Doctoral Dissertation
博士論文
Search for a heavy scalar resonance decaying into a pair of√photon-jets in pp collisions at s = 13 TeV（重心系エネルギー13 TeVの陽子陽子衝突における
光子ジェット対に崩壊する重いスカラー粒子の探索）

A Dissertation Submitted for the Degree of Doctor of Philosophy
February 2019平成31年2月博士（理学）申請
Department of Physics, Graduate School of Science,
The University of Tokyo東京大学大学院理学系研究科物理学専攻
Yuya Kano
加納 勇也

Abstract
The Standard Model of particle physics is well established, but several unsolved problems remain, suchas the hierarchy problem and the strong CP problem. They suggest the existence of physics beyond theStandard Model, but hints of new physics are not yet observed by past researches. At the energy-frontier
pp collision experiments at the LHC, it is all the more important to search for new physics in a completely
new final state. This thesis describes such a search in a novel final state called a “photon-jet” using pp√
collisions at s = 13 TeV. A photon-jet is a group of collimated photons that share close trajectories.
Photon-jets can arise from a boson with a mass of O(100) GeV decaying into lighter bosons with a massof O(1) GeV, that consecutively decays into photons. In such a case, the lighter boson is boosted, leading
to collimated photons. The existence of photon-jets is well motivated by several beyond-the-StandardModel scenarios that predict two or more new bosons, including the supersymmetry and the Peccei-Quinn
mechanism. This research aims to look into a parameter space which conventional searches had limited
sensitivity to, by employing an optimized event selection strategy based on shower shapes observed in theelectromagnetic calorimeter.
The result of a search for a new heavy scalar resonance decaying into a pair of photon-jets using pp√
collisions at s = 13 TeV is presented. The dataset was collected with the ATLAS detector in 2015 and2016, corresponding to an integrated luminosity of 36.7 fb−1 . Candidate events of a resonance decaying
into a pair of photon-jets are selected from events with two high-momentum reconstructed photons,
where each reconstructed photon corresponds to a photon-jet in the case of signal events. This searchis performed in a largely model-independent way, so that the search is sensitive to beyond-the-StandardModel scenarios leading to a pair of photon-jets in general. No significant excess of events from theStandard Model expectation is observed.
The result of a null observation is interpreted in the context of beyond-the-Standard-Model scenarios
which assume a scalar resonance decaying into a final state with photons. One scenario assumes a scalar
resonance X with a mass of O(100) GeV, with a narrow width, and produced by the gluon–gluon fusion
process; X decays into a pair of spin-0 particles a with a mass of O(1) GeV that decays into a pair ofphotons, via X → aa → 4γ. Another scenario assumes the decay of the a particle into three neutralpions, via X → aa → 6π 0 → 12γ. Upper limits on the product of the production cross section and thebranching ratios are evaluated for the region 200 GeV < mX < 2 TeV and ma < 0.01 × mX using an
asymptotic approximation. They are found to be as low as 0.2 fb for mX = 2 TeV. Some scenarios predict
photon-jets produced from a decay of a long-lived boson; the results are interpreted for such a case as
well. The results are interpreted in the context of the Next-to-Minimal Supersymmetric Standard Modelfor the process H → aa → 4γ, where H is a new scalar Higgs boson with a mass larger than 200 GeV,
and a is a pseudoscalar Higgs boson with a mass of O(1) GeV.
This thesis presents the first result of a search for a heavy (> 200 GeV) resonance decaying into photon-jets.
The important feature of this research is the optimization of the search strategy utilizing the shower shapeobserved in the electromagnetic calorimeter. This research is a new frontier of direct searches using anovel final state. This result places constraints on the subset of the parameter space of the Next-to-Minimal
Supersymmetric Standard Model that has not been looked into before.

2

Contents
1 Introduction

5

2 Theoretical overview
2.1 The Standard Model of particle physics
2.2 Supersymmetry and the NMSSM
2.2.1 Supersymmetry
2.2.2 MSSM
2.2.3 NMSSM
2.3 Photon-jets from NMSSM
2.3.1 Light state a
2.3.2 Heavy state H

11
11
11
12
14
16
21
21
23

3 ATLAS detector at the LHC
3.1 Large Hadron Collider
3.2 ATLAS detector
3.2.1 Inner Detector
3.2.2 Calorimeter3.2.3 Muon spectrometer3.2.4 Luminosity detector
3.2.5 Trigger and data acquisition
3.3 ATLAS Run 2 operation

25
25
26
26
28
30
31
32
33

4 Photon reconstruction and identification
4.1 Reconstruction
4.2 Identification
4.3 Isolation

35
35
38
42

5 Search for a resonance decaying into a pair of photon-jets
5.1 Event samples
5.1.1 Collision data
5.1.2 Simulated samples
5.2 Photon-jet signal characteristics
5.3 Event selection
5.3.1 Definition of the signal region
5.3.2 Categorization of events by the shower shape variable ∆E
5.4 Signal modelling
5.4.1 Signal mass shape
5.4.2 Signal selection efficiency
5.4.3 Category fraction
5.5 Background modelling
5.5.1 Background composition measurement
5.5.2 Background template
5.5.3 Spurious signal test
5.6 Systematic uncertainties
5.6.1 Signal mass shape modelling

44
46
46
46
47
48
48
52
56
56
61
63
65
65
69
70
74
74

3

5.7

5.8

5.9

5.6.2 Signal selection efficiency and category fraction
5.6.3 Other sources
Statistical analysis
5.7.1 Definition of the likelihood function
5.7.2 Hypothesis testing
Results
5.8.1 Results of the null hypothesis test
5.8.2 Upper limits on signal cross-section times branching ratios
5.8.3 Comparison of upper limit with that from diphoton resonance search
5.8.4 Impact of systematic uncertainties
5.8.5 Upper limits on signal visible cross-sections
Reinterpretation of the results for the case of long-lived a
5.9.1 Signal selection efficiency and category fraction parametrization
5.9.2 Reinterpretation procedure
5.9.3 Results

77
78
78
78
81
82
82
84
88
89
91
92
92
99
99

6 Interpretation of the results in the context of NMSSM
6.1 Method
6.2 Scan region
6.3 Result
6.4 Future directions

102
102
103
106
110

7 Conclusion

111

Appendix

113

A Results of selection efficiency and category fraction parametrizations for the case of longlived a113

B Distributions of shower shape variables

116

C Sensitivity estimation using simulated events for optimization of the event selection120

D Additional details to the hypothesis testing method

124

E Distributions of mγ R γ R in the signal region, with the bin widths set to the detector resolution126

4

1 Introduction
The Standard Model (SM) of particle physics describes the properties of elementary particles and theirinteractions. With the discovery of the Higgs boson in 2012, the SM is now well established. However,
several unsolved problems remain, including the hierarchy problem [1, 2] and the strong CP problem [3,4]. Several beyond-the-Standard-Model (BSM) scenarios are considered to solve these problems or realize
the grand unification. These scenarios have been tested by past researches, for instance by the conventional
direct searches conducted by the energy-frontier pp collision experiments at the Large Hadron Collider√(LHC). However, hints of new physics are not yet observed, with a wide range of results at s = 7, 8, 13 TeV
pp collisions at the LHC being consistent with the SM. The importance of a search for new physics witha completely new approach is increasing.
This thesis describes a search for new physics in a novel final state called a “photon-jet” using pp collisions√
at s = 13 TeV recorded with the ATLAS detector. A photon-jet is a group of collimated photons thatshare close trajectories (e.g. smaller than O(0.01) radians in the lab frame) [5–7]. Photon-jets canarise from a boson X with a mass of mX = O(100) GeV decaying into lighter bosons a with a mass of
ma = O(1) GeV that consecutively decays to photons, as shown in Figure 1. In such a case, the lighter
boson a is boosted with a Lorentz factor of γa = O(100), leading to collimated photons. At pp collisionsof the ATLAS experiment, an interesting feature of a photon-jet is that the collimated photons enter a small
area of the electromagnetic (EM) calorimeter (e.g. comparable to the Molière radius), typically leading
to a single EM shower cluster. Conventional searches have limited sensitivity to such final states; in thisresearch, the optimized event selection strategy, which utilizes the shower shapes in the EM calorimeter,
allows to search for these signatures with the available ATLAS pp collision dataset.
For pp collision experiments at the LHC, final states with high-energetic photons have two advantages
when searching for new physics. One is that requiring a final state containing high-energetic photons
reduces the number of background SM events significantly. Another is that the good energy and angularresolutions of the EM calorimeters of the ATLAS and CMS detectors can be used to effectively separate
signal photons and hadronic-jet backgrounds. A good example of these features is the search for SM
Higgs boson with the decay channel H → γγ: Despite the small SM branching ratio (0.2%) of the decay
H → γγ, this process was one of the main decay channel with which the Higgs boson was observed [8,9]. Thus, a search for photon-jet final states is an advantageous method to search for new physics.
The existence of photon-jets is well motivated by several BSM scenarios that predict the presence of two
or more new bosons [10–16]. The search is performed in a largely model-independent way, so that the
search is sensitive to these BSM scenarios in general. The BSM scenarios are described briefly below:

Figure 1: A schematic diagram of the process X → aa and a consecutively decaying into photons. X and a arehypothetical new bosons. When mX  ma , this process leads to a pair of photon-jets.

5• Supersymmetry provides a solution to the hierarchy problem, provides scenarios realizing the
grand unification, and provides candidates of dark matter [17]. One of the supersymmetric model,
the Next-to-Minimal Supersymmetric Standard Model (NMSSM), introduces several new Higgs
bosons [18]. This may result in the process H → aa → 4γ, where H is a new scalar Higgs
boson with a mass of mH = O(100) GeV, and a is a new pseudoscalar Higgs boson with a mass of
ma = O(1) GeV [10, 11]. This scenario is described in Section 2.3 in detail.
• The CP violation of strong interaction is experimentally not observed; this requires the CP-violatingterm to be small, leading to a fine tuning (“strong CP problem”). The Peccei-Quinn (PQ) mechanism
solves this problem by introducing a spontaneously-broken global U(1) symmetry, which leads tothe existence of a new pseudoscalar particle (“axion”) [3, 4]. One model that leads to photon-jets
is the “visible heavy QCD axion model” [12]. This introduces a mirror copy of the SM, and a Z2
symmetry between the SM sector and the copy sector, leading to a heavy axion (ma > O(0.1) GeV).
A complex scalar field is introduced to break the PQ symmetry spontaneously, resulting in a heavy
scalar boson and its decay into a pair of axions. For axions that are not so heavy (i.e. ma ∼ ΛQCD ,where ΛQCD is the QCD scale), the axions decay via mixing with the η and η 0 mesons, thus having
sizeable decay branching ratios into a → γγ and a → 3π 0 → 6γ. A similar model is implemented
with a model-independent effective field theory approach [13].
• A model incorporating a new gauge symmetry U(1)0 introduces a new vector boson Z 0 as a carrier.
The new U(1)0 symmetry may arise from a larger gauge group in grand unified theories [19].
The boson Z 0 can decay into photons either by Z 0 → sa → 4γ, Z 0 → aγ → 3γ [14], or by
Z 0 → sγ → 3γ [15], where a new scalar s and pseudoscalar a decay to pairs of photons, leading tophoton-jets.
• The Randall-Sundrum model solves the hierarchy problem with the introduction of an additional
dimension [20]. In this model, a Kaluza-Klein graviton may decay to a pair of radions [16]. In thiscase, the radion can have a sizeable decay to a pair of photons or a pair of neutral pions, leading toa photon-jet.
• A “hidden-valley-like model” [21] introduces two new real scalar fields, leading to the process
X → YY → 4π 0 → 8γ, where X and Y are new scalar bosons. When Y is considerably lighter thanX, this process leads to a pair of photon-jets.
This research is closely related to a search for an “axion-like-particle (ALP)”. Generally, a spontaneous
breaking of a global U(1) symmetry results in a massless Nambu-Goldstone boson. If the symmetry is
explicitly broken, this boson becomes massive; such a boson is called a pseudo-Nambu-Goldstone boson,and also commonly as an ALP. Because of an approximate shift symmetry, ALPs can be naturally lighter
than the electroweak scale (∼ 247 GeV). ALPs exist in several well-motivated BSM scenarios, including
the PQ mechanism discussed above. Experimental searches for ALPs have been performed by several past
researches, and they are not found yet. Constraints on couplings of ALPs and photons are placed from
eff is defined as [22]these researches, as shown in Figure 2. Here, the effective coefficient Cγγ
Γ(a → γγ) =4πα2 m3a eff 2
Cγγ
Λ2

(1)

where ma is the mass of the ALP, and Λ is the new physics scale for the suppression of the dimension-5
operators. The parameter space of ALPs is strongly constrained from e.g. axion helioscopes, cosmologicalC effobservations, and collider searches. However, the region ma & 0.1 GeV and | Λγγ | . 1 TeV−1 is weakly

6

Figure 2: Existing constraints from past observations on the ALP-photon coupling. Several of these bounds aremodel dependent. [22]constrained, as seen in Figure 2. This research can search for ALPs with smaller effective couplings, withthe sensitivity of this search strongly dependent on the production cross section of the parent particle X
in pp collisions, and the branching ratio of its decay to photons, e.g. B(X → aa → 4γ)1. This research
C eff

can search for ALPs with small effective couplings of as low as e.g. | Λγγ | & O(10−3 ) TeV−1 2 for the caseof γa ∼ 100, ma ∼ 1 GeV, and B(a → γγ) ∼ 1; this lower bound is derived from the limitation that the
ALPs need to decay before entering the ATLAS EM calorimeter, which has an inner radius of 1.5 m.
In the context of searches for a heavy (> O(100) GeV) resonant boson decaying into a final state with atleast three photons, e.g. X → aa → 4γ, there exist two related searches that have been performed in
the past, as shown in Figure 3. One of these searches is a general search for new phenomena in eventswith at least three isolated photons, performed by the ATLAS experiment at Run 1 for pp collisions at√
s = 8 TeV [23]. This search looked for a heavy boson decaying into a lighter bosons with a mass of
ma = O(10) GeV, and was sensitive to the BSM signal scenario X → aa → 4γ corresponding to theregion ma /mX & 0.08. Another search is a search for a scalar boson h decaying via h → aa → 4γ,√
performed by the ATLAS experiment at Run 1 for pp collisions at s = 7 TeV [24]. This search was
conducted before the discovery of the SM Higgs boson, assuming a SM Higgs-like boson with its massof 110 GeV < mh < 150 GeV and its decay to pseudoscalars with a mass of 100 MeV < ma < 400 MeV.
The search described in this thesis is sensitive to the region mX > 200 GeV and ma /mX < 0.01, which is

1Obviously, this search has no sensivity to ALPs if a heavy (> 200 GeV) boson that is produced by pp collisions and thathas a coupling to ALPs does not existq2This lower bound is proportional to γa B(a → γγ)/m3a .

7

Figure 3: Approximate sensitivity of existing searches for multiphoton final states for the BSM process X → aa →4γ. The lower blue region corresponds to the analysis described in this thesis. The upper green region corresponds
to the search for new phenomena with at least three isolated photons performed by ATLAS at Run 1 [23]. The grey
point corresponds to the search for a SM Higgs-like boson decaying into photon-jets performed by ATLAS at Run
1 [24].

a parameter region which has not been looked into before3.
This research shares some features with the search for a heavy (> 200 GeV) resonant boson decaying into a√pair of photons, performed by the ATLAS experiment at Run 2 for pp collisions at s = 13 TeV [25]. Thissearch looked for new physics in a final state leading to two high-energetic EM shower clusters in the EM
calorimeter, with stringent photon identification selections applied to the EM clusters using informationof shower shapes in the EM calorimeter. Because a photon-jet typically leads to a single EM cluster,
this diphoton resonance search has some limited sensitivity to photon-jet signal models; an approximate
sensitivity of this diphoton resonance search to the process X → aa → 4γ is shown in Figure 4. This
figure shows the case for a scalar boson X with a mass of mX = 800 GeV, has a narrow width, and that is
produced via the gluon–gluon fusion process (i.e. a coupling with a pair of gluons via a loop of quarks).
As seen in the figure, this diphoton resonance search loses sensitivity (i.e. upper limit on the productof the signal cross section and branching ratios) for larger values of ma , e.g. by an order of magnitude
for ma /mX > 0.003. This is because a larger value of ma leads to a smaller boost of a (i.e. smaller
γa ), leading to a wider EM shower in the EM calorimeter, thus being rejected by the stringent photon
identification selection. It is important to look into the parameter region with large values of ma in order
to search for new physics.
This thesis describes the result of the search for a heavy (> 200 GeV) resonance decaying into a pair√of photon-jets. It is performed using a dataset of pp collisions at s = 13 TeV collected with the
ATLAS detector at the LHC in 2015 and 2016, corresponding to an integrated luminosity of 36.7 fb−1 . A
3There exists an intermediate region, 0.01 < ma /mX . 0.08, which is covered by neither the three-photon search at 8 TeV
nor the search described in this thesis. The limitation here arises from the triggers for data collection. The search described in thisthesis uses the dataset collected with the diphoton trigger, as described in Section 5.1.1. For the BSM scenario X → aa → 4γ
in the region 0.01 < ma /mX . 0.08, the EM clusters of final-state photons will overlap and create wide EM clusters; these EM
clusters are typically rejected by the trigger-level photon selections.

8

Figure 4: An approximate 95% expected limit on the signal production cross section times branching ratios of
X → aa → 4γ, i.e. σX × B(X → aa) × B(a → γγ)2 . The scalar boson X is assumed to have a mass of
mX = 800 GeV, have a narrow width, and that it is produced via the gluon–gluon fusion process (i.e. a couplingwith a pair of gluons via a loop of quarks). The boson a is assumed to be spin-0. In principle, this approximationis performed based on the search result presented in the diphoton resonance search paper [25], with the selection
efficiency of the signal process X → aa → 4γ evaluated by a simulation. The details of the approximation method
is described in Section 5.8.3. The zigzag structure of the limit is due to the interpolation method used for thisapproximation; the signal selection efficiency is linearly interpolated between the mass values (mX , ma ) that thesimulated signal samples are generated.

photon-jet typically leads to a single EM shower cluster in the EM calorimeter, and thus each photon-jet
typically leads to one reconstructed photon. Candidate events of a heavy resonance decaying into a pairof photon-jets are selected from events with two high-momentum reconstructed photons, where eachreconstructed photon corresponds to a photon-jet in the case of signal events. The event selection and theevent categorization are optimized for the photon-jet events, utilizing the EM shower shape observed in
the EM calorimeter. The search is performed in a largely model-independent way, by searching for a newresonance in an invariant mass spectrum of two reconstructed photons with a smoothly-falling background.
To cover a wide range of values of mX and ma , several aspects of the search (e.g. reconstructed mass
shape of X and signal selection efficiency) are parametrized as functions of mX and ma . The shape of the
background component in the invariant mass of two reconstructed photons is evaluated with an analytic
function. An excess of events is searched for by performing an unbinned maximum-likelihood fit, withthe consideration of the signal and background components, to the invariant mass of two reconstructed
photons. This research is a new frontier of direct searches using a novel final state.
A brief review of the SM, supersymmetry, and the NMSSM is given in Section 2. This section also
describes the scenario of a new scalar Higgs boson decaying into a final state with photon-jets in the
NMSSM. The LHC and the ATLAS detector are described in Section 3. The reconstruction and theidentification of photons from pp collision data collected with the ATLAS detector are described in
Section 4. The details and the result of the search for a resonance decaying into a pair of photon-jets
are described in Section 5. The interpretation of the result in the context of the NMSSM for the process
H → aa → 4γ is described in Section 6.
The author’s contribution includes the development of the analysis strategy and the computation of the9results as described in Section 5, and the interpretation of the result in the context of NMSSM as describedin Section 6. The most important feature of this research is the development of the event selection strategy,
which was optimized using simulations of signal processes, and with a focus on large values of ma(0.003 . ma /mX < 0.01). The event selection strategy effectively utilizes the information of the shower
shape in the EM calorimeter, as described in Section 5.3.

10

2 Theoretical overview
2.1 The Standard Model of particle physics
The SM is an established theory that describes the properties of elementary particles and their interactions.
It describes three of the four fundamental forces of Nature (i.e. electromagnetic, weak, strong interactions
and excluding gravitational force) within the framework of the gauge symmetry group SU(3)C × SU(2) L ×
U(1)Y . The particle content of the SM is summarized in Tables 1 and 2. The gluon g, the weak bosons
W ± and Z, and the photon γ are the mediators of the strong, weak, and electromagnetic interactions.
The Higgs field is introduced as a scalar complex SU(2) doublet that causes the electroweak spontaneous
symmetry breaking through the Brout-Englert-Higgs mechanism, introducing masses to weak bosons andmassive fermions; this leads to the existence of a Higgs boson. With the discovery of the Higgs boson in2012 [8, 9] by the ATLAS and CMS experiments at the LHC, all of the particles have been discovered.
It is understood that the theoretical prediction of the SM is consistent with most of the various experimental
results and observations. Several unsolved problems that cannot be explained by the SM are known to exist,
including the hierarchy problem (explained in Section 2.2.1), the existence of the dark matter (explainedin Section 2.2.2), the strong CP problem, and the neutrino mass and its flavour oscillation.

2.2 Supersymmetry and the NMSSM
As described in Section 1, a resonance decaying into a pair of photon-jets can arise from the NMSSM.
The NMSSM is a minimal extension of the Minimal Supersymmetric Standard Model (MSSM), andso the MSSM must be introduced first. Supersymmetry is introduced in Section 2.2.1; the MSSM in

Table 1: The fermions (i.e. quarks and leptons) of the Standard Model. Q is the electric charge, T and T 3 are theweak isospin and its third component, Y is the weak hyper charge, and NC is the number of colour states.

Generations
 
 
t
c
u
b L
d L
s L
uR
cR
tR
d
s
b
 R
 R
 R
νe
νµ
ντ
e− L
µ− L
τ− L
eR
µR
τR


Quarks

Leptons





Q 
2/3
−1/3
2/3
−1/3


0
−1
−1

3

 T
1/2
−1/2
0
 0 
1/2
−1/2
0

T
1/2
0
0
1/2
0

Y

NC

1/3

3

4/3
−2/3

3
3

−1

0

−2

0

Table 2: The gauge bosons and the Higgs boson of the Standard Model. Q is the electric charge, T and T 3 are theweak isospin and its third component, Y is the weak hyper charge, and NC is the number of colour states.

Gauge bosons

Higgs boson

g
W±
Z
γ
h

Q
0
±1
0
0
0

11

T
0
1
0
0
1/2

T3
0
±1
0
0
−1/2

Y
0
0
0
0
1

NC
8
0
0
0
0

Section 2.2.2; the NMSSM in Section 2.2.3. Lastly,the scenario of a new scalar Higgs boson decaying
into a final state with photon-jets in the NMSSM is described in Section 2.3.
2.2.1 Supersymmetry“Supersymmetry” is defined as a symmetry relating fermions and bosons [17]. It is one of the BSM
scenarios that is considered to solve problems of the SM, especially the hierarchy problem.
In the SM, the potential of the electrically neutral part of the Higgs field is given as
V = m2H |H| 2 + λ|H| 4(2)qwhere H is a complex scalar field. The vacuum expectation value (VEV) is given by hHi = −m2H /2λ,which is known toqbe approximately 174 GeV from measurements of the weak interactions. The SM Higgs
mass is given by−2m2H and it is measured to be around 125 GeV, implying m2H = −(92.9 GeV)2 .

From a Dirac fermion f with mass m f and coupling −λ f H f¯ f , m2H receives a loop correction∆m2H = −|λ f | 2 2
Λ + ...
8π 2 UV(3)

where ΛUV is the ultraviolet momentum cutoff to regulate the loop integral. The “...” represents terms
proportional to m2f , which grow at most logarithmically with ΛUV .
From a hypothetical heavy complex scalar particle S with mass mS and coupling −λS |H| 2 |S| 2 , m2H receives
a loop correctionλS  2
2
Λ
−
2m
ln(Λ
/m
)
+
...
(4)
∆m2H =
UV
S
S
UV
16π 2
These corrections, given in Eqs. (3) and (4), are sensitive to ΛUV , which could be as high as the Planck scale.
Also, both corrections are sensitive to a possible existence of a new heavy particle. These sensitivities
lead to a possible unnatural large corrections to the Higgs potential, i.e. the hierarchy problem.
These corrections can be “neatly” cancelled by introducing supersymmetry. Because of the relative minus
sign in Eqs. (3) and (4), the corrections cancel out when each fermion has a complex scalar partner with
λS = |λ f | 2 .
The supersymmetry transformation operator Q is defined as
Q|Bosoni = |Fermioni,

Q|Fermioni = |Bosoni(5)

It follows anticommutation and commutation relationsµ{Q α, Q†β } = −2σαβ Pµ(6){Q α, Q β } = 0(7){Q†α, Q†β } = 0[Q α, P µ ] = 0[Q†α, P µ ] = 0

12

(8)
(9)
(10)

where P µ is the four-momentum generator of spacetime translations, and α, β are the spinor indices.
The irreducible representations of the supersymmetry algebra is called “supermultiplets”. A supermultiplet
is composed of fermion and boson states; these states are defined as “superpartners” of each other. A“chiral supermultiplet” is defined as a combination of a two-component Weyl fermion and a complex scalar
field. The “gauge supermultiplet” is defined as a combination of a spin-1 gauge boson and a gaugino(two-component spin-1/2 Weyl fermion with same gauge transformation properties as its partner).
In a supersymmetric model, “sparticles” (i.e. superpartners of SM particles and Higgs boson(s)) areintroduced. Considering that the sparticles are not experimentally found yet, their masses are assumed to beheavier than their superpartners. This leads to the assumption of a “soft” supersymmetry breaking. Here,“soft” means that the relationship λS = |λ f | 2 of the superpartners are maintained after the supersymmetry
breaking, to avoid the hierarchy problem. The soft supersymmetry breaking is triggered by an additionalterm Lsoft to the Lagrangian
L = LSUSY + Lsoft(11)
where LSUSY contains all gauge and Yukawa interactions in a way that preserves the supersymmetry
invariance. It can be broken down as
LSUSY = Lchiral + Lgauge + Lresidual(12)

where Lchiral is the chiral supermultiplet Lagrangian, Lgauge is the gauge supermultiplet Lagrangian, andLresidual is the residual terms.
The chiral supermultiplet Lagrangian Lchiral is given as
Lchiral = −∇µ φ∗i ∇µ φi + iψ †i σ̄ µ ∇µ ψi −

1  ijW ψi ψ j + Wi∗j ψ †i ψ †j − W i Wi∗2∂2W∂φi ∂φ j
∂W
Wi =
∂φi

Wij =

(13)

(14)
(15)

where φi is a complex scalar field, and ψi is a left-handed two-component Weyl fermion. i runs over all
gauge (i.e. colour and weak isospin) and flavour degrees of freedom of chiral supermultiplets. ∇µ is thecovariant derivative. W is the superpotential, defined as
1
1
W = L i φi + M i j φi φ j + y i jk φi φ j φk
2
6

(16)

where M i j is the symmetric mass matrix with a dimension of [mass], and y i jk is the dimension-less
Yukawa coupling that is symmetric under the interchange of i, j, k. L i are parameters with dimensions of[mass]2 which are allowed only if φi is a gauge singlet.
Denoting the largest mass scale of the soft terms Lsoft by msoft , the correction to the Higgs scalar squared
mass will be of the form
 0

λ
2
2
ln(ΛUV /msoft ) + ...
(17)∆mH = msoft
16π 2
where λ 0 is a schematic variable for various dimensionless couplings, and the “...” stands for terms
independent of ΛUV and higher loop corrections. Because the Higgs scalar squared mass should not be
unnaturally larger than the electroweak breaking scale, a condition that msoft is not much larger than the
TeV scale is derived.

13

Table 3: Chiral supermultiplets of the MSSM. The spin-0 fields are complex scalars, and the spin-1/2 fields areleft-handed two-component Weyl fermions. [17]Namesspin 0d˜L(νSU(3)C , SU(2) L, U(1)Y
3, 2, 61
3, 1, − 32
3, 1, 31

1, 2, − 21e0Hd

 (1, 1, 1) 
1, 2, + 21

1, 2, − 21spin 1/2


dL )

(u L

squarks, quarks

Q(×3 families)

ū

e
u∗R

u†R

d¯

deR∗

dR†

L

(e
ν e
eL )

(×3 families)ē

Higgs, higgsinos

Hu

e
e∗R

Hu+ Hu0

sleptons, leptons

Hd

ũ L



Hd0

Hd−

eL )
†
 eR 
eu+ H̃u0
H





e−
H
d



Table 4: Gauge supermultiplets of the MSSM. [17]Namesspin 1/2

spin 1

SU(3)C , SU(2) L, U(1)Ygluino, gluonegg(8, 1, 0)winos, W bosons
bino, B boson

e±
W

e0
W

W±

e0
B

B0

W0

(1, 3, 0)(1, 1, 0)2.2.2 MSSM
The MSSM [17] is the minimal construction of the supersymmetric model. The chiral supermultiplets of
the MSSM are given in Table 3. The gauge supermultiplets of the MSSM are given in Table 4. Contrary
to the SM, two Higgs doublets (and thus two Higgs chiral supermultiplets) are introduced, with weak
hypercharge Y = 1/2 and Y = −1/2, in order to avoid gauge anomalies. The chiral supermultiplet Hu
has the Yukawa couplings to up-type quarks, and Hd has the Yukawa couplings to down-type quarks andcharged leptons4. The details of MSSM are given in Section 2.2.2.
In the MSSM, the superpotential is given as (with the gauge (i.e. colour and weak isospin) and flavour
indices suppressed)¯ d QHd − ēye LHd + µHu Hd
WMSSM = ūyu QHu − dy(18)where only the scalar components of the objects Hu, Hd, Q, L, ū, d,¯ ē are considered. Here, yu, yd, ye are
the dimensionless Yukawa couplings in 3 × 3 matrix in family space. µ is the “µ term” with a dimensionof [mass].
The soft supersymmetry breaking terms in the MSSM is given as (with the gauge and flavour indices4This can be seen in Eq. (18) for the MSSM. This association of two Higgs doublets to quarks and leptons is necessary
because of two reasons: (1) The superpotential, defined in Eq. (16), has to be gauge invariant. (2) The superpotential has to beholomorphic in the complex fields φi by construction.

14

suppressed)
1
M3 g̃ g̃ + M2W̃ W̃ + M1 B̃ B̃ + c.c.
2˜¯ Q̃H − ē˜a L̃H + c.c.
− ū˜au Q̃Hu − da
d
e
d
d

MSSM
Lsoft
=−

(19)

†

˜¯ 2 d˜¯† − ē˜m2 ē˜†
− Q̃ m2Q Q̃ − L̃ † m2L L̃ − ū˜m2u ū˜† − dm
e
d
2
∗
2
∗
− mHu Hu Hu − mHd Hd Hd − (bHu Hd + c.c.)Here, M3, M2, M1 are the gaugino masses; m2Hu and m2Hd are the Higgs squared masses; m2Q, m2ū, m2d̄, m2 L, m2ēare the squark and slepton masses in 3 × 3 matrix in family space; au, ad, ae are the trilinear couplings incomplex 3×3 matrix in family space, with dimensions of [mass]; b is the scalar squared-mass parameter.
To suppress flavour-changing and CP-violating effects, the following limit is assumed (“soft supersymmetrybreaking universality hypothesis”) at a very high energy scale Q0 (“input scale”):2m2Q = mQ1,

m2ū = mū2 1,

m2d̄ = m2d̄ 1,

m2L = m2L 1,

au = At yu,

ad = Ab yd,

ae = Aτ yem2ē = mē2 1

Im(M1 ), Im(M2 ), Im(M3 ), Im(At ), Im(Ab ), Im(Aτ ) = 0

(20)
(21)
(22)

To suppress baryon number and lepton number violating effects, the conservation of the following symmetry is assumed (“R-parity”):
PR = (−1)3(B−L)+2s(23)
where B is the baryon number, L is the lepton number, and s is the spin. B = +1/3 is assigned to the chiralsupermultiplets Qi (i = 1, 2, 3), and B = −1/3 to ūi and d¯i . L = +1 is assigned to Li , and L = −1 to ēi .
This definition results in PR = +1 for SM particles and the Higgs bosons, and PR = −1 for sparticles.
When R-parity is exactly conserved, particles and sparticles have no mixing. Every interaction vertex
contains an even number of sparticles, and sparticles will be produced in even numbers. The lightest
sparticle (“LSP”) will be stable; each sparticle will eventually decay into a final state with odd number of
LSPs.
The VEV of neutral Higgs fields is defined asvu = hHu0 i,

vd = hHd0 i(24)v 2 = vu2 + vd2 ' (174 GeV)2(25)

tan β = vu /vd(26)

Variables v and tan β are defined as

Implications of MSSM
Supersymmetry is knwon to provide scenarios realizing the grand unification, and to provide candidatesof dark matter. They are described in this section in the context of the MSSM.
In the MSSM, from the calculation of the two-loop renormalization group equations, the quantities
α = g 2 /4π (where g is the gauge coupling) can unify at a scale MU ∼ 1.5 × 1016 GeV, as seen in Figure 5.
This is a strong hint to the grand unified theory at a scale below Planck scale.

15

Figure 5: Two-loop renormalization group evolution of the inverse gauge couplings, in the SM (dashed line) and the
MSSM (solid line) [17]. The normalization for the gauge coupling of U(1) is chosen to agree with the canonicalcovariant derivative for grand unification in SU(5) or SU(10). Here, Q is the renormalization group scale.

