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 $b=b_{\mathrm{s}}\sim {\eta }^{1/2}$, where $\eta$ defines the free factor like $(-\det \Vert g_{\lambda \mu }\Vert {)}^{\eta /2}$ 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 $b_{\mathrm{s}}$; 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.

62 pages, 8 figures. Compared to our paper arXiv:2306.11531, a soft synchronous gauge is used which allows us to do without model assumptions in specifying the measure and makes it relatively easy to provide a consistent discrete perturbative expansion. V3: a section on the Newtonian potential is added