Di-Higgs Production in the Standard Model and Beyond

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Di-Higgs Production in the Standard Model and Beyond

Hanna Br¨ annstr¨ om

Department of Physics Degree Project, 60 hp

Master’s Programme in Theoretical Physics (120 hp) Supervisor: Professor Sara Strandberg

Date: 01-06-2021

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Di-Higgs Production in the Standard Model and Beyond

Hanna Br¨ annstr¨ om

Abstract

This thesis examines how the kinetic properties of pair-produced Higgs bosons depend on whether the process is mediated by particles in the Stan- dard Model or a simplified supersymmetric model. The analysis is based on simulated data made using MadGraph. The examination within this thesis is split into two parts, one performed on the output of MadGraph (parton level) and the second performed on the output of a simplified simulation of the ATLAS detector response (reconstruction level). The first part also contains a section showing that the supersymmetric model results are consistent with the Stan- dard Model results in cases where no supersymmetric particles are present in the di-Higgs production and decay, and a section detailing the minor differences in kinematics of the Higgs bosons depending on the mass of the supersymmetric partner of the top quark. The second part of the thesis covers the decay of the Higgs boson into bottom and anti-bottom quarks, and includes a section showing that there is no difference if the decay is done using the MadSpin module or the Pythia module for MadGraph, before exploring the kinematics of the jets produced. Both when studying Higgs bosons at the parton level and when studying b-jets at the reconstruction level, it is found that there are clear differences in the kinematics between the Standard Model and the supersymmetric model. At both levels, a neural network has been designed, trained and tested.

For the parton level neural network 59% of all events are classified correctly, while at the reconstruction level 51% of all events are classified correctly. These neural network results show that it is possible to train a neural network to learn on data like this, and that with enough di-Higgs events detected, their kinematic properties could be used to indirectly infer the presence of physics beyond the Standard Model.

Keywords: di-Higgs, kinematics, stop squark, SUSY, neural networks

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Acknowledgements

I want to start by thanking my supervisor Sara Strandberg. This would not have been possible without your help.

Thank you Luca Panizzi for allowing me to use your unpublished models for this work, and for explaining things to me in an understandable way. Thank you to Tom Ingebretsen Carlson for answering my beginner questions, and to Karl Gellerstedt and J¨orgen Sj¨olin for stopping what you where doing to help a confounded master’s student. Thank you heartily to Laura Pereira, who really inspired me and encouraged me to step up my analysis and become better at machine learning.

Finally, I am so grateful for my family for being there for me. Thank you for all of the support.

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1 Introduction

1.1 Motivation

Across history, many curious scientists have studied how masses interact. From Galileo Galilei showing that things of different mass fall at the same rates, to Newtons laws unifying the gravitational theories of mass on earth with the movements of planets and stars, all the way up to general relativity, where Einstein explained how mass is tied to time and space.

All of these theories brought the scientific views of how mass interacts forward. But none of them explain what mass is, or why some particles (like the electron) has mass while others (like the photon) are massless. Theorists had considered this question before and the Higgs boson has been presented as a possible path to an explanation. The experimental results proving the exis- tance of the Higgs boson were presented by the LHC experiments ATLAS and CMS in 2012 [1, 2]. The discovery of the Higgs boson lead to a Nobel Prize to Fran¸cois Englert and Peter W. Higgs in 2013 [3].

But there is still much to learn about what gives matter mass. Learning more about the Higgs boson and how it interacts with other particles as well as with itself could lead to discovering new physics beyond the Standard Model, such as the theorized but not experimentally found particles introduced by supersymmetry.

This report details the analysis of simulated particle collisions in which pairs of Higgs bosons are produced (so-called di-Higgs production). To begin, the Standard Model is introduced along with several important concepts related to it. Then a supersymmetric extension of the Standard Model is presented, as well as the simplified version of supersymmetry used in the simplified squark model.

The methodology section details the programs that are used in the simulation and analysis of the particle collisions. The results of the simulations and their importance are explained in the results section, and the conclusion and outlook section puts the results in context and expand on how they can be used and refined in the future.

1.2 Standard Model

The theory known as the Standard Model of particle physics describes how all of the elementary particles interact with each other via the strong force, the weak force and the electromagnetic force [4]. After the theory was formulated in the 1960s, it has been consistent with experiments and even successfully predicted experimental discoveries, like the discovery of the Higgs boson in 2012 [5].

The elementary particles known as the fermions are divided into quarks and leptons, where each category consists of three generations made up of two particles per generation [4]. An overview of the fermions is shown in Table 1.

The lightest and most stable of the quarks are called up and down, and they make up the first generation and all of the protons and neutrons around us. The heavier quarks in the second and third generations are unstable and

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Quarks Generation Mass [MeV] Charge [e⁻] Spin

u 1st 2.16^+0.49_−0.26 ²₃ ¹₂

d 1st 4.67^+0.48_−0.17 −¹₃ ¹₂

s 2nd 93⁺¹¹₋₅ ²₃ ¹₂

c 2nd (1.27 ± 0.02) · 10³ −¹₃ ¹₂

b 3rd (4.18^+0.03_−0.02) · 10³ −¹₃ ¹₂

t 3rd (172.76 ± 0.30) · 10³ ²₃ ¹₂

Leptons Generation Mass [MeV] Charge [e⁻] Spin

e⁻ 1st 0.5110 ± (3.1 · 10⁻⁹) −1 ¹₂

νe 1st Massless 0 ¹₂

µ 2nd 105.6583745 ± (2.4 · 10⁻⁶) −1 ¹₂

νµ 2nd Massless 0 ¹₂

τ 3rd 1776.86 ± 0.12 −1 ¹₂

ντ 3rd Massless 0 ¹₂

Table 1: The fermions of the Standard Model, with their experimental values.

The values are given by [6, 7].

quickly decay into the lighter, more stable particles [4]. Quarks also carry one of three colour charges, and can only exist in colourless bound objects of multiple quarks (so-called hadrons) [4]. For each of the fermions there is an anti-fermion with opposite charges (such as electric charge and colour charge).

There are three generations of leptons for a total of six leptons, three of which are charged. The neutral leptons are massless in the Standard Model [5]

despite the experimental evidence of them being massive [8].

The forces explained by the Standard Model are all mediated through the exchange of particles known as bosons [4]. Each force has its own boson or bosons, with the strong force using gluons, the weak force using W⁺ bosons, W⁻bosons and Z bosons, and the electromagnetic force using photons [4]. The W⁺ and W⁻ bosons are each other’s antiparticle and are identical apart from their charge. They are often referred to together as the W boson. The Standard Model bosons are shown in Table 2. The fourth fundamental force is gravity, and it is not a part of the Standard Model [9].

