Home Page Overview Site Map Index Appendix Illustration About Contact Update FAQ



The Missing Neutrinos
Detection of Neutrino Mass
A Toy Model for Neutrino Oscillation (2019)
Neutrino Mass Terms
Cosmic Neutrinos
Neutrino Telescope
Superluminal Neutrino (see Neutrino as Tachyon)

The Missing Neutrinos

Helicity Back in the 1950s it was generally believed that neutrino has no mass and it exists only as a left-handed neutrino or right-handed anti-neutrino (see Figure 01, and Weyl spinor). Helicity is defined as the component of spin along the direction of motion, it is always perpendicular to the orbital angular momentum if there is any participating in weak interaction. For massless particle, helicity is an intrinsic property; its value is fixed in all reference frames. Otherwise, it is not a “fundamental” property for particles with mass.

Figure 01 Helicity
[view large image]

Later on it is found that there are three flavors of neutrino - the electron neutrino, muon neutrino, and tau neutrino. They are similar to each other except carrying different mass.

However, it seems that something is missing. For more than 30 years, scientists have been capturing electron-neutrinos generated by nuclear fusion in the Sun. These observations have always counted fewer neutrinos than predictions from the best models. Other observations involved the impact of cosmic ray on nuclei in the atmosphere, it is expected that the ratio of muon-neutrinos to electron-neutrinos is 2 to 1. The count inevitably has a shortfall of muon-neutrinos with a ratio of about 1.3 to 1.


Detection of Neutrino Mass

Neutrino Mixing

This case of missing neutrinos can be explained if the neutrinos possess mass in such a way that these "mass states" superimpose each other during free flight. At the end of the journey weak interaction reveals the true identity as different flavor (one of the tau-neutrino, muon-neutrino or electron-neutrino) according to the interference pattern at the time of detection.

Figure 02 shows what happens after the beta decay of a neutron in outer space. The electron neutrino turns into superimposition of three mass states. The interference pattern or composition varies as it travels through space. It can be computed statistically that the flavor ratio is 5():2():2(), i.e., there is only 5/9 chance of detecting a and hence the shortfall in the counting.

Figure 02 Neutrino Mixing

Neutrino mixing is expressed mathematically by the mixing matrix V (in analogy to the quark mixing) as shown in Figure 03, where the mixing angles ( ij = 12, 13, 23) and the phase angle are four parameters determining the amount of mixing. The neutrino states on the left of the equation are the flavor states showed up in weak interaction, while the states on the right (with the numerical subscripts)
CKM Matrix Neutrino Mixing are called mass states corresponding to free neutrino with different mass. Neutrino mixing is large in comparison to the quark mixing as shown in Figure 04. The origin of mixing is not explained by the Standard Model (SM). Indeed, the massive neutrinos are the first experimental evidence for physics beyond SM, which is now regarded as an effective theory - a low energy approximation to a deeper, still unknown theory. Neutrino mixing

Figure 03 CKM Matirx
[view large image]

Figure 04 Mixing
[view large image]

is then considered as a correction within SM providing a window to the new discovery before formulating in detail the deeper theory. (see "Theory of Neutrion Oscillation")

Neutrino Measurement Atmospheric Neutrinos Figure 05 shows the agreement between the Super-K measurement and theory with neutrino oscillation. The neutrino in the upward direction would have to travel as long as 13,000 km, i.e., the diameter of the Earth. The horizontal direction would be about 500 km, i.e., the distance to the edge of the atmosphere (see Figure 06). The Sudbury Neutrino Observatory (SNO) in Ontario measured the total number of neutrinos from the Sun as well as the number of electron-neutrinos alone, and it shows that the total is much greater. The accounting seems to balance according to oscillation (see Figure 02).

Figure 05 Measurements [view large image]

Figure 06 Super-K
[view large image]

The probability of oscillation between 2 types of flavor neutrinos (i.e., , , and ) is given by the relation:

where ij is the mixing angle, L is the distance traveled by the neutrino in meter, E stands for the energy of the neutrino in Mev, and
ji = mj2 - mi2 is the difference of the mass square in ev2. The mixing angles are determined from the amplitudes of the oscillation. The jis can be calculated from the periods.
Neutrino Mass Difference The solar neutrino measurements by SNO yields 12 ~ 30o, and 21(sun) = 5x10-5 ev2.
While those from Super-K gives 23 ~ 45o, and 23(atm) = 3.5x10-3 ev2. The short-baseline (which implies larger mass difference) LSND experiment measured the oscillation of into . It yields
()12 ~ 1 ev2 and 12 ~ 0o, which is very different from the other measurements. A sterile neutrino is required to reconcile all the data as shown in Figure 07a. Other experiments indicates 13 ~ 0o and the phase angle ~ 0o. See 2007 News.

