How Big Is The Proton?

A New Measurement Of Proton Size

A new article in the journal Nature reports a proton size four percent smaller than that found by previous methods. As the abstract explains (paragraph breaks added, footnotes omitted):

The proton is the primary building block of the visible Universe, but many of its properties—such as its charge radius and its anomalous magnetic moment—are not well understood.

The root-mean-square charge radius, rp, has been determined with an accuracy of 2 per cent (at best) by electron–proton scattering experiments.

The present most accurate value of rp (with an uncertainty of 1 per cent) is given by the CODATA compilation of physical constants. This value is based mainly on precision spectroscopy of atomic hydrogen and calculations of bound-state quantum electrodynamics (QED).

The accuracy of rp as deduced from electron–proton scattering limits the testing of bound-state QED in atomic hydrogen as well as the determination of the Rydberg constant (currently the most accurately measured fundamental physical constant).

An attractive means to improve the accuracy in the measurement of rp is provided by muonic hydrogen (a proton orbited by a negative muon); its much smaller Bohr radius compared to ordinary atomic hydrogen causes enhancement of effects related to the finite size of the proton. In particular, the Lamb shift (the energy difference between the 2S1/2 and 2P1/2 states) is affected by as much as 2 per cent.

Here we use pulsed laser spectroscopy to measure a muonic Lamb shift of 49,881.88(76) GHz. On the basis of present calculations of fine and hyperfine splittings and QED terms, we find rp = 0.84184(67) fm, which differs by 5.0 standard deviations from the CODATA value of 0.8768(69) fm.

Our result implies that either the Rydberg constant has to be shifted by −110 kHz/c (4.9 standard deviations), or the calculations of the QED effects in atomic hydrogen or muonic hydrogen atoms are insufficient.

Were Previous Proton Size Measurements Wrong?

The size of the proton measured using hydrogen atoms where the electrons have been replaced with muons is 4% smaller than prior measurements from more ordinary contexts that were previously believed to rule out this result to five standard deviations of accuracy.

The most recent estimates. . . . put the radius of the proton at around 0.8768 femtometres (1 femtometre = 10-15 metres). . . . Pohl and his colleagues fired muons from a particle accelerator at a cloud of hydrogen. Hydrogen nuclei each consist of a single proton, orbited by an electron. Sometimes a muon replaces an electron and orbits around a proton. Using lasers, the team measured relevant muonic energy levels with extremely high accuracy and found that the proton was around 4% smaller than previously thought.

QED, Used To Calculate The Result, Is Remarkably Reliable

The estimates of proton size were calculated by from the experimental data using quantum electrodynamics (QED), one of the most precisely accurate and most heavily experimentally confirmed theories in physics.

I'll let Richard P. Feynman, writing in "QED: The Strange Theory of Light and Matter" in 1985 (at page 7-8) explain just how accurate a theory it has been and how important it is to physics:

The theory of quantum electrodynamics has now lasted for more than fifty years, and has been tested more and more accurately over a wider and wider range of conditions. At the present time I can proudly say that there is no significant different between experiment and theory!

Just to give you an idea of ow the theory has been put through the wringer, I'll give you some recent numbers: experiments have Dirac's number at 1.00115965221 (with an uncertainty of about 4 in the last digit); the theory puts it at 1.00115965246 (with an uncertainty of about five ties as much). To give you a feeling for the accuracy if these numbers, it comes out something like this: If you were to measure the distance from Los Angeles to New York to this accuracy, it would be exact to the thickness of a human hair. That's how delicately quantum electrodynamics has, in the past fifty years, been checked -- both theoretically and experimentally. By the way, I have chosen only one number to show you. There are other things in quantum electrodynamics that have been measured wit comparable accuracy, which also agree very well. Things have been checked at distance scales that range from one hundred times the size of the earth down to one-hundredth the size of an atomic nucleus. These numbers are meant to intimidate you into believing that the theory is probably not too far off. . . .

I would like to again impress you with the vast range of phenomena that the theory of quantum electrodynamics describes It's easier to say it backwards: the theory describes all the phenomena of the physical world except the gravitational effect, the thing that holds you in your seats (actually, that's a combination of gravity and politeness, I think), and radioactive phenomena, which involve nuclei shifting in their energy levels.

What Could Cause This Result?

1. This result could simply be an experimental mistake, although since it has been published in the peer reviewed prestigious science journal "Nature" it is probably not an obvious mistake, if there is one. The latest result won't be viewed a reliable until it is replicated by an independent lab. But, creating muonic hydrogen and zapping it with lasers is much less expensive to do that just about any other experiment in fundamental high energy physics that hasn't already been done and is likely to be fruitful, so this could happen very quickly.