An electrically neutral LSP is an attractive candidate for non-baryonic dark matter. The neutralino (mixed
state of neutral higgsino, neutral wino, and bino) interacts only weakly with ordinary matter. The cold
dark matter density from cosmological observations is measured as [26]
ΩDM h2 ' 0.120(27)Here, ΩDM = ρDM /ρcrit , where ρDM is the average mass density of dark matter, and ρcrit is the criticaldensity leading to flat Universe. h ' 0.674 [26] is the Hubble constant in units of 100 km sec−1 Mpc−1 .
The LSP dark matter may explain the dark matter density ΩDM h2 ' 0.120 [27, 28].
2.2.3 NMSSM
One of the problems of the MSSM is the “µ problem”. In the MSSM, the Z boson mass is given as
m2Hd − m2Hu
2
mZ = p2− m2Hu − m2Hd − 2| µ| 2
1 − sin 2β2b
sin 2β = 2
mHu + m2Hd + 2| µ| 2

(28)
(29)

The parameters b, m2Hu , and m2Hd appear in the soft supersymmetry breaking Lagrangian, and so they are
expected to be around or below the order of msoft . As shown in Eq. (17), these terms should not be muchlarger than the TeV scale. In contrast, µ appears in the supersymmetry invariant Lagrangian, and it does
not have a constraint on its mass scale so far. Assuming that unnatural cancellations do not occur between
the terms in Eq. (28), a condition that µ should not be much larger than the TeV scale is derived. However,this condition that µ should be many orders of magnitude smaller than the Planck scale is “unnatural”. A
condition that the conceptually unrelated parameters, µ and msoft , should be roughly of the same order is“unnatural” as well. This µ problem necessitates a mechanism that relates µ to the soft supersymmetry
breaking.

16

The NMSSM [18] is one of the solutions to this µ problem. In the NMSSM, a new gauge-singlet chiralsupermultiplet is added to the MSSM, where S stands for both the new chiral supermultiplet and its complex
scalar component. In the simplest form of the NMSSM (“Z3 -invariant NMSSM”), the superpotential isextended as
1
WNMSSM = WMSSM + λSHu Hd + κS 3
(30)
3
with µ = 0 in WMSSM . Here, λ and κ are dimensionless Yukawa couplings. The soft supersymmetrybreaking Langrangian is extended as


1
3
NMSSM
MSSM
(31)
Lsoft
= Lsoft − λAλ SHu Hd − κ Aκ S + c.c.
3
MSSM . Here, A and A are the trilinear couplings with dimensions of [mass].
with b = 0 in Lsoftλκ

An effective µ term is given asµeff = λvs(32)

where vs = hSi is the VEV of S. The value of µeff is determined by vs , which is constrained by the otherdimensionless couplings and soft terms (see Eq. (34)). This mechanism relates µ to the soft supersymmetry
breaking, thus solving the µ problem in the NMSSM.
The Higgs potential is given as





2
2
2
VHiggs = λ Hu+ Hd− − Hu0 Hd0 + κS 2 + m2Hu + |λS| 2 Hu0 + Hu+ g2 + g2 

2
2
2
2
2
2 2
2
+ m2Hd + |λS| 2 Hd0 + Hd− + 1
Hu0 + Hu+ − Hd0 − Hd−
8


2


g2 + 0∗1
0 −∗ 2
2
2
+ −
0 0
3
H H + Hu Hd + mS |S| + λAλ Hu Hd − Hu Hd S + κ Aκ S + c.c.
+
2 u d
3

(33)

where g1 and g2 are the U(1)Y and SU(2) gauge couplings, respectively. From the minimisation conditions,the following conditions for the VEVs are obtained.
"#
2 + g2 g1
2
vu m2Hu + µ2eff + λ2 vd2 +vu2 − vd2 − vd µeff (Aλ + κvs ) = 0
4"#
2 + g2 g(34)
2vd2 − vu2 − vu µeff (Aλ + κvs ) = 0
vd m2Hd + µ2eff + λ2 vu2 + 1
4
h


i
vs mS2 + κ Aκ vs + 2κ 2 vs2 + λ2 vu2 + vd2 − 2λκvu vd − λvu vd Aλ = 0Hereafter, µeff is denoted by µ for simplicity.
NMSSM Higgs sector
Given m2Z = 21 (g12 + g22 )(vu2 + vd2 ), the NMSSM Higgs sector has six independent parameters. In this thesis,
they are chosen to be
λ, κ, Aλ, Aκ , tan β, µ(35)
The MSSM has three electrically neutral Higgs bosons: two scalar and one pseudoscalar. In contrast,
the introduction of a new gauge-singlet complex scalar field in the NMSSM leads to five neutral Higgs

17

bosons: three scalar and two pseudoscalar. Below, the mass matrices of the neutral Higgs bosons andhiggsinos are given, because they are necessary for the discussions in Section 2.3.
The mass matrix of neutral scalar Higgs bosons is given below. Here, the basis is chosen as (HSM =cos βHdR + sin βHuR, HD = − sin βHdR + cos βHuR, HS = SR ); the basis of the doublet sector is rotatedby an angle β because the upper left 2 × 2 submatrix will be approximately diagonal (i.e. the lightest
scalar will be dominantly HSM and SM-like) in the case of MHD  v.

m2Z cos2 β + λ2 v 2 sin2 2β©
­­MS2 = ­ (λ2 v 2 − m2Z ) sin 2β cos 2β
­

2
­
M A sin 2β2λµv 1 −2µ«(λ2 v 2 − m2Z ) sin 2β cos 2β2λµv 1 −

M A sin 2β
2µ

2

2
MH
D

2
− λv
2µ M A sin 2β cos 2β2− λv
2µ M A sin 2β cos 2β

2
MH
S

2
MH
= M A2 + (m2Z − λ2 v 2 ) sin2 2β
D
!

2 

λM A2
2κ
λAκ
κλ 2
2
MHS =
µ
1+
−
v sin 2β 1 −
λ
4κ µ
2
κ µ2

2µ Aλ + λκ µ
M A2 =
sin 2β

ª
®
®
®
®
®
¬

(36)

(37)
(38)
(39)

The mass matrix of pseudoscalar Higgs bosons is given below. Here, the basis is chosen as (AD =cos βHuI + sin βHdI , AS = SI ); the doublet sector is rotated by an angle β to neglect the masslessGoldstone mode.


2κ
2
µ
v ª
M
λ
A
−
λ
©
λ
A

MP2 = ­ 
(40)
®
2
µ
v
M
λ Aλ − 2κλAS«¬
M A2 S = λ2



 2
4κ
v sin 2β 3κ
Aλ +
µ
−
Aκ µ
λ
2µ
λ

(41)

These mass matrices lead to five neutral Higgs bosons, namely
H1, H2, H3, a, A

(42)

where Hi (i = 1, 2, 3) are the scalar bosons in the order of their masses, and a, A are the pseudoscalarbosons in the order of their masses.
The mass matrix of scalar Higgs bosons MS2 in the basis Hiweak = (HdR, HuR, SR ) is diagonalized by anorthogonal 3 × 3 matrix Si j which rotates Hiweak , namely
Hi = Si j H jweak(43)

The mass eigenstates of pseudoscalar Higgs bosons are given by a 2×3 matrix Pi j in the basis (HdI , HuI , SI )
as
 
 H

a
P11 P12 P13 © dI ª
(44)=
­ HuI ®
A
P21 P22 P23« SI ¬

18

The neutralino mass matrix in the basis (−iλ1, −iλ23, ψd0 , ψu0, ψS ) is given as
M1©
­­
M0 = ­­­­«

0 − g√1 vd
2
M2 g√2 vd
2
0

g1 vu
√
2
g2 vu
− √
2
µ

0

0
ª
0 ®®
−λvu ®®
®
−λvd ®
2κ
λ µ ¬

(45)

where λ1 is the U(1)Y gaugino, λ2i (i = 1, 2, 3) are the SU(2) gauginos, and ψd0 , ψu0, ψS are the neutral
higgsinos.
The charged SU(2) gauginos are λ− = √1 (λ21 + iλ22 ) and λ+ = √1 (λ21 − iλ22 ), and the charged higgsinos are22
ψu+ and ψd− . Defining ψ + = (−iλ+, ψu+ ) and ψ − = (−iλ−, ψd− ), the mass terms in the Lagrangian is givenas + 
1 + −  0 XTψ
L = − ψ ,ψ+ c.c.
X 0
ψ−
2


M2
g2 vu
X=
g2 vd µ(46)

The 2 × 2 diagonalization matrices U and V into the mass eigenstates χ− , χ+ are given asχi− = Ui j ψ −j ,χi+ = Vi j ψ +j

(47)

Considering renormalisation group equations, theoretical constraints on the values of tan β are obtained,that are applicable both to the MSSM and the NMSSM. The relationships between the Yukawa couplings
and the quark masses are given as
yt =

mt
,
v sin βyb =mbv cos β(48)

where yt , yb are the Yukawa couplings,√ and mt , mb are the masses of the top and bottom quarks. If tan β islarge, yt may become too large (& 4π) at the grand unification scale, resulting in a Landau singularity;
this imposes an approximate lower bound of tan β & 1. Similarly, considering yb , an upper bound oftan β . 80 is obtained.
The Higgs sector of the NMSSM is constrained by direct searches for new Higgs bosons, as well as themeasurements of the 125 GeV Higgs boson, that are performed by the ATLAS and CMS experiments.
The summary of the constraints, interpreted in the hMSSM scenario [29], is shown in Figure 6. The
hMSSM scenario is a MSSM scenario in which the lightest scalar Higgs boson is constrained to 125 GeV,
thus constraining the dominant radiative corrections of the Higgs sector and approximately reducing thenumber of free parameters. The constraints in Figure 6 can be naively applied to the NMSSM, leading
to an approximate constraint of M A & 500 GeV and tan β . 10–80. However, it must be noted that theconstraints in the hMSSM scenario are not directly applicable to the NMSSM. This is because, due to theintroduction of an additional gauge-singlet field S that lead to two new mass eigenstates, the Higgs sectorof the NMSSM is more complicated compared to the MSSM. It is necessary for the constraints from thepast researches to be considered quantitatively by numerical calculations.

19tan βOctober 2018
H/A→ττ
s = 13 TeV, 36.1 fb-1

40
30

JHEP 01 (2018) 055+

H → τν

s = 13 TeV, 36.1 fb-1

JHEP 09 (2018) 139+

H → tbs = 13 TeV, 36.1 fb-1

20

arXiv:1808.03599 [hep-ex]

H→ ZZ→ 4l/llν ν

s = 13 TeV, 36.1 fb-1Eur. Phys. J. C (2018) 78: 293

gg→ A→ Zh

10s = 13 TeV, 36.1 fb-1

JHEP 03 (2018) 174

H→ WW→ lν lν

ATLAS Preliminarys = 13 TeV, 36.1 fb-1
Eur. Phys. J. C 78 (2018) 24

hMSSM, 95% CL limits

43

H→ hh→ 4b,→ bb γ γ / ττ,→ WWγ γObserved
Expecteds = 8 TeV, 20.3 fb-1

Phys. Rev. D92, 092004 (2015)

H→ hh → bb γ γ2

s = 13 TeV, 3.2 fb-1
ATLAS-CONF-2016-004

h couplings [κ V, κ u, κ d]1
200

s = 13 TeV, 36.1 - 79.8 fb
ATLAS-CONF-2018-031

300 400

1000

-1

2000
mA [GeV]Figure 6: Regions of the (M A, tan β) plane in the hMSSM model that are excluded by direct searches for BSM Higgs
and the measurements of SM Higgs cross section, at 95% CL. [30]0∆m(χ̃±
1 , χ̃1) [GeV]March 201850

0
2` compressed, arXiv:1712.08119, m(χ̃02 ) = m(χ̃01 ) + 2∆m(χ̃±
1 , χ̃1 )Disappearing track, PHYS-PUB-2017-019, m(χ̃02 ) = m(χ̃01 )LEP2 χ̃±
1 excluded

20

Theoretical prediction for pure Higgsino

10
5
ATLAS Preliminary

2

√

1

All limits at 95% CL
Observed limits
Expected limits

0.5
0.2
80

s = 13 TeV, 36.1 fb−1

± 0
0 0
+ −
pp → χ̃02 χ̃±
1 , χ̃2 χ̃1 , χ̃1 χ̃1 , χ̃1 χ̃1 (Higgsino)

100

120

140

160

180

200

m(χ̃±
1 ) [GeV]
Figure 7: Exclusion of the χ̃10 and χ̃1± masses at 95% CL for the higgsino pair production with SM-boson-mediateddecays. [31]20The value of µ is constrained from searches for higgsino pair production by collider experiments (ATLAS
and LEP), as shown in Figure 7. An approximate limit of µ & 100 GeV is obtained, because the chargino
mass has a strong dependence on µ as seen in Eq. (46).
Other four parameters of the NMSSM Higgs sector, i.e. the dimensionless coupling constants λ, κ and thetrilinear coupling constants Aλ, Aκ , are weakly constrained from past researches. A subset of the regionof these four parameters can be probed by the search described in this thesis, as described in the next
section.

2.3 Photon-jets from NMSSM
In the NMSSM, the decay of a heavy scalar Higgs boson (either H2 or H3 , collectively denoted by H)
decaying into aa pair can result in a pair of photon-jets [10, 11]. The process of H produced by gluon–
gluon fusion process (gg → H) and decaying via H → aa → 4γ is schematically shown in Figure 8. Thissection describes the parameter space of the NMSSM in which a heavy scalar Higgs boson may decay
into a pair of photon-jets.
2.3.1 Light state a
When the state a is mostly singlet-like (i.e. a ' SI ), this state can be light (ma = O(1) GeV). From
Eq. (41), ma = O(1) GeV corresponds to Aκ . O(1) GeV.
A light (ma . O(1) GeV) a naturally arise in the “R-symmetry limit” λAλ, κ Aκ → 0 [32]. The R-symmetry
is a global U(1)R symmetry that exists in several well-motivated supersymmetry breaking models called
the O’Raifeartaigh models [33]. The R-symmetry is spontaneously broken by the VEVs vu, vd, vs andexplicitly broken by the gaugino mass terms, leading to the existence of an ALP (“R-axion”). In the
R-symmetry limit λAλ, κ Aκ → 0, the pseudoscalar Higgs boson a becomes the R-axion.
The pseudoscalar singlet field does not have couplings to SM fermions in the Lagrangian. The pseudoscalara has couplings to SM fermions from its doublet component (AD = cos βHuI +sin βHdI ). Naively thinking,

Figure 8: A schematic diagram of a heavy Higgs boson decaying via H → aa → 4γ. H is produced via thegluon–gluon fusion process, i.e. via a loop of quarks (mainly top quark). a decays into a pair of photons via a loopof charged higgsinos.

21

(a)

(b)

Figure 9: The branching ratios of the light pseudoscalar Higgs boson a as a function of the mixing angle θ A. The
parameters assumed here are tan β = 10, λ = 1, µ = 150 GeV, and M2 = 500 GeV. They are presented for the cases(a) ma = 0.1 GeV and (b) ma = 5 GeV. [34]a decays dominantly to a pair of SM fermions or gluons. There are two possibilities for the pseudoscalara to have a sizeable B(a → γγ) (e.g. more than 30%), as explained below.
Case 1: Decoupling limit
B(a → γγ) can be large when a is a nearly pure singlet state (“decoupling limit”). In such a case, thebranching ratio of a into a pair of SM fermions or gluons will be considerably small; a decays dominantly
into γγ via a loop of charged higgsinos [34]. The mixing angle θ A is defined by
 



cos θ A − sin θ A
a
SI
=
(49)
A
sin θ A cos θ A
cos β · HuI + sin β · HdI
Figure 9 shows the typical case of the relationship between the mixing angle θ A and the branching ratios
of a. The mass matrix of pseudoscalar Higgs bosons is given in Eq. (40). At tree level, the decoupling
limit of the pseudoscalar Higgs bosons is achieved with the following condition:µv
2λ Aλ − 2κ
λ
1
(50)
tan 2θ A =M A2 − M A2 S
In the decoupling limit, it is possible for a to be “long lived”, i.e. decaying outside of the beam pipe at the
ATLAS detector, because of its small decay width. The partial decay width is given as [35] 
2 m3 ÕÕ
G F αem
a
2 a
a
±
g
A
τχ̃i±
Γ(a → γγ) =
N
e
g
A
(τ
)
+
√
c f f f fχ̃i± χ̃i
32 2π 3
±
f
χ̃i

2
1
Ax (τx ) = τx sin−1 √
τx
τx =

4m2x
m2a

(x = f , χ̃i± )

22

2

(51)

(52)
(53)

Here, Nc is the QCD color factor, e f is the electric charge of the fermions, and gxa is the couplings of awith the fermions. The coupling constant is given as
i 
g χ̃† χ̃∓ A1 = √ λP13Ui2Vj2 − g2 P12Ui1Vj2 + P11Ui2Vj1
i j
2

(54)

In the decoupling limit, g af = 0, P13 = 1, and P12 = P11 = 0. Thus, the partial decay width is typicallyΓ(a → γγ) ∼ O(1) meV for λ = O(0.01), ma = O(1) GeV, m χ˜± = O(100) GeV. In the decoupling limit,ithe decay width of a is approximately equal to the partial decay width to photons, i.e. Γa ' Γ(a → γγ).
For mH = O(100) GeV and ma = O(1) GeV, the Lorentz factor of a is γa = O(100). This leads to an
average decay length at the lab frame of l = γa βa c/Γa ' γa βa c/Γ(a → γγ) ' O(1) cm. Here, βa is theratio of the velocity of the a and the speed of light. Thus, in the decoupling limit, it is possible for a to belong lived.
An upper limit on the BSM decay branching ratio of 125 GeV Higgs is ∼ 20% at 95% confidence level(CL), evaluated by performing global fits to experimental results from Tevatron and LHC Run 1 [36].
With this taken into consideration, a limit of roughly θ A . 1% is placed when ma < 125
2 GeV (i.e. thedecay h → aa is kinematically allowed) [37]. This experimental limit can be mitigated when taking thedecoupling limit, i.e. θ A → 0. However, in the decoupling limit P13 = 1 and P11 = P12 = 0, the triple
Higgs coupling for H1 and a is given as
!
√ 2
λ2
λκ
2κ
κ Aκ
gH1 aa ' √ (vd S11 + vu S12 ) + √ (vd S12 + vu S11 ) +(55)µ − √ S13λ222This places a limit of λ . O(0.1) [37].
Experimental constraints on ALPs need to be considered. The constraints from past observations on the
ALP-photon coupling is shown in Figure 2. For the case of the decoupling limit in Case 1, the value of theeff |/Λ = O(0.01) TeV−1 for Γ(a → γγ) = O(1) meV and m = O(1) GeV.
effective coeffcient is typically |Cγγaeff |/Λ < O(1) TeV−1 , forThis is smaller by two orders of magnitude than the upper limit, i.e. |Cγγ
ma = O(1) GeV. Thus, in the decoupling limit mentioned above, the limit from ALPs searches are
mitigated.
Case 2: Mixing with pseudoscalar meson, or dimuon loop
B(a → γγ) can be large when ma is within a limited region, namely near the masses of either a neutralpion, two muons, or an eta meson [10, 11, 38]. The branching ratios of a for a typical case is shown in
Figure 10. When ma is close to a mass of a neutral pion, it can have a large mixing with a neutral pionand then decay via a → γγ. When ma is close to a mass of an eta meson, it can have a large mixing
with an eta meson and then decay via a → γγ and a → 3π 0 → 6γ. When ma is right below the dimuon
threshold, a decays via a → γγ by a muon loop. As seen in Figure 10, the value of ma has to be in close
proximity to these values for large B(a → γγ), e.g. within O(100) MeV for B(a → γγ) > 30%. This
scenario is out of scope of this thesis, and it is not discussed further in this thesis.
2.3.2 Heavy state H√
For the process gg → H → aa to have a sizeable (e.g. around the order of O(1 fb)) cross section in s =13 TeV pp collisions, the state H has to be a mixed state of both the doublet (HD = − sin βHdR +cos βHuR )and singlet (HS = SR ) components. This is explained by the following reasons. The scalar singlet field23

Figure 10: The decay branching ratios of a (denoted by A1 ) in the NMSSM for sin θ A = 0.03 and tan β = 10. [38]does not have couplings to SM fermions in the Lagrangian. Thus, the state H has to have a large
contribution of the doublet component in order to have a sizeable coupling to the top quark. However,a doublet-like H decays dominantly into SM fermions. This is because of its weak coupling to a, i.e.
Γ(HD → t t¯, bb̄)  Γ(HD → aa). In contrast, the coupling of a singlet-like H to aa can be large, i.e.
Γ(HS → aa) ∼ Γ(HD → t t¯, bb̄). Thus, gg → H → aa has a sizeable cross section when H is a mixed
state of both the singlet and doublet components.
A large mixture of singlet and doublet components for H is realized when MHD ∼ MHS , as shownin Eq. (36). From Eqs. (37) and (38), this leads to M A ∼ 2κλ µ, with an assumption that Aκ is small(Aκ < O(1) GeV) and tan β = O(1). Considering the definition of M A as given in Eq. (39), theseassumptions lead to:
Aλ +κ
2κ 2 sin 2β
µ∼
µ
λ
λ2

(56)

In conclusion, the search described in this thesis can probe the parameter space of the NMSSM Higgs
sector, i.e. (tan β, µ, λ, κ, Aλ, Aκ ), that satisfy the relationships in Eqs. (50) and (56).

24

3 ATLAS detector at the LHC
3.1 Large Hadron Collider
The LHC [39] is an energy-frontier experiment that enables direct searches for new physics at the TeV
scale. It is a hadron collider with superconducting magnets installed in two parallel beamlines. It is placedin a tunnel with 27 km circumference and a depth of down to 175 m. Its design is to collide proton beams√
at s = 14 TeV and an instantaneous luminosity of 1034 cm−2 s−1 , and also to collide heavy ions (e.g.
Pb). It is designed to accelerate 2808 beam bunches with 1011 protons per bunch. 1232 dipole magnets
are installed that operate with 8 T magnetic field. Seven detectors are installed at four crossing points,
including the two general-purpose ATLAS [40] and CMS [41] detectors.
√
During the operation from 2009 to early 2013 (“Run 1”), the LHC performed pp collisions at s = 7, 8 TeV;√
the dataset that the ATLAS detector recorded corresponds to 5 fb−1 for s = 7 TeV and 20 fb−1 for√s = 8 TeV after applying data-quality requirements. During the operation from 2015 to 2018 (“Run√2”), the LHC performed pp collisions at s = 13 TeV; the dataset that the ATLAS detector recorded
corresponds to 140 fb−1 after applying data-quality requirements.
Before being injected to the LHC, the proton beams are accelerated with a series of accelerators, as shownin Figure 11. The initial proton beam at 50 MeV is generated with the LINAC 2; it is accelerated to
1.4 GeV with the Proton Synchrotron Booster; it is accelerated to 26 GeV with the Proton Synchrotron; it
is accelerated to 450 GeV with the Super Proton Synchrotron, then injected into the LHC main ring.
The CERN accelerator complex
Complexe des accélérateurs du CERNCMSNorth Area

LHC

GIF++

2008 (27 km)

CENF

ALICE

2015

LHCb

2016

TT20

TT40

TT41

SPS
1976 (7 km)

TI2

TT10

TI8

AWAKE

ATLAS

HiRadMat
2011

2016

TT66

TT2

AD

ELENA

1999 (182 m)

2016 (31 m)

ISOLDEBOOSTER

1992

1972 (157 m)
RIBs
p

p

2001

IRRAD/CHARM

1959 (628 m)

p

CLEAR

LINAC 2e-

LEIR

LINAC 3Ions

ions

East AreaPSnp (protons)

REX/HIE
2001/2015

n-ToF

2017

2005 (78 m)

RIBs (Radioactive Ion Beams)n (neutrons)– (antiprotons)pe- (electrons)

LHC - Large Hadron Collider // SPS - Super Proton Synchrotron // PS - Proton Synchrotron // AD - Antiproton Decelerator // CLEAR - CERN Linear
Electron Accelerator for Research // AWAKE - Advanced WAKefield Experiment // ISOLDE - Isotope Separator OnLine // REX/HIE - Radioactive
EXperiment/High Intensity and Energy ISOLDE // LEIR - Low Energy Ion Ring // LINAC - LINear ACcelerator // n-ToF - Neutrons Time Of Flight //HiRadMat - High-Radiation to Materials // CHARM - Cern High energy AcceleRator Mixed field facility // IRRAD - proton IRRADiation facility //GIF++ - Gamma Irradiation Facility // CENF - CErn Neutrino platForm

Figure 11: A schematic diagram of the accelerator complex at CERN including the LHC. [42]25Figure 12: Schematic diagram of the ATLAS detector. [40]3.2 ATLAS detector
The ATLAS (A Toroidal LHC ApparatuS) detector[40] is a general-purpose detector with a forward–
backward symmetric cylindrical geometry, as shown in Figure 12. The detector covers nearly the entire
solid angle around the collision point. The detector is 25 m in height and 44 m in length, and has an
overall weight of approximately 7000 tones. The inner tracking detector is surrounded by a 2 T thinsuperconducting solenoid magnet. The calorimeter system comprises EM and hadronic calorimeters, and
it is placed outside of the solenoid magnet. The muon spectrometer is covered by the three large air-coresuperconducting toroid magnets (one barrel and two end-caps).
ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point in the centreof the detector. The z axis is defined as the beam direction. The x axis is defined as pointing from theinteraction point to the centre of the LHC ring. The y axis is defined as pointing upwards. The azimuthal
angle φ is defined around the beam axis. The polar angle θ is defined as the angle with respect to the beamaxis. These angles are defined in radians.
p The pseudorapidity η is defined as η = − ln tan(θ/2). Angulardistance is measured in units of ∆R = (∆η)2 + (∆φ)2 5. The transverse momentum pT and the transverse
energy ET are defined in the x–y plane6.
3.2.1 Inner Detector
The inner tracking system (Inner Detector) is composed of the Pixel detector, the Semiconductor Tracker(SCT), and the Transition Radiation Tracker (TRT), as shown in Figure 13. The system is immersed in a
2 T axial magnetic field generated by the central solenoid magnet. The system provides charged-particle
tracking in the range |η| < 2.5.
The Pixel detector and the SCT cover the region |η| < 2.5. The Pixel detector uses pixel-shaped oxygenenriched n-in-n silicon semiconductor as its sensors. They are located on concentric cylinders around the
5Note that by this definition, ∆R is Lorentz invariant for a massless particle.
6The use of transverse momentum is convenient for hadron collider experiments, because those of the final state particles
arising from a parton–parton collision add up to zero, whereas the longitudinal components will not.

26

Figure 13: Schematic diagram of the inner detector. [40]beam axis in the barrel region, and on disks perpendicular to the beam axis in the end-cap regions. The
innermost barrel layer of the Pixel detector, the insertable B-layer (IBL) [43], was installed between Run
1 and Run 2. The Pixel detector has four cylinder layers in the barrel at a radius of 31–123 mm, and three
disk layers in the end-cap. The Pixel detector typically provides four measurements per track. The size ofthe pixel sensors is 50 × 250 µm2 in the IBL and 50 × 400 µm2 minimum in the other layers. The intrinsic
accuracy in the barrel (end-cap) region is 4 µm in the R − φ direction and 115 µm in the z (R) direction.
The SCT uses microstrip-shaped single-sided p-on-n silicon semiconductor as its sensors. The SCT
typically provides four two-dimensional measurement points per track using eight strip layers. The strips
have 40 mrad stereo angle and a pitch of 80 µm. The intrinsic accuracy per module in the barrel (end-cap)
region is 17 µm in the R − φ direction and 580 µm in the z (R) direction.
The TRT is a gaseous detector composed of 4 mm-diameter straw tubes confined with xenon-based activegas. A gold-plated tungsten wire of 31 µm diameter is placed in the centre of each tube and acts as theanode to collect ionized secondary electrons. The spaces between the straws are filled with polypropylene
fibres in the barrel region, and polypropylene foils in the end-cap region, to create transition radiation. The
transition radiation is useful for particle identification, especially the separation of electrons and charged
pions, because the energy of the transition radiation is proportional to the Lorentz factor γ = E/m ofthe particle. The TRT provides radially extended track reconstruction up to |η| = 2.0. It provides R − φinformation with an intrinsic accuracy of 130 µm per straw tubes. It also provides electron identification
information based on the fraction of hits (typically 30 in total) above a higher energy-deposit thresholdcorresponding to transition radiation.
The cumulative amount of material in the Inner Detector is shown in Figure 14. Here, X0 is the radiation
length.

27

Figure 14: The cumulative amount of material in the Inner Detector, including the services and thermal enclosures,in units of X0 and as a function of |η|. [43]Figure 15: Schematic diagram of the calorimeter system. [40]3.2.2 Calorimeter
The calorimeter system is composed of the EM calorimeter and the hadronic calorimeters that use different
technologies, as shown in Figure 15. The calorimeter system covers the range |η| < 4.9.
EM calorimeter
The EM calorimeter is described in detail below, because it is the most important component for the search
described in this thesis.
The EM calorimeter is a sampling calorimeter with lead absorbers and liquid argon (LAr) active medium.
It has kapton electrodes with accordion shape, which provides complete φ symmetry without cracks.
The Molière radius of the EM calorimeter is approximately 1.6 cm. The EM calorimeter is divided into
the barrel section covering |η| < 1.475 and the two end-cap sections covering 1.375 < |η| < 3.2. The
end-cap is further divided into the outer wheel region 1.375 < |η| < 2.5 and the inner wheel region
2.5 < |η| < 3.2.

28

Figure 16: Schematic diagram of the EM calorimeter at η = 0. [44]For |η| < 2.5 (i.e. the barrel and outer wheel regions), the EM calorimeter is composed of three samplinglayers in the longitudinal direction of shower depth, as shown in Figure 16. The thicknesses of the first,
second, and third layers at η = 0 are 4.3 X0 , 16 X0 , and 2 X0 , respectively. The thicknesses vary withthe |η| range, as shown in Figure 17. The first layer is segmented into high-granularity strips in the η
direction, with a typical cell size of ∆η × ∆φ = 0.003 × 0.1 for the ranges |η| < 1.4 and 1.5 < |η| < 2.4,
and a coarser cell size of ∆η × ∆φ = 0.025 × 0.1 for other regions. The second layer has a cell size of∆η ×∆φ = 0.025×0.025. This layer collects most of the energy deposited in the calorimeter by photon andelectron showers. The third layer is used to correct for energy leakage beyond the EM calorimeter fromhigh-energy showers. Placed in front of these layers, an additional thin LAr presampler layer covering|η| < 1.8 is used to correct for energy loss in material upstream of the calorimeters. As describedabove, the first sampling layer has a fine granularity in η; this allows the identification of events with twooverlapping showers originating from the decays of neutral hadrons in hadronic jets, mostly π 0 → γγ
decays. This is shown in Figure 18.
For 2.5 < |η| < 3.2 (i.e. the end-cap inner wheel region), the EM calorimeter is segmented in two sectionsin the longitudinal direction, and has a coarser lateral granularity.
√
The design goal of the energy resolution is σE /E = 10%/ E ⊕ 0.7% (where E is in units of GeV), withsimilar results obtained from measurements using electron test beams and pp collisions.
Hadronic calorimeters
Hadronic calorimeter system is composed of the Tile calorimeter, the Hadronic End-cap Calorimeter,(HEC) and the Forward Calorimeter (FCal).
The Tile calorimeter is a sampling calorimeter with steel absorbers and plastic scintillating tiles as the
sampling medium. The scintillating tiles are read out through wavelength shifting fibres and photomultiplier tubes. It is composed of the barrel region covering |η| < 1.0 and the extended barrel region

29

Figure 17: The cumulative amount of material in the EM calorimeter, and also in front of the first layer, in units of
X0 as a function of |η|. [40]Figure 18: An event display of a candidate event of a neutral pion decaying into two photons. It is shown in the R − zplane, with the yellow boxes indicating the size of the recorded energy in each cell. The four layers each correspondto the presampler, first, second, and third layers of the EM calorimeter, from bottom to top. [45]0.8 < |η| < 1.7. They are segmented in depth into three layers, with the thicknesses of approximately 1.5
interaction lengths (λ), 4.1 λ, and 1.8 λ respectively for the barrel region. The thicknesses are 1.5 λ, 2.6
λ, and 3.3 λ respectively for the extended barrel region.
The HEC is a sampling calorimeter with copper absorbers and LAr active medium. It covers the region
1.5 < |η| < 3.2. It is composed of four layers.
The FCal is divided into three modules in depth. The first module is a sampling calorimeter with copper
absorbers and LAr active medium that is optimized for EM measurements. The second and third modules
are sampling calorimeters with tungsten absorbers and LAr active medium that are optimized for hadronic
measurements. The total depth of the modules is approximately 10 λ.
3.2.3 Muon spectrometer
The magnetic bending is provided by the barrel toroid in the range |η| < 1.4, and by the two end-cap
toroids in the range 1.6 < |η| < 2.7. In the transition region 1.4 < |η| < 1.6, the magnetic bending
is provided by the combination of the barrel and end-cap toroids. The three toroids are each consisted30

Figure 19: Schematic diagram of the muon spectrometer. [40]of eight coils, assembled radially and symmetrically around the beam axis, to provide a magnetic fieldin the azimuthal direction. The field integral of the magnetic field perpendicular to the trajectory of an
infinite-momentum muon (i.e. straight trajectory) is in the range 1.5–5.5 Tm for |η| < 1.4, and in therange 1–7.5 Tm for 1.6 < |η| < 2.7.
The muon spectrometer is composed of four different detectors, as shown in Figure 19. The Monitored
Drift Tube (MDT) is a gaseous drift chamber filled with a mixed gas of Ar (93%) and CO2 (7%), andprovides a precise measurement of the muon track coordinates in the bending direction. The relative
position and deformations of the MDT chambers are monitored by precise alignment sensors based on
optical monitoring. The Cathode Strip Chamber (CSC) is a multiwire proportional chamber filled witha mixed gas of Ar (80%) and CO2 (20%) and uses strip-shaped cathodes. It is installed in the region
2 < |η| < 2.7 to withstand the high rate. The trigger chambers consist of the Resistive Plate Chamber(RPC) in the barrel region and the Thin Gap Chamber (TGC) in the end-cap region. The RPC is a gaseous
detector filled with a mixed gas (mainly C2 H2 F4 ). The TGC is a multi-wire proportional chamber filledwith a mixed gas of CO2 and n-pentan, and has small distances between the anode wires and the read-out
cathode strips of 1.4 mm. These trigger chambers provide the bunch-crossing identification and muontrack coordinates orthogonal to those of the MDT and CSC.
3.2.4 Luminosity detector
LUCID-2 (Luminosity Cherenkov Integrating Detector) [46] measures the luminosity delivered to the
ATLAS detector. It is placed in the forward region at ±17 m from the interaction point, and it measures
the inelastic pp scattering. It uses quartz as the Cherenkov radiation medium, and measures the radiatedlight with photomultiplier tubes. It was installed between Run 1 and Run 2 to replace the previous LUCID,
to cope with the increase in energy of the proton beams and the higher luminosity.
The measurement of the instantaneous luminosity relies on the detection of the pp inelastic scattering.
The measurement is performed with several methods, and the “hit-counting” method using LUCID-2 is31

described below. The “visible number of interactions” µvis is defined as the product of the efficiency ofthe measurement method, detector acceptance, and the number of interactions µ. Using the 8 PMTs (4 on+z side and 4 on −z side), µvis is evaluated from a logarithmic formula based on NHIT /NPMT , where NHIT
is the number of hits per bunch crossing, and NPMT is the number of PMTs. The logarithmic formula isbased on an assumption that the number of interactions follow a Poissonian distribution (e.g. assumingthat the probability to observe no hit in a PMT is e−µεPMT , where εPMT is the efficiency times acceptanceto observe a hit in a PMT). The number of hits NHIT is averaged over a time of roughly one minute, and
it is measured separately for each of the 3564 bunch slots in the LHC orbit. This “hit-counting” method
was used in 2016, since it can measure a high number of interactions of up to µ < 172 [46].
The calibration of the luminosity scale was performed in August 2015 and May 2016, using x–y beamseparation scans (“van der Meer scans”). It is performed using the following equation:
L=

fLHC n1 n22πΣx Σywhere L is the instantaneous luminosity, fLHC is the LHC revolution frequency, Σx and Σy are theconvoluted beam sizes obtained from the scan width of the van der Meer scans. n1 and n2 are the beam
bunch populations obtained from the measurement of the beam bunch current. Additionally, the responseof the LUCID-2 has a slight dependence on the number of beam bunches in a train, due to e.g. reflectionof signal pulses in the read-out chain that affect neighbouring bunches. This dependence is corrected by
a comparison with the number of reconstructed tracks in the Inner Detector; this correction is called the“calibration transfer”.
The uncertainty of the combined 2015+2016 integrated luminosity is evaluated as 2.1%. This is derived
following a method similar to that detailed in Ref. [47]. The main contribution (∼ 1.6%) is due tothe evaluation uncertainty of the calibration transfer. This uncertainty is estimated by reevaluating thecalibration transfer using the Tile calorimeter response instead of the number of Inner Detector tracks.
3.2.5 Trigger and data acquisition
The trigger and data acquisition system is composed of several subsystems, as shown in Figure 20. The
pp beam bunch crossing rate is roughly 40 MHz, and the trigger system is used to reduce the event rate fordata storage. The trigger system is composed of two levels, namely the first-level trigger (L1) and the HighLevel Trigger (HLT) [48]. L1 is hardware based and uses a subset of the information from the detectors;
information of the L1 calorimeter trigger (L1Calo), L1 muon trigger (L1Muon), and other subsystems areprocessed with the central trigger processor (CTP) to implement trigger menus made from combinationsof trigger selections. L1 event rate is about 100 kHz. When the L1 trigger is accepted, the detector
data read from Readout Drivers (ROD) are buffered in the Read-Out System. HLT is software-based
and receives the region-of-interest (i.e. η and φ coordinates) information from the L1, which is used forregional reconstruction of the detector data. HLT event rate is about 1 kHz. The events accepted by the
HLT are transferred to the data storage and CERN’s computing centre.
The L1Calo algorithm looks for high-pT electrons, photons,τ decaying into hadrons, hadronic jets, large
missing transverse energy, and large total transverse energy. It uses information from calorimeters with
reduced granularity (typically 0.1 × 0.1 in ∆η × ∆φ). For the electron, photon, and τ, a region-of-interest
with the size of 2 × 2 (in units of ∆η × ∆φ = 0.1 × 0.1) is looked for that has high sum of ET in the EM
calorimeter. A requirement can be placed that the sum of ET in the corresponding 2 × 2 region in the32

Figure 20: Schematic diagram of the trigger and data acquisition system.[48] L1Topo and FTK were being commissioned during 2015 and 2016.

hadronic calorimeter is smaller than a threshold (“hadronic veto”). For hadronic jets, a region-of-interest
with the size of 4 × 4 or 8 × 8 is looked for, that has a 2 × 2 core and a high sum of ET in the EM andhadronic calorimeters.
The L1Muon algorithm looks for high-pT muons using information from muon trigger chambers (i.e.
RPC and TGC). It searches for hits consistent with a muon originating from the interaction region.
The HLT algorithm uses Inner Detector tracks, finer-granularity calorimeter information, and precision
measurements from the muon spectrometer. For the electron and photon reconstruction, a similar method
as that for the offline reconstruction (see Section 4.1) is used; the sliding window algorithm is used
to look for EM cluster with high ET in the EM calorimeter, within the region-of-interest provided by
L1; a regression algorithm based on Boosted Decision Trees is used for energy correction. For photon
identification, the Loose and Tight selections (see Section 4.2) are defined using the shower shape in the
EM calorimeter.

