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Mapping between CDT correlation length and FRG scale to enable exponent comparison

Ascertain the precise functional relation between the Causal Dynamical Triangulations correlation length ξ ≈ N4^{1/4} and the Functional Renormalization Group renormalization scale k, and determine whether the ultraviolet critical exponent measured in CDT at the A–C_dS transition can be directly compared to the corresponding ultraviolet critical exponent obtained in FRG.

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Background

The paper extracts UV critical behavior near the A–C_dS transition in CDT and infers an ultraviolet critical exponent from finite-size scaling. To compare this lattice-derived exponent to the FRG ultraviolet critical exponent, a precise mapping between the CDT correlation length (set by the four-volume scaling, ξ ∝ N4{1/4}) and the FRG renormalization scale k is required.

Without a well-defined ξ–k relation, any direct numerical comparison between CDT and FRG critical exponents remains ambiguous, limiting the ability to quantitatively connect the two approaches.

References

Presently, it is unclear to us if this can be directly compared to the corresponding critical exponent obtained using the FRG approach. The main problem is that we do not know the precise connection between our correlation length ξ used to derive θ and the scale factor k used in the FRG.

— Is Lattice Quantum Gravity Asymptotically Safe? Making contact between Causal Dynamical Triangulations and the Functional Renormalization Group (2408.07808 - Ambjørn et al., 14 Aug 2024) in Section 4 (The putative UV limit)