Finite Size Effects in the Anisotropic \lambda/4!(\phi^4_1 + \phi^4_2)_d Model

TL;DR

This paper studies finite size effects in a two-field -4 model with boundary conditions, revealing the necessity of surface counterterms and analyzing the behavior of tadpole graphs to address surface divergences.

Contribution

It demonstrates that renormalization in the model requires surface counterterms and compares the behavior of tadpole graphs under different boundary conditions.

Findings

Surface counterterms are necessary for renormalization.

Tadpole graphs for DD and NN have opposite signs in their z-dependent parts.

Analysis of boundary conditions impacts surface divergence treatment.

Abstract

We consider the $\frac{\lambda}{4!}(\phi^{4}_{1}+\phi^{4}_{2})$ model on a d-dimensional Euclidean space, where all but one of the coordinates are unbounded. Translation invariance along the bounded coordinate, z, which lies in the interval [0,L], is broken because of the boundary conditions (BC's) chosen for the hyperplanes z=0 and z=L. Two different possibilities for these BC's boundary conditions are considered: DD and NN, where D denotes Dirichlet and N Newmann, respectively. The renormalization procedure up to one-loop order is applied, obtaining two main results. The first is the fact that the renormalization program requires the introduction of counterterms which are surface interactions. The second one is that the tadpole graphs for DD and NN have the same z dependent part in modulus but with opposite signs. We investigate the relevance of this fact to the elimination of surface divergences.