Massive Neutrinos: Flavor Mixing Of Leptons And Neutrino Oscillations

Flavor Mixing of Leptons and Neutrino Oscillations

Harald Fritzsch

• Publisher: WSPC
• Year: 2015
• ISBN:
  9789814704786
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

Since the discovery of neutrino oscillations neutrino physics has become an interesting field of research in physics. They imply that neutrino must have a small mass and that the neutrinos, coupled to the charged leptons, are mixtures of the mass eigenstates, analogous to the flavor mixing of the quarks. The mixing angles for the quarks are small, but for the leptons two of the mixing angles are large. The masses of the three neutrinos must be very small, less than 1 eV, but from the oscillation experiments we only know the mass differences — the absolute masses are still unknown. Also we do not know, if the masses of the neutrinos are Dirac masses, as the masses of the charged leptons and of the quarks, or whether they are Majorana masses.

In this volume, an overview of the present state of research in neutrino physics is given by well-known experimentalists and theorists. The contents — originated from talks and discussions at a recent conference addressing some of the most pressing open questions in neutrino physics — range from the oscillation experiments to CP-violation for leptons, to texture zero mass matrices and to the role of neutrinos in astrophysics and cosmology.

Since the discovery of neutrino oscillations neutrino physics has become an interesting field of research in physics. They imply that neutrino must have a small mass and that the neutrinos, coupled to the charged leptons, are mixtures of the mass eigenstates, analogous to the flavor mixing of the quarks. The mixing angles for the quarks are small, but for the leptons two of the mixing angles are large. The masses of the three neutrinos must be very small, less than 1 eV, but from the oscillation experiments we only know the mass differences — the absolute masses are still unknown. Also we do not know, if the masses of the neutrinos are Dirac masses, as the masses of the charged leptons and of the quarks, or whether they are Majorana masses.

In this volume, an overview of the present state of research in neutrino physics is given by well-known experimentalists and theorists. The contents — originated from talks and discussions at a recent conference addressing some of the most pressing open questions in neutrino physics — range from the oscillation experiments to CP-violation for leptons, to texture zero mass matrices and to the role of neutrinos in astrophysics and cosmology.

Readership: Researchers and graduate students in the field of particle physics.

Birth of Lepton Flavor Mixing

Makoto Kobayashi
KEK,

1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan

The history of the lepton flavor mixing could be traced back to the early 60s, when Maki, Nakagawa and Sakata (MNS) discussed the neutrino mixing. Their work emerged in the course of the developments of the composite model of elementary particles which was initiated by Sakata. In Sakata’s model, the weak interaction of the hadrons can be described by two types of transitions among the fundamental triplet baryons. This pattern of the weak interaction of the hadrons is similar to that of leptons provided that the neutrino consists of a single species. From this similarity, Maki, Nakagawa, Ohnuki and Sakata proposed the so-called Nagoya model, in which the fundamental triplet baryons are regarded as composite states of the leptons and a hypothetical object called B-matter. Although the Nagoya model did not make a remarkable success, when existence of two kinds of neutrinos was discovered in 1962, Maki, Nakagawa and Sakata precisely formulated lepton flavor mixing to associate leptons with the fundamental baryons in the framework of the Nagoya model. To recognize their contributions, the flavor mixing matrix of the lepton sector is called the MNS matrix. See also: M. Kobayashi, “Neutrino mass and mixing — The beginning and future”, Nucl. Phys. B (Proc. Suppl.) Vol. 235–236, (2013), pp. 4–7.

Neutrino Masses and Flavor Mixing

Harald Fritzsch

Department für Physik, Universität München,
Theresienstraße 37, 80333 München, Germany
[email protected]

We discuss the neutrino oscillations, using texture zero mass matrices for the leptons. The reactor mixing angle θ_ι is calculated. The ratio of the masses of two neutrinos is determined by the solar mixing angle. We can calculate the masses of the three neutrinos: m₁ ≈ 0.003 eV, m₂ ≈ 0.012 eV, m₃ ≈ 0.048 eV.

The flavor mixing of the quarks is parametrized by the CKM-matrix. There are several ways to describe the CKM-matrix in terms of three angles and one phase parameter. I prefer the parametrization, given below, which Z. Xing and I introduced years ago,^1,2 given by the angles θ_u, θ_d and θ:

where c_u,d ~ cos θ_u,d, s_u,d ~ sin θ_u,d, c ~ cos θ and s ~ sin θ.

