Static quark-antiquark potential for the Nambu-Goto string occupied by Bose or Fermi fields

We study the static quark-antiquark potential for a Nambu-Goto string occupied by Bose or Fermi fields. The potential is similar to the Nambu-Goto potential in regard to shape and to the ever present square-root singularity. However the presence of Bose or Fermi fields, which live in the string, produce a shift in the critical distance between quarks $r_c$ (the value of $r$ for which the potential vanishes). This would allow phenomenological values for $r_c$ to be reached without apparent difficulty.