IN
DEGREE PROJECT TECHNOLOGY, FIRST CYCLE, 15 CREDITS
,
STOCKHOLM SWEDEN 2021
Neutrino oscillations at the
ESSnuSB experiment
KENNY ANDERSSON
HANNAH WIK
Physics
Neutrino oscillations at the ESSnuSB experiment
Hannah Wik, Kenny Andersson
hannahwi@kth.se, kennyan@kth.se
SA114X Degree Project in Engineering Physics, First Level
Department of Physics
KTH Royal Institute of Technology KTH
Supervisor: Tommy Ohlsson
Abstract
This report aims to study the phenomenon of neutrino oscillations through derivation of the formulas for their transition probabilities in two flavors and expansions thereof in three flavors. By studying the series expansions of the transition probabilities in three flavors, one could get a clearer understanding of the effect of matter and the CP-violating phase, δCP on neutrino oscillations. The purpose of investigating this is to be able to
investigate the impact of matter and δCP at the potential experiment ESSnuSB. The
ESSnuSB experiment would be an extension of the linear accelerator project European Spallation Source, ESS, that is currently under construction in Lund, Sweden. The ESSnuSB aims to study neutrino oscillations at the second oscillation maximum which, through the derivations, was found at 540 km from the ESS which is consistent with other studies. The mine in Garpenberg is located 540 km from the ESS, which makes it a prime candidate for a detector to be built within.
Sammanfattning
Denna rapport har som syfte att unders¨oka fenomenet neutrinooscillationer genom h¨ arled-ningar av formlerna f¨or ¨overg˚angssannolikheterna i tv˚a smaker, samt ut¨okningen till tre smaker. Genom att studera serieutvecklingarna av ¨overg˚angsannolikheterna med tre smaker kan man ¨aven unders¨oka p˚averkan av materia och den s˚a kallade CP-fasen, δCP.
Syftet med detta var att kunna unders¨oka neutrinooscillationer, och p˚averkan som mate-ria och δCP har vid det potentiella experimentet ESSnuSB. Detta experiment kan utg¨ora
Contents
1 Introduction 3
2 Investigation 6
2.1 Problem . . . 6
2.2 Neutrino oscillations – Two flavors . . . 6
2.3 Neutrino oscillations – Three flavors . . . 9
2.4 Neutrino oscillations in matter . . . 11
2.4.1 Two flavor oscillations in matter . . . 12
2.4.2 Matter effects – Three flavors and varying density . . . 14
2.5 CP-violation in neutrino oscillations . . . 15
2.5.1 The CP-violating phase δCP . . . 17
3 The European Spallation Source 20 3.1 The ESSnuSB experiment . . . 20
3.1.1 Water-Cherenkov detector . . . 22
3.1.2 Previous theoretical studies of ESSnuSB . . . 22
3.2 Physics at ESSnuSB . . . 23
3.3 Neutrino oscillations at ESSnuSB . . . 24
3.4 Discussion . . . 27
4 Summary and Conclusion 28
Chapter 1
Introduction
To search for the answers to the fundamental questions of how the Universe works is intrinsically human. Theoretical physics is a branch of physics that attempts to do just that using mathematics as a tool. Theoretical physics looks at the largest scales through cosmology and the smallest through particle physics. This led to the development of the Standard Model during the 20th century. The Standard Model is a theory that has been used to describe the weak, strong, and electromagnetic forces and the elementary particles, of which neutrinos are a part. A fascinating concept is that some of the smallest objects in particle physics could answer questions in cosmology, such as what dark matter is or why there is an asymmetry between matter and antimatter in the Universe.
Neutrino physics has developed into an active area of research in the last decades. In 2015 Takaaki Kajita and Arthur B. McDonald were awarded the Nobel Prize in Physics for the discovery of neutrino oscillations, which implies massive neutrinos. Their discovery was evidence against the theories of the Standard Model regarding neutrinos, in which they are massless. Despite this, the Standard Model is a competent model for a lot of physics. The discovery that neutrinos have mass is the first evidence of physics beyond the Standard Model [1]. Therefore, further research into neutrino physics is significant because gaining a deeper understanding of their properties could help us find new physics beyond the Standard Model.
Neutrinos have long been some of the least understood elementary particles. Like elec-trons, neutrinos are leptons with a spin of 1/2, but unlike elecelec-trons, they carry no charge. This fact leads to the only significant interaction that neutrinos are subject to is the weak force. Neutrinos come in three flavors, where each flavor is characterized by a different lepton number, and each flavor is named after the other charged leptons with the same lepton number. This gives us the three neutrinos: electron neutrino νe, muon neutrino νµ,
and tau neutrino ντ. Neutrinos also have corresponding antiparticles, the antineutrinos
Most neutrinos that reach Earth originate in the Sun and are created as electron neutrinos through the fusion reaction
4p+→4He2++ 2e++ 2ν e.
When instruments became sensitive enough to detect electron neutrinos, only a third of the predicted number of neutrinos were observed. Several theories as to why this was the case were proposed, however all but one were refuted. The remaining theory by Bruno Pontecorvo and Vladimir Gribov stated that neutrinos are not like most other particles and oscillate among three (or more) different flavors. Evidence for this theory was found around the turn of the century when more sensitive instruments in Super-Kamiokande in Japan, and Sudbury Neutrino Observatory in Canada could be used. The results from these experiments showed excess muon- and tau neutrinos, which was evidence for the existence of neutrino oscillations [3–5].
