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arxiv: 2604.26401 · v1 · submitted 2026-04-29 · ⚛️ physics.atom-ph · hep-ph

Precision Spectroscopy of 2S-nS Transitions in Atomic Hydrogen: A Determination of the Proton Charge Radius

R. G. Bullis , W. L. Tavis , M. R. Weiss , J. Orellana Cisneros , A. J. Cheeseman , U. D. Jentschura , D. C. Yost This is my paper

Pith reviewed 2026-05-07 12:31 UTC · model grok-4.3

classification ⚛️ physics.atom-ph hep-ph
keywords proton radiushydrogen spectroscopyRydberg constanttwo-photon transitionsatomic hydrogenprecision measurement
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0 comments X

The pith

Measurements of 2S-nS transitions in hydrogen extract a proton radius of 0.8433(31) fm.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reports absolute frequency measurements of the 2S to nS two-photon transitions for n=8, 9 and 10 in a cryogenic beam of atomic hydrogen, each determined to a fractional uncertainty of 2.6 parts in 10^12. These new data are combined with the established 1S-2S transition frequency to solve simultaneously for the root-mean-square proton charge radius and the Rydberg constant. The resulting values agree with the CODATA 2022 recommendations. A reader would care because the proton radius remains a key parameter in tests of quantum electrodynamics and in reconciling spectroscopic and scattering determinations.

Core claim

We present absolute frequency measurements of 2S_{1/2}-nS_{1/2} two-photon transitions with n = 8, 9, and 10 in a cryogenic beam of atomic hydrogen. Each transition has been measured with a fractional uncertainty of approximately 2.6*10^(-12). Combining the results from this work and the 1S_{1/2}-2S_{1/2} transition frequency, we extract a root-mean-square proton radius of r_p = 0.8433(31) fm and a Rydberg frequency of cR_∞ = 3,289,841,960,252.9(9.7) kHz. These are in good agreement with the CODATA 2022 recommended values.

What carries the argument

Two-photon excitation of 2S-nS transitions in a cryogenic hydrogen beam, combined with the 1S-2S frequency to determine proton radius and Rydberg constant.

If this is right

  • The proton radius is determined independently of electron-scattering data.
  • The Rydberg constant is refined at the level of 3 parts in 10^12.
  • The results support the CODATA 2022 consensus values for both quantities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the quoted uncertainty holds, this spectroscopic route favors the smaller proton-radius value over the larger one obtained in some scattering experiments.
  • Extending the same method to still higher n or to deuterium could further tighten the radius without new assumptions about QED.
  • The agreement with CODATA suggests that residual Doppler and AC-Stark shifts in the cryogenic beam have been controlled at the 10^-12 level.

Load-bearing premise

All systematic effects in the cryogenic beam, laser stabilization, and two-photon excitation are fully characterized and stay within the quoted 2.6e-12 fractional uncertainty.

What would settle it

An independent frequency measurement of any of the 2S-8S, 2S-9S or 2S-10S transitions that differs from the reported values by more than a few times the quoted uncertainty would invalidate the extracted proton radius.

Figures

Figures reproduced from arXiv: 2604.26401 by A. J. Cheeseman, D. C. Yost, J. Orellana Cisneros, M. R. Weiss, R. G. Bullis, U. D. Jentschura, W. L. Tavis.

Figure 1
Figure 1. Figure 1: FIG. 1. Experimental apparatus. Photodiodes (PD) are view at source ↗
Figure 2
Figure 2. Figure 2: Residuals 200 kHz FIG. 2. Upper graph: Experimentally obtained lineshape of the 2S1/2 − 8S1/2 transition with active ac Stark shift mit￾igation fit with a Lorentzian function. Bottom graph: The residuals of the fit. To guide our data analysis, we have developed a line￾shape model, which takes into account the experimental geometry, the atom-laser excitation dynamics from both the spectroscopy and ac Stark … view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. ac Stark shift extrapolations for the 2S view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. dc Stark shift corrected extrapolations of the view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 view at source ↗
read the original abstract

We present absolute frequency measurements of 2S_{1/2}-nS_{1/2} two-photon transitions with n = 8, 9, and 10 in a cryogenic beam of atomic hydrogen. Each transition has been measured with a fractional uncertainty of approximately 2.6*10^(-12). Combining the results from this work and the 1S_{1/2}-2S_{1/2} transition frequency, we extract a root-mean-square proton radius of r_p = 0.8433(31) fm and a Rydberg frequency of cR_{\infty} = 3,289,841,960,252.9(9.7) kHz. These are in good agreement with the CODATA 2022 recommended values.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reports absolute frequency measurements of the 2S_{1/2}-nS_{1/2} two-photon transitions for n=8, 9, and 10 in a cryogenic beam of atomic hydrogen, each with a fractional uncertainty of approximately 2.6 × 10^{-12}. By combining these new measurements with the existing 1S_{1/2}-2S_{1/2} transition frequency, the authors extract a proton rms charge radius of r_p = 0.8433(31) fm and a Rydberg frequency of cR_∞ = 3,289,841,960,252.9(9.7) kHz, finding good agreement with the CODATA 2022 values.

