⚛️ general relativity

Constraining Lorentz Invariance Violation via Precision Atomic Spectroscopy

This paper utilizes high-precision spectroscopic data from electronic and muonic hydrogen to constrain Lorentz invariance violation, yielding a 95% confidence interval for the dimensionless coefficient $\xi$ of $[-14.3, 1.37] \times 10^5$ and revealing distinct mass-scaling regimes where the heavy nucleus or the lepton dominates the sensitivity depending on the order of the deformation.

Original author: A. S. Lemos, J. E. J. Matias, F. Dahia, I. P. Lobo

Published 2026-11-05

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CC BY 4.0

Original author: A. S. Lemos, J. E. J. Matias, F. Dahia, I. P. Lobo

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

For decades, physicists have been searching for a way to bridge the gap between two great theories of the universe. One theory, quantum mechanics, explains how the smallest particles behave, while the other, general relativity, describes how gravity shapes the cosmos. These two frameworks work perfectly on their own but clash when scientists try to combine them into a single theory of quantum gravity. The effects of this missing theory are expected to be incredibly tiny, hidden behind a barrier of energy so immense that it is known as the Planck scale. Because these effects are so small, they are usually impossible to detect with current technology. However, a growing number of researchers believe that if they look closely enough at the right systems, they might find subtle cracks in the fundamental rules of nature, such as the principle that the laws of physics remain the same regardless of how you are moving or where you are in space.

A team of researchers in Brazil recently took a fresh look at this problem by studying the simplest atoms in existence: hydrogen. They focused on two versions of this atom. The first is the standard hydrogen atom, where a single electron orbits a proton. The second is a rare, heavier version called muonic hydrogen, where the electron is replaced by a muon, a particle that is about two hundred times heavier than an electron. By comparing how these two atoms behave, the scientists hoped to spot a tiny deviation that would signal a violation of Lorentz invariance, a core symmetry of the universe that states physical laws do not change based on direction or speed. If such a violation exists, it would be a direct fingerprint of quantum gravity, revealing that space and time are not perfectly smooth but have a grainy structure at the smallest scales.

The researchers approached this challenge by treating the hydrogen atom as a precise laboratory instrument. They built a mathematical model that allowed for a slight modification in how energy and momentum relate to each other, a concept known as a modified dispersion relation. In this model, the relationship between a particle's speed and its energy is tweaked by a tiny amount, controlled by a specific number that represents the strength of the quantum gravity effect. The team then compared this model against a vast collection of the most precise measurements ever taken of hydrogen's energy levels. These measurements include the frequencies of light absorbed or emitted when the atom jumps between different energy states, recorded with an accuracy so high that the uncertainty is measured in mere kilohertz. They also included data from muonic hydrogen, which is particularly sensitive to the size of the proton, the atom's central nucleus.

To make sense of this data, the scientists had to navigate a complex web of uncertainties. The measurements of hydrogen's energy levels depend on other fundamental constants, such as the Rydberg constant, which sets the scale for atomic energy, and the radius of the proton. There is a well-known disagreement in the physics community regarding the exact size of the proton, with different experiments yielding slightly different results. The team carefully accounted for these discrepancies, treating the uncertainties in the Rydberg constant and the proton radius as variables to be adjusted alongside their new quantum gravity parameter. They used a powerful statistical method to sift through the data, looking for the combination of values that best explained all the measurements simultaneously. This approach allowed them to see if the data pointed toward a non-zero value for their quantum gravity parameter or if the results were consistent with the standard laws of physics.

The analysis revealed that the data does not support the existence of a detectable violation of Lorentz invariance at the level of precision currently available. Instead of finding a clear signal, the researchers established a strict boundary for where such a signal could possibly hide. They determined that the parameter representing the quantum gravity effect must fall within a specific range, centered near zero, with a confidence level of ninety-five percent. In simpler terms, if the effect exists, it is so small that it is currently indistinguishable from zero given the noise in the measurements. The study also highlighted how the mass of the particles involved changes the sensitivity of the experiment. For the lowest-order effects, the heavy proton in the nucleus amplifies the signal, making the whole atom act as a larger probe. However, for higher-order effects, the lightness of the orbiting particle, whether it is an electron or a muon, becomes the dominant factor.