3.3 ATLAS Run 2 operation√
The Run 2 operation of the ATLAS experiment was performed from 2015 to 2018 at s = 13 TeV forpp collisions. The peak instantaneous luminosity was as high as 1.4 × 1034 cm−2 s−1 during stable beam
conditions in 2016. The distribution of the mean number of interactions per bunch crossing (i.e. pile-up),recorded during the stable beam conditions in Run 2, is shown in Figure 21. The value of pile-up is
calculated as µ = Lbunch σinel / fLHC , where Lbunch is the per bunch instantaneous luminosity, σinel is the pp

33

500

∫Ldt=148.5 fb-1

2015: <µ> = 13.4
2016: <µ> = 25.1
2017: <µ> = 37.8
2018: <µ> = 37.0
Total: <µ> = 34.2

400
300
200

Initial 2018 calibration

Recorded Luminosity [pb-1/0.1]

ATLAS Online, 13 TeV

100
0
0

10

20

30

40

50

60

70

80

Mean Number of Interactions per Crossing

Figure 21: The mean number of interactions per bunch crossing. The data recorded by the ATLAS detector during
stable beam conditions are shown. The distribution is weighted by the luminosity. [49]inelastic cross section taken to be 80 mb, and fLHC is the LHC revolution frequency. The dataset used in
the analysis presented in this thesis was recorded during 2015 and 2016, with a pile-up of as high as 40.
The details of this dataset is described in Section 5.1.1.

34

4 Photon reconstruction and identification
4.1 Reconstruction
The reconstruction of electrons and photons proceeds in parallel [44, 50]. Electrons and photons in theregion |η| < 2.47 are reconstructed from clusters of energy deposited in the EM calorimeter cells. A
seed cluster is determined with a sliding window algorithm [51], using a window size of 3 × 5 in units of∆η × ∆φ = 0.025 × 0.025 (corresponding to the typical cell size of the EM calorimeter second samplinglayer). The window size is set wider in the φ direction compared to the η direction, considering thesolenoid magnetic field that is parallel to the beam axis; the trajectories of electrons (those originatingfrom the beam interaction region or e+ e− pairs originating from conversions of photons) are bent in the φdirection. The seed cluster is required to have a total transverse energy of larger than 2.5 GeV. Duplicate
seed clusters are removed; if two seed clusters are within 2 × 2 in units of ∆η × ∆φ = 0.025 × 0.025, theone with lower energy is removed. From MC simulations, the efficiency of seed cluster reconstruction isestimated to be larger than 99% for photons with ET > 20 GeV.
Track pattern recognition in the Inner Detector is first performed using the connected component analysis
and a neutral network classifier [52, 53] to create a track seed (i.e. a track with three hits in the silicondetectors). The track seed is extrapolated to the TRT hits using a Kalman filter formalism [54], withthe charged pion hypothesis that assumes energy loss of a minimum-ionizing particle interacting withthe detector material. When this initial track pattern recognition fails, a second track extrapolation is
performed using an electron hypothesis which allows for larger energy loss of up to 30%. The second
track extrapolation is performed only if the track seed has a transverse momentum of larger than 1 GeV
and it is within the region-of-interest7 of an EM seed cluster. The track seeds are fit with the global χ2
fitter [55]. The tracks obtained from the track fitting is loosely matched to EM seed clusters; the tracks
are extrapolated to the EM calorimeter second sampling layer, and the η and φ coordinates are comparedwith the barycentre of the EM seed clusters. A track fit using Gaussian-sum filter fitter [56] is performed
for tracks with silicon detector hits to improve the precision of the obtained track.
The reconstructed tracks are used for the reconstruction of conversion vertices of γ → e+ e− . A two-track
conversion vertex is reconstructed from two tracks with opposite charges, and it is required to be consistent
with that of a massless particle. A single-track conversion vertex is reconstructed, to reconstruct vertices
with asymmetric tracks where one of the track has a low momentum pT < 5 GeV and is not reconstructed,
or vertices with two tracks that are close to each other and cannot be separately reconstructed. A singletrack conversion vertex is reconstructed from a track with no hits in the innermost silicon layers. The
tracks of the conversion vertex is required to have a high likelihood to be electrons based on TRT hits, withtighter requirements for single-track conversion vertex and conversion vertex with no hits in the siliconsensors. When there are two or more conversion vertices associated to a seed cluster, a vertex is chosenwith the priority in the following order; a vertex with two tracks with silicon sensor hits, a vertex with two
tracks, and a vertex with single track.
An EM seed cluster with a matching track consistent with electrons originating from the beam interaction
region is classified as an electron, as shown in Figure 22. A seed cluster with one or two matching tracks
consistent with the photon conversion hypothesis in the Inner Detector is classified as a “converted photon”,
i.e. a photon converted into a e+ e− pair before leaving the Inner Detector. A seed cluster with no matching
7The region-of-interest is defined as ∆R < 0.3 with respect to the barycentre of the seed cluster that satisfies a loose shower
shape requirement (Rη > 0.65 and Rhad < 0.1, where the two variables are those defined in Table 5).

35

hadronic calorimeter

third layer∆η×∆φ = 0.05×0.0245

second layer∆η×∆φ = 0.025×0.0245

first layer (strips)

electromagnetic
calorimeterη∆η×∆φ = 0.0031×0.098

presamplerφ

TRT (73 layers)
SCT
beam axis

pixelsbeam spotd0
insertable B-layer

Figure 22: A schematic diagram illustrating the reconstruction of an electron.[50]tracks is classified as an “unconverted photon”, i.e. a photon traversing outside the Inner Detector withoute+ e− conversion. To increase the photon reconstruction efficiency, a further classification is performed
to determine an EM seed cluster as an electron or a photon by using the track pT and E/p (where E isthe cluster energy and p is the track momentum). From simulation, the efficiency to reconstruct a true
converted photon (i.e. a photon converted to a e+ e− pair within a radius of 80 cm from the Interaction
Point) as a converted photon is evaluated to be larger than 65% for a pile-up of 60 for ET > 20 GeV. The
fake rate to reconstruct a true unconverted photon as a converted photon is evaluated to be below 9% fora pileup of 60. Overall, approximately 70% of photons are classified as unconverted photons, and theremaining 30% are classified as converted photons.
From simulations, the overall efficiency to reconstruct a “prompt photon” (i.e. a photon not originatingfrom decays of hadrons) as a photon (i.e. either a converted or an unconverted photon) is estimated to
be 96% for ET > 25 GeV. The other 4% is erroneously classified as an electron. For higher ET , the two
tracks from a photon conversion becomes difficult to separate; this leads to a decrease in the efficiency ofthe photon reconstruction to approximately 90% for ET ' 1 TeV.
The energy of the photon is estimated using a clustering algorithm [51] that forms clusters around the EM
seed cluster. A window size of 3 × 7 in units of ∆η × ∆φ = 0.025 × 0.025 is used in the barrel region, anda window size of 5 × 5 in the end-cap region. The procedure for energy calibration is performed in several
steps [57]. A regression algorithm based on Boosted Decision Trees is used to correct for the energy lostin the material upstream of the EM calorimeter, the energy deposited in the cells outside of the η and φclusters, and the energy lost beyond the EM calorimeter. The algorithm is trained separately for electrons,
converted photons, and unconverted photons, using simulated data. It is performed in intervals of ET and|η|. The relative calibration between the different layers of the EM calorimeter is derived using Z → µµ
and Z → ee events and comparing collision data and simulated data. Z → µµ events are used here
because muons energy deposits are insensitive to the amount of material in front of the EM calorimeter.
The energy is corrected further for geometric effects at the boundaries of calorimeter modules and HV
settings in the regions of the calorimeter.
The overall calibration for the energy scale and energy resolution is performed by comparing Z → ee
events in collision data and simulation, with the same results applied to electrons and photons [57]. The

36

(a)

(b)

Figure 23: The corrections to energy scale and energy resolution evaluated by comparing Z → ee events in thecollision data and simulated data. [57]Figure 24: The comparison of the Z → ee invariant mass distribution obtained from collision data and that fromsimulation, with the energy scale and energy resolution corrections applied for the simulation. The distribution
from the simulation is normalized to that of the collision data. [57]corrections, αi for energy scale and ci for energy resolution, is defined by σ  data  σ  MCE
E
E data = E MC (1 + αi ),
=
⊕ ci
E
E

(57)

where i corresponds to different regions in η. The dielectron mass distribution is compared for collision
data and simulated data, and the corrections are determined by minimizing the χ2 of the two distributions.
The resulting corrections are shown in Figure 23. Several sources of systematic uncertainties are evaluated,
and the total fractional systematic uncertainty of energy scale is estimated to be at the level of O(0.1)%.
The total fractional systematic uncertainty of energy resolution is estimated to be 5–10% for 30–60 GeV,
and 20–50% for higher energy. Figure 24 shows the Z → ee invariant mass distribution obtained fromcollision data and that from simulation, with the energy scale and energy resolution corrections applied
for the simulation. The derived energy scale and resolution are cross checked that they are applicable tophotons, using Z → eeγ and Z → µµγ events in collision data and simulated data.
Reconstructed photons are required to match photons reconstructed at the trigger level, within the separa-

37

ES

f1 = Etot1

fside =
2
E3×7
S2
E7×7

wη2 =

S
E3 1

S

Rη =

r

S

S

E7 1 −E3 1

S

Rφ =

2
E3×3

ΣEi ηi
ΣEi

2

S1
S1
− Emin∆E = Emax,2S

Eratio =S

1 −E 1
Emax,1max,2
S1
S1+Emax,2
Emax,1S

E had

Rhad = ETTStrips S1

width in a 3×5 (∆η × ∆φ )region of cells in S2

2
E3×7

Hadronic
Second layer S2

ΣEi ηi2
ΣEi −

ws =

q

ΣEi (i−imax )2
ΣEi

η

φ

ws tot uses 3×2 strips (η × φ )ws 3 is defined similarly
but uses 20×2 strips

Figure 25: Schematic diagrams showing the definition of the shower shape variables. [58]tion of ∆R < 0.07.

4.2 Identification
Photon identification (ID) is based on a set of requirements placed on several discriminating variables whichcharacterize the EM shower development in the calorimeter (“shower shapes”) [44]. They are definedto efficiently select prompt photons and reject backgrounds (i.e. hadronic jets erroneously reconstructed
as photons, and photons originating from decays of hadrons). The discriminating variables (“shower
shape variables”) are defined in Table 5. A schematic representation of the shower shape variables isshown in Figure 25. Two variables quantify the shower leakage fraction in the hadronic calorimeter.
Three variables quantify the lateral shower development in the EM calorimeter second sampling layer.
One variable quantifies the fraction of energy in the EM calorimeter first sampling layer. Five variables
quantify the lateral shower development in the finely-segmented strip cells of the EM calorimeter firstsampling layer. The two of the five variables are utilized to find photon candidates with two separate
local energy maxima in the strips cells; such shower shape is characteristic of neutral hadron decays,
e.g. π 0 → γγ, as shown in Figure 18. The distributions of the shower shape variables are shown inAppendix B as Figures 88 and 89.
Several photon ID selections are defined [44]. The “Loose” selection is defined using shower shapevariables in the EM calorimeter second sampling layer and the shower leakage fraction in the hadronic
calorimeter, as shown in Table 5. It is used for photon ID selections at the trigger level. The “Tight”
selection is based on all variables, as shown in Table 5. It is the standard photon ID selection used for the
ATLAS experiment, but it is not used in the search presented in this thesis. Variations of the “Loose0"
selections are defined, as shown in Table 6. They are defined by removing selection requirements on
several variables with respect to the Tight selection. The number represents the number of selection
requirements removed with respect to the Tight selection, e.g. Loose05 represents that requirements onfive variables are removed.
The thresholds for the requirements on shower shape variables are defined differently for several intervalsof the |η| of the photon candidate, namely 0–0.6, 0.6–0.8, 0.8–1.15, 1.15–1.37, 1.52–1.81, 1.81–2.01,
2.01–2.37. The binning is defined to consider the difference in the EM calorimeter geometry and thedifference in the material upstream of the EM calorimeter. Also, the thresholds are defined differently for
an unconverted photon and a converted photon. These thresholds are optimized so that the rejection of38

Table 5: Shower shape variables used for photon ID [44]. A slight modification is made from the original table.
“Loose05” is shown as one example of the variations of the Loose0 selections.
Category
Hadronic leakage

EM middle layer

Description
Ratio of ET in the first sampling layer of the hadronic calorimeter to ET of the EM cluster (used over the ranges |η| < 0.8
or |η| > 1.52 )

Name
Rhad1

Loose


Tight


Loose05


Ratio of ET in the hadronic calorimeter to ET of the EM
cluster (used over the range 0.8 < |η| < 1.37)

Rhad







Ratio of the energy in 3 × 7 η × φ cells over the energy in7 × 7 cells centred around the photon cluster position

Rη







wη2







Rφ





ws3



wstot



Energy outside the core of the three central strips but withinseven strips divided by energy within the three central strips

fside



Difference between the energy associated with the second
maximum in the strip layer and the energy reconstructed in
the strip with the minimum value found between the first andthe second maxima∆ERatio of the energy difference between the maximum energy
deposit and the energy deposit in the secondary maximum
in the cluster to the sum of these energies

Eratio

Ratio of the energy in the first layer to the total energy of the
EM cluster

f1



q

Lateral shower width, ΣEi ηi2 /(ΣEi ) − ((ΣEi ηi ) /(ΣEi ))2 ,where Ei is the energy and ηi is the pseudorapidity of cell iand the sum is calculated within a window of 3 × 5 cells
Ratio of the energy in 3 × 3 η × φ cells over the energy of3 × 7 cells centred around the photon cluster positionr
EM strip layer

Lateral shower width,Í Í
Ei (i − imax )2 /( Ei ), where iruns over all strips in a window of 3 × 2 η × φ strips, and imax
is the index of the highest-energy strip calculated from three
strips around the strip with maximum energy depositr
Total lateral shower width,Í Í
Ei (i − imax )2 /( Ei ),where i runs over all strips in a window of 20 × 2 η × φ
strips, and imax is the index of the highest-energy strip measured in the strip layer

39

Table 6: Definition of the variations of the Loose0 selections.

Name
Tight
Loose02
Loose03
Loose04
Loose05
Loose06
Loose07

Requirements not applied with respect to Tight selection
fside, ws3
fside, ws3, ∆E
fside, ws3, ∆E, Eratio
fside, ws3, ∆E, Eratio, wstot
fside, ws3, ∆E, Eratio, wstot, wη2fside, ws3, ∆E, Eratio, wstot, wη2, Rφbackgrounds is maximized while retaining the identification efficiency of prompt photons, using simulated
events of hadronic jets and prompt photons. The thresholds used for the shower shape variables are thesame for the Tight selection and the various Loose0 selections, and they are looser for the Loose selection.
Data-driven corrections are applied to the shower shape variables [44]. The EM shower shape obtained
from MC simulations do not perfectly agree with that observed in the collision data due to the mismodelling,
especially the lateral shower development, due to e.g. imperfections of EM shower simulation in thedetector. This results in the deviations of the central values of the shower shape variable distributions;these deviations are typically 10% of the standard deviation of the distributions. The corrections areapplied as a constant shift to the value of the shower shape variable. The value of the shift is defined asa function of the ET and |η| of the photon candidate, with bin boundaries of 8, 15, 20, 25, 30, 40, 50,
60, 80, 100, 250, and 1000 GeV in ET , and a binning of 0–0.6, 0.6–0.8, 0.8–1.15, 1.15–1.37, 1.52–1.81,
1.81–2.01, and 2.01–2.37 in |η|. They are separately defined for unconverted and converted photons.
The shifts are determined by minimizing the χ2 between the distributions of the MC simulations and thecollision data, as shown in Figure 268. For ET < 50 GeV, photon candidates from Z → eeγ and Z → µµγ
events are used from the collision data and simulated data. For ET > 50 GeV, high-ET photon candidates
obtained from collision data collected using single-photon triggers, which use Loose ID selection andhave large prescaling, are used. The resulting shifts are typically 10% of the standard deviation of the
distribution; their values are, at most for ET > 60 GeV, 0.001 ± 0.0007 for Rhad , 0.008 ± 0.002 for Rη ,0.008±0.001 for Rφ , 0.0005±0.0001 for wη2 , 0.20±0.02 for wstot , 0.05±0.01 for Fside , and 0.044±0.004
for ws3 . The shifts are not defined for ∆E and Eratio , because their shifts are small (e.g. for ET > 60 GeV,
10 MeV at most for ∆E, and 0.005 at most for Eratio ), and also well below their associated uncertainties.
The photon ID efficiency to select prompt photons with the Tight selection is measured using collision
data [44], and it is shown in Figure 27. It is measured using three different events, to cover a wide rangeof ET ; photon candidates from Z → eeγ and Z → µµγ events; electron candidates from Z → ee events;and high-ET photon candidates from events collected using single-photon triggers. Figure 27 shows themeasurement with the Z → ee events and single-photon events. For Z → ee events, a correction isapplied to the distributions of the shower shape variables, to reproduce those of photons; this correction isderived from the differences in the distributions of the shower shape variables observed in simulations ofphotons and electrons. For single-photon trigger events, contamination of backgrounds (i.e. hadronic jetserroneously reconstructed as photons, and photons originating from decays of hadrons) are subtracted.
8For instance, for wstot for 60 GeV < ET < 80 GeV and |η| < 0.6 for converted photons, the resulting reduced χ2 calculated
over 500 bins is approximately 1.2.

40

(a)

(b)

Figure 26: Comparison of the ws3 distributions of Z → llγ events from collision data and simulated data withand without data-driven corrections. Photon candidates with 10 GeV < ET < 50 GeV and |η| < 2.37 (excluding
1.37 < |η| < 1.52) are used. They are shown for (a) unconverted photons and (b) converted photons. Here, ws3 isdenoted by ωs,3 . [44]

(a)

(b)

Figure 27: The photon ID efficiency to select prompt photons with the Tight selection requirement. It is shownfor |η| < 0.6 for (a) unconverted photons and (b) converted photons. For the simulated events, the efficiencies
calculated with and without the data-driven corrections to the shower shape variables are shown. The panels below
show the efficiency scale factors. [44]The “efficiency scale factors” (i.e. corrections applied to Tight selection efficiency in simulations) are
derived; they are consistent with 1 within the statistical uncertainty, and are at the level of 1% or less.
The rejection factor (i.e. the inverse of the selection efficiency) of backgrounds is estimated to be O(103 )for ET > 25 GeV; this is estimated from simulations of SM background events consisting of one photon
and one hadronic jet from the hard process.

41

Figure 28: Schematic illustration of the definition of the calorimeter cone energy ETcone . The grid represents thecells of the EM calorimeter second layer. The shaded circle represents the cone ∆R around the barycentre of the
EM cluster. The yellow filled box represent the subtracted cells corresponding to the photon EM cluster. The redfilled squares represent the energy deposits in the calorimeter. [50]4.3 Isolation
To further reject backgrounds (i.e. hadronic jets erroneously reconstructed as photons, and photons
originating from decays of hadrons), the reconstructed photons are required to be isolated from othercalorimeter energy deposits and from nearby tracks not associated with the reconstructed photon (“isolation
requirement”) [44].
The cone energy ETcone is defined as the sum of energy deposits in the calorimeters (i.e. EM and hadronic
calorimeters) in a cone of size ∆R = 0.4 around the barycentre of the photon EM cluster, subtracting theenergy associated with the photon EM cluster, as shown in Figure 28. ETcone is corrected for the leakageof the photon energy from the photon EM cluster and for the effects of pile-up [59]. The calorimeter
isolation variable of a reconstructed photon is defined as ETiso = ETcone − 0.022 × ET . It is required tosatisfy ETiso < 2.45 GeV.
To account for the small differences observed between collision data and simulation, data-driven corrections are applied to the values of ETcone . The corrections are derived using single-photon events, where theevents of collision data are selected using single-photon triggers with Loose ID selection. From the eventsin collision data, the contributions of hadronic jets are subtracted; here, the shape of ETcone of hadronic jet
contribution is taken from a control sample defined by inverting the photon ID selection. The data-driven
correction to the simulation is performed by a shift in the ETcone distribution; this shift is defined as thedifference in the peak position of the ETcone distributions of the collision data and simulated data. The
shifts are determined in bins of ET and η. The values of the shifts are 2 GeVat most.
The track isolation variable piso
T is defined as the scalar sum of the transverse momenta of tracks not
associated with the reconstructed photon in a cone of size ∆R = 0.2 around the barycentre of the photon
EM cluster. It is required to satisfy piso
T < 0.05 × ET .

42

To account for the small differences observed between collision data and simulation, data-driven corrections are applied to the track isolation efficiency. The efficiency corrections are derived using single-photon
events, where the events of collision data are selected using single-photon triggers with Loose ID selection.
From the events in collision data, the contributions of hadronic jets are subtracted; here, the shape ofthe track isolation variable piso
T of hadronic jet contribution is taken from a control sample defined byinverting photon ID selection. The track isolation efficiency correction is defined as the difference in the
track isolation efficiency between the collision data and simulated data. The efficiency corrections are
determined in bins of ET and η. The values of the corrections are 2% at most.

43

5 Search for a resonance decaying into a pair of photon-jets
A search for a new scalar resonance decaying into a pair of photon-jets is performed. Two benchmark signal
scenarios are introduced. They assume a heavy scalar boson X with a narrow width, that is produced via
gluon–gluon fusion process, and decays into a pair of spin-0 particles a. One benchmark signal scenario
assumes the decay of a into a pair of photons, i.e. X → aa → 4γ, as shown in Figure 29(a). Anotherbenchmark signal scenario assumes the decay of a into a set of three neutral pions and their subsequent
decay into photons, i.e. X → aa → 6π 0 → 12γ, as shown in Figure 29(b). For mX > 200 GeV andma ∼ O(1 GeV), the photons from the decay of a will be highly collimated and will form a photon-jet.
The two benchmark signal scenarios are introduced to interpret the search result in scenarios consisting
photon-jets with both a low multiplicity and high multiplicity of photons. Here, the decay of a is considered
to be “prompt”, i.e. having a negligible lifetime.
An overview of the analysis strategy is described below. A photon-jet can lead to a single EM clusterin the EM calorimeter, as depicted in Figure 30; in such a case, each photon-jet typically leads to one
reconstructed photon. Thus, this analysis selects events containing at least two high-ET reconstructed
photons to search for a resonance decaying into a pair of photon-jets, as shown in Figure 31; note that in
the case of a photon-jet signal event, one reconstructed photon corresponds to one photon-jet. Hereafter,a reconstructed photon is denoted by γR . The signal region is defined with event selection strategy
optimized to increase the sensitivity to photon-jet signal events. The reconstructed diphoton mass mγR γR
is defined as the invariant mass of the two reconstructed photons, and it corresponds to the value of mX in
the case of signal events. As shown in Figure 32, the mγR γR distribution of SM background events forms
a smoothly falling shape, with the rate of events decreasing logarithmically with respect to the value ofmγR γR . In contrast, the mγR γR distribution of signal events form a peaklike shape, with a width of the shapecorresponding to the detector resolution if the X boson has a narrow decay width (well below 1 GeV).
With these contrasting features of the background and signal events in mind, a “bump-hunting” technique
is employed; an excess of events from the SM background expectation is searched for by performing an
unbinned maximum-likelihood fit to the mγR γR distributions with the consideration of the background andsignal components.
The SM backgrounds that enter the signal region consist of mainly three components:• Events containing two high-ET prompt photons (γγ)• Events containing one high-ET prompt photon and one high-ET hadronic jet (γ j or jγ)

γ

g

g

a
X

a

g

γ
γ
g

γ
(a)

X

a

3π 0

a

3π 0

(b)

Figure 29: Diagrams of benchmark signal scenarios of a new scalar resonance decaying into a pair of photon-jets,
for the processes (a) X → aa → 4γ and (b) X → aa → 6π 0 .

44Figure 30: Schematic diagram of a photon-jet a → γγ entering the EM calorimeter.

Figure 31: A schematic diagram illustrating the decay of the benchmark signal scenario X → aa → 4γ in the
ATLAS detector. The cross section in the x–y plane is shown. The boson X is produced by a pp collision. It decays
to a pair of a bosons, and each a boson decays to a pair of photons. Because the a boson is highly boosted, the two
photons form a photon-jet, which creates a single EM cluster in the EM calorimeter.

Figure 32: A schematic diagram illustrating the concept of a resonance search.

45• Events containing two high-ET hadronic jets ( j j)
A hadronic jet can be erroneously reconstructed as a photon. γ j ( jγ) denotes events with the prompt
photon having a larger (smaller) ET than the hadronic jet.
The structure of this section (Section 5) is described briefly below. The descriptions of the collision dataset
and the simulated samples of signal and background events are given in Section 5.1. The experimental
characteristics of photon-jet signal events are described in Section 5.2. The event selection requirements
are optimized to increase sensitivity to photon-jet signal events, as described in Section 5.3.1. The
signal region is divided into two orthogonal event categories to increase sensitivity to photon-jet signal
events, as described in Section 5.3.2. An unbinned likelihood fit to the mγR γR distributions is performed,
simultaneously in the two event categories, and considering probability density functions of both thesignal and background components. As an input to the unbinned likelihood fit, the probability density
functions of the signal component and the background component are described in Sections 5.4 and 5.5,
respectively. The systematic uncertainties affecting the unbinned likelihood fit is described in Section 5.6.
The statistical method of the unbinned likelihood fit is described in Section 5.7. The search result andits interpretation are presented in Section 5.8. A long-lived a arises in BSM models, for instance in
NMSSM when the decay width of a is small (e.g. Γa ∼ O(1 meV)), as discussed in Section 2.3.19. Theinterpretation of the results for the case of a long-lived a in X → aa → 4γ is described in Section 5.9.

5.1 Event samples
5.1.1 Collision data
The dataset used for this search was collected under normal data-taking conditions for pp collisions at a√
center-of-mass energy of s = 13 TeV during 2015 and 2016. It corresponds to an integrated luminosityof 36.7 fb−1 after applying data-quality requirements.
The data were collected using an unprescaled diphoton trigger implemented in the HLT. This trigger
selects events with two EM clusters with ET > 35 GeV and ET > 25 GeV and satisfying the Loose photon
ID selection10. This trigger is seeded with a L1 trigger that requires two EM clusters with ET > 15 GeV
and satisfying hadronic veto11.
5.1.2 Simulated samples
Samples of the benchmark signal scenarios X → aa → 4γ and X → aa → 6π 0 were simulated with thefollowing set up. For the production of the X via gluon–gluon fusion, MadGraph5_aMC@NLO [60]Version 2.3.3 at next-to-leading order (NLO) in quantum chromodynamics (QCD) was used with the
NNPDF30NLO parton distribution function (PDF) set [61]. For the subsequent decay of the X into
aa and into photons, and also the parton-shower and hadronization simulation of initial state radiation
jets, Pythia8 [62] Version 8.210 with the A14 tune [63] was used. The samples were produced using
a narrow-width approximation (NWA) approach with the resonance widths of the X and a set to 4 MeV
9The case of a long-lived a in the scenario X → aa → 6π 0 is not considered. This is because such case arise in the NMSSM
only when ma is within some value, as described in Section 2.3.1 as Case 2. Such case is out of scope of this thesis, and is notconsidered.
10This HLT trigger is named “HLT_g35_loose_g25_loose” in the ATLAS trigger paper [48].
11This L1 trigger is named “L1_2EM15VH” in the ATLAS trigger paper [48].

46and 1 MeV, respectively. Samples for X → aa → 4γ (X → aa → 6π 0 ) were simulated for mass rangesof 200 GeV < mX < 2000 GeV and 0.1 GeV < ma < 10 GeV (0.5 GeV < ma < 10 GeV). For thesignal scenario X → aa → 4γ, samples for a that decays promptly as well as those for long-lived a were
simulated, ranging from cτ = 0 to cτ = 25 mm, where c is the speed of light and τ is the average lifetimeof the particle a at rest.
Samples of the SM background events consisting of two photons from the hard process were simulatedwith Sherpa 2.1.1 [64]. Matrix elements were calculated at leading order (LO) in QCD with up to
two additional partons. It is then merged with the Sherpa parton-shower simulation [65] using the
ME+PS@LO prescription [66]. The CT10 PDF set [67] was used in conjunction with a dedicatedparton-shower tune of Sherpa.
Samples of the SM background events consisting of one photon and one hadronic jet from the hard process
were also simulated with Sherpa 2.1.1. Matrix elements were calculated at LO in QCD with up to four
additional partons. The same merging prescription, PDF set, and parton-shower tune as for the diphoton
sample were used.
Additional interactions in the pile-up were simulated and overlayed on the simulated signal and SM
background events. These events were simulated using Pythia 8.186 [62] using the A2 tune [68] and theMSTW2008LO PDF set [69] .
All simulated event samples were produced using the ATLAS simulation infrastructure [70], using thefull Geant 4 [71] simulation of the ATLAS detector. The simulated events were passed through thedigitisation process that simulates the detector responses, then they were reconstructed with the same
software as that used for the reconstruction of the collision data.

