Characterization of dark matter based on spin in the mono-Z channel

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Characterization of dark matter based on

spin in the mono-Z channel

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Aristotle said a bunch of stuff that was wrong. Galileo and Newton fixed things up. Then Einstein broke everything again. Now we have basically got it all worked out, except for small stuff, big stuff, hot stuff, cold stuff, fast stuff, heavy stuff, dark stuff, turbulence, and the concept of time

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Abstract

So far there is no detection of dark matter in hadron colliders like the Large Hadron Collider (LHC), even though the creation of dark matter particles in colliders is in principle possible. This thesis aims to explore the correlation of the missing trans-verse energy (MET) distributions with the spin of the dark matter (DM) candidates in the mono-Z channel at the LHC, in the case of signal detection which is asso-ciated with DM. Two WIMPs are considered as DM candidates, a real scalar DM particle and a Dirac fermion DM particle. For both DM candidates the assumption that their mediator is the SM Z-boson is made. The analysis is separated into two parts. In the first part, whose purpose is to set up the technique, the Z- boson is considered as the only particle in the final state and the background is not taken into consideration. For the second part, the recasting of a CMS mono-Z-boson at 2.3 f b−1 analysis is used to study the MET in the final state at the ee and µµ channels. The frameworks that are used in this thesis are:MadGraph5_aMC@NLOfor the generation

of the events,Pythia for the hadronization of Z-boson in the final state,Delphes

to simulate the effects of the detector, MadAnalysisfor the recasting of the CMS

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Sammanfattning,

Arten av mörk materia (DM) är en av de mest populära frågorna inom områdena partikel- och astropartikelfysik. Ett stort antal teoretiska modeller har utvecklats genom åren, inklusive svagt växelverkande massiva partiklar (WIMPs) som mörk materia-kandidater. Syftet med denna avhandling är klassificeringen av möjliga Dark Matter (DM) partiklar baserat på deras spin i mono-Z-kanalen i Large Hadron Collider (LHC) vid CERN.

I princip är skapandet av DM i hadronkolliderare som LHC möjligt och signalen från en DM-kandidat verkar vara en obalans i summan av den transversella rörelsemäng-den för de observerade partiklarna, eftersom DM-partiklar undviker detektering. Emellertid kan källan till händelser med saknad transversell energi (MET) inte definieras unikt av partiklar. För detta projekt används två typer av WIMP:ar som DM-partiklar, en reell skalär partikel S0 och en Dirac-fermion χ0 , medan båda partik-larna antas ha en Z-boson som medlare. Syftet med detta projekt är att hitta om MET-fördelningarna är tillräckliga för att skilja de två partiklarnas spin.

För den första delen av analysen skapas simuleringsdata medMadGraph5_aMC@NLO.

Z-boson är den enda synliga partikeln i det slutliga tillståndet, medan bakgrund-shändelserna inte beaktas i det stadiet. För den andra delen av analysen tillåts Z-boson att sönderfalla och kanalerna för två elektroner ee och två muons µµ stud-eras. För Z-bosonens sönderfall används simulationsprogrammetPythia, medan

analysen av det slutliga tillståndet är baserat på omarbetningen av CMS mono-Z-boson-analys vid en ljusstyrka på 2,3 f b−1. Till sist används ramen MaDM för beräkn-ing av mörk materia-mängden i universum för DM-kandidaterna. För båda delarna av analysen skapas simuleringsdata baserat på kravet på en signifikans av fem σ, eftersom syftet med analysen är att undersöka om karakterisering är möjlig när en upptäckt görs.

Resultaten från den andra delen av analysen visar att för luminositeter på 140 och 300 f b−1 , är skillnaden mellan MET-fördelningar för olika massor och olika spin möjlig med en 2σ konfidensnivå (CL). Ytterligare ökning av ljusstyrkan vid LHC re-sulterar i en ökning av diskrimineringen mellan MET-fördelningarna.

Å andra sidan visar den första delen av analysen att diskriminering för MET-fördelningar mellan DM-kandidater med liknande massor är ouppnåelig för luminositeter på 100

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Acknowledgements

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Introduction

1.1 Motivation

The nature of DM is one of the most popular unanswered questions in the field of particle and astroparticle physics. A large number of theoretical models have been developed through the years including weakly interacting massive particles (WIMPs) as dark matter candidates. This thesis aims to classify possible Dark Mat-ter (DM) particles based on their characMat-teristics and give predictions about their mediators. Specifically, it focuses on the spin of DM candidates that arise from the collisions in the Large Hadron Collider (LHC) at CERN in Geneva. In principle, the production of DM in hadron colliders like the LHC is possible and the signal coming from a DM candidate would appear as an unbalance in the sum of trans-verse momenta of the observed particles, since the DM particles escape detection1_. There are several signs which support the existence of DM in the Universe. For ex-ample, the rotation velocity of galaxies, the kinematic energy in Clusters and the Bullet Cluster are evidence of DM in the Universe. The experimental research for the DM can be categorized into three classes, the direct detection, indirect detection and the detection in colliders. This study aims at the collider detection of DM in the LHC, distributions of missing transverse energy (MET) are used to test whether DM particles with different spins and different masses have statistically different MET distributions.

1.2 Standard Model

Particles physics aims to understand the elementary components of the Universe and to describe the interactions between them. Until now, the Standard Model (SM) [1] is the leading theoretical model in particle physics, as it is in great agreement with the experimental data. The particles in the SM can be categorized into two main categories based on their spin. The first category is the vector bosons, which are the carriers of the forces and their spin is equal to one. The second category is the fermions, which are the building blocks of the Universe and their spin is a half-integer number (1/2). The main difference between those groups is that fermions follow the Pauli exclusion principle which allows them to create new matter particles with inner structure. The SM can be mathematically illustrated by the SU(3)_C×

SU(2)_L×U(1)_Ysymmetry group. The SU(3)_Cdescribes the colour symmetry, which is associated with the strong force. The SU(2)_L×U(1)_Y illustrates the electroweak 1_{However, the source of events with missing transverse momentum cannot be uniquely defined by}

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2 Chapter 1. Introduction

FIGURE1.1: Standard model particles [3].

theory, with SU(2)_L describing the interactions between the left-handed particles and the gauge bosons of the SU(2)group and the U(1)_Y being associated with the hypercharge, which is defined as:

Y=2(Q−I3) (1.1)

with Q to be the charge and I3the third component of the isospin [2]. The SM con-tains left-handed and right-handed particles. The former are contained in doublets, while the latter are singlets under the SU(2)group. In the SM there are three lepton doublets, each one containing a lepton and the corresponding neutrino as can be seen in Equation 1.2 νe e , νµ µ , ντ τ (1.2) The upper components of the doublets in Equation 1.2 have isospin I3 = +1/2 and the lower components have an isospin I3 = −1/2. From the Equation 1.1 it can be seen that all left-handed leptons have YL = −1, while the right-handed leptons YR = −2, however the SM does not include right-handed neutrinos. Except for the neutrinos and the leptons, the SM also includes quarks. There are three flavours of quarks as can be seen in Equation 1.3

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1.2. Standard Model 3 the Gravity. Except for the Gravity which is not included in the SM, all the other forces have mediators which are included in the SM. The gluons (g) are the carriers of the strong force, the W± and the Z bosons are the carriers of the weak force and the photon (γ) is the carrier of the EM. All the elementary particles of the SM are illustrated in Figure 1.1.

