Measurements of Higgs boson properties in the diphoton decay channel with 36 fb$^{-1}$ of $pp$ collision data at $\sqrt{s} = 13$ TeV with the ATLAS detector

Properties of the Higgs boson are measured in the two-photon final state using 36.1 fb$^{-1}$ of proton-proton collision data recorded at $\sqrt{s} = 13$ TeV by the ATLAS experiment at the Large Hadron Collider. Cross-section measurements for the production of a Higgs boson through gluon-gluon fusion, vector-boson fusion, and in association with a vector bosonor a top-quark pair are reported. The signal strength, defined as the ratio of the observed to the expected signal yield, is measured for each of these production processes as well as inclusively. The global signal strength measurement of $0.99 \pm 0.14$ improves on the precision of the ATLAS measurement at $\sqrt{s} = 7$ and 8 TeV by a factor of two. Measurements of gluon-gluon fusion and vector-boson fusion productions yield signal strengths compatible with the Standard Model prediction. Measurements of simplified template cross sections, designed to quantify the different Higgs boson production processes in specific regions of phase space, are reported. The cross section for the production of the Higgs boson decaying to two isolated photons in a fiducial region closely matching the experimental selection of the photons is measured to be $55 \pm 10$ fb, which is in good agreement with the Standard Model prediction of $64 \pm 2$ fb. Furthermore, cross sections in fiducial regions enriched in Higgs boson production in vector-boson fusion or in association with large missing transverse momentum, leptons or top-quark pairs are reported. Differential and double-differential measurements are performed for several variables related to the diphoton kinematics as well as the kinematics and multiplicity of the jets produced in association with a Higgs boson. No significant deviations from a wide array of Standard Model predictions are observed.


Introduction
In July 2012, the ATLAS [1] and CMS [2] experiments announced the discovery of a Higgs boson [3,4] using proton-proton collisions collected at center-of-mass energies √ s = 7 TeV and 8 TeV at the CERN Large Hadron Collider (LHC). Subsequent measurements of its properties were found to be consistent with those expected for the Standard Model (SM) Higgs boson [5] with a mass m H = 125.09 ± 0.21(stat.) ± 0.11(syst.) GeV [6].
Following the modifications of the LHC to provide proton-proton collisions at a center-of-mass energy of √ s = 13 TeV, the Higgs sector can be probed more deeply: the data set collected in 2015 and 2016 allows inclusive Higgs boson measurements to be repeated with about two times better precision than to those done at √ s = 7 and 8 TeV with the Run 1 data set. The increased center-ofmass energy results in much larger cross sections for events at high partonic center-of-mass energy. This implies improved sensitivity to a variety of interesting physics processes, such as Higgs bosons produced at high transverse momentum or Higgs bosons produced in association with a top-antitop quark pair. The Higgs boson decay into two photons (H → γγ) is a particularly attractive way to study the properties of the Higgs boson and to search for deviations from the Standard Model predictions due to beyond-Standard Model (BSM) processes. Despite the small branching ratio, (2.27 ± 0.07) × 10 −3 for m H = 125.09 GeV [7], a reasonably large signal yield can be obtained thanks to the high photon reconstruction and identification efficiency at the ATLAS experiment. Furthermore, due to the excellent photon energy resolution of the ATLAS calorimeter, the signal manifests itself as a narrow peak in the diphoton invariant mass (m γγ ) spectrum on top of a smoothly falling background, and the Higgs boson signal yield can be measured using an appropriate fit to the m γγ distribution of the selected events.
In this paper, the results of measurements of the Higgs boson properties in the diphoton decay channel are presented using 36.1 fb −1 of pp collision data collected at √ s = 13 TeV by the ATLAS detector in 2015 and 2016. All the measurements are performed under the assumption that the Higgs boson mass is 125.09 GeV, and are compared to Standard Model predictions. Three types of measurements are presented in this paper and are summarized in the remainder of this section: (i) measurements of the total Higgs boson production-mode cross sections and "signal strengths"; (ii) cross sections using the SM production modes as "templates" in simplified fiducial regions; and (iii) measurements of integrated or differential cross sections in fiducial phase-space regions closely matched to the experimental selection.
The rest of this paper is organized as follows. Section 2 provides a brief description of the ATLAS detector, and Section 3 describes the selected data set. The generation of simulated event samples is described in Section 4. Section 5 gives an overview of the event reconstruction and selection, and Section 6 explains the signal and background modeling used in the measurement. The sources of systematic uncertainties are detailed in Section 7. Section 8 describes the measurement of the total Higgs boson production-mode cross sections, signal strengths, and simplified template cross sections (STXS). Similarly, Section 9 describes the measurement of the fiducial and differential cross sections. Section 10 concludes with a brief summary of the main findings.

Higgs boson production-mode cross sections and signal strengths
In this paper, cross sections times branching ratio of the Higgs to two photons B(H → γγ) are measured for inclusive Higgs boson production, as well as for several individual production processes: gluon-gluon fusion (ggH), vector-boson fusion (VBF), Higgs boson production in association with a vector boson (V H), and production of a Higgs boson in association with a top-antitop quark pair (ttH) or a single top quark (t-channel and W-associated, respectively denoted as tHq and tHW, or in their sum as "tH"). In the SM, gluon-gluon fusion is the dominant production mechanism at the LHC, contributing to about 87% of the total cross section at √ s = 13 TeV [7]. Vector-boson fusion and associated production with either a vector boson, with a top-antitop quark pair or a bottom-antibottom quark pair correspond to 6.8%, 4.0%, 0.9%, and 0.9%, respectively, of the total Higgs boson production cross section.
The data are divided into 31 categories based on the reconstructed event properties to maximize the sensitivity to different production modes and the different regions of the simplified template cross sections, which are further described in Section 1.2. The categories are defined using the expected properties of the different production mechanisms: 10 categories aimed to measure gluon-gluon fusion properties, 4 categories to measure vector-boson fusion, 8 categories that target associated production with vector bosons with different final states, and 9 categories that target associated production with a top-antitop quark pair or a single top-quark. The definition of each category was optimized using simulated events and a full summary of the categories can be found in Section 8. In the sequence of the classification, priority is given to categories aimed at selecting signal events from processes with smaller cross sections.
In order to probe the production mechanisms independently of the H → γγ branching ratio, ratios of the different production-mode cross sections normalized to gluon-gluon fusion are also reported, with their full experimental correlations. In addition, measurements of the signal strength µ, which is the ratio of the measured cross section to the SM prediction, are given for the different production processes as well as for the inclusive production. Finally, coupling-strength modifiers, which are scale factors of the tree-level Higgs boson couplings to the different particles or of the effective Higgs boson couplings to photons and gluons from loop-induced processes, are reported.

Simplified template cross sections
The measurements of cross sections separated by the production mode as presented in the previous section are extended to measurements in specific regions of phase space using the framework of the "simplified template cross sections" introduced in Refs. [7,8]. These are reported as cross section times B(H → γγ) for a Higgs boson absolute rapidity 1 |y H | less than 2.5 and with further particlelevel requirements. The different production modes are separated in a theoretically motivated way using the SM modes ggH, VBF, V H and top-quark-associated production modes as "templates". The fiducial regions are defined in a "simplified" way using the measured kinematics and topology of the Since the current data set is not large enough to probe all of the stage-1 cross sections with sufficiently small statistical uncertainties, regions with poor sensitivity or with large anti-correlations are merged together into ten regions: Six regions probe gluon-fusion Higgs boson production with zero, one, and two jets associated with them. Two regions probe VBF Higgs boson production and Higgs boson production associated with vector bosons that decay hadronically. A dedicated cross section is measured for Higgs boson production associated with vector bosons that decay via leptonic modes. The final cross section measures top-associated (ttH and tH) Higgs-boson production. To retain sensitivity to beyond the Standard Model Higgs boson production , the ≥ 1 jet, p H T > 200 GeV gluongluon fusion and p j T > 200 GeV VBF+V H regions are not merged with other regions. Here p H T and p j T denote the Higgs boson and leading jet transverse momenta, respectively, where the leading jet is the highest transverse momentum jet in a given event. However, due to their large anti-correlation, only the cross section for the summed yield of these two regions is quoted here, and thus a total of nine kinematic regions are reported. The experimental sensitivity to the difference in the yields of these two regions is expected to be small, and the corresponding result is treated as a nuisance parameter rather than a measurement. Table 1 summarizes the ten probes merged stage-1 cross sections and details which of the full 31 stage-1 cross sections were merged (middle and last column). A detailed description of the full 31 cross section proposal can be found in Appendix A.

Fiducial integrated and differential cross sections
Fiducial integrated and differential cross sections have previously been measured at √ s = 8 TeV in the H → γγ decay channel by both the ATLAS [9] and the CMS [10] Collaborations. In this paper, fiducial cross sections are determined in a variety of phase-space regions sensitive to inclusive Higgs boson production and to explicit Higgs boson production mechanisms. The measurement of these cross sections provides an alternative way to study the properties of the Higgs boson and to search for physics beyond the Standard Model. For each fiducial region of an integrated cross-section measurement or bin of a differential distribution, the H → γγ signal is extracted using a fit to the corresponding diphoton invariant mass spectrum. The cross sections are determined by correcting these yields for experimental inefficiencies and resolution effects, and by taking into account the integrated luminosity Table 1: The particle-level kinematic regions of the stage-1 simplified template cross sections, along with the intermediate set of regions used for the measurements presented in this paper. All regions require |y H | < 2.5. Jets are defined using the anti-k t algorithm with radius parameter R = 0.4 and are required to have p T > 30 GeV. The leading jet and Higgs boson transverse momenta are denoted by p j T and p H T , respectively. The transverse momentum of the Higgs boson and the leading and subleading jet is denoted as p H j j T with the subleading jet being the second highest momentum jet in a given event. Events are considered "VBF-like" if they contain at least two jets with an invariant mass of m j j > 400 GeV and a rapidity separation between the two jets of |∆y j j | > 2.8. Events are considered "V H-like" if they contain at least two jets with an invariant mass of 60 GeV < m j j < 120 GeV. All qq → Hqq VBF and V H events (with the vector boson V decaying hadronically) which are neither VBF nor V H-like are part of the "Rest" selection. For the p H T > 200 GeV gluon-gluon fusion and p j T > 200 GeV VBF + V H regions, only the sum of the corresponding cross sections is reported while the difference of the two is profiled in the fit. In total, the cross sections for nine kinematic regions are measured. The small contributions from bbH are merged with ggH. The process gg → Z H refers only to box and loop processes dominated by top and bottom quarks (see Section 4 for more details).
The inclusive fiducial region is defined at the particle level by two photons, not originating from the decay of a hadron, that have absolute pseudorapidity |η| < 2.37, excluding the region 1.37 < |η| < 1.52,2 with the leading (subleading) photon transverse momentum greater than 35% (25%) of m γγ . The two photons are required to be isolated from hadronic activity by imposing that the summed transverse momentum of charged stable particles (with a mean lifetime that satisfies cτ > 10 mm) with p T > 1 GeV, within a cone of ∆R = 0.2 centered on the photon direction, be less than 5% of the photon transverse momentum. This selection is applied to all the presented fiducial integrated and differential cross section results and the isolation criterion was tuned to mimic the detector level selection. One additional cross section and three cross-section limits are reported in smaller fiducial regions sensitive to specific Higgs boson production mechanisms: • a VBF-enhanced region with two jets with large invariant mass and rapidity separation, • a region of events containing at least one charged lepton3, • a region of events with large missing transverse momentum, • and a region of events with a topology matching the presence of a top-antitop quark pair.
The fiducial cross section for different jet multiplicities are reported and compared to several predictions. Eleven fiducial differential cross sections are reported, for events belonging to the inclusive fiducial region as a function of the following observables: • p γγ T and |y γγ |, the transverse momentum and rapidity of the diphoton system, • p j 1 T and |y j 1 |, the transverse momentum and rapidity of the leading jet, • p j 2 T and |y j 2 |, the transverse momentum and rapidity of the subleading jet, • | cos θ * |, the cosine of the angle between the beam axis and the diphoton system in the Collins-Soper frame [11], • ∆φ j j and |∆y j j |, the difference in azimuthal angle and in rapidity between the leading and subleading jets, • |∆φ γγ, j j |, the difference in azimuthal angle between the dijet system formed by the leading and subleading jets and the diphoton system, • and m j j , the invariant mass of the leading and subleading jets.
Seven additional variables are reported in Appendix C. Inclusive Higgs boson production is dominated by gluon-gluon fusion, for which the transverse momentum of the Higgs boson is largely balanced by the emission of soft gluons and quarks. Measuring p Table 2: Summary of the event generators and PDF sets used to model the signal and the main background processes. The SM cross sections σ for the Higgs production processes with m H = 125.09 GeV are also given separately for √ s = 13 TeV, together with the orders of the calculations corresponding to the quoted cross sections, which are used to normalize the samples, after multiplication by the Higgs boson branching ratio to diphotons, 0.227%. The following versions were used: P 8 version 8.212 (processes) and 8 Additional proton-proton interactions (pileup) are included in the simulation for all generated events such that the average number of interactions per bunch crossing reproduces that observed in the data. The inelastic proton-proton collisions were produced using P 8 with the A2 parameter set [68] that are tuned to data and the MSTW2008 PDF set [69]. A summary of the used signal and background samples is shown in Table 2.

Photon reconstruction and identification
The reconstruction of photon candidates is seeded by energy clusters in the electromagnetic calorimeter with a size of ∆η × ∆φ = 0.075×0.125, with transverse energy E T greater than 2.5 GeV [70]. The reconstruction is designed to separate electron from photon candidates, and to classify the latter as unconverted or converted photon candidates. Converted photon candidates are associated with the conversion of photons into electron-positron pairs in the material upstream the electromagnetic calorimeter. Conversion vertex candidates are reconstructed from either two tracks consistent with originating from a photon conversion, or one track that does not have any hits in the innermost pixel layer. These tracks are required to induce transition radiation signals in the TRT consistent with the electron hypothesis, in order to suppress backgrounds from non-electron tracks. Clusters without any matching track or conversion vertex are classified as unconverted photon candidates, while clusters with a matching conversion vertex are classified as converted photon candidates. In the simulation, the average reconstruction efficiency for photons with generated E T above 20 GeV and generated pseudorapidity |η| < 2.37 is 98%.
The energy from unconverted and converted photon candidates is measured from an electromagnetic cluster of size ∆η × ∆φ = 0.075×0.175 in the barrel region of the calorimeter, and ∆η × ∆φ = 0.125×0. 125 in the calorimeter endcaps. The cluster size is chosen sufficiently large to optimize the collection of energy of the particles produced in the photon conversion. The cluster electromagnetic energy is corrected in four steps to obtain the calibrated energy of the photon candidate, using a combination of simulation-based and data-driven correction factors [71]. The simulation-based calibration procedure was re-optimized for the 13 TeV data. Its performance is found to be similar with that of Run 1 [71] in the full pseudorapidity range, and is improved in the barrel-endcap transition region, due to the use of information from additional scintillation detectors in this region [72]. The uniformity corrections and the intercalibration of the longitudinal calorimeter layers are unchanged compared to Run 1 [71], and the data-driven calibration factors used to set the absolute energy scale are determined from Z → e + e − events in the full 2015 and 2016 data set. The energy response resolution is corrected in the simulation to match the resolution observed in data. This correction is derived simultaneously with the energy calibration factors using Z → e + e − events by adjusting the electron energy resolution such that the width of the reconstructed Z boson peak in the simulation matches the width observed in data [72].
Photon candidates are required to satisfy a set of identification criteria to reduce the contamination from the background, primarily associated with neutral hadrons in jets decaying into photon pairs, based on the lateral and longitudinal shape of the electromagnetic shower in the calorimeter [73]. Photon candidates are required to deposit only a small fraction of their energy in the hadronic calorimeter, and to have a lateral shower shape consistent with that expected from a single electromagnetic shower. Two working points are used: a loose criterion, primarily used for triggering and preselection purposes, and a tight criterion. The tight selection requirements are tuned separately for unconverted and converted photon candidates. Corrections are applied to the electromagnetic shower shape variables of simulated photons, to account for small differences observed between data and simulation. The variation of the photon identification efficiency associated with the different reconstruction of converted photons in the 2015 and 2016 data sets, due to the different TRT gas composition, has been studied with simulated samples and shown to be small. The efficiency of the tight identification criteria ranges from 84% to 94% (87% to 98%) for unconverted (converted) photons with transverse energy between 25 GeV and 200 GeV.
To reject the hadronic jet background, photon candidates are required to be isolated from any other activity in the calorimeter and the tracking detectors. The calorimeter isolation is computed as the sum of the transverse energies of positive-energy topological clusters [74] in the calorimeter within a cone of ∆R = 0.2 centered around the photon candidate. The transverse energy of the photon candidate is removed. The contributions of the underlying event and pileup are subtracted according to the method suggested in Ref. [75]. Candidates with a calorimeter isolation larger than 6.5% of the photon transverse energy are rejected. The track isolation is computed as the scalar sum of the transverse momenta of all tracks in a cone of ∆R = 0.2 with p T > 1 GeV which satisfy some loose track-quality criteria and originate from the diphoton primary vertex, i.e. the most likely production vertex of the diphoton pair (see Section 5.2). For converted photon candidates, the tracks associated with the conversion are removed. Candidates with a track isolation larger than 5% of the photon transverse energy are rejected. 13

