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Physicists Tackle the Shrinking Proton Puzzle

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Back in 2010, physicists were baffled by the results of an experiment to measure the radius of the proton using an exotic form of hydrogen. It was significantly smaller than expected: 0.00000000000003 millimeters smaller. Maybe that doesn’t seem all that significant, but at the subatomic scale it’s huge — an 8-sigma difference, or around 4% in normal-person terminology.

It’s not just that physicists like to measure everything as accurately as possible down to multiple decimal places. The size of the proton is relevant to certain predictions of quantum electrodynamics (QED), the theory that describes the interactions of light with matter, and this odd result, if it can’t be resolved, might mean something is missing in QED. Anyway, various experiments since then have sought to resolve the conundrum, with little success — the mysterious shrinking proton continues to be a baffling anomaly, according to physicists reporting on the latest experimental results at the APS April Meeting held in Denver last weekend.

I am on record as championing the Bohr model of the atom when speaking to general audiences with little to no background in physics, in which electrons move about the atomic nucleus in circular orbits. But this is a case where the Bohr model just isn’t gonna cut it. You really need the quantum mechanical description of the atom to grok the significance of the mystery of the shrinking proton. Technically, the electrons don’t really “move” around the nucleus in orbits. Electrons are really waves (although they show up as particles when you perform an experiment to determine its position), and those waves are stationary.

Sure, you can check to see where an electron is, but each time you do, it will show up in a different position — not because it’s moving, but because of the superposition of states. The electron doesn’t have a fixed position until you look at it and the wave function collapses. (If you make a ton of measurements and plot the various positions of the electron, eventually you’ll get a ghostly orbit-like pattern.) So the electron can be anywhere inside this region — including inside the actual proton, as weird as that sounds.

Given the above, just what does it mean to talk about the radius of the proton? The proton is made up of three charged quarks bound together by the strong nuclear force. However, “The proton is not a hard shell, it’s fuzzy like a cloud,” MIT’s Jan Bernauer explained at an APS press conference in Denver. “How would you describe the radius of a cloud?” Well, you’d likely talk about the density of water molecules within that cloud. The proton radius is similar, except in this case we’re talking about the distribution of the charge density. The radius of the proton is that distance at which the charge density drops below a certain energy threshold.

Once you’ve settled on how to define it, how the heck do you measure that radius? You can use the electron. Physicists usually probe the proton radius (of hydrogen atoms, in the present case) either by scattering electrons or by studying the difference between atomic energy levels known as the Lamb shift, after Willis Lamb, who first measured this shift in 1947, ultimately winning the Nobel Prize in 1955 for his work. Here’s the late Hans Bethe talking about the Lamb shift:

Per Nature:

“Physicists can measure the size of the proton by watching as an electron interacts with a proton. A single electron orbiting a proton can occupy only certain, discrete energy levels, which are described by the laws of quantum mechanics. Some of these energy levels depend in part on the size of the proton.”

There are numerous techniques for doing this, according to Argonne National Lab physicist John Arrington. There is electron scattering, a versatile tool that lets physicists probe things at different scales; at high energies, it’s great for measuring quarks and gluons, and at lower energies, it’s ideal for probing the structure of the proton. Then there’s spectroscopic techniques, in which lasers are used to zap a sample, producing a signature that can be analyzed by a spectrometer. There are many different kinds of spectroscopy, depending on the kind of laser used and what, exactly, an experiment is designed to measure. In the case of the proton radius, electron and muon spectroscopy are the most useful.

According to Arrington, in their quest to resolve the shrinking proton puzzle, physicists have thus far used electron scattering [(0.8770(60)], electron spectroscopy [0.8758(77)], and muon spectroscopy, in which the electron is replaced by its heavier sibling, the muon [0.8409(4)] to measure the radius. It’s the latter muon spectroscopy result that is problematic. The hope is that experiments using muon scattering will fill in the data gap and ultimately help resolve the conundrum.

“Assume the proton is a ball of charge, with the electron dancing through the proton,” said Randolf Pohl of the Max Planck Institute of Quantum Optics, who was one of the physicists who performed the original 2010 spectroscopic measurements, as well as the latest confirming results. “When it is in the center of the proton, it is attracted equally from all sides, with the charges all around it, so there is no net attraction between proton and muon. This shifts the whole energy state up. That is the effect we are looking at with laser spectroscopy, when we measure the difference between two energy levels: when the electron is inside and outside the proton.” In other words, they’re measuring the Lamb shift.

Pohl’s experiments used muonic hydrogen, where the electron orbiting the nucleus is actually a muon, the heavier sibling of an electron. The muon is nearly 200 times heavier, which means its orbital is much smaller and it has a much higher probability of being inside the proton — 10 million times more likely, in fact. And that means it’s 10 million times more sensitive as a measurement technique, because it is closer to the proton.

“We thought we would measure the same radius [as prior experiments] but with smaller uncertainty,” Pohl confessed. Instead, they found a significantly smaller proton radius than all the prior measurements that had been made, and the latest results only reconfirm that 2010 finding.

“Physics is like solving a jigsaw puzzle,” Bernauer said. “We have some pieces and we find new connecting pieces to broaden our knowledge. But this part doesn’t quite fit.” One possibility is that the result is due to experimental error or a misapplication of QED theory. If that’s the case, it’s not a trivial error; Pohl and his cohorts have spent years checking and re-checking the data.

