Doctoral Dissertation
博士論文

Search for resonances in light-by-lightscattering√in 14.6 fb−1 of 𝑝 𝑝 collisions at 𝑠 = 13 TeV（重心系エネルギー 13 TeV の陽子-陽子衝突データ
14.6 fb−1 を用いた光子-光子散乱における共鳴の探索）

A Dissertation Submitted for the Degree of Doctor of Philosophy
December 2022
令和 4 年 12 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

Gen Tateno
舘野元Abstract√
This thesis reports a search for resonances in light-by-light scattering using 14.6 fb−1 of 𝑠 = 13 TeV
𝑝 𝑝 collision data collected by the ATLAS experiment at the LHC. The light-by-light scattering mediatesa scattering of the beam protons. The scattered proton is detected by the ATLAS Forward Proton(AFP) spectrometer, while the ATLAS detector reconstructs the photons. At least one of the outgoingprotons is required to match the kinematics of the final state diphoton to reduce the background events.
Diphoton resonances in the mass range from 150 GeV to 1600 GeV are searched using signal andbackground distribution models with unbinned fitting. The maximum observed local significance is 2.51 at
𝑚 𝑋 = 454 GeV, and there is no excess from the standard model. Upper limits are placed on the production
cross section as a function of the resonance mass. The results are interpreted as the upper limits of the
ALP coupling constant to two photons. All the results are obtained assuming the branching ratio to two
photons is 100%. The observed upper limits of the coupling constant are about 0.04 TeV−1 in the range𝑚 𝑋 = 200 to 1000 GeV.

3

Contents
1

Introduction
1.1 Light-by-light scattering at the LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Light-by-light scattering in 𝑝 𝑝 collisions . . . . . . . . . . . . . . . . . . . . .
1.1.2 Forward proton detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Light-by-light scattering mediated by new physics . . . . . . . . . . . . . . . . . . . . .
1.3 Previous researches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 High-mass inclusive diphoton resonance searches . . . . . . . . . . . . . . . . .
1.3.2 Light-by-light scattering observation in Pb+Pb collisions . . . . . . . . . . . . .
1.3.3 Analyses using the forward proton detectors . . . . . . . . . . . . . . . . . . . .
1.4 Aim and overview of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7
7
8
10
11
13
14
15
15
16

2

Experimental Setup
2.1 LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 ATLAS detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Subdetectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 EM calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 TDAQ system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 AFP detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Calibration and performance evaluation . . . . . . . . . . . . . . . . . . . . . .

19
19
21
22
23
26
27
32

3

Data and Monte Carlo Samples
3.1 Data samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Signal MC samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Track and primary vertex reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Photon reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Proton reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37
37
38
41
42
44

4

Event Selection
4.1 Acoplanarity selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 𝜉 𝛾𝛾 selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 𝜉AFP selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Diphoton-proton matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Set operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Background selection efficiency . . . . . . . . . . . . . . . . . . . . . . . . . .

49
49
50
53
54
57
62

5

Signal Modeling
5.1 Signal fitting function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Signal modeling uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Signal efficiency and yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Cross section and yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67
67
71
72
72
74

5

Contents
6

Background Modeling
6.1 Background sample creation and background modeling . . . . . . . . . . . . . . . . . .
6.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Data-driven validation of sample and dominance of combinatorial background .
6.2.2 MC-based validation of dominance of combinatorial background . . . . . . . . .
6.3 Background template systematic uncertainty . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Background modeling uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1 Functional Decomposition smoothing . . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Spurious signal evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.3 Bias of Functional Decomposition . . . . . . . . . . . . . . . . . . . . . . . . .

79
79
83
83
85
88
90
90
91
93

7

Systematic Uncertainty
99
7.1 Photon object-related uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.2 Proton object-related uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.3 Fitting-related uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.4 Theoretical uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

8

Statistical Procedure and Results
109
8.1 Statistical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.2 Effects of systematic uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.3 Statistical tests and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.3.1 Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
8.3.2 Exclusion limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

9Discussion and Conclusion
123
9.1 Comparison with previous research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
9.2 Prospect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
9.2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
9.2.2 Event selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
9.2.3 Other analyses using the AFP detectors . . . . . . . . . . . . . . . . . . . . . . 125
9.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

A Auxiliary plots

127

B Auxiliary tables

131

C Systematic variations of 𝚫𝝃 distribution

133

D Loose 𝝃𝜸𝜸 cut effect evaluation
135
D.1 Proton energy loss fraction resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
D.2 Threshold changing test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
E Luminosity dependency of combinatorial background

139

F Variation of requirement on the number of AFP clusters

141

G Results from asymptotic formulae

143

6

1 Introduction
1.1 Light-by-light scattering at the LHC
The studies of light-by-light scattering, a process in which two photons interact and turn back into two
photons, have long attracted interest. This interaction is supposed not to occur in a vacuum in the classicalelectromagnetism. However, the quantum electrodynamics (QED), a part of the standard model (SM)of particle physics, two photons can interact with each other through a loop of charged fermions or 𝑊 ±bosons [1]. In addition to studies of this interaction, observation of the effects of new physics beyond thestandard model (BSM) mediating light-by-light scattering has been aimed in many experiments. Lightby-light scattering has been measured in photon scattering in the Coulomb field of a nucleus (Delbrück
scattering) [2–5] and in the photon-splitting process [6]. Both apply electromagnetic fields to observe lightby-light scattering. Detailed backgrounds and the other experiments concerning light-by-light scattering
are described in Ref. [7, 8].
Currently, high-energy colliders are the only place where the Schwinger limit of 1018 V m−1 is surpassed [9].
At the Large Hadron Collider (LHC) [10], the electromagnetic fields between beam hadrons are strong
enough to create fluxes of high-energy virtual photons. These photons can then interact with each other,enabling light-by-light scattering. In 2016, the ATLAS collaboration claimed the first direct evidenceof high-energy light-by-light scattering without any additional electromagnetic field applied [11]. This
experiment made use of Pb+Pb collisions, which are LHC beam collision experiments, to collide leadion beams with each other. The electromagnetic fields around the colliding lead ion can be treated as
quasi-real photons with a small virtuality [12–14], and the photons can collide with each other, as known
as ultraperipheral collision (UPC) [15, 16]. Hence, the ATLAS observation means the LHC was used
as a photon-photon (𝛾𝛾) collider. Fig. 1.1 shows the Feynman diagram of such an event. Light-by-lightscattering can occur in collisions between charged particles in general. Further progress of light-by-lightscattering measurement in the LHC is described in Sec. 1.3.2.
Historically, the modeling of photon production from a fast-moving charged particle was developed
following a discussion by Fermi [17]. The electric field produced by the charged particle is predominantly
transverse. It is, in association with the magnetic field surrounding it, described as a flux of photons withPb

Pb (∗)𝛾𝛾𝛾𝛾PbPb (∗)

Figure 1.1: Feynman diagram of SM light-by-light scattering in Pb+Pb collisions. An electron is an example of aparticle that forms a loop. The potential electromagnetic excitation of the lead ion is denoted by (∗) .

7

1 Introduction
density 𝑛(𝜔), where 𝜔 is the energy equivalent to frequency. The cross section of the electromagnetic
interaction of such photons with another particle was then approximated by the convolution of the photon
flux with the relevant real photon production cross section. This model was simplified as WeizsäckerWilliams approximation (WWA) [13, 18] to enable easier calculation of 𝑛(𝜔) and the cross section which
is given asd𝜎 = 𝜎𝛾 d𝑛,where 𝜎𝛾 is the cross section for absorption of photons with energy 𝜔. The original WWA neglects photon
virtuality. The story so far is detailed in Refs. [19, 20]. Likewise, the light-by-light scattering cross section
in a collision between charged particles is given asd𝜎 = 𝜎𝛾𝛾 d𝑛1 d𝑛2 ,where 𝜎𝛾𝛾 is the two-photons interaction cross section, and d𝑛𝑖 is the photon flux of charged particle 𝑖.
The WWA has been improved as equivalent photon approximation (EPA) [20] considering photon virtuality.
The EPA is similar to the quantum chromodynamics (QCD) factorization theory which calculates the
convolution of the parton distribution function (PDF) and the hard-process cross section to derive the crosssection in a collider experiment. The photon distribution 𝑛𝑖 is also defined as photon PDF and convoluted
with the cross section of light-by-light scattering, an example of a hard process. In the EPA, the photon
flux is given as a function of energy 𝜔 and virtuality 𝑄 2 of the photon,
"
!
#

𝜔2𝑖
(𝑄 min ) 2𝑖
𝛼 d𝜔𝑖 d𝑄 2𝑖
𝜔𝑖(𝐹E ) 𝑖 +(𝐹M ) 𝑖 ,
d𝑛𝑖 =
1−
1−
(1.1)
𝜋 𝜔𝑖 𝑄 2𝑖
𝐸𝑖
𝑄 2𝑖
2𝐸 𝑖2
where 𝛼 is the fine-structure constant, 𝐸 is the incoming charged particle energy, 𝑄 2min is the minimum
photon virtuality [20, 21]. The 𝐹E and 𝐹M depend on beam charged particle 𝑖. For example, if the particle
𝑖 is electron, (𝐹E ) 𝑖 = (𝐹M ) 𝑖 = 1.

1.1.1 Light-by-light scattering in 𝒑 𝒑 collisions
This thesis focuses on protons as the beam particles in LHC proton-proton (𝑝 𝑝) collisions. Phenomenologically, photon production from incoming proton beams can be described as elastic or inelastic, whichrefers to how each photon is produced from the proton beam. Elastic production occurs as coherent photon
radiation from the whole proton without disintegration. In inelastic production, the proton breaks up andinduces hadronic activity, which is often called dissociative production. Inelastic production can occur
in two regimes. For 𝑄 2 . 5 GeV2 , the proton excites and subsequently decays into a low-multiplicity
final state. For higher 𝑄 2 , the photon can be considered as being radiated off a parton inside the proton.
Suppose both photon productions from beam protons colliding with each other are elastic. In that case,the two-photons interaction in the 𝑝 𝑝 collision is called exclusive or elastic, referred to as EL in this thesis.
On the other hand, an event in which one photon production is inelastic is called a single-dissociative (SD)event, and an event in which both photon productions are inelastic is called a double-dissociative (DD)event. The SD and DD events are collectively called dissociative events. Fig. 1.2 shows the conceptual
Feynman diagram of EL, SD, and DD events in 𝑝 𝑝 collisions.
In the 𝑝 𝑝 collisions, the 𝑄 2min is approximated as
𝑚 2𝑝 𝜉𝑖2 +(𝑄 min ) 2𝑖 =

8



2
𝑚 𝑝 (∗)− 𝑚 2𝑝𝜉𝑖𝑖

1 − 𝜉𝑖,1.1 Light-by-light scattering at the LHC
𝑝

𝑝

𝑝∗

𝑝

𝑝∗

𝑝

𝛾

𝛾

𝛾

𝛾

𝛾

𝛾

𝑝

𝑝

𝑝

(a) EL𝑝∗𝑝𝑝(b) SD(c) DD

Figure 1.2: Feynman diagrams illustrating (a) EL (b) SD (c) DD photon-initiated event in 𝑝 𝑝 collisions.

where 𝜉 = 𝜔/𝐸, 𝑚 𝑝 is proton mass, and 𝑚 𝑝 (∗) is the invariant mass of the final state proton which can beeither intact or dissociative. The (𝐹E ) 𝑖 and (𝐹M ) 𝑖 depend on whether the 𝑖-side proton remains intact or isdissociated. In the elastic case, they are given in terms of proton form factors 𝐺 E and 𝐺 M ,(𝐹E ) 𝑖 =4𝑚 2𝑝 + (𝐺 E ) 2𝑖
4𝑚 2𝑝 + 𝑄 2𝑖,

(𝐹M ) 𝑖 = (𝐺 M ) 2𝑖 .
As a characteristic of proton, the 𝐺 E approximately satisfies(𝐺 M ) 2𝑖(𝐺 E ) 2𝑖 =
=
𝜇2𝑝

1+

𝑄 2𝑖
0.71 GeV2! −4,

where the proton magnetic moment is taken to be 𝜇2𝑝 = 7.78. In the inelastic case, the 𝐹 factors are given
in terms of proton structure function 𝐹2 (𝑥, 𝑄 2 ),
𝐹2 𝑥 𝑖 , 𝑄 2𝑖
(𝐹E ) 𝑖 =d𝑥 𝑖 ,𝑥𝑖
∫𝐹2 𝑥 𝑖 , 𝑄 2𝑖
(𝐹M ) 𝑖 =d𝑥 𝑖 ,
𝑥 𝑖3∫

where 𝑥 𝑖 is the fraction of the energy of the parton interacting with the photon over the proton energy andis approximated as
𝑄 2𝑖
𝑥𝑖 =
2 ,
𝑚 𝑝∗ 𝑖where 𝑚 𝑝∗ is the invariant mass of the dissociated proton system, in the limit that the quark mass is
negligible. The maximum photon virtuality 𝑄 2max is approximated as 1/𝑟 2𝑝 and typically assumed to be
2-4 GeV2 [22], where 𝑟 𝑝 is the proton radius.
The distribution of photon flux 𝑛 peaks at low 𝑄 because it is proportional to 1/𝑄 2 . This means the protons
with low scattering angles are dominant. Especially in the elastic photon productions, the angle is typicallyas slight as 𝑂 (10) µrad. The photon fluxes are integrated over 𝑄 2 ,∫ 𝑄max2𝑛𝑖 (𝜔𝑖 ) =
2
𝑄min

d𝑛𝑖
d𝑄 2𝑖 .
d𝜔𝑖 d𝑄 2𝑖

91 Introduction√
In the 𝑝 𝑝 collisions with the center-of-mass energy 𝑠, the luminosity of the 𝛾𝛾 collisions as a functionof center-of-mass energy of two-photons,√p√𝑚 𝛾𝛾 ≡ 𝑠ˆ ≡ 𝑠𝜉1 𝜉2 = 2 𝜔1 𝜔2 ,is calculated as a convolution of the photon spectra,d𝐿 𝛾𝛾= 𝑛1 (𝜔1 ) ⊗ 𝑛2 (𝜔2 ) .
d𝑚 𝛾𝛾
The cross section of light-by-light scattering in 𝑝 𝑝 collisions is then given by∫d𝐿 𝛾𝛾 d𝜎d𝜎
ˆ=d𝑚 𝛾𝛾 ,dΩd𝑚 𝛾𝛾 dΩwhere d𝜎/dΩˆis the cross section of light-by-light scattering as the elementary process [23].

1.1.2 Forward proton detectors
After EL or SD processes at the interaction point (IP), the intact protons remain part of the beam. They canbe detected, tagged, and have their kinematics measured by the LHC’s forward detectors. In the ATLAS
experiment [24], the ATLAS Forward Proton (AFP) detectors [25] are used as such detectors. The AFP
detectors are placed about 210 m away from the IP in the LHC beam pipe on both sides of the beam
direction. The detail of AFP is described in Sec. 2.3. The AFP can measure proton fractional energy loss
𝜉 for each side and can be used for analyses of 𝛾𝛾 collisions tagging the events with the protons.
Fig. 1.3 shows the simulated d𝐿 𝛾𝛾 /d𝑚 𝛾𝛾 as a function of 𝑚 𝛾𝛾 for each of Pb+Pb and 𝑝 𝑝 collisions. The√√𝛾𝛾 luminosity in 𝑝 𝑝 collisions at 𝑠 = 14 TeV is higher than the one in Pb+Pb collisions at 𝑠 = 5.52 TeV
in the high-mass range. On the other hand, the luminosity in Pb+Pb collisions is enhanced in the lower
mass range because it is proportional to 𝑍 4 , where 𝑍 is the atomic number of the ion. This is because an
ion form factor in the Eq. 1.1 is proportional to 𝑍 2 . The luminosities for 𝑝 𝑝 collisions with a requirementof proton tagging with forward detectors are also illustrated in the plot.
If the decay products from light-by-light scattering are detected in the ATLAS detector, the 𝜉 can becalculated from their kinematics as well.1 In the event selection, requiring the consistency of the measured
𝜉 values between the ATLAS and AFP detectors for each side reduces the background events. It improves
the sensitivity to the events of interest. The analysis in this thesis makes use of this feature as the primarystrategy for event selection. Conventional 𝛾𝛾 collision analyses specific to EL events without the forward
proton detectors usually require less track multiplicity around the interaction vertex or low calorimeter
activity instead.
The CMS-TOTEM Precision Proton Spectrometer (PPS) [27], managed by the CMS [28] and TOTEM [29]collaborations, is an AFP equivalent in the LHC. The PPS detectors are also around 200 m away fromthe IP in the CMS detector. It is proposed to add additional forward proton detectors at around 400 m
positions to enhance the proton acceptance, as also discussed for the AFP (see Fig. 1.3) [30].

1 Since AFP is also operated by the ATLAS , some papers regard AFP as part of the ATLAS detector and refer to the ATLAS

detector as the central detector. However, this thesis distinguishes the ATLAS and AFP detectors.

10

1.2 Light-by-light scattering mediated by new physics

1 ⇥ 1034

dLe↵
dM[cm 2 s 1 GeV 1 ]p
P bP b, L = 6 ⇥ 1027 cm 2 s 1 , s = 5.52 TeV
p
pp, L = 2 ⇥ 1034 cm 2 s 1 , s = 14 TeV
pp, RP 220 m
pp, RP 220 + 420 m

1 ⇥ 1033
1 ⇥ 1032
1 ⇥ 1031
1 ⇥ 1030
1 ⇥ 10291 ⇥ 1028
1 ⇥ 1027
1 ⇥ 1026

1

10

100

M

1000

[GeV]

Figure 1.3: Effective 𝛾𝛾 luminosity as a function of photon-fusion mass in Pb+Pb and 𝑝 𝑝 collisions at the LHC [26].
In the 𝑝 𝑝 case, the 𝛾𝛾 luminosity is also shown with proton tagging at 220 m and 420 m away from the IP. The “RP”
stands for the Roman Pots, containing the AFP detectors described in Sec. 2.3. Though the installation of forward
proton detectors at around 420 m has been proposed, no experiment has been realized yet.

1.2 Light-by-light scattering mediated by new physics
The cross section of SM light-by-light scattering in 𝑝 𝑝 collisions is too small to be observed in the current
LHC. However, BSM particles can enhance the cross section of light-by-light scattering and make it
observable. Although the cross section is still small, the proton tagging with the forward proton detectors
enhances the sensitivity and enables the search for high-mass BSM particles mediating light-by-lightscattering. Many BSM models predict the existence of new particles that couple with photons. The
analysis of light-by-light scattering can probe such new physics. The analysis in this thesis assumes anaxion-like particle (ALP), an extension of the axion, as a target signal process.
The QCD field has multiple vacuums mixing with each other and is naturally expected to violate 𝐶𝑃
symmetry. The 𝐶𝑃-violating term in the QCD Lagrangian isLΘ = −Θ̄

𝛼𝑠 𝜇𝜈 ˜𝐺 𝐺 𝜇𝜈 ,8𝜋

1
𝐺˜ 𝜇𝜈 ≡ 𝜖 𝜇𝜈𝜆𝜌 𝐺 𝜆𝜌 ,2where Θ̄ ∈ [−𝜋, 𝜋] is the 𝐶𝑃-violating parameter, 𝐺 𝜇𝜈 is the color field strength tensor, and 𝐺˜ 𝜇𝜈 , with
𝜀 0123 = 1, is its dual [31]. However, experimentally, 𝐶𝑃 has been measured to be preserved so far with
precision 10−10 at least [32, 33]. This unnatural fine-tuning is called strong 𝐶𝑃 problem [34, 35].
A field with a global U(1) symmetry spontaneously broken in the early universe is introduced to solve
this problem. The symmetry is called Peccei-Quinn (PQ) symmetry, and the underlying U(1) is denoted
U(1)PQ . The spontaneous symmetry breaking (SSB) selects a single vacuum from the multiple vacuums.
As a general property of global symmetry SSB, a massless Nambu-Goldstone boson (NGB) appears. If the
global symmetry is also explicitly broken, most likely due to quantum effects, the NGB acquires mass andis called pseudo-NGB (PNGB). This PNGB is called axion when the symmetry of interest is U(1)PQ [36,37], and its mass is denoted 𝑚 𝑋 . The axion is a pseudo-scalar particle with negative parity. The color
anomaly𝑎 𝛼𝑠 𝜇𝜈 ˜L 𝑎𝑔𝑔 = Ξ𝐺 𝐺 𝜇𝜈
𝑓 𝑎 8𝜋11

1 Introduction𝛾
𝑎𝛾
Figure 1.4: Feynman diagram of axion and two-photons coupling. Fermions mediate the interaction.

is induced by the axion field 𝑎, where 𝑓 𝑎 is the axion decay constant regarded as the PQ scale, and Ξ is amodel-dependent anomaly coefficient. This term also represents the potential for the axion field whose
minimum is at 𝑎 = Θ̄ 𝑓 𝑎 /Ξ, cancelling the 𝐶𝑃-violating term and thus solving the strong 𝐶𝑃 problem.
The coupling of an axion to two photons is induced by mixing with 𝜋 0 [38–40],
1
L 𝑎𝛾𝛾 = − 𝑎𝐹𝜇𝜈 𝐹˜ 𝜇𝜈 ,𝑓where 𝑓 −1 is the coupling constant,2 𝐹𝜇𝜈 is the electromagnetic field strength tensor, and 𝐹˜ 𝜇𝜈 is its dual.
Charged fermions mediate the coupling as shown in Fig. 1.4. The axion two-photon decay width iscalculated as
𝑚 3𝑋
Γ=
.
(1.2)
4𝜋 𝑓 2
As a general property of PNGB, the coupling constant is proportional to axion mass. The quantitative
relation depends on the PQ scale and the PQ charge of the fermions, which are model-dependent. However,motivated by the fact that axion has not been observed up to now, one may loosen a condition of axion
and introduce ALPs whose mass and coupling constant are independent. There is no change in otherproperties, such as Lagrangian and decay width. The ALP field is also denoted 𝑎, and the Lagrangian is
L=

1 𝜇
1
1
𝜕 𝑎𝜕𝜇 𝑎 − 𝑚 2𝑋 𝑎 2 − 𝑎𝐹 𝜇𝜈 𝐹˜𝜇𝜈 ,
2
2
𝑓

(1.3)

where the first and second terms are kinetic and mass terms, respectively.
Any BSM neutral spin-0 particle couples to fermions only via dimension-5 operators proportional to
the fermion mass, while its dominant coupling to gauge bosons is via dimension-5 operators containing
derivatives [23]. Thus, in energy regimes exceeding the top quark mass, ALPs are only accessible throughtheir coupling to the gauge bosons and the Higgs boson. In this thesis, only photon coupling is considered.
Fig. 1.5 shows the Feynman diagrams for the EL, SD, and DD processes of light-by-light scattering in 𝑝 𝑝collisions mediated by ALP. In this thesis, these processes are called signal events.
Using the 𝛾𝛾 → 𝛾𝛾 helicity amplitudes
M++++ = −4𝑠ˆ2,
𝑓 2 𝑠ˆ − 𝑚 2𝑋

!
4
𝑠ˆ2
𝑡ˆ2
𝑢ˆ 2
M++−− = 2++,𝑓 𝑠ˆ − 𝑚 2𝑋 𝑡ˆ − 𝑚 2𝑋 𝑢ˆ − 𝑚 2𝑋
M+++− = 0,
M+−+− ( 𝑠ˆ, 𝑡ˆ, 𝑢)ˆ = M++++ ( 𝑢,ˆ 𝑡ˆ, 𝑠ˆ),M+−−+ ( 𝑠ˆ, 𝑡ˆ, 𝑢)ˆ = M++++ ( 𝑡ˆ, 𝑠ˆ, 𝑢),ˆ
2 Alternatively, 𝑔 ≡ 𝑓 −1 is often used as the coupling constant.

121.3 Previous researches
𝑝

𝑝

𝑝∗

𝑝

𝛾
𝛾

𝛾

𝛾
𝛾

𝑎

𝛾

𝛾
𝑎

𝑝
(a) EL

𝑎

𝛾

𝛾
𝛾

𝑝

𝑝∗

𝑝

𝛾

𝛾

𝑝

𝑝

𝑝∗

𝑝

(b) SD(c) DD

Figure 1.5: Feynman diagrams illustrating ALP 𝑎 mediated light-by-light scattering. This process can take place inthree distinct topologies: (a) exclusive 𝑝 𝑝 → 𝑝(𝛾𝛾 → 𝛾𝛾) 𝑝, (b) single dissociative 𝑝 𝑝 → 𝑝 ∗ (𝛾𝛾 → 𝛾𝛾) 𝑝, and (c)double dissociative 𝑝 𝑝 → 𝑝 ∗ (𝛾𝛾 → 𝛾𝛾) 𝑝 ∗ .

where 𝑠ˆ, 𝑡ˆ, and 𝑢ˆ are the Mandelstam variables of the diphoton system, the elementary process unpolarizedcross section is given by


d𝜎
ˆ
1
2
2
2
2
|M
|
|M
|
|M
|
|M
|
=
+
+
+
.
++++
+−+−
+−−+
++−−
dΩ 128𝜋 2 𝑠ˆThe calculation is discussed in Ref. [23] and is valid up to 𝑚 𝛾𝛾 ∼ 4𝜋 𝑓 . In the 𝑚 𝛾𝛾 mass range where
ALP is produced resonantly, the elementary process cross section is proportional to the coupling constantsquared,𝜎ˆ ∝ 𝑓 −2 B𝑎→𝛾𝛾 ,(1.4)
where B𝑎→𝛾𝛾 is the branching ratio of ALP decaying to two photons assumed to be 100% in this analysis.
A more detailed modeling is in Ref. [22].
The ALP model described above is used for signal Monte Carlo (MC) sample generation and is assumed
to interpret the results in this thesis. However, there are other BSM models that can be examined using
light-by-light scattering, such as warped extra dimensions [41], Kaluza-Klein gravitons [42], and composite
Higgs [43]. There is also a model-independent Effective Field Theory (EFT) to represent BSM light-bylight scattering by the effective Lagrangian using dimension-8 operators,
L4𝛾 = 𝜁1 𝐹𝜇𝜈 𝐹 𝜇𝜈 𝐹𝜌 𝜎 𝐹 𝜌 𝜎 + 𝜁2 𝐹𝜇𝜈 𝐹 𝜈𝜌 𝐹𝜌𝜆 𝐹 𝜆𝜇 ,(1.5)

where the coefficients 𝜁1,2 are the parameters and predicted by any BSM model [22]. The 𝛾𝛾 → 𝛾𝛾 crosssection is calculated as
2 

d𝜎
ˆ
1  2 2
ˆ + 𝑠ˆ𝑡ˆ 48𝜁12 + 40𝜁1 𝜁2 + 11𝜁22 .
=
𝑠
ˆ
+
𝑡
dΩ 16𝜋 2 𝑠ˆAlthough searches dedicated to resonant BSM signals such as the ones introduced in Sec. 1.3.1 can also
study several different BSM models inclusively, the EFT model can take non-resonant BSM signals into
account as well.

1.3 Previous researches
ALPs have been strongly constrained by multiple observations [44]. Many such results are modelindependent and thus can be represented in the coupling constant or cross section versus mass plane of
ALP. The parameter space is loosely constrained at high masses above the GeV scale, where searches are

13

σfid × B [fb]1 Introduction

Observed CLs limit

102

Expected CLs limit
Expected ± 1 σ

10

ATLAS
s = 13 TeV, 139 fb-1
Spin-0 ModelNWA

Expected ± 2 σ

1

10−1

10−2

500

1000

1500

2000

2500

3000

mX [GeV]Figure 1.6: Exclusion limits of the fiducial cross section of resonant decay spin-0 BSM particles
into two-photons in√
NWA as a function of their mass [52]. The analysis uses 139 fb−1 of 𝑝 𝑝 collision data at 𝑠 = 13 TeV. The region
above the solid black line is excluded.

collider-based. Some studies explicitly searching for ALPs have been performed, but their mass ranges arelower than the research target in this thesis [45].
This section introduces three kinds of previous research whose methodologies are inherited in the analysisof this thesis. Each sort of analysis sets exclusion limits on a parameter space of BSM.

1.3.1 High-mass inclusive diphoton resonance searches
Some resonant BSM particles can decay into two photons. Requiring high-energy photons in the final state
reduces background events, and the excellent energy resolutions of the electromagnetic (EM) calorimetersof the ATLAS and CMS detectors enable the diphoton mass (𝑚 𝛾𝛾 ) distribution to have a narrow peakcorresponding to the resonance.3 The Higgs discovery made use of these features in the 𝐻 → 𝛾𝛾channel [46, 47]. After the discovery, high-mass diphoton resonance searches have been performed in the
ATLAS [48–52] and CMS [53–57] experiments in 𝑚 𝛾𝛾 ranges beyond 𝑂 (100) GeV. Despite the excellent
detector resolution, the peak width is dominated by the detector resolution rather than the natural width insome models. Especially the natural width is assumed to be 0 if it is smaller than a few GeV, which iscalled narrow-width approximation (NWA).
To use the narrow feature of the resonance peak effectively, the 𝑚 𝛾𝛾 distribution is modelled by a function
for each hypothetical mass of the BSM particle of interest. On the other hand, the background 𝑚 𝛾𝛾
distribution has a smoothly-falling shape, which is also parametrized with another function. Finally, thesum of these signal and background functions, the signal-plus-background (s+b) function, is fitted for each
BSM particle mass. This enhances the sensitivity to signal events compared to binning the distribution
into a histogram where the binning optimization is not trivial.
√
The latest ATLAS result uses 139 fb−1 of 𝑝 𝑝 collision data at 𝑠 = 13 TeV [52]. The exclusion limits
on the cross section of spin-0 particle resonance in NWA are shown in Fig. 1.6. The latest CMS journal√
publication uses of 35.9 fb−1 of 𝑝 𝑝 collision data at 𝑠 = 13 TeV [57]. Since these analyses search for
BSM particles inclusively, the results can be interpreted as resonant ALPs with narrow widths.
3 The ATLAS EM calorimeter also has an excellent angular resolution.

14

1/ a [TeV 1 ]1.3 Previous researches

ma [GeV]Figure 1.7: Statistically excluded regions in a plane of ALP coupling constant to two-photons and ALP mass [60].
The coupling constant is defined as √Λ𝑎−1 ≡ 4 𝑓 −1 [62]. The ATLAS light-by-light scattering analysis result using−1
2.2 nb Pb+Pb of collision data at 𝑠 = 5.02 TeV (indicated by “this paper”) is compared with the other results
derived from Refs. [44, 58, 63–65].

1.3.2 Light-by-light scattering observation in Pb+Pb collisions
After the reports of the strong evidence of light-by-light scattering [11, 58], the ATLAS achieved an
initial direct observation of high-energy light-by-light scattering [59] in 2019 by performing an exclusive√
diphoton analysis in 1.73 nb−1 of Pb+Pb collision data at 𝑠 = 5.02 TeV. The interaction does not break
the beam lead ions. Therefore, no QCD interaction occurs at the signal vertex. In the final state, twophotons are exclusive in that no other objects are associated with the signal vertex. Thus, the final state
diphoton is back-to-back in a plane orthogonal to the beam direction, and the event selection uses thischaracteristic.
The ATLAS analysis was updated using 2.2 nb−1 Pb+Pb collision data, adding the result of the ALP
search [60, 61]. Fig. 1.7 shows the exclusion limits of ALP coupling constant to two-photons derived from
Refs. [58, 60]. The limits are up to 100 GeV and do not overlap with the search region of the analysis in
this thesis.

1.3.3 Analyses using the forward proton detectors
The ATLAS observed SM 𝛾𝛾 → ℓ + ℓ − process in 14.6 fb−1 𝑝 𝑝 collision data using the AFP for the firsttime [66]. Fig. 1.8 shows the diagram of the signal event in the analysis. This result serves as proof-ofprinciple to demonstrate that the AFP system and software can be used in conjunction with the ATLAS
detector information to deliver physics results. Since the initial beam proton momenta are known, theproton energy loss fractions 𝜉𝑖 can be calculated from the measured dilepton kinematics simultaneously.
The events are required to have at least one proton with 𝜉 measured by the AFPs close to the 𝜉 calculated
from the dilepton system. This is called diphoton-proton matching. The elementary 𝛾𝛾 → ℓ + ℓ − processes
have already been measured by the ATLAS [67–69] and CMS [70, 71] collaborations without using
the forward proton detectors in the past. Thus, the 14.6 fb −1 data analysis using the AFPs has roles not
only as a verification of the SM but also as an anomaly search. In addition, the signal events are used
for the calibration and performance evaluation of the AFPs. The CMS and TOTEM s also reported theobservation of the same kind of events with proton tagging on one side of the outgoing protons [72].

151 Introduction
𝑝 (∗)

𝑝

ℓ−
𝛾
𝛾
ℓ+
𝑝

𝑝 (∗)

Figure 1.8: Feynman diagram of SM 𝛾𝛾 → ℓ + ℓ − process in 𝑝 𝑝 collisions. Lepton ℓ is electron or muon. The
potential proton dissociation is denoted by (∗) .

As another process searched for using the forward proton detectors, the CMS and TOTEM published thefirst result of a search for light-by-light scattering mediated by BSM particles in 9.4 fb−1 of 𝑝 𝑝 collisions√
at 𝑠 = 13 TeV using the PPS detectors [73]. The signal event of the analysis is the same as the one in
the research in this thesis. However, the CMS and TOTEM analysis targets only EL processes requiring
the kinematical matching of both sides of the outgoing intact protons. In this case, only electromagnetic
coupling needs to be taken into consideration. In other words, this search is independent of the strong andweak force couplings that most other LHC searches rely on, enabling clean events analysis. Such a doublematching results in extreme background events reduction. Then, a counting experiment is performed to
set exclusion limits on the parameter spaces of the BSM models. MC samples are used for the background
events contribution estimation.
The CMS-TOTEM PPS analysis results were updated using the 102.7 fb −1 data [74]. Even this updated
analysis observed only one event after all the event selection due to the strict double-matching requirement.
Both the previous and the later analyses set exclusion limits on the 𝜁1 -𝜁2 plane described in Sec. 1.2. In
addition, this analysis set the exclusion limits of the ALP coupling constant to two photons as a function of
ALP mass. In contrast to the resonance searches with 𝑚 𝛾𝛾 distribution shape fitting described in Sec. 1.3.1,
the number of events after the event selection are counted inclusively in terms of ALP mass. The signal
efficiency, the rate of signal events passing the event selection, is estimated for each hypothetical ALP
mass using signal MC samples. The exclusion limits are derived from the number of observed events andthe signal efficiency for each ALP mass. The signal efficiency and the ALP exclusion limits are shownin Fig. 1.9. The single peak structure of the efficiency reflects the PPS geometrical acceptance and is
propagated to the mass dependency of the limits.

1.4 Aim and overview of this work
This research aims to establish methods for searches for BSM light-by-light scattering and other similar
research, inspired by the observation of light-by-light scattering in 2019 [59] first. Then, ALPs with mass
ranging from 150 GeV to 1600 GeV mediating light-by-light scattering in the LHC 𝑝 𝑝 collisions arechosen as the target in the analysis as the first example. This is the first time for the ATLAS to study
light-by-light scattering in the 𝑝 𝑝 collisions. The appearance of the new detectors, AFPs, was the decisive
factor in starting this research. The analysis inherits the methodologies in Refs. [52, 60, 66] introduced in
Sec 1.3.
The 𝑝 𝑝 collision data taken in 2017 is used in this analysis. The ATLAS detector and the AFPs are used
to detect the final state of two photons and protons of the signal events, respectively. This analysis is thesecond one using the AFP after the dilepton analysis [66], and the same datasets of 14.6 fb−1 are used.

16

1.4 Aim and overview of this work
Preliminary(a) Signal efficiency(b) Exclusion limits

Figure 1.9: Results of a search for light-by-light scattering mediated by BSM particles using the CMS-TOTEM PPS
detectors [74]. (a) Signal efficiency as a function of ALP mass estimated using the signal MC samples for eachdata-taking year. The dashed lines only consider the detector efficiency, and the solid lines include the other object
and event selection efficiency. (b) The exclusion limits of ALP coupling constants to two photons as a function of
ALP mass. The region above the solid black line is excluded.

The experimental setup is detailed in Sec. 2. The analysis in this thesis uses two kinds of samples, the data
samples and the signal MC samples. The former was taken in the actual experiment in 2017 when AFPs
became operational stably. Unfortunately, the 2018 data of the AFPs failed to be recorded correctly. The
signal MC samples are used for modeling signal event distribution, evaluation of signal efficiency, andsome tests for systematic uncertainty evaluation. The details of the samples are described in Sec. 3.
Events with a high-energy back-to-back diphoton system are selected as done in the Pb+Pb light-by-lightscattering analysis [60]. The proton energy loss fractions 𝜉𝑖 are calculated from the diphoton kinematics
for each side 𝑖, assuming the initial photon virtuality is small enough. The formula is derived in Sec. 4. To
𝑖 . The 𝜉 can also be measured byclarify that the 𝜉𝑖 is calculated from the diphoton, this is denoted 𝜉 𝛾𝛾𝑖
the AFP if the outgoing proton 𝑖 remains intact and has momentum within the geometrical acceptance𝑖of the AFP. The 𝜉𝑖 measured by the AFP is denoted 𝜉AFP
. Only the events with at least one outgoing𝑖
proton within a 𝜉AFP range in which the proton reconstruction efficiency and its uncertainty are measured
𝑖 range. In
are selected. The events are also required to have at least one 𝜉𝑖 within the corresponding 𝜉 𝛾𝛾𝑖𝑖
addition, the matching between 𝜉 𝛾𝛾 and 𝜉AFP is required as done for each 𝑖 as done in Refs. [66, 74]. The
𝑖 andconcepts of the matching are illustrated in Fig. 1.10. Finally, the events with matched pair of 𝜉 𝛾𝛾𝑖
𝜉AFP
for at least one 𝑖 are selected. This is called OR selection in contrast to the selection used in the
CMS-TOTEM PPS analysis [74] requiring the matchings for both protons 𝑖, which is called AND selectionin this thesis (see Sec. 1.3.3). The event selection are detailed in Sec. 4.
The statistical model inherits one used in the high-mass diphoton resonance search [52]. The s+b fittingwith the model enhances the sensitivity to resonant ALPs. The signal distribution is modelled in Sec. 5.
The signal efficiency is also modelled as a part of signal modeling in this section. The ALP massdependency of the efficiency reflects the AFP acceptance and differs from the CMS-TOTEM PPS casebecause of the difference between OR matching and AND matching. The unit of signal yield in the final
results, signal strength, is also defined here. The background distribution is modelled in Sec. 6. For themodelling, a background sample is prepared in a fully data-driven way. This means no background MC
sample is used, unlike usual analyses. Such data-driven background sample creation method inherits the
dilepton analysis [66], but some points were improved in this analysis. The uncertainties of the signal
and background models are considered in each section. For the background modeling uncertainty, the171 Introduction

LHC magnet
AFP𝒑

C-side~-200 mProton energy loss fraction𝝃%
𝐀𝐅𝐏

LHC magnet~+70 m~-70 mmatching

Measure proton energy loss
fraction 𝝃±𝜸𝜸 from diphoton
±
𝜉!!
= 𝑚!!

𝑒 ±#!!
𝑠

𝜸

AFP

𝒑

A-side~+200 mProton energy loss fraction

g
chin
mat

𝝃$
𝐀𝐅𝐏

𝜸

EM calorimeter𝑖 and 𝜉 𝑖
Figure 1.10: Conceptual illustration of matching between 𝜉 𝛾𝛾
AFP . In the ATLAS detector illustrated at the𝑖 is calculated. The outgoing
center [75], the final state two-photons are detected by the EM calorimeter, and 𝜉 𝛾𝛾
𝑖proton is curved by the LHC magnets and enters the AFPs to have 𝜉AFP measured.

data-driven background distributions are smoothed, and spurious signals are evaluated by s+b fitting to
artificial distributions derived from the smoothing.
The systematic uncertainties originate from the preciseness of object detection, simulation, and reconstruction, for example. Such effects are summarized in Sec. 7, including the signal and background modelinguncertainties. The complete statistical modeling is discussed in Sec. 8. The systematic uncertainties areincorporated into the model as the constraints to the likelihood. Two kinds of statistical tests are performed
using popular test statistics. One is for testing BSM, and the other is for setting the exclusion limits for the
ALP parameter space. The results are shown in Sec. 9. Discussions about the interpretation of the results,
comparison to the other analyses, expected improvement in the future analysis, and how the methods
established in this research can be used in future analyses follow in the same section.

