A precision measurement science mystery - new physics or incomplete calculations?

Again, as a distraction from persistently concerning news, here is a science mystery of which I was previously unaware.

The role of approximations in physics is something that very often comes as a shock to new students.  There is this cultural expectation out there that because physics is all about quantitative understanding of physical phenomena, and the typical way we teach math and science in K12 education, we should be able to get exact solutions to many of our attempts to model nature mathematically.   In practice, though, constructing physics theories is almost always about approximations, either in the formulation of the model itself (e.g. let's consider the motion of an electron about the proton in the hydrogen atom by treating the proton as infinitely massive and of negligible size) or in solving the mathematics (e.g., we can't write an exact analytical solution of the problem when including relativity, but we can do an order-by-order expansion in powers of \(p/mc\)).  Theorists have a very clear understanding of what means to say that an approximation is "well controlled" - you know on both physical and mathematical grounds that a series expansion actually converges, for example.  

Some problems are simpler than others, just by virtue of having a very limited number of particles and degrees of freedom, and some problems also lend themselves to high precision measurements.  The hydrogen atom problem is an example of both features.  Just two spin-1/2 particles (if we approximate the proton as a lumped object) and readily accessible to optical spectroscopy to measure the energy levels for comparison with theory.  We can do perturbative treatments to account for other effects of relativity, spin-orbit coupling, interactions with nuclear spin, and quantum electrodynamic corrections (here and here).  A hallmark of atomic physics is the remarkable precision and accuracy of these calculations when compared with experiment.  (The \(g\)-factor of the electron is experimentally known to a part in \(10^{10}\) and matches calculations out to fifth order in \(\alpha = e^2/(4 \pi \epsilon_{0}\hbar c)\).).  

The helium atom is a bit more complicated, having two electrons and a more complicated nucleus, but over the last hundred years we've learned a lot about how to do both calculations and spectroscopy.   As explained here, there is a problem.  It is possible to put helium into an excited metastable triplet state with one electron in the \(1s\) orbital, the other electron in the \(2s\) orbital, and their spins in a triplet configuration.  Then one can measure the ionization energy of that system - the minimum energy required to kick an electron out of the atom and off to infinity.  This energy can be calculated to seventh order in \(\alpha\), and the theorists think that they're accounting for everything, including the finite (but tiny) size of the nucleus.  The issue:  The calculation and the experiment differ by about 2 nano-eV.  That may not sound like a big deal, but the experimental uncertainty is supposed to be a little over 0.08 nano-eV, and the uncertainty in the calculation is estimated to be 0.4 nano-eV.  This works out to something like a 9\(\sigma\) discrepancy.  Most recently, a quantitatively very similar discrepancy shows up in the case of measurements performed in 3He rather than 4He.  

This is pretty weird.  Historically, it would seem that the most likely answer is a problem with either the measurements (though that seems doubtful, since precision spectroscopy is such a well-developed set of techniques), the calculation (though that also seems weird, since the relevant physics seems well known), or both.  The exciting possibility is that somehow there is new physics at work that we don't understand, but that's a long shot.  Still, something fun to consider (as my colleagues (and I) try to push back on the dismantling of US scientific research.)