The flavor states of neutrinos can be defined as superpositions of three mass eigenstates ν1, ν2 and ν3. Every mass eigenstate can then further be described as a wave function,
which can be obtained by solving the Schr¨odinger equation for a free particle. The wave functions propagate with different velocities which yield a phase difference for each state that changes with time. Every flavor can therefore be defined as a superposition of mass eigenstates at a specific time. As time progresses the phase differences change, and if the phase differences do not match a flavor, they instead describe the probability of the neutrino being a specific flavor. This phenomenon is called neutrino oscillations, as the probabilities oscillate as the neutrinos travel [7].
The European Spallation Source (ESS) is currently being constructed and will be located in Lund, Sweden. The ESS has the main purpose to study the spallation of neutrons through a proton beam directed at a target. Certain upgrades are proposed which would allow for a project called the European Spallation Source Neutrino SuperBeam, ESSnuSB, which will be a longbaseline (distance from neutrino source to detector between 100 km -1000 km [20]) experiment. These upgrades would allow the ESSnuSB to study neutrino oscillations and it aims to determine the so-called CP-violating phase.
The ESSnuSB experiment would use two 500 kt water Cherenkov detectors. One is located a couple of hundred meters from the neutrino source, and a second one 540 km from the source in a mine in Garpenberg. This long baseline means that the CP-violating phase can be calculated with high accuracy, as the second detector will be placed at the second oscillation maximum. The ESS uses a proton linear accelerator to produce neutrons. To produce the neutrino super beam H− pulses would be used. The two electrons from the H− would be removed in an accumulator ring, which compresses the pulses to the necessary pulse length. The beam is then directed to a neutrino target, with a hadron collector, and then to a pion decay tunnel [20].
Chapter 2
Investigation
This chapter will explore neutrino oscillations and the derivation of transition probabil-ities in different environments, such as vacuum and matter. We will study the case of two and three flavors in vacuum and matter. However, due to the complex nature of the latter, it will be reviewed quite briefly for three flavors. The focus will also be on matter of constant density, as that will be the main concern later in this report.
Note: For the sake of simplicity we will use the system of natural units in all of chapter 2 (unless stated otherwise), resulting in c, ~, G and kB having the numerical value of one.
Furthermore, Greek letter indices (for example α, β) indicate flavor states and Latin letter indices (for example i, j) indicate mass states.
2.1
Problem
This report aims to review the derivation of the formulas for the transition probabilities in two and three flavors and study the impact of matter and the CP-violating phase on neutrino oscillations. In the first section, we will discuss the neutrino oscillations of two flavors and then expand to three flavors. The purpose of this is to allow for a clear understanding of neutrino oscillations which will be used in the next chapter concerning ESSnuSB.
2.2
Neutrino oscillations – Two flavors
Currently, the only way to detect a neutrino is to identify the lepton associated with it in a weak interaction process between a lepton and a neutrino [6]. However, by solving the Schr¨odinger equation for neutrinos we obtain the solution for the mass eigenstates, and not for the flavor (weak) eigenstates. From this, we can describe each flavor eigenstate as a superposition of mass eigenstates, namely
|ναi =
X
i
where P
i|ci|
2 = 1 are terms in the lepton mixing matrix as defined in equation (2.4),
and |ναi and |νii are the flavor and mass eigenstates, respectively.
Assuming that there only exist two flavors, for example α ∈ {e, µ}, we can simplify equation (2.1) and solve for |ν1i and |ν2i
|ν1i = c1,1|νµi + c2,1|νei,
|ν2i = c1,2|νµi + c2,2|νei.
(2.2)
where the terms ci,j are solved from equation (2.1) using α ∈ {e, µ}. We can force
normalization and orthogonality by writing the above equation as
|ν1i = cos θ|νµi − sin θ|νei,
|ν2i = sin θ|νµi + cos θ|νei,
(2.3)
where θ is known as the mixing angle and is obtained through experiments. Using this we can change equation (2.1) to
νµ νe = cos θ sin θ − sin θ cos θ ν1 ν2 . (2.4)
From equation (2.4), we obtain the matrix U , which is the matrix in the middle. This matrix, when described for three flavors, is known as the lepton mixing matrix (sometimes called the PMNS-matrix, or simply the MNS-matrix).
The mass eigenstates |νii are eigenstates of the Hamiltonian
ˆ
H|νii = Ei|νii, (2.5)
with energy eigenvalues Ei = p|~p|2+ m2i and mi are the eigenvalues for neutrinos at
rest [20].
Using this together with the Schr¨odinger equation results in the time-dependent equation for a mass eigenstate, i.e.
|νi(t)i = e−iEit|νi(0)i. (2.6)
Now consider that we measure an interaction between a neutrino and a lepton, we call the measured flavor α which is obtained at t = 0. Rewriting using the notation |νi(0)i = |νii,
and equation (2.1) we obtain
|ν(t)i =X
i
Uα,i† e−iEit|ν
where we define |ν(0)i = |ναi, and U†is the complex transpose. Rewriting equation (2.3) we have |νii = X β Uβ,i|νβi, (2.8)
which we can use in equation (2.7) to obtain
|ν(t)i = X
β
X
i
Uα,i† Uβ,ie−iEit
|νβi. (2.9)
From this, we observe that as time passes the flavor state we detected becomes a super-position of the flavor states. Furthermore, we note that the flavor probability (described by the coefficients in front of each state) changes as time passes, which means that we expect the neutrinos to change flavor according to the probabilities that equation (2.9) will give rise to.