Event selection and selection of the diphoton primary vertex
Events are preselected by requiring at least two photon candidates with E T > 25 GeV and |η| < 2.37 (excluding the transition region between the barrel and endcap calorimeters of 1.37 < |η| < 1.52) that fulfill the loose photon identification criteria [70]. The two photon candidates with the highest E T are chosen as the diphoton candidate, and used to identify the diphoton primary vertex among all reconstructed vertices, using a neural-network algorithm based on track and primary vertex information, as well as the directions of the two photons measured in the calorimeter and inner detector [76]. The neural-network algorithm selects a diphoton vertex within 0.3 mm of the true H → γγ production vertex in 79% of simulated gluon-gluon fusion events. For the other Higgs production modes this fraction ranges from 84% to 97%, increasing with jet activity or the presence of charged leptons. The performance of the diphoton primary vertex neural-network algorithm is validated using Z → e + e − events in data and simulation, by ignoring the tracks associated with the electron candidates and treating them as photon candidates. Sufficient agreement between the data and the simulation is found. The diphoton primary vertex is used to redefine the direction of the photon candidates, resulting in an improved diphoton invariant mass resolution. The invariant mass of the two photons is given by m γγ = 2E 1 E 2 (1 − cos α), where E 1 and E 2 are the energies of the leading and subleading photons and α is the opening angle of the two photons with respect to the selected production vertex.
Following the identification of the diphoton primary vertex, the leading and subleading photon candidates in the diphoton candidate are respectively required to have E T /m γγ > 0.35 and 0.25, and to both satisfy the tight identification criteria as well as the calorimeter and track isolation requirements. Figure 1 compares the simulated per-event efficiency of the track-and calorimeter-based isolation requirement as a function of the number of primary vertex candidates with the per-event efficiency of the Run 1 algorithm described in Ref. [76], by using a MC sample of Higgs bosons produced by gluon-gluon fusion and decaying into two photons. The re-optimization of the thresholds applied to the transverse energy sum of the calorimeter energy deposits and to the transverse momentum scalar sum of the tracks in the isolation cone, as well as the reduction of the size of the isolation cone for the calorimeter-based isolation, greatly reduces the degradation of the efficiency as the number of reconstructed primary vertices increases in comparison to the Run 1 algorithm.
In total 332030 events are selected with diphoton candidates with invariant mass m γγ between 105 GeV and 160 GeV. The predicted signal efficiency, assuming the SM and including the acceptance of the kinematic selection, is 42% (with the acceptance of the kinematic selection being 52%).

Reconstruction and selection of hadronic jets, b-jets, leptons and missing transverse momentum
Jets are reconstructed using the anti-k t algorithm [77] Figure 1: Efficiency for both photons to fulfill the isolation requirement as a function of the number of primary vertex candidates in each event, determined with a sample of simulated Higgs bosons with m H = 125 GeV, produced in gluon-gluon fusion and decaying into two photons. Events are required to satisfy the kinematic selection described in Section 5.2 for the 8 TeV (violet squares) and 13 TeV (blue circles) simulated sample. The error bars show the statistical uncertainty of the generated samples. The Run 2 (Run 1) isolation requirement is based on the transverse energy deposited in the calorimeter in a ∆R = 0.2 (∆R = 0.4) cone around the photon candidates. Both the Run 1 and Run 2 algorithms also use tracking information in a ∆R = 0.2 cone around the photon candidates. noted in Sections 8.1 and 9.3). Jets that do not originate from the diphoton primary vertex are rejected, for |η| < 2.4, using the jet vertex tagging algorithm (JVT) [83], which combines tracking information into a multivariate likelihood. For jets with p T < 60 GeV and |η| < 2.4 a medium working point is used, with an efficiency greater than 92% for non-pileup jets with p T > 30 GeV. The efficiency of the JVT algorithm is corrected in the simulation to match that observed in the data. Jets are discarded if they are within ∆R = 0.4 of an isolated photon candidate, or within ∆R = 0.2 of an isolated electron candidate.
Jets consistent with the decay of a b-hadron are identified using a multivariate discriminant, having as input information from track impact parameters and secondary vertices [84,85]. The chosen identification criterion has an efficiency of 70% for identifying jets originating from a b-quark. The efficiency is determined using a tt control region, with rejection factors of about 12 and 380 for jets originating from c-quarks and light quarks, respectively. Data-driven correction factors are applied to the simulation such that the b-tagging efficiencies of jets originating from b-quarks, c-quarks and light quarks are consistent with the ones observed in the data.
The reconstruction and calibration of electron candidates proceeds similarly as for photon candidates. Electromagnetic calorimeter clusters with a matching track in the inner detector are reconstructed as electron candidates and calibrated using dedicated corrections from the simulation and from data control samples. Electron candidates are required to have p T > 10 GeV and |η| < 2.47, excluding the region 1.37 < |η| < 1.52. Electrons must satisfy medium identification criteria [86] using a likelihood-based discriminant.
Muon candidates are primarily built from tracks reconstructed in the inner detector and the muon spectrometer, but are complemented by candidates reconstructed only in the muon spectrometer that are compatible with originating from the interaction point [87]. Muon candidates are required to have p T > 10 GeV and |η| < 2.7, and satisfy medium identification criteria based on the number of hits in the silicon detectors, in the TRT and in the muon spectrometer. For the measurements of fiducial cross sections the electron and muon selections are tightened to p T > 15 GeV.
Lepton candidates are discarded if they are within ∆R = 0.4 of an isolated photon candidate or a jet. Isolation requirements are applied to all lepton candidates. Electron candidates are required to satisfy loose criteria for the calorimeter and track isolation, aimed at a combined efficiency of 99% independently of the candidate transverse momentum. Muon candidates are similarly required to satisfy loose criteria for the calorimeter and track isolation, in this case depending on the candidate transverse momentum, and aimed at a combined efficiency ranging from 95-97% at p T = 10-60 GeV to 99% for p T > 60 GeV.
Tracks associated with both the electron and muon candidates are required to be consistent with originating from the diphoton primary vertex by requiring their longitudinal impact parameter z 0 to satisfy |z 0 sin θ| < 0.5 mm and their unsigned transverse impact parameter |d 0 | relative to the beam axis to be respectively smaller than five or three times its uncertainty.
The lepton efficiency as well as energy/momentum scale and resolution are determined using the decays of Z bosons and J/ψ mesons in the full 2015 and 2016 data set using the methods described in Refs. [86,87]. Lepton efficiency correction factors are applied to the simulation to improve the agreement with the data.
The magnitude of the missing transverse momentum E miss T is measured from the negative vectorial sum of the transverse momenta of all photon, electron and muon candidates and of all hadronic jets after accounting for overlaps between jets, photons, electrons, and muons, as well as an estimate of soft contributions based on tracks originating from the diphoton vertex which satisfy a set of quality criteria. A full description of this algorithm can be found in Refs. [88,89]. The E miss T significance is defined as E miss T / √ E T , where E T is the sum of the transverse energies of all particles and jets used in the estimation of the missing transverse momentum in units of GeV.

Signal and background modeling of diphoton mass spectrum
The Higgs boson signal yield is measured through an unbinned maximum-likelihood fit to the diphoton invariant mass spectrum in the range 105 GeV< m γγ <160 GeV for each event reconstruction category, fiducial region, or each bin of a fiducial differential cross section, as further discussed in Sections 8 and 9. The mass range is chosen to be large enough to allow a reliable determination of the background from the data, and at the same time small enough to avoid large uncertainties from the choice of the background parameterization. The signal and background shapes are modeled as described below, and the background model parameters are freely floated in the fit to the m γγ spectra.

Signal model
The Higgs boson signal manifests itself as a narrow peak in the m γγ spectrum. The signal distribution is empirically modeled as a double-sided Crystal Ball function, consisting of a Gaussian central part and power-law tails on both sides. The Gaussian core of the Crystal Ball function is parameterized by the peak position (m H + ∆µ CB ) and the width (σ CB ). The non-Gaussian contributions to the mass resolution arise mostly from converted photons γ → e + e − with at least one electron losing a significant fraction of its energy through bremsstrahlung in the inner detector material. The parametric form for a given reconstructed category or bin i of a fiducial cross section measurement, for a Higgs boson mass m H , can be written as: CB,i , and N c is a normalization factor. The non-Gaussian parts are parameterized by α ± CB,i and n ± CB,i separately for the low-(−) and high-mass (+) tails. The parameters of the model that define the shape of the signal distribution are determined through fits to the simulated signal samples. The parameterization is derived separately for each reconstructed category or fiducial region of the integrated or differential cross-section measurement. Figure 2 shows an example for two categories with different mass resolution: the improved mass resolution in the central region of the detector (defined by requiring |η| ≤ 0.95 for both selected photons) with respect to the forward region (defined by requiring one photon with |η| ≤ 0.95 and one photon with 0.95 < |η| < 2.37) results in better discriminating power against the non-resonant background and in turn in a smaller statistical error of the extracted Higgs boson signal yield. The effective signal mass resolution of the two categories, defined as half the width containing 68% (90%) of the signal events, is 1.6 (2.7) GeV and 2.1 (3.8) GeV, respectively, and the mass resolution for all used categories can be found in Appendix E.

Background composition and model
The diphoton invariant mass model for the background used to fit the data is determined from studies of the bias in the signal yield in signal+background fits to large control samples of data or simulated background events.
Continuum γγ production is simulated with the S event generator as explained in Section 4, neglecting any interference effects with the H → γγ signal. The γ j and j j backgrounds are obtained by reweighting this sample using an m γγ dependent linear correction function obtained from the fraction of γγ to γ j and γγ to j j background events in data, respectively.
For very low rate categories targeting ttH or tH events, in which the background simulation suffers from very large statistical uncertainties, various background-enriched control samples are directly obtained from the data by either reversing photon identification or isolation criteria, or by loosening or removing completely b-tagging identification requirements on the jets, and normalizing to the data in the m γγ sidebands of the events satisfying the nominal selection. For low rate categories targeting associated vector boson production, background control samples are obtained by summing the distributions from the main background processes: Those of γγ and V γγ events are obtained directly from the simulation, while the m γγ distributions of γ j and j j events are obtained from data control samples in which the nominal selection is applied except that at least one (for γ j) or both (for  The diphoton mass signal shapes of two gluon-gluon fusion categories that are later introduced in Section 8.1 are shown: ggH 0J Fwd aims to select gluon-gluon fusion events with no additional jet and at least one photon in the pseudorapidity region |η| > 0.95; ggH 0J Cen applies a similar selection, but requires both photons to have |η| ≤ 0.95 in order to have a better energy resolution. The simulated sample (labeled as MC) is compared to the fit model and contains simulated events from all Higgs boson production processes described in Section 4 with m H = 125 GeV. j j) of the two photon candidates fail to meet either the identification or isolation criteria. Except for the V γγ component, which is normalized with its theoretical cross section, the other contributions are normalized according to their relative fractions determined in data as described in the following.
The measurement of the background fractions in data is performed for each category or fiducial region. The relative fractions of γγ, γ j and j j background events are determined using a double two-dimensional sideband method [90,91]. The nominal identification and isolation requirement are loosened for both photon candidates, and the data are split into 16 orthogonal regions defined by diphoton pairs in which one or both photons satisfy or fail to meet identification and/or isolation requirements. The region in which both photons satisfy the nominal identification and isolation requirements corresponds to the nominal selection of Section 5, while the other 15 regions provide control regions, whose γγ, γ j and j j yields are related to those in the signal region via the efficiencies for photons and for hadronic jets to satisfy the photon identification and isolation requirements. The γγ, γ j and j j yields in the signal region are thus obtained, together with the efficiencies for hadronic jets, by solving a linear system of equations using as inputs the observed yields in the 16 regions and the photon efficiencies predicted by the simulation. In the V H categories, a small extra contribution from V γγ events with an electron originating from the decay of the vector boson V which is incorrectly reconstructed as a photon, is also estimated from the simulation and subtracted before applying the two-dimensional sideband method. The dominant systematic uncertainties in the measured background fractions are due to the definition of the background control regions. The yields and relative fractions of the γγ, γ j and j j backgrounds are shown in Figure 3 as a function of m γγ for the selected events. The fractions of these background sources in the inclusive diphoton sample are (78.7 +1.8 −5.2 )%, ( Figure 3: The data-driven determination of (a) event yields and (b) event fractions for γγ, γ j and j j events as a function of m γγ after the final selection outlined in Section 5. The event fractions for two differential observables, (c) p γγ T and (d) N jets defined for jets with a p T > 30 GeV are shown as well. The shaded regions show the total uncertainty of the measured yield and fraction, and the error bars show the statistical uncertainties.
fractions are systematically dominated. These results are comparable to previous results at √ s = 7 and 8 TeV [9,76]. In addition the purity is shown as a function of the p T of the diphoton system, and the number of reconstructed jets with p T > 30 GeV.
The functional form used to model the background m γγ distribution in the fit to the data is chosen, in each region, to ensure a small bias in the extracted signal yield relative to its experimental precision, following the procedure described in Ref. [3]. The potential bias (spurious signal) is estimated as the maximum of the absolute value of the fitted signal yield, using a signal model with mass between 121 and 129 GeV, in fits to the background control regions described before.
The spurious signal is required, at 95% confidence level (CL), to be less than 10% of the expected SM signal yield or less than 20% of the expected statistical uncertainty in the SM signal yield. In the case when two or more functions satisfy those requirements, the background model with the least number of parameters is chosen.
Prior to the final fit to the data, the selected model is tested against a model from the same family of functions but with one more degree of freedom (for instance, the exponential of a second-order polynomial is tested against an exponential of a third-order polynomial) to check, using only events in the diphoton invariant mass sidebands (i.e. excluding the range 121 GeV < m γγ < 129 GeV), if the data favors a more complex model. A test statistic is built from the χ 2 values and number of degrees of freedom of two binned fits to the data with the two background models. The expected distribution of the test statistic is built from pseudo-data assuming that the function with fewer degrees of freedom is the true underlying model. The value of the test statistic obtained in the data is compared to such distribution, and the simpler model is rejected in favor of the more complex one if the p-value of such comparison is lower than 5%. The background distribution of all regions is found to be well modeled by at least one of the following functions: an exponential of a first-or second-order polynomial, a power law, or a third-order Bernstein polynomial.