5.2 Photon-jet signal characteristics
Photon-jets can arise from the decays of boosted particles into photons. These boosted particles arise
when the particles are decay products of a higher-mass resonance.
For the benchmark signal scenarios, the angular separation of photons in a photon-jet depends on theratio of the masses ma /mX , when particle X is produced nearlypat rest. For instance, for X → aa → 4γ,
the distribution of ∆Rγγ is shown in Figure 33. Here, ∆Rγγ = (∆η)2 + (∆φ)2 is the angular separationof photons in the decay a → γγ. From the kinematics of boosted particles, the distribution of ∆Rγγ
has a peak at ∆Rγγ = 2/γa , where γa = Ea /ma is the Lorentz factor of the a particle. Consideringthat the energy of the a particle has a median value of Ea ∼ mX /2, the distribution of ∆Rγγ has a peakat ∆Rγγ ∼ 4 × ma /mX . The proportionality of the angular separation of photons within a photon-jet
to ma /mX holds for photon-jets in general, including X → aa → 6π 0 . Because the two scenarios
X → aa → 4γ and X → aa → 6π 0 are similar, some parts of the descriptions in the following sections
are only mentioned for X → aa → 4γ, but they apply to X → aa → 6π 0 as well.
For small values of the ratio ma /mX < 0.01, the boost of the a can lead to small angular separationsof the final-state photons (∆R . 0.04); Thus the collimated photons may enter a small area of the EM
calorimeter that is comparable to the Molière radius. In this case, the decay of the a into photons will leadto a single EM cluster, and hence to one reconstructed photon, as depicted in Figure 30. Thus, events with
two high-ET reconstructed photons can be used as a starting point for a search for a resonance decaying
to a pair of photon-jets. This analysis presents a search for a scalar resonance X decaying to a pair ofphoton-jets for the range mX > 200 GeV. Based on the estimations using simulated signal samples, it

47

Fraction of events

ATLAS Simulation
X → aa → 4γ
mX = 800 GeV
ma = 0.5 GeV
ma = 1 GeV
ma = 2 GeV
ma = 5 GeV

0.25
0.2
0.15
0.1
0.05
0
0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

∆Rγ γ , generator level

Figure 33: The distribution of ∆Rγγ for X → aa → 4γ using simulated signal samples at the generator level. [72]is found that the analysis is sensitive to the parameter region ma /mX < 0.01, which is defined as theparameter region of interest of this analysis.
For large values of the ratio ma /mX & 0.08, the final-state photons can be isolated and lead to separate EM
clusters when their angular separations are larger than roughly 0.3. As already described in Section 1, a
general search for new phenomena in events with at least three isolated photons was performed by ATLAS√
at Run 1 for pp collisions at s = 8 TeV [23]. This search was sensitive to the BSM signal scenario
X → aa → 4γ corresponding to the region ma /mX & 0.08, as shown in Figure 3.

5.3 Event selection
5.3.1 Definition of the signal region
Events containing at least two reconstructed photons are selected. The reconstructed photons with thehighest and the second highest ET are defined as the “leading” and the “subleading” reconstructed photons,
respectively. The two leading reconstructed photons are required to be within the fiducial calorimeterregion of |η| < 2.37, excluding the transition region at 1.37 < |η| < 1.52 between the barrel and end-cap
sections of the EM calorimeter. The two leading reconstructed photons are required to satisfy ET > 25 GeV
and the Loose photon ID selection. These requirements are defined as the “preselection”.
As described in Section 4.3, the two leading reconstructed photons are required to be isolated fromother calorimeter energy deposits and from nearby tracks not associated with the reconstructed photons.
The calorimeter and track isolation variables of the reconstructed photons are required to satisfy ETiso <2.45 GeV and piso
T < 0.05 × ET . The distributions of the calorimeter and track isolation variables areshown in Figure 34.
The requirement ET > 0.4 × mγR γR is applied to the leading reconstructed photon, and ET > 0.3 ×
mγR γR to the subleading reconstructed photon. This requirement increases the sensitivity to a scalar
resonance decaying into a pair of photon-jets, because such events contain reconstructed photons with48

Normalized to 1

Normalized to 1

0.5

X→aa→γ γ , m =600 GeV
X
ma=0.1 GeV
ma=0.7 GeV
ma=5 GeVγ γ MC mγ γ =500-700 GeV
j of γ j MCRmR γ γ =500-700 GeV
R R
Data mγ γ =500-700
GeV

0.45
0.4
0.35
0.3

0.5

X→aa→γ γ , m =600 GeV
X
ma=0.1 GeV
ma=0.7 GeV
ma=5 GeVγ γ MC mγ γ =500-700 GeV
j of γ j MCRmR γ γ =500-700 GeV
R R
Data mγ γ =500-700
GeV

0.45
0.4
0.35
0.3

R R

R R

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0
−40 −30 −20 −10

0

10

20

30

40

50

0
−40 −30 −20 −10

60

0

10

Econe
- 0.022ET [GeV]
T

20

30

40

50

60

piso - 0.05ET [GeV]
T

(a)

(b)

Figure 34: The distributions of (a) the calorimeter isolation variable ETcone − 0.022 × ET and (b) the track isolation
variable piso
T −0.05×ET . The histograms are plotted as the sum of leading and subleading reconstructed photons. The
events which satisfy the preselection requirements are plotted. The filled markers represent the distributions obtained
from simulated signal samples of the process X → aa → 4γ for mX = 600 GeV. The empty markers represent
the distributions obtained from simulated samples of SM processes γγ and γ j in the region 500 GeV < mγR γR <700 GeV. For the square markers, reconstructed photons originating from hadronic jets are exclusively chosen. The“X” markers represent the distributions obtained from collision dataset in the region 500 GeV < mγR γR < 700 GeV.
The vertical dotted line represents the threshold used for the requirement placed on the variable.

larger ET /mγR γR ratios compared to those from background events dominated by t-channel processes [25].
The distributions of ET /mγR γR are shown in Figure 35.
The two leading reconstructed photons are required to satisfy the Loose05 photon ID selection, defined in
Section 4.2. The Loose05 selection is a looser requirement by definition compared to the Tight selection,
which is the standard photon ID selection used for the ATLAS experiment. Based on the sensitivity
estimation using simulated samples of signal and SM background processes, this Loose05 selection isfound to provide better sensitivity to photon-jet signals, compared to the Tight selection12. This is because
photon-jet signal events with large values of ma /mX (e.g. ma /mX & 0.003 ) may lead to wider EM
shower in the EM calorimeter, and such events can be rejected by the Tight selection. The distributionsof the shower shape variables of photon-jet signal events are shown in Appendix B. Thus, the selection
efficiency of signal events increases with the use of the Loose05 selection. For instance, for X → aa → 4γ
with mass values in the region 0.003 < ma /mX < 0.006 and mX > 200 GeV, the selection efficiency isless than 5% when the Tight selection is used, and this increases to 20%–50% with the Loose05 selection(i.e. an increase of approximately a factor 4–10). Also, an increase in the number of background events
is moderate (∼ 30%) with the Loose05 selection compared to that with the Tight selection. Thus, thesensitivity to photon-jet signals is increased with the use of the Loose05 selection.
Additionally, an important point is that a “control region” (i.e. a region similar and orthogonal to the signal
region) can be defined with the use of the Loose05 selection. This is because the Loose05 selection is atighter selection compared to the Loose photon ID selection. As described in Section 5.1.1, the collisiondataset is collected with the diphoton trigger, which requires two EM clusters to satisfy the trigger-level
Loose photon ID selection. Reconstructed photons in the dataset that do not satisfy the Loose05 selection
12The details of the sensitivity estimation using simulated samples are described in Appendix C.

49

Normalized to 1

Normalized to 1

0.3

X→aa→γ γ , m =600 GeV
X
ma=0.1 GeV
ma=0.7 GeV
ma=5 GeVγ γ MC mγ γ =500-700 GeV
j of γ j MCRmR γ γ =500-700 GeV
R R
Data mγ γ =500-700
GeV

0.25
0.2

0.3

X→aa→γ γ , m =600 GeV
X
ma=0.1 GeV
ma=0.7 GeV
ma=5 GeVγ γ MC mγ γ =500-700 GeV
j of γ j MCRmR γ γ =500-700 GeV
R R
Data mγ γ =500-700
GeV

0.25
0.2

R R

R R

0.15

0.15

0.1

0.1

0.05

0.05

0
0

0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ET (1st γ ) / mγ γ
R

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ET (2nd γ ) / mγ γ
R

R R

(a)

R R

(b)

Figure 35: The distributions of ET /mγR γR for (a) leading reconstructed photons and (b) subleading reconstructed
photons. The events which satisfy the preselection requirements are plotted. The filled markers represent the
distributions obtained from simulated signal samples of the process X → aa → 4γ for mX = 600 GeV. The empty
markers represent the distributions obtained from simulated samples of SM processes γγ and γ j in the region
500 GeV < mγR γR < 700 GeV. For the square markers, reconstructed photons originating from hadronic jets
are exclusively chosen. The “X” markers represent the distributions obtained from collision dataset in the region
500 GeV < mγR γR < 700 GeV. The vertical dotted line represents the threshold used for the requirement placed on
the variable.

is used for the estimation of the rate of hadronic jets satisfying the calorimeter isolation requirement, as
described in Section 5.5.1.
Hereafter, Loose05 is denoted by Loose0 for simplicity.
Events are required to satisfy mγR γR > 175 GeV.
The signal selection efficiency ε is defined as the efficiency to select signal events including kinematic
acceptance, namelyε=Number of events that satisfy event selection requirements
Number of all generated events(58)

The signal selection efficiency evaluated using simulated samples are shown in Figure 36. The selection
efficiency increases for larger mX because of the ET , η, and isolation requirements. It also decreases forlarger ma /mX , because a larger ma /mX leads to a wider EM shower; such events are rejected by mainly
the trigger-level Loose ID selection and additionally the isolation requirement. For the parameter region
ma /mX > 0.01, the value of ε is smaller than 0.04.
The comparison of the signal selection efficiencies with the use of different photon ID selections is shownin Figure 37.

50

1.1

0.4

0.5

1.1

2.8

19.1

37.3

5.0

0.7

1.5

7.1

21.3

34.7

48.6

51.4

2.0

3.3

28.5

42.9

47.1

48.6

50.6

50.6

1.0

21.3

40.8

45.0

47.7

49.1

50.2

51.6

40
30

0.7

29.9

41.3

45.7

47.7

49.1

50.3

50.8

0.5

31.8

41.1

45.4

47.4

48.5

50.5

51.4

0.1

33.2

200

41.8

400

46.0

600

47.5

48.9

800

50.3

1000

1500

20

Selection Efficiency [%]ma [GeV]ma [GeV]

10.0

10.0

0.1

0.0

0.3

1.3

4.0

21.2

38.9

5.0

0.1

1.2

8.7

20.6

36.2

49.7

48.2

2.0

3.1

27.1

38.2

42.8

45.3

45.6

48.5

1.0

16.0

33.8

39.8

42.8

45.1

47.0

48.6

0.7

20.9

34.0

39.5

42.8

45.0

47.0

48.3

0.5

22.3

33.6

39.3

42.4

44.7

47.0

48.5

0

200

400

600

800

1000

1500

2000
mX [GeV]

45
40
35
30
25
20
15

10

51.2

2000
mX [GeV]

X → aa → 6π0, s = 13 TeV

ATLAS Simulation50

Selection Efficiency [%]X → aa → 4γ , s = 13 TeV

ATLAS Simulation

(a)

10
5
0

(b)

Selection efficiency

Figure 36: The signal selection efficiency ε as a function of mX and ma , for (a) X → aa → 4γ and (b) X → aa →6π 0 . [72]1

X→aa→γ γ , m =600 GeV
X
ma=0.1 GeV
ma=0.5 GeV
ma=0.7 GeV
ma=1 GeV
ma=2 GeV
ma=5 GeV
ma=10 GeV

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 LooseLoose'7 Loose'6 Loose'5 Loose'4 Loose'3 Loose'2

Tight

Figure 37: The comparison of the signal selection efficiencies with the use of different photon ID selections. They
are shown for X → aa → 4γ for mX = 600 GeV and different values of ma .

51

5.3.2 Categorization of events by the shower shape variable ∆E
The signal region is divided into two orthogonal event categories based on the value of the shower shapevariable ∆E of the reconstructed photons. As described in Table 5, ∆E quantifies the relative size of thesecond maximum of an EM cluster in the EM calorimeter first sampling layer. It is defined as
S1
S1∆E = Emax,2− EminS1
S1
where Emax,2is the energy of the strip cell with the second largest energy, and Eminis the energy of the
strip cell with the lowest energy located between the strips with the largest and the second largest energy.
This is depicted in Figure 38. If the strip cells with the largest and the second largest energy are locatednext to each other, then ∆E = 0. When a photon-jet from decays a → γγ and a → 3π 0 → 6γ haveseparation of photons in η larger than the segmentation of strip cells, it can lead to large values of ∆E,
as depicted in Figure 30. Thus, when ma /mX is larger, ∆E becomes larger, as shown in Figure 3913. In
contrast, a single photon entering the EM calorimeter typically has smaller values of ∆E, as seen in the
figure14. Figure 40 shows the comparison of the ∆E distributions obtained from the collision dataset andthat from simulated samples. This figure shows the level of agreement between the two.

Figure 38: A schematic diagram illustrating the definition of the shower shape variable ∆E. The energy deposit of
the EM shower in a strip cell in the first layer of the EM calorimeter is displayed as a function of η. The variable
S1
Emax,1is the energy of the strip cell with the largest energy. [72]The two orthogonal event categories are defined as the following:• Low-∆E category: The two leading reconstructed photons have values of ∆E below the threshold.
This typically corresponds to a shower shape in the first layer of the EM calorimeter with a single
peak.
• High-∆E category: At least one of the two leading reconstructed photons has a value of ∆E above
the threshold. This typically corresponds to a shower shape in the first layer of the EM calorimeterwith more than one peak.
The thresholds of the value of ∆E used for the definition of categories are the same as those used in thestandard Tight photon ID selection, which is tuned to select standard single photons and reject hadronic
jets (including π 0 → γγ events). They range in 100–250 MeV for |η| < 1.81, and in 400–500 MeV for
1.81 < |η| < 2.37, as shown in Table 7.
13The values of ∆E have wider distribution in the end-cap sector, compared to the barrel sector. This is because of several
reasons, including the fact that the width of the EM calorimeter first layer cells becomes narrower for fixed ∆η in the end-cap
sector, and the fact that the ∆η segmentation is larger for larger η (e.g. ∆η = 0.006 for 2.0 < |η| < 2.4)
14Note that the ∆E distribution of standard photons may have non-zero values, as seen in Figure 39. This is because the cells
of EM calorimeter first layer have a noise of O(10) MeV that originate from several sources, e.g. pile-up.

52

(a)

(b)

Figure 39: The distributions of ∆E of reconstructed photons in the signal region. They are displayed for X →aa → 4γ for mX = 1 TeV and ma = 0.1, 0.7, 5 GeV obtained from simulated samples. The black lines show
the distributions for reconstructed photons originating from a scalar resonance with a mass of 1 TeV produced bygluon–gluon fusion process and decaying into two photons. The last bin of the histogram shows the overflow, i.e.
the sum of events with the value of ∆E larger than 1 GeV. The distributions are displayed for reconstructed photons
entering (a) the barrel section and (b) the end-cap section of the EM calorimeter.
Table 7: Thresholds for the values of ∆E that are used for the definition of the two event categories.

|η|
Unconverted
Converted

0 − 0.6200
188

0.6 − 0.8
111
175

Threshold of ∆E [MeV]
0.8 − 1.37 1.52 − 1.81 1.81 − 2.01
140
250
400
150
176
400

2.01 − 2.37400
493

The category fraction f is defined as the fraction of signal events that appear in the low-∆E category,
namelyNumber of events in the low - ∆E categoryf =(59)
Number of events that satisfy event selection requirements
The values of f evaluated using simulated signal samples are presented in Figure 41. The value of f
decreases for larger ma /mX , because a larger ma /mX leads to larger ∆E, as described above.
Additionally, the value of f increases when ma /mX is larger than 0.01, as seen in Figure 41. This isbecause such signal events, i.e. events from the parameter region ma /mX > 0.01 and satisfying the event
selection requirements of the signal region, are heavily biased events in which the photons consistinga photon-jet have an asymmetric distribution of the momentum. A large fraction of the momentum of
the parent a particle is transferred to only one photon in the photon-jet, thus leading to a shower shapesimilar to that of a prompt photon. Such events satisfy the event selection requirement and are likely to be
included in the low-∆E category. However, this feature does not affect the search, because the parameterregion of interest is set to ma /mX < 0.01.
As seen in Figure 41, the values of f for X → aa → 4γ and X → aa → 6π 0 are similar. The main
distinction is the threshold of ma /mX for f to transition from f > 0.5 to f < 0.5. This threshold is53

s = 13 TeV, 36.7 fb-1
ET<200 GeV,|η|<1.37

5

10

Data
MC γ γMC γ j

104

106

102

102

10

10

1

1

−1

−1

200

400

600

800

10

1000

Data/MC

2
1.5
1
0.5
0

0

200

400

600

Data
MC γ γMC γ j104
103

0

s = 13 TeV, 36.7 fb-1
ET<200 GeV,1.52<|η|<1.81

105

103

10

Data/MC

Entries

Entries

106

800

0

200

400

600

200

400

600

1000

0

∆E (2nd γ ) [MeV]DataMC γ γMC γ j

106

103

102

102

10

10

1

1

−1

−1

0

200

400

600

800

10

1000

2
1.5
1
0.5
0

0

200

400

600

s = 13 TeV, 36.7 fb-1
200 GeV<ET,|η|<1.37

105

Data
MC γ γMC γ j104

Data/MC

Data/MC

Entries

Entriess = 13 TeV, 36.7 fb-1
ET<200 GeV,1.81<|η|<2.37

103

10

800

0

200

400

600

200

400

600

1000

0

Data
MC γ γMC γ j

106

103

102

102

10

10

1

1

−1

−1

200

400

600

800

10

1000

2
1.5
1
0.5
0

0

200

400

600

s = 13 TeV, 36.7 fb-1
200 GeV<ET,1.81<|η|<2.37

105

Data
MC γ γMC γ j104

Data/MC

Data/MC

Entries

Entriess = 13 TeV, 36.7 fb-1
200 GeV<ET,1.52<|η|<1.81

103

0

800

1000

0

200

400

600

200

400

600

800

1000

2
1.5
1
0.5
0

0

∆E (2nd γ ) [MeV]

800

1000

∆E (2nd γ ) [MeV]

R

(e)

1000

(d)

106

10

800
R

(c)

104

1000

∆E (2nd γ ) [MeV]

R

10

800

2
1.5
1
0.5
0

∆E (2nd γ ) [MeV]

5

1000

(b)

106

104

800
R

(a)

10

1000

∆E (2nd γ ) [MeV]

R

5

800

2
1.5
1
0.5
0

R

(f)

Figure 40: Comparison of the distributions of ∆E obtained from the collision dataset and that from the simulated
samples. The distributions are shown for subleading reconstructed photons in the signal region, with additionally
Tight photon ID selection applied to the leading reconstructed photon; this is to ensure that the subleading reconstructed photon of the γ j simulated sample is originating dominantly from hadronic jets. The ratio of the normalizationsof the two simulated samples, γγ (green) and γ j (blue), is evaluated from their production cross sections. A scaling
is applied to the histograms of simulated samples, so that the event yields of the histograms from the collisiondataset and that from simulated samples are equal. The comparison is shown for different kinematic regions of thesubleading reconstructed photon, namely: (a) ET < 200 GeV, |η| < 1.37, (b) ET < 200 GeV, 1.52 < |η| < 1.81,
54(c) ET < 200 GeV, 1.81 < |η| < 2.37, (d) 200 GeV < ET, |η| < 1.37, (e) 200 GeV < ET, 1.52 < |η| < 1.81, (f)200 GeV < ET, 1.81 < |η| < 2.37.

Fraction in the low-∆E category [%]ma [GeV]ma [GeV]

10.0

96.6

77.7

56.8

33.8

17.5

10.0

8.9

90

5.0

82.8

35.8

15.1

10.5

9.3

10.1

14.0

80
70

2.0

30.1

11.1

11.5

15.8

22.8

50.9

77.1

1.0

13.8

18.3

37.2

63.0

81.4

95.2

97.3

0.7

15.3

37.2

71.7

90.1

95.5

97.8

98.3

0.5

24.4

69.5

92.5

96.7

97.8

98.1

98.4

0.1

97.5

200

97.9

400

98.0

600

98.3

800

98.3

1000

98.5

1500

98.4

2000
mX [GeV]

X → aa → 6π0, s = 13 TeV

ATLAS Simulation

60

10.0

56.3

33.3

6.2

5.4

3.0

2.2

2.2

5.0

24.3

6.8

3.8

3.2

2.9

5.4

12.2

2.0

6.8

4.7

10.6

21.9

37.9

78.6

93.8

1.0

7.6

36.5

73.6

92.0

96.9

98.4

98.5

0.7

20.5

80.4

96.5

98.1

98.3

98.5

98.5

10
0

0.5

58.6

96.7

97.9

98.6

98.4

98.5

98.6

200

400

600

800

1000

1500

2000
mX [GeV]

70
60
50

30
20

90
80

50
40

Fraction in the low-∆E category [%]X → aa → 4γ , s = 13 TeV

ATLAS Simulation

40
30

(a)

20
10
0

(b)

Figure 41: The category fraction f as a function of mX and ma , for (a) X → aa → 4γ and (b) X → aa → 6π 0 . [72]Table 8: Number of events in the collision dataset that satisfy the successive selection requirements. Also, theselection efficiency of signal events that satisfy the successive selection requirements, for X → aa → 4γ calculatedusing the simulated samples.

Selection efficiency X → aa → 4γ, mX = 600 GeV
All triggered events
Preselection
Loose0 photon selection
Isolation
ET selection
mγR γR > 175 GeV
Low-∆E category
High-∆E category

Events in collision data
6.4 × 109
3.1 × 107
1.7 × 107
2.2 × 106
1.7 × 106
6.7 × 104
5.6 × 104
1.0 × 104ma = 0.1 GeV
0.86
0.73
0.72
0.62
0.46
0.46
0.46
0.01

ma = 2 GeV
0.78
0.66
0.62
0.53
0.43
0.43
0.05
0.39

ma /mX ' 0.0015 for X → aa → 4γ and ma /mX ' 0.0020 for X → aa → 6π 0 .
The high-∆E category is found to have a significantly smaller number of background events comparedto the low-∆E category, as seen in Table 8. This is because the events of the main source of the SM
background, γγ, are mostly contained in the low-∆E category. Hadronic jets from SM processes containing
π 0 → γγ decays mostly lead to reconstructed photons with large ∆E, but their contribution in the high-∆E
category is small due to the isolation requirements. Thus, the introduction of this categorization increases
the sensitivity to photon-jet signals, especially in the region 0.002 < ma /mX < 0.01 by a factor two15.
As seen in Figure 40, a deviation of ∆E distribution from simulation and that from collision data isobserved; this is treated as a source of systematic uncertainty of the category fraction f , as described in
Section 5.6.2.

15The details of the sensitivity estimation using simulated samples are described in Appendix C.

55

5.4 Signal modelling
As described in Section 5.3 and in detail in Section 5.7, a search for an excess of events from the SM
background expectation in the mγR γR distributions is performed by a simultaneous fit to the distributionsin the two event categories. To perform this, the parametrizations of the probability density function ofmγR γR of the signal component (“signal mass shape”), signal selection efficiency ε, and category fraction
f are necessary. This is because the simulated signal samples are generated only for a few chosen pointsof (mX , ma ); it is necessary to evaluate the signal mass shape, ε, and f for any (mX , ma ) in the parameterregion of interest of this search, mX > 200 GeV and ma /mX < 0.01.
5.4.1 Signal mass shape
As the probability density function of mγR γR of the signal component, the double-sided Crystal-Ball(DSCB) function is used. The DSCB function has a Gaussian core surrounded by asymmetric exponential
low- and high-mass tails. It is an empirical function that is known to describe a reconstructed invariant
mass of a resonance with tails arising from e.g. bremsstrahlung. It is defined as2
e−v /2



2



e−α low /2
 i n low
 h α low  n low


−
α
−
v
low
fCB mγR γR ; µCB, σCB, αlow, αhigh, nlow, nhigh = N· n low α low


2

−α
/2


e highh
 i n high
 αn high αn high − α high + v
high high(−α low ≤ v ≤ α high )(v < −α low )(α high < v)

(60)
where ν is defined as ν = (mγR γR − µCB )/σCB , and N is the normalization factor. For signal events, eachreconstructed photon corresponds to a photon-jet, and so the reconstructed mγR γR corresponds to the massof the parent particle mX .
Figure 42 shows examples of a fit of the DSCB function to the mγR γR distributions of the simulated
signal samples. The width σCB is determined by the detector resolution. For small values of ma /mX(ma /mX . 0.005), σCB increases linearly with mX from σCB = 2 GeV for mX = 200 GeV to σCB = 14 GeV
for mX = 2 TeV. For larger values of ma /mX (ma /mX & 0.005), the width σCB increases further, and also
the peak position µCB decreases. This is because of the wider angular separation of the photons inside aphoton-jet; it leads to a greater fraction of the energy of the EM shower leaking out of the window definedby the cells of the EM calorimeter to collect energy for the reconstruction of photons.
The parameters of the DSCB function are parametrized as functions of mX and ma using simulated signal
samples, so that the mγR γR shape for any (mX , ma ) values can be estimated. The results are presented
in Tables 9 and 10 for X → aa → 4γ and X → aa → 6π 0 , respectively. Here, ∆mX is defined as∆mX = µCB − mX , i.e. the shift of the peak position with respect to mX . Variables t and u are defined
as t = mX /(100 GeV) and u = 100 × ma /mX . The values of nlow and nhigh are fixed to reasonable values
prior to the parametrization, and they are chosen to be nlow = 15 and nhigh = 5. The parametrization of
the other DSCB parameters are performed by a fit to the best-fit values of each parameter, obtained by
performing DSCB function fits to the simulated signal samples.

56

X

0.07

ma = 0.1 GeV
ma = 2 GeV
ma = 5 GeV

R R

0.08

ATLAS Simulation
X→aa→4γ , m = 600 GeV

1/N dN/dmγ γ [GeV-1]

R R

1/N dN/dmγ γ [GeV-1]

0.1
0.09

0.06
0.05

0.1
0.09
0.08
0.07

ma = 0.5 GeV
ma = 2 GeV
ma = 5 GeV

0.06
0.05

0.04

0.04

0.03

0.03

0.02

0.02

0.01

0.01

0
550 560 570 580 590 600 610 620 630 640 650

ATLAS Simulation
X→aa→6π0, mX = 600 GeV

0550 560 570 580 590 600 610 620 630 640 650

mγ γ [GeV]mγ γ [GeV]

R R

R R

(a)

(b)

Figure 42: Examples of a fit of the DSCB function to the mγR γR distribution of simulated signal samples. They are
shown for mX = 600 GeV and different values of ma , for (a) X → aa → 4γ and (b) X → aa → 6π 0 . They arepresented for the signal region, i.e. the combination of the low-∆E and high-∆E categories. [72]Table 9: The result of the parametrization of the signal mass shape for X → aa → 4γ.

Parameter∆mX [GeV]σCB [GeV]
αlow
αhigh
nlow
nhigh

Parametrization function
A × (1 + Bt) × (1 + Cu + Du2 + Eu3 )A × (1 + Bt) × (1 + Cu + Du2 + Eu3 )A + Bu(A + u)B + Cu
AAA
−0.0684
0.991
1.36
0.583
15
5

B
1.2
0.68
−0.491
−1.4

C
−5.54
0.265

D
50.4
1.81

E
−30.7
−1.55

1.28

As seen in Table 9, the parametrization functions for ∆mX and σCB are chosen to be a product of two
polynomial functions; one function of mX and one function of ma /mX . The former function parametrizes
the dependence of ∆mX and σCB on mX , especially the linear increase of σCB with respect to mX . The
latter function parametrizes the dependence of ∆mX and σCB on ma /mX , which is caused by the leakageof the energy of the EM shower from the photon reconstruction window, as described above.
Figures 43, 44, and 45 show the visualization of the parametrization result for X → aa → 4γ. Those for
X → aa → 6π 0 are qualitatively the same.
The same mass shape modelling results are used for both the low-∆E and high-∆E categories. Thisis because only a small change in the mγR γR distributions due to the categorization is observed. The
comparison of the signal mass shape modelling result and the mγR γR distribution of each event category
is presented in Figure 46.

57

Table 10: The result of the parametrization of the signal mass shape for X → aa → 6π 0 .

Parametrization function
A × (1 + Bt) × (1 + Cu + Du2 + Eu3 )A × (1 + Bt) × (1 + Cu + Du2 + Eu3 )A + Bu(A + u)B + Cu
AA

0∆mX / ∆mX(ma=0)∆mX(ma=0) [GeV]Parameter∆mX [GeV]σCB [GeV]
αlow
αhigh
nlow
nhigh

−0.2
−0.4
−0.6
−0.8

A
−0.807
1.65
1.36
0.411
15
5

18

C
3.93
−0.234

D
10.9
2.35

E
−5.57
−1.49

1.34

X200GeV
X400GeV
X600GeV
X750GeV
X800GeV
X1000GeV
X1500GeV
X2000GeV

16
14
12
10

−1

8

−1.2

6

−1.4

4

−1.6

2

0

B
0.115
0.394
−0.276−0.9130

200 400 600 800 10001200 14001600 18002000∆mX [GeV]=-0.0961*(1+0.822t)
3*(1-5.64u+50.2*u2-30.4*u )(t=m /100GeV, u=100*ma/mX )X
χ2/n=50.1/46=1.09

0

0.002

0.004

0.006

0.008

0.01
ma / mX

mX [GeV]Figure 43: The parametrization of ∆mX for X → aa → 4γ. The lines represent the result of the parametrizationthat is shown in Table 9. (Left) The first two factors of the parametrization function, A × (1 + Bt). (Right) The
third factor of the parametrization function, (1 + Cu + Du2 + Eu3 ). The markers represent the best-fit values of∆mX obtained from the fit of the DSCB function to the simulated signal samples, divided by A × (1 + Bt). The bars
represent the statistical uncertainty arising from the DSCB function fit.

Figure 44: The parametrization of σCB for X → aa → 4γ. The lines represent the result of the parametrizationthat is shown in Table 9. (Left) The first two factors of the parametrization function, A × (1 + Bt). (Right) The
third factor of the parametrization function, (1 + Cu + Du2 + Eu3 ). The markers represent the best-fit values of
σCB obtained from the fit of the DSCB function to the simulated signal samples, divided by A × (1 + Bt). The bars
represent the statistical uncertainty arising from the DSCB function fit.

58

α low=1.36-0.503*(100ma/mX )χ2/n=25.2/49=0.513

1.4

αhigh

αlow

1.6

1.2
1
0.8
X200GeV
X400GeV
X600GeV
X750GeV
X800GeV
X1000GeV
X1500GeV
X2000GeV

0.6
0.4
0.2
0

0

0.002

0.004

0.006

0.008

0.01

2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

α high=(u+0.579) +1.26u(u=100*ma/mX )χ2/n=29.5/48=0.615-1.38

X200GeV
X400GeV
X600GeV
X750GeV
X800GeV
X1000GeV
X1500GeV
X2000GeV

0

0.002

0.004

ma / mX

0.006

0.008

0.01
ma / mX(a)(b)

Figure 45: The parametrization of (a) αlow and (b) αhigh for X → aa → 4γ. The lines represent the result of theparametrization that is shown in Table 9. The markers represent the best-fit values of each variable obtained fromthe fit of the DSCB function to the simulated signal samples. The bars represent the statistical uncertainty arisingfrom the DSCB function fit.

59

70
60
50
40
30
20

EventsEvents80µ [GeV]=999.2
σ[GeV]=8.069
αlow =1.312
nlow =15αhigh=1.832
nhigh=5

16
14
12
10
8

mX =1000GeV, ma=1.0GeV
low-∆E category
χ2/91=1.438

6
4

10

0
970

980

990

1000 1010 1020 1030 1040 1050

950

MC/modelling

MC/modelling

960

1.5

mγ γ [MeV]

1
0.5
950

mX =1000GeV, ma=1.0GeV
high-∆E category
χ2/83=1.523

2

0
950

µ [GeV]=999.2
σ[GeV]=8.069
αlow =1.312
nlow =15αhigh=1.832
nhigh=5

960

970

980

990 1000 1010 1020 1030 1040 1050mγ γ [GeV]

960

970

980

990

1.5

1000 1010 1020 1030 1040 1050mγ γ [MeV]

1
0.5
950

960

970

980

990 1000 1010 1020 1030 1040 1050mγ γ [GeV]

R R

R R

70
60
50
40
30
20

EventsEvents(a)µ [GeV]=997.4
σ[GeV]=8.16
αlow =1.331
nlow =15αhigh=1.981
nhigh=54
32

mX =1000GeV, ma=1.0GeV
low-∆E category
χ2/94=1.547

1µ [GeV]=997.4
σ[GeV]=8.16
αlow =1.331
nlow =15αhigh=1.981
nhigh=5
mX =1000GeV, ma=1.0GeV
high-∆E category
χ2/60=1.54

10
0
960

970

980

1.5

990

1000 1010 1020 1030 1040 1050

950

MC/modelling

MC/modelling

0950

mγ γ [MeV]

1
0.5
950

960

970

980

990 1000 1010 1020 1030 1040 1050mγ γ [GeV]
R R

960

970

980

1.5

990

1000 1010 1020 1030 1040 1050mγ γ [MeV]

1
0.5
950

960

970

980

990 1000 1010 1020 1030 1040 1050mγ γ [GeV]
R R

(b)

Figure 46: Comparison of the signal mass shape modelling result and the mγR γR distribution of the simulated signal
samples. This is presented for mX = 1000 GeV, ma = 1 GeV for (a) X → aa → 4γ and (b) X → aa → 6π 0 . The
diagrams on the left are for the low-∆E category, and those on the right are for the high-∆E category.

