Searching for Light Dark Matter and a new X17 boson with the NA64 experiment at the CERN SPS

Searching for Light Dark Matter and a new X17 boson with the NA64 experiment

at the CERN SPS

A thesis submitted to attain the degree of Doctor of Sciences of ETH Zurich

(Dr. sc. ETH Zurich)

presented by

Emilio D^EPERO MSc in Physik ETH, Zurich

born on 09.05.1992 citizen of Italy

Accepted on the reccomendation of Prof. Paolo Crivelli, examiner Prof. Sergei Gninenko, co-examiner Prof. Günther Dissertori, co-examiner

2020

Declaration of Authorship

I, Emilio DEPERO, declare that this thesis titled, “Searching for Light Dark Matter and a new X17 boson with the NA64 experiment at the CERN SPS” and the work presented in it are my own. I confirm that:

• This work was done wholly or mainly while in candidature for a research de- gree at this University.

• Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated.

• Where I have consulted the published work of others, this is always clearly attributed.

• Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work.

• I have acknowledged all main sources of help.

• Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed my- self.

Signed:

Date:

“Fall in love with some activity, and do it! Nobody ever figures out what life is all about, and it doesn’t matter. Explore the world. Nearly everything is really interesting if you go into it deeply enough. Work as hard and as much as you want to on the things you like to do the best. Don’t think about what you want to be, but what you want to do. Keep up some kind of a minimum with other things so that society doesn’t stop you from doing anything at all.”

Richard P.Feynman

EIDGENÖSSISCHE TECHNISCHE HOCHSCHULE ZÜRICH

Abstract

Institute of Particle Physics and Astrophysics Doctor of Sciences of ETH Zurich

Searching for Light Dark Matter and a new X17 boson with the NA64 experiment at the CERN SPS

by Emilio DEPERO

Il Modello Standard è stato utilizzato nel contesto della fisica delle particelle per fornire predizioni incredibilmente accurate confermate poi dai dati raccolti speri- mentalmente. La recente scoperta del bosone di Higgs presso il Large Hadron Col- lider (LHC) nel 2012 è stato un ulteriore incredibile successo di questa teoria. Nonos- tante ciò, il Modello Standard non è in grado di spiegare alcune domande aperte e pertanto rimane il bisogno di estendere questa teoria con della nuova fisica. Tra queste, l’origine della materia oscura rimane un mistero irrisolto nonostante le nu- merose ricerche effetuate sia con il LHC che dagli esperimenti ad osservazione di- retta installati sottoterra o dalle numerose osservazioni in ambito astrofisico. Recen- temente è stato proposto che la materia oscura sia parte di un Settore Oscuro, poten- zialmente popolato da particelle leggere, di masse inferiori al GeV, che potrebbe in- teragire con la materia visibile non solo tramite gravità ma anche tramite una nuova interazione mediata da un bosone oscuro. Sorprendentemente, la forza di inter- azione e la massa di questo bosone, in grado di spiegare l’abbondanza di Materia Os- cura osservata, si trovano in una regione dello spazio dei parametri che può essere esplorata tramite i nostri attuali accelleratori, in particolare nel caso di un bosone vettoriale chiamato Fotone Oscuro. L’esperimento NA64 ha come obiettivo il co- prire questo spazio di parametri dirigendo un fascio di elettroni di energia 100 GeV generato dal Super Proton Synchrotron (SPS) al CERN verso un target attivo. Questo metodo è sensibile a differenti tipi di decadimento del fotone oscuro con modifiche minime all’apparato sperimentale. In questa tesi, presentiamo l’apparato sperimen- tale usato e l’analisi dei dati raccolti durante il periodo 2016-2018. Il focus sarà sulle tecniche utilizzate per ridurre il fondo e la loro applicazione nel contesto di NA64.

Viene qui anche discussa la possibilità di usare questi dati per verificare l’esistenza di una nuova particella chiamataX17, proposta per giustificare una anomalia recen- temente osservata nello spettro nucleare del atomo⁸Be. NA64 è già in grado di lim- itare significamente le possibili spiegazioni di questa anomalia nella fisica delle par- ticelle, in questa tesi una ulteriore conferma viene data da un analisi indipendente effettuata usando i rilevatori di traccia. La regione dello spazio dei parametri ancora non coperta è difficile da accedere per via del tempo di decadimento molto corto di questa particella quando la sua forza di interazione cresce. In questa tesi viene anche presentato un nuovo apparato sperimentale ottimizato per queste ricerche.

Nell’ultima sezione verranno discusse le prospettive future di questo esperimento e diverse migliorie previste per il futuro. Il nuovo apparato permetterà di coprire completamente lo spazio di parametri in grado di giustificare la particellaX17.

EIDGENÖSSISCHE TECHNISCHE HOCHSCHULE ZÜRICH

Abstract

Institute of Particle Physics and Astrophysics Doctor of Sciences of ETH Zurich

Searching for Light Dark Matter and a new X17 boson with the NA64 experiment at the CERN SPS

by Emilio DEPERO

The Standard Model of particle physics has provided incredibly accurate predic- tions confirmed by experimental data. The recent discovery of the Higgs boson at the Large Hadron Collider (LHC) in 2012 was another remarkable landmark of this model. However, despite its success, the Standard Model cannot address some open questions and therefore new physics is required. Among those is the origin of Dark Matter which remains a mystery that has not been solved either by LHC or direct searches in the underground and astrophysical experiments. It was put forward that if Dark Matter is part of a Dark Sector, it could be light, in sub-GeV mass range, and interact with the visible matter in addition to gravity through a new interac- tion transmitted by a dark mediator(s). Surprisingly and excitingly, the coupling strengths and masses of the latter, which could explain the observed Dark Matter relic abundance, fall into the region which can be probed in experiments at current accelerators, in particular for the vector case commonly called Dark Photon. The NA64 experiment aims to effectively cover this parameter space by using a 100 GeV electron beam from the CERN Super Proton Synchrotron (SPS) directed at an active fixed target. This approach is sensitive to different decay modes of the Dark photon with minimal modifications of the setup. In this thesis, we describe the two appara- tus used and present the analysis of the data collected during 2016-2018. The focus will be on background rejection techniques and their application in the context of the NA64 experiment. The possibility of using the data collected to probe the existence of a new particle calledX17 used to justify the recently observed deviation in the

8Be nuclear spectrum is also detailed. NA64 already puts strong constraints on par- ticle physics explanations of this anomaly, which are confirmed by an independent analysis performed in this thesis using tracking detectors. The remaining region of parameter space still able to justify the existence of a new boson in this context is hard to access due to the very short decay time this particle has for large couplings.

This work will also present a new setup optimized for these searches. The future prospects of the experiment and the foreseen upgrades will be discussed in the last section. The new setup will allow us to fully cover theX17 parameter space.

Acknowledgements

In the beginning, I would like to thank my supervisor Prof. Paolo Crivelli for the valuable guidance received. In good times and in bad times, he helped me with advice, knowledge, and criticism when needed. This thesis gained tremendously from his support, and I will never stop being grateful for that.

