Studies of the spin, parity and tensor couplings of the Higgs boson in the H→ZZ∗→4ℓ\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow \gamma \gamma $$\end{document} decay processes at the LHC are presented. The investigations are based on 25fb-1\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{s}=8$$\end{document} TeV. The Standard Model (SM) Higgs boson hypothesis, corresponding to the quantum numbers JP=0+\documentclass[12pt]{minimal}
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\begin{document}$$J^{P}=0^{+}$$\end{document}, is tested against several alternative spin scenarios, including non-SM spin-0 and spin-2 models with universal and non-universal couplings to fermions and vector bosons. All tested alternative models are excluded in favour of the SM Higgs boson hypothesis at more than 99.9 % confidence level. Using the H→ZZ∗→4ℓ\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow WW^{*} \rightarrow e \nu \mu \nu $$\end{document} decays, the tensor structure of the interaction between the spin-0 boson and the SM vector bosons is also investigated. The observed distributions of variables sensitive to the non-SM tensor couplings are compatible with the SM predictions and constraints on the non-SM couplings are derived.
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An erratum to this article can be found at http://dx.doi.org/10.1140/epjc/s10052-016-3934-y.
Introduction
The discovery of a Higgs boson by the ATLAS [1] and CMS [2] experiments at the Large Hadron Collider (LHC) at CERN marked the beginning of a new era of experimental studies of the properties of this new particle. In the Standard Model (SM), the Higgs boson is a CP-even scalar particle, JCP=0++\documentclass[12pt]{minimal}
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\begin{document}$$J^{CP}=0^{++}$$\end{document}.1 Theories of physics beyond the SM (BSM) often require an extended Higgs sector featuring several neutral Higgs bosons. Such cases may include CP-mixing in the Higgs boson interactions, which could result in observable differences in the kinematics of final-state particles produced in their decays. A review of the phenomenology in the determination of Higgs boson spin and CP properties can be found in Ref. [3] and references therein.
Previous determinations of the Higgs boson spin and CP quantum numbers by the ATLAS and CMS Collaborations are reported in Refs. [4, 5]. Results on the same subject have also been published by the D0 and CDF Collaborations in Ref. [6]. All these studies indicate the compatibility of the spin and CP properties of the observed Higgs boson with the SM predictions. The ATLAS measurement excluded several alternative spin and parity hypotheses in favour of the quantum numbers predicted by the SM. In addition to the exclusion of several non-SM spin hypotheses, the CMS measurement probed the tensor structure of the Higgs boson decay to SM vector bosons in the spin-0 scenario. This paper complements the previous ATLAS study of the Higgs boson spin and parity. The new study takes advantage of improvements to the analysis strategy and to the modelling used to describe alternative spin hypotheses, and includes studies on CP-mixing for the spin-0 scenario. The improved theoretical framework is based on the Higgs boson characterisation model described in Refs. [3, 7].
The study of the spin and parity properties of the Higgs boson presented in this paper is based on the H→γγ\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow WW^{*} \rightarrow e \nu \mu \nu $$\end{document} decay channels and their combination. The H→WW∗→eνμν\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow WW^{*} \rightarrow e \nu \mu \nu $$\end{document} analysis is described in detail in a separate publication [8]. These analyses are based on 4.5 and 20.3 fb-1\documentclass[12pt]{minimal}
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\begin{document}$$^{-1}$$\end{document} of pp collision data collected by the ATLAS experiment at centre-of-mass energies of 7 and 8 TeV, respectively. For the H→WW∗→eνμν\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow WW^{*} \rightarrow e \nu \mu \nu $$\end{document} studies only the data collected at a centre-of-mass energy of 8 TeV are used. The SM hypothesis JP=0+\documentclass[12pt]{minimal}
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\begin{document}$$J^P=0^+$$\end{document} is compared to alternative spin-0 models: a pseudoscalar boson JP=0-\documentclass[12pt]{minimal}
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\begin{document}$$J^P=0^-$$\end{document} and a BSM scalar boson JP=0h+\documentclass[12pt]{minimal}
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\begin{document}$$J^P=0^+_h$$\end{document} [9, 10], which describes the interaction of the Higgs boson with the SM vector bosons with higher-dimension operators discussed in Sect. 3.1. Graviton-like tensor models with JP=2+\documentclass[12pt]{minimal}
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\begin{document}$$J^P=2^+$$\end{document} with universal and non-universal couplings [3, 7] are also considered. In these tests of fixed spin and parity hypotheses it is assumed that the resonance decay involves only one CP eigenstate.
In addition to the fixed spin and parity hypothesis tests, the possible presence of BSM terms in the Lagrangian describing the HVV vertex2 of the spin-0 resonance is also investigated. The HVV interaction is described in terms of an effective Lagrangian that contains the SM interaction and BSM CP-odd and CP-even terms [3, 7]. The relative fractions of the CP-odd and CP-even BSM contributions to the observed Higgs boson decays are constrained, and limits on the corresponding BSM tensor couplings are derived.
This paper is organised as follows. In Sect. 2 the ATLAS detector is described. In Sect. 3 the theoretical framework used to derive the spin and parity models, as well as the parameterisation used to describe the HVV coupling tensor structure, are discussed. In Sect. 4, the choice of Monte Carlo generators for the simulation of signal and backgrounds is described. The analyses of fixed spin and parity hypotheses for the three decay channels and their combination are presented in Sect. 5. Individual and combined studies of the tensor structure of the HVV interaction are presented in Sect. 6. Concluding remarks are given in Sect. 7.
The ATLAS detector
The ATLAS detector is described in detail in Ref. [11]. ATLAS is a multi-purpose detector with a forward-backward symmetric cylindrical geometry. It uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the z-axis along the beam pipe. The x-axis points from the IP to the centre of the LHC ring, and the y-axis points upward. Cylindrical coordinates (r,ϕ)\documentclass[12pt]{minimal}
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At small radii from the beamline, the inner detector (ID), immersed in a 2 T magnetic field produced by a thin superconducting solenoid located in front of the calorimeter, is made up of fine-granularity pixel and microstrip detectors. These silicon-based detectors cover the range |η|<2.5\documentclass[12pt]{minimal}
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\begin{document}$$|\eta |<2.5$$\end{document}. A gas-filled straw-tube transition-radiation tracker (TRT) complements the silicon tracker at larger radii and also provides electron identification based on transition radiation. The electromagnetic (EM) calorimeter is a lead/liquid-argon sampling calorimeter with an accordion geometry. The EM calorimeter is divided into a barrel section covering |η|<1.475\documentclass[12pt]{minimal}
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\begin{document}$$|\eta | > 2.0 $$\end{document}. The deflection is provided by a toroidal magnetic field with an integral of approximately 3 and 6 Tm in the central and end-cap regions of the ATLAS detector, respectively. The muon spectrometer is also instrumented with dedicated trigger chambers, the resistive-plate chambers in the barrel and thin-gap chambers in the end-cap, covering |η|<2.4\documentclass[12pt]{minimal}
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\begin{document}$$|\eta | <2.4 $$\end{document}.
Theoretical models
In this section, the theoretical framework for the measurements of the spin and parity of the resonance is discussed. An effective field theory (EFT) approach is adopted to describe the interaction between the resonance and the SM vector bosons, following the Higgs boson characterisation model described in Refs. [3, 7]. Three possible BSM scenarios for the spin and parity of the boson are considered:
the observed resonance is a spin-2 particle,
the observed resonance is a pure BSM spin-0 CP-even or CP-odd Higgs boson,
the observed resonance is a mixture of the SM spin-0 state and a BSM spin-0 CP-even or CP-odd state.
The third case would imply CP-violation in the Higgs sector. In the case of CP mixing, the Higgs boson would be a mass eigenstate, but not a CP eigenstate. In all cases, only one resonance with a mass of about 125 GeV is considered. It is also assumed that the total width of the resonance is small compared to the typical experimental resolution of the ATLAS detector (of the order of 1–2 GeV in the four-lepton and γγ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma \gamma $$\end{document} final states, as documented in Ref. [12]). Interference effects between the BSM signals and SM backgrounds are neglected.
