Unconstrained Hamiltonian formulation of low energy QCD

Abstract

Using a generalized polar decomposition of the gauge fields into gauge-rotation and gauge-invariant parts, which Abelianises the Non-Abelian Gauss-law constraints, an unconstrained Hamiltonian formulation of QCD can be achieved. The exact implementation of the Gauss laws reduces the colored spin-1 gluons and spin-1/2 quarks to unconstrained colorless spin-0, spin-1, spin-2 and spin-3 glueball fields and colorless Rarita-Schwinger fields. The obtained physical Hamiltonian naturally admits a systematic strong-coupling expansion in powers of λ=g^{-2/3}, equivalent to an expansion in the number of spatial derivatives. The leading-order term corresponds to non-interacting hybrid-glueballs, whose low-lying spectrum can be calculated with high accuracy by solving the Schrödinger-equation of the Dirac-Yang-Mills quantum mechanics of spatially constant fields (at the moment only for the 2-color case). The discrete glueball excitation spectrum shows a universal string-like behaviour with practically all excitation energy going in to the increase of the strengths of merely two fields, the "constant Abelian fields" corresponding to the zero-energy valleys of the chromomagnetic potential. Inclusion of the fermionic degrees of freedom significantly lowers the spectrum and allows for the study of the sigma meson. Higher-order terms in λlead to interactions between the hybrid-glueballs and can be taken into account systematically using perturbation theory, allowing for the study of IR-renormalisation and Lorentz invariance. The existence of the generalized polar decomposition used, the position of the zeros of the corresponding Jacobian (Gribov horizons), and the ranges of the physical variables can be investigated by solving a system of algebraic equations. It is exactly solvable for 1 spatial dimension and several numerical solutions can be found for 2 and 3 spatial dimensions.