Z. Phys. C - Particles and Fields 31,237 244 (1986) Part f~r Physik C and 9 Springer-Verlag 1986 A Field Configuration Closer to the True QCD Vacuum E. Elizalde and J. Soto Departament de Fisica TeOrica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain Received 19 September 1985 Abstract. An effective action for QCD at one-loop order, which is real, manifestly Lorentz and gauge invariant and which depends on an infinite family of gauge invariants (tr(F F ), tr(Fu~Fu~FpoFp~) .... ), is obtained. Moreover, an Ansatz for a vacuum con- figuration is made, whose corresponding vacuum en- ergy density is lower than the one for the Savvidy Ansatz. Both the cases of pure QCD and of QCD with massless fermions are considered. 1. Introduction The effective action functional -~ F[A,,(x)], defined in the usual way as a Legendre transform of the gener- ating functional with respect to the mean potential A~(x) [1], contains all the information about the QCD vacuum [2]. The vacuum configuration A~(x)vAc is characterized just by imposing that it minimizes the F functional, i.e., it is a solution of the equation --a fl..fitx) = ~fi(X)VAC ~r[A,(x) =0. (1.1) 5A~(x) We do not know how to calculate -a FlA,(x)] exactly, not even at one-loop order. In general, we have to content ourselves with extracting some information about it, by calculating F[A~,(x)] (usually at one- loop order) for some particular configurations ~i~(x). For instance, as F is gauge invariant, we can consider it as a function of all possible gauge in- variants --a F[A,] = C[tr(Fu,F,~), tr(F~ F~p Fp,), tr(ff,~ff~oFo~F J ..... tr(DuF, oD,F,p ) .... ]. (1.2) If we calculate -~ FIAt(x)] for a configuration of the kind -a __ 1 F ^a A,,(x)-~ ,~x~n (1.3) where ~,~=const. and rl " is some direction in color space, we obtain --a F[A,] = F[tr(Fu~Fu~), O, tr(Fu~F~oFo,F~,) ..... 0 .... ]. (1.4) Therefore, in this way information is obtained about the contribution of some of the gauge invariants only. In order that the effective action functional F[A",(x)] that one gets using (1.3) be relevant in QCD, one must assume that: i) the terms which contain ~,can be neglected, and ii) the terms of the type tr(F,~F~oFp, ) do not alter FIAt(x)] drastically - which amounts to saying that the phenomena under consideration do not depend very much on the non- abelianess of the gauge group. At one-loop order, the effective action is given by exp {C (1) [A] } = S d [B~ (x)] det (D 2) exp { - S d 4 x r 1 I2"a u,a 1 a --ab X b X 9 LS.,,.,~--~B,(x)O,~( )B,,(. )]} (1.5) where (~ab = 6 D"e D cb abc ~c ~ ,~ o o-2gf r~v ~ab (~ab ....... b ~ c ( rc)ab = if ,bc , - v u- tgt/) ,, - . (1.6) In order to work in a completely clean fashion - when we calculate -a F[Au] for some given configu- ration A~ at one loop - we ought to be absolutely sure that for this configuration the operator has only positive eigenvalues. That happens to be not the case, in practice, for the most interesting situations. In particular, in the self dual case the operator 63 has null eigenvalues, which lead to infrared diver-