Hadronic Journal, 2012, Vol.
35, No. 4, P. 349 – 363.
The radius of the proton in the self-consistent
model
Sergey G. Fedosin
Perm, Perm Region, Russia
e-mail intelli@list.ru
Based on the notion of
strong gravitation, acting at the level of elementary particles, and on the
equality of the magnetic moment of the proton and the limiting magnetic moment
of the rotating non-uniformly charged ball, the radius of the proton is found,
which conforms to the experimental data. At the same time the dependence is
derived of distribution of the mass and charge density inside the proton. The
ratio of the density in the center of the proton to the average density is
found, which equals 1.57 .
Keywords: strong gravitation; de
Broglie waves; magnetic moment; proton radius.
PACS: 12.39.Pn,
14.20.Dh
1 Introduction
Since the discovery of the proton in
1917 the question arose how to determine the radius of this elementary
particle. There are many theoretical models to estimate the radius of the
proton. Most of these models is associated with the concept of the
electromagnetic form factors as the amendment by which the scattering amplitude
of particles by proton is different from the scattering amplitude by a point
particle. The calculation of the form factors is complex and requires taking
into account many factors, including the radial density distribution of charge
and magnetic moment, the dynamics of quarks, partons and virtual particles.
There may be a variety of approaches – scattering theory, chiral perturbation
theory, lattice QCD, etc., description of which can be found in [1], [2]. Form
factors are determined from scattering experiments, depend on the energy of the
interacting particles, and allow us to find the root mean square of the charge
distribution and magnetic moment as a measure of particle’s size. Information
on the radius of the proton can be extracted from the analysis of the Lamb
shift in hydrogen and in a coupled system of a proton and a negative muon [3].
2 Other estimates of proton radius
Consider some simple methods for
determining the radius of the proton. One of them is based on the fact that in
the particles, when they are excited, standing electromagnetic waves emerge.
The maximum energy of these standing waves does not exceed the rest energy in
order to avoid the decay of particles. From this it can be derived that the de
Broglie waves are electromagnetic oscillations, detectable in the laboratory
frame in the interaction of moving particles. To describe these oscillations it
is necessary to apply the Lorentz transformations to the standing waves inside
the particles and to find their form in the laboratory reference frame [4],
[5].
In the simplest case the spherical
standing waves are modeled by two waves, one of which runs from the center to
the surface of the particle and the other at the same time is moving backwards.
We can assume that in the direction of a specified axis, for example 
, there are
two counter-propagating waves of the following form:
, 
,
, (1)
here 
, 
are the initial phases
of the oscillations with 
, 
is the amplitude of
the periodic function, 
and 
denote the angular frequency and wave number
and the primes over the variables mean that they are considered in the rest
frame of the particle.
As 
any periodic function
can be used, which satisfies the wave equation. For example, it can be the
strength or the field potential of the wave. The phases of the waves in (1)
must be shifted to 
for emerging of the
standing wave. If 
, 
, then in the center of the particle with 
there will be always a node as the absence of
visible oscillations, and (1) becomes as follows:
. (2)
As a result of oscillations (2) velocities
of charges of the particle substance and the field potentials can periodically
change inside the particle. This leads inevitably to periodic oscillations of
the field potentials also outside the particle in the surrounding space.
Now we shall assume that the
particle moves together with its standing wave along the axis in the laboratory reference frame at the
velocity 
. How are the field oscillations modified inside and outside
the particle with respect to its movement? We should express in (2) the primed
coordinates and the time inside the moving particle through the coordinates and
the time in the laboratory reference frame using the Lorentz transformations (
refers to the speed of light):
, 
, 
, 
,
. (3)
From (3) we see that as a result of
displacement of the standing wave with the particle for the external motionless
observer in the laboratory frame the wavelength and the frequency will change.
More precisely, on the observed wave additional antinodes appear, with a
wavelength between them, differing from the wavelength 
in the reference frame
of the particle. We shall stop the wave (3) for a moment with 
and shall find the
wavelengths as the spatial separation between the points of the wave in the
same phase. When 
the sine in (3) will
be zero, while when 
the phase of the sine
will change from 0 to 
. Hence we obtain:
, 
. (4)
Similarly for the wavelength of the
cosine in (3) we find:
, 
. (5)
We shall now estimate the temporal
separation between the points of the wave in one phase with , considering this separation as the corresponding period of
the wave:
, 
. (6)
, 
. (7)
From (4) − (7) we obtain the
following expressions for the velocities:
, 
. (8)
As we see from (8) the oscillations
of the wave (3) associated with the cosine, are propagating at the phase
velocity of de Broglie 
. Besides, the oscillations of the wave (3) associated with
sine, move in space at the same velocity 
as the particle
itself. The wavelength 
in (5) can be
transformed so as to bring it to the standard form for the de Broglie
wavelength. We shall associate the angular frequency of the oscillations inside
the particle, similarly to the electromagnetic wave, with the energy of
oscillations: 
, where 
is Dirac constant, 
is Planck constant. This gives the following:
. (9)
Similarly from (4) we have:
. (10)
In the limiting case when the
oscillation energy is compared with the rest energy of the particle, 
, from (9) it follows:
, (11)
where 
is the mass of particle, 
is relativistic momentum of the particle.