1.2.1 Higgs boson

The Higgs boson is an electrically neutral (Q = 0) spin zero particle [8], with a mass of 125.10 ± 0.14 GeV [6].

The Higgs boson is the excitation of the Higgs field. The Higgs field is an all permeating field that has a non-zero value even in vacuum [8].

At hadron colliders like the LHC, the main way that a single Higgs boson is produced is through gluon fusion, which is when two gluons become a Higgs boson, g + g → h, via a quark loop [8, 10]. The quark loop is required since the Higgs boson interacts with other particles in proportion to their masses, and

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Gauge bosons Related force Mass [GeV] Charge [e⁻] Spin

g Strong force Massless 0 1

γ Electromagnetic force Massless 0 1

Z Weak force 91.1876 ± 0.0021 0 1

W^± Weak force 80.379 ± 0.012 ±1 1

Scalar bosons Related force Mass [GeV] Charge [e⁻] Spin

H 125.10 ± 0.14 0 0

Table 2: The bosons of the Standard Model, with their experimental values.

The values are given by [6].

therefore cannot interact directly with the gluon since it is massless. A Feynman diagram of this process is shown in Figure 1. Due to its large mass and thus stronger interaction with the Higgs field, the top quark is the dominant quark in said loop. The gluon fusion production mode was the mode used to discover the Higgs boson [11].

(a) Gluon fusion with a top quark loop (b) Gluon fusion with a bottom quark loop Figure 1: Gluon fusion using either top or bottom quarks and anti-quarks. Here g are gluons, h are Higgs bosons, t are top quarks, b are bottom quarks, t∼ are anti-top quarks and b∼ are anti-bottom quarks.

In supersymmetric extensions of the Standard Model, there are several types of Higgs bosons [8].

1.2.2 The Brout-Englert-Higgs Mechanism

One of the goals of physics is to unify the fundamental forces. A large step towards that was achieved when the electromagnetic force and the weak force were combined into the electroweak force. A problem that arose in this unifica- tion was the fact that electromagnetism is transmitted by the massless photon, while the short ranged weak force is transmitted by W⁺, W⁻, and Z bosons,

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which all have large masses as shown in Table 2.

The Heisenberg uncertainty principle explains that a virtual particle carrying a force can “borrow” an energy ∆E using

∆E∆t ≈ h

2π (1)

where h is Planck’s constant and ∆t is the time the energy can be borrowed for [12]. The rest energy for a particle is E = mc², so in order for a particle with a large mass to be created, a large energy ∆E is required and so this virtual particle can only exist for a short time ∆t. This means that heavy particles like the W and Z bosons can only move a short distance, which is why the weak force has such limited range [12].

In order to combine the electromagnetic force and the weak force and their respective fields, it is necessary for the Higgs field to undergo spontaneous symmetry breaking while at the same time obey local gauge invariance. Insisting that both the local gauge invariance and the spontaneous symmetry breaking are true at once mathematically leads to the Higgs field being a massive gauge field and the Higgs boson being a massive scalar (i.e. spin-0) boson [8]. It is that mathematical result that is known as the Brout-Englert-Higgs mechanism, or BEH mechanism. The BEH mechanism is fundamental to the Standard Model as it introduces the Higgs field, which in turn explains why the W⁺, W⁻, and Z bosons have mass while the photon does not [13].

The Higgs field permeates all of space, and as space cooled down after the Big Bang the Higgs field underwent a phase change [12].

Spontaneous symmetry breaking means that the system loses its symmetry without external input [8]. In the Standard Model, after the spontaneous symmetry breaking of the electroweak force, the BEH potential can be written as:

V (H) = 1

2m²_HH²+ λνH³+ λH⁴ (2) where mHis the mass of the Higgs boson, H is the value of the Higgs field, λ is the Higgs boson self-coupling constant, and ν is the vacuum expectation value of the Higgs field, as explained in [11]. The fact that there is a self-coupling constant means that the Higgs boson can self-couple. An example of such a process can be seen in Figure 2. There is a relationship between the three constants given by the Standard Model [11]:

m_H=√

2λν (3)

The value of the vacuum expectation value is ν = 246 GeV, which was shown experimentally by electroweak precision tests [11, 14]. Combining that with the mass of the Higgs boson of 125.10 ± 0.14 GeV [6] gives a Standard Model estimate of the Higgs boson self-coupling constant of

λ = 0.13 (4)

So far, it has not been experimentally determined if λ agrees with this Stan- dard Model prediction.

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Figure 2: A virtual Higgs boson can decay into a pair of Higgs bosons. Here h is the Higgs boson.

1.2.3 Higgs boson pair production

As previously mentioned, the Higgs field can self-couple, or self-interact, which is the interaction taking place when a virtual Higgs boson decays into a pair of Higgs bosons as shown in Figure 2.

A way to measure the self-coupling constant called λ that appears in the parameterization of the BEH potential is to measure the cross section for the pair production of Higgs bosons.

Higgs boson pair production can occur by one of two processes shown in Figure 3. These two processes are generally referred to as the box diagram and the triangle diagram due to the shape of the Feynman diagrams of the respective process. Within the Standard Model the most likely particles to appear within the loops of these diagrams are either top and anti-top or bottom and anti-bottom quarks.

The triangle diagrams shown in Figure 3 depend on the Higgs boson self- coupling constant as they contain the vertex where a virtual Higgs boson decays into the two final Higgs bosons in the same way as was shown in Figure 2 in Section 1.2.2. The box diagrams do not depend on the self-coupling constant as the two Higgs bosons produced there never interact.

The box diagram and the triangle diagram are interfering destructively, re- sulting in the low total cross section of Higgs boson pair production. The predicted total cross section is σgg→hh = 31.05 fb at a collision energy of 13 TeV [15].

As the triangle diagram includes this self-coupling and the box diagram does not, being able to measure the cross section of Higgs boson pair production would be a way to probe λ [16]. Unfortunately, production of a pair of Higgs bosons has approximately a 1 thousand times lower rate of occurring than the single Higgs boson production in the Standard Model [15]. Moreover if two Higgs bosons are measured it does not necessarily mean that self-coupling has occurred [13]. Thus far, no events containing pairs of Higgs bosons have been experimentally observed.

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(a) Box diagram with top quarks (b) Box diagram with bottom quarks

(c) Triangle diagram with top quarks (d) Triangle diagram with bottom quarks Figure 3: The Higgs boson pair production processes. Here g are gluons, h are Higgs bosons, t are top quarks, b are bottom quarks, t∼ are anti-top quarks and b∼ are anti-bottom quarks.

