The puzzle of the proton radius


A new experiment has the physics world talking, and the subject is size — specifically the radius of the proton. The problem is, it seems to be smaller than previously thought. So why all the fuss?

To understand the workings of the Universe, from the very big to the very small, we have to observe and measure the fundamental characteristics of the material world. The advances of technology and knowledge enable measurements of higher and higher accuracy and thus either confirm our theories or lead us in new and unknown directions. One of the most important characteristics of the objects we study is their spatial extent – their size.

Unfortunately, in studying the very small, the traditional methods of using microscopes have long since passed their sell-by-date – in other words, in the world of subatomic particles, microscopes are so last year! Even modern techniques such as scanning probe microscopy, although achieving subatomic resolutions on the order of 0.1 nanometers, are unable to reveal the spatial extent of subatomic particles.

Thus when dealing with subatomic particles we are not able to directly measure the size. Instead we measure the energies and infer the size from established theoretical assumptions. This can lead to many uncertainties and therefore remains an active and ongoing area of research, with the current spotlight being the proton and its radius.

The size of the proton, specifically its radius, thus remains uncertain, especially since a discrepancy was observed between the latest measurements of the proton radius and that of the previously accepted value. This discrepancy is now even more significant due to a new experiment measuring the radius of deuteron (an atomic nucleus comprising one proton and one neutron) with the cause pointing to either experimental or theoretical error, or the possibility of new physics beyond the standard model.

The question we need to ask ourselves is, how do we measure the radius of the proton?

Well firstly, we need to remember that we do not measure the radius of the proton directly – instead we can only measure the changes in energy and then calculate the proton radius from this measured energy. This is typically done using elastic electron proton scattering, in which a beam of electrons is fired at a proton source and the scattering angles are measured. The scattering probability can be calculated as a function of proton radius, thus by measuring the scattering angle the proton radius can be found.


Another technique is that of hydrogen spectroscopy, where the light from a source is dispersed through a prism or diffraction grating to reveal the emission or absorption features of the source.  In the case of hydrogen an electric current is passed through a glass tube that contains a hydrogen gas. The spectral emission lines are due to the atomic transitions from a higher energy state to a lower energy state i.e. the energy shift. With higher resolving power we can reveal greater and greater detail with smaller and smaller energy shifts between states. These energy shifts can successfully be predicted by the famous relativistic wave equation known as the Dirac equation. Thus by comparing the observed emission lines with the theoretically predicted values, we can validate or challenge the accepted theories. However, with the ever-increasing resolving powers, the Dirac equation has been extended to include the observed complex atomic dynamics. These ‘quantum’ corrections are generally additive and are formally known as radiative corrections. The total calculated energy difference (energy shifts) takes into account all these quantum corrections, which in turn depends on the proton radius – thus from the observed energy shift the proton radius can be calculated.

In the 1930’s using the techniques of microwave spectroscopy, which allowed even greater resolution, an energy shift not predicted by quantum electrodynamics (QED) was observed. This energy shift — known as the Lamb shift after its discoverer Willis Lamb — was subsequently attributed to quantum vacuum fluctuations. As the energy shift is so small, the lamb shift, now known to high accuracy, is thus more sensitive to the proton radius.

Both these methods — electron-proton scattering and hydrogen spectroscopy — have consistently yielded similar results, with the latest CODATA (The Committee on Data for Science and Technology) value, based on a least-squares approximation method (i.e. an approximation method to find the best fit to the combined experimental data), giving a value of fm (femtometers).

However, in 2010 the measurement of the Lamb shift was made utilizing muonic hydrogen as oppose to electronic hydrogen and yielded a much smaller value than that of the combined electronic methods, giving rp =0.84814 fm (Pohl et al. 2010) and rp =0.84087 fm (Antognini et al. 2013). The more recent value, found in 2013, differs from the CODATA value by 4% (or in science talk, by 7 standard deviations (usually denoted sigma) which is 2 more than the accepted value for something to be rendered statistically significant in the physics world). These same spectroscopic experiments are used to determine both the radius of the proton and the Rydberg constant (a constant relating the wavelengths of the lines emitted in the spectra of specific elements), thus any change in the charge radius of the proton also implies a change in the Rydberg constant. The Rydberg constant is considered to be one of the most well-determined physical constants, with an accuracy of 7 parts to 1 trillion, so any change has huge implications for the standard model.

The question is, why are these different methods yielding different results? This is a problem that the physics community is still trying to reconcile, and the current failure to understand it is known as the proton radius puzzle.

How can we solve this puzzle?

First, we need to understand the difference between ‘normal’ electronic hydrogen and ‘exotic’ muonic hydrogen. The former is the most basic atom consisting of a single proton nucleus and a single electron, whereas the latter also consists of a single proton, but instead of an electron it hosts its heavier brother from the lepton family, the muon.


The sensitivities of the energy levels to the proton radius are determined by the probability that the bound lepton is within the volume of the proton. As the mass of the muon is ~ 200 times greater than the electron, the probability that it exists closer to the nucleus is much greater (8 million times) and thus the proton radius is determined to a much greater accuracy with muonic hydrogen!

But why the discrepancy between the latest muonic measurements and the value predicted by the standard model?

It could be that the QED calculations that account for the contributions from the Lamb shift are inaccurate and in need of a makeover. In addition, it could be that one of the most well-determined physical constants – the Rydberg constant – needs to be corrected. Another possibility is that the electron and muon interact differently with the proton, but this would violate the idea — known as lepton universality — that electrons, tau leptons and muons all behave the same and are produced at the same rate. However recent experiments at the Large Hadron Collider show the first clues that this may not be the case and that in fact lepton universality violation is the norm.

Maybe it’s a combination of all these things as well as the need to further explore new ideas in physics.

To add fuel to the fire, Randolf Pohl and his team at the Paul Scherrer Institute in Switzerland, recently measured the energy shift of the exotic atom deuterium and were able to determine the deuteron radius as rd =2.12562 fm, which compared with the 2010 CODATA value of rd =2.1424 fm is again smaller by ~ 7 sigma. The value is also smaller than that found from normal electronic deuterium, suggesting that there is not only a discrepancy between the electronic and muonic measurements but also the electronic experimental techniques.

Whatever the case may be, these latest results from Pohl et al. (2016) confirm that muonic atoms give a more accurate and smaller proton radius.

A6ae909030b7a848914688d15aa4cd9bas the muonic measurements are considered to be more accurate it makes sense that any new theory should be in agreement with these latest results. Maybe this new physics could be found in the unified approach, which considers the proton radius in terms of a geometric holographic solution and finds that there is no discrepancy between its predicted radius and that of the latest muonic measurements. Instead it is found to be in precise agreement within 1 sigma compared to the 7 sigma discrepancy in the standard approach.

The jury is still out and ‘officially’ the proton puzzle remains, with the hope that more precise experiments will reveal the solution.

By Amira Val Baker

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