# Vacuum Energy and Renormalization on the Edge

## TL;DR

This paper investigates how boundary conditions influence vacuum energy and renormalization in quantum field theories, revealing a rich structure of fixed points and their stability, with implications for dark energy and entanglement entropy.

## Contribution

It introduces a general framework for understanding the renormalization group flow of boundary conditions and their fixed points in quantum field theories, including implications for vacuum energy.

## Key findings

- Neumann boundary conditions are the only infrared stable fixed points.
- Boundary RG flow affects vacuum energy dependence in finite volumes.
- Entanglement entropy remains unaffected by boundary RG flow in conformal theories.

## Abstract

The vacuum dependence on boundary conditions in quantum field theories is analysed from a very general viewpoint. From this perspective the renormalization prescriptions not only imply the renormalization of the couplings of the theory in the bulk but also the appearance of a flow in the space of boundary conditions. For regular boundaries this flow has a large variety of fixed points and no cyclic orbit. The family of fixed points includes Neumann and Dirichlet boundary conditions. In one-dimensional field theories pseudoperiodic and quasiperiodic boundary conditions are also RG fixed points. Under these conditions massless bosonic free field theories are conformally invariant. Among all fixed points only Neumann boundary conditions are infrared stable fixed points. All other conformal invariant boundary conditions become unstable under some relevant perturbations. In finite volumes we analyse the dependence of the vacuum energy along the trajectories of the renormalization group flow providing an interesting framework for dark energy evolution. On the contrary, the renormalization group flow on the boundary does not affect the leading behaviour of the entanglement entropy of the vacuum in one-dimensional conformally invariant bosonic theories.