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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 develops 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 about 273 electron masses but the nucleons with three quarks have masses of almost 2000 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 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; i.e., that a particle link 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.

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)³ = 0.0654 cubic fermi (f³)

The volume of a neutron is

(4/3)π(1.1133)³ = 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)³ = 0.1551 f³

The volume of a proton is

(4/3)π(0.84)³ = 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 = 768.1684
σD = 312.9608

Masses of the Small Quarks

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.

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 48.5402 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 the 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 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 0.06980 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

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 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

(0.4756)/(0.5705) = 0.8337

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 Masses of the Large Quarks

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

(4/3)π[(1.1133)³ − (0.5725)³] = 5.0022 f³

Its mass is then

5.0022*312.9608=1565.4886 electron masses.

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

(4/3)π[(0.84)³ − (0.4756)³] = 2.0321 f³

Its mass is then

2.0321*768.1684 = 1560.0986 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 two electron masses if the anti-quarks are small.

Conclusions

Quark Mass Volume Densities
(electron masses per cubic fermi)
QuarkUpDown
Density768.1684312.9608

Radii of Up and Down Quarks
(fermi}
SizeUpDown
small0.250.3333
medium0.47560.5705
large0.841.1133

Masses of Up and Down Quarks
(electron mass)
SizeUpDown
small50.238248.5402
medium224.5973 223.0033
large1565.55251563.0214

The mass of a proton should be

48.5402 + 224.5973 + 1565.5525 = 1836.1529 electron masses

That of a neutron should be

50.2382 + 223.0033 + 1462.6670 = 1838.6840 electron masses.

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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