# A Possible Explanation of the Proton Radius Puzzle Based on the Second Flavor of Muonic Hydrogen Atoms

## Abstract

**:**

_{p}was determined by the spectroscopic method, relying on the electron energy levels in hydrogen atoms, and by the elastic scattering of electrons on protons. In 2010, and then in 2013, two research teams determined r

_{p}from the experiment on muonic hydrogen atoms and they claimed r

_{p}to be by about 4% smaller than it was found from the experiments with electronic hydrogen atoms. Since then, several research groups performed corresponding experiments with electronic hydrogen atoms and obtained contradictory results: some of them claimed that they found the same value of r

_{p}as from the muonic hydrogen experiments, while others reconfirmed the larger value of r

_{p}. The conclusion of the latest papers (including reviews) is that the puzzle is not resolved yet. In the present paper, we bring to the attention of the research community, dealing with the proton radius puzzle, the contributing factor never taken into account in any previous calculations. This factor has to do with the hydrogen atoms of the second flavor, whose existence is confirmed in four different types of atomic experiments. We present a relatively simple model illustrating the role of this factor. We showed that disregarding the effect of even a relatively small admixture of the second flavor of muonic hydrogen atoms to the experimental gas of muonic hydrogen atoms could produce the erroneous result that the proton charge radius is by about 4% smaller than its actual value, so that the larger out of the two disputed values of the proton charge radius could be, in fact, correct.

## 1. Introduction

_{p}was determined by the spectroscopic method, relying on the electron energy levels in hydrogen atoms, and by the elastic scattering of electrons on protons. The mean value of the proton charge radius, recommended by CODATA (Committee on Data of the International Science Council), was r

_{p}= (0.8775 ± 0.0051) × 10

^{−13}cm—see, e.g., the reviews by Pohl et al. [1] and by Gao and Vanderhaenghen [2], as well as references therein.

_{p}from the experiment on muonic hydrogen atoms. Because the ratio of the muon mass m

_{μ}to the electron mass m

_{e}is m

_{μ}/m

_{e}≈ 207, the average muon–proton distance in muonic hydrogen atoms is about 200 smaller than the electron–proton distance in electronic hydrogen atoms. Therefore, the shift in the energy of an S-state, caused by the finite proton size, for muonic hydrogen atoms is about 8 million times greater than for electronic hydrogen atoms. Consequently, muonic measurements should be much more sensitive to r

_{p}than the corresponding electronic measurements. The resulting proton charge radius was claimed to be r

_{p}= (0.84087 ± 0.00039) × 10

^{−13}cm, e.g., about 4% (or 5 standard deviations) smaller than the above CODATA value. This result prompted calls for a new physics model beyond the standard model.

_{p}consistent with the muonic measurements from papers [3,4]. In the same year, Xiong et al. [6] remeasured r

_{p}in the electron scattering experiment and found it to be consistent with the muonic measurements from [3,4].

_{p}, found before the year 2010, yielded the larger value. Besides, Fleurbaey et al. [7] reported the larger value of r

_{p}= (0.877 ± 0.013) × 10

^{−13}cm, obtained from the two-photon measurements in the electronic hydrogen (they measured the 1S–3S two-photon transition frequency of hydrogen using a continuous-wave excitation laser at 205 nm).

^{β/2}, where

^{2},

^{2}/(ħc) ≈ 0.007297), while for the SFHA, R(r) scales as ~ 1/r

^{2−β/2}. Consequently, for relatively large values of the linear momentum p >> p

_{0}= me

^{2}/ħ (where m is the mass of the atomic lepton, whether it is electron or muon), the corresponding wave function in the momentum representation φ(p) for the SFHA falls off much slower than for the hydrogen atoms of the first (usual) flavor. This is because φ(p) and R(r) are interconnected by the Fourier transform, so that, for the SFHA, the more rapid increase in R(r) as r decreases translates into the slower decrease in φ(p) as p increases in the range of p >> p

_{0}.

- Experimental distribution dw = F(p)dp of the linear momentum p in the ground state of electronic hydrogen atoms.

_{0}<< p << mc, i.e., in the non-relativistic part of the tail of the distribution (we note that p

_{0}/mc = α ≈ 0.007297), the experimental result, deduced by Gryzinski [15] from the analysis of atomic experiments, was F

_{exper}(p) ~ (mc/p)

^{4}, while the corresponding theoretical result by Fock [16] was F

_{theor}(p) ~ (mc/p)

^{6}. Here, F(p)dp is the probability of finding the linear momentum in the interval (p, p + dp). This means that, for the ratio F

_{theor}(p)/F

_{exper}(p) = (mc/p)

^{2}, for the values of p ~ 10p

_{0}, the discrepancy F

_{theor}(p)/F

_{exper}(p) between the experimental and theoretical results was ~200 times (!).