The transition amplitude Aνα→νβ(t) is then given by
Aνα→νβ(t) = hνβ|ν(t)i =
X
i
Uα,i† Uβ,ie−iEit. (2.10)
Finally, the transition probability is obtained as
Pνα→νβ(t) = |Aνα→νβ(t)| 2 =X i X j Uα,i† Uβ,iUα,jU † β,je −i(Ei−Ej)t. (2.11)
This formula also works for any number of flavors but with a different lepton mixing matrix U . Assuming the neutrinos to be ultrarelativistic, meaning they travel at speeds very close to c, we can use the following approximation
Ei = q |~p|2+ m2 i ≈ |~p| + m2i 2|~p| ≈ E + m2i 2E, (2.12)
where the total energy E ≈ |~p|, since we neglect the contribution of the mass to the energy as it is much smaller than the contribution of the momentum. This means that the expo-nent in equation (2.11) becomes −i(Ei− Ej)t ≈ −i∆m2ij/2Et, where ∆m2ij = m2i − m2j.
We assume that the masses are ordered, meaning that m2 > m1.
We need to make a distinction between mass hierarchy and mass order. The mass ordering and hierarchy can both be split into ’normal’ and ’inverted’. The normal mass ordering (NO) is m1 < m2 < m3 and the inverted (IO) m3 < m1 < m2. The normal hierarchy
(NH) is m1 m2 < m3 and finally the inverted hierarchy m3 m1 < m2. It is
It is often convenient to express equation (2.11) as a function of length instead of time, and as the neutrino travels at speeds close to c we can set t ≈ x. Assuming α = e and β = µ one can show that equation (2.11) becomes [1]
Pνe→νµ(x) = sin
22θ sin2 ∆m221
4E x, (2.13)
which is the transition probability. This is the probability that an electron neutrino created at point A is detected as a muon neutrino at a detector at point B, where A and B are separated by a distance x. The survival probability Pνe→νe(x) can easily
be obtained from 1 − Pνe→νµ(x). The transition from να → νβ is also known as a
channel. More specifically, the appearance channel να → νβ and the disappearance
channel να→ να.
Equation (2.13) shows us the behavior of neutrino oscillations. If we look at the equation in two parts, starting with the amplitude sin22θ. We see that it only depends on the mixing angle θ. This part of equation (2.13) describes the amplitude of the neutrino oscillations. The second part depends on the distance x that the neutrinos have traveled and their energy. This is proportional to the phase of the neutrino oscillations [1]. Using these notations one can rewrite equation (2.11) to obtain [20]
Pνα→νβ(x) = δαβ− 4
X
i>j
Re[Uα,i† Uβ,iUα,jU † β,j] sin 2 ∆m 2 ijx 4E +2X i>j
Im[Uα,i† Uβ,iUα,jU † β,j] sin ∆m2 ijx 2E . (2.14)
This equation can be used for any number of flavors. Similarly, one can use the same method to solve for the oscillation probabilities for antineutrinos and obtain [20]
Pνα→νβ(x) = δαβ− 4
X
i>j
Re[Uα,i† Uβ,iUα,jU † β,j] sin 2 ∆m 2 ijx 4E −2X i>j
Im[Uα,i† Uβ,iUα,jU † β,j] sin ∆m2 ijx 2E . (2.15)
2.3
Neutrino oscillations – Three flavors
where α = e, µ, τ and i = 1, 2, 3. Uα,i are the elements in a 3 × 3 matrix U .
From this, the unitary matrix U is given by
U = Ue1 Ue2 Ue3 Uµ1 Uµ2 Uµ3 Uτ 1 Uτ 2 Uτ 3 . (2.17)
The lepton mixing matrix U depends on three mixing angles and a CP-violating phase; θ12, θ13, θ23, and δ, respectively. For the sake of convenience, the matrix U can be
parameterized according to [1] U = c12c13 s12c13 s13e−iδ −s12c23− c12s23s13eiδ c12c23− s12s23s13eiδ s23c13 s12s23− c12c23s13eiδ −c12s23− s12c23s13eiδ c23c13 , (2.18)
where cij = cos θij and sij = sin θij. If we assume that there is a hierarchy between the
neutrino mass-squared differences ∆m2
ij = m2i − m2j, i.e.
|∆m21|2 |∆m31|2 ≈ |∆m32|2, (2.19)
we see that either m1 < m2 < m3 or m3 < m1 < m2, characterized by ∆m231 > 0
and ∆m2
31 < 0, respectively. In some experimental cases one works with distances and
energies that satisfy [8]
∆m2 21
2E x 1. (2.20)
When conducting these experiments, a satisfactory theoretical approximation can be obtained by setting ∆m221 = 0. This gives us a new, simple, formula for the transition probability using equation (2.14) [1]
Pνα→νβ(x) = 4|Uα,3| 2|U β,3|2sin2 ∆m2 31 4E x. (2.21)
This gives us the transition probabilities [1]
Figure 2.1: Electron neutrino oscillations Pνe→νµ(x), Pνe→ντ(x) and Pνe→νe(x) in a vacuum for a
short-range experiment. The values used in this figure are: ∆m2
21= 7.39 × 10−5eV 2, ∆m2
31= 2.449 × 10−3eV 2,
θ23= 51◦, θ12= 33.82◦, θ13= 8.61◦
In figure 2.1 it is clear that the neutrino probabilities are oscillating, and we see that the electron survival probability is large for short-range experiments.
