Data-Driven Background Modelling using Conditional Probabilities in Particle Physics, Tesis de Física

¡Descarga Data-Driven Background Modelling using Conditional Probabilities in Particle Physics y más Tesis en PDF de Física solo en Docsity! Prepared for submission to JHEP Non-Parametric Data-Driven Background Modelling using Conditional Probabilities A. Chisholm, T. Neep, K. Nikolopoulos, R. Owen,1 E. Reynolds2 and J. Silva School of Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, United Kingdom E-mail: andrew.chisholm@cern.ch, tom.neep@cern.ch, konstantinos.nikolopoulos@cern.ch, rhys.owen@cern.ch, elliot.reynolds@cern.ch, julia.manuela.silva@cern.ch Abstract: Background modelling is one of the main challenges in particle physics data analysis. Commonly employed strategies include the use of simulated events of the back- ground processes, and the fitting of parametric background models to the observed data. However, reliable simulations are not always available or may be extremely costly to pro- duce. As a result, in many cases, uncertainties associated with the accuracy or sample size of the simulation are the limiting factor in the analysis sensitivity. At the same time, para- metric models are limited by the a priori unknown functional form and parameter values of the background distribution. These issues become ever more pressing when large datasets become available, as it is already the case at the CERN Large Hadron Collider, and when studying exclusive signatures involving hadronic backgrounds. Two novel and widely applicable non-parametric data-driven background modelling techniques are presented, which address these issues for a broad class of searches and mea- surements. The first, relying on ancestral sampling, uses data from a relaxed event selection to estimate a graph of conditional probability density functions of the variables used in the analysis, accounting for significant correlations. A background model is then generated by sampling events from this graph, before the full event selection is applied. In the second, a generative adversarial network is trained to estimate the joint probability density function of the variables used in the analysis. The training is performed on a relaxed event selection which excludes the signal region, and the network is conditioned on a blinding variable. Subsequently, the conditional probability density function is interpolated into the signal region to model the background. The application of each method on a benchmark analysis is presented in detail, and the performance is discussed. 1Now at Rutherford Appleton Laboratory, Science and Technology Facilities Council. 2Corresponding author. Now at Physics Division, Lawrence Berkeley National Laboratory. ar X iv :2 11 2. 00 65 0v 1 [ he p- ex ] 1 D ec 2 02 1 Contents 1 Introduction 1 2 Background Modelling with Ancestral Sampling 2 2.1 Overview of Case Study: Search for H → φ(K+K−)γ 3 2.2 Event Selection, Analysis Strategy and Simulation 4 2.3 Overview of Method 5 2.4 Sampling Procedure 5 2.5 Validation 7 2.6 Signal Injection Tests 8 2.7 Systematic Uncertainties 9 2.8 Treatment of resonant background components 10 2.9 Implementation in Statistical Analysis 11 3 Background modelling with Generative Adversarial Networks 13 3.1 Overview of Case Study: Search for H → Za→ ``+ jet 15 3.2 Event Selection, Analysis Strategy and Simulation 16 3.3 Overview of Method 17 3.4 Background modelling uncertainties 18 3.5 Background model validation 19 4 Summary 22 1 Introduction The modelling of background processes is a critical element in determining the discovery potential of hadron collider experiments searching for physics beyond the Standard Model (SM). In this direction, a number of data-driven background estimation methods have been developed over the years. In simple cases, it is sufficient to estimate the expected number of background events in a given signal region, for example through single or double side-band methods. But in most cases, reliable description of the background shape in a discriminant variable is also required. For these cases, background modelling often relies on direct simulation based on Monte Carlo (MC) event generators, or on parametric methods. In many physics analyses, however, the dominant contribution to the total background cannot be modelled with sufficient