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1.3 Limitations of the Standard Model

The Standard Model of particle physics is a successful explanation for a lot of experimental results, but is regarded by many physicists as a low energy effective theory of a more general and more fundamental theory underlying it [17]. There are several reasons to believe that a more fundamental theory is in place, from electroweak symmetry breaking to the fact that there is no concrete understanding as to why the particle masses are so spread out, with six orders of magnitude separating the mass of the electron and the mass of the top quark [17]. A theory beyond the Standard Model would also allow for explanations of the nature of dark matter and dark energy, why the neutrinos have masses and why there is more matter than anti-matter in the universe.

1.4 Supersymmetry and squarks

Supersymmetry, or SUSY, is a particle physics model that aims to be the underlying theory behind the Standard Model. Each particle currently known is paired with a supersymmetric partner particle which is identical apart from the spin [7]. All of the fermion partners are bosons and can be identified by the fact that their name has an “s” added to the beginning of the particle name, meaning that the supersymmetric partner of the electron is the selectron for instance, while the boson partners are fermions and has names that ends in an extra “ino”, making e.g. the supersymmetric partner of the gluon the gluino.

All of the particles of the SUSY model and their spins are detailed in Table 3.

If the symmetry between the particles and their supersymmetric partners were exact then both particles would have the same mass and the partners would have already been detected [7]. The symmetry must be broken and the masses be different, but in order to not lose the advantages that a supersymmetric solution brings, a soft symmetry breaking term is explicitly added to the SUSY Lagrangian [17]. The new symmetry-broken Lagrangian can be written as:

L = LSU SY + LSof t (5)

where the Lagrangian term LSof t only can consist of “mass terms and couplings with a positive mass dimension” in order to not introduce any undesired quadratic divergences [9].

1.5 Simplified squark model

To simulate the impact of a supersymmetric top quark partner contributing to the loops in the Feynman diagrams for di-Higgs production, Figure 3, a simplified squark model (SSM) [19] is used. The topologies constituting the squark model, along with the Standard Model topologies, are referred to as topology 1, 10, 11, 12 and 13.

The Feynman diagram for topology 1 is shown in Figure 4. The ∼ sign in these diagrams indicates that the particle is an anti-particle. For instance, t∼

is the anti-top quark. The top and anti-top quarks can swap places within the

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Quarks Symbol Spin Squark Symbol Spin

up u ¹₂ sup eu 0

down d ¹₂ sdown de 0

strange s ¹₂ sstrange se 0

charm c ¹₂ scharm ec 0

bottom b ¹₂ sbottom eb 0

top t ¹₂ stop et 0

Leptons Symbol Spin Slepton Symbol Spin

electron e ¹₂ selectron ee 0

electron neutrino νe 1

2 electron sneutrino eνe 0

muon µ ¹₂ smuon µe 0

muon neutrino νµ 1

2 muon sneutrino νeµ 0

tau τ ¹₂ stau eτ 0

tau neutrino ντ 1

2 tau sneutrino νeτ 0

Bosons Symbol Spin Sboson Symbol Spin

gluon g 1 gluino eg ¹₂

photon γ 1 photino eγ ¹₂

Z boson Z 1 zino Ze ¹₂

W boson W^± 1 wino fW^± ¹₂

Higgs H 0 higgsino He ¹₂

Table 3: The particles and their respective spin in the Standard Model and SUSY model [7, 18].

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loop without changing the topology category. The vertices marked with a dot in the Feynman diagram has a coupling that has been altered by the squark model. This altered coupling between Higgs bosons introduces new physics and makes this topology different from the Standard Model triangle diagram shown in Figure 3c.

Figure 4: The first topology from the simplified squark model. Here g are gluons, h are Higgs bosons, t are top quarks and t∼ are anti-top quarks.

Examples of topology 10 to 13 are shown in Figure 5. These are the topologies that introduce the stop squark, sq, and the anti-stop squark, sq∼. These couplings are explained further in Section 3.1. In the same way as for Figure 4, a dot marks a coupling that has been altered in the model. Note that these are only typical examples of each topology, and that for instance stop and anti- stop squarks can swap places within the loop without changing the topology category.

The simplified squark model is designed to generate simulated data for only one topology or interference between topologies at a time. This means that in order to determine how the model works as a whole, and thus be able to predict the experimental results and compare those to the Standard Model, the results need to be added together with the appropriate cross-section weights.

In this thesis, the cross sections calculated by the simulation software Mad- Graph are used in the calculation when the individual topologies and interferences are weighted together. These cross sections are calculated using SSM coupling constants that all have a value of 1.

The equations for calculating the total cross section of the SSM are presented below. The khhh, khstst and khhstst are coefficients that modify the couplings used when the cross sections are calculated. Specifically, khhh is the coupling coefficient for the coupling between three Higgs bosons, k_hstst is the coupling coefficient for the coupling between two stop squarks and a Higgs boson and k_hhstst is the coupling coefficient for the coupling between two stop squarks and two Higgs bosons. Equation 6 shows the cross-section calculation of the pure SSM topologies (S), without any interference terms.

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(a) Topology 10 (b) Topology 11

(c) Topology 12 (d) Topology 13

Figure 5: The four last topologies from the simplified squark model. Here g are gluons, h are Higgs bosons, st are stop squarks and st∼ are anti-stop squarks.

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σ_S = k²_hhhσ₁+ k²_hststσ₁₀+ k²_hhhk_hstst² σ₁₁+ k⁴_hststσ₁₂+ k²_hhststσ₁₃ (6) In Equation 7 the cross section of the interference between the SSM topologies is presented. Equation 8 contains the interference terms between the SSM topologies and the Standard Model events (B), while Equation 9 contains the terms that determine the simultaneous interference between SSM topologies and between SSM and SM topologies.

σSS =k_hhh² khststσ1x11+ khhhk_hstst² σ1x12−10x11+ k_hstst³ σ10x12

+ k_hhhk³_hststσ_11x12+ k_hhhk_hhststσ_1x13+ k_hststk_hhststσ_10x13 + khhhkhststkhhststσ11x13+ k²_hststkhhststσ12x13

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σSB = khhhσ1xB+ khststσ10xB+ k_hstst² σ12xB+ khhststσ13xB (8)

σ_SS−SB= k_hhhk_hststσ_1x10−11xB (9) The total cross section is the sum of the Standard Model cross section, denoted σB, and the cross sections presented in Equations 6 to 9.

σSSM = σB+ σS+ σSS+ σSB+ σSS−SB (10) How this weighing is done in practice is explained in detail in Section 3.1 and Section 3.2.

1.6 Kinematic distributions

1.6.1 Four-vectors

A four-vector is defined as a vector that, when the vector is observed in another inertial reference frame, acts like:

a^µ⁰ = Λ^µ_νaν (11)

where

Λ^µ_ν =







γ −γβ 0 0

−γβ γ 0 0

0 0 1 0

0 0 0 1







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is the matrix of the coefficients of the Lorentz transformations [8].

