Muonic Ruler: Measuring Nuclei, Testing Fundamental Interactions

Half a century ago, physicists discovered that all known elementary particles and their interactions follow a clear set of rules known as the Standard Model of Particle Physics. Its development required careful comparisons between experiments and earlier theories to identify effects that didn’t quite fit and needed new explanations. In this way, quantum electrodynamics (QED) — an important pillar of the Standard Model — resolved discrepancies between theory and experiment in the energy levels of the hydrogen atom.

The Standard Model isn’t set in stone either. It answers many questions about the universe, but not all — we still don’t know why there is more matter than antimatter, or what dark energy is. So far, with the current precision of experiments and theoretical calculations, the Standard Model remains consistent with measurements. One way to search for new physics beyond the Standard Model is by colliding elementary particles at very high energies and hoping to detect new particles. This is what physicists at the Large Hadron Collider are doing. A very different approach focuses on high-precision measurements in atoms. Here, the energy is too low to directly create new particles, but atomic properties, like energy levels, may be slightly affected by new physics such as new particles or interactions. If measured with a very high precision, such discrepancies with the Standard Model may be revealed.

This low-energy approach is pursued by Prof. Dr. Aldo Antognini at PSI and ETH Zurich. His team is carrying out high-precision measurements of the sizes of atomic nuclei. These values are incredibly small — about 10,000 times smaller than the size of an atom. If an atom were the size of a truck, its nucleus would be no larger than a poppy seed. Measuring such tiny values with high precision is extremely challenging.

To overcome this challenge, Prof. Antognini’s team uses a unique facility at PSI: a beam of negatively charged muons. Muons are similar to electrons but 200 times heavier. When an atom is bombarded with muons, all its electrons are knocked away. Instead, a muon is captured by the nucleus, producing a muonic ion — a two-body system with a single muon orbiting around a nucleus. “The simplicity of this system is what allows us to perform so precise measurements,” says Prof. Antognini. Because muons are so heavy, they orbit much closer to the nucleus, making them more sensitive to its properties — especially to its charge radius. By measuring energy transitions in these muonic ions, the team determines nuclear radii about ten times more precisely than other methods.

After his groundbreaking radius measurements for the hydrogen and deuterium nuclei a decade ago, Prof. Antognini turned to a more complex nucleus — helium. Within the extended nuclear family, hydrogen (with just a single proton) and deuterium (a proton and a neutron) are the lightest members. Helium-3 and helium-4 are like their heavier cousins: helium-4 has two protons and two neutrons, while helium-3 is lighter, missing one neutron. Four years ago, Prof. Antognini measured the radius of helium-4; now, with the newly published results for helium-3, he has completed the family portrait — a full set of radius measurements for the lightest nuclei, from hydrogen to helium.

The gradual increase in complexity from nuclei with one to four interacting components makes them an ideal testing ground for modern ab initio nuclear theory, derived from symmetries of the Standard Model. As in classical mechanics, the case of two interacting bodies is easier to solve. But once three or more particles are involved, calculations become more complex. The famous three-body problem in mechanics, which has no general solution, even inspired the science-fiction novel The Three-Body Problem by Liu Cixin. Similarly, moving from hydrogen to helium adds complexity and provides a stricter test for the theory of nuclear physics. Comparing predictions to Prof. Antognini’s precise measurements will show how well the theory handles these more intricate systems.

Knowledge of the nuclear radius is also essential for testing quantum electrodynamics (QED) in atoms. The electron energy levels are predicted by quantum mechanics complemented with QED, which describes interactions between charged particles in a quantum way through the exchange of photons. On top of this, the energy levels slightly depend on the nuclear size. If the radius of the nucleus is known, its contribution to the energy levels can be well controlled, allowing a precise comparison between QED theory and experiment.

New measurements of helium-3 and helium-4 radii are particularly important in light of a recent puzzle: a significant discrepancy between theoretical energy levels in Helium atoms and their measurements, performed in the group of Frédéric Merkt at ETH Zurich. A helium atom, with its two electrons and a nucleus, is a three-body system, and calculating all its QED contributions is a major challenge for theorists. It is still too early to claim that the measurements with the Helium atom lie beyond the expectations of the Standard Model, but for sure, such comparisons help to advance the theory of atomic energy levels and bound-state QED.

The path from measuring the proton radius to determining the radii of both helium nuclei took Prof. Antognini’s team 15 years. Each experiment was a separate project, requiring the development of new tools and new ways to interact with the system. “It’s not that we just changed the gas when going from the hydrogen experiment to helium,” says Prof. Antognini. “For example, we needed to develop a completely new laser system.” Lasers are crucial to measuring energy levels in muonic ions. When hit by light with the same frequency as the transition between two levels, a muon will jump to a more energetic level and then de-excite to the ground state, emitting X-rays. By tuning the laser frequency and detecting the strong X-ray signal, Prof. Antognini can precisely determine transitions between energy levels.

With access to a unique muon beam at PSI, Prof. Antognini and his team don’t plan to stop at nuclear charge radius measurements. Their next goal is to measure the magnetic properties of nuclei, offering new ways to test nuclear theory and the effects of nuclear magnetic properties on the electron energy levels, the hyperfine interaction. Finally, comparing measurements in muonic atoms and in regular ‘electronic’ atoms may potentially expose muon-specific interactions that are beyond the Standard Model.

Aleksandra Nelson