Contents

toc

###  1 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 $\sqrt{s}=7\leavevmode\nobreak\ \text{Te\kern-1.00006ptV}$ 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 $m_{H}=125.09\pm
0.21{\mathrm{(}stat.)}\pm 0.11{\mathrm{(}syst.)}$ GeV [6].

Following the modifications of the LHC to provide proton–proton collisions at
a center-of-mass energy of $\sqrt{s}=13$ 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 $\sqrt{s}=7$ and 8 TeV with the Run 1 data set. The increased
center-of-mass 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
($H\rightarrow\gamma\gamma$) 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, $(2.27\pm 0.07)\times 10^{-3}$ 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 ($m_{\gamma\gamma}$) spectrum on top of a smoothly falling
background, and the Higgs boson signal yield can be measured using an
appropriate fit to the $m_{\gamma\gamma}$ 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 $\sqrt{s}=13$ 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.

####  1.1 Higgs boson production-mode cross sections and signal strengths

In this paper, cross sections times branching ratio of the Higgs to two
photons $B(H\to\gamma\gamma)$ 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 ($VH$), and production of a Higgs boson in
association with a top–antitop quark pair ($t\bar{t}H$) 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
$\sqrt{s}=13$ 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
$H\to\gamma\gamma$ 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 $\mu$, 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.

####  1.2 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. [8, 7]. These are reported as cross section
times $B(H\to\gamma\gamma)$ for a Higgs boson absolute rapidity 111The ATLAS
experiment uses a right-handed coordinate system with its origin at the
nominal interaction point (IP) in the center of the detector and the $z$-axis
along the beam pipe. The $x$-axis points from the IP to the center of the LHC
ring, and the $y$-axis points upward. Cylindrical coordinates $(r,\phi)$ are
used in the transverse plane, $\phi$ being the azimuthal angle around the
$z$-axis. The pseudorapidity is defined in terms of the polar angle $\theta$
as $\eta=-\ln\tan(\theta/2)$. When dealing with massive particles, the
rapidity $y=1/2\ln[(E+p_{z})/(E-p_{z})]$ is used, where $E$ is the energy and
$p_{z}$ is the $z$-component of the momentum. Angular separation is expressed
in terms of $\Delta
R=\sqrt{\left(\Delta\eta\right)^{2}+\left(\Delta\phi\right)^{2}}$. $|y_{H}|$
less than 2.5 and with further particle-level requirements. The different
production modes are separated in a theoretically motivated way using the SM
modes ggH, VBF, $VH$ 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 final state, defined by the Higgs boson, the
hadronic jets and the vector bosons or top quarks in the event, to avoid large
model-dependent extrapolations. The Higgs boson is treated as a stable final-
state particle, which allows an easy combination with other decay channels.
Similarly, vector bosons or top quarks are treated as stable particles, but
the cases of leptonic and hadronic decays of the vector boson are
distinguished.

In this paper a merged version of the so-called “stage-1” simplified template
cross-section measurements are investigated. These measurements provide more
information for theoretical reinterpretation compared to the signal strength
measurements used in Run 1 and are defined to reduce the theoretical
uncertainties typically folded into the signal strength results. In the full
stage-1 proposal, template cross sections would be measured in 31 regions of
phase space for $|y_{H}|<2.5$, where the latter requirement reflects the
experimental acceptance. The experimental categories used in this study (the
same as those used for the signal strength measurements) have been optimized
to provide the maximum sensitivity to such regions [8, 7].

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 ($t\bar{t}H$ and $tH$) Higgs-boson production.
To retain sensitivity to beyond the Standard Model Higgs boson production ,
the $\geq 1$ jet, $p_{\text{T}}^{H}>\nobreak 200\,\text{Ge\kern-1.00006ptV}$
gluon–gluon fusion and $p_{\text{T}}^{j}>\nobreak
200\,\text{Ge\kern-1.00006ptV}$ VBF+$VH$ regions are not merged with other
regions. Here $p_{\text{T}}^{H}$ and $p_{\text{T}}^{j}$ 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 LABEL:tab:STXS 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.

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 $p_{\text{T}}>30$ GeV. The leading jet and Higgs boson transverse momenta are denoted by $p_{\text{T}}^{j}$ and $p_{\text{T}}^{H}$, respectively. The transverse momentum of the Higgs boson and the leading and subleading jet is denoted as $p_{\text{T}}^{Hjj}$ 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 $m_{jj}>400$ GeV and a rapidity separation between the two jets of $|\Delta y_{jj}|>2.8$. Events are considered “$VH$-like” if they contain at least two jets with an invariant mass of $60\,\text{Ge\kern-0.90005ptV}<m_{jj}<120\,\text{Ge\kern-0.90005ptV}$. All $qq^{\prime}\to Hqq^{\prime}$ VBF and $VH$ events (with the vector boson $V$ decaying hadronically) which are neither VBF nor $VH$-like are part of the “Rest” selection. For the $p_{\text{T}}^{H}>200\leavevmode\nobreak\ \text{Ge\kern-0.90005ptV}$ gluon–gluon fusion and $p_{\text{T}}^{j}>200\leavevmode\nobreak\ \text{Ge\kern-0.90005ptV}$ VBF \+ $VH$ 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 $b\bar{b}H$ are merged with ggH. The process $gg\to ZH$ refers only to box and loop processes dominated by top and bottom quarks (see Section 4 for more details).  Process | Measurement region | Particle-level stage-1 region
---|---|---
ggH \+ $gg\to Z(\to qq)H$ | 0-jet | 0-jet
|  1-jet, $p_{\textrm{T}}^{H}<60$ GeV |  1-jet, $p_{\textrm{T}}^{H}<60$ GeV
|  1-jet, $60\leq p_{\textrm{T}}^{H}<120$ GeV |  1-jet, $60\leq p_{\textrm{T}}^{H}<120$ GeV
|  1-jet, $120\leq p_{\textrm{T}}^{H}<200$ GeV |  1-jet, $120\leq p_{\textrm{T}}^{H}<200$ GeV
|  $\geq 1$-jet, $p_{\textrm{T}}^{H}>200$ GeV |  1-jet, $p_{\textrm{T}}^{H}>200$ GeV
|  |  $\geq 2$-jet, $p_{\textrm{T}}^{H}>200$ GeV
|  $\geq 2$-jet, $p_{\textrm{T}}^{H}<200$ GeVor VBF-like |  $\geq 2$-jet, $p_{\textrm{T}}^{H}<60$ GeV
|  |  $\geq 2$-jet, $60\leq p_{\textrm{T}}^{H}<120$ GeV
|  |  $\geq 2$-jet, $120\leq p_{\textrm{T}}^{H}<200$ GeV
|  |  VBF-like, $p_{\textrm{T}}^{Hjj}<25$ GeV
|  |  VBF-like, $p_{\textrm{T}}^{Hjj}\geq 25$ GeV
$qq^{\prime}\to Hqq^{\prime}$ (VBF \+ $VH$) |  $p_{\textrm{T}}^{j}<200$ GeV |  $p_{\textrm{T}}^{j}<200$ GeV, VBF-like, $p_{\textrm{T}}^{Hjj}<25$ GeV
|  |  $p_{\textrm{T}}^{j}<200$ GeV, VBF-like, $p_{\textrm{T}}^{Hjj}\geq 25$ GeV
|  |  $p_{\textrm{T}}^{j}<200$ GeV, $VH$-like
|  |  $p_{\textrm{T}}^{j}<200$ GeV, Rest
|  $p_{\textrm{T}}^{j}>200$ GeV |  $p_{\textrm{T}}^{j}>200$ GeV
$VH$ (leptonic decays) |  $VH$ leptonic |  $q\bar{q}\to ZH$, $p_{\text{T}}^{Z}<150$ GeV
|  |  $q\bar{q}\to ZH$, $150<p_{\text{T}}^{Z}<250$ GeV, 0-jet
|  |  $q\bar{q}\to ZH$, $150<p_{\text{T}}^{Z}<250$ GeV, $\geq 1$-jet
|  |  $q\bar{q}\to ZH$, $p_{\text{T}}^{Z}>250$ GeV
|  |  $q\bar{q}\to WH$, $p_{\text{T}}^{W}<150$ GeV
|  |  $q\bar{q}\to WH$, $150<p_{\text{T}}^{W}<250$ GeV, 0-jet
|  |  $q\bar{q}\to WH$, $150<p_{\text{T}}^{W}<250$ GeV, $\geq 1$-jet
|  |  $q\bar{q}\to WH$, $p_{\text{T}}^{W}>250$ GeV
|  |  $gg\to ZH$, $p_{\text{T}}^{Z}<150$ GeV
|  |  $gg\to ZH$, $p_{\text{T}}^{Z}>150$ GeV, 0-jet
|  |  $gg\to ZH$, $p_{\text{T}}^{Z}>150$ GeV, $\geq 1$-jet
Top-associated production | top | $t\bar{t}H$
|  |  $W$-associated $tH$($tHW$)
|  |  $t$-channel $tH$($tHq$)
$b\bar{b}H$ |  merged w/ ggH | $b\bar{b}H$

####  1.3 Fiducial integrated and differential cross sections

Fiducial integrated and differential cross sections have previously been
measured at $\sqrt{s}=8$ TeV in the $H\rightarrow\gamma\gamma$ 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
$H\to\gamma\gamma$ 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 of the data. No
attempt is made to separate individual production modes in favor of presenting
fiducial regions enriched with a given production mode.

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
$\left|\eta\right|<2.37$, excluding the region
$1.37<\left|\eta\right|<1.52$,222This pseudorapidity interval corresponds to
the transition region between the barrel and endcap sections of the ATLAS
electromagnetic calorimeter, see Section 2. with the leading (subleading)
photon transverse momentum greater than 35% (25%) of $m_{\gamma\gamma}$. 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 $c\tau>10$ mm) with
$p_{\mathrm{T}}>1\leavevmode\nobreak\ \text{Ge\kern-1.00006ptV}$, within a
cone of $\Delta R=0.2$ 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 lepton333In this paper
reconstructed charged leptons denote electrons and muons.,

  * •

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:

  * •

$p_{{\mathrm{T}}}^{\gamma\gamma}$ and $|y_{\gamma\gamma}|$, the transverse
momentum and rapidity of the diphoton system,

  * •

$p_{\mathrm{T}}^{j_{1}}$ and $|y_{j_{1}}|$, the transverse momentum and
rapidity of the leading jet,

  * •

$p_{\mathrm{T}}^{j_{2}}$ and $|y_{j_{2}}|$, the transverse momentum and
rapidity of the subleading jet,

  * •

$|\cos\theta^{*}|$, the cosine of the angle between the beam axis and the
diphoton system in the Collins–Soper frame [11],

  * •

$\Delta\phi_{jj}$ and $|\Delta{y_{jj}|}$, the difference in azimuthal angle
and in rapidity between the leading and subleading jets,

  * •

$|\Delta\phi_{\gamma\gamma,jj}|$, the difference in azimuthal angle between
the dijet system formed by the leading and subleading jets and the diphoton
system,

  * •

and $m_{jj}$, 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 $p_{{\mathrm{T}}}^{\gamma\gamma}$ probes the
perturbative QCD modeling of this production mechanism which is mildly
sensitive to the bottom and charm quark Yukawa couplings of the Higgs boson
[12]. The distribution at high transverse momentum is sensitive to new heavy
particles coupling to the Higgs boson and to the top quark Yukawa coupling.
The rapidity distribution of the Higgs boson is also sensitive to the modeling
of the gluon–gluon fusion production mechanism, as well as to the parton
distribution functions (PDFs) of the colliding protons. The transverse
momentum and absolute rapidity of the leading and subleading jets probe the
perturbative QCD modeling and are sensitive to the relative contributions of
the different Higgs production mechanisms. The angular variables
$|\cos\theta^{*}|$ and $\Delta\phi_{jj}$ are sensitive to the spin and CP
quantum numbers of the Higgs boson. The dijet rapidity separation
$|\Delta{y_{jj}|}$, the dijet mass $m_{jj}$ and the azimuthal difference
between the dijet and diphoton system $|\Delta\phi_{\gamma\gamma,jj}|$ are
sensitive to the VBF production mechanism. All fiducial differential cross
sections are reported with their full statistical and experimental
correlations and are compared to several predictions.

