Resummed heat-kernel and form factors 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. Additionally, we discuss a connection between heat-kernels for Dirichlet semitransparent, Dirichlet and Robin boundary conditions.

Effective action for delta potentials: Spacetime-dependent inhomogeneities and Casimir self-energy

“…[43]; up to the fifth order and taking into account the comments in Ref. [44], they agree with our results. Alternatively, one can use integration by parts to successfully compare with Refs.…”

“…Using the results in Ref. [43], we can compute the coefficients up to the fifth order, which are also in agreement with our resummed expressions (see the comments in [44]). Note that no integration by parts is used in these comparisons.…”

Worldline approach for spinor fields in manifolds with boundaries

“…A different approach involving delta potentials, also for scalar fields, was analysed as well: firstly, a free scalar field JHEP06(2024)144 with semitransparent Dirichlet boundary conditions was studied [13,16], and semitransparent Neumann boundary conditions were analysed more recently [14]. Shortly after, the case of semitransparent Dirichlet boundary conditions for scalar fields with a potential was studied using the same method [15]. None of the aforementioned worldline approaches considered, until now, spinor fields (but see [47,48] for previous path integral approaches for Dirac particles in the presence of delta-like potentials).…”

“…An exception occurs for the free scalar field in flat space, where a resummation that leads to the correct heat-trace expansion for the Dirichlet propagator in the limit of infinite coupling is possible [13] (for a similar resummation involving the Neumann propagator, see [14]). If instead of the free scalar field one considers a potential V (x), then the aforementioned resummation for the free Dirichlet propagator can be used to compute the contribution to the heat-trace at different powers of V (x), even for strong coupling [15]. When the coupling constant is left finite, the delta-function coupling to the scalar field reproduces a semitransparent mirror, which was studied in the context of the worldline formalism in [16].…”

Effective approach to the Antoniadis-Mottola model: quantum decoupling of the higher derivative terms

“…The positive powers of this operator give zero when acting on the cosmological constant and produce surface terms when acting on the scalar curvature R. Let us note that part of the mentioned papers, refs. [6][7][8], include the discussion of the form factors of surface terms (see [9] for the latest discussions of the mathematical aspects of the problem), and there may be even interesting applications of the running of Newton constant, related to these surface terms. However, it is unclear how one can gain information about the running of the cosmological constant in the traditional covariant framework.…”