Iowa State University Capstones, Theses and Creative Components Dissertations
Fall 2020
The Current Status of the See-Saw Mechanism
Kevin Marasinghe Iowa State University
Follow this and additional works at: https://lib.dr.iastate.edu/creativecomponents
Part of the Elementary Particles and Fields and String Theory Commons
Recommended Citation Marasinghe, Kevin, "The Current Status of the See-Saw Mechanism" (2020). Creative Components. 660. https://lib.dr.iastate.edu/creativecomponents/660
This Creative Component is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Creative Components by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. The Current Status of the See-Saw Mechanism
Kevin Marasinghe December 4, 2020
Abstract The see-saw mechanism is an elegant method of resolving some of the key issues in neutrino phenomenology. In order to motivate the see-saw mechanism, the history of neutrino discovery, research, and explanation will be discussed. The current viability of the see-saw mechanism with respect to the latest experimental developments will also be addressed.
1 Contents
Acknowledgments 3
1 Neutrinos and the Development of the Standard Model4 1.1 A Radical Solution...... 4 1.2 Direct Detection...... 5 1.3 Flavors and Families...... 5 1.4 The Solar Neutrino Problem...... 6 1.5 The Higgs Mechanism and Electroweak Uni cation...... 7
2 Massive Neutrinos; Extending the Standard Model7 2.1 Solar Neutrinos Explained...... 7 2.2 Neutrino Oscillation...... 8 2.3 Extending the Standard Model...... 10 2.4 The See-Saw Mechanism...... 10 2.5 Real-World Requirements...... 11
3 The Experimental Viability of the See-Saw Mechanism 11 3.1 Neutrinoless Double Beta Decay...... 11 3.2 Normal Ordering (NO) vs. Inverted Ordering (IO)...... 12 3.3 Light Sterile Neutrinos...... 12
4 Conclusions 12
2 Acknowledgements
Thank you to my family for their unwavering support of my pursuit of an advanced degree. With- out such a support system I would never even made it to the starting line; without their love, encouragement, and hugs, I could not have summoned the motivation. Thank you to my advisor, Kerry Whisnant, for his seemingly in nite patience in dealing with the di culties I presented as a student. Thanks to Iowa State University, where I have been a student on and o for over 25 years now, for most of my education. The picture on my student ID card remains that of an 8th grade adolescent taking a summer course in trigonometry in 1995! Finally, thanks to the contingency plans that ISU instituted in response to the COVID-19 pandemic. While these were a major impediment for most people, the remote classes in my case gave me the courage to nally nish my classwork this past spring after avoiding it for so long. It is inconceivable that a virus which has destroyed so many lives could serve a constructive purpose, yet in my case it has managed to do so.
3 1 Neutrinos and the Development of the Standard Model 1.1 A Radical Solution One of the biggest drivers of the transition from what is generally known as Classical Physics, i.e. Newtonian Mechanics and Electromagetism, the twin pillars supporting the eld at the turn of the 20th century, to Modern Physics, i.e. Quantum Mechanics and Quantum Field Theory, was the discovery of spontaneous atomic radiation by Henri Becquerel. In the thirty or so years after its discovery, atomic radiation was divided into three types: alpha (α), beta (β), and gamma (γ). Each of the three types had their energy spectra measured. It was seen that while the spectra of particular α- and γ-producing processes were de ned by a narrow peak, the spectrum of β processes was broad, indicating a variable loss of energy and momentum somewhere in the process [Fig.1][4][10][14]. Furthermore, by then it was known that protons and electrons had intrinsic angular momentum, and the apparent creation of a spin-1/2 particle without a similar change in the spin of the daughter nucleus (integer to half-integer, or vice versa) seemed to violate yet another law of Classical Physics, conservation of angular momentum. To rescue these time-
(a) Alpha decay energy spectra for varying (b) Beta Decay Spectrum of Bismuth-210. isotopes. (Taken from [8].) (Adapted from [16].)
