Asymptotically Safe Gravitational Form Factors from the Proper-Time Flow Equation
We study the renormalization group flow of nonlocal form factors in four-dimensional quantum gravity within the proper-time formalism at quadratic order in the curvature expansion. We show that, in the presence of a running Newtonian coupling and a fixed negative cosmological constant, the flow equations can be integrated down to $k=0$, allowing the reconstruction of the full momentum dependence of the form factors. The running of Newton's coupling generates nontrivial momentum-dependent corrections to the effective field theory behavior, with the Planck scale setting the crossover between the IR and UV regimes. In the infrared, these corrections are organized as an expansion in powers of $q^2/m_p^2$, while in the ultraviolet they are organized in powers of $m_p^2/q^2$. Within this framework, we also analyze the properties of the dimensionless flow, identifying its fixed-point structure and constructing asymptotically safe solutions. We show that the integrated asymptotically safe solutions admit a finite and cutoff-independent $Λ\to\infty$ limit when the non-Gaussian fixed-point solution is imposed as the ultraviolet boundary condition. This removes UV logarithmic contributions and ensures independence from the renormalization scale $μ$. The resulting renormalized asymptotically safe form factors at $k=0$ display a power-law decay $\sim1/q^2$ in the ultraviolet and reproduce the expected logarithmic structure in the infrared, with the Planck scale replacing the renormalization scale.