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# What Is the Proton Radius Puzzle?

Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.

Much of modern science relies on precise basic values of universal constants, like the acceleration due to gravity or Planck's constant. Another one of these numbers we are seeking precision on is the radius of a proton. Jan C. Bernauer and Randolf Pohl decided to help narrow down the proton radius value in an attempt to refine some particle physics. Unfortunately, they instead found an issue that cannot be easily dismissed: Their finding is good to 5 sigma – a result so confident the likelihood of it happening by chance is just 1 in a million. Oh boy. What can be done to resolve this (Bernauer 34)?

## Background

We may have to look at quantum electrodynamics, or QED, one of the best-understood theories in all of science (pending this investigation) for some possible clues. It has its roots in 1928 when Paul Dirac took quantum mechanics and merged them with special relativity in his Dirac Equation. Through it, he was able to show how light was able to interact with matter, increasing our knowledge of electromagnetism as well. Through the years, QED has proven to be so successful that most experiments in the field have an uncertainty of error or less than a trillionth! (Ibid)

So naturally Jan and Randolf felt their work would just solidify another aspect of QED. After all, another experiment that proves the theory only makes it stronger. And so they went about creating a new setup. Using electron-free hydrogen, they wanted to measure the energy changes it went through as the hydrogen interacted with electrons. Based on the motion of the atom, scientists could extrapolate the proton radius size, first found using normal hydrogen in 1947 by Willis Lamb through a process now known as the Lamb Shift. This is really two separate reactions at play. One is virtual particles, which QED predicts will alter the energy levels of the electrons, and the other is proton/electron charge interactions (Bernauer 34, Baker).

Of course, those interactions are dependent on the nature of the electron cloud around an atom at a particular time. This cloud is in turn affected by the wave function, which can give the probability of an electron’s location at a particular time and atomic state. If one happens to be in an S state, then the atom processes a wave function which has a max at the atomic nucleus. This means that electrons do have a possibility of being found inside with protons. In addition, depending on the atom, as the radius of the nucleus grows then so does the chance of an interaction between protons and electrons (Bernauer 34-5).

Though not a shocker, the quantum mechanics of an electron being inside the nucleus are not a common sense issue and a Lamb Shift comes into play and helps us with measuring a proton’s radius. The electron in orbit actually does not experience the full force of the proton charge in the instances when the electron is inside the nucleus, and so therefore the total strength between the proton and electron decreases in such instances. Enter an orbital change and a Lamb Shift for the electron, which will result in an energy differential between the 2P and 1S state of 0.02%. Though the energy should be the same for a 2P and a 2S electron, it isn’t because of this Lamb Shift, and knowing it to a high precision (1/1015) gives us accurate enough data to start making conclusions. Different proton radius values account for different shifts and over an 8-year period Pohl had gotten conclusive and consistent values (Bernauer 35, Timmer, Baker).

## The New Method

Bernauer decided to use a different method for finding the radius using scattering properties of electrons as they passed by a hydrogen atom, aka a proton. Because of the electron’s negative charge and the proton’s positive charge, an electron passing by a proton would be attracted to it and have its path deviated. This deflection of course follows the conservation of momentum, and some of it will be transferred to the proton courtesy of a virtual proton (another quantum effect) from the electron to the proton. As the angle at which the electron is scattered from increases, the momentum transfer increases as well while the wavelength of the virtual proton decreases. Moreover, the smaller your wavelength, the better the resolution of the image. Sadly, we would need an infinite wavelength to fully image a proton (aka when no scattering occurs, but then no measurements would occur in the first place), but if we can get one that is just slightly bigger than a proton we can get something at least to look at (Bernauer 35-6, Baker).

Therefore, the team, using the lowest momentum possible and then extended the results to approximate a scattering of 0 degrees. The initial experiment ran from 2006 to 2007, and the next three years were devoted to analyzing the results. It even gave Bernauer a Ph. D. After the dust settled, the proton radius was found to be 0.8768 femtometers, which was in agreement with previous experiments using hydrogen spectroscopy. But Pohl decided on using a new method using a muon, which has 207 times the mass of an electron and decays within 2 * 10-6 seconds but otherwise has the same properties. They used this in the experiment instead, which allowed the muon to get 200 times closer to the hydrogen and thus get better deflection data and increase the chance of the muon going inside the proton by about a factor of 200­3, or 8 million. Why? Because the larger mass allows for a greater volume and thus allowed for more space to be covered as it traverses. And on top of this, the Lamb Shift is now 2%, much easier to see. Add a large cloud of hydrogen and you greatly increase the chances of collecting data (Bernauer 36, Pappas, Baker, Meyers-Streng, Falk).