Figure 1: Alpha vs. Beta decay honored principles, in 1933 Enrico Fermi published his theory of beta decay, in which the neutron in a nucleus would decay into a proton, and electron, and a new particle, the neutrino. The neutrino was required by experimental limits to be extremely penetrating and weakly interacting, so much so that contemporaries such as Niels Bohr were incredulous [6]. This so-called four-Fermi interaction was eventually realized to be an early example of an eective eld theory, which Fermi somewhat serendipitously stumbled into; the coupling constant, GF , was thus named after Fermi [Fig.2][19]. The new theory proved highly predictive as phenomenology for more than twenty years, despite a neutrino not having been detected directly during this time.
Figure 2: Beta decay via the four-Fermi interaction. (Taken from [19].)
4 1.2 Direct Detection In 1956, direct detection was rst achieved by Clyde L. Cowan and Frederick Reines [19]. Large tanks of water mixed with cadmium chloride were surrounded by layers of the liquid scintillator terphenyl and numerous photomultiplier tubes (the sort of experimental setup which would be repeated many times over by neutrino experiments through the years!) Incoming neutrinos were to strike a hydrogen atom (i.e. a single proton) in one of the water molecules. This would result in an energetic positron and a free neutron. The positron would almost immediately annihilate with an electron, creating two gamma rays that would cause a scintillation cascade. The neutron would hopefully be captured by a cadmium atom, which would end up in a metastable state. After a few microseconds this state would decay via gamma emission, creating another burst separated from the prior one by a characteristic interval[Fig.3][19].
Figure 3: The mechanisms used by the Cowan and Reines experiment. (Taken from [19].)
1.3 Flavors and Families The muon, which was discovered in 1936, mimics the electron in seemingly every way; except for its mass, it could be mistaken for an electron in an excited, and thus more massive, state [14]. It would even primarily decay to an electron; however, two neutrinos were always released along with it. A reasonable question to ask then was: since neutrinos lack charge, are neutrinos Majorana particles? That is, are they their own antiparticle? If they were, couldn't we expect to see a variation of the muon decay involving a photon as in [Fig.4(b)][21]? Or was there some other factor preventing this? Since the electron seemed to lack any sort of structure and was for all intents and purposes a point particle, it was seen as unlikely that a muon could be an excited electron. Thus was born the ideas of avor and family: that each fundamental particle wavefunction has a quantum number associated with its species; and that heavier, unstable fundamental particles are copies of a lighter stable particle, except for the aforementioned avor. Thus muons should be a dierent avor from electrons, and produce a dierent avor of neutrino in interactions [14]. This was shown conclusively in 1962 in experiments by Leon M. Lederman, Melvin Schwartz and Jack Steinberger [Fig.5]. It became clear that the neutrinos detected in 1956 were electron antineutrinos, and had a dierent avor from the muon neutrinos and antineutrinos produced here. Since pathways such as in [Fig.4(b)] were prohibited by this avor conservation, the question of whether or not neutrinos were Majorana particles remained open.
5 Figure 4: Two proposed decay pathways for the muon; only (a) has been observed. (Taken from [21].)
Figure 5: The 1962 experimental setup of Lederman, Schwartz, and Steinberger. Protons were collided with a beryllium target, releasing a shower of pions, some of which would quickly decay to muons and associated neutrinos. The shower was aimed at a large steel target (the decommissioned battleship Missouri) to lter out everything except the neutrinos, which hit the detector beyond. In collisions with electrons in the detector, the neutrinos produced only muons, con rming avor conservation. (Taken from [22].)
