In this paper, we first show that the four masses of the Upsilon binding states—namely, Y(1S) through Y(4S)—composed of bottom quarks and anti-bottom quarks follow logarithmic spacing. The correlation coefficient R between the experimental values and the straight line is 0.99997, indicating an extremely good fit. When the three peaks—Y(5S), Y(6S), and Y(7S), considered higher Upsilon binding states, as indicated in the recent Belle experiment—are added and plotted on the line of logarithmic spacing, the correlation coefficient R between the straight line and the experimental values for these seven “binding state levels” is 0.9998. If this line is extended to higher masses, an eighth peak in the cross-section is expected at a mass of (11,119 ± 10) MeV. In other words, it is predicted that the peak in the cross-section created by Y(8S) will be found in the Belle experiment in the future. Next, we consider why meson binding states are represented by a line with logarithmic spacing. An example is the electric field generated by a charge on a two-dimensional plane; the electric field created by a charge placed on the plane is expressed as F∼1/r, and the potential energy is expressed as V∼log(r). Therefore, we solve the two-dimensional Schrödinger equation numerically under F∼1/r and compare the results with the three-dimensional solution. We find that differences appear in the energy levels of the J=0ηc meson. Specifically, it is shown that the mass of the ηc meson is closer to the value obtained by solving the two-dimensional Schrödinger equation. Based on this ηc meson series, we predict the existence of a new ηc meson with a mass of 3955 MeV.
Muraki et al. (Wed,) studied this question.