2. A second possibility is that for some reason currently not known to science that protons compress themselves more tightly in muonic hydrogen than in ordinary hydrogen where an electron is captured by a proton.

The theory governing the way that protons form from three quarks that are bound together by gluons, called quantum chromodynamics (QCD for short), is much less well understood than QED, because it is hard to make calculations from the equations in the theory. Numerical predictions in QCD from the theory are much less accurately determined than the experimental evidence that is known to us. So, we can't know if there are subtle discrepancies between the theoretical predictions of QCD and the results actually observed.

If QCD calculations should include a term for strong force interactions between electrons and muons, on one hand, and quarks and gluons on the other, for example, and the effect was a non-linear function of mass that had a negligible effect for the small electron mass but a much bigger effect for the larger muon mass, this could produce this result without being observed in prior experiments.

But, conventional QCD equations don't have any term that would obviously predict that the size of a proton should be related to whether an electron or a muon is orbiting it, and the overall group theory structure of quantum mechanics suggests that there shouldn't be terms like this in the equation.

3. A third possibility is that muon related constants in QED have been inaccurately measured in the past. Muons were first detected in 1936 (all unattributed quotes below are to this Wikipedia source), provoking the famous quip from Nobel laureate I. I. Rabi at the time, "Who ordered that?" This was several decades after the electron has been discovered and was well understood.

QED has, however, with great success in past, treated muons as identical to electrons except that they have a different mass. Quoting Feynman again (at page 143-144 from the same source):

[T]he muon . . . is in every way exactly the same as the electron, except that its mass is much higher -- 105.8 MeV, compared to 0.511 for the electron, or about 206 times heavier. It's just as if God wanted to try out a different number for the mass! All of the properties of the muon are completely describable by the theory of electrodynamics -- . . . you just put in a different value for [mass].[fn 6]

[fn 6] The magnetic moment of the muon has been measured very accurately -- it has been found to be 1.001165924 (with an uncertainty of 9 in the last digit), while the value for the electron is 1.00115965221 (with an uncertainty of 3 in the last digit). You might be curious as to why the magnetic moment of the muon is slightly higher than that of the electron. One of the diagrams we drew had the electron emitting a photon that disintegrates into a positron-electron pair. . . There is also a small amplitude that the emitted photon could make a muon-anti-muon pair, which is heaier than the original electron. This is unsymmetrical, because when the muon emits a photon, if that photon makes a positron-electron pair that pair is lighter than the original muon The theory f quantum eletrodynamics accurately describes every electrical property of the muon as well as the electron.

Because the muon has a mass about 200 times higher than the electron . . . This has enabled us to test whether electrodynamics still behaves according to the theory at distances 200 times smaller than we've been able to test before -- although these distances are still more than eighty decimal places larger than the distances at which the theory alone might run into trouble with infinities[.]

Still, muon properties aren't quite as rigorously tested as electron properties because they are unstable and produced only in particular situations, making it much harder to conduct experiments testing their properties than would otherwise be the case.

[N]either ordinary radioactive decay events nor nuclear fission and fusion events (such as those occurring in nuclear reactors and nuclear weapons) are energetic enough to produce muons. Only nuclear fission produces single-nuclear-event energies in this range, but due to conservation constraints, muons are not produced.

In nature, muons are found mostly in cosmic rays where they are produced when protons traveling at close to the speed of light hit the atmosphere.

Some of the deviation in proton size, for example, might be due to inaccuracies in the muon mass value used to determine it, or in the constants that govern the likelihood that photons will products muon-anti-muon pairs, which while known accurately, are known far less accurately than comparable numbers for the electron and the other properties of the electron and muon like their magnetic moments.

Still, while muons are unstable, they have a relatively long life compared to other standard model particles that make them comparatively easy to study: "It is an unstable subatomic particle with the second longest mean lifetime (2.2 µs), exceeded only by that of the free neutron (~15 min)." The equality of decay rates of positive and negative muons has been established to one part in 10,000, which while accurate compared to 4%, is far less accurate than other known measurements of muon properties.

But, there is still no plausible way that these factors should account for a four percent discrepancy, and these uncertainties would have been considered when considering the accuracy of the muonic hydrogen proton size calculation.

Similarly, some theoretically possible decay modes of muons not allowed in the Standard Model, but not central to the Standard Model's integrity, should change the results only by parts per quadrillion, not four percent.