Parameters of the benchmark scenarios for spin-0 boson tensor couplings used in tests (see Eq. (1)) of the fixed spin and parity models
The EFT approach, used by the Higgs boson characterisation model, is only valid up to a certain energy scale, Λ\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda $$\end{document}. The models described in Ref. [7] assume that the resonance structure corresponds to one new boson (X(JP)\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda $$\end{document} scale is set to 1 TeV to account for the experimental results obtained at the LHC and previous collider experiments, which do not show any evidence of new physics at lower energy scales.
The case where the observed resonance has JP=1±\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow \gamma \gamma $$\end{document} decay is forbidden by the Landau–Yang theorem [13, 14] for a spin-1 particle. Moreover, the spin-1 hypothesis was already studied in the previous ATLAS publication [4] in the H→ZZ∗→4ℓ\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow WW^{*} \rightarrow e \nu \mu \nu $$\end{document} decays and excluded at a more than 99 % confidence level.
The spin-0 hypothesis
In the spin-0 hypothesis, models with fixed spin and parity, and models with mixed SM spin-0 and BSM spin-0 CP-even and CP-odd contributions are considered. In Ref. [7], the spin-0 particle interaction with pairs of W or Z bosons is given through the following interaction Lagrangian:L0V=cos(α)κSM12gHZZZμZμ+gHWWWμ+W-μ-141Λcos(α)κHZZZμνZμν+sin(α)κAZZZμνZ~μν-121Λcos(α)κHWWWμν+W-μν+sin(α)κAWWWμν+W~-μνX0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathcal L_0^{V}= & {} \left\{ \cos (\alpha ) \kappa _\mathrm{SM}\left[ \frac{1}{2}g_{HZZ} Z_{\mu } Z^ {\mu } + g_{HWW} W^{+}_{\mu } W^ {- \mu }\right] \right. \nonumber \\&\left. -\frac{1}{4}\frac{1}{\Lambda }\left[ \cos (\alpha ) \kappa _{HZZ} Z_{\mu \nu }Z^{\mu \nu }+ \sin (\alpha ) \kappa _{AZZ} Z_{\mu \nu } \tilde{Z}^{\mu \nu }\right] \right. \nonumber \\&\left. -\frac{1}{2} \frac{1}{\Lambda } \left[ \cos (\alpha ) \kappa _{HWW} W^+_{\mu \nu }W^{-\mu \nu }\right. \right. \nonumber \\&\left. \left. + \sin (\alpha ) \kappa _{AWW} W^+_{\mu \nu } \tilde{W}^{ - \mu \nu }\right] \right\} X_0. \end{aligned}$$\end{document}Here Vμ\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \ne \pi $$\end{document}, provided the corresponding coupling constants are non-vanishing. The SM couplings, gHVV\documentclass[12pt]{minimal}
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\begin{document}$$g_{HVV} \propto m^{2}_{V}$$\end{document}. Other higher-order operators described in Ref. [7], namely the derivative operators, are not included in Eq. (1) and have been neglected in this analysis since they induce modifications of the discriminant variables well below the sensitivity achievable with the available data sample.
As already mentioned, for the spin-0 studies the SM Higgs boson hypothesis is compared to two alternatives: the CP-odd JP=0-\documentclass[12pt]{minimal}
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\begin{document}$$J^P=0^-$$\end{document} and the BSM CP-even JP=0h+\documentclass[12pt]{minimal}
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\begin{document}$$J^P=0^+_h$$\end{document} hypotheses. All three models are obtained by selecting the corresponding parts of the Lagrangian described in Eq. (1) while setting all other contributions to zero. The values of the couplings corresponding to the different spin-0 models are listed in Table 1.
The investigation of the tensor structure of the HVV interaction is based on the assumption that the observed particle has spin zero. Following the parameterisation defined in Eq. (1), scenarios are considered where only one CP-odd or one CP-even BSM contribution at a time is present in addition to the SM contribution. To quantify the presence of BSM contributions in H→ZZ∗\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{\kappa }_{HVV}$$\end{document} are defined as follows:κ~AVV=14vΛκAVVandκ~HVV=14vΛκHVV,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \tilde{\kappa }_{ AVV}=\frac{1}{4} \frac{ \mathrm{v} }{\Lambda } \kappa _{AVV}\quad \mathrm{and}\quad \tilde{\kappa }_{HVV}=\frac{1}{4} \frac{ \mathrm{v} }{\Lambda } \kappa _{HVV}, \end{aligned}$$\end{document}where v\documentclass[12pt]{minimal}
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\begin{document}$$(f_{g4},\phi _{g4})$$\end{document} proposed in Refs. [3, 9, 10]. Further details of these conversions are given in Appendix A.
The BSM terms described in Eq. (1) are also expected to change the relative contributions of the vector-boson fusion (VBF) and vector-boson associated production (VH) processes with respect to the gluon-fusion (ggF) production process, which is predicted to be the main production mode for the SM Higgs boson at the LHC. For large values of the BSM couplings, at the LHC energies, the VBF production mode can have a cross section that is comparable to the ggF process [16]. This study uses only kinematic properties of particles from H→VV∗\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow V V^{*}$$\end{document} decays to derive information on the CP nature of the Higgs boson. The use of the signal rate information for different production modes, in the context of the EFT analysis, may increase the sensitivity to the BSM couplings at the cost of a loss in generality. For example the ratio of the VBF and VH production modes with respect to the ggF one can be changed by a large amount for non-vanishing values of the BSM couplings. In the studies presented in this paper the predictions of the signal rates are not used to constrain the BSM couplings.
As described in Sect. 6.2, only events with no reconstructed jets (the 0-jet category) are used in the H→WW∗→eνμν\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow WW^{*} \rightarrow e \nu \mu \nu $$\end{document} analysis for the studies of the tensor structure; hence this analysis has little sensitivity to the VBF production mode. The H→ZZ∗→4ℓ\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} analysis also has little sensitivity to this production mode since it is mainly based on variables related to the four-lepton kinematics. The Boosted Decision Tree (BDT) algorithm [17] used to discriminate signals from the ZZ∗\documentclass[12pt]{minimal}
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\begin{document}$$ZZ^{*}$$\end{document} background, described in Sects. 5.4 and 6.3, includes the transverse momentum of the four-lepton system and is trained on simulated samples of ggF-produced signals. An enhancement of the VBF production mode would improve the separation between background and signal since it predicts larger values of the transverse momentum spectrum for events produced via VBF than via ggF [3].
The spin-2 hypothesis
In the Higgs boson characterisation model [7], the description of the interaction of a spin-2 particle with fermions and vector bosons is described by the following Lagrangian:L2=-1Λ∑VκVTμνVXμν+∑fκfTμνfXμν.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathcal L_2 = - \frac{1}{\Lambda }\left[ \sum _V \kappa _V \mathcal{T}^V_{\mu \nu } X^{\mu \nu } + \sum _f \kappa _f \mathcal{T}^f_{\mu \nu } X^{\mu \nu } \right] . \end{aligned}$$\end{document}The spin-2 tensor field Xμν\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{T}^f_{\mu \nu }$$\end{document}, of any vector boson V and fermion f, as inspired by gravitation theories. The strength of each interaction is determined by the couplings κV\documentclass[12pt]{minimal}
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\begin{document}$$\kappa _f$$\end{document}. In the simplest formulation, all couplings are equal. This scenario is referred to as universal couplings (UC), while scenarios with different values of the couplings are referred to as non-universal couplings (non-UC). In the UC scenario, the production of a spin-2 particle in pp collisions is expected to be dominated by QCD processes, with negligible contributions from electroweak (EW) processes (i.e. from processes involving EW boson propagators). Simulation studies based on MadGraph5_aMC@NLO [16], which implements the Lagrangian described in Eq. (3), predict for the production cross section in the UC scenario σEW/σQCD≃3×10-4\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _{\mathrm {EW}}/\sigma _{\mathrm {QCD}}\simeq 3 \times 10^{-4}$$\end{document}. These studies also show that EW production of the spin-2 resonance would occur mainly in association with a massive EW boson (WX, ZX). Present observations do not show a dominant VH production mechanism, hence suggesting that σEW\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _{\mathrm {QCD}}$$\end{document}. This paper considers only QCD production for all the spin-2 benchmark scenarios.
