3d QFT IR divergences as UV divergences in 4d Holographic Cosmology

Abstract: In this paper we consider IR divergences in a 3d toy model field theory for 4d holographic cosmology, and we analyze them by introducing a mass term in a way that preserves a certain form of the generalized conformal structure. This allows us to compute 2- and 3-point functions at 2-loops and study their IR structure below the mass scale, from which we argue for a possible IR finiteness beyond perturbation theory, consistent with lattice results. In the holographically dual 4d cosmology, this corresponds to UV finiteness, i.e., the absence of cosmological singularities. The 3d IR field theory methods could be extended beyond this specific application.

• The paper establishes that introducing a finite mass term eliminates logarithmic IR divergences via generalized Ward identities in 3D QFT.
• Two- and three-point correlators computed up to two loops reveal saturation to finite constants, linking IR finiteness to nonsingular 4D cosmology.
• The analysis provides evidence that holographic cosmologies derived from these QFTs are UV finite, making predictions independent of deep IR behavior.

3D QFT IR Divergences as UV Divergences in 4D Holographic Cosmology

Overview

The paper "3d QFT IR divergences as UV divergences in 4d Holographic Cosmology" (2605.16587) investigates the infrared divergence structure of three-dimensional super-renormalizable quantum field theories (QFTs) possessing generalized conformal structure, with specific application to the context of holographic cosmology. The core result is a perturbative and nonperturbative analysis (up to two loops) of both two- and three-point functions in a toy model relevant to the holographic dual of 4D cosmological evolution. The key formal achievement is the derivation and application of generalized Ward identities preserving conformal structure even in the presence of a finite mass term, which facilitates control over putative IR divergences and connects these to the absence of cosmological singularities in the holographic dual.

Theoretical Context and Motivation

Holographic cosmology extends the holographic principle, concretized in the AdS/CFT correspondence, to four-dimensional cosmological spacetimes by seeking a duality between strongly coupled 4D gravity and lower-dimensional QFTs. A formalism using three-dimensional super-renormalizable QFTs with generalized conformal structure as effective duals for early-universe cosmology circumvents limitations of the usual AdS or dS geometric settings. However, such theories, particularly in three dimensions, are traditionally afflicted by severe IR divergences at the perturbative level. Lattice simulations had previously indicated possible IR finiteness [18], but a perturbative field-theoretical understanding remained incomplete.

The present work addresses this by computing loop-corrected correlators in a toy model used in holographic cosmology, emphasizing the interplay between physical and regulator mass scales, and determining their implications via the holographic map for UV structure on the cosmological side.

Toy Model and Generalized Conformal Structure

The utilized model is an SU(N) gauge theory in three dimensions coupled to adjoint scalars, with quartic interactions and a global SO(3) symmetry. Massless, it exhibits a generalized conformal structure, i.e., invariance under a combination of field and coupling re-scalings analogous to the dimensional reduction of 4D conformal invariance. The critical technical innovation is the systematic inclusion of a finite scalar mass term, viewed as a background field transforming to preserve generalized conformal symmetry, allowing for generalized dilatation Ward identities to constrain correlators even away from the massless point.

The analytical machinery developed includes:

• Formulation of Ward identities for n-point functions with finite mass, showing only two independent dimensionless parameters are permissible in correlators.
• Derivation of the scaling forms for the two-point and three-point functions, indicating their asymptotic structure in both UV ($q \gg m$) and IR ($q \ll m$).

IR Divergences and Their Resolution

Two-Point Function

The structure of the two-point function is analyzed through explicit calculation up to two loops for both the massless $m=0$ and finite mass cases. In the massless theory, logarithmic IR divergences appear perturbatively as expected. However, after introducing a genuine mass (not merely as an IR regulator), one observes:

• Logarithmic divergences cancel when the $q \to 0$ limit is taken at fixed $m$.
• The correlator saturates to a finite, momentum-independent value in the deep IR, with corrections controlled by the dimensionless ratio of the coupling to the mass, $g^2_{YM}/m$.
• The limits $m\to 0$ and $q\to 0$ do not commute; the divergence appears only if $m$ is taken to zero before the external momenta are sent to zero.

This non-commutativity and resultant IR finiteness are consistent with prior lattice results, but are shown perturbatively here in a manifestly conformal-invariant framework.

Three-Point Function

The three-point function is evaluated in the so-called "squeezed limit", relevant for cosmological applications. The same pattern emerges: the IR structure is controlled by the mass and the ratio $g^2_{YM}/m$, resulting in IR finite, saturated correlation functions. In fact, derivatives of the two-point function with respect to mass essentially give the leading structure of the three-point function, consistent with cosmological consistency relations.

Implications for Holographic Cosmology

Via the domain wall/cosmology analytic continuation, IR divergences in the 3D QFT map to UV divergences (cosmological singularities) in the 4D gravitational dual. The absence of such divergences in the QFT, confirmed by explicit calculation and Ward identity constraints, thus corresponds within the holographic map to a non-singular early-time cosmology. Specifically:

• The two- and three-point functions of the CMB-relevant operators are finite at small QFT momenta, corresponding to large distance (small time) on the cosmological side.
• Explicit mapping formulas indicate saturation of correlators to finite constants in the IR translates into regular, non-singular behavior in the cosmological context.

Numerical and Analytical Results

Perturbative expansions, asymptotics, and explicit expressions for all relevant Feynman integrals are provided, including two-loop contributions. Numerical evaluations support the analytic findings. In the IR,

• Two-point function: $q \ll m$0 for $q \ll m$1; up to corrections in $q \ll m$2.
• Three-point function (squeezed limit): $q \ll m$3 for $q \ll m$4; derivative relation to two-point function holds.
• All logarithmic divergences cancel; any residual non-analyticity is traced to the order of limits rather than fundamental pathology.

Implications and Future Directions

Theoretical Implications

This paper provides a detailed field-theoretic mechanism for the IR finiteness of 3D super-renormalizable theories with generalized conformal structure, fully consistent with non-perturbative lattice results. The mechanism is robust — generalizable to other operator insertions, higher-point functions, and potentially to other classes of models relevant in holographic cosmology. The cancellation of IR divergences in correlators, protected by generalized Ward identities, directly impacts the status of the initial cosmological singularity in holographic cosmology proposals. This work thus lends formal support to the conjecture that the cosmological singularity is absent in holographic model-building frameworks of this type.

Practical Implications

From a practical perspective, the finite, regular structure of correlators implies that predictions (e.g., for the power spectrum and higher-order non-Gaussianities) are insensitive to the deep IR completion of these theories, with all dependence reducible to the coupling and the introduced physical mass scale.

Future Directions

Several research avenues are indicated:

• Extension to three-loop and higher computations to verify resummability and the emergent functional dependence, particularly the conjectured geometric series structure.
• Full treatment of the three-point function beyond the squeezed limit with the sequence of limits $q \ll m$5, $q \ll m$6 taken properly.
• Exploration of the generalization to other models and operator classes, as well as the implications for different holographic cosmology proposals.

Conclusion

By exploiting generalized conformal Ward identities and explicit two-loop calculations, the paper demonstrates that three-dimensional super-renormalizable QFTs central to holographic cosmology are IR finite when a finite mass is present, with all putative divergences an artifact of pathological orders of limits. The findings, consistent with lattice gauge theory results, provide strong evidence that the associated four-dimensional holographic cosmologies are UV finite and nonsingular. This has significant ramifications for the theoretical underpinnings of holographic cosmology and for the calculational reliability of cosmological correlators derived from such QFT duals.