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Abstract

Zbw (mass-without-mass) model of the proton and neutron, using combined nuclear/Coulomb potentials and orbital energy equations.
1
A mass-without-mass model
of protons and neutrons
Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil
8 February 2021
Contents
Introduction .................................................................................................................................................. 1
The electron .................................................................................................................................................. 3
The muon-electron ....................................................................................................................................... 4
The proton .................................................................................................................................................... 5
The neutron .................................................................................................................................................. 7
Electron, proton, and neutron magnetic moments ...................................................................................... 9
Conclusion ................................................................................................................................................... 10
Introduction
Mass without mass’ models analyze elementary particles as harmonic oscillations whose total energy
at any moment (KE + PE) or over the cycle is given by E = ma22. One can calculate the radius or
amplitude of the oscillation directly from the mass-energy equivalence and Planck-Einstein relations, as
well as the tangential velocity formulainterpreting c as a tangential or orbital (escape) velocity.





Such models assume a centripetal force whose nature, in the absence of a charge at the center, can only
be explained with a reference to the quantized energy levels we associate with atomic or molecular
electron orbitals
1
, and the physical dimension of the oscillation in space and time may effectively be
understood as a quantization of spacetime.
1
See, for example, Feynman’s analysis of quantized energy levels or his explanation of the size of an atom. As for
the question why such elementary currents do not radiate their energy out, the answer is the same: persistent
currents in a superconductor do not radiate their energy out either. The general idea is that of a perpetuum mobile
(no external driving force or frictional/damping terms). For an easy mathematical introduction, see Feynman,
Chapter 21 (the harmonic oscillator) and Chapter 23 (resonance).
2
Figure 1: Circular and elliptical orbital motion
2
The model is based on the assumption of a pointlike charge
3
with no other properties but its charge
(zero rest mass). However, this zero-mass point charge acquires an effective mass which accounts for
half of the energy of the elementary particle: the other half of the energy is in the (electromagnetic or
nuclear) field which sustains the motion of the charge. As such, the pointlike Zitterbewegung (zbw)
charge is photon-like but, unlike a photon, it carries (electric) charge.
The motion is not necessarily circular: one may imagine elliptical orbitals, such as depicted by the polar
rose in the illustration.
4
The r() = a0·cos(k0 + 0) equation gives us the radial distance r as a function of
the phase = (t) = t of the oscillation. Thinking of r as a vector in 2D space (the plane of motion), we
get a wavefunction: 
For an electron, we get the following energy-mass calculation:

This yields the following equation:




This represents the E = KE + PE energy conservation equation. The velocity v is an orbital or tangential
velocity
5
and the mv2/2 formula for the kinetic energy is, therefore, relativistically correct.
We get the radius or amplitude of the oscillation from the E = ma22 equation:



2
Illustrations made from Wikipedia templates. For the orbital equations, see the MIT OCW reference course on
orbital motion.
3
Pointlike but not of zero dimension. See our explanation of the anomaly in the magnetic moment of an electron
in, for example, our paper on the basics.
4
The Rutherford-Bohr model considers circular orbitals only. Schrödinger’s wave equation adds non-circular (non-
spherically symmetric) orbitals as solutions.
5
This velocity is sometimes referred to as the escape velocity, but the terms are not to be used interchangeably
because they may refer to subtly different things (e.g. the velocity component with right angle to the semi-
major/minor axis of the ellipse). Needless to say, m is the relativistic mass.
3
We may interpret the positive and negative root of ħ2/m2c2 as the two possibilities that correspond to
the direction of angular momentum, which distinguishes an electron from a positron.
6
This formula misses the 1/2 factor of the effective mass m, which is half the total mass (m = E/c2) of the
elementary particle (m or E), and which explains why elementary (charged) particles are spin-1/2
particles, as shown by the calculation below:
The momentum is, of course, orbital angular momentum only. As such, we are essentially modeling
spin-zero (zero spin angular momentum) particles. We should, of course, note that a moving charge is a
current, which explains the magnetic moment
7
:


The electron
The radius formula works perfectly well for the electron. It yields the electron’s Compton radius a = rC:

The idea here is that an orbital cycle of the pointlike charge in its Zitterbewegung does not only pack the
electron’s energy (E = m·c2) but also Planck’s quantum of action (S = h).
8
For an electron, we also get the
following cycle time and electric current:

 
We can also calculate the electric current:


These values look rather phenomenal (we have a household-level current (almost 20 ampere) at the
sub-atomic scale here), but they are what they are and far from the radius-energy values one gets for
black holes (Schwarzschild radius).
6
The antimatter counterpart of an elementary particle has opposite angular momentum but shares the same form
factor. This explains why a proton and an electron are not a matter-antimatter pair: their form factors are
different. Positive and negative charge remain separate concepts, however: two electrons will, therefore, not
annihilate each other but coexist as an electron pair through a coupling of their magnetic moments, thereby
lowering total energy (which explains their stability as a pair). Formally, opposite angular momentum may also be
modelled by inversing the time sign, but time goes in one direction only:
7
The small anomaly in the magnetic moment may be explained by assuming the pointlike charge has a tiny (non-
zero) dimension itself. See our paper on the essentials.
8
The idea of an oscillation packing some amount of physical action may not be very familiar. In the context of our
model we think of physical action as the product of (i) the force that keeps the zbw charge in its orbit, (ii) the
distance along the loop, and (iii) the orbital cycle time.
4
We get the classical electron radius from the formula above:




This illustrates the interpretation of the fine-structure constant as a scaling parameter: re = rC.
9
The
harmonic oscillator model can be used to show that the elasticity or stiffness parameter k or, expressed
per unit mass, k/m in the F = kx formula is equal to:




This equation shows various oscillatory modes are possible: these modes are characterized by the
frequency (or its square
10
) which, in turn, depends on the speed of light (c) and the radius or range
parameter (a).
We can also easily calculate the magnitude of the centripetal force F= F from Newton’s force law
11
:


 
This is a huge force at the sub-atomic scale: it is equivalent to a force that gives a mass of about 106
gram (1 g = 103 kg) an acceleration of 1 m/s per second! It is, however, an electromagnetic force only.
The muon-electron
The same formulas apply to the muon-electron as well but suggest the centripetal force is of an entirely
different nature. The muon carries 105.66 MeV (about 207 times the electron energy) and has a (mean)
lifetime which is much shorter than that of a free neutron
12
but longer than that of other unstable
particles: about 2.2 microseconds (106 s).
13
This may explain why we get a sensible result when using
the Planck-Einstein relation to calculate its frequency and/or radius.
14
9
The 2019 revision of the system of SI units incorporates these new physics, which amount to what we refer to as
a realist interpretation of quantum physics. Needless to say, the fine-structure constant has other interpretations
as well. See our paper on the meaning of the fine-structure constant.
10
The energy in an oscillation is proportional to the square of the frequency (and the square of its amplitude too).
11
See p. 14-15 of our paper on the electron model for the formula for the centripetal acceleration (ac = a·ω2 =
vt2/a). In the same paper, we also comment on the rather particular behavior of the momentum function p = mv,
which resembles the mathematical particularities of the Dirac function.
12
The mean lifetime of a neutron in the open (outside of the nucleus) is almost 15 minutes!
13
The tau-electron is just a resonance (as opposed to a transient particle) with an extremely short lifetime
(2.91013 s only). Hence, the Planck-Einstein relation does not apply: it is not an equilibrium state. We think the
conceptualization of both the muon- as well as the tau-electron in terms of particle generations is unproductive.
Likewise, charged pions (π±) has no resemblance whatsoever with the neutral pion (see footnote 14).
14
The longevity of the muon-electron should not be exaggerated, however: the mean lifetime of charged pions,
for example, is about 26 nanoseconds (109 s), so that is only 85 times less. As for the 1.87 fm value, this is a radius
5
 



The ratio of the centripetal forces which keep the charge in its orbital for the electron and muon
respectively is equal to:



If a force of 0.106 N is rather humongous, then a force that is about 42,753 times as strong, may surely be
referred to as a strong force, right? Is it the nuclear force? It underscores the point about the modes of
the elementary oscillation depending on the radius or amplitude a = ħ/mc of the oscillation. We may,
indeed, rewrite the force ratio as
15
:

The proton
The proton mass is about 8.88 times that of the muon, and it is about 2.22 times smaller: we have a rather
mysterious 1/4 factor here, which needs explaining. Indeed, when applying the a = ħ/mc radius formula
to a proton, we get a value which is 1/4 of the measured proton radius: about 0.21 fm, as opposed to the
0.83-0.84 fm charge radius which was established by Professors Pohl, Gasparan and others over the past
decade.
16
 


