PreprintPDF Available
Preprints and early-stage research may not have been peer reviewed yet.

Abstract and Figures

Otto Stern's 1933 measurement of the unexpectedly large proton magnetic moment indicated to most physicists that the proton is not a point particle. At that time, many physicists modeled elementary particles as point particles, and therefore Stern's discovery initiated the speculation that the proton might be a composite particle. In this work, we show that despite being an elementary particle, the proton is an extended particle. Our work is motivated by the experimental data, which we review in section 1. By applying Occam's Razor principle, we identify a simple proton structure that explains the origin of its principal parameters. Our model uses only relativistic and electromagnetic concepts, highlighting the primary role of the electromagnetic potentials and of the magnetic flux quantum Φ = h /e. Unlike prior proton models, our methodology does not violate Maxwell's equation, Noether's theorem, or the Pauli exclusion principle. Considering that the proton has an anapole (toroidal) magnetic moment, we propose that the proton is a spherical shaped charge that moves at the speed of light along a path that encloses a toroidal volume. A magnetic flux quantum Φ = h /e stabilizes the proton's charge trajectory. The two curvatures of the toroidal and poloidal current loops are determined by the magnetic forces associated with Φ. We compare our calculations against experimental data.
Content may be subject to copyright.
THE PROTON AND OCCAM'S RAZOR
GIORGIO VASSALLO
1
, ANDRAS KOVACS
2
Abstract.
Otto Stern's 1933 measurement of the unexpectedly large proton
magnetic moment indicated to most physicists that the proton is not a point
particle. At that time, many physicists modeled elementary particles as point
particles, and therefore Stern's discovery initiated the speculation that the
proton might be a composite particle. In this work, we show that despite
being an elementary particle, the proton is an extended particle. Our work is
motivated by the experimental data, which we review in section 1.
By applying Occam's Razor principle, we identify a simple proton structure
that explains the origin of its principal parameters. Our model uses only
relativistic and electromagnetic concepts, highlighting the primary role of the
electromagnetic potentials and of the magnetic ux quantum
ΦM=h
/e.
Unlike
prior proton models, our methodology does not violate Maxwell's equation,
Noether's theorem, or the Pauli exclusion principle.
Considering that the proton has an anapole (toroidal) magnetic moment, we
propose that the proton is a spherical shaped charge that moves at the speed of
light along a path that encloses a toroidal volume. A magnetic ux quantum
ΦM=h
/e
stabilizes the proton's charge trajectory. The two curvatures of
the toroidal and poloidal current loops are determined by the magnetic forces
associated with
ΦM
. We compare our calculations against experimental data.
conversion constants for natural units:
1.9732898 ·107m'1eV 1
length
6.5821220 ·1016 s'1eV 1
time
2.99792458 ·108ms1= 1
speed
1.5192669 ·1015 Hz '1eV
frequency
8.1193997 ·1013 N'1eV 2
force
1.8755460 ·1018 C= 1
charge
relevant physical constants in natural units:
h= 2π
Planck's constant
6.62607015 ·1034J H z1
~=h
/2π= 1
reduced Planck's constant
ε0=1
4π
vacuum permittivity
µ0= 4π
vacuum magnetic permeability
c= 1
light speed in vacuum
2.99792458 ·108ms1
α1'137.036
inverse of the ne structure constant
e=±α'0.085424546 1.602176634 ·1019C
elementary charge
µN'4.552225759 ·1011 eV 15.0507837461 ·1027J T 1
nuclear magneton
µpC '1.271367397 ·1010 eV 11.41060679736 ·1026J T 1
proton magnetic moment
µp
µN'2.79284734463
CODATA proton magnetic moment to nuclear magneton ratio
mp'0.93827208816 ·109eV
proton mass
λp'6.696549362 ·109eV 11.32140985539 ·1015m
proton Compton wavelength
Ae'5.981875085 ·106eV
norm of the vector potential of the electron charge
Ve=Ae
electric potential at surface of the electron's charge
Key words and phrases.
Aharonov-Bohm electrodynamics, magnetic ux quanta, nuclear
anapole moment, Occam's razor, proton model, unied eld theory, vector potential, Zitterbe-
wegung.
1
Università degli Studi di Palermo (Dipartimento di Ingegneria), Palermo, Italy
2
BroadBit Energy Technologies.
1
THE PROTON AND OCCAM'S RAZOR 2
ΦM=h
e= 2πα1/2'73.55246020
elementary charge's magnetic ux
me=ωe'0.51099895 ·106eV
electron rest mass
ωe=me
electron's charge angular speed
Te=2π
ωe
electron Zitterbewegung period
re=ω1
e'1.956951198 ·106eV 10.3861592676 ·1012m
electron radius
rce =αre
electron charge radius
Rp,exp '4.264 ·109eV 10.8414 ·1015m
proton charge radius CODATA value
proton model parameters:
parameter set 1:
rpp =m1
p=λp
2π'1.06578893 ·109eV 10.2103089103 ·1015m
proton torus minor (poloidal)
radius
rpt =ηrpp '2.976586476 ·109eV 10.5873608214 ·1015m
proton torus major (toroidal) radius
rcp =αrpp '7.777437549 ·1012 eV 11.534698267 ·1018m
proton spherical charge radius
η=rpt
rpp =q39
5'2.792848009
proton torus aspect ratio
vpt =η1c
toroidal component of the charge speed
c
Ap=mp
e'1.098363601 ·1010 eV
the absolute value of the vector potential at the proton charge
Apt
toroidal component of the vector potential
Ap
Lp=eAptrpt =~= 1
the proton's toroidal angular momentum
eAptrpt cos (ϑ) = ±1
/2~(ϑ {π
/3,2π
/3})
measured proton spin
µp=rpt
2e'2.792848009 µN
proton model magnetic moment
Φp=h
e'73.55246020
proton model magnetic ux quantum
parameter set 2 (appendix 1):
rpp '2.3465 ·109eV 10.463 ·1015m
proton torus minor (poloidal) radius
rpt '4.2113 ·109eV 10.831 ·1015m
proton torus major (toroidal) radius
rcp '7.777437549 ·1012 eV 11.534698267 ·1018m
proton spherical charge radius
vpt =c
2
toroidal component of the charge speed
c
~= 1
the proton's toroidal angular momentum
~cos (ϑ) = ±1
/2~(ϑ {π
/3,2π
/3})
measured proton spin
µp'2.792848009 µN
proton model magnetic moment
Φp=h
e'73.55246020
proton model magnetic ux quantum
1.
