Discrete vs continuum gravitational diagrams in the soft synchronous gauge

TL;DR

This work develops a discrete Regge-calculus framework for gravity that preserves leading diffeomorphism-like directions and addresses infrared singularities with a soft synchronous gauge and a principal-value graviton propagator. By using a mixed finite-difference action that combines $\Delta^{(s)}_\lambda = i\sin p_\lambda$ and $\Delta_\lambda = T_\lambda - 1$, and setting the contraction parameter $k=1$, the authors obtain a finite, continuum-compatible diagrammatic expansion in the $\varepsilon \to 0$ limit, with ghost contributions vanishing. The electromagnetic analogy clarifies the method, and the analysis identifies how nonphysical poles depend on lattice spacings $b_s,b_t$ and how a carefully chosen principal-value prescription yields correct infrared behavior. The results suggest that discrete gravity can reproduce continuum finite diagrams at large scales, guided by a refined gauge-fixing term and a robust ghost treatment, with potential generalizations to more complex gauge-fixing schemes.

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 $n^λ(g_{λμ}-g_{λμ}^{(0)})$, $n^λ=[1,-\varepsilon(Δ^{(s)α}Δ^{(s)}_α)^{-1}Δ^{(s)β}]$, to "softly" fix the synchronous gauge $g_{0λ}=g_{0λ}^{(0)}=-δ_{0λ}$ at $\varepsilon\to0$, thus removing singularities at $p_0=0$. For the symmetric derivative $Δ^{(s)}_λ$, the propagator has a graviton pole at $\sin^2p_0=\sum^3_{α=1}\sin^2p_α$ or, at small $p_α$, at $p_0$ close to 0 or $\pm π$. This pole doubling compared to the continuum does not arise from $\sin^2(p_0/2)=\sum^3_{α=1}\sin^2(p_α/2)$ obtained from the action $\check{S}_{\rm g}$ with the usual derivative $Δ_λ= \exp (ip_λ)-1$ instead of $Δ^{(s)}_λ=i\sin p_λ$ in some terms, including in the k part of some term, and $Δ^{(s)}_λ$ 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.

Figures (3)

• Figure 1: Integration contour for calculating $\int^{+ \pi}_{- \pi} ( \Delta^{(s ) 2} + i 0 )^{- 1} \dots \mathrm{d} p_0$; *6 are poles.
• Figure 2: Location of nonphysical poles of the propagators $\oplus$ (of the quantities like ${\widecheck{G}} (n, n)$) and $\ominus$ (of the quantities like ${\widecheck{G}} ({\overline{n}}, {\overline{n}})$) for $k = 1$ or electromagnetic case (a), $k > 1$ (b) and $k < 1$ (c).
• Figure 3: Location of nonphysical poles for $k = 1$.