Statistical model
The data are interpreted following the statistical procedure summarized in Ref. The likelihood for a given reconstructed category, fiducial region, or differential bin i of the integrated or differential cross-section measurement is a marked Poisson probability distribution, where n i (N i ) is the observed (expected) number of selected candidates, f i (m j γγ ) is the value of the probability density function (pdf) of the invariant mass distribution evaluated for each candidate j, θ are nuisance parameters and G(θ) is a set of unit Gaussian constraints on a subset of the nuisance parameters, as described in the following. The likelihood for the measurements of the total Higgs boson production-mode cross sections and signal strengths is given by the product of the likelihood functions of each event reconstruction category. For the fiducial integrated and differential crosssection measurements the likelihood of all bins i in a spectrum is taken.
The number of expected candidates is the sum of the signal and background yields, denoted by N In more detail, the invariant mass distribution for each category has signal and background components, in the case of a Gaussian pdf for the effect of an uncertainty of size σ or, for cases where a negative model parameter does not make physical sense (e.g. the uncertainty in the integrated luminosity), by a factor F LN (σ, θ) = e √ ln(1+σ 2 )θ for a log-normal pdf. In both cases the corresponding component of the constraint product G(θ) is a unit Gaussian centered at zero for the nuisance parameter θ. The systematic uncertainties affecting the yield and mass resolution use the log-normal form while a Gaussian form is used for all others. When two uncertainties are considered fully correlated they share the same nuisance parameter. Systematic uncertainties with partial correlations are decomposed into their uncorrelated and fully correlated components before being assigned to nuisance parameters.
All measured Higgs boson signal yields are determined with the profile likelihood ratio test statistic whereν andθ are the values of the parameter of interest (e.g. a signal strength or a simplified template cross section) and nuisance parameters that unconditionally maximize the likelihood whileθ ν are the values of the nuisance parameters that maximize the likelihood on the condition that the parameter of interest is held fixed to a given value ν. In the asymptotic approximation, which is valid for all the results presented here, Λ(ν) may be interpreted as an increase in χ 2 from its minimum value [92] such that approximate confidence intervals are easily constructed. The total uncertainty in ν is thus obtained from the ν values such that Λ(ν) = 1 with all other parameters "profiled" (i.e. set to the values that maximize the likelihood for those values of ν). Theory uncertainties in the parameters of interest are found by fixing the nuisance parameters associated with experimental uncertainties and subtracting in quadrature the statistical uncertainty. The statistical uncertainty is similarly determined, by fixing all nuisance parameters to their best-fit values, except for those describing the background shape and normalization. The experimental uncertainty is found by subtracting in quadrature the theory and the statistical uncertainties from the total uncertainty.

Limit setting in the absence of a signal
In the absence of a significant signal yield in the measured production process categories or fiducial regions, upper limits on the corresponding signal strength or cross section are placed. For productionmode measurements, the limit is set by treating all other parameters of the fit as nuisance parameters. For the fiducial regions, each measurement is split into two orthogonal categories, one of which contains the events in the specified fiducial region and one that contains the events that are outside of it. The diphoton spectrum in both sets of events are simultaneously analyzed to extract the desired limit.
For category-based measurements the 95% CL upper limit on the parameter of interest ν is determined using the CL s prescription [94]. For this, the agreement between data and the expected yield for the hypothesized value of the parameter of interest ν is quantified by the test statistic, q ν , defined as whereν ≥ 0 is the fitted parameter of interest. The observed value of the test statistic, q obs , is determined from the ratio of the likelihood obtained by fixing the number of signal events to that predicted for a given value of the parameter of interest, to the likelihood normalized by allowing the number of signal events to float in the fit. The asymptotic behavior of Eq. (2) is well known [92]. For fiducial measurements the 95% CL upper limit are determined using a one-sided Gaussian interpretation of the observed cross section.

Systematic uncertainties
Several sources of systematic uncertainty are considered in this measurement. They can be grouped into three categories: (i) uncertainties associated with the parameterization of the signal and background when fitting the m γγ spectrum, (ii) experimental uncertainties arising either from the extraction of the signal in a given category or from migrations between categories, and (iii) theoretical and modeling uncertainties in each category, causing migrations between categories, or affect the fiducial acceptance.
The origin of the uncertainties and their treatment are discussed in detail below and summarized in Table 3.4 The analysis based on event reconstruction categories and those of fiducial cross sections treat yield and migration uncertainties differently: whereas the former incorporate them directly into the likelihood function (cf. Section 6.3), the latter incorporate them at a later stage as part of the correction factor (introduced in Section 9.2) or the luminosity. Modeling uncertainties were also estimated with different approaches as discussed further in Sections 7.3 and 7.4. A summary of the impact of the uncertainties on the measurement is given in Sections 8.2.2 and 9.5.6.

Systematic uncertainties in the signal and background modeling from fitting the m γγ spectrum
Systematic uncertainties associated with the signal and background parameterizations are treated in a similar way for all the measurements. These include systematic uncertainties in the photon energy scale and resolution, and the uncertainties due to the specific choice of background model.
The fit to the m γγ spectra is performed for a Higgs boson mass of m H = 125.09 ± 0.24 GeV [6]. The uncertainties in the photon energy scale and resolution impact the signal model, as the photon energy scale shifts the position of the peak and the assumed energy resolution broadens or narrows the signal peak relative to its nominal width. Uncertainties in the photon energy scale are included as nuisance parameters associated with Gaussian constraint terms in the likelihood functions. Uncertainties in the photon energy resolution are included as nuisance parameters, and are typically among the dominant Table 3: Summary of the sources of systematic uncertainties for results based on event reconstruction categories or fiducial integrated and differential cross sections. The columns labels "Category Likelihood" and "Fiducial Likelihood" provide an overview about which terms are part of the Likelihood ( ) or incorporated at a later stage (-). Both sets of results incorporate uncertainties associated with the Higgs boson mass, photon energy scale and resolution, and uncertainties associated with the choice of the background function into the likelihood function, either using log normal (F LN (σ i , θ i )) or Gaussian constraints (F G (σ i , θ i )) with σ i denoting the systematic uncertainty (i is the index to each of the unique nuisance parameters θ). When acting on N tot S the uncertainty value is the same for all processes, whereas the uncertainty has a different value for each signal process for the case denoted N p S . The number of nuisance parameters, N NP , for the spurious signal uncertainty varies, e.g. for the category-based results 31 independent error sources are present and for the differential measurements one source per measured bin is included.
Spurious signal Varies N spur,c θ spur,c 23 sources of systematic uncertainty in the measurement. The systematic uncertainties in the photon energy resolution and scale follow those in Refs. [71,72]. The overall energy scale factors and their uncertainties have been determined using Z → e + e − events collected during 2015 and 2016. Compared to Ref. [72], several systematic uncertainties were re-evaluated with the 13 TeV data, including uncertainties related to the observed LAr cell non-linearity, the material simulation, the intercalibration of the first and second layer of the calorimeter, and the pedestal corrections. The typical impact of the photon energy scale uncertainties is to shift the peak position by between ± 0.21% and ± 0.36% of the nominal peak position, whereas the typical impact of the photon energy resolution uncertainty is to change the width of the signal distribution by between ± 6% and ± 13% of the nominal width. The size of both uncertainties is dependent on the energy, rapidity and jet activity of the selected photon pair.
An additional uncertainty in the signal peak position is added as a nuisance parameter in the fit, reflecting the uncertainty in the measurement of the Higgs boson mass of 0.24 GeV [6]. The uncertainty in the Higgs boson mass is dominated by the statistical component, and the systematic component has contributions from both the ATLAS and the CMS muon momentum and electromagnetic energy scale uncertainties. Therefore, the correlation between this uncertainty and the photon energy scale uncertainty in the measurements presented here is considered negligible. A variation of the signal mass by ± 0.24 GeV (without including this uncertainty in the fit) is found to impact the measured global signal strength or the diphoton fiducial cross section by less than ± 0.1%.
The uncertainty due to the choice of background function is taken to be the spurious signal yield obtained when fitting the m γγ spectrum reconstructed from background-only simulated samples (or signal-suppressed control regions in data), as discussed in Section 6.

Experimental systematic uncertainties affecting the expected event yields
There are two categories of uncertainties: 1) those in the expected overall signal yield and 2) those that cause migrations of events between categories and bins, as well as into and out of the photon fiducial selection.
The sources of uncertainties in the expected signal yield consist of: • The luminosity delivered to the ATLAS experiment. The uncertainty in the combined 2015+2016 integrated luminosity is 3.2%. It is derived, following a methodology similar to that detailed in Ref.
[95], from a calibration of the luminosity scale using x-y beam-separation scans performed in August 2015 and May 2016.
• The efficiency of the diphoton trigger. Its uncertainty is estimated to be 0.4% by comparing the trigger efficiencies determined using a bootstrap method [16] in data and simulation.
• The photon identification efficiency. Its uncertainty is estimated to be 1.6% and is obtained by varying the efficiency scale factors within their uncertainties, derived from control samples of photons from radiative Z boson decays and from inclusive γ events, and of electrons from Z → e + e − decays.
• The photon track isolation efficiency. Its uncertainty is estimated to be 0.8% and is derived from measurements of the efficiency correction factors using inclusive photon control samples.
• The photon calorimeter isolation efficiency. Its uncertainty is estimated to be 0.1% and is obtained from the difference between applying and not applying corrections derived from inclusive photon events to the calorimeter isolation variable in the simulation.
Uncertainties which affect the calibration of photons, jets, and leptons cause migrations between categories and bins, as well as migrations into and out of the fiducial acceptance. These include: • The modeling of pileup in the simulation. The corresponding uncertainty is derived by varying the average number of pileup events in the simulation by an amount consistent with data. The typical size ranges from 1.4% up to 5.6% depending on the category or fiducial cross section bin.
• Uncertainties in the photon energy scale and resolution. These uncertainties cause migrations into and out of the fiducial volume or between the event reconstruction categories and impact the expected number of events. The calibration of the absolute energy scale is derived using Z → e + e − decays. The impact of the corresponding uncertainties is small, however, for all measurements, and ranges for instance between 0.2% for events with a low diphoton p T up to 1.9% for events with a high diphoton p T .
• Uncertainties in the jet energy calibration and the jet energy resolution. Uncertainties in the jet energy scale and resolution are estimated by varying the jet energies by an amount commensurate with the differences observed between 13 TeV data and simulation in the transverse momentum balance in dijet, γ + jet and Z + jet events [81, 82, 96]. The typical size of this uncertainty ranges from 2.8% to 15%.
• Uncertainties due to the efficiency of the jet vertex tagger. Such uncertainties are estimated by shifting the associated corrections applied to the simulation by an amount allowed by the data. For the measurement of the fiducial integrated and differential cross sections, uncertainties associated with the modeling of pileup jets in the simulation are estimated by recalculating the correction factor after removing 20% of pileup jets at random, which is commensurate with the observed differences in data and simulation for jets tagged as low-JVT (pileup) and high-JVT (hard scatter). Its typical size ranges from nil to 0.3%.
• Uncertainties associated with the efficiency of the b-tagging algorithm. They have been estimated to be typically of the order of 3% and are determined using tt events in 13 TeV data for jets containing the decay of a b-quark, using the method outlined in Ref. • Uncertainties associated with energy scales and resolutions of photons, jets and leptons are propagated to the E miss T uncertainty, together with the uncertainty in the contribution to E miss T from charged-particle tracks not associated with high-p T leptons, jets, or photon conversions [89]. This results in a typical migration uncertainties ranging from 4.0% to 4.8% for relevant categories or fiducial regions.

Theoretical and modeling uncertainties for results based on event reconstruction categories
The overall theoretical cross-section uncertainties affect the signal strength measurements, which are ratios of the observed to predicted event yields, but not the cross-section measurements which do not rely on absolute predictions. Modeling uncertainties that alter the kinematic properties of the events, such as the Higgs boson transverse momentum or the jet multiplicity, have an impact on both types of measurements.
The theoretical modeling uncertainties in the per-category acceptance of each production process affect the measurement of production-mode cross sections and signal strengths. Uncertainties due to the choice of parton distribution functions and the value of α S are estimated using the PDF4LHC15 recommendations [24] with the nominal PDF4LHC_nlo_30_as PDF set. Samples using the CT10 PDF set are reweighted to PDF4LHC15 to estimate these uncertainties. For the gluon-gluon fusion process, the total production-mode cross section has been calculated at N 3 LO precision in QCD and has an uncertainty of 3.9%, as determined by QCD-scale variations and including top, bottom, and charm quark mass effect uncertainties. However, the perturbative uncertainty becomes significantly larger in different kinematic regions, e.g. when requiring additional jets or high Higgs boson p T . To take this effect into account nine uncertainty sources are included: • Four sources [7] account for uncertainties in the jet multiplicities due to missing higher-order corrections: two accounting for yield uncertainties (with uncertainties up to 8.9% in each STXS region) and two accounting for migrations between jet multiplicity bins (with uncertainties up to 18% in each STXS region), using the STWZ [99] and BLPTW [99-101] predictions as an input. For more details see Table 20 of Ref. [7].
• Three uncertainty sources parameterize modeling uncertainties in the Higgs boson p T . The first two encapsulate the migration uncertainty between the intermediate and high p T region with events with at least one jet. The third uncertainty parameterizes top-quark mass effects in the gluon-gluon fusion loop, where the difference between the LO and NLO predictions is taken as an uncertainty due to missing higher-order corrections. This introduces a negligible uncertainty at low Higgs boson p T and a sizable uncertainty of the order of 30% at p T > 500 GeV.
• Two sources account for the uncertainty in the acceptance of gluon-gluon fusion events in the VBF categories, due to missing QCD higher-orders in the calculation. Such uncertainties are estimated by variations of the renormalization and factorization scales in MCFM [102] by a factor of two around the nominal scale of µ r = µ f = m H . The two sources account for the uncertainty in the overall normalization of H + 2 jet and H + ≥ 3 jet events as well as for the uncertainty due to the multivariate requirement on |∆φ γγ, j j | (cf. Section 8.1.4), which suppresses additional jet activity. The uncertainty estimation uses an extension of the Stewart-Tackmann method [103,104] and typically ranges between 20% and 32%.
The applicability of these uncertainties to P NNLOPS was tested by comparing the STWZ+BLPTW and the MCFM cross section predictions in variables relevant for the definition of the simplified crosssection bins, and reasonable agreement was found. In addition, the ggH acceptance of P NNLOPS of all categories based on BDT classifiers is compared to the acceptance derived from the MG5_ MC@NLO prediction or Refs. [46,105] which includes up to two jets at NLO accuracy using the F F merging scheme [106]. Sufficient agreement was found for all categories and no additional modeling uncertainties are assigned based on these comparisons. 5 Finally, in the categories targeting production in association with top quarks, the normalization of each of the ggH, VBF, and V H production mechanisms is assigned an uncertainty of 100%, motivated by comparisons of data with simulation in ttbb [108] and V b [109,110] productions, but this has little impact on the final results.
The uncertainty in the modeling of the parton shower, underlying event and hadronization affects all measurements (labeled as "UE/PS" in the following). It is estimated by taking the relative difference in acceptance at particle level after switching the parton showering algorithm from P 8 to H 7 in the ggH, VBF, and V H samples and from H ++ to P 8 in the ttH sample, respectively. These differences are treated as four independent uncertainty sources. Additionally, for ggH the effect of the eigenvector tunes from the AZNLO set are merged to provide one additional uncertainty component.
The theoretical modeling uncertainties in the measurement of signal strengths include all of the sources that affect the measurement of the production-mode cross sections, plus additional uncertainties in the overall normalization of each production mechanism. Uncertainties in the overall normalization of each production process from missing higher-order QCD effects and the choice of parton distribution function are specified as part of the theoretical calculations used to normalize the simulated samples. The normalization uncertainty from the H → γγ branching ratio is taken from HDECAY and PROPHECY4F.

