Neutrino oscillations at very high energy/matter density

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Master of Science Thesis

Neutrino oscillations at very high energy/matter density

Author:

Mathilde Guillaud (970423-T340) guillaud@kth.se

Department of Physics

Royal Institute of Technology (KTH)

Supervisor: Mattias Blennow

December 23, 2020

Particle and Astroparticle Physics, Department of Physics, School of Engineering Sciences

KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden

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Typeset in L^ATEX

Scientific thesis for the Master of Science degree in Engineering Physics TRITA-SCI-GRU 2020:358

©Mathilde Guillaud, 2020

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Abstract

Neutrino oscillations in matter can be studied in different regimes, depending on the energy of the incoming neutrinos and the matter density of the medium. In this thesis we investigate neutrino oscillations in dense matter at very high energy (TeV-PeV range), taking into account the absorption that the neutrinos may undergo in such dense media. This absorption phenomenon is relevant for neutrino telescope measurements of astrophysical neutrinos. We begin with a brief reminder on neutrino oscillations in vacuum and the construction of the PMNS matrix. Then, we proceed with calculations for dense matter. We then explore the accuracy of the resulting effective 2-neutrino mixing formulas. They present a good accuracy for Earth-like densities in our range of energies. We develop the calculations for oscillation probabilities in dense matter with absorption through charged-current inelastic scattering for both the two-neutrino and three-neutrino case. We find that in dense media, astrophysical neutrinos indeed undergo absorption, which reduces significantly the fluxes for each flavor, with a resonant absorption of electron-anti-neutrinos around E_res ' 6.3PeV. We discuss the impact of neutrino absorption in the Earth for neutrino telescopes measurements. We find that solar and lunar shadowing is not problematic for current telescopes but could be a good angular resolution indicator for new telescopes to come.

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Sammanfattning

Neutrinooscillationer i materia kan studeras i olika regimer beroende p˚a inkommande neutrinernas energi och densiteten hos det bakomliggande mediet. I detta examensar- bete undersöker vi neutrinooscillationer i gränsen av tät materia och mycket hög energi (TeV-PeV-intervall), och tar hänsyn till den absorption av neutriner som d˚a kan inträffa i s˚adant materia. Detta absorptionsfenomen är relevant för neutrino-teleskopmätningar av astrofysiska neutriner. Vi börjar med att kort p˚aminna oss om neutrinooscillationer i vakuum och konstruktionen av PMNS-matrisen. Vi försätter sedan med beräkningar av neutrinooscillationer i tät materia. Vi undersöker noggrannheten i resulterande effektiva 2-neutrino-blandningsformlerna. De uppvisar en god noggrannhet i jordlika materieprofi- ler i v˚art intervall av energier. Vi utvecklar beräkningarna av oscillationssannolikheterna i tät materia inklusive absorption genom laddad ström oelastisk spridning i b˚ada tv˚a- och tresmaksfallen. Vi finner att astrofysiska neutriner i tât materia absorberas, vilket minskar betydligt flödena för varje smak, med en resonansabsorption av elektron-anti- neutrino omkring E_res ' 6.3PeV. Vi diskuterar sedan effekterna av neutrinoabsorption p˚a jorden för neutrino-teleskopmätningar. Vi finner att sol- och m˚anskuggning är inte problematisk för nuvarande teleskop och kunde vara en bra vinkelupplösningsindikator för kommande teeskop.

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Chapter 1 Introduction

1.1 Introduction

Neutrinos are super light particles and long distance travellers, as they have very little chance to interact or be absorbed. Electrically neutral, they travel in straight lines from their point of interactions. This makes them ideal messengers to study astrophysical events, areas and eras which are inaccessible to photons. Astrophysical neutrinos are mostly created in the interactions of high-energy cosmic rays with massive particles or photons. If the location of the neutrino creation is close to the acceleration site of the cosmic rays, neutrinos can reveal the direction of these acceleration sites. These particles open new possibilities of probing the early Universe, which was until now opaque to photons because of the photon-matter coupling. Large experiments, called ’neutrino telescopes’ allow the observation of neutrino interactions on Earth. These telescopes have two major purposes: neutrino physics, which aims to determinate neutrino properties from their observed flavor ratio, and neutrino astrophysics, which consists in the identification and probe of neutrino sources and their properties. Experiments such as the IceCube Observatory [1], the Baikal detector NT200+ [2] or the ANTARES experiment [3] are currently probing the sky for astrophysical neutrinos and their sources.

Neutrinos were first thought to be massless, before neutrino oscillations were discovered by observation of solar neutrinos. These oscillations revealed that neutrinos are massive and their lepton flavors are mixed. First predicted in 1957-58 by Bruno Pontecorvo [4]

and recognized with the 2015 Nobel Prize for Physics (T. Kajita [5] & A. McDonald [6]), this phenomenon has been thoroughly studied both analytically and experimentally on long-baseline experiments around the globe [7–10]. Important parameters such as mixing angles and mass splittings have been determined for different set-ups with uncertainties up to 15%-30% and neutrino oscillations are now entering an era of precision measurements (e.g. [11–14]). Knowledge of these parameters is crucial for neutrino astrophysics, in order to determinate the initial neutrino flavor composition at their production sites and thus to be able to determinate the nature of these sources.

After Wolfenstein discovered the presence of matter effects in the oscillations in 1978 [15]

and theorised the MSW effect with Mikheyev and Smirnov in 1985 [16], a handful of studies have looked into the impact of matter effects on neutrino oscillations. Indeed, a matter potential arises from the elastic forward scattering of the neutrinos on the medium, which is proportional to the matter density and the energy of the neutrino.

