🔬 atomic physics

One-loop self-energy using a numerical Green function

This paper presents a numerical method using an exponential basis set to solve the radial Dirac equation, achieving high-precision calculations of the one-loop self-energy for hydrogenlike atoms in both Feynman and Coulomb gauges.

Original authors: Hugo D. Nogueira, Maen Salman, Jean-Philippe Karr

Published 2026-07-20

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CC BY 4.0

Original authors: Hugo D. Nogueira, Maen Salman, Jean-Philippe Karr

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

Imagine the universe as a giant, bustling city where every atom is a tiny apartment building. Inside these buildings, electrons zip around in specific rooms called energy levels. For a long time, scientists thought they could predict exactly how much energy an electron has just by looking at the building's blueprint. But then, they discovered a ghostly neighborhood called Quantum Electrodynamics (QED). In this neighborhood, the electron isn't just sitting still; it's constantly chatting with invisible particles of light (photons) that pop in and out of existence. These conversations change the electron's energy slightly, like a tenant constantly rearranging furniture and shifting the weight of the building. This tiny shift is called the "self-energy." It's the most important correction we need to make to understand why atoms behave the way they do, and it explains a famous mystery called the "Lamb shift," which proved that our old blueprints were missing some crucial details.

Now, imagine trying to calculate exactly how much this furniture rearranging costs. For heavy atoms (like uranium), the math is tough but manageable. But for the lightest atom of all—hydrogen, which is just one electron orbiting a single proton—the math gets incredibly tricky. It's like trying to count every single grain of sand on a beach while the wind is blowing. For decades, scientists used a "perfect" mathematical map (an analytical Green function) to solve this for hydrogen, but that map only works for simple, single-building scenarios. When scientists want to study more complex buildings, like molecules with two nuclei (two protons), that perfect map breaks down. They need a new way to calculate the energy shifts using a "numerical" approach, which is more like building a digital model of the building and simulating the physics step-by-step.

This is exactly what Hugo D. Nogueira, Maen Salman, and Jean-Philippe Karr set out to do in their paper. They wanted to see if they could use a computer to calculate the self-energy of hydrogen-like atoms using a "numerical Green function"—essentially a digital simulation of the electron's environment—instead of relying on the perfect, but limited, mathematical maps. They focused on a specific, difficult part of the calculation called the "many-potential term," which represents the electron interacting with the nucleus multiple times. To do this, they used a special set of mathematical building blocks (an exponential basis set) to solve the equations that describe the electron's behavior.

The team first tested their method on a heavy atom, hydrogen-like uranium (where the nucleus has a charge of 92). They found that their digital simulation could match the known, highly precise results with a relative uncertainty of about $10^{-5}$ . This means their method was accurate enough to be trusted. They discovered that a specific technique called "Dual Kinetic Balance" (DKB) worked the best, acting like a super-stable foundation that prevented the digital model from wobbling and losing precision.

Then, they tackled the real challenge: the hydrogen atom (where the nucleus charge is just 1). This is the hardest case because the electron moves so fast and interacts so subtly that tiny errors can blow up the whole calculation. The authors used a clever "convergence acceleration" scheme, which is like a shortcut that helps the computer stop guessing and start knowing the answer much faster. In the "Feynman gauge" (a specific way of setting up the math rules), they managed to calculate the self-energy with a relative uncertainty of $10^{-4}$ . When they switched to the "Coulomb gauge" (a different set of math rules that avoids some messy cancellations), they improved the precision even further, reaching a relative uncertainty of $10^{-5}$ .

The paper doesn't claim to have solved the problem perfectly or to have found a magic bullet for all future calculations. Instead, it suggests that this numerical approach is a promising path forward. The authors note that the main limitation right now is the need to guess the rest of the answer (extrapolation) because they can't calculate every single possible interaction. However, their results show that with enough computing power and the right mathematical tools, we can eventually apply these high-precision techniques to complex molecules, like the hydrogen ion, which are currently too difficult to study with such accuracy. It's a significant step toward understanding the quantum world in buildings that are much more complex than a single room.

Problem Statement
The bound-electron self-energy (SE) represents the largest quantum electrodynamics (QED) correction to atomic energy levels. While high-precision calculations exist for hydrogen-like systems, extending these to light non-hydrogenic systems (such as molecular hydrogen ions) presents significant challenges. Existing high-precision methods for hydrogen rely on analytical Green functions (Whittaker functions) and partial-wave decompositions that require immense computational power and complex convergence acceleration techniques. Furthermore, the "potential-expansion method," widely used for light systems, suffers from slow convergence and large numerical cancellations in the Feynman gauge, particularly for low nuclear charges ( $Z$ ). A primary obstacle for molecular systems is the lack of an analytical Dirac-Coulomb Green function, necessitating the use of numerical Green functions constructed from finite basis sets. Previous attempts using numerical Green functions (e.g., B-splines or Gaussian bases) have been limited to heavy systems and have not exceeded five-digit precision.