18

2 Experimental Setup
The experimental setup consists of the LHC, ATLAS detector, and AFPs. This section introduces them.

2.1 LHC
The LHC [10] is an energy-frontier hadron-hadron collider operated by the European Organization for
Nuclear Research (CERN) that aims to search for new physics at the TeV scale directly. It is placed acrossthe border of Switzerland and France in a tunnel with 27 km circumference and a depth of down to about√
100 m from the ground level. It is designed to collide proton beams at 𝑠 = 14 TeV and an instantaneous
luminosity of 1034 cm−2 s−1 . The LHC also accelerates heavy ions, but only 𝑝 𝑝 collisions are discussed inthis thesis. The LHC consists of straight sections and arc sections. The straight sections are used as theinsertion regions (IRs) where the experimental apparatus are inserted into the LHC tunnel. The ATLAS
detector is at an IR. To have the protons with 𝐸 = 7 TeV remain in the LHC, the dipole magnets with 8.3 T
are needed in the arc sections. The current LHC magnets cannot have a higher magnetic field because the
NbTi superconducting wire induces quenching1 . The beam proton loses its energy through synchrotron
radiation, having it complemented by radiofrequency (RF) acceleration. The energy loss is proportional
to 𝛾 4 , where 𝛾 is the beam Lorentz factor. For example, a beam proton with 𝐸 = 7 TeV loses 7 keV per
revolution (turn). During the operation from 2010 to 2012 (Run 1), the LHC performed 𝑝 𝑝 collisions at√√
𝑠 = 7 TeV and 8 TeV. During the operation from 2015 to 2018 (Run 2), the LHC worked at 𝑠 = 13 TeV.
The setups in 2017 when the data used in this research were taken are discussed in this section.
The LHC proton beams are transported in a bunch structure, and each beam bunch consists of approximately
1011 protons. The beam bunches orbit the two parallel beam pipes in opposite directions and cross at
the IP of the ATLAS detector at a frequency of 40 MHz. The beams circling the LHC clockwise andcounterclockwise are called beam 1 and beam 2, respectively. It takes about 90 µs per turn for a bunch, andits inverse (revolution frequency) is about 11 kHz. Since the 𝑝 𝑝 collision cross section is about 0.1 b, the𝑝 𝑝 collision frequency is 𝑂 (1) GHz. Thus, the averaged number of 𝑝 𝑝 collisions per bunch crossing is𝑂 (1) GHz/40 MHz ∼ 𝑂 (10). All these 𝑝 𝑝 collisions, except for the events of interest, are called pileups.
A bunch crossing or its timing referred from the detectors is called BC. Detected pileups are subdivided
into the ones originating from the same BC (in-time pileups) and different BCs (out-of-time pileups).
Although a BC is a set of 𝑝 𝑝 collision events, it is often referred to as an event, especially in physics
analyses.
The proton beams are accelerated in a series of other accelerators before being injected into the LHC.
Fig. 2.1 illustrates the CERN accelerator complex. The origins of the protons are hydrogen anions. The
initial injector, LINAC 2, accelerates them to 50 MeV. The anions are injected into the Proton Synchrotron
Booster (PSB), and the electrons are removed from the anions using the carbon foils to obtain the protonbeam. The protons are accelerated to 1.4 GeV in the PSB and injected into the Proton Synchrotron (PS)six times until the PS is filled. In the PS, the protons in the six bunches are accelerated to 26 GeV. In thisprocess, 72 of the 40 MHz beam bunches are formed by gradually inducing higher RFs. This method,
1 The critical surface limits the superconducting phase in the space of temperature, current density, and magnetic field. A

too-strong magnetic field causes heating and hence violates the superconductivity. This is called quenching.

19

2 Experimental Setup

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 2

LINAC 3Ions

ions

East AreaPSnp (protons)

REX/HIE
2001/2015

n-ToF

RIBs (Radioactive Ion Beams)

e-

LEIR

2017

2005 (78 m)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 2.1: The complex chain of particle accelerators in CERN [77]. The dark blue line illustrates the LHC. The
smaller machines are used in the chain to help boost the particles to their final energy and provide beams to a whole
set of smaller experiments, which also aim to uncover the mysteries of the universe.

called multiple splitting, is illustrated in Ref. [76]. The bunches are then injected into the Super Proton
Synchrotron (SPS) in 72 × 4 times at most until the SPS is filled. The protons are accelerated to 450 GeV
with the SPS. Finally, the bunches are injected into the LHC until the LHC is filled. Then, the protons areaccelerated to 6.5 TeV taking about 20 minutes for the Run 2 case.
The injections and extractions of the beam into and from the accelerators are controlled by the septum andkicker magnets. A septum magnet creates spaces with a strong magnetic field and without a magnetic field
in a small region, enabling the injection or extraction while the orbiting bunches are unaffected. A kicker
magnet is used to put the protons from the magnetic space into the orbit or extract the protons from theorbit to the magnetic space. The kicker magnets are magnetized only during the injection and extraction.
Their magnetization and demagnetization take a finite time, depending on the beam energy. Thus, there
exist empty bunches for that operation in the 40 MHz bunches.
The quadrupole magnets have a so-called FODO cell structure in conjunction with the dipole magnets
to repeat the focusing and defocusing of the beam in the axes orthogonal to the beam direction. Thisis called betatron oscillation, and its period is tuned not to be an integral multiple of the period of theturn in order to stabilize the beam. Position 𝑥(𝑠) in one of the transverse axes and its derivative 𝑥 0 (𝑠) aredefined for each position 𝑠 in the accelerator. The geometric emittance 𝜖 = 𝜎𝑥2 (𝑠)/𝛽 𝑥 (𝑠) is the degree202.2 ATLAS detectorof the deviation of the beam in the (𝑥, 𝑥 0 )-plane, where 𝜎𝑥2 (𝑠) is the 𝑥-direction deviation of the beam,and 𝛽 𝑥 (𝑠) is determined by the accelerator setup and is called 𝛽-function. The normalized emittanceof the beam is defined as 𝜖 n = 𝛽𝛾𝜖, where 𝛽 is velocity. This is a conserved quantity with respect to 𝑠
and is 3.5 µm in the LHC. Thus, the beam width has a negative correlation with the beam energy. Afterthe beam energy reaches 6.5 TeV, the final focusing system dynamically adjusts the 𝛽 𝑥 of the series ofquadrupole magnets closest to the IP to squeeze the beam and maximize the luminosity at the IP. After thesqueezing, 𝛽∗ ≡ 𝛽 𝑥 (IP) is 0.3-0.4 m in the standard runs in 2017. The instantaneous luminosity L at the
IP is calculated as𝑛b 𝑁b1 𝑁b2 𝑓rev𝑛b 𝑁b1 𝑁b2 𝑓rev 𝛾
𝐹≈𝐹,
4𝜋𝜎𝑥 𝜎𝑦4𝜋𝜖n 𝛽∗" 2 # − 12
𝜃 𝑐 𝜎𝑧𝐹 = 1+,2𝑑s
𝜖 n 𝛽∗𝑑≈,𝛾

L=where 𝑛b ∼ 2500 is the number of bunches per revolution, 𝑁b𝑖 is the number of particles per bunch inbeam 𝑖, 𝑓rev is the revolution frequency, 𝜎𝑥,𝑦 and 𝑑 ≈ 𝜎𝑥 ≈ 𝜎𝑦 are the transverse beam size at the IP,2 𝐹 is
the geometric luminosity reduction factor due to the crossing angle at the IP, 𝜃 𝑐 is the beam crossing angle
at the IP, and 𝜎𝑧 is the longitudinal length of the beam bunches at the IP [78]. 𝜃 𝑐 is typically 100-150 µradin the standard runs in 2017.

2.2 ATLAS detector
The ATLAS detector [24] at the LHC covers nearly the entire solid angle around the collision point. It
consists of an inner tracking detector surrounded by a thin superconducting solenoid, EM and hadroncalorimeters, and a muon spectrometer incorporating three large superconducting air-core toroidal magnets.
Fig. 2.2 shows the image of the whole ATLAS detector. To specify a position in the detectors, ATLAS
uses a right-handed coordinate system with its origin at the nominal IP in the center of the detector andthe 𝑧-axis along the beam pipe. The 𝑥-axis points from the IP to the center of the LHC ring, and the𝑦-axis points upwards. The region with 𝑧 > 0 is called A-side, and one with 𝑧 < 0 is called C-side.
Cylindrical coordinates (𝑟, 𝜙) are used in the transverse plane, 𝜙 being the azimuthal angle around the𝑧-axis. The 𝑟-direction or, more generally, a direction in the 𝑥-𝑦 plane is often referred to as the transverse
direction, while the 𝑧-direction is often referred to as the longitudinal direction.The pseudorapidity isdefinedpin terms of the polar angle 𝜃 as 𝜂 = − ln tan(𝜃/2). The angular distance is measured in units of
Δ𝑅 ≡ (Δ𝜂) 2 + (Δ𝜙) 2 .
Since the 𝑝 𝑝 collision vertices are approximately at the origin, the angles of the generated particles ortheir systems are defined in the same manner. The rapidity 𝑦 is defined as𝑦 ≡ arctanh 𝛽 𝑧 =1 𝐸 + 𝑝𝑧ln,
2 𝐸 − 𝑝𝑧

where the velocity is 𝜷 = (𝛽 𝑥 , 𝛽 𝑦 , 𝛽 𝑧 ), 𝐸 is the energy, and the momentum is 𝒑 = ( 𝑝 𝑥 , 𝑝 𝑦 , 𝑝 𝑧 ). The
pseudorapidity is then defined as the massless limit of the rapidity,𝜂≡

1 𝐸 + 𝐸 cos 𝜃𝜃ln= − ln tan .
2 𝐸 − 𝐸 cos 𝜃
2

2 The beam envelope is circular in the transverse plane at the IP.

21

2 Experimental Setup

Figure 2.2: Computer generated image of the whole ATLAS detector [75]. People are also illustrated for comparison.

In addition, the transverse energy 𝐸 T ≡ 𝐸 sin 𝜃 and transverse momentum 𝑝 T ≡in the analysis.

q

𝑝 2𝑥 + 𝑝 2𝑦 are often used

The subdetectors are introduced in Sec. 2.2.1. Since photon reconstruction mainly uses the informationfrom the EM calorimeters, Sec. 2.2.2 details them. Sec. 2.2.3 describes how the data from the subdetectors
are collected and processed.

2.2.1 Subdetectors
The inner detectors cover |𝜂| < 2.5 and track charged particles in the 2 T axial magnetic field. The innermost
layer has the high granularity silicon pixel detector, among which the insertable B-layer (IBL) [79, 80]
was installed before Run 2. The silicon pixel detector provides four measurements per track, and the IBL
usually has the first hit. The next layer is the silicon microstrip tracker (SCT), where eight measurements
are usually provided per track. The outer layer transition radiation tracker (TRT) based on straw drift tubes
covers |𝜂| < 2.0 and provides information for tracking and electron identification.
Outside the inner detector is the calorimeter system covering |𝜂| < 4.9. The EM calorimeter consists ofhigh-granularity lead/liquid-argon (LAr) calorimeters in the barrel and end cap, covering |𝜂| < 3.2. The
innermost LAr presampler covering |𝜂| < 1.8 corrects the energy. The downstream hadron calorimeterconsists of - the tile calorimeter with steel and scintillator covering |𝜂| < 1.7, - the hadron endcap
calorimeter with copper and LAr covering 1.7 < |𝜂| < 3.2, - the forward calorimeter with copper, tungsten
and LAr covering 3.2 < |𝜂| < 4.9, and measures hadronic energy.
The outermost layer is the muon spectrometer (MS), which measures the muon momentum using themagnetic field generated by the superconducting air-core toroidal magnets. The MS consists of themonitored drift tubes for |𝜂| < 2.7 and the cathode-strip chambers for the forward region for precision222.2 ATLAS detector
measurement, the resistive-plate chambers in the barrel and the thin-gap chambers in the endcap and fortriggering (see Sec. 2.2.3) in |𝜂| < 2.7.
The Luminosity Cherenkov Integrating Detector-2 (LUCID-2) [81] is mainly placed ±17 m away from the
IP and 13 cm away from the beam pipe, covering 5.6 < |𝜂| < 5.9, and measures the beam luminosity using
the pileup events BC by BC. Quartz is used as the Cherenkov radiation medium to detect the radiated lightwith the photomultiplier tubes (PMTs). ATLAS beam condition monitor (BCM) complements LUCID-2.
The usual subdetectors, such as the EM and tile calorimeters, are also used for BC-integrated measurements
based on quantities proportional to instantaneous luminosity. The integrated and instantaneous luminosities
are measured by these detectors in conjunction with some other detectors and methods [82]. The luminositysensitive detectors are calibrated in the dedicated low-luminosity runs using van der Meer (vdM) method [83,84]. The calibration is then transported to the physics data-taking regime at high luminosity. These
procedures map detector responses to luminosity.

2.2.2 EM calorimeter
High-energy photons of interest lose most of their energy in the EM calorimeters. Electrons and photons
are the primary targets for EM calorimeters and are called EM objects. The EM calorimeters are thesampling calorimeters which use LAr as the detection layer. Some hadron calorimeters on the endcap
parts also use the LAr. The set of calorimeters using LAr as the detection layer is called LAr calorimeter.
Fig. 2.3 shows the overview of the LAr calorimeter. The LAr calorimeter is subdivided into four parts,
EM barrel (EMB), EM endcap (EMEC), hadronic endcap (HEC), and forward calorimeter (FCal). The
EMB covers |𝜂| < 1.5, and the EMEC covers 1.4 < |𝜂| < 3.2. The EMEC consists of the outer and inner
wheels covering 1.4 < |𝜂| < 2.5 and 2.5 < |𝜂| < 3.2, respectively. The HEC and FCal cover higher |𝜂|regions. The photon angle used in this research is limited to |𝜂| < 2.4.
The EMB has layers of lead, copper, LAr, etc., as shown in Fig 2.4. The layers have the accordion structure
to geometrical uniformity keeping 𝜙-symmetry. When a high-energy particle enters the detector, theEM shower is generated in the lead absorption layer, and the shower ionizes the argon in the LAr layer(LAr gap). The movement of the popped electrons is detected as an electric signal. The movement of theelectrons is induced by a voltage of about 2 kV applied between a grounded stainless steel layer and acopper layer and is detected by a copper readout electrode placed between two copper anodes insulated
by kapton. Since the Ar ions have negligible mobility compared to electrons, the voltage generated bytheir movement is negligible. The LAr gap is as thin as about 2 mm, so the current obtained by a single
electron can be approximated as a rectangular pulse having the electron drift time as its length. The pulse
height of the rectangular pulse does not depend on the electrons, and the drift time depends linearly on theuniform position of the ionization. Thus, the rectangular pulses are summed into a triangular pulse with aheight proportional to the energy.
The EMB is celled in the 𝑟, 𝜂, and 𝜙 directions, as shown in Fig. 2.5. Each set of cells classified by thedivision in the 𝑟 direction is called a layer. Typically, the 0th layer (presampler), first layer (front layer),second layer (middle layer), and fourth layer (back layer) are arranged in ascending order of 𝑟. The strip
cells with high granularity in 𝜂 in the front layer are designed to distinguish a usual photon and 𝜋 0 decaying
into two photons. The second layer is thick, and EM objects drop most of their energy here. A larger unitof readout cell, Trigger Tower (TT), is used for the first-level (L1) triggering introduced in Sec. 2.2.3.
The EMEC is also 𝜙-symmetric and has the accordion structure in a different direction from the EMB
case. The mechanism to acquire the signals is the same as EMB, and the triangular pulses are generated.
Similar cell division is used as the EMB, but there are many areas where all layers from the 0th to thethird are not entirely arranged.

23

2 Experimental Setup

Figure 2.3: Computer generated image of the ATLAS LAr calorimeter [85].

47 cm

readout electrode

(~2

mm

)

outer copper layer
inner copper layer
kaptonabsorber

HV

uid
liq

liq

uid

stainless steel
glue
lead

arg

arg
on

ga
p

on
ga
p

outer copper layer

HV

Figure 2.4: Structure of the LAr EMB shown in the 𝜙-𝜂 plane [86].

242.2 ATLAS detector
Cells in Layer 3
Δϕ×Δη = 0.0245×0.05
Trigger ToweΔη = 0 r
.1

2X0

47

0m

m

η=0

16X0

982

m

m

4.3X0

15
00

1.7X0
Δϕ=0.0245x36.8m 4mx
=147.3 4mm

Square cells in
Layer 2Δϕ = 0ϕΔη = 0.025
m/8 =4Δη = 0 .69 mm.0031
Strip cells in Layer 1

37.5m

η

.0245

2008 JINST 3 S08003

TriggeTow r
Δϕ = 0er.0Figure 5.4:inSketchof a barrel
the different
layersandare clearlyvisible withganging The radiation
Figure 2.5: Cell division
the LAr
EMBmodule[24]. where
The front,
middle,back layers
are theillustrated.
of electrodes in f . The granularity in h and f of the cells of each of the three layers and of thelength is denoted 𝑋0 .
trigger towers is also shown.

R

Barrelgeometry are divided into signal acquisition for precise energy calculation and
The functions of 5.2.2
the LAr
calorimeter
The barrel electromagnetic
calorimeteris made of twocrates
half-barrels,
centredaround thetriggering. The Front-End
boards (FEBs)
are in[107]the Front-End(FECs)
mountedon z-the cryostat andaxis. Onecoversthe region
with z > through0 (0 < h the< 1.475)
and the other one
the region
read out and process
thehalf-barrel
triangularpulsesdeliveredfeedthroughs.
In the
FEBs, a triangularwith z < 0 ( 1.475 < h < 0). The length of each half-barrel is 3.2 m, their inner and outerpulse from a readoutcell is amplified by the preamplifier and shaped into a bipolar pulse by the CR-(RC)2
diameters are 2.8 m and 4 m respectively, and each half-barrel weighs 57 tonnes. As mentioned
circuit.3 The bipolaris sampled
40 MHz andwith
temporarily
in theSwitched
Capacitor Array
above,pulse
the barrel
calorimeterat
is complemented
a liquid-argonstored
presampler
detector,placed in(SCA). On the otherhand,thesurface,triangularare summed and shaped to create a bipolar TT signal infront ofits innerover thepulses
full h-range.
A half-barrel
is madeinterleaved
readoutTheelec- trigger system
the Tower Builder Board(TBB),and ofthe1024
TTaccordion-shaped
signal is sentabsorbers,
to the L1
triggerwith
system.
trodes.
Theelectrodes
arepositionedinthemiddle
of
the
gap
by
honeycomb
spacers.
The
size
analyses the signals from all the cells in conjunction with the signals from other subdetectors
and fires aof the drift gap on each side of the electrode is 2.1 mm, which corresponds to a total drift time
trigger as an L1-accept(L1A)
signal
when
the
BC
is
judgedtobeinteresting.
Only
thesignals
stored inof about 450 ns for an operating voltage of 2000 V. Once assembled, a half-barrel presents no
the SCA from the BCs where L1A is issued are digitized by the ADC in the FEBs and sent to the backend
Read Out Driver (ROD) through 80-100 m long fibers. The energy deposit in a cell is then calculated withthe optimal filtering method4 using the linearity of the pulse height with respect to the energy.
– 114 –

Beampipe
Warm
wall
Superinsulation
Cold
wall

The cell-level energy computation of the LAr calorimeter is calibrated using test pulses output from thecalibration boards in the FECs. The calibration pulses have shapes similar to the triangular pulses foreach cell. They are sent to the LAr detector to collect the responses in a similar environment as the actual
FigureElectrodeFigure5.19:
Schematic
diagramshowing
the energy,
data-taking for physics
analyses.
The input
pulse heights
are associatedwith particle
and5.20:
thecorrespondence is calibratedusing thetest beamslocated[87, 88]. Moreprecisecalibration andtheobject-level
matrix of copper plthree FCal
modulesin theend-cap
calibration are performed using 𝑝 𝑝 collisions data. An example is to use 𝑍 → 𝑒𝑒 events whose final statecryostat. The material in front of the FCal and and rods with the LAr
kinematics are known.

the shielding plug behind it are also shown. The Molière radius, R
blackregions
arefromstructuralof pileups).
the However,solidthedisk.
shape is used, the energyThecalculation
is affectedby the pileups
the previous BCsparts(out-of-time
bipolarpulse shape cancels this effect on average with the negative lobe with the same integration as the positive lobe.
cryostat.
The diagram has a larger vertical
4 The optimal filtering
algorism is designed to minimize the noise effect. This meets the demand for precise energy calculations.
scale for clarity.
3 The typical length of the triangular pulses is 400 ns, which is much larger than the BC interval, 25 ns. Thus, if this pulse

25

copper tube separated by a precision, radiation-hard plastic fibre woundrangement of electrodes and the effective Molière radius for the modules c2 Experimental Setup

Calorimeter detectors
TileCalMuon detectors

Level-1 Calo

JEP (jet, E)
CMX

Barrelsector logic

MUCTPI

L1TopoCTPCTPCORE

Level-1 Accept

Endcapsector logic

Preprocessor
nMCM
CP (e,γ,τ)CMX

Detector
Read-Out

Level-1 Muon
FE

FE

ROD

ROD

...

FE

ROD

DataFlow
Read-Out System (ROS)

Central Trigger

Level-1
RoI

Fast TracKer(FTK)

Pixel/SCTCTPOUT

High Level Trigger(HLT)
AcceptData Collection Network

Data Storage

Processors

Event
Data

Tier-0Figure 2.6: ATLAS TDAQ system in Run 2 focusing on the components of the L1 trigger system [89].

2.2.3 TDAQ system

The trigger and data acquisition (TDAQ) system selects the BCs to store the corresponding data. Since theBC frequency of 40 MHz is extremely high, not all BCs can be recorded to storage for offline analyses.
Therefore, the trigger system illustrated in Fig. 2.6 is used to select interesting BCs. The detector signals
are temporarily stored in the front-end (FE) electronics, and the same signals are also sent to the trigger
readout to judge whether the BC is worth to be processed in the following steps. If the trigger is fired, theL1A signal makes the stored data sent to ROD and processed further. The procedures up to this point are
exemplified with the LAr calorimeter case in Sec. 2.2.2. The L1 trigger system implemented in custom
hardware is used for this triggering. It uses the information from the calorimeters and the muon detectors.
The L1 trigger accepts events from the 40 MHz BCs at a rate below 100 kHz. In addition, it sends theinformation about the region of interest (RoI) for the BC to the following trigger system.

The high-level trigger (HLT) [89] then selects the BCs with algorithms implemented in software in orderto record events to disk at about 1 kHz. It uses full information from the ATLAS detector, including thefinely segmented calorimeter cells and the inner trackers in the region specified by the RoI signals.

26

2.3 AFP detectors
A-sidesector 8-1
Q6 AFP Q5 Q4 TCL4

ATLAS
Interaction Point 1
Q3 Q1
Q1 Q3

C-sidesector 1-2
TCL4 Q4 Q5 AFP Q6

P1

FAR station-217.909 m

D1

Q2

Q2

NEAR station-205.824 m

NEAR station
205.217 m

Time-of-Flight Silicon Tracker
0

plane
P3 P2 P1 P0

SiT

plane
P3 P2 P1 P0

SiT

plane
P0 P1 P2 P3

x

train
1 2

3

3

bar
2 1

D1

D2 TCL5

TCL6 ALFA

FAR station
217.302 m
Silicon Tracker Time-of-Flightplane
P0 P1 P2 P3

0

1

bar
2

3
3

TCL5 D2

1 2
train

ALFA TCL600y15σ distancebeam 2

P1

diffractive protons

15σ distancebeam 1Figure 2.7: AFP system and the related LHC beam optics between the A-side and C-side forward regions [92].
D1 and D2 are the dipole magnets that bend the beam, while Q1-6 are the quadrupole magnets that focus it. The
collimators TCL4-6 (Target Collimator Long) are also shown. The Near and Far stations are indicated. The
quantitative information about the AFPs uses their local coordinates.

2.3 AFP detectors
The AFP detectors [25] are proton spectrometers developed to study diffraction and search for new physics
mainly. They reconstruct protons from small or zero-angle scattering at the ATLAS IP. A schematic of thedetectors is presented in Fig. 2.7. The signal proton trajectories are mainly bent by the dipole magnets,
which were installed to split the incoming and outgoing beams, beam 1 and beam 2, being merged near
the IP in order for BCs for the original purpose. When a proton loses its energy in an event, its trajectoryis bent by the magnets more significantly than the usual beam protons. Fig. 2.8 shows the typical proton
trajectories with the AFPs. Such effects of the magnets allow the signal protons to reach the forward
detectors inserted near the beam. In such forward regions, the protons are measured by the AFP and
Absolute Luminosity for ATLAS (ALFA) [90, 91] detectors. Detailed descriptions and simulation results
about the LHC beam optics and the related instruments are available in Ref. [78]. In the AFPs, the positionsof the protons are measured with silicon trackers (SiTs). Therefore, the proton momenta and hence theirfractional energy loss 𝜉 = Δ𝐸/𝐸 can be measured using the information of the positions and the magnetic
fields at the LHC magnets. Fig. 2.9 shows the hit position in a SiT plane depending on the proton 𝑝 T and𝜉 as well as a distribution of track positions.
Each side of the AFP system is referred to as an arm. There are two tracking units per arm, referred toas the Near and Far stations. These subdivisions of the AFP detectors are located at 𝑧NEAR ≈ ±205 m
and 𝑧 FAR ≈ ±217 m, respectively, from the IP. Having two stations allows the system to measure not only
proton energy but also its 𝑝 T . In addition, the proton reconstruction efficiency can be measured using
the two stations as described in Sec. 3.5. A detailed description of AFP and its testing and installation isavailable in Ref. [95].
Each station has four planes with 3D silicon pixel sensors (3D-Si) that form the SiTs to measure theprotons’ trajectories. These planes are based on the IBL tracker in the ATLAS detector and have 336 × 80
pixels with 50 × 250 µm2 area each. The total area of one SiT plane is 1.7 × 2.0 cm2 , and it is 230 µm

27

2 Experimental Setup

x [mm]

Proton Trajectoryfor different energies

beam envelope*0s = 14 TeV, β = 0.55 m, beam 1

-5
beam
-10

Proton Trajectoryfor different transverse momenta

ξ < 0.012
0.012 < ξ < 0.14

-15

ξ > 0.14-20

ATLAS Simulation

beam envelope

050(a) Conceptual illustration

100

AFP
Q5
s [m]
200

150

(b) Proton trajectories

Figure 2.8: Proton trajectories from the ATLAS IP to the AFPs [93]. The horizontal axes correspond to the 𝑧-axis.
(a) Conceptual illustration of the trajectories of the proton beam (blue) and its constituent proton being bent by an
LHC dipole magnetic field (orange). The vertical axis is the 𝑥-axis. Comparisons√ for different energy and 𝑝 T valuesof the proton are illustrated. (b) Simulated proton trajectories in C-side for 𝑠 = 14 TeV and 𝛽∗ = 0.55 m. The
horizontal axis is the distance from the IP, and the vertical axis is the local coordinate of the AFPs corresponding to
the difference of the ATLAS 𝑥-coordinates from the 𝜉 = 0 case.

T

p = 2 GeV, ξ = 0Tθc = 285 µradd = 3.35 mm

p = 2 GeV, ξ = 0.07

s = 204 m

T

y [mm]

p = 1 GeV, ξ = 0.07
T

y [mm]

s = 14 TeV
β* = 0.55 m

10

20
Data 2017, run 341649

4000

track positions, C-NEAR

entriesp = 1 GeV, ξ = 0

15
3000
10

0

2000
5
1000
0

-10

ATLAS Preliminary
−20
-20

-10

0

(a) 𝑝 T and 𝜉 dependency

x [mm]

−15

−10

−5

0

0

x [mm](b) Distribution

Figure 2.9: Hit positions in an AFP SiT plane. (a) Simulated proton hit position for different assumptions of 𝑝 T and𝜉 [25]. The thick solid rectangle stands for the SiT plane. (b) Track positions of a SiT plane in the A-side Near
station in a special run in 2017 [94].

282.3 AFP detectors(a) Pixel structure(b) SiT planes

Figure 2.10: (a) Partial structure of a pixel in the AFP 3D-Si sensor [97]. The protons of interest pass through the
pixel from top to bottom. (b) The SiT planes in an AFP station [98]. The protons of interest pass through the planes
from the left back to the right front. The 3D-Si sensor is on the right front side of each plane. The hit signal is readout by the FE-I4 on the left back side and sent outside the RP through the flexes seen in front.

thick. They are tilted 14◦ with respect to the 𝑥-axis to improve the resolution in the 𝑥-direction, to pass theresolution requirement of 𝜎𝑥 = 10 µm, 𝜎𝑦 = 30 µm. Clustering in each SiT plane uses the information of
charge-weighted pixel hit distribution. Multiple pixel hits per proton improve the positional resolution
rather than a single pixel hit. However, a tilt larger than 15◦ may cause inefficiency because of the columnar
electrodes of the 3D-Si. The planes are placed only about 2 mm away from the beam center because thescattering angle of the protons of interest is shallow. Thus, they have slim edges (inactive regions) smaller
than 200 µm. Fig. 2.10 (a) shows a partial structure of a pixel. Each pixel is the n+ -in-p type and consists of
two n+ -junction columns from the front side and six p+ -ohmic columns from the back side. The isolation
between n+ columns at the surface is realized by the p-stop method, where p-type semiconductors are
partially implanted between the electrodes [96]. Protons ionize the p-type bulk, and the electrodes collectthe popped electrons.
Each 3D-Si plane is connected to a FE chip by SnAg bump-bonding. The FE-I4 is used as the FE chip.
The signals are readout for each channel, amplified and shaped, followed by the discriminators. The
processed pulses are triangular pulses with lengths proportional to the collected charge. Thus, the collectedcharge is measured using the time between exceeding and falling below the discriminator threshold,called time-over-threshold (ToT). The threshold is calibrated using the pulse injectors implemented in
the FE-I4 chips. Typically, 2000 electrons are required to send the signal to the next stage. The ToT and
the discriminator firing time are stored until an L1 trigger decision is taken as per the ATLAS TDAQ
system described in Sec. 2.2.3. The FE-I4 can also send a trigger signal using the logical OR of all fired
discriminators. However, any AFP trigger was not used in Run 2 and hence in this research. The FE-I4 isglued and wire-bonded to a flexible printed circuit (flex) to read out the signals. Fig. 2.10 (b) shows thewhole picture of the SiT planes.
The Far stations also house time-of-flight (ToF) detectors to measure the difference in the arrival times Δ𝑡of the two protons on the A-side and C-side. The 𝑧-position of the vertex is calculated by 𝑐Δ𝑡/2, where𝑐 is the light speed, and the timing resolution of 25 ps is propagated to the vertex position resolution of
about 5 mm. Fig. 2.11 shows the position of a ToF detector relative to the SiT. Each ToF detector consistsof 4 × 4 L-shaped Quartz (LQ)-bars placed at the Cherenkov angle to the beam [99]. Each LQ-bar consistsof the radiator arm and light-guide arm transparently glued together at 90◦ . The Cherenkov light emitted

29

2 Experimental Setup

Figure 2.11: The ToF and SiT detectors in an AFP Far station [99]. The protons of interest pass through the SiT
planes and the ToF radiators in order from right back to left front.

from a radiation arm is propagated to the micro-channel plate multi-anode PMT (MCP-PMT) through thelight-guide arm. The photo-electrons created in the MCP-PMT induce pulses processed by a discriminatorand high performance time-to-digital converter (HPTDC). The processed pulses are sampled in 1024 binsof about 25 ps. The ToF detectors can also send a trigger signal. However, this analysis does not consider
the ToF detectors because of their low efficiency in 2017. The reason is discussed in Sec. 3.1.
The AFP and ALFA stations are in separate Roman Pots (RPs) [90]. The AFP detectors are inserted
horizontally in the 𝑥-direction into the beam pipe by the RPs. The RP technology provides movablenear-beam devices that bring the detectors within a few mm of the beam without affecting the accelerator
vacuum. Whenever stable beams are declared, the RP systems insert the AFPs so that the SiT edge isabout 2 mm from the beam center. The nominal beam-detector distance is 15𝜎, where 𝜎 is the standard
deviation of the spatial beam distribution in the horizontal plane. Fig. 2.12 shows an AFP station with its
contents to illustrate how the RP is used. The RP covers the AFP detectors with about 1 mm thick steel toprotect them from the ultra-high vacuum in the beam pipe,5 but the window where the protons enter the
AFP and the floor separating the AFP and the beam have 300 µm thickness.
Data from the AFP are sent to the ATLAS TDAQ system through a 200 m long optical MTP fiber ribbon
to be recorded [102]. The signals for the trigger, on the other hand, are sent to the system through two
air-core coaxial cables corresponding to the Near and Far stations for each arm. The cables propagate
the signals within a time to comply with the L1 trigger latency. The L1A signals are sent back to the
AFP detectors through another optical fiber. The recorded data from the main readout are used for AFP
clustering, tracking, and proton reconstruction. The clusters are defined in each SiT plane using pixel-hit
data. The tracks and the corresponding protons are then reconstructed using the cluster information.
The reconstruction steps starting from the clustering, are conducted at a software level and described inSec. 3.5.
The primary source of the background hits is pileup protons. In general, the pileup events are subdivided
into non-diffractive dissociation (ND), single-diffractive dissociation (SD), double-diffractive dissociation
5 The inside of the RP is also highly vacuumed to prevent the RP material from being bent.

30

2.3 AFP detectors

shadow of TCL4 and TCL5 LHC beamcollimators

Roman Pot
FlangeSilicon Trackery
ToF Detector(a) Whole stationz(b) SiT and ToF detectors

x(c) Longitudinal view

Figure 2.12: AFP station [100, 101]. (a) Whole AFP station. The LHC beam penetrates the illustrated tube from theleft back to the right front. The RP is inserted into the beam pipe from right back to left front. (b) RP flange with
SiT and ToF detectors attached. This is built into the top of (a) at the same angle. (c) View from inside of the beam
pipe to the RP inserted from left to right. The collimator shadow is illustrated on the left.

𝑝

𝑝∗

𝑝

𝑝∗

𝑝

𝑝

𝑝
P

P

P

P
𝑝

𝑝
(a) ND𝑝(b) SD𝑝∗𝑝𝑝(c) DD𝑝(d) CD

Figure 2.13: Simplified Feynman diagrams of the pileup events in the 𝑝 𝑝 collisions. (a) Non-diffractive dissociation.
(b) Single-diffractive dissociation. (c) Double-diffractive dissociation. (d) Centra diffraction. P stands for pomeron.

(DD), and central diffraction (CD). Diffraction is a process where a strongly interacting color singlet
object, pomeron, is exchanged, sometimes remaining the beam proton intact [25]. Fig. 2.13 shows thesimplified diagrams of these processes. The ND is the most usual pileup with the largest cross section.
The SD and DD pileups are diffractions where single and double protons dissociate, respectively. The
CD pileup has two pomerons exchanged, called the double pomeron exchange (DPE) process. Detailed
descriptions and a diagram about diffraction are available in Ref. [103], for example. The SD pileup events
have the second dominant cross section of about 15% of the total cross section of the pileup events in the
Run 2 𝑝 𝑝 collisions. They are the dominant pileup in this research requiring AFP proton tagging. Most
tracks in Fig. 2.9 (b) are from the SD pileup events. The contributions from the out-of-time pileups are
negligible.
Protons can induce showers, where many AFP tracks are reconstructed. The showering occurs when aproton interacts with the AFP detector, RP steel window and floor, or collimator for a higher 𝜉 region.6
The average number of reconstructed tracks is slightly larger in the Far stations than in the Near stations,mainly due to the showering. If a proton showers in the Near stations, the proton does not reach the Farstation. The proton reconstruction in this research requires its associated tracks in both Near and Far
6 Which causes the most significant influence of the shower depends on the position in the plane.

31

2 Experimental Setup

Figure 2.14: The alignment in the 𝑥-direction over 20 iterations of the algorithm for the A-side Near station [104],
where the d𝑥 is the 𝑥-direction positional difference of the coordinate from the first plane (plane-0). Each line
corresponds to a plane of the station.

stations as described in Sec. 3.5, so the showering has the largest contribution to inefficiency of proton
reconstruction.
ALFA also detects protons but is only used in special low-luminosity runs. Although ALFA data are used
to measure exclusive hadrons with 𝑂 (1) GeV, SD dijets with 𝑂 (10) GeV, and scarcely DPE dijets, the
ALP stays inaccessible by this detector due to very low integrated luminosity.

2.3.1 Calibration and performance evaluation
This section briefly describes the calibration and performance evaluation of the AFP SiTs. Although theperformance study results are not published in any paper, some results and plots are available in Refs. [94,104].
The alignment of the SiTs consists of local and global alignment. The local alignment is for the relative
positions of the SiT planes in the RP in each station and is sometimes called inter-plane alignment. The
offsets of 𝑧-coordinate and rotation in 𝑥-𝑧 and 𝑦-𝑧 planes are negligible. The local alignment starts byassuming perfect alignment and that all tracks are parallel to the beam-line. Only the data events witha single cluster per plane are used for this study, and the track is reconstructed by the nominal method
described in Sec. 3.5. Then, the residuals between a track and its associated clusters are calculated, andthe alignment parameters are derived from the average of the residuals and updated. This procedure isrepeated 10 times first and is iterated 10 times further applying the selection of 𝜒2 < 2 to remove theoutliers. Fig. 2.14 shows how the relative position of other planes to a plane is determined in a station
by local alignment. Fig. 2.15 (a) shows the comparison of the 𝑦-direction residual distribution between
before and after the iteration in the first plane of the A-side Near station, while (b) shows the rotationalignment in the same manner in the same plane. The alignment parameter uncertainties are corrected bya run-by-run evaluation. The uncertainty of the local alignment is within 10 µm and is negligible in thisanalysis as described in Sec. 7.2.

32

2.3 AFP detectors(a) Residual distribution(b) 𝑥 dependency

Figure 2.15: (a) Comparison of the 𝑦-direction residual distribution between before and after the alignment procedurein the first plane (plane-0) of the A-side Near station with the statistical uncertainties [104]. (b) Comparison of the𝑥 dependency of the 𝑦-direction residual representing the AFP rotation alignment.

