Thesis
Reference
Measurement of the increase in phase space density of a muon beam through ionization cooling
DRIELSMA, François
Abstract
The Muon Ionization Cooling Experiment (MICE) collaboration seeks to demonstrate the feasibility of ionization cooling, the technique by which it is proposed to cool the muon beam at a future neutrino factory or muon collider. The position and momentum reconstruction of individual muons in the MICE trackers allows for the development of alternative figures of merit in addition to beam emittance. Contraction of the phase space volume occupied by a fraction of the sample, or equivalently the increase in phase space density at its core, is an unequivocal cooling signature. Single-particle amplitude and nonparametric statistics provide reliable methods to estimate the phase space density function. These techniques are robust to transmission losses and nonlinearities, making them optimally suited to perform quantitative measurements in MICE. These novel methods were developed for this thesis and used for the first demonstration of muon ionization cooling through a 65 mm-thick lithium hydride absorber.
DRIELSMA, François. Measurement of the increase in phase space density of a muon beam through ionization cooling . Thèse de doctorat : Univ. Genève, 2018, no. Sc. 5249
DOI : 10.13097/archive-ouverte/unige:114100 URN : urn:nbn:ch:unige-1141006
Available at:
http://archive-ouverte.unige.ch/unige:114100
Disclaimer: layout of this document may differ from the published version.
Measurement of the increase
in phase space density of a muon beam
through ionization cooling
Département de Physique Nucléaire et Corpusculaire
FACULTÉ DES SCIENCES Professeur Alain Blondel
Measurement of the increase in phase space density of a muon beam through ionization cooling
THÈSE
présentée à la Faculté des Sciences de l’Université de Genève pour obtenir le grade de docteur ès Sciences, mention Physique
par
François Drielsma
Liège (Belgique)de
Thèse n◦ 5249
GENÈVE
Beam-based detector alignment in the MICE muon beam line,arXiv:1805.06623 First observation of ionization cooling, à paraître
experiment. Experiment is the sole judge of scientific “truth”.
– Richard P. Feynman
Le travail décrit dans cette thèse de doctorat s’inscrit dans la continuité de plus d’un siècle de progrès ininterrompus en physique des particules. Durant toute la durée du XXe siècle, le développement de théories en physique des hautes énergies a suivi le développement de nouvelles technologies. C’est l’amélioration constante des instruments utilisés pour sonder l’infiniment petit qui a permis aux scientifiques d’établir les lois qui gouvernent les particules fondamentales et leurs interactions.
Le Modèle Standard s’impose aujourd’hui comme la théorie la plus pré- dictive et unifiée de l’Histoire de la physique. Son intégration de la physique quantique des champs décrit simultanément les forces électromagnétiques faibles et fortes, au sein d’un lagrangien unique. Cette théorie a été utilisée pour faire une myriade de prédictions, dont la plupart n’ont, à ce jour, toujours pas été mises en défaut. Malgré les nombreux succès indéniables de ce modèle, le neutrino n’y a pas encore trouvé de place définitive. Le neutrino est une particule indivisible neutre qui fut initialement théorisée comme ayant une masse nulle. De nombreuses expériences ont depuis démontré que les neutrinos oscillent entre leurs différents états fondamentaux et par conséquent doivent avoir une masse, aussi infime soit-elle. Ces mesures ont été utilisées pour établir le profil de cetteparticule fantôme mais plusieurs de ses caractéristiques nous échappent toujours.
La Neutrino Factory est une expérience qui a été proposée afin de répondre à toutes les questions qui subsistent au sujet du neutrino. Cette machine repose sur l’utilisation de muons, des particules qui partagent toutes les caractéristiques d’un électron pour une masse deux-cents fois supérieure. Les muons sont accélérés et accumulés dans un anneau de stockage, où chacun d’entre eux se désintègre en un électron et deux neutrinos. Cette technique permet d’obtenir un niveau de pureté au sein du faisceau de neutrino qui n’a jamais été atteint auparavant. La connaissance du faisceau, alliée à une haute luminosité, permet d’obtenir des mesures d’oscillation suffisamment
alliant les hautes énergies du Large Hadron Collider à la précision inégalée du Large Electron-Positron Collider.
La principale difficulté d’un accélérateur de muons réside dans la pro- duction d’un faisceau de petite taille. L’espace des phases occupé par les muons après leur production est trop étendu pour que le faisceau puisse être injecté sans perte majeure d’intensité. La Muon Ionization Cooling Experi- ment (MICE) a été développée afin de démontrer la faisabilité d’un nouveau système de refroidissement par ionisation. Il s’agit de la seule technique viable pour refroidir les muons instables, dont le temps de vie est de 2.2µs.
Les particules traversent un absorbeur où elles perdent de l’énergie avant que des cavités radiofréquence restaurent la composante longitudinale de l’impulsion. L’effet cummulé est la réduction de la divergence du faisceau.
Pour garantir la pureté de l’échantillon de muons analysés, MICE s’est dotée d’un arsenal de détecteurs dédiés à l’identification du type de particule.
L’Electron-Muon Ranger (EMR) est un calorimètre qui se charge du rejet des muons qui se sont désintégrés dans l’expérience, en identifiant la signature des électrons filles. Une analyse est développée pour caractériser l’efficacité du détecteur et démontrer qu’il permet d’atteindre le niveau de pureté requis.
Un niveau de contamination inférieur à 0.1 % est obtenu.
Afin de mesurer un signal de refroidissement clair, MICE est également équipée de deuxtraceurs qui mesurent la position et l’impulsion de toutes les particules avant et après avoir traversés l’absorbeur. La position de ces détecteurs est mesurée à l’aide du faisceau de muons, à 100µm et 0.002° près.
