Perturbative renormalization of $\phi_4^4$ theory on the half space $\mathbb{R}^+ \times\mathbb{R}^3$ with flow equations

TL;DR

This paper rigorously proves the renormalizability of massive $\,\phi_4^4$ theory on a half-space using flow equations, identifying five counter-terms needed for finiteness and establishing bounds for correlation functions.

Contribution

It provides a rigorous proof of renormalizability for $\,\phi_4^4$ theory on a half-space with flow equations, including explicit counter-terms and bounds.

Findings

Five counter-terms are necessary for renormalization.

Correlation functions are distributions in position space.

Uniform bounds lead to proof of renormalizability.

Abstract

In this paper, we give a rigorous proof of the renormalizability of the massive $\phi_4^4$ theory on a half-space, using the renormalization group flow equations. We find that five counter-terms are needed to make the theory finite, namely $\phi^2$, $\phi\partial_z\phi$, $\phi\partial_z^2\phi$, $\phi\Delta_x\phi$ and $\phi^4$ for $(z,x)\in\mathbb{R}^+\times\mathbb{R}^3$. The amputated correlation functions are distributions in position space. We consider a suitable class of test functions and prove inductive bounds for the correlation functions folded with these test functions. The bounds are uniform in the cutoff and thus directly lead to renormalizability.