Motivation
1.1.
A brief history of the proton model.
Before the 1970s, most scientists
viewed the proton as an elementary particle. Starting from the 1970s, scientists
working with high energy particle colliders proposed that protons and neutrons are
not elementary particles, but comprise smaller sub-particles. According to their
model, a proton and a neutron both comprise three quark sub-particles. The exis-
tence of quarks has been suggested initially in the 1960s, based on the theoretical
eorts by Gell-Mann to model baryons and mesons [12], which were observed in
a great variety during high energy nuclear experiments. The momentum distri-
bution of particles emerging from a high-energy collision is characterized by the
F2
structure function. Gell-Mann's proposition was that the
F2
structure function
probes the internal momentum distribution of sub-particles; for a particle compris-
ing
N
sub-particles, its
F2
structure function must peak at
x=1
N
. Gell-Mann's
original quark theory thus predicted the
F2
momentum distribution to peak at
x=1
3
. However, as will be shown in section 1.3, this is not the case because the
experimentally observed
F2
data peaks at
x=1
9
. This deviation from Gell-Mann's
prediction was explained away via the hypothesis that the three quarks originally
thought to form the proton are the so-called valence quarks, which are swimming
in the background of sea quarks [6]. These so-called sea quarks are a collection of
THE PROTON AND OCCAM'S RAZOR 3
quark-antiquark pairs, radiated by the three valence quarks. However, the calcu-
lations of 1970s still showed that the valence quarks together with the sea quarks
only accounted for 54% of the proton's momentum [16]. A further hypothesis was
added to supplement the momentum shortfall of the quarks; chargeless particles
called gluons were introduced into the proton model [24]. Since gluons have no
electric charge, the thinking was that they are there, but the electrons probing the
proton in deep inelastic scattering cannot see them. These hypothesized gluons
were assigned the missing proton momentum, and the resulting proton model be-
came the quark-gluon model that it is today. Despite the absence of any direct
quark observation, the quark-gluon model gained popularity during the 1970s, and
remained embraced by most theoretical physicists ever since.
According to the 1970s model of valence quarks swimming in the background of
sea quarks and gluons, there seemed to be an angular momentum decit with re-
spect to the measured angular momentum of the proton, and therefore the presence
of virtual strange quarks was also postulated during the 1990s [20].
1.2.
Experimental counter-evidences to the quark model.
Although the
quark-based model was inspired by the great variety of mesons, the proposed quark
masses do not add up the masses of observed mesons. According to quark propo-
nents, this is explained by a negative binding energy between quarks: any particle's
valence quarks masses are only a small percentage of the total particle mass, with
the bulk of the particle mass coming from virtual particles which represent the
binding force: i.e. virtual quarks and gluons. Moreover, the valence quark : virtual
quark : gluon mass ratio is allowed to vary from particle to particle in order to
match the observed masses. Now what is the physical meaning of negative binding
energy? By denition, negative binding energy means a metastable bound state.
This model implies that individual quarks should be easily observable upon the
break-up of their metastable binding. However, quark proponents also postulated
that these metastable bonds between quarks can never be dissociated. There is
a fundamental contradiction between the hypothesis of metastable quark binding
and the hypothesis of unbreakable quark bonds.
Proton-antiproton reactions provide rather direct counter-evidence. Figure 1.1
shows traces of a proton-antiproton reaction event, highlighting the produced pion
tracks. According to the quark model, a proton-antiproton pair comprises six
quarks. After a partial annihilation of quark-antiquark pairs, there can be up to
four remaining quarks, which may be organized into two pions. However, gure 1.1
shows at least eight pions emerging from the annihilation event, which contradicts
the quark model. A quark model proponent may try to explain this phenomenon by
assuming that the kinetic energy of the incoming antiproton was converted into the
production of numerous pion-antipion pairs just prior to its annihilation. However,
such an explanation is refuted by reference [7], whose authors exposed a nuclear
emulsion to antiprotons, and then analyzed the resulting tracks in the emulsion.
Their discussion of gure 2 in reference [7] clearly states that the antiproton rst
came to a rest in the emulsion, and then produced at least ve pions upon annihi-
lation with a proton. Such large number of pions emerging from proton-antiproton
reactions is impossible under the quark-antiquark annihilation model.
According to the quark model, the proton and neutron both comprise three
quarks, only diering in one quark type. The recent discovery of the nuclear electron
particle [23], which is based on numerous experimental evidences, establishes that
THE PROTON AND OCCAM'S RAZOR 4
Figure 1.1.
Proton-antiproton annihilation event. Left: bubble
chamber photograph. Right: diagram of the photo, identifying
the particles created by the annihilation event. Source: Lawrence
Berkeley National Laboratory Science Photo Library - photograph
K003/4377.
the proton-neutron dierence is the nuclear electron's presence versus absence. The
discovery of nuclear electrons thus invalidates the quark-gluon model.
1.3.
A re-interpretation of high-energy particle collision data.
Considering
the above outlined problems with the quark-based proton model, one may wonder
about the origin of the
F2
momentum-distribution data recorded in high-energy
collisions.
The production of particle-antiparticle pairs is a well established phenomenon
of high-energy collisions. Therefore, an incoming energetic electron may produce
muon-antimuon pairs upon scattering. Also, an incoming electron may be energized
into a muon upon scattering. It is thus pertinent to consider a relationship between
the
F2
data and the short-lived particles produced in preceding scattering events.
Reference [35] presents a thorough analysis of high energy scattering data from
measurements performed at the Stanford Linear Accelerator Center (SLAC), Thomas
Jeerson National Accelerator Facility (JLAB), and Hadron Electron Ring Accel-
erator (HERA). As shown in gures 1.2 and 1.3, the combined SLAC and JLAB
data of
F2
momentum distribution measurements shows clearly, without any curve
ttings, that their
F2
values peak in the vicinity of
x
=
1
9
. The JLAB
F2
values
at
x
= 0.45 and
x
= 0.25 (circled) show that JLAB data integrates well with the
original SLAC
F2
data. Is the use of single-variable
F2(x)
distributon justied, i.e.
are the energy and momentum exchange suciently high for convergence? The use
of
F2(x)
rather than
F2(x, Q2)
is justied because the SLAC data was shown to
satisfy Bjorken scaling, i.e. for
x
>0.2, the
F2
values are essentially the same for
a given
x
regardless of the
Q
amount of energy transferred between the scattering
THE PROTON AND OCCAM'S RAZOR 5
Figure 1.2.