I wish to say thank you to Prof. Andre Rubbia for giving me the opportunity to participate in this amazing project. Thanks to him I learned to never ignore the details and always aim for a deeper understanding.

Next, I would like to thank Prof. Sergei Gninenko. I always admired his enthusi- asm for physics that never failed to emerge in any situation. His tremendous amount of knowledge was invaluable times and again. The challenge of a two months beam time was much more manageable thanks to his never-ending optimism.

Next, I wish to thank Prof. Gunther Dissertori for accepting being my co-examiner for my thesis. I believe his lecture on particle physics during my bachelor was one milestone that influenced my later studies.

During this thesis, I had the pleasure of working with truly wonderful peoples.

Laura Molina Bueno and Dipanwita Banerjee are special among them for following me the closest after my supervisor. Laura in particular has been invaluable these last months, I will never forget the help and encouragement she gave me when I was close to the end. All the bachelor and master students I supervised during my PhD deserve a mention as well, it was wonderful to work with them and part of the thesis would not be possible without their help. I hope I managed to give back to them something in return.

I would like to thank the full NA64 collaboration and in particular Mikhail Kir- sanov, Vladimir Poliakov, Anton Karneyeu, and most of all Balint Radics for guiding me through all the details and technicalities encountered during this thesis. I cannot imagine NA64 without them, and I wish them the best of luck for all the work left to be done. A thanks goes also to Michael Hoesgen, Renat Dusaev, Bogdan Vasilishin, and Azzrali Vitali. Their company during beam time was a real pleasure and helped sweeten the time spent in front of the monitor. Michael Hoesgen and Renat Dusaev in particular helped me even more than that, thanks to the guidance they provided for GEM trackers and the NA64 software.

I wish to thank all the CERN staff, which made my work easier. In particular, I would like to mention Alberto Ribon and Vladimir Uzhinskiy. Without their help with the hadron models in Geant4, my work would have been nearly impossible.

I will never forget all the amazing peoples I met at ETH, in particular my col- legues Carlos Vigo Hernandez, Michael Heiss, Lars Frieder Gerchow, Gianluca Janka, Henri Sieber, Mark Raaijmakers, Lucas de Sousa Borges, Irene Cortinovis, Alexan- der Stauffer, and Johannes Wüthrich. Thank you for all the fun I had during this work, for laughing with me about my mistakes, and to let me laugh about yours in return. I discovered a lot about physics talking with you, and you always reminded me that there is always something new to be learned in this subject. I hope we will keep in touch, your friendship will always be special.

In the end, I am grateful to my parents, my relatives, my brother, my dear friends back in Italy and here in Zurich, and the special persons I met. There are no words to express how much I love all of them, and how much my character is a consequence of the impact they had on me. I will always be grateful to my mom and dad to support me in this adventure, first and foremost emotionally but also financially to avoid any distractions outside my studies. The joy of a nephew will always be an amazing memory that I will connect to this period as well, I dedicate this thesis to

Stella, a ray of sun in these difficult times. She is yet another reason to work hard for the future.

Contents

Declaration of Authorship iii

Abstract vii

Abstract ix

Acknowledgements xi

Preface 1

1 Introduction 3

1.1 The Standard Model and its challenges . . . 3

1.2 Evidence for Dark Matter . . . 4

1.3 Dark Matter candidates . . . 5

1.4 Dark Sectors . . . 7

1.4.1 Thermal Dark Matter and Dark Sector . . . 8

1.4.2 The vector portal . . . 10

1.4.3 Dark Photon production in fixed-target experiments . . . 12

Decay modes and detection . . . 13

1.4.4 Dark Sector in accelerator experiments . . . 15

1.5 The X17 anomaly . . . 16

2 The NA64 experiment 21 2.1 Experimental Technique . . . 21

2.2 Experimental setup . . . 24

2.2.1 The invisible mode setup . . . 24

2.2.2 The visible mode setup . . . 26

2.3 Detectors . . . 26

2.3.1 The Trigger system . . . 27

2.3.2 The Electromagnetic Calorimeter (ECAL) . . . 29

2.3.3 The Tungsten Electromagnetic Calorimeter (WCAL) . . . 29

2.3.4 The Hadronic Calorimeter (HCAL) . . . 30

2.3.5 The Synchrotron radiation detector (SRD) . . . 30

2.3.6 The Veto system . . . 31

2.3.7 The Tracking system . . . 33

Micromegas . . . 33

GEM detectors . . . 35

Straw detectors . . . 36

2.3.8 The data Acquisition system (DAQ) . . . 36

3 Data analysis 41 3.1 General Analysis approach . . . 41

3.2 Geant4 simulation of the experiment . . . 45

3.2.1 Reconstruction and digitization . . . 48

3.3 Monte Carlo validation usingγ+Z→µ⁺µ⁻events . . . 49

3.4 Background rejection methods . . . 50

3.4.1 Heavy charged particle rejection using synchrotron radiation . 51 3.4.2 Hadron rejection using electromagnetic shower profile . . . 56

3.5 Invisible mode analysis . . . 61

3.5.1 Selection criteria . . . 61

3.5.2 Background . . . 63

Hadronic background . . . 63

Muonic background . . . 64

Electronic background . . . 65

3.6 Visible mode analysis . . . 69

3.6.1 Selection criteria . . . 69

3.6.2 Visible mode analysis using the tracking approach . . . 70

γ+Z→µ⁺µ⁻events in the visible mode analysis using trackers 71 Signal yield correction . . . 73

3.6.3 Background . . . 75

Hadronic background . . . 75

Muonic background . . . 77

Electronic background . . . 77

4 Results 79 4.1 Exclusion limits in the(m_A⁰;e)plane . . . 79

4.2 Exclusion limits in the(y;m_χ)plane . . . 82

5 Future prospects 85 5.1 Micromegas upgrade . . . 86

5.2 Improvement to the invisible mode setup . . . 89

5.3 Improvement to the visible mode setup . . . 93

5.3.1 Invariant mass reconstruction . . . 93

Multiple scattering effects on invariant mass reconstruction . . 96

Hit separation in gas tracking detectors . . . 97

5.3.2 WCAL optimization . . . 99

Background and sensitivity . . . 99

5.4 A new approach: the Muon mode setup . . . 102

6 Conclusion 107 A Dark matter formulas and calculations 109 A.1 Calculation of cross-section using the Weizsacker-Williams approxi- mation . . . 109

A.1.1 Production rate . . . 111

A.2 The Tree-level corrections . . . 111

A.2.1 Implementation inside the Geant4 simulation . . . 113

B Physics tool 115 B.1 Synchrotron radiation . . . 115

B.1.1 Power emitted . . . 115

B.1.2 Radiation angular distribution . . . 117

B.1.3 Radiation spectrum . . . 117

B.2 Profile construction for shower profile analysis . . . 119

C Geant4 simulation 123

C.1 Simulation principle . . . 123

C.2 Interaction biasing . . . 125

C.3 Modification of the Fritiof String Model . . . 132

D Algorithms and particle tracking 137 D.1 Momentum reconstruction . . . 137

D.1.1 Three point momentum extrapolation . . . 137

D.1.2 Genfit and the Kalman filter . . . 138

D.2 Pulse reconstruction . . . 142

E Statistics tools 147 E.1 The CL_s technique in the asymptotic approximation using Asimov data set . . . 147

