Probing the nature of
matter-antimatter symmetry and
neutrino masses at future intensity
frontier experiments
THESIS
submitted in partial fulfillment of the requirements for the degree of
MASTER OFSCIENCE
in
PHYSICS
Author : Alex Mikulenko
Student ID : 2497395
Supervisor : Alexey Boyarsky
2ndcorrector : Vadim Cheianov
Probing the nature of
matter-antimatter symmetry and
neutrino masses at future intensity
frontier experiments
Alex Mikulenko
Instituut-Lorentz, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands
July 17, 2020
Abstract
We review the Standard Model of elementary particles and the three observational evidences of its incompleteness: dark matter, neutrino oscillation,
and baryon asymmetry of the universe. A particular extension of the Standard model, νMSM with only three new sterile neutrinos can in principle explain all the mentioned phenomena. This work addresses the relevant question how the
observational data constraints the parameter space of the model that can be probed by accelerator experiments. Namely, we investigate how the measurements of neutrino oscillation parameters affect the possible outcomes
obtained in SHiP experiment. We demonstrate the results of parameter reconstruction for a sterile neutrino with mass M =1 GeV.
Contents
1 Introduction 1
2 Standard Model 3
2.1 Lagrangian of the Standard Model 4
2.2 Higgs mechanism and Lagrangian in terms of observed particles 6
3 Dark matter 9
3.1 Dynamical evidences 9
3.2 Gravitational lensing 10
3.3 Cosmological evidences 13
4 Neutrino masses and oscillations 15
4.1 Discovery of neutrino oscillation 15
4.2 Description of oscillations 16
4.3 Neutrino masses 17
5 Baryon asymmetry 21
5.1 Relic baryons 21
5.2 Sakharov conditions 22
5.2.1 Baryon number violation 23
5.2.2 C and CP-violation 23
5.2.3 Nonequilibrium state 24
6 νMSM 27
6.1 νMSM Lagrangian and neutrino mixing 27
6.2 Constrains on sterile neutrino parameters 28
6.2.1 Light sterile neutrino N1 28
6.2.2 Heavy sterile neutrinos N2, N3 29
7 Data analysis 31
7.1 Accelerator experiment constraints 31
7.1.1 SHiP 31
7.1.2 Description of the method 32
7.2 Neutrino oscillation constraints 37
B Simulation results 45
B.1 SHiP bounds 45
Chapter
1
Introduction
The Standard Model is the modern theory of elementary particles and fundamen-tal interactions between them. Having been steadily developed during the last fifty years, it is formulated as a gauge theory based on a SU(3) ×SU(2) ×U(1) symmetry group. This relatively simple construction with a moderate number of particles has proved to correctly describe the nature up to the scales of 10−17cm. The last critical component of the Standard Model, the Higgs boson, has been re-cently found. Only a few phenomena remain unexplained. Some of them, such as quantum gravity or the origin of dark energy, probably require conceptually new approaches. In this work, we are interested in three phenomena Beyond the Standard Model (BSM) which can be explained only by adding new elementary particles to the theory, namely: dark matter, neutrino masses and oscillations and baryon asymmetry of the universe. This makes searching for new particles a very perspective problem.
Up to the present time, accelerator experiments aimed to increase the centre-of-mass energy to find more massive particles. While it was promising before the discoveries of W, Z, and Higgs boson, now there is no clear reason to do this further. At the same time, these experiments were insensitive to possible light feebly interacting particles. Recently, a number of new experiments were proposed to test this possibility: SHiP [1,2], MATHUSLA [3], CODEX-b [4], NA62 [5], DUNE [6]. In the near decade, direct measurements will be performed in addition to the indirect constraints from the data related to the BSM problems.
In this work, we consider the Neutrino Minimal Standard Model (νMSM) as a possible solution to explain the BSM phenomena. The model consists of new three fermions, one being a dark matter candidate and the other two explaining oscillations and baryon asymmetry. The choice of three fermions is the minimal one that does not contradict to the observational data. This ensures that the model can be easily confirmed or disproved.
The main goal of this work is to understand how accelerator experiment re-sults can be combined with the neutrino oscillation data to effectively constrain the parameter space of new particles. The structure of the thesis is the following: in the second chapter we describe the Standard Model. In the third, fourth, and fifth chapters we provide a review on the BSM problems: dark matter, neutrino oscillations, and baryon asymmetry correspondingly. In the sixth chapter, we
chapter, we provide an overview of the SHiP experiment. Finally, in the eighth chapter, we investigate how to reconstruct the parameters from different results obtained in the experiment and analyze the constraint from neutrino oscillations.
Chapter
2
Standard Model
The Standard Model is a quantum field theory [7] that treats fundamental inter-actions in the united way as a consequence of internal gauge symmetries using the Yang-Mills theory [8, 9]. This fundamental principle leads to interactions of the special form that allows a consistent description of vector particles [10].
The theory contains of a set of spin-12 fermions, typically referred to as matter, spin-1 interaction mediators and the famous spin-0 Higgs boson, a manifestation of the Higgs mechanism [11], which provides masses to all elementary particles.
Figure 2.1:Particles of the Standard Model. Leptons and quarks are fermions, organised in three generations. Gauge bosons are responsible for fundamental interaction: gluons mediate the strong interaction, W and Z-bosons carry the weak interaction, and photons are particles of the electromagnetic interaction. Credits: Wikipedia.
As a field theory, the Standard Model is completely described by its Lagrangian. However, this fundamental quantity cannot be straightforwardly constructed from the symmetry principle in terms of degrees of freedom, which are directly related to the observed particles of Fig.(2.1). This problem is gracefully resolved by the Higgs mechanism.
The fermionic sector contains of three generations of the form (U, D, l, ν)i,
where Ui and Di are quarks with electric charges +23 and −13 correspondingly,
li is a charged lepton, and νi is its neutrino. In the Standard model, all
genera-tions are equivalent, i.e., the form of interacgenera-tions between particles is the same for each generation.
The three symmetry groups SU(3) | {z }
strong
×SU(2) ×U(1)
| {z }
electroweak
act on fermions in the next way [12,13]:
• U(1): the elements of this group act as a multiplication by a phase factor: ∀Ω=eig0α ∈U(
1) : f →eig0Yfαf
(2.1) where Yf is called the weak hypercharge of f .
• SU(2): acts in the fundamental representation on the next doublets: Qi =UDi,L i,L , Li = νi,L li,L (2.2) ∀Ω =eigαiti ∈ SU(2): Q i →ΩQi, Li →ΩLi (2.3)
where ti = σ2i, i = 1, 3 are the generators of the Lie algebra of SU(2), σi are
the Pauli matrices. Only the left components of the Dirac spinors transform, while the right-handed counterparts UR, DR, lRremain invariant under this
gauge transformation. The theory does not contain νR degree of freedom,
for it would be invariant under the total symmetry group and therefore would not interact at all. The explicit separation into left and right com-ponents is dictated by the fact that weak interactions violate P-symmetry [14,15].
The SU(2) ×U(1)-sector of the Standard Model (Glashow-Salam-Weinberg theory) is the minimal mathematically consistent theory, which reduces to the Fermi theory of beta decay [16]:
LFermi = −√GF 2J µJ† µ, J µ = Jµ leptons+J µ hadrons (2.4)
Jleptonsµ = ¯νeγµ(1−γ5)e, Jhadronsµ =u¯(1−γ5)d cos θC
where GF is the Fermi coupling constant and θC ≈ 13◦ is the Cabbibo
mix-ing angle [17]. It is shown that one can reconstruct the explicit form of the Lagrangian just from the condition of unitarity of the scattering matrix in the leading order of the perturbation theory [18].