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Chapter 2

Beyond the Standard Model

2.1 Why to go Beyond the SM?

SM is the leading theory in particle physics as it predicts with great accuracy the ex-perimental data so far. However, there are some deficiencies arising and phenomena seeking an explanation. Such phenomena are the hierarchy problem and the strong CP problem, while the neutrino masses, the baryogenesis, the dark matter and the dark energy in the Universe are some of the SM deficiencies. The need for a new beyond the standard model (BSM) theory is inevitable.

2.2 Theoretical schortcomings of the SM

2.2.1 Hierarchy Problem

The Higgs mechanism is the process through which the particles gain masses. For calculations of the leading order (LO), the Higgs mechanism is in great agreement with the experimental data. However, the calculations of the Next to Leading Order (NLO) produce deficiencies. Below there is a simple example of a toy theory model where a single fermion S is coupled to a massive scalar φ and the NLO calculations of the mass are presented in order to describe their deficiency. The Langrangian of the toy theory is shown in Equation 2.1.

Lφ =iSγµ∂µS+ |∂µφ|2−m2s|φ|2−λfSSφ. (2.1) According to the Higgs mechanism the mass of the fermion is mf = λfv/

√

2 at LO, with v to be the vacuum expectation value. It is necessary to consider one-loop contributions to the fermion propagator due to the scalar particle. The renormalized mass is equal to:

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2.2. Theoretical schortcomings of the SM 5

FIGURE2.1: One loop corrections to the Higgs mass parameter in the

SM [7].

whereΛ is the cut-off of the theory, and the correction term, δmf, depends on the m_f. For small fermion masses the Lagrangian is invariant under the chiral transfor-mation. The corrections for the scalar propagator are the following

δm2_s = − λ2_f 8π Λ2₋_6m2 flog Λ mf +2m2_f +. . . (2.4) and appear to be quadratically divergent. The same problem occurs with the Higgs boson, with the Higgs potential to be V(φ) = −µφ2+λφ4and µ the mass parame-ter. The quadratic divergence of the NLO calculations adds a correction term δµ as shown in Equation 2.5 :

−µ2 → −µ2+δµ2. (2.5) The first-loop diagrams for the Higgs boson are shown in Figure 2.1. To solve this problem, an additional counter term needs to be included in order to cancel the quadratic divergence. However, that type of cancellations are highly unnatural. A second solution might be setting the cut-off scaleΛ. If ΛTeV then δµ2 µ2and that gives rise to the so-called hierarchy problem. The hierarchy problem is strongly correlated with the cut-off scale, since the value of the cut-off scale increases, the divergence of the Higgs mass increases too. A possible solution is to set a cut-off scale at 1 TeV, however in this way the SM theory is not valid for energies higher than 1 TeV, which is in contrast with the experimental data.

2.2.2 Strong CP Violation

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6 Chapter 2. Beyond the Standard Model where ˜ Fα µν = 1 2eµνρσF ρσα _(2.7)

The Langrangian term in Equation 2.6 contains a new parameter θ that leads to the CP violation, and it contributes to the neutron electric dipole moment, dn. The ex-perimental limit on θ is θ 10−9. The reason why θ is so small is known as the strong CP problem.

2.3 Evidence of BSM physics

2.3.1 Neutrino masses

Neutrinos are unique particles, as there are the only elementary fermions that are neutrally charged, and their mass is much smaller compared to the masses of the other elementary fermions [12]. The mass of neutrinos is an indication of a BSM theory because the prediction of the SM diverges from the experimental data. The mass terms in the SM theory are of the form:

νLνRmν (2.8)

with ν to be a neutrino and mν the mas of the neutrino. In the SM the mass of the

neutrinos is predicted to be mν =0 as there is no term like νLνRmνin the SM.

How-ever, experimental evidence determined that the mass of the neutrinos is different than zero, with the discovery of neutrino oscillations [13][14][15].

A possible solution to the neutrino mass problem would be to assume that neutri-nos are Majorana particles, meaning that neutrineutri-nos and anti-neutrineutri-nos are identical. However Majorana neutrinos mass terms would not allow gauge invariant mass terms within the SM. There are other possible solutions to the issue of the neutrino masses, as is the see-saw [16] mechanism but all of them are predicted in BSM theo-ries.

2.3.2 Baryogenesis

The creation of the particles started in the early stages of the Universe. It is expected that the number of matter particles should be equal to the number of antimatter particles, as everything emerged from the Big Bang. However, according to exper-imental results, it is found that the Universe contains more matter than antimatter. A measure of the matter-antimatter asymmetry of the Universe is provided by the ratio of the baryon (minus the anti-baryon) and photon number densities,

n≡ nb−ηb nγ

' nb

nγ

(2.9) where nbis the density number of baryons , nbis the density number of anti-baryons and nγ the density number of photons in the Universe. The question which arises

is what could create that asymmetry. As pointed out by Sakharov [17], a baryon asymmetry could be created in the early stages of the Universe if three necessary conditions are satisfied:

• baryon number (B) violation

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2.3. Evidence of BSM physics 7 • departure from thermal equilibrium

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8

Chapter 3

Dark Matter

Dark Matter (DM) is one of the most discussed, and unexplained at the same time, types of matter in the field of Physics. The existence of Dark Matter has been in-directly proven with several different measurements and gives evidence of a new theory Beyond the Standard Model (BSM). Dark Matter is a yet undetermined type of matter that does not emit or reflect electromagnetic radiation.

3.1 Evidence of DM

3.1.1 Galaxy clusters

The first evidence of DM was when Jan Oort in 1932 studied the orbital velocities of stars in the Milky Way. However, the name Dark Matter was given by Fritz Zwicky in the paper listed in Ref. [18]. Fritz Zwicky studied the Coma cluster, specifically he used the viral theorem to measure the kinematical energy of the cluster, then used that information to calculate the gravitational mass of the cluster. Later on, he mea-sured the luminous mass of the cluster and compared those masses. He discovered that the luminous mass was not enough to keep the system bound and named the missing mass Dark Matter.

3.1.2 Galactic Rotation Curves

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3.1. Evidence of DM 9

FIGURE3.1: Rotation curve of the NGC6503 galaxy, and contributions of different profiles [20].

3.1.3 Bullet cluster

Indications of DM can also be found in the bullet cluster, which is composed of two galaxy clusters which have collided and eventually merged. The calculation of the mass in the bullet cluster can be done with two methods: optical observations of X-ray emission and gravitational lensing [21]. When the two galaxy clusters collide, the luminous mass emits X-ray. The collision heats the system which then emits ra-diation. The X-rays provide information which can be used to predict the luminous mass and its location. The gravitational lensing can provide information about the mass of the bullet cluster through the gravitational effects. The gravitational force of the system bends the light of the background and focuses it. In other words, the bullet cluster can act as a lens through the gravitational effect, and the bend of the light can give predictions for the mass of the bullet cluster and its location. If the luminous mass is the only type of mass in the bullet cluster then the two measure-ments should predict the same amount of mass. As can be seen in Figure 3.2 the measurements of the mass have been performed with two different methods for the merging cluster 1E0657558 [22]. The coloured areas are the optical predictions of the luminous mass based on the X-rays and the green contours are the reconstructed mass from the gravitational lensing. There is a divergence between the two masses, which implies the existence of dark matter.