Combined SLAC and JLAB data of
F2
momentum
distribution measurements from electron-proton scattering, repro-
duced from [35].
Figure 1.3.
Combined SLAC and JLAB data of
F2
momentum
distribution measurements from electron-deuteron scattering, re-
produced from [35]. This scattering data shows the same
F2
mo-
mentum distribution as in the electron-proton case.
particles. In this
x
>0.2 region,
Q2
values range from 0.6 to about 30 GeV
2
. Re-
garding the
x
<0.2 region, the JLAB
Q2
values shown in Figure 1.4 are nearly the
same as the SLAC
Q2
values listed in Appendix A.1 of [35], which makes the two
results directly comparable.
One may wonder why this
F2
peak at
x
=
1
9
doesn't show up in any other lit-
erature? By 1973, mainstream theorists have essentially embraced the quark-gluon
model as adequately describing the structure of the proton. Most attempts to ex-
plain the SLAC scattering results any other way had ended, and the bulk of the
theoretical eort focused on enhancing the quark model. Dierent versions of the
quark-gluon plasma model predict either constant or rising
F2
values as
x0
. In
the 1990s, when HERA experiments began producing data in this
x
<0.1 range, it
was assumed that HERA lled the low-
x
gap left by SLAC, even though its data was
THE PROTON AND OCCAM'S RAZOR 6
Figure 1.4.
F2
structure function values for the proton (
F2
- p)
and the deuteron (
F2
- d) as a function of x and
Q2
from the JLAB
E99-118 deep inelastic scattering experiments [36].
generated from scatterings with
Q2
values tens to hundreds of times higher than
the SLAC data. Many theorists could not resist the temptation of mixing non-
comparable data in this low-
x
region of the
F2
curve, and mistakenly proclaimed
experimental support for their quark-gluon model. Reference [30] is a typical ex-
ample of such erroneous data analysis. Around 2000, the JLAB experiment began
producing scatterings with comparable
Q2
values to the SLAC experiment. By that
time, the erroneous blending of high-
Q2
HERA data with low-
Q2
SLAC data was
already a consensus procedure for obtaining the proton's
F2
curve, and mainstream
theorists had no interest in pointing out their colleagues' mistakes or discussing the
implications of the JLAB experiment.
Upon dividing the proton's mass by 9, we obtain approximately the muon mass.
This match with the
F2
peak location at
x
=
1
9
suggests that it may correspond to
electron scattering from a muon or antimuon that was produced in some preced-
ing scattering event. Our interpretation implies that one should also nd a peak
corresponding to electron-electron scattering in the very low-
x
region because an
incoming electron may also collide with a previously scattered other electron. Upon
the analysis of HERA experiments, reference [35] indeed identies yet another
F2
momentum distribution peak near
x
=
1
1836
, which corresponds to the electron
mass.
In summary, the
F2
momentum distribution data shows signatures of electron-
muon and electron-electron scattering. Consequently, the quark model is contra-
dicted by all experimental data. The absence of a reasonable proton model thus
motivates us to explore the proton's internal structure.
Antiprotons are generated in suciently energetic collisions between light and
heavy nuclei. Would a violent collision create complex structures involving many
THE PROTON AND OCCAM'S RAZOR 7
sub-particles? That would be very unlikely, and it is moreover favored by Occam's
razor principle to rstly explore simple proton structures. We thus investigate
whether a relatively simple proton model exists, which would match its experimen-
tally observed properties.
2.
How large is the proton?
2.1.
The proton's spherical charge radius.
Any particle's Compton scattering
cross-section is given by the Klein-Nishina formula, where one parameter is the
spherical charge radius. Upon tting the electron's experimental Compton scatter-
ing cross-section to the Klein-Nishina formula, in the 0.5 MeV photon energy range,
one obtains 2.82 fm spherical charge radius. This 2.82 fm electron charge radius is
referred to as the classical electron radius in the scientic literature.
Is the same method applicable for determining the proton's spherical charge
radius? Figure 2.1 shows the proton's scattering cross-section in the 1 GeV photon
energy range, which corresponds to the proton mass. There are numerous peaks
in the scattering data of gure 2.1; these correspond to photo-production of new
particles. Experimental measurements determined that the largest peak around
300 MeV corresponds to the photo-production of two neutral pions, while the peak
around 700 MeV corresponds to the photo-production of a pion and an
η
meson. In
contrast to the electron case, the scattering cross-section is now a sum of particle
photo-production and Compton scattering processes. Nevertheless, we can make
an estimation of the proton's spherical charge radius.
Figure 2.1.
The proton's interaction cross-section with high fre-
quency radiation, reproduced from [26]. The horizontal scale shows
the incoming photon energy, the left and right panels show the
cross-section at
90
°
and
130
°
scattering angles, respectively. The
red and and blue dashed lines show the Compton scattering cross-
section for the indicated spherical charge radius values.
The dashed lines on gure 2.1 show the Compton scattering cross-section at
5·1018 m
and at
1.5·1018 m
spherical charge radius values. With
5·1018 m
radius, the Compton scattering cross-section becomes larger than the experimental
values in the <200 MeV and >1000 MeV regions. Therefore, the true radius is
smaller than
5·1018 m
. In contrast, with
1.5·1018 m
radius value the Compton
scattering cross-section converges to the experimental values both in the <200 MeV
THE PROTON AND OCCAM'S RAZOR 8
and >1000 MeV regions. Therefore, light scattering measurements indicate that the
proton's spherical charge radius is approximately
1.5·1018 m
.
2.2.
The proton's apparent Zitterbewegung radius.
Numerous experiments
aim to precisely measure the proton's so-called charge radius, which is dened
as the mean radius value of its charge distribution. High-energy electron-proton
scattering experiments are one class of such measurements. As shown in table
1, one of the earliest scattering analysis based proton charge radius extraction
was published in 1963: it comprises a systematic review of scattering experiments
performed up to that date, and its authors calculated a
0.805 ·1015 m
charge
radius value. By the early 2000s, the consensus mean proton radius value increased
to
0.875 ·1015 m
, but reference [25] re-analyzes the involved measurements and
claims to have found a systemic error which caused over-estimations.
Recent measurements converge around the
0.84·1015 m
mean radius value, and
claim very small error margins of only
(5 8) ·1018 m
.