The covariant vector aµ can be found using

aµ= gµνa^ν (13)

and to find the contravariant vector from the covariant vector one uses

a^µ= g^µνaν (14)

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where g^µν= gµν is the metric matrix shown below [8].

g^µν = gµν =







1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 −1







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According to [8], for any four-vectors a^ν and b^ν the quantity calculated by a^νbν = aνb^ν = a⁰b⁰− a¹b¹− a²b²− a³b³ (16) is invariant and thus the same in every inertial reference frame.

One four-vector used in this report is the energy-momentum four-vector.

The energy-momentum four-vector can be defined as [8]

P^µ= (E

c, px, py, pz) (17) where the relativistic energy E is defined as:

E = γM c²= M c² q

1 − ^|v|_c2²

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and the relativistic momentum vectors is defined as:

p = γM v = M v q

1 −^|v|_c2²

Another form of four-vector can be found converting to spherical coordinates.

The variables pT, η and φ can be written as [20]:











p_T =q p²_x+ p²_y sinh η = _p^p^z

T

sin φ =_p^p^y

T

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In this version of the four-vectors, either the particle energy or the particle mass can be used and the particle mass will be used in this thesis. The expression for the relativistic energy found in Equation 18 can be rewritten to a formula for the rest mass:

M cγ = E

c → M = E

γc² (20)

As (^E_c, px, py, pz) has simply been rewritten, the new four-vector contains all of the same information as (M, pT, η, φ).

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1.6.2 Invariant mass

The Lorentz invariant mass of a system of n particles is given by [21]:

M²c⁴= (X

n

E_n)²− (X

n

~

p_n)²c². (21)

Here M is the invariant mass, or total mass of the system. Note that this has the same form as if one had inserted the energy momentum four-vector in Equation 17 into the general four-vector in Equation 16.

The invariant mass is the same in every reference frame, including the center of mass frame whereP

n~p_n = 0 [7]. In that center of mass frame, the invariant mass simplifies to

M²c⁴= (X

n

E_n)²= E². (22)

which can be reduced to the famous expression E = M c².

Many applications simplify relativistic kinematics by calculating the invariant mass of a collision in two different reference frames and equating them, since the invariant mass will be the same in all frames [7]. One such application is the examination of short lived particles, where the invariant mass of the decaying particle is the same as the invariant mass of the system of particles it decayed to [7].

1.7 Experimental overview

The Large Hadron Collider (LHC) is a particle collider located at the European Organization for Nuclear Research (CERN) near Geneva and the French–Swiss border. Within the 27 kilometer circumference ring, protons or heavy ions are accelerated and then collided at one of four collision locations where detectors record the results. There are currently seven different active experiments con- nected to the LHC: ALICE [22], ATLAS [23], CMS [24], LHCb [25], LHCf [26], MoEDAL [27] and TOTEM [28].

ATLAS and CMS are general purpose detectors for analysing proton-proton collisions. ATLAS and CMS use different techniques and different magnet- systems to achieve the same scientific goals [9]. One of the goals of the two detectors is to learn more about the elementary particles, including the Higgs bosons. Discovering Higgs boson pair production would be one way to learn more about the Higgs and how it interacts with matter.

The LHC is currently shut down for upgrades, but Run 3 is scheduled to start with proton collisions in 2022, and at that point the experimental search for di-Higgs production can resume.

In this thesis a simulation of the ATLAS detector is used in order to determine how well the detector would be able to measure the differences in kinematics for the Standard Model and the simplified squark model.

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1.8 Introduction summary

Relating back to the motivation, one way to learn more about mass and the fundamental physics that guide it is through the study of the Higgs boson.

Two questions that still remain unanswered are what the kinematics of di-Higgs production can tell us about the Higgs boson and potential supersymmetric particles contributing to its production and decay processes. The ATLAS project is trying to solve both of these questions using data from the Large Hadron Collider. In this thesis, the kinematics of the simplified squark model for supersymmetry is compared to the Standard Model results at both the parton and the reconstructed level. This is done through simulations of di-Higgs production events for the two models and analysis of the results.

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2 Methodology

In this methodology section all of the programs and techniques used in the generation of the results are presented. The sections are ordered by the chrono- logical order the programs are used, starting from the generation of collision data in MadGraph via preprocessing in pandas to ending with machine learning with Keras.

2.1 MadGraph

MadGraph [29] is a program to compute differential cross-sections of tree-level particle processes allowing for phenomenological investigations of particle collision events.

The article [30] introduced an automated way for MadGraph to produce Feynman diagram containing loops and do the calculation that type of Feynman diagram entails. This program is designed for loop-induced processes, which require a leading order (LO) approximation in order to account for the loop structure [30]. There are some exceptions, where the looping particle is heavy enough such that in some kinematic ranges it can be integrated out for the first order approximation and this program runs those calculations without the need for that approximation [30]. MadGraph5 [31, 32, 33, 34] is the specific version used in order to generate and analyse events in this thesis.

In order to run MadGraph, a textfile has been written for each of the different topologies and interferences from the simplified squark model, as well as a textfile for the Standard Model. This text file is given to MadGraph when it is started and acts as a script containing the inputs given to the program in order. In Table 4 all of the files are listed for creating all of the topologies and interferences in the simplified squark model. All of these files together is referred to as a set of MadGraph scripts, and one set is required for each type of simulation to be done.

In this thesis, simulation data and analysis is broadly done on two different levels of complexity. The output of MadGraph directly contains only the particles that make up the Feynman diagrams. On this level, referred to as the parton level, the Higgs bosons are analysed. In experiments like ATLAS the Higgs bosons are not measured directly however and instead it is the particles that the Higgs bosons decay to that can be detected. In order to simulate this more realistic view, the Higgs bosons are decayed into bottom and anti-bottom quarks and then a simplified simulation of the ATLAS detector response is applied. This is referred to as the reconstruction level as the simulated detector tries to reconstruct the collision event that took place. On this level the analysis is focused on the bottom and anti-bottom quarks that the detector reconstructed.

For this thesis there is one set of MadGraph scripts for parton Higgs boson data, one set for the reconstruction level data where the decay is done with MadSpin, and one set for the reconstruction level data where the decay is done in Pythia which is required for Section 3.5.1. What happens to the sets of scripts

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Background Topologies Background interferences Interferences

SM 1 1xB 1x10-11xB

10 10xB 1x11

11 12xB 1x12-10x11

12 13xB 1x13

13 10x12

10x13 11x12 11x13 12x13

Table 4: The script files required for all of the topologies and interferences required in order to simulate the simplified squark model in its entirety. The x in files like 1x11 indicates that it is the interference between the two topologies, in this case 1 and 11. The - in files like 1x10-11xB indicates that the topologies or interferences, in this case 1x10 and 11xB, consist of the same Feynman diagrams and are handled together.

and the data they produce is shown in Figure 6. An example of a MadGraph script for the parton level can be found in Appendix A.1.1 and an example for the reconstruction level can be found in Appendix A.1.2.