2.1 Lagrangian of the Standard Model 5
• SU(3): acts in fundamental representation on quark triplets. Namely, any quark field q has an internal structure of a three-component vector, compo-nents of which are usually referred to as colours:
∀Ω =eigsαaλa ∈SU(3) : q= q1 q2 q3 →Ω q1 q2 q3 (2.5)
where λa, a= 1, 8 are the Gell-Mann matrices - the generators of the Lie
al-gebra of SU(3). The SU(3) gauge theory of strong interactions, quantum chromodynamics, can explain such specific features as confinement (quarks cannot be in unbounded states) and asymptotic freedom [19, 20] (interaction weakens at high energies), due to the behaviour of the renormalization flow of the coupling constant. The existence of quarks itself follows from the so-called eightfold way [21] (organizational scheme of hadrons as bound states) and inelastic proton scattering [22] (at high energies protons behave like they are made of separate parts).
In the Yang-Mills theory, the vector fields are elements of the corresponding Lie algebras. Explicitly, we have three combinations:
Bµ Wµ =Wµiti Gµ =Gµaλa (2.6)
which transform underΩ from the corresponding group in the next way: Aµ →ΩAµΩ†+
i
gAΩ∂µΩ
† (2.7)
where A= (B, W, G).
The field tensors Fµν = (Bµν, Wµν, Gµν)are given by
Fµν =∂µAν−∂νAµ−igA[Aµ, Aν] (2.8)
The last component is the Higgs doublet H =
φ1
φ2
, which also transforms in the fundamental representation of SU(2) and is charged under U(1). In ad-dition to gauge interactions, the Higgs field is coupled to fermions via Yukawa interaction and to itself.
The gauge invariant Lagrangian of the Standard Model:
L =
∑
i ¯ Qiiγµ ∂µ−igWµ+i g0 6Bµ−igsGµ12×2 Qi+ +U¯i,Riγµ ∂µ+i 2g0 3 Bµ−igsGµUi,R+D¯i,Riγµ
∂µ−i g0 3Bµ−igsGµ Di,R + ¯Liiγµ ∂µ−igWµ−i g0 2Bµ Li+¯li,Riγµ(∂µ−ig 0 Bµ)li,R + + ∂µ−igWµ+i g0 2Bµ H 2 +µ2H†H−λ(H†H)2− −1 4BµνB µν−1 2Tr WµνW µν−1 2Tr GµνG µν− −λDij(Q¯iH)Dj,R−λUij(Q¯iH˜)Uj,R−λli(¯LiH)li,R+h.c. (2.9)
string there is also summation over generations i, j. The introduced conjugated Higgs field ˜H = 0 1
−1 0
H∗ transforms in the same SU(2)fundamental repre-sentation.
Q UR DR L lR H H˜
Y −1
6 −23 13 12 1 -12 12
Table 2.1:The values of weak hypercharges of the fields.
2.2
Higgs mechanism and Lagrangian in terms of
ob-served particles
At first sight the expression (2.9) does not contain mass terms for fermions and vector bosons, which explicitly break the symmetry. These particles acquire masses via interaction with the constant Higgs field, which presence is a consequence of spontaneous symmetry breaking, commonly referred to as the Higgs mechanism in the context of high energy physics. Explicitly, due to the special choice of positive µ2 > 0 coupling the ground state of the system has nonvanishing value of the Higgs field. In the unitary gauge, we can choose
H = √1 2 0 v+h , v2 = µ 2 λ (2.10)
where v is the vacuum expectation value (VEV) and h is the degree of freedom corresponding to Higgs boson. The three other degrees of freedom are Goldstone bosons, which were absorbed in the gauge W and B fields. Substituting this into (2.9) after diagonalization of quark and vector boson mass terms
Aµ =Bµcos θW−Wµ3sin θW Zµ =Wµ3cos θW+Bµsin θW, W ± µ = Wµ1±iW±2 √ 2 (2.11) Ui,L →WULij Uj,L Ui,R →WURij Uj,R Di,L →WDLij Dj,L Di,R →WDRij Dj,R (2.12) we obtain the actual Lagrangian describing particles of the Standard Model and their interactions: L = Lfree+ Lint Lfree =
∑
fermions ¯f iγµ ∂µ−mf f + 1 2∂ µh∂ µh− m2h 2 h 2−1 4(∂µAν−∂νAµ) 2− −1 2|∂µW + ν −∂νW + µ| 2−m2 WW+W−− 1 4(∂µZν−∂νZµ) 2+m2Z 2 Z 2 (2.13)2.2 Higgs mechanism and Lagrangian in terms of observed particles 7 Lint =
∑
f Qfe ¯fγµf Aµ+∑
quarks gs¯qγµGµaλaq+ + g cos θW∑
f [T3−Qfsin2θW]¯fLγµfL−Qf sin2θW ¯fRγµfR Zµ+ +√g 2∑
l=e,µ,τ ¯νlγµPLlWµ++ u¯ ¯c ¯t γµWµ+PLVCKM d s b +h.c. + +ig (Zνcos θW+Aνsin θW)(W − µ∂ νW+µ−W+ µ ∂ νW−µ)++cyclic permutation of W+, W−,(Z cos θW+A sin θW)
+ +g2 1 2(W − W+)2−1 2(W −)2(W+)2+ (Z cos θ W+A sin θ)2(W−W+)− −Wµ−(Zµcos θ W+Aµsin θW)Wν+(Z νcos θ W+Aνsin θW) + −gsfabc∂µGνaG µbGνc−g 2 s 4 (f abcGb µG c ν)(f adeGdµGeν)+ +gmWW−W+h+ gmZ 2 cos θW Z2h+ g 2 4 W − W+h2+ g 2 8 cos2θ W + +
∑
f gmf 2mW ¯ff h− g 4 m2h mW h3− g 2 32 m2h m2Wh 4 (2.14) Here tan θW = g 0g - the Weinberg angle, e = gg0 √
g2+g02. The masses of bosons in
terms of g, g0, and v are given by mW = gv 2 , mZ = p g2+g02 2 v= mW cos θ, mh = √ 2λv, mA=0 (2.15)
The weak isospin t3eigenvalues are given by
T3 = 1
2 for(νe, νµ, ντ, u, c, t)L T3 = −1
2 for(e, µ, τ, d, s, b)L, h T3 =0 for right fermions
and the electric charge Q generator (which reduces to a number for fermions and Higgs boson) is defined by
Q=t3−Y (2.16)
The structure constants fabc are defined by
[λa, λb] =i fabcλc (2.17)
Diagonalization of the quark mass terms leads to the Cabbibo-Kobayashi-Maskava matrix VCKM = WUL† WDL, which mixes quarks in weak interactions
Chapter
3
Dark matter
From 1930, numerous astronomical observations have revealed the fact that only a tiny fraction of matter in the universe forms stars. The remaining part does not emit light, being invisible for direct observation with telescopes, and happened to consist not only of ordinary matter, such as protons and electrons, but mostly of a completely new type of matter, dark matter.
3.1
Dynamical evidences
Galaxies and clusters are self-gravitating systems, and the motion of the objects in these systems is governed by the total gravitational potential. Several inde-pendent considerations in the framework of Newtonian gravity show that the amount of luminous matter is insufficient to explain the observational data.
1. The motion of stars in elliptical galaxies and galaxies in clusters is random. Applying the virial theorem to objects in a self-gravitating system:
htotal energyi = −hkinetic energyi = 1
2hpotential energyi (3.1) one can estimate the relation between the total mass M, the scale R, and the typical velocity V:
hV2i ∼ GMtotal
R (3.2)
Zwicky [24] in 1933 was the first to realize that the velocity dispersion V ∼ 103km/s in the Coma cluster requires a mass in 400 times higher than that of the luminous matter.
