On the gravitational diagram technique in the discrete setup

Abstract

This paper 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 [Formula: see text] at the sites. We parameterize [Formula: see text] so that the functional measure becomes Lebesgue. The discrete diagrams are free from ultraviolet divergences and reproduce (for ordinary, non-Planck external momenta) those continuum counterparts that are finite. We give the parametrization of [Formula: see text] 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 [Formula: see text], where [Formula: see text] defines the free factor [Formula: see text] 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 [Formula: see text] if its initial point is chosen close enough to the maximum point of the measure, thus justifying this choice. Discrete propagators depend on the Barbero–Immirzi parameter [Formula: see text], which determines the ratio of timelike and spacelike elementary length scales. The existing estimates of [Formula: see text] allow the propagator poles to have real energy for any (real) spatial momenta.