The global alignment is for the 𝑥-direction distance of the SiT from the beam center in each station. The
𝑥-position of the SiT is calculated as𝑥(𝑟, 𝑠) = 𝑥 pre-align + 𝑥 tracker − 𝑥 beam (𝑠) + 𝑥 RP (𝑟, 𝑠) + 𝛿𝑥corr (𝑠),where 𝑟 and 𝑠 stand for run and station, respectively. 𝑥 pre-align is the coordinate before the global alignment,
and 𝑥 tracker = −0.5 mm is the distance of the edge of the active region of the SiT from the outer side of theRP floor. 𝑥 beam is the beam position determined using the beam-based alignment (BBA) procedure [105]in dedicated runs, moving the collimators towards the beam until a sharp change in the rate is observed in
Beam Loss Monitors (BLMs) [106]. The beam position is determined from the BBA run at the beginningof 2017. The Beam Position Monitor (BPM) continuously monitors the beam position [107], and thechange over time is found to be less than 100 µm. 𝑥 RP is the inserted RP position.
Considering𝑥 AFP (𝑟, 𝑠) = 𝑥 pre-align + 𝑥 tracker − 𝑥 beam (𝑠) + 𝑥 RP (𝑟, 𝑠)
as the preliminary alignment result, 𝛿𝑥corr is determined as the in-situ corrections using the exclusive
dimuon data in the dilepton analysis [66]. From the proton energy loss fraction calculated by the dimuon
system 𝜉 𝜇𝜇 , the 𝜉AFP and hence the 𝑥-position of the proton is estimated as 𝑥 𝜇𝜇 and compared with thedetected position in the AFP 𝑥 AFP . Proton 𝑝 𝑥 is very small and almost negligible. The correction is thencalculated as the mean of the difference,𝛿𝑥 corr = 𝑥 AFP − 𝑥 𝜇𝜇 ,and is taken as the overall global alignment uncertainty. Fig. 2.16 shows the 𝑥 AFP − 𝑥 𝜇𝜇 distribution
before and after the in-situ correction and with and without the background events subtraction. The
dominant component of the background is called combinatorial background. It is estimated using the event
mixing method from which the background estimation method used in this research originates as describedin Sec. 6.1. This is subtracted from the sample to obtain the pure 𝑥 AFP − 𝑥 𝜇𝜇 distribution. The muon
momentum scale and resolution are calibrated separately as a few percent [108]. Although Fig. 2.16 shows|𝛿𝑥corr | over 400 µm, the resulting global alignment uncertainty is determined as 300 µm considering theresults of the other stations and testing various event selection criteria to be conservative enough. Finally,
the 𝑥-direction inter-station alignment is performed for each arm.
The overall 𝜉AFP resolution is measured in the dilepton analysis [66] using the dileptons as the probe.
Fig. 2.17 shows the observed 𝜉AFP − 𝜉ℓℓ distribution for each side, where 𝜉ℓℓ is the proton energy loss

33

30
25
20
15

ATLAS Preliminarys = 13 TeV (2017)
AFP A FAR
data
background

35
30
25
20
15

before alignment correction

10

10

5

5

0
Events / 200 µm

Events / 200 µm

35

30
25
20
15

−14 −12 −10 −8

−6

−4

−2

ATLAS Preliminarys = 13 TeV (2017)
AFP A FAR

Gaussian fit
mean: 429 ± 36 µm
sigma: 192 ± 32 µm

data - background

0

0
2
4
xAFP - xµµ [mm]Events / 200 µm

Events / 200 µm

2 Experimental Setup

30
25
20
15

10

10

5

5

0

0

−5

before alignment correction

−2

−1

0

1

2
3
xAFP - xµµ [mm]ATLAS Preliminarys = 13 TeV (2017)
AFP A FAR
data
backgroundafter alignment correction−14 −12 −10 −8

−6

−4

−2

0
2
4
xAFP - xµµ [mm]ATLAS Preliminarys = 13 TeV (2017)
AFP A FAR

Gaussian fit
mean: 6 ± 40 µm
sigma: 206 ± 30 µm

data - backgroundafter alignment correction

−5
−2

−1

0

1

2
3
xAFP - xµµ [mm]Figure 2.16: The 𝑥AFP − 𝑥 𝜇𝜇 distributions in the in-situ AFP alignment [109]. The left and right plots show the
distributions before and after the alignment correction, respectively. In the upper plots, the estimated background
distribution is overlaid. The background contributions are subtracted in the lower plots. The Gaussian fitting results
are shown.

fraction calculated from the dilepton system. Although the 𝜉ℓℓ uncertainty affects the result, the effect is
expected to be much smaller than the true 𝜉AFP uncertainty. The relative uncertainty of most 𝜉 value isdetermined to be about 10% as described in Sec. 3.5 in which the proton reconstruction efficiency and theacceptance are discussed.

34

Events / 0.0025

2.3 AFP detectors

70
60
50

Data 2017
Uncertainty
pp→p(γ γ →µµ)p
pp→p(γ γ →ee)p
pp→p(γ γ →µµ)p*
pp→p(γ γ →ee)p*Combinatorial bkg.

ATLAS
−1

s = 13 TeV, 14.6 fb

m ll ∉ [70, 105] GeV

40

ee+µµ, 0.02 < ξ < 0.12

30

Postfit

20
10
0

−0.02

0

0.02

0.04

ξA − ξA
ll
AFP

−0.02

0

0.02

0.04
C

ξ

AFP

− ξCllFigure 2.17: The 𝜉AFP − 𝜉ℓℓ distribution for each side observed in the dilepton analysis [66]. Dissociated proton isdenoted by ∗ .

35

3 Data and Monte Carlo Samples
This section details data and signal MC samples. The object reconstruction and event preselection using
the object information in these samples is also discussed.

3.1 Data samples
This research uses the data of 14.6 fb−1 integrated luminosity collected in 2017 during the LHC Run 2 𝑝 𝑝collisions, which is exactly the same as the one used in the dilepton analysis [66]. 2017 is the first year the
AFP was installed on both sides and recorded data suitable for diphoton-proton matching. However, in
2018, the last year of Run 2, although data was recorded with AFP, it was synchronized with the incorrectBCs. Thus, there is no diphoton-proton correlation, and it is not considered in this analysis.
The AFPs were inserted for most of the 2017 runs. There were 213 insertions corresponding to 46 fb−1
recorded by ATLAS out of 50 fb−1 delivered by LHC. Among them, the data when the AFP data acquisition(DAQ) system was in operation corresponds to 132 insertions and 32 fb−1 . Fig. 3.1 shows the changes
in integrated luminosities in 2017 corresponding to these criteria. The ATLAS global good run list
(GRL) [110] applies data quality requirements and reduces the data to 26 fb−1 . A subset GRL dedicatedto the analyses using the AFPs is then applied to reduce the data to 19.2 fb−1 . This GRL requires• all AFP stations are inserted into their nominal data-taking positions,• at least two SiT planes in each station have high voltage on,• no DAQ problems associated with the Reconfigurable Cluster Element (RCE) data processing
compute modules,• no DAQ problems associated with the Readout Subsystem (ROS) S-link cables [102].
These requirements are imposed for each lumiblock, a subdivision of a run of typically 1 minute duration
with fixed TDAQ conditions. A further tighter requirement is finally applied as the nominal GRL to selectthe 14.6 fb−1 data. This GRL requires at least three AFP SiT planes operational at high voltage for eachstation. The lumiblocks removed by this requirement are in the runs showing severe drops in AFP trackreconstruction efficiency due to the A-side Far station having fewer than three functional planes.
The data were recorded using a diphoton trigger (HLT) that required two EM clusters, clusters of EM
detector responses indicating electron or photon, with transverse energy 𝐸 T above 35 and 25 GeV, respectively, both fulfilling photon identification criteria based on shower shapes in the EM calorimeter [111],1
as done in the high-mass diphoton resonance search [52].

1 The trigger is referred to as g35_medium_g25_medium in Ref. [111].

37

Total Integrated Luminosity [fb-1]

3 Data and Monte Carlo Samples
70
60
50
40

ATLAS Preliminarys = 13 TeV
LHC Delivered
ATLAS Recorded
ATLAS and AFP Recorded
Total LHC Delivered: 50.2 fb-1
Total ATLAS Recorded: 46.9 fb-1
Total ATLAS and AFP Recorded: 32.0 fb-1

30
20
2/18 calibration

10
0

17/05 10/06 02/07 25/07 18/08 11/09 03/10 26/10 20/11
Day in 2017

Figure 3.1: Integrated luminosity recorded by ATLAS and AFP in the LHC 𝑝 𝑝 collisions in 2017 [104].

3.2 Signal MC samples
Simulated signal MC samples are used primarily to construct the model of signal 𝑚 𝛾𝛾 distribution andevaluate estimate the signal efficiency. The EL ALP signal MC samples were generated with SuperChic 4.02 [112–114] in June 2021, and the SD and DD samples were generated with SuperChic 4.14 [115]in October 2022. SuperChic is a generator for exclusive and photon-initiated production in proton andheavy ion collisions. It uses state-of-the-art formalism for cross section calculation using a structure
function approach. This approach uses the proton structure function directly bypassing any explicit
reference to photon PDF in Eq. 1.1 to reduce the uncertainty of cross section [116]. For any of the EL, SD,
and DD signals, the cross section is calculated as
0

1
𝜎=
2𝑠

∫

0
𝜇𝜇
d 𝒑 10 d 𝒑 20 dΓ  2   2  𝜌1 𝜌2𝜈𝜈 M ∗𝜇0 𝜈 0 M 𝜇𝜈 (4)𝛼 𝑄1 𝛼 𝑄2
𝛿 (𝑞 1 + 𝑞 2 − 𝑘 1 − 𝑘 2 ) ,𝐸 10 𝐸 20
𝑄 21 𝑄 22

where the subscripts 1 and 2 identify the initial proton, 𝒑 and 𝐸 are the momentum and energy of theoutgoing hadronic system, 𝛼 is the structure constant, 𝑞 and 𝑘 are the incoming and outgoing photonÎ
4-momenta with 𝑞 2 = −𝑄 2 , and dΓ = 2𝑗=1 d𝒌 𝑗 /2𝐸 𝑗 (2𝜋) 3 is the standard phase space volume. 𝑠 is the√Mandelstam variable, where 𝑠 = 13 TeV. The photon density matrix
∫ 
2 h



i
𝛼𝛽
𝛼𝛽
𝛼𝛽
𝜌𝑖 =
d 𝑚 𝑝 (∗)
𝐴𝑖 𝐹1 𝑥 𝐵,𝑖 , 𝑄 2𝑖 + 𝐵𝑖 𝐹2 𝑥 𝐵,𝑖 , 𝑄 2𝑖 ,𝑖where 𝐴, 𝐵, and 𝑥 𝐵,𝑖 are functions of photon and hadronic system kinematics, has the proton structure
functions 𝐹1,2 , which are precisely determined from previous experiments [31]. M corresponds to the𝛾𝛾 → 𝑎 → 𝛾𝛾 amplitude, where only the 𝑠-channel is considered because the contributions of 𝑡 and𝑢-channels are expected to be extremely small. The cross section is multiplied by the soft survival factor
𝑆 2 , which is the probability of no additional soft inelastic interactions between the intact protons. If suchan interaction occurs, there is a color flow between the protons, and hence both A and C-sides protons
would no longer remain intact. Fig. 3.2 shows the soft survival factor for the dilepton events illustrated inFig. 1.8. For the EL case, the survival probability is higher than the other processes because the impact
parameter is large since the photon is quasi-real. Similar values and diphoton kinematics dependency are
expected for the ALP signals.
The ALP mass 𝑚 𝑋 , coupling constant 𝑔 = 4 𝑓 −1 , and event type (EL, SD, or DD) are specified inSuperChic for each sample along with some other configuration. In this research, signal samples ranging

38

3.2 Signal MC samples
1 S

2

1 S

0.8

0.8

0.6

0.6

0.4

EL
SD
DD

EL
SD
DD

0.4
0.2

0.2
0
10

2

100
Mll [GeV]

1000

0
-5

-4

-3

-2

-1

0
yll

1

2

3

4

5

√
Figure 3.2: Soft survival factor calculated for 𝛾𝛾 → ℓℓ events in the 𝑝 𝑝 collisions at 𝑠 = 13 TeV [114]. The
diphoton mass dependency (left) and dilepton rapidity dependency (right) are illustrated for the EL, SD, and DD
processes.
Table 3.1: Summary of the used set of SuperChic parameters with their descriptions and assigned values.

Description

Value

Number of events
Center-of-mass energy
Collision type
Coupling constantDiphoton rapidity range
Final state photon 𝑝 T range
Final state photon 𝜂 range

10000√
𝑠 = 13 TeV
𝑝 𝑝 collisions
𝑓 −1 = 0.05 TeV−1𝑦 𝛾𝛾 ∈ [−2.4, 2.4]
𝑝 T ∈ [20, ∞) GeV
𝜂 ∈ [−2.4, 2.4]

from 200 GeV to 1600 GeV were generated with fixed coupling constant 𝑓 −1 = 0.05 TeV−1 . The major
configuration commonly used for the signal samples is listed in Table 3.1. The computed generation crosssections are considered in Sec. 5.2. The generated objects at this stage before the detector simulation arereferred to as the truth objects.
After the event generation by SuperChic, Pythia 8.307 [117] is used for the SD and DD signals tohadronize the dissociative system. SuperChic removes the final state intact protons of the SD signals fromits output which is the input to Pythia because Pythia reports a violation of the momentum conservation
law and does not work. Instead, Pythia automatically adds additional QCD objects not to violate theconservation laws due to the lack of intact proton and hadronizes them as well. However, these additional
impossible QCD objects are boosted enough not to contaminate the ATLAS detector. This was confirmed
by comparing the production vertices of the two final state truth photons with the first and the second
highest energy and seeing if they are identical for each event. On the other hand, they can contaminate
AFP and sometimes hadronize into a proton and behave as a fake intact proton. Therefore, such QCD
objects were removed at the truth level, and an intact proton was added artificially, having its momentum
calculated from the initial proton and photon.
The initial photon 𝑝 T on the intact side of the SD samples is exactly given as 0. Since the initial beam
proton 𝑝 T is also assumed to be 0, this means calculated intact proton 𝑝 T is 0 as well. This simplified
assumption was used because it was confirmed to have negligible effects on the final results by testing thecase where 𝑝 𝑥 and 𝑝 𝑦 are independently smeared by Gaussian distribution with the width taken from theEL sample intact protons. Fig. 3.3 shows the 𝑝 𝑥 and 𝑝 𝑦 width of intact protons in the EL samples for each𝑚 𝑋 . To be conservative, 0.23 GeV was used for the smearing test.

39

Standard deviation [GeV]

3 Data and Monte Carlo Samples

0.23

s = 13 TeV, 14.6 fb-1

0.22

Truth proton px
Truth proton py

0.21
0.2
0.19
0.18
0.17
0.16
200

400

600

800

1000

1200
1400
mX [GeV]Figure 3.3: Width of proton 𝑝 𝑥 and 𝑝 𝑦 distributions of the EL signal samples for each ALP mass.

𝑝

𝑢𝑑
𝑢
𝑢
𝛾
𝛾

𝛾
𝑎
𝛾
𝑞𝑞

𝑝

𝑞

Figure 3.4: Feynman diagram illustrating an SD signal event processed in Pythia. The inelastic photon production
is modelled as 𝑢 → 𝑢𝛾. The particles indicated by the black lines are input to Pythia with their momenta calculated
in SuperChic. The grey ovals mean hadronization. The particles indicated by the green lines are not input toPythia but considered only in SuperChic. The particles indicated by the red lines are automatically added by
Pythia. The 𝑢𝑑 object interacts with the final state 𝑢-quark to hadronize. The QCD objects 𝑞 + 𝑞𝑞 replace theintact proton that should have been one of the input particles to Pythia.

It is assumed that there is an underlying 𝑢 → 𝑢𝛾 process in inelastic photon production of the SD and DD
signal events in order for Pythia to work, even in the lower photon virtuality 𝑄 2 region where the proton
is likely to excite and dissociate without 𝑞 → 𝑞𝛾 process. This does not cause any problem because theSD and DD signal events scarcely have inelastic photons with 𝑄 2 < 5 GeV2 . Fig. 3.4 shows the diagramof an SD signal processed in Pythia.
The effect of multiple interactions in the same and neighboring bunch crossings (pileup) was modelled byoverlaying the simulated signal event with inelastic 𝑝 𝑝 events generated with Pythia 8.210 [118] using
the NNPDF2.3lo set of parton distribution functions [119] and the A3 set of tuned parameters [120], in
association with EvtGen 1.2.0 [121]. The MC events were weighted to reproduce the distribution of
the actual number of interactions per BC (𝜇) observed in the data. In contrast, the pileup protons in the
AFP detectors are neglected in the nominal signal MC samples. Effects of the pileup protons on the result
were estimated as the systematic uncertainty separately and found to have a small effect as expected as
described in Sec. 7.2.
All simulated signal events were processed using a fast simulation of the ATLAS detector [122], where thefull simulation2 of the calorimeter is replaced with a parameterization of the calorimeter response [124].
2 The full simulation of the ATLAS detector [122] is based on Geant4 [123]. This research does not use the full simulation.

40

3.3 Track and primary vertex reconstruction
The response of the AFP spectrometer is modelled by another fast simulation, where the Gaussian smearing
is applied to track positions based on the AFP spatial resolution. Each proton is selected or discarded
according to the reconstruction efficiency described in Sec. 3.5. All simulated events were reconstructedwith the same reconstruction algorithms as those used for data.

3.3 Track and primary vertex reconstruction
The energy of charged particles is deposited as hits on different layers in the ATLAS inner-detector system.
Although the charged particle tracks reconstructed using the hits are not directly used for the event selectionin this research, they are used for reconstructing the other objects. The tracking algorithm consists of threesteps.
1. Neighboring silicon-sensor hits are merged into a cluster. Then, a position in the (𝑥, 𝑦, 𝑧) space,called space-point, is defined for each cluster using the connected component analysis (CCA)
technique [125].
2. A seed-track is defined as a set of three or more space-points [126].
3. Each seed-track is extrapolated to the TRT hits using the Kalman filter [127] and incorporates thespace-points on the trajectory to construct a track object. Momentum is calculated for each track.
During the above steps, some cleaning procedures, such as overlap removal of the multiple seed-tracks andambiguity resolution of the track candidates to suppress the fake tracks, are taken. The track reconstruction
is detailed in Ref. [126].
The primary vertices are reconstructed using the reconstructed tracks. Each primary vertex is defined asa spacial position attributed to tracks requiring some criteria [128]. The two reconstructed and selected
photon objects with the highest 𝑝 T (leading and sub-leading photons) and additional information fromthe tracking systems are used to identify the diphoton production vertex from the candidate primaryvertices [129]. This is called photon pointing method. Although primary vertex information is not used inthis research, the analysis framework automatically excludes the events with the selected primary vertex
associated with less than two tracks.3 However, the EL signal event has no charged particle in the final state
in most cases, while the final state photons are sometimes converted to electron pairs due to interactions
with the detector materials, leaving tracks. A photon splitting into an electron pair is referred to as aconverted photon, while a photon without tracks is called an unconverted photon. Still, the automatic
selection efficiency is almost 100% because the photon pointing selects a pileup vertex with multiple
associated tracks. Fig. 3.5 shows the primary vertex-related distributions of an EL signal MC sample.
When the leading two photons are unconverted, the photon pointing selects a pileup vertex with many
associated tracks.
Diphoton production vertex reconstruction methods without using track information have been developed.
These methods, such as non-pointing [130, 131] and calo-vertexing [132] use only diphoton information
to reconstruct a single diphoton production vertex with about 15 mm spacial resolution. It is aimed ata search for long-lived particle decaying into a final state with diphoton at a displaced vertex from theprimary vertex. However, this method could also be efficiently used in the analysis of this research. This isdiscussed in Sec. 9.2.2. Still, such a method was not integrated into the analysis framework of this researchbecause there is no motivation to impose additional event selection since the integrated luminosity of thenominal data is as small as 14.6 fb−1 .

3 This is imposed for usual analyses for high event quality.

41

s = 13 TeV, 14.6 fb-1

250

conv + conv
conv + unconv
unconv + unconv

200

Events / 1.0

Events / 1.0 [mm]3 Data and Monte Carlo Samples
50
40

s = 13 TeV, 14.6 fb-1
conv + conv
conv + unconv
unconv + unconv

30

150

20

100

10

50
0
−50 −40 −30 −20 −10

010 20 30 40 50
zselected PV - ztrue PV [mm](a) Longitudinal positional difference

0
0

20
40
60
80
100 120 140 160
Number of tracks associated with selected PV(b) Number of tracks

Figure 3.5: The primary vertex-related distributions of the EL signal MC sample for 𝑚 𝑋 = 400 GeV. No event
selection described in Sec. 4 is applied. Each color corresponds to the combination of the conversion type of the twoleading photons. (a) The distribution of the longitudinal positional difference between the selected primary vertex
and truth signal primary vertex. (b) The distribution of the number of tracks associated with the selected primaryvertex.

3.4 Photon reconstruction
The photon reconstruction and selection inherit the high-mass diphoton resonance search [52]. Photons
are reconstructed from calorimeter signals using a dynamical, topological cell clustering-based algorithm
along with the electron reconstruction. The reconstruction, calibration, identification, and isolation ofphotons and their performances are detailed in Ref. [133]. Fig. 3.6 shows the algorithm flow of thereconstruction. The topological clusters (topo-clusters) [134, 135] are defined first. Then, the superclusters
are determined based on them, and finally, the photon objects are identified and isolated.
A topo-cluster is defined from proto-clusters. The formation of each proto-cluster begins with finding a
EM ≥ 4, where 𝜍 EM is the ratio of energy deposit at the EM scale4 to expectedcalorimeter cell with 𝜍 cellcell
EM ≥ 2 are then selected to form a cluster including the initial cell. Eachnoise. Neighboring cells with 𝜍 cellneighboring cell becomes a seed cell in the next iteration, recursively collecting each of its neighbors in
the proto-cluster. Next, all neighboring cells with positive energy are incorporated. Topo-clusters arethen formed by merging and separating within each proto-cluster and between the proto-clusters. Some
preselections are applied to the energy in each topo-cluster only from the EM calorimeter cells and its ratio
to the total energy in the topo-cluster to define an EM topo-cluster. Topo-clusters are matched to tracks orconversion vertices using the information on the positions and energy of the clusters and the positions andmomenta of the tracks. Two-track conversion vertices are reconstructed from two opposite-charge tracks
forming a vertex consistent with that of a massless particle, while single-track vertices require no hits in
the innermost sensitive layers.
A supercluster is then formed by merging some topo-clusters. Each seed topo-cluster is selected basedon 𝐸 T required to be high. Satellite topo-clusters are combined with the seed to form a supercluster, as
illustrated in Fig. 3.7. For both electrons and photons, a topo-cluster is considered a satellite topo-cluster
if it is in the Δ𝜂 × Δ𝜙 = 0.075 × 0.125 window around the seed topo-cluster. For converted photons, inaddition, a topo-cluster is added as a satellite if itself or its best-matched track shares a conversion vertex
with the seed topo-cluster. For electrons, on the other hand, a wider window is set as the acceptance region

4 The EM scale is the basic signal scale accounting correctly for the energy deposited in the calorimeter by EM showers.

42

3.4 Photon reconstruction

Figure 3.6: Algorithm flow of the photon and electron reconstruction [133].

to form a supercluster.5 After the superclustering, conversion vertices are matched to photon superclusters,
and tracks are matched to electron superclusters. An unconverted photon is defined as a cluster matched toneither an electron track nor a conversion vertex. About 20% of photons at low |𝜂| convert in the innerdetectors, and up to about 65% convert at |𝜂| ≈ 2.3.
The superclustering is performed for photons and electrons independently, so a given seed cluster can
produce both an electron and a photon. This ambiguity is resolved using a dedicated algorithm. Then,
superclusters’ energy scale and resolution are calibrated using 𝑍 → 𝑒𝑒 events considering pileup dependency. The energy scale correction is validated using 𝑍 → ℓℓ𝛾 events for photons. Subsequently, shower
shape and other discriminating variables are calculated for electron and photon identification. Photons
and electrons are defined as the superclusters having these variables.
Photon candidates are classified based on cut-based identification requirements imposed on the calorimeter
shower shape variables. The primary purpose is to discriminate photons from hadronic jets. Based on thelevels of signal acceptance and background rejection of these cuts, several photon identification working
points are prepared. The definition of the variables and working points can be found in Ref. [133]. This
research uses the tight working point. The loose working point is based on the shower shape in the middle
layer of the EM calorimeter and the energy deposited in the hadronic calorimeter. The tight working point
additionally uses information from the finely-segmented front layer of the calorimeter. It is separately
optimized for unconverted and converted photons to account for the generally broader lateral shower profile
5 This is an advantage of topo-cluster over the sliding-window algorithm used previously instead of the topo-clustering in the
ATLAS experiment [134, 136, 137]. The sliding-window algorithm uses a fixed-size window to define a cluster. Topo-clustering
is superior to sliding-window in that dynamic clusters change in size as needed to recover energy from bremsstrahlung photons orelectrons from photon conversions.

43

3 Data and Monte Carlo Samples

Figure 3.7: Superclustering algorithm for photons and electrons [133]. Seed clusters are shown in red, and satellite
clusters in blue.

of the latter. The optimization is performed in sub-ranges of photon 𝜂 and 𝐸 T using TMVA [138]. Using
the 𝑝 T -dependent tight identification cuts, photons with 𝑝 T > 25 GeV are expected to have a reconstruction
and identification efficiency of greater than 80% that reaches a plateau of about 90-95% at 40-50 GeV.
Photon candidates are also required to be isolated to prevent faking from jets further, using criteria
based on the calorimeter as implemented in the TightCaloOnly photon isolation working point definedin Ref. [133]. Other working points based on both the inner detector and calorimeter are used in theusual analyses. Any of such working points assume that the reconstruction of the primary vertex issuccessful and imposes conditions using the variables of the primary vertex. This research does not use
such a working point because the primary vertex of the EL signal is hardly reconstructed correctly, as
described in Sec. 3.3. The isolation variable is defined as the sum of 𝐸 T of the topo-clusters with positiveenergy reconstructed in the calorimeter around each photon candidate in a cone of radius Δ𝑅 = 0.4.
This energy calculation requires subtracting the contributions from the photon itself in the core windowΔ𝜂 × Δ𝜙 = 0.125 × 0.175, correcting for the leakage of the photon energy from the core window, andfinally using an event-by-event energy subtraction based on the jet area method to remove contributions
from the underlying event and pileup interactions [139]. The total calorimeter isolation energy is required
to be smaller than 0.022𝐸 T + 2.45 GeV.
The photon object selection requires the photon candidates with 𝑝 T > 40 GeV and |𝜂| < 2.37, excluding
the barrel-to-endcap transition regions of the calorimeter, 1.37 < |𝜂| < 1.52. An event preselection
requires at least two photons fulfilling all the criteria described above.

3.5 Proton reconstruction
Protons are reconstructed from their associated AFP tracks, and the tracking begins with clustering. A
cluster is defined from neighboring pixel hits in the 𝑥-direction in a 3D-Si sensor of the AFP detector.
A usual cluster has one or two pixel hits, while a cluster sometimes has more than three pixel hits due

44

3.5 Proton reconstruction

Figure 3.8: Clustering efficiency as a function of bias voltage [104]. Each color corresponds to an occupancy level.

to showering. The cluster (𝑥, 𝑦) position is determined from the charge-weighted distribution of thecomponent hits. A track is then defined by searching for a set of clusters in the same station aligned
parallel to the beam-line. Deviations of the transverse cluster positions from one another within 500 µm areallowed. The track (𝑥, 𝑦) position is determined by a linear regression fit, assuming the same uncertainties
on each cluster. Clusters from at least two different planes are required in this research. Each station was
found to have about 0.02 reconstructed tracks on average per 𝑝 𝑝 collision.
The clustering efficiency for each SiT plane is measured using clusters in the other planes in the same station
sharing the same track. The clustering rate was calculated within a window of one pixel. The clustering
efficiency depends on the applied bias voltage in the 3D-Si sensors. Fig. 3.8 shows the dependency in thefirst plane of the A-side Far station. The data are divided into three occupancy levels using the rate oftraversing tracks as a proxy. Deterioration over time was observed during the data-taking due to the ageing
effects originating from radiation damage. Therefore, the bias voltage was tuned after this observation,and the efficiency recovered. However, a SiT plane in the A-side Far station was always non-operational
during the data-taking in 2017 due to a problem with the electrical connections on the high-voltage (HV)lines.
Finally, a proton is reconstructed, searching for a pair of Near and Far station tracks within 2 mmof transverse distance. If multiple such pairs are in the same arm,
all possibilities are taken forward.
The kinematics of the proton with 4-momentum 𝐸, 𝑝 𝑥 , 𝑝 𝑦 , 𝑝 𝑧 is reconstructed by mapping the track
positions in the two stations, 𝑥 NEAR and 𝑥 FAR , to 𝐸 and 𝜃 𝑥,0 = 𝑝 𝑥 /𝑝 𝑧 assuming the positions of the SiTs
and the magnetic fields of the LHC magnet lattice are known. The 𝐸 and 𝜃 𝑥,0 are obtained by solving theequations
𝑥NEAR + 𝑥 FAR= 𝑐 1,𝑥 (𝐸) + 𝑐 2,𝑥 (𝐸)𝜃 𝑥,0 ,2
𝑥FAR − 𝑥 NEAR𝜃𝑥 ≡= 𝑐 01,𝑥 (𝐸) + 𝑐 02,𝑥 (𝐸)𝜃 𝑥,0 ,𝑧FAR − 𝑧 NEAR𝑥≡where 𝑥 NEAR (𝑥 FAR ) and 𝑧 NEAR (𝑧 FAR ) is the 𝑥 and 𝑧-coordinate of the track in the Near (Far) station,(0)
respectively, and 𝑐 𝑖,𝑥
are polynomials with respect to 𝐸 whose coefficients are determined by a protontransport simulation [140]. Additional terms,
(0)
(0)
(0)
𝑐 3,𝑥
(𝐸)𝑥 0 + 𝑐 4,𝑥(𝐸)𝑧 0 + 𝑐 5,𝑥(𝐸)𝑧 0 𝜃 𝑥,0 ,

45

102
100
98
96
94
92
90
Jul
88

A FAR (tag in A NEAR)A NEAR (tag in A FAR)C NEAR (tag in C FAR)C FAR (tag in C NEAR)ATLAS Preliminarys = 13 TeV (2017)
Aug

Sep

Oct

Nov

329716329780329835329964330025330074330079330160334264334317334350334890334907334960334993335016335022335056335082335083335131335170335177335222335282335290336506336548336567
336630336678336719336782336832336852336915336927336944336998337005337052337107337176337215337263337335337371337404337451338183338220338259338263338349338480338834338846338897338933338967338987339037340368340453probability [%]3 Data and Monte Carlo Samples

ATLAS run number

Figure 3.9: Tracking efficiency for each probe station for each run in the proton reconstruction efficiency measurement [94].

where (𝑥 0 , 𝑦 0 , 𝑧 0 ) is the vertex position, were considered originally, but they are found to be negligible.
The solved 𝐸 is interpreted as the proton energy loss fraction,i?2 Qi?2` biiBQM QM i?2 bK2 bB/2 Ui; biiBQMVX Ai Bb HbQQ#b2`pBM;  i`+F BM  ;Bp2M 6S biiBQM BM 2p2Mib rBi? 2t+iHv QM2 i`+F BM 𝐸
𝜉 BM i?2=Qp2`HT
1 − `2;BQM,`2[mB`2/ i?i i?2 TQbBiBQM Q7 i?2 i`+F BM i?2 i; biiBQM Bb AFP#2ir22M i?2 i; M/ T`Q#2 biiBQMb U#2+mb2 Q7 i?2𝐸
beam/Bz2`2M+2b BM i?2 #2K rB/i?- i?2 M2` biiBQMb r2`2 iFBM; /i i  H`;2` /BbiM+2 7`QK i?2 #2K i?M i?2 7` biiBQMbc i?2 `2;BQM mb2/
7Q` i?2 bim/v Bb −12 < ti; < −5 KKVX AM //BiBQM- i?2 TQbBiBQMb Q7 #Qi? i`+Fb ?p2 iQ 7mH}HH i?2 `2[mB`2K2Mi Q7 |tT`Q#2 − ti; | < 2 KKX
where
𝐸 beamtBb= bT2+B}2b
6.5 TeV
the initial
beamh?2
protonenergy,andM2`
𝜃 𝑥,0
𝑝 𝑥 .i?2𝑝 𝑦7+i
is i?inot `2[mB`BM;
calculatedh?2?Q`BxQMiH
i?2ishGa`mM MmK#2`X/Bz2`2M+2#2ir22M
M/is7`interpretedbiiBQMb +QK2bto7`QKbBM;H2 i`+F BM i?2 7` i; biiBQM `2/m+2b 7`+iBQM Q7 b?Qr2` 2p2Mib Km+? KQ`2 i?M i?2 bK2 `2[mB`2K2Mi 7Q` M2` i; biiBQMX

in this research but is expected to be calculated in the future. The uncertainty of the assumed magnetic
fields in the beam optics in the transport simulation has the dominant effect on proton 𝜉AFP uncertainty,
leading to about 10% for most 𝜉AFP values.

9f9Proton reconstruction efficiency was measured using a tag-and-probe method. It begins with the tracking
efficiency measurement for each station using events where there is exactly one reconstructed track (tag) in
one station of an arm and calculating the probability that a proton is reconstructed along with the existenceof a reconstructed track (probe) in the other station in the same arm. The distance of 𝑥-position of the
track in the probe station from the one in the tag station is required to be less than 2 mm as per the nominal
analysis. Fig. 3.9 shows the tracking efficiency for each probe station for each run in 2017. The slightdifference of the Far stations behaviours in the earlier runs is due to the larger distance between the RP
and the beam, causing fewer showering. The Near stations have 1% detection inefficiency, while the Farstations have 5% inefficiency, where the additional 4% is due to showering. When reconstructing a proton
using the Near and Far stations, assuming the tracking efficiency of Near and Far stations are assumed
to be uncorrelated, the maximum contribution of the detection inefficiency is 1% × 2 = 2%, and thecontribution of showering is from 4% to 4% × 2 = 8%, i.e. 6 ± 2%. Thus, the total proton reconstruction
efficiency was then determined as the 100 − (2 + 6 ± 2) = 92 ± 2%. In general, multiple protons arereconstructed for each side per bunch crossing. Though the typical number of reconstructed protons ranges
from 0 to 2, there are sometimes more than 10 per BC. The primary cause is the showering and coincident
hits from multiple pileup events.
The AFP acceptance is studied based on the proton reconstruction efficiency. The proton efficiency of
92 ± 2% is guaranteed only for 𝑥 < −3.5 mm in the AFP local coordinate and 𝜉AFP ∈ [0.035, 0.08]. The
Far stations still have high efficiency in the range 𝜉AFP ∈ [0.02, 0.035], but the Near station does notcover the range, and hence the efficiency cannot be measured by the tag-and-probe method. Although
the efficiency is expected to be still high for both stations for 𝜉AFP ∈ [0.08, 0.12],6 there were a small
number of events in the range, and the efficiency could not be derived sufficiently. As object selection inthis research, only the protons with an Near station seed track with 𝑥 < −3.5 mm is selected.7 Table 3.2
6 The existence of collimator determines the 0.12.
7 Selection of 𝜉

46

AFP ∈ [0.035, 0.08] is applied in the event selection as described in Sec. 4.3.

3.5 Proton reconstruction
Table 3.2: Photon and proton selections and the event preselection using the photons.

PhotonProtonEvent

Selection
Identification (Tight)
Isolation (TightCaloOnly)
𝑝 T > 40 GeV|𝜂| ∈ [0, 2.37] \ [1.37, 1.52]
𝑥 NEAR < −3.5 mm
At least two selected photons

summarizes the photon and proton selections and the event selection using the photons. Cut on 𝜉AFP is
imposed in the event selection described in Sec. 4.
The data from AFP ToF detectors are not used in this research as described in Sec. 2.3. This is due to very
low efficiency. The cause has not been entirely determined, but there are two main possibilities.
• The quantum efficiency decreased due to the material degradation of PMT. Since high HV wasapplied at the commissioning time, the quantum efficiency dropped at that point.
• The HV was lowered too much. Since a large number of photoelectrons were observed during thecommissioning, the HV was adjusted to prevent a decrease in quantum efficiency. However, there
was no online monitoring of efficiency in Run 2, so it was unclear how much to lower the HV tokeep sufficient efficiency.

47

4 Event Selection
This section describes event selection, introduced in Sec. 1.4, in detail. The selection flow is summarized
in Fig. 4.1. The acoplanarity selection and 𝜉 𝛾𝛾 selection is the requirement for diphoton kinematics. The
𝜉AFP is the requirement for proton kinematics. The matching is the requirement that the 𝜉 𝛾𝛾 and 𝜉AFP areclose. The selection is applied for each side from the 𝜉 𝛾𝛾 selection. Thus, there are two sets of events after
the matching selection. A set operation finally selects the events from the two samples.

4.1 Acoplanarity selection
First, the diphoton is constructed from the leading and sub-leading photons described in Sec. 3.4. The
acoplanarity of the diphoton is defined as𝛾𝛾

𝐴𝜙 ≡ 1 −|Δ𝜙 𝛾𝛾 |
,
𝜋

(4.1)

where Δ𝜙 𝛾𝛾 ∈ (−𝜋, 𝜋] is the difference of 𝜙 between the two photons. The acoplanarity is interpreted as
the degree of the diphoton not to be back-to-back in the 𝑥-𝑦 plane. Figs. 4.2 and 4.3 show the acoplanarity
distributions for the data and ALP signals after the whole preselection described in Sec. 3. The EL signals
have exclusive final state diphoton, and the momentum conservation ensures that the two photons havethe same 𝑝 T and the opposite direction.1 On the other hand, the SD and DD signals are not back-to-backbecause the proton dissociations produce other objects in the final state. Still, all the signal samples have apeak around 0, as well as the data sample. Only the event with diphoton acoplanarity less than 0.01 is
selected.

Figure 4.1: Event selection flow.
1 The relative 𝑝

T difference distributions for the data and signal MC samples are shown in Figs. A.1 and A.2. The 𝑝 T balancecharacteristic could be used for an additional event selection. However, it is not used in this analysis because too much background
reduction makes the background estimation difficult.

49Events / 0.01

4 Event Selection
22
20

×103
s = 13 TeV, 14.6 fb-1

18
16
14
12
10
8
6
4
2
0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1γγ
Diphoton acoplanarity Aφs = 13 TeV, 14.6 fb-1
EL

1

SD
DD

10−1

s = 13 TeV, 14.6 fb-1

1

EL
SD

10−1

DD

10−2
10−3

10−2

0

Events / 0.0002

Events / 0.0002

Figure 4.2: Diphoton acoplanarity distribution of the data after the preselection.

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
γγ
Diphoton acoplanarity Aφ

10−4
00.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
γγ
Diphoton acoplanarity Aφ(a) 𝑚 𝑋 = 300 GeV(b) 𝑚 𝑋 = 1200 GeV

Figure 4.3: Diphoton acoplanarity distribution of the ALP signals after the preselection. The different colors standfor the EL, SD, and SD signals.

4.2 𝝃 𝜸𝜸 selection√
Through the conservation of 4-momentum for a given beam center-of-mass energy 𝑠, the 𝜉 ± can be
determined from the photon system kinematics, where 𝜉 + ≡ 𝜉 A and 𝜉 − ≡ 𝜉 C are the energy loss fractionof proton flying towards A-side and C-side, respectively. From the definition of the proton energy loss√
fraction, the energy of initial photons is 𝜉 ± 𝑠/2. Assuming the initial photons move almost along thebeam direction, the 4-momenta of the initial photons are
√


𝜉

±

√ 
𝑠
± 𝑠
, 0, 0, ±𝜉
,22

where the first element is energy, and the last is longitudinal momentum. The ALP 4-momenta is the sumof them,
√

𝑠 +
𝛾𝛾
𝛾𝛾
𝛾𝛾 
𝜉 + 𝜉 − , 0, 0, 𝜉 + − 𝜉 − = 𝐸 𝛾𝛾 , 𝑝 𝑥 , 𝑝 𝑦 , 𝑝 𝑧 .
2

50

0.2

y = -1.5γγ
y =1.0γγ

s = 13 TeV

= -0
.5

ξCγ γ

4.2 𝜉 𝛾𝛾 selection

=

y

mγ

γγ

0.15

y γγ

γ

=
=

00

γ

16

mγ

eV

00

γ

=
00

eV

G

12

γ

=

G

0

=

eV

γ

0
10

mγ

y γγ =

γ

=

eV

0
60

G

=

γ

G

=

G
eV

0
20

y γ γ = 1.0
y = 1.5

eV

γ

0
40

mγ

γγ

eV
G

0
0

0.5

eV

G

0
80

mγ
mγ

0.05

G

14

mγ

mγ

0.1

0
0.

0.05

0.1

0.15

0.2
ξAγ γ

A , 𝜉 C ) and (𝑚 , 𝑦 ) coordinates.
Figure 4.4: The correspondence between (𝜉 𝛾𝛾
𝛾𝛾 𝛾𝛾
𝛾𝛾

As shown on the right-hand side, this is equivalent to the 4-momenta of the final state diphoton system.
The diphoton mass is
𝑚 𝛾𝛾 =q

2 − ( 𝑝 )2 − ( 𝑝 )2 − ( 𝑝 )2 =
𝐸 𝛾𝛾
𝑥
𝑦
𝑧
𝛾𝛾

𝛾𝛾

𝛾𝛾

p

𝑠𝜉 + 𝜉 − ,thus,
𝑚 𝛾𝛾 p + −√ = 𝜉 𝜉 .
𝑠

(4.2)

The diphoton rapidity is𝛾𝛾

𝐸 𝛾𝛾 + 𝑝 𝑧
11𝜉+𝑦 𝛾𝛾 = log
log − ,𝛾𝛾 =22
𝜉𝐸 𝛾𝛾 − 𝑝 𝑧
thus,
s
𝑒

𝑦𝛾𝛾

=

𝜉+
,
𝜉−

s
𝑒

−𝑦𝛾𝛾

=

𝜉−
.
𝜉+

Therefore,𝑚 𝛾𝛾
𝜉 ± = √ 𝑒 ±𝑦𝛾𝛾 .
𝑠

(4.3)

± or 𝜉 A(C) . The assumption on the above calculations is not always good approximaThis is denoted by 𝜉 𝛾𝛾𝛾𝛾𝛾𝛾𝛾𝛾tions. Especially, SD or DD events have non-zero 𝑝 𝑥 or 𝑝 𝑦 in general. Still, this definition of the 𝜉 𝛾𝛾 is
A , 𝜉 C ) and (𝑚 , 𝑦 ) coordinates isused for the event selection uniformly. The correspondence of (𝜉 𝛾𝛾𝛾𝛾 𝛾𝛾𝛾𝛾
shown in Fig. 4.4. The signal events for ALP mass 𝑚 𝑋 tend to concentrate on a mass contour 𝑚 𝛾𝛾 = 𝑚 𝑋 .
± distributions for the data and ALP signals after the acoplanarity selection.
Figs. 4.5 and 4.6 show the 𝜉 𝛾𝛾
The data are distributed in the low 𝑚 𝛾𝛾 region. In contrast, high-mass signal events are distributed inrelatively high 𝜉 regions as expected by Fig. 4.4.