Significance. If the systematic uncertainties are indeed controlled to the quoted level, this provides an important independent experimental input to the proton radius puzzle and the determination of fundamental constants. The new data from higher principal quantum numbers help to disentangle the finite nuclear size effects from other QED contributions. The agreement with CODATA is a strength, but the partial reliance on shared theoretical corrections with prior 1S-2S work limits the degree of independence.

major comments (2)
  1. The abstract and summary state a fractional uncertainty of 2.6 × 10^{-12} without providing the full error budget, data tables, or detailed characterization of systematics such as residual Doppler shifts, AC Stark shifts, second-order Doppler, Zeeman effects, and laser frequency reference chain. This is critical because an unaccounted contribution at the 10^{-12} level would affect the extracted r_p at the 0.5 sigma level, undermining the claim of agreement with CODATA.
  2. The central extraction of r_p and R_∞ subtracts theoretical QED and finite-size corrections from the measured frequencies. The manuscript should provide a clear breakdown of these corrections for the 2S-nS transitions (n=8,9,10) and compare them to those used in the 1S-2S analysis to allow evaluation of any circularity in the determination.
minor comments (2)
  1. The abstract uses informal notation '2.6*10^(-12)' instead of the standard 2.6 × 10^{-12}.
  2. The manuscript would benefit from a table summarizing the measured frequencies, their uncertainties, and the individual systematic contributions for each transition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We have addressed each major comment point by point below and revised the manuscript to improve clarity and transparency on the points raised.

read point-by-point responses

  1. Referee: The abstract and summary state a fractional uncertainty of 2.6 × 10^{-12} without providing the full error budget, data tables, or detailed characterization of systematics such as residual Doppler shifts, AC Stark shifts, second-order Doppler, Zeeman effects, and laser frequency reference chain. This is critical because an unaccounted contribution at the 10^{-12} level would affect the extracted r_p at the 0.5 sigma level, undermining the claim of agreement with CODATA.

    Authors: We agree that the abstract does not contain the full error budget and that a more prominent presentation is warranted. The main text and supplementary information already contain detailed characterizations of all listed systematics (residual Doppler, AC Stark, second-order Doppler, Zeeman, and the frequency reference chain) together with the associated data tables. To make this information immediately accessible, we have added a consolidated error-budget table in the revised manuscript that tabulates every contribution for each transition, demonstrating that the combined uncertainty remains at the quoted level with no unaccounted term large enough to shift r_p by 0.5 sigma. revision: yes

  2. Referee: The central extraction of r_p and R_∞ subtracts theoretical QED and finite-size corrections from the measured frequencies. The manuscript should provide a clear breakdown of these corrections for the 2S-nS transitions (n=8,9,10) and compare them to those used in the 1S-2S analysis to allow evaluation of any circularity in the determination.

    Authors: We agree that an explicit breakdown is necessary for evaluating independence. The revised manuscript now contains a new table that lists the individual QED and finite-nuclear-size corrections applied to each 2S-nS transition (n=8,9,10). A second column directly compares these values to the corrections used in the 1S-2S analysis. The table highlights that the leading finite-size term scales as 1/n^3, furnishing an independent constraint on r_p even though certain higher-order QED contributions are shared between the two data sets. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper reports independent experimental measurements of 2S-nS transition frequencies at 2.6e-12 fractional uncertainty and combines them with the existing 1S-2S datum. Extraction of r_p and cR_∞ proceeds by subtracting established external QED corrections (not derived or fitted within this work) to isolate the finite-size contribution. No quoted equations or steps reduce the output parameters to self-definitions, renamed fits, or load-bearing self-citations; the new data supply additional independent constraints on the same theoretical framework.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; main unstated premise is that the QED theoretical corrections for these transitions are known to better than the experimental uncertainty and that the 1S-2S frequency can be treated as an external anchor.

axioms (1)
  • domain assumption QED corrections and finite-nuclear-size shifts for 2S-nS transitions are accurate to the 10^{-12} level.
    Required to isolate r_p from the measured frequencies.

pith-pipeline@v0.9.0 · 5478 in / 1330 out tokens · 49808 ms · 2026-05-07T12:31:42.872690+00:00 · methodology

discussion (0)

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Reference graph

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