There are different theories to explain the masses and oscillations of neutrinos. The most conservative is the "New Standard Model", a slight variant of the old textbook Standard Model of particle physics. The new version is actually prettier than the old one, because now the masses of leptons work just like the masses of quarks.
I don't want to explain how this works in detail here. Instead, I just want to explain why this theory needs 4 numbers besides the 3 neutrino masses to describe the behavior of neutrino oscillations. These numbers describe something called the Pontecorvo-Maki-Nakagawa-Sakata matrix. One reason I want to explain this matrix is because I mention it in my list of fundamental physical constants. My explanation will be sketchy yet technical. However, I still think it's worth reading... especially because it leads up to an amazing fact about the difference between the past and the future.
The first key point is that the states in which a neutrino has a definite mass are not the same as the states that have a definite flavor! This is already amazing, but it's well-known that quarks work this way, so it's not too odd to assume leptons work this way too.
Since there are 3 flavors of neutrino, we describe this effect using a 3-dimensional complex Hilbert space with two orthonormal bases: the "mass eigenstate basis"
e_{1}, e_{2}, e_{3}
and the "flavor eigenstate basis".
f_{1}, f_{2}, f_{3}
They are related by a 3 × 3 unitary matrix U:
Ue_{i} = f_{i}
This matrix is called the Pontecorvo-Maki-Nakagawa-Sakata matrix — if you want to show off to your friends, say that three times fast! Besides the masses of the neutrinos, it's the numbers in this matrix that describe the phenomenon of neutrino oscillations in the New Standard Model.
Now, to describe the unitary 3 × 3 matrix U involves a total of 9 real parameters. But, without changing the physics we can redefine the mass eigenstate basis by multiplying each basis element by a phase — for a total of 3 phases. Similarly, without changing the physics we can redefine the flavor eigenstate by multiplying each basis element by a phase - for a total of 3 more phases. These phases corresponds to ways of changing the matrix U without changing the physics. So, you might think the matrix U had just 9 - 3 - 3 = 3 physically relevant entries.
But that's not quite right. If we multiply all the basis elements — that is, all the e_{i} and all the f_{i} — by the same phase, the matrix U doesn't change at all. So, there are actually 9 - 3 - 3 + 1 = 4 parameters in the matrix U which actually affect the physics of neutrinos.
That's where the number 4 comes from!
For extra fun, we can see how this would work for N generations of neutrinos. There would then be N masses. The N × N unitary matrix U would take N^{2} real parameters to describe, but there would be only
N^{2} - N - N + 1 = (N - 1)^{2}
parameters that actually affect the physics. So, N masses and (N - 1)^{2} extra numbers describing oscillations.
In fact, the Pontecorvo-Maki-Nakagawa-Sakata matrix actually affects the behavior of all leptons, not just neutrinos. Furthermore, a similar trick works for quarks — but then the matrix U is called the Cabibbo-Kobayashi-Maskawa matrix. This one was actually discovered first.
A neat thing about these matrices is that if their entries can all be made real (after multiplying by suitable phases as described above), the New Standard Model has complete symmetry under time reversal. But if we can't get all their entries to be real, time reversal symmetry is violated! Particles know the difference between past and future!
If there were only 2 generations of quarks and leptons, we could always get the matrix entries to be real. Thus, there would be no violation of time reversal symmetry.
But, violations of time reversal symmetry are also observed to occur in the physics of kaons!
(Some of these are inferred from violation of "CP symmetry". This is a symmetry that combines switching particles with their antiparticles — "charge conjugation", or C — and switching left and right — "parity", or P. The so-called "CPT theorem" says that given some reasonable-sounding assumptions, CP violation can only occur if time reversal symmetry — T — is also violated. So, experiments detecting CP violation can also be taken as evidence for a violation of time reversal symmetry. But, there's also some more direct evidence.)
This led Kobayashi and Maskawa to predict in 1973 that there were 3 generations of quarks and leptons. In 1975 they were proven correct when Perl and collaborators discovered the tau. By now we have seen all the particles in the 3rd generation: tau and tau neutrino, top and bottom quark.
Interestingly, however, the Cabibbo-Kobayashi-Maskawa matrix is very close to being real... so close that we're still not 100% sure that this matrix is the complete explanation of the difference between future and past in the standard model.
If you want to actually see the numbers in the Cabibbo-Kobayashi-Maskawa matrix, try this:
The Pontecorvo-Maki-Nakagawa-Sakata matrix is much less well known, since our only source of information about it is neutrino oscillation experiments, which are very hard to do. In particular, nobody seems to have a clue whether its entries are all real! So, we don't know if leptons violate time reversal symmetry.
For two good introductions to the problem of neutrino mass and neutrino oscillations written in the mid-1990s, see:
But beware: while the theories haven't changed drastically since then, the data is vastly better, thanks to all the experiments people are doing! Here's a more recent review article written by a theorist:
The first page shows a picture of the Pontecorvo-Maki-Nakagawa-Sakata matrix (without calling it this) and what was known about it in 2004. The paper then gives a nice quick review of some big open puzzles concerning neutrinos, and some theories people are considering. It has references to more technical papers on the current experimental data.
Also, for a big website packed with links to all the major experiments studying neutrino oscillations, try this:
© 2011 John Baez
baez@math.removethis.ucr.andthis.edu