Theoretical and modeling uncertainties for fiducial integrated and differential results
The theoretical modeling uncertainty in the detector correction factor (introduced in Section 9.2) used to measure the fiducial integrated and differential cross sections is taken to be the envelope of the following three sources: 1. The uncertainty in the relative contributions of the different Higgs boson production mechanisms. This uncertainty is estimated by varying the fraction of the ggH, VBF, V H and ttH processes by an amount commensurate with the 68% confidence levels of the measured production mode cross-section ratios [5]. The variations of each production mechanism are carried out simultaneously and include the known correlations between the measured production mode cross-section ratios. These uncertainties range from 0.1% to 31%, depending on the fiducial region or differential variable, increasing typically in bins and regions sensitive to ttH-production.
2. The uncertainty in the detector correction factor due to a possible mismodeling of the Higgs boson transverse momentum and rapidity distributions is estimated by reweighting the Higgs boson distributions in simulation to match those observed in the data. The resulting uncertainties range from 0.1% to 4.5%, increasing in fiducial regions and bins with high jet multiplicities.
3. The uncertainty in the modeling of the parton shower, underlying event, and hadronization. This uncertainty is derived as described in Section 7.3 and the size of this uncertainty ranges from from 0.1% up to 30%, with the highest uncertainties in fiducial regions with large missing transverse energy.
Typically differential measurements involving only the photon kinematics are less affected by these model uncertainties than measurements using selections on jets or missing transverse momentum.

Illustration of model errors for simplified template cross section and fiducial cross section measurements
To illustrate the difference between the two approaches of assigning theory and model errors used for category based results and the fiducial cross section results, the theoretical modeling uncertainties in the corresponding zero-jet ggF-dominated and VBF-dominated regions are compared.
The simplified template cross section defined as gg → H events with |y H | < 2.5 and no jets derives its sensitivity from the two categories requiring no jet and either one or both photons reconstructed in the barrel region of the electromagnetic calorimeter (defined by |η| ≤ 0.95). The total theory uncertainty is dominated by the uncertainty in the choice of parton distribution functions (1.5%), in the value of α S (1.4%), and in the modeling of the parton shower, underlying event, and hadronization (1.7%), and amounts to a relative uncertainty of 2.7%. The fiducial zero-jet cross section, in contrast, has only a modeling uncertainty of 0.1%, dominated by the possible mismodeling of the Higgs boson transverse momentum and rapidity distributions.
The simplified template cross section defined as qq → Hqq events with |y H | < 2.5 and no jets with p j T < 200 GeV derives its sensitivity from four VBF categories employing multivariate methods to distinguish VBF events from background processes. The total theory uncertainty is dominated by the modeling uncertainties in the parton shower, underlying event, and hadronization (9%) and in the remaining ggH contamination (5%), and amounts to a relative uncertainty of 13%. In contrast, the fiducial VBF cross section, defined by a dijet mass m j j of at least 400 GeV, a large rapidity separation ∆y j j > 2.8, and an azimuthal difference between the Higgs boson and the dijet pair of |∆φ γγ, j j | > 2.6, has only a modeling uncertainty of 4.5%, dominated by the composition variation of the fractions of the ggH, VBF, V H and ttH processes (4.5%).

Event categorization
The events satisfying the diphoton selection discussed in Section 5.2 are classified, in accord with the reconstructed event kinematics and topology, into 31 exclusive categories that are optimized for the best separation of the Higgs boson production processes and for the maximum sensitivity to the phase space regions defined by the stage 1 of the simplified template cross-section framework. A combined fit to the event reconstruction categories is then performed to determine nine simplified template cross sections (with |y H | < 2.5), as well as production-mode cross sections and signal strength interpretations of the data. The categorization proceeds from the production modes with the smallest expected cross sections to the production modes with largest expected cross sections, in the order described below. In categories with definitions based on jet properties, jets with transverse momenta greater than 30 GeV are used, unless explicitly stated otherwise.

tt H and t H enriched categories
Nine categories enriched in events produced in association with a top quark are defined to target the ttH, tHq, and tHW production modes. These categories are separated into a hadronic channel, where top quarks in the event decay to hadrons via t → W b → qq b; and a leptonic channel, where at least one top quark decays to a charged lepton via t → W b → νb. The single top quark categories are optimized for sensitivity to SM tH production, and are expected to provide additional sensitivity to anomalous values of the top quark Yukawa coupling.
Three categories target the leptonic channel by requiring the presence of at least one prompt lepton and at least one b-tagged jet with transverse momentum greater than 25 GeV. Two of these categories target tH production while the third one is optimized for ttH events. Both tH categories veto events with more than one prompt lepton. The first of these categories ("tH lep 0fwd") contains events with at most three central jets (|η| < 2.5) and a veto on forward jets (|η| > 2.5). The second tH category ("tH lep 1fwd") is defined by events with at most four central jets and at least 1 forward jet. The "ttH lep" category includes events with at least two central jets, while no requirement is applied to the forward jets. To suppress Z H events with Z → , same-flavor dilepton candidates with an invariant mass within 10 GeV of the Z boson mass are vetoed.
Six categories target the hadronic decay channel by selecting events with no prompt leptons and at least three jets, of which at least one is b-tagged. Four of these categories ("ttH had BDT1" to "ttH had BDT4") are defined by means of a boosted decision tree (BDT) trained to identify ttH signal against ggH and multijet background. The BDT exploits five kinematic variables: H T , the scalar sum of jet transverse momenta, m all jets , the mass of all jets, as well as the number of all jets, central jets (|η| < 2.5), and b-tagged jets. The training uses ttH and ggH simulated events and a data-driven multijet background sample defined by diphoton events with at least three jets and in which at least one photon fails to meet either identification or isolation requirements. Using the BDT response as a discriminating variable, events are separated into four categories with an expected fraction of ttH events (among all Higgs boson events in this category) of 95%, 89%, 86%, and 79%, respectively. Two additional hadronic categories enhanced in tH production ("tH had 4j1b" and "tH had 4j2b") are included, defined by events with exactly four jets with transverse momentum greater than 25 GeV and split by events with exactly one or two b-tagged jets, respectively. The distributions of two of the discriminating variables are shown in Figure 4.

V H leptonic enriched categories
Five categories are enriched in Higgs boson production in association with a vector boson, based on different decays of the vector bosons.
The V H dilepton category ("VH dilep") targets Z H production with Z → by requiring the presence of two same-flavor opposite-sign leptons with an invariant mass between 70 GeV and 110 GeV. Two additional V H one-lepton categories target W H production with W → ν. Events are requested to contain exactly one selected electron or muon. To suppress Z H events with Z bosons decaying to ee, in which an electron is misidentified as a photon, a veto is applied to events in which the invariant mass of the selected electron and any of the two signal photons is between 84 GeV and 94 GeV. Events are then split into two regions, where the p T of the lepton+E miss T system is higher ("VH lep High") or lower ("VH lep Low") than 150 GeV. An additional requirement on the E miss T significance, defined as E miss T / √ E T , of at least 1.0 is applied to events in the low lepton+E miss T p T category.
Two V H missing transverse momentum categories target Z H production with Z → νν and W → ν where the lepton was not reconstructed or failed to meet the selection criteria. One category ("VH MET Low") requires 80 GeV < E miss  γγ Tt for events after the first step of the V H hadronic category preselection (see text) for simulated H → γγ events produced in association with hadronically decaying vector bosons (blue) or through ggH, VBF or ttH processes (red), for the expected background from data (γ j, j j) and simulation (γγ, V γγ) control samples (green, purple), and for events from data with 105 GeV< m γγ < 120 GeV or 130 GeV< m γγ < 160 GeV (black dots with error bars showing the statistical uncertainty).

BSM enriched and V H hadronic categories
To provide sensitivity to potential beyond SM contributions, a category ("jet BSM") defined by events with a leading jet with transverse momentum greater than 200 GeV is included in the event selection. This category includes SM events in the typical VBF topology, boosted V(→ j j)H production where the vector boson is reconstructed as a single jet, as well as events produced in gluon-gluon fusion with an energetic jet.
Two V H hadronic categories target V H production with a hadronically decaying vector boson. Events are required to have at least two jets with 60 < m j j < 120 GeV. A BDT classifies the events using the following information: the dijet invariant mass, the component p γγ Tt of the diphoton ì p T transverse to its thrust axis in the transverse plane, the rapidity difference between the dijet and the diphoton system, and the cosine cos θ * γγ, j j where θ * γγ, j j is the angle between the diphoton systems momentum and the direction of motion of the diphoton-dijet system in the Collins-Soper frame. The training uses V H events as signal, and a mixture of simulated signals (everything except V H events), simulated γγ events, and γ j and j j data control samples as background. Using the BDT response as a discriminating variable, events are classified into two categories ("V H had tight" and "V H had loose") with an expected fraction of signal events due to V H production of 42% and 25%, respectively.

VBF enriched categories
Four categories are defined to enhance the sensitivity to vector boson fusion production. Events are required to contain at least two hadronic jets, and the selections applied are based on the two leading jets ( j 1 , j 2 ) in the event. The pseudorapidity separation |∆η j j | between the two leading jets is required to be greater than 2. In addition |η γγ − 0.5(η j1 + η j2 )| is required to be less than 5, with η γγ denoting the pseudorapidity of the diphoton system. The events are first split into two regions based on the value of the transverse momentum p H j j T of the vector sum of the momenta of the reconstructed Higgs boson and of the two leading jets. This variable is highly correlated with the p T of the third jet due to momentum balance. The signal in the p γγ j j T < 25 GeV "low p H j j T " region is dominated by exclusive two-jet-like events, while the signal in the p γγ j j T > 25 GeV "high p H j j T " region is dominated by inclusive ≥ 3-jet like events. This choice minimizes the otherwise large ggH jet-migration uncertainties in this phase space and is similar to a central-jet veto that separates contributions from ggH and VBF.
A BDT is then used to classify events in each region, using six kinematic variables: m j j , |∆η j j |, p γγ Tt , the absolute azimuthal difference of the diphoton and the dijet system |∆φ γγ, j j |, the minimum angular separation between either of the two signal photons and either of the two leading jets ∆R min γ j , and |η γγ − 0.5(η j1 + η j2 )|. A requirement of |∆φ γγ, j j | to be near π effectively vetoes additional jets in the event by restricting the phase space for additional emissions and, to avoid large theoretical uncertainties, the BDT does not use shape information for events with |∆φ γγ, j j | > 2.94 by merging these events into one bin. The training of the BDT uses VBF events as signal, and a mixture of simulated gluon-gluon fusion and γγ events and of γ j and j j data control samples as background. Four exclusive categories are defined with "loose" and "tight" requirements on the BDT classifier in the two p H j j T regions. The "tight" category in the p H j j T > 25 GeV region has an expected fraction of VBF events among all Higgs boson events in this category of 49%, while the "loose" category has an expected fraction of VBF events of 20%. In the p H j j T < 25 GeV region the "tight" category has an expected fraction of VBF events of 85%, whereas the "loose" category has an expected fraction of 61%. Figure 6 shows the distributions of |∆η j j | and |∆φ γγ, j j | in simulated H → γγ events, background events from simulated diphotons and data control samples of γ j and j j events, and events selected from the m γγ sidebands in data. The variables show good separation between VBF events and both gluon-gluon fusion events and background events.

Untagged categories
The remaining "untagged" events are dominated by events produced through gluon-gluon fusion and they are further split into ten categories. The untagged events are first separated by jet multiplicity into events with zero jets, exactly one jet, or at least two jets. The zero-jet events are split into two categories with either two photons in the "central" pseudorapidity region |η| < 0.95, in which the energy resolution is better ("ggH 0J Cen"), or with at least one photon in the "forward" region |η| > 0.95 which has worse energy resolution ("ggH 0J FWD"). The exclusive one-jet ("ggH 1J") and inclusive two-jet ("ggH 2J") categories are further split into regions of diphoton transverse momentum with p γγ T ∈ [0, 60) ("Low"), [60, 120) ("Med"), [120, 200) ("High") or > 200 GeV("BSM"), the latter of which is particularly sensitive to the presence of BSM physics in the loop diagrams associated with the gluon-gluon fusion production mode. Fraction of Events /0.2 Figure 6: The normalized distributions for the expected background of two kinematic variables used for the selection of the VBF categories: (a) |∆η j j | and (b) |∆φ γγ, j j | for events after the first step of the selection (see text) for simulated H → γγ events produced via vector-boson fusion (blue) and gluon-gluon fusion (red), for the expected background from data (γ j, j j) and simulation (γγ) control samples (green), and for events from data with 105 GeV< m γγ < 120 GeV or 130 GeV< m γγ < 160 GeV (black dots with error bars showing the statistical uncertainty).

Categorization summary
A summary of the selection requirements defining each category is provided in Table 4. The predicted signal efficiencies times acceptance and the event fractions per production mode for each category are given in Table 5. The fractions of signal events in each reconstructed category originating from a given simplified template cross-section region are shown in Figure 7. The defined ggH categories exhibit high purities as they are defined with a near one-to-one correspondence with the STXS regions despite small contaminations arising primarily from pileup and selection inefficiencies. The qq → Hqq bins are more ambiguous however still retain much of the diagonal structure. On the other hand, the V H leptonic and top categories are not sensitive to all of the STXS regions of interest, necessitating a merging. Finally, the fractions of signal events in each category from a given production mode are shown in Figure 8.
More information about the number of background events, the purity and the SM signal composition can be found in Tables 27 and 28 in Appendix E. Table 4: Shorthand label and event selection defining each of the 31 event reconstruction categories for the measurement of the signal strengths and simplified template cross sections. The labels denote the predominant production process or kinematic properties the category targets. Jets are required to have p T > 30 GeV unless otherwise noted. The categories are mutually exclusive and the criteria are applied in descending order of the shown categories.
Category Selection = 0 jets, one photon with |η| > 0.95 ggH 0J Cen = 0 jets, two photons with |η| ≤ 0.95 [%] [%] [%] [%] [%] [%] [%]  Figure 7: The fraction of signal events assigned to each reconstructed category (x axis and listed in Table 4) and originating from a given region (listed in Table 1) of the stage-1 simplified template cross section framework (y axis). The black lines separate the ttH and tH, V H leptonic, V H hadronic and VBF enriched, and untagged categories, along with the simplified template cross-section regions they are most sensitive to. The color shows the purity of the region per category.

Production mode measurements
Using the 31 categories, total and production mode specific signal strength measurements are carried out. Measurements of total production cross sections and simplified template cross sections are reported. The simplified template cross sections are measured in a merged scheme introduced in Section 1.2 and summarized in Table 1. In addition, the result of coupling-strength fits are reported.

Observed Data
The observed invariant mass distribution of the selected diphoton pairs of all categories as defined in Table 4, is shown in Figure 9. Figure 10 shows the invariant mass distributions for the sums of the categories most sensitive to the different production modes. In all cases, for illustration purposes, events in each category are weighted according to the expected signal (S 90 ) to background (B 90 ) ratio in a m γγ region containing 90% of the expected signal yield, using a weight of the form ln (1 + S 90 /B 90 ). The results of signal-plus-background fits to these spectra, displaying both the total sum and the background-only components, are shown, as well as the residuals between the data and the background component. Both the signal-plus-background and background-only distributions shown are obtained from the sum of the individual distributions in each category weighted in the same way as the data points. In the fit of Figure 9 a single signal strength µ affecting simultaneously all production modes has been assumed, while in the fits of Figure 10 the four signal strengths µ ggH , µ VBF , µ VH and µ ttH+tH are allowed to vary separately, as described in the following section. The observed mass peak of the Higgs boson, constrained in the fit as m H = 125.09 ± 0.24 GeV, is well within 68% CL of the Run 1 ATLAS+CMS combined measurement.

