The CODATA 2022 least-squares adjustment of the fundamental physical constants achieves statistical consistency among its most precise input data only by multiplying their quoted uncertainties by ad hoc expansion factors: 2.5 for the six input data that determine the fine-structure constant α\alpha (entering through the electron magnetic-moment anomaly and two atom-recoil measurements), 1.7 for the hydrogen and deuterium (H/D) transition-frequency data and their theoretical corrections, and 3.9 for the 16 laboratory measurements of the Newtonian constant of gravitation GG. The 2022 report's consolidated 'suggestions for future work' poses the following problem: identify the physical origins of these mutual inconsistencies — unrecognized experimental systematics or deficient theory — eliminate them by identifying the required corrections to the published input data (corrected measurements, corrected theory terms, or corrected correlations), and show that a least-squares adjustment of the corrected inputs is self-consistent without any expansion factor, including convergence of the electronic-spectroscopy and muonic-spectroscopy determinations of the proton and deuteron charge radii (currently 2.8 σ2.8\,\sigma apart). Later literature narrows parts of the problem — the 2023 electron-moment measurement favors the rubidium recoil value of α\alpha and isolates the cesium recoil value as the outlier, independent proton-radius determinations now agree with one another, even though the radii entering the 2022 adjustment remain 2.8 σ2.8\,\sigma apart, and the first replication of a high-precision GG measurement (the BIPM torsion balance operated at NIST, 2026) itself disagrees with the BIPM value at the 2.5×10−42.5\times10^{-4} relative level — but the origin of each discrepancy class remains unidentified and no un-inflated self-consistent adjustment exists, so the problem is open.

Background

The Committee on Data (CODATA) Task Group on Fundamental Constants periodically performs a least-squares adjustment (LSA): a weighted fit of all relevant experimental results and theoretical predictions, whose output is the self-consistent recommended values of the constants. The 2022 adjustment (Mohr, Newell, Taylor and Tiesinga, 2024) used 133 input data and 79 adjusted constants. Its initial, unmodified fit had χ2=109.6\chi^{2}=109.6 for 54 degrees of freedom — a 0.001% probability of arising by chance if the data were mutually consistent — so the Task Group enlarged the quoted uncertainties of three entire input-data classes until the fit became statistically acceptable. These multiplicative 'expansion factors' are the report's explicit measure of unresolved inconsistency.

The three affected classes are: (i) the six input data that determine the fine-structure constant α\alpha (the electromagnetic coupling constant, dimensionless), which enter through the experimental electron magnetic-moment anomaly aea_{e} (the fractional deviation of the electron's gg-factor from 2, measured in a one-electron Penning trap — a device that confines the particle with electric and magnetic fields) combined with its prediction from quantum electrodynamics (QED, the quantum field theory of electromagnetism), and through atom-recoil measurements of h/mh/m for rubidium and cesium (from which α\alpha follows via h/m(X)=[Ar(e)/Ar(X)] cα2/2R∞h/m(X)=[A_{\rm r}(e)/A_{\rm r}(X)]\,c\alpha^{2}/2R_{\infty}, where R∞R_{\infty} is the Rydberg constant governing hydrogen-like spectra). Their uncertainties had to be inflated by 2.5; the cesium recoil result carries a normalized residual of 4.7 in the initial fit (the normalized residual is the difference between a measured value and the value assigned by the fit, divided by the datum's quoted standard uncertainty). (ii) The 16 laboratory measurements of the Newtonian constant of gravitation GG (torsion-balance and beam-balance experiments, treated in a separate calculation); their mutual scatter requires an expansion factor of 3.9, with the largest initial residuals 7.7 (BIPM-14), 6.8 (JILA-18) and 4.8 (BIPM-01), making GG the least accurately known of the major constants. (iii) The 29 measured hydrogen and deuterium transition frequencies, which determine R∞R_{\infty} together with the proton and deuteron rms charge radii rpr_{\rm p}, rdr_{\rm d}; six of them (four 2S–8D results reported in the 1990s by a Paris collaboration of the Laboratoire Kastler-Brossel (LKB) and SYRTE laboratories, the 2018 Paris 1S–3S result, and the 2022 Colorado State 2S–8D result with residual 3.4) force an expansion factor of 1.7 applied also to the theoretical energy-level corrections and the muonic Lamb-shift data (the Lamb shift is the small radiative shift between otherwise nearly degenerate atomic levels; in muonic atoms, where a negative muon replaces the electron, the contribution of the nuclear charge distribution — the finite-nuclear-size effect — is far larger, which is why muonic spectroscopy determines the nuclear charge radii). In the same class, the radii extracted from electronic H/D spectroscopy exceed the muonic-atom values by 2.8 σ2.8\,\sigma.