3.1.4 Cosmic Microwave Background

The Cosmic Microwave Background (CMB) is radiation that was emitted after the Big Bang at the early stages of the Universe and was accidentally discovered by Penzias and Wilson [23] at Bell Labs in 1965. The Universe at the early stages was opaque, but while it started to cool down, the protons and electrons started to create hydrogen atoms which made the Universe transparent. The photons that were cre-ated at that time could travel freely in the Universe, however during the expansion they started to lose energy and transform into the CMB.

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10 Chapter 3. Dark Matter

FIGURE3.2: Deep Chandra image of the merging cluster 1E0657558. The X-ray brightness of the gas component is coded in yellow, red and blue colours. Distribution of the gravitating mass, obtained from

weak lensing reconstruction, is shown by green contours [22].

the value of the energy density in present to be: ρ0 ≈

3H₀2 8πGN

≈10−29 g

cm3 (3.1)

with H0to be the Hubble constant and GNthe Newton’s gravitational constant. The components of energy density are 70% Dark Energy, 26% Dark Matter and 4% ordi-nary mass. Therefore, the amount of DM in the Universe is:

ΩDM ≡ ρDM,0

ρ0

≈0.26 (3.2)

with the density of the DM in the present to be ρDM,0 =10−6

GeV

cm3. (3.3)

The density of DM in the Universe is 105times less dense than the density in clusters of galaxies and 106less than galaxies. The conclusion is that the Universe has regions where it is highly inhomogeneous and the DM is a crucial ingredient to that [25]. Those large inhomogeneities are responsible for the structure of the Universe as we know it. The gravity in those high dense regions causes the matter to create structure formations, which would be impossible if the Universe was homogeneous.

3.1.5 Alternative explanations

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3.2. DM Candidates 11

3.2 DM Candidates

In the SM there are no candidates for DM particles. Even the neutrinos that were assumed in the beginning to be the DM particles due to their rare interaction with the other particles, are now crossed out of the list of possible DM particles, since their speed would eliminate the density fluctuations in the early Universe and the inhomogeneity that lead to the creation of larger structures in the Universe would not be possible.

There is a list of questions that can determine if a DM candidate is a "good" candidate or not. If the DM candidate can answer positively to those questions it could be a valid option. The list of questions includes the following, as presented in Ref. [27]:

• Does it match the appropriate relic density? • Is it cold?

• Is it neutral?

• Is it consistent with Big Bang nucleosynthesis (BBN)? • Does it leave stellar evolution unchanged?

• Is it compatible with constraints on self-interactions? • Is it consistent with direct DM searches?

• Is it compatible with gamma-ray constraints? • Is it compatible with other astrophysical bounds? • Can it be probed experimentally?

There are several DM candidates, for example, the Sterile Neutrino [28] and the Ax-ions [29] which are two examples of "good" DM candidates. The general name of DM candidates that interact weakly with the SM particles is weakly interacting massive particles (WIMPs). For the purpose of this project two WIMPs are tested, the real scalar dark matter particle (RSDM or S0) and the Dirac fermion dark matter particle (DFDM or χ0). Also, the assumption that the mediator of DM particles is the SM Z-boson is made. The production diagrams in hadron colliders such as LHC for the two WIMPs which are tested in this project are shown in Figure 3.3 for the RSDM and in Figure 3.4 for the DFDM. The corresponding Lagrangian terms are described in subsection 4.2.3.

3.3 Cosmology and DM

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12 Chapter 3. Dark Matter

FIGURE3.3: Feynman diagram for the production of RSDM particle at LHC

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3.4. DM Detection 13 are the WIMPs [31], which fit perfectly the features of the CDM. At the early stages when the temperature of the universe (T) was much larger than the mass of the DM candidate T mDM, the annihilation rate and the creation rate of the DM was in an equilibrium. As the Universe started to cool down and expand at temperatures T< mDM, the density of the dark matter started to drop exponentially and the tem-perature became too low for the annihilation to keep up with the expansion rate. Consequently, the DM froze-out with the cosmological abundance (relic) observed today [32].

3.4 DM Detection

3.4.1 Direct & Indirect Detection

One of the possible detection techniques for Dark Matter is the direct detection [33]. As mentioned in the previous section, the Universe is filled with DM, since there are fluxes of DM particles that travel through the Universe. In the case where the DM particles are WIMPs, then theoretically it is possible to measure the elastic scattering of a DM particle with other SM particles. The direct detection aims to measure the re-coil energy from the DM particles scattered with a nucleus. However, the interaction rate between WIMPs and SM matter is quite small [34]. To reduce the background events and seal the detectors from the Cosmic rays or other sources that could have signal similar to dark matter, the detectors are sited deep underground. In this way it is possible to measure even the small interaction rate of DM particles with the SM particles.

The indirect detection of dark matter [35] aims to detect the annihilation of DM into SM particles. The annihilation of the WIMPs could create a secondary signal that could be measured from telescopes or detectors. The most interesting case is the Gamma rays that travel from places with high density in DM to the Earth. The gamma rays can provide information about the nature of the DM particles, since different types of DM predict different gamma ray spectra.

3.4.2 Detection at LHC

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Chapter 4

EFT and Simplified Models

4.1 EFT

4.1.1 Introduction to EFT

Effective Field Theory (EFT) introduces operators of higher dimension to the La-grangian, which are invariant under Lorentz and gauge transformations, as well as coefficients that are model-independent. The introduced operators with the higher mass dimensions can probe new physics phenomena and interactions. EFT is broadly used as it is a model-independent framework and contains simple mathematical ex-pressions of the tested interactions. However, EFT is a theory with a limitation, since there is a limit called cut-off and is symbolized withΛ, with the units of Λ being[eV]. The EFT can be used to study phenomena that their energy scale (E) is significantly smaller than the cut-off E _{Λ, while for energies that are larger than the cut-off,}

E > Λ, the EFT is not valid. EFTs use perturbation theory to expand the fraction

En/Λnin order to perform calculations with the interaction terms, while in the case where it is impossible to expand the fraction, the EFTs are not valid [39].

4.1.2 Structure of an EFT

The nature and the interactions of DM are still unknown. Therefore, to test DM can-didates or DM models it is necessary to introduce constraints in the analysis of the experimental data. Those constraints must be model-independent so that a large range of DM models can be considered. One of the most effective approaches to study DM is the EFTs as there are simple but efficient too. The EFT approach can be used independently of the DM models, but there are limits where the EFTs are valid. In the search for DM particles, EFTs include interactions between DM particles and SM particles. Below there is an interaction term shown, where a fermionic DM par-ticle "χ" interacts with a SM quark "q" and the limitation of the theory is given by the energy scale "Λ∗"

LEFT = _Λ1₂

∗

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4.2. Simplified Models 15

FIGURE4.1: Feynman diagram for the interaction term of Equation 4.1.

FIGURE4.2: Feynman diagram for a posible simplified model.

