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

Abstract

In this article we consider resummed expressions for the heat-kernel'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 heat-kernel 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. A characterization of the form factors in terms of pseudo-differential operators is given. Additionally, we discuss a connection between heat-kernels for Dirichlet semitransparent, Dirichlet and Robin boundary conditions.

24 pages, version containing all the modifications introduced in the published article. Additions: discussion of zeroth order in V, connection of semitransparent boundary conditions with Dirichlet and Robin, extended discussion of renormalization

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