The event selection is
±
𝜉 𝛾𝛾
∈




0.031 0.084
,
.
1.1
0.9

51

0.14

35
s = 13 TeV, 14.6 fb-1

0.12

30

0.1

25

Events / 0.0014

ξCγ γ

4 Event Selection

2200
2000

s = 13 TeV, 14.6 fb-1

1800

A-side1600

C-side

0.08

20

0.06

15

1200

0.04

10

800

0.02

5

400

1400
1000
600
200

0
0

0.05

0.1

ξAγ γ

0

0
0

0.02

0.04

+ vs 𝜉 −(a) 𝜉 𝛾𝛾
𝛾𝛾

0.06 0.08
0.1
0.12 0.14
Proton energy loss fraction ξγ γ±(b) 𝜉 𝛾𝛾1.2

s = 13 TeV, 14.6 fb-1

1

SD

EL
DD

0.8

Events / 0.0014

Events / 0.0014± distributions of the data after the acoplanarity selection.
Figure 4.5: 𝜉 𝛾𝛾

1.2

s = 13 TeV, 14.6 fb-1

1

SD
DD

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0
0

0.02

0.04

0
0

0.06 0.08
0.1
0.12 0.14
Proton energy loss fraction ξγAγ

Events / 0.0014

EL
SD
DD

0.06 0.08
0.1
0.12 0.14
Proton energy loss fraction ξγCγ

0.06

s = 13 TeV, 14.6 fb-1

0.05

SD

EL
DD

0.04

0.04
0.03

0.03

0.02

0.02

0.01

0.01

0
0

0.04

(b) 𝑚 𝑋 = 300 GeV, C-side

s = 13 TeV, 14.6 fb-1

0.05

0.02

Events / 0.0014(a) 𝑚 𝑋 = 300 GeV, A-side
0.07
0.06

EL

0.02

0.04

0.06 0.08
0.1
0.12 0.14
Proton energy loss fraction ξγAγ(c) 𝑚 𝑋 = 1200 GeV, A-side

0
0

0.02

0.04

0.06 0.08
0.1
0.12 0.14
Proton energy loss fraction ξγCγ(d) 𝑚 𝑋 = 1200 GeV, C-side± distributions of the signals after the acoplanarity selection. The EL, SD, and DD signal distributions
Figure 4.6: 𝜉 𝛾𝛾
are illustrated separately.

52

Events / 0.0014

4.3 𝜉AFP selection250
s = 13 TeV, 14.6 fb-1

A-side proton
C-side proton

200
150
100
50
0
0

0.02

0.04

0.06 0.08
0.1
0.12 0.14
Proton energy loss fraction ξ
AFP±
Figure 4.7: 𝜉AFP
distributions of the data after the acoplanarity selection. Each distribution has only entries forevents in which reconstructed protons are present on the corresponding arm. The shape is almost the same as before
the acoplanarity selection because the dominant combinatorial background is not correlated to the diphoton system
as described in Sec. 6.

This selection is applied for each side, and the condition is equivalent to




±
±
±
0.035 − 0.004 + 0.1𝜉 𝛾𝛾≤ 𝜉 𝛾𝛾≤ 0.08 + 0.004 + 0.1𝜉 𝛾𝛾,± iswhere 0.035 and 0.08 are the 𝜉AFP selection thresholds as described in Sec. 4.3, and the 0.004 + 0.1𝜉 𝛾𝛾± and 𝜉 ±the maximum allowed absolute difference between 𝜉 𝛾𝛾
AFP in the matching selection described inSec. 4.4, which is introduced to improve the signal and background efficiency.2

4.3 𝝃AFP selection
From this stage, protons are used for the event selection. First, for each side, if there are multiple
reconstructed protons, a unique proton with energy loss fraction 𝜉AFP closest to 𝜉 𝛾𝛾 is selected.3 MappingA , 𝜉 C ) to (𝑚 , 𝑦 ) is only possible when protons are reconstructed on both sides.
from (𝜉AFP
𝛾𝛾 𝛾𝛾
AFP
±
Figs. 4.7 and 4.8 show the 𝜉AFP
distributions for the data and ALP signals after the acoplanarity selection.
In contrast to the 𝜉 𝛾𝛾 distributions in Sec. 4.2, the distributions have entries only in the acceptable regionof the AFP detectors. High-mass signal events are distributed in relatively high 𝜉 region as expected by
Fig. 4.4. The contribution of the DD signals is minimal because each distribution has only entries forevents in which reconstructed protons are present on the corresponding arm, while the DD signals haveno intact protons in their final states. The small existing DD contribution is due to the events that the
dissociative proton systems are hadronized back to protons and coincidentally enter the AFP detectors.
±
As the event selection, the 𝜉AFPof the selected proton is required to be in the range [0.035, 0.08] in which
the proton reconstruction efficiency is evaluated with uncertainty as described in Sec 3.5. This selection isapplied for each side.
2 Too small background efficiency would make background modeling difficult.

3 This should not be done at the object selection stage in Sec. 3.5 because the background sample creation described inSec. 6 would be biased. In the signal MC samples, an efficiency of 92 ± 2% that may be wrong is applied to a proton with
𝜉AFP ∉ [0.035, 0.08] (see Sec. 3.5), but events with such a proton closest to the diphoton in terms of 𝜉 are discarded by the 𝜉AFP
selection for the side of interest.

53

0.35

Events / 0.0014

Events / 0.0014

4 Event Selections = 13 TeV, 14.6 fb-1

0.3

EL
SD

0.25

DD

0.2

0.3

EL

0.25

SD
DD

0.2
0.15

0.15
0.1

0.1

0.05

0.05

0
0

s = 13 TeV, 14.6 fb-1

0.02

0.04

0
0

0.06 0.08
0.1
0.12 0.14
Proton energy loss fraction ξA

0.02

0.04

AFP

AFP

(b) 𝑚 𝑋 = 300 GeV, C-side
Events / 0.0014

Events / 0.0014(a) 𝑚 𝑋 = 300 GeV, A-side
0.035
s = 13 TeV, 14.6 fb-1

0.03

EL
SD

0.025

DD

0.035
s = 13 TeV, 14.6 fb-1

0.03

SD

0.02
0.015

0.015

0.01

0.01

0.005

0.005
0.02

0.04

0.06 0.08
0.1
0.12 0.14
Proton energy loss fraction ξA

EL

0.025
0.02

0
0

0
0

DD

0.02

AFP

(c) 𝑚 𝑋 = 1200 GeV, A-side

0.06 0.08
0.1
0.12 0.14
Proton energy loss fraction ξC

0.04

0.06 0.08
0.1
0.12 0.14
Proton energy loss fraction ξCAFP(d) 𝑚 𝑋 = 1200 GeV, C-side± distributions of the signals after the acoplanarity selection. The EL, SD, and DD signal distributions
Figure 4.8: 𝜉AFP
are illustrated separately. Each distribution has only entries for events in which reconstructed protons are present on
the corresponding arm.

4.4 Diphoton-proton matching
Diphoton-proton matching is then applied by comparing the 𝜉 𝛾𝛾 value with the 𝜉AFP for each side as
described in Sec. 1.4. The differences++Δ𝜉 + = Δ𝜉 A ≡ 𝜉AFP
− 𝜉 𝛾𝛾,−−Δ𝜉 − = Δ𝜉 C ≡ 𝜉AFP
− 𝜉 𝛾𝛾are used for the selection. Figs. 4.9 and 4.10 show the Δ𝜉 ± distributions of the data and signal MC samples
after the acoplanarity selecion. From the widths of the signal distributions, the bare selection condition isdetermined as |Δ𝜉 ± | < 0.004. However, if the 𝜉AFP relative uncertainty of about 10% described in Sec. 3.5
is considered, this threshold is found to be unacceptably tight. The dominant source of the 10% is theuncertainty of beam optics assumed in the proton reconstruction. Fig. 4.11 shows the Δ𝜉 ± distributionsof the signal events where the beam optics configuration in the simulation is redefined as the nominal
value added or subtracted by its uncertainty. The distributions represent the nominal case, upper-shifted
case, and lower-shifted case. Fig. 4.12 shows their dependency on 𝜉 𝛾𝛾 . The bare condition fails more than
80% events for the systematically shifted samples. Some Δ𝜉 ± plots for other systematic variations are in
Appendix C.

54

Events / 0.002

4.4 Diphoton-proton matching300
s = 13 TeV, 14.6 fb-1

250

A-sideC-side

200
150
100
50
0
−0.1

−0.05

0

0.05

0.1
∆ξ

s = 13 TeV, 14.6 fb-1

0.6

EL
SD

0.5

DD

0.4

Events / 0.0002

Events / 0.0002

Figure 4.9: Δ𝜉 ± distributions of the data after the acoplanarity selection. Each distribution has only entries forevents in which reconstructed protons are present on the corresponding arm.

EL
DD

0.4
0.3

0.2

0.2

0.1

0.1
−0.005

0

0.005

0
−0.01

0.01
∆ξA

s = 13 TeV, 14.6 fb-1
EL

0

0.005

0.01
C
∆ξ

0.1
s = 13 TeV, 14.6 fb-1
EL

0.08

SD

0.08

−0.005(b) 𝑚 𝑋 = 300 GeV, C-side
Events / 0.0002

Events / 0.0002(a) 𝑚 𝑋 = 300 GeV, A-side
0.1

SD

0.5

0.3

0
−0.01

s = 13 TeV, 14.6 fb-1

0.6

DD

SD
DD

0.06

0.06
0.04
0.04
0.02

0.02
0
−0.01

−0.00500.005(c) 𝑚 𝑋 = 1200 GeV, A-side

0.01
∆ξA

0
−0.01

−0.005

0

0.005

0.01
C
∆ξ

(d) 𝑚 𝑋 = 1200 GeV, C-side

Figure 4.10: Δ𝜉 ± distributions of the signals after the acoplanarity selection. The EL, SD, and DD signal distributions
are illustrated separately. Each distribution has only entries for events in which reconstructed protons are present on
the corresponding arm.

55

0.35

Events [A.U.]

Events [A.U.]

4 Event Selections = 13 TeV
NominalSyst UP
Syst DOWN

0.3
0.25

0.3

s = 13 TeV
NominalSyst UP
Syst DOWN

0.25
0.2

0.2
0.15

0.15

0.1

0.1

0.05

0.05

0
−0.01

−0.005

0

0.005

0.01
∆ξA

0
−0.01

−0.005(a) A-side

0

0.005

0.01
C
∆ξ

(b) C-side

0.004
0.002
0
−0.002−0.004−0.006−0.008−0.01
0 0.020.040.060.08 0.1 0.120.14

0.18

0.006

0.16

0.006

0.004

0.14

0.004

0.002

0.12

0.002

0

0.1

−0.0020.08−0.002−0.0040.06−0.004−0.0060.04−0.006−0.0080.02−0.008−0.01
0
0 0.020.040.060.08 0.1 0.120.14(a) A-side nominal(b) A-side up

0.002
0−0.002−0.004−0.006−0.008−0.01
0 0.020.040.060.08 0.1 0.120.14(c) A-side down
0.18

0.006

0

Proton energy loss fraction ξAγ γ

0.2

0.008

0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0

0.16

(YHQWV

0.004

0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0

0.008

−0.01
0 0.020.040.060.08 0.1 0.120.14(YHQWV

(YHQWV

0.006

∆ξ&

Proton energy loss fraction ξAγ γ∆ξCProton energy loss fraction ξAγ γ

0.008

∆ξA

∆ξA
0.008

∆ξ&

0.006

0.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0

(YHQWV

0.008(YHQWV

Events / 0.014 / 0.0035∆ξA

Figure 4.11: Δ𝜉 distributions of the 𝑚 𝑋 = 1000 GeV simulated signal obtained with nominal and systematicallyvaried beam optics configuration. The distributions are normalized by the integrals.

0.008

0.16

0.006

0.14
0.12
0.1

0.004

0.14

0.004

0.002

0.12

0.002

0

0.1

0

−0.0020.08−0.002−0.0040.08

0.06−0.0040.06−0.0060.04−0.0060.04−0.0080.02−0.008

0.02

−0.01
0
0 0.020.040.060.08 0.1 0.120.14−0.01
00 0.020.040.060.08 0.1 0.120.14

Proton energy loss fraction ξCγ γ

Proton energy loss fraction ξCγ γ

Proton energy loss fraction ξCγ γ(d) C-side nominal(e) C-side up(f) C-side down

Figure 4.12: Δ𝜉 vs 𝜉 𝛾𝛾 2D distributions of the 𝑚 𝑋 = 1000 GeV simulated signal obtained with nominal andsystematically varied beam optics configuration.

56

Events / 0.0014

4.5 Set operation
220
200
180
160
140
120
100
80
60
40
20
0
−0.02

s = 13 TeV, 14.6 fb-1

A-sideC-side

0

0.02

0.04

0.06

0.08

0.1

0.12
∆effξ

Figure 4.13: Δeff 𝜉 ± distributions of the data after the acoplanarity selection. Each distribution has only entries forevents in which reconstructed protons are present on the corresponding arm.

Thus, the matching condition is modified as±|Δ𝜉 ± | < 0.004 + 0.1𝜉 𝛾𝛾with the correction term to consider the 10% uncertainty. This condition is rewritten as±
Δeff 𝜉 ± ≡ |Δ𝜉 ± | − 0.1𝜉 𝛾𝛾< 0.004

to make the threshold constant. Figs. 4.13 and 4.14 show the Δeff 𝜉 distributions for the data and the
ALP signal MC samples, respectively. This modified condition can cause the selection of events whose
systematic uncertainty is not evaluated enough, but the effects of such events are evaluated in Appendix D
and found to be small. The diphoton mass distribution of the data after the matching selection had been
blinded to prevent potential psychological bias until the statistical modeling and the evaluation of systematic
uncertainty were fixed. This is called the blinding strategy and is widely adopted in search experiments.

4.5 Set operation
There are two sets of events corresponding to the A-side and C-side per initial sample after the event
selection up to the matching, because the 𝜉 𝛾𝛾 selection, 𝜉AFP selection, and the matching are performedside-by-side as shown in Fig. 4.1. In this research, their union is taken as the sample after the whole event
selection. In contrast, the CMS-TOTEM PPS analysis takes the intersection of them [74]. The former isreferred to as the OR selection, while the latter is referred to as the AND selection, as described in Sec. 1.4.
One could use another set operation between the two samples, such as the XOR or NOR conditions to
combine the results with one another. Still, this research only uses the OR selection for simplicity. Fig. 4.1
is recreated as Fig. 4.15 with the quantitative information about the selection condition. Fig. 4.16 show thecut flow of the data for each of the A-side selection, C-side selection, AND selection, and OR selection,
where the data after the A(C)-side selection is the sample after the side-by-side selection which would
have been input to the set operation. No data event is observed after the AND matching selection. 441 data
events remain after the whole selection with the OR condition and requiring 𝑚 𝛾𝛾 to be in the search range[150, 1600] GeV. Table 4.1 quantifies the cut flow for the OR condition. The signal selection cut flow for
the OR selection is summarized in Tables 4.2 and 4.3.

57

0.4

s = 13 TeV, 14.6 fb-1

0.35

EL
SD

0.3

DD

0.25

Events / 0.0002

Events / 0.0002

4 Event Selection
0.45

SD

0.3

DD

0.25
0.2

0.15

0.15

0.1

0.1

0.05

0.05
−0.005

0

0.005

0
−0.01

0.01
∆effξAs = 13 TeV, 14.6 fb-1
EL

0.04

SD
DD

0.03

0.01
∆effξC

SD
DD

0.03

0.01

0.01
∆effξAs = 13 TeV, 14.6 fb-1
EL

0.01

0.005

0.005

0.04

0.02

0

0

0.05

0.02

−0.005−0.005(b) 𝑚 𝑋 = 300 GeV, C-side
Events / 0.0002

Events / 0.0002(a) 𝑚 𝑋 = 300 GeV, A-side
0.05

0
−0.01

EL

0.35

0.2

0
−0.01

s = 13 TeV, 14.6 fb-1

0.4

0
−0.01

(c) 𝑚 𝑋 = 1200 GeV, A-side−0.005

0

0.005

0.01
∆effξC(d) 𝑚 𝑋 = 1200 GeV, C-side

Figure 4.14: Δeff 𝜉 ± distributions of the signals after the acoplanarity selection. The EL, SD, and DD signal
distributions are illustrated separately. Each distribution has only entries for events in which reconstructed protons
are present on the corresponding arm.
√
Table 4.1: The cut flow of the data for 14.6 fb−1 of 𝑝 𝑝 collision at 𝑠 = 13 TeV. The values for 𝜉 𝛾𝛾 and 𝜉AFP are thenumbers of events when the set operation is performed without the latter selection.

Cut variable
Preselection
𝛾𝛾
𝐴𝜙
𝜉 𝛾𝛾
𝜉AFP
Δ𝜉eff
𝑚 𝛾𝛾 ∈ [150, 1600] GeVEvents201,723
18,597
8,626
2,624
936
441

Efficiency
9%
46%
14%
36%

The mass dependency of the selection efficiency is determined mainly at the 𝜉 𝛾𝛾 selection stage. Fig. 4.17 (a)shows the estimated efficiency of this selection on the data events for each 𝑚 𝛾𝛾 for the OR selection and theother conditions. A toy MC sample is made by generating 𝑚 𝛾𝛾 and 𝑦 𝛾𝛾 values uniformly, weighting themby the data-driven 𝑦 𝛾𝛾 distribution as Fig. 4.18, and the 𝜉 𝛾𝛾 selection is applied. It is compared with (b),A , 𝜉 C ) and (𝑚 , 𝑦 ) coordinates with the yellow OR selection range.
the correspondence between (𝜉 𝛾𝛾𝛾𝛾 𝛾𝛾𝛾𝛾The OR selection efficiency corresponds to the length of the diphoton mass contour line that overlaps theyellow region, weighted by the diphoton rapidity distribution obtained from data assuming almost all data
events are background. Sec. 4.5.1 models the selection efficiency of the background. The signal selection
efficiency is modelled in Sec. 5.2.

58

4.5 Set operation

EventsFigure 4.15: Event selection flow with quantitative selection criteria.

105
104
103
102
10
1

s = 13 TeV, 14.6 fb-1

A-sideC-sideAND
OR

Preselection

γγ

Aφ

ξγγ

ξ

AFP

Matching
Condtion

Figure 4.16: The cut flow of the data after the preselection. The values for 𝜉 𝛾𝛾 and 𝜉AFP are the numbers of events
when the set operation is performed without the latter selection.

59

4 Event Selection
Table
4.2: Signal MC sample cut flow for ALP mass from 200 GeV to 650 GeV for 14.6 fb−1 of 𝑝 𝑝 collision at√
𝑠 = 13 TeV. The values for 𝜉 𝛾𝛾 and 𝜉AFP are the numbers of events when the set operation is performed without
the latter selection.

Cut variable
Preselection
𝛾𝛾
𝐴𝜙
𝜉 𝛾𝛾
𝜉AFP
Δ𝜉eff
Cut variable
Preselection
𝛾𝛾
𝐴𝜙
𝜉 𝛾𝛾
𝜉AFP
Δ𝜉eff
Cut variable
Preselection
𝛾𝛾
𝐴𝜙
𝜉 𝛾𝛾
𝜉AFP
Δ𝜉eff
Cut variable
Preselection
𝛾𝛾
𝐴𝜙
𝜉 𝛾𝛾
𝜉AFP
Δ𝜉eff
Cut variable
Preselection
𝛾𝛾
𝐴𝜙
𝜉 𝛾𝛾
𝜉AFP
Δ𝜉eff
Cut variable
Preselection
𝛾𝛾
𝐴𝜙
𝜉 𝛾𝛾
𝜉AFP
Δ𝜉eff
60

EL
Events Efficiency
21.49
21.47
100%
11.55
54%
6.98
60%
6.98
100%
EL
Events Efficiency
16.54
16.51
100%
11.73
71%
7.34
63%
7.34
100%
Events
12.81
12.79
9.66
6.69
6.69

EL
Efficiency
100%
76%
69%
100%

EL
Events Efficiency
9.80
9.79
100%
6.51
66%
5.13
79%
5.13
100%
Events
7.82
7.81
4.35
3.30
3.30

EL
Efficiency
100%
56%
76%
100%

EL
Events Efficiency
6.91
6.90
100%
3.58
52%
2.67
75%
2.67
100%

200 GeVSD
Events Efficiency
67.69
31.65
47%
16.69
53%
4.53
27%
4.50
99%
300 GeVSD
Events Efficiency
48.77
26.56
54%
18.63
70%
5.17
28%
5.15
100%
400 GeVSD
Events Efficiency
35.93
22.14
62%
16.58
75%
5.26
32%
5.24
99%
500 GeVSD
Events Efficiency
26.89
17.62
66%
11.54
65%
4.83
42%
4.83
100%
600 GeVSD
Events Efficiency
20.97
14.51
69%
7.97
55%
4.31
54%
4.31
100%
650 GeVSD
Events Efficiency
18.78
13.32
71%
6.69
50%
4.06
61%
4.06
100%

Events
16.64
4.70
2.60
0.10
0.10

DD
Efficiency
28%
55%
4%
96%

DD
Events Efficiency
13.41
4.94
37%
3.27
66%
0.08
2%
0.08
97%
Events
10.89
4.82
3.31
0.13
0.11

DD
Efficiency
44%
69%
4%
83%

DD
Events Efficiency
8.88
4.34
49%
2.52
58%
0.09
4%
0.08
83%
Events
7.61
4.10
2.04
0.10
0.09

DD
Efficiency
54%
50%
5%
92%

DD
Events Efficiency
6.91
3.92
57%
1.83
47%
0.10
6%
0.09
91%

Events105.82
57.82
30.84
11.61
11.58

Total
Efficiency
55%
53%
38%
100%

Total
Events Efficiency
78.72
48.01
61%
33.63
70%
12.59
37%
12.57
100%
Events
59.63
39.75
29.56
12.09
12.04

Total
Efficiency
67%
74%
41%
100%

Total
Events Efficiency
45.58
31.76
70%
20.57
65%
10.05
49%
10.03
100%
Events
36.40
26.43
14.37
7.72
7.70

Total
Efficiency
73%
54%
54%
100%

Total
Events Efficiency
32.60
24.13
74%
12.09
50%
6.84
57%
6.83
100%

4.5 Set operation
Table
4.3: Signal MC sample cut flow for ALP mass from 700 GeV to 1400 GeV for 14.6 fb−1 of 𝑝 𝑝 collision at√
𝑠 = 13 TeV. The values for 𝜉 𝛾𝛾 and 𝜉AFP are the numbers of events when the set operation is performed without
the latter selection.

Cut variable
Preselection
𝛾𝛾
𝐴𝜙
𝜉 𝛾𝛾
𝜉AFP
Δ𝜉eff
Cut variable
Preselection
𝛾𝛾
𝐴𝜙
𝜉 𝛾𝛾
𝜉AFP
Δ𝜉eff
Cut variable
Preselection
𝛾𝛾
𝐴𝜙
𝜉 𝛾𝛾
𝜉AFP
Δ𝜉eff
Cut variable
Preselection
𝛾𝛾
𝐴𝜙
𝜉 𝛾𝛾
𝜉AFP
Δ𝜉eff
Cut variable
Preselection
𝛾𝛾
𝐴𝜙
𝜉 𝛾𝛾
𝜉AFP
Δ𝜉eff
Cut variable
Preselection
𝛾𝛾
𝐴𝜙
𝜉 𝛾𝛾
𝜉AFP
Δ𝜉eff

EL
Events Efficiency
6.13
6.12
100%
3.30
54%
2.25
68%
2.25
100%
EL
Events Efficiency
4.82
4.82
100%
3.14
65%
2.27
72%
2.27
100%
Events
3.78
3.77
2.81
2.11
2.11

EL
Efficiency
100%
74%
75%
100%

EL
Events Efficiency
3.00
3.00
100%
2.43
81%
1.88
77%
1.88
100%
Events
1.95
1.95
1.77
1.20
1.20

EL
Efficiency
100%
91%
68%
100%

EL
Events Efficiency
1.29
1.29
100%
1.03
80%
0.69
67%
0.69
100%

700 GeVSD
Events Efficiency
16.33
11.74
72%
6.34
54%
3.78
60%
3.77
100%
800 GeVSD
Events Efficiency
12.98
9.68
75%
6.10
63%
3.33
55%
3.32
100%
900 GeVSD
Events Efficiency
10.22
7.79
76%
5.49
70%
2.70
49%
2.69
100%
1000 GeVSD
Events Efficiency
8.32
6.52
78%
5.05
77%
2.30
45%
2.29
100%
1200 GeVSD
Events Efficiency
5.49
4.47
81%
3.91
88%
1.51
39%
1.51
99%
1400 GeVSD
Events Efficiency
3.77
3.13
83%
2.51
80%
0.92
37%
0.92
100%

Events
6.27
3.65
1.76
0.08
0.07

DD
Efficiency
58%
48%
4%
92%

DD
Events Efficiency
5.28
3.27
62%
1.98
61%
0.07
3%
0.06
92%
Events
4.39
2.91
2.04
0.07
0.06

DD
Efficiency
66%
70%
3%
91%

DD
Events Efficiency
3.73
2.55
68%
2.01
79%
0.05
2%
0.04
86%
Events
2.69
1.92
1.72
0.03
0.03

DD
Efficiency
72%
89%
2%
92%

DD
Events Efficiency
1.97
1.47
74%
1.19
81%
0.02
2%
0.02
94%

Events
28.74
21.52
11.40
6.10
6.09

Total
Efficiency
75%
53%
54%
100%

Total
Events Efficiency
23.08
17.78
77%
11.23
63%
5.67
50%
5.65
100%
Events
18.39
14.48
10.33
4.88
4.86

Total
Efficiency
79%
71%
47%
100%

Total
Events Efficiency
15.05
12.07
80%
9.49
79%
4.22
44%
4.21
100%
Events
10.13
8.34
7.40
2.75
2.74

Total
Efficiency
82%
89%
37%
100%

Total
Events Efficiency
7.03
5.89
84%
4.72
80%
1.64
35%
1.63
100%

61

0.6

s = 13 TeV

.5

0.2

= -0

0.7

y = -1.5γγ
y = -1
γγ
.0

ξCγ γ

γ

0.5

=
γ

=

eV

G

y γ γ = 1.5eVG(a) Efficiency

eV

0

0
0

y γ γ = 1.0

eVG1000 1200 1400 1600
Diphoton mass mγ γ [GeV]

0.5

eV

G

20

800

G

0
0

600

y γγ =

eV

=

400

G

0
60

40

γ

200

00

80

=

=

mγ

A-side selection

0.1

eV

00

10

=
γ
γ

0.05

AND selection

G

12

=
γ

mγ
mγ

OR selection

0.2

eV

00

=
γ

mγ

s = 13 TeV

G

14

γ

mγ

0.1

0.3

0
0.

00
16

mγ

mγ

0.4

0

=
y γγ

mγ

γγ

0.15

y

Efficiency

4 Event Selection

0.05

0.1

0.15

0.2
ξAγ γ

(b) 𝑚 𝛾𝛾 contours

Figure 4.17: (a) The estimated 𝜉 𝛾𝛾 selection efficiency of the data for the logical OR (red) and AND (blue)
requirements of the A-side and the C-side, as well as the A-side (green) requirement, depending on diphoton mass
A -𝜉 C plane. The yellow region corresponds to the 𝜉𝑚 𝛾𝛾 . (b) 𝑚 𝛾𝛾 contours in a 𝜉 𝛾𝛾𝛾𝛾 OR selection.
𝛾𝛾4.5.1 Background selection efficiency

The background selection efficiency after the acoplanarity cut is modelled as a function of diphoton mass
𝑚 𝛾𝛾 for better understanding and use in Sec. 6.4.3. The efficiency is computed numerically using someanalytical calculations with data-driven constants. The 𝜉 𝛾𝛾 selection efficiency 𝜀 𝛾𝛾 is calculated as theintegral of the probability density function of diphoton rapidity,∫
𝜀 𝛾𝛾 (𝑥 𝛾𝛾 ) = 𝐶(4.4)

𝑓 (𝑦 𝛾𝛾 )d𝑦 𝛾𝛾 ,𝐼 ( 𝑥𝛾𝛾 )√where 𝑥 𝛾𝛾 ≡ 𝑚 𝛾𝛾 / 𝑠, 𝐶 is a constant, 𝐼 is the interval of integration, and𝑓 (𝑦 𝛾𝛾 ) =












4
Õ

𝑝 𝑖 𝑦 2𝑖𝛾𝛾
for |𝑦 𝛾𝛾 | ≤ 𝑦 end

𝑖=0∫ 𝑦end Õ
4






−𝑦end


 0


𝑖=0

!
𝑝 𝑖 𝑦 2𝑖
𝛾𝛾

,

d𝑦 𝛾𝛾
for |𝑦 𝛾𝛾 | > 𝑦 end

Íwhere 𝑦 end is defined as the smallest 𝑦 𝛾𝛾 > 0 such that 4𝑖=0 𝑝 𝑖 𝑦 2𝑖𝛾𝛾 = 0. The coefficients 𝑝 𝑖 are derived byfitting the data distribution as Fig. 4.18 and summarized in Table B.2. The 𝑚 𝛾𝛾 dependency of 𝑓 (𝑦 𝛾𝛾 ) is
ignored because it is not large.

The interval 𝐼 (𝑥 𝛾𝛾 ) is defined for the OR, AND, and A-side selection. The calculation result of the C-side

62

Events / 0.054

4.5 Set operation400
s = 13 TeV, 14.6 fb-1

350
300
250
200
150
100
50
0

−2

−1

0

1
2
Diphoton rapidity yγ γ

Figure 4.18: Diphoton rapidity distribution of the data after the acoplanarity selection with the fit result of apolynomial function.
q
√
𝛾𝛾 𝛾𝛾
Table 4.4: Intervals of integration for each 𝑥 𝛾𝛾 = 𝑚 𝛾𝛾 / 𝑠 range. 𝜉¯𝛾𝛾 is defined as 𝜉min 𝜉max .

ConditionORAND
A-side

𝛾𝛾

𝐶
2
2
1

𝑥 𝛾𝛾 ∈ [0, 𝜉min )A[𝑦 A
min , 𝑦 max ]∅A[𝑦 min , 𝑦 A
max ]𝑥 𝛾𝛾 ∈ [𝜉min , 𝜉¯𝛾𝛾 )[0, 𝑦 Amax ][0, 𝑦 Cmin ]A[𝑦 A
min , 𝑦 max ]𝛾𝛾

𝑥 𝛾𝛾 ∈ [ 𝜉¯𝛾𝛾 , 𝜉max )[0, 𝑦 Cmin ][0, 𝑦 Amax ]A[𝑦 min , 𝑦 A
max ]
𝛾𝛾

𝛾𝛾

𝑥 𝛾𝛾 ∈ [𝜉max , ∞)[𝑦 Cmax , 𝑦 Cmin ]∅A[𝑦 min , 𝑦 A
max ]selection is assumed to be the same as the A-side. The end-points𝛾𝛾𝑦A
min = ln

𝜉min

,
𝑥 𝛾𝛾
𝛾𝛾
𝜉max
𝑦A
,
max = ln
𝑥 𝛾𝛾
𝛾𝛾
𝜉min
C
𝑦 min = − ln
,
𝑥 𝛾𝛾
𝛾𝛾
𝜉
𝑦 Cmax = − ln max .
𝑥 𝛾𝛾
are defined for a given 𝑥 𝛾𝛾 using Eq. 4.3 beforehand, where 𝜉min = 0.031/1.1, and 𝜉max = 0.084/0.9.
Table 4.4 shows 𝐶 and the intervals for each 𝑥 𝛾𝛾 range. The integrations for 𝐶 = 2 make use of the factthat the integrand is an even function. Fig. 4.19 shows the 𝑚 𝛾𝛾 dependency of the 𝜉 𝛾𝛾 selection efficiency
computed numerically. This matches with Fig. 4.17 (a), validating the equation. Fig. A.3 shows thedata-driven evaluation for comparison.
𝛾𝛾𝛾𝛾

The 𝑚 𝛾𝛾 dependency of the background matching selection efficiency, including the 𝜉AFP selection, isalso calculated for each of OR, AND, XOR, and NOR conditions. First of all, for a pair of diphoton andproton for a single side, the probability that the diphoton is matched with the proton is calculated as
1𝑝
𝜀 match(𝜉 𝛾𝛾 ) =∫ 𝜉𝛾𝛾 +Δ 𝜉𝑔(𝜉AFP )d𝜉AFP ,𝜉𝛾𝛾 −Δ 𝜉

63

Efficiency

4 Event Selection
0.7
0.6
0.5
0.4
0.3

s = 13 TeV

0.2

OR selection

0.1

A-side selection0

AND selection

200

400

600

800

1000 1200 1400 1600
Diphoton mass mγγ [GeV]

Figure 4.19: 𝑚 𝛾𝛾 dependency of the 𝜉 𝛾𝛾 cut efficiency computed numerically for each set operation condition.

where Δ𝜉 = 0.004 + 0.1𝜉 𝛾𝛾 . 𝑔(𝜉AFP ) is the probability density function of 𝜉AFP normalized in 𝜉AFP ∈AFP , 𝜉 AFP ], where 𝜉 AFP = 0.035, and 𝜉 AFP = 0.08, and is parametrized as
[𝜉min
max
max
min
𝑔stem (𝜉AFP )∫ 𝜉maxAFP𝑔stem (𝜉AFP )d𝜉AFP𝑔(𝜉AFP ) =
AFP

𝜉min


 0



𝜉AFP − 𝑞 3
𝑞 1 erfc −𝑞4
𝑔stem (𝜉AFP ) = 𝑞 0 +.
𝜉AFP + 𝑞 2

AFP , 𝜉 AFP ]for 𝜉AFP ∈ [𝜉minmax,
AFP , 𝜉 AFP ]for 𝜉AFP ∉ [𝜉minmax

The parameters 𝑞 𝑖 are derived by fitting the data distribution as Fig. 4.20 so that the fit result well represent
AFP , 𝜉 AFP ], assuming that the distributions for both sides are the same. The parameterthe distribution in [𝜉minmax𝛾𝛾
AFP − Δ𝜉values are summarized in Table B.2. Since the thresholds of the 𝜉 𝛾𝛾 selection are set to 𝜉min = 𝜉min𝛾𝛾
AFP + Δ𝜉, the possibility that the matching is performed on a particular side even though theand 𝜉max = 𝜉max
𝜉 𝛾𝛾 selection is not passed on the side is naturally out of consideration.
A , 𝜉 C ) is calculated for each condition,
The matching efficiency 𝜀˜match (𝜉 𝛾𝛾
𝛾𝛾


𝑁 
𝑁
1𝑝
1𝑝
OR
A
C
A
C
𝜀˜match(𝜉 𝛾𝛾, 𝜉 𝛾𝛾) = 1 − 1 − 𝜀match(𝜉 𝛾𝛾)
1 − 𝜀 match
(𝜉 𝛾𝛾
) ,

𝑁
1𝑝
1𝑝
AND
A
C
A
C
𝜀˜match (𝜉 𝛾𝛾 , 𝜉 𝛾𝛾 ) = 1 − 1 − 𝜀match (𝜉 𝛾𝛾 )𝜀 match (𝜉 𝛾𝛾 ) ,

𝑁
1𝑝
A
A
C
A
𝜀˜match(𝜉 𝛾𝛾, 𝜉 𝛾𝛾) = 1 − 1 − 𝜀match
(𝜉 𝛾𝛾
) ,
where the averaged number of reconstructed protons per event per side derived from the data is 𝑁 = 0.74.
For any condition, the matching efficiency 𝜀 match can be calculated as a function of 𝑥 𝛾𝛾 and 𝑦 𝛾𝛾 ,𝜀 match (𝑥 𝛾𝛾 , 𝑦 𝛾𝛾 ) = 𝜀˜match (𝑥 𝛾𝛾 𝑒 𝑦𝛾𝛾 , 𝑥 𝛾𝛾 𝑒 −𝑦𝛾𝛾 ).
The matching efficiency for a fixed 𝑥 𝛾𝛾 is then calculated as the weighted average of 𝜀match (𝑥 𝛾𝛾 , 𝑦 𝛾𝛾 ),∫𝜀(𝑥 , 𝑦 ) 𝑓 (𝑦 𝛾𝛾 )d𝑦 𝛾𝛾𝐼 ( 𝑥𝛾𝛾 ) match 𝛾𝛾 𝛾𝛾∫𝜀 match (𝑥 𝛾𝛾 ) =.
𝑓 (𝑦 𝛾𝛾 )d𝑦 𝛾𝛾𝐼 (𝑥 )𝛾𝛾

64

Events / 0.0014

4.5 Set operation250
s = 13 TeV, 14.6 fb-1

A-side proton
C-side proton

200
150
100
50
0
0

0.02

0.04

0.06 0.08
0.1
0.12 0.14
Proton energy loss fraction ξ
AFP

0.45

Efficiency

Efficiency

Figure 4.20: 𝜉AFP distribution of the data with the fitting result. Distributions before the 𝜉 𝛾𝛾 selection are usedbecause it is assumed that background events have no correlation between diphoton and proton.
0.18

0.4

0.16

0.35

0.14

0.3

0.12

0.25

0.1

0.2

0.08

s = 13 TeV

0.15

OR selection

0.06

0.1

AND selection

0.04

A-side selection

0.05
0

200

400

600

800

1000 1200 1400 1600
Diphoton mass mγγ [GeV]s = 13 TeV
OR selection
AND selection
A-side selection

0.02
0

XOR selection

200(a) Matching efficiency

400

600

800

1000 1200 1400 1600
Diphoton mass mγγ [GeV](b) Overall efficiency

Figure 4.21: 𝑚 𝛾𝛾 dependency of the matching and overall selection efficiency computed numerically for eachcondition.

This calculation holds when the AND, OR, or A(C)-side selection is applied. In such a case, the overall
background selection efficiency after the acoplanarity selection is∫𝜀all (𝑥 𝛾𝛾 ) = 𝜀 𝛾𝛾 (𝑥 𝛾𝛾 )𝜀 match (𝑥 𝛾𝛾 ) = 𝐶
𝜀match (𝑥 𝛾𝛾 , 𝑦 𝛾𝛾 ) 𝑓 (𝑦 𝛾𝛾 )d𝑦 𝛾𝛾 .
(4.5)
𝐼 ( 𝑥𝛾𝛾 )The efficiency for XOR and NOR conditions can be calculated as
XOR
OR
AND
𝜀 all
(𝑥 𝛾𝛾 ) = 𝜀 all(𝑥 𝛾𝛾 ) − 𝜀all(𝑥 𝛾𝛾 ),
NOR
OR
𝜀 all
(𝑥 𝛾𝛾 ) = 1 − 𝜀 all(𝑥 𝛾𝛾 ).

Fig. 4.21 shows the 𝑚 𝛾𝛾 dependency of the matching and overall selection efficiency computed numerically.
Fig. A.4 shows the data-driven evaluation of them for comparison. The efficiency is not compatible, while
the dependency shape is similar to each other. Since only the model shape affects the background estimation,
this difference is not taken seriously.

