Dark Photon decay generated by muons in the SHiP experiment

MAT-VET-F 20024

Examensarbete 15 hp Juni 2020

Dark Photon decay generated by muons in the SHiP experiment

Elizaveta Yakovleva

Teknisk- naturvetenskaplig fakultet UTH-enheten

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Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

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018 – 471 30 03

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018 – 471 30 00

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http://www.teknat.uu.se/student

Abstract

Dark Photon decay generated by muons in the SHiP experiment

Elizaveta Yakovleva

This project has investigated the muon background of the SHiP experiment to determine whether it can boost the experiment sensitivity to visible Dark Photon decay. Using Fermi-Weizsäcker-Williams approximation to muon scattering we found the probability of muons generating massive photons, using Bremsstrahlung and direct lepton pair production as an estimation of the frequency of muon EM-interactions. In this work we only considered muons with momenta above 10 GeV/c.

The number of visible Dark Photon decays was calculated for a range of the coupling constant and photon mass. The resulting range that promised visible decay has already been excluded by previous experiments, but the method could be used to further investigate enhanced production of Dark Photons from muons and electrons, and possibly also production of Axion-like particles. The work could also be used to estimate sensitivities of other experiments using muons.

Examinator: Martin Sjödin Ämnesgranskare: Richard Brenner Handledare: Richard Jacobsson

Popul¨arvetenskaplig sammanfattning

Inom partikelfysik existerar en modell f¨or alla partiklar, Standardmodellen. Denna beskriver hur de fundamentala partiklarna bygger upp och reagerar med varandra i v˚art universum. I Standard- modellen ing˚ar flera grupper av partiklar: fermionerna - hit h¨or elektroner, muoner och kvarkar, bosoner - som ¨ar kraftb¨ararna vid reaktioner mellan partiklar, och hadroner - partiklar uppbyg- gda av kvarkar, hit h¨or protoner och neutroner. Hittills har modellen p˚a ett tillfredsst¨allande s¨att beskrivit vad som har setts i experiment. Men det finns fortfarande outforskade omr˚aden i univer- sum och fr˚agor som Standardmodellen inte lyckats besvara. Inom astronomi och kosmologi finns bevisen f¨or att universum inneh˚aller mer materia ¨an den vi kan se med blotta ¨ogat - den M¨orka materian. Vi vet inte vad den h¨ar materian best˚ar av, men en hypotes ¨ar att den utg¨ors av partiklar med ingen eller l˚ag v¨axelverkan ut¨over gravitationen.

SHiP ¨ar ett experiment vid CERN som ska s¨oka efter de partiklar som skulle kunna utg¨ora M¨ork materia och partiklar som m¨ojligg¨or interaktioner mellan den M¨orka och den synliga materian.

SHiP-experimentets detektor best˚ar av en protonstr˚ale som riktas mot ett m˚al. Partiklarna som bildas vid kollisionen mellan protoner och partiklarna i m˚alet m¨oter en hadronabsorbator, som hindrar en del av dem fr˚an att passera, och en muonsk¨old som skingrar muoner. Muonerna kan i sin tur skapa nya partiklar, av vilka majoriteten ¨ar elektron-positronpar och fotoner. De partiklar som experimentet s¨oker m¨ata har l˚ag v¨axelverkan och kommer d¨arf¨or inte att p˚averkas av vare sig hadronabsorbatorn eller magnetf¨altet fr˚an muonsk¨olden utan n˚ar fram till s¨onderfallsvolymen och kan registreras av detektorn.

Det finns flera s¨att p˚a vilka M¨orka partiklar kan bildas. Ett intressant exempel ¨ar M¨orka fotoner, som skulle kunna bildas genom interaktion med virtuella massiva fotoner, det vill s¨aga fotoner med en massa skild fr˚an noll. Virtuella fotoner kan bildas i reaktioner som liknar bromsstr˚alning och direkt parbildning av leptoner. Sannolikheten f¨or att den virtuella fotonen omvandlas till en M¨ork foton beror av en ok¨and kopplingskonstant. F¨or att kunna observeras m˚aste de M¨orka fotonerna sedan antingen s¨onderfalla till eller reagera med synliga partiklar. F¨or att SHiP ska kunna m¨ata dessa h¨andelser m˚aste de M¨orka fotonerna hinna bildas och s¨onderfalla innan detektorn.

Sannolikheten att de g¨or det best¨ams av kopplingskonstanten och massan hos de M¨orka fotonerna, som ¨ar den samma som hos de virtuella fotonerna.

Det h¨ar projektet unders¨oker vilka partiklar som kan f¨orv¨antas bildas i SHiP-experimentet genom att analysera data fr˚an simulerade kollisioner. Fokus ligger sedan p˚a att se om muonerna i ex- perimentets bakgrund kan bilda tillr¨ackligt m˚anga virtuella fotoner f¨or att ¨oka sannolikheten att m¨ata M¨orka fotoner. Projektets syfte ¨ar att ge svar p˚a hur muonbakgrunden kan bidra till SHiP- experimentets k¨anslighet.

