A question with too many solutions:
Proton size puzzle
Chung-Wen Kao
Chung-Yuan Christian University,
2012. 12.14
Institute of Physics, Academic Sinica
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.
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.
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 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).
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.
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
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 = 5.5856912 +/- 0.0000022
Neutron: g = -3.8260837 +/- 0.0000018
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…
Lucky me
not to listen
to you….
Damn it, I
believe I
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.
Rosenbluth Separation Method
Within one-photon-exchange framework:
Finite size of the proton
Finite size of the proton
R. Hofstadter, Rev. Mod. Phys. 56 (1956) 214
Finite size + nuclear structure
The size of the proton?
Charge Radius
Breit Frame. q0=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 104~105 times larger than the size of the
However when the precision of measurement of the spectrum
is high enough, the finite size of the proton will become
To be more specific, let us review the spectrum of the
Spectrum of Hydrogen atom
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
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.
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
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.
How Lamb measured it?
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
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
Many correction terms
Measurement of Lamb shift
(measured directly)
(the Lamb shift
deduced from the
measured fine
structure interval
2p3/2 − 2s1/2)
Charge radius of Proton from Lamb shift measurement
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
1969. 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 2006. The latest version is Ver.6.0 called
"2010CODATA" published on 2011-06-02.
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
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!
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
The trouble is that we don’t have enough precise
value for mass of μ.
Muonic hydrogen Lamb shift
1 kHz=103 Hz
1 MHz=106 Hz
1GHz=109 Hz
1 THz=1012 Hz
1PHz=1015 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
Muonic hydrogen Spectrum
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
Surprisingly new result!
The transition frequency between 2P3/2 and 1S1/2 is
obtained to be Δν = 49881.88(77) GHz , corresponding to
an energy difference of ΔE = 206.2949(32) meV
Theory predicts a value of
ΔE = 209.9779(49) ‐ 5.2262 rp² + 0.0347 rp³ meV
[rp in fm]
This results in a proton radius of rp = 0.84184(36) fm, 4%
smaller than the previous best estimate, which has been
the average of many different measurements made over
the years.
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…..
Puzzle about the proton size
Third Zemach moment
If GE 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.
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
So we should look for other possibilities.
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 flavorconserving 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.
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 dotdashed 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.
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 pseudoscalar 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.
Example of fine tuned model of “New Physics”
C. E. Carlson and B. C. Rislow
Phys.Rev. D86 (2012) 035013
Main constraints are from muon magnetic moment (g-2) and K-decay
C. E. Carlson and B. C. Rislow
Phys.Rev. D86 (2012) 035013
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
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, 012506 (2011)
Higher order QED calculation
By Pachucki
one-loop electron self-energy
and vacuum polarization
pure recoil correction
radiative recoil correction
finite nuclear size corrections
Perturbative v.s. Perturbative
Phys. Rev. A 84, 012506 (2011)
J. D. Carroll, A. W. Thomas, J. Rafelski, G. A.
Perturbative v.s. Perturbative
Phys. Rev. A 84, 012506 (2011)
J. D. Carroll, A. W. Thomas, J. Rafelski, G. A. Miller
Thomas Walcher
Some missing contribution?
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.
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
C. Carlson and
M. Vanderhaeghen,
Phys.Rev. A84 (2011) 020102
Dispersion relation calculation
C. Carlson and
M. Vanderhaeghen,
Phys.Rev. A84 (2011) 020102
What is Polarizability?
Electric Polarizability
Excited states
Magnetic Polarizability
Polarizability is a measures of rigidity of a system and deeply relates with the
excited spectrum.
Dispersion relation calculation
C. Carlson and
M. Vanderhaeghen,
Phys.Rev. A84 (2011) 020102
Dispersion relation calculation
According to Particle data group
Recent experimental extraction
C. Carlson and
M. Vanderhaeghen,
Phys.Rev. A84 (2011) 020102
Dispersion relation calculation
C. Carlson and
M. Vanderhaeghen,
Phys.Rev. A84 (2011) 020102
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
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 offshell 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, 020101 (R)
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:1109.3779
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 modelindependent 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:1206.3030 [hep-ph]].
HBCHPT result
Miller fight back…….
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….
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:
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) 012201
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
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 fourmomentum transfers squared Q2 from 0.004 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.
J. C. Bernauer, Ph.D. thesis, Johannes
Gutenberg- UniversitLat Mainz (2010).
New Mainz precise data
Q2min=0.04 GeV2
Error bar=0.2%
J. C. Bernauer, Ph.D. thesis, Johannes
Gutenberg- UniversitLat Mainz (2010).
New Mainz precise data
J. C. Bernauer, Ph.D. thesis, Johannes
Gutenberg- UniversitLat Mainz (2010).
Parametrization for Mainz data
Inverse-Polynomial fit
We choose the following parametrization:
This parametrization gives:
<r2E>=0.782739 fm2, <r3>2=2.996667 fm3
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
My answer is : YES, WE CAN!
Our ansatz, it works!
arXiv:1108.2968 CWK and B-Y Wu
Our ansatz -- it works!
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.
Charge density difference
1/mπ=1.4 fm
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.
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
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.
Taken from talk of
Preliminary result of μd
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
Four “solutions”, but………
Any good idea? Oh~~~not
so good idea is also
Moral of this talk…..

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