Neutrinos (Part 2: Neutrino oscillations — simplified)

In my previous post on the neutrinos and the solar neutrino problem, I ended by writing that the current explanation for why there were missing neutrinos at the detector is because the neutrinos have changed state while travelling towards earth and since the detectors were calibrated on detecting only some particular states, they were missing some of them.

Discrepancies between the expected number of neutrinos of any given flavor and observations were also noted when counting neutrinos originating from other sources than the sun: for example, from nuclear reactors and from cosmic rays. This problem affects all three neutrino flavors (electron-like, muon-like or tau-like). Muon neutrinos are produced in muon decays, which affect muons produced naturally in the atmosphere by cosmic rays. Muons created in particle accelerators also produce muon neutrinos when they decay. Experiments that detect only muon neutrinos also find discrepancies.

To understand neutrino oscillation, one must think of neutrinos as waves rather than particles and have some basic knowledge of quantum mechanics. (*1*) At the start of their journey, neutrinos know definitely that they are electron-like, muon-like or tau-like what is called the different neutrino “flavors”. Because neutrinos are born in weak force interactions alone or together with other flavors of neutrinos, there is no law of Nature that dictates a certain flavor of particle should have a certain mass, so the neutrino is born knowing exactly what type of flavor but not exactly what mass it has. That is how quantum mechanics works, because of the Heisenberg uncertainty principle, you can never know both the position and the momentum (including its mass) of a particle with certainty (because momentum is mass times speed).

One theory states that neutrino flavor is a superposition of mass (Hamiltonian) eigenstates.

In the 2-flavors case (a simplification of the real 3-flavors theory), the neutrino flavors that we know from weak interactions (the weak eigenstates) are linear combinations of underlying mass eigenstates, where the mass eigenstates (denoted as ν1 and ν2) are the free particle solutions to the Schrodinger wave equation. We cannot assume a priori that the mass eigenstates and flavor eigenstates are equivalent. If they are, as we shall see momentarily, oscillation cannot take place. If they are not, then these mass eigenstates cannot be directly observed in their pure form, only weak eigenstates (via the weak interaction). The weak eigenstates are related to the mass eigenstates by a simple unitary (length preserving) matrix:

\displaystyle \begin{bmatrix}\mid\nu_e\rangle\\\mid\nu_{\mu}\rangle\end{bmatrix} = \begin{bmatrix}\cos \theta & \sin \theta \\-\sin \theta &\cos \theta \end{bmatrix} \begin{bmatrix}\mid\nu_1\rangle\\\mid\nu_2\rangle\end{bmatrix}

In other words, the weak eigenstates, aka the neutrino flavors, are coherent linear superpositions of the mass eigenstates. Here, \displaystyle \theta is referred to as the mixing angle.

We assume that the mass eigenstates for the free neutrino as a function of time are simply plane waves. Combining the superposition in the Schrodinger equation results in the equation describing that the pure flavor state at t=0 (explained in *1*) is a superposition of different flavors states at t>0. From that, it is possible to calculate the probability of flavor transition as a function of time in vacuum.

Suppose we have a weak decay producing an electron neutrino at time t=0. That neutrino expressed in terms of the mass eigenstates ν1 and ν2 would be:

\displaystyle \mid\psi(0)\rangle = \mid\nu_e\rangle = \cos\theta\mid\nu_1\rangle + \sin\theta\mid\nu_2\rangle

The two-flavor oscillation probability in vacuum as a function of distance L is:

\displaystyle P\left(\nu_e \rightarrow \nu_{\mu}\right) = \sin^22\theta\sin^2\left( \frac{\Delta m_{21}^2Lc^3}{4\hbar E} \right)

Several things can be noted from this formula. Immediately after the particle is created, at L=0, the probability is 0 that its flavor has changed (because of the second sin^2 factor). The probability varies with the distance that a neutrino travels, fluctuating periodically as L increases. It reaches a maximum every time L is such that the second factor is 1. The first factor determines what the maximum of the probability can be, and it will be nonzero if the mixing angle is nonzero.

The average of this probability across distances and energies is only a function of the mixing angle.

\displaystyle \langle P\left(\nu_e \rightarrow \nu_{\mu}\right) \rangle = \frac{1}{2} \sin^22\theta

Finally, let’s go back to the particle/wave analogy, the net result is that different mass states propagate at different speeds, and the mixing matrix (containing several mixing components) transfers this effect to flavor states. Interference between waves (representing mass states) for different flavor states is the basic cause of neutrino oscillation. So the probabilities associated with observation of specific flavor states depend in a complicated way on the (squared) mass differences between the specific mass states, as well as a particle’s energy and the distance it has traveled since creation.

Additionally, so far I’ve talked about oscillations in vacuum, but because ordinary matter is made from electrons (among other things) but not from muons and taus, electron-neutrinos have different interactions with ordinary matter than muon- and tau-neutrinos do. These interactions, which occur through the weak nuclear force, are tiny. But if a neutrino passes through a great deal of matter, such as the outer layers of the sun and then, the earth, these small effects can add up and have a big impact on the oscillations. Experiments have to take this into account to produce accurate results.

It is important to stress and maybe repeat myself again on this point. The equations presented before all use the squared mass difference and not the neutrino absolute mass. For neutrino oscillations to occur, at least one of the mass states must be non-zero. This simple statement has huge implications, for oscillations to happen, the neutrino must have mass. Furthermore the masses of the mass states must be different, else delta m is 0 and P is 0. In conclusion, careful measurements of neutrino oscillations are in fact the fastest way to learn about the properties of neutrinos and maybe determining the (absolute) neutrino masses once and for all, which has much deeper consequences and raises even more questions. Some of these questions will be addressed in the next part of this series on neutrinos.


Neutrino flavor-oscillations for dummies


Fundamentals of Neutrino Physics and Astrophysics, C. Giunti and C.W. Kim. Oxford University Press (2007)

Lecture notes from course PX435 Neutrino Physics, downloaded in March 2015

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