There’s been a lot of reporting recently on a puzzle in particle physics that I haven’t previously written about. There have been two attempts, a preliminary one in 2010 and a more detailed one reported just this month, to measure the size of a proton by studying the properties of an exotic atom, called “muonic hydrogen”. Similar to hydrogen, which consists of a proton orbited by an electron (Figure 1), this atom consists of a proton and a short-lived heavy cousin of the electron, called the muon (Figure 2). A muon, as far as we have ever been able to tell, is just like an electron in all respects except that it is heavier; more precisely, the electromagnetic force and the strong and weak nuclear force treat electrons and muons in exactly the same way. Only the first two of these forces should play a role in atoms (and neither gravity nor any force due to the Higgs field should matter either). So because we have confirmed our understanding of ordinary hydrogen with very high precision, we believe we also understand muonic hydrogen very well also. But something’s amiss.
Since the muon is about 200 times heavier than the electron, muonic hydrogen is about 200 times smaller (across) than ordinary hydrogen. [For an argument as to why, see the end of this article.] And so the proton, which has a diameter about 60,000 times smaller than ordinary hydrogen, is only 300 times smaller than muonic hydrogen. That makes the details of muonic hydrogen more sensitive to the proton’s size, and thus allows for a more precise measurement.
How is this done? On the experimental side, the experimenters use a laser whose energy per photon can be adjusted, and they measure the energy of photons needed to make muonic hydrogen transition from one state to another (similar to what is shown in Figure 3). [In quantum mechanics, atoms and other similar systems can only exist in very particular states, each one with a very particular mass, shape, size, and other properties.] Only photons of precisely the right energy will do the job for a particular transition. On the theoretical side, one uses known properties of protons, muons, and electromagnetic forces, etc., and (treating the proton’s size as completely unknown) one calculates carefully what the energy of the required laser photons is expected to be. The answer depends on the proton’s unknown size, so by requiring the calculation agrees with the measurement, one learns what the proton’s size is! Or rather, one learns this as long as one assumes
- the measurement was done accurately,
- the calculation contains no errors,
- and no important effects were left out of the calculation.
The reason for the recent excitement is that a similar technique for measuring the proton’s size has been used many times for transitions between states of ordinary hydrogen, and it gives a different answer. It is far less precise, but has been done many different ways, so the average of all the different measurements is estimated to be only 10 times less precise than the new measurements in muonic hydrogen. The value for the proton’s size obtained from ordinary hydrogen is about 4% larger than obtained from muonic hydrogen. This is illustrated in Figure 4, which is taken from a recent useful review article on the subject by Pohl, Gilman, Miller and Pachucki. [Note I have abridged their caption.] More specifically, the measurements based on transitions between states of ordinary hydrogen give an average proton radius of about 0.88 fermi (where a “fermi” = femtometer is a quadrillionth of a meter = a thousandth of a millionth of a millionth of a meter = 10-15 meters — and a meter is about 39 inches.) The uncertainty on this result is claimed to be 0.01 fermi. The measurements based on muonic hydrogen transitions give an answer of 0.84 fermi, with a much smaller uncertainty. Precise numbers (valid before the recent improved muon-based result) are given in Figure 4.
Meanwhile, there’s another strategy for measuring the proton’s size, using high-energy electrons which fly into hydrogen gas and occasionally scatter off the protons deep within the hydrogen atoms (Figure 5.) By looking at the pattern of energies and directions of the scattered electrons, physicists can infer the size of the proton. And the answer they obtain, though again not as precise as obtained from muonic hydrogen, is about as precise and gives about the same answer as that obtained from transitions between states of ordinary hydrogen: 0.88 fermi. This is illustrated in Figure 6, taken from the same recent review article.
This is not a dramatic difference — it represents only a 4% discrepancy. But the previous measurements of the proton radius were believed to be precise to about 0.9% (for a single standard deviation.) So the new measurements differ from the average of the old ones by a relatively large amount, which the authors state is about 7 standard deviations [though how this number was obtained is obscure to me currently. Presumably it is explained clearly in the published paper, which is in Science Magazine. I will try to clarify this point later today if I can.]
