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In a previous work, we established perturbative renormalizability to allorders of the massive $\phi^4_4$-theory on a half-space also called thesemi-infinite massive $\phi^4_4$-theory. Five counter-terms which are functionsdepending on the position in the space, were needed to make the theory finite.The aim of the present paper is to prove that these counter-terms are positionindependent (i.e. constants) for a particular choice of renormalizationconditions. We investigate this problem by decomposing the correlationfunctions into a bulk part, which is defined as the $\phi^4_4$ theory on thefull space $\mathbb{R}^4$ with an interaction supported on the half-space, plusa remainder which we call "the surface part". We analyse this surface part andestablish perturbatively that the $\phi^4_4$ theory in$\mathbb{R}^+\times\mathbb{R}^3$ is made finite by adding the bulkcounter-terms and two additional counter-terms to the bare interaction in thecase of Robin and Neumann boundary conditions. These surface counter-terms areposition independent and are proportional to $\int_S \phi^2$ and $\int_S\phi\partial_n\phi$. For Dirichlet boundary conditions, we prove that nosurface counter-terms are needed and the bulk counter-terms are sufficient torenormalize the connected amputated (Dirichlet) Schwinger functions. A keytechnical novelty as compared to our previous work is a proof that the powercounting of the surface part of the correlation functions is better by onepower than their bulk counterparts.

The Surface counter-terms of the $\phi_4^4$ theory on the half space $\mathbb{R}^+ \times\mathbb{R}^3$