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A Sensible Model for the Confinement
and Asymptotic Freedom of Quarks


When higher powered particle accelerators were not able to find evidence of singleton quarks it was conjectured that the force of attraction between quarks was such that it became stronger the greater the distance between them. The standard form of this conjecture became known as the bag model of quark confinement. An adjunct to this conjecture was the notion that the force between two quarks goes to zero as their separation distance goes to zero. This notion became known as asymptotic freedom. These conjectures have become accepted as fact.

Steven Weinberg in his article, "Why the Renormalization Group is a Good Thing," says,

[T]his (asymptotic freedom) explains an experimental fact which had been observed in a famous experiment on deep inelastic electron-proton scattering done by an MIT-SLAC collaboration in 1968. This was that at very high momentum transfer, in other words, at very short distances, the strong interactions seem to turn off […].

Particle physics was then left with a container bag of a mysterious nature and a force whose distance dependence is like no other known force. This force which supposedly increases with separation must after some distance decrease with distance because otherwise the effect of one quark would be enough to move stars and perhaps entire galaxies. Perhaps the proponents of such a force between quarks think that there is an exclusivity involved similar to the spin pairing of nucleons. But that cannot be because the force between nucleons is supposed to be the result of the forces between the quarks of one nucleon and the quarks of the other nucleon. The interaction force between nucleons declines with separation distance. There are thus major perplexities concerning the conjectured theory of quark confinement.

The purpose of the material presented here is show that another model, which is not dependent on mysterious components, explains everything that the Bag Model does.

But first some technical information.

The Wonderful Theorem Concerning Forces
with an Inverse Distance Squared Dependence

If a generic charge of Q is uniformly distributed over a sphere of radius R the effect of it at points of distances from the center of the sphere greater than R is the same as if the charge were concentrated at the center of the sphere. For points within the sphere the force is zero. This might be the source of asymptotic freedom.

For gravitation the generic charge is mass; for the electrostatic force it is electrical charge. There may be another type of charge associated with the so-called strong (nucleonic) force.

The theorem can be extended to cover spherical volumes; i.e., balls of charge. Likewise it also applies to spherical shells of finite thickness.

Two Concentric Spheres of Charge

Consider two spheres of radii r1 and r2 with r1<r2. Let the amounts of their charges be q1 and q2, respectively.

When the separation distance s of their centers is greater than or equal to (r1+r2) the force between them is

F = gq1q2/s²

where g is a constant. If q1 and q2 are of opposite signs then the force between them is an attraction otherwise it is a repulsion.

If s≤(r2−r1) then F=0.

For the separation distances greater than (r2−r1) and less than (r1+r2) the force is between 0 and gq1q2/(r1+r2)². Here the graph of the distance dependence of the force.

Thus for two spheres of either the same or opposite charges there is asymptotic freedom and containment of the smaller sphere within the larger.

The conventional notion of quarks identifies them with the centers of the charges and takes the charge distributions as things generated by these centers whereas the reality is that the charge distributions are the quarks and their centers are just incidental aspects of them.

The Force Required to Separate Quarks

The literature on quark confinement cites an estimate of the force required to separate quarks as being 15 tons. This supposedly puts the force involved in quark confinement into an entirely different class than the electrostatic force. All that needs to said on this matter is that the force required to separate two electrostatically charged particles goes to infinity as the separation distance goes to zero. But the difficulty of separating quarks may have to do with their structure rather than the strength of the attraction between them. Consider how ineffective firing bullets at two adjacent clouds would be in separating them. And how much less effective such probes would be at separating a cloud within another cloud.

Extension to More
General Types of Forces

Suppose the force formula is

F = gq1q2f(s)/s²

where f(s) is a declining function of separation distance. Such a force drops off faster with distance than does inverse distance squared.

If a charged sphere is located within a charged sphere it is repelled from then the system has a stable equilibrium position in which the centers of the two spheres coincide.

Such a formula would arise if the particles carrying the force decayed over time and hence with distance. This would lead to f(s) being a negative exponential function.

The Appearance of Two Concentric Spheres
of Charge to Probes by Other Particles

According to the theorem cited above the deflection of probe particles, so long as they do not penetrate the outer sphere, is the same as if the charges were concentrated at the centers of the spheres. In other words the deflections would be the same as if the spherical charges were point particles.

Breaking apart concentric spheres of quarks would be a much more difficult task that to do so for separated quarks linked only by an interaction force. Any force applied by a probe on the outer quark would also act on the inner quarks. Thus there would be no stress on the structure of concentric spheres.

The Scale Structures of the
Quarks of Protons and Neutrons

A proton consists of two up quarks and one down quark. Thus for nucleons three spheres of three different radii are needed. Since two of the quarks are of the same type that might seem like a hard prescription to fulfill. However the conventional theory argues that there are three kinds of each type of quark. It denotes the kinds by color although these kinds have nothing to do with visual color. Furthermore the conventional theory holds that any baryon contains one quark of each color and so it is color neutral, white. Stripped of the color terminology the conventional theory maintains that quarks can have one of three different attributes and any baryon contains one of each of the three attributes. Those attributes could be the radii of spherical charges. That there must be in a nucleon or other baryon quarks of three different radii suits the model presented here perfectly.

A neutron consists of two down quarks and one up quark. So the same explanation applies as for a proton or any other baryon.

Here is the conventional representation of the radial distributions of electrostatic charge for a proton and a neutron.

This spherically symmetric distribution of charge is incompatible with any planar triangular arrangement of quarks. It is compatible with nucleons being concentric spheres of charges.

The electrostatic charge of an up quark is +2/3 and that of a down quark is −1/3. The shell of positive charge that extends from 0 to a radius of 0.2 fermi would be the up quark of the neutron. The shell that extends from radius of 0.2 fermi to about 1.1 fermi is composed of the two down quark of the neutron. Possibly a proton has an inner shell of negative charge that was missed by the experimental measurement.

There could also be gaps of no charge between the quarks, including one within the innermost quark, such as is shown below.

Here is a visual depiction of nucleon structure.

Model of a Nucleon

The Scale Structure of Mesons

A positive pion consists of an up quark and an anti-down quark. In terms of the quark color terminology the color of the anti-quark is the anti-color of the color of the quark. The anti-color is basically the color complementary to the color so over all the meson in white (color neutral). If the color attribute corresponds to quark radius then the radius of the anti-quark just has to be different from the radius of the quark.

Here is a visual depiction of meson structure.

Model of a Meson


Asymptotic freedom for quarks can be explained in terms quarks being spherical shells of types of charges and some of these shells being enclosed within others. This concentric shell structure provides quark containment for the system.

The attributes of quarks which have been labeled color may be the radii of spherical quarks. Baryons are required to have quarks of each of the different colors. This could mean that baryons like protons and neutrons have to have quarks of three different radii. Mesons are required to have quarks of two different colors. This could mean that mesons need quarks of two different radii. Thus the structures of hadrons do not require a mysterious bag enclosing the quarks or a force which implausibly increases continuously in magnitude with the separation distances of the quarks. Instead the force on a quark enclosed within another quark of a different type goes abruptly from zero to a large attraction.


Kenneth Johnson, "The Bag Model of Quark Confinement," Scientific American, (July, 1979), pp. 112-121.

Steven Weinberg, "Why the Renormalization Group is a Good Thing," Proceedings, Asymptotic Realms Of Physics Cambridge, 1981, pp. 1-19.

F.J. Ynduráin, Quantum Chromodynamics, Springer-Verlag, New York, 1983.

Dedicated to Betty
My beautiful
Who lives Life with
Sensible compassion
And admirable Style.

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