2. Spherical galaxies are dominated by angular momentum. Stars move in approximately circular orbits, therefore it is possible to measure their rota-tional velocity. At the same time, this quantity is determined by to the total mass enclosed in the orbit:
V2(r) = GM(r)
Stellar Disk Dark Halo Observed
Gas
M33 rotation curve
Figure 3.1: Rotation curve V(r)for M33 [25] with the best fit M(r)model (continuous line). Also shown the stellar disk contribution, the gas contribution and the dark matter halo contribution.
3. Observation of the X-ray spectrum [26] from galaxy clusters shows that the intercluster medium is highly ionized and heated∼ 108K. It is possible to measure local parameters, such as density and temperature, and relate them to the gravitational potential under assumption of hydrostatic equilibrium:
∇p(r) = ∇
n(r)T(r)
= ∇ψ(r) (3.4)
It turned out that the intercluster gas contributes 15% to the total mass of clusters, while the galaxies - only 1%. The remaining part presumably has to be made of something different from the ordinary matter.
3.2
Gravitational lensing
General Relativity predicts deflection of light rays in gravitational field; this pro-vides a powerful method of tracing mass called gravitational lensing [27].
Consider a compact mass distribution, i.e., much smaller than the distances to the observer Dd and to the source Dds (thin lens approximation). The angular
position of the source on the sky plane β = (β1, β2) is related to the angular
position of its image θ = (θ1, θ2)via the ray-trace equation (for θ, β 1):
β=θ−2Dds c2D s ∇ Z ψ(x, y, z)dz (x,y)=Ddθ (3.5) Here ψ is the Newtonian potential and z-coordinate is directed to the sky plane; one also introduces the projected potential:
Ψ(x, y) =
Z
3.2 Gravitational lensing 11
Figure 3.2:The geometry of gravitational lensing. Credits:[27]
with Σ - surface density of the object. Gravitational lensing is actually sensitive to the projected potential.
Not only the position of the source is shifted, its shape is also distorted due to different deflection of nearby rays. This distortion is described by the magnifica-tion matrix: Aij = ∂θi ∂βj ≡1−κ−γ1 −γ2 −γ2 1−κ+γ1 −1 (3.6) There exist several types of lensing, based on different effects: strong, weak, and microlensing:
1. Strong lensing: for sufficiently dense objects with strong gravitational field (κ, γ & 1) the displacement is large and it is possible to obtain multiple images.
2. Weak lensing: κ, γ 1. In this case, distortion of the shapes of numerous galaxies on large scales is used to calculate the magnification matrix. Typi-cally these galaxies are aligned, as shown on Fig.3.4.
3. Microlensing: used for detecting small lenses - stars, black holes, planets. Gravitational lensing for these objects is extremely small to resolve displace-ment and distortion of the source. However, it is still possible to detect small variations of the brightness of the source when a lens passes between it and the observer.
Ilensed
I =det A= [(1−κ)
2−
γ21−γ22]−1
Such events are very rare and require monitoring of a large number of sources.
Figure 3.3: Examples of strong lensing. Left: the Cosmic Horseshoe, right: the Cheshire Cat. Credits: ESA/Hubble, NASA; NASA/CXC/UA/J.Irwin et al, STScI
Figure 3.4:Weak lensing. The distortion is greatly exaggerated relative to real astronom-ical systems. Credits: Wikipedia
3.3 Cosmological evidences 13
3.3
Cosmological evidences
The evolution of the universe as a whole also provides independent determina-tion of different contribudetermina-tions to the total energy density. As the two most demon-strative examples, we describe here how measurements of the redshift of distant objects and the cosmic microwave background (CMB) are sensitive to the total mass of matter and baryonic matter correspondingly.
1. A homogeneous and isotropic universe is described by the Friedmann equa-tion [28]: ˙a a 2 =H02 ΩΛ +Ωm a0 a 3 +Ωr a0 a 4 +Ωκ a0 a 2 (3.7) where a is the scale factor and the Hubble constant H0is related to the total
energy density of the universe H02 = 8πG3c2 ρat the present moment; G is the
gravitational constant, a0 is the present value of the scale factor, andΩ are
the relative contributions of the vacuum energy density, nonrelativistic mat-ter, relativistic matter/radiation, and the spacial curvature correspondingly: ΩΛ +Ωm+Ωr+Ωκ =1
For a given model{Ω}, one can find a relation between the luminosity dis-tance dL
flux= luminosity
4πd2L (3.8)
and the redshift z for a source dL(z) = (1+z)c √ ΩκH0 sinh Z z 0 √ Ωκdz0 p ΩΛ+Ωm(1+z0)3+Ωr(1+z0)4+Ωκ(1+z0)2 (3.9) It is possible to measure independently z and dL for a class of sources called
standard candles - objects with known luminosity. Analysis based on Type Ia supernovae [29] for a flatΩκ =0 model shows
Ωm =0.211±0.103, ΩΛ =0.788±0.103 (3.10)
The visible matter contribution (including the intercluster gas) is not enough to satisfyΩm ≈0.2.
2. The cosmic microwave background is anisotropic, i.e., its temperature varies for different directions of observation. Although these fluctuations are very small δT/T ∼ 10−5, their correlations on different scales are strongly af-fected by gravitational dynamics in the early universe. The analysis of this angular anisotropy is highly nontrivial [30], but the best-fit parameters [31] Ωbaryons =0.044±0.008, Ωm =0.27±0.08 (3.11)
Figure 3.5: Left:the data for Type Ia supernovae and different models. Right: 68% confi-dence curves for different samples. Credits: [29], [32]
Figure 3.6:Angular spectrum of CMB. Temperature correlatorshδT(r1)δT(r2)ican be
ex-pressed in terms of spherical harmonics with coefficients Cl. The position of the first peak
is related to the spatial curvature of the universe, the second one - to the total baryonic density. Credits: [31]
Chapter
4
Neutrino masses and oscillations
Oscillation of neutrinos, i.e., transformation of neutrino flavour during its prop-agation, has been confirmed in numerous experiments with solar, reactor, and atmospheric neutrinos. This phenomenon only can occur for particles with dif-ferent masses, implying that all three types of neutrinos cannot be massless. This is the only nonastronomical direct evidence that the Standard Model is not com-plete.
4.1
Discovery of neutrino oscillation
Neutrino oscillations were first detected as a deficiency in the electron neutrino flux produced in the sun. The energy of the sun is produced in the proton-proton chain reaction. Therefore, the total neutrino flux is directly related to the total luminosity and can be predicted with small theoretical uncertainties.
Figure 4.1: Reactions in the proton-proton chain. The % values are the branching ratios for the Sun. Credits: Wikipedia
are based on the reaction:
71Ga+
νe →71Ge+e−
and have obtained fluxesΦ significantly lower than the theoretical ones: Φ
Φth
≈0.5 (4.1)
Other experiments, Kamiokande and SNO, based on
1. ν+e− →ν+e− Kamiokande
2. νe+2H→ p+p+e− SNO, charged current
3. ν+2H→ p+n+ν SNO, neutral current
have found results that are in agreement with the assumption of neutrino oscilla-tions: Φ1 Φ1 th =0.48±0.02 Φ 2 Φ2 th =0.34±0.03 Φ 3 Φ3 th =1.0±0.2 (4.2) The important thing here is that these reactions are sensitive to different ener-gies of neutrinos, but show similar deficiency for all the experiments. This is also known as the solar neutrino problem, and it is resolved by the assumption that only 1/3 of the electron neutrino flux survives, while the remaining 2/3 oscillates into νµ, ντ.