Figure 07a Mass Difference
[view large image]

A 2011 analysis of neutrino data indicates that the best-fit model involves the usual 3 active neutrinos and 2 sterile neutrinos. The new scheme allows for CP violation, which may explain the asymmetry of matter and anti-matter among other things. See 2012 news.

These data indicate large mixing between neutrinos (in comparison to quark) and small mass (at least million times smaller relative to electron's). But the data do not provide absolute mass measurement for the neutrinos. Direct measurements of the absolute neutrino mass impose the upper bounds: < 2.2 ev, < 190 kev, and < 18.2 Mev.

Latest investigation (in 2019) reports the upper bound of the sum of neutrino masses to be : =1,3 mn 0.26 ev yielding an upper bound for the lightest neutrino mass m1 0.086 ev (see "Upper Bound of Neutrino Masses from Combined Cosmological Observations and Particle Physics Experiments", and a popular article "Scientists Share New Details About Mysterious Ghost Particle" by CNN).

Miniboone A 2007 report from the Fermilab experiment, known as MiniBooNE (for "Mini Booster Neutrino Experiment", see Figure 07b), found no evidence for the many of the muon neutrinos in the Fermilab beam oscillating into electron neutrinos (before reaching a detector 440 meters away). This study contradicts the LSND results and tends to refute the existence of the sterile neutrino. The news enables theoretical physicists to close an ugly chapter in the search for neutrino mass, because sterile neutrinos have no place in the standard model of particle physics. It would also have interfered with the growth of galaxies, changing the distribution of matter in the universe in a way that we do not observe, i.e., cosmologically, there should not be a sterile neutrino. However as the MiniBoone experiment has settled one problem, it reveals another anomaly of too many low-energy background electron neutrinos

Figure 07b Miniboone [view large image]

See 2010 news

Three Neutrinos Note that the MiniBooNE experiment has been constructed with the assistance from members of the LSND team. They would not be human if they didn't have a strong desire to see their signal confirmed and most of neutrino physics rewritten. And yet the setup is intentionally designed so that it would be almost impossible to bias the results one way or the other, and when it ruled against them, announced it openly to the world. That might not win them a Nobel prize, but it is still science at its best. The updated mass spectrum for three neutrinos is shown in Figure 07c, where it has been determined that 12 sun, and
23 atm. The fraction of each flavor state , ... is indicated by different pattern inside the bar.

Figure 07c Neutrino Mass Spectrum

Report in 2010 indicates that new data from the same MiniBooNE (with anti-neutrinos this time round) points to the existence of 4 kinds of neutrinos, raising the chances that sterile neutrinos may be real. In addition, data from WMAP show the most likely number of neutrino families in the early Universe was four, and the Chandra X-ray Observatory detected faint pulses of X-rays (from a dim dwarf galaxy) suggesting the decay of heavier neutrinos into lighter ones. But a sobering news in March 2010 from MINOS reports that the electron neutrinos do not have a high propensity of turning into sterile ones. The next batch of data, to be released in summer 2011, should give a better idea of whether this fleeting particle has finally been caught. BTW, sterile neutrino, if it exists, would be the right-handed breed (the postulated particle in the "see-saw" mechanism), which does not participate in the weak interact - that's why it is so difficult to detect. Anyway, it seems to be stirring in its grave contrary to the wish of many theoretical physicists. See 2011 news.

Daya Bay Neutrino Experiment In a paper released on 8 March 2012, the Daya Bay neutrino experiment near Hong Kong (Figure 07d) reports a measurement of the mixing angle 13 = 8.83o with a 5.2 sigma significance. The mixing angle was obtained from deficit of the expected amount of electron anti-neutrinos at distances of 0.4 - 2 km from the source and assuming 13 23. In mathematical notation it is the survival probability of the electron antineutrino = 1 - P(13) (see explicit formula).This non-zero measurement (contrary to the previous estimates which were either close to zero or below statistical significance) allows several future neutrino experiments to proceed with more confident. For example, it helps to proceed with building experiment to find out whether neutrinos behave

Figure 07d Anti-neutrino Experiment at Daya Bay

differently from anti-neutrinos. Such information may provide clues to why the universe had a preference for matter over anti-matter. See 2016 news.

