_{1}

Starting from a revised quantum electrodynamic theory by the author, an attempt is made to elaborate a particle model of quarks which in their turn form triangular neutron and proton configurations. These “quark particles” are found to be electrically neutral but contain intrinsic electrical charges of both polarities, being an order of magnitude larger than the elementary charge, e. The main interaction force between two such particles is further found to have an attractive short-range character, and it becomes nearly two orders of magnitude larger than the repulsive force which would arise from two interacting elementary charges. The spatial potential distribution of this force corresponds to an inner barrier, an intermediate well, and an outer barrier. The well is found to have a depth being nearly equal to the binding energy 8 MeV of the neutron. The distribution of the barriers and the well makes a stable position possible for the mutual particle distance. The deduced radii of the outer shell and of the core are further of the same magnitude as the known nuclear radii of the neutron and proton. All these deduced characteristic features are the same as those of the known strong force concept. This raises the question whether the present results could provide a first step in a unification of the electromagnetic and the strong nuclear forces.

From the analysis of the harmonic oscillator, Planck proved the existence of a lowest nonzero energy level of the vacuum state, the Zero Point Energy (ZPE). This state generally includes photon-like electromagnetic oscillations, both with and without associated electric charges. It manifests itself in the Casimir force arising in vacuo between two metal plates separated by a small spacing [

The ZPE and the Casimir force imply that there are sources in the vacuum which have to be included into the electromagnetic field equations. This is taken into account in an earlier Lorentz and gauge invariant treatise by the author [

Spatial integration of

The main purpose of this investigation is to show that the RQED theory can lead to interactions being the same as those due to the conventional strong force. The broken symmetry of RQED results in substantial intrinsic electric charges, even being an order of magnitude larger than the elementary charge, as will be seen in the following parts of this investigation. The mutual interaction force between the intrinsic charges of two particles is therefore expected to become both of short range and two orders of magnitude larger than that which arises from interaction between elementary charges. Such an intrinsic charge interaction thus has basic features being the same as those of the strong nuclear force described by Walker [

This paper presents a first attempt in studying the intrinsic charge effects and their relation to the strong force, in terms of a simple particle model based on a convergent generating function. It starts in Sections 2 and 3 with the restrictions and basic concepts of the present analysis, continues with the main features of a corresponding particle model in Section 4, and ends with a tentative quark model of the proton and neutron in Section 5.

With the main purpose of demonstrating the effects of the intrinsic charges, a first approach is thus made in terms of a convergent generating function of the electromagnetic field strengths. This avoids the additional complication of a divergent such function, with its related net point-charge behaviour and a related renormalisation process [

There further exists a possibility of studying a compound particle model. This would include a divergent generating function with both net and intrinsic charges, superimposed on a convergent generating function with intrinsic charges only. Also this possibility is postponed to later studies.

The coming analysis will further be restricted to a relatively simple form of generating function, treated as an Ansatz representing a set of various geometrical profiles.

A steady-state particle model will now be elaborated, to form a starting point of the present analysis. For detailed deductions of the underlying theory, reference is made to an earlier treatise [

with

With the generating function

having the amplitude

are obtained for the potentials and the local charge density. This makes the electric and magnetic field strengths and the electric charge density interconnected through the generating function. For a volume element

where

The total self-force acting on the particle due to the fields

According to Section 0 we restrict the analysis to a convergent and separable generating function given by the Ansatz

with the disposable parameters

The present theory with its nonzero

With the function

the same mass becomes

where

Treating the present particle model as a boson in a first attempt, the spin is given by [

where

Other forms of the spin would only lead to minor changes in the following numerical computations. Combination of relations (13) and (15) results in

The charge density given by relations (7) and (8) can be combined with the identity

This results in a total integrated charge q given by

For a convergent generating function with finite parts R and T as well as their derivatives, the total intrinsic charge q then vanishes. This is because the partial integration in respect to

The local and radial intrinsic charge distribution

with

differs on the other hand from zero. It gives rise to a distribution of considerable intrinsic charges of both signs. With the form (10) and the integrals

the normalized radial distribution of charge density is then given by

where

The total electromagnetic field strength in Equation (9) consists both of an electric part

with the corresponding normalized components

and

A substantial part of the contribution from

With the generating function (10) the radial part

where

When considering the total integrated force

With the generating function this leads to

where

Turning first to the polar component

The radial force component

where its radial force distribution becomes

with

according to Equations (32)-(33) (39)-(41).

The results of Equation (23), (31) and (40) for the parameter values

The field equations of the present theory as well as that by Maxwell give rise to linear general solutions. They can formally be derived from a nonlinear Lagrangian density, and their solutions can be combined to form a force density such as that of Equation (9).

But this is not the whole problem of a self-consistent approach. In the theory of a magnetized plasma, the electromagnetic field equations form an essential part of the force balance, but the latter also has to include the force of a pressure gradient. In the present approach, as well as in the conventional theory on elementary particles, there is a problem how to treat the particle interior. There are at least two questions to be raised in this connection:

• Can a self-consistent force balance be established when the simple Ansatz of Equation (10) is replaced by other more general forms?

