Sub-part-per-trillion test of the Standard Model with atomic hydrogen

This paper presents a sub-part-per-trillion precision measurement of the 2S-6P transition in atomic hydrogen that resolves discrepancies in the proton charge radius by confirming the muonic hydrogen value and validates the Standard Model and bound-state QED corrections to unprecedented levels of accuracy.

The Gist

Imagine the universe is built on a giant, incredibly complex instruction manual called the Standard Model. This manual tells us how particles like electrons and protons interact with light. For decades, scientists have been checking if the manual is written correctly by measuring the "energy levels" of the simplest atom in existence: Hydrogen.

Think of a hydrogen atom like a tiny solar system. It has a heavy sun (the proton) in the middle and a tiny planet (the electron) orbiting it. Just like planets can only orbit at specific distances, the electron can only exist at specific energy levels. When the electron jumps from one level to another, it absorbs or emits light at a very specific color (frequency).

The Mystery: The "Proton Radius Puzzle"

For a long time, scientists had a disagreement about the size of the proton (the sun).

• Group A measured the size using regular hydrogen atoms.
• Group B measured it using "muonic hydrogen," a weird version where the electron is replaced by a heavier cousin called a muon.

Group B found the proton was smaller than Group A thought. This was a huge shock, like if you measured a basketball and found it was actually the size of a marble. This disagreement, known as the "Proton Radius Puzzle," suggested that either the measurements were wrong, or the Standard Model's instruction manual had a missing page.

The Experiment: A High-Stakes Ping Pong Match

To solve this, the researchers in this paper built a super-precise "ping pong" machine for atoms.

1. The Setup: They created a beam of hydrogen atoms flying through a vacuum chamber, cooled down to near absolute zero (colder than deep space).
2. The Target: They wanted to measure a specific jump the electron makes, from the 2S level to the 6P level. This is like measuring the exact height of a specific step on a staircase.
3. The Problem: When atoms fly, they move. If you try to measure the color of light hitting a moving car, the color looks slightly different (the Doppler effect, like a siren changing pitch as an ambulance drives by). This "blur" makes it hard to measure the step height precisely.
4. The Solution: They shot laser beams at the atoms from both directions (front and back) at the same time.
   • Imagine two people throwing balls at a runner. If the runner moves forward, the ball from the front hits them harder, and the ball from the back hits them softer.
   • By combining the effects of both lasers, the "speed" errors cancel each other out. It's like a magic trick where the motion disappears, leaving only the pure, perfect measurement of the energy jump.

The "Ghost" Effects

Even with the motion canceled, there were tricky "ghost" effects that could distort the measurement:

• The Light Force: The laser light itself pushes the atoms slightly, like wind pushing a sailboat. The team had to calculate exactly how much the "wind" was pushing and correct for it.
• Quantum Interference: Sometimes, the electron takes two paths at once (a quantum superposition), and these paths can interfere with each other, distorting the signal. They used a special angle for their laser polarization (like tilting a pair of sunglasses) to cancel this out, similar to how noise-canceling headphones work.

The Result: The Manual is Correct!

After collecting data from thousands of atoms and correcting for every tiny possible error, they measured the frequency of the light jump with unprecedented precision.

• The Verdict: Their measurement matched the "smaller proton" size found by the muonic hydrogen group.
• The Implication: The "Proton Radius Puzzle" is solved! The Standard Model is correct. The proton really is smaller than the old measurements suggested.
• The Precision: They tested the theory to a precision of 0.7 parts per trillion. To visualize this: If you measured the distance from the Earth to the Moon, this experiment would be accurate to within the width of a single human hair.

Why Does This Matter?

This isn't just about counting atoms. It's a fundamental check on our understanding of reality.

• Confidence: It confirms that our best theory of physics (Quantum Electrodynamics) works perfectly, even at the tiniest scales.
• Future Hunting: Because the measurement is so precise, if there were any new, unknown particles or forces hiding in the atom, this experiment would have seen them. Since it didn't, it sets a very strict boundary for where scientists need to look next.

In short, the scientists built a super-accurate ruler, measured the smallest step in the universe, and confirmed that the universe's instruction manual is written exactly as we thought it was.

Technical Summary

1. Problem Statement

The paper addresses the "proton radius puzzle," a significant discrepancy in physics where the proton charge radius ($r_p$) derived from muonic hydrogen spectroscopy (where an electron is replaced by a muon) was found to be approximately 4% smaller than values derived from atomic hydrogen spectroscopy and electron scattering. This discrepancy, initially exceeding $5\sigma$, challenged the consistency of the Standard Model (SM) and Quantum Electrodynamics (QED).

While subsequent measurements in atomic hydrogen (e.g., 2S–4P, 1S–3S) favored the muonic value, they remained partly discrepant with each other and lacked the precision to conclusively rule out the older CODATA 2014 value or definitively test bound-state QED at the level of experimental uncertainties. The authors aimed to perform a measurement precise enough to distinguish between these conflicting values and rigorously test the SM.

2. Methodology

The researchers performed high-precision laser spectroscopy on the 2S–6P transition in atomic hydrogen using a cryogenic atomic beam.