4.2 Simplified Models

4.2.1 Introduction to Simplified Models

The simplified models which are used for the search of DM aim to extend the EFTs, so that it includes a mediator. The goal with the simplified models is to see the inner structure that is neglected with the EFT. For example, in Figure 4.1 which cor-responds to the EFT and in Figure 4.2 which corcor-responds to the simplified model, there existe an inner structure in the vertex of the EFT and a mediator "Z0" is intro-duced. The mediator is responsible for the interactions of the DM with the SM [39].

4.2.2 Structure of a Simplified Model

The simplified models introduce mediators, and in order for this to be achieved, the SM is expanded by adding new degrees of freedom. The "new" model should be kept as simple as possible, but also a well-defined theory. When a new simplified model is created there is a set of rules that can be followed as a guide [39].

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16 Chapter 4. EFT and Simplified Models

• The new interaction operators in the Lagrangian should be renormalizable, Lorentz invariant, SM gauge invariant and insure DM stability.

In addition to the SM gauge symmetries, usually a new discrete symmetry Z2 is introduced. The Z2 contains a new parity where the DM particles under the new parity are "odd" and the SM particles are "even". The Z2symmetry is used to stabilize the DM particles. An example of such symmetry is R-parity in SUSY.

4.2.3 Example of a Simplified Model

In this paragraph, a simplified model is introduced, where a "Z’" vector mediator interacts with DM particles through the s-channel diagrams. With the "Z’" to be a spin-1 neutrally charged particle and with "even parity" under the Z2symmetry. The couplings of the DM models are in principle determined by the DM representation under the SU(2) ×U(1), but they are free to parametrize further new physics.

First case: Scalar DM

For the case of a complex scalar DM particle "S" in the s-channel with mass "mS" and a mediator "Z0" with mass "m_Z0" the interaction term of the SM with the DM is given

by the Langrangian: LSDM int = −Z0 µ gSS∗ i∂µS −S i∂µS ∗ +gsZ0µZ 0µ_S∗_S _(4.2)

The collider phenomenology search of the complex scalar DM, with the vector me-diator depends on the leading decay channels of the meme-diator. Below there is the decay width of the vector mediator to SM fermions. The "Nc" stands for the color number of fermions and " f " is the SM fermions.

. ΓZ0 → f f= Nc(f) MZ0 12π v u u t1− 4m2_f M2 Z0 " |gV_f |2 1+ 2m 2 f M2 Z0 ! + |gA_f |2 1− 4m 2 f MZ0 !# . (4.3) The decay width of the mediator to complex DM particles is equal to:

Γ Z0 = g 2 sMZ0 48π 1−4m 2 s mZ0 3/2 . (4.4)

For the simpler case of a real scalar DM particle "S" under the SU(2)symmetry with a "Z0" mediator the interaction term arise from the the kinematic term.

1 2DµSD µ_S₌ 1 2∂µS∂ µ_S₊ 1 2g 2_Z0 µZ 0µ_SS _(4.5) were Dµ =∂µ−igZ 0 µ. (4.6)

The interaction term is:

L_int⊇ g2Z_µ0Z0µ_SS _(4.7)

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4.2. Simplified Models 17

Second case: Fermionic DM

The Lagrangian of a fermionic DM with a vector mediator "Z’" is:

LFDM int = −Z0 µ " χγµ gV_χ +g_χAχ+

∑

f f γµ gV_f +gA_f γ5 f # (4.8)

For the phenomenological study of the fermionic DM, the decay width of the me-diator to DM is the as in Equation 4.3, but now the mass of the DM is introduced instead of the fermion mass.

Z-boson as Mediator

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Chapter 5

Simulation framework

For this project, the analysis is separated into two parts. In the parton analysis, the final state contains two DM particles and the Z-boson, while in the detector level the Z-boson in the final state decays further. This study uses data that are gener-ated withMadGraph5_aMC@NLO[40], which is a computational program that can be

used for phenomenological studies in order to perform computations of tree-level and next-to-leading order cross sections and provide the kinematic variables. For the decay of Z-bosonPythia 8 is used. Pythia [41] is a framework for the

gen-eration of high-energy physics events that interprets hard and soft interactions and generates the decays of particles in the initial or final state and can also be used to generate particle showers. For this project, Pythia is used as an extension of MadGraph5_aMC@NLOso that the hadronization is generated.

MadAnalysis5[42] is the framework that is used for the analysis of

phenomeno-logical studies in collider physics. It is used in the recasting of the CMS analysis CMS-EXO-16-010 [43]. Also, for the recasting theDelphes3 [44] program is used

to simulate the effects of the detectors. Last theMadDM3[45] is used to calculate the

relic density of the DM candidates, whereMadDMis a numerical tool which is used

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19

Chapter 6

Analysis

6.1 Analysis introduction

This project aims to explore the different kinematic variables and their correlation with the spin of a dark matter (DM) particle. There is a variety of models which include different types of DM particles i.e. boson DM particles or fermionic DM particles. The goal of this project is to find the limit where the two kinematic distri-butions are statistically different within a 2σ confidence level (CL). In this study, we focus on mono-Z + MET in the final. Also, it is assumed that the DM particles have the Z-boson as a mediator. In the final state, there are two DM particles and one Z-boson, from which the only visible particle is the Z-boson. The MET distribution is equal to the transverse momentum PT of the Z-boson. However, for the detector level analysis, the Z-boson is allowed to decay and the MET is equal to the absolute value of the vector sum of the transverse momentum (PT) of all the visible particles

/

ET = |

∑

observed

~_P_T_|_, _(6.1)

with /ETto be the MET.

After the calculations of the MET distributions which are explained in the next two sections, follows the statistical analysis of those distributions. The statistical test which is used in this project is the χ2_test,

χ2= k

∑

i=1

(R_RSDMi−R_DFDMi)2

Max(R_RSDMi, R_DFDMi) (6.2)

In Equation 6.2 there is a difference in the denominator compared to the usual χ2test. The difference is due to the non-existing null hypothesis. There are two possible DM particles, but none of those is taken as the null hypothesis, both are possible scenar-ios. The R_RSDMiis the signal event per bin for S0and R_DFDMi is the signal event per bin for χ0_{, the degrees of freedom are equal with the number of bins. For both} Par-ton and detector level, the simulation data are generated based on the requirement of the significance to be equal to five as we are interested in the specific case where a discovery of a DM particle has been made. In the next sections, examples for both Parton and detector level analysis are given.

6.2 Parton analysis

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20 Chapter 6. Analysis

TABLE6.1: Parton Analysis simulations, information for Point 1 and Point 2 where the MET distributions for the mass point 1 and 2 are

different with 2σ CL at luminosity L= 100 f b−1

Points Point 1 Point 2

DM S0 _χ0

Mass [GeV] 20 1000

Coupling 0.0449 0.85 Cross-section [pb] 7.57×10−4 3.14×10−4

distributions are different with less than 2σ CL. In order to calculate the χ2 test, the signal or real event per bin is needed. In the Parton level, the equation for the significance is equal to:

significance= √s

s , (6.3)

where s is the signal events or real events. For the first part of the analysis, the back-ground is not taken into consideration for simplicity. Table 6.1 contains information for the two mass points that are used for the first eample i.e. the MET distributions are statistically different with a 2σ CL. The signal events can be calculated by the equation below

s =L·σ (6.4)

where L is the luminosity and σ the cross-section. Taking into account Equation 6.3 and that the significance is equal to five, it can be shown that the signal events (S) are equal with 25 events. By setting the luminosity to 100 f b−1 the cross-section is the only unknown variable in Equation 6.4 and can be easily calculated. The lumi-nosity of 100 f b−1is just an example which is used in this section, and for the next section the luminosities of Run II (140 f b−1) and the expected luminosity of Run III (300 f b−1_{) are used.}

At the first stage theMadGraph5_aMC@NLO(MG) is used to calculate the cross-section.