Despite having a few subtle differences between the different topologies, the main structure of the text-script is the same. The model file for the simplified squark model, referred to as “SQ Hsqsq diagonalCKM 4FNS NLO UFO”, is imported.

Next, the particular collision events are generated. For gluon to di-Higgs, this instruction reads “generate p p > h h [QCD] / sb” where the sbotton squark has been excluded. This instruction tells MadGraph to simulated two particles colliding that lead to two Higgs bosons being created. From the run card that MadGraph will be given later, a copy of which can be found in Appendix A.2, the fact that it is two proton beams that collide is defined. MadGraph internally calculates what particles can become two Higgs bosons among those defined in the model and calculates the cross sections of said particles being created from protons internally. In practice for all of the topologies and interferences in this thesis, the Feynman diagrams created have two gluons that are colliding.

After this general instruction, the total number of times that each altered SSM coupling appears in the Feynman diagram of the topology is specified. For generating the interference between two topologies, the total number of times the couplings appear in the Feynman diagrams of the two topologies is specified instead. This information is given in Table 5 for the topologies relevant for the simplified squark model. The number of different couplings define the topology and so is independent of specifics of the run like energy or stop mass.

The number of Standard Model couplings that appear in the Feynman diagram needs to be given as well. The strong Standard Model couplings are referred to as QCD couplings in MadGraph. In the relevant topologies there

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(a) Parton level

(b) Reconstruction level

(c) Reconstruction level not using MadSpin

Figure 6: The flowcharts of the results from the three different sets of SSM MadGraph scripts and their corresponding SM scripts. The gray boxes denote files or figures saved, the blue boxes denote programs and the larger green boxes denote which framework the programs are using. In (a) events are generated in MadGraph itself and no further programs are used before the results are turned into ROOT files. In (b) and (c) more programs are used, where (b) uses MadSpin for decaying particles and Pythia for turning the results into jets, while (c) uses Pythia for both decaying and turning the results into jets. In (c), no neural network is made. Instead, the ROOT files are compared to the ROOT files from (b) and the results are analysed in Section 3.5.1.

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Topology number HHSQSQ HSQSQ HHH QED

1 0 0 1 1

10 0 1 0 1

11 0 1 1 0

12 0 2 0 0

13 1 0 0 0

Table 5: This table details the mapping between the couplings and topologies present in the beyond Standard Model contributions to the simplified squark model. Notice that the SSM couplings are named by the component particles of stop squarks (SQ) as well as Higgs bosons (H).

are always two QCD couplings, namely the two gluon to top couplings. The electroweak Standard Model couplings, like top to Higgs and Higgs to two Higgs, are referred to as QED couplings in MadGraph. The number of couplings specified is the maximum number allowed by MadGraph, so if QED=3 is specified then Feynman diagrams with fewer than three electroweak couplings can be created as well. The Standard Model coupling information is also given in Table 5 for the topologies relevant for the simplified squark model.

The defining values for each topology or interference generated are the number of couplings of each type, as the couplings specify the Feynman diagrams. If two topologies or interferences ever end up with the exact same couplings, as is the case with the the interference between topology 1 and 12 (HHSQSQ=0+0, HSQSQ=0+2, HHH=1+0, QED=1+0) and the interference between 10 and 11 (HHSQSQ=0+0, HSQSQ=1+1, HHH=0+1, QED=1+0), then they have the same Feynman diagrams. If that is the case, then those couplings are representative of both interferences and both are generated together within a single MadGraph script file.

The last type of topology to mention are the background interferences of the topologies in Table 5. These so called “xB” versions have the same beyond Standard Model couplings as their original forms, and they are described in Table 6. The difference between these and the topologies presented earlier is that the “xB” topologies allows more electroweak couplings. Because of this, more Feynman diagrams will be created alongside the specific topology, including the Standard Model Feynman diagrams, allowing for the calculation of the interference to the Standard Model background.

These background interferences may end up having the same couplings as the interference of two non-xB topologies, as is the case with the topology 11xB (HHSQSQ=0, HSQSQ=1, HHH=1, QED=2) and the interference of 1 and 10 (HHSQSQ=0+0, HSQSQ=0+1, HHH=1+0, QED=1+1). These two have the same couplings and so define the same Feynman diagrams. If that is the case, then those couplings are representative of both the interference and the background interference and both are generated together within a single MadGraph script file.

Each set of SSM MadGraph scrips requires a script generating SM events,

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Topology number HHSQSQ HSQSQ HHH QED

1xB 0 0 1 3

10xB 0 1 0 3

11xB 0 1 1 2

12xB 0 2 0 2

13xB 1 0 0 2

Table 6: This table details the mapping between the couplings and background topologies present in the simplified squark model. Notice that the SSM couplings are named by the component particles of stop squarks (SQ) as well as Higgs bosons (H).

Coupling Variable

HHSQSQ → KHHSTST=1

HSQSQ → KHSTST=1

HHH → KHHH=1

Table 7: This is a conversion table between the couplings in a topology or interference and the coupling variables in the run card. Any coupling variables not set to one by following this conversion table are set to zero.

and for each set there also needs to be a separate script generating SM events that the SSM events can be compared to. In order to create Standard Model events, the MadGraph script reads “generate p p > h h [QCD] / sb HHSQSQ=0 HSQSQ=0 HHH=0”. Having not specified the number of QED couplings, Mad- Graph finds all valid Feynman diagrams using any number of Standard Model couplings. As all of the new physics couplings have been set to zero, only the SM Feynman diagrams will be created. That the SM events generated in this way are the same as the SM events generated by importing the model sm, where no new physics is implemented, and using “generate p p > h h [QCD]” is verified in Section 3.3.1.

After this script line defining the collision and the type of topology, the folder to write the data to is specified. Then the run card in imported as a baseline before a number of variables of the run and parameter cards are changed. The run card defines a few basic settings and cuts that will be used for all of the simulations, before any set or topology specific changes are made. The run card can be found in full in Appendix A.2. The first variables changed are the couplings to be used, located in the parameter card. The coupling variables that correspond to couplings that are a part of the topology are set to one, following Table 7, and all other couplings are set to zero.

The number of events generated is specified using the nevents variable and the width of the stop squark is set to auto. The mass of the stop squark needs to be specified, and that has been done by using MadGraphs scan function on a list of stop masses. The scan function does the entire generation process with

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the first mass specified, saves those events, then MadGraph starts over using the second mass, saves the new events in a second folder and keeps going until events have been generated and saved for each of the masses in the list. Data for the following stop masses have been simulated: 500 GeV, 600 GeV, 800 GeV, 1 TeV, 2 TeV, 3 TeV, 4 TeV, 5 TeV, 6 TeV, 7 TeV, 8 TeV, 9 TeV and 10 TeV.