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\begin{document}$$\kappa _\gamma $$\end{document} is assumed to be independent of all the other couplings. In the following, the UC scenario only refers to κq=κg\documentclass[12pt]{minimal}
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\begin{document}$$q\bar{q} \rightarrow X$$\end{document} (where q refers to light quarks), yield different polarisations for the spin-2 particle X, and hence different angular distributions of its decay products. These mechanisms are considered in the model of a graviton-like tensor with minimal couplings proposed in Refs. [9, 10], which has been studied experimentally in Ref. [4]. The EFT Lagrangian, however, also allows for more complex processes with emission of one or more additional partons. For instance, processes with one-parton emission, like qg→qX\documentclass[12pt]{minimal}
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\begin{document}$$q\bar{q}\rightarrow q\bar{q}X$$\end{document}, the spin-2 production may occur through qqX or ggX vertices, respectively, such that the polarisation of X is not uniquely determined by the initial state. Moreover, the EFT also allows for four-leg vertices like qqgX. These additional diagrams effectively change the polarisation of the particle X, compared to what is assumed by the model in Refs. [9, 10]. As a consequence, the angular distributions of the decay products become harder to separate from those expected for a scalar resonance.
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\begin{document}$$\kappa _g,~\kappa _q$$\end{document}. Presently, there are no experimental constraints on the ratio κq/κg\documentclass[12pt]{minimal}
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\begin{document}$$\kappa _q/\kappa _g$$\end{document} from observed decay modes, since the separation of jets initiated by gluons or by light quarks is experimentally difficult and has not yet been attempted in Higgs boson studies. The ratio κq/κg\documentclass[12pt]{minimal}
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\begin{document}$$\kappa _q/\kappa _g$$\end{document} can thus be regarded as a free parameter. When κq≠κg\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda $$\end{document}. At higher energies, new physics phenomena are expected to enter to regularise the anomalous ultra-violet behaviour.
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\begin{document}$$p_{\text {T}} ^X$$\end{document} tails incompatible with the observed differential distribution reported in Refs. [21, 22]. For this reason the investigated range of the κq/κg\documentclass[12pt]{minimal}
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\begin{document}$$\kappa _{q} = \kappa _{g}$$\end{document} model is referred to hereafter as the UC scenario. The κq=0\documentclass[12pt]{minimal}
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\begin{document}$$p_{\text {T}} ^X$$\end{document} of the spin-2 resonance are also shown. For the UC scenario no pTX\documentclass[12pt]{minimal}
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\begin{document}$$p_{\text {T}} ^X$$\end{document} selection is applied
The data presented in this paper were recorded by the ATLAS detector during the 2012 LHC run with proton–proton collisions at a centre-of-mass energy of 8 TeV, and correspond to an integrated luminosity of 20.3 fb-1\documentclass[12pt]{minimal}
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\begin{document}$$^{-1}$$\end{document}. For the H→γγ\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow \gamma \gamma $$\end{document} and H→ZZ∗→4ℓ\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} channels, the data collected in 2011 at a centre-of-mass energy of 7 TeV corresponding to an integrated luminosity of 4.5 fb-1\documentclass[12pt]{minimal}
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\begin{document}$$^{-1}$$\end{document}, are also used. Data quality requirements are applied to reject events recorded when the relevant detector components were not operating correctly. More than 90 % of the recorded luminosity is used in these studies. The trigger requirements used to collect the data analysed in this paper are the same as those described in previous publications [18–20]. They are only briefly recalled in the following sections.
The Monte Carlo (MC) samples for the backgrounds and for the SM Higgs boson signal are the same as those used for the analyses described in Refs. [18–20], whereas new non-SM signal samples have been simulated. An overview of the signal samples is given in Sect. 4.1.
The effects of the underlying event and of additional minimum-bias interactions occurring in the same or neighbouring bunch crossings, referred to as pile-up in the following, are modelled with Pythia 8 [23]. The ATLAS detector response is simulated [24] using either Geant 4 [25] alone or combined with a parameterised Geant 4-based calorimeter simulation [26].
SM Higgs boson and BSM signal samples
The SM Higgs boson ggF production for all analyses is modelled using the Powheg-Box [27] generator at next-to-leading order (NLO), interfaced to Pythia 8 for parton showering and hadronisation and to simulate multi-parton interactions. To improve the modelling of the SM Higgs boson pT\documentclass[12pt]{minimal}
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\begin{document}$$p_{\text {T}}$$\end{document}, a reweighting procedure is applied. This procedure applies a weight depending on the pT\documentclass[12pt]{minimal}
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\begin{document}$$p_{\text {T}}$$\end{document} of the Higgs boson to each event. The weights are chosen in order to reproduce the prediction of the next-to-next-to-leading-order (NNLO) and next-to-next-to-leading-logarithms (NNLL) dynamic-scale calculation given by the hres2.1 program [28, 29].
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\begin{document}$$H \rightarrow \gamma \gamma $$\end{document} analysis, the signal samples are generated at several values of the Higgs boson mass mH\documentclass[12pt]{minimal}
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\begin{document}$$m_H$$\end{document} around 125 GeV. The samples are used to obtain a parameterisation of the signal yields and of the invariant mass distribution of the two-photon system as continuous functions of mH\documentclass[12pt]{minimal}
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\begin{document}$$m_H$$\end{document} (both inclusively and for each category in the analysis, as described in Sect. 5.2). The spin-2 samples are generated using the MadGraph5_aMC@NLO [16] program with LO accuracy for zero, one, and two additional partons, and with subsequent matching of the matrix-element calculation with a model of the parton shower, underlying event and hadronisation, using Pythia 6 [30].
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} analysis the signal samples representing the production and decay of Higgs bosons with spin-0 and different parities are generated as follows. The SM Higgs boson production via gluon fusion at the mass mH=125.5\documentclass[12pt]{minimal}
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\begin{document}$$m_H=125.5$$\end{document} GeV is simulated using the Powheg-Box generator. For the non-SM signals, the decays of the generated Higgs bosons are simulated, according to the Higgs boson parity assumptions, using the JHU [9, 10] MC generator at leading order (LO). The spin-2 samples are generated using the MadGraph5_aMC@NLO MC generator, as for the H→γγ\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow \gamma \gamma $$\end{document} analysis.
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\begin{document}$$H \rightarrow WW^{*} \rightarrow e \nu \mu \nu $$\end{document} analysis, the SM Higgs boson signal is generated at mH=125\documentclass[12pt]{minimal}
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\begin{document}$$m_H = 125$$\end{document} GeV using the Powheg-Box Monte Carlo generator. The spin-0 BSM signal samples are generated using MadGraph5_aMC@NLO. The signal samples representing the production and decay of Higgs bosons with spin-2 are generated using the MadGraph5_aMC@NLO MC generator, as for the H→γγ\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow \gamma \gamma $$\end{document} analysis.
For studies of the tensor structure of the HVV decay, all simulated signal samples are obtained by using the matrix element (ME) reweighting method applied, as explained in the following, to a sample generated with non-zero values of the BSM couplings. The reweighting procedure is validated against samples produced at different values of the couplings, to ensure that the distributions of the CP-sensitive final-state observables and of their correlations are reproduced correctly. For the H→ZZ∗→4ℓ\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} analysis, the MC production is only performed for one set of tensor couplings: g1=1\documentclass[12pt]{minimal}
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\begin{document}$$g_4= 1+i$$\end{document}. All other configurations of couplings are obtained by reweighting this sample at generator level. The ratios of the corresponding squares of ME values calculated at LO are used as weights. To calculate these ME values, the JHUGenME [10] program is used. In the H→WW∗→eνμν\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow WW^{*} \rightarrow e \nu \mu \nu $$\end{document} analysis, only one MC sample is generated, using MadGraph5_aMC@NLO with parameters κSM=1\documentclass[12pt]{minimal}
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\begin{document}$$\cos (\alpha ) = 0.3$$\end{document}, and all other samples are obtained from it by reweighting the events on the basis of the ME amplitudes.
