O(6) algebraic theory of three nonrelativistic quarks bound by spin-independent interactions

Abstract

We apply the newly developed theory of permutation-symmetric O(6) hyperspherical harmonics to the quantum-mechanical problem of three non-relativistic quarks confined by a spin-independent 3-quark potential. We use our previously derived results to reduce the three-body Schrödinger equation to a set of coupled ordinary differential equations in the hyper-radius R with coupling coefficients expressed entirely in terms of (i) a few interaction-dependent O(6) expansion coefficients and (ii) O(6) hyperspherical harmonics matrix elements, that have been evaluated in our previous paper. This system of equations allows a solution to the eigenvalue problem with homogeneous 3-quark potentials, which class includes a number of standard Ansätze for the confining potentials, such as the Y-and ∆-string ones. We present analytic formulae for the K = 2, 3, 4, 5 shell states' eigen-energies in homogeneous three-body potentials, which formulae we then apply to the Y-and ∆-string, as well as the logarithmic confining potentials. We also present numerical results for power-law pair-wise potentials with the exponent ranging between -1 and +2. In the process we resolve the 25 year-old Taxil & Richard vs. Bowler et al. controversy regarding the ordering of states in the K = 3 shell, in favor of the former. Finally, we show the first clear difference between the spectra of ∆-and Y-string potentials, which appears in K ≥ 3 shells. Our results are generally valid, not just for confining potentials, but also for many momentum-independent permutation-symmetric homogenous potentials, that need not be pairwise sums of two-body terms. The potentials that can be treated in this way must be square-integrable under the O(6) hyperangular integral, however, which class does not include the Dirac δ-function.

Hyperspherical variables analysis of lattice QCD three-quark potentials: skewed Y-string as the mechanism of confinement?

“…1. in baryon spectroscopy the three-quark force leaves a clear signature in the second odd-parity, and higher shells of baryon resonances, and even there only in a few select states [25][26][27]. However, these three-quark force effects can easily be confused with relativistic effects.…”