Signal strengths
The signal strengths, i.e. the ratios of the measured Higgs boson production-mode cross sections times diphoton branching ratio to the SM predictions for each production mode, are measured with the extended likelihood analysis described in Section 6.3. In the likelihood the signal yield N i sig,m in each category i for a particular production mode m is expressed as the product of the integrated luminosity ∫ L dt, the signal strength µ m for that production mode, the expected SM Higgs boson production mode cross section times branching ratio to diphotons, and the acceptance times efficiency (Table 5) for signal events from that production mode in the selected category A global signal strength µ is measured assuming the ratios between different production processes to be as predicted by the SM. The profile of the negative log-likelihood ratio λ(µ) of the global signal strength of all Higgs processes µ for m H = 125.09 ± 0.24 GeV is shown in Figure 11.
The measured central value and 68% CL interval for µ is found to be: SM prediction for ggH production [17,111], which is about 10% lower than the N 3 LO calculation used here (see Section 4). The impact of the main sources of systematic uncertainty (presented in Table 3 and Section 7) in the measured global signal strength is summarized in Table 6. The distinction between yield and migration uncertainties adopted in Table 3 is used and the uncertainties are grouped into theory uncertainties, experimental uncertainties, mass resolution and scale, background shape, and luminosity.
In addition to the global signal strength, the signal strengths of the primary production processes are evaluated by exploiting the sensitivities of the analysis categories of Table 4 to specific production processes. The measured signal strengths are shown together with the global signal strengths discussed above in Figure 12 and found to be:    in the combined signal strength parameter µ. The values for each group of uncertainties are determined by subtracting in quadrature from the total uncertainty the change in the 68% CL range of µ when the corresponding nuisance parameters are fixed to their best fit values. The experimental uncertainty in the yield does not include the luminosity contribution, which is accounted for separately. The uncertainties correspond to the sources detailed in Table 3 Figure 12: Summary of the signal strengths measured for the different production processes (ggH, VBF, V H and top) and globally (µ Run−2 ), compared to the global signal strength measured at 7 and 8 TeV (µ Run−1 ) [76]. The black and orange error bars show the total and statistical uncertainties. The signal strength µ Run−1 was derived assuming the Higgs production-mode cross section based on Refs. [17,111]. Uncertainties smaller than 0.05 are displayed as 0.0. In the more recent theoretical predictions used in this analysis [7,32], the gluon-gluon fusion production-mode cross section is larger by approximately 10%. In this measurement, the bbH contributions are scaled with ggH (µ bbH = µ ggH ), and the tH and ttH productions are measured together (µ top = µ ttH+tH ). Associated production with Z or W bosons is assumed to be scaled by a single signal strength parameter (µ VH = µ ZH = µ WH ).
For Higgs boson production via V H the signal strength is assumed to be scaled by a single parameter (i.e. µ VH = µ ZH = µ WH ). The bbH contributions are scaled with ggH (i.e. µ bbH = µ ggH ), and the tH and ttH productions are measured together rather than separately (i.e. µ top = µ ttH+tH ).
The ggH signal strength is 1 σ below the Standard Model prediction, while the VBF signal strength is 2.2 σ above the prediction. The expected and observed significances Z 0 of VBF production are reported in Table 7: the significance of the observed VBF signal is close to 5 σ.
Since no significant evidence is observed for V H and top-associated Higgs boson production, upper limits at 95% CL are reported for their signal strengths, as shown in Table 8 and Figure 13. The accuracy Upper Limits of the asymptotic approximation was validated using ensembles of pseudo-experiments. Appendix F provides separate limits on µ ZH and µ WH , and Appendix G.1 shows the expected uncertainties for the inclusive and production-mode specific signal strengths reported in Figure 12. Table 9: Best-fit values and uncertainties of the production-mode cross sections times branching ratio. The SM predictions [7] with their uncertainties are shown for each production process. Uncertainties smaller than 0.05 are displayed as 0.0.

Process
Result

Production-mode cross sections
The production-mode cross sections for m H = 125.09±0.24 GeV in a region with Higgs-boson rapidity |y H | < 2.5, multiplied by the branching ratio of the Higgs boson decay to diphotons, are evaluated in the following way. The fitted value of σ top corresponds to the sum of ttH, tHq, and tHW productionmode cross sections under the assumption that their relative ratios are as predicted by the SM. The V H production-mode cross section value is fitted under the assumption that the ratio of the W H and Z H production mode cross sections is as predicted by the SM and includes both production from quark and gluon initial states. Such results are obtained through signal+background fits to the diphoton invariant mass distribution in each category by expressing, in the likelihood, the signal yield N i sig,m in each category i for a particular production mode m as N sig,m = ∫ L dt × σ SM m × B SM (H → γγ) × i m using the same notation as in Section 8.2.2.
The production-mode cross sections are summarized in Figure 14 and Table 9.
The 68% and 95% CL two-dimensional contours of σ ggH × B(H → γγ) and σ VBF × B(H → γγ) are shown in Figure   The ratios are also given in Table 10, along with their statistical, experimental and theoretical uncertainties without the normalization to the central values of the SM predictions. Both the measurements of the ggH and VBF production modes and the evaluations of the V H and top production modes agree within 1-2 σ with the SM expectations. Appendix G.2 provides the expected uncertainties for the production mode cross sections.

Simplified template cross sections
As the current data are not yet sensitive to all of the 31 regions with |y H | < 2.5 (assuming SM acceptance) of the "stage-1" scheme of the simplified template cross-section framework, simplified    Table 12: Observed and expected upper limits at 95% CL on the simplified template cross section times the Higgs to diphoton branching ratio in the BSM sensitive phase space with p j T > 200 GeV. The median expected limits are given for either the case when the true value of the cross section under study is SM-like (σ = σ SM ) or zero. The ±1 σ and ±2 σ intervals for the expected upper limit (σ = 0 fb) are also reported.

Measurement
Observed Expected Limit Expected Limit +2σ +1σ −1σ −2σ In Appendix B additional measurements are reported for a "minimally merged" set of 15 cross sections of kinematic regions defined by the requirement that the fits to expected event yields be stable even in the presence of large uncertainties or correlations.

Coupling-strength fits
Following the tree-level-motivated framework and benchmark models recommended in Ref. [17], measurements of Higgs boson coupling-strength modifiers κ j are implemented. In the narrow width approximation for the Higgs boson, the cross section σ(i → H → γγ) can be parameterized as where Γ H is the total width of the Higgs boson and Γ γγ is the partial decay width to two photons. A set of coupling-strength modifiers, ì κ, is introduced to parameterize possible deviations from the SM predictions of the Higgs boson coupling to SM bosons and fermions. For a given production process or decay mode j, a coupling-strength modifier κ j is defined such that: where all κ j values equal unity in the SM. Here, by construction, the SM cross sections and branching ratio include the best available higher-order QCD and EW corrections. This higher-order accuracy is not necessarily preserved for κ j values different from unity, but the dominant higher-order QCD corrections factorize to a large extent from any rescaling of the coupling strengths and are therefore assumed to remain valid over the entire range of κ j values considered.
Individual coupling-strength modifiers corresponding to tree-level Higgs boson couplings to different particles are introduced as well as two effective coupling-strength modifiers, κ g and κ γ , which describe the loop processes for ggH production and H → γγ decay. This is possible because BSM particles that might be present in these loops are not expected to appreciably change the kinematics of the corresponding process. The gg → H and H → γγ loop processes can thus be studied through these effective coupling-strength modifiers, providing sensitivity to potential BSM particles in the loops. In contrast, the gg → Z H process, which occurs at LO through box and triangular loop diagrams, is always taken into account by resolving the loop in terms of the corresponding coupling-strength modifiers (κ Z and κ t ). No decays to particles other than those predicted in the SM are assumed to take place. These considerations and the limited sensitivity of the data available in this analysis lead to introducing two distinct models.
In the first model, the two parameters κ g and κ γ introduced above are tested assuming that all other couplings are as in the SM. The 68% and 95% CL two-dimensional contours of both effective couplings are shown in Figure 18(a) and the best fit values and uncertainties are κ g = 0.76 +0.17 −0.14 and κ γ = 1.16 +0.14 −0.14 .
In a second model, universal coupling-strength modifiers, κ F (for all fermions) and κ V (for all bosons), are defined that resolve the gg → H and H → γγ loops: The 68% and 95% CL two-dimensional contours of both parameters are shown in Figure 18(b) and the best fit values and uncertainties are κ F = 0.64 +0.18 −0.14 and κ V = 0.92 +0.08 −0.07 . Due to the very limited sensitivity to κ b , κ τ and κ µ , the shown CLs would not change if these coupling-strength modifiers would be fixed to the SM expectation.  The SM prediction is found within the 68% CL contour for the first model and within the 95% CL contour for the second model.
Finally, a set of three ratios is constructed to probe the loop vertices (κ g , κ γ ), total width (κ H ), and the vector and top couplings (κ t and κ V respectively): κ gγ = κ g κ γ /κ H , λ V g = κ V /κ g , and λ tg = κ t /κ g . The parameter λ tg is allowed to be negative to exploit the sensitivity to the relative sign from the tH and gg → Z H processes. The expected and observed sensitivities to the relative sign are illustrated in Figure 19. The bottom quark Yukawa coupling strength is kept fixed to the top quark Yukawa coupling strength (λ bg = λ tg ); this contribution is irrelevant to the λ tg measurement as there is no sensitivity to bbH in the analysis. All other parameters are assumed to be positive without losing generality. The inclusion of κ H in the parameterization allows for non-SM decays of the Higgs boson, but this parameter is not determined directly. The best fit values of these coupling ratios are summarized in Table 13.

Measurement of fiducial integrated and differential cross sections
The measurement of fiducial integrated and differential cross sections provides an alternative way to study the properties of the Higgs boson and to search for physics beyond the Standard Model. The fiducial volumes are defined to closely mimic the detector-level photon and object selections described in Section 5. This reduces the model-dependence of the quoted cross sections in contrast to the per production mode simplified template cross-section measurements of Section 8.2.4. The cross sections are determined by correcting measured signal yields for experimental inefficiencies and resolution effects, and by taking into account the integrated luminosity of the data. Rather than separating individual production modes, fiducial regions are defined such that they are enriched with a given production mode: Fiducial cross sections are measured in a variety of phase space regions, sensitive to for instance gluon-gluon fusion Higgs boson production, vector-boson fusion production, but also to production of the Higgs boson in association with charged leptons, top quarks and neutrinos. Differential and double-differential cross sections are reported for variables related to the diphoton kinematics and the jet activity produced in the Higgs boson events. The observed signal yields are corrected for detector effects resulting in cross sections measured at the particle level. The full statistical and systematic correlations between measured distributions are determined and are available in H D along with the central values of the measured fiducial and differential cross sections to allow future comparisons and interpretations.

Particle-level fiducial definition of the Higgs boson diphoton cross sections
The fiducial volume at particle level is defined using particles with a mean lifetime cτ > 10 mm. Only photons and leptons which do not originate from the decay of hadrons are considered.7 The two highest-p T photons with |η| < 2.37 -excluding 1.37 < |η| < 1.52 -are selected as the diphoton system. The leading (subleading) photon is required to satisfy p T /m γγ > 0. 35 (0.25), where m γγ = m H = 125.09 GeV. Furthermore, for each photon the scalar p T -sum of charged particles with p T > 1 GeV within a cone of ∆R = 0.2 around the photon is required to be less than 5% of the photon p T . The lepton four-momentum is defined as the combination of an electron (or muon) and all nearby photons within ∆R < 0.1 that do not originate from the decay of a hadron. Muons are required to have p T > 15 GeV and |η| < 2.7. Electrons are required to have p T > 15 GeV and |η| < 2.47, excluding the region 1.37 < |η| < 1.52, and are rejected if the distance ∆R to a photon with p T > 15 GeV is less than 0.4. Jets are reconstructed from all particles, excluding muons and neutrinos, using the anti-k t algorithm with a radius parameter of 0.4. Unless stated otherwise, jets are required to have p T > 30 GeV, |y| < 4.4 and to be well separated from photons with p T > 15 GeV (∆R > 0.4) and electrons (∆R > 0.2). The acceptance for the VBF-enhanced fiducial region (introduced in Section 9.2) is increased by loosening the p T cut to 25 GeV. Jets are considered to originate from a b-hadron if there is a b-hadron with p T > 5 GeV within a cone of size ∆R = 0.4 around the jet.
The missing transverse momentum is defined as the vector sum of neutrino transverse momenta, for neutrinos that do not originate from the decay of a hadron. The particle-level fiducial definition is summarized in Table 14.

Fiducial integrated and differential cross sections
The cross section (σ i ) in a fiducial integrated region, and the differential cross section (dσ i /dx) in a bin of variable x, are given by where N sig i is the number of signal events as introduced in Section 6.3, ∫ L dt is the integrated luminosity of the data set, c i is a correction factor that accounts for detector inefficiency and resolution, and ∆x i is the bin width. The correction factors are determined using the simulated samples discussed in Section 4. This bin-by-bin method showed similar performance to that of the non-regularized inversion of the full migration matrix and of regularized methods [112][113][114] within the current statistical accuracy and systematic uncertainties.
The correction factor is 0.75 ± 0.03 in the diphoton fiducial region, defined to unfold all signal events to the fiducial definition of Section 9.1, which is dominated by the photon identification and isolation efficiency. The correction factor also accounts for migrations caused by detector energy resolution and migration in and out of the fiducial phase space due to detector effects. In addition, the correction factor removes a small fraction (0.5% for the diphoton fiducial region) of reconstructed H → f f γ Dalitz decays.8 The correction factor is different in fiducial regions defined by associated jet activity, for example, taking values of 0.66 and 0.87 for the ttH and VBF fiducial regions defined in the next section, respectively. For the diphoton fiducial region the uncertainty in the correction factor is dominated by the theoretical modeling uncertainty. For the ttH and VBF fiducial regions the uncertainties in the correction factors are dominated by uncertainties associated with the knowledge of the jet energy scale and energy resolution, as well as the theoretical modeling. A more complete breakdown is given in Section 9.5.6 and Table 16. T are shown. These correlations were determined from an ensemble of 100,000 bootstrapped data sets which are each reanalyzed using an unbinned maximum-likelihood fit of the diphoton invariant mass spectrum to extract the correlations. The measured differential cross sections in different observables are partially statistically correlated, since they correspond to the same data set in a given fiducial region. These correlations are obtained using a random sampling with replacement method on the detector-level data, often referred to as 'bootstrapping' [115]. Bootstrapped event samples are constructed from the data by assigning each event a weight pulled from a Poisson distribution with unit mean. All measured differential distributions are then reconstructed using the weighted events, and the signal yields in each bin of a differential distribution are determined using an unbinned maximum-likelihood fit of the diphoton invariant mass spectrum. The procedure is repeated with statistically independent weights and the correlation between two bins of different distributions is determined from the obtained cross sections. Figure 20 shows as an illustration the determined correlations between p γγ T , N jets , m j j , |∆φ j j |, and p j 1 T : the lowest p γγ T bin, reconstructing events with a Higgs boson p T between 0 and 20 GeV, is highly correlated with the zero-jet bin. The lowest p j 1 T bin, reconstructing events with a jet p T between 30 and 55 GeV, is strongly correlated with the one-jet bin. And the lowest m j j bin, reconstructing events with at least two jets and a dijet mass between 0 and 170 GeV, is strongly correlated with the two jet bin. The systematic correlations are obtained by fully correlating identical error sources described in Section 7 across bins and observables to construct the corresponding systematic covariance matrix. Knowledge of these correlations allows to simultaneously analyze all fiducial regions, differential and double differential cross sections. This is illustrated later in Section 9.5.8 with a simultaneous fit of the shown five variables of Figure 20 to set limits on new physics contributions. 55

Measurements of cross sections of fiducial integrated regions
Cross sections in five fiducial integrated regions are measured that target either specific Higgs boson production mechanisms or are sensitive to the presence of physics beyond the Standard Model. The selection criteria defining these regions are summarized in Table 14 and a description of each region follows: 1. Diphoton fiducial: This region unfolds all signal events after the selection presented in Section 5.