In all the analyses presented in this paper, the mass of the Higgs boson is fixed to 125.4 GeV [12].
Background samples
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Tests of fixed spin and parity hypotheses
The H→γγ\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} analyses are improved with respect to the previous ATLAS publication of Ref. [4]. These analyses are described in some detail in the following subsections. The spin and parity analysis in the H→WW∗→eνμν\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow WW^{*} \rightarrow e \nu \mu \nu $$\end{document} channel has also been improved, as discussed in detail in a separate publication [8]. In the following, only a brief overview of this analysis is given. The expected and observed results of the individual channels and of their combination are presented in Sect. 5.5.
Statistical treatment
The analyses rely on discriminant observables chosen to be sensitive to the spin and parity of the signal.
Expected distributions of kinematic variables sensitive to the spin of the resonance considered in the H→γγ\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{L}(\mathrm{data}~|~J^P, \mu , \vec {\theta })$$\end{document}, that depends on the spin-parity assumption of the signal is constructed as a product of conditional probabilities over binned distributions of the discriminant observables in each channel:L(data|JP,μ,θ→)=∏jNchann.∏iNbinsP(Ni,j|μj·Si,j(JP)(θ→)+Bi,j(θ→))·Aj(θ→),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathcal {L}(\mathrm{data}~|~ J^P, \mu , \vec {\theta })= & {} \displaystyle \prod _{j}^{N_{\mathrm {chann.}}} \prod _{i}^{N_{\mathrm {bins}}} P\big (N_{i,j} ~| ~\mu _{j} \cdot S^{(J^P)}_{i, j}(\vec {\theta }) \nonumber \\&+ B_{i,j}(\vec {\theta }) \big ) \cdot \mathcal {A}_{j}(\vec {\theta }) \;, \end{aligned}$$\end{document}where μj\documentclass[12pt]{minimal}
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\begin{document}$$J^{P}_{\mathrm {alt} }$$\end{document}, as discussed in Sect. 3. In order to be insensitive to assumptions on the couplings of the non-SM resonance (the alternative hypotheses) to SM particles, the numbers of signal events in each channel, for each different LHC centre-of-mass energy and for each tested hypothesis, are treated as independent parameters in the likelihood and fitted to the data when deriving results on the spin and parity hypotheses.
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\begin{document}$$\tilde{q}$$\end{document} used to distinguish between the two spin-parity hypotheses is based on a ratio of profiled likelihoods [31, 32]:q~=logLJSMP,μ^^JSMP,θ^^JSMPLJaltP,μ^^JaltP,θ^^JaltP,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \tilde{q} = \log \frac{\mathcal {L}\left( J^{P}_{\mathrm {SM} }, \hat{\hat{\mu }}_{J^{P}_{\mathrm {SM} }}, \hat{\hat{\theta }}_{J^{P}_{\mathrm {SM} }}\right) }{\mathcal {L}\left( J^{P}_{\mathrm {alt} }, \hat{\hat{\mu }}_{J^{P}_{\mathrm {alt} }}, \hat{\hat{\theta }}_{J^{P}_{\mathrm {alt} }} \right) }, \end{aligned}$$\end{document}where L(JP,μ^^JP,θ^^JP)\documentclass[12pt]{minimal}
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\begin{document}$$\hat{\hat{\theta }}_{J^P}$$\end{document} represent the values of the signal strength and nuisance parameters fitted to the data under each spin and parity hypothesis. The distributions of the test statistic for both hypotheses are obtained using ensemble tests of MC pseudo-experiments. For each hypothesis test, about 70, 000 pseudo-experiments were generated. The generation of the pseudo-experiments uses the numbers of signal and background events in each channel obtained from maximum-likelihood fits to data. In the fits of each pseudo-experiment, these and all other nuisance parameters are profiled, i.e. fitted to the value that maximises the likelihood for each value of the parameter of interest. When generating the distributions of the test statistic for a given spin-parity hypothesis, the expectation values of the signal strengths are fixed to those obtained in the fit to the data under the same spin-parity assumption. The distributions of q~\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{q}$$\end{document} is distributed such that all p-values are equally probable.
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\begin{document}$$H \rightarrow \gamma \gamma $$\end{document} analysis, and comparison to expected values for a SM Higgs boson and for a spin-2 particle with different choices of QCD couplings. a The 11 categories described in the text are displayed, corresponding to the pTγγ<300GeV\documentclass[12pt]{minimal}
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\begin{document}$$p_{\text {T}} ^{\gamma \gamma }<125~\text {GeV}$$\end{document} selection
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\begin{document}$$\begin{aligned} \mathrm{CL}_\mathrm{s}(J^{P}_{\mathrm {alt} }) = \frac{p (J^{P}_{\mathrm {alt} })}{ 1- p (J^{P}_{\mathrm {SM} })}\;. \end{aligned}$$\end{document}
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\begin{document}$$H \rightarrow \gamma \gamma $$\end{document} channel is sensitive to a possible spin-2 state. Since the spin-2 models investigated in the present paper are different from those assumed in Ref. [4], the analysis has been redesigned, to improve its sensitivity to the new models.
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\begin{document}$$p_{\text {T}}$$\end{document}, respectively. During the offline selection two photons are further required to be in a fiducial pseudorapidity region, defined by |ηγ|<2.37\documentclass[12pt]{minimal}
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\begin{document}$$p_{\text {T}} ^{\gamma \gamma }<300~\text {GeV}$$\end{document}, motivated by the assumed validity limit of the spin-2 EFT model, as explained in Sect. 3. After this selection, 17,220 events are left at a centre-of-mass energy s=7TeV\documentclass[12pt]{minimal}
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\begin{document}$$p_{\text {T}} ^{\gamma \gamma }$$\end{document} and the production angle of the two photons, measured in the Collins–Soper frame [34]:|cosθ∗|=|sinh(Δηγγ)|1+(pTγγ/mγγ)22pTγ1pTγ2mγγ2,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} |\cos \theta ^*|\ = \frac{|\sinh (\Delta \eta ^{\gamma \gamma }) |}{\sqrt{1+ ( p_{\text {T}} ^{\gamma \gamma }/m_{\gamma \gamma })^2}}\frac{2p_{\text {T}} ^{\gamma _1}p_{\text {T}} ^{\gamma _2}}{m_{\gamma \gamma }^2}, \end{aligned}$$\end{document}where Δηγγ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta \eta ^{\gamma \gamma }$$\end{document} is the separation in pseudorapidity of the two photons.
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\begin{document}$$m_{\gamma \gamma }$$\end{document} distribution is modelled in each category as the sum of one-dimensional probability density functions (pdf) for signal and background distributions:f[c](mγγ|J)=nB[c]fB[c](mγγ)+(nJ[c]+nbias[c])fS[c](mγγ)nB[c]+nJ[c]+nbias[c],\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} f^{[c]}(m_{\gamma \gamma }|J)= & {} \frac{n_B^{[c]}f_B^{[c]}(m_{\gamma \gamma })+(n_J^{[c]}+n_\mathrm{bias}^{[c]})f_S^{[c]}(m_{\gamma \gamma })}{n_B^{[c]}+n_J^{[c]}+n_\mathrm{bias}^{[c]}},\nonumber \\ \end{aligned}$$\end{document}where J is the spin hypothesis, nB[c]\documentclass[12pt]{minimal}
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\begin{document}$$f_S^{[c]}(m_{\gamma \gamma })$$\end{document} is modelled as a weighted sum of a Crystal Ball function, describing the core and the lower mass tail, and of a Gaussian component that improves the description of the tail for higher mass values. For each category, fS[c](mγγ)\documentclass[12pt]{minimal}
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\begin{document}$$f_B^{[c]}(m_{\gamma \gamma })$$\end{document} is empirically modelled as an exponential of a first- or second-degree polynomial. The choice of such a parameterisation can induce a bias (“spurious signal”) in the fitted signal yield, which is accounted for by the term nbias[c]\documentclass[12pt]{minimal}
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\begin{document}$$n_\mathrm{bias}^{[c]}$$\end{document} is constrained for each category by multiplying the likelihood function by a Gaussian function centred at zero and with a width determined by the size of the expected bias.
