Protons are the fundamental subatomic particles. Under the Standard Model, we know that protons are composite particles with three valence quarks, which, along with neutrons, form hardons. Protons have a measurable size, and historically, their root mean square charge radius was measured using two different methods, converging at around 0.877 fm (1 fm = 10−15 m). A third method was used in 2010 and found a radius 4% smaller than the standard measurement, giving a radius of 0.842. More recent studies brought those findings into questions, with two studies carried out in 2019,  one by a team led by Bezginov, and the other led by Xiong, finding a radius of 0.833 fm, with an uncertainty of ±0.010 fm. New research by physicists from the Technical University of Darmstadt and the University of Bonn analysed the data from the 1990s as well as more recent findings, to try and determine what the cause of difference was, and they found that the difference between the old and new data was based on a systematic error. The puzzle remained unsolved because many physicists doubted the truth of the smaller radius size, but this study may lead to more widespread acceptance of those results. 

How Proton Radius Was Previously Measured

Until 2010, two methods were used to determine the proton charge radius : a spectroscopy method and a nuclear scattering method. 

The spectroscopy method used the energy level of electrons orbiting the nucleus. The energy levels are very sensitive to nuclear radius. Hydrogen has a nucleus with just one proton, so this method indirectly gets to the radius of the proton. Measurements have become so precise that the proton radius is used as the limiting factor when researchers are comparing experimental results with theoretical calculations. Using this method, researchers found a proton radius of some 0.8768±0.0069 fm), with around1% relative uncertainty.

The nuclear scattering method is analogous to Rutherfor’d scattering experiments that proved the existence of the nucleus. Electrons or other small particles are fired at protons, and the degree of scattering allows us to infer the size of a proton. This method found a proton radius of around 0.8775 fm. 

The 2010 Experiment That Caused So Much COntroversy

The source of the puzzle is a 2010 experiment that challenged these earlier results. Pohl et al found that by using the scattering method but replacing normal hydrogen with muonic hydrogen, they obtained different results. They believed that because muonic hydrogen has a much higher mass than normal hydrogen, its inclination to orbit 207 times closer to the hydrogen nucleus compared to an electron, making it more sensitive to a proton’s size. 

Pohl’s team obtained a result of 0.842±0.001 fm, which is 5 standard deviations smaller than the previous measurements. Pohl’s radius results were 4% smaller than the prior results, which had been believed to be accurate within 1%. Although the 2010 results came with an uncertainty limit of just 0.1%, these made a negligible impact on the discrepancy. 

Further tests only served to heighten the discrepancy to some 7.5 standard deviations, 5% smaller than the established measurement. 

Could A Systematic Error Be Behind the Discrepancy?

In order to get a proton’s radius, scientists can choose to have electrons collide into it in accelerators, which results in the electron and proton changing their direction, a process known as elastic scattering. The bigger a proton is, the greater the frequency of collisions of this kind. The proton’s expansion tells us something about the kind and degree of elastic scattering going on. 

As the velocity of the electron beam increases, the measurements become more precise, but this comes with the risk that new particles will be created in the accelerator. Indeed, particle accelerators have often been used to deliberately create new particles by ensuring that collisions occur at very high velocities. Elastic scattering becomes rarer the higher the velocity. So, in order to more properly measure protons, the physicists realised that they would only focus on data in which the electrons had relatively low energy. 

Electron-positron annihilation is the name that we give to the phenomenon in which electrons and positrons collide, and has a range of consequences. At low energies, it annihilates both electrons and positrons and creates energetic photons. At high energies, it results in the production of new particles, such as B mesons or W and Z bosons. The researchers developed a theoretical framework that allows them to use this process to calculate a proton’s radius, while taking into account previously left out data. 

With that methodology in mind, the researchers studied the old data that had caused so much controversy. Their investigations found that protons had a radius of 0.84 fentometers, which tallies with what new measurements found, and is 5% smaller than the 0.88 fmr measurement that the old data had suggested.