F Tables 151 F.1 Multiplexing maps for Micromegas . . . 151

F.2 NA64 experiment . . . 158

Bibliography 163

List of Figures

1.1 Mass range for Dark Matter . . . 6

1.2 Lower limit ofyin dark sector . . . 10

1.3 Possible production mechanism for A⁰: Dark Bremsstrahlung with a nucleus Z a), Dark Compton b), resonance production c) and A- resonant production d). . . 11

1.4 Current exclusion limit and project for Dark Photon in the physics community . . . 17

1.5 Sketch of the setup used to detect the⁸Be anomaly. . . 18

1.6 ⁸Be anomaly . . . 18

2.1 Sketch of experimental signatures forA⁰ . . . 23

2.2 Invisible mode setup 2018 . . . 25

2.3 NA64 visible mode setup 2018 . . . 27

2.4 ECAL sketch . . . 29

2.5 WCAL sketch . . . 30

2.6 HCAL sketch . . . 31

2.7 BGO sketch . . . 32

2.8 SRD sketch . . . 32

2.9 Micromegas sketch . . . 34

2.10 Example of the readout of a multiplexing detector . . . 35

2.11 Scheme for the NA64 DAQ . . . 38

2.12 APV25 banana plot . . . 39

3.1 Flowchart of the NA64 analysis. . . 45

3.2 IWW vs tree-level energy spectra . . . 47

3.3 Dimuon spectra in ECAL for data and MC. . . 51

3.4 SR spectrum for different energy detected in the SRD . . . 53

3.5 Geometry of the BGO crystals . . . 54

3.6 SRD comparison between data and MC . . . 55

3.7 efficiency and rejection power of the SRD cut . . . 55

3.8 ECAL sketch . . . 57

3.9 Comparison between χ² distributions, electron and hadron calibra- tion run . . . 58

3.10 Comparison betweenχ²distribution for different ECAL configurations 59 3.11 Fraction of events passing theχ²cut . . . 59

3.12 ECAL vs HCAL energy deposit after a cutχ² . . . 60

3.13 ECAL vs HCAL energy deposit after a cutχ²for different ECAL con- figurations . . . 61

3.14 Effect of the cuts in invisible mode . . . 63

3.15 K⁻simulation . . . 65

3.16 ECAL vs HCAL events band . . . 66

3.17 upstream electro-hadron production upstream . . . 66

3.18 R value for the control sample . . . 67

3.19 R value comparison . . . 68

3.20 Comparison of selectedA⁰events between the calorimeter and track- ing analysis . . . 71

3.21 Hit position ofe⁻Z→e⁻Zγ;γ→µ⁺µ⁻in GEM MC-DATA . . . 72

3.22 Beam profile with different cuts . . . 74

3.23 e⁻Z→e⁻Zγ;γ→µ⁺µ⁻MC-DATA comparison in visible mode . . . 74

3.24 neutral events in visible mode . . . 77

4.1 Exclusion limits in the(m_A⁰;e) . . . 81

4.2 Exclusion limits in the(ma;gaγγ)plane for ALPS and light scalar . . . 82

4.3 Exclusion limit in thedmyplanefor scalar, pseudo-Dirac and Majorana type of light Dark Matter . . . 83

5.1 Previous version and new Micromegas design of the gas box seal . . . 86

5.2 Test of new multiplex map . . . 88

5.3 Sensitivity projection for invisible mode 2021 . . . 89

5.4 electro-nuclear interaction position . . . 90

5.5 Sketch of VHCAL in invisible mode setup 2021 . . . 90

5.6 Electro-nuclear background estimation . . . 91

5.7 Background extrapolation invisible mode . . . 92

5.8 Invariant mass reconstruction sketch . . . 94

5.9 Distance of the decay products of X17 in the 2021 setup . . . 95

5.10 2021 setup . . . 96

5.11 Invariant mass reconstruction in 2021 setup . . . 96

5.12 Hit resolution as function of the two cluster distance . . . 98

5.13 New WCAL design for 2021 . . . 100

5.14 EOT to X17 exclusion . . . 102

5.15 Sketch of muon mode setup 2021 for phase 1 . . . 104

5.16 Sketch of muon mode setup 2022 for phase 1 . . . 105

5.17 sensitivity projection for invisible mode + muon mode 2021 . . . 105

A.1 Tree level differential cross-section for different masses . . . 113

A.2 Ratio between cross-section calculated in the IWW approximation and using ETL computationK = σ^A 0 IWW/σ_ETL^A0 described in Sec.A.2 for dif- ferent Dark photon masses [150] . . . 114

B.1 Upper plot: Spectrum for a single synchrotron radiation photon emit- ted by an electron in the MC simulation. Bottom plot: Total energy deposited in the BGO for one electron event after Bremsstrahlung is included in the simulation. . . 119

C.1 Flow-chart of the MC simulation developed for the NA64 experiment. All classes used are presented. . . 124

C.2 Setup for the visible mode of 2021 as implemented in the Geant4 sim- ulation. This setup was used for the feasibility study presented in Sec.5.3. . . 125

C.3 Distribution of interaction lengths of K⁰_Linelastic scattering. . . 128

C.4 Comparison of biased and unbiasedRdistributions. . . 129

C.5 Histograms of the z position of electronuclear interactions. . . 130

C.6 MC/DATA Comparison ofπ⁻in ECAL and WCAL . . . 133

C.7 Comparison of physics list forπ⁻in WCAL energy spectrum . . . 134 C.8 Comparison of different simulated hadrons in WCAL energy spectrum 134

C.9 Comparison of different diffraction limits for hadrons in WCAL en-

ergy spectrum . . . 135

C.10 Comparison of hadron energy deposit after corrections . . . 135

D.1 Two dimensional geometrical representation of a particle trajectory (blue) deflection in a uniform magnetic field. . . 138

D.2 genfit flow diagram . . . 140

D.3 example of pulse reconstruction in NA64 . . . 143

D.4 example of pulse reconstruction with pileup in NA64 . . . 144

List of Tables

3.1 Percentage of pion and electron events for different hit multiplicity in

the SRD from the data . . . 56

3.2 Invisible mode background . . . 68

3.3 ratio between signal events observed in tracker-analysis compared to calorimeter-only analysis . . . 72

3.4 MC/DATA for the tracking procedure and vertex reconstruction . . . . 75

3.5 Visible mode background . . . 78

5.1 Error budget for the invariant mass in 2021 setup . . . 97

5.2 Possible WCAL designs and their projected experimental reach . . . . 100

C.1 Simulated interactions lengths of various hadronic processes in a thick Fe target for different biasing factors. . . 129