Le cœur de cette thèse consiste à démontrer la réduction du volume de l’espace transverse des phases, occupé par les muons, dû au refroidissement par ionisation. Des faisceaux de tailles variables sont sélectionnés et analysés en amont et en aval d’un absorbeur de 65 mm d’hydrure de lithium. Deux approches différentes sont utilisées pour mesurer un signal de refroidissement sans équivoque. La première technique consiste à reconstruire l’amplitude transverse de chacune des particules, c’est à dire leur distance par rapport au centre du faisceau. Si le nombre de particules de basses amplitudes augmente, c’est un signal clair de réduction du volume de l’espace des phases. La seconde méthode utilise les statistiques non paramétriques afin de mesurer la densité de probabilité de l’espace des phases en tout point. Cela permet de démontrer une augmentation de la densité de muons au centre du faisceau, une autre manifestation évidente du refroidissement. Une mesure quantitative est produite dans une variété de configurations. Le signal le plus significatif observé est une contraction de−15.86±1.53 (stat)±0.93 (syst) % du volume occupé par un faisceau de muons de 140 MeV/c. Ces résultats prouvent la faisabilité du refroidissement par ionisation d’un faisceau de muon, une technologie essentiel au développement de futurs accélérateurs.
Résumé i
1 Introduction 1
2 State of the art 3
2.1 Standard Model of particle physics . . . 3
2.1.1 Brief history . . . 4
2.1.2 Theoretical framework . . . 7
2.1.3 Physics beyond the Standard Model . . . 11
2.2 Neutrinos . . . 13
2.2.1 Oscillations . . . 13
2.2.2 Masses . . . 18
2.2.3 Absolute mass . . . 20
2.2.4 Hierarchy and CP-violating phase . . . 21
2.3 Neutrino factory . . . 24
2.3.1 Facility design . . . 24
2.3.2 Physics . . . 26
2.4 Muon Collider . . . 28
2.4.1 Facility design . . . 29
2.4.2 Physics . . . 31
3 Muon Ionization Cooling Experiment 33 3.1 Intent . . . 33
3.2 Beam cooling . . . 34
3.2.1 Equations of motion in trace space . . . 34
3.2.2 Twiss parameters . . . 35
3.2.3 Transverse phase space . . . 36
3.2.4 Transverse amplitude . . . 37
3.2.5 Traditional cooling techniques . . . 38
3.2.6 Ionization cooling . . . 41
3.3 Beam line . . . 43
3.3.1 Production and transport . . . 43
3.3.2 Configurations . . . 46
3.4 Experimental apparatus . . . 47
3.4.1 Magnetic Channel . . . 47
3.4.2 Scintillating fibre trackers . . . 49
3.4.3 Particle Identification detectors . . . 50
4 Electron-Muon Ranger 53 4.1 Function . . . 53
4.2 Construction . . . 55
4.2.1 Design . . . 55
4.2.2 Readout chain . . . 57
4.2.3 Internal calibration system . . . 59
4.3 Time-over-threshold response function . . . 60
4.3.1 Modelling . . . 60
4.3.2 Measurement . . . 61
4.4 Calibration . . . 63
4.4.1 Data acquisition . . . 63
4.4.2 Method . . . 63
4.5 MICE beam data . . . 65
4.5.1 Acquisition . . . 65
4.5.2 Upstream particle identification . . . 66
4.5.3 Momentum loss in TOF2 and the KL . . . 68
4.6 Reconstruction software . . . 71
4.6.1 Unpacking . . . 72
4.6.2 Space points . . . 75
4.6.3 Track . . . 77
4.7 Momentum resolution . . . 81
4.7.1 Simulations . . . 81
4.7.2 Measurement . . . 83
4.7.3 Approximation to the CSDA range . . . 85
4.8 Total charge response . . . 87
4.8.1 Birks’ law . . . 87
4.8.2 Proportionality constant . . . 89
4.8.3 Kinetic energy resolution . . . 90
4.9 Decay matching . . . 91
4.9.1 Spatial and time structure . . . 91
4.9.2 Muon lifetime . . . 94
4.9.3 Decay topology . . . 95
4.10 Electron identification variables . . . 98
4.10.1 Plane density . . . 98
4.10.2 Shower spread . . . 99
4.11 Electron rejection efficiency . . . 101
4.11.1 Plane density test . . . 102
4.11.2 Spread test . . . 102
4.11.3 Multivariate test . . . 104
4.11.4 Momentum dependence . . . 105
4.12 Pion identification . . . 106
4.12.1 Stand-alone . . . 106
4.12.2 Separate velocity measurement . . . 107
5 Beam-based detector alignment 111 5.1 Surveys . . . 111
5.2 Analysis method . . . 113
5.2.1 Module placement . . . 113
5.2.2 Sought after measurements . . . 114
5.3 Sample selection . . . 116
5.3.1 Sampling bias . . . 117
5.3.2 Criterion . . . 122
5.3.3 Outliers . . . 124
5.4 Alignment of a Monte Carlo sample . . . 125
5.4.1 Trackers internal alignment . . . 125
5.4.2 Implementation of the criterion . . . 126
5.4.3 Fitting algorithm . . . 130
5.4.4 Propagation . . . 131
5.5 Alignment of the data . . . 134
5.5.1 Single user cycle . . . 134
5.5.2 Evolution . . . 137
6 Phase space density evolution 139 6.1 Beam optics . . . 139
6.1.1 Magnetic channel settings . . . 140
6.1.2 Linear optics . . . 140
6.2 Model . . . 141
6.2.1 Expected emittance change . . . 142
6.2.2 Effect of apertures . . . 145
6.2.3 Effect of nonlinearities . . . 146
6.3 Simulation . . . 147
6.3.1 Structure . . . 147
6.3.2 Field reproduction . . . 147
6.3.3 Tracker efficiency and resolution . . . 149
6.3.4 LowpT patch . . . 151
6.4 Beam selection . . . 154
6.4.1 Tracker tracks quality . . . 154
6.4.2 Muon tagging . . . 154
6.4.3 Final samples . . . 155
6.4.4 Beam profiles . . . 155
6.4.5 Optical functions . . . 161
6.5 RMS emittance . . . 161
6.5.1 Evolution . . . 161
6.6 Amplitude . . . 164
6.6.1 Generalization . . . 164
6.6.2 Poincaré sections . . . 165
6.6.3 Correction factors . . . 167
6.6.4 Uncertainties . . . 168
6.6.5 Distributions . . . 170
6.7 Summary statistics . . . 170
6.7.1 Definitions . . . 172
6.7.2 Uncertainties . . . 175
6.7.3 Evolution . . . 176
6.7.4 Summary . . . 178
7 Nonparametric density estimation 179 7.1 Bias–variance trade-off . . . 179
7.2 Histograms . . . 181
7.2.1 Formalism . . . 181
7.2.2 Optimal binning . . . 182
7.3 k-Nearest Neighbour . . . 184