For experiments involving solar neutrinos, for which x would be significantly larger, the approximation equation (2.20) would not be accurate. In that case, one would instead use the approximation [1]
∆m2 31 2E x ≈ ∆m2 32 2E x 1. (2.23)
2.4
Neutrino oscillations in matter
In Earth-based experiments neutrinos have to travel through matter, therefore one would want to examine how matter affects neutrino oscillations. Matter can affect the oscil-lations by interacting with particles (for example electrons), which will give rise to a change in potential energy. This energy has to be incorporated in the Hamiltonian. These interactions can happen in two ways [6]:
1. A neutrino can interact with a charged lepton. However, the only charged lepton that is found abundantly on Earth is the electron. The charged current (CC) interaction potential will be proportional to the number of electrons, Ne, in the medium that the
neutrino travels in and only affect the electron (anti)neutrino. The Standard Model predicts that the potential will be [16]
VCC =
√
for νe (negative for νe), where GF is the Fermi coupling constant. This interaction is
mediated by the W boson.
2. According to the Standard Model any neutrino can interact with electrons, protons, and neutrons through the exchange of Z bosons in a neutral current (NC) interaction. This interaction potential can be calculated as [16]
VN C = −
√ 2
2 GFNn, (2.25)
where Nn is the neutron number density (positive for antineutrinos).
The Hamiltonian for a neutrino is then the sum of the vacuum Hamiltonian and the potentials. This Hamiltonian can be written in both the mass basis ˆHm and in the flavor
basis ˆHf l. The two different bases are related through the lepton mixing matrix according
to [1]
ˆ
Hf l = U ˆHmU†, Hˆm = U†Hˆf lU. (2.26)
2.4.1
Two flavor oscillations in matter
Starting with only two flavors, we have the Hamiltonian for the mass eigenstates in vacuum from equation (2.5) as ˆHm = diag(E1, E2). The Hamiltonian for the flavor
basis will then become ˆHf l = U ˆHmU†. Using this together with the time-dependent
Schr¨odinger equation and substituting ∆m2
21= ∆m2 we obtain id dt νe(t) νµ(t) = " (E + m21+m22 4E ) − ∆m2 4E cos 2θ0 ∆m2 4E sin 2θ0 ∆m2 4E sin 2θ0 (E + m2 1+m22 4E ) + ∆m2 4E cos 2θ0 # νe(t) νµ(t) . (2.27) Since only the relative phases of the relative energies matter, we can ignore the expression in the parenthesis in the diagonal as they modify the unobservable overall phase instead of the phase difference and therefore have no effect on the oscillations [8]. Thus we can ignore this term and obtain
id dt νe(t) νµ(t) =− ∆m2 4E cos 2θ0 ∆m2 4E sin 2θ0 ∆m2 4E sin 2θ0 ∆m2 4E cos 2θ0 νe(t) νµ(t) . (2.28)
Using the equations (2.24) and (2.29) we obtain the interaction potentials
Using the simplified flavor Hamiltonian from equation (2.28) together with the potentials in equation (2.29) we find the effective Hamiltonian H for propagation in matterˆe
ˆ e H = ˆHf l+ VCC 1 0 0 0 + VN C 1 0 0 1 . (2.30)
For the same reason as before, we can omit the common terms in the diagonal as they change the phase between the flavors equally, and can therefore be ignored without changing the result. We can therefore ignore the VN C term. This gives us the final
form of H for νˆe e ↔ νµ and νe ↔ ντ oscillations. For the case when we only consider νµ ↔ ντ matter does not affect the oscillations since there is no VCC term, thus we
obtainH = ˆˆe Hf l. For future calculations it is also convenient to subtract VCC/2 from the
diagonal of H.ˆe
Assuming the matter density is constant (Ne = const.) one can find the eigenstates for
ˆ e
H by diagonalizing the new Hamiltonian and obtain a new mixing matrix, similar to the case of two flavor oscillations in a vacuum. After introducing the new effective mixing angle eθ as the mixing angle and the new effective mass-squared difference ∆me2 in matter,
one obtains ˆ e H = ∆me 2 4E " − cos 2eθ sin 2eθ sin 2eθ cos 2eθ # . (2.31)
By combining equations (2.30) and (2.31) we can find a formula to calculate ∆me2 and
e
θ. By letting γ = 2√2GFNeE/∆m2, we obtain
−∆me 2 4E cos 2eθ = − ∆m2 4E cos 2θ0 + ∆m2γ 4E , ∆me2 4E sin 2eθ = ∆m2 4E sin 2θ0 (2.32)
where θ0 is the vacuum mixing angle. Using these equations one can show that [27]
tan 2eθ = sin 2θ0 cos 2θ0− γ , ∆me2 = ∆m2 q (cos 2θ0− γ)2+ sin22θ0. (2.33)
Pνe→νµ(x) = sin
22eθ sin2∆me
2x
4E . (2.34)
Using this we can also obtain the survival probability
Pνe→νe(x) = 1 − sin
22eθ sin2 ∆me
2x
4E . (2.35)
We see that if we set Ne = 0 we obtain equation (2.13), which we would expect as
it implies that the neutrino propagates in an electron-lacking environment, such as a vacuum.
When examining equation (2.33) we observe that the oscillation amplitude reaches a maximum when
√
2GFNe =
∆m2
2E cos 2θ0 (2.36)
is satisfied. Using this condition in equation (2.33) we obtain eθ = 45◦, which means that this occurs when the mixing in matter is maximal and independent of the vacuum mixing angle θ0. This is known as the Mikheyev–Smirnov–Wolfenstein (MSW) resonance
condition and it shows us that the flavor transition in matter can be very different from that of vacuum, by for example enhancing the oscillations of solar neutrinos in the Sun [1]. From this, we also see that the effective mass difference ∆me2 is at a minimum at the
resonance and becomes [20]
∆me2 = ∆m2sin 2θ0. (2.37)
2.4.2
Matter effects – Three flavors and varying density
One could further study neutrino oscillations by looking at the case of three neutrino flavors, as well as varying density in matter. The reason one would like to look at oscillations in matter with varying density is that neutrinos will likely propagate through this medium when we perform experiments on Earth. Furthermore, we can expand on the case of two flavors to three in matter. Due to the complex nature of these expansions, we will only discuss the results briefly in this report.
If we start by studying the case of two flavors in varying density. With varying densities, the Hamiltonian will become time-dependent.
where we have ignored the diagonal VN C term. From equation (2.26) we know that
ˆ
Hf l = U ˆHmU†, where ˆHm = E13x3+ 1/2 Ediag(m21, m22, m23)
We know that we can subtract a diagonal matrix from this without affecting the result since the phase difference does not change. We can therefore write the Hamiltonian
ˆ
Hm0 = ˆHm− E13x3 and use it together with the Schr¨odinger equation, which yields
id dt νe(t) νµ(t) ντ(t) = " 1 2EU diag(m 2 1, m22, m23)U † + VCC 1 0 0 0 0 0 0 0 0 # νe(t) νµ(t) ντ(t) . (2.39)
Now it is difficult to continue the calculations analytically without making any assump-tions as the calculaassump-tions can get very complicated. One can obtain an approximation by solving equation (2.39) using a series expansion. By defining
a = ∆m 2 21 ∆m2 31 , (2.40)
using current data one can find a range for the value of a. Using this and performing a series expansion up to the second-order in a and sin θ13 one obtains [15]
Pνe→νµ(∆) = a 2sin22θ 12cos2θ23 sin2A∆ A2 + 4 sin 2θ 13sin2θ23 sin2(A − 1)∆ (A − 1)2
+ 2a sin θ13sin 2θ12cos (∆ − δcp)
sin A∆ A
sin (A − 1)∆ A − 1 ,
(2.41)
where ∆ = ∆m231x/4E, and A = VCCx/2∆. A full list of all probabilities can be found
in Ref. [15], including a more detailed derivation. These expansions are not valid for low energies or very long baseline experiments.
2.5
CP-violation in neutrino oscillations
can move faster than the particle, which would appear to change the direction of motion and thus change the helicity [7].
To better describe these concepts we introduce the relativistic wave equation, the Dirac equation, namely
(i /∂ − m)ψ(x) = 0, (2.42)
where ψ(x) is a spinor field with adjoint field ψ(x) = ψ†γ0 and using the Dirac matrices
γµ, /∂ = γµ∂
µ (summation is implied).
To further explain chirality, we can use the chirality matrix γ5 ≡ iγ0γ1γ2γ3. Using this
we can find its eigenfunctions, with the eigenvalues −1 and 1, which are the chiral fields that we denote ψL and ψR, which are the left and right-handed fields [20]
γ5ψR = ψR, (2.43)
γ5ψL = ψL. (2.44)
The charge conjugation operator (C), changes the sign of the charge and converts the particle into its antiparticle [14]
C|pi = |pi. (2.45)
This can also be described using the spinor field and its adjoint
ψ(x) → ψC(x) = ξCCψ T
(x) = −ξCγ0Cψ∗(x), (2.46)
ψ(x) → ψC(x) = −ξ
CψTC†(x), (2.47)
where two consecutive transformations will result in
ψ(x) → ξCCψ T
→ |ξC|2ψ. (2.48)
Thus, we find that ξC is a phase with |ξC| = 1, this is the intrinsic charge parity of the
field.
The parity operator (P ) inverts space. For example, if the P operator is applied to a right hand, the hand would become inverted, resulting in a left hand that is upside down and backward. If we consider the P operator on a vector v, it would give us
P (v) = −v. (2.49)
ψ(x) → ψP(xP) = ξPγ0ψ(x),
ψ(x) → ψP(x
P) = ξP∗ψ(x)γ
0. (2.50)
The intrinsic parity ξP, similarly to the charge parity, transforms into the original state
after two transformations
ψ(x) → ξPγ0ψ(x) → ξP2ψ(x). (2.51)
We know that the sign of ξ2
P changes through rotations of 2π [20], therefore, we have
ξP = ±1, ±i. (2.52)
Both charge and parity are violated in weak interactions. For neutrinos, which only interact through the weak force, the charge conjugation C converts a left-handed neutrino into a left-handed antineutrino which does not exist [20]. Similarly, for the case of the parity operator (P ), which turns the left-handed neutrino into a right-handed neutrino. However, the product of charge and parity, CP-symmetry, is better defined as it converts the left-handed neutrino into its existing antiparticle, the right-handed antineutrino [1]. From the mixing matrix, see equation (2.18), one can note that neutrino oscillations depend on δCP. If this phase is nonzero, the CP-symmetry would be violated, which is
known as CP-violation [20].
For CP-transformation we can also state the following
ψ(x) → ψCP(xP) = ξCPγ0Cψ T (x) = −ξCPCψ∗(x), ψ(x) → ψCP(x P) = −ξCP∗ ψ TC†(x)γ0, (2.53)
where ξCP = ξCξP is the intrinsic CP-parity. Due to the values of |ξC|2 and ξP, stated
above, we have that ξCP is a phase, |ξCP|2 = 1, which is the CP-violating phase, δCP
[20].
