The Dual Description of Long Distance QCD and the Effective Lagrangian for Constituent Quarks

We describe long distance QCD by a dual theory in which the fundamental variables are dual potentials coupled to monopole fields and use this dual theory to determine the effective Lagrangian for constituent quarks. 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 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 of constituent quarks.