accuracy using MC simulations. Typical examples in- clude fully hadronic final states and backgrounds associated with the mis-identification of physics objects at the reconstruction level. Furthermore, the composition of the background itself, in terms of distinct scattering processes, may not be reliably known. This can lead to – 1 – considered, and a photon is sensitive to the Higgs boson coupling to the strange quark. The ATLAS collaboration has performed a search for such decays using a dataset corresponding to 36 fb−1 of √ s = 13TeV pp collisions [6]. The final state consists of a pair of oppositely charged kaons, with an invariant mass consistent with the φ mass, recoiling against an isolated photon. The main sources of background in this search are events involving inclusive direct photon production or multijet processes where a meson candidate is reconstructed from charged particles produced in a hadronic jet. Such processes are difficult to model accurately with MC event generators and represent an ideal use case for a data-driven non- parametric background modelling method. The method described above, based ancestral sampling, has been successfully deployed in several ATLAS searches for radiative Higgs boson decays to light mesons [6–9]. 2.2 Event Selection, Analysis Strategy and Simulation Events are required to contain a photon with transverse momentum, pT(γ), in excess of 35GeV, and pseudorapidity, η(γ), within |η(γ)| < 1.37 or 1.52 < |η(γ)| < 2.37. Candidate φ → K+K− decays are reconstructed from pairs of oppositely charged tracks with trans- verse momentum, pT(K), in excess of 15GeV, and absolute pseudorapidity, |η(K)|, less than 2.5. Furthermore, the highest transverse momentum track within a pair is required to satisfy pT(K) > 20GeV and the invariant mass of track pairs is required to be within the range 1.012 < m(φ) < 1.028GeV. The H → φγ candidate is formed from the combination of the photon with the highest transverse momentum and the φ→ K+K− candidate with an invariant mass closest to the φ mass. The variable I(φ) is defined to characterise the hadronic isolation of the φ→ K+K− candidates. I(φ) is defined as the scalar sum of the pT for tracks within ∆R = 0.2 (∆R = √ ∆Φ2 + ∆η2) of the leading track within a φ candidate (excluding the φ decay products), relative to the transverse momentum of the φ→ K+K− candidate. Events are retained for further analysis if ∆Φ(φ, γ) > π/2, pT(φ) > 40GeV and I(φ) < 0.5. This set of criteria define the “Signal Region” (SR) requirements. H → φγ signal events are discriminated from background events by means of a statistical analysis of the distribution of the invariant mass of selected candidates, m(φ, γ). For the purpose of demonstrating the ancestral sampling background modelling method, only the dominant contributions to the signal and background processes are explicitly simu- lated. This choice has no implications for the validation of the method and is pragmatically motivated. Inclusive Higgs boson production in pp collisions is approximated by the gluon- fusion process alone and simulated with the Pythia 8.244 MC event generator [10] with the CT14nlo PDF set [11]. Subleading contributions from the vector boson fusion process and Higgs boson production in association with vector bosons and heavy quarks are not simu- lated explicitly. The H → φγ decay is simulated directly by the Pythia 8.244 MC event generator [10] and no other Higgs boson decays are simulated. The γ + jet process alone is used as a proxy for the inclusive background to the H → φγ search, which is expected to also contain contributions from multijet events. The production of γ+ jet is simulated with the Sherpa 2.2.10 event generator [12] with the NNPDF3.0 PDF set [13]. Direct photon production with up to two additional jets is simulated at the matrix element level. – 4 – 2.3 Overview of Method The procedure is based on a sample of data events selected based on the nominal “Signal Region” requirements described in Section 2.2, modified by relaxing a number of crite- ria in order to enrich the sample in background events. The criteria which define this background-dominated sample are denoted the “Generation Region” (GR). Two additional event samples, known as “Validation Regions” (VR), are also defined to validate the back- ground modelling procedure. The definitions