This
0.84 ·1015 m
mean radius value is several orders of magnitude larger than
the above identied
rcp <5·1018 m
parameter. To understand the physical mean-
ing of the
0.84·1015 m
radius value, we again use the analogy of electron scattering
experiments. When an electron interacts with high frequency light, its scattering
cross section is given by the Klein-Nishina formula, and such scattering data reveals
the electron's 2.82 fm spherical charge radius. When an electron interacts with low
frequency light, its scattering cross section is given by the Thomson scattering for-
mula, and such scattering data reveals the electron's 386 fm Zitterbewegung radius.
In the scientic literature this electron Zitterbewegung radius is also referred to as
the electron's reduced Compton radius. By analogy, we associate the proton's
0.84 ·1015 m
radius value as an approximation of the major radius of the torus
enclosed by the proton charge trajectory.
Publication Mean proton Reference
year radius value
1963
0.805 ±0.011
fm [17]
2016
0.840 ±0.016
fm [14]
2020
0.831 ±0.019
fm [40]
2021
0.847 ±0.008
fm [10]
2022
0.840 ±0.005
fm [25]
Table 1.
Electron-proton scattering analysis based mean proton
radius measurements.
Besides the electron-proton scattering analysis, there are also spectroscopic meth-
ods for the proton's charge radius calculation [15]; all spectroscopic estimate the
impact of non-zero proton radius on the electrostatic potential of the electron's
wavefunction. Table 2 shows the results of recent proton radius measurements,
based on spectroscopic methods.
Tables 1 and 2 show remarkably similar values. Omitting the 1963 data, the
remaining recent measurements average out to
0.839 ±0.007
fm.
THE PROTON AND OCCAM'S RAZOR 9
Publication Involved Charge Reference
year particles radius
2017
e, p+0.8335 ±0.0095
fm [4]
2019
e, p+0.833 ±0.01
fm [5]
2020
e, p+0.8483 ±0.0038
fm [15]
Table 2.
Spectroscopic analysis based mean proton radius measurements.
3.
Methodology
In this work, we explore an electromagnetic proton structure which is in accor-
dance with Maxwell's equation. Our methodology is based on the recently pub-
lished electron model [22], which explains what an electron is made of, why it has
a spin, and what the origin of the quantum mechanical wavefunction is. A main
conclusion of [22] is that the electron mass comprises electromagnetic eld energy.
Given that a high-frequency electromagnetic wave can produce an electron-positron
pair, while traveling through a suciently strong electric eld, the ideas of [22] are
quite natural. More specically, by calculating the electric eld energy around the
electron's 2.82 fm spherical charge radius, one obtains 255.5 keV, which is exacty
half of the electron mass. As explained in [22], the other half of the electron mass
is magnetic eld energy. These two electromagnetic energy types continuously in-
duce each other. Such dynamics is completely analogous to the perpetual induction
within an electromagnetic wave, which may give birth to the electron-positron pair.
The 255.5 keV magnetic energy of an electron corresponds to the circular Zit-
terbewegung of its spherical charge. Such circular Zitterbewegung generates the
electron spin. The constant value of the electron spin follows from the constant
speed of its circular Zitterbewegung; as discussed in [22], it directly follows from
Maxwell's equation that the Zitterbewegung speed is the speed of light.
As outlined in table 3, our methodology is in accordance with all fundamental
physical laws. In comparison, the quark-based methodology has multiple draw-
backs: i) the quark model violates foundational laws, such as Maxwell's equation
or Noether's theorem, ii) as explained in section 1, the quark model lacks any ex-
perimental evidence, and iii) the quark model is contradicted by the commercial
Nuclear Magnetic Resonance (NMR) technology. The implications of proton NMR
data will be discussed in section 6.
Quark-based proton model This work
Violates Maxwell's equation? Yes (renormalization) No
Violates Noether's theorem? Yes (virtual particles)
1
No
Relation to NMR measurements? Contradicts
p+
NMR data Explains
p+
NMR data
Radius calculations can be veried? No (too lengthy/complex) Yes
Table 3.
A comparison between the quark-based proton model
and our present work.
At rst it seems natural to use the exact same model for describing both the elec-
tron and proton, scaling the given particle's dimensions by the appropriate particle
1
The quark-based model assigns over 98% of the proton mass to virtual particles.
THE PROTON AND OCCAM'S RAZOR 10
mass. The advantages of a simple ring-shaped proton model were indeed pointed
out by David L. Bergman in his paper The Real Proton [2]. This approach, while
works well for muons, introduces unacceptably large errors if naively used for pro-
ton modeling. Firstly, the magnetic moment of such a simple model is equal to the
nuclear magneton
µN
, while the experimental proton magnetic moment value is
approximately 2.79 times larger. Secondly, as discussed in section 2.2, the proton's
experimental Zitterbewegung radius value is
0.839 ±0.007
fm, while the scaled
positron model yields a
0.2103
fm Zitterbewegung radius from the
e+:p+
mass
ratio. The following sections present a simple proton model that overcomes these
large discrepancies while fully maintaining the conceptual framework introduced in
this section.
4.
Gaugeless electrodynamics
It's important to note that, at the Compton scale, certain quantized physical
values appear dimensionless in natural units. The elementary charge value
e=
±α
, its magnetic ux
ΦM=2π/e
, Zitterbewegung speed
c= 1,
and angular
momentum
~= 1
cannot be separated but are dierent characteristics of the same
physical entity
. As already pointed out in [22, 11], a non-linear dynamic equation
can be derived when the Maxwell's equation and the Proca equation are considered
to apply simultaneously. This equation essentially describes, using the language
of spacetime Cliord Algebra
Cl3,1(R)
, the behavior of an elementary charge that
always moves at the speed of light and is subjected to a magnetic centripetal force
that is responsible for the curvatures of its Zitterbewegung trajectory. Therefore,
the electromagnetic four-potential can be seen as the eld, a
Materia Prima
, from
which the physical entities that we call particles are generated. It is therefore
reasonable to universally apply this approach to all charged elementary particles.
Leaving behind the experimentally paradoxical hypothesis of electromagnetic
gauges [34, 9, 33], we do not assume the presence of any electromagnetic gauge,
and arrive at the simplest form of Maxwell's equation [22, 11, 3]:
2A= 0
.
The
A
notation refers to the electromagnetic four-potential
A=A+γtV
.