With this in mind, Pohl went to the Paul Scherrer Institute accelerator to fire his muons into hydrogen gas. The muons, being the same charge as electrons, would repel them and potentially push them out, allowing muon to move in and create a muonic hydrogen atom, which would exist in a highly excited energy state for a few nanoseconds before falling back to a lower energy state. For their experiment, Pohl and his team made sure to have muon in the 2S state. Upon entering the chamber, a laser would excite the muon into a 2P, which is too high an energy level for the muon to possibly appear inside the proton, but upon interacting near it and with the Lamb Shift in play, it could find its way there. The change in energy from 2P to 2S will tell us the time the muon was possibly in the proton, and from there we can calculate the proton radius (based on speed at the time and the Lamb Shift) (Bernauer 36-7, Timmer "Researchers").

Now, this only works if the laser is specifically calibrated for a jump to a 2P level, meaning it can only have a specific energy output. And after the jump to a 2P is achieved, a low energy X-ray is released when the return to the 1S level happens. This serves as a check that the muon indeed was properly sent to the right energy state. After many years of refinement and calibration, as well as waiting for a chance to use equipment, the team had enough data and was able to find a proton radius of 0.8409 ± 0.004 femtometers. Which is concerning, because it is 4% off from the established value but the method used was supposed to be 10 times as accurate as the previous run. In fact, the deviation from the established norm is over 7 standard deviations. A follow-up experiment used a deuterium nucleus instead of a proton and again orbited a muon around it. The value (0.833 ± 0.010 femtometers) was still different from the prior method to 7.5 standard deviations and agreed with the Lamb Shift method. That means it isn't s statistical error but instead means something is wrong (Bernauer 37-8, Timmer "Hydrogen", Pappas, Timmer "Researchers," Falk).

Normally, this kind of result would indicate some experimental error. Maybe a software glitch or a possible miscalculation or assumption was made. But the data was given to other scientists who ran the numbers and arrived at the same conclusion. They even went over the whole setup and found no underlying errors there. So scientists began to wonder if maybe there is some unknown physics involving muon and proton interactions. This is entirely reasonable, for muon magnetic moment doesn’t match what the Standard Theory predicts, but results from the Jefferson Lab using electrons instead of muons in the same set-up but with refined equipment also yielded a muonic value, pointing to new physics as an unlikely explanation (Bernauer 39, Timmer "Hydrogen", Pappas, Dooley).

In fact, Roberto Onofrio (from the University of Padova in Italy), thinks he might have it figured out. He suspects that quantum gravity as described in the gravitoweak unification theory (where gravity and weak forces are linked) will resolve the discrepancy. You see, as we get to a smaller and smaller scale, Newton’s gravity theory works less and less, but if you could find a way to set it proportional weak nuclear forces then possibilities arise, namely that the weak force is just a result of quantum gravity. This is because of the small Planck vacuum variations that would arise from being in a quantum situation at such a small scale. It would also provide our muon with extra binding energy beyond the Lamb Shift that would be flavor based because of the particles present in the muon. If this is true, then follow-up muon variations should confirm the findings and provide evidence for quantum gravity. How cool would it be if gravity really ties charge and mass like this? (Zyga, Resonance)

## Works Cited

Baker, Amira Val. "The Puzzle of the Proton Radius." Resonance.is. Resonance Science Foundation. Web. 10 Oct. 2018.

Bernauer, Jan C and Randolf Pohl. “The Proton Radius Problem.” Scientific American Feb. 2014: 34-9. Print.

Dooley, Phil. "The puzzle of a proton's proportions." cosmosmagazine.com. Cosmos. Web. 28 Feb. 2020.

Falk, Dan. "Proton Size Puzzle." Scientific American. Dec. 2019. Print. 14.

Meyer-Streng. "Shrinking the proton again!" innovations-report.com. innovations report, 06 Oct. 2017. Web. 11 Mar. 2019.

Pappas, Stephanie. “Mysteriously Shrinking Proton Continues to Puzzle Scientists.” Livescience.com. Purch, 13 Apr. 2013. Web. 12 Feb. 2016.

Resonance Science Foundation. "The Proton Radius Prediction and Gravitational Control." Resonance.is. Resonance Science Foundation. Web. 10 Oct. 2018.

Timmer, John. “Hydrogen Made with Muons Reveals Proton Size Conundrum.” arstechnica.com. Conte Nast., 24 Jan. 2013. Web. 12 Feb. 2016.

---. "Researchers orbit a muon around an atom, confirm physics is broken." arstechnica.com. Conte Nast., 11 Aug. 2016. Web. 18 Sept. 2018.

Zyga, Lisa. “Proton Radius Puzzle May Be Solved by Quantum Gravity.” Phys.org. ScienceX., 26 Nov. 2013. Web. 12 Feb. 2016.