All about the size: puzzle of proton charge radius Chung-Wen Kao Chung-Yuan Christian University, Taiwan March 22 nd 2012, Colloquium Institute of physics, NCTU 1

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", πρ ῶ τον. He found the protons mass at 1,836 times as great as the mass of the electron. 2

Ernest Rutherford Ernest Rutherford, 1st Baron Rutherford of Nelson, (B.1871–D. 1937) was a New Zealand physicist who became known as the father of nuclear physics. He discovered that atoms have their positive charge concentrated in a very small nucleus, and thereby pioneered the Rutherford model, or planetary, model of the atom, through his discovery and interpretation of Rutherford scattering in his gold foil experiment. He was awarded the Nobel Prize in Chemistry in He is widely credited as splitting the atom in 1917 and leading the first experiment to "split the nucleus“ in a controlled manner by two students under his direction, John Cockcroft and Ernest Walton in A plaque commemorating Rutherford's presence at the Victoria University, Manchester. 3

He is so successful even to become bill… Ernest Rutherford ( ). Nobel Prize for Chemistry in Knighted in 1914, Order of Merit in 1924, Baron in Per saltire arched Gules and Or, two inescutcheons voided of the first in fess, within each a martlet Sable. Crest: A baron's coronet. On a helm wreathed of the Colors, a kiwi Proper. Mantling: Or and Gules. Supporters: Dexter, Hermes Trismegistus (patron saint of knowledge and alchemists). Sinister, a Maori warrior. Motto: Primordia Quaerere Rerum ("To seek the first principles of things." Lucretius.) 4

Quotations of Rutherford “All science is either physics or stamp collecting.” “ If your experiment needs statistics, you ought to have done a better experiment.” The only possible conclusion the social sciences can draw is: some do, some don't” “Anyone who expects a source of power from the transformation of the atom is talking moonshine” “We haven't got the money, so we've got to think!” Don't let me catch anyone talking about the universe in my department” I've just finished reading some of my early papers, and you know, when I'd finished I said to myself, 'Rutherford, my boy, you used to be a damned clever fellow.' 5

But no Anglo-Saxon can understand relativity. Said at a dinner in 1910, teasing Ernest Rutherford, who replied, 'No, they have too much sense.' — Wilhelm Wien Of all created comforts, God is the lender; you are the borrower, not the owner. You should never bet against anything in science at odds of more than about 10~12 to 1. Quotations of Rutherford If you can't explain your physics to a barmaid it is probably not very good physics The more physics you have the less engineering you need 6

Spin of proton (1927) 7 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. The specific heats of para- and ortho-hydrogen are quite different at low temperatures. If one combines these two curves in the ratio 1 part para to 3 parts ortho, one obtains a smoothly decreasing curve that agrees well with experiment.

Spin of proton (1927) This puzle can be solved assuming the protom 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 orthohydrogen – or a spin 0, with an antisymmetric wave function, which is called parahydrogen. 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). Orthohydrogen has only odd angular momentum rotation levels because of the Pauli principle for protons, while parahydrogen 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. 8

g factor of proton (1933) 9 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 Proton: g = / Neutron: g = /

Pauli and his advice 10 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… Damn it, I cannot believe I am wrong…. Again! Hahaha… Lucky me not to listen to you….

The size of nuclei 11 The mutual Coulomb repulsion of an alpha particle and a target nucleus give rise to a predictable trajectory and led to the development of the Rutherford formula. As the Geiger- Marsden data shows, the data are in agreement with the formula for a wide range of angles. With high enough alpha energies, however, the projectile punches in close enough to the nuclear center to come into range of the nuclear strong force and the distribution of scattered alphas departs from the Rutherford formula. Eisberg, R. M. and Porter, C. E., Rev. Mod. Phys. 33, 190 (1961)

Nucleon E.M form factors Hofstadter determined the precise size of the proton and neutron by measuring their form factor in " for his pioneering studies of electron scattering in atomic nuclei and for his thereby achieved discoveries concerning the structure of the nucleons 12

Rosenbluth Separation Method Within one-photon-exchange framework: 13

Finite size of the proton 14 R. Hofstadter, Rev. Mod. Phys. 56 (1956) 214 ed-elastic: Finite size + nuclear structure ep-elastic Finite size of the proton

Hofstadter and SLAC Sometime during the year 1954, in a small informal meeting in the living room of a colleague, Hofstadter suggested the rather outlandish idea (for those days) that Stanford might undertake to construct a one-mile-long linear electron accelerator. The others present were enthusiastic, and the idea was dubbed "Project M," for "monster." It soon became even more of a monster as its proposed dimensions grew from one mile in length to two, and the number of people involved mushroomed. Eventually known as "SLAC" (Stanford Linear Accelerator Center), this pioneering accelerator was built during the mid-1960's, and had its first runs in about 1967, which soon led to the discovery of partons and in 1974, to the "November Revolution", which provided a solid basis for the quark picture of hadrons. 15

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

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 10 4 ~10 5 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. 17

Spectrum of Hydrogen atom 18

Energy scales in spectrum 19 1 kHz=10 3 Hz 1 MHz=10 6 Hz 1GHz=10 9 Hz 1 THz=10 12 Hz 1PHz=10 15 Hz 1 μeV=10 -6 eV 1 meV=10 -3 eV 1 keV=10 3 eV 1 MeV=10 6 eV 1GeV=10 9 eV 1TeV=10 12 eV

Spectrum of Hydrogen atom 20

The size of proton and Lamb shift Bound state QED started in 1947, when the Lamb shift between the 2S 1/2 and the 2P 1/2 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 r p. 21

Lamb shift (1947) In 1947, Willis Lamb discovered that the 2p 1/2 state is slightly lower than the 2s 1/2 state resulting in a slight shift of the corresponding spectral line. It was a puzzle because due to Dirac equation two states are degenerate. 22

How Lamb measured it? 23 Willis Lamb formed a beam of hydrogen atoms in the 2s 1/2 state. These atoms could not directly take the transition to the 1s 1/2 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 2p 1/2 to 2p 3/2 levels. He could then measure the allowed transition from the 2p 3/2 to the 1s 1/2 state. He used the results to determine that the zero-magnetic field splitting of these levels correspond to 1057 MHz.

