🔬 atomic physics

Improved calculation of the hyperfine structure of muonic helium

This paper presents improved theoretical calculations of the hyperfine structure in the ground state of muonic helium using second-order perturbation theory, which successfully reduces the discrepancy between theoretical predictions and recent experimental measurements.

Original authors: F. A. Martynenko, K. A. Seredina, A. P. Martynenko

Published 2026-09-16

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: F. A. Martynenko, K. A. Seredina, A. P. Martynenko

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep within the realm of atomic physics, scientists study the tiny, invisible forces that hold matter together. At the heart of this study is a simple question: how do the fundamental particles inside an atom interact with one another? To answer this, researchers often look at exotic atoms, which are like standard atoms but with a twist. In a normal helium atom, a nucleus is orbited by two electrons. In a muonic helium atom, one of those electrons is replaced by a muon, a particle that is much heavier than an electron but otherwise behaves similarly. This heavy muon orbits much closer to the nucleus, creating a unique environment where the rules of quantum mechanics can be tested with extreme precision. By measuring the energy levels of these atoms, specifically a property called hyperfine structure which arises from the magnetic interaction between the spins of the particles, physicists can check if their theories match reality. When theory and experiment disagree, it often points to a missing piece of the puzzle or a subtle effect that has been overlooked.

For years, scientists have been trying to calculate the exact energy of the ground state of muonic helium with enough precision to match new, highly accurate experiments. Recent measurements by the MuSEUM collaboration at J-PARC in Japan provided a very precise value for this energy, but when researchers compared this number to their best theoretical calculations, a small gap remained. The difference was about 0.5 megahertz, a tiny amount in the grand scheme of things, but significant enough to suggest that the theoretical models were missing some important details. Previous calculations had accounted for many effects, such as the way the particles recoil when they interact and the influence of the vacuum itself, but the discrepancy persisted. This gap meant that the current understanding of how these particles interact was not yet complete.

In this new study, researchers from Samara University in Russia set out to close that gap by looking more closely at the mathematics of the interaction. They focused on the second-order effects, which are essentially the subtle, secondary consequences of the particles interacting with each other. Imagine trying to predict the path of a ball rolling down a hill; the first calculation might tell you the general direction, but a more precise calculation must account for every tiny bump and wind gust that slightly alters the path. The team used a method called perturbation theory, which allows physicists to break down complex interactions into a series of smaller, manageable steps. They calculated new contributions that had not been fully considered in previous work, specifically looking at how the magnetic fields of the electron and muon interact with the nucleus and with each other in a more complex way than before.

The researchers divided the problem into several parts, calculating the energy shifts caused by the recoil of the particles, the exchange of virtual particles, and the specific magnetic interactions described by the Breit Hamiltonian. They performed these calculations analytically, meaning they derived exact mathematical expressions for these effects rather than relying solely on computer simulations. By carefully adding up these new contributions, they found a total correction of 0.5469 megahertz. When this new value was added to the previous theoretical estimate, the result shifted significantly. The new theoretical value for the hyperfine splitting became 4465.0511 megahertz.

This updated number brought the theory much closer to the experimental measurement of 4464.980 megahertz. The remaining difference between the two is now only 0.0711 megahertz, a dramatic improvement from the previous 0.5 megahertz gap. The authors note that this remaining difference is likely due to the approximations used in their calculations, such as simplifying certain complex functions to make the math solvable. They estimate that the uncertainty in their method is between 0.2 and 0.3 megahertz, which means the current agreement is well within the expected margin of error. This work demonstrates that by refining the theoretical models and accounting for these subtle second-order effects, scientists can achieve a level of precision that aligns closely with experimental reality. The study confirms that the current framework of quantum electrodynamics is robust, provided that all the intricate details of the particle interactions are included in the calculation.

Technical Summary: Improved Calculation of the Hyperfine Structure of Muonic Helium

Problem Statement
The hyperfine structure (HFS) of the ground state in the muonic helium atom ( $\mu e^4\text{He}$ ), a three-particle bound state consisting of an $\alpha$ particle, a muon, and an electron, has been the subject of long-standing theoretical and experimental investigation. While previous measurements in weak and strong magnetic fields yielded results around 4464.95–4465.00 MHz, a new experimental value of 4464.980(20) MHz was reported by the MuSEUM collaboration at J-PARC.

