The present report is by no means a review of the three-body quantum problem. We restrict ourselves here to a confining and symmetric potential interaction, the symmetry being broken only through the quark masses. Other aspects of the three-body problem, in particular scattering with finite-range potentials, can be found in textbooks [12] or in the
Few-Body Conference proceedings [1]. Since in this report we are dealing mostly with the technical aspects of the underlying three-body problem, we shall be rather brief on the derivation of the quark interaction from QCD. Lattice simulations, bag models, or string pictures, for instance, provide some support for the
Coulomb-plus-linear potential or its power-law approximation A + Brβ which we shall use for numerical illustrations or phenomenological applications. The question of possible three-body forces, which deserves some attention, will be raised in Chapter 9, where we discuss the links between mesons and baryons.
Throughout this review, we shall very often disregard spin-dependent forces, especially when their complexity goes beyond a simple spin spin potential (usually considered as short-range and treated perturbatively). The Lorentz nature of confinement is, of course, a key issue in quark dynamics [13]. Our goal here is to provide a solid framework for handling the three-body problem with central forces. Extensions to include spin orbit or tensor forces are straightforward, though they involve some complicated algebra.
There are, of course, other approaches to hadron spectroscopy. Bag models [14], QCD sum rules [15], and lattice simulations [16] have been discussed extensively. The differences with respect to the Non-Relativistic Quark Model (NRQM) are obvious. More interesting, perhaps, are the convergences: some static limits of bags or of lattice calculations provide us with the effective interquark potential for QQ or QQQ systems, to be compared with the empirical potentials used in NRQM. There are links between various non-perturbative quantities such as: vacuum expectation values in sum rules, string constant, slope of the linear potential, bag pressure, and energy gaps in phase transitions, which are tentatively computed on lattices. The NRQM clearly suffers from weaknesses: non-relativistic dynamics, absence of explicit quark antiquark pairs, instantaneous interaction, etc. It is easy to list in advance good reasons why it should never work. It seems more challenging to understand within QCD studies why the NRQM actually describes the hadron world so well.