2.2 MadSpin

MadSpin [35] is a tool within MadGraph5 that allows events to be decayed while accurately preserving spin correlations and finite width effects.

In this thesis, MadSpin was used to decay a Higgs boson into a bottom quark and an anti-bottom quark.

In order to run MadSpin during an analysis, two lines are added to the MadGraph file. The first one is “madspin=None”. The second line is “decay h

> b b∼”, which tells Madspin to decay each Higgs boson into a bottom quark and an anti-bottom quark. Setting MadSpin to none means that there are no spin correlations between the particle production and decay and no off-shell effects are handled [36]. One can not use the standard MadSpin method for the gluon-gluon to di-Higgs to bottom and anti-bottom quarks due to a current limitation in the internal function MadLoop which is not able to create the required matrix elements [36].

2.3 Pythia

Pythia 8.2 [37] is a tool to generate events in high energy collisions, including multi-parton interactions, beam remnants and particle decay. For this thesis the program is used, through the MadGraph interface, to turn the particle data generated by MadGraph-simulated collisions into jets and shower data.

To activate the Pythia program, the line “shower=Pythia8” is added to the MadGraph text script. The Pythia run card is given to MadGraph as well. In the cases where the data has gone through the decaying process using MadSpin, that Pythia card is the default.

For the set of MadGraph scripts that create reconstruction events where the particle decay takes place in Pythia instead of MadSpin, studied in Section 3.5.1, a modified Pythia run card is used. Four lines are added to the end of that Pythia card, which stops the decay of Higgs bosons unless it is to a bottom and an anti-bottom quark, and stops those quarks from decaying further. The version of the Pythia card used for decaying Higgs bosons can be found in full in Appendix A.3.

2.4 Delphes

A simplified detector simulation has been done using Delphes [38, 39, 40].

Delphes 3.0 was developed in order to allow for the simulation of a multipur- pose detector, where the high level objects like jets are reconstructed from the simulated detector response [38]. The Delphes 3.0 jet clustering procedures are

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done via the FastJet package [41, 42]. In this thesis, Delphes is used to create reconstructed jets of the simulated data that first was generated in MadGraph, decayed in either MadSpin or Pythia and finally hadronized in Pythia.

In order to add Delphes to the process that the generated events undergo,

“detector=Delphes” is added to the MadGraph text script.

2.5 ROOT

The data from MadGraph that has been processed through the relevant Mad- Graphs tools are saved in the form of a tree file in the ROOT file format. ROOT [43, 44] is a data analysis framework for high energy physics research developed at CERN.

For this project, ROOT is only used for saving the data produced by Mad- Graph and then the Python module root pandas [45] is used in order to turn the information in the ROOT file into a format that can be analysed in Python using the pandas module.

2.6 Python

Python [46] is an interpreted general-purpose programming language, and as such it can be used to build a wide variety of programs. Python version 3.6.8 is used to do the numerical and statistical analysis of the generated events in this thesis. The main methods used are Kolmogorov–Smirnov tests to determine the separability of two data sets, pandas for preprocessing ROOT data, and Keras for machine learning by creating a neural network that the generated data is analyzed by.

The modules and libraries used that are not explained in later sections are SciPy [47] and NumPy [48] for doing calculations, Matplotlib [49] for plotting, scikit-learn [50] for structuring and splitting dataframes and PyROOT [51] for doing ROOT-based calculations.

2.6.1 Pandas

Pandas [52, 53] is a data analysis and manipulation tool in Python [53].

For the analysis done for this thesis, the ROOT files are read for each topo- logical case and the relevant variables are turned into pandas dataframes using root pandas [45]. Once the dataframes have been made for the case at hand, whether that is parton level or reconstruction level, one dataframe is created for all of the Standard Model events and one for all of the simplified squark model events. The plotting and analysis of the neural network input variables are done at this stage.

The dataframes are then combined before being split into a training and a testing dataframe for the neural network. This splitting is done with the train test split function from the scikit-learn module [50]. The split is ran- domised but stratified by the ROOT file it was read from, meaning that the training and testing dataframes have the same number of events from each

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topology and the same number of events from each file in the cases where more than one file is created for each topology. Any unnecessary variables are then dropped from both training and testing dataframes, before the training dataframe is given to the neural network.

2.7 Kolmogorov–Smirnov test

A Kolmogorov–Smirnov test [54, 55], which is a goodness-of-fit test [56], is used in Section 3.3 and Section 3.5 to determine the probability of two sets of frequencies coming from the same distribution.

In following the proof presented in [57], let X1, X2,..., Xnbe independent and identically distributed. The Kolmogorov-Smirnov statistic Dks is then defined by

Dn = max

x |Sn(x) − F0(x)|. (23)

Here Sn is the empirical distribution function which [57] defines as

Sn(x) =







0, x < X₍₁₎,

k

n, X_(i)≤ x < X(k+1), 1, x ≥ X_(n),

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where X₍₁₎ ≤ X₍₂₎ ≤ ... ≤ X_(n) is the ordered sample. The function F₀ is the distribution function under the null hypothesis [57]. The null hypothesis here is that the two data-sets are drawn from the same distribution. There are tabulated values of what the Kolmogorov-Smirnov statistic Dks corresponds to in terms of a p-value, which in turn gives the probability that the data-set is drawn from the same distribution as the function one compares to.

In this thesis, the Kolmogorov-Smirnov test of comparing two samples in Python has been calculated with SciPy [47], which is a Python library for scientific computing. More specifically, the statistical function called ks 2samp is used, which calculates a two-sided Kolmogorov-Smirnov statistic for two samples given as arguments [58].

2.8 Machine learning

Machine learning is the capability of an AI system to discover knowledge that was not directly given by extracting patterns from a raw data set [59]. There are a number of different machine learning algorithms that are in use, and the algorithm chosen depends on the type of data and the type of pattern recognition that is desired.

Deep learning enables the program to build complex concepts out of simple ones [59]. In this project deep learning is done through the use of several layers of sequential functions within a single neural network.

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2.8.1 Neural networks

A neural network (NN), or an artificial neural network, is an information pro- cessing system that has several similarities to biological systems based on neurons [60]. A neural network is characterised by the designs between its nodes (or neurons), the learning algorithm used for determining the weight of the connections between neurons and the activation functions used [60]. The neural network created for this thesis is a multi-layer feed-forward network, and is made up of an input layer, several hidden layers and an output layer. The layers in the neural network are dense, meaning that each node in a layer is influenced by each node in the layer before it.

