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**A question with too many solutions: Proton size puzzle**

Chung-Wen Kao Chung-Yuan Christian University, Taiwan Institute of Physics, Academic Sinica

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Discovery of Proton(1917) In 1917 Rutherford proved that the hydrogen nucleus is present in other nuclei, a result usually described as the discovery of the proton. He noticed that when alpha particles were shot into nitrogen gas, his scintillation detectors showed the signatures of hydrogen nuclei. Rutherford determined that this hydrogen could only have come from the nitrogen, and therefore nitrogen must contain hydrogen nuclei. The hydrogen nucleus is therefore present in other nuclei as an elementary particle, which Rutherford named the proton, after the neuter singular of the Greek word for "first", πρῶτον. He found the protons mass at 1,836 times as great as the mass of the electron.

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Spin of proton (1927) The first indication that the proton had spin 1/2 came from the observation of an anomaly in the specific heat of the molecular hydrogen. Half a century before full quantum mechanical theory arrived. Maxwell observed that the specific heat capacity of H2 unaccountably departs from that of a diatomic gas below room temperature and begins to increasingly resemble that of a monatomic gas at cryogenic temperatures.

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Spin of proton (1927) This puzzle can be solved assuming the proton has spin one -half. If protons have spin 1/2, the two protons inside the hydrogen can have a spin 1 and hence a symmetric wave function – this is called ortho-hydrogen – or a spin 0, with an antisymmetric wave function, which is called para-hydrogen. These two protons are bound by a potential which is produced by the electrons, and they have rotation levels (vibrations also exist but are much higher). Ortho-hydrogen has only odd angular momentum rotation levels because of the Pauli principle for protons, while para-hydrogen has only even rotation levels. Taking this into account in counting the degrees of freedom of hydrogen, with a ratio 3:1 of ortho parahydrogen at room temperature.

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Spin of proton (1927) David Mathias Dennison (1900 – 1976) was an American physicist who made contributions to quantum mechanics, spectroscopy and the physics of molecular structure.

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g factor of proton (1933) Not only the electrons have the spin in the atom but also the nucleons. But the proton and the neutron have much bigger masses than the electron (saying more exactly about 1836 times bigger). And the magnetic dipole moment is inversely proportional to the mass of the particle. So the moments of the proton and the neutron are very small in comparison with the moment of the electron. Stern, Frish, and Easterman measured those tiny magnetic dipoles in 1933. Proton: g = / Neutron: g = /

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Pauli and his advice Pauli and Stern were great friends, which meant they were always arguing. Pauli had advised Stern not to measure the magnetic moment of the proton because according to the new formulated Dirac theory, the g value of point-like spin ½ particle must be 2! Lucky for Stern who didn’t follow Pauli’s advice and found that the g value of the proton is not 2 which means the proton is not point-like particle even it is very small. Only till 1960s, people could estimate the size of the proton directly by experiment of form factors… Hahaha… Lucky me not to listen to you…. Damn it, I cannot believe I am wrong…. Again!

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**Nucleon E.M form factors**

"for his pioneering studies of electron scattering in atomic nuclei and for his thereby achieved discoveries concerning the structure of the nucleons Hofstadter determined the precise size of the proton and neutron by measuring their form factor in 1961.

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**Rosenbluth Separation Method**

Within one-photon-exchange framework:

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**Finite size of the proton**

ep-elastic Finite size of the proton ed-elastic: Finite size + nuclear structure R. Hofstadter, Rev. Mod. Phys. 56 (1956) 214

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The size of the proton? Charge Radius Breit Frame. q0=0

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**The proton size and hydrogen spectrum**

There is another way to measure the proton charge radius and it can research higher precision. It is through the spectrum of hydrogen. Usually the proton is treated as a point charge since the Bohr radius is about 104~105 times larger than the size of the proton. However when the precision of measurement of the spectrum is high enough, the finite size of the proton will become measurable. To be more specific, let us review the spectrum of the hydrogen.

