# Loop Vertex Expansion for Higher Order Interactions

## TL;DR

This paper extends the loop vertex expansion technique to handle stable interactions of any high order, demonstrated through a zero-dimensional ield theory example, emphasizing a new integration approach over intermediate field representations.

## Contribution

It introduces a novel method for extending the loop vertex expansion to high-order interactions by integrating a specific field per vertex, broadening its applicability.

## Key findings

- Successfully applied to ield theory of arbitrary order
- Demonstrates the method's effectiveness in zero-dimensional models
- Provides a new perspective on field integration techniques

## Abstract

This note provides an extension of the constructive loop vertex expansion to stable interactions of arbitrarily high order, opening the way to many applications. We treat in detail the example of the $(\bar \phi \phi)^p$ field theory in zero dimension. We find that the important feature to extend the loop vertex expansion is not to use an intermediate field representation, but rather to force integration of exactly one particular field per vertex of the initial action.