Two geometric observables and their power-law falloff
The paper asks what happens to two-point Euclidean correlation functions built from geometry when gravitons are quantized in the low energy effective theory. The two operators are the local Ricci scalar curvature R and the local volume factor, the square root of the metric determinant. A two-point function here means the correlation between the value of such an operator at a source and at a sink separated by a distance x in Euclidean space.
In pure four-dimensional gravity the propagating degrees of freedom are massless gravitons, so these correlators fall off as powers of the separation rather than exponentially. The leading curvature correlator about flat space was already known to be derivatives of a delta function, meaning it vanishes everywhere away from the origin. That contact term would not be visible in numerical lattice correlation functions, so the first non-vanishing curvature contribution starts at next-to-leading order and needs a one-loop calculation. The volume correlator had not been computed in the effective theory before this work.
Why coordinate corrections are needed for gauge invariance
The central methodological point is that the source and sink of a correlator sit on a spacetime that is itself fluctuating. The separation is most conveniently written as a distance on the flat background about which the theory is expanded, so when spacetime fluctuates, the relation between that background distance and the actual relational distance changes. The authors handle this with relational observables: a shared master coordinate system X, defined to be harmonic with respect to the full metric, so that all observers refer their measurements to the same signposts.
Each component of X transforms like a scalar field under diffeomorphisms, which is why the authors call them coordinate scalars. Expanding X order by order in the gravitational coupling generates new Feynman diagrams in addition to the standard field-theory ones. The paper verifies explicitly that only after these coordinate corrections are included do the curvature and volume correlators become independent of the gauge-fixing parameters. That cancellation is presented as a strong cross-check of both the formalism and the calculation.
The curvature correlator scales as one over x to the eighth
At one loop the curvature correlator receives contributions from eight diagrams. Individually, the pieces depend on the gauge parameters alpha and beta, and a tadpole diagram that would normally vanish does not, because second-order scalar propagators introduce nontrivial momentum flow. Summing all eight diagrams cancels the alpha and beta dependence and leaves a gauge-invariant result proportional to q^4 ln(-q^2) in momentum space, plus local analytic terms proportional to q^4.
Fourier transforming to position space, the local terms become derivatives of delta functions and are dropped as contact terms. The nonlocal piece gives 3 kappa^4 / (4 pi^4 x^8), and after Euclidean continuation and rewriting in terms of Newton's constant this becomes 768 G^2 / (pi^2 x_E^8). The eighth power of the Euclidean separation is the concrete prediction a lattice calculation would need to reproduce.
A counterterm shortcut fixes the volume correlator
The volume correlator requires third-order coordinate corrections, an expansion the authors avoid by a shortcut. They note that the tree-level insertion of the O(R^2) operators produces the same linear combination of local counterterms for the curvature and volume correlators. Since no evanescent contributions appear at this order, the coefficient of the 1/epsilon pole that cancels the one-loop ultraviolet divergence also fixes the coefficient of the ln(-q^2) term.
This lets them infer the one-loop volume result from the curvature result. In momentum space it is kappa^4/(256 pi^2) ln(-q^2) plus local terms, and in position space it becomes kappa^4/(256 pi^4 x^4), or 4 G^2 / (pi^2 x_E^4). Adding the leading tree-level piece gives the full next-to-leading-order Euclidean volume correlator, 12 G / (pi x_E^2) + 4 G^2 / (pi^2 x_E^4). Note that the two terms scale differently, with the one-loop piece falling as the fourth power and the tree-level piece as the square.
The conformal mode needs care in the continuation
The abstract flags a subtlety in the analytic continuation of the conformal mode to Euclidean signature. This matters because the Euclidean correlators are the objects a lattice simulation can actually measure, and the continuation is not automatic for every mode of the metric. The paper treats this carefully rather than assuming a straightforward continuation, and the final Euclidean expressions are the ones presented as the universal predictions.
The authors also emphasize that their correlators come out positive. They attribute this to reflection positivity, which they expect to hold for the universal long-distance predictions of a more fundamental unitary theory. Positivity is offered as a reassuring consistency feature rather than as an independent derivation.
Only the nonanalytic part is a universal prediction
The effective theory is not renormalizable, so at each order new low energy constants appear that encode unknown short-distance physics. That normally destroys predictive power. What survives is the leading nonanalytic behavior from radiative corrections, which can be expressed through a small number of couplings, often just G. The curvature and volume results here are of exactly that type: the ln(-q^2) terms and their position-space power laws depend only on Newton's constant and the source-sink separation.
The local analytic terms, by contrast, are not fixed by this calculation. They Fourier transform to delta functions and derivatives of delta functions, so they are contact terms that a lattice calculation at nonzero separation would not see anyway. This is why the authors can present the power-law pieces as unambiguous predictions while leaving the local terms unspecified.
A benchmark for nonperturbative lattice gravity
The stated purpose of these expressions is comparison with nonperturbative lattice formulations of gravity. Lattice approaches aim to go beyond the effective theory and capture nonperturbative effects, typically by searching for a nontrivial fixed point that would make the theory ultraviolet-complete in Weinberg's asymptotic safety scenario. Whether or not that program succeeds, lattice methods should in principle recover the effective theory at long distances.
The concrete comparison is therefore between measured lattice two-point functions of curvature and volume and the power laws 768 G^2/(pi^2 x_E^8) for curvature and 12 G/(pi x_E^2) + 4 G^2/(pi^2 x_E^4) for volume, with G fixed in lattice units. The curvature case is the sharper test, because its leading term is a contact term invisible to the lattice and the entire visible signal is the one-loop x_E^-8 tail. The volume case carries a tree-level x_E^-2 piece alongside the one-loop x_E^-4 piece, so a lattice check must separate the two scalings rather than fit a single power.
Sources used in this answer
Euclidean Correlation Functions in Quantum Gravity
Through one-loop order, the curvature correlator is 768 G^2/(pi^2 x_E^8) and the volume correlator is 12 G/(pi x_E^2) + 4 G^2/(pi^2 x_E^4), gauge-invariant only once coordinate corrections from fluctuating spacetime are included.