1.4 The Solar Neutrino Problem Having shown that neutrinos could be detected, attempts could now be made to look for the speci c neutrinos produced by stellar fusion. Work in nuclear fusion processes had revealed how stars like our sun produced their enormous energy outputs, but the details remained to be ironed out. Many dierent processes and chains were proposed and examined, but whatever the pathway, the net reaction was always
1H 4 He + 26.73 MeV 4 1 −→2 + 2 e + 2 νe + with the energy distributed among gamma photons and the kinetic energy of the daughter products [4][10][14]. Unlike most of this energy which takes millions of years to work its way out of the core towards the surface as thermal radiation, the neutrinos arrive at Earth instantaneously by comparison. Raymond Davis, Jr. and John N. Bahcall conducted an experiment in the Homestake Gold
6 Mine in South Dakota to quantify this neutrino output and compare it with the ux predicted by the Standard Solar Model (SSM). A tank, roughly three orders of magnitude larger than the container volume used by Cowan and Reines, was lled with tetrachloroethylene and left to sit for several weeks at a time. Neutrino collisions with the chlorine-37 (24% of naturally occurring chlorine) produced the radioisotope argon-37, which has a half life of 35 days. This gas could be isolated to high purity and its radioactivity measured to very precisely determine the number of neutrino events per day. Homestake detected roughly 1/3 the neutrino ux predicted by the SSM [7]. This result was seen as probably erroneous for decades until independent con rmation in the late 1980s by Kamiokande-II, which also detected a neutrino de cit [17].
1.5 The Higgs Mechanism and Electroweak Uni cation
Figure 6: The four-Fermi interaction would be replaced by the vector-axial interaction mediated by a new vector boson. Explaining this necessarily massive new particle would form the basis for the Standard Model. (Adapted from [21].)
It had been known for some time that Fermi's theory of beta decay was inadequate from a theoretical perspective, despite its phenomenological success. It was recognized that simplifying the four-Fermi interaction into two three-way interactions connected by a boson propagator was the way forward, both for the sake of higher-energy calculations (which experiment was now approaching) and renormalization, which did not work on Fermi's theory [Fig.6]. The solution arrived in two parts: rst, Sheldon Glashow's 1961 theory of a triplet of massive vector bosons spanning the group SU(2)xU(1); second, Peter Higgs' (among others) 1964 theory of spontaneous symmetry breaking that bears his name. It was left to Steven Weinberg and Abdus Salam in 1967 to combine these ideas into the locally gauge invariant theory known as the electroweak interaction [10]. Discovery of said vector bosons W and Z followed promptly in 1983, although discovery of the heavier Higgs boson had to wait thirty more years [14]. When combined with the discovery of several new quarks during this same time period, a clear parallel was established between the quark sector and the lepton sector. Each sector had 6 avor eigenstates, which could be paired in three SU(2) doublets according to generation [Fig.7]. The weak interaction coupled the members of the doublet to each other via the exchange of a W vector boson. Additionally, in the quark sector the weak interaction coupled interactions across generations at comparatively attenuated rates; the mass eigenstate of a d-quark was thus shown to be an admixture of d0, s0, and b0 avor eigenstates, though one that was heavily weighted towards d0. This mixing between avor eigenstates and mass eigenstates is represented by the Cabibbo-Kobayashi-Masukawa (CKM) matrix, which gives transition amplitudes between various avor eigenstates as matrix elements. However, no such coupling was observed in the lepton sector; a vertex combining leptons from two dierent families did not seem to exist [21].
2 Massive Neutrinos; Extending the Standard Model 2.1 Solar Neutrinos Explained To better understand the results obtained by Homestake and Kamiokande-II, a third experiment was conducted at Creighton Mine in Sudbury, Ontario, Canada. Sudbury Neutrino Observatory
7 Figure 7: Flavor (weak) eigenstates and mass eigenstates of the three quark and lepton families, as the Standard Model was originally conceived. Note the lack of mixing in the lepton mass states. (Taken from [21].)