Neutrino oscillations have been also observed for atmospheric and reactor neutrinos. The detector KamLAND has found a deficiency of electron antineu-trinos produced at nuclear reactors at the distance 80-250 km:
ΦKamLAND
Φno osc. =0.611±0.095 (4.3)
in the range of neutrino energies Eν ∼ 3-6 MeV. Atmospheric neutrinos are
pro-duced in
π± →µ±νµ, µ →e¯νeνµ (4.4)
and can have energies of Eν ∼ 100 MeV-10 GeV; deficiency of these muon
neu-trinos has been established with, for example, the Kamiokande detector. Overall, deficiency of neutrino flux has been observed in numerous experiments for differ-ent neutrino types and energies; the results are in agreemdiffer-ent with the assumption of neutrino oscillations.
4.2
Description of oscillations
For simplicity, let us consider two neutrino flavours. Assume that in weak inter-actions electrons are coupled to νe, muons - to νµ, and these fields are
4.3 Neutrino masses 17 νe νµ =cos θ −sin θ sin θ cos θ ν1 ν2 (4.5) According to the perturbation theory, together with an electron, the interaction operator creates the state
|νei =cos θ|ν1i +sin θ|ν2i (4.6)
The evolution of the massive states is given by the Schr ¨odinger equation: |νi(t)i =e−iEt|νi(0)i =e−i
√ p2+m2
it|νi(0)i
and due to the different masses mi the relative phase between the states varies in
time. For ultrarelativistic particles E ≈ p+ m2E2 up to the total phase factor one gets: |νe(t)i =cos θ e−i m2 1 2Et|ν1i +sin θ e−i m2 2 2Et|ν2i (4.7)
It is straightforward to calculate the probability that the neutrino state pro-duced together with an electron (electron neutrino) at the distance L interacts as if it has changed its flavour:
P(νe →νe) = |hνe(t=0)|νe(t= L/c)i|2 =1−sin22θ sin2
m21−m22
4E L (4.8)
P(νe →νµ) = |hνµ(t=0)|νe(t= L/c)i|2 =sin22θ sin2
m21−m22
4E L (4.9)
These expressions show the key features of neutrino oscillation: neutrino flavours indeed oscillate with the amplitude defined by the unitary mixing matrix:
|νii = Uiα|ναi, α =e, µ, τ, i=1, 2, 3 (4.10)
and the oscillation length depending on the energy of neutrinos and the differences of squared mass eigenvalues:
∆m2
ij =m2i −m2j (4.11)
The general expression for the transition probability is given by:
P(να →νβ) =δαβ−4
∑
j>i
Re U†jαUjβUiαUiβ† sin2
∆m2 ji
4E L +2
∑
j>i
Im Ujα†UjβUiαUiβ† sin
∆m2 ji
2E L (4.12)
4.3
Neutrino masses
In the Standard Model, neutrinos are massless due to the lack of the correspond-ing right-handed Weyl spinors. The minimal extension to the Standard model is
add to the Lagrangian terms:
∆L = −λνij(¯LiH˜)(νj)R+h.c.
It is known that there exist two types of fermionic mass - Dirac and Majorana masses. All the other fermions of the Standard Model have the Dirac masses, i.e., the mass terms are of the form
LD = −m(ψ¯LψR+ψ¯RψL) (4.13)
that requires twice as more degrees of freedom ψL and ψR to satisfy relativistic
invariance. At the same time, the Majorana mass term can be constructed using a single Weyl spinor ψL only:
LM = −m 2(ψ¯
c
LψL+ψ¯LψcL) (4.14)
where ψc is the charged conjugation of ψ. In the Weyl representation, it can be rewritten as CψL = −iγ2 χ 0 ∗ = 0 −iσ2 iσ2 0 χ∗ 0 = 0 iσ2χ∗ (4.15) In other words, Majorana particles can be equivalently described by the left-handed spinor ψL or the right-handed ψcL. Due to the reduced number of degrees
of freedom, this particle appears to be its own antiparticle as a consequence of the broken U(1) symmetry - the Majorana mass term is not invariant under phase rotations ψL →eiαψL. Therefore, the Majorana mass term violates lepton number
conservation; this leads, for instance, to neutrinoless double beta decay.
Neutrinos can have Majorana masses as well; without introducing extra de-grees of freedom one can write down a gauge-invariant operator called the Wein-berg operator: ∆L = −Fij (¯LiH˜)(H˜†Lcj) Λ sym. br. =⇒ −Fij v2 2Λ¯νicνj (4.16)
However, this is an effective nonrenormilizable operator, i.e., this still implies the existence of new physics at the energy scale Λ. In particular, if the scale is sufficiently large, this can explain why neutrino masses are so small compared to the other fermions of the Standard Model.
Nondiagonal mass terms lead to the Pontecorvo-Maki-Nakagata-Sakata (PNMS) matrix [33], which is similar to the CKM matrix:
νe νµ ντ = Ue1 Ue2 Ue3 Uµ1 Uµ2 Uµ3 Uτ1 Uτ2 Uτ3 ν1 ν2 ν3 (4.17)
This matrix can be parametrized by three mixing angles θ12, θ23, θ13, and one
phase angle δ in the case of Dirac mass or, equivalently, lepton number conserva-tion. In the case of Majorana, there are two more phase angles α1, α2. Denoting
4.3 Neutrino masses 19
sin θij =sij, cos θij =cijthe parametrization takes form
U = 1 0 0 0 c23 s23 0 −s23 c23 c13 0 s13e−iδ 0 1 0 −s13eiδ 0 c13 × c12 s12 0 −s12 c12 0 0 0 1 1 0 0 0 eiα2 0 0 0 eiα2 = = c12c13 s12c13 s13e−iδ −c23s12−s23c12s13eiδ c23c12−s23s12s13eiδ s23c13 s23s12−c23c12s13eiδ −s23c12−c23s12s13eiδ c23c13 1 0 0 0 eiα2 0 0 0 eiα2 (4.18) ∆m2 sol ∆m2atm θ12 θ23 θ13 δ 7.4·10−5eV2 2.52·10−3eV2 34◦ 50◦ 8.6◦ ≈200◦
Table 4.1: Numerical values of neutrino parameters [34]. The values∆m2 are given for oscillations of solar and atmospheric neutrinos. For more detailed values see section7.2
The measured values of ∆m2are of different orders, two neutrino masses are much closer to each other compared to the splitting to the third one. Due to this situation, one distinguishes two options: the so-called normal and inverted hierarchy.
Figure 4.2: Normal and inverted hierarchies of neutrino masses. The colours represent the fractions of ναin the mass eigenstates. Credits: [35]
Chapter
5
Baryon asymmetry
In the modern universe, matter strongly dominates antimatter: stars, galaxies are composed of matter, while antimatter particles can be only found in cos-mic rays. However, in the early universe the primordial plasma was heated to large temperatures and contained both matter and antimatter. If the temperature exceeded 170 MeV, quarks were unbounded and there were rapid processes of their production and annihilation. It appears that in order to describe the ob-served amount of matter, one requires only one additional quark per 109 quark-antiquarks pairs. This exceedingly small excess naturally gives rise to the next question: what mechanism could have been responsible for the production of a small baryon asymmetry if one assumes the symmetric initial state of the uni-verse?
5.1
Relic baryons
It is quite easy to realize that antimatter is indeed absent in the universe. If other structures had been made of antimatter, this would have lead to a high annihila-tion rate on the walls between domains and therefore to a strong flux of photons with energies up to 1 GeV. Moreover, the existence of such domains is not possi-ble to explain starting from the homogeneous primordial plasma.