The MG needs as inputs the couplings and the masses to generate the output file which includes the cross-section and all the kinematic variables that are needed for the calculation of the MET.

In the MET distributions cuts are included so that the analysis is more realistic, hence the events with MET under 80 GeV are neglected. Then a limitation in the number of events per bin is also included, for example, if one bin contains less than one event then the number of events in that bin is set to zero. The cuts are summarized in Table 6.2

TABLE6.2: The allowed regions for MET and events per bin in Parton Level analysis

Quantity Allowed regions MET > 80 GeV events per bin > 1

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6.2. Parton analysis 21 efficiency. The efficiency is defined as the signal events that survived the analysis divided by the total signal events.

s =L·σ·ε. (6.5)

The cross-section must be higher than the one calculated by the Equation 6.4, as the efficiency is always smaller than one. The information for all the simulated points is contained in Table 6.3. Figure 6.1 shows the values of the Table 6.3 and their corre-lation in the mass coupling plane. Table 6.1 contains information for the couplings and cross-sections which lead to 25 signal events with the efficiency factor.

TABLE6.3: The table contains information for the simulations of the S0 (RSDM) and χ0 (DFDM) particle for luminosity 100 f b−1

corre-sponding to'25 events after the cuts

S0 χ0

Mass[GeV] coupling cross-section [pb] coupling cross-section [pb] 10 0.0421 1.07×10−3 6.05×10−3 1.2×10−3 20 0.0449 7.57×10−4 6.42×10−3 1.19×10−3 50 0.0607 4.55×10−4 3.86×10−2 8.66×10−4 100 0.0981 3.41×10−4 1×10−1 5.57×10−4 200 0.2034 2.96×10−4 1.85×10−1 4.33×10−4 500 0.8534 2.9×10−4 4.01×10−1 3.36×10−4 1000 4.4854 2.94×10−4 8.5×10−1 3.14×10−4 1500 1.56 3.04×10−4 2000 2.8 3.1×10−4

FIGURE 6.1: S0 and χ0 mass vs coupling plane for luminosity 100 f b−1.

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FIGURE 6.2: MET distributions for point 1 and point 2 with L=

100 f b−1.

TABLE6.4: Information for the chi-square test and the standard devi-ation for points 1 and 2

χ2 D.O.F. σ 10.226 4 2.088

presented in Table 6.41. The goal of the χ2 test analysis is to compare the shape of the distributions. To perform the chi-square test, the number of events per bin must be calculated. For that reason Equation 6.5 has to be multiplied by the weights of the MET distributions. The number of events per bin is given by Equation 6.6

si = L·σ·ε·Wi (6.6) where si is the number of events per bin, the index "i" symbolizes the different bins and Wirefers to the weight of each bin. The second example pertains to points three and four, whose information can be found in Table 6.5. The steps are the similar as in the first example. The MET distributions are shown in Figure 6.3 and information for the χ2 test is gathered in Table 6.6. The distributions of the Points 3 and 4 are very similar as can be seen in Figure 6.3 and the value of the χ2is 0σ CL. Table 6.7, Table 6.8 and Table 6.9 in this section compare the MET distributions for the S0_and χ0particles. The first line of the tables contains the different masses of the first DM particle in GeV and the first column contains the different masses of the second DM particle in GeV. The values within the tables represent the CL of the kinematic dis-tributions, while the values inside the parenthesis are the degrees of freedom. In the tables where the same DM particles are considered, the points with the same mass are not tested since they are the same distributions. Those tables are also symmetric, so only the upper part of them is filled. The values that have CL over 2 are high-lighted with red. The highhigh-lighted values are for mass points with large mass gaps, 1_{For simplicity we have considered only the bins in which both distributions have more than 1}

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6.2. Parton analysis 23

TABLE6.5: Parton Analysis simulations, information for Point 3 and Point 4 for L= 100 f b−1

Points Point 3 Point 4

DM S0 χ0

Mass [GeV] 50 200

Coupling 0.0607 0.1850 Cross-section [pb] 4.55×10−4 4.33×10−4

FIGURE 6.3: MET distributions for point 3 and point 4 with L= 100 f b−1.

TABLE6.6: Information for the chi-square test and the standard devi-ation for points 3 and 4

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TABLE 6.7: DFDM vs DFDM particles for luminosity 100 f b−1, the

values within the table represent the standard deviation σ of the kine-matic distributions. GeV χ0 χ0 10 20 50 100 200 500 1000 1500 2000 10 0.007(3) 0.038(3) 0.626(3) 1.228(3) 2.216(3) 2.87(3) 3.079(3) 3.139(3) 20 0.124(3) 0.832(3) 1.413(3) 2.333(3) 2.95(3) 3.149(3) 3.204(3) 50 0.295(3) 0.865(3) 1.927(3) 2.637(3) 2.864(3) 2.932(3) 100 0.187(3) 1.316(3) 2.171(3) 2.458(3) 2.554(3) 200 0.556(4) 1.508(4) 1.817(4) 1.953(4) 500 0.167(6) 0.456(6) 0.608(6) 1000 0.0005(7) 0.004(7) 1500 0(9) 2000

while the mass points with small mass gaps have CL lower than 2. The only excep-tion in the above is the 1000 GeV mass point of S0_{, which has CL values always less} than 2. The MET distribution of this point reaches energies higher than the other mass points and the overlap with the MET distributions of the mass points with small masses is very small, that can be seen from the DOF which are equal to one for small masses.

TABLE 6.8: RSDM vs DFDM particles for luminosity 100 f b−1, the values within the table represent the standard deviation σ of the

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6.3. Detector analysis 25

TABLE 6.9: RSDM vs RSDM for luminosity 100 f b−1, the values

within the table represent the standard deviation σ of the kinematic distributions. GeV S0 S0 10 20 50 100 200 500 1000 10 0.004(4) 0.323(4) 1.438(4) 2.637(4) 2.568(3) 0.619(2) 20 0.137(4) 1.170(4) 2.445(4) 2.574(3) 0.688(2) 50 0.417(4) 1.805(4) 2.465(3) 0.850(2) 100 0.642(5) 2.138(4) 1.116(3) 200 0.883(6) 1.045(5) 500 0.055(8) 1000

6.3 Detector analysis

The detector level analysis is a more realistic interpretation of the data, as it includes the detector configuration, realistic cuts and takes into account the decay of the Z-boson in the final state. The recasting is based on the CMS analysis [43], which con-tains two channels. The first channel includes two electrons in the final state while the second channel has two muons in the final state. Each case is treated differently in the analysis due to the different number of background events.

