Factorisation of 1-Loop Cosmological Correlators

• The paper demonstrates a factorisation theorem that decomposes one-loop cosmological correlators into products of tree-level and flat-space loop amplitudes at key kinematic singularities.
• It employs residue calculus, twisted cohomology, and cosmological polytope geometry to systematically classify physical factorisation channels in diverse FRW, dS, and AdS backgrounds.
• The framework bridges cosmological observables with flat-space unitarity by enforcing Steinmann-like constraints and accounting for UV renormalisation via local counterterms.

The factorisation property of one-loop cosmological correlators refers to the precise manner in which cosmological wavefunction coefficients or in-in correlators at one loop admit a decomposition into products of amplitudes, typically lower-point tree amplitudes or on-shell flat-space loop amplitudes, at certain kinematic singularities. This property establishes a deep connection between the analytic structure of cosmological observables and the unitarity cuts of flat-space quantum field theory, underlining the role of residue calculus, canonical forms, and cohomological methods in the analysis of correlators in Friedmann–Robertson–Walker (FRW), de Sitter (dS), or Anti-de Sitter (AdS) backgrounds.

1. Cohomological Foundations and Dual Structure

In the FRW wavefunction formalism, the wavefunction coefficients are defined via twisted integrals over a complex domain with hyperplane singularities, with the integrand encoding both cosmological and flat-space dynamics:

$$\psi_n^{(\ell)} = \int_0^\infty u(x)\,\Psi_n^{(\ell)}(x;X,Y)$$

where $u(x) = \prod_i x_i^\epsilon$ is the universal twist, and $\Psi_n^{(\ell)}$ involves a shifted canonical flat-space form $\Omega_n^{(\ell)}(x+X,Y)\,d^n x$ (De et al., 2024).

A crucial insight is the distinction between the FRW cohomology $H^n(M\setminus B;\nabla)$ and the dual (relative) twisted cohomology $H^n(M,B;\nabla^\dagger)$. The dual cohomology, organized in terms of cuts (sequential residues), selects a closed subspace whose forms are in one-to-one correspondence with physical factorization channels of the wavefunction into flat-space amplitudes. The intersection pairing computes weighted residues and ensures only compatible residue sequences contribute.

2. Graphical and Polytope-Based Classification

For a general one-loop $n$-site graph, physical channels are classified using the graphical tubing approach and the geometry of weighted cosmological polytopes. Each hyperplane $S_k(x+X,Y)=0$ represents a possible cut, and allowed cut sequences (tubings) are subject to three universal rules (De et al., 2024, Benincasa et al., 2024):

• Good-cut condition: Only compatible nesting of tubings (respecting minimal linear hyperplane relations) contributes.
• Degenerate-cut condition: Degenerate codimension boundaries are counted exactly once.
• One-dimensional cohomology: Only those cuts preserving a bounded chamber in the remaining coordinates survive.

In weighted cosmological polytopes, ordinary facets $E_\mathfrak{g}=0$ correspond to total energy going to zero (soft total energy), and internal facets $y_e=0$ coincide with cuts putting loop-edges on-shell. The canonical form of the polytope encodes the universal loop integrand structure, and its residues precisely yield the factorisation into tree-level correlators associated with the facets (Benincasa et al., 2024).

3. Factorisation Theorems: Residues and Cutting Rules

At singular kinematics, the correlator exhibits a logarithmic singularity whose leading residue factorises:

$\lim_{\mathcal E_B + k_e \to 0} [(\mathcal E_{\rm tot})\,\mathcal C_n^{\rm 1\text{-}loop}] = (-1)^V \sum_{\pm} \psi(\{X_v: v\in B\}, \pm k_e) \times \Res_{P_e^2 \to 0}\, \mathcal A_{\rm loop}(P_e,\{P_{\bar B}\})$

where $\psi$ is the shifted tree correlator and $\mathcal A_{\rm loop}$ is the flat-space loop amplitude continued on-shell (Pimentel et al., 2 Jan 2026).

