High Spin Baryons in Quantum Mechanical Chromodynamics

A framework of quantum mechanical chromodynamics (QMCD) is developed with the aim to place the description of the nucleon on a comparable footing with Schroedinger's quantum mechanical treatment of the hydrogen atom. Such indeed turns out to be possible upon replacing the (e--p) by a (q-qq) system, on the one hand, and the Coulomb potential by the recently reported by us exactly solvable trigonometric extension of the Cornell (TEC) potential, on the other. The TEC potential translates the inverse distance potential in ordinary flat space to a space of constant positive curvature, the 3D hypersphere, a reason for which both potentials have the SO(4) and SO(2, 1) symmetries in common. In effect, the nucleon spectrum, inclusive its Δ branch, acquire the degeneracy patterns of the electron excitations with spin in 1H without copying them, however. There are two essential differences between the N(Δ) and H atom spectra. The first concerns the parity of the states which can be unnatural for the N and Δ excitations due to compositeness of the diquark, the second refers to the level splittings in the baryon spectra which contain besides the Balmer term also its inverse of opposite sign. Our scheme reproduces the complete number of states (except the hybrid Δ(1600)), predicts a total of 33 new resonances, and explains the splittings of the N and Δ levels containing high-spin resonances. It also describes accurately the proton electric charge form factor. We here calculate the potential in momentum space (instantaneous effective gluon propagator) as a Fourier transform of the TEC potential and show that the concept of curvature allows to avoid the integral divergences suffered by schemes based on power potentials. We find a propagator that is finite at origin, likely to produce confinement. The advocated new potential picture allows for deconfinement too as effect of space flattening in the limit of infinite radius of the 3D hypersphere. The potential's SO(4)/SO(2, 1) symmetries reflect AdS5/CFT correspondence.