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\begin{document}$$p_{\text {T}} ^{\gamma \gamma }$$\end{document}) category provides strong discrimination power against the non-SM hypothesis, as visible in Fig. 2a.
To discriminate between the SM spin-0 (JSMP=0+\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{L}_{J^{P}_{\mathrm {SM} }},~\mathcal{L}_{J^{P}_{\mathrm {alt} }}$$\end{document} are built, following the general approach described in Eq. (4):-lnLJ=∑cnB[c]+nSΦJ[c]+nbias[c]-∑e∈[c]ln[nB[c]fB[c]mγγ(e)+nSΦJ[c]+nbias[c]fS[c](mγγ(e))]\documentclass[12pt]{minimal}
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\begin{document}$$\Phi _J^{[c]}$$\end{document}, for the SM Higgs boson are affected by uncertainties on the pT\documentclass[12pt]{minimal}
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\begin{document}$$p_{\text {T}} $$\end{document} spectrum of the resonance and on the size of the interference between the resonance and continuum production. The former is computed as described in Ref. [20]. The relative impact on the signal fractions is less than ±1%\documentclass[12pt]{minimal}
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\begin{document}$$\pm 13\,\%$$\end{document} for categories 10 and 11. The correction for the interference is evaluated according to Refs. [35, 36]. The systematic uncertainty is conservatively assumed to equal the correction itself, and its relative impact ranges between ±0.1%\documentclass[12pt]{minimal}
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\begin{document}$$\pm 1.8\,\%$$\end{document}.
No systematic uncertainty is assigned to the simulated pTX\documentclass[12pt]{minimal}
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\begin{document}$$p_{\text {T}} ^{X} $$\end{document} distribution of the spin-2 models. The effect of the interference between the resonance and continuum production is essentially not known, as it depends on the width, ΓX\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma _X$$\end{document}, of the resonance, which is unknown. The results presented here only hold under the assumption of a narrow width for the resonance, such that interference effects can be neglected.
Additional systematic uncertainties come from the calibration of the photon energy scale and energy resolution and affect the signal parameterisation fS[c]\documentclass[12pt]{minimal}
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\begin{document}$$f_S^{[c]}$$\end{document}. These uncertainties are evaluated as described in Ref. [12].
Spin and parity analysis in the <inline-formula id="IEq285"><alternatives><mml:math><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>W</mml:mi><mml:msup><mml:mi>W</mml:mi><mml:mrow><mml:mrow/><mml:mo>∗</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">→</mml:mo><mml:mi>e</mml:mi><mml:mi mathvariant="italic">ν</mml:mi><mml:mi mathvariant="italic">μ</mml:mi><mml:mi mathvariant="italic">ν</mml:mi></mml:mrow></mml:math><tex-math id="IEq285_TeX">\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow WW^{*} \rightarrow e \nu \mu \nu $$\end{document}</tex-math><inline-graphic xlink:href="10052_2015_3685_Article_IEq285.gif"/></alternatives></inline-formula> channel
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\begin{document}$$H \rightarrow WW^{*} \rightarrow e \nu \mu \nu $$\end{document} channel is described in detail in a separate publication [8]. In the following a brief summary is provided. The selection is restricted to events containing two charged leptons of different flavour (one electron and one muon). The eνμν\documentclass[12pt]{minimal}
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\begin{document}$$e \nu \mu \nu $$\end{document} channel is the most sensitive one [19]. The same-flavour channels (eνeν\documentclass[12pt]{minimal}
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\begin{document}$$\mu \nu \mu \nu $$\end{document}) are not expected to add much in terms of sensitivity due to the presence of large backgrounds that cannot be removed without greatly reducing the acceptance of the alternative models considered in this analysis. The leading lepton is required to have pT>22\documentclass[12pt]{minimal}
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\begin{document}$${2.5 < |\eta | < 4.5}$$\end{document}), the spin-2 analysis enlarges the acceptance by allowing for zero or one jet (selected according to the above mentioned criteria).
The major sources of background after the dilepton selection are Z/γ∗\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document} and single top) production, and W bosons produced in association with hadronic jets (W+jets), where a jet is misidentified as a lepton. The contribution from misidentified leptons is significantly reduced by the requirement of two high-pT\documentclass[12pt]{minimal}
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\begin{document}$$m_{\ell \ell }< 80$$\end{document} GeV) targets the WW background. For alternative spin models with non-universal couplings, as discussed in Sect. 3, an additional upper bound is imposed on the Higgs boson pT\documentclass[12pt]{minimal}
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\begin{document}$$p_{\text {T}} $$\end{document}, reconstructed as the transverse component of the vector sum of the momenta of the two charged leptons and the missing transverse momentum. Additionally, for events containing one jet, which include substantial top-quark and W+jets backgrounds, b-jet and Z→τ+τ-\documentclass[12pt]{minimal}
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\begin{document}$$Z\rightarrow \tau ^+\tau ^-$$\end{document} vetoes are applied, together with transverse mass requirements: the larger of the transverse masses of the two W bosons (each computed using the corresponding lepton and the missing transverse momentum) in the event is required to be larger than 50 GeV, while the total transverse mass of the WW system (defined with the two leptons and the missing transverse momentum) is required to be below 150 GeV.
Control regions (CRs) are defined for the WW, top-quark and Drell–Yan backgrounds, which are the most important ones after the topological selection described above. The CRs are used to normalise the background event yields with a fit to the rates observed in data. The simulation is then used to transfer these normalisations to the signal region (SR). The W\documentclass[12pt]{minimal}
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\begin{document}$$W$$\end{document}+jets background is estimated entirely from data, while non-WW diboson backgrounds are estimated using MC simulation and cross-checked in a validation region.
After the signal region selection, 4730 and 1569 candidate events are found in data in the 0-jet and 1-jet categories, respectively. For the latter category, the number decreases to 1567 and 1511 events when applying a selection on the Higgs boson pT\documentclass[12pt]{minimal}
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\begin{document}$$p_{\text {T}} $$\end{document} of less than 300 GeV and less than 125 GeV, respectively. In total 218 (77) events are expected from a SM Higgs boson signal in the 0-jet (1-jet) category, while about 4390 (1413) events are expected for the total background.
A BDT algorithm is used in both the fixed spin hypothesis tests and the tensor structure analyses. For spin-2 studies, the strategy follows the one adopted in Ref. [4], with the main difference being that the 1-jet channel has been added. Two BDT discriminants are trained to distinguish between the SM hypothesis and the background (BDT0\documentclass[12pt]{minimal}
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\begin{document}$$_0$$\end{document}), and the alternative spin hypothesis and the background (BDT2\documentclass[12pt]{minimal}
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\begin{document}$$m_\mathrm{T}$$\end{document}, which provide the best discrimination between signal hypotheses and backgrounds, also in the presence of one jet in the final state. All background components are used in the trainings. In total, five BDT2\documentclass[12pt]{minimal}
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\begin{document}$$_0$$\end{document} for the SM Higgs boson hypothesis.
For the spin-0 fixed hypothesis test and HWW tensor structure studies, the first discriminant, BDT0\documentclass[12pt]{minimal}
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\begin{document}$$_0$$\end{document}, is the same as the one used for the spin-2 analysis, trained to disentangle the SM hypothesis from the background. A second BDT discriminant, BDTbadhbox\documentclass[12pt]{minimal}
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\begin{document}$$\Delta p_\mathrm{T}$$\end{document} for the CP-odd analysis. The training strategy is different from the one used in the spin-2 analysis because, while the spin-2 signal is very similar to the background, the spin-0 signals are all similar to each other, while being different from the main background components. Therefore, in the latter case, training the signal hypotheses against each other improves the sensitivity. The resulting BDT variable is afterwards used in binned likelihood fits to test the data for compatibility with the presence of a SM or BSM Higgs boson.
Several sources of systematic uncertainty are considered, both from experimental and theoretical sources, and are described in detail in Ref. [8]. The correlations induced among the different background sources by the presence of other processes in the control regions are fully taken into account in the statistical procedure. The most important systematic uncertainties are found to be those related to the modelling of the WW background, to the estimate of the W+jets background (originating from the data-driven method employed) and, for the spin-2 results in particular, to the Z→ττ\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} analysis is identical to the one presented in Ref. [12]. The main improvement with respect to the previous ATLAS publication of Ref. [4] is the introduction of a BDT discriminant designed to optimise the separation between the signal and the most relevant background process.
