Theory for the Rydberg states of helium: quantum defect extensions and comparison with experiment up to $n = 102$ for the singlet and triplet $P$-states

This paper presents high-precision variational calculations for helium Rydberg states up to $n=102$ using quantum defect and $1/n$ expansions, which confirm a significant 9$\sigma$ discrepancy between theoretical predictions and experimental measurements for the $1s2s\;^3S_1$ ionization energy while providing unprecedented 20-figure accuracy for the nonrelativistic Ritz expansion.

The Gist

Imagine the helium atom as a tiny, chaotic solar system. It has a heavy sun (the nucleus) and two electrons buzzing around it. Usually, one electron stays close to the sun, while the other one gets kicked out into a very wide, distant orbit. When that outer electron is in a very high orbit, physicists call it a "Rydberg state." Think of these high orbits like the upper rungs of a giant ladder stretching far into the sky.

For a long time, scientists have been trying to measure exactly how much energy it takes to kick that outer electron completely off the ladder (ionization). They have a theoretical map of what this energy should be, and they have a ruler (experimental data) to measure what it actually is.

The Problem: A Mysterious Gap
Recently, scientists measured the energy levels of these high orbits up to rung number 102. When they compared their measurements to the best theoretical maps available, they found a stubborn, unexplained gap. The theory and the experiment disagreed by a tiny amount (about 0.5 millionths of a unit), but it was a "9-sigma" disagreement. In science, that's like flipping a coin and getting heads 9 times in a row by pure chance—it's statistically impossible. Something is missing from the map, or the ruler is slightly off.

The New Approach: Building a Better Map
The authors of this paper, G. W. F. Drake and Aaron T. Bondy, decided to rebuild the map from the ground up to see if they could find the missing piece.

1. The Foundation (The First 35 Rungs):
   First, they used super-powerful computers to calculate the exact energy of the first 35 rungs of the ladder. They didn't guess; they solved the complex math equations (Schrödinger equation) with extreme precision, accounting for how the electrons wiggle, how they spin, and how they interact with each other. They treated the nucleus as a moving target, not a fixed point, which is a crucial detail.

2. The Shortcut (Quantum Defect):
   Calculating every single rung up to 102 is like counting every grain of sand on a beach. Instead, they used a "Quantum Defect" method. Imagine the ladder has a slight bend or a "defect" in its shape near the bottom. Once you know the shape of the bottom 35 rungs perfectly, you can use a mathematical formula to predict the shape of the rest of the ladder all the way to the top. This is the "Quantum Defect" expansion.

3. The Fine-Tuning (Relativity and QED):
   The standard formula for the ladder assumes a simple world. But in reality, the electrons move fast (relativity) and interact with the vacuum of space itself (Quantum Electrodynamics or QED). The authors added these tiny, complex corrections to their predictions. They found that these corrections get smaller and smaller as you go higher up the ladder, which helped them trust their predictions for the very high rungs.

The Discovery: The Gap is Real
When they combined their ultra-precise calculations for the high rungs with the actual measurements from the lab, they calculated the energy of the starting point (the 2 3S1 state).

The result? The gap is real.

Their new, highly accurate calculation confirmed the previous finding: the experimental measurements are lower than the theoretical predictions by 0.474 MHz. The difference is so small it's hard to imagine, but it is statistically huge.

What Does This Mean?
The paper doesn't offer a solution to why the gap exists, but it confirms that the gap is not a mistake in the math or the experiment.

• It's not a calculation error: The authors checked their math with unprecedented precision (20 significant figures).
• It's not a measurement error: They used 28 different measurements to confirm the result.
• It's not just about the isotope: The gap appears in both Helium-4 and Helium-3, suggesting it's a fundamental issue with how we understand the interaction between electrons.

The Bottom Line
Think of this paper as a master carpenter checking a blueprint against a finished house. The carpenter (the authors) built a perfect model of the first 35 floors using every tool in the shed. Then, they used that model to predict what the 100th floor should look like. When they compared the prediction to the actual building, they found a discrepancy that the original blueprint couldn't explain.

This confirms that our current understanding of the laws of physics (specifically how electrons interact) might be missing a tiny, hidden piece of the puzzle. It's a "9-sigma" mystery, meaning the universe is whispering that there is something new to discover, perhaps involving new particles or forces that we haven't accounted for yet.

Technical Summary

Technical Summary: Theory for the Rydberg States of Helium

Problem Statement
Recent high-precision measurements of transition frequencies to high-lying singlet and triplet P-states of helium (up to principal quantum number $n=102$) have confirmed a persistent $9\sigma$ discrepancy between theory and experiment regarding the ionization energy of the metastable $1s2s\ ^3S_1$ state. While helium is a fundamental three-body system allowing for high-precision calculations including electron correlation, relativistic, and quantum electrodynamic (QED) corrections, the origin of this discrepancy remains unexplained. Previous theoretical work was limited to $n \le 35$, while experimental data extended to $n=102$. The challenge lies in extending theoretical predictions to these higher $n$ values with sufficient accuracy (better than $\pm 1$ kHz) to serve as reliable reference points for determining the absolute ionization energy of the lower-lying $2\ ^3S_1$ state.