- B.
- Experiments on the electron impact excitation of electronic hydrogen molecules

- C.
- Experiments on the electron impact excitation of electronic hydrogen atoms

- D.
- Experiments on the charge exchange between electronic hydrogen atoms and protons

_{e}in those calculations by m

_{μ}. So, there should exist the second flavor of muonic hydrogen atoms (SFMHA).

## 2. Model

^{5/4}{1/r

^{β/2}− ε[R

_{p}

^{2}/(5βr

^{2})]},

g(r) ≈ 4β

^{3/4}{1/r

^{β/2}− ε[R

_{p}

^{2}/(5βr)]}.

_{p}is the proton radius in units of the muonic Bohr radius a

_{0μ}= ħ

^{2}/(m

_{μ}e

^{2}), and r is the distance from the origin in units of the muonic Bohr radius a

_{0μ}. Equation (2) was simplified compared with Equation (17) from [12], using the fact that β = α

^{2}<< 1. We also note that, in Equation (17) from [12], the second term in f(r) and g(r) was proportional to the quantity:

_{0}− E

_{μ}c

^{2}. Because, in our Equation (2), the second term in f(r) and g(r) is assumed to be a relatively small correction to the first term (because ε << 1), while deriving Equation (2), we used for the shift the following approximate textbook expression (see, e.g., Flügge textbook [20]):

_{p}/5.

^{2}(r) + g

^{2}(r)].

^{2}(r)/g

^{2}(r) ~ α

^{2}<< 1, so that

_{0}(r)|

^{2}/(4π) ≈ g

^{2}(r) ≈ 16β

^{3/2}/r

^{β}− ε[32β

^{1/2}R

_{p}

^{2}/(5r

^{1+β/2})] + ε

^{2}[16R

_{p}

^{4}/(25β

^{1/2}r

^{2})].

_{p}) = b |Ψ

_{0}(R

_{p})|

^{2}R

_{p}

^{2}= b{16β

^{3/2}/R

_{p}

^{β/2}− ε(32β

^{1/2}R

_{p}

^{1 − β/2}/5) + ε

^{2}[16R

_{p}

^{2}/(25β

^{1/2})]},

_{p}) = δE(0, 0.96R

_{p}),

_{p}that would be 4% smaller than the actual value of R

_{p}.

_{1}= 1.07 × 10

^{−5}/R

_{p}

^{1.000027}≈ 1.07 × 10

^{−5}/R

_{p},

_{2}= 5.22 × 10

^{−4}/R

_{p}

^{1.000027}≈ 5.22 × 10

^{−4}/R

_{p}.

_{p}(defined as the root-mean-square radius of the proton charge distribution) in units of the muonic Bohr radius a

_{0μ}is 0.00343. The proton “sphere” radius R

_{p}would be a factor of (5/3)

^{1/2}greater than r

_{p}(it would be equal to 0.00443) if the proton would be a uniformly charged sphere (which the proton is not). The actual value of R

_{p}should be between 0.00343 and 0.00443. For further numerical estimates of ε

_{1}and ε

_{2}, we adopt the value R

_{p}≈ 0.004, so that

_{1}≈ 0.003, ε

_{2}≈ 0.13.

_{2}= 0.13 seems to be slightly more preferable (compared with ε

_{1}= 0.003). This is because it is of the same order of magnitude as the share of the SFHA in the experimental gas of the electronic hydrogen molecules, which (the share) was required for eliminating the large discrepancy (by at least of a factor of two) between the theoretical and experimental cross sections of the excitation by the electron impact [17].

_{p}is proportional to R

_{p}, the above result about the determination of R

_{p}from the energy shift is also true for r

_{p}. Namely, indeed, even a relatively small admixture of the SFMHA to the usual muonic hydrogen atoms in the experimental gas can lead to the false conclusion that the proton charge radius r

_{p}is about 4% smaller than its actual value.