of these regions are outlined in Table 1. These GR events are used to construct probability density functions of the relevant kinematic and isolation variables, parameterised to respect the most important correlations. By sampling these distributions, an ensemble of H → φγ pseudo-candidates is generated, from which a model of the m(φ, γ) distribution for background events can be derived for the SR require- ments. In addition to providing a prediction for the shape of the m(φ, γ) distribution, the normalisation of the the SR and VRs, relative to the GR, is also predicted. The absolute normalisation in the SR and VRs may be obtained directly by the model, by uniformly scaling the distributions by the ratio of the number of data events in the GR to the size of the ensemble of pseudocandidates. 2.4 Sampling Procedure Each pseudo-candidate event is described by φ and γ four-momentum vectors and an as- sociated φ hadronic isolation variable, I(φ). The generation templates which together parameterise all components of the φ and γ four-momentum vectors and the φ hadronic isolation variable are described in Table 2 and represented in Figure 1. The sampling procedure for a single pseudo-candidate proceeds as follows: 1. Correlated values for pT(φ) and pT(γ) are sampled from generation template A. 2. Based on the values of pT(φ) and pT(γ) sampled in step 1, template B is projected along the ∆Φ(φ, γ) dimension and a value for ∆Φ(φ, γ) is sampled. 3. Based on the value of ∆Φ(φ, γ) sampled in step 2, template C is projected along the ∆η(φ, γ) dimension and a value for ∆η(φ, γ) is sampled. 4. Based on the value of pT(γ) sampled in step 1, template D is projected along the I(φ) dimension and a value is sampled. Minimum pT(φ) requirement Maximum I(φ) requirement GR 35GeV Not applied VR1 40GeV Not applied VR2 35GeV 0.5 SR 40GeV 0.5 Table 1. The event selection criteria of the “Signal Region” which are modified to define the “Generation Region” (GR) and “Validation Regions” (VR). – 5 – Template Name Dimensionality Variable 1 Variable 2 Variable 3 A 2D pT(φ) pT(γ) - B 3D ∆Φ(φ, γ) pT(γ) pT(φ) C 2D ∆η(φ, γ) ∆Φ(φ, γ) - D 2D I(φ) pT(γ) - E 1D η(γ) - - F 1D φ(γ) - - G 1D m(φ) - - Table 2. The definition of the set of generation templates from which the components of the pseudo-candidates are sequentially sampled. Figure 1. Graphical representation of the sampling sequence followed in the modelling. Variables not shown explicitly are sampled in a factorised, uncorrelated, manner from an 1D template, as described in Table 2. 5. Values for η(γ) and φ(γ) are sampled from generation templates E and F, respectively. At this stage, the photon four-momentum is fully defined, with m(γ) = 0 imposed. 6. A value for m(φ) is sampled from generation template G. At this stage, the φ four- momentum is fully defined. The steps above are repeated to generate an ensemble of pseudo-candidates whose characteristics resemble those of the GR data sample. The selection requirements of the SR and two VR are imposed on the ensemble and the pseudo-candidates which are retained are used to construct distributions of composite variables built from the φ and γ four- momentum, of which m(φ, γ) is of primary interest. to.pdf – 6 – 64 19 22 ­16 13 100 45 ­7 ­3 100 13 46 ­8 100 ­3 ­16 21 ­4 100 ­8 ­7 22 11 100 ­4 19 100 11 21 46 45 64 )γ,φm( Isolationφ )γ,φ(Φ∆ )γ,φ(η∆ )φ( T p )γ( T p )γ( T p )φ( T p )γ,φ(η∆ )γ,φ(Φ∆ Isolationφ )γ,φm( 100− 80− 60− 40− 20− 0 20 40 60 80 100 C o rr e la ti o n ( % ) (a) 65 18 22 ­12 12 100 41 2 ­7 ­7 100 12 49 ­2 ­8 100 ­7 ­12 20 4 100 ­8 ­7 22 12 100 4 ­2 2 18 100 12 20 49 41 65 )γ,φm( Isolationφ )γ,φ(Φ∆ )γ,φ(η∆ )φ( T p )γ( T p )γ( T p )φ( T p )γ,φ(η∆ )γ,φ(Φ∆ Isolationφ )γ,φm( 100− 80− 60− 40− 20− 0 20 40 60 80 100 C o rr e la ti o n ( % ) (b) Figure 4. Linear correlation coefficients for pairs of variables used in the background modelling procedure, shown for the simulated data events (left) and generated pseudo-candidates (right) passing the GR selections. significance in the SR, signal contributions to the GR dataset. Furthermore, the effect on the background is found to scale linearly with the number of injected signal events. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 3 10× E v e n ts / 5 G e V Nominal model Model with signal injection Signal injected GR 50 100 150 200 250 300 ) [GeV]γ,φm( 0.9 1 1.1 In je c ti o n /N o m . (a) 0 100 200 300 400 500 600 700 E v e n ts / 5 G e V Nominal model Model with signal injection Signal injected SR 50 100 150 200 250 300 ) [GeV]γ,φm( 0.9 1 1.1 In je c ti o n /N o m . (b) Figure 5. Comparison of the m(φ, γ) distributions predicted by the model for (a) the GR and (b) the SR, derived with and without an injection of approximately 130 H → φγ signal events in the GR dataset. 2.7 Systematic Uncertainties While the tests described in Section 2.5 demonstrate that the model can provide an accurate description of the background at the level of precision associated with the statistical uncer- tainty of the validation regions, potential mismodelling beyond this level cannot be directly – 9 – excluded. It is therefore important that systematic uncertainties affecting the shape of the predicted background distributions are estimated. The strategy for incorporating systematic uncertainties in the model is motivated by the context within which the model is applied to perform the statistical analysis, namely a likelihood fit. The strategy focuses on estimating a set of complementary background shape variations which are implemented as shape variations of the nominal background probability density function. The derived shape variations are selected to capture different modes of potential deformations of the background shape. The exact size of the variation is of less importance, given that the corresponding nuisance parameters are constrained directly by the data in the likelihood fit. Moreover, the analyser may decide to leave such nuisance parameters completely free, or to add Gaussian constraint terms in the likelihood. In the latter case, care must be taken to ensure that the assigned ±1σ shape variations are large with respect to potential discrepancies between the shape of the predicted distributions and those in the data. Pairs of approximately anti-symmetric shape variations are built by performing the sampling procedure after having applied a transformation to one of the generation tem- plates. The transformations considered here include: • A translation of the photon transverse momentum distribution • A multiplicative transformation of the ∆Φ(φ, γ)/π distribution by a function of the form 1 + C × ∆Φ(φ, γ), where a pair of values (one positive and negative) for the coefficient C are chosen. Furthermore, additional alternative background models may be derived by direct trans- formations of the resulting distribution of interest. For example, an additional pair of alternative background models is derived by a multiplicative transformation of GR m(φ, γ) distribution by a linear function of m(φ, γ). These three pairs of shape variations largely span the range of possible large scale uncorrelated shape deformations of the m(φ, γ) dis- tribution, as shown in Figure 6. 2.8 Treatment of resonant background components In certain cases, in addition to the dominant backgrounds which do not exhibit resonant structures in the invariant mass distribution, non-negligible resonant contributions may also be present. One such example is the Z → µ+µ−γ process, which represents an important resonant background in the case of searches for radiative Higgs boson decays to the Υ(→ µ+µ−) bottomonium states [7, 9]. Often such processes can be described with sufficient accuracy by MC simulations and the use of such simulations to describe these subsets of the overall background is preferable. In this situation, the procedure used to build the kernel is modified, to ensure that such resonant contributions are not included in the model for the inclusive non-resonant background. During the construction of the generation templates, data events in the vicinity of the resonance are randomly discarded, with a probability which describes the likelihood that a data event with a given invariant mass was produced by the resonant process. This probability may be determined, as a function of invariant mass, by – 10 – ) [GeV]γ,φm( 50 100 150 200 250 300 E v e n ts / 1 G e V 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 310× Background Model ) Upγ( T Syst. p ) Downγ( T Syst. p Upφ∆Syst. Downφ∆Syst. Mass Tilt Up Mass Tilt Down GR (a) ) [GeV]γ,φm( 50 100 150 200 250 300 E v e n ts / 1 G e V 0 100 200 300 400 500 600 700 800 Background Model ) Upγ( T Syst. p ) Downγ( T Syst. p Upφ∆Syst. Downφ∆Syst. Mass Tilt Up Mass Tilt Down SR (b) Figure 6. Comparison of the m(φ, γ) distributions for the GR (left) and SR (right) associated with the three pairs of systematic shape variations. subtracting the distribution of the MC prediction for the resonant process from the data distribution. This procedure can mitigate the impact of resonant background contributions to the non-resonant background model [7, 9]. In the absence of such a procedure, the prediction of the model in the vicinity of the resonance may be distorted in a manner similar to the signal inject tests discussed in Section 2.6. 