(SNO) was designed as a more complete apparatus capable of detecting neutrinos of all families. It was lled with ultra pure D2O, deuterium oxide or heavy water, which was contained in a clear acrylic tank surrounded by another tank, this time full of ultra pure light (deuterium free) water. Both had sodium chloride salt made with pure chlorine-35 dissolved in solution; in this aspect it somewhat mimicked the Cowan and Reines experiment. The primary goal in using deuterium was to utilize the neutral current (NC) interaction. This could happen in one of two ways: elastic scattering o of electrons, and an inelastic process whereby a neutrino caused the ssioning of a deuterium nucleus into its constituent proton and neutron. The neutron would then be captured by a chlorine atom with release of gamma rays. Charged current (CC) interactions were also detectable as with prior observatories, with electron neutrinos interacting with the neutron in deuterium to produce a proton and an electron; and a similar elastic scattering process to the NC but for electron neutrinos only. Since the two elastic scattering (ES) processes were otherwise indistinguishable, only the net rate for all ES processes was measurable [1]. The SNO collaboration was able to show that neutrino ux across all three known avors was consistent with the standard solar model (SSM) and also that the muon and tau neutrino ux was almost exactly twice that of electron neutrino ux [Fig.8]. This con rmed once and for all that neutrinos were oscillating between avors during their ight from the Sun to Earth [4].
2.2 Neutrino Oscillation First elucidated in 1957 by Bruce Pontecorvo, the theory of neutrino oscillation was updated shortly after the discovery of the solar neutrino de cit as a possible explanation for such. The idea was inspired by and roughly analogous to quark mixing as described by the CKM matrix. However, unlike quarks, which are labeled according to their (measured) mass eigenstates (which are in turn admixtures of avor eigenstates,) neutrinos are labeled according to their avor eigenstates, which are admixtures of the mass eigenstates[Eq. 2.2.1]. Thus, a neutrino produced as a given avor will have oscillating probabilities of being in another avor eigenstate with a frequency that depends on the dierence of the masses squared of the various eigenstates . These probabilities are calculated as matrix element inner products of the Pontecorvo MakiNakagawaSakata (PMNS) matrix [Eq. 2.2.2], according to transitioning from a given initial to a given nal state[Eq. 2.2.3]. The matrix elements themselves can be characterized by six parameters, three representing the mixing angles between states (θ12, θ23, θ13) and three more representing CP-violating phases: one Dirac (δ) and two Majorana (α2, α3). If neutrinos are NOT Majorana, the Majorana phases are set to zero [5][13][18][20].
8 Figure 8: The critical result from SNO. The best t central values for observed neutrino uxes 6 − − 6 − − φe = (1.76 × 10 ) cm 2 s 1 and φµτ = (3.41 × 10 ) cm 2 s 1 show that the combined νµ and ντ ux is exactly twice that of the νe ux.(Taken from [1].)
νe Ue1 Ue2 Ue3 ν1 νµ = Uµ1 Uµ2 Uµ3 · ν2 (2.2.1) ντ Uτ1 Uτ2 Uτ3 ν3
where: Ue1 Ue2 Ue3 Uµ1 Uµ2 Uµ3 (2.2.2) Uτ1 Uτ2 Uτ3
−iδ c12c13 s12c13 s13e 1 0 0 iδ iδ iα2/2 = −s12c23 − c12s23s13e c12c23 − s12s23s13e s23c13 0 e 0 iδ iδ iα /2 s12s23 − c12c23s13e −c12s23 − s12c23s13e c23c13 0 0 e 3
−iδ 1 0 0 c13 0 s13e c12 s12 0 1 0 0 iα2/2 = 0 c23 s23 0 1 0 −s12 c12 0 0 e 0 −iδ iα /2 0 −s23 c23 −s13e 0 c13 0 0 1 0 0 e 3
cij = cos θij , sij = sin θij
2 2 X ∗ −ım2 L i 2E (2.2.3) Pα→β = |hνβ|να(t)i| = UαiUβie i
9 2.3 Extending the Standard Model Giving the neutral, chiral neutrino a mass through interaction with the Higgs eld presents more technical challenges than those which the Higgs mechanism was initially proposed to answer. The three most straightforward possibilities for modifying the SM Lagrangian are discussed here [5][18][20]: 1. The Dirac mass term.
