San José State University

applet-magic.com
Thayer Watkins
Silicon Valley,
Tornado Alley
& the Gateway
to the Rockies
USA

The Masses of Quarks in
the Concentric Spherical Shells
Model of Hadrons

Background

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 fermi.

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 1838.684 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 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

An estimate of the outer radius of the medium Down quark is 0.6092 fermi. It occupies the volume between 0.25 fermi and 0.6092 fermi. Its volume is then

(4/3)π[(0.6092)³ − (0.25)³] = 0.6731

Hence its mass is 0.6731*312.9617=210.6523 electron masses.

The mass of a negative pi meson should be

210.6523 + 47.5170 = 258.1693 electron masses.

An estimate of the outer radius of the medium Up quark is 0.45825 fermi. It occupies the volume between 0.25 fermi and 0.6092 fermi. Its volume is then

(4/3)π[(0.45825)³ − (0.3333)³] = 0.2480

Hence its mass is 0.2480*726.5599=180.1755 electron masses.

The mass of a positive pi meson should be

180.1755 + 48.5404 = 228.7159 electron masses.

Masses of the Large Quarks

The volume of the large Down quark is that between the radius of the medium Down quark of 0.6092 fermi and the radium of a neutron of 1.1133 fermi. that volume is

(4/3)π[(1.1133)³ − (0.6092)³] = 4.8330 f³

Its mass is then

4.8330*312.9617=1512.5314 electron masses.

The volume of the large Up quark is that between the radius of the medium Up quark of 0.45825 fermi and the radiu of a proton of 0.84 fermi. That volume is

(4/3)π[(0.84)³ − (0.45825)³] = 2.0796 f³

Its mass is then

2.0796*726.5599 = 1510.9653 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

The model of nucleons and pi mesons being concentric shells quarks provides and explanation of why the pi meson 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.

(To be continued.)


HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins