Renormalization of the $\Phi^4$ scalar theory under Robin boundary conditions and a possible new renormalization ambiguity

Abstract

We perform a detailed analysis of renormalization at one-loop order in the $\lambda\phi^4$ theory with Robin boundary condition (characterized by a constant $c$) on a single plate at $z=0$. For arbitrary $c\geq0$ the renormalized theory is finite after the inclusion of the usual mass and coupling constant counterterms, and two independent surface counterterms. A surface counterterm renormalizes the parameter $c$. The other one may involve either an additional wave-function renormalization for fields at the surface, or an extra quadratic surface counterterm. We show that both choices lead to consistent subtraction schemes at one-loop order, and that moreover it is possible to work out a consistent scheme with both counterterms included. In this case, however, they can not be independent quantities. We study a simple one-parameter family of solutions where they are assumed to be proportional to each other, with a constant $\vartheta$. Moreover, we show that the renormalized Green functions at one-loop order does not depend on $\vartheta$. This result is interpreted as indicating a possible new renormalization ambiguity related to the choice of $\vartheta$.