Mesonic states and vacuum replicas in potential quark models for QCD

Abstract

In this paper, we study generic non-local NJL models for four-dimensional QCD in the limit of a large number of colours. We diagonalise, through a Bogoliubov-type transformation, the Hamiltonian of the model completely in terms of the compound operators creating/annihilating mesons -bound states of dressed quarks and antiquarks -and demonstrate a one-to-one correspondence between the Bogoliubov diagonalisation condition and the Bethe-Salpeter equation for bound states. We conclude that, in the leading order in N C , the nontrivial contents of the theory is entirely encoded in the chiral angle -the solution to the mass-gap equation for dressed quarks. As a consequence, we extend the statement of existence of excited solutions to the mass-gap equation, called vacuum replicas, beyond the BCS level and prescribe a new index to the mesonic operators -the label of the vacuum state in which the corresponding meson is created.

Goldstone bosons decoupling from high-lying hadrons

“…[21,23] or in Ref. [25], where also a Hamiltonian approach to the quark-antiquark bound-state problem is developed (as a generalisation of the method applied to the 't Hooft model for QCD in two dimensions in Ref. [29]) which, after a generalised Bogoliubov-like transformation leads to the same Eq.…”

“…Skipping the details of this derivation (the interested reader can find them in Ref. [29], for the two-dimensional 't Hooft model, whereas the generalisation to the model (2) is trivial), we give it here in the final form [25]:…”

“…A similar relation holds in the GNJL models for the pion coupling to dressed quarks. Indeed, the model admits two representations: the dressed-quark and the mesonic representation which are interchangeable up to corrections suppressed by the large N C number [25], so that, in the leading in the number of colours approximation, one has two equivalent representations for the axial-vector current complying with the PCAC theorem (for the sake of simplicity, we stick to the one-flavour theory and ignore the axial anomaly, generalisation to the flavour nonsinglet axial vector current in SU(N f ), where there is no anomaly, is trivial) 1 :…”

“…In order to normalise the pion wave function in its rest frame, one is to go slightly beyond the chiral limit and to consider pionic solutions to the bound-state Eq. (21) in the form [23,25,29]:…”

“…The mass-gap equation ensures absence of anomalous Bogoliubov terms in the Hamiltonian [23]. Finally, mesons are built using the Bethe-Salpeter equation for the quark-antiquark bound states [22,23] or by a generalised bosonic Bogoliubov-like transformation applied to the operators creating/annihilating quark-antiquark pairs [25].…”