The strength and tensor structure of the Higgs boson interactions are
investigated using an effective Lagrangian, which introduces additional CP-
even and CP-odd interactions that can lead to deviations in the kinematic
properties and event rates of the Higgs boson and of the associated jets from
those in the Standard Model. This is done by a simultaneous fit to five
differential cross sections, which are sensitive to the Wilson coefficients of
four dimension-six CP-even or CP-odd operators of the Strongly Interacting
Light Higgs formulation [13]. A similar analysis was carried out at
$\sqrt{s}=8\leavevmode\nobreak\ \text{Te\kern-1.00006ptV}$ by the ATLAS
Collaboration [14].

###  2 ATLAS detector

The ATLAS detector [1] covers almost the entire solid angle about the
proton–proton interaction point. It consists of an inner tracking detector,
electromagnetic and hadronic calorimeters, and a muon spectrometer.

Charged-particle tracks and interaction vertices are reconstructed using
information from the inner detector (ID). The ID consists of a silicon pixel
detector (including the insertable B-layer [15] installed before the start of
Run 2), of a silicon microstrip detector, and of a transition radiation
tracker (TRT). The ID is immersed in a 2 T axial magnetic field provided by a
thin superconducting solenoid. The silicon detectors provide precision
tracking over the pseudorapidity interval $|\eta|<2.5$, while the TRT offers
additional tracking and substantial discrimination between electrons and
charged hadrons for $|\eta|<2.0$.

The solenoid is surrounded by electromagnetic (EM) and hadronic sampling
calorimeters allowing energy measurements of photons, electrons and hadronic
jets and discrimination between the different particle types. The EM
calorimeter is a lead/liquid-argon (LAr) sampling calorimeter. It consists of
a barrel section, covering the pseudorapidity region $|\eta|<1.475$, and of
two endcap sections, covering $1.375<|\eta|<3.2$. The EM calorimeter is
divided in three layers, longitudinally in depth, for $|\eta|<2.5$, and in two
layers for $2.5<|\eta|<3.2$. In the regions $|\eta|<1.4$ and $1.5<|\eta|<2.4$,
the first layer has a fine $\eta$ segmentation to discriminate isolated
photons from neutral hadrons decaying to pairs of close-by photons. It also
allows, together with the information from the cluster barycenter in the
second layer, where most of the energy is collected, a measurement of the
shower direction without assumptions on the photon production point. In the
range of $|\eta|<1.8$ a presampler layer allows corrections to be made for
energy losses upstream of the calorimeter. The hadronic calorimeter
reconstructs hadronic showers using steel absorbers and scintillator tiles
($|\eta|<1.7$), or either copper ($1.5<|\eta|<3.2$) or copper–tungsten
($3.1<|\eta|<4.9$) absorbers immersed in a LAr active medium.

A muon spectrometer surrounds the calorimeter. It comprises separate trigger
($|\eta|<2.4$) and precision tracking chambers ($|\eta|<2.7$) in the magnetic
field provided by three large air-core toroids.

A two-level trigger system [16] was used during the $\sqrt{s}=13$ TeV data-
taking period. Dedicated hardware implements the first-level (L1) trigger
selection, using only a subset of the detector information and reducing the
event rate to at most 100 kHz. Events satisfying the L1 requirements are
processed by a high-level trigger executing, on a computer farm, algorithms
similar to the offline reconstruction software, in order to reduce the event
rate to approximately 1 kHz.

###  3 Data set

Events were selected using a diphoton trigger requiring the presence in the EM
calorimeter of two clusters of energy depositions with transverse energy above
35 GeV and 25 GeV for the leading (highest transverse energy) and subleading
(second highest transverse energy) cluster. In the high-level trigger the
shape of the energy deposition of both clusters was required to be loosely
consistent with that expected from an electromagnetic shower initiated by a
photon. The diphoton trigger has an efficiency greater than 99% for events
that satisfy the final event selection described in Section 5.

After the application of data quality requirements, the data set amounts to an
integrated luminosity of $36.1$ fb-1, of which 3.2 fb-1 were collected in 2015
and 32.9 fb-1 were collected in 2016. The mean number of proton–proton
interactions per bunch crossing is 14 in the 2015 data set and 25 in the 2016
data set.

###  4 Event simulation

Signal samples were generated for the main Higgs boson production modes using
Monte Carlo event generators as described in the following. The mass and width
of the Higgs boson were set in the simulation to $m_{H}=125$ GeV and
$\Gamma_{H}=4.07$ MeV [17], respectively. The samples are normalized with the
latest available theoretical calculations of the corresponding SM production
cross sections, as summarized in Ref. [7] and detailed below. The
normalization of all Higgs boson samples also accounts for the
$H\rightarrow\gamma\gamma$ branching ratio of 0.227% calculated with HDECAY
[18, 19] and PROPHECY4F [20, 21, 22].

Higgs boson production via ggH is simulated at next-to-next-to-leading-order
(NNLO) accuracy in QCD using the Powheg NNLOPS program [23], with the
PDF4LHC15 PDF set [24]. The simulation achieves NNLO accuracy for arbitrary
inclusive $gg\to H$ observables by reweighting the Higgs boson rapidity
spectrum in Hj-MiNLO [25] to that of HNNLO [26]. The transverse momentum
spectrum of the Higgs boson obtained with this sample was found to be
compatible with the fixed-order HNNLO calculation [26] and the Hres2.3
calculation [27, 28] performing resummation at next-to-next-to-leading-
logarithm accuracy matched to a NNLO fixed-order calculation (NNLL+NNLO). The
Hres prediction includes the effects of the top and bottom quark masses up to
NLO precision in QCD and uses dynamical renormalization ($\mu_{\mathrm{R}}$)
and factorization ($\mu_{F}$) scales,
$\mu_{F}=\mu_{\mathrm{R}}=0.5\sqrt{m_{H}^{2}+{p_{\text{T}}^{H}}^{2}}$. The
parton-level events produced by the Powheg NNLOPS program are passed to
Pythia8 [29] to provide parton showering, hadronization and underlying event,
using the AZNLO set of parameters that are tuned to data [30]. The sample is
normalized such that it reproduces the total cross section predicted by a
next-to-next-to-next-to-leading-order (N3LO) QCD calculation with NLO
electroweak corrections applied [31, 32, 33, 34].

Higgs boson production via VBF is generated at NLO accuracy in QCD using the
Powheg-Box program [35, 36, 37, 38] with the PDF4LHC15 PDF set. The parton-
level events are passed to Pythia8 to provide parton showering, hadronization
and the underlying event, using the AZNLO parameter set. The VBF sample is
normalized with an approximate-NNLO QCD cross section with NLO electroweak
corrections applied [39, 40, 41].

Higgs boson production via $VH$ is generated at NLO accuracy in QCD through
$qq$/$qg$-initiated production, denoted as $q\bar{q}^{\prime}\to VH$, and
through $gg\to ZH$ production using Powheg-Box [42] with the PDF4LHC15 PDF
set. Higgs boson production through $gg\to ZH$ has two distinct sources: a
contribution with two additional partons, $gg\to ZHq\bar{q}$, and a
contribution without any additional partons in the final state, including box
and loop processes dominated by top and bottom quarks. In the following, the
$gg\to ZH$ notation refers only to this latter contribution. Pythia8 is used
for parton showering, hadronization and the underlying event using the AZNLO
parameter set. The samples are normalized with cross sections calculated at
NNLO in QCD and NLO electroweak corrections for $q\bar{q}^{\prime}\to VH$ and
at NLO and next-to-leading-logarithm accuracy in QCD for $gg\to ZH$ [43, 44,
45].

Higgs boson production via $t\bar{t}H$ is generated at NLO accuracy in QCD
using MG5_aMC@NLO [46] with the NNPDF3.0 PDF set [47] and interfaced to
Pythia8 to provide parton showering, hadronization and the underlying event,
using the A14 parameter set [48]. The $t\bar{t}H$ sample is normalized with a
cross section calculation accurate to NLO in QCD with NLO electroweak
corrections applied [49, 50, 51, 52].

Higgs boson production via $b\bar{b}H$ is simulated using MG5_aMC@NLO [53]
interfaced to Pythia8 with the CT10 PDF set [54], and is normalized with the
cross-section calculation obtained by matching, using the _Santander_ scheme,
the five-flavor scheme cross section accurate to NNLO in QCD with the four-
flavor scheme cross section accurate to NLO in QCD [55, 56, 57]. The sample
includes the effect of interference with the gluon–gluon fusion production
mechanism.

Associated production of a Higgs boson with a single top-quark and a $W$-boson
($tHW$) is generated at NLO accuracy, removing the overlap with the
$t\bar{t}H$ sample through a diagram regularization technique, using
MG5_aMC@NLO interfaced to Herwig++ [58, 59, 60], with the Herwig++ UEEE5
parameter set for the underlying event and the CT10 PDF set using the five-
flavor scheme. Simulated Higgs boson events in association with a single top-
quark, a $b$-quark and a light quark ($tHq$) are produced at LO accuracy in
QCD using MG5_aMC@NLO interfaced to Pythia8 with the CT10 PDF set within the
four-flavor scheme and using the A14 parameter set. The $tHW$ and $tHq$
samples are normalized with calculations accurate to NLO in QCD [61].

