A Constituent Quark Anti-Quark Effective Lagrangian Based on the Dual Superconducting Model of Long Distance QCD

Abstract

We review the assumptions leading to the description of long distance QCD by a Lagrangian density expressed in terms of dual potentials. We find the color field distribution surrounding a quark anti-quark pair to first order in their velocities. Using these distributions we eliminate the dual potentials from the Lagrangian density and obtain an effective interaction Lagrangian $L_I({\vec{x}}_1,{\vec{x}}_2;{\vec{v}}_1,{\vec{v}}_2)$ depending only upon the quark and anti-quark coordinates and velocities, valid to second order in their velocities. We propose $L_I$ as the Lagrangian describing the long distance interaction between constituent quarks. Elsewhere we have determined the two free parameters in $L_I$, $\hat{I}{\pm }_s$ and the string tension $\ddot{I}ƒ$, by fitting the 17 known levels of $b\bar{b}$ and $c\bar{c}$ systems. Here we use $L_I$ at the classical level to calculate the leading Regge trajectory. We obtain a trajectory which becomes linear at large $M^2$ with a slope $\hat{I}{\pm }^{\prime }\simeq .74{GeV}^{-1}$, and for small $M^2$ the trajectory bends so that there are no tachyons. For a constituent quark mass between 100 and 150 MeV this trajectory passes through the two known Regge recurrences of the $\ddot{I}€$ meson. In this paper, for simplicity of presentation, we have treated the quarks as spin-zero particles.

{\bf 32,UW/PT94-01