The Dual Description of Long Distance QCD and the Effective Lagrangian for Constituent Quar.pdf

The Dual Description of Long Distance QCD and the Effective Lagrangian for Constituent Quarks M. Baker∗† University of Washington, Seattle, WA 98105 5 ABSTRACT 9 9 1 We describe long distance QCD by a dual theory in which the fundamental vari- y ables are dual potentials coupled to monopole fields and use this dual theory to deter- a mine the effective Lagrangian for constituent quarks. We find the color field distribution M surrounding a quark anti-quark pair to first order in their velocities. Using these distri- 9 butions we eliminate the dual potentials and obtain an effective interaction Lagrangian 1 L (x , x ; v , v ) depending only upon the quark and anti-quark coordinates and ve- I 1 2 1 2 1 locities, valid to second order in their velocities. We propose LI as the Lagrangian v describing the long distance interaction of constituent quarks. 6 4 1. Introduction 3 5 In 1976 Mandelstam and ’t Hooft1 proposed that the force between a quark- 0 5 antiquark pair is analogous to the force between a monopole anti-monopole pair in a 9 superconductor. A dual Meissner effect prevents the electric color flux from spreading / h out as the distance between the quark anti-quark pair increases. As a result a linear po- p - tential develops which confines the quarks in hadrons. A concrete model for this “dual p superconductor” mechanism is realized by a dual theory2 defined by a Lagrangian den- e h sity L (C , B ) in which the fundamental variables are dual potentials C coupled to eff i