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\begin{document}$$ZZ^*$$\end{document} pairs. This background has the same final state as the signal events and hereafter is referred to as the irreducible background. It is estimated from simulation and normalised to the expected SM cross section calculated at NLO [37, 38]. The reducible sources of background come from Z+jets and tt¯\documentclass[12pt]{minimal}
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\begin{document}$$< m_{4\ell } <$$\end{document} 130 GeV signal region. The number of expected signal events is given for a SM Higgs boson mass of 125.5 GeV
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The choice of production and decay angles used in this analysis is presented in Fig. 3, where the following definitions are used:
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\begin{document}$$m_{34}$$\end{document}, are sensitive to the spin and parity of the resonance. In Fig. 4 the distributions of the final-state observables sensitive to the spin and parity of the decaying resonance are presented. The distributions are shown for the SM JP=0+\documentclass[12pt]{minimal}
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\begin{document}$$115~\text {GeV}<m_{4\ell }< 130~\text {GeV}$$\end{document}. The events observed in data are superimposed on each plot.
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Two approaches were pursued to develop the discriminants used to distinguish between different spin and parity hypotheses. The first uses the theoretical differential decay rate for the final-state observables sensitive to parity to construct a matrix-element-based likelihood ratio analysis (JP\documentclass[12pt]{minimal}
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\begin{document}$$J^P$$\end{document}-MELA). The second approach is based on a BDT.
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\begin{document}$$J^P$$\end{document}-MELA approach [3, 9], the probability of observing an event with given kinematics can be calculated. This probability is corrected for detector acceptance and analysis selection, which are obtained from the simulated signal MC samples. The full pdf also includes a term for incorrect pairing of the leptons in the 4μ\documentclass[12pt]{minimal}
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\begin{document}$$J^P$$\end{document} discriminant is formed for each pair of spin-parity states to be tested, by training a BDT on the variables of simulated signal events which fall in the signal mass window 115GeV<m4ℓ<130GeV\documentclass[12pt]{minimal}
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\begin{document}$$J^P$$\end{document}-MELA approach are reported. The BDT approach was used as a cross-check and produced compatible results.
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\begin{document}$$115~\text {GeV}<m_{4\ell }< 130~\text {GeV}$$\end{document} for the data (points with errors), the backgrounds (filled histograms), and for predictions for several spin and parity hypotheses. The SM hypothesis is shown by the solid line while the alternative hypotheses are shown by the dashed lines. The signal distributions are normalised to the signal strength fitted in data. a–cJP\documentclass[12pt]{minimal}
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Two general types of systematic effects impact the analyses using fixed spin and parity hypotheses: uncertainties on discriminant shapes due to experimental effects, and uncertainties on background normalisations from theory uncertainties and data-driven background estimates. The systematic uncertainties on the shape are included in the analysis by creating discriminant shapes corresponding to variations of one standard deviation in the associated sources of systematic uncertainty. The systematic uncertainties on the normalisation are included as additional nuisance parameters in the likelihood.
The list of sources of systematic uncertainty common to all ATLAS H→ZZ∗→4ℓ\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} analyses is presented in Ref. [18]. The relative impact of these sources on the final separation for all tested hypotheses is evaluated and sources affecting the final separation (given in Sect. 5.5) by less than ±0.5%\documentclass[12pt]{minimal}
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\begin{document}$$\pm 0.5\,\%$$\end{document} are neglected.
The main sources of systematic uncertainties are related to the experimental error on the Higgs boson mass, the modelling of the irreducible ZZ∗\documentclass[12pt]{minimal}
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Relative impact of the main systematic uncertainties on the expected separation (expressed in terms of numbers of standard deviations) between the SM JP=0+\documentclass[12pt]{minimal}
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As summarised in Table 6, the p-values of the combined results for the three channels show good agreement between the data and the SM hypothesis for all performed tests. All tested alternative hypotheses are rejected at a more than 99.9 % confidence level (CL) in favour of the SM hypothesis.
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Statistical treatment
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\begin{document}$$\text{ BDT }_{bad hbox}$$\end{document} are used as discriminant variables in the likelihood defined to measure the spin-0 tensor structure couplings. The only difference with respect to the spin hypothesis test is that, in this analysis, the BSM spin-0 couplings are treated as continuous variables in the test statistic.
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} process are used.
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\begin{document}$$ZZ^{*}$$\end{document} background, a kinematic BDT discriminant similar to those described in Sect. 5.4 is used as an additional observable in all fits. The BDT training is performed independently for each final state using observables with small sensitivity to parity: η4ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{BDT(}ZZ\mathrm {)} $$\end{document}.
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\begin{document}$$((\tilde{\kappa } _{AVV}/\kappa _\mathrm{SM}) \cdot \tan \alpha =0,\pm 5; \tilde{\kappa } _{HVV} =0)$$\end{document} are shown in Fig. 8.
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\begin{document}$$(\tilde{\kappa }_{AVV}/\kappa _\mathrm{SM})\cdot \tan {\alpha }$$\end{document}. The expected values are estimated for the signal strengths measured in data and assuming best-fit values for all other nuisance parameters. The 68 % and 95 % CL exclusion regions are indicated as lying above the corresponding horizontal lines. The individual distributions for H→WW∗→eνμν\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} channels are shown
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} analyses and their combinations. The distributions are shown as a function of the BSM coupling ratios κ~HVV/κSM\documentclass[12pt]{minimal}
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\begin{document}$$(\tilde{\kappa }_{AVV}/\kappa _\mathrm{SM})\cdot \tan {\alpha }$$\end{document}. The 68 % and 95 % CL exclusion regions are indicated as lying above the corresponding horizontal lines. a, b Individual H→WW∗→eνμν\documentclass[12pt]{minimal}
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\begin{document}$$\mu =1$$\end{document} and for the signal strengths obtained from the fit to data
The final pdfs used in the fits are obtained by applying linear histogram interpolation between the multi-dimensional bins of κ~HVV/κSM\documentclass[12pt]{minimal}
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\begin{document}$$(\tilde{\kappa }_{AVV}/\kappa _\mathrm{SM})\cdot \tan {\alpha }$$\end{document}. The individual likelihood functions per centre-of-mass energy (s\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{s}$$\end{document}) and final state (FS) are:LΩ¯|κ~HVVκSM,κ~AVVκSMtanα,θ¯=∏iPΩ¯i|siκ~HVVκSM,κ~AVVκSMtanα,θ¯+bi(θ¯),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\mathcal {L}\left( \bar{\Omega } \Big | \frac{\tilde{\kappa }_{HVV}}{\kappa _\mathrm{SM}}, \frac{\tilde{\kappa }_{AVV}}{\kappa _\mathrm{SM}}\tan {\alpha },\bar{\theta } \right) \nonumber \\&\quad = \prod \limits _{i} P\left[ \bar{ \Omega }_{i} \Big | s_{i} \left( \frac{\tilde{\kappa }_{HVV}}{\kappa _\mathrm{SM}}, \frac{\tilde{\kappa }_{AVV}}{\kappa _\mathrm{SM}}\tan {\alpha } , \bar{\theta } \right) + b_{i}(\bar{\theta }) \right] , \end{aligned}$$\end{document}where P is the probability density function for the data vector Ω¯\documentclass[12pt]{minimal}
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\begin{document}$$ \bar{\Omega }$$\end{document}, given the signal model s and background model b. The index i runs over all the bins of multi-dimensional histograms of observables and θ¯\documentclass[12pt]{minimal}
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\begin{document}$$\bar{\theta }$$\end{document} represents the vector of nuisance parameters corresponding to systematic uncertainties. Fits to data are performed by minimising the negative log-likelihood function with respect to the ratios of the couplings:LΩ¯|κ~HVVκSM,κ~AVVκSMtanα,θ¯=-2ln∏s∏FSLΩ¯|κ~HVVκSM,κ~AVVκSMtanα,θ¯.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&L\left( \bar{\Omega } \Big | \frac{\tilde{\kappa }_{HVV}}{\kappa _\mathrm{SM}}, \frac{\tilde{\kappa }_{AVV}}{\kappa _\mathrm{SM}}\tan {\alpha }, \bar{\theta } \right) \nonumber \\&\quad = -2 \ln \prod \limits _{\sqrt{s}} \prod \limits _\mathrm{FS} \mathcal {L}\left( \bar{\Omega } \Big | \frac{\tilde{\kappa }_{HVV}}{\kappa _\mathrm{SM}}, \frac{\tilde{\kappa }_{AVV}}{\kappa _\mathrm{SM}}\tan {\alpha }, \bar{\theta } \right) . \end{aligned}$$\end{document}The test statistic q′=-2ln(λ)\documentclass[12pt]{minimal}
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\begin{document}$$q' = -2\ln (\lambda ) $$\end{document} is defined as the profiled value of L of Eq. (12). To ensure the correctness of the statistical treatment and the absence of significant biases, a series of tests were performed before applying the fit to the data. Asimov datasets [31, 32] created from independently generated Monte Carlo samples with κ~HVV/κSM\documentclass[12pt]{minimal}
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\begin{document}$$ \pm 10$$\end{document} were injected into the analysis procedure. The tests were repeated for samples corresponding to 1 and 100 times the LHC Run-I integrated luminosity. In all cases the fitted values of coupling constants were found to be in agreement with the injected values within statistical uncertainties.