One of the most significant findings of the paper is the role of the muonic hydrogen data in clarifying the results. Because the muon is much heavier, it orbits much closer to the proton, making the atom extremely sensitive to the proton's size. This sensitivity allowed the researchers to use the muonic data to "anchor" the proton's radius in their analysis, helping to manage the tension between different measurements of the proton's size that has been a puzzle for years. The paper notes that the effect of the modified dispersion relation on the muonic Lamb shift is negligible compared to experimental uncertainties, meaning the muonic data primarily constrains the proton radius itself rather than the new physics parameter. By anchoring the proton's size with the muonic data, the team could isolate the search for quantum gravity effects more cleanly than ever before. They found that the tension between different measurements of the proton's size is not resolved by introducing a new quantum gravity effect; rather, the data sets remain mutually inconsistent, and the inclusion of the new parameter does not make the statistical adjustment acceptable. Consequently, the standard model of physics, without any modifications to Lorentz invariance, remains the statistically viable description of reality.

The study concludes that while the current generation of atomic spectroscopy is incredibly powerful, it has not yet reached the sensitivity required to see the grainy structure of spacetime predicted by quantum gravity theories. The constraints they set are remarkably close to the Planck scale, the energy level where these effects are expected to appear, but they are not quite there yet. The researchers emphasize that their work demonstrates the potential of using the simplest atoms to test the most complex theories of the universe. They suggest that if the uncertainties in the Rydberg constant and the proton radius can be reduced further in future experiments, atomic spectroscopy could become a fully capable tool for scrutinizing the Planck scale. For now, the universe continues to hold its secrets, but the search has been narrowed down with greater precision than ever before.

Technical Summary: Constraining Lorentz Invariance Violation via Precision Atomic Spectroscopy

Problem Statement
The search for observable signatures of quantum gravity (QG) remains a fundamental challenge, as QG effects are typically suppressed by the Planck scale ( $M_{Pl} \approx 1.22 \times 10^{19}$ GeV). While high-energy astrophysics has traditionally been the primary arena for testing Lorentz Invariance Violation (LIV), recent phenomenological approaches suggest that Planck-scale corrections may manifest in low-energy regimes through mechanisms such as ultraviolet/infrared (UV/IR) mixing. These corrections can alter dispersion relations, potentially influencing spacetime symmetries and uncertainty principles. However, isolating these effects in complex quantum systems is difficult due to the difficulty of distinguishing QG signals from other theoretical and experimental uncertainties. Previous studies utilizing hydrogen-like systems often failed to rigorously treat theoretical uncertainties (specifically regarding the Rydberg constant and proton radius) or relied on generalized uncertainty principles that only constrained second-order effects too faint to be detectable. Furthermore, the use of muonic atoms to probe mass-dependent scaling in LIV had not been comprehensively explored in this context.

Methodology
The authors investigate LIV by modeling a modified dispersion relation (MDR) parameterized by a dimensionless coefficient $\xi$ and suppressed by the Planck scale. The study focuses on the simplest atomic systems: electronic and muonic hydrogen.