7.3.1 Formalism . . . 184
7.3.2 Implementation . . . 186
7.4 Honeycomb . . . 186
7.4.1 Delaunay triangulation . . . 187
7.4.2 Voronoi tessellation . . . 189
7.4.3 Tessellation Density Estimators . . . 190
7.4.4 Penalised Centroidal Voronoi Tessellation . . . 191
7.4.5 Penalised Bootstrap Aggregation . . . 192
7.4.6 Bias and variance . . . 193
7.4.7 Implementation . . . 193
7.5 Contour volume . . . 195
7.5.1 Monte-Carlo method . . . 195
7.6 Choice of estimator . . . 196
7.6.1 Rate of convergence . . . 196
7.6.2 Robustness . . . 197
7.6.3 Contour resolution . . . 199
7.7 Density profiles . . . 200
7.7.1 Poincaré sections . . . 200
7.7.2 Theory . . . 200
7.7.3 Uncertainties . . . 204
7.7.4 Evolution . . . 206
7.8 Summary statistics . . . 206
7.8.1 Definitions . . . 206
7.8.2 Uncertainties . . . 208
7.8.3 Evolution . . . 208
7.8.4 Summary . . . 210
8 Conclusions 211 A Covariance matrix 213 A.1 Ellipsoid . . . 213
A.1.1 Volume . . . 214
A.1.2 Probability content . . . 214
A.2 Emittance . . . 215
A.2.1 Statistical uncertainty . . . 215
A.2.2 Measurement uncertainty . . . 216
A.3 Twiss parameters . . . 217
A.3.1 Statistical uncertainty . . . 217
A.3.2 Measurement uncertainty . . . 217
A.4 Amplitude . . . 217
A.4.1 Statistical Uncertainty . . . 218
A.4.2 Measurement uncertainty . . . 218
B Minimum chi-square polynomial estimation 219 B.1 Analytical minimisation . . . 219
B.2 Error propagation . . . 220
B.3 Practical example . . . 222
C Convex hull 223 C.1 Statistics . . . 224
D Distributions 225 D.1 Probability Density Function . . . 225
D.2 Contours . . . 228
E Lebesgue spaces 229 E.1 Definition . . . 229
E.2 Generalisedp-unitd-ball . . . 230
F Interpolation 231
F.1 Multilinear interpolation . . . 231 F.2 Simplex interpolation . . . 232
Acknowledgments 233
Bibliography 235
CHAPTER 1
Introduction
Almost ninety years have passed since Pauli first postulated the existence of the neutrino in order tosave the law of conservation of energy in his famous open letter to theradioactivepeople of the Tübingen meeting [1]. A little over twenty-five years later, Cowan and Reines published the definitive proof of its existence and a new era of particle physics was under way [2]. In Fermi’s theory of beta decay and the original formulation of the the Standard Model, the neutrinos were considered massless [3]. The observation of the solar electron neutrino flux by the Homestake experiment in the sixties challenged that notion by reconstructing a rate three times smaller than predicted [4].
Pontecorvo’s earlier suggestion that neutrinos may oscillate between their distinct flavour states – due to small but non-zero mass splittings – was then proposed as the most likely explanation for the discrepancy [5]. This theory was confirmed by multiple experiments that placed the neutrino at the fringe of the Standard Model and posed many additional questions regarding their absolute mass and their mixing parameters [6–8].
Despite the continuous international effort, several characteristics of thisghost particle are still unknown today. Its oscillation parameters have been measured to varying degrees of accuracy but its leptonic CP violating phase and its mass eigenstates ordering remain elusive [9–11]. The current resolution on the mixing matrix elements is much lower than in the quark sector and not sufficient to demonstrate unitarity at low energy [12,13]. The Neutrino Factory is a proposed state-of-the-art experiment that offers the opportunity to take accelerator neutrino physics into the precision era [14].
This machine offers to use a very high luminosity cooled muon storage ring
as a source for intense and exquisitely well understood neutrino beams [15].
The development of a Neutrino Factory also paves the way to the future of high energy circular lepton colliders. The Muon Collider is a proposed µ+µ− synchrotron based on the cooling and acceleration of muons produced from pion decays [16,17]. This accelerator would achieve the incomparable resolution of ane+e− collider and reach multi-TeV energies with a device the size of the Tevatron [18]. Muon collisions are a prime choice for the production of a compact Higgs factory, could be used to push the energy frontier and extend the current heavy neutrino searches [19, 20].
One of the main challenges associated with both facilities is the devel- opment of an operational front-end muon cooling channel [21]. A muon beam produced from the decay of pions does not naturally fit the acceptance of an accelerator without reducing its emittance. The international Muon Ionization Cooling Experiment (MICE) collaboration is set to experimentally demonstrate the viability of ionization cooling as a means to reducing the muon beam size [22]. It aims to observe a 10 % reduction in the muon beam emittance with a short section of a cooling channel [23]. The experiment has completed the acquisition of data in its Step IV configuration and is poised to produce the first muon ionization cooling measurement [24].
The current cooling section comprises a single absorber focus coil module.
The beam is sampled upstream and downstream by identical scintillating fibre trackers embedded in superconducting spectrometer solenoids. The muon purity is ensured by a comprehensive set of particle identification detectors.
The loss of a match coil in the downstream section of the experiment in September 2015 motivated the development of innovative cooling figures- of-merit. In a limited transmission environment, emittance is a misleading statistic as scraping produces apparent emittance reduction, without an underlying increase in phase space density.