The results of the matrix-element-observable fit were validated and cross-checked using a nine-dimensional matrix-element method (9D fit). The method implements a multivariate per-event extended likelihood that is sensitive to both the κ~HVV/κSM\documentclass[12pt]{minimal}
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\begin{document}$$(\tilde{\kappa }_{AVV}/\kappa _\mathrm{SM})\cdot \tan {\alpha }$$\end{document} mixing parameters and is based on nine experimental observables. The probability model is constructed with separate components for signal, the SM ZZ∗\documentclass[12pt]{minimal}
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\begin{document}$$ZZ^*$$\end{document} background and the reducible background. The background components are assumed to be independent of the Higgs boson tensor structure, so all of the sensitivity to mixing parameters comes from the signal component. Each component depends on nine experimental observables: m4ℓ\documentclass[12pt]{minimal}
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Expected and observed best-fit values of (a) κ~HVV/κSM\documentclass[12pt]{minimal}
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\begin{document}$$(\tilde{\kappa }_{AVV}/\kappa _\mathrm{SM})\cdot \tan {\alpha }$$\end{document} and 95%\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} and H→WW∗→eνμν\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow WW^{*} \rightarrow e \nu \mu \nu $$\end{document} analyses. The expected values are estimated for the signal strengths measured in data and assuming best-fit values for all other nuisance parameters. The signal strengths are treated independently per decay channel and per collision energy
The main sources of systematic uncertainty for the tensor structure measurements are the same as discussed in Sect. 5 since they are based on the same four-lepton variables. Several additional sources of uncertainty, specific to each of the methods, are also taken into account. For the matrix-element-observable fit, the uncertainty related to the Kernel Density Estimation smoothing procedure applied to signal and background multi-dimensional histograms is considered. To estimate the influence of this uncertainty on the final result, a procedure similar to the one described in Sect. 5 is employed. The impact of the different sources of systematic uncertainty on the final results is evaluated by comparing the BSM exclusion limits obtained with a specific systematic uncertainty included or excluded in the fit, while excluding all other systematic uncertainties. A similar conclusion holds in the fixed hypothesis test: the systematic uncertainties have a very limited impact on the final result. The most important uncertainties are related to the estimates of the reducible backgrounds. The relative impact of these uncertainties on the final 95 % CL exclusion limit on BSM couplings was found to be around ±1%\documentclass[12pt]{minimal}
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\begin{document}$$\pm 1\,\%$$\end{document}. The second most important group of sources of systematic uncertainty is related to the theoretical uncertainties on the production cross section of the ZZ∗\documentclass[12pt]{minimal}
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\begin{document}$$ZZ^{*}$$\end{document} background process. Their relative impact on the final result is found to be less than ±1%\documentclass[12pt]{minimal}
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\begin{document}$$\pm 1\,\%$$\end{document}. The precision of the tensor structure analysis is thus dominated by the statistical errors.
In this paper, only results based on the matrix-element-observable approach are reported. The 9D approach was used as a cross-check and produced results compatible with the matrix-element approach.
Individual and combined results
The results of the tensor structure analyses performed in the H→WW∗→eνμν\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow WW^{*} \rightarrow e \nu \mu \nu $$\end{document} channel are reported in Ref. [8] and, for completeness, they are also summarised in Table 7.
The distributions of the test statistic for fits of κ~HVV/κSM\documentclass[12pt]{minimal}
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\begin{document}$$(\tilde{\kappa }_{AVV}/\kappa _\mathrm{SM})\cdot \tan {\alpha }$$\end{document} measured in the H→ZZ∗→4ℓ\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} analysis are shown in Fig. 9.
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\begin{document}$$(\tilde{\kappa }_{AVV}/\kappa _\mathrm{SM})\cdot \tan {\alpha }$$\end{document}, together with the intervals where these couplings are excluded at above the 95%\documentclass[12pt]{minimal}
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\begin{document}$$95\%$$\end{document} CL, are reported in Table 8. The fitted values agree with the SM predictions within uncertainties.
The measurements from the H→WW∗→eνμν\documentclass[12pt]{minimal}
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\begin{document}$$(\tilde{\kappa }_{AVV}/\kappa _\mathrm{SM})\cdot \tan {\alpha }$$\end{document} are the same for the W and Z vector bosons. A common test statistic is obtained by combining the profiled likelihoods of the individual channels. The expected distributions of the likelihoods, for the signal strength values obtained from the fits to the data (μ=μ^\documentclass[12pt]{minimal}
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\begin{document}$$\mu = \hat{\mu }$$\end{document}), are presented in Fig. 10.
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} measurements are presented in Fig. 11. The asymmetric shape of the expected and observed limits in the κ~HVV/κSM\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{\kappa }_{HVV}/\kappa _\mathrm{SM}$$\end{document} results is mainly due to the interference between the BSM and the SM contributions that gives maximum deviation from the SM predictions for negative relative values of the BSM couplings.
Here the signal normalisations are treated as independent nuisance parameters of the different decay channels and the different centre-of-mass energies. The other nuisance parameters related to the experimental and theoretical uncertainties are treated as correlated when appropriate. The resulting 95 % CL exclusion regions for the combinations of H→WW∗→eνμν\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} channels are listed in Table 9.
Conclusion
Studies of the spin and parity of the observed Higgs boson in the H→ZZ∗→4ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{s}=8$$\end{document} TeV, respectively. The SM Higgs boson hypothesis, corresponding to the quantum numbers JP=0+\documentclass[12pt]{minimal}
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\begin{document}$$J^{P}=0^{+}$$\end{document}, is tested against several alternative spin and parity models. The models considered include non-SM spin-0 and spin-2 models with universal and non-universal couplings to quarks and gluons. The combination of the three decay processes allows the exclusion of all considered non-SM spin hypotheses at a more than 99.9 % CL in favour of the SM spin-0 hypothesis.
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\begin{document}$$(\tilde{\kappa }_{AVV}/\kappa _\mathrm{SM})\cdot \tan {\alpha }$$\end{document}, are compatible with the SM predictions.
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\begin{document}$$-6.0< (\tilde{\kappa }_{AVV}/\kappa _\mathrm{SM})\cdot \tan {\alpha }< 5.0$$\end{document} are excluded at the 95 % CL.