C.2 Bias gain of upstream electronuclear interaction. . . 131

C.3 MC cut efficiencies for the simulation with biask=200 and 1.79×10⁷ EOT. . . 131

F.1 original Multiplex map for the 80×_{80 mm}²Micromegas modules . . . 152

F.2 Multiplex map optimized for 2021 beam time for the 80×80 mm² Micromegas modules . . . 154

F.3 Multiplex map optimized for 2021 beam time for the 245×80 mm² Micromegas modules . . . 158

F.4 sub-detectors description . . . 159

F.5 Systematic uncertainties in the NA64 experiment . . . 160

F.6 Data cut efficiencies for calibration run 4009. . . 160

F.7 Estimated total punch-through probability after four HCAL modules (∼7λ_int) for hadrons induced shower particles. . . 161

List of Abbreviations

DM DarkMatter

MOND MOdifiedNewtonianDynamics ΛCDM LambdaColdDarkMatter SPS SuperProtonSynchrotron QFT QantumFieldTheory QED QantumEelectroDynamic DM DarkMatter

LDM LightDarkMatter

LTDM LightThermalDarkMatter SM StandardModel

WW WeizsackerWilliams ETL ExactTreeLevel EOT ElectronOnTarget MOT MuonOnTarget POT ProtonOnTarget

ECAL ElectromagneticCALorimeter (PbSc structure) HCAL HadronicCALorimeter (FeSc structure) WCAL W CALorimeter (WSc structure)

VHCAL VetoHadronicCALorimeter SRD SynchrotronRadiationDetector PCB PrintedCircuitBoard

W Tungsten calorimeter used to reject high beam divergency S_i Plastic scintillators

V_i Plastic scintillators with a hole in the middle St_i Strawtubes

VETO Three thick plastic scintillators placed after the ECAL MM MicroMegas

GEM GasElectronMultiplier FWHM FullWidthHalfMaximum MIP MinimumIonizingParticle λ_int Nuclear Interaction Length X₀ Radiation Length

TCS TriggerControlSystem

GeSiCa GemSiliconControl andAcquisition

CATCH CompassAccumulateTransfer andCacheHardware ASIC ApplicationSpecificIntegratedCircuit

ROB ReadOutBuffer HEP HighEnergyPhysics ALPs AxionLikeParticles

GDML GeometryDescriptionMarkupLanguage

Preface

Physics is a wonderful thing, I was always mesmerized by its ability to describe what surrounds us in a powerful and compact way, ultimately using this knowledge for the betterment of mankind. When I was young, equations were like an ancient magic language to me. I was not able to understand it yet, but I was absolutely impressed by knowing that people that could, were able to control a power that for a kid looked like beyond any form of comprehension, from making incredibly heavy objects flying faster than any bird or destroying an entire city by splitting something invisible for the naked eye. I always dreamed to become part of this exclusive club of magicians, and now that I am, I have to say: I got it all wrong! Physics requires a lot of thinking, a lot of reading, a lot of trial and error, in one word? A lot of work!

What follows is my attempt to make a lasting contribution to this amazing field. It won’t go to history, but I hope it will be important for some students that follow my step, and for sure it will be important to me.

So what is this thesis about? Dark Matter, if I had to use one word. One of the most prominent puzzles in all physics. With all our powerful and extremely precise models, we still have to explain more than 96% of the matter in our universe. That is embarrassing! How could we miss all that? Turns out is not simple at all when the matter that you are searching for stubbornly refuses to interact with your detectors, but I am sure that eventually, physics will prove to be even more stubborn in their measurement. This thesis was an attempt to this, an experiment to produce this elusive matter directly, and then measure its properties. Since my group does not have a noble price in its hand, it must be no mystery that we did not yet succeed, although this is not excluded for the future. I think the journey of me and the rest of my colleagues is nevertheless very instructive, it is the typical story of how an experiment is born and conducted. A "happy ending" is not always expected, and should not be assumed by the scientist that is conducting the experiment. He is there to observe, not decide. In my opinion, the experiment itself is the "happy ending"

that a scientist is seeking, when it is well performed, and that is not depending on the outcome of it. So with no further wait, we can begin our journey, at the discovery of the NA64 experiment and its search of Dark Matter using the SPS¹.

In this first chapter, our story begins, and as every story of science, it begins with a mystery: Dark Matter. Why can’t we see it? Why can’t we touch it? Those are not interesting questions, since many well-understood phenomena can’t be seen or touched. In the end, it all boils down to one question: why our models, that work so amazingly well in so many different situations, fail miserably in other situations apparently equivalent? This is what this chapter will be about, understand why we need the concept of dark matter, and justify what phenomena it could explain.

Indeed one could think that it is rather lazy to explain phenomena just by adding invisible matter to the system. "Just admit that you don’t know what is going on!", is something that I heard myself when I try to explain my work to others. That is overall an honest question, a healthy scientific skepticism about a theory that seems so arbitrary, but that I hope it will be clarified after reading this thesis.

1Super Proton Synchrotron

Chapter 1 is meant to just answer the question: what are we searching for? This is the beginning of every scientific experiment. Building an experiment to produce the Dark Photon is the next step, which will be covered in chapter 2. Turn a photon in a Dark Photon using this portal is however meaningless if we can’t prove that we did it! A robust analysis method of the data collected needs to be performed for this purpose, this is what we will explore in chapter 3 step by step, from the method to the selection criteria. In chapter 4 we will then provide what most physicists are here for, the results! No Dark Matter has been found yet, but the data acquired allows us to exclude some specific models from being a credible explanation of reality. We now know thanks to this data, that the anomalous magnetic moment of the muon cannot be explained exclusively by the U’(1). The NA64 experiment is however far from over! The stop caused by the LHC long shutdown (and unfortunately by the recent rise of Covid-19 as well) of the accelerator allowed us to look for ways to improve our setup for the upcoming challenges of 2021! The pieces of knowledge we gained for the background and the experimental condition are used to design the new version of the setup to acquire a larger number of particles and thus probe a larger number of models. In chapter 5 we will review all these changes, in particular the new setup for the visible mode of 2021, that was the last project of my Ph.D., which we will use to hopefully find (or else exclude) theX17.

With no further wait, let’s embark on this journey!

Chapter 1

Introduction

In this first chapter, we briefly review the concept of Dark Matter (DM). In Sec.1.2 we will explore why we need DM, and what are the alternatives to it. After that, we will come to realize that Dark Matter hides an extremely large number of possibilities that can be constructed using the framework of Quantum Field Theory (QFT). Dark Matter candidates are indeed very numerous, so we will focus on the most relevant ones within the scope of this thesis in Sec.1.3.

The main focus of my thesis will be the so-called Dark Sector, where Dark Matter particles can interact with the Standard Model using new undiscovered interactions.

