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 \(U\):
$$ f_i = \sum_j U_{ij} e_j $$
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 a unitary 3 × 3 matrix 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 only 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 \(N\) of neutrinos. There would then be \(N\) masses. The \(N\times N\) 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 is called the Cabbibo–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.
By now maybe you're curious to actually see these two matrices. So I'll show them to you. To do this I have to introduce some standard physics notation. The flavor eigenstates have exciting names: \(\nu_e,\nu_\mu\) and \(\nu_\tau\). The mass eigenstates have bland names: \(\nu_1,\nu_2\) and \(\nu_3\). The Pontecorvo–Maki–Nakagawa–Sakata matrix expresses the three flavor eigenstates in terms of the three mass eigenstates like this: \[ \begin{bmatrix} {\nu_e} \\ {\nu_\mu} \\ {\nu_\tau} \end{bmatrix} = \begin{bmatrix} U_{e 1} & U_{e 2} & U_{e 3} \\ U_{\mu 1} & U_{\mu 2} & U_{\mu 3} \\ U_{\tau 1} & U_{\tau 2} & U_{\tau 3} \end{bmatrix} \begin{bmatrix} \nu_1 \\ \nu_2 \\ \nu_3 \end{bmatrix} \] Here are the absolute values of the matrix elements, with error bars for 3 standard deviations of error: \[ \begin{bmatrix} |U_{e 1}| & |U_{e 2}| & |U_{e 3}| \\ |U_{\mu 1}| & |U_{\mu 2}| & |U_{\mu 3}| \\ |U_{\tau 1}| & |U_{\tau 2}| & |U_{\tau 3}| \end{bmatrix} = \left[\begin{array}{rrr} 0.801 – 0.845 & 0.513 – 0.579 & 0.143 – 0.156 \\ 0.233 – 0.507 & 0.461 – 0.694 & 0.631 – 0.778 \\ 0.261 – 0.526 & 0.471 – 0.701 & 0.611 – 0.761 \end{array}\right] \] However, the matrix entries are not in fact real! People usually write this matrix in terms of four real numbers called \(\theta_{13}, \theta_{23}, \theta_{13}\) and \(\delta_\text{CP}\). If the last one is zero, the entries of the Pontecorvo–Maki–Nakagawa–Sakata matrix are all real and CP is not violated by neutrinos. But in fact \(\delta_\text{CP} \ne 0\).
Here's how you actually write the Pontecorvo–Maki–Nakagawa–Sakata matrix in terms of these four real numbers: $$ \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta_{23} & \sin\theta_{23} \\ 0 & -\sin\theta_{23} & \cos\theta_{23} \end{bmatrix} \begin{bmatrix} \cos\theta_{13} & 0 & \sin\theta_{13}e^{-i\delta_\text{CP}} \\ 0 & 1 & 0 \\ -\sin\theta_{13}e^{i\delta_\text{CP}} & 0 & \cos\theta_{13} \end{bmatrix} \begin{bmatrix} \cos\theta_{12} & \sin\theta_{12} & 0 \\ -\sin\theta_{12} & \cos\theta_{12} & 0 \\ 0 & 0 & 1 \end{bmatrix} $$ $$ = \begin{bmatrix} \cos\theta_{12}\cos\theta_{13} & \sin\theta_{12} \cos\theta_{13} & \sin\theta_{13}e^{-i\delta_\text{CP}} \\ -\sin\theta_{12}\cos\theta_{23} - \cos\theta_{12}\sin\theta_{23}\sin\theta_{13}e^{i\delta_\text{CP}} & \cos\theta_{12}\cos\theta_{23} - \sin\theta_{12}\sin\theta_{23}\sin\theta_{13}e^{i\delta_\text{CP}} & \sin\theta_{23}\cos\theta_{13}\\ \sin\theta_{12}\sin\theta_{23} - \cos\theta_{12}\cos\theta_{23}\sin\theta_{13}e^{i\delta_\text{CP}} & -\cos\theta_{12}\sin\theta_{23} - \sin\theta_{12}\cos\theta_{23}\sin\theta_{13}e^{i\delta_\text{CP}} & \cos\theta_{23}\cos\theta_{13} \end{bmatrix} $$ It's a mess, huh? If you've ever seen 'Euler angles' as a way of describing a rotation in 3d space, this is like that — but with an extra twist, because this space is not real but complex.
Here are the best estimates of these numbers as of July 2020: $$ \begin{align} \theta_{12} & = {33.44^\circ}^{+0.78^\circ}_{-0.75^\circ} \\ \theta_{23} & = {49.0^\circ}^{+1.1^\circ}_{-1.4^\circ}\\ \theta_{13} & = {8.57^\circ}^{+0.13^\circ}_{-0.12^\circ} \\ \delta_{\textrm{CP}} & = {195^\circ}^{+51^\circ}_{-25^\circ} \\ \end{align} $$ All this data is from here:
For comparison, here are the absolute values of the entries in the Cabbibo–Kobayashi–Maskawa matrix: \[ \begin{bmatrix} |V_{ud}| & |V_{us}| & |V_{ub}| \\ |V_{cd}| & |V_{cs}| & |V_{cb}| \\ |V_{td}| & |V_{ts}| & |V_{tv}| \end{bmatrix} = \left[\begin{array}{rrr} 0.9740 & 0.2265 & 0.0036 \\ 0.2264 & 0.9732 & 0.0405 \\ 0.0085 & 0.0398 & 0.9992 \end{array}\right] \] I've left out the error bars out of laziness, but this matrix is much more accurately known than the Pontecorvo–Maki–Nakagawa–Sakata matrix. It's also much closer to the identity matrix! In other words, the quark mass eigenstates are only slightly different from their flavor eigenstates, while for neutrinos they're so different we really can't match them up in any systematic way. Nobody knows why they're so different!
Here's where I got the matrix:
For two good introductions to the problem of neutrino mass and neutrino oscillations, see:
Also, for a big website packed with links to all the major experiments studying neutrino oscillations, try this:
© 2020 John Baez
baez@math.removethis.ucr.andthis.edu