60

5.4.2 Signal selection efficiency
The signal selection efficiency ε is parametrized as a function of mX and ma using simulated signal
samples, so that its value can be estimated for any (mX , ma ). ε is defined by Eq. (58). As described in
Section 5.3.1, the value of ε increases for larger mX because of the ET , |η|, and isolation requirements.
It also decreases for larger ma /mX , because a larger ma /mX leads to wider EM shower, and such events
are rejected by the trigger-level Loose ID selection and additionally the isolation requirement. The
parametrization of ε is used for the calculation of the cross section times branching ratios from observed
number of events for the benchmark signal scenarios.
The parametrization of ε is performed with 2 different methods: (1) a function of mX for separate valuesof ma , and (2) a function of mX and ma . The first method uses the following parametrization:
ε(mX ) =A
1 + B exp



C
mX

 +D

(61)

The second method uses the following parametrization:




1



 + F + G
ε(mX , ma ) = A · [1 − B exp(−CmX )] ·  1 + D exp E ma



mX



(62)

The first method is used for the evaluation of upper limits on the production cross section times branchingratios of the benchmark signal scenario as a function of mX for different values of ma , as shown in
Figure 64(a) in Section 5.8.2. The second method is used for the evaluation of the upper limits as afunction of ma /mX , as shown in Figure 64(b) in the same section. These two different methods are usedbecause the first method gives smaller systematic uncertainties compared to the second method, and so
the first method is optimal for the evaluation of upper limits as a function of mX .
The result of the first method is shown in Figure 47 for X → aa → 4γ. That for X → aa → 6π 0
is qualitatively the same, because the two benchmark signal scenarios share similar characteristics, as
described in Section 5.2. The ratio of the value of ε obtained from simulated samples and that from theparametrization is also presented in the figure. An envelope to cover these ratios, as shown in the lower
panels of Figure 47, is taken as a source of systematic uncertainty in the parametrization of ε.
The result of the second method is shown in Figure 48. The ratio of the value of ε obtained from simulated
samples and that from the parametrization is also shown. An envelope to cover these ratios is taken as asource of systematic uncertainty in the parametrization. The evaluation of the second method is limitedma
to m≤ 0.007, because this parametrization gives large deviation from the values of ε from simulatedX
samples (∼ 80%) for larger values of ma /mX . This is because the assumption that ε can be parametrized
by Eq. (62) breaks down in this region.

61

Signal selection efficiency

Fit

0.8
0.7

ma=0.1 GeV
ε=0.32 1+exp 6.25 -1.02t(t=m /100GeV)

0.6

-1

+0.30

X

0.5
0.4

1
Fit

0.8
0.7

0.4

0.2
0.1
1500

0
0

2000

mX [GeV]
Evaluated uncertainty

1.02
1

-1.86

0.5

0.1
1000

-1

X

0.3

500

ma=1 GeV
ε=2.66 1+exp 1.91 -2.23t(t=m /100GeV)

0.6

0.2

1.04

MC

0.9

0.3

0
0
MC/fitMC

MC/fitSignal selection efficiency

1
0.9

500

1000

1500

1.05

2000

mX [GeV]
Evaluated uncertainty

1

0.98
0.96

0

0.95

500

1000

1500

2000

0

500

1000

1500

mX [GeV]

Signal selection efficiency

1
MC

0.9

Fit

0.8
ma=2 GeV
ε=0.50 1+exp 17.31 -4.41t(t=m /100GeV)

0.6

-1

+0.03

X

0.5
0.4

1
Fit

0.8
0.7

0.4

0.2
0.1
0
0

2000

mX [GeV]
Evaluated uncertainty

1.05
1

500

1000

1500

2000

mX [GeV]
Evaluated uncertainty

1.05
1

0.95

0

+0.01

0.5

0.1
1500

-1

X

0.3

1000

ma=5 GeV
ε=0.57 1+exp 36.51 -3.99t(t=m /100GeV)

0.6

0.2

500

MC

0.9

0.3

0
0
MC/fit(b)

MC/fitSignal selection efficiency

(a)

0.7

2000
mX [GeV]

0.95

500

1000

1500

2000

0

500

1000

mX [GeV]

1500

2000
mX [GeV](c)(d)

Figure 47: Parametrization of ε as a function of mX for X → aa → 4γ, for (a) ma = 0.1 GeV, (b) ma = 1 GeV,(c) ma = 2 GeV, and (d) ma = 5 GeV. The lower panels show the ratio of the value of ε obtained from simulated
signal samples and that from the parametrization. The systematic uncertainty of this parametrization is evaluatedby an envelope to cover the deviations, as shown with horizontal blue dotted lines in the lower panels. For (d), theparametrization is limited to mX > 500 GeV because the parameter region of interest is limited to ma < 0.01 × mX .

62

MC / fit

Signal selection efficiency

1
ε=0.51[1-0.57exp(-0.25t)]×{[1+0.01exp(8.59u)]-1-0.01}+0.01(t=m /100GeV, u=100ma/mX )X
mX =200 GeV
mX =600 GeV
mX =2000 GeV
Fit

0.9
0.8
0.7
0.6
0.5

1.8

mX =200 GeV
mX =400 GeV
mX =600 GeV
mX =800 GeV
mX =1000 GeV
mX =1500 GeV
mX =2000 GeV
Evaluated uncertainty

1.7
1.6
1.5
1.4
1.3

0.4

1.2

0.3

1.1

0.2

1

0.1

0.9

0
0

0.002

0.004

0.006

0.008

0.8
0

0.01
ma / mX

0.005

0.01
ma / mX

Figure 48: Parametrization of ε as a function of mX and ma for X → aa → 4γ. (Left) The comparison of theparametrization with the values of ε obtained from simulated signal samples, for the cases mX = 200 GeV, 600 GeV,
and 2000 GeV. (Right) The ratio of the value of ε obtained from simulated signal samples and that from theparametrization. The systematic uncertainty of this parametrization is evaluated by defining an envelope to cover
the deviations, as shown with horizontal dotted lines.

5.4.3 Category fraction
The category fraction f is parametrized as a function of mX and ma using simulated signal samples, sothat its value can be estimated for any (mX , ma ). f is defined by Eq. (59). As described in Section 5.3.2,
the value of f decreases for larger ma /mX , because a larger ma /mX leads to larger ∆E of reconstructed
photons. The value of f serves as an input to the simultaneous fit to the mγR γR distributions, as describedin Section 5.7.
The values of f are taken from simulated samples and a third-order spline interpolation is performed asa function of ma /mX 16. For X → aa → 4γ, f is parametrized as a function of the ratio ma /mX . Forma
1
X → aa → 6π 0 , it is parametrized as a function of m·√
, where mπ 0 is the mass of π 0 .
2X
1+(3m π 0 /m a )These variables are chosen because they are roughly proportional to the angular separation of photons
consisting a photon-jet. The results of these interpolations are presented in Figures 49 and 50. An envelope
to cover the deviations of the values of f from simulation and the interpolation is taken as a source ofsystematic uncertainty in the parametrization of f ,pand this is also presented in Figures 49 and 50. The
envelope is defined with a function of the form A f (1 − f ), which is similar to a function commonlyused to estimate the uncertainty of the mean of a binomial distribution. The result of the modelling of f ismostly the same for the two benchmark signal scenarios, X → aa → 4γ and X → aa → 6π 0 . The main
distinction is in the different trend in f with respect to mX and ma , especially the threshold of ma /mX forf to transition from f > 0.5 to f < 0.5. This threshold is ma /mX ' 0.0015 for the decay X → aa → 4γ,
and ma /mX ' 0.0020 for X → aa → 6π 0 .

16The simulated signal samples for mX = 750 GeV is used for the interpolation, because these samples have the most statistics
of generated events compared to other simulated samples.

63

X750GeV
X200GeV
X400GeV
X600GeV
X800GeV
X1000GeV
X1500GeV
X2000GeV

0.8
0.6

0.14

0.1
0.08

0.2

0.06
0.002

0.004

0.006

0.008

0.1
0.05
0
−0.05
0

0.04

0.01
ma/mX

0.002

0.004

0.006

0.008

X200GeV
X400GeV
X600GeV
X800GeV
X1000GeV
X1500GeV
X2000GeV|∆f|=0.239 f*(1-f)

0.12

0.4

0
0

∆f

|∆f|

fraction in ∆ E category

1

0.02
0
0

0.01

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

ma/mX

1

fraction in ∆E category

(a)

(b)

Figure 49: (a) Parametrization of f as a function of mX and ma for X → aa → 4γ. (b) The deviation of the valuesof f obtained from simulated samples and those from the parametrization, ∆ f . It is plotted with respect to the valueof f obtained from the parametrization (i.e. the black line in Figure (a)). An envelope to cover the values of ∆ f ispresented with a blue dotted line.

(a)

(b)

Figure 50: (a) Parametrization of f as a function of mX and ma for X → aa → 6π 0 . (b) The deviation of the valuesof f obtained from simulated samples and those from the parametrization, ∆ f . It is plotted with respect to the valueof f obtained from the parametrization (i.e. the black line in Figure (a)). An envelope to cover the values of ∆ f ispresented with a blue dotted line.

645.5 Background modelling
The SM backgrounds in this search consist of mainly three components, γγ, γ j or jγ, and j j, as describedin the beginning of Section 5. The mγR γR distribution of the sum of these components is evaluated byan analytic function, separately for each of the two ∆E categories. The parameters of these two analytic
functions are determined from a fit to the mγR γR distributions of each event category of the collision datain the region mγR γR > 175 GeV.
The analytic function to be used in the analysis is chosen from a family of functions 17 adapted from thoseused by searches for new physics signatures in dijet final states [73]:



 Ík

j
1 a
gk x; a, b j j=0,k = N 1 − x 2 x j=0 b j (log x)(63)√
The variable x is defined as x = mγR γR / s. The parameters a and b j are free parameters and N is thenormalization factor. The value of k is chosen so that the analytic function evaluates the mγR γR distributionof the background component well, i.e. the associated systematic uncertainty is minimal. This choice ismade by performing the “Spurious Signal test”, which evaluates the systematic uncertainty arising fromthe choice of the analytic function, as described in Section 5.5.3. The test uses a dataset that reproduces
the mγR γR distributions of the two event categories of the collision dataset (the “background template”),
as described in Section 5.5.2. Prior to the derivation of the background template, the measurement of thecomposition of the background component is performed, as described in Section 5.5.1.
5.5.1 Background composition measurement
The background composition is measured using the matrix method [74], which uses the difference inthe calorimeter isolation energy of photons and hadronic jets, to decompose the event yields into four
components, γγ, γ j, jγ, and j j. The method is described in detail below.
For each of the two event categories, a region is defined where the calorimeter isolation requirement isremoved and all the other requirements are identical to the signal region. The events in this region of thecollision dataset are divided into four subsets by whether both, only the leading, only the subleading, orneither of the two leading reconstructed photons satisfy the calorimeter isolation requirement, as shown in
Figure 51. The number of events in each subset is denoted by F Pass, Pass, F Pass, Fail, F Fail,Pass , and F Fail, Fail ,respectively. The number of events of each background component in the region, Fγγ, Fγ j , Fjγ , and Fj j ,can be determined by solving the following equation:
ε1 ε2
ε1 f2
f1 ε2
f1 f2
Fγγ
F Pass, Pass
© Pass, Fail ª ©
ª©
ª
ε1 (1 − ε2 )ε1 (1 − f2 )f1 (1 − ε2 )f1 (1 − f2 )­ F
® ­® ­ Fγ j ®
­ Fail, Pass ® = ­®­®
(1 − ε1 )ε2(1 − ε1 ) f2
(1 − f1 )ε2(1 − f1 ) f2
­ F
® ­® ­ Fjγ ®
Fail, Fail« F
¬ « (1 − ε1 )(1 − ε2 ) (1 − ε1 )(1 − f2 ) (1 − f1 )(1 − ε2 ) (1 − f1 )(1 − f2 ) ¬ « Fj j ¬(64)Here, ε1 and ε2 denote the rate at which a prompt photon satisfies the calorimeter isolation requirement(“isolation efficiency”) for the leading and the subleading reconstructed photon, respectively. Similarly,
17These functions are motivated by the similarity to the PDF parametrization functions, and have been known empirically to
be applicable to background probability density functions of resonance searches.

65

Figure 51: Schematic diagram illustrating the background composition measurement using the matrix method. Notethat the region in the bottom left (corresponding to F Pass, Pass ) corresponds to the signal region.

f1 and f2 denote the rate at which a hadronic jet satisfies the calorimeter isolation requirement (“isolation
fake rate”). The values of ε1, ε2, f1 and f2 are given as inputs to the matrix method.
The isolation efficiencies ε1 and ε2 are estimated using simulated samples of prompt photons. As
described in Section 4.3, the calorimeter isolation variable of reconstructed photons in simulated samples
are adjusted by applying data-driven correction factors. A systematic uncertainty for the estimation of theisolation efficiency is evaluated by comparing the nominal value with that derived without these correction
factors applied.
The isolation fake rates f1 and f2 are estimated using subsets of collision data that are similar andorthogonal to the signal region (“control regions”), with the procedures described below:• Central values: The central values of the isolation fake rates f1 and f2 are estimated using subsets
of collision data in the control region. The control regions are defined by requiring the reconstructed
photon to fail the baseline Loose05 photon selection and satisfy the Loose07 photon selection (definedin Table 6). The Loose07 photon selection is a looser photon selection requirement compared to the
Loose05.
• Systematic uncertainty arising from the definition of the control region: The systematic uncertainty of the fake rate arising from the definition of the control region is evaluated as the change inthe value obtained by alternatively using the Loose06 (defined in Table 6) instead of the Loose07.
• Systematic uncertainty arising from the difference in the ETiso distributions: A shift of approximately 1 GeV is observed for the distribution of the calorimeter isolation energy ETiso between thesignal and control regions using simulated samples of hadronic jets. The effect of this shift in the
result is accounted for by shifting the threshold of the isolation requirement by ± 1 GeV, determining
the resulting change in the fake rates, and assigning this change as a systematic uncertainty.
Table 11 summarizes the obtained values of the isolation efficiency and fake rate, together with theirassociated uncertainties.
Finally, the composition of the background components in each of the two event categories is obtained by
the following equation:
Pass,Pass
Fγγ= ε1 ε2 Fγγ

FγPass,Pass= ε1 f2 Fγ jjPass,Pass
Fjγ= f1 ε2 Fjγ

FjPass,Pass= f1 f2 Fj j
j

66

(65)

Table 11: Estimated calorimeter isolation efficiency and fake rate for events in the region where the calorimeter
isolation requirement is removed and all the other requirements are identical to the corresponding event category. The
associated statistical uncertainty is derived from the dataset used for the evaluation, and the systematic uncertainty
is derived with the procedures described in the text.

ε1
ε2
f1
f2

Low-∆E category
0.927 ± 0.006(stat) ± 0.003(syst)
0.862 ± 0.007(stat) ± 0.004(syst)
0.220 ± 0.017(stat) ± 0.079(syst)
0.209 ± 0.011(stat) ± 0.068(syst)

High-∆E category
0.874 ± 0.007(stat) ± 0.006(syst)
0.793 ± 0.009(stat) ± 0.006(syst)
0.206 ± 0.005(stat) ± 0.058(syst)
0.171 ± 0.004(stat) ± 0.051(syst)

Table 12: The measured background compositions for the two event categories. [72] Note that they add up to 1 foreach event category.

γγ
γj
jγ
jj

Low-∆E category

High-∆E category

0.930+0.027−0.031
0.051+0.021−0.0180.014+0.004−0.0050.005+0.006−0.0030.48 ± 0.16
0.32+0.08−0.09
0.108+0.001−0.0160.09+0.09−0.05

The resulting background composition is shown in Table 12.
Figure 52 presents the validation of the result of the background composition measurement. This figurecompares the distribution of the calorimeter isolation energy ETiso obtained from the collision dataset andthat obtained as a result of the background composition measurement. Both distributions agree within
their associated uncertainties.

67

1/N dN/dET [1/4.0 GeV]

ATLAS
-1
s =13 TeV, 36.7 fb

0.4
0.3

iso

low-∆E category
leading γ R

iso

1/N dN/dET [1/4.0 GeV]

0.5

Data
Matrix Method Results (total)
Real-Real component

0.2

Real-Fake component
Fake-Real component
Fake-Fake component

0.1

0.45
ATLAS
-1
s =13 TeV, 36.7 fb

0.4
0.35

low-∆E categorysubleading γ R

0.3
0.25

Data
Matrix Method Results (total)

0.2

Real-Real component

0.15

Real-Fake component

0.1

Fake-Fake component

Fake-Real component

0.05
0
−40

−20

0

20

40

60

0
−40

80
iso
ET [GeV]

−20

0

40

60

80
iso
ET [GeV]

(b)

0.3
ATLAS
-1
s =13 TeV, 36.7 fb

0.2

high-∆E category
leading γ R

iso

0.25

1/N dN/dET [1/4.0 GeV]

iso

1/N dN/dET [1/4.0 GeV]

(a)

20

Data

0.15

Matrix Method Results (total)
Real-Real component
Real-Fake component

0.1

0.2
ATLAS
-1
s =13 TeV, 36.7 fb

0.18
0.16

high-∆E categorysubleading γ R

0.14
0.12

Data

0.1

Matrix Method Results (total)
Real-Real component

0.08

Real-Fake component

Fake-Real component

0.06

Fake-Real component

Fake-Fake component

0.04

Fake-Fake component

0.05

0.02
0
−40

−20

0

20

40

60

0
−40

80
iso
ET [GeV]

(c)

−20

0

20

40

60

80
iso
ET [GeV](d)

Figure 52: Comparison of the calorimeter isolation energy ETiso distribution of the collision dataset and the expecteddistribution based on the background composition measurement results. The distributions of the γγ component(dashed) obtained from simulated samples of prompt photons, and the γ j (dotted), jγ (dot-dashed), and j j (longdashed) components obtained from the control region of the collision dataset, are added using the backgroundcomposition measured with the matrix method. The results are compared for each of the two event categories where
they are shown for (a) the leading reconstructed photon in the low-∆E category, (b) the subleading reconstructed
photon in the low-∆E category, (c) the leading reconstructed photon in the high-∆E category, and (d) the subleading
reconstructed photon in the high-∆E category. These plots present the validation of the background composition
measurement result, as described in the text. [72]68

5.5.2 Background template
The background templates are determined using both the simulation and collision data, separately for eachof the two event categories.
For the background γγ component, a simulated sample of prompt diphoton events is used.
For the background γ j, jγ, and j j components, the subsets of collision data that are similar but orthogonal
to the signal region are used. For γ j, a subset with the subleading reconstructed photon failing the default
isolation requirement and satisfying a looser one is used. Similarly, for jγ, a subset with the leading
reconstructed photon failing the default isolation requirement and satisfying a looser one is used. For j j,a subset with both leading reconstructed photons failing the default isolation requirement and satisfying
a looser one is used. The looser requirement mentioned above is defined by loosening the thresholdfor the calorimeter isolation variable to ETiso < 7 GeV. The choice of this selection for the γ j, jγ, andj j components is validated using simulated samples of SM γ j and j j events; the shape of the mγR γR
distribution for events satisfying this requirement is identical to that for events contained in the signal
region, within the statistical uncertainty.

−2

10

ATLAS
s=13 TeV, 36.7 fb-1
low-∆E category

−3

10

−4

10

10−5

Fit
Background template:Sumγ γ Fraction: 93%
γj
5%
jγ
1%
jj
1%

R R

1
10−1

1/N dN/dmγ γ [GeV-1]

R R

1/N dN/dmγ γ [GeV-1]

The resulting samples of γγ, γ j, jγ, and j j are summed to derive the background templates, scaled withthe background composition results given in Table 12. The resulting background template is presented
in Figure 53. Figure 54 shows the comparison of the background template and the mγR γR distribution
obtained from the collision dataset. This figure shows the level of agreement between the two.
1
10−1
−2

10

10−3
10−4
10−5

−6

10

10−6

10−7

10−7

10−8

10−8

−9

10

500

1000

1500

2000

10−9

2500

mγ γ [GeV]ATLAS
s=13 TeV, 36.7 fb-1
high-∆E category

500

1000

Fit
Background template:Sumγ γ Fraction: 48%
γj
32%
jγ
11%
jj
9%

1500

2000

2500

mγ γ [GeV]

R R

R R

Figure 53: Background templates obtained from the sum of the background components for each of the two event
categories. The breakdown into components (γγ, γ j, jγ, and j j) is also shown. The unbinned likelihood fit withan analytic function (Eq. (63) with k = 1) is superimposed. [72]

69

103

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mγ γ [GeV]mγ γ [GeV]

R R

R R

(a)

(b)

Figure 54: Comparison of the background template (shown in Figure 53) and the mγR γR distribution obtained fromthe collision dataset. The bin width is set to be large (250 GeV) with the purpose of blinding; because the signal
mass shape width σCB is 30 GeV at most for the parameter region of interest of this search, an excess of events
due to a possible existence of signal events is not visible in this figure. A scaling is applied to the backgroundtemplate, so that the event yields of the background template and the distribution of the collision dataset are equal.
The comparison is shown for (a) low-∆E category and (b) high-∆E category.

5.5.3 Spurious signal test
The Spurious Signal test is performed to evaluate the size of the systematic uncertainty associated tothe analytic function fit, and to decide the analytic function to be used as the background probability
density function. The functions defined in Eq. (63) are tested. The “Spurious Signal” is defined as thepotential bias toward the identification of signal events when none is present, i.e. the systematic uncertainty
associated to the choice of the background probability density function. The size of the Spurious Signal, S,
is estimated by performing an unbinned maximum-likelihood fit to the background template of each event
category, using the sum of the signal and background probability density functions; the normalizationof the signal probability density function is taken as the value of S. For the signal probability density
function, the signal mass shape modelling described in Section 5.4.1 is used. The background templates
are normalized to the number of events observed in each event category of the collision dataset. The
Spurious Signal is allowed to be negative as well as positive.
The result of the Spurious Signal test is shown in Figures 55 and 56, with the signal mass shape for
X → aa → 4γ. For this result, Eq. (63) with k = 1 is tested as the analytic function. Additionallyin Figure 56, the estimated statistical uncertainty of the 37 fb−1 dataset associated to the signal yield isshown18. It is found that this analytic function has associated Spurious Signals that are considerably small,i.e. smaller than 30% of the estimated statistical uncertainty for most of the range of mX . Thus, thisanalytic function ( Eq. (63) with k = 1 ) is chosen as the background probability density function, because
18This estimation of the statistical uncertainty is performed by calculating the uncertainty associated to S arising from the fit,
without the consideration of the weights of the background template.

70the associated systematic uncertainty is evaluated to be sufficiently small. Figure 53 shows the level ofagreement between this analytic function and the background templates.
The size of the Spurious Signal is taken as the source of systematic uncertainty associated to the backgroundmodelling. In order to have a continuous parametrization of the Spurious Signal as a function of mX , theparametrization of Spurious Signal is performed with the function defined asmX
σSpurious (mX ) = 1 − √
s

a 

mX
√
s

 b+c log m√X
s(66)

This function is chosen considering a similarity with Eq. (63). The result of the parametrization of the
Spurious Signal is presented in Figures 55 and 56.
The same parametrized result of the Spurious Signal is used as the systematic uncertainty of the backgroundmodelling when combined with the signal mass shape for X → aa → 6π 0 as well. It is checked that theresult of the Spurious Signal test performed with the signal mass shape modelling of X → aa → 4γ andthat of X → aa → 6π 0 are in agreement, within the statistical uncertainty associated to the values of the
Spurious Signal.
The test result is validated by checking that similar results are obtained when the Spurious Signal test
is performed using variations of the background templates, for which the background compositions are
shifted within the uncertainties presented in Table 12. When the fraction of the γγ component is shifted
up and those for other components are shifted down, or vice versa, the size of the resulting Spurious
Signals are consistent within the statistical uncertainty of the background templates.

71

|S|

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ma = 0.1 GeV, low-∆E category

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Statistical uncertainty
Parametrization

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500

1000

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2500

500

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2000

2500
mX [GeV](e)(f)

Figure 55: Result of the Spurious Signal test performed with the signal mass shape modelling result for X → aa →4γ. The size of the Spurious Signal is displayed as a function of mX . The bars represent the uncertainty of the
Spurious Signal, evaluated with the consideration of the weights of the background template. The black dotted line
represents the estimated statistical uncertainty of the 37 fb−1 dataset associated to the signal yield. The blue dashedline represents the parametrization of the Spurious Signal as a function of mX . (a), (b), and (c) are for the low-∆E
category and different values of ma . (d), (e), and (f) are for the high-∆E category and different values of ma .

72

S / (Stat. unc.)

S / (Stat. unc.)

1.5
ma = 0.1 GeV, low-∆E category

1
Parametrization

0.5

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1
Parametrization

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1
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1
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(c)

−1.5
0

2500

1500

2000

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0

2500

500

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2000

2500
mX [GeV](e)(f)

Figure 56: Result of the Spurious Signal test performed with the signal mass shape modelling result for X → aa →4γ. The vertical axis represents the ratio S/δS, where S is the size of the Spurious Signal, and δS is the estimatedstatistical uncertainty of the 37 fb−1 dataset associated to the signal yield. The bars represent the uncertainty of the
Spurious Signal, evaluated with the consideration of the weights of the background template, divided by δS. The
blue dashed line represents the parametrization of the Spurious Signal as a function of mX , divided by δS (Notethat this blue dashed line appears to be zigzag; this is because of the limited precision for the evaluation of thedenominator, δS). (a), (b), and (c) are for the low-∆E category and different values of ma . (d), (e), and (f) are forthe high-∆E category and different values of ma .

73

5.6 Systematic uncertainties
Several sources of systematic uncertainties that affect the analysis are taken into account. In most cases,
systematic uncertainties are found to be smaller than the statistical uncertainty, as described in Sec. 5.8.2.
5.6.1 Signal mass shape modelling
The parametrization of the signal mass shape is described in Section 5.4.1. An “injection test” is performed
to validate the signal mass shape parametrization result, with the following procedures:
1. A pseudo-dataset is generated with the same number of background events as that in the collisiondataset and with a fixed number of signal events. The background events are generated from thebackground probability density function (Eq. (63) defined in Section 5.5, with k = 1) with its
parameters determined from a fit to the observed mγR γR distributions in the collision dataset. The
signal events are generated from the DSCB function with its parameters determined from a fit to
the mγR γR distribution of the simulated signal sample.
2. A fit is performed to the mγR γR distribution of the pseudo-dataset considering both the signal andbackground components. Eq. (63) with k = 1 is used for the background probability density
function, and the signal mass shape parametrization result (Table 9 or 10) is used for the signal
probability density function19.
3. Steps 1 and 2 are repeated 1000 times. This procedure is performed separately for the low-∆E andhigh-∆E categories, and for each signal hypothesis. Examples are shown in Figure 57.
4. Step 3 is repeated several times for different numbers of generated signal events, as shown for
example in Figure 58. The average of the number of signal events obtained from these injection
tests and the number of signal events in the pseudo-dataset should be identical in an ideal case. As
shown in Figure 58, a linear function fit to the relationship of the two numbers is performed. In an
ideal case, the slope of this linear function fit should equal unity. The deviation of the value of theslope from 1 is taken as a systematic uncertainty in the extracted signal yield due to the signal mass
shape parametrization.
The values of the systematic uncertainty, evaluated as a function of mX and ma , are presented in Figure 59.
This systematic uncertainty is found to be less than 5% for most of the cases20.
The impact of the systematic uncertainty of photon energy resolution on the parametrization of the signal
mass shape width σCB is evaluated. The description of the photon energy resolution estimation is given
in Section 4.1. Variations of the simulated samples of the benchmark signal models are produced withthe photon energy resolution adjusted by one standard deviation from the nominal value in both positive
and negative directions. Using these variations of the simulated samples, the parametrization of σCB asa function of (mX , ma ) is repeated with the same procedure as described in Section 5.4.1. The relative

19Note that the signal probability density functions that are used in Steps 1 and 2 are different, as described in the text.
20This evaluated systematic uncertainty is larger than 5% for large values of ma for the low-∆E category and small values ofma for the high-∆E category. This is because the numbers are overestimated due to the limited statistics of the simulated signal
samples for these cases. The impact of this overestimated systematic uncertainty on the analysis is negligible, because most of
the signal events are expected to be in the other event category for these cases (i.e. high-∆E category for large values of ma , andlow-∆E category for small values of ma ).

74

(a)

(b)

Figure 57: Examples of the injection tests performed for the validation of the signal mass shape parametrization
results. The distribution of the extracted signal yield obtained from the fit to pseudo-datasets is shown. They are
shown for X → aa → 4γ for mX = 1 TeV, ma = 1 GeV, and for (a) the low-∆E category and (b) the high-∆E
category. For (a), the number of generated signal events is 20. For (b), the number of generated signal events is 10.

(a)

(b)

Figure 58: Examples of the injection tests performed for the validation of the signal mass shape parametrization
results. The markers represent the extracted signal yield obtained from the fit to pseudo-datasets,
averaged over√1000 trials. The error bars show the standard deviation of the 1000 trials divided by 1000, and they are not visible
in the plots because they are small. The blue line represents the result of a linear function fit. They are shown for
X → aa → 4γ for mX = 1 TeV, ma = 1 GeV, and for (a) the low-∆E category and (b) the high-∆E category.

75

(a)

(b)

(c)

(d)

Figure 59: The results of the injection tests. The black markers represent the values of (mX , ma ) for which theinjection tests are performed. The coloured bands show the values evaluated by linear interpolation between theblack markers. They are shown for X → aa → 4γ for the (a) low-∆E category, (b) high-∆E category, and for
X → aa → 6π 0 for the (c) low-∆E category, (d) high-∆E category

76

(a)

(b)

Figure 60: Impact of the systematic uncertainty of photon energy resolution on the signal mass shape width σCB .
The relative difference in the value of σCB is presented as a function of mX . The relative difference is defined as
the difference between the parametrized value of σCB using a variation of the simulated signal samples and thatfrom the nominal signal mass shape modelling result, divided by the nominal result. The lines above 0 represent
the relative differences when the photon energy resolution is adjusted by one standard deviation from the nominal
value in the positive direction. The lines below 0 represent those when the photon energy resolution is adjusted in
the negative direction21. Those for a few values of ma are displayed as examples, for (a) X → aa → 4γ and (b)X → aa → 6π 0 .

difference with respect to the nominal parametrization of σCB is shown in Figure 60. The difference
ranges from a few percent to as large as 40%, increasing with larger mX and dependent slightly on ma .
5.6.2 Signal selection efficiency and category fraction
Uncertainties in the parametrization of the signal selection efficiency ε are evaluated by an envelope tocover the deviations of the values of ε from simulation and the parametrization, as described in Sec. 5.4.2.
It is found to be less than 12%.
Uncertainties in the modelling of the category fraction f are evaluated by an envelope to cover thedeviations of the values of f from simulation and the parametrization. The absolute value of the change
in f varies as a function of ma /mX , from 0.03 at ma /mX = 0, increasing to 0.12–0.14 at aroundma /mX = 0.002, and decreasing to 0.06–0.10 at 0.002 < ma /mX < 0.01.
Uncertainties in the values of ε and f from experimental uncertainty sources are evaluated. This isperformed by comparisons between nominal and variations of the simulated signal samples with each of
the sources adjusted by its associated systematic uncertainty. It is performed for the following sources of
uncertainties:• Photon energy scale: The description of the photon energy scale estimation is given in Section 4.1.

21For ma = 2 GeV (blue line) and low mX values (mX ∼ 200 GeV), the relative difference in σCB when the photon energy
resolution is adjusted in the negative direction appears to be larger than 0. For these cases, the relative difference is considered
to be sufficiently small and thus is ignored.

77• EM shower shape modelling: The description of the data-driven correction to shower shape variables
is given in Section 4.2. The effect of the shower shape modelling uncertainty on the values of ε isevaluated by shifting the shower shape variables by their data-driven corrections. As described in
Section 4.2, the data-driven corrections are not defined for the shower shape variable ∆E. Here, theeffect of the modelling uncertainty of ∆E on the values of f are evaluated by shifting the values of∆E by a few MeV22; the resulting uncertainty is 0.008 at most.
• Calorimeter isolation requirement: This is evaluated by simulated signal samples produced without
the data-driven corrections to the calorimeter isolation variable ETiso . The details of the corrections
are given in Section 4.3.
• Track isolation requirement: This is evaluated by using simulated signal samples produced without
the track isolation efficiency corrections. The details of the corrections are given in Section 4.3.
• Pile-up: A weight (“pile-up weight”) is defined at the event-level of the simulated samples to correct
the distribution of the number of pile-up to match that of the collision data. The difference in thenumber of vertices between the collision data and simulated data is propagated as an uncertainty inthe pile-up weight. The uncertainty due to the pile-up is evaluated by shifting this weight by ±1σ.
These uncertainties are less than 1% of ε or f in almost all cases, rising to 4% for some isolation andshower shape uncertainties for larger values of ma /mX at the edge of the analysis sensitivity.
The impact on ε due to the kinematic acceptance arising from the uncertainty of the PDF used for thesimulation is evaluated. This is evaluated by variations of the PDF set used for the simulation of the signal
samples. It is less than 1% in most cases, rising to 4% for large mX around mX ∼ 2 TeV.
5.6.3 Other sources
The systematic uncertainty of the background modelling is determined from the Spurious Signal method
as described in Section 5.5. This parametrization is performed in a way that can slightly overestimate thesize of Spurious Signals, especially at the lower end of the mX range, mX = 200 GeV. The size of theparametrized Spurious Signal decreases for larger mX , and depends slightly on the ma value. It ranges
from 6 × 10−3 to 85 for the low-∆E category, and from 1 × 10−2 events to 32 for the high-∆E category.
The uncertainty in the combined 2015+2016 integrated luminosity is 2.1%. The LUCID-2 detector isused for luminosity measurements, as described in Section 3.2.4. The systematic uncertainty of themeasurement is derived following a methodology similar to that detailed in Ref. [47].

5.7 Statistical analysis
5.7.1 Definition of the likelihood function
For a given hypothesis, a fit to the reconstructed diphoton mass mγR γR distributions is performed for therange mγR γR > 175 GeV using an unbinned maximum-likelihood approach, and simultaneously for thetwo event categories (low-∆E and high-∆E). Both the signal and background components are considered.
The ratio of the two signal normalization factors for the two event categories is fixed to f : (1 − f ).
22These values correspond to the data-driven corrections that were used by the ATLAS experiment for the pp collision dataset√
collected in 2011–2012 at s = 8 TeV.