VBF-enhanced: This region retains all events with at least two jets and with an invariant dijet
mass m j j of at least 400 GeV, a large rapidity separation ∆y j j > 2.8, and an azimuthal difference between the Higgs boson and the dijet pair of |∆φ γγ, j j | > 2.6. All variables are computed using the two highest-p T jets in the event with p T > 25 GeV with matching detector-level cuts.
3. N lepton ≥ 1: This region retains events that contain at least one electron or one muon with p T > 15 GeV. For electrons the pseudo-rapidity needs to satisfy |η| < 2.47 (excluding 1.37 < |η| < 1.52) and for muons |η| < 2.7 is required. Such events are enriched in Higgs bosons produced in association with a vector boson.

High E miss T : This region retains events with missing transverse momentum E miss
T > 80 GeV and p γγ T > 80 GeV is defined to study V H production and possible contributions of Higgs boson production with dark matter particles. The simultaneous requirement that the Higgs boson system balances the missing transverse momentum reduces the fraction of selected events at detector level without particle-level E miss T > 80 GeV.

ttH-enhanced:
This region retains events with either at least one lepton and three jets or no leptons and four jets to study Higgs boson production in association with top quarks. In addition, one of the jets needs to be identified as originating from a bottom quark.
The expected composition of Higgs boson events in the Standard Model after reconstruction and at particle level is summarized in Figure 21. At particle level the VBF-enhanced fiducial region contains about 65% VBF and 32% ggH events. The particle-level N lepton ≥ 1 region is dominated by W H (47%), ttH (37%) and Z H (13%) production. The particle-level high E miss T region is populated by about equal amounts of W H, Z H, and ttH (32%, 30%, and 35%). Finally, the particle-level ttH-enhanced region contains about 80% ttH events.
The fitted invariant mass spectra for all regions are shown in Figures 22 and 23. The results of signalplus-background fits to these spectra is shown, displaying both the total sum and the background-only component as well as the residuals between the data and the background. In the diphoton fiducial region, the Higgs boson signal is clearly visible on the falling non-resonant background. In total, 1491 ± 248 (stat.) ± 64 (syst.) Higgs boson signal events are extracted. Clear evidence for Higgs boson production is observed in the VBF-enhanced region with 117 ± 26 (stat.) ± 4 (syst.) signal events, corresponding to an observed significance of 4.2 standard deviations.
The remaining three regions all show positive signal yields with large, predominantly statistical, uncertainties: 14±11, 19±11, 6±15 for the N lepton ≥ 1, high E miss T , and ttH-enhanced fiducial regions, respectively, and the error corresponds to the sum of the statistical and systematic uncertainties.  Figure 21: The expected composition of Higgs boson events in each fiducial region (a) after the reconstruction and (b) at particle-level. Details about the reconstruction can be found in Section 5 and the definition of the particle-level fiducial volume is given in Section 9.1.
[GeV] The cross section for pp → H → γγ measured in the diphoton fiducial region is σ fid = 55 ± 9 (stat.) ± 4 (exp.) ± 0.1 (theo.) fb , which is to be compared with the Standard Model prediction of 64 ± 2 fb. The gluon-gluon fusion contribution to the Standard Model prediction and its uncertainty are taken to be the N 3 LO QCD and NLO EW prediction of Refs. [7,24,[31][32][33][34] corrected for the H → γγ branching ratio and the fiducial acceptance. The fiducial acceptance is defined using the P NNLOPS prediction for gluon-gluon fusion [23]. The contributions to the Standard Model prediction from the VBF, V H, bbH and ttH production mechanisms are determined using the particle-level predictions normalized with theoretical calculations as discussed in Section 4, and are collectively referred to as X H. The measured cross section is compatible with the Standard Model prediction and the observed ggH coupling strength measured in Section 8, as the diphoton fiducial region is dominated by gluon-gluon fusion production.    The measured cross sections in the diphoton, VBF-enhanced, N lepton ≥ 1, high E miss T , and ttH-enhanced fiducial regions. The gluon-gluon fusion contribution to the Standard Model prediction of the diphoton fiducial region is taken to be the N 3 LO prediction of Refs. [7,24,[31][32][33][34] corrected for the H → γγ branching ratio and the fiducial acceptance. The gluon-gluon fusion contribution to the Standard Model for all the other regions is taken from the P NNLOPS prediction normalized with the N 3 LO prediction and includes all theory uncertainties related to gluon-gluon fusion as discussed in Section 7.3. The contributions to the Standard Model prediction from VBF, V H ttH and bbH production mechanisms are determined using the particle-level predictions described in Section 4 normalized with theoretical calculations.

Fiducial region
Measured cross section SM prediction Diphoton fiducial 55 ± 9 (stat.) ± 4 (exp.) ± 0. The cross section of the VBF-enhanced region is measured to be which is to be compared with the Standard Model prediction of 2.3 ± 0.1 fb. The gluon-gluon fusion part of the SM prediction is constructed from the P NNLOPS prediction for gluon-gluon fusion normalized with the N 3 LO in QCD and NLO EW prediction of Refs. [7,24,[31][32][33][34]. This prediction is labeled as "default MC" in the following and includes all theory uncertainties related to gluon-gluon fusion as discussed in Section 7.3.
For the N lepton ≥ 1, high E miss T , and ttH-enhanced fiducial regions, limits on the cross sections are reported at the 95% CL.9 Figure 24 and Table 15 summarize measured cross sections of the fiducial regions and limits, and compare both to the Standard Model expectations, constructed as outlined above. The P NNLOPS prediction, without any additional corrections, is also shown. The uncertainty band is estimated using a set of scale variations and includes PDF uncertainties from eigenvector variations. The Standard Model predictions of all fiducial regions are in agreement with the corresponding measured cross sections.   Figure 24: The measured cross sections or cross-section upper limits of the diphoton, VBF-enhanced, N lepton ≥ 1, high E miss T , and ttH-enhanced fiducial regions are shown. The intervals on the vertical axis each represent one of these fiducial regions. The data are shown as filled (black) circles. The error bar on each measured cross section represents the total uncertainty in the measurement, with the systematic uncertainty shown as a dark gray rectangle. Each cross section limit is shown at the 95% confidence level. The measured cross sections are compared to a range of predictions and a detailed description of each prediction can be found in the text. All comparisons include the SM predictions arising from VBF, V H, ttH, and bbH, which are collectively labeled as X H.

Measurements of cross sections of inclusive and exclusive jet multiplicities
The production of Higgs bosons in association with jets is sensitive to the theoretical modeling in QCD and to the contribution of different Higgs boson production mechanisms. In the SM, events with zero or one jet are dominated by gluon-gluon fusion production. In events with two jets the contributions from VBF and V H production modes become more important. Higgs boson production in association with top quarks (ttH) can be probed in events with the highest jet multiplicities. In BSM scenarios, the jet multiplicity distribution is sensitive to new heavy particles coupling to the Higgs boson and vector bosons. For the measurements presented here, jet multiplicity bins with zero, one, two, and at least three jets with p T larger than 30 GeV and absolute rapidity |y| < 4.4 are defined. In addition, jet multiplicity bins with a p T larger than 50 GeV are defined for zero, one, or at least two jets. The measured cross sections are compared to a range of predictions of gluon-gluon fusion production: • The parton-level N 3 LO QCD and NLO EW prediction of Refs. [7,24,[31][32][33][34]. This prediction is shown for the inclusive zero-jet cross section.
• The parton-level JVE+N 3 LO prediction of Ref. [116], which includes NNLL resummation in QCD of the p T of the leading jet which is matched to the N 3 LO total cross section. This prediction is shown for the inclusive one-jet cross section.
• The parton-level STWZ-BLPTW predictions of Refs. [99,101], which include NNLL +NNLO resummation for the p T of the leading jet in QCD, combined with a NLL +NLO resummation in QCD for the subleading jet.10 The numerical predictions for √ s = 13 TeV are taken from Ref. [7]. This prediction is shown for the inclusive zero-, one-and two-jet cross sections as well as for the exclusive zero-and one-jet cross sections.
• The parton-level NNLOJET prediction of Refs. [117,118] is a fixed-order NNLO prediction in QCD for inclusive H+one-jet production. This prediction is shown for the inclusive one-, two-jet, and three-jet cross sections as well as for the exclusive one-and two-jet cross sections.
• The parton-level G S prediction of Refs. [119,120], which provides the fixed-order loop contributions accurate at NLO in QCD in the inclusive H + zero-jet, H + one-jet, H + two-jet, and H + three-jet regions. The real-emission contributions at fixed order in QCD are provided by S [64]. This prediction is shown for the inclusive one-, two-jet, and three-jet cross sections as well as for the exclusive one-and two-jet cross sections.
• The default MC prediction (P NNLOPS normalized with the N 3 LO in QCD and NLO EW cross section) introduced in Section 9.3. This prediction is shown for all measured inclusive and exclusive jet cross sections.
• The P NNLOPS prediction which is already described in Section 4. This prediction is shown for all measured inclusive and exclusive jet cross sections.
• The MG5_ MC@NLO prediction of Refs. [46,105], which includes up to two jets at NLO accuracy using the F F merging scheme [106]. The central merging scale is taken to be 30 GeV. The generated events are passed to P 8 [29] to provide parton showering and hadronization to create the full final state (without underlying event). This prediction is shown for all measured inclusive and exclusive jet cross sections.
All predictions but NNLOJET and S (M @N ) use the NNLO PDF set following the PDF4LHC15 recommendations. The NNLOJET prediction uses the CT14 NNLO PDF set [130] and S (M @N ) uses the NNPDF3.0PDF set [47]. G S , S (M @N ), and NNLOJET apply the kinematic selection on the final-state photons. For all other predictions, the fiducial acceptance is determined using P NNLOPS. The cross sections of all parton-level predictions are multiplied with isolation correction factors to account for the efficiency of the fiducial photon isolation criterion. The additional uncertainties in the isolation correction are determined by studying multiple event generators and/or event generator tunes, and are included in the uncertainty bands of the parton-level predictions. No correction factors nor additional uncertainties to account for the impact of hadronization and the underlying event activity are applied, so the theory uncertainties in the parton-level predictions may be incomplete, but example values for such corrections and their uncertainties can be found in Table 24 in Appendix D. All other acceptance and correction factors along with their associated uncertainties can also be found in Appendix D.
No K-factors are applied to the predictions and the contributions from X H are also included in the comparison using the corresponding generators and cross sections described in Section 4. Figure 25(a) shows exclusive and inclusive zero-, one-and two-jet cross sections and the inclusive three-jet cross section for jets defined with p T > 30 GeV. Figure 25(b) shows the exclusive zeroand one-and the inclusive two-jet cross section with p T > 50 GeV. The measured cross sections are in agreement with the Standard Model predictions, although there is a 1.5 σ deficit in the N jets = 0 cross section for jets defined with p T > 30 GeV. As shown in Figure 20, there is a sizeable positive correlation between zero-jet and low-p γγ T events, and a similar deficit is observed there (cf. Section 9.5.1).

Measurements of differential and double-differential cross sections
Eleven fiducial differential cross sections are measured that characterize the Higgs boson production kinematics, the kinematics of jets produced in association with the Higgs boson, the spin and CP quantum numbers of the Higgs boson and variables sensitive to the VBF production mechanism. In addition, two double-differential cross sections are reported. The measurement of seven additional variables can be found in Appendix C.

Measurements of cross sections probing the Higgs boson production kinematics
Measuring the transverse momentum of the diphoton system, p    The data are shown as filled (black) circles. The vertical error bar on each data point represents the total uncertainty in the measured cross section and the shaded (gray) band is the systematic component. The measured differential cross sections are compared to a range of predictions and details can be found in the text. The width of the bands of each prediction reflects the total theoretical uncertainty. The small contribution from VBF, V H, ttH, and bbH is also shown as a (green) histogram and denoted by X H. differential cross sections for pp → H → γγ as a function of p γγ T and |y γγ | are shown in Figure 26. The chosen bin widths are a compromise between retaining a sufficiently significant signal and providing spectra with good granularity. Each bin is chosen such that it retains an expected significance of at least two standard deviations, estimated using the P NNLOPS and additional predictions described in Section 4 as well as using a fit to m γγ sidebands. The measurements are compared to several predictions of gluon-gluon fusion: • The default MC prediction (P NNLOPS normalized with the N 3 LO in QCD and NLO EW cross section) introduced in Section 9.3.
• H [131,132], which provides predictions at NNLO with p H T resummation at NNLL and differentially in p • R ISH+NNLO [133], which provides predictions using a p H T resummation to NNLL and matching to the one-jet NNLO differential spectrum from NNLOJET [117,118]. The shown p γγ T R ISH+NNLO prediction does include corrections from the finite top and bottom quark masses.
• SCET +MCFM8 which provides predictions at NNLO+NNLL ϕ accuracy derived by applying a resummation of the virtual corrections to the gluon form factor [134,135] and differentially in |y γγ | and | cos θ * |. 11 The underlying NNLO predictions are obtained using MCFM8 with zero-jettiness subtractions [136,137].
No additional K-factors are applied to the predictions, which all use the NNLO PDF set following the PDF4LHC15 recommendations, and the fiducial acceptance for R ISH+NNLO is determined using P NNLOPS. The SCET +MCFM8 and H predictions include the kinematic acceptance and are corrected and apply correction factors accounting for the photon isolation efficiency as described in Section 9.4. As also mentioned in Section 9.4, no correction factors to account for the impact of hadronization and the underlying-event activity are used. The SM prediction shows a slight excess at low transverse momentum and low rapidity, and shows a slight deficit at large transverse momentum. The slightly harder Higgs boson transverse momentum shown in Figure 26 is consistent with the ATLAS Run 1 measurements in both the H → γγ and H → Z Z * → 4 decay channels [9,138] and the measured zero-jet cross section. The Standard Model prediction is in agreement with the measured distributions.

Measurements of cross sections probing the jet kinematics
The transverse momentum and absolute rapidity of the leading jet, p j 1 T and |y j 1 |, as well as the transverse momentum and absolute rapidity of the subleading jet, p j 2 T and |y j 2 |, are sensitive to the theoretical modeling and to the relative contributions of the different Higgs boson production mechanisms. The transverse momentum distribution of the leading jet probes the emission of energetic quarks and gluons. In events with two jets, the contributions of VBF and V H productions become more important. The differential cross sections for pp → H → γγ as a function of p j 1 T , |y j 1 |, p j 2 T , and |y j 2 | are shown in Figure 27. The chosen bin widths are a compromise between keeping migrations between bins 11 The subscript ϕ refers to the fact that the applied resummation is to the gluon form factor.   small whilst retaining enough statistical power to measure the differential spectra. The measured p j 1 T spectrum shown in Figure 27(a) is compared to the default MC prediction as introduced in the previous section as well as to the NNLOJET and SCET (STWZ) [99,135] predictions. Both the NNLOJET and SCET predictions are corrected using isolation correction factors to account for the impact of the isolation efficiency. In addition, the NNLOJET prediction is corrected for the kinematic acceptance and the uncertainties in these corrections is included in the uncertainty bands of both NNLOJET and SCET . The first bin of the leading jet p T spectrum represents zero-jet events that do not contain any jet with p T > 30 GeV. The predicted p T distributions slightly exceed the measured distribution at low transverse momentum and all show a slight deficit at large transverse momentum. Both are compatible with the observed slightly harder Higgs boson transverse momentum distribution. The measured |y j 1 | distribution shown in Figure 27(b) is compared to the default MC and the NNLOJET predictions: Both show a slight excess at low rapidity. In Figure 27(c) the measured subleading jet p T distribution is shown. The first bin of p j 2 T represents one-jet events that do not contain two or more jets with p T > 30 GeV. The measured distribution is compared to the default MC, S (M @N ), and G S predictions, as introduced in Section 9.4. Finally, in Figure 27(d) the subleading jet rapidity distribution, |y j 2 |, is shown and compared to the expectation from the default MC, S (M @N ), and G S predictions. The SM predictions are in agreement with the measured distributions and no significant deviations are seen.