Why the consistency of these data matters is stated in the report itself: α\alpha now underlies μ0\mu_{0} and parts of the electrical SI in the unit system built on fixed constants, and the comparison of aea_{e}(exp) with aea_{e}(th) is among the most precise low-energy tests of the Standard Model — a test whose sensitivity is currently limited by the discrepant α\alpha determinations. The 2022 recommended values are α−1=137.035 999 177(21)\alpha^{-1}=137.035\,999\,177(21) and G=6.67430(15)×10−11 m3 kg−1 s−2G=6.67430(15)\times10^{-11}\,{\rm m^{3}\,kg^{-1}\,s^{-2}} (unchanged from 2018 precisely because the GG data could not be reconciled).

Problem Statement

As posed by the CODATA 2022 report (Sec. XVI.3), the problem is to identify and eliminate the sources of the mutual inconsistencies among the highest-precision determinations of the fundamental constants, so that a future least-squares adjustment of the constants is self-consistent without ad hoc uncertainty inflation. Concretely, for each of the three affected input-data classes:

(i) the fine-structure constant α\alpha: determine why the three most precise determinations — from aea_{e}(exp) with aea_{e}(th), from h/m( 87Rb)h/m(\,^{87}{\rm Rb}), and from h/m( 133Cs)h/m(\,^{133}{\rm Cs}) — are mutually inconsistent at the level requiring a 2.52.5-fold expansion of their uncertainties (the cesium determination being the 4.7σ4.7\sigma initial-fit outlier), whether through an unidentified experimental systematic or a deficient theoretical contribution, and identify the corrected measurements, theory terms, or correlations that remove the inconsistency;

(ii) the Newtonian constant of gravitation GG: identify the laboratory-specific or common systematic effects responsible for the mutual inconsistency of the 16 state-of-the-art measurements (requiring a 3.93.9-fold expansion; largest initial normalized residuals 7.7, 6.8, 4.8), and identify the corrections to those measurements, their uncertainties, or their correlations that make them converge, confirmable by independent approaches such as atom-interferometric or precision-displacement-sensor determinations;

(iii) hydrogen and deuterium spectroscopy: resolve the internal inconsistency of the 29 measured H/D transition frequencies (six data with initial normalized residuals between 2.5 and 3.4, forcing a 1.71.7-fold expansion that also covers the theoretical corrections and muonic Lamb-shift data), by identifying the unrecognized systematic or analysis issues in the discrepant measurements and/or the missing higher-order theoretical terms, and by identifying the corrected measurements and theory terms that render the Rydberg constant and the proton and deuteron charge radii consistent — including convergence of the electronic-spectroscopy and muonic-spectroscopy values of rpr_{\rm p} and rdr_{\rm d}, currently 2.8 σ2.8\,\sigma apart.

These three deliverables share one common scientific objective and one completion boundary, namely that applying the identified corrections to the published input data yields a CODATA-style least-squares adjustment with acceptable χ2\chi^{2} (standard goodness-of-fit at the stated degrees of freedom) and all normalized residuals within statistical bounds, with no expansion factor applied to any of these input classes and with the responsible systematic effects or theory deficiencies identified and independently verified; the report's recommended independent new determinations and remeasurements remain the longer-term confirmation of those corrections. The muon g−2g-2 discrepancy (4.2σ\sigma), which the Task Group excluded from the adjustment, is not part of this problem, since resolving it changes no expansion factor.

References

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