• Or has an extra force field to be added to the electromagnetic one, to establish a local balance? Could a gradient of the ZPE photon pressure serve this

purpose, as being represented by an energy density

These questions require more investigations, being outside the frame of the present considerations. Keeping these points in mind, the following sections will give a detailed example on a particle model and a related one of a neutron and proton configuration.

The particle configuration being studied here is based on a class of solutions with similar but varying profile shapes. Here a detailed illustration will be presented, as obtained from a special choice of the parameters

Earlier results [

• The interaction has a short range, because it takes place only when the spatially limited force fields overlap.

• The interaction is not only due to the electrostatic field generated by the intrinsic charges, but there is also a partly outbalancing effect by the magnetostatic part of the electromagnetic field as shown in Section 4.2.

• The spatially distributed forces due to the intrinsic charges can interact and perturb each other during a particle collision process. This complicates a rigorous treatment. In other words, there are two different types of impact processes, namely (a) those originating from divergent generating functions which lead to point-like long-range interaction of single net sources, and (b) those originating from convergent generating functions which lead to spatially distributed short- range interaction of multiple intrinsic sources.

• In this first investigation only the mutual effects of the unperturbed forces will be considered.

Due to conventional theory, the proton and neutron consists of three quarks which are coupled in a triangular configuration [

In a first approximation, we put all the three quark masses m equal to a third of the proton or neutron masses, i.e.

and Equation (15) results in

As compared to the elementary charge, the relative magnitude of the intrinsic charges of either sign becomes

where

Among the various but similar profile shapes represented by the generating function (10), we will choose the example given by the parameter values

The radial distribution of space charge density given by

The radial component

The distribution of the radial electromagnetic self-force of Equation (40) is further shown in

In the case of a long-range interaction between two particles of charge

In the analysis of the short-range interaction between two particles of the present model, the volumes of

Here

An illustration of special interest is given by

where

between two elementary charges separated by the distance

With

When applying Equations (45) and (46) to Region (3), the geometry corresponding to

Finally, in the case of Region (1), the corresponding deductions become somewhat uncertain, especially in its innermost parts which correspond to small distances between the particle centra. In any case, estimations show that the related maximum force

It should be observed that the self-force (9) and the particle interaction force

(45) are different. The former is due to the charge density (23) and the field component (29), whereas the latter arises from the density (23) of Particle (I) in combination with the reversed and displaced field component (29) of Particle (II). Even if a fully force-free state of the self-force can be attained, the particle interaction force would therefore remain to be non-zero.

The radial expansive force within the small inner core of the present model can be interpreted to exert a push on the large outer force-free shell. In its turn, this could produce an induced unbalance of the electrostatic and magnetostatic forces within the shell, to result in an integrated totally force-free state.

The distribution of the interactive force

Then there is a potential well related to Region (2), and inner and outer potential barriers to Regions (1) and (3). An approximated picture of the radial potential distribution can then be given as follows. The boundaries between these regions are defined by the zero points

as given by the average value

Here

Finally, the total height

The potential distribution

• At distances being somewhat larger than

• When

• A further decrease of

• When reaching the region below

• Due to this distribution there is a stable position of the particle interaction, near the bottom of the well.

• An impact energy of the magnitude of the well depth would be needed to release the binding between these quark particles. This is of the same magnitude as the binding energy of 8 MeV given for the neutron according to Bethe [

The main purpose of this investigation is to elaborate a model of quarks which form triangular neutron and proton configurations. The model has the following characteristic features:

• There are equal intrinsic electric charges of both polarities.

• These intrinsic charges are an order of magnitude larger than the elementary charge e.

• The electromagnetic field has both electric and magnetic components of which the latter outbalance part but not all of the former.

• The particle model which represents each of the quarks consists of an outer force-free shell defined by the spatial charge distribution, and an inner core within which there are self-forces due to a nonzero electromagnetic field in combination with an intrinsic charge density. The inner core occupies only a few percent of the total particle volume.

• The interaction which arises between two such particles is of short-range character. When gradually shrinking the distance between the particle centra, a corresponding potential distribution is traversed, in the form of one outer barrier, an intermediate well, and an inner barrier.

• The well is of particular interest, because it is associated with a large attractive force. The latter is nearly two orders of magnitude larger than the corresponding repulsive force between two elementary charges.

• The deduced depth of the well is about 7.7 MeV, being nearly of the same magnitude as the binding energy 8 MeV of the neutron.

• The obtained distribution of the barriers and the well makes a stable position of the particle distance possible, near the bottom of the well.

• The present triangular quark model of the neutron and the proton has an outer radius of about

The results of the present investigation could thus form a first step in a unification of the electromagnetic and the strong nuclear forces.

The author is indebted to MSc Yushan Zhou for a valuable work with the computations of the present analysis. The author also thanks MSc Kerstin Holm- ström for an excellent transcription of the manuscript of the present paper.

Lehnert, B. (2017) Intrinsic Particle Charges and the Strong Force. Journal of Modern Physics, 8, 1053- 1066. https://doi.org/10.4236/jmp.2017.87067