Neutrino oscillation

"Neutrino oscillation" is based on what is known as "the wave-particle duality", that is, according to quantum mechanics laws, a particle, when it is propagated, behaves like a wave.

The fundamental "weak" interaction not only can generate neutrinos (as is the case in nuclear reactions within the sun), but it can also make them observable when they interact in an experimental apparatus, producing various electrically charged particles that can be detected. The analysis of weak interaction has shown the existence of three different kinds, or "flavours" as physicists say, of neutrinos. In previous studies it has been observed that the neutrino emitted in nuclei decays or in reactions within the sun and other stars, is associated to an electron (e) and is called electron neutrino ( ). The others are the muon neutrino and the tau neutrino .

We could now have a look at the solar neutrinos to see how the assumed "oscillation" takes place. We have to consider the propagation of neutrinos (generated as ) towards the Earth, where our detector is situated. We shall see that, because of neutrino oscillation, some of our are "magically" transformed into different neutrinos (but the secret lies in the laws of quantum mechanics).

To be simple, let us consider just two neutrino families, and . This is the typical case in which the prevailing "mixing" for the is with the , closer to it in our elementary particles classification: at a mass scale for electrically charged particles associated to neutrinos, the muon is indeed the closest to the electron. What if the property of being propagated without changes does not apply to and (called flavour eigenstates), but just to some of their "mixings"[1], the so-called and , mass eigenstates? Clearly, in the case of neutrino propagation we have to argue in terms of mass eigenstate and not flavour eigenstates.

To clarify the notion of mixing, let us consider (Figure 5) a system of Cartesian coordinates where x and y are the flavour eigenstates and another system of Cartesian coordinates x'-y' (where x' and y' are the mass eigenstates) rotated by a small angle with respect to the system x-y. In the second system, a point P(x,O) on x (standing for a pure flavour eigenstate) is no longer represented by a single coordinate (x') , but takes a small component also in the other coordinate (y'). In other words, it is represented by a combination or a "mixing" of two components x' and y'. The extent of the mixing is uniquely determined by the rotational angle of one Cartesian system to the other. This angle is used as a quantitative parameter to describe the situation[2] and is called "mixing angle". If the mixing angle is small, the eigenstates are almost pure flavour eigenstates and vice versa. The flavour eigenstate generated in the Sun thus breaks up into its two mass eigenstates and ; we then follow the latter on their way towards the Earth. We see what role their possible masses[3], or rather the mass difference, do play during propagation. It should be noted that if a neutrino mass is not zero, different neutrino types will possibly have different masses and on an increasing scale, as is the case for the corresponding charged particles e, and . For this reason, any mass difference will be close to the mass of the heaviest neutrino, since the other masses are presumably much smaller, hence negligible.

For a given momentum[4] the energy associated to mass eigenstates is all the more higher as is their "rest" mass; indeed, according to the mass-energy equation, a particle "rest" mass contributes to its total energy, together with kinetic energy (associated to the momentum). Like all elementary particles, neutrino eigenstates in propagation are represented by waves, and their rate grows with energy. Thus, if neutrinos have a mass, and each neutrino type a different one, their corresponding rates (so called phases) are also different. This is not without consequences.

Even without resorting to the mathematical instruments used by physicists, necessary to reach a thorough understanding of the phenomenon, we can follow the neutrinos on their way towards the Earth and regard their mass eigenstates as waves propagated with a different rate according to the neutrino mass. If the eigenstates have the same mass, the corresponding waves reach the Earth with the same time relation (or with the same "phase"). These waves can then be combined again, resulting exactly into a , as flavour eigenstate, the one detected by experiments thanks to the weak interaction. If the masses of the eigenstates are different, the corresponding waves are propagated with a different rate, thus they would not reach the Earth with the same phase as at the start. When they are combined, the waves are no longer the pure flavour state they had when they started. There is rather a mixing. As quantum mechanics prescribes, we see a at times and a , at other times, and the probability is given by the size of the mixing. These are neutrino oscillations, whereby an observer on the Earth could see a that was never generated !

An analogy (Figure 6) can help us to better understand how a change of phase in the waves can change the flavour eigenstates. In the theory of colours we can distinguish "basic colours" (red, blue and green) and "compound colours", such as violet, a mixture of red and blue. We can use colour mixing as an analogy for neutrino oscillation.

Imagine that a given source generated a "violet" wave. Violet (analogy for a flavour eigenstate) is actually a compound colour, made up by mixing basic colours (an analogy for mass eigenstates) red and blue. The emitted wave is then made by a "red" wave (dotted curve in the figure) and by a "blue" one (continuons curve), whose initial values give the right shade of violet when they are mixed. Let us now take basic colours red and blue[5] for propagation. If the red and blue waves are propagated the same way, their overlap at every distance from the source results in the same shade of violet everywhere. If they are propagated differently, the proportion of each colour is different at each point, and so is the resulting colour seen by an observer, whose eye is globally sensitive rather than to isolated basic colours, to their mixing or overlap. Since the starting colour is actually a compound of two different basic colours (mass eigenstates) and since they are propagated differently, a compound colour (flavour self state) can be observed that is different from the starting one and can change in all points! So the word "oscillation" is not due to the wave representation of particles, but rather to the fact that the colour observed (flavour eigenstate) changes at each point as it leaves its source, following the oscillatory law. In some points, the wave could even appear completely red or completely blue to some observer.