The goal of the analysis is to explore if the MET distributions are enough to distin-guish the spin of the DM particles. The difference with the Parton level analysis is that the significance here is calculated from the Asimov formula of Equation 6.7, which takes into account background events and the error of the background events as well. The Asimov significance is given by:

Z= " 2 (s+b)ln " (s+b) b+σ_b2 b2_{+ (}_s₊_b₎_σ2 b # − b2 σ_b2ln " 1+ σ 2 bs b b+σ_b2 #!#1/2 , (6.7) where "s" is the number of signal events, "b" is the background events and "σb" is the error of the background events.

The CMS analysis provides the number of background events and the error values of the background for a luminosity of 2.3 f b−1 which can be found in Table 6.10. By renormalizing the background events, it is possible to calculate the background events for different luminosities. The renormalization process is shown in Equa-tion 6.8. b_analysis bnew = Lanalysis Lnew →bnew=banalysis Lnew L_analysis (6.8) where L_analysis = 2.3 f b−1 _{is the luminosity of the CMS analysis, L}

new is the new fixed luminosity, banalysis are the background events of the CMS analysis and bnew are the background events for the new luminosity. It is assumed that the systematic error of the background events is equal to 10% of the background and it becomes the dominant source of uncertainty for L≥10 f b−1.

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TABLE6.10: Background events and error of the background events from the CMS analysis

Channel Luminosity [ f b−1] Background events σ_b

ee 2.3 22 5.53

µµ 2.3 44 7.15

TABLE6.11: Background events for the two different channels

Channel Luminosity [ f b−1] Background events

ee 100 956.52 ee 140 1339.13 ee 300 2869.56 µµ 100 1913.04 µµ 140 2678.26 µµ 300 5739.13

contains the signal events for each luminosity and channel calculated from Equa-tion 6.7.

In the next stage, the same mass points as in the Parton level are assumed, but with different couplings, cross-sections and for different luminosity. First, a simulation is generated, where the cross-section is the solution of Equation 6.4. This corresponds to a test simulation in order to calculate the efficiency factor.

The DELPHES [46] extension program is used to simulate the effects of the detec-tors and the_MadAnalysis5is used for the recasting of the CMS analysis. At that stage, the efficiency can be extracted form the output file of MadAnalysis5. The

test simulation is only used to calculate the efficiency factor which is then used in Equation 6.5 to provide the value of the new cross-section.

The MET distributions of the Point 1 and 2 are shown in Figure 6.4 and for Point 3 and 4 in Figure 6.6. Both distributions correspond to the ee channel with a lumi-nosity of 140 f b−1_{. The standard deviation values for the Point 1 and 2 is 21.142} with 10 DOF and for the Points 3 and 4 is 3.552 with 7 DOF. The information for the couplings, cross-section and the efficiency for the mass points 1, 2, 3, 4 and for all the other mass points are presented in Table 6.13 and Table 6.14. In both tables the cross-section and coupling correspond to the number of events needed to have

TABLE6.12: Signal events for significance equal to five

Channel Luminosity [ f b−1] signal events

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6.3. Detector analysis 27 significance equal to five for the ee channel and those values are also illustrated in Figure 6.5. Table 6.15, Table 6.16 and Table 6.17 contain the values of the χ2test for a luminosity of 140 f b−1for the ee channel. The tables with the standard deviation values for luminosity 300 f b−1_{can be found in Appendix A. Both of the examples} are statistically different with more than 2σ CL. Higher luminosity provides a better resolution between the MET distributions. For the µµ channel the steps are the same as for ee channel. All the relevant information about detector level simulations for the µµ and ee channel are in Appendix A.

TABLE6.13: The table below contains information for the simulations of the χ0(DFDM) particle for the channel ee with luminosity 140 f b−1

Mass[GeV] coupling cross-section [pb] efficiency

10 0.393 5.125 0.0012 20 0.491 7.113 0.0008 50 1.1305 3.011 0.0019 100 1.3 1.72 0.0034 200 1.6997 1.259 0.0045 500 3.0908 1.012 0.0061 1000 6 0.7708 0.0077

TABLE6.14: The table below contains information for the simulations of the S0(RSDM) particle for the channel ee with luminosity 140 f b−1

Mass[GeV] coupling cross-section [pb] efficiency

10 2.5767 4.0376 0.0014

20 2.6303 2.5841 0.0022

50 3.4347 1.4499 0.004

100 5.0728 0.9127 0.0063

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FIGURE 6.4: MET distributions for point 1 and point 2 with L=

140 f b−1with Real event to be the signal events.

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6.3. Detector analysis 29

FIGURE 6.6: MET distributions for point 3 and point 4 with L= 140 f b−1. The Real event in y-axis refer to signal events.

TABLE6.15: DFDM vs DFDM. The values within the table represent the standard deviation σ and in the parenthesis are the D.O.F for the

kinematic distributions for luminosity 140 f b−1and channel ee.

ee χ0 χ0 [GeV] 10 20 50 100 200 500 1000 10 5.643(3) 8.995(3) 11.973(3) 14.641(3) 16.902(3) 19.837(3) 20 4.716(3) 6.589(3) 10.082(3) 16.833(3) 18.113(3) 50 6.452(3) 6.943(4) 15.111(4) 17.413(4) 100 8.145(5) 19.747(6) 20.903(6) 200 14.051(7) 14.720(7) 500 13.337(11) 1000

TABLE6.16: RSDM vs DFDM. The values within the table represent the standard deviation σ and in the parenthesis are the D.O.F for the

ee S0 χ0 [GeV] 10 20 50 100 200 10 7.127(3) 6.335(3) 12.370(3) 15.490(4) 20.472(3) 20 4.693(3) 3.441(3) 8.505(3) 13.584(3) 19.631(3) 50 7.544(4) 6.443(4) 6.587(4) 11.325(4) 18.345(4) 100 8.093(6) 8.236(6) 9.617(6) 15.587(6) 22.211(6) 200 9.511(7) 9.773(7) 3.552(7) 8.898(7) 17.628(7) 500 16.609(9) 16.631(9) 13.201(10) 7.7490(11) 10.095(11) 1000 18.907(9) 21.142(10) 18.303(11) 14.917(13) 11.158(14)

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30 Chapter 6. Analysis the points with small mass gaps, as it was pointed out in the Parton level too, while the mass points with small mass gaps have more DOF, which makes the distinction of the MET distributions more accurate.