2.9 Implementation in Statistical Analysis The performance of the method in practical terms is demonstrated by implementing the background model within a statistical analysis procedure. A binned maximum likelihood fit is performed to the m(φ, γ) distribution of the simulated γ + jet events used to build the background model alone. The m(φ, γ) distribution of the signal is modelled by a dou- ble Gaussian distribution with a single mean (common to both Gaussian components), two width parameters and a parameter describing the relative normalisation of the two Gaussian components. The normalisation of the signal, relative to the number of signal events predicted by the simulation, is controlled by a single free parameter, µsignal. The background distribution is built from the nominal background model in terms of a finely binned histogram, with a linear inter-bin interpolation applied. The normalisation of the background, relative to the number of events predicted by the model, is determined by a single free parameter, µbkgd. Systematic uncertainties affecting the shape of the background model, described in Section 2.7, are implemented using a moment morphing technique [14]. Each of the three shape variations described in Section 2.7 are implemented independently, each being controlled by an individual nuisance parameter. The value of the nuisance pa- rameters describing the pT(γ) shift and ∆Φ(φ, γ) deformation are constrained by Gaussian penalty terms in the likelihood, while the m(φ, γ) tilt nuisance parameter is free. The result of the fit is presented in Figure 7, where it is shown that the post-fit back- ground model is in good agreement with the m(φ, γ) distribution of the simulated γ + jet – 11 – 50 100 150 200 250 300 350 E ve nt s / 2 .5 G eV Data Background S+B Fit 50 100 150 200 250 300 ) [GeV]γ,φm( 50− 0 50 R es id ua ls Figure 9. The m(φ, γ) distribution of the initially simulated γ + jet events overlaid with the result of a binned maximum likelihood fit using the method described in Section 2 to derive the background PDF. data applications, for example in the HL-LHC, where large simulated datasets are required to match the statistical precision of the data. Figure 10. Schematic representation of a generative adversarial network. In the context of particle physics, the use of GANs has been explored in applications spanning the complete chain of event generation, simulation, and reconstruction. The use of GANs for simulating the hard scattering process was considered in Ref. [17–19], while the use of GANs for pileup description and detector simulation was explored in Ref. [20] and [21], respectively. Recent publications have also explored the idea of replacing the entire reconstructed-event generation pipeline with a GAN [17, 22–27]. Each of these applications differ in the nature of the training data used, but mostly use simulated training datasets. This solves the issue of limited simulated data samples by allowing large generated samples to be produced from much smaller datasets. However, concerns related to simulation-based mismodelling remain, which often result into some of the largest sources of uncertainty in searches and measurements at the LHC. In this section, a method is introduced which allows GANs to be trained directly – 14 – on data, solving the issue of simulation-based mismodelling. Training the GAN on data, however, presents the risk that the background model becomes contaminated by signal events. This is resolved by “blinding” the data signal region (SR) while training the GAN. Training a standard GAN using blinded data explicitly, and falsely, informs the GAN that there are no events in the SR, leading to a generative model which predicts an absence of background events in the SR. For this reason, in this article, a conditioned GAN (cGAN), shown in Figure 11, is trained to learn a generative model of the conditional probability distribution of the data, given the value of the variable used to blind the dataset. The cGAN learns the distribution of the background features conditioned on the blinding variable and, despite being given no information about the data in the SR, can interpolate or extrapolate its prediction into the SR. Figure 11. Schematic representation of a conditional generative adversarial network. 