We will describe this framework in Sec.1.4 by introducing the vector portal, which postulates the existence of an additional U(1) symmetry that generates a dark vector boson. The vector gauge boson generated by this symmetry, that we will label A⁰ in this thesis, is frequently called Dark Photon, since it plays an equivalent role of the Standard Model (SM) photon in this new symmetry. This model allows also a cross-term that couples the new Dark Photon with its SM counterpart, effectively building a portal between the two sectors. If such a model is true, we can produce this type of matter using modern accelerator. An introduction to the framework of thermal Dark Matter will then be given. We will see that the Dark Sector, and in particular the vector portal, is an interesting candidate to explain the observed relic density using the freeze-out mechanism. This further motivates using accelerators experiments to probe the mass range MeV-GeV, which is very challenging to cover using other approaches.

After this introduction, we will see how Dark Matter can be searched for using particle beams in the context of fixed-target experiments. Starting from the bench- mark model U’(1), the signal yield expected in such experiments is derived for dif- ferent decay modes. The production of Dark mediators in accelerators is not limited however to vector bosons only, but can be extended to a large group of models that predict interactions between the Dark Sector and particles from the SM. Some of these additional searches, like the one for axion and axion-like particles, were al- ready performed by NA64 as will be presented in this thesis. Another phenomenon that could be explained in this framework is the so-calledX17-anomaly, originally known by the name of⁸Be-anomaly. This refers to a deviation from the SM pre- diction in the nuclear decay spectrum of Beryllium that could be justified by the existence of a new particle. In Sec.1.5 a review of this phenomenon will be given, and we will show how particle physics explanations of this anomaly are within the sensitivity range of the NA64 experiment.

1.1 The Standard Model and its challenges

The Standard Model (SM) is the theory that to this date best describes our current un- derstanding of particles and their interactions. It was developed in the second half of

the 20th century, and reached its currently accepted formalism with the introduction of the Higgs mechanism [1] in the electro-weak interaction [2]. SM describes matter and energy in terms of interactions between elementary particles, which are con- structed as excitation of quantum fields. The theory contains 12 fermions¹and gauge bosons that mediate the 3 fundamental interactions²between particles: strong, weak and electromagnetic. These bosons are generated by symmetries described by uni- tary product groups:

SU(3)_c

| {z }

strong interaction

× SU(2)_L

| {z }

weak interaction

× U(1)_Y

| {z }

em interaction

(1.1) A complete review of the Standard Model is beyond the scope of this introduc- tion, and can be found instead in [3]. The SM is incredibly successful, e.g. anticipat- ing the existence of the W and Z boson, the gluons, and the top and charm quarks [4].

The recent discovery of the Higgs boson [5] completed the set of elementary particles predicted by the theory. Outstanding quantitative predictions were also performed within this framework, especially using Quantum ElectroDynamics (QED). One of the most famous examples is the exact value of the anomalous magnetic moment of the electron, that agrees within one part per billion with the experimental measure- ments [6].

So why are physicists still unhappy with the SM regardless of its undeniable successes? There are still many problems with the Standard Model that are not well understood and a source of debate within the scientific community. First of all it does not include a description of gravitation, being incapable to explain the General Relativity in terms of a quantum field theory. This alone underlines that SM has not the requirements to be a theory of everything and that a piece must still be missing.

On top of that, the Higgs mechanism gives rise to a Hierarchy problem [3], since its measured mass is much lighter than what anticipate by its quantum corrections unless an incredibly large fine-tuning is argued to fix this prediction.

There is however something that makes the need for a new theory even more apparent. The Standard Model explains the existence of only a tiny portion of the matter in the universe! Approximately only 5% of the total mass observed. To ex- plain this discrepancy, theΛCDM³is commonly used. It is often referred to as the

"Standard Model of Cosmology" because its predictive power is arguably compara- ble to its particle physics counterpart for cosmological observations. However, this model assumes the existence of particles which are not yet experimentally observed, and are not predicted by the Standard Model! This class of particles is called Dark Matter, one of the greatest puzzle of modern physics.

1.2 Evidence for Dark Matter

The history of Dark Matter and the birth of this term is interesting on its own, and a good example of "unequivocal accumulated evidence" in science [7,8]. The term was first used in 1937 by astronomer Fritz Zwicky⁴ to justify the velocity dispersion in Coma cluster galaxies which deviated significantly from what one would expect by simply asserting the mass from the visible matter [9]. This hypothesis was discussed

1and their corresponding anti-particles

2Gravitation, is not included in this framework, and is instead described by General Relativity.

3Stands forΛ Cold Dark Matter, whereΛ represents the cosmological constant associated with Dark Energy.

4Some weaker claim of a discrepancy was observed even before by Knut Lundmark in 1930.

more seriously in 1950 when astronomical surveys had confirmed these results with high precision. This sparked a heated debate in the scientific community, and most solutions to this problem did not include additional unknown matter, but rather the modification of gravity or sophisticated arguments based on the dynamical equilib- rium of such galaxies. In 1970, with the rising of radio astronomy, the rotation curves of galaxies were studied in detail, and two independent observations, performed by Kenneth Freeman and Vera Rubin, confirmed that the rotation velocity of objects in a galaxy becomes flat at sufficient distance from its center. The idea became very influential in the cosmological community thanks to two papers in 1974 by Einasto and Ostriker, clearly stating that the mass of galaxies has been underestimated by a factor 10 until then [10,11]. This was further strengthened by the end of the decade by a very similar analysis [12]. The existence of Dark Matter became more and more accepted in the years to come, alternative theories also arose to explain observation without the need of additional matter. Perhaps the most famous today remains the Modified Newtonian Dynamics (MOND) theory, built as a weak-field approxima- tion of some more general theory of gravity yet to be discovered. Despite an early success, this theory has always proven to be challenging to merge in the General Relativity framework, and is insufficient to justify some specific phenomena like the famous "bullet cluster", that is on the other hand very well explained by Dark Matter [13].

So what is the current situation? Today the theory is well accepted in the scien- tific community, although some debate is still present. The existence of Dark Mat- ter is the leading paradigm to explain all discrepancies observed [7]. Advances in the field of cosmology and measurement techniques have provided much more ev- idence that aid Dark Matter existence. For example, gravitational lensing, in par- ticular in the context of weak lensing, was used to characterize the mean distri- bution of Dark Matter and match it to the one predicted by large scale structure measurements [14]. Temperature anisotropies of the Cosmological Microwave Back- ground (CMB) also present structures compatible with Dark Matter, and well fitted in the ΛCDM model [15]. Many other arguments, including structure formation [16], baryon acoustic oscillation [17], and Red-shift distortions [18] have proven con- sistently to agree with the existence of Dark Matter and with the Lambda Cold Dark Matter (ΛCDM) model in particular.

While the consensus on the DM existence is now well established, its origin and exact composition is still a big mystery. In the next section, we will give an overview of what exactly defines Dark Matter, starting from its measured properties. After that, we will explore the most popular models which try to address this puzzle.

1.3 Dark Matter candidates

The first question to ask ourselves is what kind of property Dark Matter should have to satisfy the experimental constraints given by cosmology. We can define the following key characteristics [19]:

• Dark: It should not emit/absorb light, but a tiny charge is still compatible with data. This justifies e.g. models predicting "milli-charged" particles. What is effectively constrained is the ratio of charge to a power of the mass.

