More on neutrino masses, mixing and oscillation

Neutrino oscillation is a quantum mechanics process that shows up under two conditions:

· Neutrinos are massive, whatever tiny the mass is, and thus depart from the mass less particles described in the Standard Model of particles and interactions. In addition, the masses of the different mass eigenstates should not be degenerate, i.e. they should be different.

 

· The neutrino eigenstates of the mass and of the interaction (also called the neutrino family, neutrino type or neutrino generation) do not overlay and mix – a possibility that may happen a priori as a well-known consequence of quantum mechanics.

 

 

 

The oscillation probability between two families of neutrinos is equal to

                      sin²2θ sin² (1.27 Dm² L / E)

where L (in km) is the distance between the neutrinos source and the point of detection of its interaction, E (in GeV) is their energy and Dm² (in eV²) is the square of their mass difference. The angle q fixes the mixing between the mass and interaction eigenstates. The two mass eigenstates are represented by two orthogonal linear combinations of the interaction eigenstates. Arbitrarily choosing the pair n_e,n_m as families and calling n₁ and n₂ the mass eigenstates, we have:

                      n_e = cos θ n₁ + sin θ n₂
                      n_m = cos θ n₂ - sin θ n₁

If θ is null, there is no mixing, while the mixing is maximum for θ=p/4.

The probability thus oscillates periodically between 0 and a maximum of sin²2θ with a frequency depending on ∆m² as a function of the variable L/E of which the experimenter has, partly at least, the control or which can be evaluated. Thus the name neutrino oscillation. This is exemplified below in the case of a beam initially constituted of n_e. One observes that P_osc(n_e→n_m) + P_osc(n_e→n_e) =1 such that the overall number of neutrinos is conserved

 

 

 

 

 

 

 

 

 

 

It also follows from mixing that the eigenvalues of the masses are not directly measurable. The physical objects emitted in interactions and accessible to measurement are not mass but interaction eigenstates.  For example, in the two-family mixing exemplified above, the quantity named the mass of the n_e measured in the b decay of radioactive nuclei is actually an effective mass resulting from the linear combination

                      m_ne = cos² θ m_n1 + sin² θ m_n2     which is properly normalized (cos² θ + sin² θ =1)

 The current upper limit on the effective n_e mass is of the order of 1 eV/c², i.e. 500 000 times less than the electron mass and 1 billion times less than the proton and neutron masses.

In case of mixing between more than two families, the oscillation process becomes more complex. The Standard Model foresees three families, the n_e, n_m  and n_t . They have been observed experimentally, the  n_t as late as in 1999 by DONUT. The 4 LEP experiments at CERN have shown that there are only three families of neutrinos that interact with the ordinary matter (active neutrinos), at least with a mass smaller than about 50 times the mass of the proton, or 50 billion times the current upper limit on the mass of the neutrinos. If undetected types of neutrinos that do not interact with matter (sterile neutrinos) exist, they may however mix with active neutrinos and may render the pattern of mass eigenstates and the oscillation process even more complicate.

 

There exist three experimental signals of neutrino oscillation:

· The most convincing evidence : an explanation of the strong deficit in the flux of n_e emitted by the nuclear fusion engine of the Sun as measured on the Earth, though the total flux of neutrinos, including n_m  and n_t  that are not produced in the Sun, is as expected (see e.g. the sites of the Super-Kamiokande and SNO experiments). This interpretation is now fully verified with a controlled beam of man-made n_e (see the site of the KamLAND experiment). The explanation is an oscillation of the n_e into any combination of the other two active neutrinos. The domain of Dm² in which a signal is found is around 7 10^-5 eV².

 

· A fairly convincing evidence : an explanation of the strong deficit in the flux of n_m produced by cosmic ray interactions in the atmosphere and its dependence on the energy of and on the distance traveled by the neutrino (see e.g. the site of the Super-Kamiokande experiment). This interpretation is not in contradiction with the first results of an experiment performed with a controlled beam of man-made n_m (see the site of the K2K experiment). The explanation is an oscillation of the n_m preferably into n_t though oscillation into n_e or sterile neutrinos are not excluded. The domain of Dm² in which a signal is found is around 2.5 10^-3 eV². It is the verification of the interpretation of this deficit that three experiments under preparation, MINOS, OPERA and ICARUS, are aiming at.

 

· A more controversial evidence : the appearance of an excess of n_e interactions in a very pure beam of n_m  seen by one experiment, LSND, and not by another, KARMEN-II. The two results are not, however, fully statistically incompatible. The domain of Dm² in which a signal is found is above 10^-1 eV². This result is currently checked by a more sensitive experiment, MiniBOONE.

 

Note that, would all three results be genuine, the existence of three very different domains of values for ∆m² implies at least 4 mass eigenvalues and thus the existence of sterile neutrinos in addition to the three known active families.

 

The schema below represents what we think we know of the neutrinos masses and mixing in the case there are only three families of neutrinos and, therefore, the LSND signal turns out to be fake. The main unknowns are:

· What is the absolute mass scale? The lower limit of the effective n_e mass, about 1 eV/c², is still much larger than the lowest possible mass of the heaviest neutrino, about 0.05 eV/c², that is compatible with the largest value measured for Dm².

· Why are the neutrinos so much lighter that all the other particles?

· n₃ being by convention the mass eigenstate that differs most with respect to the two others, is it lighter or heavier than the two others?

· Is the n₃ eigenstate an (almost) equal mixing of  n_m and n_t or is there a small admixture of n_e?

· Neutrinos being the only electrically neutral elementary particles, are they identical to their antineutrinos or do they differ by an intrinsic property? In the former case, the apparent stringent distinction observed in nature between neutrinos and antineutrinos would be an artifact of a collusion between the complete violation of the invariance of the week interaction for space inversion and the extreme smallness of the neutrino masses.