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**Spectrum of Hydrogen atom**

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**Energy scales in spectrum**

1 kHz=103 Hz 1 MHz=106 Hz 1GHz=109 Hz 1 THz=1012 Hz 1PHz=1015 Hz 1 μeV=10-6 eV 1 meV=10-3 eV 1 keV=103 eV 1 MeV=106 eV 1GeV=109 eV 1TeV=1012 eV

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**The size of proton and Lamb shift**

Bound state QED started in 1947, when the Lamb shift between the 2S1/2 and the 2P1/2 state of the hydrogen atom was found. The Lamb shift is the splitting of an energy level caused by the radiative corrections such as vacuum polarization, electron self-energy and vertex correction. The proton charge radius is the limiting factor when comparing experiments to QED theory, so we need for a more precise measurement of rp.

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**Lamb Shift (1947) Willis Eugene Lamb, Jr.**

(B.1913 – D. 2008) In 1947, Willis Lamb discovered that the 2p1/2 state is slightly lower than the 2s1/2 state resulting in a slight shift of the corresponding spectral line. It was a puzzle because due to Dirac equation two states are degenerate.

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How Lamb measured it? Willis Lamb formed a beam of hydrogen atoms in the 2s1/2 state. These atoms could not directly take the transition to the 1s1/2 state because of the selection rule which requires the orbital angular momentum to change by 1 unit in a transition. Putting the atoms in a magnetic field to split the levels by the Zeeman effect, he exposed the atoms to microwave radiation at 2395 MHz (not too far from the ordinary microwave oven frequency of 2560 MHz). Then he varied the magnetic field until that frequency produced transitions from the 2p1/2 to 2p3/2 levels. He could then measure the allowed transition from the 2p3/2 to the 1s1/2 state. He used the results to determine that the zero-magnetic field splitting of these levels correspond to 1057 MHz.

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How Lamb measured it?

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**Lamb Shift: QED calculation**

k(n,0) is a numerical factor which varies slightly with n from 12.7 to 13.2. k(n,l) is a small numerical factor <0.05

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**Nuclear finite size in spectrum**

The nuclear finite size effects appear in the Lamb shift: The nuclear finite size effects also appear in the hyperfine splitting:

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Many correction terms

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**Measurement of Lamb shift**

(measured directly) (the Lamb shift deduced from the measured fine structure interval 2p3/2 − 2s1/2)

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**Charge radius of Proton from Lamb shift measurement**

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What is CODATA? The Committee on Data for Science and Technology (CODATA) was established in 1966 as an interdisciplinary committee of the International Council for Science. It seeks to improve the compilation, critical evaluation, storage, and retrieval of data of importance to science and technology. The CODATA Task Group on Fundamental Constants was established in Its purpose is to periodically provide the international scientific and technological communities with an internationally accepted set of values of the fundamental physical constants and closely related conversion factors for use worldwide. The first such CODATA set was published in 1973, later in 1986, 1998, 2002 and the fifth in The latest version is Ver.6.0 called "2010CODATA" published on The CODATA recommended values of fundamental physical constants are published at the NIST Reference on Constants, Units, and Uncertainty. CODATA sponsors the CODATA international conference every two years.

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Muonic hydrogen The muon is about 200 times heavier than the electron Therefore, the atomic Bohr radius of muonic hydrogen is smaller than in ordinary hydrogen. The μp Lamb shift, ΔE(2P1/2-2S1/2) ≈ 0.2 eV, is dominated by vacuum polarization which shifts the 2S binding energy towards more negative values . The μp fine- and hyperfine splitting are an order of magnitude smaller than the Lamb shift. The relative contribution of the proton size to ΔE(2P-2S) is as much as 1.8%, two orders of magnitude more than for normal hydrogen atoms. Thus, the measurement of the Lamb shift of muonic hydrogen allows a more accurate determination of the size of the proton!

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**Why not pure leptonic bound states?**

Naively one may think it would be a good idea to measure leptonic bound states such as e+e- because there is no uncertainty from hadronic physics. The reason not to measure similar thing of e+e- is because its life time is very short due to annihilation. How about measure μ+e-? Its life time is much longer. The trouble is that we don’t have enough precise value for mass of μ.