Despite extensive theoretical work using both variational methods and analytical perturbation theory, a discrepancy of approximately 0.5 MHz persisted between the calculated theoretical values (specifically those in references [11, 12, 14, 15]) and the new experimental data. Previous calculations within the authors' own perturbation theory framework accounted for recoil, vacuum polarization, and nuclear structure effects but failed to fully resolve this gap. The results obtained within the variational method indicate that the discrepancy may be primarily due to second-order corrections in perturbation theory.

Methodology
The authors employ analytical perturbation theory based on the fine-structure constant $\alpha$ and the electron-to-muon mass ratio. The system's Hamiltonian is decomposed into a main Coulomb interaction term ( $H_0$ ) and several perturbation terms ( $\Delta H$ , $\Delta H_{\text{rec}}$ , $\Delta H_{\text{vp}}$ , $\Delta H_{\text{str}}$ , $\Delta H_{\text{vert}}$ ) representing recoil, vacuum polarization, nuclear structure, and vertex corrections.

The ground-state wave function is approximated as a product of hydrogen-like wave functions for the electron and muon in the nuclear Coulomb field. The primary HFS contribution is derived from the spin-spin contact interaction. To achieve the required precision (down to hundredths of a MHz), the authors calculate new contributions in second-order perturbation theory.

Key methodological steps include:

Recoil Corrections: Calculating second-order corrections involving the electron-muon mass ratio ( $W_e/W_\mu$ ) and the fine-structure constant. This involves splitting the summation over intermediate muon states into ground ( $n=0$ ) and excited ( $n \neq 0$ ) states. The calculation utilizes the reduced Coulomb Green's function for the electron and approximates the electron Green's function with a free Green's function for excited muon states, a method validated by previous numerical integration.
Breit Hamiltonian Contributions: Evaluating second-order corrections arising from the Breit potential. This includes:
- Corrections involving two hyperfine perturbation potentials.
- Corrections involving paired contact terms of the Breit potential for muon-electron ( $\mu e$ ) and electron-nucleus ( $e\alpha$ ) interactions.
Integration Techniques: The authors perform analytical integrations over coordinates, utilizing expansions in the parameter $W_e/W_\mu$ and specific integral representations of the Green's functions (including Euler's constant and exponential integral functions).

Key Contributions and Results
The paper presents a series of new numerical corrections calculated in second-order perturbation theory, which are summarized in Table I:

Recoil Corrections (Order $\alpha^4$ ): A total recoil correction is calculated, refining previous estimates.
Breit Potential Corrections: The authors calculate a new correction arising from second-order perturbation theory terms involving the Breit Hamiltonian. This is the sum of:
- Two hyperfine potentials.
- Muon-electron contact interaction.
- Electron-nucleus contact interaction.
Total Theoretical Value: Incorporating these new corrections alongside previously calculated effects (vacuum polarization, nuclear structure, vertex corrections, etc.), the new theoretical value for the ground-state hyperfine splitting is 4465.0511 MHz.

Significance and Conclusion
The primary significance of this work is the reduction of the discrepancy between theory and experiment. The new theoretical value of 4465.0511 MHz differs from the experimental value (4464.980 MHz) by only 0.0711 MHz, a substantial improvement over the previous ~0.5 MHz discrepancy.

The authors attribute the remaining small difference of 0.0711 MHz between the theoretical value and the experimental result to the approximations made in [15] when calculating various corrections in second-order perturbation theory. They estimate the theoretical uncertainty of their current approach to be between 0.2 and 0.3 MHz.

The paper concludes that the inclusion of these specific second-order perturbative contributions, particularly those related to the Breit Hamiltonian and two-photon exchange mechanisms, is essential for achieving high-precision agreement with experimental data in muonic helium. The results support the validity of the analytical perturbation theory approach when higher-order terms are rigorously accounted for.

More like this