EUROPEAN ORGANISATION FOR NUCLEAR RESEARCH (CERN)

Measurements of Higgs boson production cross-sections are carried out in the diphoton decay channel using 139 fb -1 of 𝑝 𝑝 collision data at √ 𝑠 = 13 TeV collected by the ATLAS experiment at the LHC. The analysis is based on the definition of 101 distinct signal regions using machine-learning techniques. The inclusive Higgs boson signal strength in the diphoton channel is measured to be 1.04 +0.10 -0.09 . Cross-sections for gluon-gluon fusion, vector-boson fusion, associated production with a 𝑊 or 𝑍 boson, and top associated production processes are reported. An upper limit of 10 times the Standard Model prediction is set for the associated production process of a Higgs boson with a single top quark, which has a unique sensitivity to the sign of the top quark Yukawa coupling. Higgs boson production is further characterized through measurements of Simplified Template Cross-Sections (STXS). In total, cross-sections of 28 STXS regions are measured. The measured STXS cross-sections are compatible with their Standard Model predictions, with a 𝑝-value of 93%. The measurements are also used to set constraints on Higgs boson coupling strengths, as well as on new interactions beyond the Standard Model in an effective field theory approach. No significant deviations from the Standard Model predictions are observed in these measurements, which provide significant sensitivity improvements compared to the previous ATLAS results.

Introduction

The experimental characterization of the Higgs boson discovered by the ATLAS and CMS experiments [START_REF]Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC[END_REF][START_REF]Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC[END_REF] is not only crucial for our understanding of the mechanism of electroweak symmetry breaking [START_REF] Englert | Broken Symmetry and the Mass of Gauge Vector Mesons[END_REF][START_REF] Higgs | Broken Symmetries and the Masses of Gauge Bosons[END_REF][START_REF] Guralnik | Global Conservation Laws and Massless Particles[END_REF] but also for providing insight into physics beyond the Standard Model (SM). Despite a small Higgs boson to diphoton (𝐻 → 𝛾𝛾) branching ratio of (0.227 ± 0.007)% [START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF] in the SM, measurements in the diphoton final state have yielded some of the most precise determinations of Higgs boson properties [START_REF]Measurement of Higgs boson production in the diphoton decay channel in 𝑝 𝑝 collisions at center-of-mass energies of 7 and 8 TeV with the ATLAS detector[END_REF][START_REF]Measurements of fiducial and differential cross sections for Higgs boson production in the diphoton decay channel at √ 𝑠 = 8 TeV with ATLAS[END_REF][START_REF]Measurement of the Higgs boson mass from the 𝐻 → 𝛾𝛾 and 𝐻 → 𝑍 𝑍 * → 4ℓ channels in 𝑝 𝑝 collisions at center-of-mass energies of 7 and 8 TeV with the ATLAS detector[END_REF][START_REF]Measurements of Higgs boson properties in the diphoton decay channel with 36 fb -1 of 𝑝 𝑝 collision data at √ 𝑠 = 13[END_REF][START_REF]Measurement of the Higgs boson mass in the 𝐻 → 𝑍 𝑍 * → 4ℓ and 𝐻 → 𝛾𝛾 channels with √ 𝑠 = 13 TeV 𝑝 𝑝 collisions using the ATLAS detector[END_REF], thanks to the excellent performance of photon reconstruction and identification with the ATLAS detector.

The signature of the Higgs boson in the diphoton final state is a narrow peak in the diphoton invariant mass (𝑚 𝛾𝛾 ) distribution with a width consistent with detector resolution, rising above a smoothly falling background. The diphoton mass resolution for such a resonance is typically between 1 GeV and 2 GeV, depending on the event kinematics. The mass and event rate of the Higgs boson signal can be extracted through fits of the 𝑚 𝛾𝛾 distribution. Properties of the Higgs boson have been studied extensively in the diphoton final state by the ATLAS and CMS experiments [START_REF]Measurements of Higgs boson properties in the diphoton decay channel with 36 fb -1 of 𝑝 𝑝 collision data at √ 𝑠 = 13[END_REF][START_REF]Measurement of the Higgs boson mass in the 𝐻 → 𝑍 𝑍 * → 4ℓ and 𝐻 → 𝛾𝛾 channels with √ 𝑠 = 13 TeV 𝑝 𝑝 collisions using the ATLAS detector[END_REF][START_REF]Study of the spin and parity of the Higgs boson in diboson decays with the ATLAS detector[END_REF][START_REF]Combined measurements of Higgs boson production and decay using up to 80 fb -1 of proton-proton collision data at √ 𝑠 = 13 TeV collected with the ATLAS experiment[END_REF][START_REF]Study of the 𝐶𝑃 properties of the interaction of the Higgs boson with top quarks using top quark associated production of the Higgs boson and its decay into two photons with the ATLAS detector at the LHC[END_REF][START_REF]A measurement of the Higgs boson mass in the diphoton decay channel[END_REF][START_REF]Combined measurements of Higgs boson couplings in proton-proton collisions at √ 𝑠 = 13 TeV[END_REF][START_REF]Measurement of inclusive and differential Higgs boson production cross sections in the diphoton decay channel in proton-proton collisions at √ 𝑠 = 13 TeV[END_REF][START_REF]Measurements of Higgs boson properties in the diphoton decay channel in proton-proton collisions at √ 𝑠 = 13 TeV[END_REF][START_REF]Measurements of 𝑡 t𝐻 Production and the 𝐶𝑃 Structure of the Yukawa Interaction between the Higgs Boson and Top Quark in the Diphoton Decay Channel[END_REF].

This paper reports measurements of Higgs boson production cross-sections in the diphoton decay channel, using a data set of proton-proton collisions at √ 𝑠 = 13 TeV collected by the ATLAS experiment from 2015 to 2018, a period known as Run 2 of the Large Hadron Collider (LHC). The integrated luminosity of this data set is 139 fb -1 [START_REF]Luminosity determination in 𝑝 𝑝 collisions at √ 𝑠 = 13[END_REF][START_REF] Avoni | The new LUCID-2 detector for luminosity measurement and monitoring in ATLAS[END_REF], a roughly fourfold increase compared to the data set used in the previous ATLAS publication of such measurements in the diphoton channel [START_REF]Measurements of Higgs boson properties in the diphoton decay channel with 36 fb -1 of 𝑝 𝑝 collision data at √ 𝑠 = 13[END_REF]. Apart from the increased data set size, the most significant improvement in the sensitivity is due to redesigned and refined event selection and categorization techniques compared to Ref. [START_REF]Measurements of Higgs boson properties in the diphoton decay channel with 36 fb -1 of 𝑝 𝑝 collision data at √ 𝑠 = 13[END_REF]. Uncertainties on the modeling of continuum background have been reduced through the use of a smoothing procedure based on a Gaussian kernel [START_REF] Frate | Modeling Smooth Backgrounds and Generic Localized Signals with Gaussian Processes[END_REF]. The performance of the reconstruction and selection of the physics objects used in these measurements has also been generally improved.

The analysis is optimized to measure production cross-sections in the Simplified Template Cross-Section (STXS) framework [START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF][START_REF] Andersen | Les Houches 2015: Physics at TeV Colliders Standard Model Working Group Report[END_REF][START_REF] Berger | Simplified Template Cross Sections -Stage 1[END_REF][START_REF] Amoroso | Les Houches 2019: Physics at TeV Colliders: Standard Model Working Group Report[END_REF], in which the Higgs boson production phase space is partitioned by production process as well as by kinematic and event properties. By combining several STXS regions, the analysis provides strong sensitivity to the cross-sections of the main Higgs boson production modes, gluon-gluon fusion (ggF), vector-boson fusion (VBF), and associated production with a vector boson (𝑉 𝐻 where 𝑉 = 𝑊 or 𝑍), or a top quark pair (𝑡𝑡𝐻). The analysis is furthermore specifically optimized for the detection of single-top associated production of the Higgs boson (𝑡𝐻), which has a unique sensitivity to the sign of the top-quark Yukawa coupling. While the analysis does not reach sensitivity to the small 𝑡𝐻 event rate predicted by the SM, it can set constraints on enhanced 𝑡𝐻 rates due to potential effects from physics beyond the Standard Model (BSM). A measurement of the inclusive Higgs boson production rate within |𝑦 𝐻 | < 2.5 in the diphoton channel is also reported. Thanks to the increased integrated luminosity and an improved analysis method, a total of 28 STXS regions are measured in this analysis, compared to 10 in Ref. [START_REF]Measurements of Higgs boson properties in the diphoton decay channel with 36 fb -1 of 𝑝 𝑝 collision data at √ 𝑠 = 13[END_REF]. Uncertainties and correlations of the production mode cross-section measurements are reduced, and in particular, the uncertainties in the measurements of 𝑉 𝐻 and top-associated production modes are reduced by more than a factor of four.

Two sets of interpretations of these measurements are also performed to provide constraints on potential effects arising from BSM physics: one in terms of Higgs boson coupling strengths within the 𝜅-framework [START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF], and the other in terms of Wilson coefficients describing potential BSM interactions in the context of a Standard Model effective field theory (SMEFT) model [START_REF] Buchmüller | Effective lagrangian analysis of new interactions and flavour conservation[END_REF][START_REF] Grzadkowski | Dimension-six terms in the Standard Model Lagrangian[END_REF][START_REF] Brivio | SMEFTsim 3.0 -a practical guide[END_REF].

This paper is organized as follows. Section 2 describes the ATLAS detector, Section 3 details the data and Monte Carlo simulation samples used in this analysis, Section 4 explains the object reconstruction and event selection. The design of the measurement is discussed in Section 5, and the modelling of the diphoton mass distribution is discussed in Section 6. Systematic uncertainties are described in Section 7, and Section 8 presents the measurement results. Sections 9 and 10 respectively report the results of interpretations in the context of the 𝜅-framework and the SMEFT model. Conclusions are presented in Section 11.

ATLAS detector

The ATLAS detector [START_REF]The ATLAS Experiment at the CERN Large Hadron Collider[END_REF] at the LHC covers nearly the entire solid angle around the collision point. 1 It consists of an inner tracking detector surrounded by a thin superconducting solenoid, electromagnetic and hadronic calorimeters, and a muon spectrometer incorporating three large superconducting toroidal magnets.

The inner-detector system (ID) is immersed in a 2 T axial magnetic field and provides charged-particle tracking in the range |𝜂| < 2.5. The high-granularity silicon pixel detector covers the vertex region and typically provides four measurements per track, the first hit normally being in the insertable B-layer installed before Run 2 [START_REF]ATLAS Insertable B-Layer Technical Design Report Addendum[END_REF][START_REF] Abbott | Production and integration of the ATLAS Insertable B-Layer[END_REF]. It is followed by the silicon microstrip tracker, which usually provides eight measurements per track. These silicon detectors are complemented by the transition radiation tracker (TRT), which enables radially extended track reconstruction up to |𝜂| = 2.0. The TRT also provides electron identification information based on the fraction of hits (typically 30 in total) above a higher energy-deposit threshold corresponding to transition radiation.

The calorimeter system covers the pseudorapidity range |𝜂| < 4.9. Within the region |𝜂| < 3.2, electromagnetic calorimetry is provided by barrel and endcap high-granularity lead/liquid-argon (LAr) calorimeters, with an additional thin LAr presampler covering |𝜂| < 1.8 to correct for energy loss in material upstream of the calorimeters. Hadronic calorimetry is provided by the steel/scintillator-tile calorimeter, segmented into three barrel structures within |𝜂| < 1.7, and two copper/LAr hadronic endcap calorimeters. The solid angle coverage is completed with forward copper/LAr and tungsten/LAr calorimeter modules optimized for electromagnetic and hadronic measurements respectively. 1 ATLAS 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 upwards. Cylindrical coordinates (𝑟, 𝜙) are used in the transverse plane, 𝜙 being the azimuthal angle around the 𝑧-axis.

The pseudorapidity is defined in terms of the polar angle 𝜃 as 𝜂 =ln tan(𝜃/2). Angular distance is measured in units of Δ𝑅 ≡ √ (Δ𝜂) 2 + (Δ𝜙) 2 .

The muon spectrometer (MS) comprises separate trigger and high-precision tracking chambers measuring the deflection of muons in a magnetic field generated by superconducting air-core toroids. The field integral of the toroids ranges between 2.0 and 6.0 T•m across most of the detector. A set of precision chambers covers the region |𝜂| < 2.7 with three layers of monitored drift tubes, complemented by cathode-strip chambers in the forward region, where the background is highest. The muon trigger system covers the range |𝜂| < 2.4 with resistive-plate chambers in the barrel, and thin-gap chambers in the endcap regions.

Interesting events are selected to be recorded by the first-level trigger system implemented in custom hardware, followed by selections made by algorithms implemented in software in the high-level trigger [START_REF]Performance of the ATLAS trigger system in 2015[END_REF].

The first-level trigger accepts events from the 40 MHz bunch crossings at a rate below 100 kHz, which the high-level trigger reduces in order to record events to disk at about 1 kHz. An extensive software suite [START_REF]The ATLAS Collaboration Software and Firmware[END_REF] is used in the reconstruction and analysis of real and simulated data, in detector operations, and in the trigger and data acquisition systems of the experiment.

3 Data and simulation samples

Data

This study uses a data set of √ 𝑠 = 13 TeV proton-proton collisions recorded by the ATLAS detector during a period ranging from 2015 to 2018, corresponding to Run 2 of the LHC. After data quality requirements [START_REF]ATLAS data quality operations and performance for 2015-2018 data-taking[END_REF] are applied to ensure that all detector components are in good working condition, the data set amounts to an integrated luminosity of 139.0 ± 2.4 fb -1 [START_REF]Luminosity determination in 𝑝 𝑝 collisions at √ 𝑠 = 13[END_REF][START_REF] Avoni | The new LUCID-2 detector for luminosity measurement and monitoring in ATLAS[END_REF]. The mean number of interactions per bunch crossing, averaged over all colliding bunch pairs, was 𝜇 = 33.7 for this data set.

Events are selected if they pass either a diphoton or single-photon trigger. The diphoton trigger has transverse energy thresholds of [START_REF]Performance of electron and photon triggers in ATLAS during LHC Run 2[END_REF] GeV and 25 GeV for the leading and subleading photon candidates, respectively [START_REF]Performance of electron and photon triggers in ATLAS during LHC Run 2[END_REF], with photon identification selections based on calorimeter shower shape variables. In 2015-2016, a loose photon identification requirement was used in the trigger, while in 2017-2018, a tighter requirement was used to cope with higher instantaneous luminosity. The single-photon trigger requires the transverse energy of the leading photon be greater than 120 GeV in data collected between 2015 and 2017, with the threshold rising to 140 GeV for data collected in 2018. The photon candidate used in the trigger decision is required to pass the loose photon identification requirement mentioned above. On average, the trigger efficiency is greater than 98% for events that pass the diphoton event selection described in Section 4. The addition of the single-photon trigger improves the selection efficiency by 1% overall, and by up to 2% for high-𝑝 T Higgs boson candidates.

Simulation samples

Major Higgs boson production processes, including ggF, VBF, 𝑉 𝐻, 𝑡𝑡𝐻, and 𝑏 b𝐻, were generated using P B v2 [START_REF] Alioli | A general framework for implementing NLO calculations in shower Monte Carlo programs: the POWHEG BOX[END_REF][START_REF] Nason | A New method for combining NLO QCD with shower Monte Carlo algorithms[END_REF][START_REF] Frixione | Matching NLO QCD computations with parton shower simulations: the POWHEG method[END_REF][START_REF] Hartanto | Higgs boson production in association with top quarks in the POWHEG BOX[END_REF]. The ggF simulation achieves next-to-next-to-leading-order (NNLO) accuracy for inclusive ggF observables by reweighting the Higgs boson rapidity spectrum in H -M NLO [START_REF] Hamilton | MINLO: multi-scale improved NLO[END_REF][START_REF] Campbell | NLO Higgs boson production plus one and two jets using the POWHEG BOX, MadGraph4 and MCFM[END_REF][START_REF] Hamilton | Merging H/W/Z + 0 and 1 jet at NLO with no merging scale: a path to parton shower + NNLO matching[END_REF] to that of HNNLO [START_REF] Catani | Next-to-Next-to-Leading-Order Subtraction Formalism in Hadron Collisions and its Application to Higgs-boson Production at the Large Hadron Collider[END_REF]. The Higgs boson transverse momentum spectrum obtained with this sample is found to be compatible with the fixed-order HNNLO calculation and the H 2.3 calculation [START_REF] Bozzi | Transverse-momentum resummation and the spectrum of the Higgs boson at the LHC[END_REF][START_REF] De Florian | Transverse-momentum resummation: Higgs boson production at the Tevatron and the LHC[END_REF] performing resummation at next-to-next-to-leading-logarithm accuracy matched to a NNLO fixed-order calculation (NNLL+NNLO). The VBF process was simulated at next-to-leading-order (NLO) accuracy in QCD. The simulation of the 𝑊 𝐻 and 𝑞𝑞/𝑞𝑔 → 𝑍 𝐻 processes is accurate to NLO in QCD with up to one extra jet in the event, while the simulation for the 𝑔𝑔 → 𝑍 𝐻 process was performed at leading order in QCD. The 𝑡𝑡𝐻 and 𝑏 b𝐻 processes were simulated at NLO in the strong coupling constant 𝛼 s in the five-flavour scheme. The PDF4LHC15 sets [START_REF] Butterworth | PDF4LHC recommendations for LHC Run II[END_REF] of parton distribution functions (PDFs) were used for all the processes listed above. The NNLO set was used for ggF, and the NLO set for other processes.

The 𝑡𝐻𝑞𝑏 (𝑡𝐻𝑊) samples were produced with M G 5_ MC@NLO 2.6 [START_REF] Alwall | The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations[END_REF] in the four-flavour (five-flavour) scheme with the NNPDF3.0 PDF. The same flavour scheme was used in the matrix element calculation and the PDF. The top quark and 𝑊 boson decays were handled by M S [START_REF] Artoisenet | Automatic spin-entangled decays of heavy resonances in Monte Carlo simulations[END_REF] to account for spin correlations in the decay products. The overlap of the 𝑡𝐻𝑊 process with 𝑡𝑡𝐻 at NLO was removed by using a diagram removal technique [START_REF] Frixione | Single-top hadroproduction in association with a 𝑊 boson[END_REF][START_REF] Demartin | tWH associated production at the LHC[END_REF] The 𝑝 𝑝 → 𝑡𝐻𝑏 process has a small cross-section and was not considered in the modelling of 𝑡𝐻 production.

All generated events for the processes listed above were interfaced to P 8.2 [START_REF] Sjöstrand | A brief introduction to PYTHIA 8.1[END_REF][START_REF] Sjöstrand | An introduction to PYTHIA 8.2[END_REF] to model parton showering, hadronization and the underlying event using the AZNLO set of parameter values tuned to data [START_REF]Measurement of the 𝑍/𝛾 * boson transverse momentum distribution in 𝑝 𝑝 collisions at √ 𝑠 = 7 TeV with the ATLAS detector[END_REF]. The decays of bottom and charm hadrons were simulated using the E G 1.6.0 program [START_REF] Lange | The EvtGen particle decay simulation package[END_REF]. Systematic uncertainties related to the signal modeling are estimated using a set of samples where H 7 [START_REF] Bellm | Herwig 7.1 Release Note[END_REF][START_REF] Bellm | Herwig 7.0/Herwig++ 3.0 release note[END_REF] is used for parton showering.

Major Higgs boson production processes were also simulated using alternative generator programs. The ggF process was also generated with M G 5_ MC@NLO, using an NLO-accurate matrix element for up to two additional partons and applying the F F merging scheme to obtain an inclusive sample [START_REF] Alwall | The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations[END_REF][START_REF] Frederix | Merging meets matching in MC@NLO[END_REF]. The generation used an effective vertex with a point-like coupling between the Higgs boson and gluons in the infinite top-mass limit. The events were showered using P 8.2 with the A14 set of tuned parameters [START_REF]ATLAS Pythia 8 tunes to 7 TeV data[END_REF]. The VBF alternative sample was generated with M G 5_ MC@NLO at NLO accuracy in the matrix element. It was then showered with H 7.1.6. The 𝑉 𝐻 alternative sample was simulated with M G 5_ MC@NLO, and the simulation is accurate to NLO in QCD for zero or one additional parton merged with the FxFx merging scheme. The 𝑔𝑔 → 𝑍 𝐻 process was also simulated at LO with M G 5_ MC@NLO and showered with P 8.2. The 𝑡𝑡𝐻 alternative sample was simulated with M G 5_ MC@NLO at NLO and the parton showering was performed with P 8.2.

All Higgs boson signal events were generated with a Higgs boson mass (𝑚 𝐻 ) of [START_REF]Reconstruction of primary vertices at the ATLAS experiment in Run 1 proton-proton collisions at the LHC[END_REF] GeV and an intrinsic width (Γ 𝐻 ) of 4.07 MeV [START_REF] Andersen | Handbook of LHC Higgs Cross Sections: 3. Higgs Properties[END_REF]. The cross-sections of Higgs production processes are reported for a centre-of-mass energy of √ 𝑠 = 13 TeV and a Higgs boson with mass 𝑚 𝐻 = 125.09 GeV [START_REF]Combined Measurement of the Higgs Boson Mass in 𝑝 𝑝 Collisions at √ 𝑠 = 7 and 8 TeV with the ATLAS and CMS Experiments[END_REF]. These cross-sections [START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF][START_REF] Demartin | tWH associated production at the LHC[END_REF], shown in Table 1, are used together with the Higgs boson branching ratio to diphotons [START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF][START_REF] Djouadi | HDECAY: a program for Higgs boson decays in the Standard Model and its supersymmetric extension[END_REF][START_REF] Spira | QCD Effects in Higgs Physics[END_REF][START_REF] Djouadi | Decays of supersymmetric particles: The Program SUSY-HIT (SUspect-SdecaY-Hdecay-InTerface)[END_REF][START_REF] Bredenstein | Radiative corrections to the semileptonic and hadronic Higgs-boson decays 𝐻 → 𝑊𝑊/𝑍 𝑍 → 4 fermions[END_REF][START_REF] Bredenstein | Precise predictions for the Higgs-boson decay 𝐻 → 𝑊𝑊/𝑍 𝑍 → 4 leptons[END_REF][START_REF] Bredenstein | Precision calculations for the Higgs decays 𝐻 → 𝑍 𝑍/𝑊𝑊 → 4 leptons[END_REF] to normalize the simulated signal events. Prompt diphoton production (𝛾𝛾) was simulated with the S 2.2.4 [START_REF] Bothmann | Event generation with Sherpa 2.2[END_REF] generator. In this set-up, NLO-accurate matrix elements for up to one parton, and LO-accurate matrix elements for up to three partons were calculated with the Comix [START_REF] Gleisberg | Comix, a new matrix element generator[END_REF] and O L [START_REF] Buccioni | OpenLoops 2[END_REF][START_REF] Cascioli | Scattering Amplitudes with Open Loops[END_REF][START_REF] Denner | C : A fortran-based complex one-loop library in extended regularizations[END_REF] libraries. They were matched with the S parton shower [START_REF] Schumann | A Parton shower algorithm based on Catani-Seymour dipole factorisation[END_REF] using the MEPS@NLO prescription [START_REF] Höche | A critical appraisal of NLO+PS matching methods[END_REF][START_REF] Höche | QCD matrix elements + parton showers: The NLO case[END_REF][START_REF] Catani | QCD Matrix Elements + Parton Showers[END_REF][START_REF] Höche | QCD matrix elements and truncated showers[END_REF] with a dynamic merging cut [START_REF] Siegert | A practical guide to event generation for prompt photon production with Sherpa[END_REF] of 10 GeV. Photons were required to be isolated according to a smooth-cone isolation criterion [START_REF] Frixione | Isolated photons in perturbative QCD[END_REF]. Samples were generated using the NNPDF3.0 PDF set [START_REF] Ball | Parton distributions for the LHC run II[END_REF], along with the dedicated set of tuned parton-shower parameters developed by the S authors.

The production of 𝑉 𝛾𝛾 events was simulated with the S 2.2.4 [START_REF] Bothmann | Event generation with Sherpa 2.2[END_REF] generator. QCD LO-accurate matrix elements for up to one additional parton emission were matched and merged with the S parton shower based on the Catani-Seymour dipole factorization [START_REF] Gleisberg | Comix, a new matrix element generator[END_REF][START_REF] Schumann | A Parton shower algorithm based on Catani-Seymour dipole factorisation[END_REF] using the MEPS@LO prescription [START_REF] Höche | A critical appraisal of NLO+PS matching methods[END_REF][START_REF] Höche | QCD matrix elements + parton showers: The NLO case[END_REF][START_REF] Catani | QCD Matrix Elements + Parton Showers[END_REF][START_REF] Höche | QCD matrix elements and truncated showers[END_REF]. Samples were generated using the same PDF set and parton-shower parameters as the 𝛾𝛾 sample. The production of 𝑡 t𝛾𝛾 events was modelled using the M G 5_ MC@NLO 2.3.3 generator at LO with the NNPDF2.3 [START_REF] Ball | Parton distributions with LHC data[END_REF] PDF. The parton-showering and underlying-event simulation were performed using P 8.2.

The effect of multiple interactions in the same and neighbouring bunch crossings (pile-up) was modelled by overlaying the original hard-scattering event with simulated inelastic proton-proton (𝑝 𝑝) events generated with P 8.1 using the NNPDF2.3 PDF set and the A3 tune [START_REF]The Pythia 8 A3 tune description of ATLAS minimum bias and inelastic measurements incorporating the Donnachie-Landshoff diffractive model[END_REF]. The generated signal and background events were passed through a simulation of the ATLAS detector [START_REF]The ATLAS Simulation Infrastructure[END_REF] using the G 4 toolkit [START_REF] Agostinelli | G 4 -a simulation toolkit[END_REF]. The only exception is the prompt diphoton sample, where the generated events were processed using a fast simulation of the ATLAS detector [START_REF]The simulation principle and performance of the ATLAS fast calorimeter simulation FastCaloSim[END_REF] , where the full simulation of the calorimeter is replaced with a parameterization of the calorimeter response.

A summary of the simulated signal and background samples is shown in Table 1.

Table 1: Event generators and PDF sets used to model signal and background processes. The cross-sections of Higgs boson production processes [6, 61, 62, 68, 75-77, 80, 82, 86-93, 118, 119] are reported for a centre-of-mass energy of √ 𝑠 = 13 TeV and a Higgs boson mass of 𝑚 𝐻 = 125.09 GeV. The order of the calculated cross-section is reported in each case. The cross-sections for the background processes are omitted, since the background normalization is determined in fits to the data.

Event reconstruction and selection

Events in this analysis are selected using the following procedure. Reconstructed photon candidates are first required to satisfy a set of preselection-level identification criteria. The two highest-𝑝 T preselected photons are then used to define the diphoton system, and an algorithm is used to identify the event primary vertex. Finally, the photons are required to satisfy isolation criteria and additional identification criteria. Jets (including 𝑏-tagged jets), muons, electrons, and missing transverse energy (𝐸 miss T ) are used in the analysis in order to categorize diphoton events and measure Higgs boson properties.

Photon reconstruction and identification

Photons are reconstructed from energy deposits in the calorimeter that are formed using a dynamical, topological cell-clustering algorithm [START_REF]Electron and photon performance measurements with the ATLAS detector using the 2015-2017 LHC proton-proton collision data[END_REF]. The photon candidate is classified as converted if it is matched to either two tracks forming a conversion vertex, or one track with the signature of an electron track without hits in the innermost pixel layer; otherwise, it is classified as unconverted. The photon candidate's energy is calibrated using a procedure described in Ref. [START_REF]Electron and photon performance measurements with the ATLAS detector using the 2015-2017 LHC proton-proton collision data[END_REF].

Reconstructed photons must satisfy |𝜂| < 2.37 in order to fall inside the region of the electromagnetic (EM) calorimeter with a finely segmented first layer, and outside the range 1.37 < |𝜂| < 1.52 corresponding to the transition region between the barrel and endcap EM calorimeters. Photon candidates are distinguished from jet backgrounds using identification criteria based on calorimeter shower shape variables [START_REF]Electron and photon performance measurements with the ATLAS detector using the 2015-2017 LHC proton-proton collision data[END_REF]. A loose working point is used for preselection, and the final selection of photon candidates is made using a tight selection. The efficiency of the tight identification for unconverted (converted) photons ranges from about 84% (85%) at 𝑝 T = 25 GeV to 94% (98%) for 𝑝 T > 100 GeV.

The final selection of photons includes both calorimeter-and track-based isolation requirements to further suppress jets misidentified as photons. The calorimeter isolation variable is defined as the total energy of calorimeter clusters in a cone of size Δ𝑅 = 0.2 around the photon candidate, excluding the energy in a fixed-size window containing the photon shower; a correction is applied for leakage of photon energy from this window into the surrounding cone [START_REF]Electron and photon performance measurements with the ATLAS detector using the 2015-2017 LHC proton-proton collision data[END_REF]. Contributions from pile-up and the underlying event are subtracted [START_REF]Electron and photon performance measurements with the ATLAS detector using the 2015-2017 LHC proton-proton collision data[END_REF][START_REF] Cacciari | The Catchment Area of Jets[END_REF][START_REF] Cacciari | On the characterisation of the underlying event[END_REF][START_REF]Measurement of the photon identification efficiencies with the ATLAS detector using LHC Run-1 data[END_REF][START_REF]Measurement of the inclusive isolated prompt photon cross section in 𝑝 𝑝 collisions at √ 𝑠 = 7 TeV with the ATLAS detector[END_REF]. The calorimeter-based isolation must be less than 6.5% of the photon transverse energy for each photon candidate. The track-based isolation variable is defined as the scalar sum of the transverse momenta of tracks within a Δ𝑅 = 0.2 cone around the photon candidate. The tracks considered in the isolation variable are restricted to those with 𝑝 T > 1 GeV that are matched to the selected diphoton primary vertex described below and not associated with the photon conversion vertex, if present. Each photon must have a track isolation less than 5% of the photon transverse energy.