• Experimental Setup:

  • Source: A cryogenic beam of hydrogen atoms generated by a copper nozzle at 4.8 K.
  • Preparation: Atoms were excited from the 1S ground state to the metastable 2S state using a 243 nm two-photon preparation laser.
  • Spectroscopy: A 410 nm linearly polarized laser drove the 2S–6P transition. The experiment utilized counter-propagating laser beams to suppress the first-order Doppler shift.
  • Detection: The signal was detected via fluorescence from the rapid decay of the 6P state (predominantly Lyman-$\epsilon$ decay to 1S). Two channel electron multipliers detected photoelectrons ejected from the detector walls, covering $>20\%$ of the solid angle.
  • Velocity Selection: By periodically blocking the preparation laser, the team isolated specific velocity groups ($\tau_i$) based on time-of-flight, allowing for the extrapolation of the transition frequency to zero velocity (Doppler-free).

• Key Technical Innovations & Corrections:

  • Magic Polarization Angle: The laser polarization was set to the "magic angle" ($56.5^\circ$) to suppress distortions caused by Quantum Interference (QI) between the excitation paths of the 6P$_{1/2}$ and 6P$_{3/2}$ fine-structure levels.
  • Active Fiber Retroreflector (AFR): Used to generate high-quality, wavefront-retracing counter-propagating beams to minimize Doppler broadening and splitting.
  • Light Force Shift (LFS) Modeling: A major systematic effect arises from the diffraction of matter waves on the standing wave of the laser (Kapitza-Dirac effect). The authors developed a sophisticated quantum-mechanical model treating atoms as partially coherent matter waves to simulate and correct for LFS, which is significant due to the transverse coherence length of the beam being comparable to the laser wavelength.
  • Systematic Control: Stray electric fields were characterized and corrected for dc-Stark shifts. Other corrections included second-order Doppler shifts, blackbody radiation (BBR) shifts, Zeeman shifts, and pressure shifts.

• Data Analysis:

  • The team recorded 3,155 line scans across three measurement runs.
  • They fitted Voigt and Voigt doublet line shapes to the data, correcting for asymmetries using simulations of LFS and QI.
  • Transition frequencies were extrapolated to zero speed to obtain the Doppler-free values.

3. Key Contributions

• Unprecedented Precision: The measurement achieved a precision of 0.66 parts per trillion (ppt) for the 2S–6P fine-structure centroid, representing a sixfold improvement over their previous 2S–4P measurement.
• Resolution of the Proton Radius Puzzle: By combining the new 2S–6P frequency with the precisely known 1S–2S transition, the authors derived a proton charge radius that is 2.5 times more precise than previous atomic hydrogen determinations.
• Rigorous SM Test: The experiment provided the most stringent test of bound-state QED corrections to date, reaching the level of three-photon corrections ($\propto \alpha^5$).
• Advanced Modeling: The paper presents a comprehensive theoretical framework for modeling light-force shifts in partially coherent atomic beams, a critical advancement for future high-precision spectroscopy.

4. Results

• Measured Transition Frequency:
  $$\nu_{2S-6P} = 730,690,248,610.79(48) \text{ kHz}$$
  (Uncertainty: 0.66 ppt)

• Proton Charge Radius ($r_p$):
  $$r_p = 0.8406(15) \text{ fm}$$

  • This value is in excellent agreement with the muonic hydrogen value ($0.84060(39)$ fm).
  • It disagrees with the CODATA 2014 value by 5.5$\sigma$, effectively ruling out the larger radius previously established by atomic hydrogen and electron scattering.

• Standard Model Test:

  • Using the muonic $r_p$ as input, the SM prediction for the 2S–6P frequency is $730,690,248,610.79(23)$ kHz.
  • The difference between the experimental result and the SM prediction is $0.00(53)$ kHz.
  • This constitutes a test of the Standard Model at 0.7 ppt and bound-state QED corrections at 0.5 parts per million (ppm).

• Rydberg Constant ($R_\infty$):
  The measurement yields $R_\infty = 10,973,731.5681524(79) \text{ m}^{-1}$, which is 50% more precise than the world average and highly compatible with other determinations.

5. Significance

• Resolution of the Puzzle: The study provides definitive evidence that the "proton radius puzzle" was not due to a failure of QED or the Standard Model, but rather due to insufficient precision and systematic errors in previous atomic hydrogen measurements. The muonic hydrogen value is confirmed as correct.
• Validation of QED: The agreement between experiment and theory at the sub-ppt level validates bound-state QED calculations, including high-order terms (three-photon corrections), which were previously untested at this precision.
• Future Directions: The techniques demonstrated (cryogenic beams, magic angle polarization, advanced LFS modeling) set a new standard for atomic spectroscopy. They can be applied to deuterium and other hydrogen-like systems to further constrain new physics, such as weakly interacting bosons in the keV mass range.
• Fundamental Constants: The results contribute significantly to the determination of fundamental constants, particularly the Rydberg constant and the proton charge radius, which are essential for the redefinition of SI units and high-precision tests of fundamental physics.