Low-lying spectrum of the Y-string three-quark potential using hyper-spherical coordinates

We calculate the energies of three-quark states with definite permutation symmetry (i.e. of SU(6) multiplets) in the N=0, 1, 2 shells, confined by the Y-string three-quark potential. The exact Y-string potential consists of one term, the so-called three-string term, and three angle-dependent two-string terms. Due to this technical complication we treat the problem at three increasingly accurate levels of approximation: (1) the (approximate) three-string potential expanded to first order in trigonometric functions of hyper-spherical angles; (2) the (approximate) three-string potential to all orders in the power expansion in hyper-spherical harmonics, but without taking into account the transition(s) to two-string potentials; (3) the exact minimal-length string potential to all orders in a power expansion in the hyper-spherical harmonics, and taking into account the transition(s) to two-string potentials. We show the general trend of improvement of these approximations: the exact non-perturbative corrections to the total energy are of the order of one per cent, as compared with approximation (2), yet the exact energy differences between the [20,1+],[70,2+],[56,2+],[70,0+]-plets are shifted to 2:2:0.9, from the Bowler and Tynemouth separation rule 2:2:1, which is obeyed by approximation (2) at the one per cent level. The precise value of the energy separation of the first radial excitation (“Roper”) [56′,0+]-plet from the [70,1−]-plet depends on the approximation, but does not become negative, i.e. the “Roper” remains heavier than the odd-parity [70,1−]-plet in all of our approximations.