Event selection and selection of the diphoton primary vertex

Events are selected by first requiring at least two photons satisfying the loose identification preselection criteria. The two highest-𝑝 T preselected photons are designated as the candidates for the diphoton system. The diphoton primary vertex of the event is determined using a neural-network algorithm [START_REF]Measurement of Higgs boson production in the diphoton decay channel in 𝑝 𝑝 collisions at center-of-mass energies of 7 and 8 TeV with the ATLAS detector[END_REF]. Information about the reconstructed vertices in the event and the trajectories of the two photons, measured using the depth segmentation of the calorimeter and completed by photon conversion information if present, is used as input to the network. [START_REF]Measurement of Higgs boson production in the diphoton decay channel in 𝑝 𝑝 collisions at center-of-mass energies of 7 and 8 TeV with the ATLAS detector[END_REF]. The algorithm is trained on simulation and leads to an 8% improvement in the mass resolution for inclusive Higgs boson production, relative to the default primary vertex selection [START_REF]Reconstruction of primary vertices at the ATLAS experiment in Run 1 proton-proton collisions at the LHC[END_REF], and results in better analysis sensitivity. Its performance was validated using studies of 𝑍→𝑒𝑒 events in data and simulation, in which the electrons were treated as photon candidates and their track information ignored. This performance is weakly dependent on the event pile-up, and its residual dependence is well described by simulation.

The two preselected photon candidates are required to satisfy the tight identification criteria and the isolation selection described above. Finally, the highest-𝑝 T and second-highest-𝑝 T photon candidates are required to satisfy 𝑝 T /𝑚 𝛾𝛾 > 0.35 and 0.25, respectively. As discussed in Sections 5 and 6, events that fail the tight identification or the isolation selection are used as a control sample for background estimation and modelling purposes.

The trigger, photon and event selections described above are used to define the events that are selected for further analysis for Higgs boson properties. In total, about 1.2 million events are selected in this data set with a diphoton invariant mass between 105 and 160 GeV. The predicted efficiency for a SM Higgs boson signal with |𝑦 𝐻 | < 2.5 is 39%.

Reconstruction and selection of hadronic jets, 𝒃-jets, leptons, top quarks and missing transverse momentum

Jets are reconstructed using a particle-flow algorithm [START_REF]Jet reconstruction and performance using particle flow with the ATLAS Detector[END_REF] from noise-suppressed positive-energy topological clusters [START_REF]Topological cell clustering in the ATLAS calorimeters and its performance in LHC Run 1[END_REF] in the calorimeter using the anti-𝑘 𝑡 algorithm [START_REF] Cacciari | The anti-𝑘 𝑡 jet clustering algorithm[END_REF][START_REF] Cacciari | FastJet user manual[END_REF] with a radius parameter 𝑅 = 0.4. Energy deposited in the calorimeter by charged particles is subtracted and replaced by the momenta of tracks that are matched to those topological clusters. The jet four-momentum is corrected for the non-compensating calorimeter response, signal losses due to noise threshold effects, energy lost in non-instrumented regions, and contributions from pile-up [START_REF]Jet energy scale and resolution measured in proton-proton collisions at √ 𝑠 = 13 TeV with the ATLAS detector[END_REF]. Jets are required to have 𝑝 T > 25 GeV and an absolute value of rapidity 𝑦 less than 4.4. A jet-vertex-tagger (JVT) multivariate discriminant [START_REF]Performance of pile-up mitigation techniques for jets in 𝑝 𝑝 collisions at √ 𝑠 = 8 TeV using the ATLAS detector[END_REF] is applied to jets with 𝑝 T < 60 GeV and |𝜂| < 2.4, to suppress jets from pile-up; in the |𝜂| range beyond 2.5, a forward version of the JVT [START_REF]Forward jet vertex tagging using the particle flow algorithm[END_REF] is applied to jets with 𝑝 T < 120 GeV. Jets with |𝜂| < 2.5 containing 𝑏-hadrons are identified using the DL1r 𝑏-tagging algorithm and its 60%, 70%, 77% and 85% efficiency working points, which are combined into a pseudo-continuous 𝑏-tagging score [START_REF]ATLAS 𝑏-jet identification performance and efficiency measurement with 𝑡 t events in 𝑝 𝑝 collisions at √ 𝑠 = 13 TeV[END_REF].

Electrons are reconstructed by matching tracks in the ID to topological clusters formed using the same dynamical, topological cell-clustering algorithm as in the photon reconstruction [START_REF]Electron and photon performance measurements with the ATLAS detector using the 2015-2017 LHC proton-proton collision data[END_REF]. Electron candidates are required to have 𝑝 T > 10 GeV and |𝜂| < 2.47, excluding the EM calorimeter transition region of 1.37 < |𝜂| < 1.52, and must satisfy the medium identification selection based on a likelihood discriminant using calorimeter shower shapes and track parameters [START_REF]Electron and photon performance measurements with the ATLAS detector using the 2015-2017 LHC proton-proton collision data[END_REF]. Isolation criteria are applied to electrons, using calorimeter-and track-based information. The reconstructed track matched to the electron candidate must be consistent with the diphoton vertex, which is ensured by requiring its longitudinal impact parameter 𝑧 0 relative to the vertex to satisfy |𝑧 0 sin 𝜃| < 0.5 mm. In addition, the electron track's transverse impact parameter with respect to the beam axis divided by its uncertainty, |𝑑 0 |/𝜎 𝑑 0 , must be less than 5.

Muons are reconstructed by matching tracks from the MS and ID subsystems. In the pseudorapidity range of 2.5 < |𝜂| < 2.7, muons without an ID track but whose MS track is compatible with originating from the interaction point are also considered. Muon candidates are required to have 𝑝 T > 10 GeV and |𝜂| < 2.7, and must satisfy the medium identification requirements [START_REF]Muon reconstruction and identification efficiency in ATLAS using the full Run 2 𝑝 𝑝 collision data set at √ 𝑠 = 13 TeV[END_REF]. Muons are required to satisfy calorimeterand track-based isolation requirements that are 95%-97% efficient for muons with 10 ≤ 𝑝 T ≤ 60 GeV and 99% efficient for 𝑝 T > 60 GeV. Muon tracks must satisfy |𝑧 0 sin 𝜃| < 0.5 mm and |𝑑 0 |/𝜎 𝑑 0 < 3.

Top quark candidates are reconstructed and identified using a boosted decision tree (BDT) discriminant, using the same procedure as in Ref. [START_REF]Study of the 𝐶𝑃 properties of the interaction of the Higgs boson with top quarks using top quark associated production of the Higgs boson and its decay into two photons with the ATLAS detector at the LHC[END_REF] applied to the particle-flow jets described above. The BDT targets both leptonic top quark signatures, in which the top quark decays to a 𝑊 boson that decays to an electron or a muon, and hadronic signatures in which the 𝑊 boson decays to hadrons or to a 𝜏-lepton.

An overlap removal procedure is performed in order to avoid double-counting objects. First, electrons overlapping with any photons (Δ𝑅 < 0.4) that pass the isolation and identification requirements are removed. Jets overlapping with the selected photons (Δ𝑅 < 0.4) and electrons (Δ𝑅 < 0.2) are removed. In the calculation of the Δ𝑅 between a jet and another object, the jet rapidity is used. Electrons overlapping with the remaining jets (Δ𝑅 < 0.4) are removed to match the requirements imposed when measuring isolated electron efficiencies. Finally, muons overlapping with photons or jets (Δ𝑅 < 0.4) are removed.

The missing transverse momentum is defined as the negative vector sum of the transverse momenta of the selected photon, electron, muon and jet objects, plus the transverse momenta of remaining low-𝑝 T particles, estimated using tracks matched to the diphoton primary vertex but not assigned to any of the selected objects [START_REF]Performance of missing transverse momentum reconstruction with the ATLAS detector using proton-proton collisions at √ 𝑠 = 13 TeV[END_REF]. Its magnitude is denoted by 𝐸 miss T .

Finally, an event veto is applied to suppress the overlap between the selection described here and that of the search for Higgs boson pair production in the 𝑏 b𝛾𝛾 final state [START_REF]Search for Higgs boson pair production in the two bottom quarks plus two photons final state in pp collisions at √ 𝑠 = 13 TeV with the ATLAS detector[END_REF], to facilitate the statistical combination of the two results at a later stage. Most of the vetoed events would enter the 𝑡𝑡𝐻 and 𝑡𝐻 classes defined in Section 5. This veto has a negligible impact on the analysis results.

5 Design of the measurement

Overview

The analysis is designed to measure the production cross-sections in the STXS framework [START_REF] Berger | Simplified Template Cross Sections -Stage 1[END_REF]. The regions considered in this paper are based on the Stage 1.2 STXS binning. They are defined in the Higgs boson rapidity range of |𝑦 𝐻 | < 2.5, separately for mutually exclusive Higgs boson production processes: the 𝑔𝑔 → 𝐻 process, which includes both ggF production and 𝑔𝑔 → 𝑍 𝐻 production followed by a hadronic decay of the 𝑍 boson; the electroweak 𝑞𝑞 → 𝐻𝑞𝑞 process, encompassing both VBF production and 𝑞 q → 𝑉 𝐻 production followed by a hadronic decay of the vector boson; the 𝑉 (→ leptons)𝐻 process, corresponding to 𝑝 𝑝 → 𝑉 𝐻 production followed by a leptonic decay of the vector boson (in the case of 𝑍 𝐻, including both decays to charged leptons and to neutrinos); and top-associated 𝑡𝑡𝐻 and 𝑡𝐻 production. The Higgs boson decay information is not used in the definition of STXS regions. For each process, non-overlapping fiducial regions are defined. These are based on the kinematics of the Higgs boson and of the associated jets and 𝑊 and 𝑍 bosons, as well as the numbers of jets, leptons and top quarks. Jets are reconstructed at the particle level from all stable particles with a lifetime greater than 10 ps, excluding the decay products of the Higgs boson and leptons from 𝑊 and 𝑍 boson decays, using the anti-𝑘 𝑡 algorithm with a jet radius parameter 𝑅 = 0.4, and must have a transverse momentum larger than 30 GeV.

Compared to the Stage 1.2 STXS definition, two sets of modified STXS regions are defined: a set of analysis regions which is used in the design of the analysis strategy, and is defined below; and a set of measurement regions, in which some analysis regions are merged, which are used to present the measurement results and are defined at the beginning of Section 8.4. The 45 STXS analysis regions are listed in Figure 1. They follow the Stage 1.2 definitions with the following modifications:

• The 𝑏 b𝐻 production mode is experimentally difficult to separate from 𝑔𝑔 → 𝐻, and these two production modes have similar selection acceptance and efficiency. The two modes are therefore measured as a single process, with each STXS region of the combined process corresponding to the sum of 𝑔𝑔 → 𝐻 and 𝑏 b𝐻 contributions.

• For 𝑔𝑔 → 𝐻 and 𝑞𝑞 → 𝐻𝑞𝑞 processes, STXS regions requiring two or more jets are not split by the transverse momentum of the system consisting of the Higgs boson and two highest-𝑝 T jets, 𝑝

𝐻 𝑗 𝑗

T , since the measurement does not provide sufficient sensitivity to this split. In addition, the STXS region defined by 𝑚 𝑗 𝑗 ≥ 700 GeV, where 𝑚 𝑗 𝑗 is the invariant mass of the two highest-𝑝 T jets, is split into two bins corresponding to 𝑚 𝑗 𝑗 above or below 1 TeV. An additional splitting at 𝑚 𝑗 𝑗 = 700 GeV is also introduced in the 𝑝 𝐻 T ≥ 200 GeV region of the 𝑞𝑞 → 𝐻𝑞𝑞 process. • The 𝑔𝑔 → 𝑍 𝐻 and 𝑞 q → 𝑍 𝐻 production modes with a leptonic 𝑍 boson decay similarly cannot be distinguished by the analysis selections, and are therefore considered as a single 𝑝 𝑝 → 𝑍 𝐻 process. In addition, each region of this process is split into separate regions for charged (𝑝 𝑝 → 𝐻ℓℓ) and neutral (𝑝 𝑝 → 𝐻𝜈 ν) dileptons.

• Production of 𝑡𝐻 is split into separate 𝑝 𝑝 → 𝑡𝐻𝑊 and 𝑝 𝑝 → 𝑡𝐻𝑞𝑏 contributions, since the two processes have different acceptances for the analysis selections. The 𝑠-channel 𝑝 𝑝 → 𝑡𝐻𝑏 process is neglected due to its small cross-section.

• The 𝑉 (→ leptons)𝐻 regions are not separated according to the number of jets in the event.

Categorization

The events passing the selection described in Section 4 are classified into mutually exclusive event categories, each targeted towards a particular STXS region. 2 This follows a technique similar to the one used in Ref. [START_REF]Measurements of Higgs boson properties in the diphoton decay channel with 36 fb -1 of 𝑝 𝑝 collision data at √ 𝑠 = 13[END_REF], but the definition of the categories has been improved significantly. The categorization in Ref. [START_REF]Measurements of Higgs boson properties in the diphoton decay channel with 36 fb -1 of 𝑝 𝑝 collision data at √ 𝑠 = 13[END_REF] was implemented sequentially over production modes, in order of increasing cross-section. In the present analysis, the categories are instead defined using a unified technique covering all processes simultaneously, and are designed to maximize a global criterion of sensitivity in the measurement of the cross-sections in all STXS regions. The technique proceeds in several steps. First, simulated Higgs boson production event samples are used to train a multiclass BDT to separate signal events coming from different STXS regions. This multiclass BDT classifier outputs one discriminant value for each STXS region. The output discriminant values are then used to assign signal events to various STXS classes. Each of these detector-level classes targets events from a particular STXS region defined at the particle level. Finally, each class is further divided into multiple categories using a binary multivariate classifier. This classifier is trained to separate signal from continuum background and Higgs boson events from other STXS regions in each class.

The inputs to all the classifiers are variables describing the kinematic and identification properties of the reconstructed particles presented in Section 4:

• the kinematics of the diphoton system;

• the numbers of reconstructed jets, 𝑏-jets, electrons, muons and top quarks;

• the kinematics of the system composed of the two photons and one or more jets, if jets are present, and of the system composed of the two highest-𝑝 T jets in the event, if at least two jets are present;

• the kinematics of the reconstructed leptons and top quarks;

• the reconstruction score of the top quarks, computed from the kinematics of the top quark decay products as described in Ref. [START_REF]Study of the 𝐶𝑃 properties of the interaction of the Higgs boson with top quarks using top quark associated production of the Higgs boson and its decay into two photons with the ATLAS detector at the LHC[END_REF];

• other event quantities such as the missing transverse momentum.

Among the top-associated production processes, the 𝑡𝐻𝑞𝑏 mode can be separated from both 𝑡𝑡𝐻 and 𝑡𝐻𝑊 due to differences in kinematics and event topology, in particular the presence of a forward jet and the absence of a second well reconstructed top quark candidate in the event.

In order to avoid distorting the smoothly falling shapes of the background 𝑚 𝛾𝛾 distributions, any variable found to have a linear correlation coefficient of 5% or more with 𝑚 𝛾𝛾 in the signal or background training samples is removed from the list of inputs to the binary classifiers. The training variables used in the analysis are summarized in Tables 2 and3. The multiclass BDT used in the initial step of the classification is trained on a data set obtained by merging the ggF, VBF, 𝑉 𝐻, 𝑡𝑡𝐻 and 𝑡𝐻 signal samples described in Section 3.2. A weight is applied to the events in each STXS region so that the regions have equal event yields in the training sample. This configuration improves the performance of the discrimination. For each event, the output of the BDT consists of a set of class discriminants 𝑦 𝑖 , where the index 𝑖 runs over the 45 STXS regions defined in Table 1. This output is then normalized into the parameters 𝑧 𝑖 = exp(𝑦 𝑖 )/ 𝑗 exp(𝑦 𝑗 ), a procedure also known as a softmax layer. The training is performed by minimizing the cross-entropy of the 𝑧 𝑖 with respect to the true STXS region assignments 3 using the LightGBM package [START_REF] Ke | LightGBM: A Highly Efficient Gradient Boosting Decision Tree[END_REF].

|𝑦𝐻 | < 2.5 𝑔𝑔 → 𝐻 + 𝑔𝑔 → 𝑍 (→ 𝑞 q)𝐻 + 𝑏 b𝐻 𝑝 𝐻 T < 200GeV 0-jet 𝑝 𝐻 T < 10 GeV 𝑔𝑔 → 𝐻, 0-jet, 𝑝 𝐻 T < 10 GeV 𝑝 𝐻 T ≥ 10 GeV 𝑔𝑔 → 𝐻, 0-jet, 10 ≤ 𝑝 𝐻 T < 200 GeV 1-jet 𝑝 𝐻 T < 60 GeV 𝑔𝑔 → 𝐻, 1-jet, 𝑝 𝐻 T < 60 GeV 60 ≤ 𝑝 𝐻 T < 120 GeV 𝑔𝑔 → 𝐻, 1-jet, 60 ≤ 𝑝 𝐻 T < 120 GeV
Z → + - pp → H , pV T < 75 GeV Z → ν ν pp → Hν ν, pV T < 75 GeV 75 ≤ pZ T < 150 GeV Z → + - pp → H , 75 ≤ pV T < 150 GeV Z → ν ν pp → Hν ν, 75 ≤ pV T < 150 GeV 150 ≤ pZ T < 250 GeV Z → + - pp → H , 150 ≤ pV T < 250 GeV Z → ν ν pp → Hν ν, 150 ≤ pV T < 250 GeV pZ T ≥ 250 GeV Z → + - pp → H , pV T ≥ 250 GeV Z → ν ν pp → Hν ν, pV T ≥ 250 GeV
A second training phase is then performed to optimize the classification procedure in terms of the analysis sensitivity itself. The sensitivity is estimated as the inverse determinant |𝐶 | -1 of the covariance matrix of the measurement of the signal event yields in each region. This D-optimality (determinant) criterion leads in particular to a reduction of the expected statistical uncertainty of the measurement, and is suggested by the fact that |𝐶 | -1 is a known measure of the information provided by the measurement [START_REF] Lindley | On a Measure of the Information Provided by an Experiment[END_REF]. The classification procedure is performed so that events are assigned to the STXS class 𝑖 corresponding to the maximum value of 𝑤 𝑖 𝑧 𝑖 , where the 𝑤 𝑖 are a set of per-class weights. These weights are initially set to 1, 3 The cross-entropy loss function is computed as

- 𝑛 𝑘=1 𝜔 𝑘 45 𝑖=1
𝛿 𝑖,𝑘 ln(𝑧 𝑖 ), where 𝑘 runs over the 𝑛 events in the training sample, 𝜔 𝑘 are event weights applied to balance the class yields as described in the text, 𝑖 runs over the classes, and 𝛿 𝑘,𝑖 has a value of 1 if class 𝑖 is the correct assignment for event 𝑘, and 0 otherwise. , and then iteratively updated so as to maximize |𝐶 | -1 : for each value of the 𝑤 𝑖 , a simulated data set is generated for each region by mixing events from each signal sample in proportion to their SM production cross-sections, together with a sample of simulated continuum background events normalized to data in the control region 95 ≤ 𝑚 𝛾𝛾 < 105 GeV. A simplified statistical model approximating the full model described in Section 6 is then used to estimate |𝐶 | -1 , and the procedure is iterated until a maximum is found for |𝐶 | -1 .

𝜙 * 𝛾𝛾 = tan 𝜋-|Δ𝜙 𝛾𝛾 | 2 √ 1 -tanh 2 Δ𝜂 𝛾𝛾 2 , cos 𝜃 * 𝛾𝛾 = (𝐸 𝛾 1 + 𝑝 𝛾 1 𝑧 ) •(𝐸 𝛾 2 -𝑝 𝛾 2 𝑧 )-(𝐸 𝛾 1 -𝑝 𝛾 1 𝑧 ) •(𝐸 𝛾 2 + 𝑝 𝛾 2
Figure 2 shows distributions of the weighted multiclass discriminant output 𝑤 𝑖 𝑧 𝑖 for four representative STXS classes, illustrating the discrimination provided by the multiclass BDT. While events with high BDT output values for a given region tend to be selected in the corresponding class, this does not manifest itself as a sharp cut, due to the interplay between the selections for the different classes. Compared to the simple selection based only on the 𝑧 𝑖 , the selection based on the 𝑤 𝑖 𝑧 𝑖 provides both higher purity and higher selection efficiency for classes associated with rare processes such as 𝑡𝐻, 𝑡𝑡𝐻, 𝑉 𝐻 and VBF, as well as production at high values of 𝑝 𝐻 T or 𝑚 𝑗 𝑗 . This leads to measurements with generally smaller uncertainties and lower correlations. This multiclass training allows the selection of target process events that otherwise would fail a requirement based on detector-level quantities corresponding to the STXS region definition. For example, in the STXS region 𝑔𝑔 → 𝐻, 1-jet, 𝑝 𝐻 T < 60 GeV, detector-level events that originate from the target process but have no reconstructed jets would fail requirements defined by the number of jets and 𝑝 𝐻 T ; however, those events could be selected by the multiclass discriminant. For this STXS region, 20% of events from the target process have no reconstructed jets. The recovery of these events leads to a reduction of about 6% in the measurement uncertainty. It is also robust against pile-up in the determination of jet multiplicity in 𝑔𝑔 → 𝐻. After the classes are defined, binary classifiers are then trained and used to further divide each class into multiple categories, to improve the measurement sensitivity. For each of the classes targeting 𝑔𝑔 → 𝐻, 𝑞𝑞 → 𝐻𝑞𝑞 and 𝑉 (→ leptons)𝐻 processes, a binary BDT classifier is trained to distinguish between simulated signal events of the corresponding STXS region and both simulated continuum background events and Higgs boson events from other STXS regions.

For the 𝑡𝑡𝐻 and 𝑡𝐻𝑊 classes, a binary BDT classifier is trained to separate 𝑡𝑡𝐻 signal and the continuum background using all events assigned to various 𝑡𝑡𝐻 classes targeting different 𝑝 𝐻 T regions. Similarly, a binary BDT classifier is trained to separate 𝑡𝐻𝑊 signal and the continuum background using events assigned to the 𝑡𝐻𝑊 class.

To enhance the sensitivity to the sign of the top-Yukawa coupling modifier 𝜅 𝑡 (defined in more detail in Section 9), a specialization is introduced for the 𝑡𝐻𝑞𝑏 class. First, the class is divided into two sub-classes based on a neural-network (NN) binary classifier that separates 𝑡𝐻𝑞𝑏 production with 𝜅 𝑡 = 1 from 𝑡𝐻𝑞𝑏 production with 𝜅 𝑡 = -1. In each sub-class, the events are then further divided into categories based on NN binary classifiers trained to separate the corresponding 𝑡𝐻𝑞𝑏 signal events from continuum background events and Higgs boson events from other processes.

The binary classifiers used to suppress continuum background processes in the 𝑡𝑡𝐻, 𝑡𝐻𝑊, and 𝑡𝐻𝑞𝑏 classes are trained on events from control regions in data, which provide larger event yields than the available simulated background samples. These regions are defined using the same selections as the classes, but reversing the photon identification requirement, the photon isolation requirement, or both.

In each class, events are then assigned to categories corresponding to ranges of binary classifier output values. Up to three categories are defined in this way, depending on the targeted STXS region. The category boundaries in the BDT output are determined by scanning over all possible values and finding the set that maximizes the sum in quadrature of the expected significance values in all categories. The expected significance is computed as 𝑍 = √ 2((𝑆 + 𝐵) ln(1 + 𝑆/𝐵) -𝑆) [START_REF] Cowan | Asymptotic formulae for likelihood-based tests of new physics[END_REF], where 𝑆 and 𝐵 are the expected signal yield and background yield in the targeted STXS region in the smallest range of 𝑚 𝛾𝛾 around the signal peak position that contains 90% of signal events. The background 𝐵 includes contributions from continuum background and Higgs boson events from other STXS regions. The continuum background is computed from the 𝑚 𝛾𝛾 distribution in simulation, normalized to the data control region 95 ≤ 𝑚 𝛾𝛾 < 105 GeV. A class is split into two categories if this leads to an improvement of more than 5% in the expected significance, and into three categories if a further improvement of at least 5% relative to the two-category configuration can be achieved. The categories are referred to as High-purity, Med-purity and, in the case of a 3-category split, Low-purity in order of decreasing BDT output values. No events are removed at the categorization stage, since the lower-purity categories bring non-negligible contributions to the analysis sensitivity. Figure 3 shows binary BDT discriminant distributions as well as category boundaries for four representative STXS classes.

The categorization for the 𝑡𝐻𝑞𝑏 class follows a different procedure, which aims to maximize both the sensitivity to a 𝑡𝐻𝑞𝑏 signal and the sensitivity to the sign of 𝜅 𝑡 . A boundary is placed in the NN classifier that separates the 𝑡𝐻𝑞𝑏 signal with 𝜅 𝑡 = 1 from the 𝑡𝐻𝑞𝑏 signal with 𝜅 𝑡 = -1. Different boundaries are also placed in the two binary NN classifiers that separate 𝑡𝐻𝑞𝑏 signals from continuum background. These boundaries are determined simultaneously. Finally, a low-purity top category is formed by grouping together the events with the lowest binary classifier output values in both the 𝑡𝑡𝐻 and 𝑡𝐻 classes.

The entire categorization procedure results in the definition of 101 categories in total. The expected signal and background yields in these categories are summarized in Table 4. The expected signal purity, defined as the expected signal yield divided by the expected yield from both the signal and background processes, in the smallest 𝑚 𝛾𝛾 window containing 90% of signal events, ranges from 0.03% to 78%. Figure 4 shows the contributions to the expected event yields from each of the 28 merged STXS regions defined in Section 8.4. The contributions are shown as fractions of events originating from each STXS region, in groups of analysis categories targeting the same region. They are obtained as a weighted sum of the fractions for each category in the group, with weights given by the signal-over-background ratio 𝑓 in each category as defined in Table 4. Figure 4: Contributions of STXS regions to the expected event yields in groups of analysis categories. The vertical axis lists the 28 merged STXS regions defined in Section 8.4, while the horizontal axis lists groups of analysis categories that target the same STXS region, weighted by their 𝑓 value as given in Table 4. Entries correspond to the percentage of the signal yield in each group of analysis categories (on the 𝑥-axis) that is contributed by a given STXS region (on the 𝑦-axis).

Entries with a value below 1% are not shown. The entries in each column, corresponding to the same group of analysis categories, add up to 100 (%), except for rounding effects and values below 1%.

Modelling of diphoton mass distributions

The 𝑚 𝛾𝛾 distribution in each category is described by an extended probability density function (pdf) in which the signal and background shapes are analytic functions of 𝑚 𝛾𝛾 . As in the previous measurement [START_REF]Measurements of Higgs boson properties in the diphoton decay channel with 36 fb -1 of 𝑝 𝑝 collision data at √ 𝑠 = 13[END_REF], the analytic functions are defined over the range of 105 < 𝑚 𝛾𝛾 < 160 GeV. The analysis results are obtained by a simultaneous fit of these pdfs to the 𝑚 𝛾𝛾 distributions in the categories defined in Section 5.2. Systematic uncertainties related to signal yield, signal shape and background modelling are incorporated into the likelihood model as nuisance parameters. For each of these nuisance parameters, a Gaussian or log-normal constraint pdf is included in the likelihood function. Gaussian constraints are used for uncertainties related to the background modelling, the peak position of the signal 𝑚 𝛾𝛾 distribution, and the Higgs boson mass. Log-normal constraints are used for other uncertainties, including multiplicative uncertainties in expected signal yields and in the 𝑚 𝛾𝛾 mass resolution. Asymmetric log-normal forms are used when the corresponding uncertainties are themselves asymmetric. The Higgs boson mass 𝑚 𝐻 is assumed to be 125.09 ± 0.24 GeV, as measured in Ref. [START_REF]Combined Measurement of the Higgs Boson Mass in 𝑝 𝑝 Collisions at √ 𝑠 = 7 and 8 TeV with the ATLAS and CMS Experiments[END_REF].

The effects of interference between the 𝐻 → 𝛾𝛾 signal process and continuum 𝛾𝛾 production lead to a small change in the expected Higgs boson production rate (a 2% reduction in the inclusive rate [START_REF] Campbell | Interference in the 𝑔𝑔 → ℎ → 𝛾𝛾 On-Shell Rate and the Higgs Boson Total Width[END_REF]) as well as a shift in the signal peak position that is small compared to the uncertainty in 𝑚 𝐻 [START_REF] Dixon | Bounding the Higgs Boson Width Through Interferometry[END_REF]. Both effects are neglected.

In each category 𝑖, the normalization of the background pdf is a free parameter in the fit, as well as the parameters describing the shapes of the background pdfs, as discussed in Section 6.2 below. The normalization of the signal pdf is expressed as

𝑁 𝑖 = ∑ 𝑡 (𝜎 𝑡 × 𝐵 𝛾𝛾 ) 𝜖 𝑖𝑡 L 𝐾 𝑖 (𝜽 yield ) + 𝑁 spur,𝑖 𝜃 spur,𝑖 (1)
where the sum runs over STXS regions, (𝜎 𝑡 × 𝐵 𝛾𝛾 ) is the measurement parameter for region 𝑡, 𝜖 𝑖𝑡 describes the efficiency for events from region 𝑡 to be reconstructed in category 𝑖, and L is the integrated luminosity of the fitted sample. The factor 𝐾 𝑖 (𝜽 yield ) corresponds to multiplicative corrections to the signal yields from systematic uncertainty effects detailed in Section 7, as a function of nuisance parameters collectively denoted by 𝜽 yield ; 𝑁 spur,𝑖 is the value of the background modelling uncertainty described in Section 6.2, implemented as an additive correction to the signal yield proportional to the nuisance parameter 𝜃 spur,𝑖 . The values of the measurement parameters are obtained from a maximum-likelihood fit to the data.

Modelling of the signal shape

The signal component in each category corresponds to the sum of the contributions from each STXS region, which are all assumed to follow the same 𝑚 𝛾𝛾 distribution in this category. The shape is described using a double-sided Crystal Ball (DSCB) function [START_REF] Oreglia | A Study of the Reactions 𝜓 → 𝛾𝛾𝜓[END_REF][START_REF]Search for Scalar Diphoton Resonances in the Mass Range 65-600 GeV with the ATLAS Detector in 𝑝 𝑝 Collision Data at √ 𝑠 = 8[END_REF], consisting of a Gaussian distribution in the region around the peak position, continued by power-law tails at lower and higher 𝑚 𝛾𝛾 values. An intrinsic shape difference between the DSCB function and signal 𝑚 𝛾𝛾 distribution is found to cause only a negligible bias in the estimated signal yield [START_REF]Measurements of Higgs boson properties in the diphoton decay channel with 36 fb -1 of 𝑝 𝑝 collision data at √ 𝑠 = 13[END_REF].