78

Systematic uncertainties are treated with nuisance parameters in the likelihood function, where each is a
floating parameter constrained by a Gaussian function.
A per-event likelihood Li is defined for the two event categories, where i = 1 is for the low-∆E category
and i = 2 is for the high-∆E category. It is defined as
Li (x j ; µ, mX , ma, NB,i, k i, θ) = NS,i (µ, mX , ma, θ S ) + σSpurious,i θ Spurious,i · fS x j ; mX , ma, θ resolution(67)+ NB,i fB (x j ; k i )where each variable is defined as the following:√√• x j = mγR γR, j / s is the observed value of mγR γR divided by s for the j-th event in the i-th event
category• µ = σX × B is the parameter of interest, which is defined as the product of the signal production
cross section and the branching ratios• ki is the shape parameters of the background probability density function for the i-th category• θ collectively designates the nuisance parameters used to describe the systematic uncertainties• NS,i is the normalization of the signal probability density function for the i-th category• θ S is the subset of the nuisance parameters affecting NS,i
• σSpurious,i and θ Spurious,i are the value of the background modelling uncertainty (i.e. Spurious Signal)
and its associated nuisance parameter for the i-th event category• fS is the signal probability density function• θ resolution is the nuisance parameter associated to the systematic uncertainty of the photon energy
resolution• NB,i is the normalization of the background probability density function for the i-th event category• fB is the background probability density function
The normalization of the signal probability density function NS,i is defined asÖNS,i (µ, mX , ma, θ S ) = Lluminosity · µ · ε(mX , ma ) · fi (mX , ma, θ migration ) ·(1 + σnormalization,l )θnormalization, ll(68)
where each variable is defined as the following:• Lluminosity is the integrated luminosity of the collision data
• ε is the signal selection efficiency described in Sec. 5.4.2• fi are the fraction of signal events in the i-th category• θ migration is the subset of θ S for the systematic uncertainty sources that affect the category fraction,
as described in Section 5.6.2 (i.e. category fraction parametrization, photon energy scale, EM
shower shape modelling, calorimeter isolation, track isolation, and pile-up)79• θ normalization is the subset of θ S for the systematic uncertainty sources that affect the normalizationof the signal probability density function, as described in Sections 5.6.1, 5.6.2, and 5.6.3 (i.e. signal
mass shape modelling, selection efficiency parametrization, photon energy scale, EM shower shapemodelling, calorimeter isolation, track isolation, pile-up, PDF, and luminosity)• σnormalization,l is the size of the systematic uncertainties affecting the signal normalization
The fraction of events fi is defined as
f1 (mX , ma, θ migration ) = f (mX , ma ) +Õσmigration,m θ migration,m

(69)

m

f2 (mX , ma, θ migration ) = 1 − f1 (mX , ma, θ migration )(70)

where each variable is defined as the following:• f is the nominal value of the category fraction, described in Section 5.4.2• σmigration,m are the values of the systematic uncertainties affecting the category fraction
The signal probability density function fS is defined as
fS x j ; mX , ma, θ resolution = fCB mγR γR, j ; p CB (mX , ma, θ resolution )

(71)

0 ,
where fCB is a double-sided Crystal Ball function defined in Eq. (60). Its parameters p CB = {µCB, σCB
0 is defined as σ 0αlow, αhigh, nlow, nhigh } are the DSCB parameters described in Section 5.4.1 σCBCB =θresolution
σCB · (1 + σresolution ), where σCB is the nominal width and σresolution is the value of the systematic
uncertainty affecting the signal mass shape width due to the photon resolution uncertainty.

The background probability density function fB is defined as ai
1b +c log x j
2· xj i ifB (x j ; ai, bi, ci ) = Ni 1 − x j(72)

where Ni is the normalization factor, and k i = {ai, bi, ci } are the free parameters of the probability density
function.
The overall likelihood, including extended and constraint terms, areL(µ, mX , ma, NB,1, NB,2, k 1, k 2, θ) =
(73)
!
#
"


2 Ö

Ö
 Ö

 ni

θ l2
e−(NS,1 +NS,2 +N B,1 +N B,2 ) 


L
(x
;
µ,
m
,
m
,
N
,
k
,
θ)
exp
−
i
j
X
a
B,i
i



(n1 + n2 )!
2
 i=1  j=1
 l






Î
θl2
, is
where ni is the number of events in the i-th category in the dataset. The last factor,
l exp − 2
the Gaussian function to constrain the nuisance parameters associated to the systematic uncertainties.
The “visible cross-section” is defined as the product of the production cross-section, the branching ratios,
and the signal selection efficiency. Additionally to the calculation of limits on the product of the production
cross-section and the branching ratios, the calculation of the upper limits on the visible cross-section is80performed. In this case, the parameter of interest µ is defined as µ = σX × B × ε(mX , ma ), and thenormalization factor of the signal probability density function NS,i , defined in Eq. (68), is redefined asÖNS,i (µ, mX , ma, θ S ) = Lluminosity · µ · fi (mX , ma, θ migration ) ·(1 + σnormalization,l )θnormalization, l(74)lwhere θ normalization is defined as the subset of θ S which affects the normalization of the signal probability
density function but excluding those associated to the signal selection efficiency (i.e. signal mass shapemodelling uncertainty and the integrated luminosity uncertainty).
5.7.2 Hypothesis testing
The p-values are calculated with the profile likelihood ratio as the basis for the test statistic and utilizing
an asymptotic approximation [75]. The details of the test statistic and the asymptotic approximation are
given in Appendix D.
The calculation of p-values for the background-only hypothesis (p0 ) is performed from mX = 200 GeV to
mX = 2.7 TeV with a scan step of 1 GeV, for the benchmark signal scenario involving a scalar particle X
with a narrow width decaying via X → aa → 4γ. Since the samples for the benchmark signal scenario
were simulated for the mX values in the range 200 GeV < mX < 2 TeV, the results of the parametrizationsof the signal mass shape, category fraction f , and systematic uncertainties are extrapolated for the mX
values in the range 2 TeV < mX < 2.7 TeV.
The upper limits at the 95% CL on the production cross-section times the product of branching ratios
are calculated following the CLs modified frequentist method [76]. The “observed upper limit” and the“expected upper limit” are evaluated separately, where the observed limit is defined as a limit evaluatedusing the events in the signal region of the collision dataset, and the expected limit is defined as themedian of the limit assuming a background-only hypothesis. Upper limits are determined separately forthe two benchmark signal models, X → aa → 4γ and X → aa → 6π 0 , as a function of the valuesof (mX , ma ). The assumptions inherent in the use of the asymptotic approximation are validated bysampling distributions of the test-statistic using pseudo-experiments. This comparison is performed fora few signal mass points. The expected upper limits obtained from the asymptotic approximation arewithin 5% of those calculated with the pseudo-experiments sampling for most of the values of mX and ma
tested. Due to the small number of events in collision data in the region mγR γR > 1 TeV in the high-∆E
category, larger deviations are observed for mX > 1 TeV and large ma . The deviation is smaller than
5% at (mX , ma ) = (1 TeV, 10 GeV), and increases to 20% for (mX , ma ) = (1.5 TeV, 10 GeV) and 30% for(mX , ma ) = (2 TeV, 10 GeV), with the asymptotic approximation giving stronger limits. The details of theCLs method are given in Appendix D.
The evaluation of the upper limit at the 95% CL on the visible cross-section is performed as a function of
the values of (mX , f ). For the probability density function of the signal component, the signal mass shapemodelling result for the decay X → aa → 4γ is used.

81s=13 TeV, 36.7 fb-1
low-∆ E category

103

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R R

R R

Figure 61: The observed mγR γR distributions in the two event categories of the signal region. The result of abackground-only fit assuming µ = 0 is superimposed. The ±1σ uncertainty of the fit result, due to the uncertainties
in the fit function parameter values, is shown as a shaded band. In the lower panels, the ratios of the observed event
yield and the expectation from the fit result are presented. The blue arrows represent the bins with this ratio larger
than two, i.e. out of the range of the plot.

5.8 Results
5.8.1 Results of the null hypothesis test
The observed mγR γR distributions in the signal region is shown in Figure 61. The result of a backgroundonly fit assuming µ = 0 is superimposed in the figure.
The result of the p0 values, expressed in local significance (i.e. the number of standard deviations awayfrom the mean of a normal distribution) is presented in Figure 62. Two different regions can be observedin this plot, above and below the threshold at ma ∼ 0.0015 × mX . These are a result of the categorizationof events based on the shower shape variable ∆E. For ma . 0.0015 × mX , a larger fraction of signal
events is expected in the low-∆E category, and for ma & 0.0015 × mX , a larger fraction of signal events
is expected in the high-∆E category.
The largest local deviation from the background-only hypothesis is found to be 2.7σ, corresponding to mX= 729 GeV and ma = 0.1 GeV for X → aa → 4γ. The width of the signal mass shape for mX = 729 GeV
and ma = 0.1 GeV is σCB = 6 GeV, and thus this deviation appears as a small area in Figure 62. A small
excess of events is also observed centred around mX = 1.1 TeV and ma = 7 GeV, which corresponds to a
local deviation of 2.2σ.

82

(a)

(b)

Figure 62: Results of the search for deviations from the background-only hypothesis in the observed distributionsof the mγR γR , expressed in significance. They are presented as a function of mX and ma for the benchmark signal
scenario involving a scalar particle X with narrow width decaying via X → aa → 4γ. Only the values of local
significance that are positive are shown. The vertical axis represents ma for (a), and ma /mX for (b). [72]To quantify the statistical significance of obtaining the observed maximum local deviation (i.e. the “global
significance”), a test with pseudo-experiments is performed. Using the probability density function
obtained from the background-only fit (as presented in Figure 61), 40 sets of pseudo-datasets with thesame number of events as the collision dataset are generated. From these pseudo-datasets, the distributionof the maximum local significance is obtained by calculating the maximum local significance in the searchregion defined as mX values from 200 GeV to 2.7 TeV and ma values from 0.1 GeV to 0.01 × mX . The
result is presented in Figure 63. It is found that the observed maximum local deviation, 2.7σ, is lesssignificant than the median of the largest local deviation obtained in pseudo-experiments23. Thus, it isfound that the mγR γR distributions are consistent with the background-only hypothesis.

23Because the number of pseudo-datasets is 40, the uncertainty in the evaluation of the median is not small (∼ 0.07). However,
the important conclusion is that the observed maximum local deviation is not significant, and that the mγR γR distributions are
consistent with the background-only hypothesis.

83

Figure 63: The distribution of the maximum local significance, calculated from pseudo-experiments. The observed
maximum local significance, which is 2.7σ, is indicated by an arrow. The corresponding global p-value is 0.65,
which corresponds to a negative global significance.

5.8.2 Upper limits on signal cross-section times branching ratios
Since it is found that the mγR γR distribution of the collision dataset is consistent with the background-only
hypothesis, the 95% CL observed and expected upper limits on the product of the production cross-section
and the branching ratios are evaluated. They are presented in Figures 64 and 65.
Figure 64 shows the limits for the cross-section of the production via gluon–gluon fusion of a high-massscalar particle X with a narrow width, times the branching ratios into a pair of a particles and the subsequent
decay of each a into a pair of photons, σX × B(X → aa) × B(a → γγ)2 . Figure 64(b) illustrates the two
features of this search. First, when the ratio ma /mX is larger than a threshold of roughly 0.0015, moresignal events are expected in the high-∆E category, which has a significantly better signal-to-backgroundratio compared to the low-∆E category, thus leading to stronger upper limits. Second, for larger values ofma /mX , the decrease in the signal selection efficiency leads to weaker upper limits.
Figure 65 shows the limits for the cross-section times product of branching ratios for the decay of the ainto three neutral pions, σX × B(X → aa) × B(a → 3π 0 )2 . This result shows features similar to thatshown in Figure 64, with slight differences arising mainly from the different trend of the category fraction
f with respect to the values of mX and ma .
Figures 66 and 67 show the same results as Figures 64 and 65, together with the bands representing thestatistical uncertainty associated to the expected limits.

84

102

ATLAS
s = 13 TeV, 36.7 fb-1
X→aa→4γ NWA

Obs.

Exp. ma = 0.1 GeV

Obs.

Exp. ma = 1 GeV

Obs.

Exp. ma = 2 GeV

Obs.

Exp. ma = 5 GeV

Obs.

Exp. ma = 10 GeV

10

95% CL limit on σX×B(X→aa→4γ ) [fb]95% CL limit on σX×B(X→aa→4γ ) [fb]

103

1
10−1
10−20

500

1000

1500

2000

104
103

ATLAS
s = 13 TeV, 36.7 fb-1
X→aa→4γ NWA

Obs.
Exp. m = 200 GeV
X
Obs.
Exp. m = 400 GeV
X
Obs.
Exp. m = 600 GeV
X
Obs.
Exp. m = 800 GeV
X
Obs.
Exp. m = 1000 GeV
X
Obs.
Exp. m = 2000 GeV
X
Expected ±1σ for mX = 200 GeV
Expected ±2σ

102
10
1
10−10

0.001 0.002 0.003 0.004 0.005 0.006 0.007
ma/mX

mX [GeV](a)(b)

95% CL limit on σX×B(X→aa→6π0) [fb]Figure 64: The observed and expected upper limits on the production cross-section times the product of branchingratios for the benchmark signal scenario involving a scalar particle X with a narrow width decaying via X → aa →4γ, σX × B(X → aa) × B(a → γγ)2 . (a) The limits presented as a function of mX . The limits for ma = 5 GeV and
10 GeV do not cover as large a range as the other mass points, since the region of interest is limited to ma < 0.01×mX .
(b) The limits presented as a function of ma /mX . The statistical uncertanity bands for mX = 200 GeV are shown as
an example. [72]

103
102

ATLAS
s = 13 TeV, 36.7 fb-1
X→aa→6π0 NWA

Obs.

Exp. ma = 0.5 GeV

Obs.

Exp. ma = 1 GeV

Obs.

Exp. ma = 2 GeV

Obs.

Exp. ma = 5 GeV

Obs.

Exp. ma = 10 GeV

1000

1500

10
1
10−1
10−20

500

2000
mX [GeV]Figure 65: The observed and expected upper limits on the production cross-section times the product of branchingratios for the benchmark signal scenario involving a scalar particle X with narrow width decaying via X → aa → 6π 0 ,σX × B(X → aa) × B(a → 3π 0 )2 . The limits for ma = 5 GeV and 10 GeV do not cover as large a range as the othermass points, since the region of interest is limited to ma < 0.01 × mX . [72]

85

(a)

(b)

(c)

(d)

(e)

(f)

Figure 66: The observed and expected upper limits on the production cross-section times the product of branchingratios for the benchmark signal scenario involving a scalar particle X with a narrow width decaying via X →aa → 4γ, σX × B(X → aa) × B(a → γγ)2 . They are shown for different values of ma : (a) ma = 0.1 GeV, (b)ma = 0.5 GeV, (c) ma = 1 GeV, (d) ma = 2 GeV, (e) ma = 5 GeV, and (f) ma = 10 GeV.

86

(a)

(b)

(c)

(d)

(e)

Figure 67: The observed and expected upper limits on the production cross-section times the product of branchingratios for the benchmark signal scenario involving a scalar particle X with narrow width decaying via X → aa → 6π 0 ,σX × B(X → aa) × B(a → 3π 0 )2 . They are shown for different values of ma : (a) ma = 0.5 GeV, (b) ma = 1 GeV,(c) ma = 2 GeV, (d) ma = 5 GeV, and (e) ma = 10 GeV.

87

5.8.3 Comparison of upper limit with that from diphoton resonance search
As discussed in Section 1, this research shares some features with the search for a heavy (> 200 GeV)
resonant boson decaying into a pair of photons, performed by the ATLAS experiment at Run 2 for pp√
collisions at s = 13 TeV [25]. This search looked for new resonances leading to a final state with two
photonlike EM clusters in the calorimeter. The notable difference, compared to the search described inthis thesis, is the event selection strategy; the diphoton search uses the standard Tight photon ID selection(the “diphoton selection”), and does not introduce an event categorization.
A comparison of the expected upper limit on signal cross section times branching ratios, σX × B(X →aa) × B(a → γγ)2 , of the search described in this thesis, and that derived for the diphoton selection, isshown in Figure 68. The expected upper limit for the diphoton selection is evaluated with the following
prodedures. The signal selection efficiency is evaluated using simulated signal samples, and a linear
interpolation is performed to evaluate that for the (mX , ma ) values for which the simulated signal samples
are not produced. The parametrizations of the signal mass shape and the evaluation of the systematic
uncertainties are substituted with those used for the low-∆E category. The expected limits are calculated
by performing a fit to the mγR γR distribution of the events in the collision dataset that satisfy the diphoton
selection. There is a difference of approximately as much as a factor 20 between the two expected
limits. In conclusion, this research is able to search for photon-jet signal events with large ma (e.g.
0.001 . ma < 0.01 for X → aa → 4γ) with a sensitivity that was never achieved before.

Figure 68: Comparison of the expected upper limit on signal cross section times branching ratios, σX × B(X →aa) × B(a → γγ)2 , of the search described in this thesis, and that derived for the conventional diphoton selection.
For details, see text.

885.8.4 Impact of systematic uncertainties
The impact of systematic uncertainties on the analysis results are shown in Figures 69 and 70.
Figure 69 shows the impact of systematic uncertainties on the unconditional fit to the collision data,
where the parameter of interest µ = σX × BR is treated as a free parameter, for X → aa → 4γ. The
markers represent the “pull”. The pull is defined as the value of the nuisance parameters θ obtained froman unconditional fit. The bars of the markers represent the ±1σ uncertainty of the nuisance parameters
arising from the fit. The bar charts represent the ratio of the “impact” ∆µ and the total uncertainty σtot .
The impact ∆µ is defined as the deviation of the value of µ with respect to the nominal result, when an
unconditional fit is repeated with one of the nuisance parameters shifted by 1σ of its error and treatedas a fixed parameter. The variable σtot is the total uncertainty of µ from the nominal unconditional fit.
In principle, the ratio ∆µ/σtot corresponds to the contribution of the systematic uncertainty to the overall
uncertainty of the determination of the µ. The sources of systematic uncertainties are sorted with the sizeof |∆µ|, from the largest to the smallest. The results for X → aa → 6π 0 are qualitatively similar.
Table 13 summarizes the impact ∆µ/σtotal . Additionally, the contribution of the statistical uncertainty
divided by σtotal is presented. The contribution of the statistical uncertainty is evaluated by repeating
an unconditional fit with all the nuisance parameters fixed to their best-fit values, and obtaining the fit
error of µ. It is found that the analysis uncertainty is basically dominated by statistical uncertainty. The
exception is for low values of mX , e.g. mX = 200 GeV, for which the impact of the background modelling
uncertainty (i.e. Spurious Signal) is similar to that of the statistical uncertainty.
Figure 70 shows the impact of systematic uncertainties on the 95% CL expected upper limits on σX ×
B(X → aa) × B(a → γγ)2 . The expected limits evaluated with the nuisance parameters fixed to theirnominal values are compared with those for the nominal result. Both results are in agreement, except
for the low mass region mX ∼ 200 GeV. The impact of the systematic uncertainties on the expected
limit decreases with the resonance mass mX from 51% at most for mX = 200 GeV to 5% at most for
mX > 800 GeV. This is because the parametrization of the size of spurious signals slightly overestimates
the values at the lower end of the mX range, as described in Section 5.5.3.

Table 13: The impact of the sources of systematic uncertainties for the case of an unconditional fit. The sum inquadrature of the individual components differs from 100% due to small correlations between the components.
The values here are for X → aa → 4γ. The mass points (mX , ma ) = (200 GeV, 0.3 GeV), (600 GeV, 0.9 GeV)
correspond to those values for which the systematic uncertainty of the category fraction f is the highest.

mX [GeV], ma [GeV]
Statistical
Spurious signal (low-∆E)
Spurious signal (high-∆E)
Category fraction f
Signal mass resolution
Signal mass shape (low-∆E)
Signal mass shape (high-∆E)(200, 0.1)

(200, 0.3)

66%
74%
6%
7%
7%
3%
–

72%
37%
67%
19%
2%
1%
–

89

(200, 2) (600, 0.1)|∆µ|/σtotal
86%
9%
55%
9%
5%
–
1%

99%
13%
2%
3%
13%
5%
3%

(600, 0.9)(600, 5)

94%
5%
24%
25%
12%
4%
3%

98%
3%
22%
7%
1%
3%
2%

Pull
+1σ Postfit impact-1σ Postfit impact

mX=200 GeV ma=0.1 GeV
X→aa→4γ−1∆σ/ ∆σtot−0.5

Spurious signal (low-∆E)
Fraction modelling
Photon energy resolution
Spurious signal (high-∆E)
PileupSignal modelling (low-∆E)
Luminosity
Efficiency modelling

0

0.5

1

mX=200 GeV ma=2 GeV
X→aa→4γ−1

Spurious signal (high-∆E)
Fraction modelling
Spurious signal (low-∆E)
Photon energy resolution∆σ/ ∆σtot
−0.5

0

0.5

−1

0

1
2
(θ-θ0)/∆θ

1

Calorimeter isolation
Efficiency modelling
PileupLuminosity
Shower shape

Track isolation
Calorimeter isolation
PDF
Shower shape
Signal modelling (high-∆E)
Photon energy scale

PDF
Track isolation
Signal modelling (high-∆E)
Signal modelling (low-∆E)
Photon energy scale

−2

−1

0

−2

1
2
(θ-θ0)/∆θ

(a)

(b)

mX=600 GeV ma=0.1 GeV
X→aa→4γ−1∆σ/ ∆σtot
−0.5

0

0.5

1

Photon energy resolution
Spurious signal (low-∆E)
Signal modelling (low-∆E)
Photon energy scale
Shower shape
Fraction modelling

mX=600 GeV ma=5 GeV
X→aa→4γ−1

Spurious signal (high-∆E)
Fraction modelling
Pileup
Spurious signal (low-∆E)
Photon energy scale∆σ/ ∆σtot
−0.5

0

0.5

−1

0

1
2
(θ-θ0)/∆θ

1

Track isolation
PDF
Signal modelling (low-∆E)

Track isolation
Signal modelling (high-∆E)
PDF
PileupCalorimeter isolation
Luminosity
Signal modelling (high-∆E)
Efficiency modelling
Shower shape
Photon energy resolution

Calorimeter isolation
Efficiency modelling
Spurious signal (high-∆E)
Luminosity

−2

−1

0

−2

1
2
(θ-θ0)/∆θ

(d)

(e)

Figure 69: The impact of the sources of systematic uncertainties for the case of an unconditional fit. The markers
represent the pull. The bar charts represent the ratio of the impact ∆µ and the total uncertainty σtot . They arepresented for X → aa → 4γ, for (a) (mX , ma ) = (200 GeV, 0.1 GeV), (b) (mX , ma ) = (200 GeV, 2 GeV), (c)(mX , ma ) = (600 GeV, 0.1 GeV), and (d) (mX , ma ) = (600 GeV, 5 GeV).

90

(a)

(b)

Figure 70: Comparison of the expected 95% CL upper limits on the cross-section times the branching ratio for
X → aa → 4γ without the consideration of systematic uncertainties, and those for the nominal analysis result. (a)ma = 0.1 GeV and (b) ma = 2 GeV.

5.8.5 Upper limits on signal visible cross-sections
The visible cross-section is defined as the product of the signal production cross-section, branching ratios,
and the signal selection efficiency, i.e. σX × B(X → aa) × B(a → γγ)2 × ε for X → aa → 4γ, as definedalready in Section 5.7.1. The 95% CL observed limit on the visible cross-section for X → aa → 4γ ispresented in Figure 71, as a function of mX and f . The limits are calculated using the likelihood function
defined in Eq. (74). The limits are first calculated as a function of (mX , ma ), and are translated to afunction of (mX , f ) based on the relationship of ma /mX and f for X → aa → 4γ (shown in Figure 49(a)of Section 5.4.3). Here, the values of ma /mX are limited to ma /mX < 0.005, so that the relationship ofma /mX and f becomes a one-to-one correspondence.
The limits on visible cross-sections are considered to be model independent, and that they can be used toestimate the upper limit on signal cross-section times branching ratios for a given signal hypothesis if ε andf can be evaluated. One thing to note is that the evaluation of the limits on signal visible cross-sections are
performed with an assumption that the size of ma is small (approximately ma . mX × 0.01). The limit on
the visible cross-section will differ depending on the signal mass shape, especially the width σCB . Froma comparison of the limits on visible cross-sections for ma /mX = 0.002 and those for ma /mX = 0.01
(which correspond to same values of f as seen in Figure 49(a)), these two limits differ by 20%, with theone for ma /mX = 0.002 giving stronger (i.e. smaller) limit. This is because of the increase in the size ofσCB with respect to ma /mX , as described in Section 5.4.1.
The upper limits on the signal visible cross-sections are utilized for the reinterpretation of the results forthe case of long-lived a particles, as described in the next section.

91

(a)

Figure 71: The observed upper limit on the visible cross-section for the benchmark signal scenario involving a scalarparticle X with narrow width decaying via X → aa → 4γ, σX × B(X → aa) × B(a → γγ)2 × ε. [72]

5.9 Reinterpretation of the results for the case of long-lived a
For the benchmark signal scenario X → aa → 4γ, the case of a long-lived a is considered, as described in
the beginning of Section 5. This is because a long-lived a arises in BSM models, for instance in NMSSM
when the decay width of a, Γa , is small (e.g. Γa ∼ O(1 meV)), as discussed in Section 2.3.1.
5.9.1 Signal selection efficiency and category fraction parametrization
The decay radius r of the particlepa is defined as the distance of the position of its decay and the interaction
point in the x–y plane (i.e. r = x 2 + y 2 ). Its value is approximately r = γa βa ct sin θ, where γa is theLorentz factor of a and t is the decay time of the a at its rest frame. βa is the velocity of the particle a
divided by c, and it is approximately βa = 1 when γa = O(100). Hereafter, the factor βa c is omitted
mX 1
mX
for simplicity. The expected value of r can be approximated by the equation 2m, because γa ∼ 2m.
a ΓaaThe decay radius of up to a few meters is considered, corresponding to a small decay width satisfying themX 1
condition 2m. O(1 m).
a ΓaConsidering that the inner radius of the EM calorimeter is REM = 1.5 m, the important case is when r
is larger than r ∼ 10 cm. Two features arise compared to the case of the prompt decay of the particle
a: Firstly, the separation of photons entering the EM calorimeter decreases linearly with respect to r, as
depicted in Figure 72. Secondly, the selection efficiency is low for particle a not decaying before entering
the surface of the EM calorimeter, i.e. r > REM or | tanr θ | > ZEM . Here, ZEM = 3.7 m is the distance
between the end-cap section of the EM calorimeter and the interaction point. It is low because of the
Loose0 photon selection requirement: It requires a considerable portion of the EM shower to be depositedin the first layer of the EM calorimeter (namely f1 > 0.005 for the shower shape variable f1 defined inTable 5), and so events with particle a not decaying before entering the EM calorimeter are rejected from92

Figure 72: Schematic diagram of the separation of the photons originating from a → γγ at the inner surface of the
EM calorimeter, ∆R.

the event selection requirements. In this analysis, such events are considered to be completely excluded
from the signal region as a conservative approach.
The signal selection efficiency ε for the benchmark signal scenario X → aa → 4γ for the values(mx, ma, Γa ) can be evaluated by the following equation:
ε(mX , ma, Γa ) =∫p(ET,1, η1, t1, ET,2, η2, t2 |mX , ma, Γa )α(ET,1, η1, ET,2, η2 )2ÖεγR (ET,i, ηi, ti ) dET,i dηi dti

i=1

(75)
Here, ET,i, ηi, ti for i = 1 (i = 2) are the ET , η, t of the particle a with the higher (lower) ET . The
function p(ET,1, η1, t1, ET,2, η2, t2 |mX , ma, Γa ) is the probability density function of the variables. The
variable εγR (ET, η, t) is defined as the selection efficiency of a reconstructed photon originating from thephoton-jet a → γγ with the values of (ET, η, t).
The function α(ET,1, η1, ET,2, η2 ) takes the value of either 1 or 0; α = 1 when the two a particles are bothwithin the kinematic acceptance of the analysis, and α = 0 when they are not. Namely, α = 1 when thefollowing requirements are satisfied:• ET,1 > 0.4 × mX
• ET,2 > 0.3 × mX
• |ηi | < 1.37 OR 1.52 < |ηi | < 2.37 (i = 1, 2)
The parametrization of εγR (ET, η, t) is performed using simulated signal samples in a binned region of(ET, η). The parameter space of (ET, η) is binned in 4 × 4 = 16 regions; |η| < 0.6, 0.6 < |η| < 1.37, 1.52 <|η| < 1.81, and 1.82 < |η| < 2.37 in η, and 50 < ET < 70 GeV, 70 < ET < 100 GeV, 100 < ET <200 GeV, and 200 GeV < ET in ET . The binning of η is defined considering the difference in the detector
configuration in each region of η. The binning of ET is defined considering the dependence of the EM
shower development, hence the Loose0 selection efficiency, on ET .
The variable εγR in a given (ET, η) bin is parametrized with the following function:εγR (ET, η, t) =A+D
1 + exp B(ν + C)

93

(76)

Selection efficiency

Selection efficiency2|η|<0.6, 50<ET /GeV<70mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.8
1.6
1.4
1.2

-1

y=0.615 1+exp170(x-0.0273) +0.137

1

2
1.6
1.4
1.2
0.8

0.6

0.6

0.4

0.4

0.2

0.2
0.01

0.02

0.03

0.04

0.05

0
0

0.06 0.07 0.08m
γ tsinθ2 a 1ETREM

y=0.868 1+exp150(x-0.0282) -1+0.0823

1

0.8

0
0

|η|<0.6, 200<ET /GeV
mX = 800 GeV, ma = 1 GeV
mX = 800 GeV, ma = 5 GeV
mX = 800 GeV, ma = 10 GeV
mX = 1500 GeV, ma = 1 GeV
mX = 1500 GeV, ma = 5 GeV
mX = 1500 GeV, ma = 10 GeV

1.8

0.01

0.02

2

1.52<|η|<1.81, 50<ET /GeV<70mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.8
1.6
1.4
1.2

-1

y=0.564 1+exp142(x-0.03) +0.193

1

2
1.6
1.4
1.2
0.8
0.6

0.4

0.4

0.2

0.2
0.02

0.03

0.06 0.07 0.08m
γ tsinθ2 a 1ET
REM

0.04

0.05

0.06 0.07 0.08m
γ tcosθ2 a 1ETZEM-1

y=0.816 1+exp123(x-0.0335) +0.101

1

0.6

0.01

0.05

1.52<|η|<1.81, 200<ET /GeV
mX = 800 GeV, ma = 1 GeV
mX = 800 GeV, ma = 5 GeV
mX = 800 GeV, ma = 10 GeV
mX = 1500 GeV, ma = 1 GeV
mX = 1500 GeV, ma = 5 GeV
mX = 1500 GeV, ma = 10 GeV

1.8

0.8

0
0

0.04

(b)
Selection efficiency

Selection efficiency

(a)

0.03

0
0

0.01

0.02

(c)

0.03

0.04

0.05

0.06 0.07 0.08m
γ tcosθ2 a 1ET
ZEM(d)Figure 73: The result of the parametrization of εγR (ET, η, t). The value of εγR is shown as a function of v.
Examples in four (ET, η) regions are given: (a) |η| < 0.6, 50 < ET < 70 GeV, (b) |η| < 0.6, 200 GeV < ET , (c)1.52 < |η| < 1.81, 50 < ET < 70 GeV, and (d) 1.52 < |η| < 1.81, 200 GeV < ET . The results of other (ET, η)regions are given in Figure 86 in Appendix A.

where the variable ν is defined as
γa t sin θ
2ma


1−
(|η| < 1.37)

 ET

REM


ν=

γa t cos θ
2ma


1−
(1.52 < |η| < 2.37)

ZEM
 ET

(77)

This variable ν is roughly proportional to the separation of the photons originating from a → γγ at the
inner surface of the EM calorimeter, as depicted in Figure 72. The result of the parametrization of εγR isshown in Figure 73.
As described above, the value of εγR (ET, η, t) is sufficiently small when the particle a decays after passing
the inner surface of the EM calorimeter (i.e. γa t sin θ > REM for |η| < 1.37 and γa t cos θ > ZEM for1.52 < |η| < 2.37). As a conservative approach, the value of εγR (ET, η, t) is treated as 0 in such cases, i.e.
ν < 0.

94

Similarly, the category fraction f for X → aa → 4γ for the values (mx, ma, Γa ) can be evaluated by thefollowing equation:∫
1f (mX , ma, Γa ) =p(ET,1, η1, t1, ET,2, η2, t2 )α(ET,1, η1, ET,2, η2 )(78)
ε(mX , ma, Γa )
Ö×εγR (ET,i, ηi, ti ) fγR (ET,i, ηi, ti ) dET,i dηi dti
i

The variable fγR (ET, η, t) is defined as the fraction of reconstructed photons with the value of ∆E smaller
than the threshold, for reconstructed photons originating from the photon-jet a → γγ and satisfying theevent selection requirements. The variable fγR in a given (ET, η) bin is parametrized with the following
function:AD
fγR (ET, η, t) =+(79)
1 + exp B(v + C) 1 + exp E(v + F)
The result of the parametrization is shown in Figure 74.
Using Eq. (75), the signal selection efficiency ε(mX , ma, Γa ) can be estimated using the simulated signal
sample of the decay X → aa → 4γ for the case of the prompt decay of a and the same values of mX and
ma :
1 Õ j
j
j
j
j
ε(mX , ma, Γa ) = Í jw α(ET,1, η1, ET,2, η2 )
(80)
jw
j
∫ Ö
j
j
×
εγR (ET,i, ηi , ti )Γa e−Γa ti dti
iÍ
Here, j denotes the j-th event of the simulated signal sample, j is the summation for all events in thejjsimulated signal sample, w j is the weight associated to the simulated event, and ET,i, ηi are the values ofET,i and ηi of the simulated event. Similarly, from Eq. (78), the value of f (mX , ma, Γa ) can be estimatedwith the following equation:
Õ
1
j
j
j
j
w j α(ET,1, η1, ET,2, η2 )(81)
f (mX , ma, Γa ) =ε(mX , ma, Γa ) j
∫ Ö
j
j
j
j
×
εγR (ET,i, ηi , ti ) fγR (ET,i, ηi , ti )Γa e−Γa ti dti
i

The result of the estimation of the values of ε and f is shown in Figure 75, for some of the values of(mX , ma ). Also in the figure, the values of ε and f calculated using the simulated signal samples of
X → aa → 4γ are shown. A deviation of the value of ε obtained from the estimation and that obtained
from the simulated signal sample is shown in Figure 76(a). For 0.1 < ε, this deviation is at the level of 8%,
and for 0.03 < ε < 0.1, this is at the level of 20%. For ε < 0.03, this deviation is considerably large andso the estimation method described in this section is considered to be unreliable; thus, the reinterpretationof the results is not performed for a signal hypothesis resulting in ε < 0.03. A comparison of the valueof f obtained from the estimation, festimation , and that obtained from the simulated signal sample, fMC , isshown in Figure 76(b). The relationship between the two values is evaluated with a function defined as
fMC = max [0, min(A, B + C · festimation )](82)

From the fit, the parameters are determined as A = 0.98, B = −0.056, and C = 1.3. These deviations of
the estimated values of ε and f are taken into account in the analysis, as described in Sec. 5.9.2.