This work aims to demonstrate the reduction of a muon beam’s transverse phase space volume in lithium hydride by using a robust particle amplitude reconstruction method and novel non-parametric techniques. Theses methods allow for the observation of an unequivocal cooling signature and for the quantification of the change in phase space volume.
CHAPTER 2
State of the art
The work of this doctorate thesis and its application to future research facilities is in the continuity of 120 years of uninterrupted progress in particle physics. Throughout the XXth century, the development of theories in high energy physics consistently followed the invention of new technologies. It is the ever improving machines available to probe the infinitesimal scale that allowed scientists to observe symmetries and establish models of the underlying laws that govern particles and their interactions.
This chapter sets the landscape of particle physics in which the Muon Ionization Cooling Experiment was proposed. It presents the current state- of-the-art in subatomic physics and the many successes and challenges of the Standard Model. Neutrino oscillations are discussed as a compelling example of physics beyond the Standard Model. Future muon-based facilities that could help push the boundaries of high energy physics are introduced.
2.1 Standard Model of particle physics
The Standard Model stands today as the most unified and successful theory ever postulated in the history of physics. Its implementation of quantum field theory conjointly accounts for the electromagnetic, weak and strong forces within a single fundamental Lagrangian. It has been used to make countless predictions throughout the XXth century, the large majority of which are yet to be faulted. A short review of the genesis and theoretical framework of the model are presented.
2.1.1 Brief history
In the fifth century BC, Democritus had already surmised that all things were constructed from a set of identical and fundamental building blocks that he then calledatoms, i.e. indivisible ones [25]. The instruments available at the time and in the many centuries that followed did not allow one to probe the constituents of matter to confirm or infer that conception of nature.
In the early XVIIth century, Galileo Galilei revolutionizes science as a whole, rejecting the ancient paradigm ofdevelopment-by-accumulation of accepted facts, and develops a set of rigorous methods based on the sole observation of nature [26]. A few decades later, Isaac Newton theorizes the concept of force in a mathematical model that successfully explains the motion of macroscopic objects, on Earth and in the solar system. He suggests the existence of a universal attractive force between massive objects that he calls gravity [27]. Electrodynamics, developed shortly after by many and synthesized by James Clerk Maxwell, initially fits perfectly into the framework of classical mechanics [28].
Throughout the XIXth century, physicists gradually establish the reality of atoms. The chemist John Dalton proposes his modern atomic theory that matter is composed of a limited set of atoms of defined mass, size and other properties. They combine to form chemical compounds and are rearranged through chemical reactions [29]. Sir Joseph John Thomson discovers the electron by observing its deflection in a cathode tube in the presence of an electric field. To explain the neutral charge of atoms, he suggests that electrons are uniformly distributed across the positive atom in the same way that raisins are spread around a plum pudding [30].
At the end of the century, some physicists are growing in confidence that they have reached a global understanding of the fundamental laws that govern our Universe. The many successes of the theory of gravitation, the laws of electromagnetism and thermodynamics lead some scientists, such as Albert A. Michelson, to believe that “most of the grand underlying principles have been firmly established and that further advances are to be sought chiefly in the rigorous application of these principles” [31]. However, as Karl Popper later puts it, “the game of science is, in principle, without end. He who decides one day that scientific statements do not call for any further test, and that they can be regarded as finally verified, retires from the game” [32].
At the turn of the XXth century, there is a consensus that Thomas Young’s interference experiment demonstrates that light behaves strictly as a wave [33]. In an attempt to find a law that governs the radiation spectrum of black bodies, Max Planck “sacrifices any of his previous convictions about physics” and suggests that light is in fact quantized [34]. Albert Einstein takes his idea very seriously and proposes the existence of a quantum of light, the photon, which behaves like a particle. He subsequently uses this
approach to explain the photoelectric effect [35]. Louis de Broglie takes the concept of wave-particle duality further and postulates in his PhD thesis that electrons and all matter have wave properties [36]. In 1926, Erwin Schrödinger publishes a series of revolutionary papers in which he derives the wave equation for time-independent systems and shows that it gives the observed energy levels for a hydrogen-like atom [37]. This confirms de Broglie’s postulate and suggests that the evolution of a subatomic system can be described by the propagation of complex wave functions.
Special relativity is conceptualized by Albert Einstein in the same time frame as quantum mechanics and poses an additional challenge [38]. Since photons have rest mass zero, they must travel at the speed of light, something that ordinary quantum mechanics cannot account for. Dirac postulates in 1927 that particles are quantized excited states of underlying fields capable of interacting with each other. Quantum electrodynamics (QED) describes the photon as a realization of the electromagnetic field. The interaction between two charged particles can be thought of as the exchange of photons between them [39]. This is historically the first gauge theory in particle physics, a form of field theory with a Lagrangian invariant under Lorentz transformations, on which the entire standard model is constructed.
The initial limitation of the formalism resides in perturbation theory, on which it relies to make prediction. The first order development of QED gives approximate explanations of observations but higher order corrections are found to be divergent. Renormalization is invented independently by two groups, to deal with infinities, and showed to be equivalent by Freeman Dyson.
Renormalization removes the concept of fundamental charge and mass and replaces it by experimental constants. The strength of the coupling between electrons and photons is considered screened by a cloud of electron-positron pairs and its value increases with the energy used to probe it [40].
The nuclear model of the atom is developed by Rutherford who realizes that the scattering of alpha particles on gold atoms suggests that most of the positive charge is concentrated in a very small fraction of the atom.
The neutron is introduced to explain the existence of isotopes, i.e. atoms of identical charge but different mass [41]. After the undeniable success of QED, physicists start to search for similar gauge theories that would explain the behaviour of the strong interaction that keep the protons together inside the nucleus and the weak interaction that causes the neutron to decay.
The observation of cosmic rays in photographic emulsions and cloud chambers in the thirties and forties produces a plethora of particles with distinct properties [42]. The large variety of those particles, initially organised in terms ofisospinas an analogy with electrons, suggests a more fundamental symmetry. Murray Gell-Mann points out in 1963 that this pattern may be understood if all the strongly interacting particles are composites [43]. At the time, threequarksnamedup,downandstrangewere sufficient to explain
the observed cosmic rays. In 1968, deep inelastic scattering experiments at the Stanford Linear Accelerator Center demonstrate that the proton contains point-like constituents and is not an elementary particle [44,45].