The parameters of the Crystal Ball function in each category are obtained by a fit to a mixture of the ggF, VBF, 𝑉 𝐻, 𝑡𝑡𝐻 and 𝑡𝐻 samples, described in Section 3.2, in proportion to their SM cross-sections. A shift of 0.09 GeV is applied to the position of the signal peak to account for the difference between the reference Higgs boson mass used in this analysis (𝑚 𝐻 = 125.09 GeV) and the mass for which the samples were generated (𝑚 𝐻 = 125 GeV). Simulated signal 𝑚 𝛾𝛾 distributions and their corresponding DSCB functions are shown for two groups of categories in Figure 5.

Modelling of the continuum background shape

The background in the selected diphoton sample mainly consists of continuum 𝛾𝛾 production, 𝛾 𝑗 and 𝑗 𝑗 production where one or more jets in the event are misidentified as photons. In the categories targeting 𝑉 (→ leptons)𝐻 production, the main contribution is from the 𝑉 𝛾𝛾 processs, while in categories targeting 𝑡𝑡𝐻 and 𝑡𝐻 production the main contributions are from 𝑡 t𝛾𝛾 and other processes involving top quarks. The modelling of this continuum background follows the same procedure as in previous analyses [START_REF]Measurements of Higgs boson properties in the diphoton decay channel with 36 fb -1 of 𝑝 𝑝 collision data at √ 𝑠 = 13[END_REF]. This procedure involves two main steps: first, a background 𝑚 𝛾𝛾 template is constructed from a combination of simulation samples and data control samples; secondly, a background function is selected from a number of candidate functions, using the spurious-signal test, with the goal of identifying an analytic function that is flexible enough to fit the 𝑚 𝛾𝛾 distribution in data and results in a sufficiently small potential bias compared to the statistical uncertainty.

In categories targeting the 𝑔𝑔 → 𝐻 and 𝑞𝑞 → 𝐻𝑞𝑞 processes, the template is defined as a combination of 𝛾𝛾, 𝛾 𝑗, and 𝑗 𝑗 processes, each of which is weighted according to its fraction in the selected analysis category. The fractions of these processes are determined by a data-driven method, known as the double two-dimensional sideband method [START_REF]Measurement of the production cross section of pairs of isolated photons in 𝑝 𝑝 collisions at 13 TeV with the ATLAS detector[END_REF], which uses three control regions in data in which one (for the 𝛾 𝑗 process) or both (for the 𝑗 𝑗 process) photons fail to satisfy the identification and/or isolation criteria, respectively. The fraction of the total background that is from the 𝛾𝛾 process ranges between 75% and 95%, the fraction from the 𝛾 𝑗 process is between 2% and 25%, and the 𝑗 𝑗 process contributes less than 6%.

While a simulation sample is used to model the 𝛾𝛾 process in this study, it is computationally prohibitive to generate sufficiently large samples of 𝛾 𝑗 and 𝑗 𝑗 production events passing analysis selections due to their large cross-sections and the high jet-rejection performance of the ATLAS photon identification algorithms. To avoid this issue, the 𝑚 𝛾𝛾 shapes of the 𝛾 𝑗 or 𝑗 𝑗 components are obtained from data control samples defined by inverting the identification requirement of one or both photons as described above. The ratio of the 𝑚 𝛾𝛾 distribution of the 𝛾 𝑗 and 𝑗 𝑗 components to that of the simulated 𝛾𝛾 sample is well described by a linear function of 𝑚 𝛾𝛾 . A linear fit to the ratio of these distributions is therefore used to provide an 𝑚 𝛾𝛾 -dependent weight that is applied to the 𝛾𝛾 sample to obtain a final template that also accounts for the 𝛾 𝑗 and 𝑗 𝑗 components. Changing the fraction of the 𝛾 𝑗 and 𝑗 𝑗 components within the uncertainties of their determination is found to have a negligible impact on the spurious-signal test described below.

For categories targeting the 𝑉 (→ leptons)𝐻 STXS regions, the background template is built using simulated events of 𝑉 𝛾𝛾 and prompt 𝛾𝛾 production. Since the available yields for the latter are not sufficient to build the template directly, the following procedure is followed: first the ratio of the 𝑚 𝛾𝛾 distribution of both samples to that of the 𝑉 𝛾𝛾 sample alone is fitted to a linear function of 𝑚 𝛾𝛾 ; the fit result is then applied to the 𝑚 𝛾𝛾 distribution of the 𝑉 𝛾𝛾 sample as an 𝑚 𝛾𝛾 -dependent weight to obtain the final template describing both contributions. For categories targeting the 𝑡𝑡𝐻 and 𝑡𝐻 processes, the diphoton events are primarily from 𝑡 t𝛾𝛾 production. As such, a sample of simulated 𝑡 t𝛾𝛾 events is used to construct the background template for those categories. Contributions from processes with jets misidentified as photons are not considered in categories targeting 𝑉 𝐻, 𝑡𝑡𝐻 and 𝑡𝐻 STXS regions as they do not significantly alter the background shape. The background templates constructed for four categories targeting the 𝑔𝑔 → 𝐻, 𝑞𝑞 → 𝐻𝑞𝑞 , 𝑉 𝐻 and 𝑡𝑡𝐻 processes are shown as examples in Figure 6. While the background template and data 𝑚 𝛾𝛾 distribution have slightly different shapes in some categories, the selected background analytic functions are flexible enough to absorb these small differences.

The background templates are defined over the range 105 < 𝑚 𝛾𝛾 < 160 GeV with 220 uniform-width bins. A template smoothing procedure based on a Gaussian kernel [START_REF] Frate | Modeling Smooth Backgrounds and Generic Localized Signals with Gaussian Processes[END_REF] is applied to analysis categories where the average bin occupancy in the background template is at least 20 entries. This procedure suppresses statistical fluctuations in the background templates, decreasing the systematic uncertainty on the modeling of the background. A study using pseudo-experiments generated with known template shapes was performed to verify that the smoothing procedure does not introduce a significant bias in the estimate of the spurious signal.

Three families of analytic functions are tested as candidates to model the 𝑚 𝛾𝛾 distribution for a given analysis category. They include power-law functions, Bernstein polynomials [START_REF] Bernstein | Démonstration du Théorème de Weierstrass fondée sur le calcul des Probabilités[END_REF], and exponential functions of a polynomial. These functions and the number of degrees of freedom tested are summarized in Table 5. The coefficients of these functions are considered to be independent across categories, regardless of the functions chosen, and in all cases are treated as free parameters in the fits to data. The main criterion used to Power law

𝑚 𝑎 𝛾𝛾 1 PowerLaw Bernstein polynomial (1 -𝑥) 𝑛 + 𝑎 1 𝑛𝑥(1 -𝑥) 𝑛-1 + • • • + 𝑎 𝑛 𝑥 𝑛 𝑛 = 1-5 Bern1-Bern5 Exponential exp(𝑎𝑚 𝛾𝛾 ) 1 Exp Exponential of second-order polynomial exp(𝑎 1 𝑚 𝛾𝛾 + 𝑎 2 𝑚 2 𝛾𝛾 ) 2 ExpPoly2 Exponential of third-order polynomial exp(𝑎 1 𝑚 𝛾𝛾 + 𝑎 2 𝑚 2 𝛾𝛾 + 𝑎 3 𝑚 3 𝛾𝛾 ) 3 ExpPoly3
select the functional form in each category is a bias test performed by fitting the background template using a model with free parameters for both the signal and the background event yields. In this fit, the background - template is normalized to the number of events observed in data in this category. The potential bias due to the mis-modelling of background 𝑚 𝛾𝛾 distribution is estimated from the fitted signal yield (the spurious signal). This test is performed for 𝑚 𝐻 values ranging from 123 GeV to 127 GeV, in steps of 0.5 GeV. In order to avoid accidentally small bias values at the nominal Higgs boson mass of 𝑚 𝐻 = 125.09 GeV, the maximum absolute value of fitted signal yield |𝑆 spur | over the range 123 < 𝑚 𝐻 < 127 GeV is considered as the potential bias. For categories where the original background 𝑚 𝛾𝛾 templates (before normalization to the data yields) have at least 20 entries per bin on average, the background functions are required to yield a value of |𝑆 spur | that is smaller than either 10% of the total expected Higgs boson signal yield (𝑆 exp ) or 20% of the statistical uncertainty of the fitted signal yield (𝜎 exp ). If multiple functions pass these requirements, the one with the smallest number of degrees of freedom is chosen.

[GeV]
An additional check is performed for the nine categories in which a fit of the analytic function to the background template is found to yield a 𝜒 2 𝑝-value that is below 1%. 4 For each of these categories, a set of samples is randomly drawn from the background template, each with a number of events equal to the observed yield in the data sidebands. The fit of the analytic function and the computation of the 𝜒 2 are then repeated for each sample. In all nine categories, more than 90% of the samples yield a 𝜒 2 𝑝-value above 5%, which shows that the low 𝑝-values observed in the fit to the nominal background template are related to features compatible with statistical fluctuations. For categories where the average number of entries per bin is less than 20, candidate background functions are limited to Exp, ExpPoly2 and ExpPoly3 (as defined in Table 5) in order to avoid unphysical fits due to large statistical fluctuations in the sidebands. The function is chosen using a Wald test [START_REF] Wald | Sequential Tests of Statistical Hypotheses[END_REF]: first the quantity In all cases, the |𝑆 spur | value of the selected background function provides an estimate of the possible bias in the fitted signal yield introduced by the intrinsic difference between the background 𝑚 𝛾𝛾 shape and the selected function. It is used to define the systematic uncertainty for the background modelling in category 𝑖, denoted as 𝑁 spur,𝑖 in Eq. ( 1). The selected functional form for each category is shown in Table 6.

𝑞 12 = -2 ln 𝐿 1 /𝐿 2 is

Systematic uncertainties

Systematic uncertainties considered in this analysis can be grouped into two categories: uncertainties in the modelling of the 𝑚 𝛾𝛾 distribution for the signal and background processes, and uncertainties in the expected signal yields in each category arising from either experimental or theory sources. These systematic uncertainties are incorporated into the likelihood model of the measurement as nuisance parameters, as explained in Section 6. More details about the uncertainties are provided in this section.

Experimental systematic uncertainties

Experimental systematic uncertainties relevant to the modelling of the signal 𝑚 𝛾𝛾 distribution include the uncertainties in the energy scale and energy resolution of photon candidates, as well as in the Higgs boson mass. The photon energy scale uncertainties are propagated to the peak position of the signal DSCB shape, with an impact that is usually less than 0.3% relative to the peak position value, depending on the category. The photon energy resolution uncertainties are propagated to the Gaussian width of the signal DSCB shape, with a relative impact between 1% and 15%, depending on the category. The estimation and implementation of the photon energy scale and resolution uncertainties follow the procedure outlined in Ref. [START_REF]Electron and photon performance measurements with the ATLAS detector using the 2015-2017 LHC proton-proton collision data[END_REF]. The total uncertainty in the measured Higgs boson mass, 0.24 GeV, is considered as an additional uncertainty of the peak position of the signal DSCB shape. The modelling of the background 𝑚 𝛾𝛾 distribution with an analytic function can produce a potential bias in the fitted signal yield. An uncertainty in the modelling of the background, computed using the spurious-signal method described in Section 6.2, is included as an additive contribution to the signal yield in each category as shown in Eq. ( 1). This uncertainty is considered to be uncorrelated between different categories. Out of the 101 analysis categories, 46 categories have a background modelling uncertainty that is no more than 10% of the background statistical uncertainty, and only 2 categories (𝑞𝑞 → 𝐻𝑞𝑞 , ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T ≥ 200 GeV, High-purity and 𝑝 𝑝 → 𝐻𝜈 ν, 0 ≤ 𝑝 𝑉 T < 75 GeV, High-purity) have a background modelling uncertainty that is at least 50% of the background statistical uncertainty.

Experimental uncertainties affecting the expected signal yields include: the efficiency of the diphoton trigger [START_REF]Performance of electron and photon triggers in ATLAS during LHC Run 2[END_REF], the photon identification efficiencies [START_REF]Electron and photon performance measurements with the ATLAS detector using the 2015-2017 LHC proton-proton collision data[END_REF], the photon isolation efficiencies, the impact of the photon energy scale and resolution uncertainties on the selection efficiency [START_REF]Electron and photon performance measurements with the ATLAS detector using the 2015-2017 LHC proton-proton collision data[END_REF], the modelling of pile-up in the simulation, which is evaluated by varying by ±9% the value of the visible inelastic cross-section used to reweight the pile-up distribution in the simulation to that in the data [START_REF]Measurement of the Inelastic Proton-Proton Cross Section at √ 𝑠 = 13 TeV with the ATLAS Detector at the LHC[END_REF], the jet energy scale and resolution [START_REF]Jet energy scale and resolution measured in proton-proton collisions at √ 𝑠 = 13 TeV with the ATLAS detector[END_REF], the efficiency of the jet vertex tagger, the efficiency of the 𝑏-tagging algorithm [START_REF]ATLAS 𝑏-jet identification performance and efficiency measurement with 𝑡 t events in 𝑝 𝑝 collisions at √ 𝑠 = 13 TeV[END_REF], the electron [START_REF]Electron and photon performance measurements with the ATLAS detector using the 2015-2017 LHC proton-proton collision data[END_REF] and muon [START_REF]Muon reconstruction and identification efficiency in ATLAS using the full Run 2 𝑝 𝑝 collision data set at √ 𝑠 = 13 TeV[END_REF] reconstruction, identification and isolation efficiencies, the electron [START_REF]Electron and photon performance measurements with the ATLAS detector using the 2015-2017 LHC proton-proton collision data[END_REF] and muon [START_REF]Muon reconstruction and identification efficiency in ATLAS using the full Run 2 𝑝 𝑝 collision data set at √ 𝑠 = 13 TeV[END_REF] energy and momentum scale and resolution, as well as the contribution to 𝐸 miss T from charged-particle tracks that are not associated with high-𝑝 T electrons, muons, jets, or photons [START_REF]Performance of missing transverse momentum reconstruction with the ATLAS detector using proton-proton collisions at √ 𝑠 = 13 TeV[END_REF]. The uncertainty in the combined 2015-2018 integrated luminosity is 1.7% [START_REF]Luminosity determination in 𝑝 𝑝 collisions at √ 𝑠 = 13[END_REF], obtained using the LUCID-2 detector [START_REF] Avoni | The new LUCID-2 detector for luminosity measurement and monitoring in ATLAS[END_REF] for the primary luminosity measurements.

Theory modelling uncertainties

The main theory uncertainties arise from missing higher-order terms in the perturbative QCD calculations, the modelling of parton showers, the PDFs and the value of 𝛼 s . For measurements of the 𝑡𝑡𝐻 and 𝑡𝐻 processes, the modelling of heavy-flavour jets in the ggF, VBF, and 𝑉 𝐻 processes is also important.

QCD uncertainties for each production mode are estimated by varying the renormalization and factorization scales used in the event generation, and the resulting variations in the predicted event yield in each STXS regions are considered as uncertainties.

For the 𝑔𝑔 → 𝐻 processes, the QCD uncertainty model is implemented using four components [START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF][START_REF] Liu | Reducing theoretical uncertainties for exclusive Higgs-boson plus one-jet production at the LHC[END_REF][149][START_REF] Boughezal | Combining resummed Higgs predictions across jet bins[END_REF] accounting for modelling uncertainties in the jet multiplicity, three describing uncertainties in the modelling the 𝑝 𝐻 T distribution, one [START_REF] Stewart | Theory uncertainties for Higgs and other searches using jet bins[END_REF][START_REF] Gangal | Next-to-leading-order uncertainties in Higgs+2 jets from gluon fusion[END_REF] accounting for the uncertainty in the distribution of the 𝑝 𝐻 𝑗 𝑗 T variable, four accounting for the uncertainty in the distribution of the 𝑚 𝑗 𝑗 variable, and six covering modelling uncertainties in STXS regions with high 𝑝 𝐻 T (> 300 GeV). Following the principles of the Stewart-Tackmann procedure [START_REF] Stewart | Theory uncertainties for Higgs and other searches using jet bins[END_REF], two components account for uncertainties in the inclusive 𝑔𝑔 → 𝐻 event yields, while the others describe migration uncertainties in the fraction of events passing the selections defining the STXS regions. The model provides a full description of the uncertainty in each STXS region, in which the uncertainty components are each assigned to one nuisance parameter that is treated as statistically independent from the others. These uncertainties are typically less than 22% of the expected signal yield in analysis categories targeting the 𝑔𝑔 → 𝐻 process.

For each of the 𝑊 𝐻, 𝑞𝑞/𝑞𝑔 → 𝑍 𝐻, and 𝑔𝑔 → 𝑍 𝐻 processes, the QCD uncertainty model includes four independent components to account for uncertainties in the distribution of 𝑝 𝑉 T , and two independent components for uncertainties in the jet multiplicity distribution. For 𝑞𝑞 → 𝐻𝑞𝑞 (VBF and 𝑉 (→ hadrons)𝐻) processes, the QCD uncertainty model includes a similar set of independent components: two for the modelling of the jet multiplicity and the 𝑝 𝐻 𝑗 𝑗 T distribution, one for migration between the 𝑝 𝐻 T < 200 GeV and 𝑝 𝐻 T ≥ 200 GeV regions, and six for the modelling of the 𝑚 𝑗 𝑗 distribution. These uncertainties are less than 10% of the expected signal yield in analysis categories targeting these processes, with the exception of the 𝑔𝑔 → 𝑍 𝐻 process where the uncertainty can be as large as 26%.

For the 𝑡𝑡𝐻 process, QCD uncertainties include five components covering migrations between 𝑡𝑡𝐻 STXS regions with different 𝑝 𝐻 T . These uncertainties are less than 10% of the expected signal yield in the 𝑡𝑡𝐻 STXS regions in their targeted analysis categories. In the case of 𝑡𝐻𝑊, 𝑡𝐻𝑞𝑏 and 𝑏 b𝐻, an overall QCD uncertainty is used, taking the value from Ref. [START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF].

To check the robustness of the uncertainty model, a comparison between the efficiency factors of the nominal Higgs signal sample and the alternative sample generated by M G 5_ MC@NLO, as described in Section 3.2, is performed for the VBF, 𝑉 𝐻, and 𝑡𝑡𝐻 processes. The differences between the efficiency factors of the nominal and alternative samples are significantly larger than the uncertainties from QCD scale variations and can reach values of up to 20% in some phase-space regions of the VBF process. The differences between the efficiency factors of the nominal and alternative samples are therefore considered as an additional systematic uncertainty. A similar comparison was not performed for the 𝑔𝑔 → 𝐻 process since the corresponding alternative samples are already used in the derivation of the QCD uncertainty model described above.

The modelling of the parton shower, underlying event, and hadronization is assessed separately for each Higgs boson production mode by comparing the acceptance times efficiency factors of simulated signal samples showered by P 8 with those of samples showered by H 7. The uncertainties estimated from the differences between these factors typically do not exceed 20%, and increase with jet multiplicity.

Uncertainties on the PDFs and the value of 𝛼 s are taken from Ref. [START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF] for the 𝑡𝐻𝑊, 𝑡𝐻𝑞𝑏 and 𝑏 b𝐻 processes. For other modes, the uncertainties are estimated using the PDF4LHC15 recommendations [START_REF] Butterworth | PDF4LHC recommendations for LHC Run II[END_REF]. Their effects are usually small compared to the those of the two other main sources of uncertainty mentioned at the start of this subsection.

In categories targeting the 𝑡𝑡𝐻 and 𝑡𝐻 processes, the predicted ggF, VBF and 𝑉 𝐻 yields are each assigned a conservative 100% uncertainty (correlated between categories), which is due to the theoretical uncertainty associated with the radiation of additional heavy-flavour jets in these Higgs boson production modes. This is supported by measurements using 𝐻→𝑍 𝑍 * →4ℓ [START_REF]Measurements of the Higgs boson inclusive and differential fiducial cross sections in the 4ℓ decay channel at √ 𝑠 = 13 TeV[END_REF], 𝑡 t𝑏 b [START_REF]Measurements of fiducial cross-sections for 𝑡 t production with one or two additional 𝑏-jets in 𝑝 𝑝 collisions at √ 𝑠 = 8 TeV using the ATLAS detector[END_REF], and 𝑉 𝑏 [START_REF]Measurements of the production cross-section for a 𝑍 boson in association with 𝑏-jets in proton-proton collisions at √ 𝑠 = 13 TeV with the ATLAS detector[END_REF][START_REF]Measurement of the cross-section for 𝑊 boson production in association with 𝑏-jets in 𝑝 𝑝 collisions at √ 𝑠 = 7 TeV with the ATLAS detector[END_REF] events. The impact of this uncertainty on the results is generally negligible compared to the statistical uncertainties, since the contributions from non-𝑡𝑡𝐻 processes are generally low.

A total uncertainty of 2.9% is assigned to the 𝐻 → 𝛾𝛾 decay branching ratio, based on calculations from the HDECAY [START_REF] Djouadi | HDECAY: a program for Higgs boson decays in the Standard Model and its supersymmetric extension[END_REF][START_REF] Spira | QCD Effects in Higgs Physics[END_REF][START_REF] Djouadi | Decays of supersymmetric particles: The Program SUSY-HIT (SUspect-SdecaY-Hdecay-InTerface)[END_REF] and P 4 [START_REF] Bredenstein | Radiative corrections to the semileptonic and hadronic Higgs-boson decays 𝐻 → 𝑊𝑊/𝑍 𝑍 → 4 fermions[END_REF][START_REF] Bredenstein | Precise predictions for the Higgs-boson decay 𝐻 → 𝑊𝑊/𝑍 𝑍 → 4 leptons[END_REF][START_REF] Bredenstein | Precision calculations for the Higgs decays 𝐻 → 𝑍 𝑍/𝑊𝑊 → 4 leptons[END_REF] programs, which also includes the uncertainty arising from its dependence on quark masses and 𝛼 s .

Theory uncertainties, such as missing higher-order QCD corrections and PDF-induced uncertainties, affect both the expected signal yields from each production process and the signal acceptance factors (𝜖 𝑖𝑡 in Eq. ( 1)) in each category. Uncertainties in signal acceptance factors are included in all the measurements presented in this paper. Signal yield uncertainties, including the uncertainty in the 𝐻 → 𝛾𝛾 branching ratio, are included only for the measurement of the Higgs boson signal strength and interpretations within the 𝜅-framework and SMEFT models, which rely on comparisons between the observed event yields and their SM predictions. Uncertainties on the parton shower, underlying event, and hadronization effects are included in all the measurements, without a separation into yield and acceptance components. In addition, cross-section measurements spanning multiple STXS regions require assumptions about the expected event yields in each region, as explained in Section 8.4, which introduces a weak dependence on the signal yield uncertainties.

Table 7 shows the expected experimental and theoretical systematic uncertainties of the cross-section measurements in the SM hypothesis, computed as described in Section 8.1.

Results

Results are presented in terms of several descriptions of Higgs boson production: the overall signal strength of Higgs boson production measured in the diphoton decay channel (Section 8.2), separate cross-sections

< ±1 < ±1 < ±1 ±1.3 < ±1 ±5.6 Jet/𝐸 miss T < ±1 ±6.8 < ±1 ±2.2 ±3.5 ±22 Flavour tagging < ±1 < ±1 < ±1 < ±1 ±1.5 ±3.4 Leptons < ±1 < ±1 < ±1 < ±1 < ±1 ±1.8 Higgs boson mass < ±1 < ±1 < ±1 < ±1 < ±1 < ±1
for the main Higgs boson production modes (Section 8.3), and cross-sections in a set of merged STXS regions defined for each production process (Section 8.4).

Statistical procedure

The results for each measurement reported in this paper are obtained by expressing the signal event yields in each analysis category in terms of the measurement parameters, and fitting the model to the data in all categories simultaneously. Both positive and negative values are allowed for all parameters, unless otherwise indicated. Best-fit values are reported along with uncertainties corresponding to 68% confidence level (CL) intervals obtained from a profile likelihood technique [START_REF] Cowan | Asymptotic formulae for likelihood-based tests of new physics[END_REF]. The endpoints of the intervals are defined by the condition -2 ln Λ(𝜇) = 1, where Λ(𝜇) is the ratio of the profile likelihood at a value 𝜇 of the parameters of interest to the profile likelihood at the best-fit point. Similarly, 95% CL intervals are defined by the condition -2 ln Λ(𝜇) = 3.84. In some cases, uncertainties are presented as a decomposition into separate components: the statistical component is obtained from a fit in which the nuisance parameters associated with systematic uncertainties are fixed to their best-fit values; the systematic component, corresponding to the combined effect of systematic uncertainties, is computed as the square root of the difference between the squares of the total uncertainty and the statistical uncertainty. Uncertainty components corresponding to smaller groups of nuisance parameters are obtained by iteratively fixing the parameters in each group, subtracting the square of the uncertainty obtained in this configuration from that obtained when the parameters are profiled, and taking the square root.

Expected results for the SM are obtained from a fit to an Asimov data set [START_REF] Cowan | Asymptotic formulae for likelihood-based tests of new physics[END_REF][START_REF]and The LHC Higgs Combination Group, Procedure for the LHC Higgs boson search combination in Summer 2011[END_REF] built from the likelihood model with signal and background components. The nuisance parameters of the likelihood model are determined in a fit to the observed data where the STXS parameters defining the signal normalization in each category are left free. The STXS parameters are set to their SM expectations to generate the Asimov data set. Compatibility with the Standard Model is assessed from the value of the profile likelihood ratio of the model in data under the SM hypothesis; a 𝑝-value for compatibility with the SM is computed assuming that the profile likelihood follows a 𝜒 2 distribution with a number of degrees of freedom equal to the number of parameters of interest [START_REF] Cowan | Asymptotic formulae for likelihood-based tests of new physics[END_REF]. In the case of cross-section measurements, uncertainties in the SM predictions are not accounted for in the 𝑝-value computation.

Overall Higgs boson signal strength

To quantify the overall size of the Higgs boson signal in the diphoton channel, the inclusive signal strength, 𝜇, defined as the ratio of the observed value of the product of the Higgs boson production cross-section and the 𝐻 → 𝛾𝛾 branching ratio (𝜎 × 𝐵 𝛾𝛾 ) in |𝑦 𝐻 | < 2.5 to that of its SM prediction, is measured by simultaneously fitting the 𝑚 𝛾𝛾 distributions of the 101 analysis categories. The signal strength 𝜇 is treated in the likelihood function as a single parameter of interest which scales the expected yields in all STXS regions and is found to be 𝜇 = 1.04 +0.10 -0.09 = 1.04 ± 0.06 (stat.) +0.06 -0.05 (theory syst.) +0.05 -0.04 (exp. syst.).

The overall 𝑚 𝛾𝛾 distribution of the selected diphoton sample is shown in Figure 7. The events are weighted by the ln(1 + 𝑆/𝐵) of their category, where 𝑆 and 𝐵 are the expected signal and background yields within the smallest 𝑚 𝛾𝛾 window containing 90% of the signal events, shown in Table 4. This choice of event weight is designed to enhance the contribution of events from categories with higher signal-to-background ratio in a way that approximately matches the impact of these events in the categorized analysis of the data.

Table 8 further breaks down the impact of systematic uncertainties on the signal-strength measurement.

The leading sources of experimental systematic uncertainty are the photon energy resolution uncertainty (2.8%) and photon efficiency uncertainty (2.6%), while the leading sources of theoretical uncertainty are the QCD scale uncertainty (3.8%) and 𝐻 → 𝛾𝛾 branching ratio uncertainty (3.0%).

Production cross-sections

The mechanism of Higgs boson production is probed by considering the ggF, VBF, 𝑊 𝐻, 𝑍 𝐻, 𝑡𝑡𝐻, and 𝑡𝐻 production processes separately. The measurement is reported in terms of the (𝜎 × 𝐵 𝛾𝛾 ) value in each case, with the cross-sections defined in |𝑦 𝐻 | < 2.5. As in the STXS region definition, the contribution from the 𝑏 b𝐻 process is included in the ggF component. Figure 8 shows the 𝑚 𝛾𝛾 distributions for analysis categories targeting different production modes separately. The same weighting procedure as in Figure 7 is used, except that the signal yield only includes the contributions from the targeted production process, while other signal production processes are included in the background yield.

The best-fit values of the production cross-sections and their uncertainties are summarized in Figure 9 and Table 9. A negative best-fit value is observed for the cross-section of the 𝑍 𝐻 process, which corresponds to a total observed event yield that is below the background expectation. The 𝑝-value for compatibility of the -Data

Figure 7: The inclusive diphoton invariant mass distribution of events from all analysis categories. The data events (dots) in each category are weighted by ln(1 + 𝑆/𝐵), where 𝑆 and 𝐵 are the expected signal and background yields in this category within the smallest 𝑚 𝛾𝛾 window containing 90% of the signal events. The expected signal is considered inclusively over all STXS regions. The fitted signal-plus-background pdfs from all categories are also weighted and summed, shown as the solid line. The blue dotted line represents the weighted sum of the fitted background functions from all categories. The error bars on the data points are computed following Ref. [START_REF] Bohm | Introduction to statistics and data analysis for physicists[END_REF].

cross-section measurement and the SM prediction is 55%. The correlations between these measurements are shown in Figure 10. Compared to Ref. [START_REF]Measurements of Higgs boson properties in the diphoton decay channel with 36 fb -1 of 𝑝 𝑝 collision data at √ 𝑠 = 13[END_REF], correlations between measurements are reduced, and in particular, the anti-correlation between the ggF and VBF measurements is now -13%, corresponding to a 30% reduction. This is driven by a reduction in the ggF contamination in categories targeting the VBF process, mainly resulting from the use of the D-optimality criterion in the categorization. An anti-correlation of -37% is observed between the 𝑊 𝐻 and 𝑍 𝐻 measurements, mainly due to contamination by 𝑞𝑞 → 𝐻ℓ𝜈 events in the categories targeting the 𝑝 𝑝 → 𝐻𝜈 ν process. This correlation is mitigated by the separation of the 𝑝 𝑝 → 𝐻ℓℓ and 𝑝 𝑝 → 𝐻𝜈 ν processes that is introduced in the analysis categorization.

Similarly, 𝑡𝑡𝐻 contamination in the categories targeting 𝑡𝐻 lead to an anti-correlation of -44% between these two processes.

The largest theoretical systematic uncertainty in these measurements arises from the modelling of the parton showering and underlying event, and its impact on the measured cross-sections ranges from 38% for the 𝑡𝐻 process to 14% for the VBF process and to 3%-4% for the 𝑔𝑔 → 𝐻 and 𝑊 𝐻 processes. For the 𝑔𝑔 → 𝐻 process, the leading experimental systematic uncertainty is the photon energy resolution uncertainty (3%). For the VBF and 𝑡𝑡𝐻 processes, the leading experimental uncertainty is related to the properties of jets and missing transverse momentum, reaching 5.4% for VBF. For other processes, the leading experimental systematic uncertainty is the background modelling uncertainty, ranging from 3.7% for 𝑊 𝐻 to 24% for 𝑡𝐻.

