Resummed heat-kernel and form factors for surface contributions: Dirichlet semitransparent boundary conditions

In this article we consider resummed expressions for the heat-kernel’s (HK’s) trace of a Laplace operator, the latter including a potential and imposing Dirichlet semitransparent boundary conditions on a surface of codimension one in flat space. We obtain resummed expressions that correspond to the first and second order expansion of the HK in powers of the potential. We show how to apply these results to obtain the bulk and surface form factors of a scalar quantum field theory in d = 4 with a Yukawa coupling to a background. Additionally, we discuss a connection between HKs for Dirichlet semitransparent, Dirichlet and Robin boundary conditions.