Genom att unders¨oka de fotoner och elektron-positronpar som bildas av muonerna i simuleringar kunde projektet ber¨akna antalet f¨orv¨antade virtuella fotoner under SHiP-experimentets hela k¨orn- ing. F¨or olika v¨arden p˚a kopplingskonstanten mellan M¨orka och synliga fotoner, samt olika v¨arden f¨or foton-massan, kunde antalet f¨orv¨antade synliga M¨orka fotons¨onderfall ber¨aknas. Det visar sig att omr˚adet inom vilket vi f¨orv¨antar oss dessa s¨onderfall redan unders¨okts av tidigare experiment utan resultat, och vi kan anta att detsamma kommer g¨alla f¨or SHiP. D¨aremot finns m˚anga fler partiklar kvar att unders¨oka som bildas vid SHiP, vilket kan g¨oras med den framtagna metoden.

Contents

1 Introduction 5

1.1 Hypothesis . . . . 5

1.2 Theory . . . . 5

1.2.1 The Standard Model . . . . 5

1.2.2 Particle interactions . . . . 6

1.2.3 Dark Matter and the Hidden Sector . . . . 7

1.2.4 Measuring invisible particles . . . . 8

1.2.5 Production of Dark Photons . . . . 9

1.2.6 Dark Photon production through muon scattering . . . . 12

1.2.7 Lifetime acceptance . . . . 13

1.3 Methods . . . . 14

1.3.1 The experiment . . . . 14

1.3.2 The simulation . . . . 15

2 Results 17 2.1 Particles generated in the SHiP experiment . . . . 17

2.2 Muons and their daughter particles . . . . 17

2.3 The weight function . . . . 21

2.4 Number of expected events . . . . 24

3 Discussion 25

4 Conclusion 26

1 Introduction

The Standard Model is a theory of the fundamental particles and their interactions in the universe.

Everything that it predicts has been experimentally verified. However, this is not a complete theory of the whole universe, for example, it does not cover Dark Matter. We know that Dark Matter exists, because there is proof of this within the field of astrophysics and cosmology, such as the Bullet cluster, but its composition is still unknown, and it has yet to be measured in experiments.

There are many theories of what Dark Matter might be. One such theory is that it consists of either very massive or very small, feebly interacting particles, which would explain why they have not yet been measured. With a minimal extension of the Standard Model, assuming we can use the same formalism as for ordinary matter, the Standard Model can be used to model Dark Matter particles. If these particles could be measured in experiment Dark Matter could indeed be included in the Standard Model.

The SHiP experiment is a fixed target proton collider experiment. SHiP searches for the particles that could make up Dark Matter and particles propagating possible interactions between Dark Matter particles. One such example is Dark Photons. These could be created from Bremsstrahlung, light meson decay or quark annihilation. In this project we seek to find the probability of Dark Photons being created from virtual photons in the SHiP experiment. To be more specific, we want to see if muons can increase the rate of Dark Photons by giving rise to virtual photons, which in turn may convert into Dark Photons.

1.1 Hypothesis

The muons created by the proton beam in the SHiP experiment might lead to the creation of Dark Photons that will decay and reach the detector, where the decay could be measured.

• Can we enhance or boost the measurements of Dark Matter by using the muon flux?

• What is the probability of the creation of Dark Photons?

• Dark Photons need to mix with virtual photons in order to be created. How often can photons become virtual photons and how many reach the detector?

1.2 Theory

1.2.1 The Standard Model

The particles included in the Standard Model are the fermions, bosons and hadrons. The fermions are spin-¹₂ particles that include the quarks and leptons (such as the electron e⁻, muon µ⁻ and the neutrinos), which are divided into three families, and their corresponding anti-particles (positron e⁺, positive muon µ⁺, antineutrinos). Bosons are the spin-1 gauge bosons, serving as force carrier (such as photons), and one spin-0 particle - the Higgs boson, which gives mass to the particles within the theory. These Standard Model particles are treated as elementary, meaning they are point particles without inner structure. The hadrons are composite particles. They consist of quark bound states, which are called baryons (3 quarks), antibaryons (3 antiquarks) and mesons (a quark-antiquark pair). Examples of hadrons are the neutron n and proton p (nucleons) and the pions (π⁺, π⁻, π⁰).

The forces that govern the fundamental particles are the weak and strong interaction, the elec- tromagnetic interaction and gravity (this is however very weak in comparison and neglected for simplicity). The strong interaction binds quarks together and is the reason why quarks cannot be observed as individual particles, but only in their bound states. The weak interaction only works when two particles are very close, due to the large mass of the force carriers, the W and Z bosons.

The electromagnetic interaction works through the exchange of photons between charged particles.

The Standard Model is not a complete theory of particles in the universe. There are still unexplored areas of the universe that have hitherto been invisible; high energies and large masses, light masses and weakly coupled particles. New Physics theories, like the Grand Unified Theory, Supersymmetry and String Theory explore the hidden areas beyond the Standard Model.

To explain our universe and find the particles predicted by the New Physics theories one would have to recreate how the universe looked in the very beginning of its existence. The experiments with particle accelerators strive to recreate the conditions of the early universe. With the help of cosmology we can look back at even earlier stages of the universe as well as measure phenomena that could help explain the unanswered questions within particle physics. Cosmological experiments can predict the mass density needed in the universe for it to continue expanding as observed. In order to account for all matter in the universe there needs to be nonluminous, nonbaryonic matter. This is denoted as Dark Matter and since it does not interact with visible matter it exists within what is called the Hidden Sector [1].