An upper limit on the rate of 𝑡𝐻 production is obtained by treating the normalization of other Higgs boson production processes as nuisance parameters. Using the CL s method [START_REF] Read | Presentation of search results: the 𝐶 𝐿 𝑆 technique[END_REF], this excludes a 𝑡𝐻 production The data (black dots) are weighted by ln(1 + 𝑆/𝐵) where 𝑆 and 𝐵 are respectively the expected signal and background yields in the smallest 𝑚 𝛾𝛾 window containing 90% of the signal events. In this calculation, only Higgs boson events from the targeted production processes are considered as signal events. Higgs boson events from other processes as well as the continuum background events are considered as background. The fitted signal-plus-background pdfs from the relevant categories are summed, and represented by a solid line. The blue dotted line represents the weighted sum of the fitted continuum background pdfs, while the dashed line combines the contributions of continuum background and other Higgs boson events. The error bars on the data points are computed following Ref. [START_REF] Bohm | Introduction to statistics and data analysis for physicists[END_REF]. The weighted combination of categories with low event counts leads in some cases to data errors that are highly asymmetric and change by large amounts from point to point. rate of 10 times its SM prediction or greater at 95% CL while the expected 95% CL limit is 6.8 times the SM 𝑡𝐻 production cross-section.

Cross-sections in STXS regions

A measurement of the cross-sections defined in the STXS scheme is performed using the analysis regions described in Section 5. In order to avoid large uncertainties and large absolute correlations between the measurements, a set of measurement regions is obtained by merging some of the analysis regions as follows:

• For the 𝑔𝑔 → 𝐻 process, within the phase space of ≥ 2-jets, 𝑚 𝑗 𝑗 < 350 GeV, the two regions with 𝑝 𝐻 T < 60 GeV and 60 ≤ 𝑝 𝐻 T < 120 GeV are merged into one region corresponding to 𝑝 𝐻 T < 120 GeV. Within ≥ 2-jets, 𝑝 𝐻 T < 200 GeV, the three bins defined in the 𝑚 𝑗 𝑗 variable are merged into a single 𝑚 𝑗 𝑗 > 350 GeV region. Finally, the 𝑝 𝐻 T > 650 GeV bin is merged with the neighbouring 450 ≤ 𝑝 𝐻 T < 650 GeV bin to form a single region corresponding to 𝑝 𝐻 T ≥ 450 GeV. • For the 𝑞𝑞 → 𝐻𝑞𝑞 process, the 0-jet and 1-jet regions, as well as the regions corresponding to 𝑚 𝑗 𝑗 < 60 GeV and 120 < 𝑚 𝑗 𝑗 < 350 GeV, are combined into a new region, referred to as 𝑞𝑞 → 𝐻𝑞𝑞 , ≤ 1-jet and 𝑉 𝐻-veto. The regions corresponding to ≥ 2-jets, 𝑝 𝐻 T ≥ 200 GeV, 350 < 𝑚 𝑗 𝑗 < 1000 GeV, are merged into a single region.

( )

Figure 9: Cross-sections times 𝐻 → 𝛾𝛾 branching ratio for ggF+ 𝑏 b𝐻, VBF, 𝑉 𝐻, 𝑡𝑡𝐻, and 𝑡𝐻 production, normalized to their SM predictions. The values are obtained from a simultaneous fit to all categories. The error bars and shaded areas show respectively the total and systematic uncertainties in the measurements. The grey bands show the theory uncertainties in the predictions, including uncertainties due to missing higher-order terms in the perturbative QCD calculations, the PDFs and the value of 𝛼 s , as well as the 𝐻 → 𝛾𝛾 branching ratio uncertainty.

• For both the 𝑞 q → 𝑊 𝐻 and 𝑝 𝑝 → 𝑍 𝐻 processes, only the two regions 𝑝 𝑉 T < 150 GeV and 𝑝 𝑉 T ≥ 150 GeV are retained, removing the intermediate splits at 𝑝 𝑉 T = 75 GeV and 𝑝 𝑉 T = 250 GeV. For 𝑝 𝑝 → 𝑍 𝐻 processes, no distinction is made between regions with charged leptons and regions with neutrinos.

• The 𝑡𝐻𝑞𝑏 and 𝑡𝐻𝑊 regions are merged into a single 𝑡𝐻 region. This scheme is based on the expected analysis sensitivity under the SM hypothesis, independently of the observed data, and is illustrated in Figure 11. The merging reduces the number of regions for which a measurement is reported to 28 in the scheme described above. The 101 categories in which the measurement is performed, described in Section 5.2, remain unchanged. The acceptance factors for merged STXS regions are computed as weighted averages of those for the original STXS regions, with the weights corresponding to the expected cross-sections in the SM. The acceptance uncertainty of the merged STXS region is then calculated from the uncertainties in the acceptance factors and expected cross-sections of the original STXS regions.

Results are shown in Table 10 and Figure 12. The correlation matrix of the measurements is shown in Figure 13. The correlation between most STXS region measurements is small, and the largest correlation is -51%, observed for the measurements of STXS regions 𝑔𝑔 → 𝐻, ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 350 GeV, 𝑝 𝐻 T < 200 GeV and 𝑞𝑞 → 𝐻𝑞𝑞 , ≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T < 200 GeV. The Higgs boson production processes in these STXS regions have similar event topologies and are intrinsically difficult to separate. The relative uncertainties in the measurements range from 20% to more than 100%. Smaller uncertainties are associated with the 0-jet and 1-jet regions of 𝑔𝑔 → 𝐻, as well as the 200 ≤ 𝑝 𝐻 T < 300 GeV region of 𝑔𝑔 → 𝐻 and the 𝑚 𝑗 𝑗 ≥ 700 GeV region of 𝑞𝑞 → 𝐻𝑞𝑞 . Larger uncertainties occur especially in regions of high 𝑝 𝐻 T and 𝑝 𝑉 T , as well as the low-𝑚 𝑗 𝑗 regions of 𝑞𝑞 → 𝐻𝑞𝑞 . The systematic component of the uncertainties is everywhere smaller than the statistical component, but reaches similar values for the 0-jet regions of 𝑔𝑔 → 𝐻. No significant deviations from the SM expectation are observed and the 𝑝-value for compatibility of the measurements and the SM predictions is 93%. Results in a finer set of 33 STXS measurements regions are also presented in Table 13 of Appendix A.

Table 10: Best-fit values and uncertainties for the production cross-section times 𝐻 → 𝛾𝛾 branching ratio (𝜎 𝑖 × 𝐵 𝛾𝛾 ) in each STXS region. The values for the 𝑔𝑔 → 𝐻 process also include the contributions from 𝑏 b𝐻 production. The total uncertainties are decomposed into statistical (Stat.) and systematic (Syst.) uncertainties. The uncertainties for the 𝑝 𝑝 → 𝐻ℓℓ/𝜈 ν, 𝑝 𝑉 T < 150 GeV region are truncated at the value for which the model pdf becomes negative. SM predictions [START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF] are also shown for each quantity with their total uncertainties.

STXS region (𝜎

Interpretation of the results in the 𝜿-framework

Event rates for processes involving Higgs bosons can be expressed in terms of modifiers applied to the SM Higgs boson couplings, based on the leading-order contributions to each process [START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF]. These coupling modifiers affect Higgs boson production cross-sections and decay partial widths and therefore provide a consistent framework for Higgs boson coupling measurements in both production and decay.

Multiplicative modifiers are introduced for Higgs boson couplings to the 𝑊 and 𝑍 bosons, denoted respectively by 𝜅 𝑊 and 𝜅 𝑍 , and for couplings to the charm, bottom and top quarks and muons and 𝜏-leptons, respectively 𝜅 𝑐 , 𝜅 𝑏 , 𝜅 𝑡 , 𝜅 𝜇 and 𝜅 𝜏 . Couplings to other SM particles are assumed to be equal to their SM predictions. Two parameterizations are considered for the 𝑔𝑔 → 𝐻 and 𝐻 → 𝛾𝛾 processes:

• A resolved parameterization in which they are assumed to proceed through the same loop amplitudes as in the SM at leading order. Their rates are expressed using the modifiers defined above, involving the particles appearing in the loop.

• An effective parameterization that makes no assumption about the internal structure of the interactions.

Event rates are expressed using modifiers to the effective couplings of the Higgs boson to the gluon and the photon, respectively denoted by 𝜅 𝑔 and 𝜅 𝛾 .

The 𝑔𝑔 → 𝑍 𝐻 loop process is always described in the resolved parameterization. The SM corresponds to the case 𝜅 𝑊 = 𝜅 𝑍 = 𝜅 𝑡 = 𝜅 𝑏 = 𝜅 𝑐 = 𝜅 𝜏 = 𝜅 𝜇 = 1, and in addition 𝜅 𝑔 = 𝜅 𝛾 = 1 when effective parameterizations are used. The 𝜅 𝑍 modifier is assumed to be positive, without loss of generality, since all predictions are invariant under a simultaneous flip of the sign of each 𝜅 modifier. Sensitivity to the sign of the other modifiers is obtained through interference between processes involving different combinations of modifiers. These include, in particular, the 𝐻 → 𝛾𝛾 and 𝑔𝑔 → 𝐻 loops and the 𝑔𝑔 → 𝑍 𝐻 and 𝑡𝐻 processes.

The Higgs boson production cross-section in STXS region 𝑖 followed by a 𝐻→ 𝛾𝛾 decay is written in the narrow-width approximation as

𝜎 𝑖 • 𝐵 𝛾𝛾 = 𝜎 𝑖 (𝜿) • Γ 𝛾𝛾 (𝜿) Γ 𝐻 (𝜿)
where 𝜎 𝑖 is the production cross-section in region 𝑖, 𝐵 𝛾𝛾 and Γ 𝛾𝛾 are respectively the Higgs boson branching ratio and partial width into the 𝛾𝛾 final state, and Γ 𝐻 is the total width of the Higgs boson. The parameterizations 𝜎 𝑖 (𝜿), Γ 𝛾𝛾 (𝜿) and Γ 𝐻 (𝜿) are shown in Table 14 in Appendix B. They are similar to the ones used in Ref. [START_REF]Combined measurements of Higgs boson production and decay using up to 80 fb -1 of proton-proton collision data at √ 𝑠 = 13 TeV collected with the ATLAS experiment[END_REF], except for the parameterization of the 𝑡𝐻𝑊 and 𝑡𝐻𝑞𝑏 processes. These have been updated to reflect the fact that the acceptance factors in each analysis category depend on the 𝜅 𝑡 and 𝜅 𝑊 modifiers because of changes in the process kinematics caused by interference effects between the different parton-level processes contributing to 𝑡𝐻𝑊 and 𝑡𝐻𝑞𝑏. Separate parameterizations are therefore used in each analysis category for 𝑡𝐻𝑊 and 𝑡𝐻𝑞𝑏. The Higgs boson total width Γ 𝐻 is expressed as a function of the 𝜅 modifiers, assuming no contributions from Higgs boson decays other than the ones present in the SM. The SM predictions of 𝐵 𝛾𝛾 and the 𝜎 𝑖 are taken from Ref [START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF]. These include the highest-order available computations in both the QCD and electroweak couplings.

Two specific models of coupling modifications are considered in this section, and two additional models are described in Appendix B. The first model focuses on the 𝜅 𝑡 coupling modifier. Two configurations are used for the 𝑔𝑔 → 𝐻 and 𝐻 → 𝛾𝛾 loop processes: in the first case, both are described using their resolved parameterization as a function of 𝜅 𝑡 ; in the second case, both are described using the effective couplings 𝜅 𝑔 and 𝜅 𝛾 . All other 𝜅 modifiers are fixed to their SM values, in particular the effect of 𝜅 𝑊 in the resolved parameterization of the 𝐻 → 𝛾𝛾 loop. These two models also allow the sign of 𝜅 𝑡 to be probed, with sensitivity coming from interference effects in certain amplitudes. These occur in the 𝑡𝐻 and 𝑔𝑔 → 𝑍 𝐻 processes, as well as in the 𝐻 → 𝛾𝛾 process when its parameterization is resolved in terms of 𝜅 𝑡 and other coupling modifiers.

The negative log-likelihood scans for both configurations are shown in Figure 14. In both cases, good The 𝐻 → 𝛾𝛾 and 𝑔𝑔 → 𝐻 loops are either parameterized as a function of 𝜅 𝑡 (blue) or fixed to their SM expectation (orange). In the latter case, sensitivity to the sign of 𝜅 𝑡 is provided by the 𝑡𝐻 process, and to a lesser degree by the 𝑔𝑔 → 𝑍 𝐻 process. The solid curves correspond to observed data, and the dotted curves to an Asimov data set generated under the SM hypothesis. agreement with the SM expectation of 𝜅 𝑡 = 1 is seen. When the 𝐻 → 𝛾𝛾 and 𝑔𝑔 → 𝐻 loops are resolved, negative values of 𝜅 𝑡 are excluded with a significance of 6.7𝜎 or above. When effective loop couplings are used, an exclusion of 2.2𝜎 or above is observed through the sensitivity provided by the 𝑡𝐻 process, with a smaller contribution from the 𝑔𝑔 → 𝑍 𝐻 process. Values of 𝜅 𝑡 outside of the range 0.87 < 𝜅 𝑡 < 1.20 are excluded at 95% CL in the first case (0.85 < 𝜅 𝑡 < 1.19 expected under the SM hypothesis), as are values outside 0.65 < 𝜅 𝑡 < 1.25 in the second case (0.71 < 𝜅 𝑡 < 1.29 expected).

In the second model, the 𝑔𝑔 → 𝐻 and 𝐻→ 𝛾𝛾 loop processes are described using the effective modifiers 𝜅 𝑔 and 𝜅 𝛾 . Both modifiers are assumed to be positive, since the measurement provides no sensitivity to their signs. Other modifiers are fixed to their SM values. The measurement in the plane of (𝜅 𝑔 , 𝜅 𝛾 ) is shown in Figure 15. The best-fit values are 𝜅 𝑔 = 1.01 + 0.11 -0.09 𝜅 𝛾 = 1.02 + 0.08 -0.07 .

A linear correlation coefficient of -79% between the parameters is observed.

Interpretation framework

The Standard Model effective field theory (SMEFT) framework provides a model-independent setting to describe deviations from SM predictions. New effective interactions involving Standard Model particles are introduced in the Lagrangian to describe the effect of physics beyond the SM occurring above a high scale Λ. These interactions are considered order by order in the mass dimension 𝑑 of the relevant operators, with leading-order effects occurring at 𝑑 = 6 when assuming that the lepton number 𝐿 and baryon number 𝐵 are conserved.

The effective Lagrangian up to dimension 6 is written as

L = L SM + ∑ 𝑘 𝑐 𝑘 Λ 2 𝑂 𝑘
where the sum runs over the dimension-6 operators 𝑂 𝑘 describing effective interactions in the SMEFT.

The 𝑐 𝑘 are the corresponding Wilson coefficients, which are considered as the measurement parameters of the model. Subleading contributions with dimension 8 and above are neglected, and only operators with even CP quantum numbers that conserve 𝐵 and 𝐿 are considered. The selected operators are expressed in the Warsaw basis [START_REF] Buchmüller | Effective lagrangian analysis of new interactions and flavour conservation[END_REF][START_REF] Grzadkowski | Dimension-six terms in the Standard Model Lagrangian[END_REF]. The 𝑈 (3) 5 -symmetric model of fermion flavour [START_REF] Brivio | SMEFTsim 3.0 -a practical guide[END_REF] is considered, assuming separate global flavour symmetries for each fermion type over the three fermion generations. In the cases where Wilson coefficients can have complex values, only their real parts are considered. The SM corresponds to all 𝑐 𝑘 set to 0. The 𝑐 𝑘 are defined for a scale Λ = 1 TeV.

The 𝑐 𝑘 are determined through an interpretation of the STXS results presented in Appendix A. This is achieved by expressing as functions of the 𝑐 𝑘 the signal-strength parameters

𝜇 𝛾𝛾 𝑖 = 𝜎 𝑖 • 𝐵 𝛾𝛾 𝜎 SM 𝑖 • 𝐵 SM 𝛾𝛾 ,
where 𝑖 runs over the 33 STXS regions listed in Table 13, and 𝜎 SM 𝑖 and 𝐵 SM 𝛾𝛾 are the SM predictions for the production cross-section in STXS region 𝑖 and the 𝐻 → 𝛾𝛾 branching ratio, respectively. The 33 STXS regions used here correspond to a finer binning than the 28 regions for which results were reported in Section 8.4, and provide better granularity especially at high 𝑚 𝑗 𝑗 values in the 𝑔𝑔 → 𝐻 and 𝑞𝑞 → 𝐻𝑞𝑞 processes.

SMEFT effects are modelled as single insertions of the operators 𝑂 𝑘 in each Higgs boson production and decay amplitude. These amplitudes are therefore linear in the 𝑐 𝑘 , so that the production cross-sections and decay widths are at most quadratic functions of the 𝑐 𝑘 . The signal strengths are thus written as

𝜇 𝛾𝛾 𝑖 = 1 + ∑ 𝑘 𝐴 𝑖→𝐻 𝑘 𝑐 𝑘 + ∑ 𝑘𝑙 𝐵 𝑖→𝐻 𝑘𝑙 𝑐 𝑘 𝑐 𝑙 1 + 𝑘 𝐴 𝐻 → 𝛾𝛾 𝑘 𝑐 𝑘 + 𝑘𝑙 𝐵 𝐻 → 𝛾𝛾 𝑘𝑙 𝑐 𝑘 𝑐 𝑙 1 + 𝑘 𝐴 Γ 𝑘 𝑐 𝑘 + 𝑘𝑙 𝐵 Γ 𝑘𝑙 𝑐 𝑘 𝑐 𝑙 (2)
where

𝐴 𝑖→𝐻 𝑘 , 𝐴 𝐻 → 𝛾𝛾 𝑘
and 𝐴 Γ 𝑘 are respectively the coefficients describing the linear 𝑐 𝑘 -dependence of the production cross-section 𝜎 𝑖 , the partial decay width Γ 𝛾𝛾 and the total width Γ 𝐻 . Similarly, the 𝐵 𝑖→𝐻 𝑘𝑙 , 𝐵 𝐻 → 𝛾𝛾 𝑘𝑙 and 𝐵 Γ 𝑘𝑙 coefficients describe the quadratic dependence of the same quantities on the 𝑐 𝑘 . Two SMEFT parameterizations are considered in the following: a linear parameterization including only the effect of the 𝐴 coefficients and a linear+quadratic parameterization including both the 𝐴 and 𝐵 terms. In the linear case, Eq. ( 2) is linearized to first order in the 𝑐 𝑘 so that

𝜇 𝛾𝛾 𝑖 = 1 + ∑ 𝑘 𝐴 𝑖→𝐻 𝑘 + 𝐴 𝐻 → 𝛾𝛾 𝑘 -𝐴 Γ 𝑘 𝑐 𝑘 . (3)
𝑞𝑞

( q𝑟 𝛾 𝜇 𝜏 𝐼 𝑞 𝑟 ) ( q𝑠 𝛾 𝜇 𝜏 𝐼 𝑞 𝑠 ) 𝑐 𝑊 𝜖 𝐼 𝐽 𝐾 𝑊 𝐼 𝜈 𝜇 𝑊 𝐽 𝜌 𝜈 𝑊 𝐾 𝜇 𝜌 𝑐 (3) 𝑞𝑞 ( q𝑟 𝛾 𝜇 𝜏 𝐼 𝑞 𝑠 ) ( q𝑠 𝛾 𝜇 𝜏 𝐼 𝑞 𝑟 ) 𝑐 𝐻 (𝐻 † 𝐻) 3 𝑐 (1) 𝑞𝑞 ( q𝑟 𝛾 𝜇 𝑞 𝑟 ) ( q𝑠 𝛾 𝜇 𝑞 𝑠 ) 𝑐 𝐻 (𝐻 † 𝐻) (𝐻 † 𝐻) 𝑐 (1)
𝑞𝑞

( q𝑟 𝛾 𝜇 𝑞 𝑠 ) ( q𝑠 𝛾 𝜇 𝑞 𝑟 ) 𝑐 𝐻𝐷 𝐻 † 𝐷 𝜇 𝐻 * 𝐻 † 𝐷 𝜇 𝐻 𝑐 (3) 𝑙𝑞 ( l𝑟 𝛾 𝜇 𝜏 𝐼 𝑙 𝑟 ) ( q𝑠 𝛾 𝜇 𝜏 𝐼 𝑞 𝑠 ) 𝑐 𝐻𝐺 𝐻 † 𝐻 𝐺 𝐴 𝜇𝜈 𝐺 𝐴𝜇𝜈 𝑐 (1) 𝑙𝑞 ( l𝑟 𝛾 𝜇 𝑙 𝑟 ) ( q𝑠 𝛾 𝜇 𝑞 𝑠 ) 𝑐 𝐻𝑊 𝐻 † 𝐻 𝑊 𝐼 𝜇𝜈 𝑊 𝐼 𝜇𝜈 𝑐 𝑒𝑒 ( ē𝑟 𝛾 𝜇 𝑒 𝑟 ) ( ē𝑠 𝛾 𝜇 𝑒 𝑠 ) 𝑐 𝐻𝐵 𝐻 † 𝐻 𝐵 𝜇𝜈 𝐵 𝜇𝜈 𝑐 𝑒𝑢 ( ē𝑟 𝛾 𝜇 𝑒 𝑟 ) ( ū𝑠 𝛾 𝜇 𝑢 𝑠 ) 𝑐 𝐻 𝑊𝐵 𝐻 † 𝜏 𝐼 𝐻 𝑊 𝐼 𝜇𝜈 𝐵 𝜇𝜈 𝑐 𝑒𝑑 ( ē𝑟 𝛾 𝜇 𝑒 𝑟 ) ( d𝑠 𝛾 𝜇 𝑑 𝑠 ) 𝑐 𝑒𝐻 (𝐻 † 𝐻) ( l 𝑝 [𝑌 † 𝑒 ] 𝑝𝑞 𝑒 𝑞 𝐻) 𝑐 𝑢𝑢 ( ū𝑟 𝛾 𝜇 𝑢 𝑟 ) ( ū𝑠 𝛾 𝜇 𝑢 𝑠 ) 𝑐 𝑢𝐻 (𝐻 † 𝐻) ( q 𝑝 [𝑌 † 𝑢 ] 𝑝𝑞 𝑢 𝑞 𝐻) 𝑐 𝑢𝑢 ( ū𝑟 𝛾 𝜇 𝑢 𝑠 ) ( ū𝑠 𝛾 𝜇 𝑢 𝑟 ) 𝑐 𝑑𝐻 (𝐻 † 𝐻) ( q 𝑝 [𝑌 † 𝑑 ] 𝑝𝑞 𝑑 𝑞 𝐻) 𝑐 𝑑𝑑 ( d𝑟 𝛾 𝜇 𝑑 𝑟 ) ( d𝑠 𝛾 𝜇 𝑑 𝑠 ) 𝑐 𝑒𝑊 ( l 𝑝 𝜎 𝜇𝜈 [𝑌 † 𝑒 ] 𝑝𝑞 𝑒 𝑞 )𝜏 𝐼 𝐻 𝑊 𝐼 𝜇𝜈 𝑐 𝑑𝑑 ( d𝑟 𝛾 𝜇 𝑑 𝑠 ) ( d𝑠 𝛾 𝜇 𝑑 𝑟 ) 𝑐 𝑒𝐵 ( l 𝑝 𝜎 𝜇𝜈 [𝑌 † 𝑒 ] 𝑝𝑞 𝑒 𝑞 )𝐻 𝐵 𝜇𝜈 𝑐 (1)
𝑢𝑑

( ū𝑟 𝛾 𝜇 𝑢 𝑟 ) ( d𝑠 𝛾 𝜇 𝑑 𝑠 ) 𝑐 𝑢𝐺 ( q 𝑝 𝜎 𝜇𝜈 𝑇 𝐴 [𝑌 † 𝑢 ] 𝑝𝑞 𝑢 𝑞 ) 𝐻 𝐺 𝐴 𝜇𝜈 𝑐 (8)
𝑢𝑑

( ū𝑟 𝛾 𝜇 𝑇 𝐴 𝑢 𝑟 ) ( d𝑠 𝛾 𝜇 𝑇 𝐴 𝑑 𝑠 ) 𝑐 𝑢𝑊 ( q 𝑝 𝜎 𝜇𝜈 [𝑌 † 𝑢 ] 𝑝𝑞 𝑢 𝑞 )𝜏 𝐼 𝐻 𝑊 𝐼 𝜇𝜈 𝑐 𝑙𝑒 ( l𝑟 𝛾 𝜇 𝑙 𝑟 ) ( ē𝑠 𝛾 𝜇 𝑒 𝑠 ) 𝑐 𝑢𝐵 ( q 𝑝 𝜎 𝜇𝜈 [𝑌 † 𝑢 ] 𝑝𝑞 𝑢 𝑞 ) 𝐻 𝐵 𝜇𝜈 𝑐 𝑙𝑢 ( l𝑟 𝛾 𝜇 𝑙 𝑟 ) ( ū𝑠 𝛾 𝜇 𝑢 𝑠 ) 𝑐 𝑑𝐺 ( q 𝑝 𝜎 𝜇𝜈 𝑇 𝐴 [𝑌 † 𝑑 ] 𝑝𝑞 𝑑 𝑞 )𝐻 𝐺 𝐴 𝜇𝜈 𝑐 𝑙𝑑 ( l𝑟 𝛾 𝜇 𝑙 𝑟 ) ( d𝑠 𝛾 𝜇 𝑑 𝑠 ) 𝑐 𝑑𝑊 ( q 𝑝 𝜎 𝜇𝜈 [𝑌 † 𝑑 ] 𝑝𝑞 𝑑 𝑞 )𝜏 𝐼 𝐻 𝑊 𝐼 𝜇𝜈 𝑐 𝑞𝑒 ( q𝑟 𝛾 𝜇 𝑞 𝑟 ) ( ē𝑠 𝛾 𝜇 𝑒 𝑠 ) 𝑐 𝑑𝐵 ( q 𝑝 𝜎 𝜇𝜈 [𝑌 † 𝑑 ] 𝑝𝑞 𝑑 𝑞 )𝐻 𝐵 𝜇𝜈 𝑐 (1)
𝑞𝑢

( q𝑟 𝛾 𝜇 𝑞 𝑟 ) ( ū𝑠 𝛾 𝜇 𝑢 𝑠 ) 𝑐 (3)
𝐻 𝑙

(𝐻 † 𝑖 ← → 𝐷 𝐼 𝜇 𝐻) ( l𝑟 𝜏 𝐼 𝛾 𝜇 𝑙 𝑟 ) 𝑐 (8) 𝑞𝑢 ( q𝑟 𝛾 𝜇 𝑇 𝐴 𝑞 𝑟 ) ( ū𝑠 𝛾 𝜇 𝑇 𝐴 𝑢 𝑠 ) 𝑐 (1)
𝐻 𝑙

(𝐻 † 𝑖 ← → 𝐷 𝜇 𝐻) ( l𝑟 𝛾 𝜇 𝑙 𝑟 ) 𝑐 (1)
𝑞𝑑

( q𝑟 𝛾 𝜇 𝑞 𝑟 ) ( d𝑠 𝛾 𝜇 𝑑 𝑠 ) 𝑐 𝐻 𝑒 (𝐻 † 𝑖 ← → 𝐷 𝜇 𝐻) ( ē𝑟 𝛾 𝜇 𝑒 𝑟 ) 𝑐 (8) 𝑞𝑑 ( q𝑟 𝛾 𝜇 𝑇 𝐴 𝑞 𝑟 ) ( d𝑠 𝛾 𝜇 𝑇 𝐴 𝑑 𝑠 ) 𝑐 (3)
𝐻 𝑞

(𝐻 † 𝑖 ← → 𝐷 𝐼 𝜇 𝐻) ( q𝑟 𝜏 𝐼 𝛾 𝜇 𝑞 𝑟 ) 𝑐 𝑙𝑒𝑑𝑞 ( l 𝑗 𝑝 [𝑌 † 𝑙 ] 𝑝𝑞 𝑒 𝑞 ) ( d𝑟 [𝑌 𝑑 ] 𝑟 𝑠 𝑞 𝑗 𝑠 ) 𝑐 (1) 𝐻 𝑞 (𝐻 † 𝑖 ← → 𝐷 𝜇 𝐻) ( q𝑟 𝛾 𝜇 𝑞 𝑟 ) 𝑐 (1) 𝑞𝑢𝑞𝑑 ( q 𝑗 𝑝 [𝑌 † 𝑢 ] 𝑝𝑞 𝑢 𝑞 )𝜖 𝑗 𝑘 ( q𝑘 𝑟 [𝑌 † 𝑑 ] 𝑟 𝑠 𝑑 𝑠 ) 𝑐 𝐻 𝑢 (𝐻 † 𝑖 ← → 𝐷 𝜇 𝐻) ( ū𝑟 𝛾 𝜇 𝑢 𝑟 ) 𝑐 (1) 𝑞𝑢𝑞𝑑 ( q 𝑗 𝑝 [𝑌 † 𝑑 ] 𝑝𝑠 𝑢 𝑞 )𝜖 𝑗 𝑘 ( q𝑘 𝑟 [𝑌 † 𝑢 ] 𝑟 𝑞 𝑑 𝑠 ) 𝑐 𝐻 𝑑 (𝐻 † 𝑖 ← → 𝐷 𝜇 𝐻) ( d𝑟 𝛾 𝜇 𝑑 𝑟 ) 𝑐 (8) 𝑞𝑢𝑞𝑑 ( q 𝑗 𝑝 𝑇 𝐴 [𝑌 † 𝑢 ] 𝑝𝑞 𝑢 𝑞 )𝜖 𝑗 𝑘 ( q𝑘 𝑟 𝑇 𝐴 [𝑌 † 𝑑 ] 𝑟 𝑠 𝑑 𝑠 ) 𝑐 𝐻 𝑢𝑑 (𝐻 † 𝑖𝐷 𝜇 𝐻) ( ū 𝑝 𝛾 𝜇 [𝑌 𝑢 𝑌 † 𝑑 ] 𝑝𝑞 𝑑 𝑞 ) 𝑐 (8) 𝑞𝑢𝑞𝑑 ( q 𝑗 𝑝 𝑇 𝐴 [𝑌 † 𝑑 ] 𝑝𝑠 𝑢 𝑞 )𝜖 𝑗 𝑘 ( q𝑘 𝑟 𝑇 𝐴 [𝑌 † 𝑢 ] 𝑟 𝑞 𝑑 𝑠 ) 𝑐 𝑙𝑙 ( l𝑟 𝛾 𝜇 𝑙 𝑟 ) ( l𝑠 𝛾 𝜇 𝑙 𝑠 ) 𝑐 (1) 𝑙𝑒𝑞𝑢 ( l 𝑗 𝑝 [𝑌 † 𝑒 ] 𝑝𝑞 𝑒 𝑞 )𝜖 𝑗 𝑘 ( q𝑘 𝑟 [𝑌 † 𝑢 ] 𝑟 𝑠 𝑢 𝑠 ) 𝑐 𝑙𝑙 ( l𝑟 𝛾 𝜇 𝑙 𝑠 ) ( l𝑠 𝛾 𝜇 𝑙 𝑟 ) 𝑐 (3) 𝑙𝑒𝑞𝑢 ( l 𝑗 𝑝 𝜎 𝜇𝜈 [𝑌 † 𝑒 ] 𝑝𝑠 𝑒 𝑞 )𝜖 𝑗 𝑘 ( q𝑘 𝑟 𝜎 𝜇𝜈 [𝑌 † 𝑢 ] 𝑟 𝑞 𝑢 𝑠 ) 0.5 1 1.5 2 2.5 3 3.5 4
Rel. uncertainty c (1) Hq c (3) H l c H u c Hd (8) qu c (8) qd 0.1 × c (8) ud 0.

