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of Up and Down Quarks and Estimates of the the Outer Radii of the Small, Medium and Large Up and Down Quarks |
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This material develops an alternative to the conventional model of hadron structure. In this model quarks are spherical shells of mass. Nucleons and pi mesons are concentric shells of such quarks. This model provides an explanation of why the charged pi mesons composed of only two quarks have masses of only about 273 electron masses but the nucleons with three quarks have masses of almost 2000 electron masses.
The explanation is roughly that a nucleon has a spherical structure. Quarks are spherical shells of mass and a nucleon is three concentric spheres. A meson is two such concentric spherical shells. The quarks which make up a pi meson occupy a sphere of a radius that is about one half of the radius of a nucleon. They then occupy a volume which is about one eighth of the volume of the nucleon. If volume mass densities are the same then the meson would have one eighth the mass of a nucleon. Thus if the nucleon has a mass equal to two thousand electron masses then the meson would have a mass of 250=2000/8 electron masses. The full explanation, given below, is only a bit more complicated.
It is worth noting that the concentric shell model involves the spin pairing of the quarks in nucleons which contributes to a binding energy for their stability. The spin pairing of concentric structures is the ultimate in spin pairing.
Hadron is a term coined to cover nucleons (protons and neutrons) and mesons such as pi particles. The conventional theory of hadron structure has quarks being charged point particles. A charged point particle would require an infinite amount of energy to create. There is not enough energy in the entire Universe to create even one charged point particle.
Instead quarks can be spherical shells of charge and mass. Outside of their shells they have the same effect as if their charges and masses were concentrated at their centers. A nucleon or meson is made up of concentric quarkic spherical shells.
For details on this theory of concentric quarkic spheres see Quarkic Structures.
This means that quarks come in three radius sizes: small, medium and large. Conventional theory talks about there being three attributes for point particle quarks which it labels as color. This so-called color attribute could be radius size.
The radial distribution of the charges of nucleons has been determined experimentally; i.e.,
According to the quark theory of nucleonic structure a neutron is composed of two Down quarks and one Up quark. A proton on the other hand has two Up quarks and a Down quark. An Up quark has an electrostatic charge of +2/3 whereas a Down quark has a charge of −1/3.
There are only three possible radial arrangements of the quarks in a neutron: UDD, DUD and DDU, where the left repesents the center of the nucleon. The DUD arrangement violates the apparent rule for particle linkages due to spin pairing; i.e., that a particle links to no more than one particle of the same kind and no more than one of the opposite kind. The DDU arrangement would result in a positive magnetic moment for the neutron contrary to observation. Thus the arrangement must be UDD for the neutron. Likewise it must be DUU for the proton.
The above empirical charge distribution for a neutron is entirely consistent with the UDD for the concentric shell model. The distribution for a proton should have a radial range of negative charge. Experimental probing of protons with electrons at SLAC found that some electrons bounced back. This was interpreted as evidence of electrons impinging upon hard cores of quarks. It could also be interpreted as evidence of electrons impinging upon a range of negative charge.
The above distribution of charge in a neutron indicates that the positive Up quark is located between 0 and 0.25 fermi in radius. The two negatively charged Down quarks are located between radii of 0.25 fermi and 1.1133 ferm, the radius of a neutron.
This means a small Up quark occupies a volume of
The volume of a neutron is
Thus the volume occupied by the medium and large Down quarks is (5.7800−0.0654)=5.7146 f³.
Let σU and σD be the volume mass density of the Up and Down quark materials, respectively. The units for these densities are electron masses per cubic fermi.
The mass of a neutron is 1838.684 electron masses. Therefore
It is established elsewhere on the basis of magnetic moments that the scale of an Up quark is (3/4) the scale of the corresponding Down quark. That means that a proton should have a negatively charged small Down quark occucupying the space between its center and 0.3333 fermi. This volume occupied is
The volume of a proton is
Thus the volume occupied by the medium and large Up quarks is (2.4827−0.1551)=2.3276 f³.
The mass of a proton is 1836.1529 electron masses. Therefore
The conditions to be satisfied are
2.3276σU + 0.1551σD = 1836.1529
0.0654σU + 5.7146σD = 1838.684
The solutions for densities in units of electron masses per cubic fermi are
The mass of the small Down quark is then 312.9608*0.1551=48.5402 electron masses. The mass of the small Up quark is 768.1684*0.0654=50.2382 electron masses.
The masses of the medium and large quarks have to be estimated by a procedure that will be given later.
The positive pi meson is said to be composed of a medium Up quark and a small Down antiquark. Any Down antiquark is properly an anti-Down quark. It is their Downness which is contrary rather than their quarkness.
