Discrete gravitational diagram technique and corrections to the Newtonian potential

Abstract

Starting from simplicial Regge gravity, we use a bell-shaped form of the measure obtained using functional integration over connection. A "hypercubic" structure is considered (some variables are frozen), it is described by the metric $g_{\lambda \mu}$ at the sites. The metric is parameterized to make the measure Lebesgue. The linear part of this parametrization leads to a discrete form of standard Feynman diagrams that approximates finite continuum diagrams and is finite for infinite ones; the nonlinear part gives new vertices and diagrams. The maximum of the measure is at the edge length scale $b = b_{\rm s} \sim \eta^{1 / 2}$, where $\eta$ defines the free factor like $ ( - \det \| g_{\lambda \mu} \| )^{ \eta / 2}$ in the measure and should be a large parameter to ensure true action upon integration over connection. For general perturbative expansion (including both that for the measure and S matrix) to be free of increasing powers of $\eta$, its starting point must be at $b$ sufficiently close to $b_{\rm s}$; this appears to be a dynamic mechanism for establishing $b$ as an optimal starting point of the perturbative expansion. We use a discrete version of the soft synchronous gauge in the principal value type prescription we discuss in a recent paper (with a refined finite-difference form of the action to match the analytical properties of the propagator to the continuum case). This allows one to fix the timelike length scale at a low level for which the measure is known in closed form. This technique is applied to Newton's potential. Some of new diagrams, including potentially large ones, are mutually cancelled. The S matrix expansion is analyzed to consist of standard diagrams. These diagrams form series with a small parameter $\eta^{- 1}$; one-loop diagrams calculated in the literature represent the leading order.