• Collisionless: Dark Matter self-interaction to mass ratio is constrained by obser- vation of cluster mergers and the ellipticity of galactic halos. The cross-section itself is not however necessarily small. Assuming a mass similar to the one of

a proton, the constraint amounts to a cross-section in the order of a few barn, very similar to the one observed for the strong interaction.

• Classical: Dark Matter is observed to be confined on the galactic scales of a few kpc in dwarf galaxies. Hence, their de Broglie length must be smaller than that to have a coherent Dark Matter halo. This argument is typically used to put a lower limit on the DM mass. If the Dark Matter candidate is a fermion, con- straints are stronger as Pauli blocking limits the density to at most the phase space density to f = gh⁻³ whereg is the number of internal degrees of free- dom.

• Fluid: For macroscopic DM, with a mass much larger than the solar massM, tidal disruption is expected to break the stability in a globular cluster. This limit is typically placed around 10⁶M.

In summary, while the cosmological and Galactic Dark Matter density is known to a good degree, very weak constraints exist on possible interaction strength (in addition to gravity) and the exact mass of Dark Matter. The huge mass scale that spans over the possible region is depicted in Fig.1.1, where we see several possible DM candidates in relation to their mass and the different techniques to probe them.

A few experimental anomalies are labeled in red to see mismatches between theory and experiment that could potentially be explained by Dark Matter. We can see that most of them are in the MeV-GeV scale, which is the one covered by the NA64 experiment. Even the problem of the cosmological scale, like small-scale structure, are potentially well explained by this class of models [20].

zeV aeV feV peV neV meV eV keV MeV GeV TeV PeV

zeV aeV feV peV neV meV eV keV MeV GeV TeV PeV QCD Axion

Ultralight Dark Matter

Pre-inflationary Axion

post-inflationary Axion

Hidden Dark sector Hidden Thermal relics/WIMPless Dark Matter

Asymmetric Dark Matter Freeze-In Dark Matter

SIMPs / ELDERS

Coherent Field Searches, Direct Detection, Nuclear and Atomic physics, Accelerators Microlensing

WIMP

Black Holes

X17-anomaly Muon g-2 Small-Scale Structure

Anomalies experiments

Models

FIGURE1.1: Mass range for Dark Matter and mediator particle can- didates, experimental anomalies, and search techniques.

Because of the relatively loose constraints, it comes to no surprise that a vast amount of models have been theorized for the explanation of DM. The ones de- picted in Fig.1.1 are only a subset of what is currently considered. Models can also be extended to include scales not admitted in their original incarnation. It is the case for Axion Like Particles (ALPs), that are an extension of the QCD Axion model origi- nally developed to solve the strong CP problem [21]. Contrary to Axions, their mass

m_ais taken as an independent parameter from their main couplingg_aγγ. ALPs mass can be in the MeV-GeV scale, in the range where the NA64 experiment is sensitive.

As we will show in Sec.4.1, we already used our data to constrain a portion of this parameter space [22].

A description of each single model of DM is out of the scope of this thesis. For a complete review, one can refer to [3, 19, 20, 23–25]. Here, we provide a short description of the mainstream possibilities currently considered for Dark Matter:

• WIMP: For a long time, Weakly Interacting Massive Particles (WIMP) were considered the most probable candidate by the scientific community. Their main characteristic is tree-level interaction with the W and Z boson, but not with gluons or photons. Historically, their mass is in the 10 GeV-TeV, although more recent models extended the mass range to lower masses as well. The so-called WIMP miracle [26], provided a very effective and natural way to introduce Dark Matter in the thermal history of the early universe in a way that predicted precisely the current relic density. However, after an extensive amount of research, accelerators and direct searches have to this date failed to provide evidence for their existence [27]. These models have yet to be ruled out completely and remain the most studied type of Dark Matter.

• Axions and Axion-like: Axion and Axion-like particles are obtained by intro- ducing a new pseudoscalar fieldawith coupling to photons, gluons, or Higgs bosons [28]. Originally Axions were motivated by the strong CP problem and they were predicted to be extremely light (<eV). The model was later extended to offer a possible explanation for Dark Matter at higher masses. Several ex- periments are based on the "light shining through a wall" concept or using a magnetic helioscope [29] already put some stringent limits on Axions. Axion- like particles on the other hand can have a mass large enough to allow testing using accelerators.

• Sterile Neutrino: While Standard model neutrinos are not a good fit to explain Dark Matter as they are produced very hot in the early thermal bath, Sterile neutrinos are on the other hand a viable explanation. A subset of these models can also explain the light neutrinos masses as a consequence of the see-saw mechanism, although they are not favored as Dark Matter explanation [25].

• Dark Sector: By the name of Dark Sector, we describe a very large class of models characterized by particles not charged directly under strong, weak, or electromagnetic force [24]. Dark Matter can still interact with the SM parti- cles through so-called "portal interactions" constrained by the symmetries of the SM. This class of models, specifically the ones charged under a new U’(1) symmetry, are the focus of the NA64 experiment, and they will be described in more detail in the next sections.

1.4 Dark Sectors

Dark sectors are very interesting candidates to explain the origin of Dark Matter (see [20,24] for recent reviews). On top of reproducing the observed DM abundance in freeze-out or freeze-in scenarios, many experimental anomalies currently observed can be explained inside this framework. The anomalous magnetic moment of the muon [30], the proton charge radius⁵ [32] and more recently the X17-anomaly [33,

5Recent new measurements suggest that this puzzle can be solved without new physics [31].

34] have been suggested as possible hints of its existence [24]. The definition of Dark Sector is extremely broad, and therefore accommodates many possible models. Its physics however, can be explored effectively and in a systematic way by using spe- cific portal interactions as a classification scheme. The existence of a mediator acting as a portal is not a necessity to create a Dark Sector, but a small interaction with the SM allows a signature in particle physics experiments as well as a mechanism to compute the observed relic density [35,36]. The gauge and Lorentz symmetries restrict the ways in which the mediator can couple to SM particles. We can classify them using their spin and parity. By excluding dimensions operator larger than 5 we obtain four renormalizable possibilities [24]:

L ⊃

− ^e

2 cosθ_WB_µνF^0µν, vector portal (µφ+λφ²)H^†H, Higgs portal y_nLHN, neutrino portal

a

f_aF_µνF˜^µν, axion portal

(1.2)

Here, H is the Higgs doublet, L is a lepton doublet of any generation, Bµν ≡

∂_µB_ν −∂_νB_µ is the hypercharge field strength tensor, F_µν ( ˜F_µν) is the (dual) field- strength tensor of the SM photon field, θ_W is the weak mixing angle, and F^0µν ≡

∂µF_ν⁰−∂νF_µ⁰ is the field strength tensor of a Dark U⁰(1)vector boson. In this work, we will limit the discussion to the vector portal, as it is the most viable for thermal mod- els of Light Dark Matter (LDM). If we assume the DM mediator to be a vector boson arising from an additional U’(1) gauge group under which the LDM is charged, we can derive a terme/2 cosθ_WB^µνF⁰µνthat is invariant both on this symmetry and the standard U(1) from QED. This can be used to explain the phenomenology of a large class of models, such as scenarios where the Dark Photon couples preferentially to Baryonic (B), Leptonic (L), or (B - L) currents. One such case, where the coupling to baryons is disfavored, is the protophobicX17 gauge boson [37], that will be in- troduced at the end of this chapter. Other possibilities include the LDM possessing a Majorana mass in the absence of an exact U(1)symmetry [38], or the existence of a rich sector where many particles exist on top of the Dark mediator and the LDM [39].

