BCS solutions and effective quark energies of the QCD Hamiltonian in the Coulomb gauge

Abstract

The search of theoretical approximations adequate to the description of the non-perturbative regime of QCD, that is the low-energy portion of the hadron spectrum, requires the adoption of notions more frequently applied to other, more conventional, quantum many body systems, like the atomic nucleus, solid state systems, etc. The identification and the use of effective degrees of freedom is one of the notions whose limitations we shall explore in this work. To start with the choice of the basis to be adopted to represent effective fermionic degrees of freedom is a crucial first step towards the goal of constructing an effective Hamiltonian from which the low energy portion of the hadronic spectrum may emerge as its solution. Here, we propose the use of the harmonic oscillator basis and emphasize its advantages compared to the basis of free momentum states. The starting Hamiltonian is the effective Coulomb plus linear potential, which we have used in previous calculations. We proceed by pre-diagonalizing the Hamiltonian to build a single particle spectrum, namely: the spectrum of effective quark degrees of freedom, starting from arbitrary chosen quark masses, which are later on re-normalized. Then, quark-pair correlations are described in the context of the well-known Bogoliubov transformations. The resulting quasiparticle states are then used to construct meson-like states. The dependence of the results upon the parameters which enter in the calculations is explored in detail.