This makes the choice of the framework important for the calculations: energy regions,

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density profiles and properties of the medium (e.g. (un)polarized, moving, fluctuating).

The first studies on the MSW effect established effective 2-flavor formulas for these oscillations [15, 16], but the Standard Model is composed of (at least) 3 active flavors of neutrinos. Thus, 3-flavor formulas are needed to obtain precision measurements. In 1988, Zaglauer and Schwarzer [17] extended the 2-flavor formulas developped by Wolfenstein, Smirnov and Mikheev to 3-flavor formulas. They underlined the importance of the 3^rd generation and the MSW effects it undergoes in matter, as well as the influence of the CP-violating phase. These amplitudes can be calculated numerically but analytical expressions are helpful to understand the dependence of the different parameters on one another and the physics behind this phenomenon. Most of the calculations are made using a series expansion, considering the matter effects as a perturbation to the vacuum Hamiltonian. Different perturbation parameters were chosen depending on the framework: ∆m²₂₁/∆m²₃₁ for Freund [18], Blennow and Smirnov [19] and Akhmedov et al. [20]

(who also used sin(θ₁₃)), whereas Denton et al. [21] chose to use ∆m²₂₁/∆m²_ee. However, direct calculation and exact formulas are also used by Kimura et al. [22] for matter with constant density. Still, these studies used Standard Model interactions at low energy (only forward elastic scattering is considered), which does not take into account the absorption and inelastic scattering of neutrinos in matter.

This thesis will focus on oscillations in the high energy/matter density limit, which is mainly encountered for astrophysical neutrinos with energies in the TeV-PeV scale. The previous studies mainly focused on the oscillation aspect of the propagation. Blennow and Ohlsson developed an effective case of two-neutrino oscillations [23], as well as Luo [24], and Xing and Zhu [25] established sum rules and relations between matter and vacuum parameters for three-neutrino oscillations. However, in this regime, neutrinos can be subject to absorption due to interactions with the matter background. This absorption effectively leads to a non-Hermitian addition to the neutrino oscillation Hamiltonian.

This thesis aims to calculate the quantum eigenstates for this non-Hermitian Hamilto- nian and deduce the implications for the neutrino oscillation experiments such as neutrino telescopes studying astrophysical high energy neutrinos passing through the Earth.

1.2 Outline of the Thesis

This thesis is organised as follows.

A theoretical background on astrophysical neutrinos, neutrino oscillations in vacuum and the PMNS matrix is given in chapter 2. Chapter 3 is an overview of the theoretical results for oscillations in dense matter. It starts with a derivation of the standard calculations for the mixing matrix and eigenstates in matter, with the dense matter limit taken after calculation, then follows a derivation starting from an adapted ”dense matter”

framework with the kinematic terms as the perturbation. The accuracy of the resulting effective formulas is discussed. Chapter 4 uses these previous calculations to derive the oscillation probabilities in very dense matter taking the absorption of neutrinos into account. Chapter 5 discusses the results and the implication for astrophysical neutrino observations at neutrino telscopes on Earth.

Natural units and Normal Mass Ordering (NMO) are used throughout this thesis.

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1.3 Author’s Contribution

The calculations from Chapters 2 to 3 are synthesized from the different papers cited in References. The calculations from section 3.1 are mostly inspired by Freund [18], Blennow and Smirnov [19] and Akhmedov et al. [20]. Calculations from section 3.2 are inspired by Blennow and Ohlsson [23] and Luo [24]. In Chapter 4, the corrected cross-section is defined from the values determined by Gandhi et al. [26] and the graphs are partly inspired from the IceCube collaboration [27]. I did all the calculations from Chapter 4 and the appendix, made all the graphs and tables unless stated otherwise and wrote this manuscript.

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Chapter 2 Neutrinos

This chapter gives a brief overview of neutrino properties and the derivation of the oscillations in vacuum. The two-neutrino and three-neutrino case are rapidly explained. For more details, the reader is invited to review the calculations from Thomson’s book [28].

2.1 Standard Model and neutrinos

Neutrinos were first proposed around 1930 due to a problem arising in nuclear beta decay.

A disturbing discrepancy in the outgoing momenta, between theory and observation, led Pauli [29] to theorise a massless, electrically neutral particle which was ejected alongside the electron. He initially named the particle ’neutron’, which was to be changed later for ’neutrino’ by Fermi at the 1932 Paris conference and the 1933 Solvay conference, to distinguish it from the massive ’neutron’ discovered by Chadwick in 1932. This theory of a three-body nuclear beta decay was not experimentally proven successful before 1956 by Reines and Cowan [30]. Indeed, neutrinos cannot be directly detected as they do not interact with detectors, but one can observe their presence through the results of their weak interaction.

The first evidence for neutrino oscillation was known as the ’solar neutrino problem’, which corresponded to a discrepancy between the predicted and observed number of electron neutrinos coming from the Sun. This discrepancy was later explained by the oscillation between flavors, first theorized by Pontecorvo [4] in 1957.

The Standard Model as we know it contains three active neutrino flavors: ν_e, ν_µ, ν_τ. Some Beyond the Standard Model (BSM) theories are constructed with more active flavors and/or passive neutrinos. Some theories consider the addition to the Standard Model of right-handed neutrinos or sterile neutrinos, which could solve the problems of the neutrino mass and/or dark matter. Currently, the three-neutrino oscillations model provides the best explanation for the observed results in atmospheric, solar [5, 6], accel- erator [9, 10] and reactor [7, 8] neutrino experiments. If the magnitude of the differences of the squared masses (∆m²₂₁ = m²₂ − m²₁ and ∆m32 = m²₃ − m²₂) have been measured within 3σ, the sign of ∆m²₃₁ remains unknown, which leaves two possible orderings for the neutrino masses (Normal Mass Ordering (NMO): m₁ < m₂ < m₃ and Inverted Mass Ordering (IMO): m3 < m1 < m2). The best fits to these mass difference can be found in Table 2.1, with data provided by the Particle Data Group [14]. Throughout this thesis, we will assume NMO for the calculations.