65

5 Signal Modeling
5.1 Signal fitting function
This section describes the signal modeling strategy. In this research, signal modeling assumes that thevariance of diphoton mass (𝑚 𝛾𝛾 ) distribution of reconstructed signal events is dominated by the resolutionof the ATLAS detector, and the ALP natural width is negligible. This assumption is called NWA, as
described in Sec. 1.3.1. Under this assumption, the signal 𝑚 𝛾𝛾 distribution is modelled by a double-sided
Crystal Ball (DSCB) function [141, 142],2
𝑒 −0.5𝑡2𝑒 −0.5𝛼L  𝑛L 𝛼L 𝑛L − 𝛼 − 𝑡L
𝑓 𝑋 𝑚 𝛾𝛾 ; 𝒙 𝑋 (𝑚 𝑋 ) = 𝑁 ·
𝑛L 𝛼L

2


𝑒 −0.5𝛼R  𝑛R 𝛼R 𝑛R 𝑛 𝛼 − 𝛼R + 𝑡R R

if − 𝛼L ≤ 𝑡 ≤ 𝛼R
if 𝑡 < −𝛼L,
if 𝑡 > 𝛼R(5.1)

𝒙 𝑋 (𝑚 𝑋 ) ≡ {𝜇CB (𝑚 𝑋 ), 𝜎CB (𝑚 𝑋 ), 𝛼L , 𝛼R , 𝑛L , 𝑛R } ,𝑚 𝛾𝛾 − 𝜇CB (𝑚 𝑋 )𝑡≡,𝜎CB (𝑚 𝑋 )as per the high-mass diphoton resonance search [52]. 𝜇CB and 𝜎CB , depending on hypothetical ALP mass𝑚 𝑋 , represent the peak and width of the Gaussian core of the function, 𝑁 is a normalization parameter, 𝛼L(𝛼R ) is the position of the junction between the Gaussian and power law on the low (high) mass side inunits of 𝑡, and 𝑛L (𝑛R ) is the exponent of this power law.1 This function and its derivative are continuous
for all 𝑡 and converge to 0 for 𝑡 → ∞ with 𝑛L,R > 0.
The parameters of the DSCB are determined in a multi-step procedure. First, the 𝑚 𝛾𝛾 distribution of
the signal MC sample for each mass point 𝑚 𝑋 after the whole event selection is fitted with the DSCB
function, yielding a set of DSCB parameters. The fits are shown in Figs. 5.1 and 5.2. Next, linear or
constant functions,𝜇CB (𝑚 𝑋 ) = 𝑚 𝑋 + Δ𝑚(𝑚 𝑋 ) = 𝑚 𝑋 + 𝑝 Δ𝑚0 + 𝑝 Δ𝑚1 𝑚 𝑋 ,𝜎CB (𝑚 𝑋 ) = 𝑝 𝜎CB 0 + 𝑝 𝜎CB 1 𝑚 𝑋 ,𝛼L = 𝑝 𝛼L , 𝛼R = 𝑝 𝛼R
𝑛L = 𝑝 𝑛L , 𝑛R = 𝑝 𝑛R
are fit to the 𝑚 𝑋 dependency of each parameter to parametrize the mass-dependent evolution, where theconstants 𝑝 are the parameters. The parametrization is shown in Fig. 5.3 and is summarized in Table 5.1.
The parameterization models the resonance peak for arbitrary hypothetical ALP mass. Each uncertainty inthe table is considered a systematic uncertainty of the corresponding parameter as described in Sec. 8.1.
1 Although 𝛼
L,R and 𝑛L,R are also expected to depend on 𝑚 𝑋 , the dependency is ignored because the effect is negligible.

67

6
4
2
0

Events / 1.26 [GeV]

1.4
1.2
1
0.8
0.6
0.4
0.2
0

s = 13 TeV, 14.6 fb

1.8
s = 13 TeV, 14.6 fb
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
4
3
2
1
0
270
280
290

Ratio

Ratio

Events / 0.93 [GeV]

5 Signal Modeling180 185 190 195 200 205 210 215 220
Diphoton mass mγ γ [GeV](b) ALP mass 300 GeV
Events / 1.91 [GeV]

Events / 1.58 [GeV](a) ALP mass 200 GeV2

s = 13 TeV, 14.6 fb

1.5
1
0.5

1.4
1.2

370

380

390

400 410 420 430
Diphoton mass mγγ [GeV]

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

2
1.5
1
0.5
0

s = 13 TeV, 14.6 fb

Diphoton mass mγ γ [GeV]

1
0.8
0.6
0.4

460 470 480 490 500 510 520 530 540
Diphoton mass mγγ [GeV](d) ALP mass 500 GeV
1.2
0.8
0.6
0.4
0.2

0

0

4
3
2
1
0

2
1.5
1
0.5
0

580

600
620
640
Diphoton mass mγγ [GeV](e) ALP mass 600 GeV

Diphoton mass mγ γ [GeV]s = 13 TeV, 14.6 fb

1

0.2

560

Diphoton mass mγ γ [GeV]s = 13 TeV, 14.6 fb(c) ALP mass 400 GeV
Events / 2.40 [GeV]

Events / 2.24 [GeV]

Ratio

Ratio

1
0

Ratio

Ratio

0

2

300
310
320
330
Diphoton mass mγγ [GeV]

600

620

640

660
680
700
Diphoton mass mγγ [GeV](f) ALP mass 650 GeV

Diphoton mass mγ γ [GeV]
Diphoton mass mγ γ [GeV]
Figure 5.1: Fitting results of signal
MC samples after the whole event selection using the
DSCB function, shown
here for different signal masses ranging from 200 GeV to 650 GeV. The bottom panel of each plot shows the ratio of
MC sample distribution compared with the fit function.

68

1

Events / 2.89 [GeV]

Events / 2.56 [GeV]

5.1 Signal fitting functions = 13 TeV, 14.6 fb

0.8
0.6
0.4
0.2

0.6
0.4
0.2

3

2
1
0

s = 13 TeV, 14.6 fb

0

Ratio

Ratio

0

1
0.8

640

660

680

2
1
0

700
720
740
760
Diphoton mass mγγ [GeV]

740

Diphoton mass mγ γ [GeV]

1
0
820

Events / 3.54 [GeV]2

s = 13 TeV, 14.6 fb

840

860

880

900 920 940 960 980
Diphoton mass mγγ [GeV]

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

s = 13 TeV, 14.6 fb

0.3

Events / 4.85 [GeV]

Diphoton mass mγ γ [GeV]

Diphoton mass mγ γ [GeV]21
0920 940 960 980 1000 1020 1040 1060 1080
Diphoton mass mγγ [GeV](d) ALP mass 1000 GeV
0.2

s = 13 TeV, 14.6 fb

Diphoton mass mγ γ [GeV]

0.15

0.2
0.1

0.1

0.05

0

0

1.5
1
0.5
0
1100

1.5
1
0.5
0

Ratio

Events / 4.20 [GeV]Ratio

s = 13 TeV, 14.6 fb

800
820
840
860
Diphoton mass mγγ [GeV]3(c) ALP mass 900 GeV
0.4

780

(b) ALP mass 800 GeV

Ratio

Events / 3.22 [GeV]Ratio(a) ALP mass 700 GeV
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

760

1150

1200
1250
1300
Diphoton mass mγγ [GeV](e) ALP mass 1200 GeV

1300

1350

1400
1450
1500
Diphoton mass mγγ [GeV](f) ALP mass 1400 GeV

Diphoton mass mγ γ [GeV]
Diphoton mass mγ γ [GeV]
Figure 5.2: Fitting results of signal
MC samples after the whole event selection using the
DSCB function, shown
here for different signal masses ranging from 700 GeV to 1400 GeV. The bottom panel of each plot shows the ratioof MC sample distribution compared with the fit function.

69

0.4

σ [GeV]X∆m = µ - m [GeV]

5 Signal Modelings = 13 TeV, 14.6 fb-1

0.2

16
14

0

12

−0.2

10

−0.4

s = 13 TeV, 14.6 fb-1

8

−0.6
−0.8

6

−1

4

−1.2

2
200

400

600

800

1000

1200 1400
mX [GeV]200

400

s = 13 TeV, 14.6 fb-1

5

2.2

4.5

2

4

1.8

3.5

1.6

3

1.4

2.5

1.2

2

1

1.5
200

400

600

800

1000

800

1000

1200 1400
mX [GeV]

10001200 1400
mX [GeV]

10001200 1400
mX [GeV](b) 𝜎CBαR

αL(a) Δ𝑚 = 𝜇CB − 𝑚 𝑋
2.4

600

s = 13 TeV, 14.6 fb-1

200

1200 1400
mX [GeV]400800(d) 𝛼RnRnL(c) 𝛼L600-1

s = 13 TeV, 14.6 fb

20

3

s = 13 TeV, 14.6 fb-1

2.5
2

15

1.5

10

1
0.5

5

0
−0.5

0
200

400

600

800

(e) 𝑛L

1000

1200 1400
mX [GeV]

200

400

600

800

(f) 𝑛R

Figure 5.3: The mass dependence of the DSCB parameters. The linear or constant functions (red lines) are obtained
from a fit to the DSCB fit results of each mass point (blue points).

70

5.1 Signal fitting function
Table 5.1: Signal model parameters. Parameters modelled by linear function have the subscripts followed by the
number 𝑖, which means the coefficient of the 𝑖-th order.

Parameter

Uncertainty

9.33 × 10−2

𝑝 Δ𝑚0
𝑝 Δ𝑚1
𝑝 𝜎CB 0𝑝 𝜎CB 1
𝑝 𝛼L
𝑝 𝛼R
𝑝 𝑛L
𝑝 𝑛R

𝛿Δ𝑚0 ≡ 8.18 × 10−2
𝛿Δ𝑚1 ≡ 1.26 × 10−4
𝛿 𝜎CB 0 ≡ 8.94 × 10−2
𝛿 𝜎CB 1 ≡ 1.39 × 10−4
𝛿 𝛼L ≡ 2.89 × 10−2𝛿 𝛼𝑟 ≡ 3.53 × 10−2
𝛿 𝑛L ≡ 1.75 × 10−1
𝛿 𝑛R ≡ 5.84 × 10−2

−4.59 × 10−4
1.00
7.78 × 10−3
1.54
2.28
3.31
7.69 × 10−114
s = 13 TeV, 14.6 fb-1

12

Ninterpolation/Noriginal

Number of signal eventsValues = 13 TeV, 14.6 fb-1

1.04
1.02

10
8

1

6
0.98

4

0.96

2
0

200

400

600

800

1000

1200 1400
mX [GeV](a) Signal yields of original fittings

200

400

600

800

1000

1200 1400
mX [GeV](b) Relative difference of signal yields

Figure 5.4: (a) The signal yields of the original fits. The non-smooth behaviour at around 650 GeV originates fromthe 𝜉 𝛾𝛾 cut efficiency. (b) The signal yields of fittings using the fixed parameters relative to (a) with maximum
errors.

The parametrized model does not entirely match the original fit results in Fig. 5.3. Therefore, the effect ofthe difference on the resulting signal yield is studied by comparing the original fit with the fit using the
DSCB function whose parameters are fixed to the interpolation results. Fig. 5.4 (a) shows the signal yieldsof original fittings. (b) shows the signal yields of fittings using the fixed parameters relative to (a). The
difference is slight for any ALP mass 𝑚 𝑋 .

5.1.1 Signal modeling uncertainty
One of the systematic uncertainties of the signal model is evaluated. The systematic uncertainty of signal
modeling is evaluated for each hypothetical ALP mass, comparing the resulting signal yield with the input
number of events in the signal MC sample. The input signal events are weighted variously to check if theresulting yield follows the weight accordingly, following a linear relation. The yield and number of eventsare unitized by signal strength, which is the number of events (normalized to 14.6 fb−1 ) of the nominal
signal MC samples for the corresponding ALP mass after the whole event selection. Fig. 5.5 shows theresults of the signal injection test performed by fitting the DSCB function to the signal MC distribution foreach ALP mass hypothesis. The parameters of the DSCB function, except the signal yield, are fixed to theinterpolated values to distinguish the sources of the systematic uncertainty. The input and output signal

71

5 Signal Modeling
Table 5.2: Signal modeling uncertainty for each ALP mass.

ALP mass [GeV]𝛿smodel

200
300
400
500
600
650
700
800
900
1000
1200
1400

−0.0034−0.0017−0.0006−0.0006−0.0027−0.0045−0.0062−0.0022−0.0013−0.0008−0.0010−0.0212strengths are compared for each ALP mass hypothesis. The fitting result of a linear function
𝜇out (𝜇in ) = (1 + 𝛿smodel )𝜇in
is represented by the red lines, where 𝛿smodel is the parameter. Table 5.2 lists the fitting results of the linear
function. The maximum deviation from 1 is taken as the signal modeling uncertainty,
𝛿smodel ' 2.1%.

5.2 Signal efficiency and yield
The signal yield 𝑁 𝑋 (𝑚 𝑋 ) in the results of this analysis is defined as the sum of EL, SD, and DD signal
yields,Õ𝑁 𝑋 (𝑚 𝑋 ) =𝑁𝑖 (𝑚 𝑋 ),𝑖 ∈ {EL,SD,DD}

𝑁𝑖 (𝑚 𝑋 ) = 𝐿 int 𝜎𝑖 (𝑚 𝑋 )𝜀 𝑖 (𝑚 𝑋 ),where 𝐿 int = 14.6 fb−1 is the integrated luminosity, 𝑁𝑖 is the signal yield, 𝜎𝑖 is the signal cross sectionin the phase space limited to the region specified in Table 3.1, and 𝜀 𝑖 is the selection efficiency, for each
event type 𝑖.

5.2.1 Efficiency
The efficiency 𝜀 𝑖 is parametrized as a function of ALP mass 𝑚 𝑋 with the evaluation of uncertainties to get
results for arbitrary ALP mass. It is calculated as the fraction of the number of remaining events after thewhole event selection over the total events for each ALP signal MC sample for each event type 𝑖,𝜀 𝑖𝑚𝑋 =

72

𝑖
𝑁𝑚
𝑋

,
𝑖
𝐿 int 𝜎𝑚𝑋

Ideal1.4

1.6

s = 13 TeV, 14.6 fb-1
Ideal

1.4

Observed

1

1

0.6

0.4

0.4

0.4

0.2

0.2

0.2

-1

s = 13 TeV, 14.6 fb
Ideal(b) ALP mass 300 GeV

1.6

-1

s = 13 TeV, 14.6 fb
Ideal

1.4

Observed

1.2
1

1

0.8

0.6

0.4

0.4

0.2

0.2

0.2

1.4

0
0

s = 13 TeV, 14.6 fb
Ideal
Observed

1.2(e) ALP mass 600 GeV

1.6
1.4

-1

s = 13 TeV, 14.6 fb
Ideal
Observed

1.2

1

0.8

0
0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength
Output signal strength

Output signal strength

1.6

-1

(f) ALP mass 650 GeV

1.6

1

0.6

0.4

0.4

0.2

0.2

0.2

0
0

0
0

Observed

1.2

1.4

s = 13 TeV, 14.6 fb-1
Ideal
Observed

1.2

1

0.8

1.6
1.4

1

0.8

Ideal
Observed

1

0.8

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0
0

0
0

(j) ALP mass 1000 GeV

s = 13 TeV, 14.6 fb-1

1.2

0.6

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength(i) ALP mass 900 GeV
Output signal strength

Ideal

Output signal strength

Output signal strength

1.4

0
0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength(h) ALP mass 800 GeV
1.6

Observed1

0.6

s = 13 TeV, 14.6 fb-1

Ideal

0.8

0.4

(g) ALP mass 700 GeV

s = 13 TeV, 14.6 fb-1

1.4

0.6

1.6

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength

1.2

0.8

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength

Observed

10.6(d) ALP mass 500 GeV

Ideal

0.8

0.4

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strengths = 13 TeV, 14.6 fb-1

1.2

0.6

0
0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength(c) ALP mass 400 GeV

1.6
1.4

Observed

1.2

0.8

0
0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength
Output signal strength

1.4

0
0

Output signal strength

Output signal strength(a) ALP mass 200 GeV

1.6

Observed

1

0.6

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength

Ideal

0.8

0.6

0
0

s = 13 TeV, 14.6 fb-1

1.2

0.8

0.8

1.6
1.4

Observed

1.2

1.2

Output signal strengths = 13 TeV, 14.6 fb-1

Output signal strength

1.6

Output signal strength

Output signal strength

5.2 Signal efficiency and yield

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength(k) ALP mass 1200 GeV

00

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength(l) ALP mass 1400 GeV

Figure 5.5: The signal injection test results with statistical uncertainties for each ALP mass hypothesis. The input
and output signal strengths are compared using the red fitted lines. The error bars represent the statistical uncertainty
for 14.6 fb−1 .

73

5 Signal Modeling
Table 5.3: Signal efficiency parametrization.

Parameter𝑖 = EL𝑖 = SD𝑖 = DD

𝑝 𝑖0
𝑝 𝑖1
𝑝 𝑖2
𝑝 𝑖3
𝑝 𝑖4
𝑝 𝑖5
𝑝 𝑖6
𝑝 𝑖7

−1.66 × 10−1

−1.89 × 10−2

2.22 × 10−3
−2.66 × 10−6
1.49 × 10−9−3.50 × 10−13
4.00 × 10−1
6.94 × 102
1.23 × 102

2.30 × 10−4
3.34 × 10−7
−4.20 × 10−10
1.05 × 10−13−1.56 × 10−3
2.31 × 10−5
−1.14 × 10−8

𝑖 , and 𝜎 𝑖𝑖where 𝜀 𝑖𝑚𝑋 , 𝑁 𝑚
𝑚𝑋 are the efficiency, signal yield, and cross section. The 𝜎𝑚𝑋 is output from𝑋
SuperChic and known for each ALP mass. Since the signal MC samples are generated at discrete ALP
mass points 𝑚 𝑋 , these efficiencies are discrete functions of 𝑚 𝑋 at this point. The mass dependency ofsignal efficiency is parametrized, fitting the functions
!
4


Õ
𝑖
EL EL EL𝜀 EL (𝑚 𝑋 ) =
𝑝 EL
𝑚
×
1
−
𝐺
(𝑚
;
𝑝
,
𝑝
,
𝑝
)
,
𝑋
𝑖
𝑋
7
5
6

𝜀 SD (𝑚 𝑋 ) =

𝑖=0
4
Õ

𝑖
𝑝 SD
𝑖 𝑚𝑋,

𝑖=0

𝜀 DD (𝑚 𝑋 ) =

2
Õ

𝑖
𝑝 DD
𝑖 𝑚𝑋,

𝑖=0

where

"


2#
1 𝑚𝑋 − 𝑏
𝐺 (𝑚 𝑋 ; 𝑎, 𝑏, 𝑐) = 𝑎 exp −
2
𝑐

(5.2)

is the Gaussian function. The parameters are listed in Table 5.3. Fig. 5.6 shows the computed efficiency
values, and the functions are overlaid (orange). The envelopes are also illustrated to cover the difference
between the computed efficiency and the functions. To derive the envelopes, the linear (blue) and cubic(cubic) interpolations are performed, and the upper (cyan) and lower (pink) envelopes are set to coverthese interpolation results. The parametrization of the envelopes are𝛿 𝜀EL = 0.05 + 0.4𝑒 −0.05(𝑚𝑋 −130 GeV)𝛿down
𝜀EL = 0.05

(5.3)

up
𝛿 𝜀SD = 0.04 + 0.4𝑒 −0.045(𝑚𝑋 −130 GeV)up
𝛿 𝜀DD = 0.4

𝛿down
𝜀SD = 0.04
𝛿down
𝜀DD = 0.4

(5.4)

up

(5.5)

in terms of the ratio to the nominal parametrization. They are used as the systematic uncertainty as
described in Sec. 8.1.

5.2.2 Cross section and yield
The signal yield for the fixed ALP coupling constant 𝑓 −1 = 0.05 TeV−1 , assumed when generating thesignal MC samples, called standard coupling constant, is also parametrized. Such signal yield is referred
to as the standard signal yield 𝑁std , and the signal strength is redefined as the quantity 𝜇 satisfying
𝑁 𝑋 (𝑚 𝑋 ) = 𝜇(𝑚 𝑋 )𝑁std (𝑚 𝑋 ),

74

(5.6)

0.6

Efficiency ratio

Efficiency

5.2 Signal efficiency and yields = 13 TeV, 14.6 fb-1

0.5
0.4

1.4

MC

0.2

Cubic interpolation

Linear interpolation

Cubic interpolation
Function

1.3

Upper envelope
Lower envelope

1.1

Function
Upper envelope

0.1

1

Lower envelope

200

400

600

800

1000 1200 1400 1600
mX [GeV]

200

400

600

800

1000 1200 1400 1600
mX [GeV](b) EL ratio
Efficiency ratio(a) EL efficiency
Efficiency

Linear interpolation

1.2

0.3

0.25

MC

s = 13 TeV, 14.6 fb-1

s = 13 TeV, 14.6 fb-1

0.2

1.4

MC

s = 13 TeV, 14.6 fb-1

Linear interpolation
Cubic interpolation
Function

1.3

Upper envelope
Lower envelope

1.2

0.15
MC
Linear interpolation

0.1

1.1

Cubic interpolation
Function
Upper envelope

0.05

1

Lower envelope

0

200

400

600

800

1000 1200 1400 1600
mX [GeV]

200

0.025

MC

s = 13 TeV, 14.6 fb-1

Linear interpolation
Cubic interpolation

0.02

Function
Upper envelope
Lower envelope

0.015

2.4
2.2

600

800

1000 1200 1400 1600
mX [GeV](d) SD ratioMCs = 13 TeV, 14.6 fb-1

Linear interpolation

2

Cubic interpolation
Function

1.8

Upper envelope

1.6

Lower envelope

1.4
1.2

0.01

1

0.8

0.005
0

Efficiency ratio

Efficiency(c) SD efficiency

400

0.6
0.4
200

400

600

800

1000 1200 1400 1600
mX [GeV](e) DD efficiency

200

400

600

800

1000 1200 1400 1600
mX [GeV](f) DD ratio

Figure 5.6: (a)(c)(e) Signal efficiency derived from the signal MC samples. The orange line is its parametrization.
The envelopes are also shown. The cyan and magenta lines represent the envelopes. (b)(d)(f) The ratio of the signal
efficiency taken with respect to the nominal efficiency modeling function.

75

5 Signal Modeling
Table 5.4: Signal cross section parametrization in fb unit for coupling constant 𝑓 −1 = 0.05 TeV.

Parameter𝑖 = EL𝑖 = SD𝑖 = DD2.87983000−0.00256053
4.60166000−0.00838870
0.02282670−0.00000639

9.85015−0.00417
19.75920−0.01123
2.65697−0.00162

2.04608−0.00191
3.90991−0.00809
0.004710.00115

Events

Cross section [fb]

𝑝 𝑖0
𝑝 𝑖1
𝑝 𝑖2
𝑝 𝑖3
𝑝 𝑖4
𝑝 𝑖5

EL

s = 13 TeV, 14.6 fb-1

Total

10

EL

EL fit8SD

SD fit6

DD

DD fit

4

Total

2

DD

Ratio

1

10−1
200

400

600

800

s = 13 TeV, 14.6 fb-1

12

SD

10

14

1000 1200 1400 1600
mX [GeV]

0
0.6
0.4
0.2
0

200

(a) Standard cross section

400

600

800

1000 1200 1400 1600
mX [GeV](b) Standard yield

Figure 5.7: (a) Signal cross sections and their parametrization for coupling constant 𝑓 −1 = 0.05 TeV. (b) Signal
yield parametrization for the same coupling constant. The lower panel shows the ratio of the number of events foreach event type compared to the total number of events.

where 𝑁 𝑋 is the signal yield of interest. The standard signal yield is calculated asÕ
𝑖𝑁std (𝑚 𝑋 ) =𝑁std(𝑚 𝑋 ),𝑖 ∈ {EL,SD,DD}𝑖𝑖
𝑁std (𝑚 𝑋 ) = 𝐿 int 𝜎std
(𝑚 𝑋 )𝜀 𝑖 (𝑚 𝑋 ),𝑖 (𝑚 ) is the cross section corresponding to the fixed coupling constant, and 𝜀 (𝑚 ) is parametrized
where 𝜎std
𝑋𝑖𝑋
as described in Sec. 5.2.1. The cross sections are parametrized as
𝑖

𝑖

𝑖

𝑖
𝜎std
(𝑚 𝑋 ) = 𝑝 𝑖0 𝑒 𝑝1 𝑚𝑋 + 𝑝 𝑖2 𝑒 𝑝3 𝑚𝑋 + 𝑝 𝑖4 𝑒 𝑝5 𝑚𝑋 ,where the parameters are listed in Table 5.4. The sumÕ𝜎std (𝑚 𝑋 ) =𝑖
𝜎std(𝑚 𝑋 )𝑖 ∈ {EL,SD,DD}

𝜎std is called standard cross section. Now that both cross section and efficiency are parametrized tobe continuous, signal yield is also parametrized as the product of them and 𝐿 int . Fig. 5.7 shows the𝑖 , 𝑁 𝑖 , 𝜎 , and 𝑁 . The ratio of signal yield between EL and SD is compatible
parametrization of 𝜎std
std
std
std
with the estimations from the SuperChic developers illustrated in Fig. 5.8 for comparison, roughly
supporting the implementation of the event selection in this research, although the selection criteria areslightly different.

76

C/TOT

5.2 Signal efficiency and yield

1.2

Superchic 4.1 + PYTHIA 8.2
1 with S2

.. __
. _

EL
SD
DD

500

1000

ALP
0.8

No charged particles with
p >0.5 GeV and | η | < 2.5

µµ

T

0.6

0.4

0.2

0

0

1500

2000

2500
mALP [GeV]

Figure 5.8: The ratio of the number of signal events for coupling constant 𝑓 = 0.05 TeV−1 for each event typecompared to the total number of events [115]. SuperChic and Pythia are used, and no detector simulation is
performed. The object and event selection similar to this research is applied at the truth level. The dimuon events
illustrated in Fig. 1.8 are also studied for comparison. The 𝑆 2 stands for the soft survival factor described in Sec. 3.2.

77

6 Background ModelingIn this section, the modeling of the diphoton mass distribution of the background events is describedwith systematic uncertainty. The dominant background arises from the coincident matching between
the diphoton and proton objects from different 𝑝 𝑝 collision vertices. Such events are referred to as
combinatorial backgrounds, while all non-combinatorial background events are called single-vertex
backgrounds. The sources of the diphoton objects in the combinatorial backgrounds are, for example,
inclusive diphoton production and fake jets. On the other hand, the dominant source of the backgroundprotons in this research using the OR selection (see Sec. 4.5) is the single-diffraction as described inSec. 2.3.
A background sample is prepared by a fully data-driven method to use as a template to determine thebackground model in Sec. 6.1. The background model is constructed in the same section. The validationof the data-driven method is described in Sec. 6.2. It is also proved that the single-vertex backgrounds
are negligible. Signal contamination in the background sample is considered a systematic variation of
the template in Sec. 6.3, and the systematic uncertainty of the background model is determined using thenominal and varied templates in Sec. 6.4.

6.1 Background sample creation and background modeling
A fully data-driven method is used to make a background template inheriting the procedure in the dilepton
analysis [66] as described in Sec. 1.4. First, the nominal data sample after the object selection and theevent preselection described in Sec. 3 is prepared. The amount of combinatorial background depends
on instantaneous luminosity and hence the measured number of 𝑝 𝑝 collisions 𝜇 in the BC, as shown in
Fig. 6.1, which represents the 𝜇 dependency of the selection efficiency after the acoplanarity selection.
Therefore, the preselected events are divided into the bins of the 𝜇 in the BC from which each event
originates. Fig. 6.2 shows the number of events in each bin. The bin width is 1, and almost all events
are in the bins with more than 100 entries, while the exceptional events are ignored in the backgroundestimation.
In each bin, the events are processed as illustrated in Fig. 6.3. The sample has a set of proton objects foreach event because the unique proton selection for each side has not been performed yet at this stage. The
colorful boxes indicate the proton sets. The procedure is as follows.
1. The event selection up to the 𝜉 𝛾𝛾 selection with the OR selection is applied to the data sample ineach bin. Since only the diphoton object information is used in this selection, this is referred to as
the diphoton selection. The set of selected events necessarily contains the events that can pass thewhole event selection with the OR selection.1
2. The set of protons in each event is then replaced by the sets from the other events before the diphoton
selection to create multiple events with the same object information except for protons. Since the

1 The diphoton selection can be skipped because the whole selection is applied after all, but it is applied in order to reduce theamount of computation in the following processes.

79

Selection efficinecy

6 Background Modeling0.08

s = 13 TeV, 14.6 fb-1

0.07
0.06
0.05
0.04
0.03
0.02
0.01
0

20

25

30

35

40

45

50

55

µ

Events / 1.0

Figure 6.1: The ratio of the number of data events after the whole event selection to the selection up to the acoplanarity
selection in each coarse 𝜇 bin. The statistical uncertainty is also illustrated.

s = 13 TeV, 14.6 fb-1

8000
7000
6000
5000
4000
3000
2000
1000
0
0

10

20

30

40

50

60

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80
µ

Figure 6.2: Distribution of the number of 𝑝 𝑝 collisions 𝜇 in the BC from which each data event originates.

assigned set of protons is from a different BC for each created event, the diphoton and protons are
not correlated. This procedure is referred to as the decorrelation by multiple reassignments.
3. The whole selection is applied to these many events. Only the events with diphotons and protons
coincidentally matched remain, embodying the definition of combinatorial background.
4. Since a set of events originating from the same preselected event has the same diphoton mass value,such events are unified as one event with the weight calculated as the fraction of the selected eventsin the set.
Finally, the unified events are merged with respect to the 𝜇 bins to almost finalize the background sample.
The difference in the method from the dilepton analysis [66] is that they mixed the proton sets withoutbinning the events with respect to 𝜇 and reassigned only 100 proton sets to each event.
Although the backgrounds in data are expected to come from almost the combinatorial background, there
could be non-negligible contamination of single-vertex backgrounds and signal events. However, thedecorrelation method suppresses such contamination because the matching selection eliminates most ofthem. The suppression of signal contamination is quantified in Sec. 6.3. One could just shift the sets ofprotons to the next events one by one to decorrelate the diphotons and protons, but the original data size isregarded to be too small to use for the background modeling. Fig. 6.4 compares the 𝑚 𝛾𝛾 distribution ofsuch a simple background sample and the nominal fully data-driven background sample after the whole
event selection. The nominal background template is much smoother than the former one because events80

6.1 Background sample creation and background modeling
Diphotonselection

Event 1

Event 2

Event 3

Event 4

Event 5Whole event
selection

Proton set
replacement

Protons

Unification

Event 2

Protons

Event 2

Event 2

Protons

Event 2

Event 2

Protons

Event 2

Event 2

Protons

Event 2

Event 4

Protons

Event 4

Event 4

Protons

Event 4

Event 4

Protons

Event 4

Event 4

Protons

Event 4

Event 5

Protons

Event 5

Event 5

Protons

Event 5

Event 5

Protons

Event 5

Event 5

Protons

Event 5

Event 2
(weight 0.5)

Protons

Protons

Protons

Protons

Event 5(weight 0.25)

Figure 6.3: Algorithm flow of background sample creation. A 𝜇 bin with five preselected events is illustrated forsimplicity. The yellow and grey boxes stand for the selected and not selected events, respectively. Each event has a
set of proton objects represented by colorful boxes.

which should have been excluded by the matching selection in the simpler method remain with weights
less than 1 in the nominal method. The distribution shape uncertainty is also evaluated as
sÕ
𝑤 2𝑖
𝑖

for each 𝑚 𝛾𝛾 bin, where 𝑤 𝑖 is the weight of the event with index 𝑖. The uncertainty is smaller than theformer case, whose shape uncertainty is identical to the statistical uncertainty, the square root of the bincontent. In the diphoton mass range from 150 GeV to 1600 GeV, the data after the diphoton selection has
3977 events, and the number of events is reduced to 441 after the whole event selection. However, thedata-driven background sample, even after the whole selection, holds the information of 3964 events outof the 3977 events. The higher the integrated luminosity and the tighter the selection after the diphoton
selection, the more powerful this background sample creation method becomes.
The weight of each event in the nominal background sample is slightly modified finally. Since the amountof combinatorial background and 𝜇 are positively correlated, for example, if a proton set from an event in
a BC with a higher luminosity is assigned to a diphoton from an event in a BC with a lower luminosity,
the coincident matching rate becomes overestimated. Each event in the background sample is weighted by
𝜇 𝛾𝛾 /𝜇 𝑝 to consider the effect of the slight difference of the 𝜇 between the event from which the diphoton
originates, 𝜇 𝛾𝛾 , and the event from which the protons originate, 𝜇 𝑝 , assuming the linearity between
the amount of combinatorial background and 𝜇. This weight has uncertainty within the 𝜇 bin. The 𝜇dependency and the systematic uncertainty evaluation of the weight 𝜇 𝛾𝛾 /𝜇 𝑝 are detailed in Appendix E,
and the effect of the systematic uncertainty is found to be negligible.
The function form𝑎𝑓b (𝑥; {𝑎, 𝛼0 }) ≡ 𝑓0 (𝑥; {𝑎, 𝛼0 }) = 1 − 𝑥 1/3 · 𝑥 𝛼0 ,

(6.1)

√
is used for the background function fitting in the mass range [150, 1600] GeV. 𝑥 = 𝑚 𝛾𝛾 / 𝑠 is a transformation that improves the numerical stability of the fit. This function form is an element of a family of81Events / 29.4 [GeV]

Events / 29.4 [GeV]

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2

10

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Events / 29.00 [GeV]

Figure 6.4: The 𝑚 𝛾𝛾 distribution of the background samples after the whole event selection with the shape uncertainty.
(a) Background sample where the proton set in each event of the data is shifted to the next event. (b) The nominal
background sample.

s = 13 TeV, 14.6 fb-1

102
10
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10−1

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Diphoton mass mγ γ [GeV]

Figure 6.5: Comparison of 𝑚 𝛾𝛾 distribution between the fully data-driven background after the whole selection
with shape uncertainty (black) and function fitted to it (blue) in the diphoton mass range [150, 1600] GeV.

functions 𝑓 𝑘 traditionally used in the high-mass diphoton resonance searches [50, 52] to describe the 𝑚 𝛾𝛾
spectrum, where 𝑎 Í𝑘𝑖𝑓 𝑘 (𝑥; {𝑎, 𝛼𝑖 | 𝑖 ∈ Z ∪ [0, 𝑘]}) = 1 − 𝑥 1/3 · 𝑥 𝑖=0 𝛼𝑖 (log 𝑥 ) .
Although the latest high-mass diphoton resonance search [52] used 𝑓1 , this research uses the simpler 𝑓0
because the background modeling uncertainty is expected to be dominated by the statistical uncertainty
considering the low luminosity, 14.6 fb−1 . The parameters 𝑎 and 𝛼0 are free parameters in this researchas described in Sec. 8.1. The fit result is shown in Fig. 6.5. The best-fit parameters for the nominal
background template are 𝑎 = 15.9 and 𝛼0 = −0.828, which are used as the initial values in the latter
analyses.

82

6.2 Validation
Data

Nominalobject selection andevent preselection

Nominaldiphoton selection
Sample 1

Sample 2 - Sample 1

Object selection andevent preselection(loose photon ID andno photon isolation)

Whole event selection(0.01 < A < 0.5)

Background sample creation(0.01 < A < 0.5)

Data
for validation

Background sample
for validation

Sample 2
CompareFigure 6.6: The algorithm flow of the data-driven validation of the fully data-driven background creation method in
the acoplanarity range [0.01, 0.5].

6.2 Validation
The data-driven background sample creation method described in Sec. 6.1 above is validated by comparing
data and the background sample in validation regions. The agreement on the comparison supports thatthe combinatorial background is dominant. Sec. 6.2.1 defines the validation regions and compares thedata and background sample. Furthermore, Sec. 6.2.2 examines some possible single-vertex backgrounds
using MC samples to see if there is a non-negligible contribution.

6.2.1 Data-driven validation of sample and dominance of combinatorial backgroundThe background sample is compared with the data in terms of the 𝑚 𝛾𝛾 distribution in a backgroundrich region to validate the background sample creation method. Fig. 6.6 summarizes the procedure.
The loose photon identification working point is used, and the photon isolation requirement is removed(“Sample 2”). Events that are identical to the nominal data events that pass the event selection up to the𝜉 𝛾𝛾 selection (“Sample 1”) are excluded from the validation samples. The acoplanarity range is changed
𝛾𝛾
to 0.01 < 𝐴 𝜙 < 0.5. The full nominal selection, except these criteria, is applied to the samples to create
the data sample for validation for comparison. On the other hand, the background sample is made with thesame selection by the same method as the nominal one. Fig. 6.7 shows the comparison between the data
and the background samples in the validation region. The 𝜒2 test and the Kolmogorov-Smirnov test are
performed for these distributions. The resulting reduced 𝜒2 and Kolmogorov test probability are 0.649
and 0.859, respectively. These results show the good compatibility of the background sample with thedata and validate the background sample creation method.
Further tests are performed to validate the dominance of combinatorial background in a similar data-driven
way. The background sample is compared with the data in another validation region closer to the signal
region. A validation region is defined by the selection partially different from the nominal one in that
• Only one of the leading and the sub-leading photons is required to pass the tight ID criteria, and theother photon is required not to pass the tight identification and pass the loose identification,𝛾𝛾• 0.01 < 𝐴 𝜙 < 0.5,
• 𝑚 𝛾𝛾 ∈ [130, 1600] GeV.
Since the first point makes the sample mutually exclusive from the nominal data, there is no need to takeany difference set as illustrated in Fig. 6.6. The process is as simple as replacing the “Sample 2 - Sample 1”

83

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Events / 73.5 [GeV]

Figure 6.7: Comparison between the data and background sample distributions in the background-rich region.
(a) Unzoomed and (b) zoomed versions of the lower panels are represented for better visibility. The shape uncertainty
and statistical uncertainty of the background sample are also illustrated. The shape uncertainty is smaller than thestatistical uncertainty as described in Sec. 6.1.
103

s = 13 TeV, 14.6 fb-1
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1000 1200 1400 1600
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Figure 6.8: Comparison of 𝑚 𝛾𝛾 distributions between the data and background sample in the validation region with
the non-back-to-back requirement and only one tight photon.

in the figure with the sample made following the first condition. Fig. 6.8 shows the 𝑚 𝛾𝛾 distribution of thedata and background sample in the validation region. The 𝜒2 test is performed for these distributions. The
resulting reduced 𝜒2 is 0.628.2 Fig. 6.9 compares the acoplanarity distribution, where the 𝜒2 test and theKolmogorov-Smirnov test are performed. The resulting reduced 𝜒2 and Kolmogorov test probability are
0.917 and 0.969, respectively. There is no positive excess of the data from the background sample. Thismeans there is no significant single-vertex background near the signal region. The number of backgroundand data events are 1534.3 and 1465, respectively. They are compatible within the 2𝜎 uncertainty assuming
the Poisson statistics.

2 The Kolmogorov-Smirnov test is not performed because the number of events is insufficient in some bins.

84Events / 0.0245

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Data / Pred.

10

1.2
1
0.8
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5γγDiphoton acoplanarity AφFigure 6.9: Comparison of acoplanarity distribution between the data and background sample in the validation
region with the non-back-to-back requirement and only one tight photon.

6.2.2 MC-based validation of dominance of combinatorial backgroundAn event with a diphoton and protons from the same vertex in their final state is called a single-vertex
background event. Since the diphoton and protons are correlated, such an event is not a combinatorial
background. If there is non-negligible single-vertex background, especially having a resonance-like peakin 𝑚 𝛾𝛾 distribution, the background modeling would be biased. In this case, the distribution of such
events needs to be added to the background template. In addition, the contribution of such events to thebackground sample itself should be evaluated since the background sample is fully data-driven. Thissection evaluates the effect of single-vertex background and whether it is negligible. The MC samples ofthe single-vertex background considered possibly dominant are generated and analyzed at the truth level.
Two generators are used to create different kinds of MC samples.

Test with SuperChic
SuperChic 4.13 is used to generate the MC events. Selection similar to the nominal one is applied to thefinal state of two central particles regarding them as the two photons even if they are not actually photons.
The “photon” objects are required to satisfy• |𝜂| ∈ [0, 1.37] ∪ [1.52, 2.37],• 𝑝 T > 40 GeV,𝛾𝛾
• 𝐴 𝜙 < 0.01,
A ∈ [0.031/1.1, 0.084/0.9] ∨ 𝜉 C ∈ [0.031/1.1, 0.084/0.9],• 𝜉 𝛾𝛾𝛾𝛾
as per the nominal selection. The following events are considered as possible single-vertex backgrounds.
• Dilepton production events analyzed in the dilepton analysis [66], 𝑝 𝑝 → 𝑝(𝛾𝛾 → ℓ + ℓ − ) 𝑝 (∗) . The
EL and SD processes are generated separately. 100,000 events are generated for each channel. For
the latter case, the matching efficiency of the side to which the dissociative proton directs is set as 0.
Likewise, it is assumed that the DD contribution is negligible.