TABLE6.17: RSDM vs RSDM. The values within the table represent

the standard deviation σ and in the parenthesis are the D.O.F for the kinematic distributions for luminosity 140 f b−1and channel ee.

ee S0 S0 [GeV] 10 20 50 100 200 10 2.752(8) 7.685(8) 12.824(9) 20.282(9) 20 7.773(10) 13.820(10) 21.451(10) 50 8.193(11) 17.655(11) 100 10.323(23) 200

6.4 Relic Density

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31

Chapter 7

Conclusion

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32

Appendix A

A.1 Hadron analysis simulation data

TABLEA.1: The table below contains information for the simulations of the χ0(DFDM) particle for the channel µµ with luminosity 140 f b−1

Mass[GeV] coupling cross-section [pb] efficiency significance

10 0.43 6.147 0.0019 5.09 20 0.4961 7.273 0.0017 5.35 50 1.13 2.995 0.0038 4.98 100 1.4 2.295 0.0053 5.364 200 1.9 1.967 0.0063 5.36 500 3.4 1.472 0.0075 4.85 1000 6.6 1.127 0.0098 4.89

TABLEA.2: The table below contains information for the simulations of the S0(RSDM) particle for the channel µµ with luminosity 140 f b−1

10 2.9713 5.3688 0.002 5

20 3.0277 3.4238 0.0033 5

50 4.0297 1.9957 0.0057 5

100 5.8072 1.1962 0.0095 5

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A.1. Hadron analysis simulation data 33

TABLEA.3: The table below contains information for the simulations of the χ0_{(DFDM) particle for the channel ee with luminosity 300 f b}−1

10 0.393 5.125 0.0012 5.33 20 0.49 7.113 0.0008 4.98 50 1.1305 3.011 0.0019 5.23 100 1.3 1.72 0.0034 5.23 200 1.6997 1.259 0.0045 4.96 500 3.09 1.005 0.0055 4.86 1000 6 0.7708 0.0077 5.17

TABLEA.4: The table below contains information for the simulations of the S0(RSDM) particle for the channel ee with luminosity 300 f b−1

10 2.553 3.9652 0.0014 5

20 2.6066 2.5377 0.0022 5

50 3.4038 1.4239 0.004 5

100 5.0271 0.8963 0.0063 5

200 9.7785 0.683 0.0083 5

TABLEA.5: The table below contains information for the simulations of the χ0(DFDM) particle for the channel µµ with luminosity 300 f b−1

10 0.415 5.711 0.002 5.03 20 0.496 7.273 0.0017 5.39 50 1.131 3.004 0.0035 4.8 100 1.4 2.295 0.0053 5.4 200 1.90 1.967 0.0063 5.4 500 3.4 1.472 0.0075 4.89 1000 6.6 1.127 0.0098 4.93

TABLEA.6: The table below contains information for the simulations of the S0(RSDM) particle for the channel µµ with luminosity 300 f b−1

10 2.9562 5.3145 0.00214 5

20 3.0123 3.3892 0.0033 5

50 4.0093 1.9755 0.0057 5

100 5.7778 1.1841 0.0095 5

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34 Appendix A. Appendix A

A.2 Hadron analysis plots for mass coupling plane and

sig-nificance

The z-axis in the figures is the significance. The different colours indicate different significance values in the mass coupling plane

FIGUREA.1: S0 and χ0 mass vs coupling plane for the channel µµ and luminosity 140 f b−1,the Z axis is the significance.

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A.3. Hadron analysis tables with the standard deviation values 35

FIGUREA.3: S0 and χ0 mass vs coupling plane for the channel µµ and luminosity 300 f b−1,the Z axis is the significance.

A.3 Hadron analysis tables with the standard deviation

val-ues

The tables in that section compare the MET distributions for two DM particles. The first line of the tables contains the different masses of the first DM particle in GeV and the fist column contains the different masses of the second DM particle in GeV. The values within the tables represent the standard deviation σ of the kinematic distributions and in the parethesis are the DOF.

TABLEA.7: DFDM vs DFDM. The values within the table represent the standard deviation σ and in the parenthesis are the D.O.F for the

kinematic distributions for luminosity 140 f b−1and channel µµ.

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TABLEA.8: RSDM vs DFDM. The values within the table represent the standard deviation σ and in the parenthesis are the D.O.F for the

µµ S0 χ0 [GeV] 10 20 50 100 200 10 13.146(3) 16.626(3) 22.706(3) 29.904(3) 33.023(3) 20 11.255(4) 13.768(4) 20.379(4) 29.200(4) 32.591(4) 50 7.078(4) 11.583(4) 15.664(4) 25.318(4) 28.752(4) 100 15.838(4) 11.761(4) 17.402(4) 25.168(4) 30.323(4) 200 18.553(8) 15.046(8) 9.127(8) 14.662(8) 23.184(8) 500 29.787(10) 28.526(11) 21.478(12) 15.238(13) 10.492(13) 1000 31.517(10) 30.959(11) 24.736(12) 20.665(13) 12.446(13)

TABLE A.9: RSDM vs RSDM. The values within the table represent the standard deviation σ and in the parenthesis are the D.O.F for the

µµ S0 χ0 [GeV] 10 20 50 100 200 10 6.673(10) 14.126(10) 25.566(10) 31.862(10) 20 6.673(10) 11.116(11) 22.728(11) 31.041(11) 50 14.126(10) 11.116(11) 13.285(12) 23.2007(12) 100 25.566(10) 22.728(11) 13.285(12) 14.472(14) 200 31.862(10) 31.041(11) 23.2007(12) 14.472(14)

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A.3. Hadron analysis tables with the standard deviation values 37

ee S0 χ0 [GeV] 10 20 50 100 200 10 11.311(3) 10.132(3) 18.599(3) 23.075(3) 30.244(3) 20 7.867(3) 6.064(3) 13.040(3) 20.332(3) 29.025(3) 50 11.858(4) 10.260(4) 10.653(4) 17.267(4) 27.265(4) 100 12.656(6) 12.861(6) 14.859(6) 32.898(6) 21.683(3) 200 14.587(7) 14.957(7) 6.143(7) 13.810(7) 26.274(7) 500 21.268(8) 22.697(9) 19.928(10) 13.541(10) 18.538(10) 1000 28.170(9) 31.436(10) 27.397(11) 22.689(13) 17.473(14)

TABLEA.12: RSDM vs RSDM. The values within the table represent the standard deviation σ and in the parenthesis are the D.O.F for the

ee S0 S0 [GeV] 10 20 50 100 200 10 5.106(8) 12.038(8) 19.401(9) 30.056(9) 20 5.106(8) 12.336(10) 20.890(10) 31.793(10) 50 12.038(8) 12.336(10) 13.008(11) 26.423(11) 100 19.401(9) 20.890(10) 13.008(11) 16.152(13) 200 30.056(9) 31.793(10) 26.423(11) 16.152(13)

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µµ S0 χ0 [GeV] 10 20 50 100 200 10 21.068(3) 27.280(3) 33.603(3) 43.734(3) 47.744(3) 20 17.075(4) 20.658(4) 30.191(4) 42.940(4) 47.892(4) 50 5.014(5) 8.806(5) 16.289(5) 32.490(5) 39.672(5) 100 23.547(4) 17.751(4) 25.941(4) 37.180(4) 44.663(4) 200 23.547(4) 17.751(4) 25.941(4) 37.180(4) 44.663(4) 500 44.047(10) 42.262(11) 32.134(12) 23.165(13) 16.496(13) 1000 46.566(10) 45.799(11) 36.839(12) 30.993(13) 19.287(13)

TABLEA.15: RSDM vs RSDM. The values within the table represent the standard deviation σ and in the parenthesis are the D.O.F for the

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39

List of Figures

1.1 Standard model particles [3]. . . 2

2.1 One loop corrections to the Higgs mass parameter in the SM [7]. . . 5

3.1 Rotation curve of the NGC6503 galaxy, and contributions of different profiles [20]. . . 9

3.2 Deep Chandra image of the merging cluster 1E0657558. The X-ray brightness of the gas component is coded in yellow, red and blue colours. Distribution of the gravitating mass, obtained from weak lensing reconstruction, is shown by green contours [22]. . . 10