An anti-Down quark has the same volume and mass as a small Down quark. The mass of a small Down quark has a mass of 48.5402 electron masses. The mass of a positive pi meson is 273.1315 electron masses. Therefore the mass of a medium Up quark is 224.5973 electron masses. The volume occupied by a medium Up quark is then its mass divided by its density; i.e., (224.5973)/(768.1684) = 0.2924 f³
This number divided by (4/3)π gives 0.06980 which is the difference in the cube of the outer radius of the medium Up quark and the cube of its inner radius. Its inner radius is the same as the outer radius of the small Down quark; i.e., 0.3333 fermi. Its cube is 0.03780. Adding this to 0.06980 give the cube of the radius of the medium Up quark; i.e., 0.1076. The cube root of this number, 0.4756 fermi, is the outer radius of the medium Up quark.
The negative pi meson is composed of a medium Down quark and a small anti-Up quark. A small anti-Up quark has the same volume and mass as a small Up quark. A small Up quark has a mass of 50.2382 electron masses. The mass of a negative pi meson is 273.1315 electron masses. Therefore the mass of a medium Down quark is 223.0033 electron masses. This divided by the density of Down quark material gives the volume of medium Down quark as 0.7126 cubic fermi.
This number divided by (4/3)π gives 0.1701 which is the difference in the cube of the outer radius of the medium Down quark and the cube of its inner radius. Its inner radius is the same as the outer radius of the small Up quark; i.e., 0.25 fermi. Its cube is 0.0156. Adding this to 0.1701 give the cube of the radius of the medium Down quark; i.e., 0.1857. The cube root of this number, 0.5705 fermi, is an estimate of the outer radius of the medium Down quark.
The above analysis made use of the result based upon the magnetic moments of the nucleon that the size of an Up quark should be three quarters of the size of the corresponding Down quark. The ratio of the radii of the medium quarks is
This is not 0.75 but it is reasonably close enough to represent a notable confirmation of the theory of the concentric shell model of hadrons. To satisfy the three quarters ratio the outer radius of the medium Down quark would have to be (4/3)(0.4756)=(0.6341) fermi.
The simple way to determine the mass of a large quark is the mass of a nucleon less the mass of a pi meson. For the mass of the large Down that is (1838.6840−273.1315)=1565.5525 electron masses. For the large Up quark it is (1836.1529−273.1315)=1563.0214 electron masses.
There is another method for the calculation of the masses of the large quarks. The volume of the large Down quark is that between the radius of the medium Down quark of 0.5705 fermi and the radius of a neutron of 1.1133 fermi. That volume is
Its mass is then
The volume of the large Up quark is that between the radius of the medium Up quark of 0.4756 fermi and the radius of a proton of 0.84 fermi. That volume is
Its mass is then
In 1950 a particle was found that was thought to be a neutral pi meson. Its mass was estimated by deduction to be 264 electron masses in contrast to the 273 electron masses. There was likely to be a much larger margin of uncertainty for the 264 figure tha the 273 figure which came from direct measurement of the curvature of the trajectories of the charged particles in a magnetic field.
The neutral pi meson was conjectured to be an Up quark combined with an anti-Up quark or a Down quark combined with an anti-Down quark. Such combinations suggest particle and anti partcle annhilation. The half life of the neutral pi meson is only 8.4x10−17 second compared to 2.6x10−8 second for the charged meson.
The difference between a neutral pi meson and a positive pi meson is that one carries an anti-Up quark where the other carries an anti-Down quark. According to the previous estimates that amounts to about two electron masses if the anti-quarks are small.
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Quark Mass Volume Densities (electron masses per cubic fermi) | ||
|---|---|---|
| Quark | Up | Down |
| Density | 768.1684 | 312.9608 |
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Outer Radii of Up and Down Quarks (fermi) |
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|---|---|---|
| Size | Up | Down |
| small | 0.25 | 0.3333 |
| medium | 0.4756 | 0.5705 |
| large | 0.84 | 1.1133 |
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Masses of Up and Down Quarks (electron mass) |
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|---|---|---|
| Size | Up | Down |
| small | 50.2382 | 48.5402 |
| medium | 224.5973 | 223.0033 |
| large | 1565.5525 | 1563.0214 |
The mass of a proton should be
That of a neutron should be
Thus the masses of the nucleons can be represented as the sums of their quarkic components.
The discreptancy beween the masses of the pi meson and those of the nucleons does not require the existence of hypothetical gluons flitting in and out of existence within the hadrons.
(To be continued.)
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