHT < 200

VBF cuts

pj1T [200, ∞]

BSM

pj1T [0, 200]

pHjjT [0, 25]

pHjjT [25, ∞]

(+) ! 3j

≃ 2j

VBF (EW qqH incl. V H →qqH)

≥ 2-jet VBF cuts ≥ 2-jet VH cuts Rest(+)

gg → ZHqq → V H

V H

= 0-jet

≥ 1-jet

pVT [0, 150]

pVT [150, ∞]

(+)

(H+ leptonic V )

pVT [0, 150]

pVT [150, 250]

= 0-jet

≥ 1-jet

pVT [250, ∞]

W → ℓν

(+)

Z → ℓℓ + νν

= 0-jet

≥ 1-jet(+)

pVT [250, ∞]

pVT [0, 150]

pVT [150, 250]

(+)

(EW qqH)

ggF bbH tHttHVBF

(H+ leptonic V )

V H

qq →WH

qq → ZH

gg → ZH

VBF

H+ had. V

(Run1-like)

ATLAS Preliminary

Constructed from figures in arXiv:1610.07922

Figure 1: The stage-1 simplified template cross section regions [3, 6] targeted by the H → γγ and H → Z Z∗ → 4`analysis categories.

of the Higgs boson to down-type quarks and leptons. Removing these operators reduces the set to eight;as a final reduction, the operator that primarily affects the Higgs field normalization is neglected, as itcauses a global change in rates and is not well constrained given the current precision of the data.

The operator reduction described above results in an effective Lagrangian consisting of the SMLagrangiansupplemented by the operators listed in Table 2. The leading vertices produced by these operators are givenin the table. The HEL implementation recasts the Wilson coefficients of Eq. (1) into the dimensionlessparameters indicated in the table with monospace typing. In this framework the precision electroweakparameter S is given by S = cWW + cB [12]; given the strong experimental constraint on this parametercWW + cB is assumed to be zero and the orthogonal combination cWW − cB is taken as a parameter in thestudy. The parameters considered in this note are therefore cG, cA, cu, cHW, cHB, and cWW − cB.

To determine the relationship between the STXS measurements and the EFT parameters, the cross sectionis separated into SM, SM-BSM interference, and BSM components:

σ = σSM + σint + σBSM.

The dependence of the cross section on the couplings can then be expressed asσ

σSM= 1 +

∑i

Ai ci +∑i j

Bi j ci cj,

where σint/σSM =∑

i Ai ci and σBSM/σSM =∑

i j Bi j ci cj . The ci cj term, which is suppressed by 1/Λ4,is included because it is the leading term that does not depend on the SM amplitude (the dimension-8

4

Table 2: The operators included in the fit for EFT parameters and their relevant vertices.

Operator Expression HEL coefficient Vertices

Og |H |2GAµνGAµν cG =

m2W

g2s

cg Hgg

Oγ |H |2BµνBµν cA =m2

W

g′2cγ Hγγ,HZ Z

Ou yu |H |2ulHuR+ h.c. cu = v2cu Htt

OHW i (DµH)† σa (DνH)Waµν cHW =

m2W

g cHW HWW,HZ Z

OHB i (DµH)† (DνH) Bµν cHB =m2

W

g′ cHB HZ Z

OW i(H†σaDµH

)DνWa

µν cWW =m2

W

g cW HWW,HZ Z

OB i(H†DµH

)∂νBµν cB =

m2W

g′ cB HZ Z

operators suppressed by this factor are subleading interference terms). The inclusion of this term preventsnegative cross sections and leads to a better fit of the data. An equivalent expression is used for the partialand total decay widths of the Higgs boson.

The Ai and Bi j coefficients have been calculated using a Madgraph leading-order generation of σ andσSM for each HEL parameter in each STXS bin, neglecting uncertainties, and documented in Ref. [14].The fit described here uses separate parameterizations for VBF and associated production with a vectorboson, followed by the hadronic decays of the vector bosons. The small contributions from gg → ZHand tH production are not explicitly calculated and therefore use the same EFT parameterizations as ggF,H``, or ttH. The relationship between EFT parameters and STXS regions is illustrated in Figure 2, whichdemonstrates the effect of the cA, cHW, cHB, and cWW − cB parameters on a set of STXS regions. Theseregions form a reduced set of the STXS regions used in the study and are obtained by merging all ggFregions, which are affected in the same way by the parameters, and by merging all VH categories withpVT > 150 GeV, since there is currently little sensitivity to these regions in the data. The value of eachparameter in Figure 2 corresponds to the ≈ +1σ expected value assuming the SM.