Measurements of cross sections probing spin and CP
The absolute value of the cosine of the angle between the beam axis and the photons in the Collins-Soper frame [11] of the Higgs boson, | cos θ * |, can be used to study the spin of the Higgs boson. The    azimuthal angle between the two leading jets, ∆φ j j ,12 in events containing two or more jets is sensitive to the charge conjugation and parity properties of the Higgs boson interactions with gluons and weak bosons in the gluon-gluon fusion and the VBF production channels, respectively [139][140][141][142].
The differential cross sections for pp → H → γγ as a function of | cos θ * | and ∆φ j j are shown in Figure 28. For a scalar particle | cos θ * |, shows a strong drop around 0.6 due to the fiducial requirement on the photon system, whereas for a spin-2 particle, an enhancement would be present in precisely this region. The charge conjugation and parity properties of the Higgs boson are encoded in the azimuthal angle between the jets: For example, in gluon-gluon fusion, its distribution for a CP-even coupling has a dip at ± π 2 and present peaks at 0 and ±π, whereas for a purely CP-odd coupling it would present as peaks at ± π 2 and dips at 0 and ±π. For VBF the SM prediction for ∆φ j j is approximately constant with a slight rise towards ∆φ j j = ±π. Any additional anomalous CP-even or CP-odd contribution to the interaction between the Higgs boson and weak bosons would manifest itself as an additional oscillatory component, and any interference between the SM and anomalous couplings can produce distributions peaked at either ∆φ j j = 0 or ∆φ j j = ±π [139,141,142]. The shape of the distribution is therefore sensitive to the relative contribution of gluon-gluon fusion and vector-boson fusion, as well as to the tensor structure of the interactions between the Higgs boson and gluons or weak bosons. This is exploited in Section 9.5.8 to set limits on new physics contributions. To quantify the structure of the azimuthal angle between the two jets, a ratio is defined as , which is motivated by a similar ratio presented in Ref. [141]. The measured ratio in data as determined by measuring |∆φ j j | in three bins is This value can be compared to the SM prediction from the default MC simulation. The predicted value is A SM |∆φ j j | = 0.44 ± 0.01, consistent with the measured ratio.
In summary, the measured | cos θ * | and ∆φ j j distributions are consistent with Standard Model predictions for a CP-even scalar particle.

Cross sections probing the VBF production mode
The distribution of the dijet rapidity separation, |∆y j j |, the azimuthal angle between the dijet and diphoton systems, |∆φ γγ, j j |, and the invariant mass of the leading and subleading jets, m j j for events with two or more jets are sensitive to the differences between the gluon-gluon fusion and VBF production mechanisms. In vector-boson fusion, the t-channel exchange of a W/Z boson typically results in two moderate-p T jets that are well separated in rapidity. Furthermore, quark/gluon radiation in the rapidity interval between the two jets is suppressed in the VBF process when compared to the gluongluon fusion process, because there is no color flow between the two jets. The |∆φ γγ, j j | distribution for VBF production is therefore expected to be steeper and more peaked towards |∆φ γγ, j j | = π than for gluon-gluon fusion. 12 To preserve the sign information, the azimuthal angles of the jets are ordered according to the jet with the highest rapidity. This definition of ∆φ j j is invariant under a redefinition of the ordering by choosing the opposite beam axis, as explained in Ref. [139].   Figure 28: The differential cross sections for pp → H → γγ as a function of (a) | cos θ * | and (b) ∆φ j j are shown and compared to the SM expectations. The data and theoretical predictions are presented in the same way as in Figure 26. In addition, the SCET +MCFM8 prediction and the S (M @N ) and G S predictions, described in the text, are displayed in (a) and (b), respectively.
The differential cross sections for pp → H → γγ as a function of |∆y j j |, |∆φ γγ, j j |, and m j j are shown for events with at least two jets with p T > 30 GeV in Figure 29. These variables are used to discriminate between gluon-gluon fusion and the VBF production of the Higgs boson and enter the multivariate classifier introduced in Section 8.1.4 that defines the categories used for the simplified template cross-section and coupling measurements. The measured distributions are in agreement to the default MC, S (M @N ), and the G S predictions. The accuracy of the fixed-order parton-level prediction from G S breaks down in the lowest bin of π − |∆φ γγ, j j | and the measured cross section moderately exceeds the SM predictions at high m j j values.

Double-differential cross sections
The double-differential cross section for pp → H → γγ as a function of p γγ T and N jets , for jets with p T > 30 GeV, and p γγ T and | cos θ * | are shown in Figure 30. These cross sections are sensitive to the modeling of the Higgs boson kinematic, its production mechanisms, and its spin-CP properties. Both double-differential cross sections are in agreement with the Standard Model expectation.

Impact of systematic uncertainties on results
A summary of the uncertainties in the measured cross sections of the fiducial regions are shown in Table 16. As an example concerning the differential measurements, a breakdown of the systematic uncertainties in the differential cross sections as a function of p γγ T and N jets is shown in Figure 31.       Figure 30: The double-differential cross section for pp → H → γγ as a function of (a) p γγ T and N jets , for jets with p T > 30 GeV, and (b) p γγ T and | cos θ * | separating the two regions of | cos θ * | < 0.5 and | cos θ * | > 0.5 from each other. The data and theoretical predictions are presented in the same way as in Figure 26. Table 16: The expected uncertainties, expressed in percent, in the cross sections measured in the diphoton fiducial, VBF-enhanced, N lepton ≥ 1, ttH-enhanced, and high E miss T regions. The fit systematic uncertainty includes the effect of the photon energy scale and resolution, and the impact of the background modeling on the signal yield. The theoretical modeling uncertainty is defined as the envelope of the signal composition, the modeling of Higgs boson transverse momentum and rapidity distribution, and the uncertainty of parton shower and the underlying event (labeled as "UE/PS") as described in Section 7.4.

Source
Uncertainty in fiducial cross section Diphoton VBF-enhanced the uncertainty in the fitted signal yield, due to the background modeling and the photon energy resolution, is typically more important than the uncertainty in the correction factor due to the theoretical modeling. The jet energy scale and resolution uncertainties become increasingly important for high-jet multiplicities and in the ttHand VBF-enhanced phase space. [GeV]

Compatibility of measured distributions with the Standard Model
The compatibility between the measured distributions and the Standard Model is tested by comparing the first and second moments of the measured distributions. Figure 32 shows the first and second moments (mean and RMS) of the distributions and compares them to the moments of the default MC prediction, as calculated from the measured and predicted cross-section bins. The theory uncertainties are constructed as outlined in Section 7.3. The measured Higgs boson transverse momentum has somewhat higher first and second moments than the Standard Model prediction, which is consistent with the previous observations [9,138]. The leading-jet p T spectrum shows the same feature. In addition a χ 2 test is carried out for all distributions reported in Section 9: The resulting p-values are reported in Table 17, which confirms that within the current uncertainties the data are in agreement with the SM predictions.
(b) Figure 32: (a) The ratio of the first moment (mean) of each differential distribution predicted by the Standard Model to that observed in the data. The SM moment is calculated by using the default MC distributions for gluon-gluon fusion and the other production mechanisms. (b) The ratio of the second moment (RMS) of each differential distribution predicted by the Standard Model to that observed in the data. The intervals on the vertical axes each represent one of the differential distributions. The band for the theoretical prediction represents the corresponding uncertainty in that prediction (see text for details). The error bar on the data represents the total uncertainty in the measurement, with the gray band representing only the systematic uncertainty.

Search for anomalous Higgs-boson interactions using an effective field theory approach
The strength and tensor structure of the Higgs-boson interactions can be investigated using an effective field theory approach, which introduces additional CP-even and CP-odd interactions that change the event rates and the kinematic properties of the Higgs boson and associated jet spectra from those in the Standard Model. The parameters of the effective field theory are probed using a fit to five of the most relevant differential cross sections. The effective Lagrangian of Ref. [143] is used which adds dimension-six operators of the Strongly Interacting Light Higgs formulation [13] to the Standard Model interactions. The H → γγ differential cross sections are mainly sensitive to the operators that affect the Higgs boson interactions with gauge bosons and the relevant terms in the Lagrangian can be specified by  [143] can also modify Higgs-boson interactions but are not considered here due to the lack of sensitivity of the H → γγ decay channel. Combinations of some of the CP-even operators have been constrained using global fits to experimental data from LEP and the LHC [143,145,146].
The effective Lagrangian has been implemented in FeynRules [144].13 Parton-level event samples are produced for specific values of Wilson coefficients by interfacing the universal file output from FeynRules to the MG5_ MC@NLOevent generator [46]. Higgs bosons are produced via gluon-gluon fusion with up to two additional partons in the final state using leading-order matrix elements.
The generated events are passed to P 8 [29] to provide parton showering, hadronization and underlying event and the zero-, one-and two-parton events are merged using the MLM matching scheme [148] to create the full final state. Event samples containing a Higgs boson produced either in association with a vector boson or via vector-boson fusion are produced using leading-order matrix elements and passed through the P 8 generator. For each production mode, the Higgs boson mass is set to 125 GeV and events are generated using the NNPDF2.3LO PDF set [47] and the A14 parameter set [48]. All other Higgs boson production modes are assumed to occur as predicted by the SM. Event samples are produced for different values of a given Wilson coefficient. The particle-level differential cross sections are produced using R [149]. The P method [150] is used to 13 The implementation in Ref. [144] involves a redefinition of the gauge boson propagators that results in unphysical amplitudes unless certain physical constants are also redefined. The original implementation did not include the redefinition of these physical constants. However, the impact of redefining the physical constants is found to be negligible on the predicted cross sections across the range of Wilson coefficients studied. The relative change in the predicted Higgs boson cross sections as functions of the different Wilson coefficients is also found to agree with that predicted by the Higgs characterization framework [147], with less than 2% variation across the parameter ranges studied.
interpolate between these samples, for each bin of each distribution, to provide a parameterization of the effective Lagrangian prediction. The parameterization function is determined using 11 samples when studying a single Wilson coefficient, whereas 25 samples are used when studying two Wilson coefficients simultaneously. As the Wilson coefficients enter the effective Lagrangian in a linear fashion, second-order polynomials are used to predict the cross sections in each bin. The method was validated by comparing the differential cross sections obtained with the parameterization function to the predictions obtained with dedicated event samples generated at the specific point in parameter space.
The model implemented in FeynRules fixes the Higgs boson width to be that of the SM, Γ H = 4.07 MeV [17]. The cross sections are scaled by Γ H /(Γ H + ∆Γ), where ∆Γ is the change in partial widths due to a specific choice of Wilson coefficient. The change in partial widths is determined for each Higgs coupling using the partial-width calculator in MG5_ MC@NLOand normalized to reproduce the SM prediction from H [18].
The leading-order predictions obtained from MG5_ MC@NLOare reweighted to account for higherorder QCD and electroweak corrections to the SM process, assuming that these corrections factorize from the new physics effects. The differential cross section as a function of variable X for a specific choice of Wilson coefficient, c i , is given by where the summation j is over the different Higgs boson production mechanisms, 'MG5' labels the interpolated MG5_ MC@NLOprediction and 'ref' labels a reference sample for SM Higgs boson production. For the reference sample the default MC simulation is used.
The measured differential cross sections of p γγ T , N jets , m j j , |∆φ j j |, and p j 1 T are compared in Figure 33(a) to the SM hypothesis and to two non-SM hypotheses, specified byc g = 2 × 10 −4 andc HW = 0.05, respectively. The new CP-odd gluon-gluon fusion operator results in a large increase in rate and the additional CP-even W H operator leads to a larger number of Higgs boson with sizeable p T and an increased number of zero-jet events.
The ratios of the expected differential cross sections to the SM predictions for some representative values of the Wilson coefficients are shown in Figure 33(b). The impact of thec g andc g coefficients are presented for the gluon-gluon fusion production: it displays a large change in the overall crosssection normalization. Thec g coefficient also changes the shape of the ∆φ j j distribution, which is expected from consideration of the tensor structure of CP-even and CP-odd interactions [139,141]. In contrast, the impact of thec HW andc HW coefficients are presented specifically for the VBF+VH production channel: one expects large shape changes in all of the studied distributions and the ∆φ j j distribution is known to discriminate between CP-even and CP-odd interactions in the VBFproduction channel [140].
Limits on Wilson coefficients are set by constructing a likelihood function where ì σ data and ì σ pred are k-dimensional vectors from the measured and predicted differential cross sections of the five analyzed observables, C = C stat + C syst + C theo is the k × k total covariance  Figure 33: (a) The measured differential cross sections as a function of p γγ T , N jets , m j j , |∆φ j j |, and p j 1 T are compared to the SM hypothesis and two non-SM hypotheses withc g = 1 × 10 −4 andc HW = 0.05, respectively. (b) Ratios of differential cross sections, as predicted for specific choices of Wilson coefficient, to the differential cross sections predicted by the SM: the impact of non-zeroc g andc g is shown relative to the SM ggH prediction, while the impact of non-zeroc HW andc HW is shown relative to the SM VBF+V H prediction. matrix defined by the sum of the statistical, systematic and theoretical covariances, and |C| denotes its determinant. The theory covariance is constructed as outlined in Section 7.3 and includes no additional uncertainty to account for the factorization assumption in Eq. 9.5.8. Based on this likelihood, one can construct a χ 2 test to test the compatibility of the five distributions with the SM and a probability of 93% is found. In what follows, the likelihood function is numerically maximized to determine L max and confidence limits for one or several Wilson coefficients are determined via with L(c i ) denoting the likelihood value evaluated for a given Wilson coefficient value c i , and f (x) denoting the distribution of the test statistic. The coverage of the confidence limit is determined using ensembles of pseudo-experiments. Form factors are sometimes used to regularize the change of the cross section above a momentum scale Λ FF . This was investigated by reweighting the VBF+VH samples using form-factor predictions from VBFNLO [151]. The impact on thec HW andc HW limits is negligible for Λ FF >1 TeV.
In Table 18, the observed and expected 95% CL limits for four Wilson coefficient fits are given. The limit forc g is derived by fixing all other Wilson coefficients to zero. This additional interaction can interfere with the corresponding SM interaction and destructive interference causes the gluon-gluon fusion production-mode cross section to be zero atc g ∼ −2.2 × 10 −4 . Thec g coefficient is also derived after setting all Wilson coefficients to zero. Due to the CP conjugate structure of the interaction, no interference with the SM process is possible. The 95% CL limit forc HW is obtained after settinḡ c H B =c HW to ensure that the partial width for H → Z γ is unchanged from the SM prediction (Values of |c HW −c H B | > 0.03 lead to a very large decay rate for the H → Zγ process that is contradicted by the experimental constraints reported by ATLAS [152,153]) and setting all other Wilson coefficients to zero. Finally, the 95% CL limit forc HW is given after settingc H B =c HW to ensure a SM decay rate for H → Z γ and all other Wilson coefficients to zero. The observed limits are improved by about a factor of two compared to the Run 1 analysis of Ref. [14]. Figure 34 shows the 68% and 95% confidence regions obtained from scanningc HW andc HW simultaneously, with the other two Wilson coefficients set toc H B =c HW andc H B =c HW . All other Wilson coefficients are fixed at zero. Thec HW andc HW Wilson coefficients produce large shape changes in all distributions, as shown in Figure 33, and the obtained limits are strongest when fitting all five distributions simultaneously. The shape of the observed 68% confidence regions thus results from both shape and yield differences between data and expectations: the operators proportional toc HW can destructively interfere with the SM contributions, a negative value ofc HW reduces the overall predicted cross section in the zero-jet and the lowest m j j bins, where deficits are observed in the data. The operators proportional toc HW can only increase the cross section from its SM value and can increase the predicted cross sections in the higher jet bins and the tails of the distributions (cf. Figure 33). If only shape information is used to constrain the Wilson coefficients, the reported limits onc HW andc HW weaken by about 20% and 50%, respectively. As also shown in Figure 34, these results display significant improvements on similar limits obtained from the Run-1 analysis [14]. All reported results assume that QCD effects and new physics effects factorise. This assumption cannot be avoided with the current state-of-the-art implementation of the effective Lagrangian of Ref. [143]. The full statistical and systematic correlations between measured distributions and all measured fiducial and differential cross sections are available in H D to allow future interpretations with better models. Table 18: Observed allowed ranges at 95% CL for thec g andc HW Wilson coefficients and the CP-conjugate coefficients. Limits onc g andc g are each derived with all other Wilson coefficients set to zero. Limits onc HW andc HW are derived withc H B =c HW andc H B =c HW , respectively.