3.3 Feynman diagram for the production of RSDM particle at LHC . . . . 12

3.4 Feynman diagram for the production of DFDM particle at LHC . . . . 12

4.1 Feynman diagram for the interaction term of Equation 4.1. . . 15

4.2 Feynman diagram for a posible simplified model. . . 15

6.1 S0_{and χ}0_{mass vs coupling plane for luminosity 100 f b}−1 _{. . . 21}

6.2 MET distributions for point 1 and point 2 with L= 100 f b−1. . . 22

6.3 MET distributions for point 3 and point 4 with L= 100 f b−1. . . 23

6.4 MET distributions for point 1 and point 2 with L= 140 f b−1_{with Real} event to be the signal events. . . 28

6.5 S0 and χ0 mass vs coupling plane for the channel ee and luminosity 140 f b−1. The points correspond to significance'5 . . . 28

6.6 MET distributions for point 3 and point 4 with L= 140 f b−1. The Real event in y-axis refer to signal events. . . 29

A.1 S0 and χ0 mass vs coupling plane for the channel µµ and luminosity 140 f b−1,the Z axis is the significance . . . 34

A.2 S0 and χ0 mass vs coupling plane for the channel ee and luminosity 300 f b−1,the Z axis is the significance . . . 34

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40

List of Tables

6.1 Parton Analysis simulations, information for Point 1 and Point 2 where the MET distributions for the mass point 1 and 2 are different with 2σ CL at luminosity L= 100 f b−1 . . . 20 6.2 The allowed regions for MET and events per bin in Parton Level analysis 20 6.3 The table contains information for the simulations of the S0 _(RSDM)

and χ0(DFDM) particle for luminosity 100 f b−1corresponding to'25 events after the cuts . . . 21 6.4 Information for the chi-square test and the standard deviation for

points 1 and 2 . . . 22 6.5 Parton Analysis simulations, information for Point 3 and Point 4 for

L= 100 f b−1 _{. . . 23} 6.6 Information for the chi-square test and the standard deviation for

points 3 and 4 . . . 23 6.7 DFDM vs DFDM particles for luminosity 100 f b−1, the values within

the table represent the standard deviation σ of the kinematic distribu-tions. . . 24 6.8 RSDM vs DFDM particles for luminosity 100 f b−1, the values within

the table represent the standard deviation σ of the kinematic distribu-tions. . . 24 6.9 RSDM vs RSDM for luminosity 100 f b−1, the values within the table

represent the standard deviation σ of the kinematic distributions. . . . 25 6.10 Background events and error of the background events from the CMS

analysis . . . 26 6.11 Background events for the two different channels . . . 26 6.12 Signal events for significance equal to five . . . 26 6.13 The table below contains information for the simulations of the χ0

(DFDM) particle for the channel ee with luminosity 140 f b−1 . . . 27 6.14 The table below contains information for the simulations of the S0

(RSDM) particle for the channel ee with luminosity 140 f b−1 . . . 27 6.15 DFDM vs DFDM. The values within the table represent the standard

deviation σ and in the parenthesis are the D.O.F for the kinematic distributions for luminosity 140 f b−1and channel ee. . . 29 6.16 RSDM vs DFDM. The values within the table represent the standard

deviation σ and in the parenthesis are the D.O.F for the kinematic distributions for luminosity 140 f b−1and channel ee. . . 29 6.17 RSDM vs RSDM. The values within the table represent the standard

deviation σ and in the parenthesis are the D.O.F for the kinematic distributions for luminosity 140 f b−1and channel ee. . . 30 A.1 The table below contains information for the simulations of the χ0

(DFDM) particle for the channel µµ with luminosity 140 f b−1 . . . 32 A.2 The table below contains information for the simulations of the S0

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LIST OF TABLES 41 A.3 The table below contains information for the simulations of the χ0

(DFDM) particle for the channel ee with luminosity 300 f b−1 . . . 33 A.4 The table below contains information for the simulations of the S0

(RSDM) particle for the channel ee with luminosity 300 f b−1 . . . 33 A.5 The table below contains information for the simulations of the χ0

(DFDM) particle for the channel µµ with luminosity 300 f b−1 . . . 33 A.6 The table below contains information for the simulations of the S0

(RSDM) particle for the channel µµ with luminosity 300 f b−1 . . . 33 A.7 DFDM vs DFDM. The values within the table represent the standard

deviation σ and in the parenthesis are the D.O.F for the kinematic distributions for luminosity 140 f b−1and channel µµ. . . 35 A.8 RSDM vs DFDM. The values within the table represent the standard

deviation σ and in the parenthesis are the D.O.F for the kinematic distributions for luminosity 140 f b−1and channel µµ. . . 36 A.9 RSDM vs RSDM. The values within the table represent the standard

deviation σ and in the parenthesis are the D.O.F for the kinematic distributions for luminosity 140 f b−1and channel µµ. . . 36 A.10 DFDM vs DFDM. The values within the table represent the standard

deviation σ and in the parenthesis are the D.O.F for the kinematic distributions for luminosity 300 f b−1and channel ee. . . 36 A.11 RSDM vs DFDM. The values within the table represent the standard

deviation σ and in the parenthesis are the D.O.F for the kinematic distributions for luminosity 300 f b−1and channel ee. . . 37 A.12 RSDM vs RSDM. The values within the table represent the standard

deviation σ and in the parenthesis are the D.O.F for the kinematic distributions for luminosity 300 f b−1and channel ee. . . 37 A.13 DFDM vs DFDM. The values within the table represent the standard

deviation σ and in the parenthesis are the D.O.F for the kinematic distributions for luminosity 300 f b−1and channel µµ. . . 37 A.14 RSDM vs DFDM. The values within the table represent the standard

deviation σ and in the parenthesis are the D.O.F for the kinematic distributions for luminosity 300 f b−1and channel µµ. . . 38 A.15 RSDM vs RSDM. The values within the table represent the standard

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Characterization of dark matter based on spin in the mono-Z channel

M

T

30 HP

Characterization of dark matter based on

spin in the mono-Z channel

Abstract

Sammanfattning,

Acknowledgements

Contents

Chapter 1

Introduction

1.1

Motivation

1.2

Standard Model

Chapter 2

Beyond the Standard Model

2.1

Why to go Beyond the SM?

2.2

Theoretical schortcomings of the SM

2.3

Evidence of BSM physics

Chapter 3

Dark Matter

3.1

Evidence of DM

3.2

DM Candidates

3.3

Cosmology and DM

3.4

DM Detection

Chapter 4

EFT and Simplified Models

4.1

EFT

4.2

Simplified Models

∑

Chapter 5

Simulation framework

Chapter 6

Analysis

6.1

Analysis introduction

∑

∑

6.2

Parton analysis

6.3

Detector analysis

6.4

Relic Density

Chapter 7

Conclusion

Appendix A

Appendix A

A.1

Hadron analysis simulation data

A.2

Hadron analysis plots for mass coupling plane and

sig-nificance

A.3

Hadron analysis tables with the standard deviation

val-ues

List of Figures

List of Tables

References