To perform the fit, the procedures of Refs. [4] and [5] are used to relate the yields in each data categoryto ratios of measured STXS values to those predicted by the SM. The SM predictions are modelled usingthe best available calculations and generators [9] and take into account the SM theoretical uncertainties(described in Section 4). The ratios are then expressed as quadratic functions of the HEL parameters usingthe Madgraph generation described above. The result is an effective parameterization of the data yieldsin terms of the HEL parameters.

The procedure potentially introduces a model dependence because of the assumption that each parameter-ization does not vary within its STXS region. Theoretical uncertainties covering such model dependenceare not included. Higher-order effects introduce dependencies on additional EFT couplings, such asthe top-quark loop in gluon fusion that causes an explicit cu dependence in the gluon-fusion regions.Higher-order QCD corrections can affect the momentum transfer distribution within a given STXS region,modifying the dependence on EFT couplings that are sensitive to this distribution (such as the EFT para-meters sensitive to the HVV couplings). These effects are not evaluated here and the relations betweenthe STXS and the HEL parameters do not account for such uncertainties.

5

4l)→B(H)γγ→B(H ggF

<25 GeV3j

Tp

Hqq→ qq

25 GeV≥3j

T p

Hqq→ qq VH-like

Hqq→qq rest

Hqq→qq

>200 GeVj

Tp

Hqq→ qq W

Tlow p

ν HlW

Thigh p

ν HlZ

Tlow p Hll

Z

Thigh p Hll Htt

Rat

io w

ith r

espe

ct to

SM

1

10

cG = 0.00003 cHW = 0.04

cA = 0.0003 cHB = 0.15

cu = 0.25 cWW - cB = 0.06

-1= 13 TeV, 36.1 fbs

ATLAS Preliminary

Figure 2: The values of modified STXS regions, relative to the SM, for ≈ +1σ expected SM values of cG (3× 10−5),cA (3 × 10−4), cu (0.25), cWW − cB (0.06), cHW (0.04), and cHB (0.15). The STXS regions have been modified bymerging all ggF regions, which are affected in the same way by all parameters, and by merging VH categories withpVT > 150 GeV (“high pWT ” and “high pZ

T ”).

4 Statistical model

The statistical treatment follows the procedures described inRefs. [9, 15–20]. The six parameters of interest(POIs) are defined to be cG, cA, cu, cHW, cHB, and cWW−cB (see Table 2), and are fit simultaneously usinga likelihood as described below.

For the H → Z Z∗ → 4` channel, the number of events observed in each analysis bin is treated as anindependent Poisson-distributed value. For the H → γγ categories, the likelihood is given by a Poissonterm multiplied by a likelihood binned in diphoton invariant mass for categories with more than 220 dataevents. In other H → γγ categories an unbinned likelihood is used. The full likelihood is given by theproduct of likelihoods for each channel and category.

Degrees of freedom related to the data-driven background estimations or the systematic uncertainties areincluded in the fit as nuisance parameters in the likelihood function. These parameters are constrained bydata in the fit and in most cases include auxiliary constraints. Common nuisance parameters are used forshared sources of experimental uncertainty, such as the uncertainties on the integrated luminosity or thejet energy scale and resolution.

The sources of theoretical uncertainty on the SM cross sections of the STXS regions are missing higher-order perturbative corrections, uncertainties in the PDFs, uncertainty in αS , and uncertainties in themodelling of the underlying event and parton shower (UE/PS). In relating measurement categories toSTXS regions, migration uncertainties across ggF kinematic regions are accounted for using a generalizedcovariance matrix [4, 21]. These nuisance parameters and those associated with UE/PS uncertainties arecorrelated between the H → γγ and H → Z Z∗ → 4` channels. Theoretical and parametric uncertaintieson the decay branching fractions [3, 22–25] are included, with correlations following the strategy ofRef. [15].

