Abstract

In a previous paper, the present author proposed the infinite sub-layer quark model, in which the proton and the neutron are composed of an infinite number of point-like (structure-less) quarks ?_? and anti-quarks  at an infinite sub-layer level. The limit particle ?_? has all one-half quantum numbers of spin , isospin , third component of isospin , and fractional electric charge , where e is the electron charge. This fermion will behave as if it was lepton, since the baryon number vanishes at an infinite sub-layer level. Thus, this ultimate particle is considered as constituting the non-baryonic cold dark matter. A pair of an infinite number of ?_? and  quarks would be produced in the first moments after the Big Bang and form the nucleons and remain as the dark matter for all time, stable against decay. It was then shown that  is violated in the doublet of ?_? and  quarks to account for the asymmetry of the number of particles and antiparticles in the present universe. Furthermore, it was shown that the Higgs bosons are composed of ?_? and  dark matter particles. In this paper, we will show that the Cantor set is constructed from an infinite number of point-like quarks ?_? and anti-quarks which constitutes the nucleon and the dark matter. It is also shown that the color charges in quantum chromodynamics (QCD) of ?_? and  quarks vanish and these limiting particles of dark matter have no color force with the strong interaction and subject only to the weak interaction and gravity.

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