1.2.2 Particle interactions

Particle interactions and their mediators, the force carriers, are modelled by the Standard Model.

In electromagnetic interactions a photon propagates the interaction between two charged particles, or a charged particle and a nucleus. During Bremsstrahlung a charged particle is slowed down by the electromagnetic field of a nucleus and forced to emit photons, this is illustrated in the right Feynman diagram in Figure 1, [2].

A Standard Model photon can become virtual, meaning that it has borrowed energy from vacuum or a nucleus and has become massive. This can happen in Bremsstrahlung, in direct pair production or annihilation of the particle-antiparticle pair. The invariant mass of particles is the same in any reference frame. This means that if new particles are created from particle decay, the invariant mass of the created particles must be the same as that of the initial particle. That is why, if an electron-positron pair is created from a photon, the photon in question must have had the same mass as the particle pair, and this massive photon must also be virtual. See the left Feynman diagram of Figure 1.

Figure 1: Feynman diagrams of pair production of electron positron pair and Bremsstrahlung [3].

1.2.3 Dark Matter and the Hidden Sector

The Hidden Sector consists of matter that we cannot measure due to its very weak or non-existing interactions with matter in the Standard Model. Evidence of its existence can be found within the field of astrophysics and cosmology. The Bullet cluster was an early discovery of two clusters of galaxies that once collided with each other. Similar clusters with this phenomenon have since been discovered and analysed with greater precision. During the collisions the galaxies themselves have a very low probability of colliding, but the gas of each cluster will interact with the other, heat up and gather in the space where the two clusters passed through each other. The gas emits heat radiation that can be used to reconstruct the events of the collision, see Figure 2. However, when measuring the bending of light around the galaxy clusters one can see that the majority of the mass is still within the galaxy clusters and has passed through the collision mostly unaffected. This invisible mass, with possible feeble interactions, that exists in the universe is called Dark Matter [4].

Figure 2: The Bullet cluster with Dark Matter map (blue) and heat radiation from the heated gas (pink) between the clusters of galaxies. [5]

There are several theories of what Dark Matter might be, the particles predicted by an extension of the Standard Model into the Hidden Sector is only one of them. The particles in the Hidden Sector are theoretically either weakly coupled, meaning they do not interact much with each other, or very massive particles. These particles might have interactions mirroring those of the Standard Model, for example a dark electromagnetic interaction. A hypothesis is that if those interactions exist they have a much weaker coupling constant, making Dark Matter feebly interacting. In order for us to measure the hidden particles these have to interact with the particles of the Standard Model, for example through kinetic mixing, when a Dark Matter particle becomes a Standard Model particle. These mixing particles could provide a portal between the Standard Model particles and the Hidden Sector, with different portals leading to different particles: dark scalars (Dark Higgs- particles), heavy neutrinos, axion-like particles, or combinations of them. One such portal that is of interest in this project are the ones created by Dark Photons [6], [7].

1.2.4 Measuring invisible particles

Dark Matter can be seen through scattering events, when a Dark Matter particle scatters against a Standard Model particle, or through decay into a Standard Model particle. In order to measure such an event in an experiment like SHiP the lifetime of the Dark Matter particle cannot be too long, or it will miss the detector, nor too short or it will never reach the detector. In calculations of the possibility of measuring Dark Matter the lifetime acceptance gives the probability of the scattering event or decay happening within the span of the experiment. The coupling constant, ε, of a Dark Matter particle tells us how likely it is to interact or mix with a Standard Model particle.

This constant must be very small for light particles.

The Dark Photon portal can be created through interactions with virtual massive photons. The

Figure 3: Feynman diagrams of particle interactions with the possible creation of a virtual photon that converts into a Dark Photon through kinetic mixing with strength ε: (from the top, left to right) Bremsstrahlung, light neutral meson decay, quark/gluon scattering and particle-antiparticle annihilation [6].

virtual photon has a very short lifetime and will in most cases immediately decay into a particle- antiparticle pair, for example an electron and a positron. There might be a possibility that, instead of decaying, the virtual photon will turn into a Dark Photon through a coupling interaction called kinetic mixing. The Dark Photon is weakly coupled and might have a much longer lifetime than the virtual photon, it would then travel a long way before eventually turning back into a virtual Standard Model photon.

In order to find the Dark Photons one must first find the Standard Model photons. Photons with enough energy to create a particle-antiparticle pair could be created in the SHiP experiment. The sources of virtual massive photons that this project will investigate are muons, which will either create photons through Bremsstrahlung or through direct pair production of leptons and antileptons [6], [7].

1.2.5 Production of Dark Photons

The production of a Dark Photon would come from processes involving the exchange of photons, which in the virtual state turns into Dark Photons through kinetic mixing with strength ε, where ε  1. Virtual photons may be produced in processes similar to Bremsstrahlung (from muons or electrons), light neutral meson decay (for example π⁰ decay), quark annihilation and direct pair production of leptons, when a particle and anti-particle pair are formed (see Figure 3). In this project we only used the cases of muon Bremsstrahlung and direct pair production as a measurement of the total production of photons from muons.