LDirac = mD[νLNR + N RνL] = mDνν with ν = νL + NR (2.3.1)
Inserting a term such that neutrinos gain mass via the same pathway as the charged leptons requires postulating the existence of a right-handed species of neutrino. Without also modi- fying the Lagrangian to include a right-handed version of the vector bosons, there exists no mechanism by which to detect these particles, either by their presence or their absence. It also leaves open the question of why the mass scale separations in the lepton sector are more extreme than the quark sector by a factor of one million. 2. The Majorana mass term.
1 c c 1 c LMajorana,Left = ML[νLν + ν νL] = MLνν with ν = νL + ν 2 L L 2 L (2.3.2) 1 c 1 L = M [N N c + N N ] = M νν with ν = N + N c Majorana,Right 2 R R R R R 2 R R R
The physics allowed by this term is in nitely more interesting and theoretically fecund; however, it comes with its own sets of problems. In particular, lepton number conservation
is always violated. The inclusion of a nonzero Majorana mass ML associated with the known neutrinos makes the Lagrangian nonrenormalizable. This seemingly prevents the use of Majorana terms to provide mass to the left-handed neutrinos. However, a mass state that slightly mixes with right-handed counterparts (and thus has mixed helicity) can obtain a small mass through the see-saw mechanism, discussed below.
3. Mixed mass terms.
1 1 1 T c ML mD νL (2.3.3) Lmixed = mDνν + MLνν + MRνν = νL N R c +h.c 2 2 2 mD MR NR
where T ML mD mD MR is the Dirac-Majorana neutrino mass matrix. The use of both terms is possible, and it can be done in three dierent ways. We will cover the simplest method (Type I See-Saw Mechanism) here.
2.4 The See-Saw Mechanism By far the most appealing approach is known as the see-saw mechanism. There are many variants of this mechanism, but at the base of all of them is the subtle interaction between Dirac and Majorana terms in the SM Lagrangian. The combination of the two types of terms can be written as a matrix inner product in a basis of left and right components; after the Higgs mechanism is applied, these terms all acquire coupling constants. The Majorana mass of the left-handed neutrinos is set to zero to allow renormalization. Diagonalizing the mass matrix one obtains the usual free Lagrangian for two particles ν1 and ν2, with masses m1 and m2, respectively [5][13][18][20]:
2 MD m1 ' ⇒ light neutrino (left-handed) MR
m2 ' MR ⇒ heavy neutrino (right-handed)
10 As can be seen, something nice happens to the resulting diagonalized mass eigenstates: the left- handed neutrino gains mass inversely proportional to the right-handed Majorana mass. When the Majorana mass of the right-handed neutrino is assumed to be very large (TeV), the left-handed neutrino mass can be made arbitrarily small. This solves quite a few problems: 1. The initial problem of how to incorporate a massive neutrino into the Standard Model.
2. The mass hierarchy problem, at least for leptons. The existence of a much larger scale than the Dirac (electron) mass necessitates the existence of the much smaller scale (by the same order dierence) where neutrinos lie. 3. Lepton number remains an approximate symmetry at the scale of everyday life, as heretofore observed; however, it should still be observably broken in small amounts. Lepton number- violating interactions also become a favorable pathway via the heavy species; this is helpful in constructing Grand Uni ed Theories (GUTs,) and in cosmology (e.g. explaining leptoge- nesis). 4. This SM extension can now be treated as an eective theory, since the Majorana mass term establishes a scale for higher order corrections, allowing it to be renormalized. However, in any future GUT, the issue of what happens at this scale would have to be addressed.
2.5 Real-World Requirements In order for the see-saw mechanism to be a viable model for the masses of neutrinos, several conditions must be met:
1. Majorana Neutrinos. There should be evidence that neutrinos are Majorana particles; speci cally, they must self-annihilate. 2. Normal vs. Inverted Ordering. For speci c ordering of mass eigenstates, a lower bound is placed on the mass of the lightest Majorana-type neutrino species.
3. No Light Sterile Neutrinos. Any light sterile neutrinos cannot possibly be the heavy right-handed neutrino we want to see. This complicates the model somewhat, though it is still possible with tweaking.
