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Estimates of the Mass Densities
of Up and Down Quarks and
Estimates of the the Outer Radii
of the Small, Medium and Large
Up and Down Quarks

Background

This material developes an alternative to the conventional model of hadron structure. In this model nucleons and pi mesons are concentric shells of quarks. It provides an explanation of why the charged pi mesons composed of only two quarks have masses of only 273 electron masses but the nucleons with three quarks have masses at or above 1836 electron masses.

Hadron is a term coined to cover nucleons (protons and neutrons} and mesons such as pi particles. The conventional theory 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. Outside of their shells they have the same effect as if their charges concentrated at their centers. A nucleon or meson is made up of concentric quarkic spheres.

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 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.

Volumes Occupied
by Quarks in a Neutron

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 neutroni.

This means a small Up quark occupies a volume of

(4/3)π(0.25)² = 0.0654 cubic fermi (f³)

The volume of a neutron is

(4/3)π(1.1133)² = 5.7800 f³

Thus the volume occupied by the medium and large Down quarks is (5.7800−0.0654)=5.7146 f³.

Mass Densities

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

0.0654σU + 5.7146σD = 1838.684

Volumes Occupied
by Quarks in a Proton

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

(4/3)π(0.3333)² = 0.1551 f³

The volume of a proton is

(4/3)π(0.84)² = 2.4827 f³

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

2.3276σU + 0.1551σD = 1836.1529

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

σU = 726.5599
σD = 312.9617

Masses of the Small Quarks

The mass of the small Down quark is then 312.9617*0.1551=48.5404 electron masses. The mass of the small Up quark is 726.5599*0.0654=47.5170 electron masses.

Masses of the Medium Quarks

The masses of the medium and large quarks have to be estimated by a procedure that will be given later.

The Positive Pi Meson

The positive pi meson is composed of a medium Up quark and a small Down antiquark. A small Down antiquark has the same volume and mass as a small Down quark. The mass of a small Down quark has a mass of 48.5404 electron masses. The mass of a positive pi meson is 273 electron masses. Therefore the mass of a medium Up quark is 224.4506 electron masses. The volume occupied by a medium Up quark is then its mass divided by the density; i.e.,

224.4506/726.5599 = 0.3089 f³

This number divided by (4/3)π gives 0.0738 which is the difference in the cube of the outer radius of the medium Up quarck 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 0.0738 give the cube of the radius of the medium Up quark; i.e., 0.1108. The cube root of this number, 0.4804 fermi, the outer radius of the medium Up quark.

The Negative Pi Meson

The negative pi meson is composed of a medium Down quark and a small Up antiquark. A small Up antiquark has the same volume and mass as a small Up quark. A small Up quark has a mass of 47.5170 electron masses. The mass of a negative pi meson is 273 electron masses. Therefore the mass of a medium Down quark is 225.483 electron masses. This divided by the density of Down quark material gives the volume of medium Down quark as 0.7205 cubic fermi.

This number divided by (4/3)π gives 0.1720 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 0.1720 give the cube of the radius of the medium Down quark; i.e., 0.1856. The cube root of this number, 0.5725 fermi, the outer radius of the medium Down quark.

Masses of the Large Quarks

The volume of the large Down quark is that between the radius of the medium Down quark of 0.5725 fermi and the radius of a neutron of 1.1133 fermi. That volume is

(4/3)π[(1.1133)³ − (0.5725)³] = 4.9940 f³

Its mass is then

4.9940*312.9617=1562.9307 electron masses.

The volume of the large Up quark is that between the radius of the medium Up quark of 0.4804 fermi and the radiu of a proton of 0.84 fermi. That volume is

(4/3)π[(0.84)³ − (0.4804)³] = 2.0183 f³

Its mass is then

2.0183*726.5599 = 1466.4209 electron masses.

The Neutral Pi Meson

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 one electron mass if the anti-quarks are small and about thirty electron masses if they are medium.

Conclusions

Radii of Up and Down Quarks
(fermi}
SizeUpDown
small0.250.3333
medium0.48040.5725
large0.841.1133

Masses of Up and Down Quarks
(electron mass)
SizeUpDown
small47.517048.5404
medium224.4506225.483
large1466.42091562.9307

(To be continued.)


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