0.4 - 0.2 - 0 0.2 0.4 )/SM i (c STXS σ ∆ 0.4 - 0.2 - 0 0.2 0.4 )/SM i (c STXS σ ∆ 0.4 - 0.2 - 0 0.2 0.4 )/SM i (c STXS σ ∆ ATLAS Simulation = 1 TeV Λ = 125.09 GeV, H , m γ γ → H -1 =13 TeV 139fb s < 1 0 H T H 0 J p → g g 1 0 ≥ H T H 0 J p → g g < 6 0 H T H 1 J p → g g < 1 2 0 H T p ≤ H 1 J 6 0 → g g < 2 0 0 H T p ≤ H 1 J 1 2 0 → g g < 6 0 H T < 3 5 0 , p JJ 2 J m ≥ H → g g < 1 2 0 H T p ≤ < 3 5 0 , 6 0 JJ 2 J m ≥ H → g g < 2 0 0 H T < p ≤ < 3 5 0 , 1 2 0 JJ 2 J m ≥ H → g g < 2 0 0 H T < 7 0 0 , p JJ m ≤ 2 J 3 5 0 ≥ H → g g < 2 0 0 H T < 1 0 0 0 , p JJ m ≤ 2 J 7 0 0 ≥ H → g g < 2 0 0 H T 1 0 0 0 , p ≥ JJ 2 J m ≥ H → g g < 3 0 0 H T p ≤ H 2 0 0 → g g < 4 5 0 H T p ≤ H 3 0 0 → g g 4 5 0 ≥ H T H p → g g 1 J ≤ H q q → q q 2 J V H h a d ≥ H q q → q q 2 J V H v e to ≥ H q q → q q < 2 0 0 H T < 7 0 0 , p JJ m ≤ 2 J 3 5 0 ≥ H q q → q q < 2 0 0 H T < 1 0 0 0 , p JJ m ≤ 2 J 7 0 0 ≥ H q q → q q < 2 0 0 H T > 1 0 0 0 , p JJ 2 J m ≥ H q q → q q 2 0 0 ≥ H T < 7 0 0 , p JJ m ≤ 2 J 3 5 0 ≥ H q q → q q 2 0 0 ≥ H T < 1 0 0 0 , p JJ m ≤ 2 J 7 0 0 ≥ H q q → q q 2 0 0 ≥ H T > 1 0 0 0 , p JJ 2 J m ≥ H q q → q q < 1 5 0 V t p ν H l → q q 1 5 0 ≥ V t p ν H l → q q < 1 5 0 V t H ll p 1 5 0 ≥ V t H ll p < 6 0 H T tt H p < 1 2 0 H T p ≤ tt H 6 0 < 2 0 0 H T p ≤ tt H 1 2 0 < 3 0 0 H T p ≤ tt H 2 0 0 3 0 0 ≥ H T tt H p tH c HG = 0.005 c u G = 0.1 cuH = 1. 0 c G = 1. 0 c (3 )′ q q = 0.2 40 - 20 - 0 20 40 cHW = 0. 5 cHWB = 1.0 cHB = 1.0 c (3 ) Hq = 0.1 cHu = 0. 2 c (1 ) Hq = 0.2 ) γ γ → B ( H
0.1 × c H □ c H D c (3) qq 0.1 × c (3)′ qq c (1)′ qq c ′ ll c ′ uu c

Simultaneous measurement of SMEFT parameters

In this section, multiple SMEFT parameters are left free to vary simultaneously. The information present in the STXS measurement does not, however, allow constraints to be placed simultaneously on all the SMEFT parameters listed in Table 11. In addition, both the constrained and the unconstrained degrees of freedom generally consist in combinations of parameters, since predictions in each STXS region are affected by multiple SMEFT operators.

Unconstrained directions can be removed from consideration without loss of generality, since the corresponding measurement information is in any case negligible. This allows the number of measurement parameters to be reduced without incurring model-dependence, and also avoids the probing of regions of parameter space beyond the bounds of SMEFT validity that occurs when confidence intervals along unconstrained directions extend beyond these bounds. Finally, this also avoids numerical issues in maximum-likelihood fits, since non-linear minimization algorithms can fail in cases where the local curvature of the likelihood function is too low.

The flat directions are identified by performing a principal component analysis of the information matrix 𝐶 -1 SMEFT of the SMEFT parameter measurement. The information matrix is computed using the linear model with the assumption that the probability distribution function of the STXS measurement is approximately Gaussian. It is obtained as

𝐶 -1 SMEFT = 𝑃 𝑇 𝐶 -1 STXS 𝑃
where 𝐶 -1 STXS is the information matrix of the STXS measurement, computed in an Asimov data set generated under the SM hypothesis, and 𝑃 is the matrix representing the linear relation between the 𝜇 A rotation is then performed to align the measurement parameters with the eigenvectors EV𝑛 of 𝐶 -1 SMEFT . The unconstrained degrees of freedom of the measurement are identified with the eigenvectors corresponding to eigenvalues 𝜆 𝑛 of 𝐶 -1 SMEFT with magnitude 𝜆 𝑛 < 0.005. In the limit of a Gaussian measurement, each 𝜆 𝑛 is the inverse square of the measurement uncertainty along the direction of the corresponding EV𝑛, so the threshold for 𝜆 𝑛 corresponds to an uncertainty of about 14, which approximately corresponds to the region of SMEFT validity defined previously. The unconstrained EV𝑛 parameters are fixed to 0 in the model, while the remaining 12 EV𝑛 are considered as the measurement parameters. Their components along the 𝑐 𝑘 SMEFT parameters are shown in Figure 18. The full decomposition is shown in Table 17 of Appendix C. The EV1 parameter is mainly sensitive to the total event rates; EV2 and EV8 to the difference between the rate of 𝑔𝑔 → 𝐻 and of the other production modes; EV3 and EV7 to the high-𝑝 𝑉 T regions of the 𝑝 𝑝 → 𝑉 𝐻 processes; EV4 and EV5 to the high-𝑝 𝐻 T regions of the 𝑡𝑡𝐻 process; and EV6 to the rate of the 𝑞𝑞 → 𝐻𝑞𝑞 process. Best-fit values and confidence intervals for each EV𝑛 parameter are shown in Table 12 and -0.53 -0.02 0.00 0.23 0.71 -0.40 0.02 0.02 0.04 0.01 -0.00 0.00 -0.00 -0.00 -0.00 0.00 0.00 -0.00 -0.00 -0.00 0.00 -0.00 -0.00 -0.00 0.00 -0.00 -0.00 0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.85 -0.02 0.00 -0.14 -0.44 0.25 -0.01 -0.01 -0.02 -0.01 0.00 -0.00 0.00 0.01 0.00 -0.00 -0.00 -0.00 -0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 -0.00 0.00 0.00 -0.01 0.05 -0.01 -0.17 0.03 -0.04 0.00 0.00 0.00 -0.01 -0.01 0.00 -0.00 -0.98 -0.07 0.02 0.03 0.00 0.00 0.01 0.00 0.01 0.00 0.03 0.00 0.01 0.00 0.00 0.01 0.00 -0.00 0.00 -0.00 -0.01 0.68 -0.06 -0.08 0.01 -0.04 0.00 -0.01 -0.00 0.13 -0.07 -0.01 0.00 0.08 0.00 -0.00 -0.01 0.00 0.00 0.14 0.01 0.27 0.06 0.56 0.02 0.26 0.04 0.01 0.17 0.04 -0.00 -0.00 0.00 -0.02 0.64 -0.09 -0.24 0.04 -0.06 0.00 0.00 0.00 0.15 -0.09 -0.01 -0.00 0.05 0.02 -0.01 -0.02 0.00 0.00 -0.19 -0.02 -0.28 -0.04 -0.52 -0.01 -0.27 -0.03 -0.01 -0.16 -0.03 0.00 -0.00 -0.00 0.01 -0.24 0.01 -0.90 0.21 -0.14 0.01 0.01 0.01 -0.11 0.01 0.01 -0.00 0.15 0.10 -0.03 -0.08 0.00 0.01 0.03 0.00 0.05 0.00 0.08 0.00 0.05 0.00 0.00 0.03 0.00 -0.00 -0.00 -0.00 -0.00 0.03 0.03 0.09 0.15 0.32 0.00 0.00 0.00 0.02 0.01 0.05 0.00 -0.10 0.83 -0.25 -0.31 -0.04 -0.05 0.00 0.00 0.00 -0.01 0.00 0.00 0.00 -0.00 0.00 0.00 -0.00 -0.00 -0.00 0.00 -0.00 0.27 0.38 0.02 0.06 0.10 0.00 0.02 0.00 -0.78 0.37 0.07 -0.00 0.01 -0.04 0.00 0.09 0.00 0.00 0.09 -0.00 -0.03 -0.06 -0.04 0.00 -0.03 0.00 0.00 -0.02 -0.00 -0.00 -0.00 -0.00 -0.00 0.01 0.03 0.09 -0.38 -0.65 -0.00 -0.08 -0.01 -0.17 0.03 -0.08 0.00 -0.02 0.13 0.04 -0.56 0.09 0.12 0.02 -0.00 -0.01 0.18 -0.02 -0.00 -0.01 -0.00 -0.00 -0.01 -0.00 -0.00 -0.00 0.00 -0.00 -0.00 0.06 0.02 -0.09 -0.13 -0.00 0.37 -0.00 0.05 -0.02 -0.00 0.00 -0.00 0.01 0.02 -0.14 0.02 0.03 -0.05 0.00 0.04 -0.89 0.06 0.00 0.03 0.00 -0.00 0.02 0.00 0.00 0.00 0.00 0.00 -0.01 0.04 0.01 -0.03 -0.05 -0.00 -0.02 0.00 -0.10 0.03 -0.00 0.00 0.00 0.06 -0.05 0.11 0.01 0.02 -0.95 -0.00 0.15 0.05 0.11 -0.01 0.13 -0.01 -0.01 0.09 -0.00 0.00 -0.00 0.00 -0.00 -0.01 -0.15 0.01 -0.20 -0.36 -0.00 0.02 -0.01 -0.13 -0.16 -0.06 0.00 -0.00 0.37 -0.30 0.69 0.10 0.14 0.14 0.00 -0.02 -0.05 -0.01 0.00 -0.02 0.00 0.00 -0.01 -0.00 -0.00 0.00 0.00 illustrated in Figure 19. No significant deviation from the SM is observed. In the linear parameterization, expected signal yields can become negative for some values of the SMEFT parameters, leading in some cases to a negative value of the model pdf, which invalidates the profile-likelihood computation. In these cases, the bounds of the confidence intervals are truncated at the point at which the pdf reaches 0. Results in the linear+quadratic parameterization are not affected since the expected signal yields are always positive by construction.

Profile likelihood scans for selected EV𝑛 parameters are shown for illustration in Figure 20. For some parameters, such as EV1, a broad shape is seen in the expected scan in the SM hypothesis for the linear+quadratic parameterization. This is caused in part by the presence of two degenerate minima, due to the quadratic dependence of the expected yields on the SMEFT parameters. The degeneracy is partially lifted by the fact that the observed data do not exactly correspond to the SM expectation, which leads to narrower profiles in the observed scans. Similar scans are performed for eigenvectors corresponding to unconstrained directions with eigenvalues below 0.01, with the measured EV𝑛 also free to vary in the fits. The scans show that the measurement sensitivity in each of these directions is negligible.

The correlation matrix of the measurement is shown in Figure 21. Non-zero values outside the diagonal are due to differences between the observed results and their expectations, and to the fact that the information matrix used in the principal component analysis is not an exact representation of the measurement, due to non-Gaussian effects. In the linear parameterization, these include in particular the effect of low expected event counts in some categories and the non-linear impact of some systematic uncertainties. In the linear+quadratic case, larger correlations are observed due to the effect of the quadratic terms, which are not considered in the principal component analysis. These correlations also contribute to the larger uncertainties reported in Table 12 for some EV𝑛 parameters in the linear+quadratic parameterization, compared to the linear parameterization, and to the wider contours visible in Figure 20 for the linear+quadratic case. Linear parameterization results including corrections to the propagators of off-shell 𝑊 and 𝑍 bosons, the Higgs bosons and the top quarks, as implemented in the SMEFT generator [START_REF] Brivio | SMEFTsim 3.0 -a practical guide[END_REF] are shown in Appendix C.3.

A Additional production mode cross-section and STXS measurement results

Table 13 shows STXS results with a higher granularity than the baseline results presented in Section 8.4. A total of 33 regions are measured, with the following changes compared to the 28 regions in the baseline measurement:

• For the 𝑔𝑔 → 𝐻 process, within the phase space of ≥ 2-jets, 𝑚 𝑗 𝑗 < 350 GeV, the two regions with 𝑝 𝐻 T < 60 GeV and 60 ≤ 𝑝 𝐻 T < 120 GeV are kept separate. The same also applies to the three bins in the 𝑚 𝑗 𝑗 variable within the ≥ 2-jets, 𝑝 𝐻 T < 200 GeV region, which are not merged. Compared to the STXS analysis regions defined in Section 5.1, the only merging that is performed is that of the 𝑝 𝐻 T > 650 GeV bin with the neighbouring 450 ≤ 𝑝 𝐻 T < 650 GeV bin. • For the 𝑞𝑞 → 𝐻𝑞𝑞 process, the 0-jet and 1-jet regions are merged into a single ≤ 1-jet bin, and the 𝑚 𝑗 𝑗 < 60 GeV and 120 < 𝑚 𝑗 𝑗 < 350 GeV regions are also combined into a new 𝑉 𝐻-veto region, but the two sets are not merged together as in the baseline results. The three regions in the 𝑚 𝑗 𝑗 variables within ≥ 2-jets, 𝑝 𝐻 T ≥ 200 GeV, are also not merged. The correlation matrix of the measurement is shown in Fig. 22. Table 13: Best-fit values and uncertainties for the production cross-section times 𝐻 → 𝛾𝛾 branching ratio (𝜎 𝑖 × 𝐵 𝛾𝛾 ) in each STXS region. The values for the 𝑔𝑔 → 𝐻 process also include the contributions from 𝑏 b𝐻 production. The total uncertainties are decomposed into components for data statistics (Stat.) and systematic uncertainties (Syst.). SM predictions [START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF] are also shown for each quantity with their total uncertainties.

B.3 Generic parameterization using ratios of coupling modifiers

In this model, the effective parameterization of the 𝑔𝑔 → 𝐻 and 𝐻→ 𝛾𝛾 processes is used, and a common coupling modifier 𝜅 𝑉 = 𝜅 𝑊 = 𝜅 𝑍 is introduced for couplings to both 𝑊 and 𝑍 bosons. The 𝜅 𝜏 parameter is fixed to 1 and 𝜅 𝑏 = 𝜅 𝑡 is assumed. The total width of the Higgs boson is expressed using the effective parameterization Γ 𝐻 = 𝜅 2 𝐻 Γ SM 𝐻 , where Γ SM 𝐻 is the SM value of the width and 𝜅 𝐻 is a coupling modifier.

The measurement parameters are

𝜅 𝑔𝛾 = 𝜅 𝑔 𝜅 𝛾 /𝜅 𝐻 𝜆 𝑉 𝑔 = 𝜅 𝑉 /𝜅 𝑔 𝜆 𝑡𝑔 = 𝜅 𝑡 /𝜅 𝑔 ,
the first corresponding to the modifier for the 𝑔𝑔 → 𝐻→ 𝛾𝛾 process, which is taken as a reference, and the others two to ratios of coupling modifiers that can be measured without assumptions about the total width of the Higgs boson. The 𝜆 𝑡𝑔 parameter is allowed to take positive or negative values, while the other two parameters are positive by construction. Results are shown in Table 15. The negative log-likelihood scan of the 𝜆 𝑡𝑔 parameter is shown in Figure 24. Sensitivity to the sign of 𝜆 𝑡𝑔 is provided by the 𝑡𝐻 and 𝑔𝑔 → 𝑍 𝐻 processes, and leads to exclusion of the region 𝜆 𝑡𝑔 < 0 with a significance of 2.1𝜎.

C Effective field theory interpretation C.1 Measurement of single SMEFT parameters

This appendix presents the complete results of the single-parameter SMEFT measurements described in Section 10.2, and illustrated in Figure 17. The SMEFT parameters corresponding to each operator in Table 11 are individually measured, in each case while fixing the other SMEFT parameters to 0 as in the SM. Confidence intervals at 68% and 95% CL are computed both in observed data and in an Asimov data set generated under the SM hypothesis. Results for the parameters 𝑐 𝑘 are reported in Table 16, except those where the confidence intervals extend beyond the region |𝑐 𝑘 | ≤ 20 where the SMEFT framework is considered valid.

C.2 Simultaneous measurement of SMEFT parameters

This appendix presents the complete results of the simultaneous measurement of SMEFT parameters described in Section 10.3, shown in part in Table 12 and illustrated in Figure 19. The measurement parameters EV𝑛 are shown in Table 17. Confidence intervals at 68% and 95% CL for the EV𝑛 parameters defined in Table 17 are computed both in observed data and in an Asimov data set generated under the SM hypothesis. Results for the linear SMEFT parameterization are shown in Table 18 and for the linear+quadratic parameterization in Table 19.

Table 17: Measurement directions corresponding to the 12 largest eigenvalues of the Fisher information matrix of the SMEFT interpretation of the STXS measurement, shown as a decomposition in terms of Wilson coefficients in the Warsaw basis. The information matrix is obtained from the covariance matrix 𝐶 -1 STXS of the STXS measurement, computed using an Asimov data set generated in the SM hypothesis, propagated to the SMEFT measurement using the linear parameterization. Each linear combination is normalized to unit Euclidean norm. Only Wilson coefficients with a coefficient larger than 0.01 are shown.

Eigenvalue Eigenvector

350000

-0.53𝑐 𝐻𝐺 -0.02𝑐 𝑢𝐺 + 0.23𝑐 𝐻𝑊 + 0.71𝑐 𝐻𝐵 -0.4𝑐 𝐻 𝑊𝐵 + 0.02𝑐 𝑊 + 0.02𝑐 𝑢𝑊 + 0.04𝑐 𝑢𝐵 34000 -0.85𝑐 𝐻𝐺 -0.02𝑐 𝑢𝐺 -0.14𝑐 𝐻𝑊 -0.44𝑐 𝐻𝐵 +0.25𝑐 𝐻 𝑊𝐵 -0.01𝑐 𝑊 -0.01𝑐 𝑢𝑊 -0.02𝑐 𝑢𝐵 +0.01𝑐 (3) 𝐻 𝑞 110 -0.01𝑐 𝐻𝐺 + 0.05𝑐 𝑢𝐺 -0.17𝑐 𝐻𝑊 + 0.03𝑐 𝐻𝐵 -0.04𝑐 𝐻 𝑊𝐵 -0.01𝑐 (3) 𝐻 𝑙 -0.98𝑐 (3) 𝐻 𝑞 -0.07𝑐 𝐻 𝑢 + 0.02𝑐 𝐻 𝑑 + 0.03𝑐 (1) 𝐻 𝑞 + 0.01𝑐 (1) 𝑞𝑞 + 0.03𝑐 (3) 𝑞𝑞 + 0.01𝑐 𝑢𝑢 20 -0.01𝑐 𝐻𝐺 + 0.68𝑐 𝑢𝐺 -0.06𝑐 𝑢𝐻 -0.08𝑐 𝐻𝑊 + 0.01𝑐 𝐻𝐵 -0.04𝑐 𝐻 𝑊𝐵 + 0.13𝑐 (3) 𝐻 𝑙 -0.07𝑐 𝑙𝑙 -0.01𝑐 𝐻𝐷 + 0.08𝑐 (3) 𝐻 𝑞 + 0.14𝑐 𝐺 + 0.01𝑐 (1) 𝑞𝑞 + 0.27𝑐 (1) 𝑞𝑞 + 0.06𝑐 (3) 𝑞𝑞 + 0.56𝑐 (3) 𝑞𝑞 + 0.02𝑐 𝑢𝑢 + 0.26𝑐 𝑢𝑢 + 0.04𝑐 (8) 𝑢𝑑 + 0.17𝑐 (8) 𝑞𝑢 + 0.04𝑐 (8) 𝑞𝑑 2.9 -0.02𝑐 𝐻𝐺 + 0.64𝑐 𝑢𝐺 -0.09𝑐 𝑢𝐻 -0.24𝑐 𝐻𝑊 + 0.04𝑐 𝐻𝐵 -0.06𝑐 𝐻 𝑊𝐵 + 0.15𝑐 (3) 𝐻 𝑙 -0.09𝑐 𝑙𝑙 -0.01𝑐 𝐻𝐷 +0.05𝑐 (3) 𝐻 𝑞 +0.02𝑐 𝐻 𝑢 -0.02𝑐 (1) 𝐻 𝑞 -0.19𝑐 𝐺 -0.02𝑐 (1) 𝑞𝑞 -0.28𝑐 (1) 𝑞𝑞 -0.04𝑐 (3) 𝑞𝑞 -0.52𝑐 (3) 𝑞𝑞 -0.01𝑐 𝑢𝑢 -0.27𝑐 𝑢𝑢 -0.03𝑐 (8) 𝑢𝑑 -0.16𝑐 (8) 𝑞𝑢 -0.03𝑐 (8) 𝑞𝑑 1.8 -0.24𝑐 𝑢𝐺 + 0.01𝑐 𝑢𝐻 -0.9𝑐 𝐻𝑊 + 0.21𝑐 𝐻𝐵 -0.14𝑐 𝐻 𝑊𝐵 + 0.01𝑐 𝑢𝐵 -0.11𝑐 (3) 𝐻 𝑙 + 0.01𝑐 𝐻𝐷 + 0.15𝑐 (3) 𝐻 𝑞 + 0.1𝑐 𝐻 𝑢 -0.03𝑐 𝐻 𝑑 -0.08𝑐 (1) 𝐻 𝑞 + 0.03𝑐 𝐺 + 0.05𝑐 (1) 𝑞𝑞 + 0.08𝑐 (3) 𝑞𝑞 + 0.05𝑐 𝑢𝑢 + 0.03𝑐 (8) 𝑞𝑢 0.89 +0.03𝑐 𝑢𝐺 + 0.03𝑐 𝑢𝐻 + 0.09𝑐 𝐻𝑊 + 0.15𝑐 𝐻𝐵 + 0.32𝑐 𝐻 𝑊𝐵 + 0.02𝑐 (3) 𝐻 𝑙 + 0.01𝑐 𝑙𝑙 + 0.05𝑐 𝐻𝐷 -0.1𝑐 (3) 𝐻 𝑞 + 0.83𝑐 𝐻 𝑢 -0.25𝑐 𝐻 𝑑 -0.31𝑐 (1) 𝐻 𝑞 -0.04𝑐 𝐻 𝑒 -0.05𝑐 (1) 𝐻 𝑙 0.075 +0.27𝑐 𝑢𝐺 +0.38𝑐 𝑢𝐻 +0.02𝑐 𝐻𝑊 +0.06𝑐 𝐻𝐵 +0.1𝑐 𝐻 𝑊𝐵 +0.02𝑐 𝑢𝑊 -0.78𝑐 (3) 𝐻 𝑙 +0.37𝑐 𝑙𝑙 +0.07𝑐 𝐻𝐷 + 0.01𝑐 (3) 𝐻 𝑞 -0.04𝑐 𝐻 𝑢 + 0.09𝑐 (1) 𝐻 𝑞 + 0.09𝑐 𝐺 -0.03𝑐 (1) 𝑞𝑞 -0.06𝑐 (3) 𝑞𝑞 -0.04𝑐 (3) 𝑞𝑞 -0.03𝑐 𝑢𝑢 -0.02𝑐 (8) 𝑞𝑢 0.038 +0.01𝑐 𝑢𝐺 +0.03𝑐 𝑢𝐻 +0.09𝑐 𝐻𝑊 -0.38𝑐 𝐻𝐵 -0.65𝑐 𝐻 𝑊𝐵 -0.08𝑐 𝑢𝑊 -0.17𝑐 (3) 𝐻 𝑙 +0.03𝑐 𝑙𝑙 -0.08𝑐 𝐻𝐷 -0.02𝑐 (3) 𝐻 𝑞 + 0.13𝑐 𝐻 𝑢 + 0.04𝑐 𝐻 𝑑 -0.56𝑐 (1) 𝐻 𝑞 + 0.09𝑐 𝐻 𝑒 + 0.12𝑐 (1) 𝐻 𝑙 + 0.02𝑐 𝐺 + 0.18𝑐 (3) 𝑞𝑞 -0.02𝑐 (3) 𝑞𝑞 0.027 +0.06𝑐 𝑢𝐻 + 0.02𝑐 𝐻𝑊 -0.09𝑐 𝐻𝐵 -0.13𝑐 𝐻 𝑊𝐵 + 0.37𝑐 𝑢𝑊 + 0.05𝑐 (3) 𝐻 𝑙 -0.02𝑐 𝑙𝑙 + 0.02𝑐 𝐻 𝑑 -0.14𝑐 (1) 𝐻 𝑞 + 0.02𝑐 𝐻 𝑒 + 0.03𝑐 (1) 𝐻 𝑙 -0.05𝑐 𝐺 + 0.04𝑐 (1) 𝑞𝑞 -0.89𝑐 (3) 𝑞𝑞 + 0.06𝑐 (3) 𝑞𝑞 + 0.03𝑐 𝑢𝑢 + 0.02𝑐 (8) 𝑞𝑢 0.011 +0.04𝑐 𝑢𝐻 -0.03𝑐 𝐻𝐵 -0.05𝑐 𝐻 𝑊𝐵 -0.02𝑐 𝑢𝑊 -0.1𝑐 (3) 𝐻 𝑙 +0.03𝑐 𝑙𝑙 +0.06𝑐 𝐻 𝑢 -0.05𝑐 𝐻 𝑑 +0.11𝑐 (1) 𝐻 𝑞 + 0.01𝑐 𝐻 𝑒 + 0.02𝑐 (1) 𝐻 𝑙 -0.95𝑐 𝐺 + 0.15𝑐 (1) 𝑞𝑞 + 0.05𝑐 (3) 𝑞𝑞 + 0.11𝑐 (3) 𝑞𝑞 + 0.13𝑐 𝑢𝑢 -0.01𝑐 (1) 𝑞𝑢 + 0.09𝑐 (8) 𝑞𝑢 0.0067 -0.01𝑐 𝑢𝐺 -0.15𝑐 𝑢𝐻 + 0.01𝑐 𝐻𝑊 -0.2𝑐 𝐻𝐵 -0.36𝑐 𝐻 𝑊𝐵 + 0.02𝑐 𝑢𝑊 -0.13𝑐 (3) 𝐻 𝑙 -0.16𝑐 𝑙𝑙 -0.06𝑐 𝐻𝐷 + 0.37𝑐 𝐻 𝑢 -0.3𝑐 𝐻 𝑑 + 0.69𝑐 (1) 𝐻 𝑞 + 0.1𝑐 𝐻 𝑒 + 0.14𝑐 (1) 𝐻 𝑙 + 0.14𝑐 𝐺 -0.02𝑐 (1) 𝑞𝑞 -0.05𝑐 (3) 𝑞𝑞 -0.01𝑐 (3) 𝑞𝑞 -0.02𝑐 𝑢𝑢 -0.01𝑐 (8) 𝑞𝑢

C.3 Results including SMEFT propagator corrections

This appendix presents results similar to those in Section 10, but with SMEFT corrections applied to the mass and width parameters of off-shell SM particles, as implemented in the SMEFT generator [START_REF] Brivio | SMEFTsim 3.0 -a practical guide[END_REF]. These corrections are applied to the propagators of the 𝑊 and 𝑍 boson, the Higgs boson and the top quark in each process, at first order in the SMEFT. These corrections are only available for the linear SMEFT parameterization.

Table 20 shows the observed results with the propagator corrections included, for comparison with the ones in Table 18. The EV𝑛 parameters are defined in the same way as for the baseline linear parameterization. Differences from the baseline results are visible in the measurement of EV3, due to the impact of 𝑊 and 𝑍 propagator corrections on the 𝑞 q → 𝑉 𝐻 processes. Small changes in the principal components of the measurements due to the propagator corrections also lead to changes in the uncertainties in other parameters, in particular EV1. This also leads to generally larger correlations between the measurements than in the baseline linear parameterization, as shown in Figure 25.

ttH+tH ttH pH T < 60

 60 GeV ttH, pH T < 60 GeV 60 ≤ pH T < 120 GeV ttH, 60 ≤ pH T < 120 GeV 120 ≤ pH T < 200 GeV ttH, 120 ≤ pH T < 200 GeV 200 ≤ pH T < 300 GeV ttH, 200 ≤ pH T < 300 GeV pH T ≥ 300 GeV ttH, pH T ≥ 300 GeV

Figure 1 :

 1 Figure1: Summary of the STXS regions considered in the analysis design. The left part of the plot shows the selections applied to particle-level quantities in simulated signal events, with the selections applied sequentially along the branches of the graph. The final selection for each region is indicated by a box, and the name of each region, used in the rest of this paper, is shown on the right.

Figure 3 :

 3 Figure 3: Binary BDT discriminant distributions in four representative STXS classes. The binary BDT discriminant distribution is shown for simulated signal events in the target STXS region (solid) and in other STXS regions (dashed), and by the events in the diphoton mass sidebands (105 ≤ 𝑚 𝛾𝛾 < 120 GeV or 130 ≤ 𝑚 𝛾𝛾 < 160 GeV) representing background (dots). The vertical lines delimit the categories defined in the analysis within each class.

  𝑔𝑔 → 𝐻, 1-jet, 120 ≤ 𝑝 𝐻 T < 200 GeV

Figure 5 :

 5 Figure 5: Shape of the signal 𝑚 𝛾𝛾 distribution for two groups of categories. 5(a) compares the signal 𝑚 𝛾𝛾 shapes for the two categories targeting the 𝑔𝑔 → 𝐻, 1-jet, 120 ≤ 𝑝 𝐻 T < 200 GeV region. 5(b) compares the signal 𝑚 𝛾𝛾 shapes for three High-purity categories targeting different 𝑝 𝐻 T regions of the 𝑡𝑡𝐻 process. The markers represent distributions in MC samples with 𝑚 𝐻 = 125 GeV, while the solid lines represent the corresponding fitted DSCB functions.

Table 5 :

 5 Summary of the functions used for the modelling of the continuum background component. The free parameters used to define the function shape are denoted as 𝑎 or 𝑎 𝑖 , and their total number by 𝑁 pars . For the definition of the Bernstein polynomials, 𝑥 = (𝑚 𝛾𝛾 -𝑚 min )/(𝑚 max -𝑚 min ), where 𝑚 min = 105 GeV and 𝑚 max = 160 GeV are respectively the lower and upper bounds of the fitted 𝑚 𝛾𝛾 range. Type Function 𝑁 pars Acronym

Figure 6 :

 6 Figure 6: The diphoton invariant mass 𝑚 𝛾𝛾 distribution in data (black points) and continuum background templates (histograms) in four representative STXS categories. The data are shown excluding the region 120 < 𝑚 𝛾𝛾 < 130 GeV containing the signal. In panels 6(a) and 6(b), stacked histograms corresponding to the 𝛾𝛾 (white), 𝛾 𝑗 (green) and 𝑗 𝑗 (magenta) background contributions are shown. In panel 6(c), the histogram represents contributions from 𝑉 𝛾𝛾 and other sources of prompt 𝛾𝛾 production. In 6(d), the histogram corresponds to simulated 𝑡𝑡𝛾𝛾 events. The templates do not represent the background shapes used in the analysis, but are used to identify flexible functions used to model the background in each category as described in the text.

Figure 8 :

 8 Figure 8: Combined diphoton invariant mass distributions for categories targeting the same production processes.The data (black dots) are weighted by ln(1 + 𝑆/𝐵) where 𝑆 and 𝐵 are respectively the expected signal and background yields in the smallest 𝑚 𝛾𝛾 window containing 90% of the signal events. In this calculation, only Higgs boson events from the targeted production processes are considered as signal events. Higgs boson events from other processes as well as the continuum background events are considered as background. The fitted signal-plus-background pdfs from the relevant categories are summed, and represented by a solid line. The blue dotted line represents the weighted sum of the fitted continuum background pdfs, while the dashed line combines the contributions of continuum background and other Higgs boson events. The error bars on the data points are computed following Ref.[START_REF] Bohm | Introduction to statistics and data analysis for physicists[END_REF]. The weighted combination of categories with low event counts leads in some cases to data errors that are highly asymmetric and change by large amounts from point to point.

Figure 10 :

 10 Figure 10: Correlation matrix for the measurement of production cross-sections of the Higgs boson times the 𝐻 → 𝛾𝛾 branching ratio.

Figure 11 :

 11 Figure 11: Summary of the 28 regions for which STXS measurements are reported.

Figure 12 :

 12 Figure12: Best-fit values and uncertainties for STXS parameters in each of the 28 regions considered, normalized to their SM predictions. The values for the 𝑔𝑔 → 𝐻 process also include the contributions from 𝑏 b𝐻 production. The error bars and shaded areas show the total and systematic uncertainties in the measurements, respectively. The uncertainties for the 𝑝 𝑝 → 𝐻ℓℓ/𝜈 ν, 𝑝 𝑉 T < 150 GeV region are truncated at the value for which the model pdf becomes negative. The grey bands around the vertical line at 𝜎 𝛾𝛾 /𝜎 𝛾𝛾 𝑆 𝑀 =1 show the theory uncertainties in the predictions, including uncertainties due to missing higher-order terms in the perturbative QCD calculations, the PDFs and the value of 𝛼 s , as well as the 𝐻 → 𝛾𝛾 branching ratio uncertainty. The 𝑝 T and 𝑚 𝑗 𝑗 values in the region definitions are indicated in GeV.

Figure 13 :

 13 Figure 13: Correlation matrix for the measurement of STXS parameters in each of the 28 regions considered.

Figure 14 :

 14 Figure14: Negative log-likelihood scans as a function of 𝜅 𝑡 in a model where other coupling modifiers are fixed to their SM values. The 𝐻 → 𝛾𝛾 and 𝑔𝑔 → 𝐻 loops are either parameterized as a function of 𝜅 𝑡 (blue) or fixed to their SM expectation (orange). In the latter case, sensitivity to the sign of 𝜅 𝑡 is provided by the 𝑡𝐻 process, and to a lesser degree by the 𝑔𝑔 → 𝑍 𝐻 process. The solid curves correspond to observed data, and the dotted curves to an Asimov data set generated under the SM hypothesis.

Figure 15 :

 15 Figure15: Negative log-likelihood contours at 68% (dashed line) and 95% CL (solid line) in the (𝜅 𝑔 , 𝜅 𝛾 ) plane, assuming that all other coupling-strength modifiers take their SM values. All other 𝜅 modifiers are fixed to their SM values. The best-fit point is indicated by a cross while the SM prediction is indicated by a star.

Figure 16 :

 16 Figure 16: Relative impact of the most relevant SMEFT operators on the STXS regions and 𝐻 → 𝛾𝛾 decay mode in the linear SMEFT model. Coloured bars indicate the relative impact of SMEFT parameters on the expected cross-section in the corresponding region. The impacts are computed for the parameter values shown on the right, relative to the SM prediction. The parameters are defined for a scale Λ = 1 TeV. Three sets of operators with similar impacts on the measurement are shown in separate panels: those with impact mainly on the 𝑔𝑔 → 𝐻 and 𝑡𝑡𝐻 processes (second from top), the 𝐻 → 𝛾𝛾 decay (third from top), and VBF and 𝑉 𝐻 processes (bottom). The expected total relative uncertainty in the measurement of the signal strength in each STXS region is shown in the top panel, as an indication of the experimental sensitivity of each region. The 𝑝 T and 𝑚 𝑗 𝑗 values in the region definitions are indicated in GeV, and the 0J, 1J and 2J shorthands refer respectively to the 0-jet, 1-jet and ≥ 2-jets selections.

  𝛾𝛾 𝑖 and the 𝑐 𝑘 in the linear SMEFT parameterization, with components given by 𝑃 𝑖𝑘 = 𝐴 𝑖→𝐻 𝑘 + 𝐴 𝐻 → 𝛾𝛾 𝑘 -𝐴 Γ 𝑘 in the notation of Eq. (3).

Figure 18 :

 18 Figure 18: Components of the EV𝑛 parameters (𝑦-axis) along each of the Warsaw-basis Wilson coefficients (𝑥-axis). The EV𝑛 are normalized to unit Euclidean norm. Coefficients below 0.01 are not shown. The Warsaw-basis Wilson coefficients are defined for a scale Λ = 1 TeV. The information-matrix eigenvalues (𝜆 𝑛 ) corresponding to each eigenvector are shown on the right side of the plot.

Figure 19 :Figure 20 :Figure 21 :

 192021 Figure19: Results of the EV𝑛 parameter measurement in data, in the linear (blue) and linear+quadratic (orange) parameterizations of the SMEFT. All the EV𝑛 parameters are free to vary in the fits. The ranges shown correspond to 68% CL (solid) and 95% CL (dashed) intervals. Cases where the intervals are truncated due to a negative model pdf are indicated by hashes.

Figure 22 :

 22 Figure 22: Correlation matrix for the measurement of STXS parameters in each of the 33 regions considered.

Figure 24 :

 24 Figure 24: Negative log-likelihood scan as a function of 𝜆 𝑡𝑔 = 𝜅 𝑡 /𝜅 𝑔 in the coupling-modifier ratio model described in the text. The solid curve corresponds to observed data, and the dotted curve to an Asimov data set generated under the SM hypothesis.

Figure 25 :

 25 Figure 25: Observed linear correlation coefficients of the EV𝑛 parameters in the linear SMEFT parameterization including corrections to 𝑊, 𝑍, Higgs boson and top quark propagators.

Table 2 :

 2 Training variables used as input to the multiclass BDT. The dagger symbol † denotes variables that have two versions with different jet 𝑝 T requirements. One version of such a variable is defined using jets with 𝑝 T > 25 GeV, and the other version is defined using jets with 𝑝 T > 30 GeV. Both versions are used in the training of the multiclass BDT. The two highest-𝑝 T photons are denoted as 𝛾 1 and 𝛾 2 , the two highest-𝑝 T jets as 𝑗 1 and 𝑗 2 , the two highest-𝑝 T top quarks as 𝑡 1 and 𝑡 2 and the most forward jet as 𝑗 𝐹 . Δ𝑅(𝑊, 𝑏) is the Δ𝑅 between the 𝑊 and 𝑏 components of a top-quark candidate. 𝑗 𝑗 , 𝑚 𝑗 𝑗 , and Δ𝑦, Δ𝜙, Δ𝜂 between 𝑗 1 and 𝑗 2 , 𝑝 T,𝛾𝛾 𝑗 1 , 𝑚 𝛾𝛾 𝑗 1 , 𝑝 T,𝛾𝛾 𝑗 𝑗 †, 𝑚 𝛾𝛾 𝑗 𝑗 Δ𝑦, Δ𝜙 between the 𝛾𝛾 and 𝑗 𝑗 systems, minimum Δ𝑅 between jets and photons, invariant mass of the system comprising all jets in the event, dilepton 𝑝 T , di-𝑒 or di-𝜇 invariant mass (leptons are required to be oppositely charged), 𝜂, 𝜙 of top-quark candidates, 𝑚 𝑡 1 𝑡 2 Number of jets †, of central jets (|𝜂| < 2.5) †, of 𝑏-jets † and of leptons, 𝑝 T of the highest-𝑝 T jet, scalar sum of the 𝑝 T of all jets, scalar sum of the transverse energies of all particles ( 𝐸 T ), 𝐸 miss 𝑏) of 𝑡 2 , 𝜂 𝑗 𝐹 , 𝑚 𝛾𝛾 𝑗 𝐹 Average number of interactions per bunch crossing.

				𝛾𝛾
		𝑝 †	𝜂 𝛾 1 , 𝜂 𝛾 2 , 𝑝	T , 𝑦 𝛾𝛾 ,
	T, 𝐸 miss T , 𝑝 T and transverse mass of the lepton + 𝐸 miss T	system,
			𝑝 T , T	significance,
	𝐸 miss T	-𝐸 miss T (primary vertex with the highest 𝑝 2 T,track ) > 30 GeV
		Top reconstruction BDT of the top-quark candidates,
			Δ𝑅(𝑊,

Table 3 :

 3 Training variables used for the binary classifiers. The sets of classes to which the classifiers are applied are specified in the first column, and the corresponding variables in each case are listed in the second column. The asterisk symbol * denotes 𝑡𝐻 training variables that are only used for the classifiers suppressing the continuum background. Other 𝑡𝐻 training variables are used in all three 𝑡𝐻 classifiers. The 𝛾𝛾 and 𝑗 𝑗 notations refer to the systems composed of the two highest-𝑝 T photons and jets, respectively. The two highest-𝑝 T photons are denoted as 𝛾 1 and 𝛾 2 , the two highest-𝑝 T top quarks as 𝑡 1 and 𝑡 2 , and the most forward jet as 𝑗 𝐹 . The differences in 𝜂 and 𝜙 between 𝛾 1 and 𝛾 2 are denoted respectively as Δ𝜙 𝛾𝛾 and Δ𝜂 𝛾𝛾 . Δ𝑅(𝑊, 𝑏) is the Δ𝑅 between the 𝑊 and 𝑏 components of a top-quark candidate.

	STXS classes		Variables
			All multiclass BDT variables,
	Individual STXS classes from 𝑔𝑔 → 𝐻	𝒑	𝛾𝛾 T projected to the thrust axis of the 𝛾𝛾 system (𝑝 𝜂 𝛾𝛾 -𝜂 𝑗 𝑗 Δ𝜂 𝛾𝛾 , 𝜂 Zepp = 2	𝛾𝛾 Tt ),
	𝑞𝑞 → 𝐻𝑞𝑞
	𝑞𝑞 → 𝐻ℓ𝜈
	𝑝 𝑝 → 𝐻ℓℓ
	𝑝 𝑝 → 𝐻𝜈 ν

𝑚 𝛾𝛾𝑡 1 Number of jets with 𝑝 T > 25 GeV, Number of 𝑏-jets with 𝑝 T > 25 GeV * ; Number of leptons * , 𝐸 miss T significance *

  𝜂, 𝜙 of 𝛾 1 and 𝛾 2 , 𝑝 T , 𝜂, 𝜙 and 𝑏-tagging scores of the six highest-𝑝 T jets, 𝜂 𝛾𝛾 , 𝑝 T , invariant mass, BDT score and Δ𝑅(𝑊, 𝑏) of 𝑡 1 , 𝑝 T , 𝜂 of 𝑡 2 , 𝑝 T , 𝜂 of 𝑗 𝐹 , Angular variables: Δ𝜂 𝛾𝛾𝑡 1 , Δ𝜃 𝛾𝛾𝑡 2 , Δ𝜃 𝑡 1 𝑗 𝐹 , Δ𝜃 𝑡 2 𝑗 𝐹 , Δ𝜃 𝛾𝛾 𝑗 𝐹 Invariant mass variables: 𝑚 𝛾𝛾 𝑗 𝐹 , 𝑚 𝑡 1 𝑗 𝐹 , 𝑚 𝑡 2 𝑗 𝐹 ,

				𝑧 )
			𝑚 𝛾𝛾 + √	𝑚 2 𝛾𝛾 +( 𝑝 T ) 2 𝛾𝛾
		Number of electrons and muons.
	all 𝑡𝑡𝐻 and 𝑡𝐻𝑊 STXS classes	𝑝 T , 𝐸 miss T , 𝐸 miss T significance, 𝐸 miss T	azimuthal angle,
	combined	Top reconstruction BDT scores of the top-quark candidates,
		𝑝 T , 𝜂, 𝜙 of the two highest-𝑝 T leptons.
		𝑝	𝛾𝛾 T /𝑚 𝛾𝛾 ,
	𝑡𝐻𝑞𝑏

Table 4 :

 4 Expected signal (𝑆) and background (𝐵) yields in each category, within the smallest mass window containing 90% of signal events, the half-width of which is given by 𝜎. The signal purity 𝑓 = 𝑆/(𝑆 + 𝐵) and expected significance 𝑍 = √ 2((𝑆 + 𝐵) ln(1 + 𝑆/𝐵) -𝑆) are also shown. Only the signal process corresponding to the targeted STXS region is considered in the signal yield. 𝑚 𝑗 𝑗 < 350 GeV, 𝑝 𝐻 T < 60 GeV, Med-purity 43.1 4360 3.39 0.98 0.65 ≥ 2-jets, 𝑚 𝑗 𝑗 < 350 GeV, 𝑝 𝐻 T < 60 GeV, Low-purity 47.5 16 800 3.51 0.28 0.37 ≥ 2-jets, 𝑚 𝑗 𝑗 < 350 GeV, 60 ≤ 𝑝 𝐻 𝑚 𝑗 𝑗 < 350 GeV, 60 ≤ 𝑝 𝐻 𝑚 𝑗 𝑗 < 350 GeV, 120 ≤ 𝑝 𝐻 𝑚 𝑗 𝑗 < 350 GeV, 120 ≤ 𝑝 𝐻

				𝜎	𝑓					𝜎	𝑓
	Category	𝑆	𝐵			𝑍	Category	𝑆	𝐵			𝑍
				[GeV] [%]					[GeV] [%]
	𝑔𝑔 → 𝐻						≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T ≥ 200 GeV, High-purity	1.31 2.19	2.48	37 0.81
	0-jet, 𝑝 𝐻 T < 10 GeV	695 26 000	3.43	2.6	4.3	≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T ≥ 200 GeV, Med-purity	1.40 9.22	2.49	13 0.45
	0-jet, 𝑝 𝐻 T ≥ 10 GeV	1440 47 000	3.41	3.0	6.6	≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T ≥ 200 GeV, Low-purity	1.16 65.5	2.54	1.7 0.14
	1-jet, 𝑝 𝐻 T < 60 GeV, High-purity	168	4250	3.20	3.8	2.6	≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T ≥ 200 GeV, High-purity 2.51 3.02	2.43	45	1.3
	1-jet, 𝑝 𝐻 T < 60 GeV, Med-purity	197 11 500	3.38	1.7	1.8	≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T ≥ 200 GeV, Med-purity 1.49 47.4	2.54	3.0 0.22
	1-jet, 60 ≤ 𝑝 𝐻 T < 120 GeV, High-purity	186	3310	3.10	5.3	3.2	≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T ≥ 200 GeV, High-purity	5.65 1.57	2.39	78	3.3
	1-jet, 60 ≤ 𝑝 𝐻 T < 120 GeV, Med-purity	180	7780	3.37	2.3	2.0	≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T ≥ 200 GeV, Med-purity	2.96 6.31	2.55	32	1.1
	1-jet, 120 ≤ 𝑝 𝐻 T < 200 GeV, High-purity	23.0	182	2.61	11	1.7	𝑞𝑞 → 𝐻ℓ𝜈
	1-jet, 120 ≤ 𝑝 𝐻 T < 200 GeV, Med-purity ≥ 2-jets, 𝑚 𝑗 𝑗 < 350 GeV, 𝑝 𝐻 T < 60 GeV, High-purity ≥ 2-jets, T < 120 GeV, High-purity ≥ 2-jets, T < 120 GeV, Med-purity	40.7 23.5 49.1 93.9	717 1050 901 6440	3.00 3.08 3.03 3.30	5.4 2.2 0.72 1.5 5.2 1.6 1.4 1.2	𝑝 𝑉 T < 75 GeV, High-purity 𝑝 𝑉 T < 75 GeV, Med-purity 75 ≤ 𝑝 𝑉 T < 150 GeV, High-purity 75 ≤ 𝑝 𝑉 T < 150 GeV, Med-purity 150 ≤ 𝑝 𝑉 T < 250 GeV, High-purity	1.91 4.91 2.59 20.2 2.62 2.05 2.08 12.4 1.74 2.06	3.17 3.28 3.02 3.23 2.78	28 0.81 11 0.57 56 1.6 14 0.58 46 1.1
	≥ 2-jets, T < 200 GeV, High-purity	15.5	74.8	2.64	17	1.7	150 ≤ 𝑝 𝑉 T < 250 GeV, Med-purity	0.16 2.90	3.17	5.2 0.09
	≥ 2-jets, T < 200 GeV, Med-purity	22.7	343	2.97	6.2	1.2	𝑝 𝑉 T ≥ 250 GeV, High-purity	1.36 1.79	2.41	43 0.91
	≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T < 200 GeV, High-purity	4.31	47.5	2.72	8.3 0.62	𝑝 𝑉 T ≥ 250 GeV, Med-purity	0.02 3.12	3.15 0.78 0.01
							𝑝 𝑝 → 𝐻ℓℓ
							𝑝 𝑉 T < 75 GeV, High-purity	1.14 1.82	3.25	39 0.78
							𝑝 𝑉 T < 75 GeV, Med-purity	1.06	215	3.29 0.49 0.07
							75 ≤ 𝑝 𝑉 T < 150 GeV, High-purity	1.07 1.58	3.08	40 0.77
							75 ≤ 𝑝 𝑉 T < 150 GeV, Med-purity	0.02 1.81	3.06	1.2 0.02
							150 ≤ 𝑝 𝑉 T < 250 GeV, High-purity	0.71 1.79	2.78	28 0.50
		2.49	142	3.37	1.7 0.21	150 ≤ 𝑝 𝑉 T < 250 GeV, Med-purity	0.10 16.5	2.88 0.62 0.03
	200 ≤ 𝑝 𝐻 T < 300 GeV, High-purity	15.3	38.0	2.28	29	2.3	𝑝 𝑉 T ≥ 250 GeV	0.27 2.06	2.48	12 0.18
	200 ≤ 𝑝 𝐻 T < 300 GeV, Med-purity	29.4	236	2.64	11	1.9	𝑝 𝑝 → 𝐻𝜈 ν
	300 ≤ 𝑝 𝐻 T < 450 GeV, High-purity 300 ≤ 𝑝 𝐻 T < 450 GeV, Med-purity 300 ≤ 𝑝 𝐻 T < 450 GeV, Low-purity 450 ≤ 𝑝 𝐻 T < 650 GeV, High-purity 450 ≤ 𝑝 𝐻 T < 650 GeV, Med-purity 450 ≤ 𝑝 𝐻 T < 650 GeV, Low-purity 𝑝 𝐻 T ≥ 650 GeV	1.52 6.75 4.66 1.00 0.800 0.830 0.220	2.13 17.7 43.1 1.25 2.00 10.7 1.08	2.02 2.16 2.46 1.85 1.98 2.19 1.73	42 0.95 28 1.5 9.8 0.70 45 0.81 29 0.53 7.2 0.25 17 0.20	𝑝 𝑉 T < 75 GeV, High-purity 𝑝 𝑉 T < 75 GeV, Med-purity 𝑝 𝑉 T < 75 GeV, Low-purity 75 ≤ 𝑝 𝑉 T < 150 GeV, High-purity 75 ≤ 𝑝 𝑉 T < 150 GeV, Med-purity 75 ≤ 𝑝 𝑉 T < 150 GeV, Low-purity 150 ≤ 𝑝 𝑉 T < 250 GeV, High-purity	0.60 1.15 1020 170 0.87 2630 0.58 2.30 1.83 17.8 2.18 288 0.92 2.00	3.50 0.35 0.05 3.57 0.11 0.04 3.67 0.03 0.02 2.97 20 0.37 3.26 9.3 0.43 3.44 0.75 0.13 2.75 32 0.61
	𝑞𝑞 → 𝐻𝑞𝑞						150 ≤ 𝑝 𝑉 T < 250 GeV, Med-purity	0.75 2.54	2.94	23 0.45
	0-jet, High-purity	0.330	25.0	3.33	1.3 0.07	150 ≤ 𝑝 𝑉 T < 250 GeV, Low-purity	0.26 11.7	3.28	2.2 0.08
	0-jet, Med-purity	1.27	471	3.35 0.27 0.06	𝑝 𝑉 T ≥ 250 GeV, High-purity	0.67 1.55	2.46	30 0.50
	0-jet, Low-purity	10.7 18 800	3.48 0.06 0.08	𝑝 𝑉 T ≥ 250 GeV, Med-purity	0.05 1.97	3.05	2.6 0.04
	1-jet, High-purity	1.08	2.78	2.99	28 0.61	𝑡𝑡𝐻
	1-jet, Med-purity	3.50	26.1	3.11	12 0.67
	1-jet, Low-purity	2.88	145	3.24	2.0 0.24	𝑝 𝐻 T < 60 GeV, High-purity	3.04 4.01	3.18	43	1.4
	≥ 2-jets, 𝑚 𝑗 𝑗 < 60 GeV, High-purity	0.350	2.10	2.71	14 0.24	𝑝 𝐻 T < 60 GeV, Med-purity	2.78 13.3	3.37	17 0.74
	≥ 2-jets, 𝑚 𝑗 𝑗 < 60 GeV, Med-purity	0.670	19.0	2.79	3.4 0.15	60 ≤ 𝑝 𝐻 T < 120 GeV, High-purity	4.30 4.09	3.06	51	1.9
	≥ 2-jets, 𝑚 𝑗 𝑗 < 60 GeV, Low-purity	1.92	243	2.93 0.78 0.12	60 ≤ 𝑝 𝐻 T < 120 GeV, Med-purity	2.99 8.61	3.31	26 0.97
	≥ 2-jets, 60 ≤ 𝑚 𝑗 𝑗 < 120 GeV, High-purity	3.45	6.34	2.65	35	1.3	120 ≤ 𝑝 𝐻 T < 200 GeV, High-purity	4.65 3.52	2.73	57	2.1
	≥ 2-jets, 60 ≤ 𝑚 𝑗 𝑗 < 120 GeV, Med-purity	4.99	43.0	2.85	10 0.75	120 ≤ 𝑝 𝐻 T < 200 GeV, Med-purity	1.66 4.16	2.93	29 0.77
	≥ 2-jets, 60 ≤ 𝑚 𝑗 𝑗 < 120 GeV, Low-purity	2.99	87.3	3.01	3.3 0.32	200 ≤ 𝑝 𝐻 T < 300 GeV	3.39 2.26	2.46	60	1.9
	≥ 2-jets, 120 ≤ 𝑚 𝑗 𝑗 < 350 GeV, High-purity	2.98	24.4	2.93	11 0.59	𝑝 𝐻 T ≥ 300 GeV	2.73 1.66	2.12	62	1.8
	≥ 2-jets, 120 ≤ 𝑚 𝑗 𝑗 < 350 GeV, Med-purity	6.73	204	2.94	3.2 0.47	𝑡𝐻
	≥ 2-jets, 120 ≤ 𝑚 𝑗 𝑗 < 350 GeV, Low-purity	8.78	1360	2.99 0.64 0.24
	≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T < 200 GeV, High-purity	2.52	2.75	2.96	48	1.4	𝑡𝐻𝑞𝑏, High-purity	0.55 2.16	3.04	20 0.36
	≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T < 200 GeV, Med-purity ≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T < 200 GeV, Low-purity	9.15 5.97	34.7 106	3.06 3.27	21 5.3 0.57 1.5	𝑡𝐻𝑞𝑏, Med-purity 𝑡𝐻𝑞𝑏, BSM (𝜅𝑡 = -1)	0.14 2.78 0.12 1.86	3.45 3.25	4.9 0.09 6.0 0.09
	≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T < 200 GeV, High-purity	2.91	3.00	2.90	49	1.5	𝑡𝐻𝑊	0.16 6.91	2.74	2.3 0.06
	≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T < 200 GeV, Med-purity	5.60	22.7	3.11	20	1.1	Low-purity top	5.18 65.8	3.32	7.3 0.63
	≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T < 200 GeV, High-purity	10.8	3.89	3.01	74	4.2
	≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T < 200 GeV, Med-purity	10.7	19.0	3.23	36	2.3

≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T < 200 GeV, Med-purity 15.4 380 3.02 3.9 0.78 ≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T < 200 GeV, Low-purity 10.5 1080 3.31 0.97 0.32 ≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T < 200 GeV, High-purity 2.34 33.3 2.84 6.6 0.40 ≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T < 200 GeV, Med-purity 4.23 136 3.07 3.0 0.36 ≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T < 200 GeV, Low-purity 3.34 429 3.26 0.77 0.16 ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T < 200 GeV, High-purity 1.14 14.5 2.97 7.3 0.30 ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T < 200 GeV, Med-purity 2.52 47.5 3.10 5.0 0.36 ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T < 200 GeV, Low-purity

  computed from the maximum likelihood values 𝐿 1 and 𝐿 2 of background-only fits to the data sideband regions using respectively the Exp and ExpPoly2 descriptions of the backgrounds. The ExpPoly2 model is chosen if the 𝑝-value computed from 𝑞 12 is less than 0.05, assuming that 𝑞 12 follows a 𝜒 2 distribution with one degree of freedom. Similarly, the ExpPoly3 form is chosen over ExpPoly2 if the 𝑝-value for the corresponding Wald test is 0.05 or less. For 32 out of the 101 categories, the Wald-test-based condition was used and the Exp function was selected in each case.