The charm, bottom and top quarks have now been observed at various American laboratories and complete the three generations of building blocks that account for all known composite particles, the hadrons [46–50].
µ+ π+
e+
•
• •
•
•
•
•
π− • π+
K0 K+
K¯0 K−
π0 η s= 1
s= 0
s=−1
q=−1 q= 0 q= 1
Figure 2.1: (Left) Pion decay chain π+ → νµ+ (µ+ → e++ ¯νµ+νe) in the liquid hydrogen 2m Bubble Chamber at CERN. (Right) Gell-Mann’s ‘Eightfold Way’
representation of spin-0 mesons as a function of strangeness,s, and charge,q.
The existence of hadrons composed of three identical quarks, e.g. ∆++, was troublesome as it violated Pauli’s exclusion principle. Acolour charge was introduced to differentiate their quantum states and was the basis for the introduction of eight bicolour spin-one particles, the gluons, to mediate the strong force between quarks. The new symmetries introduced were adapted to the framework of non-abelian gauge theories developed in 1953 by Yang and Mills to form quantum chromodynamics, the first important piece of the standard model puzzle [51].
On the weak front, the observation of the continuous beta decay spectrum of atoms by Lise Meitner and Otto Hahn pushes Wolfgang Pauli to postulate the existence of another particle involved in the interaction [1, 52]. The newly postulated neutrino enters as the final fermion of Enrico Fermi’s model that applies the quantum field framework to the theory of weak interaction [3]. The interaction has the interesting property that it only seems to involve left-handed fermions and right-handed antifermions and as such violates parity conservation. The comparatively much longer life time of the neutron to the π0 suggests a much smaller coupling constant than QED. The suppressed scale of the coupling is explained by the large mass scale of mediating bosons, calledW± andZ0. Schwinger suggests they might also be the gauge bosons of a gauge theory, and indeed that there might be a unified theory of weak and electromagnetic interactions [53].
The electroweak theory is the other central piece of the standard model.
The existence of theZ0 boson is confirmed by the observation of neutrino
neutral current interactions in the Gargamelle bubble chamber at CERN, a major step towards the verification of the theory [54]. The UA1 and UA2 experiments on the Spp¯S proton-antiproton collider at CERN definitively observe of theW± andZ0 bosons [55,56].
IP•
ν
µγ
γ e− e− e−
e+
e+ e+
Figure 2.2: (Left) Beta-decay spectrum of 210Bi [57]. (Right) Neutrino neutral current interaction,νµ+e−→νµ+e−, in CERN’s Gargamelle bubble chamber.
In the sixties, the final loophole resides in the electroweak unification.
The underlying symmetry must somehow be spontaneously broken to justify the difference between the massive weak bosons and the massless photon.
The so-called Brout-Englert-Higgs mechanism was developed simultaneously by three groups of theorists that postulated that the masses might be given to bosons if the symmetry of the electroweak Lagrangian is not a symmetry of the vacuum [58,59]. The vacuum is filled with a massive scalar field, the Higgs field, that couples to all massive particles to give them mass. The existence of the Higgs boson was demonstrated in 2012 at the LHC and completed the zoo of fundamental particles that make up matter [60,61].
2.1.2 Theoretical framework
Quantum field theory provides the mathematical framework for the Standard Model (SM). The system Lagrangian dictates the dynamics and kinematics of particle fields that permeate space-time. The gauge theory is built upon a set of postulated symmetries by developing the most general renormalizable Lagrangian from its particle content that observes these symmetries [62].
Noether’s theorem states that every differentiable symmetry of the La- grangian of a physical system has a corresponding conservation law [63].
Lorentz invariance, consisting of time, translation and rotation invariance, is the global Poincaré symmetry group central to special relativity. It is neces- sary to guarantee energy, momentum and angular momentum conservations.
Each term of the SM Lagrangian must obey Lorentz invariance.
The fundamental objects of the SM are quantum fields, defined every- where in spacetime. Contrary to classical fields, they are operator valued, i.e. they do not represent a physical quantity but rather act upon quantum
states of the system. There is an operator for every point in spacetime that is merely used to exhibit some aspect of the state, at the point to which they belong. A single quantum field is associated with each particle type included in the theory. The excited states of the field are interpreted as particles in the classical framework. The amount of energy required to produce an excited state of the field is related to the massof the associated particle.
Figure2.3 compiles the Standard Model particles and their properties.
Quarks Leptons
Fermions Bosons
125.1 GeV 0
0
H
0Higgs boson
u c t γ
d s b g
ν
eν
µν
τZ
0e µ τ W
±up charm top photon
down strange bottom gluon
eneutrino µneutrino τneutrino Zboson
electron muon tau Wboson
4.7 MeV 1.274 GeV 173.1 GeV 0
2.2 MeV 94.6 MeV 4.176 GeV 0
0 0 0 91.2 GeV
0.511 MeV 105.7 MeV 1.777 GeV 80.4 GeV
+23 +23 +23 0
−13 −13 −13 0
0 0 0 0
−1 −1 −1 ±1
1
2 1
2 1
2 1
1
2 1
2 1
2 1
1
2 1
2 1
2 1
1
2 1
2 1
2 1
Figure 2.3:Particles of the Standard Model. The figures in each tile represent the rest mass, the charge and the spin of the particle in natural units. The fermions that exhibit both helicities are represented with a slant across them.
The internal symmetry group of the SM is defined as a local
SU(3)C⊗SU(2)L⊗U(1)Y (2.1) gauge symmetry [64]. The SU(3)C group represents the colour symmetry observed by the system. Particles that participate in the strong interaction, the quarks, carry a colour charge that is conserved by the system. The gauge bosons of the strong interaction are the gluon field tensors,Ga, witharunning over the eight possible representation of the group. The SU(2)L×U(1)Y
electroweak symmetry group corresponds to the conservation of weak isospin, T, and weak hypercharge,YW. The electroweak gauge bosons areW1,W2
andW3 representingSU(2)L andB representingU(1)Y. The fermions that make up all matter are each represented by fermion fieldsψ.