If CP is not conserved the oscillation probabilities of neutrinos and antineutrinos differ, which would be a violation of the CP-symmetry. As mentioned, the mixing matrix would allow us to observe this phenomenon [8].
2.5.1
The CP-violating phase δ
CPfor neutrinos, as there is no evidence for the existence of right-handed neutrinos or left-handed antineutrinos. We find that the left-handedness of neutrinos is symmetric in the so-called CP-symmetry, as right-handed antineutrinos exist. However, since this symmetry was defined physicists have discovered that the decay of kaons violates it. Experiments such as the ESSnuSB could determine if this violation exists for neutrinos as well, which would be the first evidence of CP-violation in the leptonic sector [1].
The CP-phase is responsible for the so-called CP-asymmetry, which the ESSnuSB aims to determine. The importance of studying this CP-asymmetry is that it could potentially explain the asymmetry of matter and antimatter in the Universe [18].
Given the oscillation probabilities for neutrinos να and νβ, and antineutrinos να and
νβ, as Pα→β and Pα→β, respectively, one would expect Pα→β − Pα→β = 0 if neutrino
oscillations were CP-symmetric. This is not the case [17]. Using equations (2.14) and (2.15) we obtain
ACPαβ = Pα→β − Pα→β = 4
X
i>j
Im[Uα,i† Uβ,iUα,jU † β,j] sin
∆m2 ijx
2E , (2.54) which is non-zero if the CP-violating phase δCP is non-zero. For three flavors in a vacuum,
we can use the Jarlskog invariant, i.e.
J = sin θ12cos θ12sin θ23cos θ23sin θ13cos θ13sin δCP. (2.55)
Applying this we obtain [17]
ACP = 16J sin∆m 2 21x 4E sin ∆m232x 4E sin ∆m231x 4E . (2.56)
From this, we see that if the phase is different from either 0 or π we have CP-violation in neutrino oscillations. If δCP ∈ {0, π}, then the value signifies to what degree/
the CP-symmetry is violated.
In matter we instead use equation (2.54) with the series expansions for neutrinos and antineutrinos. When conducting experiments, such as at the ESSnuSB, it is this CP-asymmetry that is used to determine δCP. Statistics on the obtained data can then be
used to obtain a value for δCP.
A problem with trying to determine δCP when conducting experiments in matter is
searching for δCP [12]. It would still be beneficial to make this effect even smaller. To
ignore this induced CP-violation we can instead regard
∆ACP(δCP) = ACP(δCP) − ACP(0), (2.57)
which can be used to extract δCP from experimental data. We note that ∆ACP(δCP)
behaves similarly to ACP(δ
CP) in vacuum as both vanish when δCP = 0. Therefore, by
using ∆ACP(δ
CP) when conducting experiments we can get better statistics as we are
Chapter 3
The European Spallation Source
3.1
The ESSnuSB experiment
To study neutrino oscillations one could study reactor experiments as well as accelerator experiments. Nuclear power plants create large amounts of antineutrinos in their fission reactions. These antineutrinos are of low energy which means that the detectors can be placed at short distances. This also means that only the disappearance channel of electron neutrinos can be studied, as the energies for muon or tau neutrinos are too low to produce muons or taus, which can be detected. The antineutrinos produced in the reactor experiments are detected through the following process [20]
νe+ p → n + e+.
Accelerator experiments involve a neutrino beam produced by pion decay, muon decay, or a beam dump. Furthermore, there are short baseline accelerator experiments and long-baseline experiments.
Parameter Value
Average beam power 5 MW
Proton kinetic energy 2.0 GeV
Average macro-pulse current 62.5 mA
Macro-pulse length 2.86 ms
Pulse repetition rate 14 Hz
Maximum accelerating cavity surface field 45 MV/m
Maximum linac length 352.5 m
Annual operating period 5000 h
Reliability 95 %
Table 3.1: Parameter values relevant for the ESS [18].
Table 3.1 shows the main parameters of the ESS facility, without the upgrades for the ESSnuSB experiment [9].
To do so, half a megaton water-Cherenkov neutrino detector would be placed at the second neutrino oscillation maximum between (300 - 600) km from Lund. This detector would be placed underground in existing mines, Zinkengruvan at 360 km or the Garpen-berg mine at 540 km from the ESS. The detector is placed underground to protect from the cosmic background radiation [9]. This section will focus on the option of using the mine in Garpenberg. However, it is worth noting that the option of using Zinkgruvan still is a viable candidate. As it is closer it would be able to detect 2.25 times more events, yielding better statistics [25]. At 360 km the second oscillation maximum is not as domi-nating as the first maximum. This gives slightly worse statistics at the second maximum as compared to the option of the mine in Garpenberg but also gives extra statistics for both the first and second maximum. Furthermore, more logistics are needed to rebuild Zinkgruvan to make it compatible to be able to place a detector in it, which is not needed for the mine in Garpenberg. Overall, if the experiment is conducted over 10 years, the mine in Garpenberg is a slightly better candidate for the location of the detector as it would allow us to determine δCP to 5σ for 62 % of the possible values of δCP, as compared
to Zinkgruvan which would allow us to do this for 56 % of possible values [13].
