Constraints on Yukawa-type New Forces from the Lamb Shift in Muonic Hydrogen and Deuterium

Technical Summary: Constraints on Yukawa-type New Forces from the Lamb Shift in Muonic Hydrogen and Deuterium

Problem Statement
The "Proton Radius Puzzle," arising from the discrepancy between proton charge radii measured in normal hydrogen versus muonic hydrogen, has motivated investigations into physics beyond the Standard Model, specifically modified gravity and "fifth forces." These interactions are frequently modeled as Yukawa-type potentials mediated by light scalar or vector bosons. While various experiments have attempted to constrain the coupling strength ($\alpha$) and interaction range ($\lambda$) of such forces, the parameter space remains largely unconstrained, particularly for short-range interactions. Previous theoretical efforts to map radius discrepancies to energy shifts in atomic systems relied on coordinate-space approximations (e.g., assuming spherical nuclear charge distributions or truncating potentials). These approximations involve computationally intensive six-fold integrals and inherently limit the precision of the resulting bounds, failing to capture the full physical behavior of the Yukawa potential.

Methodology
The authors develop an analytical framework to calculate the energy shift ($\Delta E_{Lamb}$) induced by a Yukawa-type potential without relying on coordinate-space approximations.

1. Momentum-Space Transformation: The authors transform the required six-fold coordinate-space integral into a one-dimensional momentum-space convolution. By expressing the atomic charge densities ($\rho_{2s}, \rho_{2p}$) and the Yukawa potential ($V$) via their Fourier transforms, the energy shift is reduced to a direct integral of the product of atomic form factors and the potential's momentum-space representation.
2. Formal Derivation:
   • The atomic densities for $2s$ and $2p$ orbitals are derived, and their Fourier transforms are calculated analytically.
   • The nuclear charge distribution is modeled using both Gaussian and Dipole models to test robustness. The Fourier transform of the Yukawa potential is convolved with the nuclear form factor $F(q)$.
   • The final energy shift expression is formulated as a one-dimensional integral over momentum $q$, incorporating the difference in atomic form factors and the nuclear form factor.
3. Data Application: The framework is applied to the latest spectroscopy data for hydrogen and deuterium. The authors utilize the measured discrepancies in proton and deuteron radii between normal and muonic systems (based on 2013–2019 measurements) to calculate the residual energy shift $\Delta(\Delta E_{Lamb})$. This experimental residual is compared against the theoretical shift derived from the Yukawa potential to set upper limits on the coupling constant $\alpha$ for a range of interaction lengths $\lambda \in [10^{-4}, 10^6]$ fm.

Key Contributions

• Analytical Precision: The paper eliminates the need for coordinate-space approximations by providing an exact analytical formulation in momentum space. This allows for the direct incorporation of empirical nuclear charge distributions via form factors.
• Model Independence: The study demonstrates that the derived constraints are robust and independent of the specific nuclear charge distribution model (Gaussian vs. Dipole). This is attributed to the universal behavior of form factors in the low-momentum limit ($q \to 0$), where $F(q) \approx 1 - q^2 r_p^2/6$.
• Identification of Resonance-like Cancellation: The authors identify a critical phenomenon where the energy shifts of the $2s$ and $2p$ orbitals cancel each other out at specific Yukawa ranges. This results in a vanishing net energy shift, causing the constraint on the coupling constant $\alpha$ to diverge.

Results

• Constraints on $\alpha$ and $\lambda$: The study derives constraints on the Yukawa coupling constant over the range $\lambda \in [10^{-4}, 10^6]$ fm. The constraints derived from the hydrogen system are found to be more stringent than those from deuterium, primarily because the radius discrepancy in hydrogen is smaller, leading to tighter bounds on allowable new physics.
• Resonance Phenomenon: A sharp "resonance-like" peak is observed where the upper limit on $\alpha$ diverges to infinity.
  • For Hydrogen, this occurs at $\lambda \approx 1.9 \times 10^5$ fm.
  • For Deuterium, it occurs at $\lambda \approx 7.9 \times 10^4$ fm.
  • The authors provide an analytical derivation showing this cancellation arises from the finite size of the nucleus. In the point-nucleus limit, the integral is strictly positive; however, the finite-size correction (proportional to the nuclear radius squared) introduces a term that can exactly cancel the residual Yukawa attraction difference between the $2s$ and $2p$ states.
  • This divergence marks a transition between attractive (negative $\alpha$) and repulsive (positive $\alpha$) interaction regimes. At the critical range, the potential becomes "invisible" to the $2s-2p$ Lamb shift transition.
• Spectroscopic Resolution Limits: The calculated transition frequency shifts induced by these new forces are found to be on the order of $10^1$ Hz in the short-range limit. This is well below the current spectroscopic resolution (several kHz), indicating that the primary bottleneck for probing these interactions is the precision of current spectroscopic measurements, not the theoretical framework.

Significance
The paper establishes that atomic spectroscopy provides a robust and independent probe for Yukawa-type fifth forces, comparable to collider and neutron scattering experiments but applicable to different parameter regimes. The authors conclude that while their method offers a highly precise framework for constraining new physics, the current inability to detect these forces is due to the inherent precision limits of general spectroscopy rather than a lack of theoretical sensitivity. The identification of the resonance-like cancellation provides a critical boundary condition for future searches, distinguishing between attractive and repulsive interaction regimes and highlighting the necessity for higher precision in both electronic and muonic systems to uncover or rule out fifth-force interactions in the attometer regime.