A Relativistic Flux-tube Model for Hybrid Mesons
  A number of authors have considered potential models for hybrid mesons. These frequently involve approximating the vibrating flux-tube by a set of beads, and making an adiabatic approximation which gives rise to a static inter-quark potential which has an effective repulsive 1/r potential. We show that this approximation is almost certainly wrong. Since the beads are presumably massless, the correct approximation requires the solution of a Klein-Gordon-like equation treating the beads and the quarks on the same footing. We show how to solve this in the one-bead case for massless quarks, and find the spectrum has an unexpected degeneracy. We generalise this to the N-bead case, which can still be solved exactly for massless quarks, and show how to renormalize the energy to obtain a plausible spectrum. We give a generic method to solve the equation for massive quarks, and use this to derive a different non-relativistic equation. The results show a smooth behaviour both with respect to the quark mass and the number of beads. 
Peter J. S. Watson