The electric charges and currents then correspond to a scalar eld on a spherical
surface. As required by Maxwell's equation, this charged surface is moving at
light speed, and is characterized by a vector potential
A,
an electric potential
V,
a
current
I=αA
/2π
, a mechanical momentum
mc=eA
, and an angular speed
ω=
eA
. Respecting both magnetic and electric Aharonov-Bohm relations, the charge's
electromagnetic four potential
A=A+γtV
is a nilpotent vector
A2
= 0
. These
laws may be considered as a powerful tool for modeling the structure and properties
of elementary charged particles. Prior to [22, 11], gaugeless electrodynamics has
been already introduced and explored by other authors [1, 18, 19, 21, 27, 28, 29,
31, 32, 37, 39]. Most of these preceding works introduce the electromagnetic scalar
eld as an additional entity besides charges and currents, rather than the entity
that actually produces the apparent charges and currents. A notable exception is
the work of Giuliano Bettini [3] that recognizes the electromagnetic sources as the
partial derivatives of the scalar eld.
The elementary charge is characterized by a simple Lagrangian
L
that denes
the action
S
:
(4.1)
L=eA·ceV
THE PROTON AND OCCAM'S RAZOR 11
S=ˆLdt
The stationary action condition
δS= 0
is a consequence of the Aharonov-Bohm
relations
S=ˆ(eA·ceV )dt =eˆA·dl eˆV dt = 0
δS= 0
As
A
and
c
are parallel vectors for a freely moving charge, it's possible to
substitute the dot product with the product of their modulus:
L=eAc eV =eA dl
dt eV
If the radius of the charge's Zitterbewegung trajectory is
r
, the dierential of
the displacement
dl
can be substituted by the product
rdϕ
:
dl =rdϕ
L=eAr
dt eV
Consequently, the following simple conditions guarantee that the action
S
is always
zero:
eAr =~= 1
dt =eV =eA =1
r
r1=
dt =ω=m
In natural units, the elementary particle's mass-energy is equal to its Zitterbe-
wegung angular speed, to the inverse of its Zitterbewegung radius, and to the value
its Zitterbewegung momentum
eA
.
5.
Proton model
5.1.
Proton geometric structure.
We develop our proton model in agreement
with the above considerations. While the natural choice for a proton model is a
simple scaled positron model, it leads to some unacceptable discrepancies with
the experimental data, that we have already pointed out.
Measurements of the proton's anapole magnetic moment have been claimed since
1997 [38]. In such experiments, electron-proton coupling interactions are used for
mapping out the proton's various magnetic modes. Since the anapole magnetic
moment is generated by a toroidal charge current, these experiments suggest that
the proton's charge moves on a toroidal surface.
We therefore consider a model where the stationary proton charge follows a
toroidal Zitterbewegung trajectory, similar to a toroidal coil winding. The toroidal
volume enclosed by the proton charge trajectory has a minor (poloidal) proton
radius
rpp
which remains to be determined. The major (toroidal) radius of the
THE PROTON AND OCCAM'S RAZOR 12
enclosed toroidal volume is
rpt
, and it is the mean distance of the proton charge
from its geometric center.
At the Compton radius scale, the universal value of the magnetic ux quantum
h
/e
induces a centripetal magnetic force that constrains an elementary particle to
follow either a circular Zitterbewegung path (positron case) or helicoidal Zitterbe-
wegung path (proton case).
In this article, we describe two ways of applying our methodology to a toroidal
proton model. These two approaches agree on the spherical charge radius value,
but yield slightly dierent toroidal charge radius values. One approach, which rep-
resents the perspective of one author, is described in sections 5.2-5.5 of this article
2
.
The other approach, which represents the perspective of the other author, is de-
scribed in the rst appendix. We invite readers to analyze these two approaches,
and debate the pros and cons for each one.
5.2.
The proton's electromagnetic energy and charge radius.
The proton
charge is assumed to follows a closed helicoidal trajectory similar to a toroidal coil
winding. The main geometric parameters of this model, such as its spherical charge
radius or its toroidal and poloidal radii, can be found by starting from the proton
mass-energy value and imposing the quantization of the angular momentum and of
the experimental magnetic moment value.
The electric energy
WpE
of the proton is calculated by integrating the energy
density of its electric eld down to its spherical charge radius
rcp
:
WpE =e2
8πˆ
rcp
1
r4·4πr2dr =e2
2ˆ
rcp
1
r2dr =e2
2
1
r
rcp
=e2
2rcp
=α
2rcp
We assume that the electric energy is equal to one half of the proton mass, as
required by Maxwell's equation for any electromagnetic wave. Introducing
rpp
as
the reduced Compton wavelength of the proton,
WpE =mp
2=1
2rpp
we calculate the proton charge radius:
α
2rcp
=1
2rpp
rcp =αrpp
rcp =α
mp'7.777437549 ·1012 eV 11.534698267 ·1018m
The other half of the proton mass comprises its magnetic energy
WpM
:
WpM =1
2ΦMIp=1
2·2π
e·α
2πAp=1
2eAp=mp
2
The charge radius
rcp
implies a potential
Vp
at the surface of the charge. Con-
sequently it is in agreement with the electric Aharonov-Bohm 5.1equation if we
assume that the charge spans an angle
in a time
dt
moving at light speed
c= 1
:
2
This approach is the perspective of Giorgio Vassallo
THE PROTON AND OCCAM'S RAZOR 13
=cdt
rpp
=dt
rpp
Vp=e
rcp
(5.1)
=eVpdt
dt =eVp=e2
rcp
=α
rcp
=1
rpp
=mp
5.3.
Proton torus aspect ratio.
Assuming that the torus volume enclosed by
the proton charge trajectory has a minor radius
rpp
equal to the proton reduced
Compton wavelength
λp/2π
and that, using natural units, is equal to the inverse of
the proton mass [2], it's possible to nd the major radius imposing the quantization
of the proton angular momentum
Lp
and the experimental value of the magnetic
moment
µp
Lp=mpvptrpt =~= 1
vpt =rpp
rpt
Multiplying the toroidal component
Ipt
of the proton current by the enclosed
area
πr2
pt
we get the proton magnetic moment:
Ipt =Ipvpt =α
2πApvpt =α
2πAp
rpp
rpt
Iptπr2
pt =µp
α
2πAp
rpp
rpt
πr2
pt =µp
remembering that
α=e2
and that
eAp=mp=r1
pp
, we can write
e
2rpt =µp
rpt =2µp
e
µN=e
2mp
rpt
rpp
=v1
pt =2µp
emp=µp
µN
=r39
5
This means that the torus aspect ratio
η=p39
/5
is a value that is equal to the
ratio of proton's magnetic moment and the nuclear magneton
µN
.
The toroidal component of a charge displacement equal to one Compton wave-
length
λp
is equal to
λpvpt
. The aspect ratio
η
implies that the proton charge
travels along a path length of 39
λp
after 5 turns around the torus center:
39λpvpt = 5 ·2πrpt
THE PROTON AND OCCAM'S RAZOR 14
Figure 5.1.
