The Two-loop Massless phi^4 Model in Non-translational Invariant Domain

TL;DR

This paper investigates a massless scalar phi^4 theory in a four-dimensional space with boundary conditions, demonstrating how to perform perturbative renormalization up to two loops, including bulk and surface counterterms.

Contribution

It provides a method to implement two-loop renormalization in a non-translational invariant scalar field theory with boundary conditions, highlighting the necessity of surface counterterms.

Findings

Bulk counterterms suffice at one-loop level.

Surface counterterms are needed at two-loop level.

Renormalization is successfully achieved with boundary conditions.

Abstract

We study the $\frac{\lambda}{4!}\phi^{4}$ massless scalar field theory in a four-dimensional Euclidean space, where all but one of the coordinates are unbounded. We are considering Dirichlet boundary conditions in two hyperplanes, breaking the translation invariance of the system. We show how to implement the perturbative renormalization up to two-loop level of the theory. First, analyzing the full two and four-point functions at the one-loop level, we shown that the bulk counterterms are sufficient to render the theory finite. Meanwhile, at the two-loop level, we have to introduce also surface counterterms in the bare lagrangian in order to make finite the full two and also four-point Schwinger functions.