The surface counter-terms of the ϕ 4 4 theory on the half space R + × R 3

In a previous work, we established perturbative renormalizability to all orders of the massive ϕ 4 4 -theory on a half-space also called the semi-infinite massive ϕ 4 4 -theory. Five counter-terms which are functions depending on the position in the space, were needed to make the theory finite. The aim of the present paper is to establish that for a particular choice of the renormalization conditions the effective action consists of a part which is independent of the boundary conditions (Dirichlet, Neumann and Robin) plus a boundary term in the case of the Robin and Neumann boundary conditions. The key idea of our method is the decomposition of the correlators into a bulk part, which is defined as the scalar field model on the full space R 4 with a quartic interaction restricted to the half-space, plus a remainder which we call “the surface part.” We analyse this surface part and establish perturbatively that the ϕ 4 4 theory in R + × R 3 is made finite by adding the bulk counter-terms and two additional counter-terms to the bare interaction in the case of Robin and Neumann boundary conditions. These surface counter-terms are position independent and are proportional to ∫_Sϕ² and ∫_Sϕ∂_nϕ. For Dirichlet boundary conditions, we prove that no surface counter-terms are needed and the bulk counter-terms are sufficient to renormalize the connected amputated (Dirichlet) Schwinger functions. A key technical novelty as compared to our previous work is a proof that the power counting of the surface part of the correlators is better by one scaling dimension than their bulk counterparts.