One can estimate numerically what would be the relic number density of baryons in the baryon symmetric universe. For nonrelativistic particles at tem-perature T, the number density in chemical equilibrium µB = µB¯ = 0 is given
by nB =nB¯ = m pT 2π 32 e−mpT (5.1)
The inverse lifetime of an antibaryon in the medium is given by
Γ= hσannvinB, hσannvi ≈ 10−30m2 ≈25 GeV−2 (5.2)
Once this quantity is lower than the expansion rate of the universe H - the Hub-ble parameter - the annihilation process ceases. Assuming this to happen in the
H=1.66√g∗ T 2
MPl
, g∗(1 MeV .T.100 MeV) = 10.25 (5.3)
where MPl = G−1/2 = 1.2·1019GeV and g∗ is the effective number of degrees
of freedom, the freeze-out temperature Γ(Tfr) = H(Tfr) and the number density
turn out to be Tfr =21 MeV (5.4) nB(Tf r) = H(Tfr) hσannvi =7.8·10−24GeV−3 =3.5·10−18nγ (5.5) where nγ = 2ζ(3) π2 T
3is the number density of photons. The baryon-to-photon ratio
ηB ∼10−18 is many orders of magnitude lower than the actual value ηB ∼10−9.
Evidently, the universe is not symmetric between matter and antimatter; sur-prisingly, that at the temperatures T & 200 MeV the number density of u and d quarks was of the same order as the number density of photons
nquarks = 7 8 2ζ(3) π2 T 3 (5.6) therefore, the baryon-to-photon ratio at the same time represents what fraction of quarks eventually forms baryons after annihilation of the quark-antiquark pairs:
NB
Nquarks
∼ηB ∼10−9 (5.7)
5.2
Sakharov conditions
To produce baryon asymmetry dynamically in a system with the symmetric initial state, three Sakharov conditions should be satisfied:
1. Baryon number violation. Indeed, if the baryon number operator
B=
∑
quarks
Z
d3x ¯qγ0q (5.8)
commutes with the Hamiltonian of the system [B, H] = 0, its eigenvalue remains zero.
2. C and CP-violation. These symmetries change baryons into antibaryons: C[¯qγ0q]C−1= ¯qcγ0qc = −¯qγ0q (5.9) CP [¯qLγ0qL+ ¯qRγ0qR](CP )−1= ¯qRcγ0qcR+ ¯qcLγ0qcL = −¯qγ0q (5.10)
If any of these symmetries is present, processes with particles and antiparti-cles occur with the same probability. Therefore, it is not possible to obtain a nonzero baryon number evolving from the symmetric initial statehBi = 0.
5.2 Sakharov conditions 23
3. Nonequilibrium state. If the system is in thermodynamic equilibrium, the chemical potentials µB and µB¯ are zero. In general, equilibrium processes
wash out the generated baryon asymmetry.
Surprisingly, there are mechanisms in the Standard Model to satisfy all the conditions. However, for the given values of the Higgs boson mass and CP vio-lation, it is not possible to have successful baryogenesis [36] and new physics is still required.
5.2.1
Baryon number violation
Symmetries of classical systems can be anomalously broken in quantum mechan-ics due to the appearance of quantum corrections. In particular, the baryon cur-rent
jµB = 1
3quarks
∑
¯qγµq (5.11)has to be conserved as a consequence of the SM Lagragian invariance under si-multaneous phase rotation of all quark fields q → eiαq. This is true in the per-turbative description - baryon number is a conserved quantity indeed in particle decays or interactions. However, in general this current and the corresponding lepton current have anomalous divergence:
jBµ =jµL =3 g 2 16π2W i µνW i,µν (5.12)
and it is possible to produce baryons and leptons in sufficiently strong SU(2) fields: ∆B ≡B(tf) −B(ti) = Z tf ti dt Z d3x 3g 2 16π2W i µνW i,µν (5.13)
This integral is nonzero for transitions between special field configurations that differ in their topological properties. They are separated with potential bar-riers; the field configurations corresponding to maximums of these barriers are known as sphalerons. In the early hot plasma, transitions with nonconservation of baryon number occur due to thermal fluctuations. It is known that this process is in equilibrium at temperature T &100 GeV.
5.2.2
C and CP-violation
The weak interaction completely violates C-symmetry. The Yukawa sector of the Standard Model provide a source of CP-violation in the form of a complex phase δ in CKM matrix. Indeed, a unitary matrix 3×3 has 9 independent parameters: three rotational angles and six complex phases. Five complex phases are non-physical and can be eliminated by independent phase rotations of quarks, while the last one has physical consequences.The parametrization of the CKM matrix is: U = 1 0 0 0 c23 s23 0 −s23 c23 c13 0 s13e−iδ 0 1 0 −s13eiδ 0 c13 × c12 s12 0 −s12 c12 0 0 0 1 (5.14)
unless δ =0 (which is not the case).
CP transformation changes Yukawa couplings to their complex conjugated, therefore the Lagrangian of the Standard Model is not CP invariant. Actually, the observation of CP-violation in kaon decays was the reason to introduce the third quark generation - for a 2×2 matrix the abovementioned consideration does not work.
The other possible sources of CP-violation are: a phase in the PNMS matrix (neutrino sector) or more complex structure of the Higgs sector.
5.2.3
Nonequilibrium state
In the primordial plasma the Higgs potential gets thermal correction. Conse-quently, the vacuum expectation value and the masses of particles depend on temperature. Depending on the form of the potential, there can be a continuous changing of VEV or phase transition of the first order.
Figure 5.1:Possible dependence of the Higgs potential on temperature. Left: the first or-der phase transition, right: continuous transition. Temperature increases upwards. Cred-its: [28]
In the first order phase transition, the expectation value cannot change simul-taneously in all points. Instead, the transition reminds boiling of water: bubbles of finite size with nonzero VEV appear and then expand. This is a highly non-equilibrium process. Thus, in principle in the Standard Model the third Sakharov condition could be fulfilled; however, for this the mass of Higgs boson has to be
mh.70 GeV (5.15)
which contradicts to the experimental data.
First order phase transition can take place in extensions with more complex structure of the Higgs sector. Moreover, such a complicated mechanism is not
5.2 Sakharov conditions 25
necessary at all and to satisfy the condition one can just introduce particles whose interaction is too weak for maintaining equilibrium.
Chapter
6
ν
MSM
The Neutrino Minimal Standard Model (νMSM) [37, 38] is an extension of the Standard Model with three right-handed neutrino fields - sterile neutrinos or heavy neutral leptons (HNLs) with masses below the electroweak scale. This model has a parameter region to describe consistently all the three phenomena considered in this thesis. Two (at least) sterile neutrinos are necessary to explain the two mass differences∆m2soland∆m2atmof active neutrinos, however, none of them can be a dark matter candidate due to their short lifetime. Therefore, the third sterile neutrino with the mass O(1 keV)should be added as a dark matter particle, which contribution to neutrino masses is negligible.