The free particle Lagrangian defines the evolution of the fermion and
gauge fields in the absence of interaction through Lkin=iψ /¯∂ψ−1
4BµνBµν−1
2trWµνWµν−1
2trGµνGµν, (2.2) with∂/=γµ∂µ and γµ the Dirac matrices, ¯ψ=ψ†γ0,Fµν =∂µAν−∂νAµ, F = B, W, G the field strength tensors and the trace accounting for the summation over the different representations of the boson fields. In the absence of interaction, the principle of stationary action requires that
∂Lkin
∂ψ − ∂
∂xµ ∂L
∂(∂µψ)
= 0, (2.3)
which yields the Dirac equation of a free massless field, i /∂ψ = 0. The evolution of theU(1)Y gauge field reads∂µBµν = 0 which is analogous to the evolution of the electromagnetic field tensor in Maxwell’s equations.
In the interaction picture fermions may scatter or be created or destroyed through the exchange of gauge bosons. This changes the free kinematic derivative,∂µ, to its dynamic form
Dµ=∂µ−igsGaµTa+ig01
2YWBµ+ig1
2τWµ, (2.4) withgs,g0 andgthe coupling constants of the strong and electroweak forces, Ta the tensor generators of theSU(3)C symmetry andτ the Pauli matrices generators of theSU(2)L symmetry. The second term, related to the strong force, exists only for quarks. The electroweak couplings are present for all fermions. The last term maximally violates parity and only couples to left- handed fermions. At this point the Lagrangian describes a theory of massless fermions interacting through the exchange of massless gauge bosons.
One could give mass to all fermions in the theory by adding a self-coupling term to each of them by hand. There is no way to give mass to gauge bosons that way because it would violate Lorentz invariance. The Higgs mechanism postulates the existence of a scalar boson,φ, with Lagrangian
Lh= (Dµφ)†(Dµφ)−V(φ), (2.5) where Dµφ does not include the strong coupling term. Consider a com- plex scalar field in the spinor representation ofSU(2)L, φ= φ+ φ0T. Renormalizability ofSU(2)L⊗SU(1)Y requires the potential to be of the form
V(φ) =−µ2φ†φ+λ φ†φ2
, (2.6)
withλ > 0. Ifµ2 >0, the field admits an infinity of minimums atφ†φ6= 0. Although the field is globally symmetric about zero, the symmetry is spontaneously broken to a non-zero vacuum expectation value (VEV) by
the potential. Since the potential depends only on the combinationφ†φ, the VEV is arbitrarily chosen as
hφi= 1
√2 0
v
, (2.7)
withv=µ/√
λ. Due to the observed conservation of electric charge, only a neutral scalar field can acquire a VEV. TheU(1)Y symmetry is unbroken by the scalar VEV, i.e. it yields a breaking scheme
SU(2)L⊗U(1)Y →U(1)Q, (2.8) withQ=T3+YW/2 the electric charge. It is the symmetry of quantum electrodynamics and is by construction the true realized vacuum symmetry.
One can parametrize the second component of the scalar field asv+hwith hthe Higgs boson, a perturbation around the VEV. Using this expression in the first term of equation2.5yields
(Dµφ†)(Dµφ) = v2 8
g2 (Wµ1)2+ (Wµ2)2
+ (gWµ3−g0Bµ)2
. (2.9) Considering that the physical bosons realized in vacuum are expressed as
Wµ±=√12 Wµ1±iWµ2 Zµ=√ 1
g2+g02 gWµ3−g0Bµ Aµ=√ 1
g2+g02 g0Wµ3+gBµ
, (2.10)
the coupling term in equation2.9can be rewritten (Dµφ†)(Dµφ) = 1
2 gv
2 2
Wµ†Wµ+1 2
vp g2+g02
2
!2
Zµ†Zµ, (2.11) giving mass to theW± andZ bosons but leaving the photon massless.
The fermions acquire mass in the Standard Model through the so-called Yukawa Lagrangian which describes the coupling between the scalar fieldφ and the fermion fields of the form
LYuk=−ψyφψ,¯ (2.12)
withy the Yukawa couplings. As the scalar potential has a minimum in vacuum athφi 6= 0, it introduces a term
− yv
√2ψψ¯ ≡ −mψψ,¯ (2.13)
which is a mass term for each fermion and function of their individual Yukawa couplings, free parameters of the Standard Model.
2.1.3 Physics beyond the Standard Model
Despite the many undeniable successes of the Standard Model (SM), it is only an effective theory. It is an excellent description of the scale that is currently accessible by modern accelerators to a great level of accuracy but it is expected to be the low-energy limit of aTheory of Everything, albeit restricted to three of the four fundamental forces [65]. Newton’s classical equations of motion, once believed to be fundamental, were later shown to be a large scale approximation of quantum mechanics by Ehrenfest’s work on the correspondence principle [66]. Fermi’s theory of weak interactions, which provides an excellent framework for the description of beta decays, is a low-energy limit of an electroweak theory that encompasses both weak and electromagnetic forces [67].
The SM contains 19 free parameters1, including three coupling con- stants, one for each of the forces it includes. It is theorised that the three constants converge into a single number at the so-called Grand Unified Theory (GUT) scale of ΛGUT '1016GeV [68]. Several unification routes have been studied, including the simplest unified Lie group which contains the SM,SU(5)⊃SU(3)×SU(2)×U(1), proposed by Georgi and Glashow in 1974 [69]. In those groups, the coupling constants converge approximately but not exactly at large scale. The Minimal Supersymmetric Standard Model (MSSM) is an extension that couples boson partners to all fermions and vice versa [70]. It is an attractive theoretical extension because it achieves gauge coupling unification, but it has been severely challenged by the absence of supersymmetric partners at the LHC [71]. Figure2.4shows the running of coupling constants in the SM and the MSSM. The simple symmetry at the GUT scale is spontaneously broken to produce three distinct forces of vari- able strength at lower energy. Larger symmetry groups achieve unification without the need for supersymmetry.