An illustration of the toroidal proton geometry. The brown curve is the
Zitterbewegung trajectory, the blue arrow represents the poloidal proton radius
(rpp)
, and the purple arrow represents the toroidal proton radius
(rpt)
.
X,Y,Z values are multiples of
rpp '0.21
fm
39 ·2πrppvpt = 5 ·2πrpt
rpp
rpt
vpt =5
39
vpt =rpp
rpt
v2
pt =5
39
vpt =η1=r5
39
5.4.
Proton charge and Lorentz force.
The proton's charge has a mechani-
cal momentum
mpc
equal to the product of the elementary charge and its vector
potential:
eAp=mpc
The proton charge is subjected to the magnetic Lorentz force:
Fp=ec×Bp=edAp
dt =mp
dc
dt
Fp=mp
c2
rpp
=mp
rpp
=r2
pp
The force vector
Fp
has a component
Fpt
that is always directed towards the
torus' center
Fpt =edApt
dt =eAp
dvpt
dt =mp
dvpt
dt
Fpt =mp
v2
pt
rpt
=v2
pt
rpprpt
=r2
pp
r3
ptrpp
=rpp
r3
pt
THE PROTON AND OCCAM'S RAZOR 15
Figure 5.2.
Proton radial charge density distribution
f(r) = 1
e4πr2ρ(r) = 1
e
dq(r)
dr
. Radial charge density equation from [13] have been
used. The vertical red line marks a value that in the proton model is the toroid
major radius (~0.587 fm). The two vertical blue lines are the toroid's internal and
external radius. The cross indicate the radius value (~0.625 fm) that contains half
of the proton charge.
The magnetic ux density
Bp
seen by the proton charge is one half the averaged
valued of the magnetic ux density obtained dividing the magnetic ux
ΦM=2π/e
by an area equivalent to the toroid cross section
πr2
pp
Bp=1
2
ΦM
πr2
pp
=2π
eπr2
pp
=1
er2
pp
5.5.
The proton charge radius interpretation.
Observing the proton on
time scales much larger than the time required for a complete Zitterbewegung
turn around the torus center, which is
T2=2πrpt/vpt '3.44 ·1023s
, the charge
appears conned inside a radius
Rp
equal to the sum of the two torus radii,
rpt
and
rpp
, and the proton's spherical charge radius
rcp =αrpp
:
Rp=rpt +rpp +rcp '0.7992 ·1015m
Recent experimental measurements of the proton radius, which are listed in
tables 1 - 2, average out to the following value:
Rp,exp '(0.839 ±0.007) ·1015m
Remembering however that
Rp,exp
is dened as the mean radius value of the
charge distribution, this value should be approximated by the charge distribution's
toroidal radius
rpt '0.587 ·1015m
. Consequently there is either a relatively large
error of about
30%
in the model of sections 5.2-5.5, or an incomplete interpretation
of this particular experimental value.
THE PROTON AND OCCAM'S RAZOR 16
6.
Proton spin and gyromagnetic factor
6.1.
The proposed electromagnetic model based interpretation of pro-
ton spin.
Analogously to the electron case, the absolute value
Lp
of the proton's
toroidal angular momentum
Lp
is equal to the reduced Planck constant
3
:
Lp=~
In the presence of an external magnetic eld
BE
, we can write the vector
Lp
as the sum of two vectors. One vector is parallel to
BE
, and the second one is
orthogonal to it:
Lp=Lpk+Lp
Lpk=Lpcos (θ)
Lp=Lpsin (θ)
where
θ
is the angle between the vectors
µp
and
BE
. The proton is therefore
subjected to a torque
τ
:
τ=
µp×BE
=µpBEsin (θ)
Consequently, the proton's toroidal structure will be in a Larmor precession,
with angular frequency
ωpP
:
(6.1)
τ=
dLp
dt
=Lpsin (θ)
dt =Lpsin (θ)ωpp.
µpBE=~ωpp
What we call proton spin
sp
is the measured component of its angular momen-
tum vector
Lp
along the external magnetic eld
BE:
sp=~cos (θ)
Figure 6.1 illustrates the precessing proton structure under an external magnetic
eld
BE
.
The measurable angular momentum transitions are also universally quantized to
~
value. This implies the following for the quantization of the angle
θ:
4Lpk=±~=θπ
3,2π
3,cos (θ) = ±1
2
sp=±~
2
The two spin values are characterized by two dierent energy levels,
EL
and
EH
;
3
Under the approach of sections 5.2-5.5, the toroidal angular momentum is
~
according to the
Lp=rpt ×mpvpt
formula, where
mp
is the full proton mass. Under the approach of appendix
1, the toroidal angular momentum is
~
according to the
Lp=rpt ×mptvpt
formula, where
mpt
is the mass component in the toroidal direction that corresponds to the toroidal current loop.
See the explanation of appendix 1 about splitting the proton mass into the toroidal and poloidal
current loop components.
THE PROTON AND OCCAM'S RAZOR 17
ωpp
Lp
Be
Figure 6.1.
The Larmor precessing proton in external magnetic
eld
BE
. The proton's toroidal angular momentum vector
Lp
precesses with angular frequency
ωpp
.
θ=π
3=EL=~ωpp
θ=2π
3=EH=~ωpp
4E=EHEL= 2~ωpp
This energy gap
4E
is equal to
~ωNM Rp
, where
ωNM Rp
is the proton's NMR
angular frequency. Therefore:
(6.2)
ωNM Rp = 2ωpp = 2µpBE=e
mp
µp
µN
BE
This linear relationship between the applied
BE
magnetic eld and the resulting
4E
energy gap is the basis of NMR technology.
The value of
ωNM Rp
can be written as a function of the gyromagnetic factor
gp
:
(6.3)
ωNM Rp =e
2mp
gpBE
The proton's gyromagnetic-factor
gp
is therefore
gp= 2 µp
µN
= 5.585696018
Our calculation precisely matches the CODATA value of the proton's experi-
mentally measured gyromagnetic factor, which is 5.5856946893.
We note that all of the above applies completely analogously to the electron,
whose angular momentum value is also
~
. The exact same Larmor precession arises
when the electron is placed under an external magnetic eld
BE
, and thus the
measured value of its angular momentum becomes
se=±~
2
[22]. This phenomenon
THE PROTON AND OCCAM'S RAZOR 18
is the basis of Electron Spin Resonance (ESR) technology: the measured energy gap
4E
is then equal to
~ωESR
, where
ωESR = 2µBBE
and
µB
is the Bohr magneton.