3.1 Overview of Case Study: Search for H → Za → ``+ jet Searches for additional scalar or pseudo-scalar particles in the Higgs sector are a major part of the LHC physics programme. In particular, the possibility for light pseudo-scalar particles, produced in the decays of the observed Higgs boson, feature in several beyond the SM theories [28], including the two-Higgs-doublet model (2HDM) and the 2HDM with an additional scalar singlet. Searches typically focus on Higgs boson decays into pairs of the light scalars, or into a Z boson and a light scalar. To-date, several searches have been performed, focusing primarily either on masses of the light resonance in excess of 4 GeV, or considering only leptonic decays of lighter resonances. Recently, the ATLAS Collaboration published the first search for Higgs boson decays to a Z boson and a light hadronically decaying resonance [29]. The Z boson was recon- structed from its leptonic decays to electrons and muons. Masses of the light resonance a between 0.5 GeV and 4 GeV were considered and the hadronic decay of the resonance was reconstructed inclusively as a jet. This is a particularly interesting case study to apply this approach, for two reasons: First, the cross-section of the main background, Z + jets, is such that in the case of a O(100 fb−1) dataset, it is not feasible to generate a simulated event sample with comparable statistical power. Second, the decaying resonance is identi- fied using multi-variate methods, which require a detailed modelling of a large number of – 15 – correlations between the relevant kinematic and jet substructure variables. The sensitiv- ity of the published analysis is limited by the background systematic uncertainties, which originate predominantly from the insufficient size of the simulated data samples used. By suppressing these uncertainties, through large background samples derived directly from the data, one may expect to first approximation a fourfold improvement on the obtained 95% confidence level upper limit on the lowest light resonance masses considered, and more significant improvements at higher masses. In what follows, this analysis is used as a case study to implement a cGAN-based multivariate background modelling method. The study described here is closely aligned with the ATLAS analysis, and the event selection is summarised below. One of the main differences with respect to the ATLAS analysis is that for simplicity only the signal with an a mass of 0.5GeV is considered here. Furthermore, in this study, only Z → µ+µ− decays are considered, while the ATLAS analysis also considered Z → e+e− decays. This practical simplification is inconsequential for the purposes of demonstrating the method. 3.2 Event Selection, Analysis Strategy and Simulation Events are required to contain two oppositely charged muons with transverse momenta pT > 5 GeV, and a hadronic jet with transverse momentum pT > 20 GeV, reconstructed using the anti-kt algorithm with a distance parameter of 0.4. At least one muon is required to have pT > 27 GeV to model the threshold imposed by the trigger, and the invariant mass of the muons is required to be within 10GeV of the Z boson mass. This selection results in a large background arising from Z + jets events, which is mitigated using charged particle track-based jet substructure information. Tracks with ∆R < 0.4 of the jet are selected if they have pT > 0.5 GeV and if their transverse and longitudinal impact parameters are compatible with the particle being produced at the primary vertex. Events are required to have exactly two tracks passing these requirements. Four substructure variables are formed using these tracks: ∆R between the highest pT track and the jet axis; the ratio of the pT of the highest pT track to the vector sum of