The reason for placing the detector at the second neutrino oscillation maximum is to decrease systematic errors. As the aim of the ESSnuSB experiment is to study CP-violation, the neutrino-antineutrino asymmetry in vacuum is approximately 0.3 sin δCP at
the first maximum, whereas for the second maximum it becomes 0.75 sin δCP, which gives
a higher sensitivity to CP-violation. However, due to the second maximum being located further away, there will be a decrease in statistics compared to the first maximum at the same energy. Thus, the ESSnuSB requires a very intense neutrino beam to give similar results for the second maximum as the first one for lower neutrino energy [10].
The effects of matter on the νµ ↔ ντ channel is smaller than those of ντ ↔ νeand νµ↔ νe,
which means that the νµ ↔ ντ channel is irrelevant for experiments in the ESSnuSB [1].
The channel which will be studied is the νµ ↔ νe channel, as interference between two
what the ESSnuSB aims to determine [15]. To determine the CP-violating phase it is necessary to study the appearance channel. For the disappearance channel the imaginary part of equations (2.14) and (2.15) would become zero.
As discussed in section 2.4.1, to determine the CP-violating phase one would need to measure the change in the transition probabilities for νµ→ νe and νµ→ νe. However, to
collect the same statistics from the antineutrinos the ESSnuSB would need to run with antineutrinos about 4 times as long. The time interval one should run the neutrino and antineutrino channels have been discussed. The suggested options have been 2 and 8 years, and 5 years each. However, by measuring the sensitivity of each method one can determine that 2 and 8 years would be the better option due to lower sensitivity and thus a more accurate result [22].
3.1.1
Water-Cherenkov detector
To detect neutrinos from the ESS, a 500 kt water-Cherenkov detector will be used. It allows the observation of neutrino interactions through the trajectories of the charged leptons that are produced [20]. Several other experiments have used water-Cherenkov detectors, such as Super-Kamiokande and SNO. This detector could also be used to study proton decay, supernova, solar and atmospheric neutrinos.
For a water-Cherenkov detector to work the neutrinos need to interact with the water and we have previously established that the neutrinos interact with matter. Neutrinos interact with electrons according to the following process
να+ e− → να+ e−.
This is an elastic scattering process, and thus the only reaction that this causes is a change in the distribution of the energy and momentum of each particle in the process. A similar process exists for antineutrinos [20].
3.1.2
Previous theoretical studies of ESSnuSB
Several theoretical studies and comparisons have been carried out relating to the ESS-nuSB. ’A comparative study between ESSnuSB and T2HK discovery in determining the leptonic CP-phase’, discussed the long-baseline experiments ESSnuSB and T2HK (Tokai to Hyper-Kamiokande), which both aim to determine the δCP [12]. The focus was to
compare the two experiments as they study different oscillation maxima. T2HK studies the first and ESSnuSB the second. The work aimed to study the impact of the neutrino mass ordering degeneracy and the leptonic mixing angle θ23 octant degeneracy on the
detection of the CP-violation and precision of these experiments. They concluded that for the ESSnuSB the neutrino mass ordering degeneracy did not have a significant im-pact on the determination of δCP [12]. It is also expected the ESSnuSB will be able to
determine the neutrino mass hierarchy at a confidence level of at least 3σ [9].
ESSnuSB would be able to determine the value of δCP with a 5σ confidence level for
60 % of its possible values and detect CP-violation in the lepton sector.
3.2
Physics at ESSnuSB
For experiments such as the ESSnuSB, one could approximate the density of the matter to be constant. The neutrinos would not move through strong variations in density, such as through the core of the Earth, which is why the density is assumed to be constant. An approximate value for the density of the mantle is ρmantle ≈ 4.5 g/cm3 and for the
crust ρcrust≈ 3 g/cm3 [12].
As discussed above, the νµ → νe transition probability will be used to study physics at
the ESSnuSB.
The formula that is relevant for our calculations is equation (2.41), is stated below [15] Pνe→νµ(∆) = a 2sin22θ 12cos2θ23 sin2A∆ A2 + 4 sin 2θ 13sin2θ23 sin2(A − 1)∆ (A − 1)2
+ 2a sin θ13sin 2θ12cos (∆ − δCP)
sin A∆ A sin (A − 1)∆ A − 1 , (3.1) where ∆ = ∆m2
31x/4E, and A = VCCx/2∆, δCP is positive for neutrinos and negative
for antineutrinos, and a is given by equation (2.40). The first term in equation (3.1) is known as the ’solar’ term, the second is the ’atmospheric’ term and the last is the term that includes the CP interference [12]. The table below displays the values of different parameters that will be used to study the neutrino oscillations at the ESSnuSB.
Parameter Value Error
∆m221/10−5eV2 7.39 +0.21−0.20 ∆m322/10−3eV2 2.449 +0.032−0.030 θ12 33.82◦ +0.78 ◦ −0.76◦ θ13 8.61◦ +0.13 ◦ −0.13◦
Average neutrino energy 0.36 GeV
Table 3.2: Parameter values relevant for neutrino oscillations [18, 23].
The value of θ23is not known to an acceptable accuracy as we do not know which octant
it lies in; we do not know whether θ23 > 45◦ or θ23 < 45◦ is true. Usually one assumes
3.3
Neutrino oscillations at ESSnuSB
To study neutrino oscillations through the ESSnuSB we will need to take into account the impact of matter on the neutrino oscillations, which was discussed in section 2.3 as well as the implication of CP-violation, discussed in section 2.4. Matter will impact neutrino and antineutrino oscillations in different ways, for cos 2θ0 > 0 and ∆m2 > 0 the
oscillations of antineutrinos will be suppressed, but it could enhance those of neutrinos. This is a similar effect to that of CP-violations, which makes it difficult to differentiate the impact of matter on neutrino oscillations from CP-violations [1].
