Low-energy spectrum of SU(3) Yang-Mills Quantum Mechanics

Abstract

The SU(3) Yang-Mills Quantum Mechanics of spatially constant gluon fields is considered in the unconstrained Hamiltonian approach using the "flux-tube gauge". The Faddeev-Popov operator, its determinant and inverse, are rather simple, but show a highly non-trivial periodic structure of six Gribov-horizons separating six Weyl-chambers. The low-energy eigensystem of the obtained physical Hamiltonian can be calculated (in principle with arbitrary high precision) using the orthonormal basis of eigenstates of the corresponding harmonic oscillator problem with the same non-trivial Jacobian only replacing the chromomagnetic potential by the 16-dimensional harmonic oscillator potential. This turns out to be integrable and its eigenstates be made out of orthogonal polynomials of the 45 components of eight irreducible symmetric tensors. The calculations in this work have been carried out in all sectors $J^{PC}$ up to spin $J=11$, and up to polynomial order $10$ for even and $11$ for odd parity. The low-energy eigensystem of the physical Hamiltonian of $SU(3)$ Yang-Mills Quantum Mechanics is found to converge nicely when truncating at higher and higher polynomial order (equivalent to increasing the resolution in functional space). Our results are in good agreement with the results of Weisz and Ziemann (1986) using the constrained Hamiltonian approach. We find excellent agreement in the $0^{++}$ and $2^{++}$ sectors, much more accurate values in other sectors considered by them, e.g. in the $1^{--}$ and $3^{--}$ sectors , and quite accurate "new results" for the sectors not considered by them, e.g. $2^{--}, 4^{--}, 5^{--}, 3^{++}$.