Meson-meson bound states in a(2+1)-dimensional strongly coupled lattice QCD model

Abstract

We consider bound states of two mesons (antimesons) in lattice quantum chromodynamics in an Euclidean formulation. For simplicity, we analyze an SU(3) theory with a single flavor in $2+1$ dimensions and two-dimensional Dirac matrices. For a small hopping parameter \ensuremath{\kappa} and small plaquette coupling ${g}_{0}^{\ensuremath{-}2},$ such that $0l{g}_{0}^{\ensuremath{-}2}\ensuremath{\ll}\ensuremath{\kappa}\ensuremath{\ll}1,$ recently we showed the existence of a (anti)mesonlike particle, with an asymptotic mass of the order of $\ensuremath{-}2\mathrm{ln}\ensuremath{\kappa}$ and with an isolated dispersion curve---i.e., an upper gap property persisting up to near the meson-meson threshold which is of the order of $\ensuremath{-}4\mathrm{ln}\ensuremath{\kappa}.$ Here, in a ladder approximation, we show that there is no meson-meson (or antimeson-antimeson) bound state solution to the Bethe-Salpeter equation up to the two-meson threshold. Remarkably the absence of such a bound state is an effect of a potential which is nonlocal in space at order ${\ensuremath{\kappa}}^{2},$ i.e., the leading order in the hopping parameter \ensuremath{\kappa}. A local potential appears only at order ${\ensuremath{\kappa}}^{4}$ and is repulsive. The relevant spectral properties for our model are unveiled by considering the correspondence between the lattice Bethe-Salpeter equation and a lattice Schr\odinger resolvent equation with a nonlocal potential.