95

|η|<0.6, 50<ET /GeV<70mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.6
1.4
1.2

Fraction of events with small ∆E

Fraction of events with small ∆E

2
1.8

-1

y=1 1+exp350(x-0.00697) +0.965 1+exp-56.2(x-0.0337)

-1

1
0.8
0.6
0.4
0.2
0
0

2

|η|<0.6, 200<ET /GeV
mX = 800 GeV, ma = 1 GeV
mX = 800 GeV, ma = 5 GeV
mX = 800 GeV, ma = 10 GeV
mX = 1500 GeV, ma = 1 GeV
mX = 1500 GeV, ma = 5 GeV
mX = 1500 GeV, ma = 10 GeV

1.8
1.6
1.4
1.2

-1

y=1 1+exp271(x-0.00649) +1 1+exp-43.8(x-0.043)

0.8
0.6
0.4
0.2

0.01

0.02

0.03

0.04

0.05

0
0

0.06 0.07 0.08m
γ tsinθ2 a 1ET
REM

0.01

0.02

2

1.52<|η|<1.81, 50<ET /GeV<70mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.8
1.6
1.4
-1

y=1 1+exp338(x-0.00913) +0.866 1+exp-80.4(x-0.0362)

-1

1
0.8
0.6
0.4
0.2
0
0

0.03

0.04

0.05

0.06 0.07 0.08m
γ tsinθ2 a 1ETREM(b)Fraction of events with small ∆E

Fraction of events with small ∆E

(a)

1.2

-1

1

2

1.52<|η|<1.81, 200<ET /GeV
mX = 800 GeV, ma = 1 GeV
mX = 800 GeV, ma = 5 GeV
mX = 800 GeV, ma = 10 GeV
mX = 1500 GeV, ma = 1 GeV
mX = 1500 GeV, ma = 5 GeV
mX = 1500 GeV, ma = 10 GeV

1.8
1.6
1.4
1.2

-1

y=1 1+exp287(x-0.00719) +1 1+exp-43.9(x-0.0522)

-1

1
0.8
0.6
0.4
0.2

0.01

0.02

0.03

0.04

0.05

0.06 0.07 0.08m
γ tcosθ2 a 1ET
ZEM

(c)

0
0

0.01

0.02

0.03

0.04

0.05

0.06 0.07 0.08m
γ tcosθ2 a 1ET
ZEM(d)Figure 74: The result of the parametrization of fγR (ET, η, t). The value of fγR is shown as a function of v.
Examples in four (ET, η) regions are given: (a) |η| < 0.6, 50 < ET < 70 GeV, (b) |η| < 0.6, 200 GeV < ET , (c)1.52 < |η| < 1.81, 50 < ET < 70 GeV, and (d) 1.52 < |η| < 1.81, 200 GeV < ET . The results of other (ET, η)regions are given in Figure 87 in Appendix A.

96Selection efficiency

Selection efficiency

0.8
mX = 200 GeV
ma = 0.1 GeV
ma = 1 GeV
ma = 2 GeV
Estimation
MC

0.7
0.6
0.5
0.4

0.8

0.6
0.5
0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

20

40

60

80

100

120

0

140 160mX 1[cm]
2maΓ amX = 800 GeV
ma = 1 GeV
ma = 5 GeV
ma = 10 GeV
Estimation
MC

0.7

0

20

40

1.8
mX = 200 GeV
ma = 0.1 GeV
ma = 1 GeV
ma = 2 GeV
Estimation
MC

1.6
1.4
1.2
1

1.4
1.2

0.6

0.4

0.4

0.2

0.2
40

60

140 160mX 1[cm]
2maΓ a

1

0.6

20

120

80

100

120

140 160mX 1[cm]
2maΓ a(c)mX = 800 GeV
ma = 1 GeV
ma = 5 GeV
ma = 10 GeV
Estimation
MC

1.6

0.8

0

100

1.8

0.8

0

80

(b)
Category fraction

Category fraction

(a)

60

0

0

20

40

60

80

100

120

140 160mX 1[cm]
2maΓ a(d)mX 1
Figure 75: The values of ε and f for X → aa → 4γ. They are shown as functions of the expression 2m
. The
a Γa
lines represent the values evaluated using Eq. (80) and (81). The markers represent the values evaluated using thesimulated signal samples for the values (mX , ma, Γa ). (a) and (b) show the results of ε for the case of mX = 200 GeV
and mX = 800 GeV, respectively. (c) and (d) show the results of f for the same.

97

Category fraction from MC

MC / estimation

1.5

mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 800 GeV, ma = 1 GeV
mX = 800 GeV, ma = 5 GeV
mX = 800 GeV, ma = 10 GeV
mX = 1500 GeV, ma = 1 GeV
mX = 1500 GeV, ma = 5 GeV
mX = 1500 GeV, ma = 10 GeV

1.4
1.3
1.2
1.1
1
0.9

mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 800 GeV, ma = 1 GeV
mX = 800 GeV, ma = 5 GeV
mX = 800 GeV, ma = 10 GeV
mX = 1500 GeV, ma = 1 GeV
mX = 1500 GeV, ma = 5 GeV
mX = 1500 GeV, ma = 10 GeV

1.6
1.4
1.2

y = max( 0 , min( 0.976 , -0.0556 + 1.27*x ) )

1
0.8
0.6
0.4

0.8
0.7

2
1.8

0.2
0

0.1

0.2

0.3

0.4

0
0

0.5

Selection efficiency from estimation

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Category fraction from estimation

(a)

(b)

Figure 76: Comparison of the values of ε and f estimated from Eqs. (80) and (81), and those obtained from thesimulated signal samples. (a) The ratio of the values of ε obtained from simulated signal samples and from theestimation, shown as a function of the estimated value. (b) The value of f obtained from simulated signal samples
shown as a function of the estimated value. The dotted line represents the function used to evaluate the relationshipbetween the two values.

985.9.2 Reinterpretation procedure
The upper limit on the visible cross-section as a function of (mX , f ), shown in Figure 71, is considered tobe model-independent, i.e. applicable to signal models with photon-jets in the final state in general. Thisis because the effect of the difference in the signal mass shape for different signal models on the limit on
visible cross-section is small.
For the benchmark signal scenario X → aa → 4γ for the case of the long-lived a, the values of ε and f
can be estimated using Eqs. (80) and (81), respectively. Using these values, the upper limit on the visible
cross-section can be approximated by taking the corresponding value in Figure 71 as a function of (mX , f ).
Finally, the upper limit on the signal cross-section times branching ratios can be obtained by dividing theupper limit on the visible cross-section by the value of ε.
As discussed in Section 5.9.1 and shown in Figure 76(a), there exists a deviation of the value of ε obtained
from the estimation and that obtained from the simulated signal sample. For 0.1 < ε, this deviation isat the level of 8%, and for 0.03 < ε < 0.1, this is at the level of 20%. For ε < 0.03, this deviation isconsiderably large. To take this into account, the following procedure is taken as a conservative measure.
When the value of ε obtained from Eq. (80), εestimation , is smaller than 0.03, the signal hypothesis is ignoredfrom the analysis and the upper limit on the signal cross-section times branching ratios is not evaluated.
When 0.03 < εestimation < 0.1, the value of ε is reduced by a factor 0.8 (i.e. ε = εestimation × 0.8). When
0.1 < εestimation , the value of ε is reduced by a factor 0.92 (i.e. ε = εestimation × 0.92).
The systematic uncertainty of ε arising from sources of experimental uncertainties (i.e. photon energy
scale, isolation selection efficiency, shower shape modelling, pileup) are negligibly small compared to theuncertainty of the estimation procedure described above.
To take into account the ambiguity of the estimation of the value of f as shown in Figure 76(b), thefollowing procedure is taken as a conservative measure. The upper limit on the visible cross-section forfestimation and that for fMC obtained from Eq. (82) are compared, and the limit with larger value is taken as
the resulting value.
As discussed in Section 5.8.5, the limit on the visible cross-section slightly depends on the signal mass
shape, especially the width σCB , and the limit may increase (i.e. become weaker) by approximately 20%
for larger ma /mX . To take this into account, the resulting upper limit on the signal cross-section times
branching ratios is increased by a factor 1.2 when f < 0.5 as a conservative approach.
These procedures will result in the overestimate of the resulting upper limit on the signal cross-section
times branching ratios, as a conservative measure. As a validation of this procedure, the comparison of theupper limit for the case of the prompt decay of the particle a obtained from the estimation procedure, andthat obtained from the direct computation in Sec. 5.8.2, is performed. The result is shown in Figure 77.
This figure presents the level of agreement of the two methods. It is found that the upper limits obtained
from the estimation method are overestimated for some values of (mX , ma ), because of the conservative
approach as described above.
5.9.3 Results
The 95% CL observed upper limit on the product of the production cross-section and the branching ratios
for X → aa → 4γ for the case of the long-lived a is shown in Figure 78. It shows the upper limit on99

95% CL limit on σX×BR(X→aa→4γ ) [fb]

103
102

τa = 0 (prompt)ma = 0.1 GeV
ma = 2 GeV
ma = 5 GeV
ma = 10 GeV
Estimation
Nominal

10

1
10−1
10−20

200 400 600 800 1000 1200 1400 1600 1800 2000
mX [GeV]Figure 77: The comparison of the observed upper limits on the production cross-section times the product ofbranching ratios for X → aa → 4γ evaluated using two different methods. The solid lines represent the result
obtained from the reinterpretation procedure described in Sec. 5.9.2. The dotted lines represent the result obtained
from the direct computation described in Sec. 5.8.2.

σX × B(X → aa) × B(a → γγ)2 . The limit is calculated based on the result shown in Figure 71, with thereinterpretation procedure described in Section 5.9.2.
mX 1
A relatively stronger limit is placed for long-lived a around 2m∼ 40 cm, compared to that for prompta Γadecays. This is because of a mild increase (up to a factor 2) in the signal selection efficiency comparedmX 1
to that for prompt decays, as seen in Figure 75. For larger life time 2m> 50 cm, the limit becomesa Γaweaker due to the decrease in the signal selection efficiency.

100

102
10

95% CL limit on σX×B(X→aa→4γ ) [fb]95% CL limit on σX×B(X→aa→4γ ) [fb]103ma = 0.1 GeV
τa = 0 mm (prompt)
τa = 0.1 mm (Γa = 2 meV)τa = 0.5 mm (Γa = 400 µ eV)
τa = 1 mm (Γa = 200 µ eV)

1
10−1
10−20

200 400 600 800 1000 1200 1400 1600 1800 2000

103

ma = 1 GeV
τa = 0 mm (prompt)
τa = 1 mm (Γa = 200 µ eV)
τa = 2 mm (Γa = 100 µ eV)
τa = 5 mm (Γa = 40 µ eV)
τa = 10 mm (Γa = 20 µ eV)

102
10

1
10−1
10−20

200 400 600 800 1000 1200 1400 1600 1800 2000

mX [GeV]mX [GeV]

103
102
10

(b)
95% CL limit on σX×B(X→aa→4γ ) [fb]95% CL limit on σX×B(X→aa→4γ ) [fb](a)ma = 2 GeV
τa = 0 mm (prompt)
τa = 2 mm (Γa = 100 µ eV)
τa = 5 mm (Γa = 40 µ eV)
τa = 10 mm (Γa = 20 µ eV)
τa = 20 mm (Γa = 10 µ eV)

1
10−1
10−20

200 400 600 800 1000 1200 1400 1600 1800 2000

103

ma = 5 GeV
τa = 0 mm (prompt)
τa = 5 mm (Γa = 40 µ eV)
τa = 10 mm (Γa = 20 µ eV)
τa = 20 mm (Γa = 10 µ eV)
τa = 50 mm (Γa = 4 µ eV)

102
10

1
10−1
10−20

200 400 600 800 1000 1200 1400 1600 1800 2000

mX [GeV]mX [GeV](d)
95% CL limit on σX×B(X→aa→4γ ) [fb]

(c)
103
102
10

ma = 10 GeV
τa = 0 mm (prompt)
τa = 10 mm (Γa = 20 µ eV)
τa = 20 mm (Γa = 10 µ eV)
τa = 50 mm (Γa = 4 µ eV)

1
10−1
10−20

200 400 600 800 1000 1200 1400 1600 1800 2000
mX [GeV](e)

Figure 78: The observed upper limits on the production cross-section times the product of branching ratios forthe benchmark signal scenario involving a scalar particle X with a narrow width decaying via X → aa → 4γ,
σX × B(X → aa) × B(a → γγ)2 . (a) ma = 0.1 GeV, (b) ma = 1 GeV, (c) ma = 2 GeV, (d) ma = 5 GeV, and (e)ma = 10 GeV.

101

6 Interpretation of the results in the context of NMSSM
In the NMSSM, the process H → aa → 4γ can result in a pair of photon-jets, as described in Section 2.3.
Here, H is a scalar Higgs boson with a mass larger than 125 GeV, and a is a pseudoscalar Higgs bosonwith a mass of O(1) GeV. H is assumed to be produced by the gluon–gluon fusion process.
The interpretation of the search result (described in Section 5.8) is performed for the process H → aa →4γ24. A scan of the NMSSM parameters is performed to determine the parameter space in which thesignal hypothesis is excluded by the search result.

6.1 Method
The public computation package NMSSMTools 5.3.1 [77, 78] is used for the computation of the masses,
couplings, and decay widths of all Higgs bosons, as well as the masses and decays of the sparticles. The
Higgs spectrum is computed including full one-loop, Yukawa-driven two-loop, and pole corrections [79].
Experimental constraints from LEP, Tevatron, LHC experiments, and constraints from B and K physics arechecked. HiggsBounds 5.3.2 [80–84] is used to check the experimental constraints from LEP, Tevatronand LHC experiments for the Higgs sector. The production cross sections of the Higgs particles at 13 TeV
pp collisions are computed using SusHi 1.6.1 [85, 86], interfaced with LHAPDF 6.2.1 [87] and using
MMHT2014nnlo68cl [88] as the PDF.
For the computation results in this section, NMSSM parameters that are not directly related to the Higgs
sector are fixed to reasonable values, namely
2M1 = M2 =M3
= 4 TeV3
At = −20 TeV

Ab = Aτ = 1.5 TeV(83)m L̃ = 800 GeV
mQ̃ = 10 TeV
M1, M2, M3 are the gaugino masses, At , Ab, Aτ are the trilinear soft couplings, m L̃ is the soft masses ofthe sleptons chosen to be degenerate (for simplicity), and mQ̃ are the squark soft masses chosen to bedegenerate (for simplicity).
As described in Eq. (35) of Section 2.2.3, the NMSSM Higgs sector has six independent parameters, which
are chosen to be µ, tan β, λ, κ, Aλ, Aκ in this thesis. In the following computation results in this section,µ and tan β are fixed to chosen values for simplicity. As discussed already in Section 2.3, the values oftan β are constrained to 1 . tan β . 10–80, considering that the top quark Yukawa coupling should not
become too large at the GUT scale, and the exclusion limit in the (M A, tan β) plane for the hMSSM model(shown in Figure 6). From the LEP limit for the higgsino mass (shown in Figure 7), an approximate limitof µ & 100 GeV is obtained. Also, a value of µ close to the electroweak scale is considered to be “natural”

24The process H → aa → 6π 0 → 12γ is not considered. This is because this process arise in the NMSSM only when ma is
within some value, as described in Section 2.3.1 as Case 2. Such case is out of scope of this thesis, and is not considered.

102

(derived from the value of m2Z in Eq. (28)). Thus, taking these constraints into consideration, the valuesof µ and tan β are chosen to be the following:µ = 110 GeV(84)

tan β = 5Note that with the parameter sets in Eqs. (83) and (84), the value of ∆m( χ̃1±, χ̃10 ) = m χ̃1± − m χ̃0 , i.e. the mass1

difference between the lightest chargino and the lightest neutralino, is typically ∆m( χ̃1±, χ̃10 ) = 1.2 GeV.
Thus, the parameter sets that are tested (Eqs. (83) and (84)) are not excluded by past searches shown in
Figure 7.

6.2 Scan region
A parameter scan is performed in the scan region defined as the following:
0.028 < λ < 0.043
3.2λ < κ < 4.4λ2κ2κµ × 1.08 < Aλ <µ × 1.14
λλ
0.1 GeV < Aκ < 0.7 GeV(85)

The scan step of λ is 0.001. The scan steps of κ and Aλ are defined as 1/20 of their scan regions. The scanstep of Aκ is 0.005 GeV.
The rationale for the choice of this scan region is described below.
Scan region of AλFor the pseudoscalar a to have a sizeable B(a → γγ) (e.g. more than 30%), a has to be a nearly puresinglet state (i.e. the decoupling limit), as described in Section 2.3. In such a case, the branching ratio ofa into a pair of SM fermions or gluons is considerably small; a can dominantly decay into γγ via a loopof charged higgsinos.
The mass matrix of pseudoscalar Higgs bosons, in the basis (AD = cos βHuI + sin βHdI , AS = SI ) after
rotating away the Goldstone mode, is given in Eq. (40). This is rewritten below for convenience:


2κ
2
M
λ
A
−
µ
v ª
λ
©
λ
A

MP2 = ­ (86)®
λ Aλ − 2κµ v
M A2 Sλ«¬
2µ Aλ + λκ µsin 2β 24κ
v sin 2β 3κ22
M AS = λ Aλ +
µ
−
Aκ µ
λ
2µ
λ
M A2 =At tree level, the decoupling limit of the pseudoscalar Higgs bosons is given in Eq. (50) as:
2λ Aλ − 2κµv
λtan 2θ A =1
M A2 − M A2 S

103

(87)
(88)

(89)

Figure 79: The computational results of B(a → γγ) with respect to Aλ and ma , computed in the region definedwith Eq. (85).

Thus, the following condition is considered:
Aλ −

2κ
µ∼0
λ

(90)

6κ
2
Note that in this case, M A2 ∼ λ sin
2β µ .

The values of B(a → γγ) as a function of Aλ − 2κ
λ µ is shown in Figure 79. The value of B(a → γγ)µ∼
0;
however,theactualdecoupling limit is achieved when Aλ − 2κ
increases to 1 when Aλ − 2κλλ µ ∼90 GeV, as seen in Figure 79, because of the loop corrections to the mass matrix. Thus, to take this shift
into account, the scan region of Aλ is defined as Aλ = 2κλ µ× (1.08–1.14).
A notable feature of Figure 79 is the change in the distribution of B(a → γγ) with respect to ma . For
ma > 2mτ , the decay a → ττ opens up, and thus the region of Aλ in which B(a → γγ) is near 1 becomes
smaller compared to ma < 2mτ . For ma = 9.4 GeV or 10 GeV, B(a → γγ) decreases due to the mixingof a with a pseudoscalar bottomium.
Scan region of Aκ
In the decoupling limit, ma ∼ M AS . As discussed in Section 5.8, the photon-jet search result is sensitive
roughly to the region ma < 0.01×mX . For simplicity, the value of ma is limited to the region ma < 10 GeV
in the following discussions in this section. Thus, applying the limit 0 < M A2 S < (10 GeV)2 in Eq. (88),the value of Aκ is limited to O(0.1 GeV). This relationship is shown in Figure 80.

104

Figure 80: The computational results of ma projected in the (Aκ , λ) plane, computed in the region defined with
Eq. (85). The plotted points are limited to the region 0 < ma < 10 GeV.

Scan region of λ and κσ(ggF → H → aa) is the product of the production cross section of the heavy Higgs via gluon–gluon
fusion process and the branching ratio into aa pair. For the value of σ(ggF → H → aa) to be sizeable(e.g. around the order of O(1 fb)), it is necessary for the heavy Higgs to have both a large ggH coupling(e.g. ∼ 10% of the SM Higgs-like boson with the same mass) and a large B(H → aa) (e.g. more than 0.1).
Thus, the BSM scalar Higgs in question has to be a mixed state of the doublet and singlet components, as
described in Section 2.3.
The mass matrix of electrically neutral scalar Higgs bosons in the basis (HSM = cos βHdR +sin βHuR, HD =− sin βHdR + cos βHuR, HS = SR ) is given in Eq. (36). This is rewritten below for convenience:2
M A sin 2β
2
2
2
2
2
2
2
2
© MZ cos β + λ v sin 2β (λ v − mZ ) sin 2β cos 2β 2λµv 1 −
ª
2µ
­
®
­
λv
2 sin 2β cos 2β ®
2MMH−MS2 = ­ (λ2 v 2 − m2Z ) sin 2β cos 2β
(91)
®
2µ A
D
­
®
2

­
®
sin 2β22− λv
MH2λµv 1 − M A2µ
2µ M A sin 2β cos 2β
S
«
¬
2
MH
= M A2 + (m2Z − λ2 v 2 ) sin2 2β
D
!
2 


2
λM
λA
κλ
2κ
κ
A
2
MH
=
µ
1+
−
v 2 sin 2β 1 −
S
λ
4κ µ
2
κ µ2

(92)
(93)

2κ
κ
When M A  v, MHD ∼ M A. When 1  λA
4κµ , MHS ∼ λ µ. For the heavy scalar Higgs to be a mixed stateof the doublet and singlet components, MHD ∼ MHS . Under these assumptions, the following relationshipis obtained:Aλ +κ
2κ 2 sin 2β
µ∼
µ
λ
λ2

(94)

Considering the decoupling limit of the pseudoscalar Higgs bosons, Eq. (90),κ3∼λ 2 sin 2β

105

(95)

(a)

(b)

Figure 81: The computational result of (a) σ(ggF → H → aa) and (b) mH projected in the (λ, κ) plane, computedin the region defined with Eq. (85). For details, see text.

For tan β = 5, this relationship leads to κ ∼ 3.9λ.
Figure 81(a) shows the values of σ(ggF → H → aa) projected in the (λ, κ) plane. The maximum of thetwo values, σ(gg → H2 → aa) and σ(gg → H3 → aa), is plotted. It is found that, for a fixed value of λ,σ(ggF → H → aa) is maximum at κ ∼ 3.9λ. Additionally, the maximum value of σ(ggF → H → aa)
decreases for smaller λ. This is because B(H → aa) decreases, which is because the coupling constantof H and a is proportional to κ.
Figure 81(b) shows the values of mH projected in the (λ, κ) plane. The value of mH is chosen to be mH2 if
σ(gg → H2 → aa) > σ(gg → H3 → aa), or mH3 otherwise. It is found that mH is roughly proportionalto λκ .
Note that from Eqs. (93) and (95), the following relationship is derived:
MHS ∼

3
µ
sin 2β(96)

Thus, when considering a large σ(gg → X → aa → 4γ) (e.g. around the order of O(1 fb)), the value of
MHS is roughly determined by two variables, µ and tan β.

6.3 Result
The result of the computational scan is shown in Figure 82. The scan region is that defined in Eq. (85).
The selections applied to the plotted points are the following:• The derived parameters are not excluded by measurements from past experiments at 95% CL25.
This is checked with the use of NMSSMTools and HiggsBounds, as described in Section 6.1.
• 0 < ma < 10 GeV
25This excludes the measurement result of muon g − 2. A dark matter relic density smaller than the observed value is allowed.
The SM Higgs mass is allowed to be in the range 122–128 GeV, considering the uncertainty of the calculation of its mass.

106

(a)

(b)

(c)

Figure 82: The computational result of σ(gg → H → aa → 4γ) with respect to ma . Only the parameter sets with
B(a → γγ) > 0.3 are plotted. The dotted line shows the observed upper limit on σ(gg → H → aa → 4γ) at 95%
mH 1
CL for the case of a prompt decay of the particle a. The z axis shows the values of (a) Γa , (b) 2m, and (c) mH .
a Γa
For (b), the value of z-axis is plotted as 104 cm if it is larger than 104 cm. For detail, see text.

• B(a → γγ) > 0.3
For the figure, the maximum of the two values, σ(gg → H2 → aa → 4γ) and σ(gg → H3 → aa → 4γ),
is plotted. The value of mH is chosen to be mH2 if σ(gg → H2 → aa) > σ(gg → H3 → aa), or mH3
otherwise.
mH 1
For the z axis of Figure 82(b), the variable 2m
is plotted; this is approximately equal to the average ofa Γathe γa βa cτa of the a at the lab frame. In these figures, the observed limit for the case of mH = 860 GeV
and the prompt decay of a (i.e. Γ1a = 0) is plotted with a dotted line. As described in Section 5.8,
the observed limit for the prompt decay case of a is applicable (within a few tens of %) to the cases of
mH 1
2m a Γa . 1 m. Thus, from these figures, it is found that the photon-jet search result is sensitive to someparameter sets in the region 2 GeV . ma . 5 GeV.

Figure 83 shows the parameter region rejected by the photon-jet search at 95% CL. In this figure, the 4dimensional parameter space (λ, κ, Aλ, Aκ ) is projected in the 3-dimensional space (λ, κ, Aκ ) (i.e. projected

107

(a)

(b)

(c)

Figure 83: The 4-dimensional parameter space (λ, κ, Aλ, Aκ ) projected in the 3-dimensional space (λ, κ, Aκ ) (i.e.
projected in the Aλ direction). The colours of the markers show the value of the z axis, i.e. Aκ , to help the readabilityof the plots. (a) The parameter region with 0 < ma < 10 GeV. (b) The subset of (a) which is not excluded bymeasurements from past experiments at 95% CL. (c) The subset of (b) which is excluded by the photon-jet searchat 95% CL.

in the Aλ direction). Figure 83(a) shows the parameter sets with 0 < ma < 10 GeV. Figure 83(b) shows
the subset of Figure 83(a) which is not excluded by measurements from past experiments at 95% CL25.
Figure 83(c) shows the subset of Figure 83(b) which is excluded by the photon-jet search at 95% CL.
Figure 84 shows the results shown in Figure 83(c) projected in the (λ, κ) plane or the (λ, Aκ ) plane.
From the result presented in Figure 83(c), it is summarized that the photon-jet search result excludes a
subset of the (λ, κ, Aλ, Aκ ) parameter space for the case µ = 110 GeV and tan β = 5, namely:• 0.030 < λ < 0.040• κ ∼ 3.9λ 26
26One thing to note from Figure 84(a) is that the parameter space rejected by the photon-jet search result is centred aroundtwo values, κ/λ ' 3.9 and κ/λ ' 3.7. This is explained by the fact that the observed upper limit for the case of mX = 800 GeV
is stronger by a factor 2 compared to that of mX = 860 GeV , due to the statistical fluctuation, as seen in Figure 78.

108

Aκ [GeV]

κ
0.16

0.55
0.5

0.15
0.45
0.14
0.4
0.13

0.35

0.12

0.3

0.11

0.25
0.03 0.032 0.034 0.036 0.038 0.04 0.042

0.03 0.032 0.034 0.036 0.038 0.04 0.042

λ

λ

(a)

(b)

mH 1
× [cm]
2ma Γ aFigure 84: The region of parameter space (λ, κ, Aλ, Aκ ) excluded by the photon-jet search at 95% CL, projected inthe (a) (λ, κ) plane and (b) (λ, Aκ ) plane.
200
180

Surviving past experiments

160

and BR(a→γ γ )>0.9

140

Rejected by the analysis

120
100
80
60
40
20
0
0

1

2

3

4

5

6

7

8

9

10

ma [GeV]
mH 1
Figure 85: The excluded parameter sets by the photon-jet search at 95% CL in the ma, 2m
plane. The blacka Γamarkers are the parameters sets that are not excluded by measurements from past experiments at 95% CL andB(a → γγ) > 0.9. The red markers are the subset of the black markers which are excluded by the photon-jet search.

• Aλ ∼ 2κ
λ µ such that B(a → γγ) ∼ 1• Aκ ∼ O(0.1 GeV) which is corresponding to 2 GeV . ma . 5 GeV
The limit λ < 0.040 is derived because the region λ > 0.040 is rejected by the past measurement, namely
the SM Higgs cross-section measurement at the LHC experiments. A larger value of λ leads to a larger
B(h → aa), where h is the SM-like Higgs boson, and thus a smaller cross section of h → Z Z than themeasured value.

mH 1
Figure 85 shows the parameter sets rejected by the photon-jet search result in the ma, 2m
plane.
Γa a
mH 1
One thing to note is that 2m
is roughly proportional to the inverse of m2a . Another thing to note isa Γathat the region which the photon-jet search result is sensitive to is found to be 2 GeV . ma . 5 GeV andmH 1
2m a Γa . 1 m.

109

Finally, it should be mentioned here the dependence of the excluded parameter space on the values of(µ, tan β). The discussion in this section is limited to the case of µ = 110 GeV and tan β = 5, forsimplicity. A similar result is inferred for other cases of (µ, tan β), with the following two points taken
into consideration. Firstly, as in Eq. (96), the value of mH is determined by the values of (µ, tan β) forthe case of a sizeable σ(gg → H → aa → 4γ). The observed upper limit on σ(gg → X → aa → 4γ)
depends on the value of mX , and thus the excluded space of the parameters (λ, κ, Aλ, Aκ ) depends on thevalues of (µ, tan β). Secondly, the limit on (m A, tan β) space for the hMSSM model, as in Figure 6, isnaively applicable to the NMSSM, and this needs to be taken into account.

6.4 Future directions
As described in the previous section, the photon-jet search result excludes a subset of the (λ, κ, Aλ, Aκ )parameter space, depending on the values of (µ, tan β).
The excluded region for the parameters λ, κ, and Aλ can be expanded by improving the upper limiton σ(gg → X → aa → 4γ), e.g. by increasing the statistics of the collision dataset or improving
the analysis strategy. This is because the parameters λ, κ, and Aλ are strongly related to the values
B(H → aa), σ(gg → H), and B(a → γγ), respectively.
Other regions of Aλ can be checked by performing a search for a scalar resonance decaying into other final
states, e.g. X → 4µ, j j, 4τ, or 4b.
The excluded region for the parameter Aκ can be expanded by improving the analysis strategy of thephoton-jet search. The parameter Aκ is strongly related to the value of ma . The upper limit for theparameter Aκ (corresponding to ma . 5 GeV in the previous section) can be expanded by improving theanalysis strategy of the photon-jet search, so that the search is sensitive to less collimated photon-jets,
i.e. larger ma /mX . The lower limit for the parameter Aκ (corresponding to 2 GeV . ma in the previous
section) can be expanded by developing an analysis strategy for a long-lived a. The sensitivity of themH 1photon-jet search results is in the region 2m. 1 m, because it requires the particle a to decay withina Γathe inner radius of the EM calorimeter. To search for a longer-lived a, it is necessary to search for particles
that travel long distances (e.g. > O(10) m) before decaying via a → γγ. The FASER experiment [89]is a proposal to build a detector far from the ATLAS pp interaction point, that may be able to search forsuch long-lived particles.

110

7 Conclusion
A search for a heavy scalar resonance decaying into a pair of photon-jets (i.e. groups of collimated photons√
that share close trajectories) is performed using pp collisions at s = 13 TeV. The pp collision dataset
was collected with the ATLAS detector in 2015 and 2016, corresponding to an integrated luminosity of
36.7 fb−1 . The search is performed in a largely model-independent way, so that the search is sensitive tothe BSM scenarios leading to a pair of photon-jets in general. Candidate events of a resonance decaying
into a pair of photon-jets are selected from events with two high-ET reconstructed photons, where eachreconstructed photon corresponds to a photon-jet in the case of signal events. The event selection and theevent categorization are optimized for the signal events, utilizing the EM shower shape observed in the
EM calorimeter. To cover a wide range of values of mX and ma , several aspects of the search (e.g. mγR γR
shape of signal events, signal selection efficiency, and category fraction) are parametrized as functionsof mX and ma . The shape of the background component in the mγR γR distributions is evaluated with ananalytic function. An excess of events is searched for by performing an unbinned maximum-likelihood
fit, with the consideration of the signal and background components, to the mγR γR distributions of the two
event categories. The observed mγR γR distributions are found to be consistent with the SM backgroundexpectation, with no significant excess of events being observed.
The results of a null observation are interpreted in the context of BSM scenarios which assume a scalar
resonance decaying into a final state with photons. One scenario assumes a scalar resonance X with amass of O(100) GeV, with a narrow width, and produced by the gluon–gluon fusion process; X decays into
a pair of spin-0 particles a with a mass of O(1) GeV that decays into a pair of photons, via X → aa → 4γ.
Another scenario assumes the decay of the a particle into three neutral pions, via X → aa → 6π 0 → 12γ.
Upper limits on the cross-sections of the two BSM scenarios, σX × B(X → aa) × B(a → γγ)2 andσX ×B(X → aa)×B(a → 6π 0 )2 , are evaluated for the region 200 GeV < mX < 2 TeV and ma < 0.01×mX ,using an asymptotic approximation. They are found to be as low as 0.2 fb for mX = 2 TeV. Some scenarios
predict photon-jets produced from a decay of a long-lived boson; the results are interpreted for such a caseas well.
The results are interpreted in the context of the NMSSM for the process H → aa → 4γ, where H is a
new scalar Higgs boson with a mass larger than 200 GeV, and a is a pseudoscalar Higgs boson with amass of O(1) GeV. It is possible for the cross section of the process to be sizeable (e.g. around the orderof O(1 fb)); this possibility arise when the physical state H is a mixed state of the doublet and singlet
field components, and the physical state a is a decoupled state with dominant singlet field component. It
is found that the photon-jet search result excludes a subset of the parameter space of the NMSSM; for thecase of tan β = 5 and µ = 110 GeV, a subset of the parameter space with 2 GeV < ma < 5 GeV and an
ma 1
expected decay radius of a of less than 1 m (i.e. 2m< 1 m) is excluded.
H ΓaThis is the first result of a search for a heavy (> 200 GeV) resonance decaying into photon-jets, performed
to search for new physics at the energy-frontier pp collision experiment at the LHC. This research is a
new frontier of direct searches using a novel final state. This result places constraints on the subset of theparameter space of the NMSSM that have not been looked into before.