3 The Experimental Viability of the See-Saw Mechanism 3.1 Neutrinoless Double Beta Decay Since the elegance the see-saw mechanism displays in explaining several fundamental issues in high energy physics only emerges if neutrinos have a mixed Dirac/Majorana character, one aspect of neutrinos that needs to be observed experimentally has to be something revealing a characteristic feature of Majorana fermions in general. The obvious feature to exploit is the fact that Majorana fermions should annihilate against other Majorana fermions of the same type. With neutrinos, this is di cult to engineer. If, for example, an attempt was made to collide two neutrino beams, the extreme smallness of the interaction cross sections would quickly render the notion of detecting such events fanciful, as the neutrino does not respect the electromagnetic interaction. There is the further complication of oscillation: even if two neutrinos are produced using identical processes, there is no guarantee that the resulting mixed states generated at a distance from the location of the neutrinos' creation could be prepared identically. The proposal of the existence of neutrinoless double beta decay (0νββ) [Fig.9a.] allows for the possibility of an indirect detection of Majorana neutrinos via lepton number violation. Further- more, the signature of such an interaction would be easy to spot [Fig.9b.]. Even so, this presents extreme technical challenges. Normal double beta decay is itself not a readily observed process, and 0νββ is further suppressed by the Higgs eld interaction. Experiments such as GERDA [11], MAJORANA [3], and CUORE [12] have so far failed to detect a signal. Current experimental bounds suggest that 0νββ events must have a half-life of ≥ 1026 yr.
11 (b) The energy spectrum of electrons (a) Schematic diagram of 0νββ. Interac- emerging from a double beta decay. The tion with the Higgs eld ips the chirality spike at the rightmost (most energetic) end of the neutrino current. is the characteristic signal of 0νββ.
Figure 9: Neutrinoless Double Beta Decay (0νββ) (taken from [21].)
3.2 Normal Ordering (NO) vs. Inverted Ordering (IO) The ambiguity in mass eigenstate ordering has a large impact on establishing the scale of the
Majorana mass. In particular, it impacts the calculation of the mass matrix element hmeei. It can be shown that in the case of Inverted Ordering (IO), the value of this element has a lower bound:
q IO 2 2 2 hmeeimin = 1 − |Ue3| ∆mA 1 − 2 sin θ12
With , 2 − eV2, and 2 |Ue3| = 0.143 → 0.156 ∆mA = 2.510 × 10 3 sin θ12 = 0.310
IO hmeeimin = (18.57 → 18.64) meV
This corresponds to 0νββ events that have a half-life of ≥ 1028 yr [9]. Should IO become favored by experiment, pushing the lower bound of 0νββ past this point would rule out Majorana neutrinos [20]. For now NO remains favored, (Fig. 10) though it is by no means a foregone conclusion. It should be pointed out that NO has its own lower bound; however, this bound is an order of magnitude less in matrix element magnitude, and thus two orders of magnitude less in half-life of 0νββ events. Current experimental techniques are not capable of approaching this value.
3.3 Light Sterile Neutrinos The experimental discovery of a 4th generation light sterile (right-handed) neutrino would be disappointing for the see-saw mechanism. While a heavy sterile neutrino would not be ruled out, much of the aesthetic reasoning behind it would become moot. A light sterile neutrino would be of mainly Dirac type, and we would be left with an as-yet-unexplained hierarchy problem in the lepton sector. There is no current evidence for a light sterile neutrino [2].
4 Conclusions
While phenomenology is not short on dierent way to tweak and extend the original see-saw idea, so far experimental evidence has not shown up. See-saw Type I models as described here haven't necessarily been ruled out; (0νββ) searches are still ongoing and may yet yield results. One thing currently in its favor is that no new ideas have been proposed that are as alluring; hence, the search continues.
12 Figure 10: (Taken from [15].)
13 References
[1] Q. R. Ahmad and et al. Direct Evidence for Neutrino Flavor Transformation from Neutral-Current Interactions in the Sudbury Neutrino Observatory. Physical Review Letters, 89(1):011301, July 2002. [2] M. Aker and et al. Bound on 3+1 active-sterile neutrino mixing from the rst four-week science run of KATRIN. arXiv e-prints, page arXiv:2011.05087, November 2020.