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2.2 Oscillations in vacuum

Neutrinos were initially believed to have zero-mass, however experiments have shown that these particles have a very small mass. These masses are revealed by the phenomenon of neutrino oscillations. These arise if one considers the flavour eigenstates (ν_e, ν_µ, ν_τ) to be a linear superposition of mass eigenstates (ν₁, ν₂, ν₃), which have definite mass. The mass eigenstates are the stationary states of the free particle Hamiltonian:

H |ψi = E |ψi (2.1)

and

ψ(x, t) = φ(x)e^−iEt . (2.2)

Since the flavor eigenstates are a superposition of the mass eigenstates, which do not propagate at the same speed, one can expect a change in the flavor of the neutrino after propagation on a certain distance and time.

2.2.1 Two-neutrino oscillations in vacuum

For simplicity let us explain first the two-flavor case, which can display the main features of neutrino oscillations. Let us take the electron neutrino alongside with a neutrino of flavor a. These weak eigenstates can be written as superpositions of the mass states ν₁ and ν2.







|ν₁i = cos(θ) |ν_ei − sin(θ) |ν_ai

|ν₂i = sin(θ) |ν_ei + cos(θ) |ν_ai

(2.3)

with the coefficients in sin and cos to preserve the orthonormality of the states. θ is then the mixing angle. After propagation (with k = 1, 2):

|ν_k(t)i = |ν_ki e^{i( ¯}^p^k^·¯^x−E^k^t)= |ν_ki e^−ip^k^·x. (2.4)

For the example, let us assume the beam of neutrinos is composed of electron neutrinos at t = 0. It interacts and is detected at a time T and at a distance L along its direction of flight (with φ_1,2 = E_1,2T − p_1,2.L and ∆φ₁₂ = φ₁− φ₂):

|ψ(0, 0)i = |ν_ei = cos(θ) |ν₁i + sin(θ) |ν₂i

|ψ(L, T )i = cos(θ) |ν₁i e^−iφ¹ + sin(θ) |ν₂i e^−iφ²

= e^−iφ¹

cos²(θ) + e^i∆φ¹²sin²(θ) |ν_ei − (1 − e^i∆φ¹²) cos(θ) sin(θ) |ν_ai

= c_e|ν_ei + c_a|ν_ai .

(2.5)

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Now assuming that L ' T (that is to say v ' c and that p ' E_ν), we obtain:

P (ν_e→ ν_a) = c_ac^∗_a = (1 − e^i∆φ¹²)(1 − e^−i∆^φ12) cos²(θ) sin²(θ)

= sin²(2θ) sin²

∆φ12

2

(2.6)

which gives the following survival and oscillation probabilities:











P (ν_e→ ν_a) = sin²(2θ) sin²

(m²₁−m²₂)L 4Eν

P (ν_e→ ν_e) = 1 − sin²(2θ) sin²

(m²₁−m²₂)L 4Eν

.

(2.7)

This shows that the neutrinos are indeed oscillating between flavors, if sin(2θ) 6= 0 and

∆m² = m²₁− m²₂ 6= 0.

2.2.2 Three-neutrino oscillations in vacuum

In the three-neutrino case, the eigenstate vectors are linked by the Pontecorvo-Maki- Nakagawa-Sakata (PMNS) matrix U_{P M N S}:





 ν_e ν_µ ν_τ







=







U_e1 U_e2 U_e3 U_µ1 U_µ2 U_µ3 U_{τ 1} U_{τ 2} U_{τ 3}







| {z }

UP M N S





 ν₁ ν₂ ν₃







. (2.8)

The unitary condition U U^†= I implies that:







U_e1 U_e2 U_e3 U_µ1 U_µ2 U_µ3 U_{τ 1} U_{τ 2} U_{τ 3}













U_e1^∗ U_µ1^∗ U_{τ 1}^∗ U_e2^∗ U_µ2^∗ U_{τ 2}^∗ U_e3^∗ U_µ3^∗ U_{τ 3}^∗







=







1 0 0 0 1 0 0 0 1







, (2.9)

which gives nine relations such as:







U_e1U_1e^∗ + U_e2U_e2^∗ + U_e3U_e3^∗ = 1 U_e1U_µ1^∗ + U_e2U_µ2^∗ + U_e3U_µ3^∗ = 0.

(2.10)

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Similarly to the two-neutrino case, assuming the neutrino beam is initially electron- flavored:

|ψ(L, T )i = U_e1^∗(U_e1|ν_ei + U_µ1|ν_µi + U_{τ 1}|ν_τi)e^−iφ¹ +U_e2^∗(U_e2|ν_ei + U_µ2|ν_µi + U_{τ 2}|ν_τi)e^−iφ² +U_e3^∗(U_e3|ν_ei + U_µ3|ν_µi + U_{τ 3}|ν_τi)e^−iφ³

= c_e|ν_ei + c_µ|ν_µi + c_τ|ν_τi .

