Hydrogenic rotational levels with spin-0 or spin-1/2 constituent particles

Technical Summary: Hydrogenic Rotational Levels with Spin-0 or Spin-1/2 Constituent Particles

Problem Statement
Current theoretical descriptions of two-body atomic systems often rely on the Dirac equation for spin-1/2 particles or the Klein-Gordon equation for scalar (spin-0) particles. However, these differential equations become inadequate when the masses of the orbiting particle and the nucleus are comparable, as is the case in muonic, kaonic, and antiprotonic atoms. In such regimes, neither equation accurately describes the energy levels of the system, and no single differential equation exists that can precisely model these two-body systems with arbitrary mass ratios and magnetic moments. Furthermore, existing calculations often lack a unified framework capable of handling both spin-0 and spin-1/2 constituents with arbitrary masses and magnetic moments in rotational states ($L > 1$).

Methodology
The authors employ Nonrelativistic Quantum Electrodynamics (NRQED) to provide a unified description of two-body systems. The core of the methodology involves:

1. Unified Formalism: Deriving and collecting formulas applicable to rotational states ($L > 1$) for systems with spin-0 or spin-1/2 constituents, arbitrary masses, and arbitrary magnetic moments.
2. NRQED Expansion: Expressing the energy spectrum as a sum of spin and angular momentum couplings ($E_{NS}, E_{S1}, E_{S2}, E_{SS}, E_{LL}$) expanded in powers of the fine-structure constant $\alpha$.
   • $E^{(2)}$: The nonrelativistic energy obtained by numerically solving the Schrödinger equation with a potential $V(r)$ that includes nonperturbative electron vacuum polarization (EVP) up to the two-loop (Wichman-Kroll) level.
   • $E^{(4)}$: The leading relativistic correction derived from the expectation value of the Breit Hamiltonian, which includes recoil effects, finite charge radii, and magnetic moment interactions.
   • $E^{(5)}$ and $E^{(6)}$: Higher-order corrections including recoil effects dependent on constituent spins. Explicit formulas are provided for cases where $s_1=0, s_2=1/2$ and $s_1=s_2=1/2$, as well as for spin-0 pairs.
   • Polarizability: Inclusion of electric dipole polarizability corrections ($E^{(6)}_{pol}$) for both the orbiting particle and the nucleus.
3. Implementation: These formulas are implemented in an updated version of the PbarSpectr code. The code utilizes the Johnson algorithm for solving the radial Schrödinger equation, modified to handle $r \ln r$ terms in the small-$r$ expansion.
4. Scope of Corrections: The calculations include corrections up to order $\alpha^6$ exactly. Higher-order QED corrections ($E^{(7)}$ and $E^{(8)}$) are included in the infinite nuclear mass limit. Three-loop vacuum polarization and certain higher-order EVP corrections are currently omitted, with their uncertainties estimated conservatively.

Key Contributions

• Extension of PbarSpectr: The code has been extended from its original scope (spin-1/2 particle + spinless nucleus) to handle arbitrary mass ratios and magnetic moments for both spin-0 and spin-1/2 constituents.
• Analytical Formulas: The paper presents exact analytical expressions for the $\alpha^6$ recoil corrections ($E^{(6)}$) for various spin configurations ($s_1=0, s_2=1/2$ and $s_1=s_2=1/2$) in rotational states with $L > 1$. These formulas reduce to Klein-Gordon or Dirac limits for infinitely heavy nuclei but remain valid for arbitrary mass ratios.
• Unified Treatment: The work provides a consistent framework for calculating energy levels across different exotic atoms (muonic, kaonic, antiprotonic) without switching between different relativistic equations.

Results
The authors present state-of-the-art theoretical predictions for several systems, comparing their results with previous calculations and experimental data:

• $\mu^{20}\text{Ne}$: Calculations for muonic neon show excellent convergence in powers of $\alpha$. The dominant uncertainty arises from the uncalculated three-loop vacuum polarization. The results suggest that with improved theoretical precision (including three-loop EVP), these transitions could determine the nuclear electric dipole polarizability, though current experimental precision (order of 10 eV) is insufficient for 9-digit precision requirements.
• $\bar{p}^{28}\text{Si}$ and $\bar{p}^{29}\text{Si}$: For antiprotonic silicon, the NRQED approach demonstrates excellent convergence. The antiproton spin-orbit interaction is the dominant spin coupling. The calculated isotope shift between $^{29}\text{Si}$ and $^{28}\text{Si}$ for the $6h \to 5g$ transition is $69.87(14)$ eV, serving as a test of the NRQED approach.
• $K^{19}\text{F}$ (Kaonic Fluorine): Comparisons with previous calculations (Ref. [15]) show small differences (0.01–0.15 eV) that grow with decreasing principal quantum number. The authors attribute these discrepancies to the exact mass dependence of relativistic corrections, which is not fully captured by reduced-mass approximations in previous works. The uncertainty in the kaon mass currently limits the precision of these predictions.
• $K^{20}\text{Ne}$ (Kaonic Neon): For the two spin-0 system, discrepancies with previous theoretical predictions increase as the principal quantum number decreases (up to ~0.07 eV). The authors conclude that finite nuclear mass effects require an explicit two-body treatment rather than simple reduced-mass substitution in relativistic corrections.

Significance and Claims
The paper claims that the NRQED approach offers a more rigorous and accurate framework for exotic atoms with comparable constituent masses compared to traditional Dirac or Klein-Gordon approaches. The authors argue that:

1. Theoretical Accuracy: Theoretical accuracy can be substantially improved (by approximately two orders of magnitude) by incorporating three-loop vacuum polarization and one-loop EVP corrections into the $E^{(5)}$ term.
2. Future Applications: Once these theoretical improvements are made and experimental precision is enhanced, precision spectroscopy of muonic and hadronic atoms could be used for:
   • High-accuracy determinations of nuclear charge radii.
   • Determination of nuclear electric dipole polarizabilities.
   • Testing the existence of hypothetical long-range hadronic interactions.
3. Mass Determination: For pionic and kaonic atoms, the current limiting factor is the precision of the orbiting scalar particle's mass. The authors suggest that atomic spectroscopy could be used to determine these masses more accurately once other uncertainties are reduced.

The work positions the updated PbarSpectr code and the derived formulas as essential tools for interpreting upcoming experiments (e.g., at J-PARC and by the PAX collaboration) and for probing physics beyond the Standard Model via precision spectroscopy.