## 3. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Pohl, R.; Gilman, R.; Miller, G.A.; Pachucki, K. Muonic Hydrogen and the Proton Radius Puzzle. Annu. Rev. Nucl. Part. Sci.
**2013**, 63, 175–204. [Google Scholar] [CrossRef] - Gao, H.; Vanderhaenghen, M. The Proton Charge Radius. Rev. Mod. Phys.
**2022**, 94, 015002. [Google Scholar] [CrossRef] - Pohl, R.; Antognini, A.; Nez, F.; Amaro, F.D.; Biraben, F.; Cardoso, J.M.R.; Covita, D.S.; Dax, A.; Dhawan, S.; Fernandes, L.M.P.; et al. The Size of the Proton. Nature
**2010**, 466, 213. [Google Scholar] [CrossRef] [PubMed] - Antognini, A.; Nez, F.; Schuhmann, K.; Amaro, F.D.; Biraben, F.; Cardoso, J.M.R.; Covita, D.S.; Dax, A.; Dhawan, S.; Diepold, M.; et al. Proton structure from the measurement of 2S-2P transition frequencies of muonic hydrogen. Science
**2013**, 339, 417. [Google Scholar] [CrossRef] [PubMed] - Bezginov, N.; Valdez, T.; Horbatsch, M.; Marsman, A.; Vutha, A.C.; Hessels, E.A. A measurement of the atomic hydrogen Lamb shift and the proton charge radius. Science
**2019**, 365, 1007–1012. [Google Scholar] [CrossRef] [PubMed] - Xiong, W.; Gasparian, A.; Gao, H.; Dutta, D.; Khandaker, M.; Liyanage, N.; Pasyuk, E.; Peng, C.; Bai, X.; Ye, L.; et al. A small proton charge radius from an electron–proton scattering experiment. Nature
**2019**, 575, 147–151. [Google Scholar] [CrossRef] [PubMed] - Fleurbaey, H.; Galtier, S.; Thomas, S.; Bonnaud, M.; Julien, L.; Biraben, F.; Nez, F.; Abgrall, M.; Guéna, J. New Measurement of the 1S-3S Transition Frequency of Hydrogen: Contribution to the Proton Charge Radius Puzzle. Phys. Rev. Lett.
**2018**, 120, 183001. [Google Scholar] [CrossRef] [PubMed] - Karr, J.P.; Marchand, D. Progress on the proton radius puzzle. Nature
**2019**, 575, 61–62. [Google Scholar] [CrossRef] [PubMed] - Karshenboim, S.G.; Korzinin, E.Y.; Shelyuto, V.A.; Ivanov, V.G. Theory of Lamb shift in atomic hydrogen. J. Phys. Chem. Ref. Data
**2015**, 44, 031202. [Google Scholar] [CrossRef] - Simon, G.; Schmitt, C.; Borkowski, F.; Walther, V.H. Absolute electron-proton cross sections at low momentum transfer measured with a high pressure gas target system. Nucl. Phys.
**1980**, A333, 381. [Google Scholar] [CrossRef] - Perkins, D.H. Introduction to High Energy Physics; Addison-Wesley: Menlo Park, CA, USA, 1987; Sect. 6.5. [Google Scholar]
- Oks, E. High-Energy Tail of the Linear Momentum Distribution in the Ground State of Hydrogen Atoms or Hydrogen-like Ions. J. Phys. B At. Mol. Opt. Phys.
**2001**, 34, 2235–2243. [Google Scholar] [CrossRef] - Oks, E. Alternative Kind of Hydrogen Atoms as a Possible Explanation of the Latest Puzzling Observation of the 21 cm Radio Line from the Early Universe. Res. Astron. Astrophys.
**2020**, 20, 109. [Google Scholar] [CrossRef] - Oks, E. Two Flavors of Hydrogen Atoms: A Possible Explanation of Dark Matter. Atoms
**2020**, 8, 33. [Google Scholar] [CrossRef] - Gryzinski, M. Classical Theory of Atomic Collisions. I. Theory of Inelastic Collisions. Phys. Rev.
**1965**, 138, A336. [Google Scholar] [CrossRef] - Fock, V. Zur Theorie des Wasserstoffatoms. Z. Physik
**1935**, 98, 145. [Google Scholar] [CrossRef] - Oks, E. Experiments on the Electron Impact Excitation of Hydrogen Molecules Indicate the Presence of the Second Flavor of Hydrogen Atoms. Foundations
**2022**, 2, 697–703. [Google Scholar] [CrossRef] - Oks, E. Experiments on the Electron Impact Excitation of the 2s and 2p States of Hydrogen Atoms Confirm the Presence of their Second Flavor as the Candidate for Dark Matter. Foundations
**2022**, 2, 541–546. [Google Scholar] [CrossRef] - Oks, E. Analysis of Experimental Cross-Sections of Charge Exchange between Hydrogen Atoms and Protons Yields another Evidence of the Existence of the Second Flavor of Hydrogen Atoms. Foundations
**2021**, 1, 265–270. [Google Scholar] [CrossRef] - Flügge, S. Practical Quantum Mechanics; Springer: Berlin, Germany, 1974; problem 75. [Google Scholar]

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**MDPI and ACS Style**

Oks, E.
A Possible Explanation of the Proton Radius Puzzle Based on the Second Flavor of Muonic Hydrogen Atoms. *Foundations* **2022**, *2*, 912-917.
https://doi.org/10.3390/foundations2040062

**AMA Style**

Oks E.
A Possible Explanation of the Proton Radius Puzzle Based on the Second Flavor of Muonic Hydrogen Atoms. *Foundations*. 2022; 2(4):912-917.
https://doi.org/10.3390/foundations2040062

**Chicago/Turabian Style**

Oks, Eugene.
2022. "A Possible Explanation of the Proton Radius Puzzle Based on the Second Flavor of Muonic Hydrogen Atoms" *Foundations* 2, no. 4: 912-917.
https://doi.org/10.3390/foundations2040062