Explicit graphical rules for cuts directly enumerate all factorisation channels, with each sequence of residues corresponding to a unique factorisation into subamplitudes (e.g., a one-loop amplitude splits into products of tree and loop amplitudes upon sequential cutting of the total energy and internal edge hyperplanes) (De et al., 2024). This procedure is underpinned by intersection theory and closes under kinetic differentiation.

In Mellin-Barnes and in-in approaches, the one-loop correlator decomposes as:

$$G^{\rm 1\text{-}loop}(k_i) = G^{\rm fact}(k_i) + G^{\rm bkg}(k_i)$$

where the entire nonlocal signal content (oscillatory features stemming from massive exchanges) resides in $G^{\rm fact}$, a sum over products of tree-level pieces and a universal spectral measure, while $G^{\rm bkg}$ contains only local and UV-divergent contributions, removable by counterterms (Qin, 2024, Qin et al., 2023, Qin et al., 2023).

4. Explicit Example: Four-Site One-Loop Graph

A canonical case is the four-site loop (square), whose hyperplane structure encodes all allowed cuts:

• 1-cut (total energy): Residue yields the one-loop flat-space box amplitude.
• 2-cuts (single-edge or loop-facet): Further residues yield factorisation into tree × one-loop or triangle × tree channels.
• 3-cuts: Complete factorisation into three tree amplitudes.

Intermediate cuts are regulated by compatibility conditions and Steinmann-like relations, with degenerate channels counted via unique $d\log$ combinations. The cohomological closure guarantees the dimension of the physical subspace matches the count of physical factorisation channels (De et al., 2024).

5. Analytic Structure, OPE, and Factorisation in Various Contexts

Factorisation extends beyond FRW and flat space:

• In AdS/dS, the worldsheet (scattering equations) formulation shows one-loop correlators inherit the factorisation structure of tree-level puncture collisions, with residues corresponding to products of lower-point correlators and differential operators encoding on-shell exchanges (Gomez et al., 2021).
• The OPE (operator product expansion) interpretation applies in the squeezed or soft limit, with nonanalytic terms in the correlator expansion mapping precisely to partial-wave blocks and anomalous dimensions extracted from loop-induced corrections (Heckelbacher et al., 2022, Qin et al., 2023).
• For arbitrary 1PI one-loop inflation diagrams, the general factorisation theorem yields nonanalytic "clock" signals as products of tree-like components and bubble signals, both in leading and subleading (higher-momentum-ratio) asymptotics (Qin et al., 2023, Qin et al., 2023).

6. Geometric and Combinatorial Constraints

The analytic integrity of the factorisation structure is maintained by Steinmann-like vanishing conditions: no two incompatible (overlapping) facets can simultaneously produce singularities, as encoded in the polytope boundary structure (Benincasa et al., 2024). This ensures a consistent causal and combinatorial organisation of channels analogous to flat-space unitarity but adapted to cosmological settings.

Canonical forms, polytope adjoints, and the mutual closure of physical residues establish a mathematical foundation for the correspondence between cosmological and flat-space factorisation, enabling the enumeration and calculation of all physical channels without explicit computation of the entire integrand.

7. Generalisations, Limitations, and UV Renormalisation

The factorisation property holds for general one-loop ($n$-site, arbitrary topology) graphs, with UV divergences restricted to bubble topologies and renormalised by local counterterms mimicking flat-space procedure (Qin, 2024, Pimentel et al., 2 Jan 2026). For higher-point loops and mixed cosmological backgrounds, the structure generalises via the dual cohomology and polytope approaches.

Mass ranges and background symmetries influence the detailed implementation; for instance, the worldsheet formula is strictly valid within the complementary series of dS (or BF-bound in AdS), ensuring convergence and manifest conformal invariance (Gomez et al., 2021).

The factorisation property thus provides a unified theoretical and computational framework, merging residue calculus, cohomological structure, and geometric representation, for understanding the analytic and physical bases of one-loop cosmological correlators across quantum cosmology, inflationary, and holographic regimes.