Table 6 :

 6 Selected background functional form, number of observed data events in the range 105 < 𝑚 𝛾𝛾 < 160 GeV (𝑁 data ), and modelling uncertainty (𝑁 spur ) for each of the 101 analysis categories. The last column indicates whether the Wald test is used to determine the functional form, as described in the text. 𝑚 𝑗 𝑗 < 350 GeV, 120 ≤ 𝑝 𝐻 T < 200 GeV, High-purity ExpPoly2 728 0.004 ≥ 2-jets, 𝑚 𝑗 𝑗 < 350 GeV, 120 ≤ 𝑝 𝐻 T < 200 GeV, Med-purity PowerLaw 3007 0.983 ≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T < 200 GeV, High-purity Exp 432 0.487 ≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T < 200 GeV, Med-purity ExpPoly2 3084 1.33 ≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T < 200 GeV, Low-purity Exp 7999 5.78 ≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T < 200 GeV, High-purity Exp 302 0.560 ≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T < 200 GeV, Med-purity Exp 1033 1.44 ≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T < 200 GeV, Low-purity Exp 3187 4.32 ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T < 200 GeV, High-purity Exp 113 0.192 ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T < 200 GeV, Med-purity Exp 332 0.804 ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T < 200 GeV, Low-purity

	Category	Function	𝑁data 𝑁spur Wald	Category	Function	𝑁data 𝑁spur Wald
	𝑔𝑔 → 𝐻				≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T ≥ 200 GeV, High-purity	Exp	18 0.189
	0-jet, 𝑝 𝐻 T < 10 GeV 0-jet, 𝑝 𝐻 T ≥ 10 GeV 1-jet, 𝑝 𝐻 T < 60 GeV, High-purity 1-jet, 𝑝 𝐻 T < 60 GeV, Med-purity 1-jet, 60 ≤ 𝑝 𝐻 T < 120 GeV, High-purity 1-jet, 60 ≤ 𝑝 𝐻 T < 120 GeV, Med-purity	ExpPoly2 191623 ExpPoly2 349266 ExpPoly2 32644 ExpPoly2 85229 Exp 26236 ExpPoly2 56669	64.8 50.4 20.7 24.9 23.7 21.3	≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T ≥ 200 GeV, Med-purity ≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T ≥ 200 GeV, Low-purity ≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T ≥ 200 GeV, High-purity ≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T ≥ 200 GeV, Med-purity ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T ≥ 200 GeV, High-purity ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T ≥ 200 GeV, Med-purity	Exp Exp Exp Exp Exp Exp	84 0.513 595 0.721 19 0.110 411 0.193 23 1.30 56 0.329
	1-jet, 120 ≤ 𝑝 𝐻 T < 200 GeV, High-purity	ExpPoly2	1570	1.48	𝑞𝑞 → 𝐻ℓ𝜈
	1-jet, 120 ≤ 𝑝 𝐻 T < 200 GeV, Med-purity ≥ 2-jets, 𝑚 𝑗 𝑗 < 350 GeV, 𝑝 𝐻 T < 60 GeV, High-purity ≥ 2-jets, 𝑚 𝑗 𝑗 < 350 GeV, 𝑝 𝐻 T < 60 GeV, Med-purity ≥ 2-jets, 𝑚 𝑗 𝑗 < 350 GeV, 𝑝 𝐻 T < 60 GeV, Low-purity ≥ 2-jets, 𝑚 𝑗 𝑗 < 350 GeV, 60 ≤ 𝑝 𝐻 T < 120 GeV, High-purity ≥ 2-jets, 𝑚 𝑗 𝑗 < 350 GeV, 60 ≤ 𝑝 𝐻 T < 120 GeV, Med-purity ≥ 2-jets, PowerLaw ExpPoly2 ExpPoly2 ExpPoly2 ExpPoly2 120357 6163 8513 31163 ExpPoly2 7582 ExpPoly2 48362 1020	5.33 1.51 13.6 15.7 2.26 6.21 1.09	0 ≤ 𝑝 𝑉 T < 75 GeV, High-purity 0 ≤ 𝑝 𝑉 T < 75 GeV, Med-purity 75 ≤ 𝑝 𝑉 T < 150 GeV, High-purity 75 ≤ 𝑝 𝑉 T < 150 GeV, Med-purity 150 ≤ 𝑝 𝑉 T < 250 GeV, High-purity 150 ≤ 𝑝 𝑉 T < 250 GeV, Med-purity 𝑝 𝑉 T ≥ 250 GeV, High-purity 𝑝 𝑉 T ≥ 250 GeV, Med-purity 𝑝 𝑝 → 𝐻ℓℓ 0 ≤ 𝑝 𝑉 T < 75 GeV, High-purity 0 ≤ 𝑝 𝑉 T < 75 GeV, Med-purity 75 ≤ 𝑝 𝑉 T < 150 GeV, High-purity 75 ≤ 𝑝 𝑉 T < 150 GeV, Med-purity 150 ≤ 𝑝 𝑉 T < 250 GeV, High-purity 150 ≤ 𝑝 𝑉 T < 250 GeV, Med-purity	Exp Exp Exp Exp Exp Exp Exp Exp Exp PowerLaw Exp Exp Exp Exp	40 0.277 158 0.609 15 0.069 104 0.255 17 0.128 21 0.150 16 0.237 27 0.054 12 0.027 1620 2.28 13 0.015 18 0.016 14 0.059 136 0.194
	200 ≤ 𝑝 𝐻 T < 300 GeV, High-purity	Exp	420	1.68	𝑝 𝑉 T ≥ 250 GeV	Exp	14 0.311
	200 ≤ 𝑝 𝐻 T < 300 GeV, Med-purity	Exp	2296 0.714	𝑝 𝑝 → 𝐻𝜈 ν
	300 ≤ 𝑝 𝐻 T < 450 GeV, High-purity 300 ≤ 𝑝 𝐻 T < 450 GeV, Med-purity 300 ≤ 𝑝 𝐻 T < 450 GeV, Low-purity 450 ≤ 𝑝 𝐻 T < 650 GeV, High-purity 450 ≤ 𝑝 𝐻 T < 650 GeV, Med-purity 450 ≤ 𝑝 𝐻 T < 650 GeV, Low-purity 𝑝 𝐻 T ≥ 650 GeV 𝑞𝑞 → 𝐻𝑞𝑞	Exp Exp Exp Exp Exp Exp Exp	25 0.407 186 0.259 422 0.121 15 0.138 25 0.391 109 0.031 14 0.448	0 ≤ 𝑝 𝑉 T < 75 GeV, High-purity 0 ≤ 𝑝 𝑉 T < 75 GeV, Med-purity 0 ≤ 𝑝 𝑉 T < 75 GeV, Low-purity 75 ≤ 𝑝 𝑉 T < 150 GeV, High-purity 75 ≤ 𝑝 𝑉 T < 150 GeV, Med-purity 75 ≤ 𝑝 𝑉 T < 150 GeV, Low-purity 150 ≤ 𝑝 𝑉 T < 250 GeV, High-purity 150 ≤ 𝑝 𝑉 T < 250 GeV, Med-purity	Exp Exp ExpPoly3 18084 1174 6897 Exp 16 0.407 12.3 4.13 9.95 Exp 124 1.30 Exp 2019 1.96 Exp 16 0.121 Exp 17 0.184
	0-jet, High-purity	Exp	176 0.180	150 ≤ 𝑝 𝑉 T < 250 GeV, Low-purity	Exp	87 0.644
	0-jet, Med-purity	ExpPoly2	3238	4.73	𝑝 𝑉 T ≥ 250 GeV, High-purity	Exp	15 0.237
	0-jet, Low-purity	ExpPoly2 133314	49.7	𝑝 𝑉 T ≥ 250 GeV, Med-purity	Exp	18 0.201
	1-jet, High-purity	Exp	19 0.125	𝑡𝑡𝐻
	1-jet, Med-purity	Exp	187 0.361
	1-jet, Low-purity	PowerLaw	1040	1.97	𝑝 𝐻 T < 60 GeV, High-purity	Exp	35 0.040
	≥ 2-jets, 𝑚 𝑗 𝑗 < 60 GeV, High-purity	Exp	17 0.499	𝑝 𝐻 T < 60 GeV, Med-purity	Exp	96 0.192
	≥ 2-jets, 𝑚 𝑗 𝑗 < 60 GeV, Med-purity	Exp	157 0.489	60 ≤ 𝑝 𝐻 T < 120 GeV, High-purity	Exp	34 0.038
	≥ 2-jets, 𝑚 𝑗 𝑗 < 60 GeV, Low-purity	PowerLaw	1978	1.29	60 ≤ 𝑝 𝐻 T < 120 GeV, Med-purity	Exp	74 0.274
	≥ 2-jets, 60 ≤ 𝑚 𝑗 𝑗 < 120 GeV, High-purity	Exp	53 0.165	120 ≤ 𝑝 𝐻 T < 200 GeV, High-purity	Exp	39 0.018
	≥ 2-jets, 60 ≤ 𝑚 𝑗 𝑗 < 120 GeV, Med-purity	Exp	329 0.520	120 ≤ 𝑝 𝐻 T < 200 GeV, Med-purity	Exp	37 0.057
	≥ 2-jets, 60 ≤ 𝑚 𝑗 𝑗 < 120 GeV, Low-purity	PowerLaw	709	1.15	200 ≤ 𝑝 𝐻 T < 300 GeV	Exp	23 0.261
	≥ 2-jets, 120 ≤ 𝑚 𝑗 𝑗 < 350 GeV, High-purity	Exp	214	1.08	𝑝 𝐻 T ≥ 300 GeV	Exp	19 0.180
	≥ 2-jets, 120 ≤ 𝑚 𝑗 𝑗 < 350 GeV, Med-purity	ExpPoly2	1671	1.07	𝑡𝐻
	≥ 2-jets, 120 ≤ 𝑚 𝑗 𝑗 < 350 GeV, Low-purity	PowerLaw	11195	6.34
	≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T < 200 GeV, High-purity	Exp	25 0.162	𝑡𝐻𝑞𝑏, High-purity	Exp	17 0.371
	≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T < 200 GeV, Med-purity	Exp	260 0.443	𝑡𝐻𝑞𝑏, Med-purity	Exp	19 0.320
	≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T < 200 GeV, Low-purity	Exp	753	1.17	𝑡𝐻𝑞𝑏, BSM (𝜅𝑡 = -1)	Exp	14 0.496
	≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T < 200 GeV, High-purity	Exp	25 0.670	𝑡𝐻𝑊	Exp	38 0.070
	≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T < 200 GeV, Med-purity ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T < 200 GeV, High-purity	Exp Exp	166 0.713 48 1.47	Low-purity top	Exp	500 0.870
	≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T < 200 GeV, Med-purity	Exp	142 0.270

Table 7 :

 7 Expected contributions from the main sources of systematic uncertainty to the total uncertainty in the measurement of the cross-section times 𝐻 → 𝛾𝛾 branching ratio for each of of the main Higgs boson production processes. The uncertainty from each source (Δ𝜎) is shown as a fraction of the total expected cross-section (𝜎).

	ggF + 𝑏 bH	VBF	𝑊 𝐻	𝑍 𝐻	𝑡𝑡𝐻	𝑡𝐻

Table 8 :

 8 Summary of the leading sources of systematic uncertainty in the measurement of the Higgs boson signal strength.

	Uncertainty source	Δ𝜇 [%]
	Theory uncertainties
	Higher-Order QCD Terms	±3.8
	Branching Ratio	±3.0
	Underlying Event and Parton Shower	±2.5
	PDF and 𝛼 𝑠	±2.1
	Matrix Element	±1.0
	Modeling of Heavy Flavor Jets in non-𝑡 t𝐻 Processes < ±1
	Experimental uncertainties
	Photon energy resolution	±2.8
	Photon efficiency	±2.6
	Luminosity	±1.8
	Pile-up	±1.5
	Background modelling	±1.3
	Photon energy scale	< ±1
	Jet/𝐸 miss T Flavour tagging	< ±1 < ±1
	Leptons	< ±1
	Higgs boson mass	< ±1

Table 9 :

 9 Best-fit values and uncertainties for the production cross-sections of the Higgs boson times the 𝐻 → 𝛾𝛾 branching ratio. The total uncertainties are decomposed into statistical (Stat.) and systematic (Syst.) uncertainties. SM predictions are shown for the cross-section of each production process. These are obtained from the total cross-sections and associated uncertainties reported in Ref.[START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF], multiplied by an acceptance factor for the region |𝑦 𝐻 | < 2.5 computed using the Higgs boson simulation samples described in Section 3.2.

	Process	Value	Uncertainty [fb]	SM pred.
	(|𝑦 𝐻 | < 2.5)	[fb] Total Stat. Syst.	[fb]
	ggF+𝑏 b𝐻	106	+10 -10	+8 -8	+6 -6	102 +6 -6
	VBF	9.5 +2.2 -1.9	+1.5 -1.4	+1.7 -1.4	7.9 +0.2 -0.2
	𝑊 𝐻	4.2 +1.5 -1.4	+1.5 -1.4	+0.4 -0.2	2.8 +0.1 -0.1
	𝑍 𝐻	-0.4 +1.1 -1.0	+1.1 -1.0	+0.2 -0.3	1.8 +0.1 -0.1
	𝑡𝑡𝐻	1.0 +0.4 -0.3	+0.3 -0.3	+0.1 -0.1	1.1 +0.1 -0.1
	𝑡𝐻	0.5 +0.8 -0.6	+0.7 -0.6	+0.3 -0.2	0.19 +0.01 -0.02

  𝑖 × 𝐵 𝛾𝛾 ) 𝑔𝑔 → 𝐻, ≥ 2-jets, 𝑚 𝑗 𝑗 < 350 GeV, 𝑝 𝐻 T < 120 GeV 4

		ATLAS γ γ → H	H m	-1 =13 TeV, 139 fb s |<2.5 H = 125.09 GeV |y
			Obs + Tot. Unc.	Syst. unc.	SM + Theo. unc.	p-value = 93%
	𝑔𝑔 → 𝐻, 0-jet, 𝑝 𝐻 T < 10 GeV 𝑔𝑔 → 𝐻, 0-jet, 𝑝 𝐻 T ≥ 10 GeV 𝑔𝑔 → 𝐻, 1-jet, 𝑝 𝐻 T < 60 GeV 𝑔𝑔 → 𝐻, 1-jet, 60 ≤ 𝑝 𝐻 T < 120 GeV 𝑔𝑔 → 𝐻, 1-jet, 120 ≤ 𝑝 𝐻 T < 200 GeV 𝑔𝑔 → 𝐻, ≥ 2-jets, 𝑚 𝑗 𝑗 < 350 GeV, 120 ≤ 𝑝 𝐻 T < 200 GeV 𝑔𝑔 → 𝐻, 200 ≤ 𝑝 𝐻 T < 300 GeV 𝑔𝑔 → 𝐻, 300 ≤ 𝑝 𝐻 T < 450 GeV 𝑔𝑔 → 𝐻, 𝑝 𝐻 T ≥ 450 GeV 𝑞𝑞 → 𝐻𝑞𝑞 , ≤ 1-jet and 𝑉 𝐻-veto 𝑞𝑞 → 𝐻𝑞𝑞 , 𝑉 𝐻-had 𝑞𝑞 → 𝐻𝑞𝑞 , ≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T < 200 GeV T < 200 GeV T < 200 GeV T ≥ 200 GeV 300 ≥ H T ttH, p < 300 H T p ≤ ttH, 200 < 200 H T p ≤ ttH, 120 < 120 H T p ≤ ttH, 60 < 60 H T ttH, p 150 ≥ V t , p ν ν Hll/ → pp < 150 V t , p ν ν Hll/ → pp 150 ≥ V t , p ν Hl → qq < 150 V t , p ν Hl → qq 200 ≥ H T 1000, p ≥ jj 2-jets, m ≥ Hqq', → qq' 200 ≥ H T < 1000, p jj m ≤ 2-jets, 350 ≥ Hqq', → qq' < 200 H T 1000, p ≥ jj 2-jets, m ≥ Hqq', → qq' < 200 H T < 1000, p jj m ≤ 2-jets, 700 ≥ Hqq', → qq' < 200 H T < 700, p jj m ≤ 2-jets, 350 ≥ Hqq', → qq' 2-jets, VH-had ≥ Hqq', → qq' 1-jet and VH-Veto ≤ Hqq', → qq' 450 ≥ H T H, p → gg < 450 H T p ≤ H, 300 → gg < 300 H T p ≤ H, 200 → gg < 200 H T 350, p ≥ jj 2-jets, m ≥ H, → gg < 200 H T p ≤ < 350, 120 jj 2-jets, m ≥ H, → gg < 120 H T < 350, p jj 2-jets, m ≥ H, → gg < 200 H T p ≤ H, 1-jet, 120 → gg < 120 H T p ≤ H, 1-jet, 60 → gg < 60 H T H, 1-jet, p → gg < 200 H T p ≤ H, 0-jet, 10 → gg < 10 H T H, 0-jet, p → gg	Value [fb] 10 58 16 11 1.6 2.8 1.6 0.04 +0.13 Uncertainty [fb] Total Stat. Syst. +4 -4 +4 -4 +2 -1 +9 -8 +7 -7 +5 -4 +5 -5 +5 -5 +2 -2 +4 -3 +3 -3 +2 -2 +0.9 -0.9 +0.9 -0.8 +0.4 -0.2 +4 -3 +3 -3 +1 -1 +1.0 -1.0 +1.0 -1.0 +0.3 -0.2 +2 -2 +2 -2 +1 -1 +0.4 -0.4 +0.4 -0.4 +0.2 -0.1 -0.11 +0.12 -0.11 +0.03 -0.03 0.09 +0.06 -0.05 +0.06 -0.05 +0.02 -0.01 6 +6 -5 +6 -5 +2 -1 0.19 +0.85 -0.73 +0.83 -0.71 +0.17 -0.17 1.5 +0.9 -0.7 +0.7 -0.6 +0.6 -0.3 0.8 +0.5 -0.4 +0.4 -0.3 +0.2 -0.1 1.2 +0.4 -0.4 +0.3 -0.3 +0.3 -0.2 0.04 +0.12 -0.10 +0.12 -0.10 +0.02 -0.02 Tot. Stat. Syst. SM prediction [fb] 15 +2 -2 47 +4 -4 15 +2 -2 10 +1 -1 1.7 +0.3 -0.3 7 +1 -1 2.1 +0.5 -0.5 2.0 +0.5 -0.5 1.0 +0.2 -0.2 0.24 +0.06 -0.06 0.04 +0.01 -0.01 6.6 +0.2 -0.2 1.16 +0.04 -0.04 1.22 +0.04 -0.04 0.58 +0.02 -0.02 1.00 +0.03 -0.03 0.167 +0.005 -0.005 0.7 -0.9 + 1.1 0.7 -0.9 + 0.1 -0.2 ( ) + 0.6 -0.8 + 1.2 0.6 -0.8 + 0.1 -0.1 ( ) + 0.5 -0.6 + 0.6 0.5 -0.6 + 0.1 -0.1 ( ) + 0.5 -0.6 + 0.8 0.5 -0.6 + 0.0 -0.1 ( ) + 0.7 -0.8 + 0.8 0.7 -0.8 + 0.1 -0.1 ( ) + 0.9 -1.1 + 0.4 0.9 -1.1 + 0.2 -0.2 ( ) + 0.2 -0.9 + -0.6 0.2 -0.9 + 0.0 -0.1 ( ) + 0.9 -1.1 + 1.6 0.9 -1.1 + 0.1 -0.1 ( ) + 0.7 -0.8 + 1.8 0.7 -0.8 + 0.1 -0.2 ( ) + 0.5 -0.6 + 1.6 0.5 -0.6 + 0.2 -0.3 ( ) + 0.6 -0.7 + 0.2 0.6 -0.7 + 0.1 -0.1 ( ) + 0.35 -0.43 + 1.19 0.30 -0.33 + 0.19 -0.28 ( ) + 0.6 -0.8 + 1.4 0.6 -0.7 + 0.2 -0.4 ( ) + 0.6 -0.8 + 1.2 0.5 -0.6 + 0.2 -0.5 ( ) + 0.6 -0.7 + 0.2 0.6 -0.7 + 0.1 -0.1 ( ) + 0.8 -0.9 + 0.9 0.8 -0.8 + 0.2 -0.3 ( ) + 1.1 -1.4 + 2.1 1.1 -1.4 + 0.2 -0.4 ( ) + 0.5 -0.5 + 0.2 0.4 -0.5 + 0.1 -0.1 ( ) + 0.4 -0.4 + 1.6 0.4 -0.4 + 0.1 -0.2 ( ) + 0.9 -0.9 + 1.0 0.8 -0.8 + 0.4 -0.3 ( ) + 0.5 -0.5 + 1.3 0.4 -0.5 + 0.1 -0.1 ( ) + 0.5 -0.5 + 0.6 0.5 -0.5 + 0.1 -0.2 ( ) + 0.5 -0.5 + 1.0 0.5 -0.5 + 0.1 -0.2 ( ) + 0.34 -0.38 + 1.11 0.30 -0.30 + 0.16 -0.22 ( ) + 0.35 -0.36 + 1.07 0.34 -0.34 + 0.11 -0.14 ( ) + 0.17 -0.18 + 1.23 0.15 -0.15 + 0.09 -0.10 ( ) + 0.27 -0.28 + 0.67 0.25 -0.25 + 0.10 -0.13 + ( )
	𝑞𝑞 → 𝐻ℓ𝜈, 𝑝 𝑉 T < 150 GeV tH	1 -	0	T ≥ 200 GeV 1	2	0.27 +0.11 -0.09 1.4 +0.6 -0.6 3	+0.10 -0.08 +0.6 -0.6 4 3 -4 + 2	+0.05 -0.04 +0.1 -0.1 5 3 -4 + (	-+	0.166 +0.005 -0.005 0.79 +0.02 -0.02 6 1 1 )
	𝑞𝑞 → 𝐻ℓ𝜈, 𝑝 𝑉 T ≥ 150 GeV 𝑝 𝑝 → 𝐻ℓℓ/𝜈 ν, 𝑝 𝑉 T < 150 GeV	2 -			0	2		0.20 +0.13 -0.11 -0.29 +0.40 -0.08 4 6	+0.13 -0.11 +0.39 -0.08	+0.02 -0.01 +0.07 -0.00 γ γ B ⋅ σ ( 8	0.121 +0.005 -0.005 0.45 +0.02 -0.02 10 SM ) γ γ B ⋅ σ )/(
	𝑝 𝑝 → 𝐻ℓℓ/𝜈 ν, 𝑝 𝑉 T ≥ 150 GeV							0.04 +0.10 -0.08	+0.10 -0.08	+0.02 -0.02	0.09 +0.01 -0.01
	𝑡𝑡𝐻, 𝑝 𝐻 T < 60 GeV							0.22 +0.21 -0.18	+0.21 -0.18	+0.03 -0.01	0.27 +0.04 -0.04
	𝑡𝑡𝐻, 60 ≤ 𝑝 𝐻 T < 120 GeV							0.32 +0.23 -0.20	+0.23 -0.20	+0.04 -0.02	0.40 +0.05 -0.05
	𝑡𝑡𝐻, 120 ≤ 𝑝 𝐻 T < 200 GeV							0.18 +0.18 -0.15	+0.17 -0.15	+0.04 -0.02	0.29 +0.04 -0.04
	𝑡𝑡𝐻, 200 ≤ 𝑝 𝐻 T < 300 GeV							0.14 +0.09 -0.07	+0.09 -0.07	+0.01 -0.01	0.12 +0.02 -0.02
	𝑡𝑡𝐻, 𝑝 𝐻 T ≥ 300 GeV							0.06 +0.05 -0.04	+0.05 -0.04	+0.01 -0.01	0.06 +0.01 -0.01
	𝑡𝐻							0.4	+0.8 -0.6	+0.7 -0.6	+0.2 -0.2	0.19 +0.01 -0.02

𝑔𝑔 → 𝐻, ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 350 GeV, 𝑝 𝐻 T < 200 GeV 2 𝑞𝑞 → 𝐻𝑞𝑞 , ≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 𝑞𝑞 → 𝐻𝑞𝑞 , ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 𝑞𝑞 → 𝐻𝑞𝑞 , ≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 𝑞𝑞 → 𝐻𝑞𝑞 , ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻

Table 11 :

 11 Wilson coefficients 𝑐 𝑖 and corresponding dimension-6 SMEFT operators O 𝑖 used in this analysis. The notations follow that of Ref.[START_REF] Brivio | SMEFTsim 3.0 -a practical guide[END_REF]. Hermitian conjugates of non-Hermitian operators are implicitly considered in addition to the expression shown in the table. The operators indicated by a checkmark are the ones included in the measurement, due to having a significant impact on STXS cross-sections or on the 𝐻 → 𝛾𝛾 branching ratio.

	Coeff.	Operator		Incl. Coeff.	Operator	Incl.
	𝑐 𝐺	𝑓 𝐴𝐵𝐶 𝐺 𝐴𝜈 𝜇 𝐺	𝐵𝜌 𝜈 𝐺	𝐶 𝜇 𝜌	𝑐

  Figure17: Summary of the 68% CL (solid lines) and 95% CL (dashed lines) intervals for individual measurements of SMEFT parameters observed in data. In each case, SMEFT parameters other than the one measured are fixed to 0. Blue and green curves correspond respectively to the linear and linear+quadratic SMEFT parameterizations. For presentation purposes, some parameters are scaled by a factor indicated below the parameter name. Results are not shown for coefficients 𝑐 𝑘 where one or more of the intervals extend beyond the |𝑐 𝑘 | ≤ 20 region, which is considered to be the region of validity of the SMEFT framework.

	ATLAS			Linear best-fit (obs.)	68% (obs.)	95% (obs.)
	s	= 13TeV, 139 fb -1		Linear+quadratic best-fit (obs.)	68% (obs.)	95% (obs.)
	γ → H	γ	H , m	= 125.09 GeV,	Λ	= 1 TeV
							×	1

Table 12 :

 12 Summary of the EV𝑛 parameter measurements in the linear and linear+quadratic parameterizations. The ranges correspond to 68% CL intervals. All the EV𝑛 parameters are free to vary in the fits. The upper bound of the intervals reported for EV9 and EV12 in the linear parameterization (shown in bold text) are truncated at the value for which the model pdf becomes negative.

	Parameter		Linear	Linear+quadratic
		Value	Uncertainty Value	Uncertainty
	EV1	-0.0008 +0.0017 -0.0018	0.0043 +0.0067 -0.0095
	EV2	0.0004 +0.0059 -0.0055	-0.0061 +0.0084 -0.0086
	EV3	0.039	+0.095 -0.10	0.035	+0.11 -0.081
	EV4	-0.035	+0.25 -0.22	-0.079	+0.29 -0.35
	EV5	-0.22	+0.59 -0.62	0.29	+0.30 -0.69
	EV6	0.19	+0.81 -0.80	0.011	+0.79 -0.47
	EV7	-1.7	+1.0 -0.96	-0.91	+1.2 -0.53
	EV8	-0.65	+3.5 -3.2	-1.2	+2.5 -1.0
	EV9	7.5	+2.5 -5.2	1.7	+1.4 -1.6
	EV10	0.48	+6.7 -8.5	0.42	+0.46 -0.60
	EV11	-5.6	+9.4 -9.6	0.045	+0.47 -0.21
	EV12	2.6	+12 -13	1.2	+0.81 -1.0

  STXS region (𝜎𝑖 × 𝐵 𝛾𝛾 )

		Value	Uncertainty [fb]	SM prediction
		[fb]	Total Stat.	Syst.	[fb]
	𝑔𝑔 → 𝐻, 0-jet, 𝑝 𝐻 T < 10 GeV	10	+4 -4	+4 -4	+2 -1	15 +2 -2
	𝑔𝑔 → 𝐻, 0-jet, 𝑝 𝐻 T ≥ 10 GeV	59	+9 -8	+7 -7	+5 -4	47 +4 -4
	𝑔𝑔 → 𝐻, 1-jet, 𝑝 𝐻 T < 60 GeV	17	+6 -5	+5 -5	+2 -2	15 +2 -2
	𝑔𝑔 → 𝐻, 1-jet, 60 ≤ 𝑝 𝐻 T < 120 GeV	13	+4 -4	+3 -3	+2 -2	10 +1 -1
	𝑔𝑔 → 𝐻, 1-jet, 120 ≤ 𝑝 𝐻 T < 200 GeV	1.9	+1.0 -0.9	+0.9 -0.9	+0.4 -0.2	1.7 +0.3 -0.3
	𝑔𝑔 → 𝐻, ≥ 2-jets, 𝑚 𝑗 𝑗 < 350 GeV, 𝑝 𝐻 T < 60 GeV	0.3	+3.1 -2.9	+2.9 -2.8	+1.1 -0.5	2.7 +0.6 -0.6
	𝑔𝑔 → 𝐻, ≥ 2-jets, 𝑚 𝑗 𝑗 < 350 GeV, 60 ≤ 𝑝 𝐻 T < 120 GeV	1.3	+2.5 -2.5	+2.4 -2.4	+0.7 -0.6	4.1 +0.9 -0.9
	𝑔𝑔 → 𝐻, ≥ 2-jets, 𝑚 𝑗 𝑗 < 350 GeV, 120 ≤ 𝑝 𝐻 T < 200 GeV	2.2	+1.1 -1.1	+1.1 -1.0	+0.3 -0.2	2.1 +0.5 -0.5
	𝑔𝑔 → 𝐻, ≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T < 200 GeV	2.7	+1.6 -1.5	+1.5 -1.5	+0.5 -0.3	1.4 +0.3 -0.3
			+0.7 -0.8	+0.7 -0.7	+0.2 -0.3	0.3 +0.10 -0.10
			+0.6 -0.6	+0.6 -0.5	+0.2 -0.2	0.28 +0.08 -0.08
	𝑔𝑔 → 𝐻, 200 ≤ 𝑝 𝐻 T < 300 GeV	1.5	+0.5 -0.4	+0.4 -0.4	+0.2 -0.1	1.0 +0.2 -0.2
	𝑔𝑔 → 𝐻, 300 ≤ 𝑝 𝐻 T < 450 GeV	0.01	+0.13 -0.12	+0.13 -0.11	+0.03 -0.04	0.24 +0.06 -0.06
	𝑔𝑔 → 𝐻, 𝑝 𝐻 T ≥ 450 GeV	0.08	+0.06 -0.05	+0.06 -0.05	+0.02 -0.01	0.04 +0.01 -0.01
	𝑞𝑞 → 𝐻𝑞𝑞 , ≤ 1-jet	0.7	+4.9 -4.4	+4.8 -4.2	+1.2 -1.4	4.9 +0.2 -0.2
	𝑞𝑞 → 𝐻𝑞𝑞 , 𝑉 𝐻-veto	4	+3 -3	+3 -2	+1 -1	1.67 +0.05 -0.05
	𝑞𝑞 → 𝐻𝑞𝑞 , 𝑉 𝐻-had	0.4	+0.9 -0.8	+0.9 -0.7	+0.2 -0.2	1.16 +0.04 -0.04
	𝑞𝑞 → 𝐻𝑞𝑞 , ≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T < 200 GeV	1.2	+0.9 -0.7	+0.8 -0.7	+0.5 -0.2	1.22 +0.04 -0.04
	𝑞𝑞 → 𝐻𝑞𝑞 , ≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T < 200 GeV	1.1	+0.6 -0.5	+0.5 -0.4	+0.4 -0.2	0.58 +0.02 -0.02
	𝑞𝑞 → 𝐻𝑞𝑞 , ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T < 200 GeV	1.4	+0.5 -0.4	+0.4 -0.4	+0.4 -0.2	1.00 +0.03 -0.03
	𝑞𝑞 → 𝐻𝑞𝑞 , ≥ 2-jets, 350 ≤ 𝑚 𝑗 𝑗 < 700 GeV, 𝑝 𝐻 T ≥ 200 GeV	0.12	+0.15 -0.12	+0.15 -0.12	+0.03 -0.03	0.100 +0.003 -0.003
	𝑞𝑞 → 𝐻𝑞𝑞 , ≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T ≥ 200 GeV -0.009 +0.057 -0.041	+0.057 -0.041	+0.006 -0.008	0.067 +0.002 -0.002
	𝑞𝑞 → 𝐻𝑞𝑞 , ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T ≥ 200 GeV	0.28	+0.11 -0.09	+0.10 -0.09	+0.05 -0.04	0.166 +0.005 -0.005
	𝑞𝑞 → 𝐻ℓ𝜈, 𝑝 𝑉 T < 150 GeV	1.4	+0.6 -0.6	+0.6 -0.6	+0.1 -0.1	0.79 +0.02 -0.02
	𝑞𝑞 → 𝐻ℓ𝜈, 𝑝 𝑉 T ≥ 150 GeV	0.20	+0.13 -0.11	+0.13 -0.11	+0.02 -0.01	0.121 +0.005 -0.005
	𝑝 𝑝 → 𝐻ℓℓ/𝜈 ν, 𝑝 𝑉 T < 150 GeV	-0.29	+0.40 -0.07	+0.39 -0.07	+0.07 -0.00	0.45 +0.02 -0.02
	𝑝 𝑝 → 𝐻ℓℓ/𝜈 ν, 𝑝 𝑉 T ≥ 150 GeV	0.04	+0.10 -0.08	+0.10 -0.08	+0.02 -0.02	0.09 +0.01 -0.01
	𝑡𝑡𝐻, 𝑝 𝐻 T < 60 GeV	0.21	+0.21 -0.18	+0.21 -0.18	+0.03 -0.01	0.27 +0.04 -0.04
	𝑡𝑡𝐻, 60 ≤ 𝑝 𝐻 T < 120 GeV	0.32	+0.23 -0.20	+0.23 -0.20	+0.04 -0.02	0.40 +0.05 -0.05
	𝑡𝑡𝐻, 120 ≤ 𝑝 𝐻 T < 200 GeV	0.18	+0.18 -0.15	+0.17 -0.15	+0.04 -0.02	0.29 +0.04 -0.04
	𝑡𝑡𝐻, 200 ≤ 𝑝 𝐻 T < 300 GeV	0.14	+0.09 -0.07	+0.09 -0.07	+0.01 -0.01	0.12 +0.02 -0.02
	𝑡𝑡𝐻, 𝑝 𝐻 T ≥ 300 GeV	0.06	+0.05 -0.04	+0.05 -0.04	+0.01 -0.01	0.06 +0.01 -0.01
	𝑡𝐻	0.36	+0.76 -0.60	+0.72 -0.57	+0.24 -0.17	0.19 +0.01 -0.02