6.2.
The quark model based interpretation of proton spin.
When Otto
Stern measured the proton's
µp= 2.793µN
magnetic moment in 1933, most physi-
cists assumed that this measured value is the absolute value of the proton's internal
magnetic moment vector. The quark model based magnetic moment calculations
were developed under this assumption. With the recent advent of NMR technology,
the operators of NMR equipment have recognized that under applied magnetic eld
the proton is subjected to Larmor precession. However, quark proponents never
revised their calculations, which do not consider quarks being in Larmor precession.
Any Larmor precession implies that the absolute value of individual quarks' angular
momentum vector must be larger than the
~
2
value assumed in those calculations.
Therefore, the quark model based spin interpretation is fundamentally contradicted
by the NMR technology.
One may also wonder why the hypothetical quarks' magnetic moments always
add up to the same value of
2.793µN
, which is measured experimentally. To ex-
plain this constant value, the quark model based proton spin interpretation also
requires that the three valence quarks remain in isotropic spin entanglement, which
means that their individual spin measurements are always correlated. Specically,
n
particles are said to be isotropically spin-correlated, if a measurement made in
an
arbitrary
direction
θ
on
one
of the particles allows us to predict with certainty
the spin value of each of the other
n1
particles for the same direction
θ
. Such
spin-correlation is required to maintain a constant value of the measured proton
magnetic moment. It was pointed out to pioneering quark proponents that their
requirement is in contradiction with the Pauli exclusion principle. This issue lead
Oscar Greenberg to postulate in 1964 that quarks also have color charge; the
purpose of this color charge hypothesis was to remove the contradiction with re-
spect to the Pauli exclusion principle. However, Paul O'Hara recently proved that
Greenberg's postulate does not remove the contradiction with the Pauli exclusion
principle: the isotropic spin entanglement of three particles is a mathematical im-
possibility if their spins are individually measurable. This mathematical contra-
diction holds regardless of the presence or absence of color charges. Paul O'Hara's
proof can be found in the second appendix.
In summary, the quark model based proton spin interpretation contains the
above explained these fundamental contradictions. Each of these contradictions
invalidates the quark model.
7.
Conclusions
We have presented a proton model that describes the physical origin of numer-
ous proton parameters, such as its mass, its spin, charge radius, magnetic moment.
Despite our model's simplicity, our calculations are in a relatively fair agreement
with experimental values. The proton's spherical electric charge generates a Zitter-
bewegung current over a toroidal surface.
A consistent application of Maxwell's equation thus lead to the discovery of the
proposed proton model. The strong similarities with the electron model suggest
a universal applicability of fundamental physical laws. Both the electron and the
proton comprise an electromagnetic wave, whose formulation can be derived by
solving Maxwell's equation. These solutions must not neglect the eects of general
THE PROTON AND OCCAM'S RAZOR 19
relativity, as demonstrated in reference [22] and in our present work. Our proton
mass calculation demonstrates that Maxwell's equation remains valid at least down
to
1018 m
, which is the length scale of the proton's spherical charge radius.
Based on our results, the proton may regain its elementary particle status. The
main dierence between an electron and a proton is the topology of their Zitter-
bewegung: a circular Zitterbewegung current in the electron case and a toroidal
Zitterbewegung current in the proton case. It remains to be understood why only
these two topologies lead to a stable particle.
Acknowledgements
The authors thank Paul O'Hara for insightful discussions of the Pauli exclusion
principle, William Stubbs for insightul discussions of high-energy electron-proton
scattering data and Giuliano Bettini for some essential suggestions.
Appendix 1: An alternative approach to the proton geometry
calculation
4
Proton geometry.
We calculate the proton radii by applying the toroidal Zitter-
bewegung model, which was introduced in section 5.1 and is illustrated in gure
??
.
The total electric energy of the proton is calculated by integrating the energy
density of its electric eld down to its spherical charge radius:
We=e2
32π20ˆ
rcp
1
r4·4πr2dr =e2
8π0ˆ
rcp
1
r2dr =e2
8π0
1
r
rcp
=e2
8π0rcp
In accordance with Maxwell's equation, the electric an magnetic elds induce
each other within the proton. Therefore its electric and magnetic energies must
be equal:
We=Wm
. From the 938.272 MeV proton mass value we get
We=
Wm=
469.136 MeV. We can now calculate the proton's spherical charge radius
rcp
:
rcp =e2
8π0We
= 1.5347 ·1018 m
This calculated
rcp
value is remarkably similar to the experimental value dis-
cussed in section 2.1.
To transform the circular Zitterbewegung model of the positron [22] into a
toroidal geometry, the naive approach is to view the positron from rotating ref-
erence frame. Such reference frame transformation must take into account the
relativistic Thomas precession eect which arises in a rotating reference frame.
This eect reduces the apparent lab-frame speed of a circularly orbiting object in
proportion to its Lorentz boost factor:
βlab =β0
γlab
where
0
is the true rotation speed,
lab
is the apparent rotation speed in the lab
frame, and
γlab =p1β2
lab1
. When
βlab =1
2
, we get
β0= 1
: this limiting value
corresponds to the true rotation speed being the speed of light. This result means
that in the
βlab <1
2
regime the toroidal charge distribution is in fact a rotating
4
This approach is the perspective of Andras Kovacs
THE PROTON AND OCCAM'S RAZOR 20
scaled positron because we can make a rotational change of reference frame which
transforms the charge current back to the positron's ring shaped Zitterbewegung.
The rotating scaled positron will eventually loose its rotational energy by interacting
with other particles, and will thus transform back into an ordinary scaled positron.
Therefore, the
βlab <1
2
regime does not correspond to a stable proton particle. On
the other hand, the limiting
βlab =1
2
value stays invariant under any rotational
reference frame transformation, and therefore it corresponds to a truly toroidal
charge current, which retains the same geometry in any reference frame. Since
the proton retains its basic properties in all reference frames, this
βlab =1
2
value
corresponds to its toroidal Zitterbewegung speed:
vt=c
2
.
It follows from Maxwell's equation that electromagnetic waves propagate at the
speed of light, which means that the spherical charge moves at a Zitterbewegung
speed which is always the speed of light [22]. In the toroidal geometry, the Zit-
terbewegung speed vector comprises toroidal and poloidal components, which are
perpendicular to each other:
v2
t+v2
p=c2
Since we already know the toroidal Zitterbewegung speed, we can calculate the
poloidal one as well from he above relationship:
vp=c
2
.