(2.11)

Using the unitary relations (2.9) as well as the identity

| z1+ z2+ z3 |²≡| z1 |² + | z2 |² + | z3 |² +2Re{z1z^∗₂ + z1z^∗₃ + z2z^∗₃}, (2.12) we get the following oscillation probability:

P (ν_e → ν_µ) = 2Re{U_e1^∗U_µ1U_e2U_µ2^∗ [e^i(φ²^−φ¹⁾− 1]}

+ 2Re{U_e1^∗U_µ1U_e3U_µ3^∗ [e^i(φ³^−φ¹⁾− 1]}

+ 2Re{U_e2^∗U_µ2U_e3U_µ3^∗ [e^i(φ³^−φ²⁾− 1]}

(2.13)

and noting that:

Re{e^i(φ²^−φ¹⁾− 1} = cos(φ_j− φ_i) − 1 = −2 sin²

(m²_j − m²_i)L 4Eν

= −2 sin²(∆_ji), (2.14) we obtain

P (ν_e→ ν_e) = 1 − 4|U_e1|²|U_e2|²sin²(∆₂₁)

− 4|Ue1|²|Ue3|²sin²(∆31) − 4|Ue2|²|Ue3|²sin²(∆32) .

(2.15)

The PMNS matrix is used in its standard parametrization:

U_{P M N S} = U₂₃(θ₂₃)I_δU₁₃(θ₁₃)I_δ^∗U₁₂(θ₁₂). (2.16) U_ij(θ_ij) are the rotation matrices in the ij-plane with angle θ_ij and I_δ ≡ diag(1, 1, e^iδ).

Thus, the weak-charged current vertex for a lepton a = e, µ, τ and a neutrino of type k = 1, 2, 3 takes the form:

− ig_W

√2 l¯_aγ^µ1

2(1 − γ⁵)U_akν_k . (2.17) The PMNS matrix can be described in terms of 3 real parameters (the 3 mixing angles) and a single phase. If this matrix were real, it could be described as three rotations between the flavor states, however, it is unitary and not real, which induces six additional

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degrees of freedom which are written as complex phases. If the neutrinos are considered as Dirac particles, five out of the six phases can be absorbed by redefining the lepton and neutrino phases. This shows starting from the standard parametrization (the rotation matrices are complex). Details are found in Appendix A.1:







e^iγ² 0 0

0 c₂₃e^iα² s₂₃e^−iβ² 0 −s₂₃e^iβ² c₂₃e^−iα²













c₁₃e^iα³ 0 s₁₃e^−iβ³ 0 e^iγ³ 0

−s₁₃e^iβ³ 0 c₁₃e^−iα³













c₁₂e^iα¹ s₁₂e^−iβ¹ 0

−s₁₂e^iβ¹ c₁₃e^−iα³ 0

0 0 e^iγ¹







=







eîa 0 0 0 eîb 0 0 0 eîc













c₁c₃ s₁c₃ s₃e^−iδ

−s₁c₂− c₁s₂s₃eîδ c₁c₂ − s₁s₂s₃eîδ s₂c₃ s₁s₂− c₁c₂s₃eîδ −c₁s₂− s₁c₂s₃eîδ c₂c₃













eîx 0 0 0 eîy 0 0 0 eîz





 ,

with











a = γ₂+ α₃+ α₁− β₁ b = α₂+ γ₃

c = β₂+ γ₃











x = β₁ y = −α₁

z = γ₁− γ₃− α₂− α₃− β₂

(2.18)

δ = α1− α2− β1− β2+ β3− γ3 .

The phases a, b, c can be absorbed in the definition of the leptons and the phases x, y, z in the definition of the neutrinos since they are considered as Dirac particles. Only remains the CP-phase δ. The final matrix can then be expressed in the standard parametrization (2.16). In this thesis, possible Majorana masses are not considered as they are irrelevant for neutrino oscillations both in vacuum and matter.

In Table 2.1 are listed the best fit values for the mixing parameters in vacuum for both Normal Mass Ordering and Inverted Mass Ordering.

∆m²₂₁[10⁻⁵eV²] ∆m²₃₁[10⁻³eV²] θ₁₂/^◦ θ₁₃/^◦ θ₂₃/^◦ δ/^◦ NMO 7.39^+0.21_−0.20 2.51^+0.031_−0.030 33.82^+0.78_−0.76 8.61^+0.13_−0.13 48.3^+1.2_−1.9 222⁺³⁸₋₂₈ IMO 7.39^+0.21_−0.20 −2.44^+0.032_−0.032 33.82^+0.78_−0.76 8.65^+0.13_−0.12 48.6^+1.1_−1.5 285⁺²⁴₋₂₆

Table 2.1: Best fit values with one standard deviation for the mixing parameters in vacuum.

Data provided by the Particle Data Group [14]

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2.3 Astrophysical high-energy neutrinos

In this thesis we will focus mainly on high-energy astrophysical neutrinos, that is to say neutrinos with energy in the TeV-PeV range, created outside of the Earth’s atmosphere.

These neutrinos are the ones that are relevant to neutrino telescopes observations. These neutrinos are created in the interaction of high-energy cosmic rays with photons or heavy particles near cosmic ray acceleration sites. When detected on Earth, these neutrinos can provide information on their sources’ energy spectrum and composition. Neutrinos are believed to be mainly produced in pion decays created in cosmic ray interactions:

π → µ + ν_µ µ → e + ν_e+ ν_µ .

(2.19)

These processes are considered to give rise to a neutrino flux with flavor ratio 1 : 2 : 0 (flavor ratios are expressed as ν_e : ν_µ : ν_τ), which gives after long-baseline oscillations a ratio close to 1 : 1 : 1 when arriving on Earth. However, neutrinos can also be produced in muon-damped sources which give an initial 0 : 1 : 0 ratio (∼ 0.19 : 0.43 : 0.38 on Earth), neutron beams with initial 1 : 0 : 0 ratio (∼ 0.55 : 0.19 : 0.26 on Earth) or even charm sources, as discussed by Choubey and Rodejohann [31] and the IceCube collaboration [32].