How Lamb measured it? 24

Who is Lamb? Willis Eugene Lamb, Jr. (B.1913 – D. 2008) was an American physicist who won the Nobel Prize in Physics in 1955 "for his discoveries concerning the fine structure of the hydrogen spectrum". Lamb and Polykarp Kusch were able to precisely determine certain electromagnetic properties of the electron. 25

Lamb Shift: QED calculation 26 k(n,0) is a numerical factor which varies slightly with n from 12.7 to k(n,l) is a small numerical factor <0.05

Nuclear finite size in spectrum 27 The nuclear finite size effects appear in the Lamb shift: The nuclear finite size effects also appear in the hyperfine splitting:

Measurement of Lamb shift 28 (measured directly) (the Lamb shift deduced from the measured fine structure interval 2p 3/2 − 2s 1/2 )

Charge radius of Proton from Lamb shift measurement 29

What is CODATA? 30 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.

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(2P-2S) ≈ 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! 31

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Muonic hydrogen Lamb shift 33 1 kHz=10 3 Hz 1 MHz=10 6 Hz 1GHz=10 9 Hz 1 THz=10 12 Hz 1PHz=10 15 Hz

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 34

Muonic hydrogen Spectrum

36 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! Surprisingly new result!

The transition frequency between 2P 3/2 and 1S 1/2 is obtained to be Δν = (77) GHz, corresponding to an energy difference of ΔE = (32) meV Theory predicts a value of ΔE = (49) ‐ r p ² r p ³ meV [r p in fm] This results in a proton radius of r p = (36) fm 4% smaller than the previous best estimate, which has been the average of many different measurements made over the years. 37

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Puzzle about the proton size

Third Zeemach moment If G E is dipole form

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.

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

New Physics? 43 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

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. 44

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)

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

Perturbative v.s. Perturbative 47 Phys. Rev. A 84, (2011) J. D. Carroll, A. W. Thomas, J. Rafelski, G. A. Miller

Perturbative v.s. Perturbative 48 Phys. Rev. A 84, (2011) J. D. Carroll, A. W. Thomas, J. Rafelski, G. A. Miller

TPE and Lamb shift

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

Dispersion relation calculation 51 C. Carlson and M. Vanderhaeghen, Phys.Rev. A84 (2011)

Dispersion relation calculation 52 C. Carlson and M. Vanderhaeghen, Phys.Rev. A84 (2011)

Dispersion relation calculation 53 C. Carlson and M. Vanderhaeghen, Phys.Rev. A84 (2011)

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

Reevaluation of TPE C. Carlson and M. Vanderhaeghen Phys.Rev. A84 (2011)

Off-shell effects 56 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. 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, α E and β M. This forces a serious constraint upon any parameterization of off-shell behavior. This constraint, proportional to the measured α E and β M, leads to much smaller values of the crucial parameter than desired in the work of Miller and Thomas. Carlson and Vanderhaeghen, arXiv: G. A. Miller, A. W. Thomas, J. D. Carroll, J. Rafelski, Phys. Rev. A 84, (R) (2011).

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. Of course there is still possible that some subtle effects have been neglected. But it is unlikely. Hence, we should consider the last possibility…. 57

Controversy about Third Zeemach 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:

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)

Controversy about Third Zeemach moment However De Rújula still argued that there is possible to make third Zeemach 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 χ 2 per degree of freedom.

New Mainz precise data 61 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 Q 2 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 =0.879(5)stat.(4)syst.(2)model(4)group fm =0.777(13)stat.(9)syst.(5)model(2)group fm. J. C. Bernauer, Ph.D. thesis, Johannes Gutenberg- UniversitLat Mainz (2010).

New Mainz precise data Q 2 min =0.04 GeV 2 Error bar=0.2% J. C. Bernauer, Ph.D. thesis, Johannes Gutenberg- UniversitLat Mainz (2010).

New Mainz precise data 63 J. C. Bernauer, Ph.D. thesis, Johannes Gutenberg- UniversitLat Mainz (2010).

Parametrization for Mainz data 64 J. C. Bernauer, Ph.D. thesis, Johannes Gutenberg- UniversitLat Mainz (2010).

Inverse-Polynomial fit 65 = fm 2, 2 = fm 3 We choose the following parametrization: This parametrization gives:

So, is it possible……? 66 Here we face a simple question: Is it possible to find a G E (Q 2 ) which can accommodate all existing ep data and still give a large third Zeemach moment and same value for the charge radius of the proton as CODATA’s value? My answer is : YES, WE CAN!

Our ansatz, it works! 67 arXiv: CWK and B-Y Wu

Our ansatz -- it works! 68 R<0.2%

More parameter sets 69 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.

Charge density difference 70 Inverse-polynomial Ours 1/m π =1.4 fm

But you get to pay anyway…. Adding a “lump” at G E seems a nice solution, simple and it violates no experimental constraints. However if one calculates 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. 71

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 Q 2 on G E 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 72

Morale of this talk….. 73 Amen!