111

Acknowledgements
I would like to express my sincere gratitude to my supervisors, Professor Masaya Ishino and Associate
Professor Yasuyuki Okumura, for their continuous support for this research and providing thoughtful
insight. I would like to thank Professor Sachio Komamiya, who was my supervisor for the first two yearsof my PhD course, for his support and encouragement.
The academic researchers at the University of Tokyo International Center for Elementary Particle Physics
have provided me with precious feedback and various perspectives, and I am grateful of their support. I
appreciate the colleagues of the ATLAS Collaboration, especially those of the analysis team and the relatedgroups who have assisted me with completing this work. I thank my fellow colleagues for stimulating
discussions, and for all the fun that we had during the course of the three years.
I would like to thank my thesis committee members for their insightful discussions and hard questions
during the defence.
Last but not the least, I would like to thank my family and friends for their support and understanding thathave helped me with completing this research.

112

Appendix
A Results of selection efficiency and category fraction parametrizations
for the case of long-lived a
The signal selection efficiency and category fraction parametrizations for the case of long-lived a aredescribed in Section 5.9.1. A few examples are shown in Figures 73 and 74, and all the results in all(ET, η) regions are shown in Figures 86 and 87. These results can be utilized for reinterpretation of thephoton-jet search results to a BSM signal scenario leading to photon-jet final states in general.

113

1.2

y=0.615 1+exp170(x-0.0273) -1+0.137

1|η|<0.6, 70<ET /GeV<100mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.6
1.4
1.2

y=0.69 1+exp188(x-0.0266) -1+0.145

1

1.2

y=0.764 1+exp179(x-0.0277) -1+0.129

1

1.4
1.2

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0.04

0.05

0
0

0.06 0.07 0.08m
γ tsinθ2 a 1ET
REM

0.01

0.02

2
1.6
1.4
1.2

0.04

0.05

0
0

0.06 0.07 0.08m
γ tsinθ2 a 1ET
REM

0.4
0.2
0.01

0.02

(b)
0.6<|η|<1.37, 50<ET /GeV<70mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.8

0.03

y=0.647 1+exp144(x-0.0281) -1+0.189

1

2
1.6
1.4
1.2

0.04

0.05

0
0

0.06 0.07 0.08m
γ tsinθ2 a 1ETREM

y=0.681 1+exp162(x-0.0281) -1+0.194

12

0.6<|η|<1.37, 100<ET /GeV<200
mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.8
1.6
1.4
1.2

y=0.756 1+exp142(x-0.0288) -1+0.176

1

2

1.2

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0
0

0
0

0
0

0.01

0.02

1.52<|η|<1.81, 50<ET /GeV<70mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.6
1.4
1.2

0.05

0.06 0.07 0.08m
γ tsinθ2 a 1ETREM

y=0.564 1+exp142(x-0.03) -1+0.193

12

1.52<|η|<1.81, 70<ET /GeV<100mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.8
1.6
1.4
1.2

0.2
0.01

0.02

0.03

0.04

0.05

0
0

0.06 0.07 0.08m
γ tsinθ2 a 1ETREM

y=0.64 1+exp159(x-0.0314) -1+0.168

12

1.52<|η|<1.81, 100<ET /GeV<200
mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.8
1.6
1.4
1.2

y=0.648 1+exp183(x-0.0303) -1+0.207

1

1.2

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0
0

0
0

0
0

0.02

1.81<|η|<2.37, 50<ET /GeV<70mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.6
1.4
1.2

0.05

0.06 0.07 0.08m
γ tcosθ2 a 1ETZEM

y=0.659 1+exp145(x-0.0224) -1+0.137

12

1.81<|η|<2.37, 70<ET /GeV<100mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.8
1.6
1.4
1.2

0.2
0.01

0.02

0.03

0.04

0.05

0
0

0.06 0.07 0.08m
γ tcosθ2 a 1ETZEM

y=0.66 1+exp215(x-0.0225) -1+0.162

12

1.81<|η|<2.37, 100<ET /GeV<200
mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.8
1.6
1.4
1.2

y=0.764 1+exp160(x-0.0228) -1+0.147

1

1.2

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0
0

0
0

0
0

(m)

0.01

0.02

0.03

0.04

0.05

0.06 0.07 0.08m
γ tcosθ2 a 1ET
ZEM(n)0.06 0.07 0.08m
γ tcosθ2 a 1ETZEM

y=0.927 1+exp117(x-0.0237) -1+0.084

1

0.4

0.06 0.07 0.08m
γ tcosθ2 a 1ETZEM

0.05

1.81<|η|<2.37, 200<ET /GeV
mX = 800 GeV, ma = 1 GeV
mX = 800 GeV, ma = 5 GeV
mX = 800 GeV, ma = 10 GeV
mX = 1500 GeV, ma = 1 GeV
mX = 1500 GeV, ma = 5 GeV
mX = 1500 GeV, ma = 10 GeV

1.4

0.6

0.05

0.04

2

0.8

0.04

0.03

1.6

0.8

0.03

0.02

1.8

0.8

0.02

0.01

(l)

0.8

0.01

y=0.816 1+exp123(x-0.0335) -1+0.101(k)Selection efficiency

2
1.8

0.04

(j)
Selection efficiency

(i)

0.03

Selection efficiency

0.01

0.06 0.07 0.08m
γ tsinθ2 a 1ET
REM

1

0.4

0.06 0.07 0.08m
γ tcosθ2 a 1ETZEM

0.05

1.52<|η|<1.81, 200<ET /GeV
mX = 800 GeV, ma = 1 GeV
mX = 800 GeV, ma = 5 GeV
mX = 800 GeV, ma = 10 GeV
mX = 1500 GeV, ma = 1 GeV
mX = 1500 GeV, ma = 5 GeV
mX = 1500 GeV, ma = 10 GeV

1.4

0.6

0.05

0.04

2

0.8

0.04

0.03

1.6

0.8

0.03

0.02

1.8

0.8

0.02

0.01

(h)

0.8

0.01

y=0.885 1+exp123(x-0.0301) -1+0.0912(g)
Selection efficiency

2
1.8

0.04

(f)
Selection efficiency

(e)

0.03

Selection efficiency

0.06 0.07 0.08m
γ tsinθ2 a 1ET
REM

0.06 0.07 0.08m
γ tsinθ2 a 1ET
REM

1

0.6

0.05

0.05

0.6<|η|<1.37, 200<ET /GeV
mX = 800 GeV, ma = 1 GeV
mX = 800 GeV, ma = 5 GeV
mX = 800 GeV, ma = 10 GeV
mX = 1500 GeV, ma = 1 GeV
mX = 1500 GeV, ma = 5 GeV
mX = 1500 GeV, ma = 10 GeV

1.4

0.8

0.04

0.04

1.6

0.8

0.03

0.03

1.8

0.8

0.02

0.02

(d)

0.8

0.01

0.01

(c)
0.6<|η|<1.37, 70<ET /GeV<100mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.8

0.03

Selection efficiency

0.03

Selection efficiency

0.02

y=0.868 1+exp150(x-0.0282) -1+0.0823

1
0.8

0.01

|η|<0.6, 200<ET /GeV
mX = 800 GeV, ma = 1 GeV
mX = 800 GeV, ma = 5 GeV
mX = 800 GeV, ma = 10 GeV
mX = 1500 GeV, ma = 1 GeV
mX = 1500 GeV, ma = 5 GeV
mX = 1500 GeV, ma = 10 GeV

1.6

0.8

Selection efficiency

Selection efficiency

1.4

2
1.8

0.8

(a)

Selection efficiency|η|<0.6, 100<ET /GeV<200
mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.6

0.8

0
0

Selection efficiency

2
1.8

Selection efficiency

1.4

2
1.8

Selection efficiency|η|<0.6, 50<ET /GeV<70mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.6

Selection efficiency

Selection efficiency

2
1.8

0.2
0.01

0.02

0.03

0.04

(o)

0.05

0.06 0.07 0.08m
γ tcosθ2 a 1ET
ZEM

0
0

0.01

0.02

0.03

0.04

0.05

0.06 0.07 0.08m
γ tcosθ2 a 1ET
ZEM(p)

Figure 86: The result of the parametrization of εγR (ET, η, t). The value of εγR is presented as a function of v. The
result of the parametrization is presented for each of the 16 bins of the (ET, η) parameter space of the particle a:|η| < 0.6 and (a) 50 < ET < 70 GeV, (b) 70 < ET < 100 GeV, (c) 100 < ET < 200 GeV, (d) 200 GeV < ET ,0.6 < |η| < 1.37 and (e) 50 < ET < 70 GeV, (f) 70 < ET < 100 GeV, (g) 100 < ET < 200 GeV, (h) 200 GeV < ET ,1.52 < |η| < 1.81 and (i) 50 < ET < 70 GeV, (j) 70 < ET < 100 GeV, (k) 100 < ET < 200 GeV, (l) 200 GeV < ET ,1.81 < |η| < 2.37 and (m) 50 < ET < 70 GeV, (n) 70 < ET < 100 GeV, (o) 100 < ET < 200 GeV, (p)200 GeV < ET .

114

0.4
0.2
0
0

y=1 1+exp321(x-0.00643) -1+0.956 1+exp-56.8(x-0.0321) -1

1
0.8
0.6
0.4
0.2

0.01

0.02

0.03

0.04

0.05

0
0

0.06 0.07 0.08m
γ tsinθ2 a 1ETREM

0.6<|η|<1.37, 50<ET /GeV<70mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.4
1.2

y=0.922 1+exp428(x-0.00713) -1+0.927 1+exp-60.8(x-0.03) -1

1
0.8
0.6
0.4

0.01

0.02

0.6
0.4

0.04

0.05

0
0

0.06 0.07 0.08m
γ tsinθ2 a 1ETREM

2

0.6<|η|<1.37, 70<ET /GeV<100mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.6
1.4
1.2

y=1 1+exp284(x-0.00627) -1+1 1+exp-46.6(x-0.0359) -1

1

0.05

0.06 0.07 0.08m
γ tsinθ2 a 1ET
REM

0.8
0.6
0.4

0.01

0.02

1.2

y=1 1+exp338(x-0.00913) -1+0.866 1+exp-80.4(x-0.0362) -1

1
0.8
0.6
0.4

2

0.05

0.06 0.07 0.08m
γ tsinθ2 a 1ET
REM

1.6
1.4
1.2

1.2

0.4

0.05

0.06 0.07 0.08m
γ tcosθ2 a 1ETZEM

y=1 1+exp325(x-0.00612) -1+1 1+exp-43.8(x-0.042) -1

1
0.8
0.6
0.4

0.01

0.02

1.81<|η|<2.37, 50<ET /GeV<70mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.4
1.2

0.04

0.05

0.06 0.07 0.08m
γ tcosθ2 a 1ETZEM

y=0.69 1+exp343(x-0.0101) -1+1 1+exp-35.9(x-0.0219) -1

1
0.8
0.6
0.4

2

1.81<|η|<2.37, 70<ET /GeV<100mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.8
1.6
1.4
1.2

1.4
1.2

y=1 1+exp263(x-0.00741) -1+1 1+exp-44.5(x-0.047) -1

0.4

y=0.915 1+exp264(x-0.00922) -1+0.898 1+exp-66.1(x-0.0233) -1

1

0.02

0.8
0.6
0.4

(m)

0.01

0.02

0.03

0.05

0.04

0.05

0.06 0.07 0.08m
γ tcosθ2 a 1ET
ZEM

(n)

0.04

2

0.05

0.06 0.07 0.08m
γ tsinθ2 a 1ETREM

0.6<|η|<1.37, 200<ET /GeV
mX = 800 GeV, ma = 1 GeV
mX = 800 GeV, ma = 5 GeV
mX = 800 GeV, ma = 10 GeV
mX = 1500 GeV, ma = 1 GeV
mX = 1500 GeV, ma = 5 GeV
mX = 1500 GeV, ma = 10 GeV

1.6
1.4
1.2

y=1 1+exp289(x-0.00609) -1+1 1+exp-37.9(x-0.0461) -1

1
0.8
0.6
0.4

0.01

0.02

0.03

0.04

2

0.05

0.06 0.07 0.08m
γ tsinθ2 a 1ETREM

1.52<|η|<1.81, 200<ET /GeV
mX = 800 GeV, ma = 1 GeV
mX = 800 GeV, ma = 5 GeV
mX = 800 GeV, ma = 10 GeV
mX = 1500 GeV, ma = 1 GeV
mX = 1500 GeV, ma = 5 GeV
mX = 1500 GeV, ma = 10 GeV

1.8
1.6
1.4
1.2

y=1 1+exp287(x-0.00719) -1+1 1+exp-43.9(x-0.0522) -1

1
0.8
0.6
0.4
0
0

0.06 0.07 0.08m
γ tcosθ2 a 1ET
ZEM

0.01

0.02

0.03

0.04

0.05

0.06 0.07 0.08m
γ tcosθ2 a 1ETZEM(l)1.81<|η|<2.37, 100<ET /GeV<200
mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.4

y=0.885 1+exp276(x-0.0072) -1+0.846 1+exp-47.8(x-0.02) -1

1
0.8
0.6
0.4
0
0

0.06 0.07 0.08m
γ tcosθ2 a 1ET
ZEM

0.04

2

0
0

0.05

0.03

1.6
1.2

0.03

0.2
0.01

1.8

0
0

0.04

0.02

(h)

0.6

0.2

0.03

0.01

1.8

0
0

0.06 0.07 0.08m
γ tsinθ2 a 1ET
REM

0.8

0.2
0.02

0.05

1

0.2
0.01

0.4

(k)
Fraction of events with small ∆E

2

0.04

1.52<|η|<1.81, 100<ET /GeV<200
mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV(j)Fraction of events with small ∆E

(i)

0.03

0.03

2

0
0

1.6

0.02

1.6

0
0

0.04

0.6

0.2
0.01

1.8

0
0

1.8

y=1 1+exp283(x-0.0059) -1+1 1+exp-39.8(x-0.0391) -1

0.6

0.2

0.03

y=1 1+exp271(x-0.00649) -1+1 1+exp-43.8(x-0.043) -1

1
0.8

(d)

0.8

0.2
0.02

1.2

0
0

0.06 0.07 0.08m
γ tsinθ2 a 1ET
REM

1

0.2
0.01

1.4

(g)
1.52<|η|<1.81, 70<ET /GeV<100mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.8

0.05

0.6<|η|<1.37, 100<ET /GeV<200
mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV(f)
1.52<|η|<1.81, 50<ET /GeV<70mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.4

0.04

Fraction of events with small ∆E

2
1.6

0.03

0.04

1.4

0
0

0.04

0.03

2

0
0

1.8

0.02

1.6

0
0

0.03

|η|<0.6, 200<ET /GeV
mX = 800 GeV, ma = 1 GeV
mX = 800 GeV, ma = 5 GeV
mX = 800 GeV, ma = 10 GeV
mX = 1500 GeV, ma = 1 GeV
mX = 1500 GeV, ma = 5 GeV
mX = 1500 GeV, ma = 10 GeV

1.6

0.2
0.01

1.8

0.2

0.02

2
1.8

(c)

0.2

Fraction of events with small ∆E

Fraction of events with small ∆E

0.03

1.8

(e)

Fraction of events with small ∆E

0.8

0.2
0.01

y=1 1+exp297(x-0.00621) -1+1 1+exp-46(x-0.0374) -1

1(b)
Fraction of events with small ∆E

Fraction of events with small ∆E

2
1.6

1.2

0.2

(a)
1.8

1.4

Fraction of events with small ∆E

0.6

1.2

|η|<0.6, 100<ET /GeV<200
mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.6

Fraction of events with small ∆E

0.8

1.4

2
1.8

Fraction of events with small ∆E

y=1 1+exp350(x-0.00697) -1+0.965 1+exp-56.2(x-0.0337) -1

1

Fraction of events with small ∆E

1.2|η|<0.6, 70<ET /GeV<100mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.6

Fraction of events with small ∆E

1.4

2
1.8

Fraction of events with small ∆E|η|<0.6, 50<ET /GeV<70mX = 200 GeV, ma = 0.1 GeV
mX = 200 GeV, ma = 1 GeV
mX = 200 GeV, ma = 2 GeV
mX = 300 GeV, ma = 0.1 GeV
mX = 300 GeV, ma = 1 GeV
mX = 300 GeV, ma = 3 GeV

1.6

Fraction of events with small ∆E

Fraction of events with small ∆E

2
1.8

2

1.81<|η|<2.37, 200<ET /GeV
mX = 800 GeV, ma = 1 GeV
mX = 800 GeV, ma = 5 GeV
mX = 800 GeV, ma = 10 GeV
mX = 1500 GeV, ma = 1 GeV
mX = 1500 GeV, ma = 5 GeV
mX = 1500 GeV, ma = 10 GeV

1.8
1.6
1.4
1.2

y=0.878 1+exp302(x-0.00891) -1+0.864 1+exp-48.7(x-0.0282) -1

1
0.8
0.6
0.4
0.2

0.01

0.02

0.03

0.04

(o)

0.05

0.06 0.07 0.08m
γ tcosθ2 a 1ET
ZEM

0
0

0.01

0.02

0.03

0.04

0.05

0.06 0.07 0.08m
γ tcosθ2 a 1ET
ZEM(p)

Figure 87: The result of the parametrization of fγR (ET, η, t). The value of fγR is presented as a function of v. The
result of the parametrization is presented for each of the 16 bins of the (ET, η) parameter space of the particle a:|η| < 0.6 and (a) 50 < ET < 70 GeV, (b) 70 < ET < 100 GeV, (c) 100 < ET < 200 GeV, (d) 200 GeV < ET ,0.6 < |η| < 1.37 and (e) 50 < ET < 70 GeV, (f) 70 < ET < 100 GeV, (g) 100 < ET < 200 GeV, (h) 200 GeV < ET ,1.52 < |η| < 1.81 and (i) 50 < ET < 70 GeV, (j) 70 < ET < 100 GeV, (k) 100 < ET < 200 GeV, (l) 200 GeV < ET ,1.81 < |η| < 2.37 and (m) 50 < ET < 70 GeV, (n) 70 < ET < 100 GeV, (o) 100 < ET < 200 GeV, (p)200 GeV < ET .

115

B Distributions of shower shape variables
The definitions of the shower shape variables are given in Table 5 of Section 4.2. The distributions of theshower shape variables are shown in Figures 88 and 89. They are shown for simulated events of photons
and hadronic jets, with no selections applied.

Figure 88: The distributions of the shower shape variables for unconverted photons in the region |η| < 0.6 andET > 20 GeV [90]. Those for reconstructed photons originating from prompt photons (shaded histogram) andbackgrounds (i.e. hadronic jets, and photons originating from√hadronic decays) (unshaded histogram) are shown.
3 , w
These distributions are calculated using simulated events at s = 7 TeV. Here, ws3 is denoted by wη1stot istot
denoted by wη1 , and fside is denoted by Fside .

116

Figure 89: The distributions of the shower shape variables for converted photons in the region |η| < 0.6 andET > 20 GeV [90]. Those for reconstructed photons originating from prompt photons (shaded histogram) andbackgrounds (i.e. hadronic jets, and photons originating from√hadronic decays) (unshaded histogram) are shown.
3 , w
These distributions are calculated using simulated events at s = 7 TeV. Here, ws3 is denoted by wη1stot istot
denoted by wη1 , and fside is denoted by Fside .

117

10−1
10−2

X→aa→γ γ , m =600 GeV
X
ma=0.1 GeV
ma=0.7 GeV
ma=5 GeVγ γ MC mγ γ =500-700 GeV
j of γ j MCRmR γ γ =500-700 GeV
R R
Data mγ γ =500-700
GeV

Normalized to 1

Normalized to 1

1

R R

1

10−1

10−2

−3

10

10−3

10−4
10−50 0.005 0.010.0150.020.025 0.030.035 0.040.045 0.05

410−0.86

0.88

0.9

0.92

0.94

0.96

0.98

1
Rη

1

Normalized to 1

Normalized to 1

Rhad

10−1

1

10−1

10−2
10−2
10−3

−4
100.0060.0070.0080.009
0.01 0.0110.0120.0130.0140.0150.016wη2

3
10−0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1
Rφ

Figure 90: The distributions of the shower shape variables defined using the information of the EM calorimetersecond layer. The histograms are plotted as the sum of leading and subleading reconstructed photons. The events
which satisfy the preselection, isolation, and the ET selection requirements are plotted. The filled markers represent
the distributions obtained from simulated signal samples of X → aa → 4γ for mX = 600 GeV. The empty
markers represent the distributions obtained from simulated samples of SM processes γγ and γ j in the region
500 GeV < mγR γR < 700 GeV. For the square markers, reconstructed photons originating from hadronic jets
are exclusively chosen. The “X” markers represent the distributions obtained from collision dataset in the region
500 GeV < mγR γR < 700 GeV. The vertical dotted line represents the threshold used for the Tight photon ID
selection for unconverted photons in the region |η| < 0.6. The first bins include the underflows, and the last bins
include the overflows.

Figures 90 and 91 show the distributions of the shower shape variables for the process X → aa → 4γ
as well as photons and hadronic jets. They are shown for events in a region similar to the signal region(defined in Section 5.3.1), with the Loose photon ID selection applied instead of the Loose05 photon ID
selection.

118

X→aa→γ γ , m =600 GeV
X
ma=0.1 GeV
ma=0.7 GeV
ma=5 GeVγ γ MC mγ γ =500-700 GeV
j of γ j MCRmR γ γ =500-700 GeV
R R
Data mγ γ =500-700
GeV

10−1

Normalized to 1

Normalized to 1

1

R R

10−2

10−1

10−2

10−3

10−40

1

10−3

2

4

6

8

10

12

4
10−0.3

14

0.4

0.5

0.6

0.7

0.8

0.9
ws3

1

Normalized to 1

Normalized to 1

wstot

10−1
10−2

1

10−1

10−2
−3

10

10−3

10−4
10−50

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

10−40

1

Normalized to 1

Normalized to 1

1

10−1

600

800

1000

10−1

10−2

10−3

10−3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

400

1

10−2

10−40

200

∆E [MeV]

fside

10−40

1

Eratio0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
f1

Figure 91: The distributions of the shower shape variables defined using the information of the EM calorimeterfirst layer. The histograms are plotted as the sum of leading and subleading reconstructed photons. The events
which satisfy the preselection, isolation, and the ET selection requirements are plotted. The filled markers represent
the distributions obtained from simulated signal samples of X → aa → 4γ for mX = 600 GeV. The empty
markers represent the distributions obtained from simulated samples of SM processes γγ and γ j in the region
500 GeV < mγR γR < 700 GeV. For the square markers, reconstructed photons originating from hadronic jets
are exclusively chosen. The “X” markers represent the distributions obtained from collision dataset in the region
500 GeV < mγR γR < 700 GeV. The vertical dotted line represents the threshold used for the Tight photon ID
selection for unconverted photons in the region |η| < 0.6. The first bins include the underflows, and the last bins
include the overflows.

119C Sensitivity estimation using simulated events for optimization of theevent selection
As described in Section 5.3.1, the Loose05 photon ID selection is used for the definition of the signal
region. Also, an event categorization based on the values of the shower shape variable ∆E is introduced,
as described in Section 5.3.2. These choices are made based on estimations of the “sensitivity” of theanalysis strategy to X → aa → 4γ signal events. The estimations are performed using simulated event
samples.
Here, the “sensitivity” is quantified by evaluating the expected value of the signal production cross section
times branching ratios, σX × B(X → aa) × B(a → γγ)2 , that correspond to an expected discovery
significance of 3σ. The expected significance, in units of Gaussian σ, is estimated with an asymptotic
approximation [75]:
r h

i
s
Z = 2 (s + b) log 1 +
−s
(97)
b
Here, s is the yield of signal√ events, and b is the yield of background events. Note that for s  b, thisexpression leads to Z ' s/ b, which is a commonly used expression for the estimation of the discovery
significance. Using this expression, the value of σX × B(X → aa) × B(a → γγ)2 which corresponds to
Z = 3 is evaluated.
The sensitivity estimation follows the following procedure, which is schematically illustrated in Figure 92:
1. The integrated luminosity assumed for this evaluation is 35 fb−1 27.
2. The value of s is estimated using simulated signal samples. For a given σX × B(X → aa) × B(a →γγ)2 , the mγR γR distribution of signal events that satisfy event selection requirements is calculatedusing simulated signal samples. The value of s is defined as the number of events within the region
maverage − 2mstd. dev. < mγR γR < maverage + 2mstd. dev. , where maverage and mstd. dev. are the average andstandard deviation of the signal mγR γR distribution.
3. The value of b is estimated using simulated samples of SM γγ events. The mγR γR distributionof background events that satisfy event selection requirements is calculated using the simulated
samples. The normalization of the mγR γR distribution is defined so that the event yield in the region
mγR γR > 175 GeV is equal to that observed in the collision dataset28. The value of b is defined as
the number of events within the region maverage − 2mstd. dev. < mγR γR < maverage + 2mstd. dev. .
Comparison of estimated sensitivity for different photon ID selections
The comparison of estimated sensitivities for signal regions defined using different photon ID selections
is shown in Figure 93. As seen in the figure, a photon ID selection which is less stringent results inbetter sensitivity (i.e. lower σX × B(X → aa) × B(a → γγ)2 that corresponds to Z = 3); using Loosephoton selection or Loose05 selection results in better sensitivity compared to using Tight selection or
27As the integrated luminosity, 35 fb−1 is used in this evaluation instead of 36.7 fb−1 , which corresponds to the collisiondataset used in this thesis. This is simply because this evaluation was performed before the data taking in 2016 was completed.
Nevertheless, the impact of this difference in the result of this evaluation is negligible.
28With this method, the event yield of the sum of all background components, i.e. γγ, γ j, and j j, is estimated. This method
relies on the fact that the overall shape of the mγR γR distribution is approximately the same for each background component,
γγ, γ j, and j j, as seen in Figure 53.

120

Figure 92: A schematic representation of the procedure for the sensitivity estimation. This figure shows anexample for X → aa → 4γ, mX = 750 GeV, ma = 1 GeV, and σX × B(X → aa) × B(a → γγ)2 = 1 fb.
The histograms are stacked, with the red histogram representing the signal component and the black histogram
representing the background component. The blue vertical dotted lines show the region maverage − 2mstd. dev. <mγR γR < maverage + 2mstd. dev. . For details, see text.

Loose04 selection. Thus, the Loose05 selection is used for the definition of the signal region, as describedin Section 5.3.129.
Comparison of estimated sensitivity for different categorization definitions
The comparison of estimated sensitivities for event categorization defined using different shower shapevariables is shown in Figure 94. Here, the categorization is defined similarly to that described in
Section 5.3.2. Theqdiscovery significance resulting from the combination of two event categories is

estimated by Z = Z12 + Z22 , where Z1 and Z2 are the discovery significances estimated for the first
and the second event category, respectively. As seen in the figure, the event categorization using ∆E
results in the largest sensitivity (i.e. lowest σX × B(X → aa) × B(a → γγ)2 corresponding to Z = 3).
Thus, the shower shape variable ∆E is used for the definition of the event categorization, as described in
Section 5.3.2.
Figure 95 shows the “gain” in sensitivity by introducing the event categorization using ∆E. Here, the“gain” is quantified by evaluating the ratio of the values of σX × B(X → aa) × B(a → γγ)2 corresponding
to Z = 3 before and after introducing the event categorization (i.e. the ratio of the results in Figure 93(c)and Figure 94(a)). The introduction of event categorization based on ∆E results in a sensitivity gain of
approximately as much as a factor 2.

29As seen in Figure 93, using the Loose 0 5 selection for the signal region definition results in a slightly worse sensitivity
compared to that defined using the Loose selection, for large values of ma /mX (e.g. worse by approximately a factor 2 forma /mX = 0.01). However, as described in Section 5.3.1, an important point is that a control region (i.e. a region similar andorthogonal to the signal region) can be defined with the use of the Loose 0 5 selection. This is because the Loose 0 5 selection isa tighter selection compared to the Loose photon ID selection. As described in Section 5.1.1, the collision dataset is collectedwith the diphoton trigger, which requires two EM clusters to satisfy the trigger-level Loose photon ID selection. Reconstructed
photons in the dataset that do not satisfy the Loose 0 5 selection is used for the estimation of the rate of hadronic jets satisfying
the calorimeter isolation requirement, as described in Section 5.5.1

121

(a)

(b)

(c)

(d)

Figure 93: The estimation of the values of σX × B(X → aa) × B(a → γγ)2 corresponding to Z = 3. The signal
region is defined similarly to that described in Section 5.3.1, with different photon ID selections used; (a) Tight
selection, (b) Loose selection, (c) Loose05 selection, and (d) Loose04 selection.

122

(a)

(b)

(c)

(d)

Figure 94: The estimation of the values of σX × B(X → aa) × B(a → γγ)2 corresponding to Z = 3. The event
categorization is defined similarly to that described in Section 5.3.2, with different shower shape variables used forthe definition of the categories; (a) ∆E, (b) Eratio , (c) fside , and wstot . The result using ws3 is similar to that using
wstot , and so is not shown here.

123

Figure 95: The gain in sensitivity by introducing the event categorization using the ∆E variable. The gain isquantified by evaluating the ratio of the values of σX × B(X → aa) × B(a → γγ)2 corresponding to Z = 3 before
and after introducing the event categorization (i.e. the ratio of the results in Figure 93(c) and Figure 94(a)).

D Additional details to the hypothesis testing method
The definition of the likelihood is described in Section 5.7.1, and the hypothesis testing method is describedin Section 5.7.2. The definitions of the test statistics, the details of the asymptotic approximations, andthe definition of the CLs method are described below.
Define the profile likelihood ratio as
λ(µ) =

ˆL(µ, θ̂)L( µ̂, θ̂)(98)Here, L(µ, θ) is the overall likelihood defined in Eq. (73) for a given signal hypothesis with mass values of(mX , ma ). The parameter of interest µ = σ × BR is the product of the signal production cross section andthe branching ratios. θ collectively designates all free parameters excluding µ (i.e. NB,1, NB,2, k 1, k 2 , andθ in Eq. (73)). θ̂ˆ denotes the conditional maximum-likelihood estimators, i.e. values of θ that maximizes L
for a given µ. Thus, θ̂ˆ is a function of µ. µ̂ and θ̂ are the unconditional maximum-likelihood estimators.
The calculation of p-values for the background-only hypothesis (p0 )The test statistic q0 is defined as [75]ˆ
 − 2 log λ(0) = −2 log L(0, θ̂)q0 =L( µ̂, θ̂)0Let µ0 be the expected value of µ. The value of p0 is calculated as∫ ∞p0 =f (q0 | µ0 = 0) dq0
q0,obs124µ̂ ≥ 0(99)µ̂ < 0(100)

q0,obs is the observed value of q0 . f (q0 | µ0) is the probability density function of q0 under the assumptionof µ0. The discovery significance is defined as
Z = Φ−1 (1 − p0 )(101)Here, Φ is the cumulative distribution of a standard Gaussian distribution.
Under the asymptotic approximation µ̂ ∼ N(µ0, σ), the p0 value is approximated by [75]√p0 = 1 − Φ( q0,obs )(102)

Thus, the discovery significance is approximated by√
Z = q0,obs(103)

The calculation of CLs limits
The test statistic q̃µ is defined as [75]ˆL(µ, θ̂)



−
2
log


ˆ

L(0, θ̂)



ˆ
q̃µ =
L(µ, θ̂)


−
2
log



L( µ̂, θ̂)



0


µ̂ ≤ 00 ≤ µ̂ < µ(104)µ < µ̂

CLs , the modified frequentist method, is defined as [76]∫∞f (q̃µ | µ0 = µ) d q̃µq̃µ,obsCLs (µ) = ∫ ∞f (q̃µ | µ0 = 0) d q̃µ
q̃

(105)

µ,obs

The 95% CL upper limit on µ is evaluated by taking the value of µ when CLs = 0.05.
Under the asymptotic approximation µ̂ ∼ N(µ0, σ), the cumulative distribution function of q̃µ is approximated as [75]p
µ − µ0
µ2


q̃
−
0
<
q̃
≤
Φ

µ,obs
µ,obs

∫ q̃µ,obs
σ

σ2


0
2
0
f (q̃µ | µ ) d q̃µ =(106)µ −2µµµ2© q̃µ,obs − σ 2 ª−∞< q̃µ,obs
®

Φ ­
2µ

σ2
σ
 «
¬
The value of σµ is approximated by the Asimov method [75], and it is approximated asµ p' q̃µ, A(107)σ
The value of q̃µ, A is evaluated by calculating the value of q̃µ for an “Asimov dataset”. The Asimov dataset
is a dataset that when one uses it to obtain the maximum-likelihood estimators µ̂ and θ̂, one obtains thetrue parameter values. In this research, the Asimov dataset is defined as a weighted dataset that is obtained
from a background-only fit to the mγR γR distributions of the collision data, i.e. the blue lines shown in
Figure 61.
The expected (i.e. median) upper limit is evaluated by substituting q̃µ,obs with q̃µ, A, thus resulting inp
expectedCLs(µ) = 2 1 − Φ( q̃µ, A)
(108)

125

103
102

s=13 TeV, 36.7 fb-1
low-∆ E category
Data
Background-only fit

105
104
103
102

10

10

1

1

10−1

10−1

10−2

Data / fit

Events / GeV

104

10−2

103

2
1.5
1
0.5
0

mγ γ [GeV]

Data / fit

Events / GeV

105

R R

103

s=13 TeV, 36.7 fb-1
high-∆ E category
Data
Background-only fit

103

2
1.5
1
0.5
0

mγ γ [GeV]
R R

103

mγ γ [GeV]R R

mγ γ [GeV]
R R

Figure 96: The observed mγR γR distributions in the two event categories of the signal region. The result of abackground-only fit assuming µ = 0 is superimposed. In the lower panels, the ratios of the observed event yield andthe expectation from the fit result are presented. The blue arrows represent the bins with this ratio larger than two,i.e. out of the range of the plot.

E Distributions of mγ R γ R in the signal region, with the bin widths set to thedetector resolution
Figure 96 shows the mγR γR distributions in the signal region obtained from collision data (i.e. the same
result as those shown in Figure 61). In this figure, the bin widths are set to be equal to the energy resolution
Xof the detector. Here, the detector resolution is estimated as 0.991 × (1 + 0.680 × 100mGeV) GeV, i.e. theparametrized result of σCB shown in Table 9 in the limit ma → 0.

126

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