A node is a point within one of these neural network layers where input parameters are given and an output parameter is calculated [61]. The input for the first layer of nodes are the examples given to the NN to train on, while the input for every following layer is the output of the layer before it. These input variables are altered by the model weight of the node the output came from and the bias of the node at hand, usually by doing the calculation of the activation function using the variable z =P

jw_jx_j+ b, where w_j are the model weights, x_j are the input variables and b is the bias of the node where the calculation takes place [61]. It is these model weights and biases that are changed as the neural network is training in order to try and more accurately reach the correct classification as the final output of the model.

Each layer of the neural network has an activation function, which converts the input signal into an output signal [60]. These activation functions introduce non-linearity to the system, which allows for learning from a larger variety of data sets [60]. Two different types of activation functions are used for the neural networks in this thesis: The Rectified Linear Unit activation function ReLU is used for the first and middle layers while sigmoid is used for the final layer.

The ReLU function is a widely used activation function defined as:

f (x) =

(0, x < 0,

x, otherwise, (25)

f⁰(x) =

(0, x < 0,

1, otherwise, (26)

where f⁰(x) is the derivative of f (x) [60]. The sigmoid activation function, used in the final layer to turn the input into an estimated classification as either of the two classes, is defined as [59, 60]:

f (x) = 1

1 + e^−x, (27)

f⁰(x) = f (x)(1 − f (x)). (28) Deep learning is a type of machine learning using a neural network which learns directly from a data set [60]. If a NN is deep is determined by the number of layers it is made up of, where a more traditional neural networks had two or

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three layers and a modern deep neural network could have hundreds of layers [60]. The neural networks in this thesis are on the shallower side with five layers.

2.8.2 Keras

Keras [62] is a deep learning API that runs using Tensorflow [63], which is an end-to-end machine learning platform. The neural network used in this thesis to classify collision events is built in Keras. It is a sequential model, consisting of several dense layers of neurons. The sigmoid activation function makes sure that the final result can be interpreted as part of one of the two classes; either the event is classified as a Standard Model event or a simplified squark model event. The neural network is using supervised learning, as it is given desired outputs to compare the results to.

Once the model has been defined it needs to be compiled. During compila- tion, the loss function to use is set as binary crossentropy, which is a measure- ment of how well a classification model performs compared to the right answers given. The binary crossentropy function is defined as:

J (θ) = −1 N

N

X

i=1

yilog (ˆyi(θ)) + (1 − yi) log (1 − ˆyi(θ)) (29)

where N is the number of events, ˆyiis the calculated result and yiis the desired result.

Most deep learning algorithms involve an optimization step, in order to minimize the loss function. The optimizer used for this neural network is SGD, or stochastic gradient decent. SGD is similar to the more traditional batch gradient decent in that it uses the gradient of the loss function J (θ) in order to change the parameterization variable θ so that the loss function becomes lower [64]. The difference between batch gradient decent and SGD is that while the former only updates the parameter for each run through of all of the training events, the later updates with each training example x⁽ⁱ⁾and its class y⁽ⁱ⁾, which in turn speeds up computation [64]. The formula for the SGD parameter update is given below, with θ being the parameter, J (θ) the loss function and η_LR the learning rate [64].

θ = θ − ηLR· ∇θJ (θ; x⁽ⁱ⁾, y⁽ⁱ⁾) (30) The compiled model needs to be fitted to the training set, which is done with the Keras function model.fit. The fit function is given the pandas dataframe of the training set events, containing the data from the relevant four-vectors, as well as an array containing information about which class the training set truly belongs to, which the fitting function will train against and try to reach. The fit function is given a sample weight, which is an array of the normalized weights of the events. This sample weight variable tells the neural network how important it is to classify an individual event correctly. The last arguments given to the fit function is the instruction that it should shuffle the order of the events between

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each epoch, and the number of epochs (or passes over the data set) it should train for.

After the neural network is trained, the model is saved and can then be used to test how accurate the model is using events that it was not trained on, or for classifying events where the class in not known.

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3 Results

3.1 Coupling coefficients

To be able to run the simplified squark model data generation, the coupling coefficients introduced in Section 1.5 are needed. There are in fact three different relevant non-SM couplings needed for this thesis; the coupling between two Higgs bosons and two stop squarks, the coupling between one Higgs boson and two stop squarks, and the coupling between three Higgs bosons. Each of the three non-SM couplings needs a coupling coefficient in order to calculate the results of the SSM as a whole.

The coupling coefficients used in this thesis are presented in Table 8. Insert- ing these coupling coefficients and the cross sections generated by the simplified squark model topologies for a stop mass of 1 TeV into Equations 6 to 10, gives a total SSM cross section of 0.053 pb, which is the largest di-Higgs production cross section allowed by experiments [65].

Coupling coefficient Value

khhh 0.255

k_hstst 10.0

khhstst 10.0

Table 8: This table contains the values of the coupling coefficients used in calculating the SSM model results.

When the simplified squark model is used to generate Standard Model processes the Standard Model coupling values are used.

3.2 Cross sections and weights

For the rest of the thesis, the cross section of an event times the relevant coupling coefficients will be considered the weight of the event. When analysing events at parton level, the production cross section of each individual event, given by the “Event.Weight” variable in the MadGraph ROOT output, is used to calculate the overall normalization of the sample. When examining events at reconstruction level, the average of the parton cross section for each topology is used for all events of that topology. This is done as it simplifies the code and the production cross section is the same for a topology at both the parton level and the reconstruction level.

Plotted in Figure 7 are all of the cross sections calculated for each event in all of the topologies in the Standard Model and the simplified squark model cases, where it is assumed that the stop squark has a mass of 1 TeV. In Figure 7a one can clearly see that all of the events have the same cross section. This is due to the fact that all of the events are created in the same way from the same topology and so the cross section calculated would not be expected to vary significantly.

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In Figure 7b, the simplified squark model cross sections are plotted. Here there are multiple different topologies involved, which leads to more varied weights than in the Standard Model case. Note that in the stop squark case there are events and topologies with negative cross sections. The negative cross section indicates that those events lower the probability of that event being measured for the squark model.

(a) Standard Model (b) Simplified squark model Figure 7: The cross sections for (a) the Standard Model and (b) the simplified squark model. A stop squark mass of 1 TeV is assumed in the simplified squark model case.

Whenever a sample is weighted by the corresponding event cross sections and coupling coefficients, the negative weights have the intuitive effect of making the values associated to events with negative weights less likely to occur.

3.3 Parton level

In order to look at the parton level of the reaction, ROOT files are produced by MadGraph when no other programs act on the simulated data, except ExRoot making the ROOT file itself. This parton level analysis is what is depicted in the flowchart in Figure 6a.

As each topology is run separately, there is a ROOT file for each of the different individual topologies, for each xB topology and for each interference.