A successful GUT would unify electromagnetic, weak and strong inter- actions under a unique symmetry group but would fail to include gravity.
General relativity has been developed in a different theoretical framework by Albert Einstein and produces exquisite predictions at the scale at which gravity dominates [72,73]. As it turns out, quantum field theory and general relativity cannot be reconciled in regions of extremely high mass and small scale, e.g. within black holes or soon after the Big Bang. A Theory of Every- thing must provide a quantum gravitational understanding of the Universe that includes both theories. String Theory is the most significant theory to have emerged as a single-explanatory framework [74]. It postulates that the most fundamental structures of the universe are vibrating one-dimensional strings. It is expected to describe all interactions at the Planck scale, i.e.
1Straightforward extensions of SM with massive neutrinos need 7 more parameters, 3 masses and 4 mixing matrix parameters, for a total of 26 parameters.
Figure 2.4:Running coupling constants in the Standard Model (left) and with the introduction of supersymmetry (right).
ΛP '1019GeV. At lower energies, strings would manifest themselves as point-like particles in effective theories, e.g. the SM.
Beside gravity, other critical phenomena cannot be explained by the SM in its current form. Cosmological observations seem to suggest that the amount of visible matter represents about 5 % of the energy present the Universe. Darkmatter and energy is necessary to explain the coalescence of galaxies and the rate of expansion of the Universe. It must participate to the gravitational force but interact very weakly with matter through SM processes, as it is yet to be observed in a detector. A related issue is the baryon asymmetry. Matter was produced in a significantly larger fraction than antimatter during the Big-Bang, yet no mechanism in the minimal SM can explain this asymmetry. Neutrino oscillations constitute the third most prominent challenge to the SM and are described in detail in section2.2.
Recent precision tests have shown non-negligible deviations from the SM predictions that could hint at new physics. The measurement of B meson decays at BaBar, Belle, CMS and LHCb have revealed a combined deviation of over 5σfor several rare processes [75]. Most intriguing is the apparent violation of lepton flavour universality exhibited by decays of the formb→sl+l− which significantly favour electron over muon production.
The precise measurement of the proton radius in a muonic-hydrogen (muon- proton system) has also challenged previous well-established observations by measuring a distance seven standard deviations below the previous measurements [76]. The author himself, Randolf Pohl, believes that the tension arises from a misgauged Rydberg constant in previous experiments, rather than new physics. State-of-the-art muon magnetic moment experiment have also shown∼3.5σtension with the SM value [77]. Recent computations by Morishima and others successfully explain the observed discrepancy by including general relativistic effects of the Earth’s gravitational field [78].
2.2 Neutrinos
Neutrino oscillations are arguably the only significant sign of physics beyond the Standard Model. In its minimal form, the Standard Model does not include neutrino masses. A massless particle travels at the speed of light and as such is not localized in time, i.e. its quantum field is a constant of free motion. An overwhelming wealth of experiments in the last fifty years have unequivocally demonstrated that neutrinos oscillate between their flavour eigenstates. This calls for an extension of the model Lagrangian to include a neutrino mass term. There is no certainty on which mechanism is correct as there is more than one way to give masses to neutral leptons. This uncertainty along with many other unknown neutrino parameters has motivated the development of a plethora of current and future experiments. This section gives a review of the evidence for neutrino masses, the theoretical mechanism that could give rise to them and a description of the open issues.
2.2.1 Oscillations
The observation of the solar neutrino flux by the Homestake experiment in a deep mine in South Dakota was historically the first indication that neutrinos oscillate [4]. Raymond Davis designed an experiment consisting of a large tank of dry-cleaning fluid in which neutrinos emitted by nuclear fusion in the Sun could interact via the inverse beta decay process, i.e.
νe+37Cl+→37Ar++e−. (2.14) The gaseous radioactive argon was regularly collected and its activity mea- sured to determine the number of neutrino interactions. The measurement was consistently very close to one-third of John Bahcall’s prediction. This gave rise to thesolar neutrino problem. Many subsequent radiochemical and water Cherenkov experiments confirmed the deficit [6,7,79–81].
Bruno Pontecorvo, a nuclear physicist who instigated several direct measurements for neutrinos, was the first to suggest that neutrinos may come in electron and muon flavours [82]. The existence of a second neutrino type was confirmed by the observation of the charged current production of muons at the Alternating Gradient Synchrotron [83]. In a 1957 breakthrough paper, he proposed that the two flavours may oscillate between each other in an analogy with the previously observed CP-violatingK0 oscillations [5,84].
For this to happen, the neutrinos cannot have zero-mass and therefore do not travel at the speed of light. In the two-neutrino model developed at the time, the flavour eigenstates are a rotation of the mass eigenstates through
νe
νµ
=
Ue1 Ue2
Uµ1 Uµ2
ν1
ν2
= cosθ sinθ
−sinθ cosθ ν1
ν2
, (2.15)
withν1, ν2 the mass eigenstates andθthe mixing angle.
The neutrino oscillation scheme is represented in figure2.5. When pro- duced in a weak process, the neutrino of flavourαis in a superimposition of the two mass eigenstates. Since theνi, i= 1,2 are mass eigenstates, their propagation can be described by plane wave solutions of the Schrödinger equation of the form
|νi(t)i=|νi(0)iexp (−i(Eit−pixi)/~), (2.16) witht the propagation time, Ei, pi and xi the energy, three-momentum and spacial position of the mass eigenstate, respectively, and~the reduced Planck constant. In the ultrarelativistic limit |pi| ≡pimic, the energy can be expanded at first order to
Ei=q
p2ic2+m2ic4'pic+m2ic3 2pi
'E+m2ic4
2E , (2.17) withE the total energy of the neutrino, now approximatively identical in each eigenstate. In the limitv'c, the timetrelates to the distance through L/c. Dropping the phase factor, the wave function after a distanceLreads
|νi(L)i=|νi(0)iexp
−im2ic3L 2~E
. (2.18)
Source
•
W να
`¯α
ν
• W
νβ
`¯β
L
Target
Figure 2.5:A neutrino of flavourαtravels a distanceLfrom its source to produces a charged leptonlβ in its interaction with the target.