Well, 7 standard deviations (or “sigmas”) is far more than the 3 to 5 standard deviations that are typically considered impressive. However, one should remember that statistically significant results simply imply discovery of either (a) something about nature or (b) a human error… with the latter always more likely. The famous faster-than-light neutrinos disagreed with Einstein’s prediction by 6 standard deviations, and all that meant was that someone made a mistake.
Now the most exciting and revolutionary explanation (but, in my current view, very unlikely) for this disagreement would be a new force or other new phenomenon that affects muons and not electrons (or affects exotic small atoms more strongly than ones of ordinary size.) Its presence would foul up the proton size measurement by changing muonic hydrogen slightly, in a way that is not accounted for in the theory calculation that is needed for extracting the proton’s size.
[Note Added: It is extremely unlikely indeed that there could be an unknown force powerful enough to actually change the proton's size! Such a force would have to counter the strong nuclear force that sets the size of the proton, and anything so large would have been observed decades ago, in other experiments. No; instead, what is most plausible is a wimpy force whose small effects subtly change the relationship between the measurement and the calculation that is done to infer the proton's size from that measurement. Then the proton has only one size, but the size wasn't properly extracted from one of the measurements because a small effect was left out.]
But a number of my colleagues have shown, in various ways, that to introduce a new force that would affect muonic hydrogen by this amount, and not muck up previous measurements of the properties of muons or electrons or protons or neutrons, and not show up in the decays of other well-studied particles, is extremely difficult; it requires arranging multiple cancellations between multiple effects, at a level that is highly unpalatable. Maybe I’ll write about this at a later time, but I warn you that you won’t like what you see. You can find relevant references in section 4.5 of the Pohl et al. review.
On top of this difficulty, the discrepancy is not (in my personal view) as crisp as it looks at first glance. The calculations required to extract the proton’s size are very complicated indeed, and have many ingredients and steps; the measurements are not very sensitive to the proton’s size, so highly precise measurements and a heck of a lot of theoretical processing are both needed before one can extract the desired information. (In the Pohl et al. review, the authors, merely to summarize the various calculations that are needed and have been performed, require well over a dozen densely written and highly technical pages.) At certain stages along the way, experts might perhaps have been a bit 0verly optimistic about how well their methods work, and if so, perhaps the resulting theoretical uncertainty in one of the proton’s size measurements is underestimated. Given the complexities and subtleties, which lie outside my expertise, the best I can do is warn you that the Pohl et al. review does point out some potential concerns. Here’s a quotation: “To summarize, the apparently simple problem of determining the slope of the form factor at Q2 = 0 [which is needed to extract the proton radius from electron-proton scattering] has numerous potential pitfalls. The weight of the evidence at this point continues to favor a larger radius, about 0.88 fm, but suggests that claiming an uncertainty at the 0.01 fm level is optimistic.” This is not the only issue where questions are raised, though without firm conclusions. It does mean one should remain cautious. A discrepancy of 7 standard deviations sounds like a lot; but if you underestimated an important source of systematic uncertainties by a factor of 2 or 3, your 7 standard deviations could drop to an unimpressive 3.5 or 2 in a big hurry. So a very careful reconsideration of the uncertainties is necessary.
Fortunately many additional related measurements (discussed in section 5 of the Pohl et al. review) are being planned, and collectively these should help clarify and perhaps resolve this puzzle. Meanwhile the relevant theorists are clearly working hard to check previous calculations and assure that the limitations on their methods are properly accounted for.
At this point I am not optimistic that this discrepancy has a profound origin. But hey — you never really know for certain until the situation is satisfactorily settled — so we’ll need to keep a close eye on this puzzle as it evolves. There’s a very low but non-zero probability that it is of great importance. And if it is, I personally doubt anyone has yet thought of the reason.