Table 2:  Summary of the event generators and PDF sets used to model the
signal and the main background processes. The SM cross sections $\sigma$ for
the Higgs production processes with ${m_{H}=125.09}$ GeV are also given
separately for ${\sqrt{s}=13}$ 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: Pythia8 version 8.212
(processes) and 8.186 (pile-up overlay); Herwig++ version 2.7.1; Powheg-Box
version 2; MG5_aMC@NLO version 2.4.3; Sherpa version 2.2.1

Process | Generator | Showering | PDF set | $\sigma\ [\mathrm{pb}]$ | Order of calculation of $\sigma$
---|---|---|---|---|---
$\sqrt{s}=13$ TeV
ggH | Powheg NNLOPS | Pythia8 | PDF4LHC15 | 48.52 | N3LO(QCD)+NLO(EW)
VBF | Powheg-Box | Pythia8 | PDF4LHC15 | 3.78 | NNLO(QCD)+NLO(EW)
$WH$ | Powheg-Box | Pythia8 | PDF4LHC15 | 1.37 | NNLO(QCD)+NLO(EW)
$q\bar{q}^{\prime}\to ZH$ | Powheg-Box | Pythia8 | PDF4LHC15 | 0.76 | NNLO(QCD)+NLO(EW)
$gg\to ZH$ | Powheg-Box | Pythia8 | PDF4LHC15 | 0.12 | NLO+NLL(QCD)
$t\bar{t}H$ |  MG5_aMC@NLO | Pythia8 | NNPDF3.0 | 0.51 | NLO(QCD)+NLO(EW)
$b\bar{b}H$ |  MG5_aMC@NLO | Pythia8 | CT10 | 0.49 | 5FS(NNLO)+4FS(NLO)
$t$-channel $tH$ |  MG5_aMC@NLO | Pythia8 | CT10 | 0.07 | 4FS(LO)
$W$-associated $tH$ |  MG5_aMC@NLO | Herwig++ | CT10 | 0.02 | 5FS(NLO)
$\gamma\gamma$ | Sherpa | Sherpa | CT10 |  |
$V\gamma\gamma$ | Sherpa | Sherpa | CT10 |  |

The generated Higgs boson events are passed through a Geant4 [62] simulation
of the ATLAS detector [63] and reconstructed with the same analysis software
used for the data.

Background events from continuum $\gamma\gamma$ production and $V\gamma\gamma$
production are simulated using the Sherpa event generator [64], with the CT10
PDF set and the Sherpa default parameter set for the underlying-event
activity. The corresponding matrix elements for $\gamma\gamma$ and
$V\gamma\gamma$ are calculated at leading order (LO) in the strong coupling
constant $\alpha_{\mathrm{S}}$ with the real emission of up to three or two
additional partons, respectively, and are merged with the Sherpa parton shower
[65] using the Meps@lo prescription [66].

The very large sample size required for the modeling of the $\gamma\gamma$
background processes is obtained through a fast parametric simulation of the
ATLAS detector response [67]. For $V\gamma\gamma$ events the same full
detector simulation as for the signal samples is used.

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 Pythia8 with the A2 parameter set
[68] that are tuned to data and the MSTW2008lo PDF set [69]. A summary of the
used signal and background samples is shown in Table 2.

###  5 Event reconstruction and selection

####  5.1 Photon reconstruction and identification

The reconstruction of photon candidates is seeded by energy clusters in the
electromagnetic calorimeter with a size of $\Delta\eta\times\Delta\phi=$
0.075$\times$0.125, with transverse energy $E_{\mathrm{T}}$ 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 $E_{\text{T}}$ above 20 GeV and generated pseudorapidity
$|\eta|<2.37$ is 98%.

The energy from unconverted and converted photon candidates is measured from
an electromagnetic cluster of size $\Delta\eta\times\Delta\phi=$
0.075$\times$0.175 in the barrel region of the calorimeter, and
$\Delta\eta\times\Delta\phi=$ 0.125$\times$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 $Z\to e^{+}e^{-}$
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
$Z\to e^{+}e^{-}$ 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 $\Delta R=$ 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 $\Delta R=0.2$
with $p_{\mathrm{T}}>1$ 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.

####  5.2 Event selection and selection of the diphoton primary vertex

Events are preselected by requiring at least two photon candidates with
$E_{\text{T}}>25$ GeV and $|\eta|<2.37$ (excluding the transition region
between the barrel and endcap calorimeters of $1.37<|\eta|<1.52$) that fulfill
the loose photon identification criteria [70]. The two photon candidates with
the highest $E_{\text{T}}$ 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 $H\to\gamma\gamma$ 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 $Z\to e^{+}e^{-}$ 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
$m_{\gamma\gamma}=\sqrt{2E_{1}E_{2}\left(1-\cos\alpha\right)}$, where $E_{1}$
and $E_{2}$ are the energies of the leading and subleading photons and
$\alpha$ 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 $E_{\text{T}}/m_{\gamma\gamma}>$ 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.

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 $\Delta R=0.2$ ($\Delta R=0.4$) cone around the photon
candidates. Both the Run 1 and Run 2 algorithms also use tracking information
in a $\Delta R=0.2$ cone around the photon candidates.

In total 332030 events are selected with diphoton candidates with invariant
mass $m_{\gamma\gamma}$ 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%).

####  5.3 Reconstruction and selection of hadronic jets, $b$-jets, leptons and
missing transverse momentum

Jets are reconstructed using the anti-$k_{t}$ algorithm [77] with a radius
parameter of 0.4 via the FastJet package [78, 79]. The inputs to the algorithm
are three-dimensional topological clusters of energy deposits in the
calorimeter cells [74]. Jets are corrected on an event-by-event basis for
energy deposits originating from pileup [80], then calibrated using a
combination of simulation-based and data-driven correction factors, which
correct for different responses to electromagnetic and hadronic showers of the
calorimeter and inactive regions of the calorimeter [81, 82]. Jets are
required to have $p_{\mathrm{T}}>25$ GeV for $|\eta|<2.4$. The jet selection
is tightened to $p_{\mathrm{T}}>30$ GeV within $|y|<$ 4.4 for most event
reconstruction categories and the measurement of fiducial integrated and
differential cross sections (with exceptions noted in Sections 8.1 and 9.3).
Jets that do not originate from the diphoton primary vertex are rejected, for
$|\eta|<2.4$, using the jet vertex tagging algorithm (JVT) [83], which
combines tracking information into a multivariate likelihood. For jets with
$p_{\mathrm{T}}<60$ GeV and $|\eta|<2.4$ a _medium_ working point is used,
with an efficiency greater than 92% for non-pileup jets with $p_{\mathrm{T}}>$
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 $\Delta
R=0.4$ of an isolated photon candidate, or within $\Delta R=0.2$ 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 $t\bar{t}$ 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 $p_{\mathrm{T}}>10$
GeV and $|\eta|<2.47$, excluding the region $1.37<|\eta|<1.52$. 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 $p_{\mathrm{T}}>10$ GeV and $|\eta|<2.7$, 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
$p_{\text{T}}>15$ GeV.

Lepton candidates are discarded if they are within $\Delta R=0.4$ 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 $p_{\text{T}}=$ 10–60
GeV to 99% for $p_{\text{T}}>60$ 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 $|z_{0}\sin\theta|<0.5$
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 $J/\psi$ mesons in the full 2015
and 2016 data set using the methods described in Refs. [87, 86]. Lepton
efficiency correction factors are applied to the simulation to improve the
agreement with the data.

The magnitude of the missing transverse momentum
$E_{\text{T}}^{\mathrm{miss}}$ 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
$E_{\text{T}}^{\mathrm{miss}}$ significance is defined as
$E_{\text{T}}^{\mathrm{miss}}/\sqrt{\sum E_{\text{T}}}$, where $\sum
E_{\text{T}}$ 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.

###  6 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_{\gamma\gamma}<$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 $m_{\gamma\gamma}$ spectra.

####  6.1 Signal model

The Higgs boson signal manifests itself as a narrow peak in the
$m_{\gamma\gamma}$ 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
$(m_{H}+\Delta\mu_{\mathrm{CB}})$ and the width $(\sigma_{\mathrm{CB}})$. The
non-Gaussian contributions to the mass resolution arise mostly from converted
photons $\gamma\to e^{+}e^{-}$ 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:

$f_{i}^{\mathrm{sig}}(m_{\gamma\gamma};\Delta\mu_{\mathrm{CB},i},\sigma_{\mathrm{CB},i},\alpha_{\mathrm{CB},i}^{\pm},n_{\mathrm{CB},i}^{\pm})={\cal
N}_{c}\begin{cases}e^{-t^{2}/2}&-\alpha_{\mathrm{CB},i}^{-}\leq
t\leq\alpha_{\mathrm{CB},i}^{+}\\\
\left(\frac{n_{\mathrm{CB},i}^{-}}{|\alpha_{\mathrm{CB},i}^{-}|}\right)^{n_{\mathrm{CB},i}^{-}}e^{-|\alpha_{\mathrm{CB},i}^{-}|^{2}/2}\left(\frac{n_{\mathrm{CB},i}^{-}}{\alpha_{\mathrm{CB},i}^{-}}-\alpha_{\mathrm{CB},i}^{-}-t\right)^{-n_{\mathrm{CB},i}^{-}}&t<-\alpha_{\mathrm{CB},i}^{-}\\\
\left(\frac{n_{\mathrm{CB},i}^{+}}{|\alpha_{\mathrm{CB},i}^{+}|}\right)^{n_{\mathrm{CB},i}^{+}}e^{-|\alpha_{\mathrm{CB},i}^{+}|^{2}/2}\left(\frac{n_{\mathrm{CB},i}^{+}}{\alpha_{\mathrm{CB},i}^{+}}-\alpha_{\mathrm{CB},i}^{+}-t\right)^{-n_{\mathrm{CB},i}^{+}}&t>\alpha_{\mathrm{CB},i}^{+}\end{cases},$
(1)

where
$t=(m_{\gamma\gamma}-m_{H}-\Delta\mu_{\mathrm{CB},i})/\sigma_{\mathrm{CB},i}$,
and ${\cal N}_{c}$ is a normalization factor. The non-Gaussian parts are
parameterized by $\alpha_{\mathrm{CB},i}^{\pm}$ and $n_{\mathrm{CB},i}^{\pm}$
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 $|\eta|\leq 0.95$ for both selected photons) with respect to the
forward region (defined by requiring one photon with $|\eta|\leq 0.95$ and one
photon with $0.95<|\eta|<2.37$) 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.

Figure 2:  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 $|\eta|>0.95$; ggH 0J Cen applies a
similar selection, but requires both photons to have $|\eta|\leq 0.95$ 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.

####  6.2 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 $\gamma\gamma$ production is simulated with the Sherpa event
generator as explained in Section 4, neglecting any interference effects with
the $H\to\gamma\gamma$ signal. The $\gamma j$ and $jj$ backgrounds are
obtained by reweighting this sample using an $m_{\gamma\gamma}$ dependent
linear correction function obtained from the fraction of $\gamma\gamma$ to
$\gamma j$ and $\gamma\gamma$ to $jj$ background events in data, respectively.

