PHYSICAL REVIEW D VOLUME 29, NUMBER 8 15 APRIL 1984 Color screening in classical Yang-Mills theories with sources C. H. Lai and C. H. Oh Department of Physics, National University of Singapore, Kent Ridge, Singapore 051 1 (Received 20 September 1983) We show that color screening of external sources by Yang-Mills fields is gauge independent, and a sufficient condition for color screening to occur is given. A gauge-invariant conserved total color of the system is constructed. I. INTRODUCTION In the past few years there has been some interest in the problems of the Yang-Mills (YM) fields interacting with external sources.' The main motivation is that the in- sights and experience~ gained at the classical level will il- luminate our understanding of the fully quantized YM theories, particularly the nonperturbative aspects.' Color screening is one of those properties that can be envisaged at the classical level. Recently, questions have been raised on whether color screening is a gauge artifact3 and wheth- er total-screening solutions found previously4~5 are really completely ~ c r e e n i n ~ . ~ , ~ The purpose of this paper is to point out that color screening is complete and gauge in- dependent, at least at the classical level. A sufficient con- dition for the YM fields to screen external sources is ex- plicitly stated. The crux of the color-confinement problem is the defi- nition of the total color and what one means by color screening. For the non-Abelian gauge field interacting with an external source, the Noether charge due to the global symmetry ceases to be a conserved quantity when the symmetry becomes localized. In contrast, the total electric charge in the Abelian case is always conserved whether the U(1) symmetry is global or local. This is be- cause the external current j, is an invariant irrespective of whether U(1) is global or local, so that one always has Wj, =O. Thus when the non-Abelian symmetry is local- ized, the Noether charge associated with the global sym- metry can remain conserved only if we restrict the gauge transformations to a specified class, that is, the gauge transformation must be independent of x, at large dis- tances. This is discussed in the next section. On the other hand, one can introduce a color direction in the internal group space at each space-time point so that a meaningful total color charge can be defined. We present this point of view in Sec. 111. Once the total color of the whole system (external sources plus the YM fields) and the total color of the external source have been clarified, it is a simple matter to determine whether a YM configuration can completely screen the external source in a gauge- independent manner. 11. NOETHER COLOR In the presence of an external source current j i , the SU(2) Yang-Mills (YM) equations are4 (la) (lb) where ua are the Pauli matrices and our metric is g.. 11 - - -goo= 1. The external current j, is gauge- covariantly conserved, D,j,=O , (2) and for static sources, j;=O, it then follows that The above equations of motion can be derived from the Lagrangian density Under the gauge transformations we have A,-UA,U-'-a,UU-' , F,~-uF,~u-~ , 1 j,,-uj,u- . Although the equations of motion are gauge covariant, the Lagrangian density (4) is in general not gauge invariant. However, it is still globally invariant and one can con- struct the Noether current4 Expression (7a) indicates that the contributions to the to- tal color current comes from the external color current as well as from the YM field. It can also be derived directly from the Lagrangian density (4) by The total color charges arising from Eq. (7) are I=Ia(ua/2i)= 1 d3x ~'(x) 1805 @ 1984 The American Physical Society