The formula (11) defines de Broglie
wavelength with the help of particle momentum. We shall note that de Broglie
wrote (11) on condition that the energy of the particle 
is equal to the energy
of the wave accompanying the particle.
According to the obtained expression
(9), the wavelength must be present in the particle also at low
excitation energy 
In this case as the excitation energy
decreases, the wavelength should increase.
As a rule in the experiments only 
is found from (11), and not the wavelength 
from (9).
This can occur because among the
number of interacting particles at the same time there are particles with
different excitation energies 
and different , so that
the wave phenomena are blurred. The same is true for the waves with wavelength 
in (10). Only for the most actively
interacting particles, the excitation energies of which are close to the rest energy of the
particles, the limiting value of the wavelength is reached equal to the de
Broglie wavelength. Thus this wavelength is revealed in the experiment. When 
we can also predict for the particles the wave
phenomena with the critical wavelength 
. In
particular, 
is the Compton wavelength, discovered in the
Compton effect.
According to our point of view,
emerging of de Broglie wave should be treated as a purely relativistic effect,
which arises as a consequence of the Lorentz transformation of the standing
wave, moving with the particle.
As a result, we have to assume that
the wave-particle duality is realized in full only in those particular
particles, the excitation energies of which reach their rest energies. In this
case the difference of particles and field quanta, if they are treated from the
point of view of their wave properties, becomes minimal. At low excitation
energies the particles can not emit their energy greatly, and the amplitudes of
the oscillations of the field potentials near the particles are small. Then the
particles would interact with each other not in the wave way, but rather in the
usual way, and the wave phenomena become invisible.
If we assume that the length of the
standing wave is equal to 
, where 
is the radius of the
proton, then from the equality of the wave energy and the rest energy of the
proton we obtain:
, 
, 
m,
here 
is the oscillation frequency, 
is the mass of the
proton.
Another way to estimate the radius
of the proton assumes that the difference between the rest energy of the
neutron and the proton is due to the electrical energy of the proton charge. In
this case, it should be:
, (12)
where 
is the mass of the
neutron, 
is the elementary
charge, 
is the vacuum
permittivity.
In (12) for the case of the uniform
distribution of the charge in the volume of the proton 
, as a result the estimation of the proton radius gives the
value of 
m.
In [6] and [7], the radius of the
proton was found from the condition that the limiting angular momentum of the strong
gravitation field inside the proton is equal in magnitude to the spin of the
proton. This leads to the following formula:
m. (13)
In (13) the strong gravitational
constant 
is used. According to
[4], this constant is determined from the equation of electric force and the
force from the strong gravitation field, acting in the hydrogen atom on the
electron with the mass 
, which is located in the ground state on the Bohr radius 
:
, 
m3∙kg–1∙s–2, (14)
In addition to the attractive forces
from gravitation and the charges of the nucleus and the electron, in the
hydrogen atom the electron substance in the form of the rotating disc is
influenced by the repulsive forces acting away from the nucleus. One of these
forces is the electric force of repulsion of the charged substance of the
electron cloud from itself. In the rotating non-inertial reference frame in
which an arbitrary part of the electron substance is at rest, there is also the
force of inertia in the form of the centrifugal force, which depends on the
velocity of rotation of this substance around the nucleus. In the first approximation,
these forces are equal in magnitude, which leads to (14).
We shall remind that the idea of
strong gravitation was introduced into science in the works of Abdus Salam and
his colleagues [8], [9] as the alternative explanation of the strong interaction
of the particles. Assuming that hadrons can be represented as Kerr-Newman black
holes, they estimated the strong gravitational constant as 
m3∙kg–1∙s–2.
With the help of the strong
gravitation constant (14) we can express the fine structure constant:
.
Another estimate of the radius of
the proton follows from the equality of the rest energy and the absolute value
of the total energy, which, taking into account the virial theorem, is
approximately equal to the half of the absolute value of the strong gravitation
energy associated with the proton [4]:
.
(15)
If we take for the case of the
uniform mass distribution, then from (15) it follows that 
m.
All of the above estimates are based
on the classical approach to the proton as to the material object of small size
in the form of the ball with the radius 
. It is assumed that the strong gravitation acts at the level
of elementary particles in the same way as ordinary gravitation at the level of
planets and stars.
In the Standard model of elementary
particles and in quantum chromodynamics it is assumed that the nucleons and other
hadrons consist of quarks, and baryons have three quarks, while mesons have two
quarks. Instead of the strong gravitation, the action of gluon fields is
assumed to hold the quarks in hadrons. Quarks are considered to be charged
elementary particles, therefore as the radius of the proton the charge and
magnetic root mean square radii are considered. These radii are determined by
the electric and magnetic interactions of the proton and can differ from each
other.