According to "Nature News in 12 August 2015", 4 experiments are being planned to probe further into the properties of neutrino
Planned Neutrino Experiment (Figure 07e). Two of them from China and India are designed to resolve the structure of the mass spectrum. The purpose of the other two experiments in U.S. and Japan is to detect the difference between the neutrino/anti-neutrino and hopefully to decipher the paradox of matter/anti-matter imbalance in this universe.

Figure 07e Neutrino Experiment Plans [view large image]

The IceCube neutrino detector in Antarctica reported in 2016 that there is no signal for the hypothesized particle called the 'sterile neutrino' or at least the sterile neutrino doesn't exist at the mass range that physicists had hoped, based on anomalies from several experiments over the past three decades.


A Toy Model for Neutrino Oscillation (2019)

The conventional derivation of neutrino oscillation considers the interference of the so-called "mass eigen-state" traveling waves during the flight from the source to the detector (see "Propagation and Interference of Neutrino Mass States"). This toy model treats the interference pattern as superposition of the stationary (neutrino) flavor state wave functions similar to the superposition of the up and down spin states of an electron (see another example of time varying superposition in "Square Well Potential"). The detector collapses the superposition to register a definite flavor state depending on the proportional probability between these states. The model is based on non-relativistic quantum mechanics starting from the Schrodinger's equation :

Derivation of the above formulism is adopted closely from Chapter 18 in "Quantum Theory" by David Bohm, 1951.

Neutrino Mass Neutrino Energy In the model, it is assumed that the state of neutrino in a weak interaction process can be described by the Hamiltonian H0 with stationary energy
En = mnc2 corresponding to the three flavor states (Figure 07f). The small perturbation prepares the "superposition of wave functions" for its journey in free flight. To an external observer taking measurement of high energy neutrino in the range of Gev, the rest mass energy mc2 has to be replaced by the relativistic mass Mc2 = mc2 / (1 - v2/c2)1/2 ~ Ek + m2c4/2Ek, where Ek = pc >> mc2 is the kinetic energy (Figure 07g).

Figure 07f Neutrino Mass and Flavor States

Figure 07g Neutrino Kinetic Energy [view large image]

N.B. : The relativistic effect doesn't apply to the magnitude in Eq.(2) as it is measured by counting the particle number of each species (flavors).

As shown in Figure 07g, the (kinetic) energy Ek of neutrino covers a wide range from a few ev to more than 1016 ev. It indicates that this amount of Ek is not inherent, i.e., it is not within the scope of the governing Hamiltonian. For example, cosmic ray collision with nucleus would produce secondary particles like pions, muons, and electrons, which then decay into neutrinos (Figure 07g insert). The decaying particles have been accelerated to high energy already prior to neutrino production (see "Sources of High Energy Neutrinos"). Anyway,

Neutrino Oscillation Neutrino Mixing

Figure 07h Theory, 2-States Oscillation

Figure 07i 3-States Mixing

Thus, the model provides an alternate view of neutrino oscillation (Figure 07h). It also implies that the various flavors (or generations, families) of fermions are the different states of the same particle, i.e., they are not different species (Figure 07i).
The 3 mass states correspond to 3 different superpositions of wave functions with initial state s = 1, 2, 3 respectively. (Figure 07f).

Neutrinos with Sterile New Scjeme Sterile neutrino is similar to the class of particles having the characteristics of small mass, charge less, and rare interaction with other particles. The LSND experiment in the 1990s purported to have detected an oscillation of (m2)12 ~ 1 ev2 and 12 ~ 0o. It leaded to the proposed existence of a 4th neutrino (sterile for its secretive identity) to reconcile with other measurements (see Figure 07j). Figure 07k shows a new scheme according to the toy model and new data. That result was subsequently nullified by the MiniBooNE experiment in 2007. Since then it has reincarnated in

Figure 07j Neutrinos with Sterile [view large image]

Figure 07k New Scheme

various forms as the right-handed breed, heavy neutral lepton (also see "Seesaw"), and dark matter. Meanwhile, MiniBooNE has re-affirmed the existence of sterile neutrino in 2010 and 2018 (see "Sterile Neutrino’s Resurrection" and a "footnote").
According to this toy model, the number of generations (in conventional definition) could be more than three. There could be the 4th or nth depending on the energy of the perturbation. The sterile neutrino could be just a neutrino existing in higher energy state. The rare form of perturbation and high energy level combine to make its appearance so secretive (as shown by the observed data, see above, and also footnote). In addition, this approximate formulation will break down or not working very well if the perturnation is too much.