In order to find the probability of the creation of Dark Photons from muons one must take into account three probabilities: the probability of photons becoming virtual, the probability of muon Bremsstrahlung or direct lepton pair production from muons, and the probability of kinetic mixing between ordinary matter and Dark Matter [6],

P (Dark P hoton) = P (Bremsstrahlung + pair production)P (virtual photon)P (mixing). (1)

In this project we want to investigate if the proton beam creates enough muons that could produce measurable Dark Photons. The processes of interest are the electromagnetic interactions when muons scatter off atoms in the hadron absorber and muon shield (see Figure 4). The probability of an event producing a Dark Photon is represented by its cross-section, describing the interaction between the proton and a nucleus in the target (N ) that creates a muon µ and anything else, denoted X. The muon later interacts with a nucleus in the muon shield (N⁰) and creates a Dark Photon A⁰,

σ(pN → µX, µN⁰ → A⁰µN⁰) [cm²]

The rate of events is given by the cross-section multiplied by the luminosity ρ_event= σL [N/s],

where the luminosity for fixed target experiments is given by the flux of particles, Φ, times the nucleon density (number of protons and neutrons), ρ_N, and the target length, l,

L = Φ · ρ_N· l. (2)

Number of events in the experiment is the rate integrated over the runtime, t, Nevents= σ

Z Ldt.

Assuming all protons on target (N_pot) interact this results in the number of events Nevents(pN −−→ X) = σ^inel _inel

Z

Ldt = Npot, (3)

where σ_inel is the total inelastic cross-section.

From the simulation we can find the rate of muons per protons on target. The remaining task is to calculate the branching ratios (fractions) for muons that could create Dark Photons. Using the

branching ratio (χ(x → y) = ^σ(x→y)_σ

tot ) for processes producing Dark Photons, A⁰, the number of events becomes:

N_events(pN → A⁰X) = N_pot· χ(pN → A⁰X) =

= N_pot·σ(pN → µX)

σ_inel ·σ(µN⁰ → µA⁰N⁰) σ(µN⁰→ µγN⁰) =

= Npot· χ(pN → µX) · χ(µN⁰ → µA⁰N⁰)

(4)

We make the assumption that we can factorise the probability density function f for muons to undergo electromagnetic interactions resulting in photons (we add together Bremsstrahlung and direct pair production) and the effect of producing virtual photons (massive photons with mass m_A⁰) through the weight function δ:

dN_A⁰

dpⁱ_µ (m_A⁰, ε) ≈ df (µN⁰ → γµN⁰, e⁺e⁻µN⁰)

dpⁱ_µ ·dδ(γ ⇒ γ^∗(m_A⁰))

dpⁱ_µ · ε², (5)

where ε is the strength of kinetic mixing, pⁱ_µ is the momentum of an individual muon, γ is a Standard Model photon and γ^∗(m_A⁰) is the massive photon.

The probability density function for the muon interactions with the material of the absorber and the muon shield, f , and the normalisation to the number of protons on target may be obtained through simulation, allowing to integrate the complete flux of muons to get the expected total production of virtual photons.

The probability density function δ for Dark Photon production will depend on the momentum of the initial muon, as well as the mass of the Dark Photon. For this reason we need to know δ as a function of the Dark Photon momentum and integrate over the possible momentum range given by the muon momentum.

Not all events will be observable. Some particles will gain a large momentum and a large scattering angle, and will miss the detector volume. Particles with too long or too short lifetime will decay outside the decay volume. Taking into account the geometric detector acceptance (depending on Dark Photon production angle), and the lifetime acceptance (depending on the Dark Photon’s proper lifetime τ , which in turn depends on the p_A⁰ and m_A⁰) we obtain the number of observed events:

N_obs = N_event· P_vertex(θ_A⁰, p_A⁰) · P (A⁰ → visible). (6)

Thus we have to derive the probability density function δ as a differential function of both Dark Photon momentum and production angle α⁰ with respect to the muon, similar to what has been done for electrons and protons [8]–[12]:

dδ

dp_A⁰dα⁰(mA⁰) (7)

We will attempt to obtain δ by applying the Fermi-Weizs¨acker-Williams approximation to muon scattering against nuclei.

1.2.6 Dark Photon production through muon scattering

Fermi-Weizs¨acker-Williams (FWW) approximation of initial state radiation from muon scattering against nuclei (Energy in GeV, momentum in GeV/c, mass in GeV/c², c=1):

dδ

dz dp²_⊥ =αQED

2πH

 1 + (1 − z)²

z − 2z(1 − z) 2m²_µ+ m²_A0

H − z²2m⁴_µ H²

!

+ 2z(1 − z)(1 + (1 − z)²)m²_µm²_A0

H² + 2z(1 − z)²m⁴_A0

H²



(8)

where

H(z, p²_⊥) = p²_⊥+ (1 − z)m²_A0 + z²m²_µ (9) Here we use the following notations: m_µ- muon mass, p⊥- transverse momentum of the Dark Pho- ton with respect to the initial state muon, z - fraction of the initial muon momentum carried away by the Dark Photon parallel to the initial state muon (z = p_k/pⁱ_µ), and the constant characterizing the strength of electromagnetic interaction α_QED ≈ 1/137. Note that this expression will neglect final state radiation from the scattered muon as a source of Dark Photon production, and it will neglect Bremsstrahlung produced by muon scattering against atomic electrons.