As the flavour eigenstates are a linear combination of the mass eigenstates with evolving parameters, it is possible to observe a neutrino change its flavour during its propagation. The probability of a neutrino originally of flavour αto be later observed at target in the same flavour is
Pα→α=|hνα|να(L)i|2= 1−sin2(2θ) sin2
1.27∆m2L E
GeV eV2·km
, (2.19) with ∆m2=m22−m21. The effect of the oscillation is maximal whenL' E/∆m2. The appropriate choice of baseline,L/E, allows for the observation
of the most significant disappearance of the emitted flavour or appearance of other neutrino flavours.
The survival probability of an electron neutrino is represented as a function ofL/E in the two-neutrino paradigm in figure 2.6 for θ = 33.5°
and ∆m2 = 7.5×10−5eV2. At a solar distance L ' 1.5×108km, the oscillations of MeV neutrinos are in theLE/∆m2 regime for which the probability effectively integrates to
Pα→α'1−1
2sin2(2θ)'0.575. (2.20) This is the ratio observed by solar neutrino experiments sensitive to sub-MeV neutrinos. At higher energies, the contribution of matter effects inside the sun become significant and reduces this ratio. The solar neutrino problem was definitely solved much later at Sudbury National Observatory. The detector consisted of heavy water sensitive to all neutrino flavours through neutral current interactions. It observed the expected total flux and showed that the multi-MeV electron neutrinos only make up a third of it [85].
100 101 102
0.2 0.4 0.6 0.8 1
L/E [km/MeV]
Pe→e
Figure 2.6:Survival probability of an electron neutrino in the two-neutrino model as a function of the baselineL/Eforθ= 33.5° and ∆m2= 7.5×10−5 eV2.
The observation of CP-violating processes – not allowed by the 2×2 Cabbibo quark mixing matrix – lead Kobayashi and Maskawa to surmise the existence of a third generation of fermions [86]. The discovery of the τ lepton and its neutrino at the Stanford Linear Accelerator Centere+e− colliding ring in 1975 confirmed its existence and prompted Maki, Nakagawa and Sakata to extend the dimensions of the neutrino mixing matrix to its current form [87]:
UPMNS=
Ue1 Ue2 Ue3 Uµ1 Uµ2 Uµ3 Uτ1 Uτ2 Uτ3
, (2.21)
which is most commonly decomposed as
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
, (2.22) withcij = cosθij andsij = sinθij andδ the CP-violating phase. The total number of neutrino generations was corroborated by the precise measurement of theZboson width by multiple experiments located on the Large Electron Positron collider at CERN [88]. Tau neutrino interactions were directly observed at the turn of the XXIst century by a dedicated experiment at Fermilab, DONUT [89]. Figure2.7gives a schematic representation of the mixing angles between the two neutrino eigenbases.
ν1
ν2
ν3
νe
νµ
ντ
θ12
θ12
θ13
θ23
θ13
θ23
Figure 2.7: Neutrino mixing angles represented as a product of Euler rotations:
(νe, νµ, ντ)T =UPMNS(ν1, ν2, ν3)T. Some representative values of the angles are used based on the best-fit values summarised in table2.1.
In the three-neutrino model, the probability of a neutrino of energyE produced in flavourαto be later observed in flavourβ after a propagation distanceL reads
Pα→β=|hνβ|να(L)|2=
X
i
Uαi∗Uβie−im2ic3L/2~E
, (2.23) or, in terms of the mass splittings,
Pα→β=δαβ − 4X
i>j
< Uαi∗UβuUαjUβj∗ sin2 ∆m2ijc3L 4~E
!
+ 2X
i>j
= Uαi∗UβuUαjUβj∗
sin ∆m2ijc3L 2~E
!
, (2.24)
with ∆m2ij =m2i −m2j [90]. This formula now represents a superposition of two oscillations of frequencies driven by two mass splitting ∆m221 and
∆m23l ≡ ∆m232 ' ∆m231 ∆m212 and amplitudes determined by three mixing angles θ12, θ13 andθ23. The small splitting ∆m221 is defined to be positive but ∆23l may be of either sign. Figure2.8 shows the two possible orderings. The mass eigenstate with the smallest fraction ofνeis the heaviest in the normal ordering scenario and the lightest in the inverted ordering.
Normal Ordering (NO) Inverted Ordering (IO)
Figure 2.8:The two possible orderings of the neutrino mass eigenstates.
Equation 2.24 is usually well approximated by its two-neutrino form in equation 2.20, due to the relative small scale of ∆m221 and θ13. The
∆m221 mass splitting and theθ12 mixing angle are the leading factors in the long-baseline solar oscillationsνe ↔νx measurements. The splitting was measured accurately by the KamLAND experiment, a kiloton scintillator tanker in Kamioka that observed the disappearance of MeV nuclear reactor
¯
νe at a mean distanceL0'180 km [91].
The large mass splitting ∆23land θ23drive the atmospheric oscillations νµ↔νx. Multi-GeV muon neutrinos are produced in the upper atmosphere by pion and muon decays. The neutrinos do not oscillate before reaching the surface of the Earth but do when they cross it. The observation of the disappearance ofνµ coming from below the Super-Kamiokande 50 kT water Cherenkov detector demonstrated this second regime of oscillations [92].
Precision measurements of the mixing parameters were performed by the T2K and NOνA collaborations by observing the disappearance of accelerator νµ produced by dumping protons on targets at accelerator facilities and also measuring the appearance of the oscillatedνe [93,94].
The small mixing angleθ13 has its largest impact onνedisappearance at short-baseline, before the solar oscillations dominate. Reactor neutrinos are an ideal candidate to measure this angle as they are produced in large quantities in commercial nuclear reactors. The oscillation maximum of an MeV neutrino occurs at a distance of order km from the source and constitutes