For the ESSnuSB the detector will be placed at the second oscillation maximum. This is due to the larger interference of the CP-violation phase which will thus improve the statistics of determining its value. Using equation (3.1) above, we can plot the neutrino oscillations in matter with the specific values of the ESSnuSB. To calculate Vcc we used
[15] V (x) ≈ 7.56 × 10−14 ρ(x) g/cm3 Ye(x) [eV], (3.2)
where we use ρ(x) = ρcrust ≈ 3 g/cm3 together with the values in table 3.2. The Ye(x)
term is the number of electrons per nucleon and is typically 0.5 [15], which is the value used in this paper.
To determine the oscillation maxima of the neutrino oscillations at the ESSnuSB one can look at the leading term in equation (3.1), it is determined by ∆ = ∆m2
31x/4E. Using
this we have that [21]
∆m231x 4E = (2n − 1) π 2, x = (2n − 1)π 2 4E ∆m2 31 , (3.3)
Implementing our parameter values from table 3.2, equation (3.3) gives us
Figure 3.1: Electron neutrino oscillations Pνe→νµ(x) in matter, assuming the energy of each neutrino to
be is 0.36 GeV [18], using the values in table 3.2, and θ23= 45◦ for maximal mixing.
From figure 3.1 we can determine the location of the first and second maxima of neutrino oscillations in matter. The first maximum is located at around 200 km, whereas the second one is located at around 540 km. At the first maximum the CP-interference terms, which includes δCP, have a smaller magnitude compared to the solar and atmospheric
terms. At the second maximum, however, the magnitude is almost the same as the leading atmospheric term. This means that the CP-term has a much larger impact on the total probability, which gives more accurate data when trying to determine δCP. The
matter effects that can have similar impacts on the probability as the CP-term are lower at the second maximum. Thus, the ESSnuSB would benefit from having the detector 540 km from the source in Lund [18].
Furthermore, by looking at equation (3.1) and expanding the cosine term in the CP-interference term we find (− for neutrinos, + for antineutrinos)
cos (∆ ∓ δCP) = cos(∆) cos(δCP) ± sin(∆) sin(δCP). (3.5)
The second term in the equation above is the CP-violating term, apparent by the ±. Multiplying in the rest of the CP-interference term from equation (3.1) with the last term above we obtain
± 2a sin θ13sin 2θ12
sin A∆ A
sin (A − 1)∆
It is possible to further inspect the difference between the first and second maxima. The larger the amplitude of sin(δCP) is, the more impact it has on the oscillations, meaning
it is easier to obtain an accurate value of δ. Using the values in table 3.2 we find that the amplitude at the second maximum is more than ten times larger than that of the first maximum.
Finally, when measuring the probabilities in the experiment one can use statistics on the collected data of the νµ→ νe and νµ→ νe channels to calculate the value of δCP within
3.4
Discussion
We analyzed the advantages of extending the ESS facility with the ESSnuSB experiment, including placing a detector near the second oscillation maximum in the mine in Garpen-berg located at a distance of around 540 km from the ESS. Since the mine already exists, there are potentially fewer costs involved in the construction of the detector as it has to be underground to minimize disturbances. Furthermore, the environmental impact can be reduced by using the existing mine structures. The calculations of the second oscillation maximum in both vacuum and matter (see figure 3.1) makes it clear that the distance to the second oscillation maximum is around 540 km, which makes the option of Garpenberg preferable. However, Zinkgruvan remains a good potential candidate for the location of the detector.
One of the main objectives with the ESSnuSB is to accurately determine the value of δCP. Since the second oscillation maximum provides better data compared to the first
oscillation maximum as the CP-violating term has a larger impact the ESSnuSB has the potential to achieve its objective. Furthermore, the impact of matter decreases at the second oscillation maximum, as it has a similar effect as the CP-violating term one could more accurately determine the value without the interference of matter effects. Combining this with the 8-2 configuration, which reduces the sensitivity of the experiment we obtain the most accurate values [22]. By locating a detector further away than the second maximum one encounters other problems, such as the neutrino detection rate which decreases as 1/x2. This means that one would need much stronger detectors or
sources to obtain enough data [24].
Chapter 4
Summary and Conclusion
The purpose of this report was to review the formulas for neutrino oscillations in two flavors as well as discuss the case of three flavors. We also discussed the behavior of neutrino oscillations in matter. The reason for reviewing the derivation of these formulas was to gain an understanding of how the neutrino oscillations occur, and what they are. The reason neutrino oscillations and their interaction with matter were studied was to discuss the proposed experiment ESSnuSB and the effect matter and the CP-violating phase have on such oscillations.
Through the derivations of the transition probabilities for the neutrino oscillations with two and three flavors, we gained an understanding of how neutrinos behave and how they defy the predictions of the Standard Model. The existence of physics beyond the Standard Model raises more questions to be answered, which is often the case when discoveries are made. The study of neutrino oscillations could solve the problem of asymmetry of matter and antimatter in the Universe, and as well give rise to new questions or solve other future problems.
Throughout this work, we investigated neutrinos and their behavior in matter to see how this can impact the neutrino oscillations that the ESSnuSB experiment aims to study. The matter effects on neutrino oscillations mimic the behavior of the CP-interference term. Therefore, one would like to reduce these effects to achieve an accurate value for δCP.
The opportunity that the ESS has to study neutrino oscillations is unique in its accuracy. The ESSnuSB experiment would therefore be optimal for studying the δCP since the
Acknowledgment
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