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Seemingly Relatively Low
Mass of the Pi Mesons
Compared to the Mass
of a Proton
In 1934 the Japanese physicist Hideki Yukawa published an article in which he argued that there should exist an elementary particle with a mass of about 200 timeselectron. Shortly thereafter a particle with a mass of 207 times that of an electron was found in the debris of cosmic rays interaction with Earth's atmosphere. This made Yukawa's prediction seem to be amazingly accurate. But that particle did not have the right properties and later was deemed to be a heavy electron subsequently labeled a muon.
Still later (1947) a particle of mass 270 times that of an electron was found. There were ones of positive charge and ones of negative charge. They were callled the positive and negative pi mesons. Later their mass was refined to 273 times the mass of an electron. Later still a neutral pi meson was found of a slighly different mass, 264 times that of an electron, from that of the charged pi mesons.
When the quark theory of nucleonic structure was proposed by Murray Gell-Mann in 1964 the pi mesons were included as being made up of a quark and an antiquark. A positive pi meson was conjectured to be composed of an Up quark and an anti-Down quark. An Up An Up-quark has an electrostatic charge of +2/3 and a Down quark of −1/3. An anti-Down quark has a charge of +1/3=−(−1/3) so the charge of a positive pi meson is 1=2/3+1/3.
The problem is the mass of a pi meson with two quarks is only 273 elecron masses whereas a proton with three quarks has a mass 1836 electron masses.
The conventional explanation of this discreptancy in particle mass is that only a small proportion, say 23%, of a proton's mass is derived from the mass of its constituent quarks. The rest supposedly comes from hypothetical virtual particles called gluons that flit in and out of existence in the proton.
The conventional theory 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.,
From the charge distribution for the neutron we find the radius of the small Up quark is 0.25 fermi. The radius of the large Up quark is the same as the radius of a proton; i.e., 0.84 fermi. A resonable assumption is that the radius of the medium Up is a fraction ρ of the radius of the large Up quark and the radius of the small quark is the same fraction of the medium quark. Thus
This makes the radius of the medium quark the geometric mean of the radii of radii of the larger and smaller quarks.
The volume contained within the small and medium quarks is proportional to the cube of the radius of the medium Up quark. The volume relative to the volume of the proton is proportional to the cube of the radius of the medium quark relative to the radius of the large quark; i.e., 0.1624. If there is uniform mass volume density this figure mutiplied times the mass of the proton should give the mass of a pi meson
This only about 9 percent larger than the measured mass of a pi meson of 273 electron masses.
It is found elsewhere that the scale of a Down quark is (4/3) the scale of the corresponding Up quark. This makes the radius of the small Down quark 0.333 fermi. The radius of the large Down quark is the same as the radius of the neutron; i.e., 1.1133 fermi. Then
The figure of 0.5472 cubed gives the relative volume within the radius of the medium Down quark; i.e., 0.1636. Thus the mass of a pi meson made up of a Down quark and its anti-quark is given by
This is 10 percent higher than the measured value of 273.
The low value of the mass of the pi meson is explanable in terms of the volume of the small and medium quarks relative to the volume of the large quarks. There is no need to rely upon gluons to explain the relative difference in the masses of nucleons and mesons.
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