2.3.1 Candidate sources

Neutrino sources (or cosmic ray acceleration sites) have not yet been clearly identified and the candidates are numerous and non exhaustive. The currently most favored candidate sources are Active Galactic Nuclei (AGN), blazars, Gamma-Ray Bursts (GRB), Super- Massive Black Holes (SMBH), star-burst galaxies, supernova remnants or even galaxy clusters [3, 26, 32, 33]. The observed flux of astrophysical neutrinos is, for now, quite diffuse and isotropic with no hint of concentration towards a particular direction [1].

However, searches for point-like sources are ongoing in the different neutrino telescope experiments around the globe. In 2018, the IceCube Observatory even identified the blazar TXS 0506+056 as a likely source of extragalactic neutrinos.

2.3.2 Neutrino telescopes

Neutrino telescopes are very large detectors which aim to identify the products of interaction of neutrinos with matter in the detector. Since the neutrino are very weakly interacting, these experiments have to be built on very large volumes. For example, the IceCube detector is a cubic-kilometer neutrino telescope, which is located in Antarctica.

Most of these telescopes are Cherenkov detectors, which use the Cherenkov radiation emitted by the products of interaction of the neutrinos when travelling through the detector. In order to collect such radiations, the detectors need a medium which is trans- parent to these radiations and in which the speed of light can be lower than the speed of the product particle (this latter condition is primordial for emission of Cherenkov light).

The neutrino telescopes ANTARES and Baikal use water in which photodetectors are arranged in arrays, whereas the IceCube telescope uses ice. The RICE experiment is also a Cherenkov detector, however it aims to detect the radio emissions from the neutrino

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interactions in ice.

Still, the Baksan telescope or MACRO experiment are not Cherenkov detectors, but a scintillation telescope with a muon detector for Baksan and liquid scintillation counters with nuclear track detectors for MACRO. The ANITA experiment aims to detect radio chirps of Askaryan emission from neutrino interaction in ice.

This list of neutrino telescopes is non exhaustive.

2.3.3 Astrophysical neutrino signal

In the Cherenkov telescopes, the expected signals from astrophysical neutrinos are the following. On one hand, muon-neutrinos are expected to produce a muon, through a charged-current (CC) interaction, which will fly out of the detector in straight line, giving rise to a ’track-like’ signature for these events. Moreover, these tracks allow a reconstruction of the incoming direction of the neutrino with accuracy up to ∼ 1^◦. On the other hand, neutral-current (NC) interactions for any flavor, as well as CC interaction for electron- and tau-neutrinos will give rise to hadronic showers with dimensions smaller than the detector spacing, which is observed by a ’bulk’ around the interaction point.

This allows initial direction reconstruction only up to ∼ 15^◦− 30^◦. This makes it hard to discriminate electron- and tau-neutrinos events. Muon-neutrinos are then favored as they give the clearest signature and the best direction accuracy.

2.3.4 Background signal

Though the detection of muon-neutrinos seems then quite straightforward, there is a strong background that need to be taken into account, with signatures that can be mis- taken for an astrophysical muon-neutrino. Indeed, the interaction of cosmic rays with the Earth’s atmosphere can create particles which release so-called ’atmospheric’ neutrinos when they decay. From all the particles created in these showers, only the muons and neutrinos will reach the detector. Above the horizon, the atmospheric neutrinos are often accompanied by the muon that was created alongside them, which makes it easier to detect them as a background signal. However, below the horizon, the neutrinos arrive alone since the muon was absorbed before reaching the detector, which makes it hard to discriminate them from astrophysical neutrinos. Still, we will see later that this problem is not preponderant when reaching the TeV-PeV scale, as the Earth absorbs part of the flux of neutrinos, which makes the ’above-horizon’ events the most accurate and the ones with the less unidentified background. Another criterion for event selection is that the neutrino interaction needs to happen inside the detector, to avoid mistaking an atmospheric muon for a muon created from an astrophysical neutrino outside the detector and to enhance the atmospheric neutrino exclusion.

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Chapter 3 Interaction with matter

After 1978, neutrinos are known to undergo matter effects, which affects the flavor oscillation process. Many studies have looked into the impact of different medium densities on the oscillation parameters and proposed different methods to retrieve analytically such parameters. In this chapter we look into two different methods for deriving the oscillation parameters in matter, and more particularly for matter-dominated media (defined in 3.2).

3.1 Processes

When travelling through matter, at low energies, neutrinos can undergo elastic forward scattering on neutrons, protons and electrons, through the following processes. There are neutral-current (NC) processes for the three flavors (Fig.3.1) and charged-current (CC) processes for the electron-flavor (Fig.3.2). Only these interactions are considered in most of the studies that develop calculations for oscillations in matter (e.g. [18–20, 23]), which is why we will keep this formalism for the calculations in matter in this chapter.

(NC): ν + n → ν + n , ν + p → ν + p , ν + e⁻ → ν + e⁻. ν

X

ν

X Z⁰

¯ ν

X

¯ ν

X Z⁰

Figure 3.1: Feynman diagrams for the neutral-current elastic scattering of both neutrinos and anti-neutrinos on an X particle with X = e⁻, n, p.

