|San José State University|
& Tornado Alley
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 […].
Weinberg's interpretation of the experimental results is not the only one plausible. But it was the one generally accepted by the physics community. But that left particle physics 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. It would have to be consistent with a potential energy function something like one of the following.
For the first case the force between the quarks switches from attraction to repulsion at limits and then declines with the reciprocal of separation distance squared. For the second case the force between the quarks continues to be an attraction beyond the limits but declines thereafter with the reciprocal of separation distance squared.
The case which applies would depend upon whether the quarks involved are of the same type or different types.
The conventional theory of quarks also involves their asymptotic freedom. But that asymptotic freedom leaves no mechanism for determining the arrangement and separation distances of the three quarks in each nucleon.
The conventional model has quarks as charged point particles. There is no getting around the fact that there is not enough energy in the entire Universe, even if all of its masses were converted to energy, to create even one charged point particle. It would have to have infinite energy.
Furthermore, the Larmor Proposition that accelerating or decelerating charges radiate electromagnetic waves applies to point particles. That means point quarks would be radiating energy. Since they would radiating energy from an infinite source they effectively would be continuously creating energy.
The conventional theory of the quarkic structure of nucleons appears to be a desparate, contrived attempt for explaining a single fact: That no evidence of a singleton quark has been found. Once the community of physicists accepted it as an adequate explanation of that fact it was presented as being physically true despite the only logical implication being that it might be physically true. So, despite this logical failing, generation after generation of students of physics have been told that this desparately contrived explanation for the nonexistence of evidence for singleton quarks is as physically true as the existence of protons and neutrons.
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, including asymptotic freedom.
But first some technical information.
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 quite likely is 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 nuclear strong force. There are some serious problems with the so-called nuclear strong force and there may be an entirely different force between nucleons.
The theorem can be extended to cover spherical volumes; i.e., balls of charge. Likewise it also applies to spherical shells of finite thickness.
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
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 are the graphs of the distance dependence of the force.
where the first graph is for repelling like particles and the second for attracting unlike particles.
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 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 between them.
Consider how ineffective crashing together structures consisting of concentric spheres would be at separating any of those spheres. And even if an outer sphere were destroyed the result would not be a single sphere but instead a structure with one less concentric spheres. Thus if the outer Down quark of a neutron were destroyed the result would not be an isolated quark but something in the nature of a meson. So it should be no surprise that an isolated quark has not been found.
Nucleons have also been subjected to probing by energetic electrons. Such probing would be like firing bullets at clouds. They would be as ineffective as trying to separate a cloud within another cloud.
Suppose the force formula is
where f(s) is a declining function of separation distance. Such a force drops off faster with distance than does inverse distance squared.
If such a charged sphere is located within another such 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.
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.
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 standard 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 as in the conventional theory. . It is however compatible with nucleons being concentric spheres of charges. Furthermore there is no mechanism in the conventional model of three point particle quarks for determining definite spherical boundaries for the nucleons.
The electrostatic charge of an Up quark is +2/3 and that of a Down quark is −1/3. From the above diagram 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 then 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 of charge distribution. .
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.
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.
So far the model of nucleons as three concentric spherical shells of charges is just a plausible alternative to the conventional model involving quarks as point particles contained in a bag. Analysis elsewhere derives the result that the ratio of the radius of an Up quark to the radius of the corresponding Down quark is equal to 3/4.
The outer radius of the large Up quark is equal to the radius of a proton. Likewise the outer radius of the large Down quark is equal to the radius of a neutron. Thus according to the model the ratio of the radius of a proton to that of a neutron should be 3/4. The radius of a neutron is 1.1133 fermi (10−15 m). The accepted radius of the proton had been 0.86 fermi, but a new experimental method puts it at 0.84 fermi. The ratio of 0.84 to 1.1133 is 0.7545. That would be solid confirmation of the concentric shells model of nucleons. Even the ratio of 0.86 to 1.1133 of 0.7725 is good confirmation of the model.
With this confirmation of the model we know the radius of the medium Up quark is three quarters of the radius of the medium Down quark. It is likewise for the small quarks. It happens that from the radial distribution of charge for the neutron we have an estimate of the radius of the small Up quark.
It appears that the outer radius of the small Up quark is about 0.3 fermi. That would make outer radius of the small Down quark 0.4 fermi. The volume contained in a sphere of radius 0.3 fermi is 0.1131 fermi³. If it contais 2/3 of a unit electrostatic charge then the charge density is 5.8946 e/f³.
For a small Down quark with its −1/3 charge contained within a sphere of radius 0.4 fermi the charge density is −3.906 e/f³.
There must be a volume of 0.1131 fermi³ in the spherical shell of the large Up quark. With an outer radius of 0.84 fermi the inner radius must be 0.778 fermi to contain that volume.
In the large Down quark there has to be a volume of 0.2681fermi³ between its outer radius of 1.1133 fermi and its inner radius. Its inner radius must be 1.0958 fermi to achieve that. Thus the shell thicknesses for the large quarks are relatively small.
It is worth noting that a neutron having a radial structure of an Up quark at the center and two outer Down quarks satisfies the exclusivity rule of a particle being linked with no more than one of its like particles and no more than one of its unlike particles.
There is substantial evidence that the interaction of nucleons can be explained by their having nucleonic charges with a neutron having a nucleonic charge which is opposite in sign and smaller in magnitude than that of a proton. See Nucleus for details. If the nucleonic charge of the proton is taken to be +1 then that of the neutron is −2/3.
Cosider what the nucleonic charges of the Up and Down quarks have to be. Let νU and νD be their nucleonic charges. Then
First multiply the first equation by 2/3 to obtain
Then add this equation to the second of the above two equations to obtain
This equation and the first equation of the two equations above determine that
Through their nucleonic charges two like quarks are repelled from each other and two unlike quarks are attracted to each other. For point particle quarks this dictates linear U-D-U and D-U-D for the two nucleons which are incompatible with asymptotic freedom. Hollow concentric spherical shell quarks would guarantee asymptotic freedom.
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 effect of a spherical distribution of charge is, outside of the sphere, the same as if the charge were concentrated at the center of the sphere. This explains any evidence of quarks being point particles and the appearance of asymptotic freedom.
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 if the two separate. For a quark enclosed within another of the same type the force goes abruptly from zero to a large repulsion if the two separate.
When the spherical shells of charges are concentric their centers are close together and the shells experience no force. This explains asymptotic freedom.
The three concentric spherical shells model of nucleons implies the ratio of the proton radius to the neutron radius should be 3/4. To a close approximation that is the case, thus providing substantial confirmation of the model.
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.
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