6.1
νMSM Lagrangian and neutrino mixing
The model addsN = 3 right-handed fermions NI, I = 1,N, which are singlets
under all gauge groups of the Standard Model. The most renormilizable and gauge-invariant Lagrangian is given by
L = LSM+i ¯NIγµ∂µNI − FαI(¯LαH˜)NI − MI 2 N¯ c INI+h.c. (6.1) HereLSM is the Standard Model Lagrangian, FαI are the Yukawa couplings, and
MI are the masses of sterile neutrinos. Without loss of generality, one can choose
the Majorana mass matrix in diagonal form. After spontaneous symmetry break-ing ¯LαH˜ → √v2¯ν the total mass matrix appears of the form:
L = LSM+i ¯NIγµ∂µNI− 1 2 ¯νβ N¯ c J 03×3 √ 2vFβI √ 2vFα J −MIδI J ναc NI +h.c. (6.2) If the Dirac masses that mix sterile and active neutrinos are much smaller than the Majorana ones vFαI MI, the seesaw mechanism [39] gives rise to the active
neutrino mass matrix
mαβ =
∑
I
FαIFβI
v2
2MI (6.3)
gauge states of active neutrinos να are dominated by the corresponding mass
eigenstates νi, there is also a small contribution of the sterile neutrino mass
eigen-states:
να =Uαiνi+θαINIc (6.4)
where|θαI| 1 are mixing angles. Consequently, sterile neutrinos interact
super-weakly, i.e., they act exactly as the active ones but with the suppressed coupling GF|θ|2instead of the Fermi constant GF. The mixing angles can be estimated as:
θαI ∼ FαIv MI , θ 2 I =
∑
α |θαI| 2 (6.5) where FI is the typical value of the Yukawa couplings FαI. For a single sterileneutrino contribution one can estimate mν ∼
q ∆m2
atm ∼θ2IMI −→ θ2I ∼10−101 GeV
MI (6.6)
6.2
Constrains on sterile neutrino parameters
6.2.1
Light sterile neutrino N
1The abundance of dark matter in compact objects implies that if the dark matter particle is fermion, its mass has to be
mDM &400 eV (6.7)
This is known as the Tremaine-Gunn bound [40, 41]. Therefore, the interesting mass range for the dark matter candidate is of keV scale and above. A sterile neutrino, say N1, with a mass much smaller that the electron mass M1 me
decays into 3 active neutrinos N → 3ν and a neutrino with a photon N → νγ with the corresponding decay widths [42,43]:
ΓN→3ν= [5·1018s]−1 M1 1 keV 5 θ21 (6.8) ΓN→νγ = [2·1021s] −1 M1 1 keV 5 θ21 (6.9)
The first decay mode is dominant and therefore determines the lifetime of the sterile neutrino. The requirement for N1 to be stable on the 1017s timescale (the
age of the Universe) leads to a conservative constraint
θ12.50 1 keV M1
5
(6.10) The second decay mode leads to a narrow line E = M1
2 in the X-ray flux from
6.2 Constrains on sterile neutrino parameters 29
45], however, the origin of this line is still a matter of discussion. Nevertheless, nonobservation of more evident lines has put a stronger constraint:
θ12.1.8·10−5 1 keV M1
5
(6.11) This constraint implies that the dark matter candidate N1 cannot contribute
significantly to the active neutrino mixing mass for M1 & 2 keV. It can be easily
shown by substituting this into (6.6)
θ21M1.10−4eV 2 keV M1 4 =0.02 q ∆m2 atm 2 keV M1 4 (6.12) and taking into account strong dependence on M1. This proves the necessity of
introducing a dark matter sterile neutrino N1independently from the other two
N2, N3, which in this context are referred to as the heavy sterile neutrinos.
Al-though N1couples to matter very feebly, it can be produced in weak processes in
the primordial plasma in sufficient amounts to explain the observed dark matter abundance; the expression for the abundance is given by [46]:
ΩNh2 ∼0.1 θ12 10−8 MI 1 keV 2 (6.13)
6.2.2
Heavy sterile neutrinos N
2, N
3In addition to neutrino oscillation, the heavy sterile neutrinos N2, N3can explain
baryon asymmetry [47]. For this, their mass should lie in the range 150 MeV < M2,3 <100 GeV. Below the lower bound, Big Bang nucleosynthesis is affected; in
other words, there is a constraint on their lifetime:
τ <τBBN ≈0.1 s (6.14)
The upper bound comes from the requirement that N2, N3 cannot enter
equilib-rium before Tsph = 100 GeV, otherwise the third Sakharov condition will not be
satisfied and sphalerons will wash out any asymmetry.
First of all, sterile neutrinos clearly violate lepton number conservation due to their Majorana mass. Second, the mixing matrix is an additional source of CP violation. However, it is not seen in the tree-level processes, while the higher orders are suppressed by extra powers of the Yukawa couplings, making CP vi-olation extremely inefficient. This can be avoided assuming mass degeneracy ∆M23 M2, M3, which leads to oscillations between N2and N3[48]. This
mech-anism can produce large lepton asymmetry, which is then to be converted into baryon asymmetry by sphalerons. The calculation of the resulting ηB is
0.1
0.5
1
5
10
50
10
-1410
-1210
-1010
-8M
1, keV
θ
1 2X-ray
constraint
Tremaine
-Gunn
bound
3.5keV
line
Figure 6.1:Constraints on N1parameters. The X-ray constraint and the 3.5 keV line data
are taken from [50].
Figure 6.2: Constraints on N2 and N3 parameters. The red bound is obtained from the
requirement not to spoil the Big Bang nucleosynthesis (BBN). Below the seesaw bound active neutrino masses become smaller than
q ∆m2
atm. BAU stands for the constraint from
Chapter
7
Data analysis
7.1
Accelerator experiment constraints
7.1.1
SHiP
The Search for Hidden Particles (SHiP) experiment [1,2] is a promising example of the future generation of intensity frontier experiments. It was expected that to-gether with the Higgs boson, the Large Hadron Collider will discover a multitude of new particles predicted in a number of theories, such as supersymmetry. How-ever, nothing was found. Under the circumstances, it becomes plausible that new particles are not seen not due to their large mass, but because they might inter-act with matter very feebly. The corresponding parameter space of such particles remains unexplored. Instead of increasing the energy of collisions, which is the main aim of energy frontier experiments, to probe the parameter space of feebly in-teracting particles, one needs to increase the number of events. It can be achieved by increasing the intensity of the beams in experiments, and this is what the in-tensity frontier stands for.
SHiP is based on the Super Proton Synchrotron that accelerates protons to 400 GeV. The proton beam will collide with a Molybdenum and Tungsten target, producing a large number of hadrons. When decaying, they can produce a tiny amount of new long-lived particles. The main idea is as follows: if all the Stan-dard Model particles are removed and a large volume is placed next to the target, long-lived particles can decay within the volume and the decay products can be registered.
The SHiP facility contains of a target, a hadron absorber for pions and kaons, a muon shield to remove a large flux of muons produced in pion decay, hidden particle decay volume and detectors. The pressure in the decay volume is main-tained sufficiently low (10−6bar) to reduce background events, namely, neutrino scattering. If screening is effective enough to ensure that the expected number of events in the decay volume is close to zero, anything reaching the detector becomes a signal of new physics.
We use the SHiP detector in our work just as a relevant example; the follow-ing analysis can be easily applied to other experiments, where it is possible to distinguish different decay modes of new long-lived particles.
Figure 7.1:Overview of the experimental and detector area for SHiP. Credits: [2]
7.1.2
Description of the method
In this section, we consider a single HNL. From the phenomenological point of view, it has only four parameters: its mass M and three real mixing angles Ue2, U2µ,
Uτ2, where Uα2 = |θα|2. For simplicity, we assume that it is possible to determine
independently the mass of HNL M from kinematics. For qualitative analysis, we choose M = 1 GeV. For this value of mass, a sufficient number of decay channels are open. On the other hand, it still possible to describe decays into mesons perturbatively.
The SHiP detector can distinguish charged particles and photons, providing a set of results for different decay modes. HNL decays through the Fermi interac-tion, therefore has specific relations between different decay modes. We use the results of the comprehensive analysis given in [51].