All flavors give the same contribution through the neutral-current interactions and only the electron-neutrino has an additional channel. Smirnov and Blennow [19] define

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the corresponding effective Hamiltonian as:

H_int= GF

√2νγ¯ ^µ(1 − γ₅)ν{¯eγ_µ(g_V + g_Aγ₅)e + ¯pγ_µ(g_V^p + g_A^pγ₅)p + ¯nγ_µ(gⁿ_V + g_Aⁿγ₅)n}. (3.1)

where gV and gA are the vector and axial coupling constants.

(CC): ν_e+ e⁻ → ν_e+ e⁻. ν_e

e⁻ ν_e

e⁻

W⁻

¯ ν_e

e⁻

¯ ν_e

e⁻ W⁻

Figure 3.2: Feynman diagrams for the charged-current elastic scattering of both neutrinos and anti-neutrinos on an electron.

3.2 Framework

Let us consider the previously defined PMNS matrix in its standard parametrization:

UP M N S = U23(θ23)IδU13(θ13)I_δ^∗U12(θ12).

In vacuum, the flavor evolution is given by the Schr¨odinger-like equation ( where |ν_fi is the flavor eigenstates vector):

id

dt |νfi = M M^†

2E |νfi . (3.2)

Now bearing in mind that |ν_fi = U_{P M N S}|ν_mi, the Hamiltonian in vacuum is then:

H₀ = 1

2EU_{P M N S}M_diag² U_{P M N S}^† (3.3)

and M_diag² = diag(m²₁, m²₂, m²₃) (with m²₁, m²₂, m²₃ considered to be real).

When travelling through matter, the neutrinos can be subject to forward scattering at low energies with charged-current contributions for the electron neutrino and neutral- current contributions for the three flavors.

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As discussed in [19], the contributions to the neutral current from the electrons and the protons of the medium (for an electrically neutral medium) are equal and of opposite signs, which makes them cancel each other out in the effective potential. Then remain the charged-current from the electrons and the neutral-current from the neutrons. The matter potential for a neutrino flavor a can be written:

V_a=√

2G_F(δ_an_e− 1

2n_n), (3.4)

with G_F the Fermi coupling constant, n_eand n_nare respectively the densities of electrons and neutrons in the medium.

In a normal (electrically neutral) medium, the muon-neutrinos and tau-neutrinos interact the same way through neutral-current: Vµ− Vτ = 0. Writing Vµ= Vτ = V⁰:

V_m = diag(V_e, V_µ, V_τ) = diag(V_e− V⁰, 0, 0) + V⁰I₃. (3.5)

The neutral current contributions are the same for ν_e, ν_µ and ν_τ. A real diagonal matrix just adds an overall phase to the Hamiltonian and can be neglected when computing the oscillations probabilities. Only the charged-current forward scattering remains. The effective potential then takes the form:

V =ˆ







V 0 0 0 0 0 0 0 0







, (3.6)

with V = V_e− V⁰ =√

2G_Fn_e.

In the flavor basis, the Hamiltonian becomes:

Hf = _2E¹ UP M N SM_diag² U_{P M N S}^† + ˆV + V⁰I3

= _2E¹ U_{P M N S} diag(0, ∆m²₂₁, ∆m²₃₁) U_{P M N S}^† + diag(V, 0, 0) + (^m_2E²¹ + V⁰)I₃

= H +˜ ^m²¹^+2EV_2E ⁰I3.

(3.7)

Note that this equation holds for neutrinos. The equation for anti-neutrinos can be obtained with: U → U^∗, V → −V .

As discussed by Luo [24], the magnitude of A = 2EV /∆m²₃₁ can highlight three different regimes: the vacuum-dominated regime (A α), the resonance regime (atmospheric: A ∼ 1, solar: A ∼ α) and the matter-dominated regime (A α, 1).

We will treat the later case in this thesis.

We will first start with the standard derivation of the oscillation parameters in matter and take the limit A → ∞ at the end of the derivation, and then compare with the results obtained with the kinematic term considered as a perturbation to the matter potential.

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In order to find the new mixing parameters and oscillation probabilities in dense matter, we will use perturbation theory to diagonalise the Hamiltonian from (3.7) such that:

H_f = 1

2EU_{P M N S}^dm M_DU_{P M N S}^†dm , (3.8) were U_{P M N S}^dm is the mixing matrix in very dense matter and M_D is a diagonal matrix with the effective squared masses in matter.

3.3 Series expansion to first order in α

For this first method, we will follow the standard derivation for oscillations in matter (with perturbation theory and series expansion to first order in α), and take the dense matter limit, that is to say EV → ∞, at the end of the calculations.

3.3.1 Neutrino mixing in matter

The expansion parameter ∆m²₂₁/∆m²₃₁ will be denoted α. Also, ˆV is invariant under 2-3 rotations so the effective Hamiltonian can be written (for simplicity U_ij(θ_ij) → U_ij)

H =˜ 1 2EU₂₃I_δ

U₁₃I_δ^∗U₁₂ diag(0, ∆m²₂₁, ∆m²₃₁) U₁₂^† I_δU₁₃^† + diag(2EV, 0, 0)

I_δ^∗U₂₃^† , (3.9) where I_δ is invariant under 1-2 rotations and I_δ^∗I_δ = I₃,

H =˜ ∆m²₃₁ 2E U₂₃I_δ

U₁₃U₁₂ diag(0, α, 1) U₁₂^†U₁₃^† + diag(2EV

∆m²₃₁, 0, 0)

I_δ^∗U₂₃^† . (3.10)

With A = 2EV /∆m²₃₁, we obtain

H =˜ ∆m²₃₁

2E U₂₃I_δM I_δ^∗U₂₃^† , with M = U₁₃U₁₂ diag(0, α, 1) U₁₂^†U₁₃^† + diag(A, 0, 0) , (3.11) and where