For very low rate categories targeting $t\bar{t}H$ 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 $m_{\gamma\gamma}$ 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
$\gamma\gamma$ and $V\gamma\gamma$ events are obtained directly from the
simulation, while the $m_{\gamma\gamma}$ distributions of $\gamma j$ and $jj$
events are obtained from data control samples in which the nominal selection
is applied except that at least one (for $\gamma j$) or both (for $jj$) of the
two photon candidates fail to meet either the identification or isolation
criteria. Except for the $V\gamma\gamma$ 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 $\gamma\gamma$, $\gamma
j$ and $jj$ 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 $\gamma\gamma$, $\gamma j$ and
$jj$ 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 $\gamma\gamma$, $\gamma j$ and $jj$ 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 $VH$ categories, a small extra contribution from
$V\gamma\gamma$ 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 $\gamma\gamma$,
$\gamma j$ and $jj$ backgrounds are shown in Figure 3 as a function of
$m_{\gamma\gamma}$ for the selected events. The fractions of these background
sources in the inclusive diphoton sample are ($78.7\,^{+1.8}_{-5.2}$)%,
($18.6\,^{+4.2}_{-1.6}$)% and ($2.6\,^{+0.5}_{-0.4}$)%, respectively. The
uncertainties in the measured background fractions are systematically
dominated. These results are comparable to previous results at $\sqrt{s}$ = 7
and 8 TeV [9, 76]. In addition the purity is shown as a function of the
$p_{\mathrm{T}}$ of the diphoton system, and the number of reconstructed jets
with $p_{\mathrm{T}}>30$ GeV.

((a))  ((b))

((c))  ((d))

Figure 3: The data-driven determination of (a) event yields and (b) event
fractions for $\gamma\gamma$, $\gamma j$ and $jj$ events as a function of
$m_{\gamma\gamma}$ after the final selection outlined in Section 5. The event
fractions for two differential observables, (c)
$p_{{\mathrm{T}}}^{\gamma\gamma}$ and (d) $N_{\mathrm{j}ets}$ defined for jets
with a $p_{\mathrm{T}}>30$ 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.

The functional form used to model the background $m_{\gamma\gamma}$
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
$121\leavevmode\nobreak\
\text{Ge\kern-1.00006ptV}<m_{\gamma\gamma}<129\leavevmode\nobreak\
\text{Ge\kern-1.00006ptV}$), if the data favors a more complex model. A test
statistic is built from the $\chi^{2}$ 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.

####  6.3 Statistical model

The data are interpreted following the statistical procedure summarized in
Ref. [92] and described in detail in Ref. [93]. An extended likelihood
function is built from the number of observed events and invariant diphoton
mass values of the observed events using the analytic functions describing the
distributions of $m_{\gamma\gamma}$ in the range 105–160 GeV for the signal
and the background.

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,

$\mathcal{L}_{i}=\mathrm{Pois}(n_{i}|N_{i}(\bm{\theta}))\cdot\prod_{j=1}^{n_{i}}f_{i}(m_{\gamma\gamma}^{j},\bm{\theta})\cdot
G(\bm{\theta}),$

where $n_{i}$ ($N_{i}$) is the observed (expected) number of selected
candidates, $f_{i}(m_{\gamma\gamma}^{j})$ is the value of the probability
density function (pdf) of the invariant mass distribution evaluated for each
candidate $j$, $\bm{\theta}$ are nuisance parameters and $G(\bm{\theta})$ 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 cross-section 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 $N_{i}^{\mathrm{sig}}$ and $N_{i}^{\mathrm{bkg}}$, and the
fitted spurious signal yield,
$N_{i}^{\mathrm{spur}}\cdot\theta_{i}^{\mathrm{spur}}$,

$N_{i}=N_{i}^{\mathrm{sig}}+N_{i}^{\mathrm{bkg}}+N_{i}^{\mathrm{spur}}\cdot\theta_{i}^{\mathrm{spur}}.$

In more detail, the invariant mass distribution for each category has signal
and background components,

$\displaystyle
f_{i}(m_{\gamma\gamma}^{j})=\left[(N_{i}^{\mathrm{sig}}+N_{i}^{\mathrm{spur}}\cdot\theta_{i}^{\mathrm{spur}})\cdot
f_{i}^{\mathrm{sig}}(m_{\gamma\gamma}^{j},\bm{\theta}_{i}^{\mathrm{sig}})+N_{i}^{\mathrm{bkg}}\cdot
f_{i}^{\mathrm{bkg}}(m_{\gamma\gamma}^{j},\bm{\theta}_{i}^{\mathrm{bkg}})\right]/N_{i},$

where $\bm{\theta}_{i}^{\mathrm{sig}}$ and $\bm{\theta}_{i}^{\mathrm{bkg}}$
are nuisance parameters associated with systematic uncertainties affecting the
resolutions and positions (Section 7.1) of the invariant mass distributions of
the signal $f_{i}^{\mathrm{sig}}$ (further detailed in Section 6.1) or the
shape of the background $f_{i}^{\mathrm{bkg}}$ (as explained in Section 6.2),
respectively.

Systematic uncertainties are incorporated into the likelihood function by
multiplying the relevant parameter of the statistical model by a factor

$F_{\mathrm{G}}(\sigma,\theta)=(1+\sigma\cdot\theta)$

in the case of a Gaussian pdf for the effect of an uncertainty of size
$\sigma$ or, for cases where a negative model parameter does not make physical
sense (e.g. the uncertainty in the integrated luminosity), by a factor

$F_{\mathrm{LN}}(\sigma,\theta)=e^{\sqrt{\ln(1+\sigma^{2})}\theta}$

for a log-normal pdf. In both cases the corresponding component of the
constraint product $G(\theta)$ is a unit Gaussian centered at zero for the
nuisance parameter $\theta$. 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

$\Lambda(\nu)=-2\ln{\mathcal{L}(\nu,\hat{\bm{\theta}}_{\nu})\over\mathcal{L}(\hat{\nu},\hat{\bm{\theta}})},$
(2)

where $\hat{\nu}$ and $\hat{\bm{\theta}}$ 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 while
$\hat{\bm{\theta}}_{\nu}$ 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 $\nu$. In the asymptotic approximation, which is
valid for all the results presented here, $\Lambda(\nu)$ may be interpreted as
an increase in $\chi^{2}$ from its minimum value [92] such that approximate
confidence intervals are easily constructed. The total uncertainty in $\nu$ is
thus obtained from the $\nu$ values such that $\Lambda(\nu)=1$ with all other
parameters “profiled” (i.e. set to the values that maximize the likelihood for
those values of $\nu$). 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.

####  6.4 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 production-mode 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 $\nu$ is determined using the $\mathrm{CL}_{\mathrm{s}}$ prescription
[94]. For this, the agreement between data and the expected yield for the
hypothesized value of the parameter of interest $\nu$ is quantified by the
test statistic, $q_{\nu}$, defined as

$\displaystyle
q_{\nu}=\bigg{\\{}\begin{matrix}\Lambda(\nu)&0<\hat{\nu}\leq\nu\\\
0&\nu<\hat{\nu}\end{matrix}\,,$ (3)

where $\hat{\nu}\geq 0$ is the fitted parameter of interest. The observed
value of the test statistic, $q_{\mathrm{o}bs}$, 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. (3) 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.

###  7 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
$m_{\gamma\gamma}$ 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 3.444The breakdown of uncertainties differs from
those used in the Run 1 measurement Ref. [76] as more updated recommendations
for experimental and theory uncertainties are used.

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.

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 ($\checkmark$) 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 (${F_{\mathrm{LN}}(\sigma_{i},\theta_{i})}$) or Gaussian constraints (${F_{\mathrm{G}}(\sigma_{i},\theta_{i})}$) with $\sigma_{i}$ denoting the systematic uncertainty ($i$ is the index to each of the unique nuisance parameters $\theta$). When acting on $N_{\mathrm{S}}^{\mathrm{tot}}$ the uncertainty value is the same for all processes, whereas the uncertainty has a different value for each signal process for the case denoted $N_{\mathrm{S}}^{\mathrm{p}}$. The number of nuisance parameters, $N_{\mathrm{NP}}$, 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.  |  | Systematic uncertainty source | $N_{\mathrm{NP}}$ | Constraint | Category | Fiducial
---|---|---|---|---|---|---
|  |  |  |  | Likelihood | Likelihood
Theory |  | ggH QCD | 9 | $N_{\mathrm{S}}^{\mathrm{ggH}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
Missing higher orders (non-ggH) | 6 | $N_{\mathrm{S}}^{\mathrm{p}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
$B(H\to\gamma\gamma)$ | 1 | $N_{\mathrm{S}}^{\mathrm{tot}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
| PDF | 30 | $N_{\mathrm{S}}^{\mathrm{p}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
$\alpha_{\mathrm{S}}$ | 1 | $N_{\mathrm{S}}^{\mathrm{p}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
UE/PS | 5 | $N_{\mathrm{S}}^{\mathrm{p}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
Experimental |  Yield | Heavy flavor content | 1 | $N_{\mathrm{S}}^{\mathrm{p}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
Luminosity | 1 | $N_{\mathrm{S}}^{\mathrm{tot}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
Trigger | 1 | $N_{\mathrm{S}}^{\mathrm{tot}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
Photon identification | 1 | $N_{\mathrm{S}}^{\mathrm{p}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
Photon isolation | 2 | $N_{\mathrm{S}}^{\mathrm{p}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
Migration | Flavor tagging | 14 | $N_{\mathrm{S}}^{\mathrm{p}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
Jet | 20 | $N_{\mathrm{S}}^{\mathrm{p}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
Jet flavor composition | 7 | $N_{\mathrm{S}}^{\mathrm{p}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
Jet flavor response | 7 | $N_{\mathrm{S}}^{\mathrm{p}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
Electron | 3 | $N_{\mathrm{S}}^{\mathrm{p}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
Muon | 11 | $N_{\mathrm{S}}^{\mathrm{p}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
Missing transverse momentum | 3 | $N_{\mathrm{S}}^{\mathrm{p}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
Pileup | 1 | $N_{\mathrm{S}}^{\mathrm{p}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
|  | Photon energy scale | 40 | $N_{\mathrm{S}}^{\mathrm{p}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | -
Mass |  |  ATLAS-CMS $m_{H}$ | 1 | $\mu_{\mathrm{CB}}\,F_{\mathrm{G}}(\sigma_{i},\theta_{i})$ | ✓ | ✓
| Photon energy scale | 40 | $\mu_{\mathrm{CB}}\,F_{\mathrm{G}}(\sigma_{i},\theta_{i})$ | ✓ | ✓
| Photon energy resolution | 9 | $\sigma_{\mathrm{CB}}\,F_{\mathrm{LN}}(\sigma_{i},\theta_{i})$ | ✓ | ✓
Background |  | Spurious signal | Varies | $N_{\mathrm{spur},c}\,\theta_{\mathrm{spur},c}$ | ✓ | ✓

####  7.1 Systematic uncertainties in the signal and background modeling from
fitting the $m_{\gamma\gamma}$ 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 $m_{\gamma\gamma}$ spectra is performed for a Higgs boson mass
of $m_{H}=125.09\pm 0.24$ 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 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 $Z\rightarrow e^{+}e^{-}$
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
$\pm\leavevmode\nobreak\ 0.21\%$ and $\pm\leavevmode\nobreak\ 0.36\%$ 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 $\pm\leavevmode\nobreak\ 6\%$ and $\pm\leavevmode\nobreak\ 13\%$ 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
$\pm\leavevmode\nobreak\ 0.24$ 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 $\pm\leavevmode\nobreak\ 0.1\%$.