Theoretical Framework: The authors construct a two-body LIV Hamiltonian for a nucleus and a lepton. By transforming to the center-of-mass frame, they derive the leading-order energy shifts for different deformation orders ( $n$ ).
- For $n=1$ (linear deformation), the effect scales with mass, amplifying the signal via the heavy nucleus.
- For $n=2$ (quadratic), the effect is mass-invariant.
- For $n \geq 3$ (higher-order), the effect scales inversely with mass, making the light lepton the primary probe.
  The analysis focuses on the linear case ( $n=1$ ), deriving a master equation for the energy shift $\Delta E$ that accounts for reduced mass dynamics and momentum expectation values.
Data and Fitting: The study utilizes a global statistical adjustment of high-precision spectroscopic data, including:
- Electronic hydrogen transitions (e.g., $1S-2S$ , $2S-2P$ , $8D-2S$ ) with uncertainties on the order of kHz.
- The Lamb shift in muonic hydrogen ( $\mu H$ ).
- Determinations of the Rydberg constant ( $R_\infty$ ) from Rydberg states.
The authors employ a Markov Chain Monte Carlo (MCMC) analysis to simultaneously estimate the Rydberg constant shift ( $\Delta R_y$ ), the proton charge radius ( $r_p$ ), and the LIV parameter ( $\xi$ ). Crucially, the analysis treats theoretical uncertainties (specifically for the $1S$ state and muonic Lamb shift) as nuisance parameters with Gaussian priors. To address the known tension between different experimental datasets (resulting in high $\chi^2$ values), the authors globally rescale uncertainties using the Birge factor ( $R_B = \sqrt{\chi^2_{red}}$ ) before determining confidence intervals.

Key Contributions

Rigorous Uncertainty Treatment: Unlike previous works, this analysis explicitly incorporates theoretical uncertainties in the Rydberg constant and proton radius as adjustable parameters, preventing them from biasing the LIV constraints.
Mass Scaling Analysis: The paper elucidates how the sensitivity to LIV depends on the deformation order $n$ . It demonstrates that for linear deformations ( $n=1$ ), the heavy nucleus amplifies the signal, whereas for higher orders, the lepton dominates.
Muonic Hydrogen Integration: The study integrates muonic hydrogen data to break degeneracies between the proton radius and other parameters, leveraging the fact that the muonic Lamb shift is largely insensitive to MDR effects at current precision levels.
Global Fit Approach: By analyzing multiple transition configurations (including and excluding specific datasets like muonic hydrogen or Rydberg states), the authors map parameter degeneracies and test the robustness of the constraints against dataset inconsistencies.

Results
The global fit to the complete dataset yields a best-fit value for the LIV parameter of $\xi = (-6.5 \pm 2.1) \times 10^5$ at the 68.3% confidence level. However, the analysis reveals significant tension within the datasets, leading to a high reduced chi-squared ( $\chi^2_{red} \approx 3.57$ ) even after including LIV.

After applying the Birge factor to rescale uncertainties, the 95% confidence level constraint is determined to be:
$\xi \in [-14.3, 1.37] \times 10^5$
In this rescaled scenario, the standard physics hypothesis ( $\xi = 0$ ) lies within $1.6\sigma$ of the best fit, indicating that the data does not provide statistically significant evidence for Lorentz violation. The analysis further shows that the preference for a negative $\xi$ in the complete fit largely stems from the tension between the proton radius values derived from electronic versus muonic hydrogen; the LIV parameter effectively absorbs part of this "proton radius tension."

Significance and Claims
The paper claims to provide the most robust constraint on the dimensionless LIV parameter $\xi$ derived from the simplest atomic systems (electronic and muonic hydrogen). By using these minimal systems, the authors argue they circumvent ambiguities inherent in heavier atoms with complex nuclear structures.

The significance of the result is twofold:

Planck-Scale Sensitivity: The derived bounds approach Planck-scale sensitivity, demonstrating that tabletop atomic spectroscopy can probe quantum gravity effects.
Methodological Robustness: The study highlights that current limitations in the independent determination of the Rydberg constant and proton radius, combined with theoretical uncertainties, prevent atomic spectroscopy from being fully conclusive regarding UV/IR mixing effects. The authors conclude that more precise determinations of these fundamental constants are necessary for atomic spectroscopy to fully scrutinize the Planck scale. The paper does not claim to have detected LIV but rather establishes a rigorous framework for constraining it, showing that current data is consistent with standard physics once systematic tensions are accounted for.

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