Discrete vs continuum gravitational diagrams in the soft synchronous gauge

TL;DR

This paper investigates the differences between discrete and continuum gravitational diagrams in a specific gauge, proposing a soft gauge fixing method to handle singularities and comparing propagator behaviors.

Contribution

It introduces a soft synchronous gauge fixing approach for discrete gravity, analyzes the resulting propagator poles, and compares discrete and continuum cases, including ghost contributions.

Findings

Propagator pole doubling in discrete gravity compared to continuum.

A principal value prescription resolves singularities and ensures diagram finiteness.

Ghost contributions vanish in the soft gauge fixing limit.

Abstract

Due to the non-renormalizability of gravity, the perturbative expansion has sense, say, for its discrete simplicial (Regge calculus) version. A finite-difference form of gravity action has diffeomorphism symmetry at leading order over metric variations from site to site, and we add a term bilinear in $n^\lambda(g_{\lambda\mu}-g_{\lambda \mu}^{(0)})$, $n^\lambda=[1,-\varepsilon(\Delta^{(s)\alpha}\Delta^{(s)}_\alpha)^{-1}\Delta^{(s)\beta}]$, to "softly" fix the synchronous gauge $g_{0\lambda}=g_{0\lambda}^{(0)}=-\delta_{0\lambda}$ at $\varepsilon\to0$, thus removing singularities at $p_0=0$. For the symmetric derivative $\Delta^{(s)}_\lambda$, the propagator has a graviton pole at $\sin^2p_0=\sum^3_{\alpha=1}\sin^2p_\alpha$ or, at small $p_\alpha$, at $p_0$ close to 0 or $\pm \pi$. This pole doubling compared to the continuum does not arise from $\sin^2(p_0/2)=\sum^3_{\alpha=1}\sin^2(p_\alpha/2)$ obtained from the action $\check{S}_{\rm g}$ with the usual derivative $\Delta_\lambda = \exp (ip_\lambda)-1$ instead of $\Delta^{(s)}_\lambda=i\sin p_\lambda$ in some terms, including in the k part of some term, and $\Delta^{(s)}_\lambda$ in the 1-k part of that term. Given the propagator $\check{G}(n,\overline{n})$, we form a principal value type propagator $[\check{G}(n,n)+\check{G}(\overline{n},\overline{n})]/2$ by analytically continuing from real $n=\overline{n}$. Singularities are roughly resolved as $p_0^{-j}\Rightarrow[(p_0+i\varepsilon)^{-j}+(p_0-i\varepsilon)^{-j}]/2$ leading to separate diagram finiteness at $\varepsilon\to0$. We find that k=1 is needed for this prescription to work properly and match the continuum case. The gauge-fixing term needed for this propagator and its finiteness are considered, the ghost contribution is found to vanish at $\varepsilon\to0$. We use these results in arXiv:2601.03228. Calculations are illustrated by the electromagnetic (Yang-Mills) case.