Functions to define the weight function as a function of dark photon momentum and dark photon production angle:

z(pⁱ_µ, p_A⁰, α⁰) ≈ p_A⁰ pⁱ_µ√

α⁰²+ 1 p²_⊥(pⁱ_µ, pA⁰, α⁰) ≈ (pⁱ_µα⁰z)²,

(10)

where pⁱ_µ is the momentum of the initial state muon, pA⁰ momentum of the Dark Photon, and α⁰ is the production angle of the Dark Photon.

We define the probability density function as a differential function of Dark Photon momentum and Dark Photon angle by a change of variables:

dδ

dp_A⁰dα⁰ = dδ

dz dp²_⊥ ·dp²_⊥ dα⁰ · dz

dp_A⁰ (11)

where

dp²_⊥

dα⁰ ≈ 2α⁰pⁱ_µ²z² dz

dp_A⁰ ≈ 1 pⁱ_µ√

α⁰²+ 1

(12)

Integration gives the total δ function for any momentum transfer and angle of the Dark Photon:

δ(Eⁱ_µ, mA⁰, pⁱ_µ) =

Z max(p_A0) min(p_A0)

Z

SHiP

dδ

dp_A⁰dα⁰dpA⁰dα⁰ (13) Range of integration min(p_A⁰) and max(p_A⁰) is assumed to be 10% and 90% of the incoming muon momentum. The range of integration in α⁰ is over the geometric acceptance of the SHiP detector.

Probability of producing Dark Photon with given mass mA⁰ and kinetic mixing ε by an initial muon of given momentum pⁱ_µ:

f (mA⁰, pⁱ_µ, ) = ²δ(mA⁰, pⁱ_µ) (14) 1.2.7 Lifetime acceptance

Similar to the relation between the decay time of excited atomic states and the line shapes in spectroscopy, the proper lifetime τ in the rest system of an unstable particle is related to the uncertainty in its mass state, the width Γ of its mass:

∆E ∆τ ≤ ~ Γ τ ∼ ~ τ ∼ ~

Γ

(15)

The lifetime distance in the lab system of a particle with a relativistic gamma factor γ is given by t v = γ τ β c = E

mτ p Ec = p

mτ c, (16)

where β = ^v_c, γ = (1 − β²)^−1/2 and c is the speed of light.

The probability that the dark photon decays within a range l₁ < l₂,

P (p_A⁰, m_A⁰, ) = Rl2

l1 e^−γcτl dl R∞

0 e^−γcτl dl

= Z l2

l1

e^−γcτl

γcτ dl. (17)

The lifetime τ will be given as above, with Γ being the line width of a Dark Photon decaying to leptons:

Γ(A⁰ → l⁺l⁻) = 1

3α_QEDm_A⁰² s

1 −4m²_l m²_A0



1 +2m²_l m²_A0



(18)

Below a mass of 0.5 GeV/c² we assume that it can only decay to muons and electrons, and electrons only below 2mµ.

1.3 Methods

1.3.1 The experiment

SHiP (Search for Hidden Particles) is a proposed experiment at the SPS (Super Proton Synchrotron) accelerator at CERN. Its purpose is to explore possible Dark Matter particles and associated interactions, assumed to be very weakly interacting particles. These particles can be measured by the experiment both through visible decay, when the Dark Matter particles decay into visible matter, and scattering events, when a Dark Matter particle interacts with and recoils off visible matter.

Figure 4: The SHiP detector: a. target, b. hadron absorber, c. muon shield, d. decay volume, e.

detector [13].

The SHiP experiment detector consists of a proton beam dump sending protons towards a target.

The base plan is to collect 2 · 10²⁰protons on target, which could be achieved in 5 years of operation at the nominal luminosity. In order to achieve background suppression the particles that are created in the collisions are filtered by a hadron absorber, which prevents most of the particles from passing, and a muon shield that deflects muons. The muon shield is followed by a decay volume and the detector itself. The decay volume consists of a vacuum volume with the purpose of minimising the risk of neutrino interactions producing decays which may mimic Hidden Sector particle decays.

The particles that the SHiP experiment seeks to measure are feebly interacting particles and will therefore not be affected by the hadron absorber or the magnetic field of the muon shield but will

reach the spectrometer where the particle identification will be made.

The muons that interact in the muon shield or make it to the decay volume can in turn create new particles, some of which might be of interest for the experiment. In order to reach the detector the particles that are created in the target can have an angle of at most 40 mrad relative to the centre of the target. The length of acceptance is the length of the experiment where particle decay can lead to a signal, which is mostly in the decay volume. In the lifetime acceptance of the Dark Photon (17) the limits of the integral will be l₁ = 40.0, l₂ = 90.0 [m].

If the experiment is to measure any Dark Photons there needs to be many photons generated that the Dark Photons could interact with through kinetic mixing. The SHiP experiment will look at several scenarios that can generate large amounts of photons:

1. Proton Bremsstrahlung

2. Proton inelastic scattering: generating new particles, such as quarks, baryons and mesons that decay into photons, which could potentially be massive virtual photons.