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**Muonic hydrogen Lamb shift**

1 kHz=103 Hz 1 MHz=106 Hz 1GHz=109 Hz 1 THz=1012 Hz 1PHz=1015 Hz

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Hard work for 40 years This kind of measurement has been considered for over 40 years, but only recent developements in laser technology and muon beams made it feasible to carry it out The experiment is located at a new beam‐line for low‐energy (5keV) muons of the proton accelerator at the Paul Scherrer Institute (PSI) in Switzerlan

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**Muonic hydrogen Spectrum**

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**Surprisingly new result!**

To match the 2010 value with the CODATA value, an additional term of 0.31meV would be required in the QED equation. This corresponds to 64 times its claimed uncertainty!

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**Surprisingly new result!**

The transition frequency between 2P3/2 and 1S1/2 is obtained to be Δν = (77) GHz , corresponding to an energy difference of ΔE = (32) meV Theory predicts a value of ΔE = (49) ‐ rp² rp³ meV [rp in fm] This results in a proton radius of rp = (36) fm, 4% smaller than the previous best estimate, which has been the average of many different measurements made over the years.

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**It becomes headline news in Taiwan…..**

There is a team led by Prof Liu, Yi-Wei who is faculty member in National Tsing-Huang University participating this experiment. When their article appeared at “Nature” , this becomes a headline news in newspaper…..

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**Puzzle about the proton size**

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Third Zemach moment If GE is dipole form

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**Four possibilities The experimental results are not right.**

The relevant QED calculations are incorrect. There is, at extremely low energies and at the level of accuracy of the atomic experiments, physics beyond the standard model appears. A single-dipole form factor is not adequate to the analysis of precise low-energy data.

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However………… In view of the excellent accuracy of the muonic experiment and the very good agreement of the about half dozen electronic experiments, it is highly unlikely that the reason for the deviation is due to a problem on the experimental side. So we should look for other possibilities.

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New Physics? V. Barger Cheng-Wei Chiang, Wai-Yee Keung, and Danny Marfatia explore the possibility that new scalar, pseudoscalar, vector, axial-vector, and tensor flavor-conserving nonuniversal interactions may be responsible for the discrepancy. They consider exotic particles that among leptons, couple preferentially to muons and find that the many constraints from low energy data disfavor new spin-0, spin-1 and spin-2 particles as an explanation. Phys.Rev.Lett.106:153001,2011

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New Physics? The 95% C. L. range of / required to reproduce the muonic Lamb shift is indicated by the green shaded region. The black solid, red dashed and blue dot-dashed lines are the upper limits for vector, scalar and spin-2 particles, respectively, from a combination of n−208Pb scattering data and the anomalous magnetic moment of the muon. The black dotted curve is the upper bound obtained from atomic X-ray transitions. All bounds are at the 95% C. L. Phys.Rev.Lett.106:153001,2011

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There is still hope…… There are ways to relax some of the bounds at the expense of introducing complication. Since the contributions of scalars and pseudo-scalars are opposite in sign, allowing both a scalar boson and a pseudo-scalar boson with appropriately tuned couplings can lead to a cancellation that permits a rather large muonic coupling. Then, although the hadronic couplings are highly restricted, the muonic Lamb shift can be accommodated . Another possibility is that the new interaction violates isospin or CP, so that additional freedom is garnered. Phys.Rev.Lett.106:153001,2011

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**Example of fine tuned model of “New Physics”**

C. E. Carlson and B. C. Rislow Phys.Rev. D86 (2012) Main constraints are from muon magnetic moment (g-2) and K-decay

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**C. E. Carlson and B. C. Rislow**

Phys.Rev. D86 (2012)

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**Incorrect QED calculation?**

Although QED is very successful theory, nevertheless, to calculate the energy levels of the bounded electron is much more complicated than the usual computation of the cross sections of the scattering processes. It is not entirely impossible the previous QED calculation is not good enough. Furthermore, the calculation of Lamb shift is also involved with the hadronic uncertainty, more prudent exam is needed.