1.4.1 Thermal Dark Matter and Dark Sector

To justify the Dark Matter density observed today, a model that describes the pro- duction of such particles in the early universe is commonly used. Such a model is not only useful to describe the dynamics of the universe but can also provide some constraints on the properties of Dark Matter, allowing us to restrict our searches to a class of models that obeys these conditions. Arguably the most popular frame- work of this kind describes Dark Matter as a thermal relic generated in the early universe thermal bath. In this environment, DM is produced in the direct annihila- tionSM+SM→DM+DM. For this interaction to take place, the energy of the SM particles needs to be sufficient to produce the mass of the DM candidate. After the universe cools down to a temperatureTbelow the dark matter particle massmDM, the number of Dark Matter particles becomes Boltzmann suppressed, dropping as e^−T/mDM [25]. Such mechanism is called "freeze-out" and is based on the decoupling of DM particles from the SM caused by the universe cooling down to a tempera- ture where the DM production becomes inefficient. The exact relic density is then obtained by solving numerically the Boltzmann equation:

dn

dt =−3Hn− hσ_Avi(n²−n²_eq) (1.3) wherenis the number density of DM particles, H is the Hubble parameter,hσ_Avi is the thermally averaged cross-section, andn_eqis the Dark Matter number density in thermal equilibrium. One can use the above formula to calculate the observed relic density, which will depend on the exact value of the cross-section of direct annihi- lation. As the annihilation cross-section is in many theories determined exclusively by the mass mχ, the two parameters can be matched using this framework. If we assume a weak interaction between DM and SM, for example, we get the formula [25]⁶:

σ_Av =k g⁴_weak

16π²m²_χ (1.4)

which can be used to match the massm_χ directly to the relic density observed in the present universe. WIMPs make an excellent Dark Matter candidate for this reason: a new particle in the mass range of 100 GeV- 1 TeV interacting weakly can easily account for all Dark Matter observed. There are many different caveats to this model that we skipped here, and many alternative mechanisms were proposed [25, 27, 28, 40, 41]. Nevertheless, WIMPs remain very popular candidates in the freeze-out framework, but the many negative results obtained by direct detection experiments and so far no sign of Supersymmetry (SUSY) at LHC would suggest that a wider range of masses should be explored as well. This provides an excellent motivation for the Dark Sector, where a Lightest Thermal Dark Matter (LTDM) is stable and produced in the early universe via one of the portals described in the previous section. The Dark Sector accommodates a wider range of masses compared to WIMPs (see Fig.1.4). Particles in the mass range 1 MeV - 10 GeV are however hard to probe using direct detection experiments, as the energy recoil from the scattering of a thermal relic with a nucleus becomes too small to be measured by standard detectors⁷. Accelerator experiments are on the other hand sensitive to these masses, and have the advantage of a production rate independent of the exact details of the Dark Sector predicted from the freeze-out mechanism [20]. This cross-section can be computed for a generic Dark Sector portal. We assume a mediator MED to be present together with a generic LTDM with massmLTDM which can account for the relic density. If we defineg_SMas the SM-mediator coupling we obtain:

hσvi= ¹ 6π

g²_Dg²_SMm²_LTDMv²

(m²_MED−4m²_LTDM)²+m²_MEDΓ²_MED (1.5) where we assumedv candm_e m_LTDM. If we further assume to be away from the resonance regionmMED ≈ 2mLTDMandmMED _ΓMED, the cross-section only depends on the mass of the LTDM and the dimensionless parameter:

y= ^g

2Dg²_SM 16π²

m_LTDM m_MED

4

(1.6) To be compatible with a freeze-out mechanism, a lower bound must be defined on the direct annihilation cross-section, which can be translated in a minimum value

6An additional factor ofv² multiplies the right-hand side of the equation if we assume P-wave annihilation.

7Methods to lower the threshold are however being developed, which would allow also such ex- periments to extend their sensitivity to sub-GeVparticles [42].

of y. A larger value of y is also possible and describes models where the direct annihilation is not the dominant process that describes the relic abundance [20]. This way we obtained a well-defined parameter space where to search for Dark Matter and a powerful method (accelerators) to explore it. In the next section, we will study the vector portal in more detail, and develop the most relevant formulas needed for the production and detection of the mediator. In the study that follows, we will assume a Dirac spinor as LTDM. Accelerator experiments, however, can probe many possible candidates, like complex scalars and axially coupled Majorana fermions.

A detailed classification of mediators and LTDM in the Dark Sector can be found in [43]. The lower limit of y predicted for several interesting cases is presented in Fig.1.2.

FIGURE 1.2: Lower limit of y motivated by thermal freeze-out as function of the mass of the LTDM in the context of the Dark Sector.

The limit is shown for scalar relic, Majorana fermion, Pseudo-Dirac fermion (black line). The scenario of asymmetric fermion Dark Matter is also shown (grey line), which is a common variation of the classical

thermal-origin framework [20].

1.4.2 The vector portal

The vector portal is built by introducing an additional U’(1) symmetry to the stan- dard model Lagrangian:

L_DS =L_SM−¹

4F_µν⁰ F^0µν+ ^mA0 2 A⁰_µA^0µ

| {z }

U’(1)

+iχ∂¯ µχ−mχχχ¯ −eDχγ¯ ^µA⁰_µχ

| {z }

Dirac spinor

+ ^e 2F_µν⁰ F^µν

| {z }

mixing term

(1.7) This new symmetry introduced an additional gauge vector boson A⁰. Here we also added a Dirac spinor fieldχthat is coupled to A⁰ by a coupling constantα_D. This is for the completeness of the model, which is also supposed to contain an LDM that justifies the relic density. Indeed the new parameters introduced in the Lagrangian can be used to describe the value ofy:

FIGURE 1.3: Possible production mechanism for A⁰: Dark Bremsstrahlung with a nucleus Z a), Dark Compton b), resonance

production c) and A-resonant production d).

y=α_De²(m_χ/m_A⁰)⁴ (1.8) which can be used to cast results in the parameter space relevant for a freeze-out scenario. The part of the Lagrangian over the first parenthesis is the one generated by the new U’(1) symmetry and is the most relevant to compute the physics of the portal. In particular, the term multiplied byerepresents the kinetic mixing between γand A⁰, as it multiplies the two field tensors generated by the two U(1) groups.