The uncertainty due to the choice of background function is taken to be the
spurious signal yield obtained when fitting the $m_{\gamma\gamma}$ spectrum
reconstructed from background-only simulated samples (or signal-suppressed
control regions in data), as discussed in Section 6.

####  7.2 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 $\gamma$ events, and of electrons from $Z\to
e^{+}\,e^{-}$ 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 $Z\to e^{+}\,e^{-}$
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 $p_{\mathrm{T}}$ up to 1.9% for events with a high diphoton
$p_{\mathrm{T}}$.

  * •

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, $\gamma+{\mathrm{j}et}$ and $Z+{\mathrm{j}et}$ events [96, 81, 82]. 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 $t\bar{t}$ events in 13 TeV data for jets containing the decay of a
$b$-quark, using the method outlined in Ref. [97]. The corresponding
uncertainties in the identification of jets originating from $c$-quarks, light
quarks and gluons are taken directly from Run 1 studies [97], with additional
uncertainties to cover the extrapolation to Run 2 conditions.

  * •

Uncertainties in the electron [86] and muon [87] reconstruction,
identification and isolation efficiencies. They have been obtained from
dilepton decays of $Z$ bosons and $J/\psi$ mesons collected in Run 2, using a
tag-and-probe technique. The typical size of these uncertainties is about 0.6%
for electrons and about 0.5% for muons in the relevant categories or fiducial
regions.

  * •

Uncertainties in the electron [98] and muon [87] energy and momentum scale and
resolution. They are determined from comparisons between the reconstructed
invariant mass in data and simulation of dileptons from decays of $Z$ bosons
or $J/\psi$ mesons. The impact is negligible for all measurements.

  * •

Uncertainties associated with energy scales and resolutions of photons, jets
and leptons are propagated to the $E_{\text{T}}^{\mathrm{miss}}$ uncertainty,
together with the uncertainty in the contribution to
$E_{\text{T}}^{\mathrm{miss}}$ from charged-particle tracks not associated
with high-$p_{\text{T}}$ 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.

####  7.3 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 $\alpha_{\mathrm{S}}$ 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 N3LO 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 $p_{\text{T}}$. 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 [100, 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 $p_{\text{T}}$. The first two encapsulate the migration uncertainty
between the intermediate and high $p_{\text{T}}$ 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
$p_{\text{T}}$ and a sizable uncertainty of the order of 30% at
$p_{\text{T}}>500$ 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 $\mu_{r}=\mu_{f}=m_{H}$. The two sources account
for the uncertainty in the overall normalization of $H+2\leavevmode\nobreak\
{\mathrm{j}et}$ and $H\ +\geq 3\leavevmode\nobreak\ {\mathrm{j}et}$ events as
well as for the uncertainty due to the multivariate requirement on
$|\Delta\phi_{\gamma\gamma,jj}|$ (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 Powheg NNLOPS was tested by
comparing the STWZ+BLPTW and the MCFM cross section predictions in variables
relevant for the definition of the simplified cross-section bins, and
reasonable agreement was found. In addition, the ggH acceptance of Powheg
NNLOPS of all categories based on BDT classifiers is compared to the
acceptance derived from the MG5_aMC@NLO prediction or Refs. [46, 105] which
includes up to two jets at NLO accuracy using the FxFx merging scheme [106].
Sufficient agreement was found for all categories and no additional modeling
uncertainties are assigned based on these comparisons. 555Recent measurements
of QCD and electroweak (VBF) $Z$-boson production in association with two jets
in Ref. [107] show large deviations between the data and the predictions for
the QCD $Zjj$ background at large $m_{jj}$. These differences are
significantly larger than the 30–40% uncertainties assigned here to the ggH
background in the experimental categories targeting VBF Higgs boson
production. Increasing this uncertainty to 100% results in an increase in the
theory uncertainty in the VBF signal strengths or simplified template cross
sections by a factor of about two, while the increase in the total uncertainty
is about 10%, as it is dominated by the statistical component.

Finally, in the categories targeting production in association with top
quarks, the normalization of each of the ggH, VBF, and $VH$ production
mechanisms is assigned an uncertainty of 100%, motivated by comparisons of
data with simulation in $t\bar{t}b\bar{b}$ [108] and $Vb$ [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 Pythia8 to Herwig7
in the ggH, VBF, and $VH$ samples and from Herwig++ to Pythia8 in the
$t\bar{t}H$ 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 $H\rightarrow\gamma\gamma$ branching ratio is taken from
HDECAY and PROPHECY4F.

####  7.4 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. 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, $VH$ and $t\bar{t}H$ 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 $t\bar{t}H$-production.

  2. 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. 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.

####  7.5 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 $gg\to H$ events with
$\left|y_{H}\right|<2.5$ 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
$\left|\eta\right|\leq 0.95$). The total theory uncertainty is dominated by
the uncertainty in the choice of parton distribution functions (1.5%), in the
value of $\alpha_{\mathrm{S}}$ (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 $qq\to Hqq$ events with
$\left|y_{H}\right|<2.5$ and no jets with $p_{\mathrm{T}}^{j}<200$ 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 $m_{jj}$ of at least 400 GeV, a large
rapidity separation $\left|\Delta y_{jj}\right|>2.8$, and an azimuthal
difference between the Higgs boson and the dijet pair of
$|\Delta\phi_{\gamma\gamma,jj}|>2.6$, has only a modeling uncertainty of 4.5%,
dominated by the composition variation of the fractions of the ggH, VBF, $VH$
and $t\bar{t}H$ processes (4.5%).

###  8 Measurement of total production-mode cross sections, signal strengths,
and simplified template cross sections

####  8.1 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.

#####  8.1.1 $t\bar{t}H$ and $tH$ enriched categories

Nine categories enriched in events produced in association with a top quark
are defined to target the $t\bar{t}H$, $tHq$, and $tHW$ production modes.
These categories are separated into a hadronic channel, where top quarks in
the event decay to hadrons via $t\to Wb\to qq^{\prime}b$; and a leptonic
channel, where at least one top quark decays to a charged lepton via $t\to
Wb\to\ell\nu b$. 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 $t\bar{t}H$ 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 ($|\eta|<2.5$)
and a veto on forward jets ($|\eta|>2.5$). 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 $ZH$
events with $Z\to\ell\ell$, 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 $t\bar{t}H$ signal
against ggH and multijet background. The BDT exploits five kinematic
variables: $H_{\mathrm{T}}$, the scalar sum of jet transverse momenta,
$m_{\mathrm{all\ jets}}$, the mass of all jets, as well as the number of all
jets, central jets ($|\eta|<2.5$), and $b$-tagged jets. The training uses
$t\bar{t}H$ 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 $t\bar{t}H$ 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.

((a))  ((b))

Figure 4: The normalized distributions for the expected background of two
kinematic variables used for the selection of the hadronic $t\bar{t}H$
categories: (a) $H_{\mathrm{T}}$ and (b) $m_{\mathrm{all\,jets}}$ for events
after the first step of the selection (see text) for simulated
$H\to\gamma\gamma$ events produced via $t\bar{t}H$ (blue) and gluon–gluon
fusion (red), for the expected background derived from the data control region
(green) described in the text and events from data with 105
GeV$<m_{\gamma\gamma}<$ 120 GeV or 130 GeV$<m_{\gamma\gamma}<$ 160 GeV (black
dots with error bars showing the statistical uncertainty).

#####  8.1.2 $VH$ 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 $VH$ dilepton category (“VH dilep”) targets $ZH$ production with
$Z\rightarrow\ell\ell$ by requiring the presence of two same-flavor opposite-
sign leptons with an invariant mass between 70 GeV and 110 GeV. Two additional
$VH$ one-lepton categories target $WH$ production with $W\to\ell\nu$. Events
are requested to contain exactly one selected electron or muon. To suppress
$ZH$ 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
$p_{\text{T}}$ of the lepton+$E_{\mathrm{T}}^{\mathrm{m}iss}$ system is higher
(“VH lep High”) or lower (“VH lep Low”) than 150 GeV. An additional
requirement on the $E_{\mathrm{T}}^{\mathrm{m}iss}$ significance, defined as
$E_{\text{T}}^{\mathrm{miss}}/\sqrt{\sum E_{\text{T}}}$, of at least 1.0 is
applied to events in the low lepton+$E_{\mathrm{T}}^{\mathrm{m}iss}$
$p_{\text{T}}$ category.

Two $VH$ missing transverse momentum categories target $ZH$ production with
$Z\to\nu\nu$ and $W\to\ell\nu$ where the lepton was not reconstructed or
failed to meet the selection criteria. One category (“VH MET Low”) requires
$80\leavevmode\nobreak\
\text{Ge\kern-1.00006ptV}<E_{\mathrm{T}}^{\mathrm{m}iss}<150\leavevmode\nobreak\
\text{Ge\kern-1.00006ptV}$ and $E_{\mathrm{T}}^{\mathrm{m}iss}$ significance
greater than 8. The other category (“VH MET High”) requires
$E_{\mathrm{T}}^{\mathrm{m}iss}>150\leavevmode\nobreak\
\text{Ge\kern-1.00006ptV}$ and $E_{\mathrm{T}}^{\mathrm{m}iss}$ significance
greater than 9, or $E_{\mathrm{T}}^{\mathrm{m}iss}>250\leavevmode\nobreak\
\text{Ge\kern-1.00006ptV}$.

#####  8.1.3 BSM enriched and $VH$ 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 $V(\to jj)H$ 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 $VH$ hadronic categories target $VH$ production with a hadronically
decaying vector boson. Events are required to have at least two jets with
$60<m_{jj}<120$ GeV. A BDT classifies the events using the following
information: the dijet invariant mass, the component
$p_{\mathrm{T}t}^{\gamma\gamma}$ of the diphoton $\vec{p}_{\mathrm{T}}$
transverse to its thrust axis in the transverse plane, the rapidity difference
between the dijet and the diphoton system, and the cosine
$\cos\theta^{*}_{\gamma\gamma,jj}$ where $\theta^{*}_{\gamma\gamma,jj}$ 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 $VH$
events as signal, and a mixture of simulated signals (everything except $VH$
events), simulated $\gamma\gamma$ events, and $\gamma j$ and $jj$ data control
samples as background. Using the BDT response as a discriminating variable,
events are classified into two categories (“$VH$ had tight” and “$VH$ had
loose”) with an expected fraction of signal events due to $VH$ production of
42% and 25%, respectively.