3. Photons generated in deep inelastic scattering from the induced showers of quarks and gluons In this project we investigate if there could be yet another way for Dark Photons to be generated, which would boost the sensitivity of the experiment. There are many electrons and muons generated by proton collisions, which in turn generate photons. These particles are currently considered as background in the experiment [14]–[20].

1.3.2 The simulation

The experimental facility of SHiP is not yet built and has to be simulated in order to estimate what it will measure. For the analysis of the simulation the programming language ROOT will be used in combination with Python, called pyROOT. The simulations are created beforehand in Pythia8 (the event generator) and Geant4 (detector simulation). Particles are generated with Monte Carlo simulations, based on the well-known physics describing the probability of each particle to interact in a material, the cross-section. The code for analysing the simulations is run in SHiP’s own simulation environment, called FairShip.

How the simulation works: The virtual reality of the simulation is generated in the event generator Pythia8. The detector geometries, the events, the particles created in each event and the hits when a particle reaches a detector are simulated in the detector simulator, Geant4. The information about the particle is stored in tracks, snapshots of a particle in the detector volume; dynamic properties, such as momentum, energy, spin, etc.; static properties, such as charge, mass, life time, etc. The simulation follows every particle from the moment it is created until it disappears from the simulation volume or decays into other particles [21], [22].

The simulation models the experimental setup of SHiP, which will filter out the background with the hadron absorber. All geometries of the experiment - the target, the hadron absorber, the magnets and the particle detector - are simulated with its specific material, size and position of the real experiment. The interesting signal, which consists of feebly interacting particles, should reach the detector mostly undisturbed. Some background cannot be stopped by the hadron absorber, that is muons and neutrinos. The muons can create photons in scattering events with a nucleus.

If these photons are virtual they could potentially lead to the creation of Dark Photons, which is the signal we are looking for.

We have also looked at a simpler simulation which models a thin 1 mm wolfram target followed by 50 thin silicon sheets to examine the different particles that might be created from the proton beam. Every silicon sheet is considered a detector by the simulation.

2 Results

2.1 Particles generated in the SHiP experiment

The proton collisions generate a lot of different particles in the target. Using the simpler simulation with a wolfram target and no hadron absorber or muon shield we could investigate what kind of particles could be created. Some common particles that would be generated by the experiment can be seen in Table 1 divided by the number of protons on target. In the SHiP experiment the hadron absorber would stop most of these particles, except for muons and neutrinos. The SHiP experiment has a longer target, which will generate more particles per each proton as more protons will interact in the target. In Table 1 we can see that muons and electrons are commonly generated, and they could be further investigated. Unlike the simulation of the experimental background this simulation had no momentum cut, but saved all generated particles.

Name Number of particles/proton on target

muons (both positive and negative) 0.0056

electrons 0.069

positrons 0.031

photons 0.29

neutrons 0.13

protons 0.043

π⁻ 0.098

π⁺ 0.096

π⁰ 0.11

ρ⁰ 0.0016

η 0.014

Ω 0.00014

K_L⁰ 0.0073

K_S⁰ 0.0074

K⁺ 0.0093

K⁻ 0.0059

Λ 0.0078

Σ⁺ 0.0015

Σ⁻ 0.0015

Σ⁰ 0.0018

Table 1: Most common particles generated in the experiment per protons on target.

2.2 Muons and their daughter particles

From the simulation of the SHiP experimental setup we investigated data from 100 000 events, meaning 100000 protons on target with a mixture of minimum biased enriched di-muon events.

These events generated 122939 unweighted muons, normalised to the total amount of 2 · 10²⁰ protons on target this corresponds to 7.41 · 10¹⁶ muons with momentum above 10 Gev/c. Their momenta can be seen in Figure 5. Note that each proton has a momentum of 400 GeV/c, which is the limit of how much energy can be transferred to a daughter particle (particle generated in

scattering event by another particle, the mother).

Most muons have a momentum smaller than 300 GeV/c, and the mean value for all muons is around 38 GeV/c. The apparent excess at 210-240 GeV/c in Figure 5 is within the margins of statistical error of the measurement. The simulation has only taken into account particles that are generated in the proton collision with a momentum larger than 10 GeV/c. Less energetic particles would quickly lose their energy through continuous energy loss (dE/dt) and never reach the detector. A muon with momentum below 10 GeV/c loses 2 GeV/c per meter to electromagnetic showers of delta electrons, low-energy electrons, before reaching the muon shield.

Muons can undergo electromagnetic processes during interactions with a nucleus. In the experiment the muons with momenta > 10 GeV/c will not be stopped by the hadron absorber, but will reach the muon shield, where they can interact with the iron nuclei and generate daughter particles.

This is illustrated by Figure 6 where the shape of the muon shield is clearly marked by the dots of photon, electron and positron start positions.

Figure 5: Momentum of muons generated from 100 000 protons, normalised to 2 · 10²⁰ protons on target.

Figure 6: Muon interactions in muon shield, where the x-axis is along the width of the shield and z-axis goes parallel with the length of the shield.

The daughter particles are mainly electrons, positrons and photons. The photons are created through muon Bremsstrahlung and can receive enough momentum that they will become massive.

The positrons are created through direct pair production with electrons, which have a larger mo- mentum than delta-electrons and have been created by a virtual photon. In Figure 7 the angles at which the muon interacts are illustrated along with the energies of the muons that generate photons or electron-positron pairs. Most of the interactions have a low scatter angle. The lower plots show that most daughter particles receive a significant momentum from the mother muon.