Figure 7.2:HNL decays through a) charged and b) neutral currents. Credits: [51]
The decay width of a heavy lepton into decay products, denoted as Yα, takes
the next form:
Γ(N →Yα) =U
2
7.1 Accelerator experiment constraints 33
where α is the type of active neutrino in which N has oscillated. In other words, the decay products Yαin total have nonzero α-lepton number. The factorΓ(Nα →
Yα)is just the decay width of active neutrino να that assumed to have mass M =
MN. These factors appear to depend only on the well-known Fermi coupling
constant GF, meson form factors for hadronic decays, and on kinematics (i.e.,
masses of particles). We consider Γ(Nα → Yα) just as a set of constants. The
explicit expressions used in the thesis are given in Appendix A, the numerical values can be found in TableA.2.
The functional dependence of branching ratios on the parameters of HNL turns out to be very simple:
Br(N →Yα) =
Uα2ΓYα
U2
eΓe+Uµ2Γµ+Uτ2Γτ
(7.2) where we suppressedΓ(Nα →Yα)to justΓYα and denoted
Γα =
∑
Yα
Γ(Nα →Yα) (7.3)
where the sum is taken over all channels with lepton number α. Instead of Uα2we define new parameters: U2=Ue2+Uµ2+Uτ2and
Xe = U 2 e U2, Xµ = U2µ U2, Xτ =1−Xe−Xµ = Uτ2 U2 (7.4)
It is easy to notice that the branching ratios do not depend on the U2:
Br(N →Yα) =
XαΓYα
XeΓe+XµΓµ+XτΓτ
(7.5) and therefore are functions of only two independent parameters Xe and Xµ.
De-termination of these parameters can be performed just using experimental data for decay branching ratio, independently on the absolute scale of U2.
Once Xe and Xµ are determined, one can estimate U2 in order to obtain the
total event number Nevents as observed in the experiment. Assuming that this
quantity is a random variable with the Poisson distribution function, one can choose the interval of values for the appropriate confidence level and estimate U2 for this interval. We show the dependence on U2using the data from [52]. For not very large Neventsthere are two solutions for U2; in our case M =1 GeV the larger
value of mixing angle is ruled out by previous experiments. For sufficiently large Nevents one can use Gauss distribution approximation
confidence level nσ : Nevents±n √
Nevents
In general, there are more scrupulous analyses of the confidence intervals for the Poisson distribution, e.g. [53]. We do not deepen into the question further. We stated the method to reconstruct U2 and hereafter we consider Nevents as a free
parameter which replaces U2, since we are mainly interested in the determination of Xe, Xµ.
X
e
/X
μ
=1
X
τ
=1
10
-910
-710
-510
-310
1000
10
510
710
910
U
2N
eventsU
e/ μ 2lim
U
τ 2lim
.
Figure 7.3: The total event number as a function of U2 for pure mixings. Xe = 1 and
Xµ =1 cases are shown as a single line. The limits on Ue2, Uµ2, U
2
τ are taken from [54]. The
total number of events is bounded by 106(109for pure τ-mixing).
We assume that the SHiP detector distinguishes different charged particles: e, µ, π±. In addition, we conservatively assume that it can detect an appearance of hard photons, without specifying the number and energy. The considered 10 decay modes (and one invisible) for HNL with the mass 1 GeV are listed below, together with the corresponding processes:
• Clear modes: 1. ee : N → (νe+νµ+ντ)ee 2. µµ: N → (νe+νµ+ντ)µµ 3. eµ : N → (νe+νµ)eµ 4. eπ : N →eπ 5. µπ : N →µπ 6. π+π− : N →να(ρ →π + π−) • Modes with photons:
7. eπ photons : N →e(ρ →ππ0) 8. µπ photons : N →µ(ρ→ππ0) 9. π−π+photons : N →να(η 27% → . . .) N →να(ω 90% → . . .) 10. photons : (N →να(π0 →2γ) N →να(η 72% → . . .) N →να(ω 8% →. . .) • Invisible mode: N →νανβ¯νβ
7.1 Accelerator experiment constraints 35
The algorithm is the following:
1. We set Xe, Xµ and the number of expected HNLs ¯N. For these parameters
we compute the expected number of visible decay modes: ¯
Ni = N Br¯ i(Xe, Xµ, Xτ) (7.6)
i ∈ {ee, µµ, eµ, eπ, µπ, π+π−, eπ ph., µπ ph., π+π−ph., ph.} Here, for example,
Br1 =
XeΓ(Ne →νeee) +XµΓ(Nµ →νµee) +XτΓ(Nτ →ντee)
XeΓe+XµΓµ+XτΓτ
(7.7) and similarly for the other channels. HNL as a Majorana particle can decay into charge conjugated products. However, taking this into account leads only to a total factor 2, which cancels in the expression for branching ratio. Therefore, we do not consider charge conjugated channels.
2. To simulate the results of the experiment, we take a set of random variables Ni with Poisson distribution and the corresponding mean values ¯Ni. A
re-alization of this set is treated as a possible output.
Now the question is how to reconstruct the underlying parameters of the model that would leads to similar results. For this we apply Pearson’s chi squared test. Namely, we compute the function
χ2(¯n, xe, xµ) =
∑
i Ni− ¯nBri(xe, xµ, 1−xe−xµ) 2 ¯nBri(xe, xµ, 1−xe−xµ) (7.8) where xe, xµ, and ¯n are our guesses for Xe, Xµ, and ¯N correspondingly. Sincewe are not interested in constraint of ¯n, we define: ∆χ2(x
e, xµ) = χ2n(xe, xµ) −χ2min (7.9)
where χ2min is the minimal value of χ2 with respect to all variables and χ2(xe, xµ)is the minimal value with respect to ¯n for the given xe, xµ.
According to the chi squared test, two-parametric ∆χ2(xe, xµ) is bounded
from above by
CL 1σ : ∆χ2<2.30 CL 2σ : ∆χ2<6.18 3. For the best fit parameters xbe, xµb, ¯n
b we apply a separate chi squared test. In
this point, χ2minis just a sum of 10 independent random variables and there-fore has the 10-parametric chi squared distribution. Using the cumulative distribution function, we can compute the probability not to exceed χ2min to answer is it in principle correct to describe the results with the model of HNL.
1σ CL
2σ CL
X
e=0.4
X
μ
=0.4
x
e
b
=0.45
x
μ
b
=0.31
0.1
0.3
0.5
0.7
0.9
0.1
0.3
0.5
0.7
0.9
x
e
x
μ
N
obs
=79
n
b
=97.68
Prob =98%
Figure 7.4: A typical result of the algorithm described in the text for ¯N = 100 HNLs. Xe and Xµ are the initial parameters for the simulation. Nobs is the number of the
ob-served events, i.e., without the invisible decays. xbe, xbµ (the green point), and ¯nbare the best-fit values. The probability limits for 1σ and 2σ are Prob . 32% and Prob . 4.6% correspondingly. More examples are given in AppendixB. The minimization algorithm can fail and the obtained results may be not correct. This is a minor problem and can be avoided by increasing the computational precision; we aim to show the applicability of the method itself.