Alternatively, perhaps physicists need to make some tweaks to update the underlying QED theory to account for these unusual results. While QED is still correct, there might be something a little different about the muon’s properties that needs to be taken into account

Finally, there’s the most exciting — and hence least likely — explanation: that this is due to new physics beyond the Standard Model, possibly even indirect evidence for the “superpartners” predicted by supersymmetry theories. Or there could be another force carrier to augment the photon. Those as-yet-undetected hypothetical particles could be altering the interactions between muon and proton.

However, “The most boring explanation is the most possible: someone messed up the experiment,” Pohl said. “I would like to see more data before claiming there is something beyond the Standard Model.” But if, even after more experiments are done, the discrepancy still holds up, “Then I will get excited too.”


Antognini, Anton et al. (2013) “Proton Structure from the Measurement of 2S-2P Transition Frequencies of Muonic Hydrogen,” Science 339(6118): 417-420.

Distler, Michael O., Bernauer, Jan C., and Walcher, Thomas. (2011) “The RMS Charge Radius of the Proton and Zemach Moments,” Phys. Lett. B 696:343-347.

Lamb, Willis E.; Retherford, Robert C. (1947). “Fine Structure of the Hydrogen Atom by a Microwave Method,” Physical Review 72 (3): 241–243.

Pohl, R. et al. (2010) “The Size of the Proton,” Nature 466, 213-217.

Pohl, Randolf et al. (2013). “Muonic hydrogen and the proton radius puzzle,” submitted to Annu. Rev. Nucl. Part. Sci., January 5, 2013.


Jennifer Ouellette About the Author: Jennifer Ouellette is a science writer who loves to indulge her inner geek by finding quirky connections between physics, popular culture, and the world at large. Follow on Twitter @JenLucPiquant.

The views expressed are those of the author and are not necessarily those of Scientific American.

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  1. 1. jtdwyer 9:36 am 04/19/2013

    Interesting results. Please see
    - comment 11 RogerPink 01:21 AM 1/25/13 begins:
    “… I’m saying the Muon, orbiting closer to the nucleus, has an increased probability of being INSIDE the nucleus (than an electron would). This increased probability of presence of a negatively charged particle inside the nucleus will interfere with the Proton’s Electromagnetic Interactions with each other (the muon effectively shields the protons from each other a little in the nucleus). Such an increased in shielding between protons in the nucleus (who repel each other)would reduce the radius of the nucleus a small amount.”

    The discussion continues – to include many other comments. The findings you report seem to confirm Roger’s suggested explanation…

    Link to this
  2. 2. jtdwyer 10:47 am 04/19/2013

    Reading your article more closely, it seems that Roger’s explanation has not been considered. As you explain:
    “Pohl’s experiments used muonic hydrogen, where the electron orbiting the nucleus is actually a muon, the heavier sibling of an electron. The muon is nearly 200 times heavier, which means its orbital is much smaller and it has a much higher probability of being inside the proton — 10 million times more likely, in fact. And that means it’s 10 million times more sensitive as a measurement technique, because it is closer to the proton.”

    While the muon’s ‘orbit’ is closer to the proton than an electron would be, as you say – it’s also more frequently manifested within the proton’s boundaries, then potentially reducing the range of proton’s motion through its charge interaction, affecting its spectroscopically measured size.

    In this case, the muon is not only a ‘more accurate’ measure of proton size – because of its much greater mass it interacts with the proton differently than an electron, affecting the proton’s actual size.

    BTW, if I remember correctly, the experiment fired a muon beam at the hydrogen gas to force the replacement of electrons by hydrogen. A laser spectroscope is used to determine the proton’s size. See

    Link to this
  3. 3. sault 12:01 pm 04/19/2013

    I’m a little confused. Since a proton is made from 2 “up” quarks, each with a charge of +2/3 and 1 “down” quark, with a charge of -1/3, wouldn’t the charge inside the proton vary somewhat and not be exactly even in the “center” in all directions? It’s unclear from what I’ve read whether quarks behave like electrons when they are bound in a proton and have a probabilistic distribution of positions they can take. Does trying to determine the “radius” of the proton collapse quark wave functions like what happens with electrons when we try to determine their position? Would this collapse lower the quark’s energy somehow, leading to a smaller observed proton radius?

    I’m certain this is something I just don’t understand and I trust the physicists that have been doing this for decades WAY more than my armchair ramblings!

    Link to this
  4. 4. Jennifer Ouellette 12:53 pm 04/19/2013

    @Sault: It’s a confusing subject — there was a lot of back and forth on those points in the press briefing — and I’m afraid the best answer I, personally, can give at the moment is the catchall cop-out, “Because QUANTUM MECHANICS!” Which really isn’t helpful. I’ll see if I can’t find you a better answer.

    @jtdwyer: good points all, and hopefully people will read the older comment thread where much of this is discussed in more detail. RE: final comment on using laser spectroscopy to measure proton’s size, I thought I was clear on that point, but reading over the post this AM, I see that I was not. :) Have deleted that sentence to avoid further confusion.

    Link to this
  5. 5. jtdwyer 2:27 pm 04/19/2013

    The quarks within the proton are not separable, being bound by the strong force – the proton’s charge is +1. As I understand, the muon’s probabilistic location can range within the proton’s probabilistic location (IMO thereby proximally interacting with the proton to further constrain its probabilistic location). I hope this is correct, and that it helps…

    Jennifer Ouellette – Thanks!
    I also see that I made another error – the corrected statement is:
    “… the experiment fired a muon beam at the hydrogen gas to force the replacement of electrons by muons.”

    Link to this
  6. 6. jtdwyer 9:42 am 07/21/2013

    Also see – especially the comment by Policarpo Ulianov, who has written a report on the subject.

    Link to this

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