The 1/4 factor is puzzling, and there may be no obvious way to explain it. However, geometry offers a way
out. We have a 1/4 factor between the volume of a sphere (V = 4πr2) and the surface area of a circle (A =
πr2) and, hence, one might intuitively think of an oscillation in three rather than just two dimensions only.
In other words, the oscillator would be driven by two (perpendicular) forces rather than just one. We can
model this by thinking of two oscillators which, according to the equipartition theorem, should each pack
half of the total energy of the proton. This spherical view of a proton as opposed to the planar picture
of an electron fits nicely with packing models for nucleons.
The frequency of each of the oscillators would be equal to = E/2ħ = mc2/2ħ: each of the two
and, hence, should be multiplied by 2 to get the CODATA value (more or less) for the Compton wavelength of the
muon (1.1734441101014 m 0.0000000261014 m).
15
We did not find any easy interpretation of this ratio in terms of the fine-structure constant, however. Hence, the
mass or energy of the elementary particles may be considered to be fundamental constants of Nature themselves.
16
For the exact references and contextual information on the (now solved) ‘proton radius puzzle’, see our paper
on it: https://vixra.org/abs/2002.0160, in which we also make some remarks on the (anomalous) magnetic
moment of the proton.
6
perpendicular oscillations would, therefore, pack one half-unit of only.
17
This, then, gives us the
experimentally established value for the proton radius:
 

The force along one of the two axes or planes of oscillation inside of a proton is equal to:



Hence, we get a force of 4,532 N inside of a muon and a force of 89,349 N inside of a proton. Compensating
for the 1/4 factor (which we loosely refer to as the different form factor of the proton oscillation), we find
the force inside of a proton is almost 5 times stronger than the force inside of a muon. Hence, we may
conclude that the force inside of a muon-electron and a proton (and neutron, which we think of as a
proton-electron combination
18
) are of the same nature. However, a muon-electron is, clearly, not the
antimatter counterpart of a proton !
The orbital energy equation for the nuclear field is given by
19
:


Can we calculate a out of this? Maybe. How? Perhaps by evaluating potential and kinetic energy at the
periapsis, where the distance between the charge and the center of the radial field is closest. However,
the limit values vπ = c (for rπ 0) and rπ = 0 (for vπ c) are never reached and should, therefore, not be
used: neither the kinetic nor the potential energy seems to reach the zero value and we can, therefore,
probably not simplify any further.
20
We, therefore, prefer the simpler zbw approach as outlined above:



We may now substitute this value for a in the orbital energy equation
21
:
17
This explanation is similar to our explanation of one-photon Mach-Zehnder interference, in which we assume a
photon is the superposition of two orthogonal linearly polarized oscillations (see p. 32 of our paper on basic
quantum physics, which summarizes an earlier paper on the same topic).
18
See our paper on the nuclear force and the neutron hypothesis.
19
The reader can/should check the physical dimensions:





Note that we have a plus (+) sign in the equation because the potential energy in the orbital energy equation is
zero at the center and, therefore, always positive. For more details, see our paper on the nuclear force.
20
One can, however, calculate other interesting properties of the orbitals, such as the eccentricity (see the above-
mentioned MIT OCW reference course on orbital motion).
21
We use the definition (cf. the 2019 revision of SI units) of the fine-structure constant:
 formula.
7



Re-arranging yields:


It is a nice formula. In the next section, we will find a similar formula for the neutron.
The neutron
We think of the (free) neutron as a composite (non-stable) particle consisting of a proton and an
electron. However, we will soon qualify this statement: the reader should effectively think in terms of
pointlike charges hererather than in terms of a massive proton and a much less massive electron!
Both the proton and the electron carry the elementary (electric) charge but we think both are bound
in a nuclear as well as in an electromagnetic oscillation. In order to interpret v as an orbital or tangential
velocity, we must, of course, choose a reference frame. Let us first jot down the orbital energy equation
for the nuclear field, however
22
:

Figure 2: Two opposite charges in elliptical orbitals around the center of mass
23
22
A dimensional check of the equation yields:





We recommend the reader to regularly check our formulas: we do make mistakes sometimes!
23
Illustration taken from Wikipedia. For the orbital equations, see the MIT OCW reference course on orbital
motion.
8
The mass factor mN is the equivalent mass of the energy in the oscillation
24
, which is the sum of the
kinetic energy and the potential energy between the two charges. The velocity v is the velocity of the
two charges (qe+ and qe) as measured in the center-of-mass (barycenter) reference frame
25
and may be
written as a vector v = v(r) = v(x, y, z) = v(r, , ), using either Cartesian or spherical coordinates.
We have a plus sign for the potential energy term (PE = akeqe2/mr2) because we assume the two charges
are being kept separate by the nuclear force.
26
The electromagnetic force which keeps them together is
the Coulomb force:
The total energy in the oscillation is given by the sum of nuclear and Coulomb energies and we may,
therefore, write:



The latter substitution uses the definition of the fine-structure constant once more.
27
Dividing both sides
of the equation by c2, and substituting mN and mC for m/2 using the energy equipartition theorem,
yields:



It is a beautiful formula, and we could/should probably play with it some more by, for example,
evaluating potential and kinetic energy at the periapsis, where the distance between the charge and the
center of the radial field is closest. However, the limit values vπ = c (for rπ 0) and rπ = 0 (for vπ c) are
never reached and should, therefore, not be used. We are sure one of our readers will find ways to get a
specific value for the radius a, which should be, hopefully, very near to 0.84 fm (the proton/neutron
diameter. It should, in fact, be slightly larger because of the energy difference between a proton and a
neutron, which is of the order of about 1.3 MeV, which is about 2.5 times the energy of a free electron.
28
24
We will use the subscripts xN and xC to distinguish nuclear from electromagnetic mass/energy.
25
This relates to the point we made in regard to the nature of the ‘proton’ in the neutron: it is not like the massive
proton at the center of the hydrogen atom, with the electron orbiting around it.
26
We have a minus sign in the same formula in our paper on the nuclear force because the context considered two
like charges (e.g. two protons). As for the plus (+) sign for the potential energy in the electromagnetic orbital
energy, we take the reference point for zero potential energy to be the center-of-mass and we, therefore, have
positive potential energy here as well.
27
One easily obtains the keqe2 = ħc identity from the
 formula.
28
We note there is no CODATA value for the neutron radius. This may or may not be related to the difficulty of
measuring the radius of a decaying neutral particle. As for the instability of the free neutron, its lifetime is very
9
Till then, we must assume we may apply the mass-without-mass formula for the proton radius to the
neutron too:



Electron, proton, and neutron magnetic moments
The magnetic moment of an elementary ring current is the current (I) times the surface area of the loop
(πa2). The current is the product of the elementary charge (qe) and the frequency (f), which we can
calculate as f = c/2πa, i.e. the velocity of the charge
29
divided by the circumference of the loop. We
write:



Using the Compton radius of an electron (ae = ħ/mec), this yields the correct magnetic moment for the
electron
30
: 
The CODATA value for the magnetic moment of a proton is equal to:
μ = 1.410606797361026 J·T1 0.00000000060 J·T1
The electron-proton scattering experiment by the PRad (proton radius) team at Jefferson Lab measured
the root mean square (rms) charge radius of the proton as rp = 0.831 ± 0.007stat ± 0.012syst fm.
When applying the a = μ/0.24…1010 equation to the proton, we get the following radius value:


We interpret this to be the effective mm (magnetic moment) radius because we think of the proton
oscillation as a spherical (3D) oscillationin contrast to the electron oscillation, which is a plane (2D)
long as compared to the muon-electron, so we may effectively assume that the oscillation must very nearly pack
one unit of Planck’s quantum of action.
29
We assume a pointlike charge with zero rest mass. Its (orbital/tangential) velocity is, therefore, equal to the
speed of light (c). This ring current model, which was first proposed by the British chemist and physicist Alfred
Lauck Parson (1915) and which was mentioned rather prominently in the Solvay Conferences, amounts to a mass-
without-mass model of elementary particles (John Wheeler, 1957, 1960, 1961). It analyzes them as harmonic
oscillations whose total energy at any moment (KE + PE) or over the cycle is given by E = ma22. One can then
calculate the radius or amplitude of the oscillation directly from the mass-energy equivalence and Planck-Einstein
relations, as well as the tangential velocity formulainterpreting c as a tangential or orbital velocity, indeed:





30
The calculations do away with the niceties of the + or sign conventions as they focus on the values only. We
also invite the reader to add the SI units so as to make sure all equations are consistent from a dimensional point
of view. The CODATA values were taken from the NIST website.
10
oscillation. If we multiply this effective radius with 2, we get a value for the proton radius which fits
into the 0.831 0.007 interval:

Conversely, if we divide the magnetic moment that is associated with the above-mentioned radius value
(0.83 fm) by the same factor 2, we get the actual magnetic moment of a proton.
31
μp = 1.410606797361026 J·T1 = (1.9948992641026 J·T1)/2
Let us now turn to the neutron. The CODATA value for the neutron magnetic moment is:
μn = 0.966236511026 JT1
The difference (2.376843307361026 JT1) corresponds to a radius of about 1 fm:

 
This is the right order of magnitude, and we think the difference with the actual neutron radius may,
once again, be explained by the 2 factor: the negative charge inside of the neutron will be in a three-
dimensional oscillation itself, and the effective radius of the electron inside of the neutron (which we
denote as ae) will, therefore, be smaller:

We take this to confirm our neutron model.
Conclusion
Mass-without-mass models of elementary particles model the oscillation of a pointlike charge. Potentials
and forces depend on and/or act on a charge: the elementary charge e± (e, e, ep) or its antimatter
counterpart. We think there are only two forces/potentials: electromagnetic and nuclearor some
combination thereof.
32
We also think forces/potentials/particles have a field- or light-particle counterpart: the photon or the
neutrino (as applicable to proton/neutron Verwandlung reactions).
Of course, we are very much aware that we offer a non-conventional analysis here which breaks away
from common ideas on several critical points. Most importantly, perhaps, we think leptons do partake in
nuclear interactions, which explains deep electron orbitals
33
, our model of the muon as a (potentially)
pure nuclear oscillation (neutrinos carry the (excess) energy of a muon decay reaction) and, perhaps, why
low-energy nuclear reactions (transmutation of nucleons by laser irradiation) can possibly take place.
We effectively think the classification of particles into generations or into baryons and leptons are too
general to be useful: we just have two forces/potentials, and combinations thereof. Incidentally, we also
31
For a discussion of the anomalies in the magnetic moment, see our paper on the topic.
32
We develop a model for the deuteron nucleus in the above-mentioned paper too!
33
See, for example, Andrew Meulenberg and Jean-Luc Paillet, Deep Electron Orbitals and the Dirac Equation,
January 2020.
11
think the quark hypothesis might not be very useful: at best, they are temporary non-equilibrium states
and, as such, mathematical abstractions. We get the following table of elementary matter- and light-
particles
34
:
Table 1: Elementary particle classification according to form factors
2D oscillation
3D oscillation
electromagnetic force
e (electron/positron)
orbital electron (e.g: 1H)
nuclear force
(muon-electron/antimuon)
p (proton/antiproton)
composite
pions (π/ π0)?
e.g: n (neutron),
D+ (deuteron)
corresponding field particle
(photon)
(neutrino)
This is nice and complete: each theoretical/mathematical/logical possibility corresponds to a physical
reality, with spin distinguishing matter from antimatter for particles with the same form factor.
So what is our theory of reality, then? We think physic reality and our logical representation of it blends
as part of the sensemaking process. Wittgenstein was wrong on language, but his intuition was quite
correct: Die Welt ist die Gesamtheit der Tatsachen, nicht der Dinge. (Wittgenstein, TLP, 1.1)
We, therefore, stick to classical quantum physics or, as we refer to it, to a realist interpretation of it.
Indeed, we think the criticism of H.A. Lorentz of the new theories, before he left the scene, was quite
apt:
“I would like to draw your attention to the difficulties in these theories. We are trying to
represent phenomena. We try to form an image of them in our mind. Till now, we always tried
34
We think of the tau-lepton as a resonance or a very short-lived transient. It is, therefore, not an elementary
particle in our view but only an intermediary reaction product.
12
to do using the ordinary notions of space and time. These notions may be innate; they result, in
any case, from our personal experience, from our daily observations. To me, these notions are
clear, and I admit I am not able to have any idea about physics without those notions. The image
I want to have when thinking physical phenomena has to be clear and well defined, and it seems
to me that cannot be done without these notions of a system defined in space and in time.”
35
We conclude with Wittgenstein’s last and final proposition in his Tractatus Logico-Philosophicus (TLP):
Wovon man nicht sprechen kann, darüber muss man schweigen.”
Brussels, 8 February 2021
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See our brief history of quantum-mechanical ideas.
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