Figure 7: The following graphs are for muon interactions that generate photons or direct pair production of electrons and positrons: a) Scatter angle of mother muon. b) Energy of incoming mother muon. c) Momentum of daughter particles. d) Fraction of muon energy carried away by daughter particle.

When the muon interacts and creates a particle both the muon and the daughter particle will leave the vertex (point of interaction) with an angle with respect to the trajectory of the incoming muon.

See Figure 8 for an illustration of the angles. These angles are relatively small, as can be seen in Figure 9. In the plot in 9.b) it is clear that most muons interact at early stages of the muon shield. From these two plots we can make the approximation that the daughter particles will point at the detector and that we can integrate over all muons when implementing the weight function (13). We can also integrate over −40 < α⁰ < 40 [mrad] and still cover a majority of the particles that can reach the detector. Note that the kinematics changes slightly when the photon is massive, therefore the angles in the bottom plots are an approximation.

Figure 8: Angles at muon interaction generating a massive photon. Here µ and µ⁰are the incoming muon and outgoing muon, γ^∗ is the massive photon, the dotted line is the z-axis.

Figure 9: The following graphs are for muon interactions that generate photons or direct pair production of electrons and positrons: a) The angle of incoming muon against beam axis. b) Position of interaction in z (along beam axis). c) Scatter angle against incoming muon of daughter photon. d) Angle of daughter photon against beam axis.

2.3 The weight function

The probability density function or the weight function for production of Dark Photons that was derived in section 1.1.5 is illustrated as a function of Dark Photon momentum and the production angle in Figure 10. This is only for one value of muon momentum, 64.4 GeV which is the mean momentum of muons that generate photons and direct pair production of leptons (see Figure 7.b)).

Note that this is only the probability of producing virtual massive photons as we have not yet multiplied with the coupling constant ε.

(a) Contour plot of weight function

(b) 3D plot of weight function

Figure 10: Weight function for creation of virtual massive photons (or Dark Photons with ε = 1) with mass mA⁰ = 100 MeV/c² as a function of photon production angle and photon momentum.

Here the momentum of the mother muon is 64.4 GeV/c.

Integrating the weight function over the angle we obtain the momentum dependence. The weight as a function of all muon momenta in the simulation and the possible Dark Photon momenta for each muon gives the plot in Figure 11. Dark Photon momentum will be dominated by low energies.

The plot is based on a simulation of 100 000 events.

(a) Contour plot of weight function (b) 3D plot of weight function

Figure 11: Weight function for creation of virtual massive photons (or Dark Photons with ε = 1) with mass m_A⁰ = 100 MeV/c² as a function of muon momentum and photon momentum.

To find the number of expected events in the SHiP experiment we integrate over the production angle and possible Dark Photon momenta for each muon and some different masses for the Dark Photon, then normalise to 2 · 10²⁰ protons on target. Table 2 shows the number of expected events for m_A⁰ = 2, 5, 10, 50, 100, 500 MeV/c² integrated over both 0 < α⁰ < 0.04 and 0 < α⁰ < 0.1 to see how many events we lose with the geometric acceptance of the detector. The result for m_A⁰ = 10, 100, 500 MeV/c² integrated over 0 < α⁰ < 0.04 is given in Figure 12. The apparent peaks at 30 − 35 MeV/c could be fluctuations or statistical error.

Table 2: Number of expected virtual photons of masses m_A⁰ = 2, 5, 10, 50, 100, 500 MeV/c² with the weight function integrated twice over 0 − 0.1 rad and 0 − 0.04 rad

masses [GeV/c²] 0.002 0.005 0.01

0 - 0.1 [rad] 3.784 · 10¹⁰ 3.782 · 10¹⁰ 3.771 · 10¹⁰ 0 - 0.04 [rad] 3.016 · 10¹⁰ 3.014 · 10¹⁰ 3.003 · 10¹⁰

masses [GeV/c²] 0.05 0.1 0.5

0 - 0.1 [rad] 3.454 · 10¹⁰ 3.089 · 10¹⁰ 1.909 · 10¹⁰ 0 - 0.04 [rad] 2.688 · 10¹⁰ 2.329 · 10¹⁰ 1.219 · 10¹⁰

Figure 12: Number of expected Dark Photon events, with ε = 1 and 0 < α < 0.04 rad, for the masses a) m_A⁰ = 10 MeV/c², b) m_A⁰ = 100 MeV/c², c) m_A⁰ = 500 MeV/c²

To make sure the Dark Photons will decay within the limits of the decay volume we multiply by the lifetime acceptance, which depends on the mass of the Dark Photon and the coupling constant ε. Figure 13 shows that there is only within certain limits that the mass and coupling constant give a lifetime acceptance that allows us to measure the Dark Photon decay.

Figure 13: Lifetime acceptance as a function of

Dark Photon mass and coupling constant ε. Figure 14: Number of expected Dark Photon de- cays in SHiP as a function of the coupling con- stant ε for different masses of the Dark Photon.

2.4 Number of expected events

After multiplying with the lifetime acceptance the number of expected events is given by Figure 14.