the track pT; angularity(2) [30]; and the modified correlation function U1(0.7) [31]. A neural network is used to separate signal from background events on the basis of substructure variables described above. During the neural network training, only signal events with tracks that originate from the decay of the a are included. A requirement is placed on the output of the MLP which has an efficiency of 96% and 2.5% for H → Za signal and Z + jets background events, respectively. A binned likelihood fit to the invariant mass distribution of the Z → µ+µ− candidate and jet, for events passing this selection, is used to estimate σ(pp→ H)×B(H → Za). The signal and mock data are modelled using simulation, and the background is modelled using the cGAN approach. Inclusive Higgs boson production in pp collisions is approximated by the gluon-fusion process alone and simulated with the Pythia 8.244 MC event generator with the CT14nlo PDF set. For the H → Za search, Z + jets production is expected to represent the dominant background. Contributions such as tt̄ production are present only at a negligible level owing to the requirement of an opposite-charge same-flavour dilepton with an invariant mass consistent with the Z boson mass. For the purposes of this study, – 16 – 3.5 Background model validation The performance of the cGAN is illustrated in Figure 12, which shows the substructure variables in the m``j SR. The cGAN is able to model these distributions accurately, despite these data events not being included in the cGAN training. Figure 13 shows the substruc- ture variables for the low and high m``j sidebands separately, demonstrating that the cGAN has successfully learnt the dependence of the substructure variables on m``j. The obtained correlation matrix from the cGAN is compared to the data correlation matrix in Figure 14. Excellent agreement is observed. The variation of the top five cGANs about their ensemble is shown in Figure 15(a), and the resulting uncertainties from the principle component analysis are shown in Figure 15(b). (a) (b) (c) (d) Figure 12. Jet substructure variables in mock data and modelled by the cGAN, which are used as inputs to the classification MLP. The error bars on the markers in the lower panels represent the statistical uncertainty on the mock data. A profile likelihood fit to them``j distribution with freely floating signal and background – 19 – (a) (b) (c) (d) Figure 13. Jet substructure variables in mock data and modelled by the cGAN, which are used as inputs to the classification MLP, shown for the low and high m``j sidebands separately. The error bars on the markers in the lower panels represent the statistical uncertainty on the mock data. normalisations is used to estimate the signal in the SR from a fit to the background-only dataset. The extracted signal normalisation is −0.003 ± 0.010 times its predicted value, assuming a SM Higgs boson cross section [34] and a branching fraction B(H → Za) = 100%. This is compatible with the lack of signal in the dataset used in the fit. The values of all fitted parameters are given in Table 4, along with their uncertainties. The m``j distribution after the likelihood fit is shown in Figure 16. This case study demonstrates that the proposed background modelling method is ca- pable of accurately modelling a set of correlated variables and their correlations, and of interpolating over significant distances in the conditioning variable. The performance of this method is expected to improve further in the case of smaller distances in the condi- tioning variable, for example when the signal exhibits a narrow resonance in the blinding – 20 – ­9 5 13 ­10 100 59 ­49 ­64 100 ­10 ­44 4 100 ­64 13 ­30 100 4 ­49 5 100 ­30 ­44 59 ­9 LeadTrackR∆ (AllTracks) T p (LeadTrack) T p 2 ang U1(0.7) Hm H m U1(0.7) 2 ang (AllTracks) T p (LeadTrack) T p LeadTrack R∆ 100− 80− 60− 40− 20− 0 20 40 60 80 100 C o rr e la ti o n ( % ) (a) ­9 5 13 ­10 100 59 ­49 ­64 100 ­10 ­44 4 100 ­64 13 ­29 100 4 ­49 5 100 ­29 ­44 59 ­9 LeadTrackR∆ (AllTracks) T p (LeadTrack) T p 2 ang U1(0.7) Hm H m U1(0.7) 2 ang (AllTracks) T p (LeadTrack) T p LeadTrack R∆ 100− 80− 60− 40− 20− 0 20 40 60 80 100 C o rr e la ti o n ( % ) (b) Figure 14. Correlation matrix (a) from the data and (b) from the cGAN. (a) (b) Figure 15. 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