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How to calculate it? To calculate the theoretical shift corresponding to the measured transition, some have used perturbation theory with non-relativistic wave-functions to predict the size of the contributing effects, including relativistic effects. Alternatively one can use the Dirac equation for the muon with the appropriate potential as an effective approximation to the two-particle Bethe-Saltpeter equation to calculate the perturbed wave-functions. But the result is close to the previous ones. Phys. Rev. A 84, (2011)

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**Higher order QED calculation**

one-loop electron self-energy and vacuum polarization two-loops three-loops pure recoil correction radiative recoil correction finite nuclear size corrections By Pachucki

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**Perturbative v.s. Perturbative**

J. D. Carroll, A. W. Thomas, J. Rafelski, G. A. Miller Phys. Rev. A 84, (2011)

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**Perturbative v.s. Perturbative**

Phys. Rev. A 84, (2011) J. D. Carroll, A. W. Thomas, J. Rafelski, G. A. Miller

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**Some missing contribution?**

Thomas Walcher arXiv: Walcher argued that the external "Uehling potential" derived from for the matrix element of the undressed self-energy function with the unperturbed wave function is a leading order approximation only. If we solve the wave equation with the external "Uehling potential" all contributions due to higher order loop exchanges are missing in the wave function. He then claimed the effect is large enough to explain the proton sixe puzzle, but he just performed some very simplified model calculation.

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TPE and Lamb shift

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**Dispersion relation calculation**

The imaginary part of TPE is related to the structure functions measured in DIS Dispersion relations:

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**Dispersion relation calculation**

C. Carlson and M. Vanderhaeghen, Phys.Rev. A84 (2011)

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**Dispersion relation calculation**

C. Carlson and M. Vanderhaeghen, Phys.Rev. A84 (2011)

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**What is Polarizability?**

Excited states Electric Polarizability Magnetic Polarizability Polarizability is a measures of rigidity of a system and deeply relates with the excited spectrum.

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**Dispersion relation calculation**

C. Carlson and M. Vanderhaeghen, Phys.Rev. A84 (2011)

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**Dispersion relation calculation**

According to Particle data group Recent experimental extraction C. Carlson and M. Vanderhaeghen, Phys.Rev. A84 (2011)

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**Dispersion relation calculation**

C. Carlson and M. Vanderhaeghen, Phys.Rev. A84 (2011)

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**Reevaluation of TPE C. Carlson and M. Vanderhaeghen**

Phys.Rev. A84 (2011) To explain the Proton size puzzle one needs find (ELamb) = 0.322(46)meV

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**Off-shell effects in TPE**

G.Miller and A.W. Thomas have directly calculated the elastic box diagram contribution to the muonic hydrogen Lamb shift, using several models for the off-shell contributions to the proton vertices. They find that, for a choice of parameters, the effect is large enough to explain the discrepancy between the muonic and electronic measurements of the proton radius. G. A. Miller, A. W. Thomas, J. D. Carroll, J. Rafelski, Phys. Rev. A 84, (R) (2011). One may work with it anyway, but one should compare the resulting T1 and T2 to known expansions of the Compton amplitudes beyond the pole terms, which are given at low energy and momentum in terms of the electric and magnetic polarizabilities, αE and βM. This forces a serious constraint upon any parameterization of off-shell behavior. This constraint, proportional to the measured αE and βM, leads to much smaller values of the crucial parameter than desired in the work of Miller and Thomas. Carlson and Vanderhaeghen, arXiv:

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Recent development We calculate the amplitude T1 for forward doubly-virtual Compton scattering in heavy-baryon chiral perturbation theory, to fourth order in the chiral expansion and with the leading contribution of the N form factor. This provides a model-independent expression for the amplitude in the low-momentum region, which is the dominant one for its contribution to the Lamb shift. It allows us to significantly reduce the theoretical uncertainty in the proton polarisability contributions to the Lamb shift in muonic hydrogen. We also stress the importance of consistency between the definitions of the Born and structure parts of the amplitude. Our result leaves no room for any effect large enough to explain the discrepancy between proton charge radii as determined from muonic and normal hydrogen. Michael C. Birse and Judith A. McGovern Eur. Phys.J. A 48, 120 (2012) [arXiv: [hep-ph]].

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HBCHPT result

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Miller fight back…….

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**No enough deviation found!**

So far, there has been no enough deviation of the previous QED calculation to be found yet. Even weak interaction is also considered but the effect is far too small. Some people believed off-shell effect in TPE could solve the problem, however, it is too early to judge. Of course there is still possible that some subtle effects have been neglected. But it is unlikely. Hence, we should consider the last possibility….