For Dark Photon masses between 0.002−0.005 GeV/c² and a coupling constant of order 10⁻⁵−10⁻⁴ we expect to see more than 2 events in the detector.

Previous experiments that have searched for Dark Photons have already covered a range of coupling constants and masses with their sensitivity. With the SHiP experiment the sensitivity will go beyond those ranges. See Figure 15 for the nominal sensitivity of SHiP, which only includes Dark Photons generated directly from the proton beam and not from secondary electrons or muons.

Figure 15: Sensitivity of various experiments to Dark Photon visible decays with marked area of expected visible Dark Photon decay established in this project (blue square) [23].

3 Discussion

One of the big questions within physics today is the nature of Dark Matter. Experiments have searched for it at the high energy frontier with the help of hadron colliders and in direct searches for cosmic Dark Matter through scattering, but it is evident that a wider search in different ranges of masses and coupling constants needs to be implemented, [23]. Lately the search for hidden particles in sub-GeV ranges has become popular. SHiP can promise a large nominal sensitivity in a range beyond any previous experiments in the low mass and coupling range. The range has been established similarly to the method used in this project. The wide sensitivity in the mass range is due to the large energy of the proton collisions, thanks to the energy of the SPS accelerator. The sensitivity in the ε range is due to the acceptance of the SHiP experiment and the large number of proton collisions. It is of great interest to know if this sensitivity could be boosted further into the unexplored area.

The ranges for the mass of the Dark Photon and its coupling constant predicted by the current investigation of the capability of SHiP’s background muon flux to generate visible Dark Photon decay has already been ruled out by previous experiments, which can be seen in Figure 15. The range is however not too far from the range of the SHiP experiment, which means that it might still bring a contribution to the SHiP sensitivity to Dark Photons. There is room for improvement in this project, and the analysis could be applied to other particles as well.

In this project the simulation did not take into account the muons which were created with energies below 10 GeV/c, nor the electromagnetic showers of electrons. This was due to the project using files created for background studies of SHiP. In order to save computational time these simulations

did not include muons or other particles that would not get past the hadron absorber. There are however files with these low energy muons, with a cut at 1 GeV/c, and these could be interesting to investigate further. The same goes for electromagnetic showers, which are also time consuming to simulate, but could contribute to the sensitivity. These particles could also give rise to virtual massive photons and as such be sources of Dark Photons. Since m_A⁰ is light the particle could be generated by a muon or electron with low momentum, and since the coupling constant would be low the Dark Photon would pass through both the hadron absorber and the muon shield unaffected and reach the decay volume.

Another possibility is to search for axion-like particles in the experiment, as they are proposed mediators between the Standar Model and Dark Matter and appear as three different couplings.

Our method could be applied to the photon-coupling.

The project has used several simplifications. Firstly the production angles were calculated for massless photons, the kinematics differ slightly in the range of masses considered. Secondly we also approximated that all Dark Photons would be generated in the early stages of the muons shield and with small angles relative to the detector, as we saw that many, but not all, daughter particles had small production angles (see Figure 9). Finally FWW is only one possible approximation to use, and does not have to be the ultimate choice. It is necessary to verify the result by comparing it with another approach of finding the probability density function of massive photon production, for example by modifying the formulae used for Bremsstrahlung [24].

The Dark Photon is an important possible component of the Dark Matter model as it propagates an interaction between Dark Matter particles and between the Hidden Sector and the visible particles.

It is of great interest to continue the search for this particle, and this method could prove useful in that search. The experiment NA64 will investigate Dark Photon events using the SPS accelerator at CERN with an electron beam at 100 GeV, [25]. The method we have established in this project could be of use as they complement the electron beam with muons.

4 Conclusion

The project has investigated the muon background of the SHiP experiment, specifically looking at muons with momenta > 10 GeV/c undergoing electromagnetic interactions in the muon shield.

Using the FWW approximation to muon scattering we could find a plausible weight function for massive photons generated in the electromagnetic interactions of the muon. By simulating 100000 protons on target and scaling it to SHiP’s nominal luminosity of 2 · 10²⁰protons on target we could establish the rates of muons per proton on target and of muon scattering events. The rates were then multiplied by the weight function integrated over SHiP’s geometric acceptance, the lifetime acceptance of a Dark Photon and the coupling constant, ε, to give the predicted number of visible Dark Photon decays.

The result of the project predicts that the muon background of the SHiP-experiment could give rise to more than 2 visible Dark Photon decays for photon masses between 0.002 − 0.005 GeV/c² and a coupling constant of order 10⁻⁵− 10⁻⁴. This range has, however, already been excluded by previous experiments. Therefore we do not expect any contribution to the SHiP sensitivity directly from the muon background.

There is still potential in using the established method to further investigate the background of

SHiP and the contribution of other particles. There is a potential contribution from muons with momenta < 10 GeV/c, electrons and axion-like particles.

Acknowledgement

Thank you to my supervisor Richard Jacobsson for the help with the theory and derivation of formulae used in this project, and above all for introducing me to the SHiP experiment and the depth of particle physics, of which I have only seen the surface.

In this project I have had great use of the courses studied within my program of Engineering Physics at Uppsala University. Thanks to this project I have extended my knowledge of particle physics, writing in python and planning a project of significant scale.

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