M =







A + s²₁₃+ αs²₁₂c²₁₃ αs₁₂c₁₂c₁₃ s₁₃c₁₃− αs₁₃c₁₃s²₁₂ αs₁₂c₁₂c₁₃ αc²₁₂ −αs₁₃s₁₂c₁₂ s₁₃c₁₃− αs₁₃c₁₃s²₁₂ −αs₁₃s₁₂c₁₂ c²₁₃+ αs²₁₂s²₁₃







. (3.12)

We will try to diagonalise M with successive rotations in the three planes (1-2, 1-3 and 2-3). The zeroth order terms in α do not form a diagonal matrix. We will perform a 1-3 rotation to eliminate the 1-3 and 3-1 terms. We will denote c⁰₁₃ = cos(θ₁₃⁰ ) and

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s⁰₁₃ = sin(θ⁰₁₃) with θ₁₃⁰ the angle of the new rotation. The detailed calculations can be found in Appendix B.1.1. This leads to

tan(2θ⁰₁₃) = sin(2θ₁₃) cos(2θ13) −_(1−αs^A2

12)

. (3.13)

Now let us determine the different terms of the new matrix (we will denote αs²₁₂= β). We will consider, without loss of generality, θ₁₃ and θ⁰₁₃ to lie in the first quadrant ([0 − ^π₂]), which implies sin(2θ₁₃) ≥ 0 and sin(2θ⁰₁₃) ≥ 0.







Mˆ1,1 = ¹₂ 1 + A + β −p

(A − cos(2θ13)(1 − β))²+ sin²(2θ13)(1 − β)² Mˆ_3,3 = ¹₂ 1 + A + β +p

(A − cos(2θ₁₃)(1 − β))²+ sin²(2θ₁₃)(1 − β)² (3.14) The 2-2 term is not affected by the rotation, so ˆM_2,2 = αc²₁₂ and







Mˆ_1,2, ˆM_2,1 = αs₁₂c₁₂cos(θ₁₃⁰ − θ₁₃) Mˆ_2,3, ˆM_3,2 = αs₁₂c₁₂sin(θ₁₃⁰ − θ₁₃) .

(3.15)

Let us re-write θ₁₃⁰ = θ_13m.

The effective Hamiltonian is now (writing U_13m = U₁₃(θ_13m)) H =˜ ∆m²₃₁

2E U₂₃I_δU_13mM Uˆ _13m^† I_δ^∗U₂₃^† , (3.16) with

M =ˆ







Mˆ_1,1 αs₁₂c₁₂cos(θ_13m− θ₁₃) 0

αs₁₂c₁₂cos(θ_13m− θ₁₃) αc²₁₂ αs₁₂c₁₂sin(θ_13m− θ₁₃) 0 αs₁₂c₁₂sin(θ_13m− θ₁₃) Mˆ_3,3





 .

(3.17) Let us now perform a 1-2 rotation to get a clear separation of the zeroth-order term in a diagonal matrix and the first-order terms in another matrix. We obtain

tan(2θ⁰₁₂) = α sin(2θ₁₂) cos(θ_13m− θ₁₃)

αc²₁₂− ˆM_1,1 . (3.18)

We will consider θ₁₂ and θ⁰₁₂ to lie in the first quadrant ([0 − ^π₂]), which implies sin(2θ₁₂) ≥ 0 and sin(2θ⁰₁₂) ≥ 0.











M˜_1,1 = ¹₂

Mˆ_1,1+ αc²₁₂− q

(αc²₁₂− ˆM_1,1)²+ α²sin²(2θ₁₂) cos²(θ_13m− θ₁₃)

M˜_2,2 = ¹₂

Mˆ_1,1+ αc²₁₂+ q

(αc²₁₂− ˆM_1,1)²+ α²sin²(2θ₁₂) cos²(θ_13m− θ₁₃)

M˜_3,3 = Mˆ_3,3

(3.19)

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Let us write θ⁰₁₂= θ_12m. The effective Hamiltonian is now (writing U₁₂(θ_12m) = U_12m) H =˜ ∆m²₃₁

2E U₂₃I_δU_13mU_12mM U˜ _12m^† U_13m^† I_δ^∗U₂₃^† , (3.20) with

M =˜







M˜_1,1 0 0 0 M˜_2,2 0 0 0 M˜_3,3







+ αs₁₂c₁₂sin(θ_13m− θ₁₃)







0 0 −s_12m

0 0 c_12m

−s_12m c_12m 0







. (3.21)

The goal now is to diagonalise ˜M , such that ˜M = W DW^† with D = diag(λ₁, λ₂, λ₃).

Let us expand ˜M as a series of parameter α up to first order (the λ_i are the eigenvalues of ˜M and the v_i its eigenvectors):











M˜ = M˜⁽⁰⁾+ α ˜M⁽¹⁾+ O(α²) λ_i = λ⁽⁰⁾_i + αλ⁽¹⁾_i + O(α²)

v_i = v⁽⁰⁾_i + αv⁽¹⁾_i + O(α²) ,

(3.22)

which gives the following unperturbed eigenvalues λ⁽⁰⁾_i and eigenvectors u⁽⁰⁾_i :







λ⁽⁰⁾₁ = M˜_1,1 ; λ⁽⁰⁾₂ = M˜_2,2 ; λ⁽⁰⁾₃ = M˜_3,3 u⁽⁰⁾₁ = (1, 0, 0)^T ; u⁽⁰⁾₂ = (0, 1, 0)^T ; u⁽⁰⁾₃ = (0, 0, 1)^T .