85

6 Background Modeling– 𝑝 𝑝 → 𝑝(𝛾𝛾 → 𝑒 + 𝑒 − ) 𝑝 (∗) . The electron to photon misidentification rate is assumed to be
2.5% for |𝜂| < 0.6, 5.5% for |𝜂| < 1.37, and 16% for the other 𝜂 region considering Ref. [136].
The product of the rates of the two particles is used as the event weight.
– 𝑝 𝑝 → 𝑝(𝛾𝛾 → 𝜇+ 𝜇 − ) 𝑝 (∗) . Since the misidentification rate should be much lower than the
dielectron case, this kind of event is neglected.
– 𝑝 𝑝 → 𝑝(𝛾𝛾 → 𝜏 + 𝜏 − ) 𝑝 (∗) . The tauon-to-photon misidentification rate is calculated as the
product of tauon to neutral pion (𝜋 0 ) decay branching ratio, 𝜋 0 -to-photon misidentification
rate, and photon isolation efficiency for 𝜋 0 . The product of the rates of the two particles isused as the event weight.
• SM light-by-light scattering, 𝑝 𝑝 → 𝑝(𝛾𝛾 → 𝛾𝛾) 𝑝 (∗) . 100,000 events are generated. Since the SD
process of this kind of event is not implemented in this generator, SD contribution is just assumed
to be the same as EL. Therefore, the event weight is 2.
• Central exclusive production (CEP). In this research, the CEP is defined as gluon-initiated events
without proton dissociation. The detailed definition of CEP is in Ref. [112]. Since the SD processof this kind of event is not implemented in this generator, SD contribution is just assumed to be thesame as EL. Therefore, the event weight is 2.
– 𝑝 𝑝 → 𝑝(𝑔𝑔 → 𝛾𝛾) 𝑝 (∗) . 100,000 events are generated. There is no additional event weight.
Ref. [23] supports that this event is negligible.
– 𝑝 𝑝 → 𝑝(𝑔𝑔 → 𝑞 𝑞)
¯ 𝑝 (∗) . 50000 events are generated. The quark-to-photon misidentification
rate, photon identification efficiency multiplied by isolation efficiency, is assumed to be 0.02%.
Thus, the event weight is 4 × 10−8 .
– 𝑝 𝑝 → 𝑝(𝑔𝑔 → 𝑔𝑔) 𝑝 (∗) . 50000 events are generated. The same misidentification rate as thequark case is assumed. The event weight is 4 × 10−8 .
Fig. 6.10 shows the 𝑚 𝛾𝛾 stack distribution of the above single-vertex background events. Its ratio comparedto the combinatorial background estimated with the nominal background sample is shown in Fig. 6.11.
Since the contribution is less than the 1% of the combinatorial background, any of these single-vertex
backgrounds are neglected.
Test with Pythia
Pythia 8.306 is used to generate the usual pileup events. Although they do not have intact protons intheir final state, the partons could be hadronized into protons. The generated events are as follows.
• SD pileup with diphoton final state, 𝑞 𝑞¯ → 𝛾𝛾 or 𝑔𝑔 → 𝛾𝛾. 100,000 events are generated. The
nominal selection is applied to the photons and protons. Only 0.17 events remain after the event
selection, including the matching, indicating this kind of event and the double dissociative diffraction
assumed to be rare is negligible.
• SD pileup with dijet final state, 𝑔𝑔 → 𝑗 𝑗. Two photons or jets with the highest 𝑝 T are used todefine diphoton, where the jet is selected probabilistically, assuming the product of jet-to-photon
misidentification rate and photon isolation efficiency is 0.01. After generating 5 billion events, no
event remained after the nominal selection. The upper limit of the number of events after the event
selection is 0.74, calculated simply as the number of events at 14.6 fb−1 inferred from the generation
cross section divided by 5 billion. This is negligible, and the DD pileup assumed to be rare is also
considered negligible.

86Events / 29.4 [GeV]

6.2 Validation

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CEP gg→qq
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Figure 6.10: 𝑚 𝛾𝛾 stack distribution of the single-vertex background events. Each kind of event is normalized to14.6 fb−1 . The legends are in ascending order in terms of the weighted integral. “SD A(C)-side” means A(C)-side
proton is dissociative.

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Figure 6.11: 𝑚 𝛾𝛾 stack distribution of the single-vertex background events divided by the combinatorial background
distribution taken from the nominal background sample. The legends are in ascending order in terms of the weighted
integral. “SD A(C)-side” means A(C)-side proton is dissociative.

87

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−0.02 −0.015 −0.01 −0.005

00.005 0.01 0.015 0.02
∆effξA0−0.02 −0.015 −0.01 −0.005(a) A-side

00.005 0.01 0.015 0.02
∆effξC(b) C-side

Figure 6.12: Δeff 𝜉 distribution for the signal samples for ALPs with mass 400 GeV and coupling constant 𝑓 −1 =0.05 TeV−1 after the acoplanarity selection.

• ND pileup with dijet production. The procedure after the generation is the same as the SD → 𝑗 𝑗case. After generating 50 billion events, no event survives the nominal selection. The upper limitof the number of events after the event selection is 9.33. This number is large, but it is neglectedconsidering that this kind of background does not produce any resonance-like peaks in the 𝑚 𝛾𝛾
distribution and that the number of events contributes only a few % compared to the combinatorial
background.
Table B.1 shows each process’s configurations in Pythia.

6.3 Background template systematic uncertainty
If there are signal events in the data, the fully data-driven background sample is contaminated by such events.
This section evaluates the contributions of such events to the background template. The effect of signal
contamination is estimated using the nominal signal MC samples with coupling constant 𝑓 −1 = 0.05 TeV−1 ,which is regarded to have been excluded by the high-mass diphoton resonance search [52] (see Sec. 1.3.1).
The proton sets in the signal samples are replaced by proton sets randomly extracted from the nominal data
event-by-event to reproduce the data-driven proton sets reassignment in the nominal background sample
creation method. These samples are called mixed signal samples. Fig. 6.12 compares Δeff 𝜉 distribution
after the acoplanarity selection for the 400 GeV signal samples with and without the replacement foreach side. When the replacement decorrelates diphotons and protons, the peak around Δeff 𝜉 = −0.003 is
flattened out. This leads to much fewer selected events after the matching selection, Δeff 𝜉 < 0.004, and is
why the nominal background sample creation method suppresses the signal contamination.
However, a small number of signal events can contaminate the background sample even after the matching
selection. Fig. 6.13 compares diphoton mass distribution after the whole selection for the 400 GeV signal
samples with and without the proton sets replacement. A small resonant peak remains even with thereplacement and after the matching selection. A merge of the mixed signal sample with the nominal
background sample is created for each ALP mass. Fig. 6.14 shows the 𝑚 𝛾𝛾 distributions of such merged
samples with the nominal distribution for comparison.

88Events / 2.0 [GeV]

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Figure 6.13: Diphoton mass distribution for the signal samples for ALPs with mass 400 GeV and coupling constant𝑓 −1 = 0.05 TeV−1 . The distributions with and without the proton set replacement after the whole event selection arecompared. The distribution just after the acoplanarity selection is illustrated as the sample before the event selection
using the proton information.

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800

1000 1200 1400 1600
Diphoton mass mγ γ [GeV]

Figure 6.14: Diphoton mass distributions of the samples made by combining the background and mixed signal
samples. The whole selection is applied. Each legend stands for the mass of the ALP, whose events are added to thebackground sample. The ALP coupling constant 𝑓 −1 = 0.05 TeV−1 is assumed.

89

6 Background Modeling

6.4 Background modeling uncertainty
In this section, the systematic uncertainty of the background model, Eq. 6.1, is evaluated using thebackground templates in Fig. 6.14. First, the templates are smoothed in Sec. 6.4.1 in order to remove thestatistical uncertainty and focus on the systematic uncertainty. Then, in Sec. 6.4.2, the inflexibility of the
background model is evaluated as the signal yield uncertainty originating from the background modeling,
assuming the smoothed lines are the true background distribution. The evaluated systematic uncertainty isthen modified considering the bias of this assumption in Sec. 6.4.3.

6.4.1 Functional Decomposition smoothing
The statistical uncertainty is desired to be removed when evaluating the systematic uncertainty of the
background modeling so as not to overestimate the latter. The data-driven Functional Decomposition (FD)method [143] is used to smooth the background distribution to remove the statistical uncertainty. The
FD method is originally implemented for resonance searches and generally takes multiple assumptions
about the resonance peak shapes. However, if no signal shape is assumed, the process becomes much
simpler and behaves as a background distribution smoother. The background template from the diphoton
mass 𝑚 0 = 130 GeV is input to the tool, and the FD method expresses the background distribution using a
combination of complete orthogonal functions,𝑓˜(𝑧) =∞
Õ𝑐 𝑛 𝐸˜ 𝑛 (𝑧),𝑛=0where 𝑐 𝑛 is the real coefficient, and𝑧= 𝑚 𝛾𝛾 − 𝑚 0  𝛼,
𝜆𝐸˜ 𝑛 (𝑧) =𝑛Õ𝑑 𝑛𝑚 𝐹˜𝑚 (𝑧),𝑚=0√𝐹˜𝑛 (𝑧) = 2𝑒 −𝑛𝑧 ,√𝑑 𝑛𝑚 = 𝑛(−1)


𝑛+𝑚

 𝑚−1
𝑛
2𝑚 Ö 𝑚 + 𝑖 Ö 𝑖 + 𝑚,
𝑛 + 𝑚 𝑖=1 𝑚 − 𝑖 𝑖=𝑚+1 𝑖 − 𝑚

𝐸˜ 𝑛 , 𝐸˜ 𝑚 = 𝛿 𝑛𝑚 ,∫ ∞˜𝑎,˜ 𝑏˜ =d𝑧 𝑎(𝑧)˜ 𝑏(𝑧).
0

The hyperparameters 𝜆 and 𝛼 are positive scalars. 𝜆 is the scale parameter similar to the bandwidth
in the usual kernel density estimation and has the same dimension as mass, while 𝛼 is dimensionless.
The exponential family is used for the bases because the simple exponential is an entropy-maximizing
distribution, i.e. most delocalized, and they are suitable for representing the smooth-falling background
distribution. Although the background is modelled in the mass range [150, 1600] GeV, the 𝑥 0 is set to
130 GeV with the 20 GeV margin because the FD results have an oscillating behaviour around the leftend. The coefficients {𝑑 𝑛𝑚 } for the orthogonalization can be calculated recurrently and used to derive
recurrence relations among 𝐸˜ 𝑛 , which are used for the computation of 𝐸˜ 𝑛 .
The input sample is processed as the sequence of 𝑚 𝛾𝛾 values in 𝑚 𝛾𝛾 ∈ [130, 1600] GeV, 𝑚 𝛾𝛾,𝑖 , and they
are converted to {𝑧 𝑖 }. The conventional approach to obtain the coefficients {𝑐 𝑛 } is the maximum-likelihood

90

6.4 Background modeling uncertainty
method, but the FD uses another method for better performance as follows. Each coefficient is computed
as
𝑀
1 Õ ˜
𝑐𝑛 =
𝐸 𝑛 (𝑧 𝑖 ) ,𝑀 𝑖=1
where 𝑀 is the number of events, and this is referred to as the decomposition of the sample. The expressionof the sample is truncated toNÕΩ̃(𝑧) =
𝑐 𝑛 𝐸˜ 𝑛 (𝑧),𝑛=0which corresponds to the distribution smoothing. The N , the degree to which the FD describes the input
sample in detail, is another hyperparameter. Too large N causes a large modeling bias, i.e. overfitting,while if N is too small, the variance will increase.
The hyperparameters 𝜆, 𝛼, and N are optimized so that
𝐷 KL 𝑓˜k Ω̃ + 𝐷 KL Ω̃k P̃
is minimized, where
𝐷 KL 𝑎k˜ 𝑏˜ ≡

∫ ∞




𝑎(𝑧)
˜
𝑎(𝑧)
˜ log
d𝑧
˜
𝑏(𝑧)
−∞

is the Kullback-Leibler (KL) divergence, which is a measure of the difference between the two distributions√
given as its arguments. The first term requires the FD result to express the sample well. P̃ = 2𝑒 −𝑧 is a
simple exponential function, so the second term suppresses the overfitting. The maximum allowed N is
set to 8 in this research. Finally, the obtained Ω̃(𝑧) is transformed to a function of 𝑚 𝛾𝛾 ,Ω(𝑚 𝛾𝛾 ) =

N
Õ
𝑛=0


𝑐𝑛


d𝑧
𝐸˜ 𝑛 (𝑧).
d𝑚 𝛾𝛾Fig. 6.15 shows this FD smoothing result of the nominal background sample, and Fig. 6.16 shows all the
FD results for the background templates in Fig. 6.14. The resulting hyperparameters for the nominal caseare 𝜆 = 106, 𝛼 = 0.445, and N = 6.

6.4.2 Spurious signal evaluation
The inflexibility of the background model, Eq. 6.1, is evaluated using the smoothed lines, assuming thenominal FD smoothing result is the true background distribution. This method inherits the previous
diphoton analyses [52, 129, 144]. The signal-contaminated cases are also tested to see the effect of thepotential bias of the fully data-driven background sample creation method due to the signal contamination.
Samples with a large number of events weighted to have the same integral as the smoothed lines, so-called
Asimov samples, are created from the lines one by one. A primitive MC method is used to generate thesamples. Each of randomly generated 𝑚 𝛾𝛾 values following a uniform distribution in the mass range[150, 1600] GeV is selected if Ω(𝑚 𝛾𝛾 ) is smaller than a value randomly generated following anotheruniform distribution in [0, Ω(150 GeV)]. The event generation is repeated until 20 million 𝑚 𝛾𝛾 values
are selected. The signal-plus-background (s+b) function,
𝑁 𝑋 𝑓 𝑋 (𝑚 𝛾𝛾 ; 𝜇CB , 𝜎CB , 𝑛R , 𝑛L , 𝛼R , 𝛼L ) + 𝑁b 𝑓b (𝑚 𝛾𝛾 ; 𝑎, 𝛼0 )(6.2)

is then fitted to the Asimov data for each hypothetical ALP mass 𝑚 𝑋 , where the signal function 𝑓 𝑋 andbackground function 𝑓b are defined in Eqs. 5.1 and 6.1, respectively. The 𝑁 𝑋 and 𝑁b are the signal andbackground yield, respectively. The signal function parameters are fixed to the values in Table 5.1 in order91

Events / 29.4 [GeV]

6 Background Modelings = 13 TeV, 14.6 fb-1

102

Background sample
FD smoothing

10
1

Ratio

10−1

2
1.5
1
0.5
0

200

400

600

800

1000 1200 1400 1600
Diphoton mass mγ γ [GeV]

Events / 29.4 [GeV]

Figure 6.15: Comparison of 𝑚 𝛾𝛾 distribution between the nominal background sample with statistical uncertainty(black) and the FD smoothing result (red).

Nominals = 13 TeV, 14.6 fb-1

200 GeV102

300 GeV
400 GeV
500 GeV
600 GeV10

650 GeV
700 GeV
800 GeV
900 GeV

11000 GeV
1200 GeV
1400 GeV
1600 GeV

−1

Ratio

10

2.5
2
1.5
1

200

400

600

800

1000 1200 1400 1600
Diphoton mass mγ γ [GeV]

Figure 6.16: FD smoothing results of the background templates for the nominal and the signal-contaminated cases.
Each legend stands for the mass of the ALP, whose events are added to the background sample. The ALP coupling
constant 𝑓 −1 = 0.05 TeV−1 is assumed.

92

6.4 Background modeling uncertainty
0.5

s = 13 TeV, 14.6 fb-1

2

1.5

NSS/ δS

NSS/δS2.5

s = 13 TeV, 14.6 fb-1

0.4
0.3

1

0.2

0.5

0.1

0

0

−0.5

−0.1

−1

−0.2

−1.5

−0.3

−2

−0.4

−2.5

200

400

600

800

1000 1200 1400 1600
mX [GeV]

−0.5

200

(a) Nominal background sample

400

600

800

1000 1200 1400 1600
mX [GeV](b) FD driven Asimov data

Figure 6.17: 𝑚 𝑋 dependency of SS relative to the statistical uncertainty determined using the s+b function and(a) the nominal background sample (b) the FD driven Asimov data.
Table 6.1: Spurious signal envelope parametrization.

Parameter

Value

𝑝0
𝑝1
𝑝2

0.019
1.2
−0.0054

to distinguish the sources of systematic uncertainties, while the background function parameters are set to
be variable. Since it is assumed that the FD smooth line is the true background distribution, the fitting
result of 𝑁 𝑋 should be exactly 0. However, it can actually be non-zero because the background distribution
shape model has only two free parameters. This incompleteness of the background model is expressed as
the non-zero 𝑁 𝑋 values, called spurious signal (SS) and denoted 𝑁SS , for each ALP mass hypothesis.
Fig. 6.17 (b) shows the SS evaluation results relative to its uncertainty 𝛿𝑆. Traditionally, |𝑁SS |/𝛿𝑆 < 50%
is set as the threshold for considering the background model to be flexible.3 The SS result for the nominal
background sample illustrated in Fig. 6.17 (a) does not pass the criterion because the statistical fluctuation
deteriorate the result. On the other hand, the result after the smoothing (b) is much better than (a) andpass the test. The unstable fitting behaviour of the negative 𝑁SS results does not cause any problem in thisresearch because the test statistics do not use such values as described in Sec. 8
An envelope of |𝑁SS | is taken as the bare background modeling systematic uncertainty as a function of𝑚 𝑋 . Fig. 6.18 shows the |𝑁SS | and |𝑁SS |/𝛿𝑆 for all the Asimov samples as well as the envelope, which isparametrized as a function of ALP mass. The parametrization is obtained from a fit of a function(6.3)

𝛿SS (𝑚 𝑋 ) = 𝑝 0 + 𝑝 1 𝑒 𝑝2 𝑚𝑋 ,to the local maxima, where the parameters are listed in Table 6.1.

6.4.3 Bias of Functional Decomposition
The systematic uncertainty evaluation with the SS test assumes that the smooth distribution created by
the FD method is identical to the true background distribution. However, obviously, this assumption
3 Since no assumption is made on the signal production cross section, there is no strict requirement on |𝑁

SS |/𝛿𝑆.

93

SS

|N |

6 Background Modeling

10

Nominals = 13 TeV, 14.6 fb-1

200 GeV1

300 GeV
400 GeV

10−1

500 GeV
600 GeV

10−2

650 GeV
700 GeV−3

10

800 GeV
900 GeV−41000 GeV

101200 GeV−51400 GeV

101600 GeV

NSS/ δS

200

400

600

800

1000 1200 1400 1600 1800 2000
mX [GeV](a) Absolute number of SS

0.5
0.4

Nominals = 13 TeV, 14.6 fb-1

200 GeV

0.3

300 GeV

0.2

400 GeV

0.1

600 GeV

500 GeV0

650 GeV−0.1

800 GeV−0.21000 GeV

700 GeV
900 GeV−0.31200 GeV−0.4

1600 GeV−0.5

1400 GeV

200

400

600

800

1000 1200 1400 1600 1800 2000
mX [GeV](b) Relative number of SS
Figure 6.18: (a) Absolute number of SS as functions of 𝑚 𝑋 evaluated for the Asimov samples. The thick red lineis the result of a fit of an analytical function to the local maxima of the graphs. (b) Number of SS relative to thestatistical uncertainty obtained from the Asimov samples.

94

6.4 Background modeling uncertainty
should not be completely correct. Therefore, the background modeling systematic uncertainty, Eq. 6.3,
is modified considering such bias evaluated using some toy MC samples. First, the following three 𝑚 𝛾𝛾
distribution functions are prepared in the range [130, 1600] GeV.
1. The background function, Eq. 6.1. The parameters are determined by fitting the function to thenominal data just after the acoplanarity selection.
2. The background function 1 multiplied by the 𝜉 𝛾𝛾 OR selection efficiency, Eq. 4.4. This has a
non-smooth shape illustrated in Fig. 4.19 and hence is an extreme case to be conservative in the bias
evaluation.
3. The background function 1 multiplied by the 𝜉 𝛾𝛾 and matching OR selection efficiency, Eq. 4.5.
This function is more realistic than the function 2.
Assuming that each of these distribution functions is the true background distribution, the difference in the
SS evaluation results between the cases where the distribution is smoothed and not smoothed by the FD
method is investigated. A similar approach is taken in Ref. [144].
Fig. 6.19 shows the procedure as well as the nominal SS evaluation procedure. For each of the three
distribution functions, there are two different paths as follows.
• The case without the FD smoothing is checked. An Asimov sample with 20 million events andintegral normalized to the nominal background sample in [150, 1600] GeV is created from theassumed distribution function. The generation method is the same as described in Sec. 6.4.2. The
SS is evaluated for the sample as per the nominal SS evaluation.
• The case with the FD smoothing is checked. 100 toy samples with the same numbers of eventsas the nominal background sample are generated in [130, 1600] GeV from the same distributionfunction with the same method as the Asimov sample creation. These toys are also normalized
to the nominal background sample in terms of the integral. The toy samples are then smoothedby the FD method, and an Asimov sample is created from each smooth line in [150, 1600] GeV.
Subsequently, the SS evaluation is performed for each Asimov sample. The average of the 100 SS
results is then calculated.
The results of the two evaluations are illustrated in order in Figs. 6.20 and 6.21.4 The SS result of thefunction 2 without the FD smoothing has a non-smoothed shape as expected, while such behaviour is not
visible for the FD smoothing case.
The two results are compared with each other for each initial distribution function. Fig. 6.22 shows thedifference between these results in terms of the number of SS relative to the statistical uncertainty. An
analytical function5Õ𝛿FD (𝑚 𝑋 ) =𝑝 𝑖 𝑚 𝑖𝑋
𝑖=0

is used as the envelope. The coefficients are listed in table 6.2. The absolute signal yield uncertainty
originating from the background modeling is then modified as
𝛿BG (𝑚 𝑋 ) = 𝛿SS (𝑚 𝑋 ) (1 + 𝛿FD (𝑚 𝑋 )) ,(6.4)

where 𝛿SS is taken from Eq. 6.3 and Table 6.1.

4 Some mass points are removed because the fit did not converge, but this does not cause any problem in this research as

long as the fit results for the nominal data converge since the background modeling uncertainty has nothing to do with local
convergence problems.

95

6 Background Modeling

Nominal background sample

Smoothing by Functional Decomposition
Distribution function
Smoothdistribution

Smooth distribution(×100)

Asimov sample generation in [150, 1600] GeV

Toy sample generation in [130, 1600] GeV (×100)

AsimovsampleAsimovsampleToy sample (×100)

Asimov sample(×100)

s+b fit for each hypothetical ALP masss+b fit result(× mass points)

s+b fit result(× mass points ×100)

s+b fit result(× mass points)

Merging

SS results
SS results

SS results(×100)
Averaging

Take difference
SS results

Figure 6.19: The procedures of the nominal SS evaluation (blue) and the FD smoothing bias evaluation (red).
Ellipses represent objects, and rectangles represent processes. The FD bias study procedure is repeated for threedifferent background functions.
Table 6.2: FD bias parameters.

Parameter
𝑝0
𝑝1
𝑝2
𝑝3
𝑝4
𝑝5

96

Value
0.460
−0.00185
5.64 × 10−06−7.05 × 10−09
3.81 × 10−12−7.54 × 10−16s = 13 TeV, 14.6 fb-1
Function 1
Function 2
Function 3

10

NSS/ δSSS|N |6.4 Background modeling uncertainty1
s = 13 TeV, 14.6 fb-1

0.8

Function 1
Function 2
Function 3

0.6
0.4

1

0.2
0

10−1

−0.2
−0.4

10−2

−0.6

10−3

−0.8
200

400

600

800

−1
200

1000 1200 1400 1600
mX [GeV](a) Absolute number of SS

400

600

800

1000 1200 1400 1600
mX [GeV](b) Relative number of SS

s = 13 TeV, 14.6 fb-1
Function 1
Function 2
Function 3

10

NSS/δSSS|N |Figure 6.20: SS evaluation results for the FD bias evaluation derived without the FD smoothing. (a) Absolute
number of SS as a function of 𝑚 𝑋 . (b) SS relative to the statistical uncertainty.
1
s = 13 TeV, 14.6 fb-1

0.8

Function 1
Function 2
Function 3

0.6
0.4

1

0.2
0

−1

10

−0.2
−0.4

10−2

−0.6

10−3

−0.8
200

400

600

800

−1
200

1000 1200 1400 1600
mX [GeV](a) Absolute number of SS

400

600

800

1000 1200 1400 1600
mX [GeV](b) Relative number of SS

SS

|/δS
|NFD - NnoFD
SS

Figure 6.21: SS evaluation results for the FD bias evaluation derived with the FD smoothing. (a) Absolute numberof SS as a function of 𝑚 𝑋 . (b) SS relative to the statistical uncertainty.

s = 13 TeV, 14.6 fb-1

0.45

Function 1
Function 2
Function 3

0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0

200

400

600

800

1000 1200 1400 1600
mX [GeV]Figure 6.22: FD smoothing bias defined as the difference in SS evaluation results between the with and without FD
smoothing cases in terms of the number of SS relative to the statistical uncertainties. The thick orange line is theenvelope.

97

7 Systematic Uncertainty
Various sources of systematic uncertainty in this research are considered, quantified, and summarized inthis section. They are categorized into photon object-related, proton object-related, fitting-related, andtheoretical uncertainty, and described in Sec. 7.1, 7.2, 7.3, and 7.4, respectively.
To evaluate the effect of each systematic uncertainty, basically, the value of a parameter that is supposed tohave uncertainty is shifted up and down by the amount of its uncertainty (referred to as shifting in thisresearch), and the signal MC samples are recreated. The relative difference in each of• mean of 𝑚 𝛾𝛾 distribution,• standard deviation of 𝑚 𝛾𝛾 distribution,• number of events normalized to 14.6 fb−1
from the nominal signal MC sample are evaluated for each ALP mass 𝑚 𝑋 after the whole selection.
In most cases, the maximum relative difference considering all the ALP mass points is taken as thesystematic uncertainty for the corresponding source. The uncertainty is propagated to 𝜇CB , 𝜎CB (seeEq. 5.1), and signal yield, respectively. Non-negligible contributions among all the systematic uncertainties
are summarized in Table 7.1 with the quantitative information, while negligible sources are not listed.
The following two uncertainties, which do not belong to any aforementioned category, are also considered.
• Integrated luminosity uncertainty 𝛿lumi . For the 2017 dataset, the uncertainty is 2.4% [82]. This isdirectly used as the relative uncertainty of signal yield.
• Proton beam energy uncertainty. The 6.5 TeV beam energy in 2017 has 0.1% uncertainty [145],
which is negligible.

7.1 Photon object-related uncertainty
The following sources are considered for the photon object-related systematic uncertainty.
• Photon energy scale uncertainty originating from the photon energy calibration described in Sec. 3.4.
After the shifting, the resulting distributions are compared with the nominal sample in Fig. 7.1 for𝑚 𝑋 = 1000 GeV. The mean values of the modified 𝑚 𝛾𝛾 distributions subtracted by 𝑚 𝑋 are plotted asa function of 𝑚 𝑋 and compared to those of the nominal 𝑚 𝛾𝛾 distributions in Fig. 7.2. The maximum
relative difference is regarded as the uncertainty of 𝜇CB .
• Photon energy resolution uncertainty originating from the photon energy calibration described inSec. 3.4. This primarily affects the width of 𝑚 𝛾𝛾 distribution. The shift is applied as the changeof the degree of photon kinematics smearing in the simulation. The resulting distribution after
the shifting is compared with the nominal distribution in Fig. 7.3. The standard deviations of thenominal and systematically varied 𝑚 𝛾𝛾 distributions are plotted as a function of 𝑚 𝑋 in Fig. 7.4. The
maximum relative difference is regarded as the uncertainty of 𝜎CB .
• Photon-related efficiency uncertainty. This primarily affects the signal yield. The MC-to-datacorrection scale factor of event weight is varied to consider this uncertainty.

99

7 Systematic Uncertainty

Table 7.1: Summary of the non-negligible systematic uncertainty considered in this research.

SymbolSystematic uncertainty description

Value𝒎𝜸𝜸 scale uncertainty
𝛿scalePhoton energy scale±0.1%

𝒎𝜸𝜸 resolution uncertainty
𝛿 𝜎scale𝛿res𝛿 𝜎PRW

Photon energy scale
Photon energy resolution
Pileup reweighting+1.6 %−2.1+14.1 %
−4.8
+2.1 %−1.4

Signal yield uncertainty
𝛿lumi
𝛿ID
𝛿iso
𝛿PRW
𝛿optics𝛿glob
𝛿 𝜀AFP𝛿Nclust𝛿smear𝛿0.1 𝜉𝛾𝛾
𝛿smodel𝛿BG

Integrated luminosity
Photon identification efficiency
Photon isolation efficiency
Pileup reweighting
Beam optics
AFP global alignment
Proton reconstruction efficiency
Required number of AFP clusters
AFP track smearing
Selection correction
Signal modeling linearity
Background modeling (absolute uncertainty)±2.4%+1.6 %−1.5±1.9%+2.7 %−2.6+0.8 %−3.4+10.0 %
−8.6
+3.0 %
−2.2
+0 %
−6.6
+2.1 %
−1.5
±2.2%
±2.1%
Eq. 6.4

Signal efficiency uncertainty
𝛿 𝜀EL
𝛿 𝜀SD
𝛿 𝜀DD

EL signal efficiency
SD signal efficiency
DD signal efficiency

Eq. 5.3
Eq. 5.4
Eq. 5.5Signal cross section uncertainty
𝛿 𝑆 2 EL𝛿 𝑆 2 SD𝛿 𝑆 2 DDEL signal soft survival factor
SD signal soft survival factor
DD signal soft survival factor

±2%
±10%
±50%

Signal model parameter uncertainty
𝛿Δ𝑚0,1
𝛿 𝜎CB 0,1
𝛿 𝛼L,R𝛿 𝑛L,R

100

Signal model parameter 𝑝 Δ𝑚0,1 (absolute uncertainty)
Signal model parameter 𝑝 𝜎CB 0,1 (absolute uncertainty)
Signal model parameter 𝑝 𝛼L,R (absolute uncertainty)
Signal model parameter 𝑝 𝑛L,R (absolute uncertainty)

Table 5.1
Table 5.1
Table 5.1
Table 5.1

Events / 2.0 [GeV]

7.2 Proton object-related uncertainty

0.4

s = 13 TeV, 14.6 fb-1

0.35

NominalSyst UP
Syst DOWN

0.3
0.25
0.2
0.15
0.1
0.05

0
900 920 940 960 980 1000 1020 1040 1060 1080 1100
Diphoton mass mγγ [GeV]

×10−3

3

Variation∆m [GeV]Figure 7.1: Invariant mass distributions of the 𝑚 𝑋 = 1000 GeV simulated signal after the selection obtained with
the nominal and systematically shifted photon energy scales.

-1

s = 13 TeV, 14.6 fb
NominalSyst UP
Syst DOWN

2

1.5
1

1
0.5
0

s = 13 TeV, 14.6 fb-1

0
−1

Syst UP
Syst DOWN

−0.5

−2

−1

−3
200

400

600

800

1000 1200 1400 1600
mX [GeV](a) 𝑚 𝛾𝛾 − 𝑚 𝑋

200

(b)

D

400

600

up/down

𝑚 𝛾𝛾

E

800

1000 1200 1400 1600
mX [GeV]− 𝑚 nominal𝛾𝛾

.

𝑚 nominal𝛾𝛾

Figure 7.2: (a) The mean of the 𝑚 𝛾𝛾 distribution subtracted by 𝑚 𝑋 as a function of 𝑚 𝑋 for the signal samples after
the whole selection for the nominal and systematically shifted photon energy scales. (b) The relative difference ofthe mean between the shifted sample and the nominal sample after the whole selection.

– Photon identification efficiency uncertainty 𝛿ID (see Sec. 3.4).
– Photon isolation efficiency uncertainty 𝛿iso (see Sec. 3.4).
– Photon trigger efficiency uncertainty [111] (see Sec. 3.1).
• Pileup reweighting described in Sec. 3.2. A shifting in the pileup reweighting of the simulationis performed to cover the difference between the predicted and measured inelastic 𝑝 𝑝 cross section [146]. This primarily affects the signal yield. The 𝜇 distribution derived from the data is shiftedby ±3% before reweighting the MC sample according to the distribution.

7.2 Proton object-related uncertainty
The proton object-related systematic uncertainty is described in this section. The position of the truth
proton is smeared to create an AFP track object in the nominal AFP simulation as described in Sec. 3.2.

101

Events / 2.0 [GeV]

7 Systematic Uncertainty

0.45
0.4
0.35

s = 13 TeV, 14.6 fb-1
NominalSyst UP
Syst DOWN

0.3
0.25
0.2
0.15
0.1
0.05
0
900 920 940 960 980 1000 1020 1040 1060 1080 1100
Diphoton mass mγ γ [GeV]30

Variation

Standard deviation

Figure 7.3: Invariant mass distributions of the 𝑚 𝑋 = 1000 GeV simulated signal after the selection obtained withnominal and systematically shifted photon energy resolutions.

s = 13 TeV, 14.6 fb-1

NominalSyst UP
Syst DOWN

25

0.2

Syst UP
Syst DOWN

0.15

20

0.1

15

0.05

10

0

5
0

s = 13 TeV, 14.6 fb-1

−0.05
200

400

600

800

1000 1200 1400 1600
mX [GeV](a) Standard deviation

200

400

600

800

1000 1200 1400 1600
mX [GeV](b) Relative difference

Figure 7.4: (a) The standard deviation of the 𝑚 𝛾𝛾 distribution as a function of 𝑚 𝑋 for the signal samples after thewhole selection for the nominal and systematically shifted photon energy resolutions. (b) The relative difference ofthe standard deviation between the shifted sample and the nominal sample after the whole selection.

102

7.2 Proton object-related uncertainty
However, in tests for some systematic uncertainty sources, the simulation starts by creating cluster objects(see Sec. 3.5) with similar smearing. In this section, such samples are referred to as the clustered MC
samples. The following systematic uncertainty sources are considered.
• Beam optics and beam angle uncertainty in the proton transport simulation (see Sec. 3.5). Fig. 7.5
shows the Δeff 𝜉 distributions of the nominal and systematically shifted signal MC samples for the1000 GeV ALP. Despite the significant shift, most signal events are selected because of the correction
term described in Sec. 4.4. Fig. 7.7 (a) shows that the beam optics systematic uncertainties are
below 5% for all the mass points.
• AFP alignment uncertainty.
– AFP global alignment uncertainty is 300 µm as described in Sec. 2.3.1. Fig. 7.6 shows the Δeff 𝜉
distributions of the nominal and systematically shifted signal MC samples for the 1000 GeV
ALP. Although the dominant source of the 𝜉AFP uncertainty described in Sec. 3.5 is the beam
optics uncertainty, the global alignment uncertainty becomes the dominant source of the signal
yield uncertainty, after applying the correction term in the event selection (see Sec. 4.4) asshown in Fig. 7.7. Such a difference in behaviour from the beam optics uncertainty is due tothe correlation between the A-side and C-side, which can be found by comparing Figs. 4.11
A
C
and C.1. The global alignment variation shifts 𝜉AFP
and 𝜉AFP
towards the same direction,changing the selection efficiency drastically at some particular ALP mass points.
– AFP local alignment uncertainty (see Sec. 2.3.1).
– Roman pot rotation uncertainty. The pots are rotated about the 𝑧-axis by 4 mrad to evaluate
the uncertainty.
• Proton reconstruction efficiency uncertainty.
– The nominal efficiency is 92 ± 2% as described in Sec. 3.5.
– Potential efficiency deterioration by showering from signal protons. The tag-and-probe measurement of the nominal proton reconstruction efficiency might not be fully sensitive to such
effects. Therefore, the following three configurations in the proton reconstruction process areconservatively tightened, and the decrease in the number of events is regarded as the showering
effect.
∗ The allowed transverse distance between AFP SiT clusters to reconstruct an AFP track isdecreased from the default of 500 µm to 400 µm. The clustered MC samples are used.
∗ The allowed transverse distance between tracks in the Near and Far stations to reconstructa proton is tightened from the default of 2 mm to 1 mm.
∗ The minimum required number of clusters to reconstruct an AFP track is increased fromthe default of 2 to 3 for the data sample to quantify the decrease of proton reconstruction
efficiency as detailed in Appendix F. The derived value is used as the proton reconstruction
efficiency and applied for the shift as per the nominal efficiency uncertainty case.
• AFP track smearing uncertainty. A Gaussian width 𝜎𝑥 = 10 µm was used for smearing the truth
proton position to reconstruct a track. This reflects the spacial resolution of the AFP 3D-Si sensors.
However, the dilepton analysis [66] found that 𝜎𝑥 = 50 µm width provided reasonable modeling of
the signal for the selection criteria 𝜉AFP ∈ [0.035, 0.08] and |Δ𝜉 | < 0.005. Thus, the configurationis changed to 𝜎𝑥 = 50 µm to consider this as a source of the systematic uncertainty.1 Fig. 7.8 showsthat most signal events pass the matching selection Δeff 𝜉 < 0.004 even after the shifting.
• Pileup proton overlay uncertainty. The nominal signal MC samples do not have pileup proton objects
even at the truth level because generating pileup events with reasonable uncertainty is difficult.
1 This is because the origin of the protons is the same as the dilepton analysis in that they are photon-induced. Although

the true 𝜉 distribution is different from the dilepton analysis, the 𝜎𝑥 = 50 µm is used because this systematic uncertainty is not
dominant even if the variation doubles.

103

0.3

s = 13 TeV, 14.6 fb-1
NominalSyst UP
Syst DOWN

0.25

Events [A.U.]

Events [A.U.]

7 Systematic Uncertainty
0.3

0.2

0.15

0.15

0.1

0.1

0.05

0.05
−0.005

0

0.005

0
−0.01

0.01
∆effξA

NominalSyst UP
Syst DOWN

0.25

0.2

0
−0.01

s = 13 TeV, 14.6 fb-1

−0.005(a) A-side

0

0.005

0.01
∆effξC(b) C-side

0.16

s = 13 TeV, 14.6 fb-1

0.14

NominalSyst UP
Syst DOWN

0.12

Events [A.U.]

Events [A.U.]

Figure 7.5: Δeff 𝜉 distributions of the 𝑚 𝑋 = 1000 GeV simulated signal before the selection obtained with nominaland systematically varied beam optics configuration.
0.16

0.12
0.1

0.08

0.08

0.06

0.06

0.04

0.04

0.02

0.02
−0.00500.005(a) A-side

0.01∆effξA

NominalSyst UP
Syst DOWN

0.14

0.1

0
−0.01

s = 13 TeV, 14.6 fb-1

0
−0.01

−0.005

0

0.005

0.01
∆effξC(b) C-side

Figure 7.6: Δeff 𝜉 distributions of the 𝑚 𝑋 = 1000 GeV simulated signal before the selection obtained with nominaland systematically varied global alignment.

Therefore, the reconstruction probability of protons that should have failed to be reconstructed
due to the existence of pileup might not be fully considered in the nominal proton reconstruction
efficiency estimation. To consider this possibility, pileup effects on clustering and tracking results
are taken from the data, and the cluster and track objects are added to the nominal signal sample
and subsequently used in proton reconstruction to see the effect.
• Clustered MC uncertainty. The results of the nominal and clustered MC samples are compared.
• Selection correction uncertainty 𝛿0.1 𝜉𝛾𝛾 . The potential failure of uncertainty evaluation due to thedifference of 𝜉 𝛾𝛾 and 𝜉AFP selection thresholds is mentioned in Sec. 4.4 and evaluated in Sec. D.

7.3 Fitting-related uncertainty
This section describes the systematic uncertainty originating from the function fittings.
• Signal model parameter uncertainty (see Table 5.1).
• Signal modeling linearity uncertainty 𝛿smodel . This is described in Sec. 5.1.1.

104

0.03

s = 13 TeV, 14.6 fb-1

0.02

Syst UP
Syst DOWN

0.01
0Variation

Variation

7.3 Fitting-related uncertaintys = 13 TeV, 14.6 fb-1

0.15

Syst UP
Syst DOWN

0.1
0.05

−0.01
−0.02

0

−0.03

−0.05

−0.04
−0.05

−0.1
200

400

600

800

1000 1200 1400 1600
mX [GeV]

200(a) Beam optics uncertainty

400

600

800

1000 1200 1400 1600
mX [GeV](b) Global alignment uncertainty

0.18
s = 13 TeV, 14.6 fb-1

0.16

Nominal
Syst

0.14
0.12

Events [A.U.]

Events [A.U.]