[3] I. J. Arnquist and et al. The Majorana Demonstrator's Search for Double-Beta Decay of 76Ge to Excited States of 76Se. arXiv e-prints, page arXiv:2008.06014, August 2020. [4] V. Barger, D. Marfatia, and K. Whisnant. The Physics of Neutrinos. Princeton University Press, 2012. [5] Samoil M. Bilenky. Neutrinos: Majorana or Dirac? Universe, 6(9):134, August 2020. [6] Laurie M. Brown. The idea of the neutrino. Physics Today, 31(9):23, 1978. [7] Bruce T. Cleveland, Timothy Daily, Jr. Raymond Davis, James R. Distel, Kenneth Lande, C. K. Lee, Paul S. Wildenhain, and Jack Ullman. Measurement of the solar electron neutrino ux with the homestake chlorine detector. The Astrophysical Journal, 496(1):505526, mar 1998. [8] C. Cozzini, G. Angloher, C. Bucci, F. von Feilitzsch, D. Hau, S. Henry, Th. Jagemann, J. Jochum, H. Kraus, B. Majorovits, and et al. Detection of the natural decay of tungsten. Physical Review C, 70(6), Dec 2004. [9] P. F. de Salas, D. V. Forero, S. Gariazzo, P. Martínez-Miravé, O. Mena, C. A. Ternes, M. Tór- tola, and J. W. F. Valle. 2020 global reassessment of the neutrino oscillation picture, 2020. [10] M. Fukugita and T. Yanagida. Physics of Neutrinos and Applications to Astrophysics. Springer-Verlag, 2003.
[11] GERDA collaboration and et al. Final Results of GERDA on the Search for Neutrinoless Double-β Decay. arXiv e-prints, page arXiv:2009.06079, September 2020. [12] A. Giachero and et al. New results from the CUORE experiment. arXiv e-prints, page arXiv:2011.09295, November 2020.
[13] Carlo Giunti. Neutrino Flavor States and the Quantum Theory of Neutrino Oscillations. In Heriberto Castilla-Valdez, Omar Miranda, and Eli Santos, editors, Particles and Fields: XI, volume 1026 of American Institute of Physics Conference Series, pages 319, July 2008. [14] David Gri ths. Introduction to Elementary Particles. Wiley-VCH, 2nd edition, 2008.
[15] Kevin J. Kelly, Pedro A. N. Machado, Stephen J. Parke, Yuber F. Perez-Gonzalez, and Renata Zukanovich Funchal. Back to (Mass-)Square(d) One: The Neutrino Mass Ordering in Light of Recent Data. arXiv e-prints, page arXiv:2007.08526, July 2020.
[16] G. J. Neary and John Douglas Cockcroft. The α-ray spectrum of radium e. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 175(960):7187, 1940. [17] Yuichi Oyama. Results from kamiokande-ii experiment. Nuclear Physics B - Proceedings Supplements, 13:362 367, 1990. [18] Particle Data Group, P. A. Zyla, and et al. Review of Particle Physics. Progress of Theoretical and Experimental Physics, 2020(8):083C01, August 2020.
[19] Fred Reines and Clyde Cowan Jr. Reines-cowan experiments: Detecting the poltergeist. Technical report, Los Alamos National Laboratory, 1997. [20] Werner Rodejohann. Neutrino-Less Double Beta Decay and Particle Physics. International Journal of Modern Physics E, 20(9):18331930, January 2011.
14 [21] R. C. Slansky, S. A. Raby, T. Goldman, G. T. Garvey, and Necia Grant Cooper. The oscillating neutrino: An introduction to neutrino masses and mixing. Technical report, Los Alamos National Laboratory, 1997.
[22] B. Vasiliev. Neutrino as speci c magnetic Υ-quantum. Journal of Modern Physics, 8(3):338 348, 2017.
15