Gauge Symmetry Breaking in Gravity and Auxiliary Effective Action

At first, we consider the path integral method for the covariant symmetry breaking in gravity. We replace the scalar fields, instead of the degrees of freedom which have been removed by gauge fixing constraints. Finally the specific ghost degree of freedom, remains excluded via one of the constraints. Secondly, we define an auxiliary effective action. Entering an auxiliary field, we will have a new dynamic field separated from the fundamental field.