Methodology
The authors calculate the one-loop self-energy in hydrogen-like atoms using a numerical Green function obtained by solving the radial Dirac equation within an exponential basis set. The methodology involves:

Numerical Green Function Construction: The Green function is expressed as a sum over Dirac eigenvectors and eigenvalues derived from a finite basis set. The authors employ an exponential basis set ( $\pi_\mu(x) = x^{\ell+1}e^{-\alpha_\mu x}$ ), similar to that used in their previous work on molecular ions, to accommodate future extensions to two-center potentials.
Potential Expansion and Subtraction: The SE correction is separated into zero-, one-, and many-potential terms. The divergent zero- and one-potential terms are renormalized and evaluated in momentum space. The finite many-potential term, which is the most numerically challenging, is calculated using the numerical Green function. To isolate this term, the authors subtract the zero- and one-potential contributions (calculated analytically or numerically via the same basis) from the full Green function.
Kinetic Balance Schemes: The authors compare three kinetic balance schemes for the basis set: No Kinetic Balance (NKB), Restricted Kinetic Balance (RKB), and Dual Kinetic Balance (DKB). They find that DKB significantly improves precision and eliminates spurious states, particularly for $\kappa > 0$ .
Convergence Acceleration: To address the slow convergence of the partial-wave expansion, the authors implement a subtraction scheme where an analytically calculable approximation of the many-potential Green function is subtracted and added back in momentum space.
Gauge Choice: Calculations are performed in both the Feynman and Coulomb gauges. The Coulomb gauge is utilized to mitigate numerical cancellations that plague the Feynman gauge at low $Z$ .
Numerical Implementation: All integrals (radial and energy) are performed numerically using high-precision arithmetic (MPFUN2020) and specific quadrature schemes (tanh-sinh and exp-sinh). The authors optimize the algorithm for calculating the one-potential Green function ( $G^1_\kappa$ ) to reduce computational complexity from $O(n_q^3 n_G^3)$ to $O(\max(n_G, n_q)^4)$ .

Key Results

Hydrogen-like Uranium ( $Z=92$ ): Using the DKB exponential basis set, the authors achieved a relative uncertainty of approximately $10^{-5}$ for the ground state self-energy in the Feynman gauge. The results show a clear cancellation of errors between the zero-, one-, and bound-state contributions, validating the numerical subtraction approach. The DKB scheme was found to be superior to NKB and RKB, improving precision by two orders of magnitude.
Hydrogen Atom ( $Z=1$ ):
- Feynman Gauge: By applying the convergence acceleration scheme, the authors extended calculations to the hydrogen atom, achieving a relative uncertainty of $10^{-4}$ for the total SE correction. The primary limitation was the instability of the extrapolation to infinite partial waves due to the large relative uncertainty of the remainder.
- Coulomb Gauge: Calculations in the Coulomb gauge avoided the numerical cancellations seen in the Feynman gauge, allowing for higher precision in individual partial-wave contributions ( $10^{-5}$ to $10^{-6}$ ). This improved the precision of the total SE correction to $10^{-5}$ relative uncertainty.
Basis Set Performance: The exponential basis set with DKB performed comparably to the Gaussian basis sets used in previous high-precision studies, despite the authors not performing a full optimization of the nonlinear parameters.

Significance and Claims
The paper demonstrates the feasibility of performing high-precision self-energy calculations in light one-electron systems using a purely numerical Green function approach. The authors claim that their method represents a substantial step toward extending SE calculations to non-hydrogenic systems, specifically molecular hydrogen ions, where analytical Green functions are unavailable.

The work highlights that:

Numerical Green functions in exponential bases can achieve precision comparable to analytical methods for heavy atoms and are viable for light atoms when combined with convergence acceleration and gauge optimization.
The Coulomb gauge offers a distinct advantage for low- $Z$ systems by reducing numerical cancellations, thereby improving accuracy.
The primary limitation of the current approach is the extrapolation to an infinite number of partial waves, which remains the dominant source of uncertainty.

The authors conclude that while further optimization of basis set parameters is required to fully exploit the advantages of the Coulomb gauge, the current results validate the numerical framework for future applications to complex molecular systems.

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