Renormalization group flow equations connected to the $n$PI effective action

In this paper we derive a hierarchy of integral equations from the 4PI effective action which have the form of Bethe-Salpeter equations. We show that the vertex functions defined by these equations can be used to truncate the exact renormalization group flow equations. This truncation has the property that the flow is a total derivative with respect to the flow parameter. We also show that the truncation is equivalent to solving the $n$PI equations of motion. This result establishes a direct connection between two non-perturbative methods.