Coefficient
Observed 95% CL limit Expected 95% CL limit  Figure 34: The observed 68% (dark) and 95% (light) confidence level regions from the simultaneous fit to thē c HW andc HW Wilson coefficients. The values ofc H B andc H B are set to be equal toc HW andc HW , respectively, and all other Wilson coefficients are set to zero, except forc H B andc H B which are set to be equal toc HW andc HW , respectively. The SM expectation at (0, 0) is also shown, together with the Run-1 confidence regions reported in Ref. [14].

79
Measurements of Higgs boson cross sections in the Higgs boson to diphoton decay channel are performed using pp collision data recorded by the ATLAS experiment at the LHC. The data were taken at a center-of-mass energy of √ s = 13 TeV and correspond to an integrated luminosity of 36.1 fb −1 . All measurements assume a Higgs boson mass of 125.09 ± 0.24 GeV. The measured signal strength relative to the Standard Model expectation is found to be: −0.14 .
Signal strengths of the main production modes are measured separately via event reconstruction categories that are designed to be sensitive to the specific production modes. They are found to be: The total uncertainties of both the global and the production mode signal strengths is dominated by their respective statistical uncertainties. The global signal strength measurement improves on the precision of the previous ATLAS measurement in the diphoton channel by a factor of two [154]. The ggH (VBF) signal strength is measured to be 1 σ below (2 σ above) the Standard Model expectation. The precision of the coupling-strength measurements involving top quarks improves by about a factor of three compared to the previous ATLAS measurement in the diphoton channel [154]. These improvements result from a combination of the larger Higgs boson sample collected at √ s = 13 TeV, from the use of multivariate techniques to target the VBF, V H, and top-quark associated production modes more efficiently, from the improved precision of the ggH Standard Model theory predictions, and from a significant reduction of some of the experimental uncertainties such as the photon energy resolution. Production mode cross-section measurements for a Higgs boson of rapidity |y H | < 2.5 for gluon-gluon fusion, vector-boson fusion, and Higgs boson production in association with vector bosons or a top quark pair are reported: Higgs boson coupling-strength modifiers are reported and two models are investigated: the first one reports results on effective coupling-strength modifiers for Higgs boson production in gluon-gluon fusion and decay, κ g and κ γ , respectively. They are found to be: κ g = 0.76 +0.17 −0.14 , and κ γ = 1.16 +0.14 −0.14 .
The second model resolves the Higgs boson production and decay loops in terms of the more fundamental fermionic and vector boson couplings under the assumption of universal coupling-strength modifiers for all fermions and vector bosons, namely κ V and κ F , respectively. They are found to be: The fiducial cross sections for different jet multiplicities is reported and compared to several theoretical predictions. Eleven differential cross sections and two double-differential cross sections are reported for events belonging to the inclusive diphoton fiducial region, as a function of kinematic variables of the diphoton system or of jets produced in association with the Higgs boson. The reported cross sections are sensitive to the Higgs boson production kinematics, the jet kinematics, the spin and CP quantum numbers of the Higgs boson, and the VBF production mechanism. All measured differential cross sections are compared to predictions and no significant deviation from the Standard Model expectation is observed. The full statistical and systematic correlations between measured distributions are determined and are available in H D along with the central values of the measured fiducial and differential cross sections to allow future comparisons and interpretations.
The strength and tensor structure of the Higgs boson interactions is investigated using five differential variables and an effective Lagrangian, which introduces additional CP-even and CP-odd interactions. No significant new physics contributions are observed and the reported 68% and 95% limits on such contributions have improved by a factor of two in comparison to the previous ATLAS measurement.
The measurements presented in this paper lay the foundation for further studies. All reported results are statistically limited and their precision will further improve with the full data set to be recorded during Run 2 of the LHC.

Appendix A Simplified template cross-section framework
As introduced in Section 1.2, this paper includes cross-section measurements using the so called "stage-1" of the simplified template cross-section framework [7,8]. In the full stage-1 proposal, template cross sections are defined in 31 regions of phase space with |y H | < 2.5. These regions have been chosen to minimize the dependence on theoretical uncertainties and isolate possible BSM effects, while maximizing the experimental sensitivity. The 31 regions of particle-level phase space corresponding to the stage 1 of the template cross-section approach are the following [7,8]: • Gluon-gluon fusion (11 regions). Gluon-gluon fusion events, together with gg → Z H events followed by hadronic decays of the Z boson, are split according to the number of jets in the event in 0, 1, and ≥ 2-jet events. Jets are reconstructed from all stable particles14 with lifetime greater than 10 ps using the anti-k t algorithm [77] with a jet radius parameter R = 0.4 and must have p T > 30 GeV. The region containing two or more jets is split into two, with one of the two subregions ("VBF-like") containing events with a topology similar to vector-boson fusion events (invariant mass of the leading-p T jet pair m j j > 400 GeV, and rapidity separation between the two jets |∆y j j | > 2.8). The one-jet and non-VBF-like two-jet regions are further split according to the transverse momentum of the Higgs boson in "low" (0-60 GeV), "medium" (60-120 GeV), "high"  and "BSM" (> 200 GeV) regions. The VBF-like events are further split into exclusive two-jet-like and inclusive three-jet-like events through a requirement on the transverse momentum p H j j T of the system formed by the two photons and the two leading-p T jets (p H j j T < 25 GeV or p H j j T > 25 GeV, respectively). The separation between events with zero, one, or two or more jets probes perturbative QCD predictions. Events containing a very high transverse momentum Higgs-boson of more than 200 GeV are sensitive to BSM contributions, such as those from loop-induced amplitudes mediated by hypothetical particles heavier than the top-quark.
• Vector-boson fusion (5 regions). Vector-boson fusion events, and V H events followed by hadronic V-boson decays, are first split according to the p T of the leading jet. Events that contain at least one jet with a transverse momentum greater than 200 GeV, which are sensitive to BSM contributions, are measured separately in a "VBF BSM" category. The remaining events are separated into VBF-like events, V H-like events, and events that have a ggH-like topology (referred to as "Rest"). VBF-like events satisfy the same m j j and |∆y j j | requirements as for the gluon-gluon fusion VBF-like category and are similarly split into "two-jet" and "≥ 3-jet" events by requiring p H j j T < 25 GeV or p H j j T > 25 GeV, respectively. V H-like events are selected by requiring that they have at least two jets and an invariant mass of 60 GeV < m j j < 120 GeV.
• Associated production with vector bosons decaying to leptons (11 regions). V H events are first split according to their production mode (qq → W H, qq → Z H, or gg → Z H). Events are separated further into regions of the vector boson transverse momentum p V T , and of jet multiplicity. For gg → Z H, two regions are defined with p V T ("low": 0-150 GeV, and "high": > 150 GeV). The "high-p V T " gg → Z H region is further split into zero-jet and ≥ 1-jet regions. Regions sensitive to BSM contributions with p V T > 250 GeV are defined for the qq → V H production modes and two further p V T regions are defined ("low": 0-150 GeV, and "high": 150-250 GeV). The "high-p V T " qq → V H region is finally split into zero-jet and ≥ 1-jet regions.
• Associated production with top and bottom quarks (4 regions). ttH, t-channel tH, W-associated tH, and bbH events are classified according to their production mode, with no further separation into specific regions of phase space. Finally, the acceptances are shown for the rarer production processes: ttH, t-channel tH, W-associated tH, and bbH. All STXS regions require |y H | < 2.5 and are determined using the samples summarized in Table 2.   Table 1 and Appendix A. All regions require |y H | < 2.5. The leading jet transverse momentum is denoted by p j T . In total, the cross sections for fifteen kinematic regions are measured.

Process
Measurement region Particle-level stage 1 region GeV, ≥ 1-jet Top-associated production top ttH tHW tHq bbH merged w/ ggH bbH

B Minimally merged simplified template cross sections
In this appendix, the measurement for a minimally merged set of fifteen simplified template cross section regions is presented. The merged regions are defined in Table 20 and the extracted cross  sections are summarized in Table 21 and Figure 35.   Table 20. For illustration purposes, the central values have been divided by their SM expectations but no additional SM uncertainties have been folded into the measurement. The uncertainties in the SM predicted cross sections are shown in gray in the plot. The fitted value of σ(top) corresponds to the sum of the ttH, tHq, and tHW production-mode cross sections under the assumption that their relative ratios are as predicted by the SM. The σ(VH, leptonic) cross-section values are determined under the assumption that the ratio of the W H and Z H production mode cross sections is as predicted by the SM and includes production from both the quark and gluon initial states. The bbH contributions are merged with ggH.

C Additional unfolded differential cross sections
This appendix presents additional measurements and comparisons to theoretical predictions to those discussed in Section 9.5. Figure 36 shows differential cross sections as a function of p γγ Tt , the orthogonal component of the diphoton momentum when projected on the axis given by the difference of the 3-momenta of the two photons, as well as |∆y γγ |, the rapidity separation of the two photons. Figure 37 shows differential cross sections as a function of H T , the scalar sum of all reconstructed jets in a given event with p T > 30 GeV, the absolute value of the azimuthal difference |∆φ j j | between the leading and subleading jet in events with at least two jets, and the vectorial sum of the transverse momentum of the diphoton system and the leading and subleading jet system, p γγ j j T , in events with at least two jets. Figure 38 displays measurements of the beam-thrust-like variables τ C, j1 and τ C, j . For a given jet, τ is defined by where y is the jet rapidity and m is the jet mass. The variable τ C, j1 refers to the highest-τ jet, and τ C, j is the scalar sum of τ for all jets with τ > 8 GeV. For large jet rapidities, τ corresponds to the small light-cone component of the jet, p + jet = E jet − p z,jet , while the sum is closely related to the beam-thrust global event shape [156], as measured in the diphoton rest frame. Figure 39 shows the first and second moments of each of the additional differential distributions. The data are compared to a variety of theoretical predictions. In general, the SM predictions are in agreement with the measured distributions.

D Diphoton acceptance, photon isolation and non-perturbative correction factors for parton-level gluon-gluon fusion calculations
This appendix presents the diphoton acceptance factors that are applied to parton-level calculations of Higgs production via gluon-gluon fusion, in order to correctly account for the diphoton selection criteria applied to the Higgs-boson decay products, are shown in Table 22 for the fiducial and differential cross sections presented in Section 9.5 and Appendix C. Multiplicative isolation efficiency and nonperturbative correction factors that account for the efficiency of the photon isolation criterion and the impact of hadronization and underlying-event activity are presented in Tables 23 and 24, respectively. The isolation efficiency is defined as the fraction of selected diphoton events (i.e. within the kinematic acceptance) that also satisfy the isolation criteria, and is determined using samples before including hadronization and the underlying-event activity. The non-perturbative correction factors are defined as the ratios of cross sections produced with and without hadronization and the underlying event.
The default non-perturbative correction is taken as the central value of an envelope formed from multiple event generators and/or event generator tunes, with the uncertainty taken to be the maximal deviation observed in the envelope. Table 25 also provides the combined non-perturbative and isolation correction with a total uncertainty that takes into account the correlations between the uncertainties of   Figure 36: The differential cross sections for pp → H → γγ as a function of (a) p γγ Tt and (b) |∆y γγ | are shown and compared to the SM expectations. The data are shown as filled (black) circles. The vertical error bar on each data point represents the total uncertainty in the measured cross section and the shaded (gray) band is the systematic component. The SM prediction, defined using the P NNLOPS prediction for gluon-gluon fusion and the default MC samples for the other production mechanisms, is presented as a hatched (blue) band, with the width of the band reflecting the total theoretical uncertainty (see text for details). The small contribution from VBF, V H ttH and bbH is also shown as a (green) histogram and denoted by X H. The default MC has been normalized with the N 3 LO prediction of Refs. [7,24,[31][32][33][34]. In addition, the H and SCET +MCFM8 predictions, described in Section 9.5, are displayed in (a) and (b), respectively. both factors. Note though that no non-perturbative correction factors are applied to the SM predictions presented in this paper.
A summary of the binning of all differential variables is given in Table 26. Table 27 summarizes the number of expected signal events and measured background events in the smallest interval expected to contain 90% of the expected SM signal events, together with the expected signal purity and local significance in the same interval, for each of the event reconstruction categories. The definition of the categories can be found in Table 4 in Section 8.1.6. Table 28 summarizes the fractions of signal events from the different production modes expected in each reconstruction category, as illustrated in Figure 8.   Figure 36. In addition, the NNLOJET prediction is displayed in (a), and the S and G S predictions are displayed in (b) and (c). More details of these predictions can be found in Section 9.5.1.   Figure 38: The differential cross sections for pp → H → γγ as a function of (a) τ C, j1 and (b) τ C, j are shown and compared to the SM expectations. The data and theoretical predictions are presented in the same way as in Figure 36. In addition, the NNLOJET prediction is displayed in (b).      for the diphoton fiducial region and all differential variable bins in this analysis. The isolation efficiency is defined as the probability for both photons to fulfill the isolation criteria (as described in Section 9.1) for events that satisfy the diphoton kinematic criteria. Regions of phase space where no reliable estimate could be obtained are listed as '100' without uncertainties. Uncertainties are assigned in the same way as for the non-perturbative correction factors: by varying the fragmentation and underlying-event modeling. These factors can be multiplied by the kinematic acceptance factors (see Table 22) to extrapolate an inclusive gluon-gluon fusion Higgs prediction to the fiducial volume used in this analysis. The range of each bin is given in Table 26.  Table 24: Non-perturbative correction factors in percent accounting for the impact of hadronization and the underlying-event activity for the diphoton fiducial region and all differential variable bins. Uncertainties are evaluated by deriving these factors using different generators and tunes as described in the text. The range of each bin is given in Table 26.    Table 25: Combined non-perturbative (Table 24) and particle-level isolation correction factors (Table 23)      Exponential of a second order polynomial ggH 1J Low Exponential of a second order polynomial ggH 0J Fwd Exponential of a second order polynomial ggH 0J Cen Exponential of a second order polynomial Table 30: Observed and expected 95% CL limits for the signal strengths of the VH associated production processes. The observed asymptotic limit on µ VH is compared to that obtained using an ensemble of pseudoexperiments (PEs). Separate observed limits obtained from toys are reported for µ ZH and µ WH . These are shown for the background-only case (µ i = 0), together with the ±1 σ and ±2 σ intervals.

F Limits on µ ZH and µ WH using pseudo-experiments
As discussed in Section 8.2.2, Table 30 shows the observed and expected limits for µ VH , and separately for µ ZH and µ WH , as obtained using the asmptotic approximation and ensembles of pseudoexperiments.

G Summary of couplings results
In this Appendix the expected and observed central values and uncertainties of signal strength measurements, production mode cross section measurements, and simplified template cross section measurements from Section 8.2 and Appendix B are summarized. Table 31 summarizes the observed and expected signal strengths for inclusive production and for various production modes.   Table 32 summarizes the observed and expected cross sections times diphoton branching ratio for various production modes, in the fiducial region |y H | < 2.5. Table 33 summarizes the observed and expected simplified template cross sections times diphoton branching ratio, in the fiducial region |y H | < 2.5.  Table 34 summarizes the observed and expected minimally merged simplified template cross sections times diphoton branching ratio, in the fiducial region |y H | < 2.5.

H Observed and expected correlation maps
This appendix summaries the observed and expected correlations between the parameters of interest of each of the measurements presented in Section 8.2 are given. The observed and expected correlations for the production-mode cross sections and production mode cross-section ratios are shown in Figures 40 and 41. The observed and expected correlations for the simplified template cross sections and minimally merged simplified template cross sections are shown in Figures 42 and 43.