The estimate of the proton charge radius can be made with the help of
the experiments on the scattering of charged particles on the proton target
[10]. In such experiments the total cross sections of interaction of the
particles 
are found. For the
case of the protons scattering on nucleons with energies more than 10 GeV we
can assume that 
, and 
m2. Hence
we obtain 
m.
3 The self-consistent model
Our aim will be to find a more exact
value of the radius of the proton by using classical methods. In the calculations
we shall use only the tabular data on the mass, charge and magnetic moment of
the proton. The proton will be considered from the standpoint of the theory of
infinite nesting of matter [11], in which the analogue of the proton at the
level of stars is a magnetar or a charged neutron star with a very large
magnetic and gravitational field. Similarly to the magnetar, the substance of
the proton must be magnetized and held by a strong gravitation field.
To take into account the
non-uniformity of the substance density inside the proton we shall use the
simple formula in which the substance density changes linearly increasing to
the center:
, (16)
where 
is the central
density, 
is the current radius,
is the coefficient
which should be determined.
Formula (16) should be considered as
a first approximation to the actual distribution of the density of matter
inside the proton. Approximate linear dependence of the density of matter in
neutron stars has been shown in [12], and we assume that this is also true for
the proton as an analogue of the neutron stars.
To estimate the values 
and the radius we shall consider the
integral for the proton mass in the spherical coordinates:
. (17)
For accurate calculation of state of
neutron stars, and thus protons as their analogues we should consider the curvature
of spacetime in a strong gravitational field, as well as the contribution of
the energy of the gravitational field to the total mass-energy. We shall assume
that in (16), in dependency of matter density on the radius all relativistic
effects are taken into account, and the mass of the proton (17) is the
gravitational mass from the point of view of a distant observer.
In (17) there are three unknown
quantities, to obtain which two more equations are required. We shall assume
the virial theorem to be valid and equate the rest energy of the proton to the
half of the absolute value of the energy of the static field of strong
gravitation:
, (18)
where 
is the energy density of the strong
gravitation field according to [4],
is the gravitational
acceleration or strength of gravitational field.
In (18), the integration of the
energy density of the field should be done both inside and outside of the
proton. The value inside the proton can
be conveniently found by integrating the equation for the strong gravitation
field 
, which is part of the equations of the Lorentz-invariant
theory of gravitation [13]. After integrating over the spherical volume with
the radius 
, and then using the Gauss theorem, that is making transition
to integrating over the area of the indicated sphere inside this proton, in
view of (17) we obtain:
,
. (19)
Outside the proton the gravitational
acceleration is equal to:
.
(20)
Substituting (19) and (20) in (18),
we obtain the relation:
. (21)
In (21) we can eliminate the value using (17), which give
the dependence of 
on 
in the form of the
quadratic equation:
.
The analysis of this equation shows
that it has the following solution:
, (22)
on condition that when 
, then accordingly 
.
We shall now turn to the magnetic moment
of the proton. As in [4], we assume that the magnetic moment of the proton is
equal to the magnetic moment, which is formed due to the maximum rapid rotation
of the charged substance of the proton. In spherical coordinates, the magnetic
moment can be approximately calculated as the sum of the elementary magnetic
moments of the separate rings with their radius 
, which have the magnetic moment due to the current 
flowing in them from
the rotation of the charge:
(23)
The angular velocity 
of the maximum
rotation of the proton can be found from the condition of limiting rotation,
with the equality of the centripetal force and the gravitation force at the
equator: 
. Further we believe that for the charge density and the
substance density the equation 
holds, and we use
(17). This gives the following:
. (24)
4 Conclusions
The relation (24) together with (22)
allow us to find the radius of the proton 
m, as well as the
value 
. From (17) we obtain then the central substance density 
kg/m3,
which exceeds the average density of the proton 1.57 times. The maximum angular
velocity of rotation of the proton in view of (23) is equal to 
rad/s. At the same
time, if the spin of the proton in the approximation of the uniform density of
substance would be equal to the standard value for the spin of the fermion 
, then the angular velocity of rotation 
rad/s would correspond
to this spin.
For comparison with the experimental
data we shall point to the results of calculations of electron scattering from
[14], where the charge radius 
m is obtained taking
into account only the scattering on protons, 
m – taking into
account the data on the pion scattering, and 
m – taking into
account the data on the neutron scattering. In [3] the charge radius 
m was found in the
study of the coupled system of the proton and the negative muon. The study of
the scattering cross section of polarized photons by protons [15] gives the
charge radius 
m and the magnetic radius 
m. The charge radius 
m and the magnetic
radius 
m of the proton are
listed on the site of Particle data group
[16]. In the database CODATA [17] the proton charge radius is equal to 
m.
The value 
m obtained in the
framework of the self-consistent model is close to the experimental values of
the radius of the proton, which confirms the possibility of applying the idea
of strong gravitation to describe the strong interaction of elementary
particles.
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