Figure 7.7: The relative difference in the number of signal events between the shifted and nominal signal samples
after the whole selection. The upper and lower (a) beam optics and (b) global alignment systematic variations areapplied to the signal samples.

0.18

s = 13 TeV, 14.6 fb-1

0.16

Nominal
Syst

0.14
0.12

0.1

0.1

0.08

0.08

0.06

0.06

0.04

0.04

0.02

0.02

0
−0.01

−0.0050(a) A-side

0.005

0.01
∆effξA0−0.01−0.005

0

0.005

0.01
∆effξC(b) C-side

Figure 7.8: Δeff 𝜉 distributions of the 𝑚 𝑋 = 1000 GeV simulated signal before the selection obtained with nominaland systematically varied track smearing width in the AFP simulation.

• Background modeling uncertainty 𝛿BG . Eq. 6.4 is used as the absolute signal yield uncertainty
originating from background modeling.
• The potential bias of the s+b fitting is evaluated with a signal injection test, as shown for the casewithout backgrounds in Sec. 5. Asimov samples are generated following the s+b function in Eq. 6.2.
In contrast to Sec. 6.4.2, the Asimov sample has only one event for each fine mass point, every 2 GeV,
but is weighted to follow the distribution function. The signal and background parameters for thegeneration are set as determined in Sec. 5 and 6.1, except for 𝑁 𝑋 , which is varied to test the linearityof the fit result. The signal function parameters 𝜇CB , 𝜎CB , 𝑛L,R , and 𝛼L,R are fixed to distinguish thesources of systematic uncertainties, while the other parameters are set to be variable when fitting.
The signal strength 𝜇 defined in Eq. 5.6 is compared between the input and out values in Fig. 7.9 foreach hypothetical ALP mass. The maximum relative difference from 1 is only 𝛿sbfit = 0.006. Thus,
this kind of fitting uncertainty is neglected.

105

s = 13 TeV, 14.6 fb-1
Ideal

1.4

s = 13 TeV, 14.6 fb-1
Ideal

1.4

Observed

1.2

1.6

1

1.6

Observed

Observed

1

1

0.8

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0
0

0
0

s = 13 TeV, 14.6 fb-1
Ideal

1.4

Observed

1.2
1

0.8

(b) ALP mass 300 GeV
1.6

s = 13 TeV, 14.6 fb-1
Ideal

1.4

0
0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength
Output signal strength(a) ALP mass 200 GeV

Output signal strength

Output signal strength

0.6

1.6

Observed

1

1

0.8

0.8
0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

s = 13 TeV, 14.6 fb
Ideal(e) ALP mass 600 GeV

1.6

-1

s = 13 TeV, 14.6 fb
Ideal

1.4

Observed

1.2

0.8

1

0.8

1

0.8
0.6

0.4

0.4

0.4

0.2

0.2

0.2

0
0

s = 13 TeV, 14.6 fb
Ideal
Observed

1.2(h) ALP mass 800 GeV
1.6
1.4

-1

s = 13 TeV, 14.6 fb
Ideal
Observed

1.2

1

0.8

0
0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength

1.6
1.4

Ideal
Observed

1

1

0.8

0.6

0.6

0.6

0.4

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0
0

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(j) ALP mass 1000 GeV

s = 13 TeV, 14.6 fb-1

1.2

0.8

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength(i) ALP mass 900 GeV
Output signal strength

1.4

-1

Output signal strength

Output signal strength(g) ALP mass 700 GeV
1.6

Ideal
Observed

0.6

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strengths = 13 TeV, 14.6 fb-1

1.2

0.6

0
0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength(f) ALP mass 650 GeV

1.6
1.4

Observed

1.2

1

0
0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength

Output signal strength

-1

1.4

0
0

Output signal strength

Output signal strength(d) ALP mass 500 GeV

1.6

Ideal

1.2

1.2

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strengths = 13 TeV, 14.6 fb-1

1.4

Observed

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength(c) ALP mass 400 GeV

1.6

0.6

0
0

Ideal

1.2

0.8

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strengths = 13 TeV, 14.6 fb-1

1.4

1.2

0.8

Output signal strength

1.6

Output signal strength

Output signal strength

7 Systematic Uncertainty

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength(k) ALP mass 1200 GeV

00

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Input signal strength(i) ALP mass 1400 GeV

Figure 7.9: The s+b signal injection test results. The input and output signal strengths are compared for eachhypothetical ALP mass. The error bars represent the statistical error for 14.6 fb−1 .

106

7.4 Theoretical uncertainty

7.4 Theoretical uncertainty
The dominant source of theoretical uncertainty of the ALP signal events is the uncertainty of the soft
survival factor 𝑆 2 described in Sec. 3.2, estimated to be 2%, 10%, and 50% for the EL, SD, and DD signal
cross sections, respectively, and the other theoretical uncertainties, such as experimental uncertainty due
to structure function and higher order corrections, are found to be negligible compared to the soft survival
factor, concluded referring to Ref. [147]. The small uncertainty of EL 𝑆 2 is due to its large 𝑝 𝑝 impact
parameter, resulting in small model variation of the soft QCD interaction [148]. The uncertainty of SD
and DD 𝑆 2 are estimated in the evaluation of the short-distance component of 𝑆 2 [149].

107

8 Statistical Procedure and Results
In this section, a statistical model is built, and the search results and exclusion regions for the signal eventsare presented through statistical tests. First, the statistical model, including the systematic uncertainty
effect, is constructed in Sec. 8.1 based on the maximum likelihood method. The effects of the systematic
uncertainty on the s+b fitting results are evaluated in Sec. 8.2. Then, statistical tests are performed using
dedicated test statistics. The search results and the 95% exclusion limits in the 𝑚 𝑋 - 𝑓 −1 plane are describedin Sec. 8.3 with the test methods. The 441 data events after the whole selection in 𝑚 𝛾𝛾 ∈ [150, 1600] GeV,
which is compatible with the prediction from the background sample, 429 ± 8 events, are used for thetests.

8.1 Statistical model
The bare s+b function in Eq. 6.2 does not consider the effect of systematic uncertainty. Another statisticalmodel is constructed in this section to incorporate systematic uncertainty. The s+b function is rewritten
as




F 𝑚 𝛾𝛾 ; 𝝈 𝑋 , 𝑚 𝑋 , 𝑁b , 𝒂, 𝜽 = 𝑓 𝑋 𝑚 𝛾𝛾 ; 𝒙 𝑋 (𝑚 𝑋 , 𝜽 CB ) · 𝑁 𝑋 𝝈 𝑋 ; 𝑚 𝑋 , 𝜽 𝑁𝑋 + 𝑓b 𝑚 𝛾𝛾 ; 𝒂 · 𝑁b ,(8.1)

where 𝑓 𝑋 , 𝑓b , 𝑁 𝑋 , and 𝑁b stand for the probability density function (PDF) and yield as concisely used in
Eq. 6.2, and 𝒂 = {𝑎, 𝛼0 } denotes the background shape parameters. 𝜽 = 𝜽 CB , 𝜽 𝑁𝑋 is a set of nuisanceparameters for systematic uncertainties, where𝜽 CB = 𝜃 𝜇CB 0 , 𝜃 𝜇CB 1 , 𝜃 𝜎CB 0 , 𝜃 𝜎CB 1 , 𝜃 𝛼L , 𝜃 𝛼R , 𝜃 𝑛L , 𝜃 𝑛R , 𝜃 scale , 𝜽 𝜎CB ,𝜽 𝜎CB = 𝜃 𝑘 | 𝑘 ∈ 𝑆 𝜎CB ,𝑆 𝜎CB = {𝜎scale , res, 𝜎PRW , }affect the signal model parameters, and𝜽 𝑁𝑋 = {𝜽 1 , 𝜽 2 , 𝜃 BG } ,𝜽 𝑖 = {𝜃 𝑘 | 𝑘 ∈ 𝑆𝑖 } ,𝑆1 = lumi, ID, iso, PRW, optics, glob, 𝜀 AFP , Nclust, smear, 0.1𝜉 𝛾𝛾 , smodel ,𝑆2 = 𝜀EL, 𝜀SD, 𝜀DD, 𝑆 2 EL, 𝑆 2 SD, 𝑆 2 DD
affect signal yield.
The signal model 𝑓 𝑋 is the DSCB function (Eq. 5.1) as discussed in Sec. 5. The systematic uncertainty

109

8 Statistical Procedure and Results
𝜽 CB of its parameters 𝒙 𝑋 are considered in the fitting,1

𝒙 𝑋 (𝑚 𝑋 , 𝜽 CB ) = 𝜇CB (𝑚 𝑋 , 𝜃 𝜇CB 0,1 ), 𝜎CB (𝑚 𝑋 , 𝜃 𝜎CB 0,1 ), 𝛼L,R (𝜃 𝛼L,R ), 𝑛L,R (𝜃 𝑛L,R ) ,𝜇CB (𝑚 𝑋 , 𝜃 𝜇CB 0,1 ) = 𝑚 𝑋 + 𝑝 Δ𝑚0 + 𝛿Δ𝑚0 𝜃 𝜇CB 0 + 𝑝 Δ𝑚1 + 𝛿Δ𝑚1 𝜃 𝜇CB 1 𝑚 𝑋 𝐾scale (𝜃 scale ) ,

 Ö


𝜎CB (𝑚 𝑋 , 𝜃 𝜎CB 0,1 ) = 𝑝 𝜎CB 0 + 𝛿 𝜎CB 0 𝜃 𝜎CB 0 + 𝑝 𝜎CB 1 + 𝛿 𝜎CB 1 𝜃 𝜎CB 1 𝑚 𝑋
𝐾 𝑘 (𝜃 𝑘 ),𝑘 ∈𝑆 𝜎CB

𝛼L,R (𝜃 𝛼L,R ) = 𝑝 𝛼L,R + 𝛿 𝛼L,R 𝜃 𝛼L,R ,𝑛L,R (𝜃 𝑛L,R ) = 𝑝 𝑛L,R + 𝛿 𝑛L,R 𝜃 𝑛L,R ,
where


up
𝐾 𝑘 (𝜃 𝑘 ) = 𝐾 𝜃 𝑘 ; 𝛿 𝑘 , 𝛿down
,
𝑘

 h 
i 𝜃
𝐾 𝜃; 𝛿up , 𝛿down = 𝑟 𝜃; 𝛿up , 𝛿down,(
1 + 𝛿up ,if 𝜃 > 0𝑟 𝜃; 𝛿up , 𝛿down =.
down
1/(1 − 𝛿), if 𝜃 < 0(8.2)

The subscripts up and down stand for the upper and lower uncertainty. 𝐾 denotes a function characterizing
the effect of the systematic uncertainties, which are generally asymmetric. The expression of 𝑟 is anapproximation ensuring that the modifications to the signal event yield for 𝜃 = ±1 correspond to the ±1𝜎variations used to define the uncertainties. The actual expression for 𝑟 is interpolated smoothly between
the cases 𝜃 > 0 and 𝜃 < 0 using HistFactory [150] to avoid numerical problems at 𝜃 = 0.
The signal yieldÖ
𝑁 𝑋 𝝈 𝑋 ; 𝑚 𝑋 , 𝜽 𝑁𝑋 = 𝐿 int 𝜎ˆ 𝑋 (𝝈 𝑋 ; 𝑚 𝑋 , 𝜽 2 )𝐾 𝑘 (𝜃 𝑘 ) + 𝛿BG (𝑚 𝑋 )𝜃 BG

(8.3)

𝑘 ∈𝑆1

is characterized by signal cross sections
𝝈 𝑋 = {𝜎EL , 𝜎SD , 𝜎DD } ,Õ𝜎ˆ 𝑋 (𝝈 𝑋 ; 𝑚 𝑋 , 𝜽 2 ) =𝜎𝑖 (𝑚 𝑋 )𝜀 𝑖 (𝑚 𝑋 )𝐾 𝜀𝑖 (𝑚 𝑋 , 𝜃 𝜀𝑖 )𝐾𝑆 2 𝑖 (𝜃 𝑆 2 𝑖 ) ,𝑖 ∈ {EL,SD,DD}𝑖
𝜎𝑖 (𝑚 𝑋 ) = 𝜇𝜎std(𝑚 𝑋 ),

where



up
𝐾 𝑘 (𝑚 𝑋 , 𝜃 𝑘 ) = 𝐾 𝜃 𝑘 ; 𝛿 𝑘 (𝑚 𝑋 ), 𝛿down
(𝑚
)
𝑋 ,
𝑘

𝑖 is described in Sec. 5.2.2.2
𝜇 is signal strength, and 𝜎std

The background model 𝑓b is the function of Eq. 6.1 as discussed in Sec. 6. Its parameters 𝒂 = {𝑎, 𝛼0 } andbackground yield 𝑁b are all treated as free parameters in the fit.
The overall likelihood, including extended and Gaussian constraint terms, is given by
"𝑀
#
Ö
Ö



2
𝐿 𝜇, 𝝂; 𝑚 𝑋 , 𝑚 𝛾𝛾,𝑖 = 𝑒 − ( 𝑁𝑋 +𝑁b )F 𝑚 𝛾𝛾,𝑖 ; 𝝈 𝑋 (𝜇), 𝑚 𝑋 , 𝑁b , 𝒂, 𝜽𝑒 − 𝜗 /2 ,
𝑖=1

(8.4)

𝜗 ∈𝜽

1 The correlation among the signal parameters is ignored because their systematic uncertainty has a negligible effect on the

results as described in Sec. 8.2.
2 The correlation among the soft survival factors is found to be negligible. The test was performed using an additional
constraint, 𝜃 𝑆 2 EL = 𝜃 𝑆 2 SD = 𝜃 𝑆 2 DD , in the evaluation of the systematic uncertainty effects in Sec. 8.2. The difference in the
result from the nominal evaluation was negligible, so it was concluded that the presence or absence of the correlation does notaffect the results.

110

8.2 Effects of systematic uncertainty
where 𝝂 ≡ {𝑁b , 𝒂, 𝜽 }, 𝑀 is the number of events in the dataset, and 𝑚 𝛾𝛾,𝑖 is the diphoton mass for a given
event 𝑖. Unbinned fitting is performed for the statistical tests.

8.2 Effects of systematic uncertainty
The effect of each systematic uncertainty on the signal strength is evaluated using the background sample
with a similar size as the data represented in Fig. 6.4 (a). It can validate the method’s implementationby comparing the results with the relative qualitative impact of each source of systematic uncertainty
expected from the statistical model in Sec 8.1 and the systematic uncertainty in Table 7.1. First, the s+b
fit using the likelihood in Eq. 8.4 is performed with all the parameters not fixed, called unconditional fit.
The set of the resulting value and error is referred to as pull for each nuisance parameter 𝜗 ∈ 𝜽. Then, thefollowing procedure is repeated for each 𝜗.
Starting from the resulting parameters of the unconditional fit, the s+b fit is performed again,
fixing 𝜗 to the upper edge of the error bar of the pull and floating other parameters. The sameis performed for the lower edge of the error bar. The differences in the signal strength 𝜇 of thetwo fits from the unconditional fit Δ𝜇 are regarded as the effect of the systematic uncertainty
corresponding to 𝜗.
Fig. 8.1 shows the pulls (lower panel) and effects (upper panel) of each nuisance parameter distinguished by
the horizontal axis. The results are obtained by the fits setting the hypothetical ALP mass as 𝑚 𝑋 = 550 GeV
and 300 GeV. The systematic uncertainties are ranked according to the effects, and the source of thesystematic uncertainty corresponding to each rank is shown in Table 8.1 and Table 8.2, respectively.
The two mass points are in the mass ranges with locally concentrated events and no excess of events,
respectively. Fig. 8.2 compares the 𝑚 𝛾𝛾 distribution of the sample and the background-only fit (fixing
𝜇 = 0) result. The dominant systematic uncertainty for 𝑚 𝑋 = 550 GeV is the AFP alignment systematics.
At the 300 GeV part without event excess, the rank of background modeling uncertainty, which is absolute
uncertainty (see Eq. 8.3), is increased, and the rank of the other signal yield uncertainties is decreased, as
expected. Asymmetric behaviours for some nuisance parameters such as the second bin in Fig. 8.1 (a) are
due to the asymmetric input systematic values listed in Table 7.1. Some systematic uncertainties show thesame sign of effects for both the upper and lower shifting tests. Two cases can cause such behaviour. The
first case is that the input systematic uncertainty is entirely one-sided, in which the interpolation algorithmat 𝜃 = 0 in Eq. 8.2 the factor causes such a specific behaviour. The second case is that the systematic
uncertainty is a signal modeling parameter uncertainty corresponding to 𝜽 CB . In this case, there is no needfor the effects to have different signs.
In many other analyses, the background distribution in the blinded kinematic region (see Sec. 4.4), referred
to as signal region (SR), is estimated using background MC samples, so the difference from the data ismeasured in another region close to the SR, referred to as control region (CR), and reflected by adding
a constant term to 𝜗 in Eq. 8.4. This is often expressed as “the auxiliary measurements constrain thenuisance parameters”. In such cases, the pull information is used to quantify the level of the constraints.
However, since no CR is used in this analysis because the background distribution is determined in a fully
data-driven way, there is no such CR-originated constraint. The pulls in Fig. 8.1 are almost 0 ± 1 for all thenuisance parameters, so it is confirmed before unblinding the SR that there is no significant constraint.
For 𝑚 𝑋 = 550 GeV, the conditional fit result is 𝜇 = 0.683 ± 0.348. On the other hand, if all thenuisance parameters are fixed, i.e. the bare function Eq. 6.2 is used, the result becomes 0.685 ± 0.337.
This√ error corresponds to the effect of statistical uncertainty, and the systematic uncertainty effect is0.3482 − 0.3372 ≈ 0.087. Thus, the statistical uncertainty dominates.

111

0.06

∆µ

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8 Statistical Procedure and Results

s = 13 TeV, 14.6 fb-1
mX = 550 GeV
Pseudo-data background sample

0.04
0.02

0.03
0.01
0

0

−0.01

−0.02
−0.06

-1σ

−0.02

5

10

15

20

1σ
-1σ

−0.03

Pull

1σ

Pull

−0.04

2
1
0
−1
−2

s = 13 TeV, 14.6 fb-1
mX = 300 GeV
Pseudo-data background sample

0.02

25
30
Systematic rank(a) 𝑚 𝑋 = 550 GeV

2
1
0
−1
−2

5

10

15

20

25
30
Systematic rank(b) 𝑚 𝑋 = 300 GeV

Events / 29.00 [GeV]

Figure 8.1: The ranking plots created with hypothetical ALP mass (a) 550 GeV corresponding toSystematicthe massrankrangewith events locally concentrated, and (b) 300 GeV corresponding to the range with almost no event, created using
the background sample represented in Fig. 6.4 (a). The nuisance parameters’ pulls and effects are displayed indescending order rank corresponding to the horizontal axis starting from 1. The source of the systematic uncertainty
corresponding to each rank is listed in Table 8.1 for (a) and Table 8.2 for (b). A red (blue) bar shows the effectevaluated by fixing the nuisance parameter to the upper (lower) edge of the error bar of the pull.

102

s = 13 TeV, 14.6 fb-1
χ2/DOF = 0.505

10Background-only fit

1

10−1

Ratio

10−2

2
1
0

200

400

600

800

1000 1200 1400 1600
Diphoton mass mγ γ [GeV]

Figure 8.2: Diphoton mass distribution of the pseudo-data background sample represented in Fig. 6.4 (a) and its
background-only fit result.

112

8.2 Effects of systematic uncertainty

Table 8.1: Ranking of the effects of systematic uncertainty on the signal strength for hypothetical ALP mass 550 GeV.
The pseudo-data background sample represented in Fig. 6.4 (a) is used.

Rank

Nuisance parameter

Systematic uncertainty source

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30

𝜃 glob
𝜃 Nclust
𝜃 𝑆 2 SD
𝜃 optics
𝜃 𝜀AFP
𝜃 PRW
𝜃 lumi𝜃 𝜀EL
𝜃 0.1 𝜉𝛾𝛾
𝜃 smodel𝜃 smear
𝜃 𝜀SD
𝜃 iso
𝜃 BG
𝜃 ID
𝜃 𝑆 2 EL𝜃 res
𝜃 𝑛R
𝜃 𝑆 2 DD
𝜃 𝜀DD
𝜃 scale𝜃 𝛼R
𝜃 𝜎scale
𝜃 𝑛L
𝜃 𝛼L
𝜃 𝜎CB 0𝜃 𝜎PRW𝜃 𝜎CB 1
𝜃 Δ𝑚0
𝜃 Δ𝑚1

AFP global alignment
Required number of AFP clusters
SD signal soft survival factor
Beam optics
Proton reconstruction efficiency
Pileup reweighting
Integrated luminosity
EL signal efficiency
Selection correction
Signal modeling linearity
AFP track smearing
SD signal efficiency
Photon isolation efficiency
Background modeling (absolute uncertainty)
Photon identification efficiency
EL signal soft survival factor
Photon energy resolution
Signal model parameter 𝑝 𝑛R (absolute uncertainty)
DD signal soft survival factor
DD signal efficiency
Photon energy scale
Signal model parameter 𝑝 𝛼R (absolute uncertainty)
Photon energy scale
Signal model parameter 𝑝 𝑛L (absolute uncertainty)
Signal model parameter 𝑝 𝛼L (absolute uncertainty)
Signal model parameter 𝑝 𝜎CB 0 (absolute uncertainty)
Pileup reweighting
Signal model parameter 𝑝 𝜎CB 1 (absolute uncertainty)
Signal model parameter 𝑝 Δ𝑚0 (absolute uncertainty)
Signal model parameter 𝑝 Δ𝑚1 (absolute uncertainty)

113

8 Statistical Procedure and Results

Table 8.2: Ranking of the effects of systematic uncertainty on the signal strength for hypothetical ALP mass 300 GeV.
The pseudo-data background sample represented in Fig. 6.4 (a) is used.

114

Rank

Nuisance parameter

Systematic uncertainty source

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30

𝜃 scale
𝜃 BG
𝜃 res
𝜃 𝑛R
𝜃 Δ𝑚0
𝜃 glob
𝜃 Δ𝑚1
𝜃 𝜎CB 0
𝜃 𝜎scale
𝜃 𝛼R
𝜃 Nclust
𝜃 𝑆 2 SD𝜃 𝜀AFP𝜃 𝜎PRW
𝜃 𝜀EL
𝜃 PRW
𝜃 𝜎CB 1
𝜃 lumi𝜃 0.1 𝜉𝛾𝛾
𝜃 smear𝜃 smodel
𝜃 iso
𝜃 𝜀SD
𝜃 ID
𝜃 𝛼L
𝜃 𝑆 2 EL
𝜃 𝑛L
𝜃 optics
𝜃 𝑆 2 DD𝜃 𝜀DD

Photon energy scale
Background modeling (absolute uncertainty)
Photon energy resolution
Signal model parameter 𝑝 𝑛R (absolute uncertainty)
Signal model parameter 𝑝 Δ𝑚0 (absolute uncertainty)
AFP global alignment
Signal model parameter 𝑝 Δ𝑚1 (absolute uncertainty)
Signal model parameter 𝑝 𝜎CB 0 (absolute uncertainty)
Photon energy scale
Signal model parameter 𝑝 𝛼R (absolute uncertainty)
Required number of AFP clusters
SD signal soft survival factor
Proton reconstruction efficiency
Pileup reweighting
EL signal efficiency
Pileup reweighting
Signal model parameter 𝑝 𝜎CB 1 (absolute uncertainty)
Integrated luminosity
Selection correction
AFP track smearing
Signal modeling linearity
Photon isolation efficiency
SD signal efficiency
Photon identification efficiency
Signal model parameter 𝑝 𝛼L (absolute uncertainty)
EL signal soft survival factor
Signal model parameter 𝑝 𝑛L (absolute uncertainty)
Beam optics
DD signal soft survival factor
DD signal efficiency

s = 13 TeV, 14.6 fb-1
χ2/DOF = 1.11

102

Background-only fit

10

1.5

s = 13 TeV, 14.6 fb-1

1
0.5

1
0

10−1

Ratio

µ

Events / 29.00 [GeV]

8.3 Statistical tests and results

−0.5
−1

2
1
0

200

−1.5

400

600

800

1000 1200 1400 1600
Diphoton mass mγ γ [GeV]

200

400

(a) Background-only fit

600

800

1000 1200 1400 1600
mX [GeV](b) Signal strength

Figure 8.3: Fit results to the unblinded data. (a) Background-only fit result with the data distribution. (b) Unconditional fit results 𝜇ˆ as a function of 𝑚 𝑋 .

8.3 Statistical tests and results
In this section, statistical tests are performed using the unblinded data to search for a BSM resonance andset the exclusion limits using the likelihood in Eq. 8.4, abbreviated as𝐿(𝜇, 𝝂) ≡ 𝐿 𝜇, 𝝂; 𝑚 𝑋 , 𝑚 𝛾𝛾,𝑖 ,for each hypothetical ALP mass 𝑚 𝑋 . The signal strength 𝜇 is called the parameter of interest (POI) to
distinguish it from the other parameters 𝝂 and is used to define the hypothesis. To define the test statistics,
the profile likelihood ratio [151] for the hypothetical signal strength 𝜇hypo is defined as
𝐿 𝜇hypo , 𝝂ˆˆ (𝜇hypo )𝜆(𝜇hypo ) =,𝐿 ( 𝜇,ˆ 𝝂ˆ )
where 𝜇ˆ and 𝝂ˆ are the maximum-likelihood estimators (MLEs) of the unconditional fit, and 𝝂ˆˆ (𝜇hypo ) is
the MLEs for the conditional fit fixing the signal strength as 𝜇hypo . Fig. 8.3 shows the 𝑚 𝛾𝛾 distribution of
the data, in comparison to its background-only fit result (fixing 𝜇 = 0), and the resulting signal strength of
the unconditional fit for each 𝑚 𝑋 .3
The test statistics used in Sec. 8.3.1 and 8.3.2 are based on a statistic
𝑡 𝜇hypo = −2 ln 𝜆(𝜇hypo ).
It follows the 𝜒2 distribution if the true signal strength 𝜇true = 𝜇hypo or noncentral 𝜒2 distribution if
𝜇true ≠ 𝜇hypo . This leads to the use of so-called asymptotic formulae [151] instead of toy MC samples
with a large number of events to obtain the distributions of the test statistics. Thus, the test statistics
based on 𝑡 𝜇hypo have been conventionally used in the ATLAS collaboration. However, in this research, theasymptotic formulae are not used for the nominal results because it is not valid for the number of data
3 The fit result becomes unstable in a higher mass range, especially above 1000 GeV. This is because there are local mass

ranges with no event. The fits are judged to be converged if events with extremely small weights are added every 1 GeV to
the sample. However, the fit results are still unstable because the fit result converges not on a differentiable minimum but on
a non-smooth point due to the artificial modification of the likelihood to prevent negative PDF. Still, this means the unstablebehaviour is seen when 𝜇ˆ < 0, and there is no problem because the test statistic does not use the unconditional fit result when𝜇ˆ < 0.

115

p0

Significance

8 Statistical Procedure and Results

10−1

s = 13 TeV, 14.6 fb-1

2.5
2
1.5
1
0.5

10−2

s = 13 TeV, 14.6 fb-1

200

400

600

800

1000 1200 1400
mX [GeV]0200(a) 𝑝-value

400

600

800

1000 1200 1400
mX [GeV](b) SignificanceFigure 8.4: Test results for null hypothesis 𝜇 = 0 using the unblinded data as functions of 𝑚 𝑋 . (a) 𝑝-value 𝑝 0 .
(b) Significance 𝑍0 .

events corresponding to 14.6 fb−1 , although the results using the asymptotic formulae are in Appendix G
for comparison. Instead, a primitive method is used in which the toy MC samples normalized to 14.6 fb −1
are generated based on the statistical model described in Sec. 8.1 to create the distributions of the test
statistics assuming the null hypotheses.

8.3.1 SignificanceFor the search for a BSM resonance, the null hypothesis 𝜇hypo = 0 is used. The test statistic is defined as𝑡 0 𝜇ˆ ≥ 0
𝑞0 =.
0 𝜇ˆ < 0𝑞 0 is set as 0 when 𝜇ˆ < 0 because the negative signal strength is not physical when searching for a BSM
resonant contribution. 1000 toy MC samples were made to create the distribution following the PDF
𝑓 (𝑞 0 | 0), where 𝑓 (𝑞 0 | 𝜇true ) is the conditional PDF of 𝑞 0 under the condition 𝜇 = 𝜇true .
Then, the 𝑝-value 𝑝 0 and significance 𝑍0 are calculated as
∫ ∞

𝑝0 =
𝑓 𝑞 00 | 0 d𝑞 00 ,𝑞0
−1

𝑍0 = Φ

(1 − 𝑝 0 ) ,for each 𝑚 𝑋 corresponding to the toy MC samples, where
1𝑥
Φ(𝑥) ≡
1 + erf √2
2
is the cumulative distribution function of the normalized Gaussian distribution. Fig. 8.4 shows the results
as functions of hypothetical ALP mass 𝑚 𝑋 . No excess of significance from 3 is observed, meaning noevidence of a BSM resonance. The maximum observed significance is 2.51 at 𝑚 𝑋 = 454 GeV. Fig. G.1
shows the results using the asymptotic formulae for comparison. Fig. 8.5 shows the ranking plot createdby the same method as in Sec. 8.2, tested at 𝑚 𝑋 = 454 GeV.
The local significance of 2.51 is relatively large. Still, the look-elsewhere effect (LEE) [152, 153],
considering that many trials increase the probability of a low-probability event occurring, should also be116

8.3 Statistical tests and results

Table 8.3: Ranking of the effects of systematic uncertainty on the signal strength for hypothetical ALP mass 454 GeV.
The unblinded data is used.

Rank

Nuisance parameter

Systematic uncertainty source

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30

𝜃 glob
𝜃 Nclust
𝜃 𝑆 2 SD
𝜃 optics
𝜃 𝑛R
𝜃 𝜀EL
𝜃 PRW
𝜃 lumi𝜃 𝜀AFP
𝜃 BG
𝜃 0.1 𝜉𝛾𝛾
𝜃 smodel
𝜃 iso
𝜃 𝜀SD
𝜃 smear𝜃 ID
𝜃 scale𝜃 𝛼R𝜃 𝑆 2 EL𝜃 res
𝜃 𝑆 2 DD
𝜃 𝜀DD
𝜃 Δ𝑚0
𝜃 Δ𝑚1
𝜃 𝜎PRW𝜃 𝜎CB 0
𝜃 𝜎scale𝜃 𝜎CB 1
𝜃 𝛼L
𝜃 𝑛L

AFP global alignment
Required number of AFP clusters
SD signal soft survival factor
Beam optics
Signal model parameter 𝑝 𝑛R (absolute uncertainty)
EL signal efficiency
Pileup reweighting
Integrated luminosity
Proton reconstruction efficiency
Background modeling (absolute uncertainty)
Selection correction
Signal modeling linearity
Photon isolation efficiency
SD signal efficiency
AFP track smearing
Photon identification efficiency
Photon energy scale
Signal model parameter 𝑝 𝛼R (absolute uncertainty)
EL signal soft survival factor
Photon energy resolution
DD signal soft survival factor
DD signal efficiency
Signal model parameter 𝑝 Δ𝑚0 (absolute uncertainty)
Signal model parameter 𝑝 Δ𝑚1 (absolute uncertainty)
Pileup reweighting
Signal model parameter 𝑝 𝜎CB 0 (absolute uncertainty)
Photon energy scale
Signal model parameter 𝑝 𝜎CB 1 (absolute uncertainty)
Signal model parameter 𝑝 𝛼L (absolute uncertainty)
Signal model parameter 𝑝 𝑛L (absolute uncertainty)

117

∆µ

8 Statistical Procedure and Results
0.06

s = 13 TeV, 14.6 fb-1
mX = 454 GeV

0.04
0.02
0
−0.02

1σ

−0.04

Pull

−0.06
2
1
0
−1
−2

-1σ

5

10

15

20

25
30
Systematic rank

Figure 8.5: The ranking plot created with a hypothetical ALP mass of 454 GeV. The nuisance parameters’ pulls
and effects are displayed in descending order rank corresponding to the horizontal axis starting from 1. The sourceof the systematic uncertainty corresponding to each rank is listed in Table
8.3. A redrank(blue) bar shows the effect
Systematic
evaluated by fixing the nuisance parameter to the upper (lower) edge of the error bar of the pull.

large in this case. This is because the 𝑚 𝛾𝛾 resonance peak of the expected signal events is very narrowcompared to the mass range where the BSM resonance is searched, and many hypothetical 𝑚 𝑋 are tested.
The statistical significance of the maximum local deviation is evaluated as follows.4 The maximum 𝑞 0
with respect to 𝑚 𝑋 , 𝑞 0,max , is calculated for each toy MC sample and filled into a histogram to createthe
distribution following the PDF of 𝑞 0,max under the condition that 𝜇 = 0 for any 𝑚 𝑋 , 𝑓max 𝑞 0,max . Then,
using the observed 𝑞 0,max in the data, 𝑞 0,max = 6.03, the global 𝑝-value 𝑝 glob and global significance 𝑍glob
are calculated as
∫ ∞


0
0
𝑝 glob =𝑓 𝑞 0,max
d𝑞 0,max,
𝑞0,max−1

𝑍glob = Φ
1 − 𝑝 glob .

Fig. 8.6 shows the 𝑓max 𝑞 0,max and the observed 𝑞 0,max . The observed 𝑞 0,max is found to be smaller thanthe median of the distribution, so it is concluded that the data 𝑚 𝛾𝛾 distribution is consistent with thebackground-only hypothesis.

8.3.2 Exclusion limits
To make the exclusion limits in the 𝑚 𝑋 - 𝑓 −1 plane, null hypotheses,
𝜇hypo ∈ {0.10, 0.20, 0.40, 0.70, 1.00, 1.50, 2.00, 3.00, 4.00, 5.00, 6.20, 7.40, 8.70, 9.99} ,4 The procedure is the same as the one used in Ref. [154], although the process is documented only internally.

118

(8.5)

Sample / 0.40

8.3 Statistical tests and results
100
80
60
40
20
0
0

2

4

6

8

10

12

14

16

18 20
q0,max
Figure 8.6: The PDF 𝑓max 𝑞 0,max (red) and the observed 𝑞 0,max = 6.03 (blue). The 𝑝-value is 𝑝 glob = 0.56.

are tested using the test statistic



ˆ

𝐿
𝜇
,
𝝂
ˆ
(𝜇
)
hypo
hypo





−2 ln




ˆˆ (0)
𝐿0,𝝂
𝑞˜ 𝜇hypo =𝑡 𝜇hypo0𝜇ˆ < 0.
0 ≤ 𝜇ˆ ≤ 𝜇hypo
𝜇ˆ > 𝜇hypo

If 𝜇ˆ < 0, the unconditional fit is replaced by the conditional fit assuming 𝜇 = 0, while if 𝜇ˆ > 𝜇hypo , thetest statistic is set as 0 because this case is not worth considering in the context of rejecting the hypothesisthat there is a BSM contribution (𝜇hypo > 0). Each null hypothesis is tested for each hypothetical 𝑚 𝑋 .
To derive the observed exclusion limits, 1000 toy MC samples were made every 4 GeV of the hypothetical
ALP mass 𝑚 𝑋 to create a distribution following the PDF 𝑔( 𝑞˜ 𝜇hypo | 𝜇hypo ), where 𝑔( 𝑞˜ 𝜇hypo | 𝜇true ) is theconditional PDF of 𝑞˜ 𝜇hypo under the condition 𝜇 = 𝜇true . In addition, the same number of samples weremade with the same interval to create a distribution following the PDF 𝑔( 𝑞˜ 𝜇hypo | 0), which corresponds tothe alternative hypothesis. Then,
∫ ∞
CLs =

𝑞˜ 𝜇hypo
𝑔 𝑞˜ 0𝜇hypo | 𝜇hypo d𝑞˜ 0𝜇hypo∫ ∞𝑞˜ 𝜇hypo
𝑔 𝑞˜ 0𝜇hypo | 0 d𝑞˜ 0𝜇hypo

is used for the tests rather than the 𝑝-value to prevent non-physical results [155, 156]. The root of an
equation of 𝜇hypo , CLs = 0.05, is regarded as the observed 95% upper limits 𝜇95(obs.) of signal strength 𝜇.
The solid black line in Fig. 8.7 represents the results.
The observed limits are roughly within the 2𝜎 confidence intervals centered on the expected limits, which
are calculated only from the nominal statistical model. To derive the expected limits, as in the case of
the observed limits, 10000 toy samples were made every 40 GeV to create a distribution following the
PDF 𝑔( 𝑞˜ 𝜇hypo | 𝜇hypo ), and the same number of samples with the same interval were made to a distribution
following the PDF 𝑔( 𝑞˜ 𝜇hypo | 0). Then, the CLs value is calculated for each pair of the samples to create
the distribution of CLs with 10000 entries for each 𝜇hypo and 𝑚 𝑋 . The median, Φ(±1)-quantiles, and119

Signal strength

8 Statistical Procedure and Results

s = 13 TeV, 14.6 fb-1
Observed

10

Expected
Expected ± 1 σ
Expected ± 2 σ

1

200

400

600

800

1000 1200 1400
mX [GeV]Figure 8.7: The expected and observed upper limits of signal strength 𝜇 at 95% confidence level as functions ofhypothetical ALP mass 𝑚 𝑋 . The 1𝜎 and 2𝜎 confidence intervals are also illustrated by the colorized bands.

Φ(±2)-quantiles of the distribution are recorded, and they are denoted CLs , CL±1𝜎, and CL±2𝜎
. The
s
s
equations of 𝜇hypo ,CLs = 0.05,
CL∓1𝜎= 0.05,sCL∓2𝜎= 0.05,s±1𝜎±2𝜎
are solved to obtain 𝜇95(exp.) , 𝜇95(exp.), and 𝜇95(exp.), corresponding to the expected limit, its 1𝜎 confidence
interval, and 2𝜎 confidence interval, respectively. The dashed line and the colorized bands in Fig. 8.7
represent them.

A sharp downward swing around 200 GeV extends far beyond the 2𝜎 band, but this is due to the differencein the number of samples created for the observed and expected limits. If the same number of samples
is used, the relation between the band and observed limits looks more reasonable, as found comparing
Figs. 8.9 and A.5. The step of the lower edge of the 2𝜎 band around 650 GeV is caused by the small
number of 𝜇hypo points (see Eq. 8.5) and the imprecise interpolation of 𝜇hypo dependencies in CLs .
The upper limits 𝜇95 are converted to the limits of signal yield 𝑁 𝑋 and cross section 𝜎𝑋 as
𝑁95 = 𝜇95 𝑁std ,𝜎95 = 𝜇95 𝜎std ,where 𝑁std and 𝜎std are defined in Sec. 5.2.2, and
𝜎𝑋 =

Õ

𝜎𝑖 .

𝑖 ∈ {EL,SD,DD}

Fig. 8.8 shows these results. Furthermore, according to Eq. 1.4, the limits of the ALP coupling constant𝑓 −1 are calculated as
√
−1
−1
𝑓95
= 𝜇95 𝑓std,
−1 = 0.05 TeV −1 is the specified couplingassuming the ALP’s branching ratio to two photons is 100%. 𝑓std
constant when generating the signal MC samples. Fig. 8.9 shows the results. The ALP natural width

120

102

2

Cross section σX [fb]Signal yield

8.3 Statistical tests and results

s = 13 TeV, 14.6 fb-1

10

Observed
Expected
Expected ± 1 σ
Expected ± 2 σ

s = 13 TeV, 14.6 fb-1
Observed
Expected
Expected ± 1 σ
Expected ± 2 σ

10

10

1
200

400

600

800

1000 1200 1400
mX [GeV]200(a) Signal yield limits

400

600

800

1000 1200 1400
mX [GeV](b) Cross section limits

Coupling constant f-1 [TeV-1]

Figure 8.8: The expected and observed upper limits of (a) signal yield 𝑁 𝑋 and (b) signal cross section 𝜎𝑋 at 95%
confidence level as functions of hypothetical ALP mass 𝑚 𝑋 . The 1𝜎 and 2𝜎 confidence intervals are also illustrated
by the colorized bands.

s = 13 TeV, 14.6 fb-1
Observed
Expected

1

Expected ± 1 σΓ=Expected ± 2 σ

3G

eV
Γ=
2
Γ = GeV
1G
eV

10−1

200

400

600

800

1000 1200 1400
mX [GeV]Figure 8.9: The expected and observed upper limits of the ALP coupling constant 𝑓 −1 at 95% confidence level asfunctions of hypothetical ALP mass 𝑚 𝑋 . The 1𝜎 and 2𝜎 confidence intervals are also illustrated by the colorizedbands. Contours of the ALP natural width Γ expressed in Eq. 1.2 are illustrated by the smooth blue solid lines.

is Γ = 1 GeV for 𝑚 𝑋 ≈ 1400 GeV and Γ = 3 GeV for 𝑚 𝑋 ≈ 1600 GeV. Such widths are small enoughto be ignored because the detector resolution dominates them. The exclusion of a coupling constant𝑓 −1 much larger than the illustrated Γ contours is not justified in this research because the narrow-width
approximation in the signal modeling does not hold. Fig. G.2 and G.3 show the results using the asymptotic
formulae for comparison, although these are only for the expected limits.5

5 This is because 𝜎 cannot be calculated when the signal strength 𝜇ˆ is negative due to the instability of the unconditional fit
𝜇ˆ(see Appendix G).