Figure 5 shows the distributions of $m_{jj}$ and
$p_{\mathrm{T}t}^{\gamma\gamma}$ in signal and background events and in events
selected in data from the $m_{\gamma\gamma}$ sidebands. The variables show
good separation between $VH$ events and both the other signal events and
background events.

((a))  ((b))

Figure 5: The normalized distributions of two kinematic variables used for the
selection of the $VH$ hadronic categories: (a) $m_{jj}$ and (b)
$p_{\mathrm{T}t}^{\gamma\gamma}$ for events after the first step of the $VH$
hadronic category preselection (see text) for simulated $H\to\gamma\gamma$
events produced in association with hadronically decaying vector bosons (blue)
or through ggH, VBF or $t\bar{t}H$ processes (red), for the expected
background from data ($\gamma j$, $jj$) and simulation ($\gamma\gamma$,
$V\gamma\gamma$) control samples (green, purple), and for events from data
with 105 GeV$<m_{\gamma\gamma}<$ 120 GeV or 130 GeV$<m_{\gamma\gamma}<$ 160
GeV (black dots with error bars showing the statistical uncertainty).

#####  8.1.4 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 $|\Delta{\eta_{jj}|}$ between the two
leading jets is required to be greater than 2. In addition
$|\eta_{\gamma\gamma}-0.5(\eta_{j1}+\eta_{j2})|$ is required to be less than
5, with $\eta_{\gamma\gamma}$ denoting the pseudorapidity of the diphoton
system. The events are first split into two regions based on the value of the
transverse momentum $p_{\mathrm{T}}^{Hjj}$ 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 $p_{\mathrm{T}}$ of the third jet due to momentum
balance. The signal in the
$p_{\mathrm{T}}^{\gamma\gamma{}jj}<25\leavevmode\nobreak\
\text{Ge\kern-1.00006ptV}$ “low $p_{\mathrm{T}}^{Hjj}$” region is dominated by
exclusive two-jet-like events, while the signal in the
$p_{\mathrm{T}}^{\gamma\gamma{}jj}>25\leavevmode\nobreak\
\text{Ge\kern-1.00006ptV}$ “high $p_{\mathrm{T}}^{Hjj}$” region is dominated
by inclusive $\geq 3$-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: $m_{jj}$, $|\Delta{\eta_{jj}|}$, $p_{\mathrm{T}t}^{\gamma\gamma}$,
the absolute azimuthal difference of the diphoton and the dijet system
$|\Delta\phi_{\gamma\gamma,jj}|$, the minimum angular separation between
either of the two signal photons and either of the two leading jets $\Delta
R^{\mathrm{min}}_{\gamma j}$, and
$|\eta_{\gamma\gamma}-0.5(\eta_{j1}+\eta_{j2})|$. A requirement of
$|\Delta\phi_{\gamma\gamma,jj}|$ to be near $\pi$ 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 $|\Delta\phi_{\gamma\gamma,jj}|>2.94$ 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 $\gamma\gamma$
events and of $\gamma j$ and $jj$ data control samples as background. Four
exclusive categories are defined with "loose" and "tight" requirements on the
BDT classifier in the two $p_{\mathrm{T}}^{Hjj}$ regions. The "tight" category
in the $p_{\mathrm{T}}^{Hjj}>25\leavevmode\nobreak\ \text{Ge\kern-1.00006ptV}$
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 $p_{\mathrm{T}}^{Hjj}<25\leavevmode\nobreak\
\text{Ge\kern-1.00006ptV}$ 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 $|\Delta{\eta_{jj}|}$ and
$|\Delta\phi_{\gamma\gamma,jj}|$ in simulated $H\to\gamma\gamma$ events,
background events from simulated diphotons and data control samples of $\gamma
j$ and $jj$ events, and events selected from the $m_{\gamma\gamma}$ sidebands
in data. The variables show good separation between VBF events and both
gluon–gluon fusion events and background events.

((a))  ((b))

Figure 6: The normalized distributions for the expected background of two
kinematic variables used for the selection of the VBF categories: (a)
$|\Delta{\eta_{jj}|}$ and (b) $|\Delta\phi_{\gamma\gamma,jj}|$ for events
after the first step of the selection (see text) for simulated
$H\to\gamma\gamma$ events produced via vector-boson fusion (blue) and
gluon–gluon fusion (red), for the expected background from data ($\gamma j$,
$jj$) and simulation ($\gamma\gamma$) control samples (green), and for events
from data with 105 GeV$<m_{\gamma\gamma}<$ 120 GeV or 130
GeV$<m_{\gamma\gamma}<$ 160 GeV (black dots with error bars showing the
statistical uncertainty).

#####  8.1.5 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 $|\eta|<0.95$, in which the energy resolution is better (“ggH 0J Cen”),
or with at least one photon in the “forward” region $|\eta|>0.95$ 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 $p_{{\mathrm{T}}}^{\gamma\gamma}\in[0,60)$
(“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.

