Bethe-Salpeter equations forq $\bar q$ andqqq systems in the instantaneous approximation

Abstract

Bethe-Salpeter (B.S.) equations are formulated in a general way for unequal massq−q andqqq systems with pairwiseq−q andq ₁ q ₂ inter-actions of the single-gluon exchange (QCD) and long-range (confining) types with a common colour (~λ⁽¹⁾·λ⁽²⁾) and spin (~γ ⁽¹⁾_μ γ ⁽²⁾_μ ) 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 particleq ₃ when a givenq ₁ q ₂ pair is in interaction, is then used for a reduction of these B.S. equations to the threedimensional level. The latter equations which are written down in closely analogous forms forq−q andqqq systems for the general case of unequal mass kinematics, represent the main results of this paper, and are capable of straightforward extension to theqq−q−q system as well.