A team of theoretical physicists from Johannes Gutenberg University Mainz (JGU) has once again made significant advancements in calculating the electric charge radius of the proton, as initially published in 2021. This time, they achieved an impressively precise result entirely through theoretical calculations, without relying on experimental data.

These latest calculations also lean toward a smaller value for the proton’s size. Moreover, these physicists have introduced a stable theoretical prediction for the magnetic charge radius of the proton. You can explore these groundbreaking findings in three preprints accessible on the arXiv server.

All atomic nuclei are composed of protons and neutrons, but many aspects of these fundamental particles remain elusive. One key puzzle has been determining the precise size of the proton. In 2010, a novel technique involving laser spectroscopy of muonic hydrogen produced a surprising result. In this unique form of hydrogen, the electron was replaced by its heavier counterpart, the muon, which offered a more sensitive means of probing the proton’s dimensions.

The outcome of these experiments yielded a substantially smaller proton radius compared to measurements conducted with “regular” hydrogen and the conventional electron-proton scattering method. This discrepancy has raised a profound question among physicists: does it indicate the presence of new physics beyond the Standard Model, or does it merely reflect inherent uncertainties within the various measurement techniques?

## Has the proton radius puzzle been solved?

Recent years have provided mounting evidence that the smaller experimental value for the proton’s radius is likely the correct one, suggesting that there may not be any new physics underlying the proton radius puzzle. Theoretical calculations are playing a pivotal role in decisively addressing this question. Back in 2021, a team led by Prof. Dr. Hartmut Wittig from the Mainz Cluster of Excellence PRISMA+ made significant strides by conducting lattice calculations with impressive precision, adding another reliable piece to the puzzle of the smaller proton radius.

Hartmut Wittig elaborates on the progress, noting that Miguel Salg, a doctoral student in his research group, has achieved notable results that further enhance and extend their earlier calculations. Two years ago, the Mainz research group had calculated the isovector radius, not the proton radius itself, by combining experimental data for the neutron radius. Now, through refining their techniques, increasing statistical significance, and better constraining systematic errors, they have achieved the remarkable feat of relying entirely on theoretical calculations, without the need for experimental data.

Furthermore, they’ve conducted direct calculations to verify the accuracy of their 2021 results, affirming their earlier findings. In the context of the proton radius puzzle, the growing body of evidence suggests that the smaller value indeed accurately characterizes the proton’s size, as stated by Hartmut Wittig.

The calculations conducted by the Mainz physicists are rooted in the theory of quantum chromodynamics (QCD), which describes the fundamental forces within atomic nuclei. These forces bind quarks, the building blocks of matter, to form protons and neutrons, and are mediated by exchange particles known as gluons. To mathematically model these processes, the scientists in Mainz employ lattice field theory.

In this approach, quarks are distributed across discrete points in space-time, resembling a crystal lattice. Special simulation methods are then employed, often utilizing supercomputers, to calculate properties of nucleons, starting with electromagnetic form factors. These form factors elucidate the distribution of electric charge and magnetization within the proton, ultimately allowing for the determination of the proton’s radius.

## First stable theory prediction for the magnetic charge radius

Beyond the electric charge radius, which has been the primary focus, the proton presents another intriguing puzzle related to its magnetic charge radius. The researchers in Mainz have also delved into this aspect, employing the framework of quantum chromodynamics (QCD). To simplify, one could envision these distinct radii as how an incoming electron perceives the expansion of electric or magnetic charge distributed within the proton during the scattering process, as explained by Hartmut Wittig.

Significantly, the Mainz research group has, for the first time, computed a stable theoretical prediction for the magnetic charge radius, relying solely on theoretical calculations. Additionally, from their precise knowledge of the electric and magnetic form factors, they’ve successfully derived the Zemach radius of the proton through QCD, a crucial parameter for experimental measurements involving muonic hydrogen. This accomplishment underscores the remarkable progress in the accuracy and capabilities of lattice QCD calculations, as highlighted by Hartmut Wittig.