Low Energy Eeective Actions with Composite Fields

We investigate a Wilson real space renormalization group approach for theories in which composite elds are needed at low energies. It furnishes a sequence of eeective actions S which depend on an UV cut-oo. We adopt Wilsons fundamental postulate that these eeective actions should be local. This is our basic guiding principle on how to construct \blockspins" , i.e. the elds which appear in the eeective actions. Given a fundamental high energy theory which does not contain composite elds we gradually integrate out high frequency modes in order to lower the cut-oo. Eventually appearing nonlocalities at some cut-oo value c indicate the necessity to introduce new composite degrees of freedom into the theory. An analysis based on Symanzik's innnite set of Bethe-Salpeter equations for all n-point functions shows that a local low energy eeective action containing composite elds can be constructed at the compositeness scale c. Further integration of high frequency modes generates new nonlocalities which can be absorbed into the composite degrees of freedom. There are indications from an 1 N expansion that this suuces already to eliminate high energy degrees of freedom from the composite eld so that no separate integration is needed to achieve this. In general the composite eld will have self interactions.