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\begin{document}$$(\tilde{\kappa }_{AVV}/\kappa _\mathrm{SM})\cdot \tan {\alpha }$$\end{document} couplings have the same values for the HWW and HZZ processes. As a result of this combination, the regions outside of -0.73<κ~HVV/κSM<0.63\documentclass[12pt]{minimal}
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\begin{document}$$-2.18< (\tilde{\kappa }_{AVV}/\kappa _\mathrm{SM})\cdot \tan {\alpha }< 0.83$$\end{document} intervals are excluded at the 95 % CL. The corresponding expected not-excluded intervals at the 95 % CL, assuming the SM Higgs boson hypothesis and the signal strength values measured in data, are -0.55<κ~HVV/κSM<4.80\documentclass[12pt]{minimal}
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\begin{document}$$-2.33< (\tilde{\kappa }_{AVV}/\kappa _\mathrm{SM})\cdot \tan {\alpha }< 2.30$$\end{document}.
Acknowledgments
We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently. We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC, Australia; BMWFW and FWF, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP, Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT, Chile; CAS, MOST and NSFC, China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic; DNRF, DNSRC and Lundbeck Foundation, Denmark; EPLANET, ERC and NSRF, European Union; IN2P3-CNRS, CEA-DSM/IRFU, France; GNSF, Georgia; BMBF, DFG, HGF, MPG and AvH Foundation, Germany; GSRT and NSRF, Greece; RGC, Hong Kong SAR, China; ISF, MINERVA, GIF, I-CORE and Benoziyo Center, Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; FOM and NWO, Netherlands; BRF and RCN, Norway; MNiSW and NCN, Poland; GRICES and FCT, Portugal; MNE/IFA, Romania; MES of Russia and NRC KI, Russian Federation; JINR; MSTD, Serbia; MSSR, Slovakia; ARRS and MIZŠ, Slovenia; DST/NRF, South Africa; MINECO, Spain; SRC and Wallenberg Foundation, Sweden; SER, SNSF and Cantons of Bern and Geneva, Switzerland; NSC, Taiwan; TAEK, Turkey; STFC, the Royal Society and Leverhulme Trust, United Kingdom; DOE and NSF, United States of America. The crucial computing support from all WLCG partners is acknowledged gratefully, in particular from CERN and the ATLAS Tier-1 facilities at TRIUMF (Canada), NDGF (Denmark, Norway, Sweden), CC-IN2P3 (France), KIT/GridKA (Germany), INFN-CNAF (Italy), NL-T1 (Netherlands), PIC (Spain), ASGC (Taiwan), RAL (UK) and BNL (USA) and in the Tier-2 facilities worldwide.
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To compare the exclusion limits obtained in this analysis to other existing studies, the final results of this analysis are also expressed in terms of effective cross-section fractions (fg2,ϕg2)\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} channels and for their combination. The symbol “n.a.” denotes the absence of 95 % CL sensitivity
When, in addition to the SM term, only one CP-even or CP-odd BSM contribution is present, the conversion between the parameterisation used in this analysis and the (fgi,ϕgi)\documentclass[12pt]{minimal}
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\begin{document}$$(f_{gi},\phi _{gi})$$\end{document} parameterisation is given by Eq. (13) rewritten in the following way:fgi=ri121+ri12;(i=2,4),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} f_{g_i}=\frac{r_{i1}^2}{1+r_{i1}^2}; \quad (i=2,4), \end{aligned}$$\end{document}where r41\documentclass[12pt]{minimal}
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\begin{document}$$r_{41}$$\end{document} and r21\documentclass[12pt]{minimal}
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\begin{document}$$r_{21}$$\end{document} are chosen such that:r212=σHVVσSMk~HVVkSM2,andr412=σAVVσSMk~AVVkSM2tan2α.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} r_{21}^2= & {} \frac{\sigma _{HVV}}{\sigma _\mathrm{SM}} \left( \frac{ \tilde{k}_{HVV}}{k_\mathrm{SM}} \right) ^2, \quad \mathrm{and} \nonumber \\ r_{41}^2= & {} \frac{\sigma _{AVV}}{\sigma _\mathrm{SM}} \left( \frac{ \tilde{k}_{AVV}}{k_\mathrm{SM}} \right) ^2 \tan ^2{\alpha }. \end{aligned}$$\end{document}The numeric coefficients σSM\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _\mathrm{SM}$$\end{document}, σHVV\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _{HVV}$$\end{document} and σAVV\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _{AVV}$$\end{document} are effective cross sections of the HVV interaction calculated when only each of the κSM\documentclass[12pt]{minimal}
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\begin{document}$$\kappa _\mathrm{SM}$$\end{document}-, κHVV\documentclass[12pt]{minimal}
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\begin{document}$$\kappa _{HVV}$$\end{document}- and κAVV\documentclass[12pt]{minimal}
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\begin{document}$$\kappa _{AVV}$$\end{document}-related terms is present in the Lagrangian.
For consistency with previous measurements reported in Ref. [5], the expected and observed results of the current analysis of the H→WW∗→eνμν\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow WW^{*} \rightarrow e \nu \mu \nu $$\end{document} and H→ZZ∗→4ℓ\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} channels and for their combination are expressed in terms of fgi\documentclass[12pt]{minimal}
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\begin{document}$$f_{gi}$$\end{document} and ϕgi\documentclass[12pt]{minimal}
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\begin{document}$$H \rightarrow ZZ^{*} \rightarrow 4 \ell $$\end{document} decay, (fg2ZZ,ϕg2ZZ)\documentclass[12pt]{minimal}
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\begin{document}$$(f_{g2}^{ZZ} ,\phi _{g2} ^{ZZ})$$\end{document} and (fg4ZZ,ϕg4ZZ)\documentclass[12pt]{minimal}
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\begin{document}$$(f_{g4}^{ZZ},\phi _{g4}^{ZZ})$$\end{document}. These parameters are denoted hereafter by (fg2,ϕg2)\documentclass[12pt]{minimal}
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\begin{document}$$(f_{g2},\phi _{g2})$$\end{document} and (fg4,ϕg4)\documentclass[12pt]{minimal}
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\begin{document}$$(f_{g4}, \phi _{g4})$$\end{document}. The corresponding results are presented in Tables 10 and 11.
To obtain these results, the effective cross sections σSM\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _\mathrm{SM}$$\end{document}, σHVV\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _{HVV}$$\end{document} and σAVV\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _{AVV}$$\end{document} of the HZZ interaction are calculated using the MadGraph5_aMC@NLO Monte Carlo generator [16] at leading order. The ratios of cross sections used in the calculation are: σHVV/σSM=0.349\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _{HVV}/\sigma _\mathrm{SM} = 0.349$$\end{document} and σAVV/σSM=0.143\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _{AVV}/\sigma _\mathrm{SM} = 0.143 $$\end{document}, respectively.
In the following, for brevity, only the JP\documentclass[12pt]{minimal}
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\begin{document}$$J^P$$\end{document} label is used to indicate the spin and CP quantum numbers.
In this paper the symbol V is used to describe a massive SM vector boson, namely either a W or a Z boson.
Here channel can be used to indicate different categories in the same final state when producing results for individual decay channels, or different final states when combining them.
Throughout this section, the following variables are used: pTℓℓ\documentclass[12pt]{minimal}
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\begin{document}$$p_{\text {T}} ^{\ell \ell }$$\end{document} and mℓℓ\documentclass[12pt]{minimal}
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\begin{document}$$m_{\ell \ell }$$\end{document} are the transverse momentum and the invariant mass of the two-lepton system, respectively, Δϕℓℓ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta \phi _{\ell \ell }$$\end{document} is the azimuthal angular difference between the two leptons, mT\documentclass[12pt]{minimal}
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\begin{document}$$m_\mathrm{T}$$\end{document} is the transverse mass of the reconstructed Higgs boson decay system, ΔpT\documentclass[12pt]{minimal}
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\begin{document}$$\Delta p_\mathrm{T}$$\end{document} is the absolute value of the difference between the momenta of the two leptons and Eℓℓνν=pTℓ1-0.5pTℓ2+0.5pTmiss\documentclass[12pt]{minimal}
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\begin{document}$$ E_{\ell \ell \nu \nu } =p_{\text{ T }}^{{\ell }_1} -0.5p_{\text{ T }}^{{\ell }_2}+0.5 p_{\text{ T }}^{miss}$$\end{document}, where pTmiss\documentclass[12pt]{minimal}
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\begin{document}$$p_{\text{ T }}^{miss}$$\end{document} is the missing transverse momentum.