Here e is taken as an a priori parameter that controls the strength of the kinetic mixing. It can in principle take any value, but being naturally generated inside loops of heavy fields charged under both symmetries, it is expected to be small, in the rangee∼10⁻⁸−10⁻²[44].

The channels that allow the production of this new boson A⁰ are similar to the ones that can be used forγas they both are the generators of a U(1) symmetry. In first order they are the following:

• Dark Bremsstrahlung: The reaction e⁻Z → e⁻ZA⁰ where A⁰ is emitted after a virtual photon is exchanged with a target nucleusZ.

• Dark Compton: The reactione⁻γ → e⁻A⁰, where A⁰ is produced as a conse- quence of the interaction of a photon and an electron.

• Dark Resonance: The reaction l⁻l⁺ → A⁰ where two leptons annihilate and produce anA⁰ as a consequence of a resonance.

• A-resonant Emission: The emission of A⁰ is also possible without a resonance with the additional emission of a photon via the interactionl⁻l⁺→ A⁰γ.

The Feynman diagrams of all processes mentioned above are depicted in Fig.1.3.

Other channels might be possible with minimal extension of the model, or one could find a significant yield of production not only in the scattering of an electron

but also using another type of primary. An example would be using the electro- magnetic portion of a large hadron shower to compute the yield of A⁰, which was suggested as a possible future method to search for low massA⁰in [45].

When a high energy particle hits a thick target, all of the mentioned mechanisms will be relevant to some degree. In most fixed-target experiments, however, the Dark Bremsstrahlunge⁻Z→ e⁻ZA⁰is used as the "Golden Channel" of production because of its good yield compared to the other channels. The other types of produc- tions mechanism described are considered corrections to a production rate mostly dominated by this interaction. A dedicated setup, however, might make other con- tributions relevant as well. An example is found in [46], where the non-resonant and resonant production are used to improve significantly the signal yield for a specific mass range.

After the production mechanism is chosen, some first estimate on the signal yield can be performed. However, successfully producing the particle is not by itself suf- ficient without a way to detect it. The question is: what happens to A⁰ after it is produced inside a target? Since a coupling between standard matter and dark mat- ter was theorized in the U’(1) model, theA⁰will be able to decay in ae⁺e⁻pair (or in more massive leptons) provided that its mass is sufficiently large, but it could also decay in particles of the dark sector ¯χχ. It is important to know that there is no mean- ingful constraint that prevents both branching ratios to be of the same magnitude.

Likewise, models with more complicated decays are possible as well. An example is the one described in [47], where the decay chainA⁰ →χ₁χ₂(χ₂ →χ₁e⁺e⁻)domi- nates.

1.4.3 Dark Photon production in fixed-target experiments

In this section, the Dark-Bremsstrahlung channel will be described by studying the specific case of an electron impacting on a fixed-target. The main formulas to calcu- late the rate of production and detection are developed starting from the Lagrangian.

In terms of production rate inside a fixed-target, the U’(1) model is only defined by the mass of the mediatorm_A⁰ and the strength of the couplinge, and hence the pa- rameter space of these hypotheses is characterized by the plane(m_A⁰;e). The mixing term generates the interaction:

L_int=eeA⁰_µJ_em^µ (1.9)

between the Dark photon and the ordinary matter. The Dark Photon is produced as Dark Bremsstrahlung in the processe⁻Z→e⁻ZA⁰.

Calculating this cross-section is challenging due to complicated integrals involv- ing nuclear effects. We instead assume that the mass ofA⁰is large enough to treat the γexchanged in the reaction as a physical photon instead of a virtual one, which al- lows us to reduce the problem to the scattering of the electron with a physical photon emitted by a nucleus. This is called the Weizsacker-Williams (WW) approximation [48], and yields to the result [44]:

dσ

dxdcosθ_A⁰ = ^8Z

2α³e²E²₀ U² Log×

"

(1−x+x²/2)− ^x(1−x)m²_A0E₀²xθ²_A0

U²

#

(1.10) whereE₀is the energy of the incoming electron,E_A⁰is the energy of the emitted A⁰,θ_A⁰ is the angle of emission in the lab frame, and Zis the atomic number of the nucleus.x= E_A⁰/E₀is the fraction of original energy transferred to the Dark Photon,

theLog ∼5−10 is a factor accounting for atomic screenings and nuclear size effects (see Appendix.A.1). The function U defines the virtuality of the incoming electron in the intermediate state of the Bremsstrahlung, defined as:

U=E₀²xθ²_A0+m²_A0

1−x

x +m²_ex (1.11)

We continue by performing the angular integral on Eq.1.10:

dσ dx ≈ ^8Z

2α³e²x m²_A0

1+ ^x

2

3(1−x)

Log (1.12)

Now we can apply this formula to compute the yield inside a target. If we as- sume an electron with energyE₀impacts a thick target with total radiation lengthT, we derive:

dN

dx = N_eN₀X₀ A

Z _E0

E_A0

dE₁ E₁

Z _T

0 dtI(E₁;E₀;t)×E₀dσ dx⁰

x⁰=E_A0/E1

(1.13) where N0 is the Avogadro’s number, X0 is the radiation length of the target, A is the target atomic mass, andI is the energy distribution of electrons after passing through t radiation lengths. The above integral is still fairly complicated, mainly an accurate description of the electron energy distribution aftert radiation lengths is not an easy task⁸. A common approach is the thin target approximation, where I ≈ δ(E₁−E₀)is used as the target is assumed to be thin enough that no electro- magnetic shower (em-shower) is triggered. In fixed-target experiments, however, it is desirable to block the incoming e⁻ completely to suppress the background, which means a thick target is used instead. This requires some additional care in parametrizingI as illustrated in Appendix.A.1.1. Here we report the final result, the rate at which theA⁰is produced is approximated by:

N_A⁰ ' N_EOT×C⁰e² m²_e m²_A0

(1.14) This allows us to calculate the number of Dark Photons produced as a function of the accumulated EOT (Electrons On Target). The dimensionless factorC⁰ ≈ 10 is obtained after solving the integral. One has to be careful with this formula since many approximations were used to derive it and is expected to be accurate within an order of magnitude [44]. However, it provides us with some very useful scaling with the parameters of the model and allows us to understand the sensitivity of the experiment plotted in the(m_A⁰;e)space. To compute the exact sensitivity with high precision, a detailed MC-simulation should be used instead.

Decay modes and detection

After the production ofA⁰, a mechanism to detect it is needed. To address this prob- lem, we need to understand what happens toA⁰after it is emitted inside the target.

In our model a Dirac field is also present, so the Dark Photon can in principle de- cay in a pair of particles generated by this field, which are stable LTDMs. However, a decay to SM leptons is also possible, since the kinetic mixing mechanism can act in both directions. This provides two different decay channels, which will define the precise detection strategy. We compute the two decay rates starting from the interaction Lagrangian in Eq.1.9. We obtain:

8In practice, this is solved by MC-simulation as we will see in chapter 3