The result of the statistical combination is a likelihood function L(α, θ), where α represents the parametersof interest and θ the set of nuisance parameters. A statistical test of hypothetical α values is carried out

6

using the profile likelihood ratio,

λ(α) =L(α, ˆθ(α))

L(α, θ). (2)

In the numerator, the nuisance parameters are set to the values ˆθ(α) that maximize the likelihood functionfor a given set of values of the parameters of interest α. In the denominator, both the parameters of interestand the nuisance parameters are set to the values α and θ, respectively, that maximize the likelihoodfunction. Asymptotically, −2 ln λ(α) is distributed as a χ2 distribution with n degrees of freedom, wheren is the dimensionality of the vector α. The results are based on likelihood evaluations and give 68%confidence level (C.L.) intervals assuming the asymptotic approximation [26].

The compatibility with the Standard Model is quantified using the p-value, defined as the probability toobtain a value of the test statistic that is at least as high as the observed value. It is obtained from the valueof −2 ln λ(α = αSM) assuming an asymptotic χ2 distribution with a number of degrees of freedom equalto the number of parameters of interest [26].

5 Results

The results of the fit are shown in Figure 3 and Table 3 for the data and for an Asimov dataset [26]generated under the SM hypothesis. The correlations and likelihood scans are shown in Figures 4 and 5,respectively. The quadratic dependence of each STXS region on the EFT parameters leads to degeneratesolutions in general; when these are well separated we focus on the solution that is most consistent withthe SM (i.e. the value of the EFT parameter closest to zero). The p-value representing the consistency ofthe 6-parameter with respect to the SM expectation is 47%.

The central values of the parameters are consistent with the patterns observed in the STXS measure-ments [9]: the ratio of branching fractions B(H → γγ)/B(H → 4`) is measured to be below that of theSM, leading to a positive cA sinceB(H → γγ)/BSM (H → γγ)−1 ∝ −cA to first order in the coefficients;the measured VBF cross section is above that of the SM, leading to positive cWW and negative cHW, sinceσ(VBF)/σSM (VBF) − 1 ≈ cWW − 4cHW; and the measured ttH cross section is below that of the SM,leading to negative cu since σ(ttH)/σSM (ttH) − 1 ∝ cu to first order.

Compared to the SM expectation, the sensitivity to the dimension-6 HVV couplings (cHW, cHB, cWW− cB)is higher because the measured VBF cross-section is higher than (though still compatible with) the SMprediction [9]. This also leads to a positive correlation between cHB and cWW − cB, and a more negativecorrelation between cHB and cHW than expected in the SM. There is a large anti-correlation betweencWW − cB and cHW from the VH categories, where the corresponding Ai coefficients are large and enterwith the same sign [14]. For the ttH coupling cu the observed sensitivity is lower than expected becausethe low observed yield [9] effectively removes the degeneracy and gives a flatter likelihood profile.

A previous global fit to data from the Tevatron and Run 1 of the LHC found similar constraints on theseparameters [12]. The results presented here show improved sensitivity to cG and cA, as expected. TheHVV couplings can be further constrained by including other VBF Higgs measurements and H → WWdecays. Tighter constraints on cu can be obtained by including measurements of ttH production in otherHiggs boson decay channels.

7

Parameter value-2 0 2

γγ → ZZ* and H →Observed HEL constraints with H

-1 = 13 TeV, 36.1 fbs

ATLAS Preliminary ]-4cG [ 10

]-4cA [ 10

cu

]-1cHW [ 10

]-1cHB [ 10

]-1cWW - cB [ 10

(a)

Parameter value-2 0 2

γγ → ZZ* and H →SM expected HEL constraints with H

-1 = 13 TeV, 36.1 fbs

ATLAS Preliminary ]-4cG [ 10

]-4cA [ 10

cu

]-1cHW [ 10

]-1cHB [ 10

]-1cWW - cB [ 10

(b)

Figure 3: The (a) observed and (b) SM predicted best-fit values and 68%C.L. intervals for each of the six parameters.

Table 3: The fit values and 68% C.L. uncertainties from the fit for six EFT parameters using the Stage 1 STXSbinning to relate the data to the parameters. The observed and SM expected results are shown.