(3.23)

Developing perturbation theory gives

( ˜M₀+ α ˜M₁)(u⁽⁰⁾_i + αu⁽¹⁾_i + · · · ) = (λ⁽⁰⁾_i + αλ⁽¹⁾_i + · · · )(u⁽⁰⁾_i + αu⁽¹⁾_i + · · · ) . (3.24)

Identifying the powers of α:











α⁰ : M˜₀u⁽⁰⁾_i = λ⁽⁰⁾_i u⁽⁰⁾_i

α¹ : M˜₀u⁽¹⁾_i + ˜M₁u⁽⁰⁾_i = λ⁽⁰⁾_i u⁽¹⁾_i + λ⁽¹⁾_i u⁽⁰⁾_i α² : · · · .

(3.25)

Using the first and second equations gives for the leading-order perturbation to the eigenvalues and the perturbations χ_iku⁽⁰⁾_k to the eigenvectors u⁽⁰⁾_i (for i, k = 1, 2, 3 and k 6= i):

λ⁽¹⁾_i = u⁽⁰⁾_i ^†M˜₁u⁽⁰⁾_i = 0 , and χ_ik = u⁽⁰⁾_k ^†M˜₁u⁽⁰⁾_i

λ⁽⁰⁾_i − λ⁽⁰⁾_k . (3.26)

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Thus,











u⁽¹⁾₁ = −sin(2θ₁₂) sin(θ_13m− θ₁₃)s_12m 2( ˜M_1,1− ˜M_3,3) u⁽⁰⁾₃

u⁽¹⁾₂ = sin(2θ₁₂) sin(θ_13m− θ₁₃)c_12m 2( ˜M_2,2− ˜M_3,3) u⁽⁰⁾₃

u⁽¹⁾₃ = −sin(2θ₁₂) sin(θ_13m− θ₁₃)s_12m

2( ˜M_3,3− ˜M_1,1) u⁽⁰⁾₁ + sin(2θ₁₂) sin(θ_13m− θ₁₃)c_12m 2( ˜M_3,3− ˜M_2,2) u⁽⁰⁾₂ ,

(3.27) that is to say, to first order in α:











W = I₃+ α sin(2θ₁₂) sin(θ_13m− θ₁₃) 2







0 0 − s12m

M˜_3,3− ˜M_1,1

0 0 c_12m

M˜_3,3− ˜M_2,2

− s_12m M˜_1,1− ˜M_3,3

c_12m

M˜_2,2− ˜M_3,3 0





 D = diag( ˜M_1,1, ˜M_2,2, ˜M_3,3) .

(3.28) We will be simplifying these expressions in the next section, in the high energy/matter density regime. The effective Hamiltonian is now

H =˜ ∆m²₃₁

2E U₂₃I_δU_13mU_12mW DW^†U_12m^† U_13m^† I_δ^∗U₂₃^† . (3.29) These calculations are inspired by Akhmedov et al., Freund, Smirnov and Blennow [18–

20].

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3.3.2 High energy/matter density limit

This regime is characterized by EV → ∞, with E the energy of the neutrinos and V the matter potential arising from the charged-current interaction of the electron-neutrino with matter. We will take this limit on the results found in the previous subsection. We will use

A = 2EV /∆m²₃₁ 1. (3.30)

Considering that the energies we are interested in range from TeV to PeV, A_minEarth ' 1.5 × 10² 1 at E = 1TeV in Earth-like matter [20], which satisfies the approximation we use. Taking the standard values provided by the Particle Data Group [14] from Table 2.1, one gets α ' 4.10⁻² and β = αs²₁₂' 10⁻² 1.

We will use these approximations to simplify the expression of ˆM_1,1, ˆM_3,3, ˜M_1,1, ˜M_2,2 and ˜M_3,3. The details can be found in Appendix B.1.3. Starting with ˆM_1,1 and ˆM_3,3:







Mˆ_1,1 ' c²₁₃+ αs²₁₂s²₁₃ Mˆ_3,3 ' A + s²₁₃+ αs²₁₂c²₁₃ .

(3.31)

We already mentioned that β 1 so 1 − β ' 1. We will also use that θ₁₂ ' 33^◦ so cos(2θ₁₂) ' 0.5 and θ₁₃ ' 8^◦ so c²₁₃ ' 0.98. Using this for ˜M_1,1, ˜M_2,2 and ˜M_3,3:











M˜_1,1 ' αc²₁₂

M˜_2,2 ' Mˆ_1,1 = c²₁₃+ αs²₁₂s²₁₃ M˜_3,3 = Mˆ_3,3 ' A + s²₁₃+ αs²₁₂c²₁₃ .

(3.32)

Taking the limit A → ∞:







M˜_3,3− ˜M_2,2 = A − cos(2θ₁₃)(1 − β) ' A

M˜_3,3− ˜M_1,1 = A + s²₁₃(1 − β) − α cos(2θ₁₂) ' A ,

(3.33)

and we can see that the first-order corrections to the eigenvectors approach zero when A goes to infinity. The eigenvalues of ˜M in matter to leading order in α are











λ₁ = αc²₁₂

λ₂ = c²₁₃+ αs²₁₂s²₁₃

λ₃ = A + s²₁₃+ αs²₁₂c²₁₃ ' A .

(3.34)

When we take the limit A → ∞, the effective Hamiltonian becomes (to first-order in α)

H =˜ ∆m²₃₁

2E U₂₃I_δU_13mU_12mDU_12m^† U_13m^† I_δ^∗U₂₃^† = ∆m²₃₁

2E U_{P M N S}^dm DU_{P M N S}^dm† . (3.35)

Neutrino oscillations at very high energy/matter density