7.2 Neutrino oscillation constraints 37
7.2
Neutrino oscillation constraints
The known values of the neutrino mixing parameters provide another constraint on possible Xe, Xµ. It comes from the Casas-Ibara parametrization [55] of the
Yukawa couplings FαI of νMSM. We rewrite the seesaw mechanism (6.3):
mν =
v2 2 FM
−1FT (7.10)
where we have defined 3×3 matrices of Yukawa couplings F, sterile neutrino mass matrix M, and active neutrino mass matrix mν. The latter can be
diagonal-ized with the PNMS matrix U; we denote the diagonaldiagonal-ized active mass matrix as Dν. At the same time, M ≡DM
Dν =UTmνU = v2 2 U TFD−1 MFTU 1= v 2 2 D −1/2 ν U TFD−1/2 M D −1/2 M FUD −1/2 ν (7.11)
From this expression it follows that
R= √v 2D −1/2 M FTUD −1/2 ν , R TR=1 (7.12)
is a complex-valued orthogonal matrix. The most general expression for F is given by FT = √ 2 v D 1/2 M RD1/2ν U † (7.13)
with arbitrary orthogonal R. Mixing angles for the first HNL are Uα2= v 2|F α1|2 2M12 = 1 M1 (RD 1/2 ν U †) 1α 2 (7.14) We are not interested in the absolute scale, therefore, we can choose any value of M1. Since R is orthogonal, the set of R contains all basis rotation matrices,
therefore it does not matter which HNL to choose in the expression. Considering all R, we can find the set of possible values of Xe, Xµ consistent with neutrino
oscillations. We take the smallest neutrino mass mν = 10−4eV, which does not
contradict to the possible contribution of the DM sterile neutrino. ∆m2
sol ∆m2atm δ
NH (7.39±0.21)10−5eV2 (2.525±0.033)10−3eV2 (215±36)◦ IH (7.39±0.21)10−5eV2 (2.512±0.033)10−3eV2 (284±28)◦
The parametrization of PNMS matrix is given by (4.18). For active neutrino masses we use DNHν = mν 0 0 0 q m2 ν+∆m2sol 0 0 0 q m2 ν+∆m 2 sol+∆m2atm (7.15)
NH (33.82±0.77)◦ (49.6±1.1)◦ (8.61±0.13)◦ IH (33.82±0.77)◦ (49.8±1.1)◦ (8.65±0.13)◦
Table 7.1: The neutrino oscillation parameters for the normal (NH) and inverted (IH) hierarchy [34]. DIHν = q m2 ν+∆m 2 atm 0 0 0 q m2 ν+∆m2sol+∆m2atm 0 0 0 mν (7.16)
According to (7.14), we need only values R1i. From the explicit expression of R as a multiplication of three rotations R23R13R12 (similarly to CKM and PNMS
parametrisation), it can be seen that we actually need only two complex angles ω12and ω13 for parametrization:
(R)1i = (cos ω12cos ω13, sin ω12cos ω13, sin ω13) (7.17)
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
ex
μFigure 7.5:Points on Xe, Xµplane for the normal hierarchy. The real and imaginary parts
7.2 Neutrino oscillation constraints 39
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
ex
μFigure 7.6:Points on Xe, Xµplane for the inverted hierarchy.
If we include uncertainties of the neutrino parameters, these ellipses become larger. For neutrino oscillations, we define χ2(xe, xµ)as
χ2 = (θ12− hθ12i)
2
σθ2
12
+. . . (7.18)
where hθ12i, σθ12 are the mean value and the uncertainty of θ12 correspondingly
and the sum is over θ12, θ23, θ13, δ, ∆m2sol, and ∆m2atm with the same notation.
The values θ12, . . . are taken such that to minimize χ2over all the parameters that
provide access to the point (xe, xµ). Since this has large computational cost, we
0 2.5 5.0 7.5 10.0 12.5
Figure 7.7: χ2 density plot within 2σ CL (χ2 < 12.59) for neutrino oscillations, normal hierarchy. The green line is an analogous curve taken from [54].
0 2.5 5.0 7.5 10.0 12.5
7.3 Combined constraints 41
7.3
Combined constraints
To consider correctly both constraints, we use the next criterion. Let F1, F2 be
the cumulative distribution functions of χ2 for SHiP and neutrino oscillations correspondingly. The 2σ test on χ2in fact states that
F(χ2) <95.4% →1−F(χ2) >4.6% (7.19) We interpret a point xe, xµas a valid one if it satisfies the requirement
[1−F1(∆χ21(xe, xµ))] × [1−F2(χ22(xe, xµ))] >4.6% (7.20)
This represents the well-known rule of probability multiplication: the probability to get both χ2 large should to be small. The best-fit point is evidently the point that maximize the left part of (7.20).
We show the combined constraint only for the normal hierarchy, the case of inverted hierarchy is completely analogous. One can see that in general neutrino oscillations just cut off the parameter space - everything out the bound is forbid-den. In particular, if the pure SHiP bound lies completely inside the oscillation bound, the combined constraint is just the same at the SHiP one.
1σ CL
2σ CL
X
e
=0.3
X
μ
=0.3
x
e
b
, x
μ
b
SHiP 1σ
SHiP 2σ
Osc. NH
N
=100
N
obs
=85
n
=106.14
Prob =93%
Figure 7.9: Limits on the parameter space taking into account neutrino oscillations. The combined 1σ and 2σ bounds and pure bounds from neutrino oscillation and SHiP are shown. The resulting constraint appears to be the intersection of the two bounds. More examples are given in AppendixB
Appendix
A
HNL decay widths
The expressions for decay width are taken from [51]. Here GF is the Fermi
cou-pling constant, θW is the Weinberg angle, Vudis the element of the CKM matrix,
MN is the HNL mass, and xi = MmNi .
• Leptonic decays: Γ(Nα →νανβ¯νβ) = (1+δαβ) G2FM5N 768π3 (A.1) Γ(Ne →eνµµ¯) ≈ G2FM5N 192π3 h 1−8x2µ+8x6µ−x8µ−12x4µln x2µi (A.2) Γ(Nα →ναlβ¯lβ) = G2FM5N 192π3 C1 (1−14x2l −2x4l −12x6l) q 1−4x2l+ +12x4l(x4l −1)L(xl) +4C2 x2l(2+10x2l −12x4l) q 1−4x2l+ +6x4l(1−2x2l +2x4l)L(xl) (A.3) • Hadronic decays Γ(Nα →l − α π +) = G2F|Vud|2M3N 16π f 2 π[(1−x 2 l)2−x2π(1+x 2 l)] q λ(1, x2π, x2l) (A.4) Γ(Nα →l − α ρ +) = G2F|Vud|2M3N 16πm2 ρ g2ρ[(1−x2l)2+x2ρ(1+x2l) −2x4ρ] q λ(1, x2ρ, x2l) (A.5) Γ(Nα →ναh0P) = G2FM3N 32π f 2 h(1−x2h)2, h0P = {π0, η} (A.6) Γ(Nα →ναh0V) = G2FM3N 32πm2hg 2 hκ2h(1+2x2h)(1−x2h)2, h0V = {ρ0, ω} (A.7)
L(x) = ln1−3x
2− (1−x2)√1−4x2
x2(1+√1−4x2) (A.8)
λ(a, b, c) = a2+b2+c2−2ab−2bc−2ac (A.9) while the values of the numerous parameters are given in TableA.1[51,56]
C1 C2
β6=α 14(1−4 sin2θW+8 sin4θW) 12sin2θW(2 sin2θW−1)
β=α 14(1+4 sin2θW+8 sin4θW) 12sin2θW(2 sin2θW+1)
Meson m[MeV] f [MeV] g[GeV2] κ
π 139.6 (±) 135.0(0) 130.2 — — η 548 81.7 — — ρ 770 — 0.162 1−2 sin2θW ω 782 — 0.153 43sin2θW
Table A.1:The parameters used in the expressions of the decay widths
1015Γ [GeV] ee µµ eµ eπ µπ Ne 13.4 2.41 21.0 41.8 0 Nµ 2.87 11.26 21.0 0 41.8 Nτ 2.87 2.41 0 0 0 π+π− eπ ph. µπ ph. π+π− ph. ph. Ne 5.99 17.8 0 2.70 25.4 Nµ 5.99 0 15.5 2.70 25.4 Nτ 5.99 0 0 2.70 25.4
Table A.2:Decay widths of the visible decay modes for MN = 1 GeV. Nα corresponds to
pure mixing Uα2 = 1. The invisible decay width isΓ(Nα → 3ν) = 22.9. In our analysis
