> Is there a set of neutrino masses that is minimally-illuminating of physics beyond the standard model?

There are no neutrino masses at all in the Standard Model. You can put them in by hand, but then you break half the nice ideas used to construct the Standard Model in the first place. Or you can put them in more properly, but then there are perhaps hundreds of distinct ways to do it, and we don't know which one is realized in nature. In other words, the state we're at with neutrinos is roughly where we were at with everything else in the 1960s. In some sense neutrinos have been "new physics" for decades -- we didn't understand them 30 years ago and we don't now.

As for specific mass measurements, there are theories that predict exceptionally nice values for the neutrino mass, but also plenty that don't have any such order. It's certainly possible that knowing all the neutrino masses to high precision wouldn't help at all. After all, we know the quark masses to high precision, and that precision hasn't helped us find an origin for the pattern in their masses. We just have to take them as free parameters.

There are a few interesting things that we could get from looking at neutrino masses.

The first has to do more with the nature of the mass than the mass itself. In the standard model, electrons, muons, and taus get their mass from the Higgs field. There's a way for neutrinos to get their mass in other ways, but it requires them to be their own antiparticles. And this gives a satisfactory answer as to why their masses are so tiny, and suggests some new particles (although at electroweak unification scale, so not anything we're going to achieve with a collider any time soon). There are a number of double-beta decay experiments trying to measure the Majorana mass of neutrinos.

The other would be if the neutrino hierarchy is "inverted". The tau is heavier than the muon is heavier than the electron. Right now, with neutrinos, we can only measure the difference in masses. And so it's not clear if the neutrino with the most election portion[1] is the lightest or if the neutrino with the most tau portion is. The latter would be "inverted" from what we expect, and trying to figure out why might be interesting, though I don't know that it immediately implies new physics.

There're also other things related to masses that are interesting. Neutrino oscillations are determined by the differences in mass. Looking at these, we might be able to discover more generations of neutrino (beyond electron, mu, and tau), which would be new physics.

[1] the mass eigenstates of the neutrino (that is, the things with well-defined masses) are not weak force eigenstates, so they contain mixtures of the electron, mu, and tau neutrinos

Q. So, an atom decays and gives off some particles including a neutrino. So, we look at the mass-energy arithmetic before and after the decay and see that it all adds up but does need the tiny mass-energy of a neutrino.

That fact, that small difference, seems curious, maybe toward new physics? That is, somehow maybe the mass-energy amounts are, once again in science, whole number multiples of something small. If so, then we can look for how the other particles are whole number multiples??

I have to expect that 99+% of physics students have already thought of this.

Is there anything curious about that tiny bit of mass, e.g., why it has to be there at all?

It's a bit different than that. The decay of a neutron into a proton and an electron conserved charge, mass-energy (to an expected degree), and momentum. However, spin was not conserved. The neutrino was dreamed up as kind of a placeholder for the spin. However, it turned out that it was a real thing!

The mass-energy arithmetic should not be the thing you look at for a couple of reasons. First, it's quite difficult to measure with exactitude. Second, the binding energy for particles and their constituents plays a part that is easily within error bounds.

Thanks.

> First, it's quite difficult to measure with exactitude.

I wondered about something like that -- the mechanism really is exact to tiny accuracy, no fuzz, but it's super tough actually to measure that accurately, or some such. If my startup works, I'll return to physics!!! I promise!! Thanks.

If there are more generations of neutrinos, would that also imply extra generations of the charged leptons?

It's a reasonable hypothesis. But the new physics might also lie in why we don't see more than 3 generations of charged leptons or quarks.

In the standard model neutrino masses are 0....

Adding masses is a trivial extension, though, by giving them masses in the same way that electrons and other leptons do.