Non-perturbative Dynamics of the Heavy-Light Quark System

We obtain the non-perturbative interaction kernel of the heavy-light quark system in a gauge-invariant approach. The potential and the sum rules limit are discussed. Some insight is given in the light quark interaction. INTRODUCTION The dynamics of a system composed by two heavy quarks is well understood in terms of a potential interaction (static plus relativistic corrections) obtained from the semirelativistic reduction of the QCD dynamics (see e.g. [1]). If at least one of the quarks is light the system behaves relativistically and a pure relativistic treatment becomes necessary (via Dirac or Bethe–Salpeter equations). A lot of phenomenologically justified relativistic equations can be found in the literature but up to now we miss a relativistic treatment which follows directly from QCD. Our work goes in this direction. In order to simplify the problem we focus on the heavy-light mesons in the non-recoil limit (i.e. infinitely heavy antiquark). Only at the end we will briefly discuss the two-body case. Our starting point is the quark-antiquark gauge-invariant Green function taken in the infinite mass limit of one particle. The only dynamical assumption is on the behaviour of the Wilson loop (i.e. on the nature of the non-perturbative vacuum). The gauge invariance of the formalism guarantees that the relevant physical information are preserved at any step of our derivation. In this way we obtain a QCD justified fully relativistic interaction kernel for the quark in the infinite mass limit of the antiquark. This kernel reduces in some region of the physical parameters to the heavy quark mass potential, and leads in some other region to the heavy quark sum rules results, providing in this way an unified description. We discuss our result with respect to the old-standing problem of the Lorentz structure of the Dirac kernel for a confining interaction. A derivation of the main results summed up here and a complete reference list can be found in [2,3]. 1) Alexander von Humboldt Fellow. THE RELATIVISTIC INTERACTION IN THE ONE-BODY LIMIT The quark-antiquark Green function in the Feynman–Schwinger representation can be written as a quantomechanical path integral over the quark trajectories (z1(t1) and z2(t2)) (quenched approximation), Ginv(x, u, y, v) = 1 4 〈 TrP (iD/ (1) y +m1) ∫ ∞