The toy model could be applied to the quark as well. However, comparison of the mixing matrix for neutrino and quark in the table below shows that the mixing of the quark flavor state wave functions is minimal. It is the consequence of large energy gap mns and small transition probability uns according to Eq.(4).


Mixing Matrix for Neutrino and Quark

Returning now to the original proposition of the un-perturbed Schrodinger's equation :

Such discrepancy may not be a problem as the potential curve would be very different in the real world. It would have at least a finite barrier height to allow for the escape of the particle (by quantum tunneling). The potential energy should be provided by a source of weak interaction similar to the mass in gravitational or charge in electro-static interaction (see "Energy of A Particle").

See a graphical illustration below (not to scale):

footnote : Scientists working on the MiniBooNE project have published data from its 15-year run in 2018, showing many more electron neutrinos being detected than would be expected under the Standard Model. The implication of the paper is that some of the muon neutrinos are flipping to sterile neutrinos before switching identity again to electron neutrinos (Quotes from "Has US Physics Lab Found a New Particle ?").
A comment : this alleged sterile neutrino can be identified to the heavy right-handed Majorana (virtual) neutrino in a conventional theory called "Seesaw Mechanism" to endow mass to neutrinos (also see Figure 08).


Neutrino Mass Terms


Cosmic Neutrinos


Neutrino Telescope

Neutrino Source By definition, telescope is an instrument for making distant objects to be nearer and larger. The original optical telescope has been expanded to peer into radio and gamma-ray ranges. Now in the 21st century, astronomers are ready to capture signals in the form of neutrinos. The primary neutrino source in the sky comes from the Sun. It produces neutrino in 3 of the nuclear reactions inside the core - from the p-p reaction H1 + H1 D2 + e+ + , and from the CNO cycle N13 (or O15) C13 (or N15) + e+ + .

Figure 12 Neutrino Source

These are Gev neutrino in experiment to determine the neutrino mass. More interesting neutrinos lie in the Tev range from exotic objects further away. Such neutrinos are mostly produced by the collision of high energy protons with photons or nuclei as shown in Figure 12. These astronomical neutrino sources include:
Neutrino Telescope 1. X-ray Binaries - The protons attain high energy during the accretion process, and produce neutrino flux within the accreting matter.
2. Supernova Remnants - Protons are accelerated to high energy in the expanding shell. Interaction of these protons with the matter in the shell gives rise to neutrinos and photons.
3. Active Galactic Nuclei (AGN) - High Energy protons may be accelerated by shock waves associated with the accretion flow into the black hole or in the inner regions of jets. These will then produce neutrinos by interacting with ambient radiation or gas in the environment.
4. Gamma-ray Bursts (GRB) - GRBs are the most violent phenomena in the universe involving tens of seconds long gamma-ray flashes. They could be related to black hole formation through coalescence of a binary system of either a black hole-neutron star or a neutron star - neutron star. Protons are accelerated to high energy in the fireball. They collide with the GRB gamma-rays to produce 100 Tev neutrinos.
5. Cosmic Rays - Nobody know the source of the ultra-high energy cosmic rays. Recent observations have found gamma-ray signals associated with at least 2 supernova remnants. An observation of neutrinos would provide a clear indication of proton acceleration with the directions identifying the source.

Figure 13 Neutrino Telescope

Unlike the electromagnetic waves (in all forms), neutrinos pass through dust and gas and travel in inter-galactic space unimpeded. Thus, their detection is valuable to study astronomical objects otherwise obstructed by whatever intervening. They may be hard to catch but are worth the effort.
Since neutrino rarely interacts with matter, it requires a huge volume of ice to capture a few such as the ICECUBE neutrino observatory located under kilometers of ice at the South Pole (to be completed by 2011). Depending on the flavor of the neutrino, it releases an electron, muon, or tau upon striking a proton or neutron inside the atomic nucleus (Figure 13). Eventually the secondary particle emits visible light with different optical signatures revealing the identity of the neutrino as shown in the Figure. The neutrino telescope is buried deep underground to insure that the detection is not marred by the electrons, muons, or taus produced above ground. The detection ultimately yields information about the direction and energy of the incoming neutrino.