Bethe-Salpeter equations forq qq̅ and qqq systems in the instantaneous approximation

Abstract

Bethe-Salpeter (B.S.) equations are formulated in a general way for unequal mass qq̅ and qqq systems with pairwise qq̅ and q₁ q₂ inter-actions of the single-gluon exchange (QCD) and long-range (confining) types with a common colour (˜λ⁽¹⁾·λ⁽²⁾) and spin (˜ ϒμ⁽¹⁾ ϒμ⁽²⁾) dependence for both. Spin reductions of these equations are achieved in a four dimensionally convariant manner by the method of Gordon decomposition which exhibits the structure of the spin-spin, spin-orbit and tensor terms in an elegant and compact fashion for any pairwise interaction, in preference to the usual procedure of reduction, to large and small components. The instantaneous approximation (IA), with its standard definition for a two-body system, and an extended one for the three-body system through a matching ansatz for the off-shell energy of the spectator particle q₃ when a given q₁ q₂ pair is in interaction, is then used for a reduction of these B.S. equations to the three dimensional level. The latter equations which are written down in closely analogous forms for qq̅ and qqq systems for the general case of unequal mass kinematics, represent the main results of this paper, and are capable of straightforward extension to the qqq̅q̅ system as well.