Methodology
The authors employ a hybrid theoretical approach combining direct variational calculations with asymptotic expansions:

1. Direct Variational Calculations ($n \le 35$): High-precision nonrelativistic wave functions and energies are generated using Hylleraas coordinates. The Hamiltonian includes mass polarization terms arising from nuclear motion. To maintain accuracy for high-lying Rydberg states, the authors utilize "triple basis sets" where each combination of powers in the basis functions is included three times with independently optimized nonlinear parameters. Calculations are performed for both infinite nuclear mass and the finite mass of the $^4$He isotope, utilizing Bailey's double-quadruple precision arithmetic for $n > 16$.
2. Relativistic and QED Corrections: Matrix elements for relativistic (order $\alpha^2$ Ry) and QED (order $\alpha^3$ Ry) corrections are calculated using the nonrelativistic wave functions. These include Breit interaction terms, spin-orbit and spin-other-orbit interactions, self-energy, vacuum polarization, and finite nuclear size effects. Finite nuclear mass corrections are handled via mass scaling and recoil terms.
3. Quantum Defect and $1/n$ Expansions:
   • Nonrelativistic Energy: For the nonrelativistic binding energies, the authors apply the Ritz quantum defect expansion (containing only even powers of $1/(n-\delta)$). This is justified by Hartree's proof for short-range potentials.
   • Relativistic/QED Corrections: For these corrections, the authors utilize $1/n$ expansions.
   • Mass Polarization Analysis: A critical component of the methodology is the identification of second-order mass polarization (recoil) terms. The authors demonstrate that these terms exhibit a leading $1/n^2$ dependence with state-independent coefficients. Consequently, these terms must be subtracted from the input energies before performing the quantum defect fit to avoid contaminating the $1/n^{*2}$ expansion.
4. Extrapolation: The quantum defect parameters derived from the $n \le 35$ data are used to extrapolate nonrelativistic energies to $n=102$. The $1/n$ expansions for relativistic and QED corrections are added to these extrapolated values to obtain total theoretical ionization energies.

Key Contributions

• Extension of Theoretical Range: The work extends high-precision theoretical ionization energies for helium P-states from the previous limit of $n=35$ up to $n=102$.
• Identification of Recoil Terms: The paper identifies and quantifies second-order mass polarization terms that vary as $1/n^2$. This provides a formal theoretical justification for the phenomenological use of an effective reduced-mass Rydberg constant, $R_M^{(+)}$, which accounts for the scattering of a Rydberg electron from a $He^+$ core rather than a bare nucleus.
• Verification of Ritz Expansion: The nonrelativistic Ritz expansion is verified to an unprecedented 20-figure accuracy for the nonrelativistic energies, confirming its validity in the nonrelativistic, infinite-nuclear-mass limit.
• Refined Ionization Energy: By combining the extrapolated theoretical ionization energies with 28 measured transition frequencies, the authors derive a new value for the ionization energy of the $1s2s\ ^3S_1$ state.

Results

• The calculated ionization energy for the $1s2s\ ^3S_1$ state is determined to be 1152 842 742.705(16) MHz.
• This result differs from the previous theoretical prediction [4] by 0.474 $\pm$ 0.052 MHz, confirming the previously observed $9\sigma$ disagreement between theory and experiment.
• The analysis shows that the discrepancy is statistically significant and consistent across both singlet and triplet measurements.
• The calculated values agree extremely well with a direct conventional quantum defect fit to the experimental data (using $R_M^{(+)}$), validating the extrapolation method used.
• The paper notes that the discrepancy is similar for $^3$He and does not appear in the $2\ ^3S_1 - 2\ ^1S_0$ isotope shift, suggesting the issue is not dependent on the specific isotope or neutron number.

Significance
The paper claims that its primary significance lies in two areas:

1. Validation of Quantum Defect Theory (QDT): The close agreement between the authors' method (using QDT only for nonrelativistic energies) and standard experimental QDT fits provides strong confirmation of the Ritz expansion and QDT validity in the low-$Z$ region where relativistic effects are manageable.
2. Confirmation of the Discrepancy: The work provides an unequivocal confirmation of the $9\sigma$ discrepancy in the ionization energy of the $2\ ^3S_1$ state. Since the discrepancy persists despite high-precision calculations and is not explained by isotope effects, the authors suggest (citing Cong et al.) that it may point to new physics, such as a light scalar boson affecting the electron-electron interaction, though this remains speculative pending further theoretical calculations of higher-order terms (specifically $m\alpha^7$ terms) and parallel studies in other systems like $Li^+$.