121

9 Discussion and Conclusion
This work is one of the first analyses to correlate information from the forward proton detectors in termsof the kinematics with the central detector in collider experiments at the energy frontier, serving as ademonstration of the proton tagging for background rejection in a search for a BSM particle. In this section,
the results are compared with the previous research. Then, the prospect is described while showing thesignificance of this work. Finally, this research is concluded.

9.1 Comparison with previous research

Coupling constant f-1 [TeV-1]

The limits of the coupling constant in Fig. 8.9 can be compared with the CMS-TOTEM results [74] inFig. 1.9. Fig. 9.1 shows the comparison of the expected limits. The upper limits exceed the CMS results
for 𝑚 𝑋 < 700 GeV and 𝑚 𝑋 > 1100 GeV. One reason for this difference is the set operation in the event
selection. This research uses the OR selection, while the CMS-TOTEM uses the AND selection. The
signal efficiency of the analyses is qualitatively the same as Fig. 4.19, so the CMS-TOTEM analysis has
sensitivity in a limited mass range, while this analysis is sensitive in a wider range. In addition, this research
generally has more stringent limits considering the difference of the integrated luminosity, 14.6 fb−1 forthis research and 102.7 fb −1 for the CMS-TOTEM analysis. One reason for such higher sensitivity is that
this research performs the resonance search with unbinned fits making use of the narrowness of the 𝑚 𝛾𝛾
distribution, while the CMS-TOTEM just performs the counting experiment with a single bin.

s = 13 TeV, 14.6 fb-1

1

This research (14.6 fb-1)-1
CMS-TOTEM (102.7 fb )10−1

200 400 600 800 1000 1200 1400 1600 1800 2000
mX [GeV]Figure 9.1: The comparison of the expected exclusion limits of ALP coupling constant 𝑓 −1 from the CMS-TOTEM
analysis results [74].

123

9 Discussion and Conclusion

9.2 Prospect
9.2.1 Experimental setup√
The physics data-taking of the LHC has been restarted with 𝑠 = 13.6 TeV in 2022 (Run 3). The Run 3 isplanned to operate until 2025 to collect the data with up to 300 fb−1 , which is about 20 times higher than
the integrated luminosity used
√ in this research. This reduces the statistical uncertainty that was dominant
in this research to about 1/ 20 times. There is no significant change in the instantaneous luminosity, andthe luminosity leveling [157], an operation that dynamically adjusts the 𝛽-function during the run, hasbeen introduced. Thus, the combinatorial background rate is comparable to this research. Furthermore,high-luminosity LHC (HL-LHC) experiment is planned to take place using the LHC from 2029. In√
HL-LHC, the data-taking is expected to perform at 𝑠 = 14 TeV to collect up to 4000 fb−1 of integrated
luminosity with the instantaneous luminosity of 5 × 1034 cm2 s−1 .
Higher instantaneous luminosity means more pileup events and a concomitant increase in trigger rate.
However, the trigger rate is limited due to finite storage capacity. Therefore, the ATLAS collaboration
renovates the detector and trigger system in the Phase-I upgrade before Run 3 and Phase-II upgrade before
HL-LHC to cope with the high luminosity of HL-LHC. For example, the EM calorimeter has a new
trigger readout system in the Phase-I upgrade. The higher granularity of the cell division for the trigger
readout was realized to improve the object identification in the triggers. The author has been engaged in
the performance evaluation and calibration of this EM trigger readout system [158–160].
The AFP ToF detectors have been reinstalled to take the correct data this time. Furthermore, a plan toinstall new AFP SiT detectors in different positions from the SiTs used in Run 2 is being discussed for
HL-LHC. The CMS-TOTEM is also planning to increase the PPS stations.

9.2.2 Event selection
This research has a lot of room for optimization of the event selection conditions. Since this researchis the first search for new particles using AFP, the event selection was not aggressively optimized while
the establishment of the background estimation etc., has been focused. Especially, the correction 0.1𝜉 𝛾𝛾deteriorates the sensitivity. This was introduced considering the large beam optics uncertainty, but it
is expected to be overestimated. For example, the Δ𝜉 distribution of the dilepton analysis for 𝜉AFP ∈[0.02, 0.12] in Fig. 2.17 is as narrow as 0.002. But it does not realize unless 𝜉AFP = 0.02 for all protons, aslong as the beam optics uncertainty of 𝜉AFP is 10%. Thus, this uncertainty is just initial and preliminarily
and is being improved.
The acoplanarity selection can also be improved. Since the SD and DD signal samples were generated
very late, as described in Sec. 3.2, the selection optimization carries over to the subsequent analysis.
The OR selection has the advantage of covering a wide mass range than the AND selection, but the
AND selection contributes very strongly to background reduction and becomes valuable as the integrated
luminosity increases. Then, it is conceivable to combine the AND and XOR selection results, which aremutually exclusive, to increase the sensitivity compared to the OR selection. Moreover, it is also possible
to combine such results and the usual high-mass diphoton resonance search with results imposing the
NOR condition on 𝜉 𝛾𝛾 to the latter. An initial study was performed in Ref. [161], and this approach ispromising.
The primary vertex information can also be used for the event selection. For example, one can use theinformation about the number of high-energy tracks around the primary vertex or the tracks’ 𝜂 for the
event selection. Furthermore, as the primary vertex of an EL signal event where both two photons are

124

LB—JLLB JL — Draft November 3, 201813— Draft November 3, 201

9.2 Prospect

p

p
`

`˜

˜01
`˜

˜01
`

p

p

9.2: pair-production
Feynman diagramof8.theExclusiveevent→ 𝜒˜ 1 ℓ 𝜒˜ 1 ℓ in the𝑝 collisions[162].
FIG.
pair-production
of 𝑝W
boson
pairs
via
photon–photon fusion in the `⌫FIG. 8. Figure
Exclusiveof sleptons`˜ via𝛾𝛾photon–photon
fusion.
Each slepton
decaysdirectly to a lepton and neutralinostate.
˜01 .
0

0

unconverted cannot be selected correctly as described in Sec. 3.3, it could be efficient to use the primaryvertex reconstructed by the non-pointing or calo-vertexing mentioned in the same section can be used
for the dedicated selection in combination with the photon pointing. In addition, the vertex position
information from the AFP ToF detectors can also be used. To select events, one can compare the primaryvertex position with the ToF vertex information. But this is feasible basically only when using the AND
selection condition because the ToF vertexing needs protons on both A and C-sides.

9.2.3 Other analyses using the AFP detectors
AFP detectors are suited for analyses whose sensitivity has been limited by pileup and underlying events.
Possible dark matter search using the AFP detectors is described in Refs. [162, 163]. It targets mildly
compressed slepton 𝑙˜ and neutralino dark matter 𝜒˜ 10 . Fig. 9.2 shows the Feynman diagram of the signal
event. The primary strategy is to constrain the kinematics using the relation,2
𝑊miss= 𝑝 2miss ,𝑝 miss = 𝑝 𝛾1 + 𝑝 𝛾2 − 𝑝 ℓ1 − 𝑝 ℓ2 ,p
2𝑝 𝛾1 + 𝑝 𝛾2 = 𝑚 2𝛾𝛾 = 𝑠𝜉1 𝜉2 ,where 𝑊miss and 𝑝 miss are the mass and 4-momentum of the invisible 2 𝜒˜ 10 system, 𝑝 𝑥𝑖 is the 4-momentumof the 𝑖-th particle 𝑥, and the last relation uses Eq. 4.2. Ref. [162] expects good sensitivity with 100 fb −1of luminosity, but Ref. [163] expects it is barely sensitive even with 300 fb−1 . The former focuses on thephoton-initiated diboson production (𝛾𝛾 → 𝑊𝑊) [164] as the background, while the latter determines thedominant backgrounds to be the SD dilepton events (see Fig. 1.8) and pileup events. However, the AFP
ToF detectors are not considered in these evaluations. If the ToF is used, the ToF vertex is matched to theprimary vertex reconstructed by the ATLAS detector responses, and the pileup background is reduced
roughly by a factor of 10. This is the same as the matching selection in this research in that it requiresconsistency between the ATLAS detector information and the AFPs information. Thus, the multiple
assignment procedure for the background sample creation described in Sec. 6.1 could be applied to model
background. The difference from this research is that the matching is not performed for each side.
As exemplified above, there is a lot of possible application of the methodology developed in this research
to BSM searches using AFP. On the other hand, the AFP can also be used to identify the event type whenan excess from SM is observed in another BSM search not using the AFP. If the signal event is photon
initiated, the event can have associated forward protons and possibly fewer tracks around the primaryvertex. Therefore, information such as 𝜉AFP , ToF vertex position, and track multiplicity can be used to see
if the excess originates from photon-induced events.

125

9 Discussion and Conclusion

9.3 Conclusion
A diphoton resonance search has been performed using the ATLAS and AFP detectors with the 14.6 fb −1√
data of the LHC 𝑝 𝑝 collisions in 2017 at 𝑠 = 13 TeV. A diphoton trigger was used, and the photons
were required to have high transverse momentum and to be back-to-back in the 𝑥-𝑦 plane. In the signal
event, photons originating from the beam protons scatter with each other, mediated by an ALP. Thus, thisresearch is also a light-by-light scattering search. The protons remain intact or dissociate. Intact protons
that lose roughly 2% to 10% of their energy in the event are detected by AFP, and their fractional energy
loss 𝜉 are measured. The 𝜉 is measured from the diphoton system as well, and the difference between thetwo 𝜉 values is used for the event selection to reduce the background events.
The search was performed with statistical tests on the diphoton mass distributions of the samples. The
target mass range is [150, 1600] GeV. The signal and background PDFs are modelled to fit them into thesamples. The signal modeling was performed using the signal MC samples generated by SuperChic. The
DSCB function was used to model the resonant peak, and each parameter was determined with systematic
uncertainty by interpolating the fit results to the MC samples. The background modeling was performed
in a fully data-driven manner. The dominant source of background is the coincident matching between thediphoton object and protons from diffractive pileup events. The background sample was made from thedata reassigning the proton sets to diphoton from the other events. It is used to model the backgroundand evaluate the systematic uncertainty of the modeling. Background events other than the combinatorial
background are found to be negligible.
Various systematic uncertainties were studied. The AFP global alignment is the source of the dominantsystematic uncertainty of signal yield of a resonant peak. Most of the uncertainty in 𝜉 measured by AFP
is due to a lack of knowledge about the magnetic fields between the ATLAS detector and the AFPs, anda correction was made to the event selection conditions to suppress this effect. Statistical uncertainty is
finally found to be the dominant source of total uncertainty.
The results were expressed as the significance and exclusion limits. The statistical tests were performed
for each hypothetical ALP mass point 𝑚 𝑋 . The maximum observed local significance was 2.51 at
𝑚 𝑋 = 454 GeV, and there was no excess from the standard model. The exclusion limits were set for thesignal yield, signal cross section, and ALP coupling constant with two photons assuming the branchingratio is 100%. The observed upper limits of the coupling constant were about 0.04 TeV−1 in the mass range𝑚 𝑋 = 200 to 1000 GeV. The limits are compared with the previous research where both outgoing protons
are required to match with the diphoton kinematics. This research has more sensitivity in a wider range
than the previous research because requiring at least one proton matching extends the mass range withhigh acceptance. Moreover, the statistical modeling specialized in diphoton resonance search makes use of
the narrow-width of the resonant 𝑚 𝛾𝛾 peak and enhances the sensitivity. This research is the first analysis
specialized for BSM search using the AFP. There is a lot of possibility of optimizing event selection
criteria for future analysis with higher integrated luminosity and applying the methodology developed inthis research to other analyses.

126

A Auxiliary plots

Events / 0.01

Auxiliary plots are collected in this appendix. Detailed plots not covered in each section are shown below.
Each plot is referenced in each section.

8000

s = 13 TeV, 14.6 fb-1

7000
6000
5000
4000
3000
2000
1000
0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Diphoton ∆p /p
T

T1

1

Events / 0.001

Events / 0.001

Figure A.1: The distribution of 𝑝 T difference between the detector level two photons relative to the leading photon’s𝑝 T in the data before the acoplanarity selection.
s = 13 TeV, 14.6 fb-1
EL

0.8

SD
DD

0.6

0.2

s = 13 TeV, 14.6 fb-1

0.18

EL

0.16

SD

0.14

DD

0.12
0.1

0.4

0.08
0.06

0.2

0.04
0.02

0
0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Diphoton ∆p /pT(a) 𝑚 𝑋 = 300 GeV

T1

00.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Diphoton ∆p /pTT1(b) 𝑚 𝑋 = 1200 GeV

Figure A.2: The distribution of 𝑝 T difference between the detector level two photons relative to the leading photon’s𝑝 T in the signal MC samples before the acoplanarity selection.

127ξγ γ acceptance (OR) / A < 0.01

A Auxiliary plots2
1.8

s = 13 TeV, 14.6 fb-1

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

200

400

600

800

1000 1200 1400 1600
Diphoton mass mγ γ [GeV]

Figure A.3: 𝑚 𝛾𝛾 dependency of the 𝜉 𝛾𝛾 cut efficiency derived from the data for the OR condition dividing the 𝑚 𝛾𝛾
distribution after the 𝜉 𝛾𝛾 cut by the distribution just after the acoplanarity cut.

Matching (OR) / A < 0.01

Matching (OR) / ξγ γ acceptance (OR)0.35
s = 13 TeV, 14.6 fb-1

0.3

0.25
0.2

0.15
0.1

s = 13 TeV, 14.6 fb-1

0.25
0.2
0.15
0.1
0.05

0.05
0

0.3

200

400

600

800

1000 1200 1400 1600
Diphoton mass mγ γ [GeV](a) Matching efficiency

0

200

400

600

800

1000 1200 1400 1600
Diphoton mass mγ γ [GeV](b) Overall efficiency

Figure A.4: (a) 𝑚 𝛾𝛾 dependency of the matching selection efficiency derived from the data for the OR condition
dividing the 𝑚 𝛾𝛾 distribution after the matching by the distribution just after the 𝜉 𝛾𝛾 cut. (b) 𝑚 𝛾𝛾 dependency of theoverall selection efficiency derived from the data and the multi-shifted data for the OR condition dividing the 𝑚 𝛾𝛾
distribution after the matching by the distribution just after the acoplanarity cut.

128

Coupling constant f-1 [TeV-1]

s = 13 TeV, 14.6 fb-1

1

Observed
Expected
Expected ± 1 σ
Expected ± 2 σ

10−1

10−2
200

400

600

800

1000 1200 1400
mX [GeV]Figure A.5: The expected and observed upper limits of the ALP coupling constant 𝑓 −1 at 95% confidence level asfunctions of hypothetical ALP mass 𝑚 𝑋 . The 1𝜎 and 2𝜎 confidence intervals are also illustrated by the colorizedbands. The bands are made from 1000 toy MC samples generated for each 𝜇hypo and 𝑚 𝑋 .

129

B Auxiliary tables
Auxiliary tables are collected in this appendix. Detailed tables not covered in each section are shown
below. Each table is referenced in each section.

Table B.1: Pythia configuration for single-vertex background estimation.

Parameter
Beams:eCM
Diffraction:doHard
Diffraction:sampleType
PhaseSpace:pTHatMin
PhaseSpace:mHatMin
PromptPhoton:ffbar2gammagamma
PromptPhoton:gg2gammagamma
SigmaDiffractive:PomFlux
PDF:PomSet
HardQCD:all

SD → 𝛾𝛾
13000.
on
4
40.
130.
on
on

Process
SD → 𝑗 𝑗
13000.
on
4
40.
130.

ND → 𝑗 𝑗
13000.
on
4
40.
130.

1
4
on

1
4
on

Table B.2: Background efficiency model parameters.

𝑖=0
𝑖=1
𝑖=2
𝑖=3
𝑖=4

𝑝
295
−41.7
−8.76
4.54
−0.593𝑞
16.9
14.4
0.03472.92
0.00404131

C Systematic variations of 𝚫𝝃 distribution
The global alignment uncertainty affects the signal yield by shifting the Δ𝜉 distributions. Fig. C.1 shows
the Δ𝜉 ± distributions of the signal events where the global alignment configuration in the simulation isredefined as the nominal value added or subtracted by its uncertainty. The distributions represent thenominal case, upper-shifted case, and lower-shifted case.

0.35

s = 13 TeV, 14.6 fb-1

0.3

NominalSyst UP
Syst DOWN

0.25

Events [A.U.]

Events [A.U.]

The AFP track smearing uncertainty described in Sec 7.2 also affects the signal yield by changing thewidth of the Δ𝜉 distributions. Fig. C.2 shows the Δ𝜉 ± distributions of the signal events where the AFP
track smearing configuration in the simulation is redefined as the nominal value added or subtracted by its
uncertainty. The nominal case and the systematically variated case are compared.

0.35
0.3
0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0
−0.01

−0.005

0

0.005

0
−0.01

0.01
∆ξA

s = 13 TeV, 14.6 fb-1
NominalSyst UP
Syst DOWN−0.005(a) A-side

0

0.005

0.01
C
∆ξ

(b) C-side

0.35

s = 13 TeV, 14.6 fb-1

0.3

NominalSyst

Events [A.U.]

Events [A.U.]

Figure C.1: Δ𝜉 distributions of the 𝑚 𝑋 = 1000 GeV simulated signal obtained with nominal and systematicallyvaried global alignment parameters. The distributions are normalized by the integrals.

0.25

0.3
0.25

Nominal
Syst

0.2

0.2
0.15

0.15

0.1

0.1

0.05

0.05

0
−0.01

s = 13 TeV, 14.6 fb-1

−0.0050(a) A-side

0.005

0.01
∆ξA

0
−0.01

−0.005

0

0.005

0.01
C
∆ξ

(b) C-side

Figure C.2: Δ𝜉 distributions of the 𝑚 𝑋 = 1000 GeV simulated signal obtained with nominal and systematicallyvaried AFP track smearing configurations in the AFP simulation. The distributions are normalized by the integrals.

133

D Loose 𝝃 𝜸𝜸 cut effect evaluation
For the signal event simulation, the 𝜉AFP resolution for a given true 𝜉 originates from some modeling of
the AFP resolution and uncertainty. However, the tail of the distribution may not actually follow the modelprimarily constructed using Gaussian distributions. The 𝜉 𝛾𝛾 selection is set looser than the 𝜉AFP selection
as described in Sec. 4. This can cause the situation that the non-negligible number of events are selected
at the tail of 𝜉AFP distribution, considering the events where 𝜉 𝛾𝛾 is outside of the 𝜉AFP selection range but
inside the 𝜉 𝛾𝛾 selection range. Fig. D.1 shows such a situation. Such events at the tail of 𝜉AFP distribution
are possible not to be modelled well. That is, the evaluated systematic uncertainty could be incorrect for
the events. Therefore, additional systematic uncertainty is needed to take this situation into account.

D.1 Proton energy loss fraction resolution
Suppose the 𝜉 𝛾𝛾 resolution is much worse than the 𝜉AFP resolution. In that case, the additional systematic
uncertainty does not need to be set because the well-modelled photon kinematics uncertainty dominates
the Δ𝜉 uncertainty. The 𝜉 𝛾𝛾 and 𝜉AFP resolutions are investigated for each ALP mass. Fig. D.2 and D.3
shows the distributions of relative difference of reconstructed 𝜉 𝛾𝛾 and 𝜉AFP from the true values. The
𝜉AFP resolution is better than the 𝜉 𝛾𝛾 resolution for the nominal case. However, since the difference is nottremendous, the situation of interest cannot be naively ignored.

D.2 Threshold changing test
The signal MC samples for the nominal and systematically varied cases are used to compare the numberof events remaining when the entire selection is applied for each of the nominal 𝜉 𝛾𝛾 selection and the 𝜉 𝛾𝛾!(( selectoin!!"# selection
matching

matching

0.02

0.03
!$%&'

0.04

0.05

0.06

0.07

0.08

0.09 !!"#!$%&'

Figure D.1: 𝜉AFP distributions for events where 𝜉 𝛾𝛾 is outside of the 𝜉AFP selection range but inside the 𝜉 𝛾𝛾 selection
range. The green parts show the events which are selected by the matching criteria.

135

1
0.8

Events / 0.004

Events / 0.004

D Loose 𝜉 𝛾𝛾 cut effect evaluations = 13 TeV, 14.6 fb-1

A-sideC-side

0.6
0.4
0.2
0
−0.1−0.08−0.06−0.04−0.02 0(a)

reco − 𝜉 true
𝜉 𝛾𝛾
𝛾𝛾

0.02 0.04 0.06 0.08 0.1
reco
true
true
(ξγ γ - ξγ γ ) / ξγ γ

2.4
s = 13 TeV, 14.6 fb-1
2.2
A-side2
C-side
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−0.1−0.08−0.06−0.04−0.02 0

0.02 0.04 0.06 0.08 0.1
reco
true
true
(ξ
-ξ )/ ξ
AFP

(b)

true
/𝜉 𝛾𝛾

reco − 𝜉 true
𝜉AFP
AFP



AFP

AFP

true
/𝜉AFP

0.14

Events / 0.004

Events / 0.004

Figure D.2: Distributions of the relative difference of reconstructed (a) 𝜉 𝛾𝛾 and (b) 𝜉AFP from the truth values
evaluated using the 200 GeV ALP signal MC sample.
s = 13 TeV, 14.6 fb-1

A-side

0.12

0.3

0.25

C-side0.1

s = 13 TeV, 14.6 fb-1

A-sideC-side

0.2

0.08
0.15

0.06

0.1

0.04

0.05

0.02
−0.1−0.08−0.06−0.04−0.02 0(a)

reco − 𝜉 true
𝜉 𝛾𝛾
𝛾𝛾

0.02 0.04 0.06 0.08 0.1
reco
true
true
(ξγ γ - ξγ γ ) / ξγ γ



true
/𝜉 𝛾𝛾

0
−0.1−0.08−0.06−0.04−0.02 0

0.02 0.04 0.06 0.08 0.1
reco
true
true
(ξ
-ξ )/ ξAFP(b)reco − 𝜉 true
𝜉AFP
AFP



AFP

AFP

true
/𝜉AFP

Figure D.3: Distributions of the relative difference of reconstructed (a) 𝜉 𝛾𝛾 and (b) 𝜉AFP from the truth values
evaluated using the 1400 GeV ALP signal MC sample.

selection with the same condition as the 𝑥𝑖 AFP selection. The latter has fewer events left, and its selection
range [0.035, 0.08] is regarded as the safe range in which the uncertainty modeling is correct, as assumed
in the dilepton analysis [66]. The difference in the number of events between the two cases is regarded as
the event loss in case the 𝜉 𝛾𝛾 selection is tightened and interpreted as the number of events where the 𝜉AFP
may be mismodelled. For each ALP mass 𝑚 𝑋 , the event loss fractions are shown in Fig. D.4.
Only 2.2% of all events are possible to be mismodelled in terms of the 𝜉AFP distribution when considering
only the nominal case. The 2.2% events may not have a reliable systematic uncertainty. It can be naturally
assumed that it is conservative enough if the signal yield uncertainty of such events is evaluated as 100%.
That is, the nominal 𝜉 𝛾𝛾 selection described in Sec. 4.2 can be used, and 2.2% of the signal yield uncertainty
should be considered.
Although the event loss fractions of the samples with some systematic variations are larger than the nominal
case, they should not be considered as the uncertainty originating from the difference of the selection range
between the nominal 𝜉 𝛾𝛾 selection and 𝜉AFP selection. For example, when the systematic variation for the
track smearing configuration is applied, the 𝜉AFP distribution becomes more expansive, and the matchingrange tends to select a part of the distribution closer to its center. Therefore, even if the number of failed136

Event loss fraction

D.2 Threshold changing test

0.14
s = 13 TeV, 14.6 fb-1

0.12

NominalGlobal alignment (down)
Global alignment (down)
AFP track smearing
Beam optics (down)
Beam optics (up)RP rotation
Transverse distance Near vs Far
Number of clusters for tracking
Transverse distance of clusters
Pileup proton overlay
Simulation from clustering

0.1
0.08
0.06
0.04
0.02
0

200

400

600

800 1000 1200 1400 1600 1800 2000 2200
mX [GeV]Figure D.4: Event loss fractions when changing the 𝜉 𝛾𝛾 selection from the nominal one to the condition identical tothe 𝜉AFP selection. Samples with systematic variations (see Sec. 7) are also tested.

!(( selectoin!!"# selection
matching

matching

0.02

0.03
!$%&'

0.04

0.05

0.06

0.07

0.08

0.09 !!"#!$%&'

Figure D.5: 𝜉AFP distributions for events where 𝜉 𝛾𝛾 is outside of the 𝜉AFP selection range but inside the 𝜉 𝛾𝛾 selection
range. The green parts show the events which are selected by the matching criteria.

events increases by the selection threshold change, it only means that the number of events with better
𝜉AFP modeling increases. Fig. D.5 shows such a situation. In the other case, when the systematic variation
for AFP global alignment or the beam optics configuration is applied, the 𝜉AFP distribution is shifted. The
number of failed events increases only when the peak is systematically shifted towards the 𝜉AFP selection
range. Fig. D.6 shows such a situation. The core of the distribution is more likely to be selected than
the nominal problematic case, so even if the number of failed events increases by the selection thresholdchange, it only means that the number of events with better 𝜉AFP modeling increases.

137

D Loose 𝜉 𝛾𝛾 cut effect evaluation!(( selectoin!!"# selection
matching

matching

0.02

0.03
!$%&'

0.04

0.05

0.06

0.07

0.08

0.09 !!"#!$%&'

Figure D.6: 𝜉AFP distributions for events where 𝜉 𝛾𝛾 is outside of the 𝜉AFP selection range but inside the 𝜉 𝛾𝛾 selection
range. The green parts show the events which are selected by the matching criteria.

138

E Luminosity dependency of combinatorial
backgroundThe event selection efficiency after the diphoton selection depends on the number of interactions per BC 𝜇
as described in Sec. 6.1. In addition, the selection efficiency differs depending on diphoton kinematics
because the matching uses the diphoton information. The efficiency of the selection requiring only
A(C)-side matching is modelled as

𝜇
±
±
±
𝜀A,C (𝜇, 𝜉 𝛾𝛾
) = 1 − 1 − 𝜀 A,C (1, 𝜉 𝛾𝛾
) ≈ 𝜇𝜀 A,C (1, 𝜉 𝛾𝛾),± ) is the efficiency per 𝑝 𝑝 collision depending only on the diphoton kinematics. The
where 𝜀A,C (1, 𝜉 𝛾𝛾
matching efficiency is low enough for the approximation to hold. Using them, the OR selection efficiency
can be written as±±±±
𝜀 OR (𝜇, 𝜉 𝛾𝛾
) = 𝜀 A (𝜇, 𝜉 𝛾𝛾
) + 𝜀C (𝜇, 𝜉 𝛾𝛾
) − 𝜀 AND (𝜇, 𝜉 𝛾𝛾),± ) is the AND selection efficiency. If 𝜀±±
where 𝜀AND (𝜇, 𝜉 𝛾𝛾
AND (𝜇, 𝜉 𝛾𝛾 ) is negligible, 𝜀 OR (𝜇, 𝜉 𝛾𝛾 ) canbe regarded to be proportional to 𝜇. The event weight 𝑅 ≡ 𝜇 𝛾𝛾 /𝜇 𝑝 discussed in Sec. 6.1 uses thisapproximation.
± ) is convex upward and monoIn general, as 𝜇 increases, the proportionality breaks down. 𝜀 OR (𝜇, 𝜉 𝛾𝛾
tonically increases with respect to 𝜇. Using the fact that it converges to 1 when 𝜇 → ∞, the uncertaintyof the event weight is calculated. Fig. E.1 shows 𝜀 OR (𝜇) for a diphoton in case the efficiency is notproportional to 𝜇. 𝜇low and 𝜇high are min{𝜇 𝛾𝛾 , 𝜇 𝑝 } and max{𝜇 𝛾𝛾 , 𝜇 𝑝 }, respectively. The left (a) caseis that the efficiency saturates for both 𝜇 𝛾𝛾 and 𝜇 𝑝 . The lengths of blue and red lines, in this case, are𝛿/𝜇high and 𝛿/𝜇low , respectively, where 𝛿 = 𝜇high − 𝜇low . The more general case (b) is that the efficiency is
between the proportionality and saturation for 𝜇 𝛾𝛾 and 𝜇 𝑝 . The lengths of blue and red lines, in this case,are 𝜀 low 𝛿/𝜇high and 𝜀low 𝛿/𝜇low , respectively, where 𝜀 low = 𝜀OR (𝜇low ). Since 𝜀low < 1, the systematicdeviation is maximum in case of (a). Therefore, if 𝜇 𝑝 ≥ 𝜇 𝛾𝛾 , the event weight has

𝛿

=1−𝑅

𝜇high
of upper uncertainty. On the other hand, if 𝜇 𝑝 < 𝜇 𝛾𝛾 , the event weight has
𝛿

= 𝑅−1

𝜇low
of lower uncertainty. Thus, the event weight with the systematic uncertainty is 𝑅 +(1−𝑅)
for 𝑅 ≤ 1 and
−0
𝑅 +0
for
𝑅
>
1.
This
means
that
the
systematic
uncertainty
of
𝑅
can
be
conservatively
evaluated in− (𝑅−1)any case by comparing the cases with and without the event weight 𝑅 applied.
Fig. E.2 compares the background templates with only the upward and the downward shifts according
to these uncertainties with the nominal template. The uncertainty is about 0.5%. This uncertainty isneglected because the function fitting in this research is insensitive to the number of background eventsand non-local changes in the distribution shape.

139

E Luminosity dependency of combinatorial background

𝜀

𝜀

1

1

0

𝜇low 𝜇high0𝜇(a) Fast convergence𝜇low 𝜇high𝜇(b) Slow convergence

Events / 29.4 [GeV]

Figure E.1: 𝜀 OR (𝜇) for a diphoton. Two different ramps represent the badness of the assumption that the efficiency
is proportional to 𝜇.

102

s = 13 TeV, 14.6 fb-1

10

Nominal
Down
Up

1

10−1

Ratio

1.01
1

0.99
200

400

600

800

1000 1200 1400 1600
Diphoton mass mγ γ [GeV]

Figure E.2: The background templates with only the upward and the downward shifts considering the 𝜇 𝛾𝛾 /𝜇 𝑝
uncertainties compared with the nominal template.

140

F Variation of requirement on the number of AFP
clusters
In the proton object-related systematic uncertainty evaluation, changing the minimum required number ofclusters to reconstruct a track is found to have relatively significant effects on the signal yield. Thus, theeffect is evaluated from the data with the real clustering rather than the clusters obtained just by smearing
the truth proton positions in the clustered MC samples. However, the resulting difference between thenominal and changed cases is huge because one of the A-side AFP Far station planes was not working
permanently. Fig. F.1 shows the fraction of events with at least one reconstructed proton for each run
number1 in data for each side. The efficiency of the A-side is low when the change is applied. The test
requires at least three clusters to reconstruct a track, while the nominal case requires at least two. In
addition, proton reconstruction for each side requires tracks from both Near and Far stations for bothcases. Therefore, protons cannot be reconstructed on A-side unless clusters are in all three working planesof Far station. This is why the A-side efficiency is very low in this case.

s = 13 TeV, 14.6 fb-1
Nominal data

Data with variation

0.5
0Fraction of events with ≥1 proton

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1

Ratio

Ratio

Fraction of events with ≥1 proton

Thus, the relative change of the number of events between the nominal and changed cases after the wholeselection is used as the proton reconstruction efficiency uncertainty, where the final set operation in theevent selection requires only the C-side selection result. Fig. F.2 shows the cut flow of the changed data.
The decrease of events on the C-side is 6.6% of the original number of events in this research illustrated inFig. 4.16. This value is used as the systematic uncertainty of proton reconstruction efficiency.

330000 332000 334000 336000 338000 340000
Run number(a) A-side

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1

s = 13 TeV, 14.6 fb-1
Nominal data

Data with variation

0.5
0330000 332000 334000 336000 338000 340000
Run number(b) C-side

Figure F.1: Fraction of events with at least one reconstructed proton for each run number in data for (a) A-side and(b) C-side. The red points show the nominal data, and the blue points show the changed data.

1 Run number is the identifier of the LHC 𝑝 𝑝 collision run.

141

Events

F Variation of requirement on the number of AFP clusters

105
104
103
102
10
1

s = 13 TeV, 14.6 fb-1

A-sideC-sideAND
OR

Preselection

γγ

Aφ

ξγγ

ξ

AFP

Matching
Condtion

Figure F.2: Cut flow of the changed data after the preselection. The values for 𝜉 𝛾𝛾 and 𝜉AFP are the numbers ofevents when the set operation is performed without the latter selection.

142

G Results from asymptotic formulae
In this section, the results in Sec. 8.3 are replaced by the asymptotic formulae for comparison. The
formulae are derived and detailed in Ref. [151].
The significance and 𝑝-value are calculated as√
𝑍0 = 𝑞 0 ,𝑝 0 = 1 − Φ (𝑍0 ) ,by the asymptotic formulae. Fig. G.1 shows the results. The maximum observed significance is 2.53 at
𝑚 𝑋 = 456 GeV.
𝑁𝜎
The limits of the signal strengths 𝜇95(exp.)
are calculated as the root of the equation of 𝜇hypo ,𝜇hypo = 𝜎𝜇ˆ Φ−1 (1 − 0.05Φ(𝑁)) + 𝑁 ,p0

Significancepwhere 𝜎𝜇ˆ = 𝜇hypo / 𝑞˜ 𝜇hypo ,A , and 𝑞˜ 𝜇hypo ,A is the calculated 𝑞˜ 𝜇hypo from an Asimov sample generated
assuming 𝜇true = 0 [165]. This is converted to the limits of the signal strength, cross section, and coupling
constant, shown in Figs. G.2 and G.3.

10−1

s = 13 TeV, 14.6 fb-1

2.5
2
1.5
1
0.5

10−2

s = 13 TeV, 14.6 fb-1

200

400

600

800

1000 1200 1400
mX [GeV](a) 𝑝-value

0

200

400

600

800

1000 1200 1400
mX [GeV](b) SignificanceFigure G.1: Test results for null hypothesis 𝜇 = 0 using the unblinded data as functions 𝑚 𝑋 . (a) 𝑝-value 𝑝 0 .
(b) Significance 𝑍0 . The asymptotic formulae are used.

143s = 13 TeV, 14.6 fb-1
Expected
Expected ± 1 σ
Expected ± 2 σ

10Cross section σX [fb]Signal yield

G Results from asymptotic formulae102

s = 13 TeV, 14.6 fb-1
Expected
Expected ± 1 σ
Expected ± 2 σ

10

1
1

200

400

600

800

1000 1200 1400
mX [GeV]200(a) Signal yield limits 𝑁95

400

600

800

1000 1200 1400
mX [GeV](b) Cross section limits 𝜎95

Coupling constant f-1 [TeV-1]

Figure G.2: Expected exclusion limits of signal yield and cross section.

s = 13 TeV, 14.6 fb-1

1

Expected
Expected ± 1 σ
Expected ± 2 σ

10−1

10−2
200

400

600

800

1000 1200 1400
mX [GeV]Figure G.3: Expected exclusion limits in the 𝑚 𝑋 - 𝑓 −1 plane.

144

Acknowledgements
I would like to express my gratitude to the referees Prof. Toshinori Mori, Prof. Takeo Moroi, Prof. Yutaka Ushiroda, Associate Prof. Wataru Ootani, and Associate Prof. Taku Gunji, who review this thesis. I
would also like to thank Prof. Junichi Tanaka (my supervisor) and Assistant Prof. Yuji Enari, for their careful advice on this thesis. I also thank the SPRING GX program of the University of Tokyo for supporting
this research.
Everyone at ICEPP gave me a lot of advice for my research. Prof. Tanaka, Assistant Prof. Enari, AssociateProf. Yasuyuki Okumura, and Assistant Prof. Tatsuya Masubuchi have always consulted me about thisresearch. The Ref. [22] shared by Prof. Shoji Asai was one of the triggers for me to start this research.
Prof. Tanaka and Associate Prof. Koji Terashi helped me get used to physics analysis for about a year after I
joined the analysis team. Dr. Takuya Nobe told me about the ATLAS EFT framework, and Dr. Shion Chen
gave me ideas for future analysis using AFP. Assistant Prof. Masahiko Saito consulted me about statistical
test methods. Associate Prof. Ryu Sawada and Associate Prof. Tetsuro Mashimo provided advice on
computing. All the secretaries also helped me a lot.
I would also like to thank the members of the analysis team to which I belong. As an analysis contact,
Dr. André Sopczak (Prague CTU) allowed me to proceed with the analysis smoothly. Dr. Peter J Bussey(University of Glasgow), as one of the earliest members of the team, gave me various comments on theanalysis. Dr. Rafał Staszewski (Krakow IFJ PAN), based on his experience with the dilepton analysis [66],gave me a lot of helpful advice for almost all phases of the analysis. Research Prof. Marek Taševský
(Prague AS) kindly answered many questions about the AFP and SuperChic. Mr. Denis-Patrick Odagiu,
a CERN summer student, made the prototype of my analysis software. Several other undergraduate andmaster’s students also participated in this analysis and produced exciting results.
The editorial board members Dr. Simone Pagan Griso (Berkeley LBNL), Associate Prof. Deepak Kar(University of Witwatersrand), and Dr. Mustafa Schmidt (the University of Wuppertal) provided insightfulcomments and made the analysis robust during the ATLAS approval stage of this research. The analysis
sub-group conveners, Dr. Danijela Bogavac (CERN) and Dr. Christian Sander (DESY), and the analysis
group conveners, Dr. Flavia De Almeida Dias (Nikhef, Amsterdam) and Dr. Tamara Vazquez Schroeder(CERN) provided many comments on this analysis and approved it smoothly. As members of the dilepton
analysis [66], Dr. Lydia Beresford (DESY) and Dr. Jesse Liu (University of Cambridge) gave me many
comments and answered my questions.
Dr. Lucian Harland-Lang (University College London), the author of SuperChic, updated SuperChic for this research and answered many questions about it. Research Prof. Taševský often intermediated us. Dr. Ewelina Maria Lobodzińska (DESY), Dr. Aleksandra Dimitrievska (Berkeley LBNL),
William Patrick Mccormack (Berkeley LBNL), and Dr. Pagan Griso proceeded the implementation ofSuperChic in the ATLAS software.
Dr. Chris Hayes (the University of Michigan), Dr. Mungo Davide (University of Toronto) and Dr. Chris Meyer(Indiana University) supported the introduction of analysis tools. Dr. Ahmed Tarek (Michigan SU) and
Dr. Anthony Morley (CERN) answered many questions about the tool and sample productions. Nicolas Berger (LAPP Annecy) has generously supported me as the author of the function fitting tool. Dr. Kiley Kennedy (Princeton University) introduced some materials as an analyser of calo-vertexing.

145

G Results from asymptotic formulae
Dr. Krzysztof Ciesla (Krakow AGH-UST) and Dr. Staszewski answered software questions related to AFP
and helped with updates. Mr. Adrien Auriol (University of Birmingham) and Miss Maura Barros (LIP)
helped me to include the pileup effect in the AFP simulation for the systematic uncertainty evaluation.
KEK-IINAS Iwate Collider School 2022 taught me much about MC sample generation methodology.
Especially, Associate Prof. Kentarou Mawatari (Iwate University) showed me how to generate signal MC
samples using MadGraph. Also, Mr. Atsuya Niki (University of Tokyo), whom I met at the school,
invited me to a seminar and gave me a chance to study.
Finally, my family has contributed greatly to my research life.
Many other people also helped me. This research was completed thanks to all the above people. I appreciate
all the contributions. Thank you very much.

146

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