# Graph rules for the linked cluster expansion of the Legendre effective action

## TL;DR

This paper derives graph rules for the linked cluster expansion of the Legendre effective action in Euclidean dimensions, enabling exact solutions of the functional renormalization group equation with long-range interactions.

## Contribution

It introduces a novel set of graph rules for the linked cluster expansion of the Legendre effective action, including articulation vertex weights, applicable to long-range, scale-dependent interactions.

## Key findings

- Derived and proved graph rules in Euclidean dimensions.
- Computed articulation vertex weights from labeled tree graphs.
- Facilitated exact solutions of the functional renormalization group equation.

## Abstract

Graph rules for the linked cluster expansion of the Legendre effective action $\Gamma[\phi]$ are derived and proven in $D\geq 2$ Euclidean dimensions. A key aspect is the weight assigned to articulation vertices which is itself shown to be computable from labeled tree graphs. The hopping interaction is allowed to be long ranged and scale dependent, thereby producing an in principle exact solution of $\Gamma[\phi]$'s functional renormalization group equation.