On the gravitational diagram technique in the discrete setup

Abstract

This article is in the spirit of our work on the consequences of the Regge calculus, where some edge length scale arises as an optimal initial point of the perturbative expansion after functional integration over connection. Now consider the perturbative expansion itself. To obtain an algorithmizable diagram technique, we consider the simplest periodic simplicial structure with a frozen part of the variables ("hypercubic"). After functional integration over connection, the system is described by the metric $g_{\lambda \mu}$ at the sites. We parameterize $g_{\lambda \mu}$ so that the functional measure becomes Lebesgue. The discrete diagrams are finite and reproduce (for ordinary, non-Planck external momenta) those continuum counterparts that are finite. We give the parametrization of $g_{\lambda \mu}$ up to terms, providing, in particular, additional three-graviton and two-graviton-two-matter vertices, which can give additional one-loop corrections to the Newtonian potential. The edge length scale is $\sim \sqrt{ \eta }$, where $\eta$ defines the free factor $ ( - \det \| g_{\lambda \mu} \| )^{ \eta / 2}$ in the measure and should be a large parameter to ensure the true action after integration over connection. We verify the important fact that the perturbative expansion does not contain increasing powers of $\eta$ if its initial point is chosen close enough to the maximum point of the measure, thus justifying this choice.