Renormalization of the Φ 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
λ
ϕ
4
theory with Robin boundary condition (characterized by a constant
c
) on a single plate at
z=0
. For arbitrary
c≥0
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
ϑ
. Moreover, we show that the renormalized Green functions at one-loop order does not depend on
ϑ
. This result is interpreted as indicating a possible new renormalization ambiguity related to the choice of
ϑ
.