Operator Fit result (observed) Fit result (SM expected)

Og cG = −0.05+0.27−0.28 × 10−4 cG = 0.00+0.38

−0.26 × 10−4

Oγ cA = 0.3+1.9−1.8 × 10−4 cA = 0.0+2.8

−2.2 × 10−4

Ou cu = −0.50+0.45−0.81 cu = 0.00+0.24

−0.28

OHW cHW = −0.052 ± 0.028 cHW = 0.000+0.041−0.043

OHB cHB = 0.026 ± 0.077 cHB = 0.00+0.14−0.16

OW , OB cWW − cB = 0.078 ± 0.049 cWW − cB = 0.000+0.057−0.074

8

-0.39 -0.43 -0.13 -0.86 0.19 1

-0.031 -0.35 -0.0056 -0.44 1

0.24 0.33 0.089 1

0.23 0.07 1

0.63 1

1

cG cA cu cHW cHB cWW-cB

cWW-cB

cHB

cHW

cu

cA

cG (X,Y

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1 PreliminaryATLAS

-1 = 13 TeV, 36.1 fbs

4l→ ZZ* →H and γ γ →H

(a)

-0.31 -0.069 0.088 -0.89 -0.14 1

-0.46 -0.4 -0.11 -0.19 1

0.4 0.17 -0.077 1

0.3 0.24 1

0.69 1

1

cG cA cu cHW cHB cWW-cB

cWW-cB

cHB

cHW

cu

cA

cG (X,Y

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1 PreliminaryATLAS

-1 = 13 TeV, 36.1 fbs

4l→ ZZ* →H and γ γ →H

(b)

Figure 4: The (a) observed and (b) SM expected correlation coefficients between the HEL parameters.

6 Summary

An initial fit has been performed for EFT parameters using combined ATLAS Higgs cross sectionmeasurements. The constraints on the parameters that describe interactions between the Higgs bosonand either gluons or photons are stronger than those from a prior fit to a similar parameter set [12]. Thesix-parameter fit demonstrates a general procedure for moving beyond κ parameter fits into a more generalEFT framework for combined Higgs boson measurements. As more measurements are included and morerobust theoretical tools become available, the fits will expand to larger operator sets and will further exploitkinematic distributions to tighten the constraints on EFT parameters.

9

cG

-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-310×

)λ-2

ln(

0

1

2

3

4

5

6

7

8ObservedSM expected

PreliminaryATLAS

-1 = 13 TeV, 36.1 fbs

4l→ ZZ* →H and γ γ →H

(a)

cA

-0.4 -0.2 0 0.2 0.4 0.6-310×

)λ-2

ln(

0

1

2

3

4

5

6

7

8ObservedSM expected

PreliminaryATLAS

-1 = 13 TeV, 36.1 fbs

4l→ ZZ* →H and γ γ →H

(b)

cu

-2 -1.5 -1 -0.5 0 0.5

)λ-2

ln(

0

1

2

3

4

5

6

7

8ObservedSM expected

PreliminaryATLAS

-1 = 13 TeV, 36.1 fbs

4l→ ZZ* →H and γ γ →H

(c)

cHW

-0.1 -0.05 0 0.05 0.1

)λ-2

ln(

0

1

2

3

4

5

6

7

8ObservedSM expected

PreliminaryATLAS

-1 = 13 TeV, 36.1 fbs

4l→ ZZ* →H and γ γ →H

(d)

cHB

-0.2 -0.1 0 0.1 0.2

)λ-2

ln(

0

1

2

3

4

5

6

7

8ObservedSM expected

PreliminaryATLAS

-1 = 13 TeV, 36.1 fbs

4l→ ZZ* →H and γ γ →H

(e)

cWW - cB

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

)λ-2

ln(

0

1

2

3

4

5

6

7

8ObservedSM expected

PreliminaryATLAS

-1 = 13 TeV, 36.1 fbs

4l→ ZZ* →H and γ γ →H

(f)

Figure 5: The observed (solid) and SM expected (dashed) profiled negative log-likelihood scans for the six-parameterfit. The parameters are (a) cG, (b) cA, (c) cu, (d) cHW, (e) cHB, and (f) cWW − cB.

10

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