There are 13 different masses examined for the stop squark, from 500 GeV to 10 TeV, and each mass has its own set of ROOT files.

From each ROOT file, data values like the angle between the four-vectors of the two outgoing Higgs bosons and their invariant mass are calculated. Those values from each event for each case are weighted together by the cross section times the coupling coefficients of each specific event. The result is the total histogram of for instance the invariant mass for the gluon-gluon to di-Higgs process.

The kinematic distributions such as pxand φ given by the MadGraph output are plotted. As all of these measurements are given for each outgoing particle there are two graphs for each such kinematic distribution, one for each Higgs

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boson.

In this section 20 thousand events per topology case are used. Unless specified, the stop squark is assumed to have a mass of 1 TeV. The reason that this analysis start out examining the pure Higgs boson result, despite the fact that Higgs bosons are not measured directly, is in order to get a sense of the accuracy of the method in the simplest possible case. It is reasonable to assume that decay and even more so a simulated detector would add noise and complexities that limit how well the distinction between the SM and the SSM can be made, and so looking at the Higgs case gives us an upper limit in terms of accuracy.

3.3.1 The Standard Model comparisons

There are two different ways to simulate the Standard Model data. The first is the Standard Model created by the MadGraph model sm, which is a Standard Model often used in MadGraph calculations. The second is the simplified squark model with all of the non-Standard Model coupling constants set to zero such that no supersymmetric particles take part in the process. If everything is correct then these two approaches should give the same result.

The two Standard Model approaches need to be shown to be equivalent before one of them can be used as the Standard Model baseline. In order to show that, the four-vector components are plotted for both of the outgoing Higgs bosons in both models. The results for the first Higgs boson are shown in Figure 8 while the results for the second Higgs boson are shown in Figure 9.

Examining these graphs, there is no clear disparities between the two different models.

In order to get a clearer understanding of how different the two calculations for the Standard Model are, Kolmogorov-Smirnov tests have been performed for the four-vector data above as well as a few calculated variables. The results are shown in Figure 10. Most of the p-values are significantly higher than 0, indicating that the null-hypothesis that the two are the same cannot be rejected.

The variable with a p-value closest to zero is the mass calculated for the second Higgs boson, where the p-value is 0.01786. This p-value indicates that it is less than a 1.8% chance that the two sets are drawn from the same underlying distribution and that the differences observed are simply statistical fluctuations.

This p-value is low, but higher than 0.01 which is the standard cut-off for when the two sets should be considered as representing different distributions.

While the observed p-values do not constitute a proof that the distributions are identical, they do indicate that the different models produce similar enough results that either one of them can be used as the Standard Model for comparison purposes. The simplified squark model version of the Standard Model is used and referred to as the Standard Model going forward.

3.3.2 Four-vector components

As described in Section 1.6.1, the output from MadGraph are the four-vectors of the generated particles. The first two four-vector components are the mo-

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(a) px (b) py

(c) pz (d) E

Figure 8: The four-vector kinematic plots for the first Higgs boson using the Standard Model and the Standard Model from the simplified squark model.

The last bin contains overflow events. The energy of a Higgs boson can never be smaller than the Higgs boson rest mass of 125 GeV, so the plot of the energy starts at that value. These plots show the case where the stop squark has a mass of 1 TeV.

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(a) px (b) py

(c) pz (d) E

Figure 9: The four-vector kinematic plots for the second Higgs boson using the Standard Model and the Standard Model from the simplified squark model.

The last bin contains overflow events. The energy of a Higgs boson can never be smaller than the Higgs boson rest mass of 125 GeV, so the plot of the energy starts at that value. These plots show the case where the stop squark has a mass of 1 TeV.

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Figure 10: This plot shows the p-values corresponding to the probabilities that the variables plotted from the two different SM samples are drawn from the same underlying distributions. In the cases where MadGraph creates data for each Higgs boson, those variables are called h1 and h2 for the first and second Higgs boson respectively. This plot shows the case where the stop squark has a mass of 1 TeV.

(a) First Higgs boson (b) Second Higgs boson

Figure 11: The absolute value of the momentum in the x-direction for the Higgs bosons from the Standard Model and the simplified squark model is plotted.

The last bin contains overflow events. These plots show the case where the stop squark has a mass of 1 TeV.

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(a) First Higgs boson (b) Second Higgs boson

Figure 12: The absolute value of the momentum in the y-direction for the Higgs bosons from the Standard Model and the simplified squark model is plotted.

The last bin contains overflow events. These plots show the case where the stop squark has a mass of 1 TeV.

mentum along the x- and y-directions, denoted as px and py. The px and py

components given by MadGraph are generated for both Higgs bosons, and the absolute of those components are shown in Figure 11 and Figure 12 respectively.

At a glance, the p_xand p_y components for the Standard Model and the simplified squark model are clearly distinguishable, with the Standard Model having a broad momentum distribution while the distribution for the simplified squark model is more peaked towards zero. Doing a Kolmogorov–Smirnov test on the two data sets gives:

px First Higgs boson Second Higgs boson

D_ks 0.1393 0.1393

pks 0.0 0.0

p_y First Higgs boson Second Higgs boson

Dks 0.1388 0.1388

pks 0.0 0.0

With p-values of 0.0 < 0.01, it is concluded that the px and py values for the SM and SSM cases are from different underlying distributions and so can be distinguished.

A more through examination of the p_x values also reveals that the p_x value for the first Higgs boson is the same size and opposite sign of the p_xvalue for the second Higgs boson in both the SM and SSM cases. The same can also be seen for the p_y values. That means that the sum of the momenta in the x-direction, and the sum of the momenta in the y-direction, for the two outgoing Higgs bosons is zero, which is as expected since momentum is conserved and the total momentum in the x- and y-directions was zero for the incoming gluons.

The third of the four-vector components is the momentum in the z-direction, also known as the pz component. The pz components given by MadGraph are generated for both Higgs bosons, and the absolute of those components are

Di-Higgs Production in the Standard Model and Beyond

Di-Higgs Production in the Standard Model and Beyond

Hanna Br¨ annstr¨ om

Di-Higgs Production in the Standard Model and Beyond

Hanna Br¨ annstr¨ om

Abstract

Acknowledgements

Contents

1 Introduction

1.1 Motivation

1.2 Standard Model

1.3 Limitations of the Standard Model

1.4 Supersymmetry and squarks

1.5 Simplified squark model

1.6 Kinematic distributions

1.7 Experimental overview

1.8 Introduction summary

2 Methodology

2.1 MadGraph

2.2 MadSpin

2.3 Pythia

2.4 Delphes

2.5 ROOT

2.6 Python

2.7 Kolmogorov–Smirnov test

2.8 Machine learning

3 Results

3.1 Coupling coefficients

3.2 Cross sections and weights

3.3 Parton level