The angle θ_u describes the mixing between the quarks “u–c,” the angle θ_d the mixing between the quarks “d–s” and the angle θ the mixing among the heavy quarks “t, c–b, s.” The three angles have been determined by the experiments:

Presumably the flavor mixing angles are not fixed values, but functions of the quark masses. If the masses change, the mixing angles will also change. For example, the Cabibbo angle θ_C

13° could be given by the ratio of the quark masses:

This relation works very well:

Such a relation can be derived, if the quark mass matrices have “texture zeros,” as shown by S. Weinberg and me in 1977.^3–5

Let me discuss a simple example, using only four quarks: u, d–c, s. Their mass matrices have a zero in the (1,1)-position:

These mass matrices can be diagonalized by a rotation. The rotation angles are:

The Cabibbo angle is given by the difference:

In the complex plane this relation describes a triangle. The phase parameter is unknown, however it must be close to 90°, since the Cabibbo angle is given by the ratio m_d/m_s:

Thus the triangle is close to a rectangular triangle.

For six quarks the “texture zero” mass matrices for the quarks of charge (2/3) and of charge (–1/3) are:

We can calculate the angles θ_u and θ_d as functions of the mass eigenvalues:

Using the observed mass values for the quarks, we find:

The experimental values agree very well with the theoretical results:

Now we consider the flavor mixing of the leptons. The neutrinos, emitted in weak decays, are mixtures of different mass eigenstates. This leads to neutrino oscillations — at least two neutrinos must have finite masses.

The lepton flavor mixing is described by a 3 × 3 unitary matrix U, analogous to the CKM mixing matrix for the quarks. It can be parametrized in terms of three angles and three phases. I use a parametrization, introduced by Z. Xing and me:⁶

where c_l,ν ~ cos θ_l,ν s_l,ν ~ sin θ_l,ν, c ~ cos θ and s ~ sin θ. The angle θ_ν is the solar angle θ_sun, the angle θ is the atmospheric angle θ_at, and the angle θ_l is the “reactor angle.” The phase matrix P_v = Diag{e^iρ, e^iσ, 1} is relevant only, if the neutrino masses are Majorana masses.

The neutrino oscillations are described by two large mixing angles:

The reactor angle θ_l is much smaller: θ_l

13°.

We assume t...

Table of contents

• Cover
• Half Title
• Advanced Series on Directions in High Energy Physics
• Title Page
• Copyright
• Preface
• Contents
• 1. Birth of Lepton Flavor Mixing
• 2. Neutrino Masses and Flavor Mixing
• 3. Fermion Mass Matrices, Textures and Beyond
• 4. General Lepton Textures and Their Implications
• 5. Status and Implications of Neutrino Masses: A Brief Panorama
• 6. Neutrino Masses and SO10 Unification
• 7. Relating Small Neutrino Masses and Mixing
• 8. Predictions for the Dirac CP Violation Phase in the Neutrino Mixing Matrix
• 9. Sterile Neutrinos in E6
• 10. Phenomenology of Light Sterile Neutrinos
• 11. Neutrino–Antineutrino Mass Splitting in the Standard Model: Neutrino Oscillation and Baryogenesis
• 12. The Strong CP Problem and Discrete Symmetries
• 13. Neutrino Interaction with Background Matter in a Noninertial Frame
• 14. Seesaw Models with Minimal Flavor Violation
• 15. Generating Majorana Neutrino Masses with Loops
• 16. Three-Neutrino Oscillation Parameters: Status and Prospects
• 17. On the Majorana Neutrinos and Neutrinoless Double Beta Decays
• 18. Dirac or Inverse Seesaw Neutrino Masses from Gauged B – L Symmetry
• 19. Searching for Radiative Neutrino Mass Generation at the LHC
• 20. Lepton-Flavor Violating Signatures in Supersymmetric U(1)′ Seesaw
• 21. From Electromagnetic Neutrinos to New Electromagnetic Radiation Mechanism in Neutrino Fluxes
• 22. Lepton Number Violation and the Baryon Asymmetry of the Universe
• 23. Status of the MAJORANA DEMONSTRATOR: A Search for Neutrinoless Double-Beta Decay
• 24. Towards Neutrino Mass Spectroscopy Using Atoms/Molecules
• 25. Detection Prospects of the Cosmic Neutrino Background
• 26. Supernova Bounds on keV-mass Sterile Neutrinos
• 27. Precision Calculations for Supersymmetric Higgs Bosons
• 28. Minimal Supersymmetric Standard Model with Gauged Baryon and Lepton Numbers
• 29. From the Fourth Color to Spin-Charge Separation — Neutrinos and Spinons
• 30. Measurement of the Underlying Event Activity Using Charged-Particle Jets in Proton–Proton Collisions at √s = 2.76 TeV