𝑔𝑔 → 𝐻, ≥ 2-jets, 700 ≤ 𝑚 𝑗 𝑗 < 1000 GeV, 𝑝 𝐻 T < 200 GeV -0.2 𝑔𝑔 → 𝐻, ≥ 2-jets, 𝑚 𝑗 𝑗 ≥ 1000 GeV, 𝑝 𝐻 T < 200 GeV -0.3

Table 14 :

 14 Parameterization of Higgs boson production cross-sections 𝜎 𝑖 , the partial decay widths Γ 𝛾𝛾 , Γ 𝑔𝑔 and Γ 𝑍 𝛾 of the 𝐻→ 𝛾𝛾, 𝐻 → 𝑔𝑔 and 𝐻 → 𝑍𝛾 decays, respectively, and the total width Γ 𝐻 , normalized to their SM values, as functions of the coupling-strength modifiers 𝜅. The coefficients for 𝜎(𝑡𝐻𝑊) and 𝜎(𝑡𝐻𝑞𝑏) include acceptance effects that differ between analysis categories as described in the text. Other coefficients are derived following the methodology in Refs.[START_REF] De Florian | Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector[END_REF][START_REF] Andersen | Handbook of LHC Higgs Cross Sections: 3. Higgs Properties[END_REF]. 𝑡 -1.903 𝜅 𝑍 𝜅 𝑡 -0.011 𝜅 𝑍 𝜅 𝑏 + 0.003 𝜅 𝑡 𝜅 𝑏 𝑊 + 0.072 𝜅 2 𝑡 -0.674 𝜅 𝑊 𝜅 𝑡 + 0.009 𝜅 𝑊 𝜅 𝜏 + 0.008 𝜅 𝑊 𝜅 𝑏 -0.002 𝜅 𝑡 𝜅 𝑏 -0.002 𝜅 𝑡 𝜅 𝜏 𝑡 -0.125 𝜅 𝑊 𝜅 𝑡 + 0.003 𝜅 𝑊 𝜅 𝑏

	Production	Main	Effective
				Resolved modifier
	cross-section	interference modifier
	𝜎(ggF)	𝑡-𝑏	𝜅 2 𝑔	1.040 𝜅 2 𝑡 + 0.002𝜅 2 𝑏 -0.038 𝜅 𝑡 𝜅 𝑏 -0.005 𝜅 𝑡 𝜅 𝑐
	𝜎(VBF)	-	-	0.733 𝜅 2 𝑊 + 0.267 𝜅 2 𝑍
	𝜎(𝑞 q → 𝑍 𝐻)	-	-	𝜅 2 𝑍
	𝜎(𝑔𝑔 → 𝑍 𝐻)	𝑡-𝑍	-	2.456 𝜅 2 𝑍 + 0.456 𝜅 2
	𝜎(𝑊 𝐻)	-	-	𝜅 2 𝑊
	𝜎(𝑡𝑡𝐻)	-	-	𝜅 2 𝑡
	𝜎(𝑡𝐻𝑊)	𝑡-𝑊	-	𝐴 𝜅 2 𝑡 + 𝐵 𝜅 2 𝑊 + 𝐶 𝜅 𝑡 𝜅 𝑊 , category-dependent
	𝜎(𝑡𝐻𝑞𝑏)	𝑡-𝑊	-	𝐴 𝜅 2 𝑡 + 𝐵 𝜅 2 𝑊 + 𝐶 𝜅 𝑡 𝜅 𝑊 , category-dependent
	𝜎(𝑏 b𝐻)	-	-	𝜅 2 𝑏
	Partial and total decay widths
	Γ 𝛾𝛾	𝑡-𝑊	𝛾 𝜅 2	1.589 𝜅 2
	Γ 𝑔𝑔	𝑡-𝑏	𝜅 2 𝑔	1.111 𝜅 2 𝑡 + 0.012 𝜅 2 𝑏 -0.123 𝜅 𝑡 𝜅 𝑏
	Γ 𝑍 𝛾	𝑡-𝑊		1.118 𝜅 2 𝑊 + 0.004 𝜅 2
				0.581 𝜅 2 𝑏 + 0.215 𝜅 2 𝑊 + 0.063 𝜅 2 𝜏
				+ 0.026 𝜅 2 𝑍 + 0.029 𝜅 2 𝑐 + 0.0023 𝜅 2 𝛾
	Γ 𝐻	-	𝜅 2 𝐻	+ 0.0004 𝜅 2 𝑠 + 0.00022 𝜅 2 𝜇
				+ 0.082 (Γ 𝑔𝑔 /Γ SM 𝑔𝑔 )
				+ 0.0015 (Γ 𝑍 𝛾 /Γ SM 𝑍 𝛾 )

B.

2 Parameterization with universal coupling modifiers to weak gauge bosons and fermions

  In this model, two universal coupling modifiers are considered: 𝜅 𝑉 = 𝜅 𝑊 = 𝜅 𝑍 which modifies Higgs boson couplings to gauge bosons, and 𝜅 𝐹 = 𝜅 𝑡 = 𝜅 𝑏 = 𝜅 𝑐 = 𝜅 𝜏 = 𝜅 𝜇 , modifying couplings to fermions. The 𝑔𝑔 → 𝐻, 𝐻 → 𝛾𝛾 and 𝑔𝑔 → 𝑍 𝐻 loops are described using their resolved parameterizations as a function of 𝜅 𝑉 and 𝜅 𝐹 . The measurement in the plane of (𝜅 𝑉 , 𝜅 𝐹 ) is shown in Figure23. Only the region 𝜅 𝐹 > 0 is considered, since 𝜅 𝐹 < 0 was excluded with a significance larger than 4𝜎 in analyses of the Run 1 data set[START_REF]Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC 𝑝 𝑝 collision data at √ 𝑠 = 7 and 8 TeV[END_REF]. The best-fit values in data are 𝜅 𝑉 = 1.02 + 0.06 Negative log-likelihood contours at 68% CL (dashed line) and 95% CL (solid line) in the (𝜅 𝑉 , 𝜅 𝐹 ) plane of modifiers applied to Higgs boson couplings to gauge bosons (𝜅 𝑉 ) and fermions (𝜅 𝐹 ). Loop processes and the Higgs boson total width Γ 𝐻 are parameterized as a function of 𝜅 𝑉 and 𝜅 𝐹 . The best-fit point is indicated by a cross, and the SM prediction by a star.

							-0.05
							𝜅 𝐹 = 1.00 + 0.16 -0.13 .
	A linear correlation coefficient of 77% between the parameters is observed.
	F
	κ	1.8	ATLAS	SM
			s	-1 = 13 TeV, 139 fb	Observed best fit
		1.6	H	→	γ	γ	Observed 68 % CL Observed 95 % CL
		1.4
		1.2
		1
		0.8
		V 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 κ 0.6
	Figure 23:

Table 15 :

 15 Best-fit values and uncertainties in the coupling-modifier ratio model. The second column expresses the measured parameters in terms of the coupling modifiers. The SM corresponds to 𝜅 𝑔𝛾 = 𝜆 𝑡𝑔 = 𝜆 𝑉 𝑔 = 1. 𝜅 𝑔 𝜅 𝛾 /𝜅 𝐻 1.02 ± 0.06 𝜆 𝑉 𝑔 𝜅 𝑉 /𝜅 𝑔 1.01 ± 0.11 𝜆 𝑡𝑔 𝜅 𝑡 /𝜅 𝑔 0.95 + 0.15

	Definition in terms
	Parameter	Result
	of 𝜅 modifiers
	𝜅 𝑔𝛾
		-0.16

Table 16 :

 16 Measurement results for each SMEFT parameter individually, obtained from profile-likelihood scans in which other SMEFT parameters are fixed to 0. Confidence intervals at 68% and 95% CL are reported in data (observed) and in an Asimov dataset generated under the SM hypothesis (expected). Results in the linear and linear+quadratic SMEFT parameterizations are shown, for a scale Λ = 1 TeV. Confidence intervals for each parameter 𝑐 𝑘 are reported, except if they extend beyond the |𝑐 𝑘 | ≤ 20 region where the SMEFT framework is considered valid.

					Observed				Expected
	Parameter		linear		linear+quadratic		linear	linear+quadratic
			Value	Uncertainty	Value	Uncertainty	Uncertainty	Uncertainty
				68% CL 95% CL		68% CL 95% CL 68% CL 95% CL 68% CL 95% CL
	𝑐 𝐻𝑊	-0.0035 +0.0071 -0.0077	+0.014 -0.016	-0.0034 +0.0071 -0.0073	+0.014 -0.014	+0.0070 -0.0075	+0.013 -0.015	+0.0072 -0.0072	+0.014 -0.014
	𝑐 𝐻𝐵	-0.0011 +0.0023 -0.0025	+0.0044 -0.0050	-0.0011 +0.0023 -0.0023	+0.0046 -0.0046	+0.0022 -0.0024	+0.0043 -0.0049	+0.0023 -0.0023	+0.0046 -0.0046
	𝑐 𝐻 𝑊𝐵	0.0020 +0.0044 -0.0042	+0.0090 -0.0079	0.0019 +0.0042 -0.0041	+0.0083 -0.0081	+0.0043 -0.0041	+0.0088 -0.0077	+0.0042 -0.0042	+0.0083 -0.0082
	𝑐 𝐻𝐺	0.0011 +0.0030 -0.0028	+0.0062 -0.0053	0.0011 +0.0029 -0.0028	+0.0059 -0.0055	+0.0030 -0.0027	+0.0061 -0.0052	+0.0029 -0.0028	+0.0059 -0.0054
	𝑐 𝑊	-0.047	+0.098 -0.11	+0.19 -0.21	-0.047	+0.098 -0.11	+0.19 -0.21	+0.096 -0.10	+0.18 -0.21	+0.096 -0.10	+0.18 -0.21
	𝑐 𝐺	0.32	+1.5 -1.2	+3.4 -2.0	0.077	+0.13 -0.30	+0.22 -0.40	+1.5 -1.1	+3.4 -1.9	+0.18 -0.20	+0.28 -0.30
	𝑐 𝑢𝑊	-0.039	+0.080 -0.087	+0.15 -0.18	-0.039	+0.080 -0.087	+0.15 -0.18	+0.079 -0.083	+0.15 -0.17	+0.079 -0.083	+0.15 -0.17
	𝑐 𝑢𝐵	-0.021	+0.043 -0.046	+0.082 -0.094	-0.021	+0.043 -0.046	+0.082 -0.094	+0.042 -0.045	+0.080 -0.092	+0.042 -0.045	+0.080 -0.092
	𝑐 𝑢𝐺	0.030	+0.078 -0.078	+0.16 -0.14	0.030	+0.077 -0.078	+0.16 -0.15	+0.079 -0.074	+0.16 -0.14	+0.078 -0.075	+0.16 -0.14
	𝑐 𝑢𝐻	-0.29	+1.4 -1.5	+2.7 -3.0	-0.30	+1.4 -1.6	+2.6 -3.3	+1.4 -1.5	+2.7 -3.0	+1.4 -1.5	+2.5 -3.3
	𝑐 𝑑𝐻	0.63	+1.4 -1.3	+2.9 -2.5	0.61	+1.3 -1.3	+2.5 -2.7	+1.4 -1.3	+2.8 -2.4	+1.3 -1.4	+2.5 -2.7
	𝑐 𝑒𝐻	5.8	+13 -12	-	1.9	+5.6 -5.7	+8.9 -8.9	+13 -12	-	+9.1 -5.2	+12 -8.4
	𝑐 (3) 𝐻 𝑞	-0.027	+0.091 -0.081	+0.19 -0.15	-0.037	+0.096 -0.21	+0.17 -0.34	+0.10 -0.089	+0.20 -0.17	+0.085 -0.12	+0.16 -0.29
	𝑐 (1) 𝐻 𝑞	1.9	+1.7 -2.0	+3.1 -4.1	0.029	+0.20 -0.22	+0.35 -0.37	+2.0 -2.3	+3.6 -4.8	+0.30 -0.28	+0.44 -0.41
	𝑐 (3) 𝐻 𝑙 𝑐 (1) 𝐻 𝑙	-0.15 -	+0.28 -0.28 -	+0.52 -0.58 -	-0.15 4.4	+0.28 -0.28 +6.8 -6.9	+0.52 -0.58 +12 -12	+0.26 -0.28 +13 -15	+0.50 -0.57 -	+0.26 -0.28 +16 -7.8	+0.50 -0.57 -
	𝑐	𝐻 𝑢	-0.97	+0.79 -0.67	+1.7 -1.2	-0.14	+0.30 -0.24	+0.51 -0.41	+0.96 -0.82	+2.0 -1.5	+0.32 -0.43	+0.49 -0.61
	𝑐	𝐻 𝑑	3.4	+2.2 -2.6	+4.0 -5.4	0.070	+0.33 -0.36	+0.55 -0.60	+2.7 -3.1	+4.9 -6.5	+0.51 -0.44	+0.73 -0.67
	𝑐 𝐻	0.68	+1.5 -1.4	+3.1 -2.7	0.63	+1.4 -1.4	+2.7 -2.7	+1.5 -1.4	+3.0 -2.6	+1.4 -1.4	+2.8 -2.9
	𝑐 𝐻𝐷	-0.21	+0.42 -0.44	+0.79 -0.91	-0.21	+0.42 -0.45	+0.79 -0.93	+0.41 -0.43	+0.77 -0.88	+0.40 -0.43	+0.76 -0.89
	𝑐 (3) 𝑞𝑞 𝑐 (3) 𝑞𝑞 𝑐 (1) 𝑞𝑞 𝑐 (1) 𝑞𝑞	0.72 0.042 2.0 0.097	+3.4 -2.8 +0.37 -0.28 +14 -11 +0.79 -0.60	+7.3 -5.0 +0.83 -0.50 -+1.7 -1.0	-0.20 -0.30 -0.20 -0.50	+0.55 -0.18 +0.52 -0.19 +0.69 -0.30 +0.92 -0.31	+0.69 -0.32 +0.67 -0.34 +0.90 -0.52 +1.2 -0.57	+3.2 -2.6 +0.38 -0.29 +14 -11 +0.79 -0.61	+6.8 -4.7 +0.84 -0.49 -+1.8 -1.0	+0.29 -0.31 +0.21 -0.44 +0.44 -0.45 +0.39 -0.73	+0.43 -0.46 +0.36 -0.60 +0.67 -0.68 +0.66 -1.0
	𝑐	𝑙𝑙	0.30	+0.53 -0.56	+1.1 -1.0	0.30	+0.52 -0.56	+1.1 -1.0	+0.55 -0.51	+1.1 -0.98	+0.54 -0.51	+1.1 -0.99
	𝑐 𝑢𝑢	1.4	+13 -9.9	-	-0.25	+0.85 -0.37	+1.1 -0.64	+13 -9.9	-	+0.53 -0.56	+0.81 -0.84
	𝑐 𝑢𝑢 𝑐 (1) 𝑞𝑢	0.098 -	+0.80 -0.61 -	+1.8 -1.1 -	-0.50 -0.30	+0.92 -0.31 +1.1 -0.48	+1.2 -0.57 +1.4 -0.81	+0.81 -0.61 -	+1.8 -1.1 -	+0.39 -0.72 +0.68 -0.70	+0.66 -1.0 +1.0 -1.0
	𝑐 (8) 𝑞𝑢 𝑐 (1) 𝑞𝑑	0.15 -	+1.3 -0.97 -	+2.8 -1.7 -	-1.8 0.75	+2.6 -0.82 +0.94 -2.4	+3.3 -1.5 +1.7 -3.2	+1.3 -0.97 -	+2.8 -1.7 -	+0.84 -2.3 +1.5 -1.5	+1.5 -3.0 +2.3 -2.3
	𝑐 (8) 𝑞𝑑	0.53	+5.5 -4.3	+12 -7.5	-2.3	+4.8 -2.3	+6.4 -3.9	+5.6 -4.3	+12 -7.5	+2.4 -4.1	+4.0 -5.8
	𝑐 (1) 𝑢𝑑 𝑐 (8) 𝑢𝑑	-0.53	-+5.5 -4.3	-+12 -7.5	0.75 -2.5	+0.93 -2.4 +5.1 -2.0	+1.7 -3.1 +6.7 -3.6	-+5.6 -4.4	-+12 -7.6	+1.5 -1.5 +2.4 -4.1	+2.3 -2.3 +4.0 -5.7

Table 18 :

 18 Measured values of the EV𝑛 parameters in data (observed) and in an Asimov data set generated under the SM hypothesis (expected). The linear SMEFT parameterization is used. Numbers in bold script indicate that the uncertainty band is truncated at the value for which the model pdf becomes negative.

	Model parameter		Observed		Expected
			Uncertainty	Uncertainty
		Value
			68% CL 95% CL 68% CL 95% CL
	EV1	-0.0008 +0.0017 -0.0018	+0.0032 -0.0037	+0.0016 -0.0018	+0.0031 -0.0036
	EV2	0.000	±0.006	+0.012 -0.010	+0.006 -0.005	+0.011 -0.010
	EV3	0.04	±0.10	+0.18 -0.21	+0.09 -0.10	+0.18 -0.20
	EV4	-0.04	+0.25 -0.22	+0.5 -0.4	+0.24 -0.21	+0.5 -0.4
	EV5	-0.2	±0.6	+1.2 -1.3	±0.6	+1.1 -1.3
	EV6	0.2	±0.8	+1.7 -1.6	+0.8 -0.7	±1.5
	EV7	-1.7	±1.0	+2.0 -1.3	+1.1 -1.0	+2.2 -2.1
	EV8	-0.7	+3.5 -3.2	+7 -6	+3.9 -3.4	+8 -7
	EV9	7.5	+2.5 -5.2	+2.5 -11	+5 -5	+10 -11
	EV10	0	+7 -9	+8 -19	+5 -7	+9 -16
	EV11	-6	+9 -10	+18 -19	±10	±19
	EV12	3	+12 -13	+12 -25	±12	±24

Table 19 :

 19 Measured values of the EV𝑛 parameters in data (observed) and in an Asimov data set generated under the SM hypothesis (expected). The linear+quadratic SMEFT parameterization is used.

	Model parameter		Observed		Expected
			Uncertainty	Uncertainty
		Value
			68% CL 95% CL 68% CL 95% CL
	EV1	0.004 +0.007 -0.010	+0.014 -0.049	+0.14 -0.01	+0.20 -0.03
	EV2	-0.006 +0.008 -0.009	+0.017 -0.030	+0.006 -0.095	+0.014 -0.16
	EV3	0.04	+0.11 -0.08	+0.37 -0.21	+0.14 -0.11	+0.40 -0.27
	EV4	-0.08	+0.29 -0.35	+0.5 -1.2	+0.23 -0.48	+0.5 -1.1
	EV5	0.29	+0.30 -0.69	+0.7 -2.2	+0.5 -0.6	+0.9 -1.7
	EV6	0.0	+0.8 -0.5	+1.7 -1.0	+0.9 -0.7	+2.2 -1.2
	EV7	-0.9	+1.2 -0.5	+1.4 -1.5	+0.7 -1.7	+1.2 -3.3
	EV8	-1.2	+2.5 -1.0	+10 -1.8	+3.1 -1.7	+9.0 -2.4
	EV9	1.7	+1.4 -1.6	+4.8 -3.3	+3.9 -1.8	+8.4 -2.9
	EV10	0.4	+0.5 -0.6	±1.2	+0.9 -0.5	+1.8 -0.8
	EV11	0.05	+0.47 -0.21	+1.6 -0.5	+0.7 -0.4	+1.8 -0.6
	EV12	1.2	+0.8 -1.0	+2.3 -2.1	+2.4 -1.1	+4.8 -1.8

Table 20 :

 20 Observed values of the EV𝑛 parameters in data together with their 68% CL and 95% CL intervals in data for the linear SMEFT parameterization including corrections to the 𝑊, 𝑍, Higgs boson and top quark propagators as described in the text. Numbers in bold script indicate than the uncertainty band is truncated at the value for which the probability distribution function of the fit becomes negative.

			Uncertainty
	Model parameter Value
			68% CL 95% CL
	EV1	0.0001 +0.0031 -0.0038	+0.006 -0.007
	EV2	0.000	±0.006	+0.012 -0.010
	EV3	0.05	±0.12	+0.22 -0.26
	EV4	-0.03	+0.25 -0.23	+0.5 -0.4
	EV5	-0.2	±0.6	+1.2 -1.3
	EV6	0.2	±0.8	+1.7 -1.6
	EV7	-2.0	+1.1 -0.9	+2.2 -0.9
	EV8	-0.5	+3.4 -3.2	+7 -6
	EV9	8.2	+2.5 -5.8	+2.5 -12
	EV10	1	+7 -9	+8 -19
	EV11	-5	+9 -10	+18 -19

In this paper, categories refers to event groupings defined from reconstructed quantities, while regions refers to the particle-level selections defined in the STXS framework.

The 𝜒 2 is computed in a background template uniformly binned over 105 < 𝑚 𝛾𝛾 < 160 GeV. It has

bins, and the number of degrees of freedom used in the computation is 54 -𝑁 pars , where the 𝑁 pars is the number of free function parameters. The normalization of the template removes one degree of freedom. The background 𝑚 𝛾𝛾 templates before the smoothing procedure are used.

Results are derived using both parameterizations, and their difference is considered to be indicative of the impact of the neglected higher-order terms in the SMEFT expansion.

The values of the 𝐴 and 𝐵 coefficients are generally obtained using the SMEFT [START_REF] Brivio | SMEFTsim 3.0 -a practical guide[END_REF][START_REF] Brivio | The SMEFTsim package, theory and tools[END_REF] and SMEFT@NLO [START_REF] Degrande | Automated one-loop computations in the standard model effective field theory[END_REF] programs. The coefficients are obtained by setting SMEFT parameters to non-zero values (one parameter at a time to compute 𝐴 coefficients, and in pairs to compute 𝐵 coefficients), and comparing the cross-sections obtained in this case with the ones for all coefficients set to 0. Events corresponding to each STXS region are selected using a RIVET routine [162]. SMEFT@NLO is used to obtain predictions for 𝑔𝑔 → 𝐻 and 𝑔𝑔 → 𝑍 𝐻 loop processes, while SMEFT is used for all other processes. The dependence of the Higgs boson total decay width on the SMEFT parameters is computed by considering all decays of the Higgs boson with up to four particles in the final state. The 𝐴 coefficients for the 𝐻→ 𝛾𝛾 decay are taken from an analytic calculation [START_REF] Dawson | Electroweak corrections to Higgs boson decays to 𝛾𝛾 and 𝑊 + 𝑊in standard model EFT[END_REF], which includes NLO electroweak corrections that are not implemented in the programs mentioned above. The coefficients are obtained for the full phase space of the decay, relying on the fact that they are only weakly dependent on acceptance, as discussed below. The 𝐵 coefficients for the 𝐻→ 𝛾𝛾 decay are computed using SMEFT .

The SMEFT operators considered in the analysis are shown in Table 11. Initially, 60 operators are considered but only 34 are found to have significant impact in at least one STXS region or on the 𝐻 → 𝛾𝛾 branching ratio, defined by a value above 0.01 for the corresponding 𝐴 coefficient. Only these 34 operators are considered in the measurements presented in this paper. The impact of the most relevant SMEFT parameter in the measured STXS regions is summarized in Figure 16. The acceptance and efficiency factors which are applied to the observed event yields to obtain the 𝜇 𝛾𝛾 𝑖 parameters, as shown in Eq. ( 1), can depend on the SMEFT parameters due to modifications of the Higgs signal characteristics within each STXS bin. The effect was studied for the main Warsaw basis operators affecting the measurement, and setting the corresponding Wilson coefficients to 1 individually is generally found to have an impact below 10% on the acceptance and efficiency factors. These changes are therefore neglected in the analysis.

Measurements of single SMEFT parameters

In the measurements presented in this section, one SMEFT parameter at a time is left free to vary, while the others are fixed to 0 as in the SM. This provides a measure of the sensitivity of the analysis for individual Wilson coefficients in the Warsaw basis, but the restrictive nature of this model limits the applicability of the measurements in probing effects beyond the SM.

The measurement results are summarized in Figure 17, and full results are provided in Table 16 in Appendix C. The SMEFT framework used in this measurement is considered valid only for parameter values of 𝑂 [START_REF]Measurements of Higgs boson properties in the diphoton decay channel with 36 fb -1 of 𝑝 𝑝 collision data at √ 𝑠 = 13[END_REF] or less, and confidence intervals that extend outside |𝑐 𝑘 | ≤ 20 are therefore not shown. Uncertainties in the parameter values range from below ±0.01 to above the |𝑐 𝑘 | = 20 threshold used to define the region of SMEFT validity. All values are compatible with the SM within measurement uncertainties. For parameters where the sensitivity mainly derives from inclusive event yields, such as 𝑐 𝐻 𝐺 , the linear and linear+quadratic parameterizations provide similar results. Conversely, operators with sensitivity to the high-𝑝 𝐻 T bins of the 𝑡𝑡𝐻 or 𝑔𝑔 → 𝐻 processes and the high-𝑝 𝑉 T regions of 𝑞 q → 𝑉 𝐻 show markedly smaller confidence intervals for the linear+quadratic case than for the linear case. The significant impact of the quadratic terms of the SMEFT parameterization in these cases may be indicative of significant effects from missing higher-order terms in the SMEFT expansion.

Conclusion

Higgs boson production is measured in the diphoton decay channel using 139 fb -1 of 13 TeV proton-proton collision data, corresponding to the full data set collected by ATLAS during Run 2 of the LHC.

The overall Higgs boson signal strength relative to its SM prediction is measured to be 𝜇 = 1.04 +0. 10 -0.09 = 1.04 ± 0.06 (stat.) +0.06 -0.05 (theory syst.) +0.05 -0.04 (exp. syst.).

in good agreement with the SM.

Cross-sections for ggF + 𝑏 b𝐻, VBF, 𝑊 𝐻, 𝑍 𝐻, 𝑡𝑡𝐻 and 𝑡𝐻 production are reported, with relative uncertainties of 10% for ggF + 𝑏 b𝐻, 22% for VBF, and 35% for 𝑊 𝐻 and 𝑡𝑡𝐻. An upper limit of ten times the SM prediction is set for the 𝑡𝐻 process. This represents the most stringent experimental constraint on 𝑡𝐻 production, superseding the previous ATLAS result from Run 2. A fine-grained description of Higgs boson production is provided by cross-section measurements in 28 phase-space regions defined within the STXS framework, including additional measurements at high values of 𝑝 𝐻 T and 𝑚 𝑗 𝑗 compared to previous analyses. These measurements benefit from significant analysis improvements compared to previous ATLAS results [START_REF]Measurements of Higgs boson properties in the diphoton decay channel with 36 fb -1 of 𝑝 𝑝 collision data at √ 𝑠 = 13[END_REF]. A detailed classification of selected events into 101 separate categories based on multi-class machine learning techniques is used, and the uncertainties relative to the modeling of the continuum have been reduced through the use of Gaussian kernel smoothing.

Results are interpreted in models of Higgs boson coupling modifiers. All couplings are found to be compatible with their SM values. Sensitivity to the sign of the 𝜅 𝑡 modifier to the top quark coupling in the 𝑡𝐻 process leads to an exclusion of the 𝜅 𝑡 < 0 region with a significance of 2.2𝜎. An interpretation in the framework of SM effective field theory is used to set constraints on physics effects beyond the SM. Individual Wilson coefficients are measured while fixing the others to 0. A simultaneous measurement of the linear combinations of Wilson coefficients that the STXS measurements are sensitive to is also performed. All results are in agreement with SM expectations.

Appendix B Additional 𝜿-framework interpretations B.1 Parameterization of STXS cross-section parameters and the 𝑯→ 𝜸𝜸 branching ratio

Table 14 presents the modifiers that are applied to the STXS cross-section parameters 𝜎 𝑖 , the partial decay widths Γ 𝛾𝛾 , Γ 𝑔𝑔 and Γ 𝑍 𝛾 of the 𝐻→ 𝛾𝛾, 𝐻 → 𝑔𝑔 and 𝐻 → 𝑍𝛾 decays, respectively, and the total width Γ 𝐻 . The symbols Γ SM 𝑔𝑔 and Γ SM 𝑍 𝛾 denote the SM predictions for Γ 𝑔𝑔 and Γ 𝑍 𝛾 respectively. Modifiers for the 𝑔𝑔 → 𝐻 and 𝐻 → 𝛾𝛾 loop processes are given either in terms of the effective modifiers 𝜅 𝑔 and 𝜅 𝛾 or a resolved parameterization in terms of the modifiers to tree-level Higgs boson couplings, as specified in the description of each model. The total width Γ 𝐻 is expressed as a function of the 𝜅 modifiers, assuming no contributions from Higgs boson decays other than the ones present in the SM, except in the model in Appendix B.3 in which an effective description in terms of the 𝜅 𝐻 modifier is used instead.