4. Yet another possibility is that the calculations or experiments failed to properly consider the possibility that particles of Muonium were present in the experiment, a possibility that should lead to a lower observed average proton size (since some supposedly measured protons are really muonium), and doesn't call for any deviations from the Standard Model.

A positive muon, when stopped in ordinary matter, can also bind an electron and form an exotic atom known as muonium (Mu) atom, in which the muon acts as the nucleus. The positive muon, in this context, can be considered a pseudo-isotope of hydrogen with one ninth of the mass of the proton. Because the reduced mass of muonium, and hence its Bohr radius, is very close to that of hydrogen, this short-lived "atom" behaves chemically — to a first approximation — like hydrogen, deuterium and tritium.

Experiments measuring isolated muons, as opposed to muons in atoms, wouldn't detect Muonium (or its inverse, a positron that has captured a muon).

The trouble is that experiments calibrated to find a proton size where it was predicted to be in muonic hydrogen didn't identify any muonic hydrogen. If there were really both muonic hydrogen and Muonium present, it should have produced two alternate proton sizes, rather than a single consensus value with no results in the predicted range.

5. Could it be gravity?

Gravity is 200 times more significant in a proton-electron hydrogen atom than it is in a proton-muon hydrogen atom. A 4% difference due to gravitational attraction, would be equivalent to a 0.0002% impact of gravity in ordinary hydrogen.

But, to the extent that Newton's formula for the gravitational attraction between two masses (GMm/r^2) is a ballpark approximation of the strength of gravity in this context, the gravitational effects predicted are vastly smaller than those needed to produce the discrepancy observed.

Mass has far more of an impact in a hydrogen or muonic hydrogen atom via the fact that it is incorporated in electromagnetic calculations than it does via gravity between protons and either electrons or muons at the distances involved in this experiment.

6. Could this result involve a running coupling constant?

The strength of the electric charge is intimately related to the electron and muon's magnetic moment, and is very similar for the two particles in a predictable way, as explained above.

But, one of the fundamental concepts motivating grand unification theories in physics, is that the coupling constants of the electromagnetic force, weak force and strong force are not truly fundamental. Instead, they are "running constants" that converge in high energy contexts.

The magnitude of the difference between the high energy and low energy coupling constants of the electroweak force is on the order of 4%.

This experiment could be creating a high energy situation that has changed not the size of the proton, but the magnitude of the coupling constant being used to estimate its size in some way. In contrast, perhaps prior muon experiments were based on lower energy situations where the ordinary electromagnetic coupling constant was appropriate to use.

7. Could this be evidence of a compound electron and muon?

One of the puzzles of quantum physics is that quarks have charges one-third as strong as electrons and muons. Protons and neutrons are made of three quarks with charges of +/- 1/3 each. Electrons and positrons and W particles have charges of +/- 1 each.

There has been no experimental evidence so far to suggest that electrons or muons are compound like protons and neutrons, however.

Compound muons might behave differently from compound electrons in muonic hydrogen in a way that a fundamental particle muon might not differ from a fundamental electron. For example, perhaps a compound muon might have different predicted energy levels than a fundamental particle muon.

8. Were old experiments simply inaccurate?

The old measures of proton size were not terribly accurate. And, experience suggests that error estimations in terms of standard deviations routinely overestimate the actual accuracy of measurements in physics. Experimental error can account for a significant share of the deviation of the previously measured values.

Conclusion

The muonic hydrogen estimates of proton size in this experiment strongly point to some sort of physics beyond the Standard Model.

In and of itself, this isn't that surprising a thing to discover. Several other experiments have found discrepancies between the Standard Model and experiment. Almost all theoretical physicists agree that is some form of fundamental physics out there not described fully by the Standard Model.

And, it is also worth observing that many of the deviations from the Standard Model already observed involve strange quarks, which bear the same relationship to one of the kinds of quarks that make up ordinary protons and neutrons, that muons bear to electrons. Given the strangeness of the strange quark, it would hardly be that remarkable if the muon has some strange properties of its own.

But, almost none of the deviations from the Standard Model seriously considered by theoretical physicists have predicted that proton size measured in ordinary hydrogen would differ from proton size measured in muonic hydrogen for any reason.

Indeed, many of the models that extend the Standard Model predict just as strongly as the Standard Model that the observed difference in proton sizes should not exist, for the very reasonable reason that there was no experimental motivation before now to develop a theory that would predict this result.

Since many theoretical extensions of the Standard Model have tight internal constraints that aren't easily adjusted to reflect this result, a seemingly modest experimental discrepancy in apparent proton size could rule out of large share of previously plausible fundamental physics theories.