The proton's toroidal and poloidal radii.
The electron's and positron's circu-
lar Zitterbewegung structure is discussed in reference [22]. In order to determine
the proton's toroidal and poloidal radii, we must briey review the positron's mag-
netic energy and magnetic ux calculation. The positron's canonical momentum,
generated by the vector potential
A
at its spherical charge surface, is
p=eA
. The
corresponding positron angular momentum is
= eArZBW
, where
rZBW
is the cir-
cular Zitterbewegung radius. The electron's and positron's Zitterbewegung radius is
experimentally determined by Thomson scattering:
rZBW = 0.3861592676·1012 m
.
This
rZBW
value is referred to in the scientic literature as the reduced Compton
radius.
By setting
= ~
, we obtain the norm of the vector potential at the positron's
spherical charge surface:
A=~
erZBW
.
Once we know the vector potential, it is possible to determine the magnetic
ux produced by the rotating elementary charge by applying the circulation of the
vector potential
A
:
φ=˛λ
Adλ =ˆ2π
0
~
erZBW
rZB W = 2π~
e=h
e4.135 667 ·1015 V·s
i.e.
the magnetic ux crossing the Zitterbewegung loop is quantized. Now it is
possible to calculate the magnetic energy stored in the positron current loop:
Wm=1
2φIpositron =1
2·2π~
e·ec
2πrZBW
=~c
2rZBW 255.5 keV
which is equal to half the positron rest energy, thereby satisfying the
We=Wm
requirement of an electromagnetic wave. This result demonstrates the correctness
THE PROTON AND OCCAM'S RAZOR 21
of setting the intrinsic angular momentum value to
~
. We note that although the
above equation refers to a static current loop, the result stays the same in the case
of a circulating elementary charge. To see this, we evaluate the current interaction
part of the electromagnetic Lagrangian density:
Lint =JM·AM=JA =Ipositron
πr2
charge ·~
erZBW 1.352 604 ·1027 J·m3
By integration over the volume described by the spherical charge trajectory, it
is possible to recompute the positron's energy:
Wpositron =˚V
J A dV =Ipositron
πr2
charge ·~
erZBW ·2π2rZB W r2
charge =φIpositron 511 keV
Considering the above expression, we can take the toroidal volume, and divide
it into two halves. The spherical positron charge is in one of those halves, and
thus the integration volume becomes half of the toroid volume, while the eective
current between the integration endpoints is now twice as large. The integration
result for
Wpositron
remains invariant. By repeating this halving of the toroidal
volume segments, we see that the total magnetic energy remains invariant as we
approach the circulating spherical charge scenario.
The recognition that the magnetic ux of a Zitterbewegung loop is quantized
to
h
e
is a central result of [22]. We also show that this magnetic ux quantization
is equivalent to the electric charge quantization. Since the proton's charge is the
elementary charge
e
, the
h
e
magnetic ux quantization must hold true for the proton.
How is the proton's
Wm=
469.136 MeV magnetic eld energy divided between its
toroidal and poloidal current loops? The proton's magnetic moment measurement
is in fact its toroidal magnetic moment measurement. For an elementary particle,
its measured magnetic moment is given by the
µ=e~
2m
formula, where the only
non-constant factor is the particle mass. Since the proton mass is derived from
electromagnetic induction, its excess toroidal magnetic moment is inversely pro-
portional to its toroidal magnetic mass. Therefore the proton's toroidal magnetic
energy is:
Wmt =Wm/µp
µN=469.136
2.79285 MeV = 167.978 MeV
The remaining poloidal magnetic eld energy is:
Wmp =WmWmt = 301.158 MeV
Under the toroidal proton geometry there are two Zitterbewegung loops: a
toroidal loop and a poloidal loop. Since both current loops are generated by the
elementary charge
e
, the
h
e
magnetic ux quantization holds for each current loop.
Thus we can calculate the magnetic energy values by applying the
h
e
magnetic ux
quantization condition:
Wmt =1
2φItoroidal =1
2·2π~
e·evt
2πrpt
=~vt
2rpt
Wmp =1
2φIpoloidal =1
2·2π~
e·evp
2πrpp
=~vp
2rpp
THE PROTON AND OCCAM'S RAZOR 22
We thus evaluate the toroidal and poloidal Zitterbewegung radii from the above
equations:
rpt =~vt
2Wmt
= 0.831 fm
rpp =~vp
2Wmt
= 0.463 fm
The obtained
0.831
fm toroidal radius value has a 99% match with the experi-
mentally measured
0.839 ±0.007
fm proton charge radius value. We predict that
future measurements will further converge to the above calculated
0.831
fm toroidal
radius value.
0
0.375
0.750
1.125
1.500
0 0.5 1 1.5 2
Charge density (fm-3)
Radial distance (fm)
rpt rpt+rpp
Figure 7.1.
The proton's radial charge density distribution. The
blue curve shows the proton's charge density estimation, based on
the data reported in [8], and the dashed lines indicate the calcu-
lated mean radius and furthest radius.
The proton's radial charge distribution.
The toroidal proton geometry implies
that the circulating proton charge is radially distributed between
rpt rpp rcp
and
rpt +rpp +rcp
distance from its center, i.e. its charge reaches up to 1.296 fm
radial distance.
The proton's radial charge distribution has been measured via high-energy electron-
proton scattering experiments [8], and gure 7.1 combines the results from several
such experiments. An incoming particle sees the proton torus from a random
angle, and will have non-zero probability of hitting it up to 1.3 fm distance.
Our calculation implies that a non-zero charge density should be measured up
to 1.3 fm radial distance, and implies practically zero charge density beyond that.
As can be seen in gure 7.1, the experimental data validates our calculation of
rpp
. We note that these measurements increasingly underestimate the proton's
true charge density when
r > rpp
: the proton is not spherical, and therefore an
THE PROTON AND OCCAM'S RAZOR 23
energetic electron ying by at e.g. 1 fm radial distance has some chance of missing
the torus. The chance of missing the torus becomes high as
r
1.3 fm.
In summary, we calculated the proton's
rcp
,
rpt
, and
rpp
radii without any pa-
rameter ttings, and found that each of them matches well with experimental data.
Appendix 2: Isotropic spin entanglement
5
By denition,
n
particles are said to be isotropically spin-correlated (ISC), if a
measurement made in an
arbitrary
direction on
one
of the particles allows us to
predict with certainty the spin