#####  8.1.6 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 $qq\to Hqq$
bins are more ambiguous however still retain much of the diagonal structure.
On the other hand, the $VH$ 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 $p_{\text{T}}>30$ GeV unless otherwise noted. The categories are mutually exclusive and the criteria are applied in descending order of the shown categories.  Category |  Selection
---|---
tH lep 0fwd |  $N_{\mathrm{lep}}$ = 1, $N_{\mathrm{jets}}^{\mathrm{cen}}$ $\leq$ 3, $N_{b\mathrm{-tag}}$ $\geq$ 1, $N_{\mathrm{jets}}^{\mathrm{fwd}}$ = 0 ($p_{\mathrm{T}}^{\mathrm{jet}}>$ $25\text{\,}\mathrm{GeV}$)
tH lep 1fwd |  $N_{\mathrm{lep}}$ = 1, $N_{\mathrm{jets}}^{\mathrm{cen}}$ $\leq$ 4, $N_{b\mathrm{-tag}}$ $\geq$ 1, $N_{\mathrm{jets}}^{\mathrm{fwd}}$ $\geq$ 1 ($p_{\mathrm{T}}^{\mathrm{jet}}>$ $25\text{\,}\mathrm{GeV}$)
ttH lep |  $N_{\mathrm{lep}}$ $\geq$ 1, $N_{\mathrm{jets}}^{\mathrm{cen}}$ $\geq$ 2, $N_{b\mathrm{-tag}}$ $\geq$ 1, $Z_{\ell\ell}$ veto ($p_{\mathrm{T}}^{\mathrm{jet}}>$ $25\text{\,}\mathrm{GeV}$)
ttH had BDT1 |  $N_{\mathrm{lep}}$ = 0, $N_{\mathrm{jets}}\geq$ 3, $N_{b\mathrm{-tag}}$ $\geq$ 1, $\mathrm{BDT}_{\mathrm{ttH}}>0.92$
ttH had BDT2 |  $N_{\mathrm{lep}}$ = 0, $N_{\mathrm{jets}}\geq$ 3, $N_{b\mathrm{-tag}}$ $\geq$ 1, $0.83<\mathrm{BDT}_{\mathrm{ttH}}<0.92$
ttH had BDT3 |  $N_{\mathrm{lep}}$ = 0, $N_{\mathrm{jets}}\geq$ 3, $N_{b\mathrm{-tag}}$ $\geq$ 1, $0.79<\mathrm{BDT}_{\mathrm{ttH}}<0.83$
ttH had BDT4 |  $N_{\mathrm{lep}}$ = 0, $N_{\mathrm{jets}}\geq$ 3, $N_{b\mathrm{-tag}}$ $\geq$ 1, $0.52<\mathrm{BDT}_{\mathrm{ttH}}<0.79$
tH had 4j1b |  $N_{\mathrm{lep}}$ = 0, $N_{\mathrm{jets}}^{\mathrm{cen}}$ = 4, $N_{b\mathrm{-tag}}$ = 1 ($p_{\mathrm{T}}^{\mathrm{jet}}>$25\text{\,}\mathrm{GeV}$$)
tH had 4j2b |  $N_{\mathrm{lep}}$ = 0, $N_{\mathrm{jets}}^{\mathrm{cen}}$ = 4, $N_{b\mathrm{-tag}}$ $\geq$ 2 ($p_{\mathrm{T}}^{\mathrm{jet}}>$25\text{\,}\mathrm{GeV}$$)
VH dilep |  $N_{\mathrm{lep}}\geq 2$, $$70\text{\,}\mathrm{GeV}$\leq m_{\ell\ell}\leq$110\text{\,}\mathrm{GeV}$$
VH lep High |  $N_{\mathrm{lep}}=1$, $|m_{e\gamma}-$89\text{\,}\mathrm{GeV}$|>$5\text{\,}\mathrm{GeV}$$, $p_{\text{T}}^{\ell+E_{\text{T}}^{\text{miss}}{}}>$150\text{\,}\mathrm{GeV}$$
VH lep Low |  $N_{\mathrm{lep}}=1$, $|m_{e\gamma}-$89\text{\,}\mathrm{GeV}$|>$5\text{\,}\mathrm{GeV}$$, $p_{\mathrm{T}}^{\ell+E_{\text{T}}^{\text{miss}}{}}<$150\text{\,}\mathrm{GeV}$,E_{\mathrm{T}}^{\mathrm{m}iss}{}\ \text{significance}>1$
VH MET High |  $$150\text{\,}\mathrm{GeV}$<E_{\text{T}}^{\text{miss}}{}<$250\text{\,}\mathrm{GeV}$,E_{\mathrm{T}}^{\mathrm{m}iss}{}\ \text{significance}>9$ or $E_{\text{T}}^{\text{miss}}{}>$250\text{\,}\mathrm{GeV}$$
VH MET Low |  $$80\text{\,}\mathrm{GeV}$<E_{\text{T}}^{\text{miss}}{}<$150\text{\,}\mathrm{GeV}$,E_{\mathrm{T}}^{\mathrm{m}iss}{}\ \text{significance}>8$
jet BSM |  $p_{\mathrm{T,j1}}>$200\text{\,}\mathrm{GeV}$$
VH had tight |  $$60\text{\,}\mathrm{GeV}$<m_{\mathrm{jj}}<$120\text{\,}\mathrm{GeV}$$, $\mathrm{BDT}_{\mathrm{VH}}>0.78$
VH had loose |  $60\leavevmode\nobreak\ \text{Ge\kern-0.80005ptV}<m_{\mathrm{jj}}<$120\text{\,}\mathrm{GeV}$$, $0.35<\mathrm{BDT}_{\mathrm{VH}}<0.78$
VBF tight, high $p_{\mathrm{T}}^{Hjj}$ |  $|\Delta{\eta_{jj}|}>2$, $|\eta_{\gamma\gamma}-0.5(\eta_{\mathrm{j1}}+\eta_{\mathrm{j2}})|<5$, $p_{\mathrm{T}}^{Hjj}>$ $25\text{\,}\mathrm{GeV}$, $\mathrm{BDT}_{\mathrm{VBF}}>0.47$
VBF loose, high $p_{\mathrm{T}}^{Hjj}$ |  $|\Delta{\eta_{jj}|}>2$, $|\eta_{\gamma\gamma}-0.5(\eta_{\mathrm{j1}}+\eta_{\mathrm{j2}})|<5$, $p_{\mathrm{T}}^{Hjj}>$ $25\text{\,}\mathrm{GeV}$, $-0.32<\mathrm{BDT}_{\mathrm{VBF}}<0.47$
VBF tight, low $p_{\mathrm{T}}^{Hjj}$ |  $|\Delta{\eta_{jj}|}>2$, $|\eta_{\gamma\gamma}-0.5(\eta_{\mathrm{j1}}+\eta_{\mathrm{j2}})|<5$, $p_{\mathrm{T}}^{Hjj}<$ $25\text{\,}\mathrm{GeV}$, $\mathrm{BDT}_{\mathrm{VBF}}>0.87$
VBF loose, low $p_{\mathrm{T}}^{Hjj}$ |  $|\Delta{\eta_{jj}|}>2$, $|\eta_{\gamma\gamma}-0.5(\eta_{\mathrm{j1}}+\eta_{\mathrm{j2}})|<5$, $p_{\mathrm{T}}^{Hjj}<$ $25\text{\,}\mathrm{GeV}$, $0.26<\mathrm{BDT}_{\mathrm{VBF}}<0.87$
ggH 2J BSM |  $\geq 2$ jets, $p_{\mathrm{T}}^{\gamma\gamma}\geq$200\text{\,}\mathrm{GeV}$$
ggH 2J High |  $\geq 2$ jets, $p_{\mathrm{T}}^{\gamma\gamma}\in[120,200]$ GeV
ggH 2J Med |  $\geq 2$ jets, $p_{\mathrm{T}}^{\gamma\gamma}\in[60,120]$ GeV
ggH 2J Low |  $\geq 2$ jets, $p_{\mathrm{T}}^{\gamma\gamma}\in[0,60]$ GeV
ggH 1J BSM |  $=1$ jet, $p_{\mathrm{T}}^{\gamma\gamma}\geq$200\text{\,}\mathrm{GeV}$$
ggH 1J High |  $=1$ jet, $p_{\mathrm{T}}^{\gamma\gamma}\in[120,200]$ GeV
ggH 1J Med |  $=1$ jet, $p_{\mathrm{T}}^{\gamma\gamma}\in[60,120]$ GeV
ggH 1J Low |  $=1$ jet, $p_{\mathrm{T}}^{\gamma\gamma}\in[0,60]$ GeV
ggH 0J Fwd |  $=0$ jets, one photon with $|\eta|>0.95$
ggH 0J Cen |  $=0$ jets, two photons with $|\eta|\leq 0.95$
Table 5: Signal efficiencies times acceptance, $\epsilon$, and expected signal event fractions per production mode, $f$, in each category for ${\sqrt{s}=13}$ TeV and $m_{H}=125.09$ GeV. The second-to-last row shows the total efficiency per production process summed over the categories. Values labeled as ’nil’ correspond to efficiencies or fractions that are smaller than 0.05%. The total number of expected signal events, $N_{\mathrm{S}}$, in the last row corresponds to an integrated luminosity of $36.1\text{\,}{\mathrm{fb}}^{-1}$.  | ggH | VBF | $WH$ | $ZH$ | ttH | $b\bar{b}H$ | $tHq$ | $tHW$ | All
---|---|---|---|---|---|---|---|---|---
Category |  $\epsilon$[%] |  $f$[%] |  $\epsilon$[%] |  $f$[%] |  $\epsilon$[%] |  $f$[%] |  $\epsilon$[%] |  $f$[%] |  $\epsilon$[%] |  $f$[%] |  $\epsilon$[%] |  $f$[%] |  $\epsilon$[%] |  $f$[%] |  $\epsilon$[%] |  $f$[%] | $N_{\mathrm{S}}$
ggH 0J Cen | 8.9 | 97.3 | 1.2 | 1.1 | 1.4 | 0.4 | 1.9 | 0.4 | nil | nil | 8.2 | 0.9 | nil | nil | nil | nil | 333.5
ggH 0J Fwd | 15.5 | 97.0 | 2.4 | 1.2 | 3.0 | 0.5 | 3.7 | 0.4 | nil | nil | 14.7 | 0.9 | 0.2 | nil | 0.1 | nil | 579.5
ggH 1J Low | 7.2 | 90.5 | 5.7 | 5.7 | 5.0 | 1.7 | 4.4 | 1.0 | 0.1 | nil | 9.1 | 1.1 | 0.5 | nil | 0.2 | nil | 289.9
ggH 1J Med | 3.6 | 83.5 | 6.4 | 11.7 | 4.2 | 2.6 | 4.1 | 1.6 | 0.1 | nil | 1.9 | 0.4 | 0.6 | nil | 0.3 | nil | 156.2
ggH 1J High | 0.7 | 76.0 | 1.9 | 17.5 | 1.1 | 3.4 | 1.4 | 2.7 | 0.1 | 0.1 | 0.3 | 0.3 | 0.2 | nil | 0.1 | nil | 31.5
ggH 1J BSM | nil | 72.4 | 0.1 | 16.9 | 0.1 | 6.0 | 0.2 | 4.2 | nil | 0.3 | nil | nil | nil | 0.1 | nil | nil | 2.2
ggH 2J Low | 1.8 | 79.1 | 2.7 | 9.6 | 3.7 | 4.5 | 4.1 | 3.1 | 2.2 | 1.1 | 5.4 | 2.3 | 3.9 | 0.3 | 1.9 | nil | 81.1
ggH 2J Med | 1.5 | 77.6 | 3.1 | 12.2 | 3.2 | 4.4 | 3.8 | 3.2 | 2.6 | 1.5 | 1.6 | 0.7 | 4.5 | 0.4 | 2.4 | nil | 72.4
ggH 2J High | 0.6 | 75.8 | 1.3 | 12.8 | 1.4 | 4.9 | 1.9 | 4.0 | 1.4 | 2.0 | 0.1 | 0.1 | 2.2 | 0.4 | 1.6 | 0.1 | 29.2
ggH 2J BSM | 0.2 | 76.2 | 0.3 | 10.3 | 0.4 | 4.9 | 0.6 | 4.6 | 0.6 | 3.0 | 0.1 | 0.2 | 0.8 | 0.6 | 1.3 | 0.2 | 7.6
VBF Hjj Low loose | 0.2 | 32.3 | 4.5 | 66.7 | 0.1 | 0.3 | 0.1 | 0.3 | nil | nil | 0.1 | 0.2 | 0.3 | 0.1 | nil | nil | 19.4
VBF Hjj Low tight | nil | 12.9 | 4.2 | 86.7 | nil | 0.1 | nil | 0.1 | nil | nil | nil | nil | 0.3 | 0.1 | nil | nil | 13.8
VBF Hjj High loose | 0.3 | 69.9 | 1.4 | 23.8 | 0.4 | 2.2 | 0.5 | 1.8 | 0.4 | 0.9 | 0.4 | 0.7 | 1.8 | 0.6 | 0.5 | nil | 16.5
VBF Hjj High tight | 0.3 | 47.0 | 3.4 | 48.2 | 0.2 | 1.2 | 0.4 | 1.3 | 0.4 | 0.8 | 0.2 | 0.3 | 4.4 | 1.2 | 0.6 | nil | 20.2
VHhad loose | 0.3 | 67.2 | 0.3 | 4.9 | 2.4 | 14.6 | 2.9 | 11.0 | 0.6 | 1.6 | 0.2 | 0.4 | 0.8 | 0.3 | 0.8 | 0.1 | 16.5
VHhad tight | 0.2 | 52.4 | 0.1 | 3.4 | 3.0 | 23.8 | 3.5 | 18.0 | 0.6 | 1.9 | nil | 0.1 | 0.5 | 0.2 | 1.0 | 0.1 | 12.3
jet BSM | 0.4 | 59.9 | 2.4 | 25.8 | 1.6 | 5.9 | 1.9 | 4.4 | 2.0 | 3.0 | 0.1 | 0.1 | 3.1 | 0.6 | 5.1 | 0.2 | 26.7
VHMET Low | nil | 11.9 | nil | 0.4 | 0.1 | 23.4 | 0.6 | 63.2 | nil | 0.5 | nil | 0.3 | nil | 0.2 | nil | nil | 0.6
VHMET High | nil | 1.3 | nil | 0.1 | 0.3 | 22.8 | 1.4 | 66.2 | 0.3 | 8.3 | nil | nil | 0.1 | 0.6 | 0.8 | 0.7 | 1.3
VHlep Low | nil | 11.4 | nil | 1.1 | 4.4 | 68.0 | 0.8 | 8.1 | 1.3 | 8.5 | 0.2 | 0.9 | 1.8 | 1.6 | 2.2 | 0.4 | 6.4
VHlep High | nil | 0.2 | nil | nil | 1.2 | 76.5 | 0.1 | 4.6 | 0.6 | 16.2 | nil | nil | 0.3 | 1.2 | 1.6 | 1.3 | 1.5
VHdilep | nil | nil | nil | nil | nil | nil | 1.4 | 95.8 | 0.1 | 4.0 | nil | nil | nil | nil | 0.1 | 0.2 | 0.9
tHhad 4j2b | nil | 23.8 | nil | 2.8 | nil | 1.6 | 0.1 | 13.5 | 0.6 | 39.0 | 0.1 | 8.2 | 1.2 | 10.5 | 0.3 | 0.6 | 0.6
tHhad 4j1b | nil | 35.4 | nil | 4.0 | 0.1 | 4.3 | 0.3 | 7.9 | 2.2 | 36.3 | 0.2 | 2.2 | 3.8 | 8.5 | 2.6 | 1.3 | 2.5
ttHhadBDT4 | nil | 7.0 | nil | 0.8 | nil | 1.4 | 0.2 | 4.5 | 4.8 | 79.4 | nil | 0.3 | 1.9 | 4.3 | 4.7 | 2.4 | 2.5
ttHhadBDT3 | nil | 3.5 | nil | 0.5 | nil | 1.0 | nil | 3.1 | 1.3 | 86.1 | nil | 0.5 | 0.3 | 3.1 | 1.1 | 2.2 | 0.6
ttHhadBDT2 | nil | 3.6 | nil | 0.3 | nil | 0.8 | nil | 1.6 | 3.8 | 89.3 | nil | 0.2 | 0.6 | 1.8 | 3.4 | 2.4 | 1.8
ttHhadBDT1 | nil | 1.2 | nil | 0.1 | nil | 0.1 | nil | 0.7 | 3.4 | 95.0 | nil | 0.1 | 0.2 | 0.7 | 2.5 | 2.1 | 1.4
ttHlep | nil | nil | nil | nil | nil | 0.2 | nil | 0.1 | 5.6 | 96.0 | nil | 0.1 | 0.4 | 1.0 | 5.0 | 2.6 | 2.4
tHlep 1fwd | nil | 1.8 | nil | 0.2 | nil | 1.4 | nil | 0.9 | 2.1 | 79.4 | nil | 0.2 | 2.6 | 13.5 | 2.3 | 2.6 | 1.1
tHlep 0fwd | nil | 4.1 | nil | 0.2 | 0.1 | 5.6 | nil | 2.8 | 1.9 | 75.7 | nil | 0.9 | 1.5 | 8.2 | 2.1 | 2.5 | 1.0
Total $\epsilon$ [%] | 41.8 | - | 41.3 | - | 37.6 | - | 40.5 | - | 39.1 | - | 42.8 | - | 38.9 | - | 44.5 | - | 41.8
Events | 1518.4 | 119.1 | 37.1 | 25.2 | 16.0 | 14.8 | 2.2 | 0.5 | 1733.2
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 LABEL:tab:STXS) of the stage-1 simplified template cross
section framework ($y$ axis). The black lines separate the $t\bar{t}H$ and
$tH$, $VH$ leptonic, $VH$ 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.  Figure
8: The expected composition of the selected Higgs boson events, in terms of
the different production modes, for each reconstructed category.

