Derivation of Dielectric Model of Confimenent in QCD

After the gauge invariant gluon condensation, gluons remain as massless excitations. When an effective theory desribing the condensation and the excitaion is constructed, a constraint has to be imposed for the vacuum to be stable. The constraint implies the perfect dia-electricity and assures the solution of color flux tube when qurks are introduced. Thus the dia-electric model of Kogut-Susskind [5] and ’t Hooft [6] are derived by the stability of the condensed vacuum. PACS nimber; 11.15-q, 12.38Aw, 12.38Lg. Introduction; Understanding of the confinement mechanism based on QCD Lagrangian remains as a fundamental problem in the strong interaction regime. Besides lattice QCD [1], active researches have been performed along the line of the dual superconductivity with the abelian gauge fixing [2, 3], infrared structure of the gluon propagators [4], e.t.c.. In continuum QCD, the key point will be how to extract the effect of the gluon condensation. In this letter, we propose an approach to construct the effective Lagrangian and derive the dielectric model of Kogut-Susskind [5] and ’t Hooft [6]. We asssume that gluons condense in gauge invariant form, so the gauge invariance is not broken in the vacuum and gluons which are excitations above the condensed vacuum remain massless. Because of masslessness, they may condense further in color singlet form. Therefore, when we construct the effective Lagrangian describing the interaction between the condensation and excitation field, non-trivial conditions have to be imposed to assure the the stability of the vaccum. Remarkably, these conditions imply the perfect dia-electricty of the vacuum and guarantee the existence of the color flux tube of infinite length when quarks are introduced. As for the gluon condensation, the magneteic type condensation > 0 has been established [7] but the advantage of the present work is that we do not have to ask the precise mechanism of the condensation, the magnetic monopole[8] or the gluon pairs[9] e.t.c.. Effective Lagrangian Leff ; To measure the singlet condensation, any gauge invariant operator can be used but we choose for simplicity the normal oredred form of G2μν ; φ(x) = NGμν(x) = Gμν(x)− <0|Gμν(x)|0>. (|0> is the