Volume 221, number 3,4 PHYSICS LETTERSB 4 May 1989 PROPAGATORS IN HIGHER DIMENSIONAL SCALE INVARIANT GAUGE THEORIES M. HORTACSU and J. KALAYCI Physws Department, Faculty of Sctences and Letters, Techmcal Umverstty of Istanbul, 80626 Maslak-lstanbul, Turkey Recmved8 January 1988 We calculate the propagator m the scale mvanant gaugetheories in d= 6 in the background of classical solutmns. We find that the propagatorbetween different fields is not zero Several authors suggested alternative models for pure gauge theories in even higher dimensions with dimen- sionless coupling constants [ 1,2 ]. These models have lagrangians composed of powers of the field strength F~, which is raised to the power d/2 in d dimensions. Such terms also exist in the expansions of the string scattering amplitudes in powers of the string constant [3,4]. Since these models are conformally invariant, as standard Yang-Mills (YM) theory is in four dimensions, the classical solutions for such models are of the same form as the YM solutions in four dimensions. Tchrakian and collaborators found the spherically symmetric gauge field configurations with fimte action [ 5 ], instantons in 4k dimensions [ 6 ], and merons in all even dimensions [ 7 ]. Sagllo~lu found instanton solutions in six dimensions [2]. We, collaborating with Dibekqi, found monopole solutions also in six dimensions [ 8 ]. One has to note that these models have a meaning only in the classical sense. Since such models lack the quadratic term, a quantum theory based on such a lagrangmn will not have a propagator for the basic fields A ~. This would make the perturbative solution of the model in terms of Feynman diagrams unfeasible. In a previous publication [ 9 ], one of us (M.H.) coupled this model to a spinor field to induce a propagator for the A ~ field. Although this may be an acceptable algorithm, one wonders whether such a model cannot "live on its own", that is whether it can be quantxzed without couphng it to another field. A possible way may be an expansion around solutions of the classical equations of motion. One may induce the kinetic term in such a background. A similar situation exists in string theory where from the pure cubic action [ 10 ], the kinetic term is generated in this fashion [ 11 ]. In this note, we try to get the propagator for the model in six dimensions, by expanding around the instanton solution, given in ref. [ 2 ]. We find a remarkable result that then the propagator between certain different fields do not vanish. The field loses its identity and becomes "schizophrenic". To be more explicit on this point we have to introduce some formalism. We use fields defined as a v v TO) -A~a (1) where e *~' are the chiral spinor generators of the Lorentz group in six dimensions as well as generators of the internal group, which is taken to be SU(4) [2 ]. They satisfy the commutation relations [ a,~a, o,~ ] = 2i ( a~.~ao, + Ja~,a~,, -a,~o,a~, -aa, a~,o,) • (2) Here all the group as well as Lorentz indices go from zero to five. We find that, for instance, n 0 (Aol,Ao,) #0 for n=0 ..... 5, (3) 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division ) 319