####  8.2 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 LABEL:tab:STXS. In addition, the result of coupling-
strength fits are reported.

#####  8.2.1 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 $m_{\gamma\gamma}$ region
containing 90% of the expected signal yield, using a weight of the form
$\ln\left(1+S_{90}/B_{90}\right)$. 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 $\mu$ affecting simultaneously all
production modes has been assumed, while in the fits of Figure 10 the four
signal strengths $\mu_{\mathrm{ggH}}$, $\mu_{\mathrm{VBF}}$,
$\mu_{\mathrm{VH}}$ and $\mu_{\mathrm{ttH+tH}}$ are allowed to vary
separately, as described in the following section. The observed mass peak of
the Higgs boson, constrained in the fit as $m_{H}=125.09\pm
0.24\leavevmode\nobreak\ \text{Ge\kern-1.00006ptV}$, is well within 68% CL of
the Run 1 ATLAS+CMS combined measurement.

Figure 9: Weighted diphoton invariant mass spectrum observed in the 2015 and
2016 data at 13 TeV. Each event is weighted by the $\ln(1+S_{90}/B_{90})$
ratio of the expected signal ($S_{90}$) and background ($B_{90}$) of the 90%
signal quantile in the category to which it belongs to. The values of $S_{90}$
and $B_{90}$ used for each category are shown in Table 27 of Appendix E. The
error bars represent 68% confidence intervals of the weighted sums. The solid
red curve shows the fitted signal-plus-background model when the Higgs boson
mass is constrained to be $125.09\pm 0.24$ GeV. The background component of
the fit is shown with the dotted blue curve. The signal component of the fit
is shown with the solid black curve. Both the signal-plus-background and
background-only curves reported here are obtained from the sum of the
individual curves in each category weighted by the logarithm of unity plus the
signal-to-background ratio. The bottom plot shows the residuals between the
data and the background component of the fitted model.

((a))

((b))

((c))

((d))

Figure 10: Weighted diphoton invariant mass spectra observed in the 13 TeV
data for events belonging to: (a) “untagged” categories and the “jet BSM”
category, in which the expected signal is produced mainly through gluon–gluon
fusion, (b) VBF categories, (c) $VH$ categories and (d) $t\bar{t}H$
categories. Each event is weighted by the $\ln(1+S_{90}/B_{90})$ ratio of the
expected signal ($S_{90}$) and background ($B_{90}$) of the 90% signal
quantile in the category it belongs to. The values of $S_{90}$ and $B_{90}$
used for each category are shown in Table 27 of Appendix E. The error bars
represent 68% confidence intervals of the weighted sums. The solid red curve
shows the fitted signal-plus-background model when the Higgs boson mass is
constrained to be $125.09\pm 0.24$ GeV. The background component of the fit is
shown with the dotted blue curve. The signal component of the fit is shown
with the solid black curve. Both the signal-plus-background and background-
only curves reported here are obtained from the sum of the individual curves
in each category weighted by the logarithm of unity plus the signal-to-
background ratio. The bottom plot shows the residuals between the data and the
background component of the fitted model.

#####  8.2.2 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
$N_{\mathrm{sig,}m}^{i}$ in each category $i$ for a particular production mode
$m$ is expressed as the product of the integrated luminosity $\int
L\,\mathrm{d}t$, the signal strength $\mu_{m}$ for that production mode, the
expected SM Higgs boson production mode cross section times branching ratio to
diphotons, and the acceptance times efficiency $\epsilon$ (Table 5) for signal
events from that production mode in the selected category
($N_{\mathrm{sig,m}}^{i}=\mu_{m}\times\int
L\,\mathrm{d}t\times\sigma_{m}^{\mathrm{SM}}\times
B(H\to\gamma\gamma)\times\epsilon_{m}^{i}$).

A global signal strength $\mu$ is measured assuming the ratios between
different production processes to be as predicted by the SM. The profile of
the negative log-likelihood ratio $\lambda(\mu)$ of the global signal strength
of all Higgs processes $\mu$ for $m_{H}=125.09\pm 0.24$ GeV is shown in Figure
11.

Figure 11: Observed negative log-profile likelihood $\Lambda$ of the global
signal strength $\mu$. The three likelihood contours shown correspond to all
theory and experimental nuisance parameters fixed (Stat.), all experimental
nuisance parameters fixed (Theo.), and with all nuisance parameters floating
(Total). The intersections of the solid curves and horizontal lines at
$\Lambda=1$ and $\Lambda=4$ indicate the 1 and 2 $\sigma$ confidence intervals
of the corresponding result.

The measured central value and 68% CL interval for $\mu$ is found to be:

$\displaystyle\mu$ $\displaystyle=0.99\ ^{+0.15}_{-0.14}=0.99\ \pm
0.12\,\mathrm{(stat.)}\ ^{+0.06}_{-0.05}\,\mathrm{(exp.)}\
^{+0.07}_{-0.05}\,\mathrm{(theo.)}\ ,$

well compatible with the SM prediction ($\mu=1$). This result confirms the
ATLAS Run-1 diphoton signal strength measurement of $\mu=1.17\pm
0.23\,\mathrm{(stat.)}\ ^{+0.10}_{-0.08}\,\mathrm{(exp.)}\
^{+0.12}_{-0.08}\,\mathrm{(theo.)}$ with around a factor of two improvement in
each component of the uncertainty. The Run-1 result was obtained using the
NNLO SM prediction for ggH production [17, 111], which is about 10% lower than
the N3LO 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.

Table 6:  Main systematic uncertainties $\sigma_{\mu}^{\mathrm{syst.}}$ in the combined signal strength parameter $\mu$. 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 $\mu$ 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.  Uncertainty Group | $\sigma_{\mu}^{\mathrm{syst.}}$
---|---
Theory (QCD) | 0.041
Theory ($B(H\to\gamma\gamma)$) | 0.028
Theory (PDF+$\alpha_{S}$) | 0.021
Theory (UE/PS) | 0.026
Luminosity | 0.031
Experimental (yield) | 0.017
Experimental (migrations) | 0.015
Mass resolution | 0.029
Mass scale | 0.006
Background shape | 0.027

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:

$\displaystyle\mu_{\mathrm{ggH}}$ $\displaystyle=0.81\ ^{+0.19}_{-0.18}=0.81\
\pm 0.16\,\mathrm{(stat.)}\ ^{+0.07}_{-0.06}\,\mathrm{(exp.)}\
^{+0.07}_{-0.05}\,\mathrm{(theo.)}$ $\displaystyle\mu_{\mathrm{VBF}}$
$\displaystyle=2.0\ ^{+0.6}_{-0.5}=2.0\ \pm 0.5\,\mathrm{(stat.)}\
^{+0.3}_{-0.2}\,\mathrm{(exp.)}\ ^{+0.3}_{-0.2}\,\mathrm{(theo.)}$
$\displaystyle\mu_{\mathrm{VH}}$ $\displaystyle=0.7\ ^{+0.9}_{-0.8}=0.7\ \pm
0.8\,\mathrm{(stat.)}\ ^{+0.2}_{-0.2}\,\mathrm{(exp.)}\
^{+0.2}_{-0.1}\,\mathrm{(theo.)}$ $\displaystyle\mu_{\mathrm{top}}$
$\displaystyle=0.5\ ^{+0.6}_{-0.6}=0.5\ ^{+0.6}_{-0.5}\,\mathrm{(stat.)}\
^{+0.1}_{-0.1}\,\mathrm{(exp.)}\ ^{+0.1}_{-0.0}\,\mathrm{(theo.)}$

For Higgs boson production via $VH$ the signal strength is assumed to be
scaled by a single parameter (i.e.
$\mu_{\mathrm{VH}}=\mu_{\mathrm{ZH}}=\mu_{\mathrm{WH}}$). The $b\bar{b}H$
contributions are scaled with ggH (i.e.
$\mu_{\mathrm{bbH}}=\mu_{\mathrm{ggH}}$), and the $tH$ and $t\bar{t}H$
productions are measured together rather than separately (i.e.
$\mu_{\mathrm{top}}=\mu_{\mathrm{ttH+tH}}$).

Figure 12:  Summary of the signal strengths measured for the different
production processes (ggH, VBF, $VH$ and top) and globally
($\mu_{\mathrm{Run-2}}$), compared to the global signal strength measured at 7
and 8 TeV ($\mu_{\mathrm{Run-1}}$) [76]. The black and orange error bars show
the total and statistical uncertainties. The signal strength
$\mu_{\mathrm{Run-1}}$ 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 [32, 7], the gluon–gluon fusion production-mode cross section is
larger by approximately 10%. In this measurement, the $b\bar{b}H$
contributions are scaled with ggH ($\mu_{\mathrm{bbH}}=\mu_{\mathrm{ggH}}$),
and the $tH$ and $t\bar{t}H$ productions are measured together
($\mu_{\mathrm{top}}=\mu_{\mathrm{ttH+tH}}$). Associated production with $Z$
or $W$ bosons is assumed to be scaled by a single signal strength parameter
($\mu_{\mathrm{VH}}=\mu_{\mathrm{ZH}}=\mu_{\mathrm{WH}}$).

The ggH signal strength is $1\leavevmode\nobreak\ \sigma$ below the Standard
Model prediction, while the VBF signal strength is $2.2\leavevmode\nobreak\
\sigma$ 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 $5\leavevmode\nobreak\ \sigma$.

Since no significant evidence is observed for $VH$ 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 of the asymptotic
approximation was validated using ensembles of pseudo-experiments. Appendix F