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**Controversy about Third Zemach moment**

De Rújula pointed out that if the proton’s third Zemach moment is very large then the discrepancy between the two results can be removed. He used some “toy model” and obtained the result: His number is 13 times larger than the experimental extraction of Friar and Sick, who use electron-proton scattering data to determine:

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**Controversy about Third Zeemach moment**

Clöt and Miller show that published parametrizations, which take into account a wide variety of electron scattering data, cannot account for the value of the third Zemach moment found . They concluded that enhancing the Zemach moment significantly above the dipole result is extremely unlikely. Phys.Rev. C83 (2011)

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**Controversy about Third Zeemach moment**

However De Rújula still argued that there is possible to make third Zemach large and not ruled out by the ep data. His argument based on the fact that There is no data corresponds to the A result based on those data has to be an extrapolation of data with a large spread and a poor χ2 per degree of freedom.

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**J. C. Bernauer, Ph.D. thesis, Johannes Gutenberg- UniversitLat Mainz (2010).**

New Mainz precise data New precise results of a measurement of the elastic electron-proton scattering cross section performed at the Mainz Microtron MAMI are presented. About 1400 cross sections were measured with negative four-momentum transfers squared Q2 from to 1 (GeV/c)2 with statistical errors below 0.2%. The electric and magnetic form factors of the proton were extracted by fits of a large variety of form factor models directly to the cross sections. The form factors show some features at the scale of the pion cloud. The charge and magnetic radii are determined to be <r2E>=0.879(5)stat.(4)syst.(2)model(4)group fm <r2M>=0.777(13)stat.(9)syst.(5)model(2)group fm.

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**J. C. Bernauer, Ph.D. thesis, Johannes Gutenberg- UniversitLat Mainz (2010).**

New Mainz precise data Q2min=0.04 GeV2 Error bar=0.2%

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**J. C. Bernauer, Ph.D. thesis, Johannes Gutenberg- UniversitLat Mainz (2010).**

New Mainz precise data

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**Parametrization for Mainz data**

J. C. Bernauer, Ph.D. thesis, Johannes Gutenberg- UniversitLat Mainz (2010). Parametrization for Mainz data

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**Inverse-Polynomial fit**

We choose the following parametrization: This parametrization gives: <r2E>= fm2, <r3>2= fm3

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So, is it possible……? Here we face a simple question: Is it possible to find a GE(Q2) which can accommodate all existing ep data and still give a large third Zemach moment and same value for the charge radius of the proton as CODATA’s value? My answer is : YES, WE CAN!

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Our ansatz, it works! arXiv: CWK and B-Y Wu

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Our ansatz -- it works! R<0.2%

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More parameter sets As K2 decreases K1 increase. As K3 increases K1 increases. As K4 increases K1 decreases. To obtain small K1 we need small K2, K3 and large K4.

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**Charge density difference**

Ours Inverse-polynomial 1/mπ=1.4 fm

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**But you get to pay anyway….**

Adding a “lump” at GE seems a nice solution, simple and it violates no experimental constraints. However if one calculates <rn> for n>2, then he will obtain enormous numbers compared with the smooth ones. It may jeopardize the expansion associated with α so the formula should be modified.

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Future measurements PSI is working on the similar measurement of muonic Helium and muonic Deuterium. If there is some QED effect to be neglected, the new data may be helpful to nail down this neglected effect. However, if there is no such effect to be discovered, then our suggestion might be the only “solution” Since there is no simple relation between proton form factors and the from factors of the deuteron and helium nuclei.

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**Preliminary result of μd**

Taken from talk of Preliminary result of μd

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Conclusions The puzzle of the size of the proton is under intensive studies. We propose an extremely simple solution for this puzzle. If one put a “lump” at extreme low Q2 on GE then the third Zemach moment is large enough and the charge radius is almost same with CODATA. Our ansatz does not conflict with the current data. Our ansatz causes a long but small oscillatory tail for the charge density. However higher moments become huge and requires more study…..

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**Four “solutions”, but………**

Any good idea? Oh~~~not so good idea is also welcome!!!!

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Moral of this talk….. Amen!

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