On Cosmological Correlators at One Loop

Abstract: We study equal-time in-in correlators of massless scalar fields in flat space at one loop. Using the time-ordered decomposition of correlators together with a cosmological analogue of the Baikov representation, we systematically construct relatively simple loop integrals and make manifest why, in this setting, loop corrections to correlators are simpler than those of wavefunction coefficients. As benchmark examples, we analyse the bubble and triangle diagrams. The bubble exhibits a UV divergence that can be removed by a local counterterm, while the triangle yields a finite result, which we evaluate explicitly in terms of dilogarithms using an integral transform for the Laplacian Green's function. We classify the kinematic singularities of these diagrams using Landau analysis, identifying novel types of singular behaviour, and validate this analysis against the explicit results. Finally, we derive a factorisation property of one-loop cosmological correlators at singular kinematics, relating them to flat-space loop amplitudes and lower-point tree-level correlators.

• The paper presents an efficient one-loop computation method for cosmological correlators using time-ordering and a cosmological Baikov representation.
• It uncovers the detailed analytic structure, classifying both physical and spurious singularities via comprehensive Landau analysis.
• It achieves an explicit dilogarithmic evaluation of triangle integrals, confirming factorisation theorems and matching numerical validations.

One-Loop Cosmological Correlators: Analytic Structure, Singularities, and Explicit Computation

Introduction and Motivation

The manuscript "On Cosmological Correlators at One Loop" (2601.00952) investigates one-loop corrections to equal-time in-in correlators of massless scalar fields in flat three-dimensional space. The context is the rapidly developing theory of cosmological correlators, crucial for interpreting primordial data in CMB and LSS analyses and for constraining models of early universe physics, including inflation. In contrast to amplitude computations in Minkowski space, cosmological correlators involve a time-dependent background, which both breaks Lorentz invariance and introduces technical challenges. At the one-loop level, these difficulties are exacerbated by the convolution of explicit time integrations and the absence of standard parametrisations familiar from flat-space S-matrix calculations.

This work makes technical and conceptual progress by (1) establishing an efficient framework for computing one-loop cosmological correlators directly (not via wavefunction coefficients), (2) performing a complete analytic and singularity analysis of the bubble and triangle diagrams, and (3) expressing the finite triangle integral in terms of dilogarithms using advanced integral transforms, notably the Whittaker integral representation of the Laplacian Green's function.

Framework: Time-Ordered Decomposition and Cosmological Baikov Representation

A central result is the identification of a systematic decomposition of the loop integral based on time-ordered diagrams. Each loop diagram is written as a sum over time-orderings, which in turn uniquely specify rational functions in the loop integrand. This rational structure is optimally paired with an explicit change of integration variables—from the three-momentum components $\vec{q}_1$ to “loop energies” (norms of momenta) $q_i$—which constitutes a cosmological version of the Baikov representation.

The Jacobian for the change of variables is proportional to the volume of the corresponding momentum simplex; specifically, the product of loop energies $q_1 \cdots q_V$ multiplies a dimension-dependent power of the simplex’s Gram determinant $\mathcal{V}^2(q_1,\ldots,q_V)$. The integration region is mapped from all of $\mathbb{R}^3$ to the real positive volume region of these loop energies, constrained by triangle (or tetrahedron) inequalities. Strikingly, for in-in correlators, the denominator of the integrand, after time-ordering and partial fraction, depends only on $V-1$ linearly independent combinations of loop energies, which directly simplifies the class of special functions (e.g., hypergeometric or polylogarithmic) required for their analytic expression.

Figure 2: The bubble Feynman graph with momenta orientations, whose Baikov measure is determined by triangle inequalities.

For the bubble, this structure reduces the integral to an iterated one-dimensional integral, while for the triangle, it reduces a three-dimensional integral to an effectively two-dimensional one. The case of the wavefunction involves more complicated rational structures and higher transcendental weight functions, as the denominator generically depends on all loop energies.

Analytic and Landau Singularity Structure

A highlight of the study is an exhaustive classification of singularities using Landau analysis, both physical (pinch and end-point) and on other analytic sheets. The authors derive Landau equations suitable for the cosmological setting: since energy is, in general, not globally conserved in in-in correlators (due to the time-slicing), the analogues of energy constraints are imposed only where time-ordering requires. The resulting equations, enriched by the combinatorics of admissible “tubings” (consistent subgraphs corresponding to different classical propagation channels), capture all leading and subleading singular loci in the external kinematic—energy—variables.

For the triangle, four distinct physical singularities are identified, corresponding to partial energy vanishing (e.g., $X_2+X_3+k_1=0$). Their geometric meaning is clarified via pinched graphs: a subset of vertices (together with attached legs) are drawn to infinite past, and the graph develops “tubes” (reduced to effective lines) that communicate between widely separated time regions, resulting in factorisation. These limits are precisely those expected to control physical discontinuities, as further confirmed by explicit multidimensional residue calculations (Leray theory).

Figure 4: The triangle Feynman graph, with arrows indicating loop momenta; reduction to $(u,v,w)$ variables simplifies the singularity and measure structure.

Explicit Computation: Dilogarithmic Structure of the Triangle Integral

The main technical result is an explicit, closed-form evaluation of the triangle diagram, accomplished via the cosmological Baikov representation and the Whittaker integral transform. The triangle's reduced two-fold integral, after direct integration of the measure, exposes a square root of a quartic polynomial $\Omega(u,v)$ (arising from the tetrahedron volume constraint), obstructing direct dilogarithmic integration.

To overcome this, the authors express the measure as an auxiliary complex contour (θ-) integral, transmuting the irreducible square root into a kernel $1/(x\cos\theta + y\sin\theta + z)$, with the remaining $(u,v)$ integrations now fully rational. This allows all $u, v$ integrals to be performed in terms of logarithms and dilogarithms for each fixed $\theta$, and then the $\theta$-integral is reduced to a residue sum in the complex $w = e^{i\theta}$ plane. The nontrivial part of the analysis is the precise assignment of branches and analytic continuations, which is achieved via a careful study of the location of physical and spurious singularities and associated branch cuts.

Figure 5: Real (solid) and imaginary (dashed) parts of the triangle integral $I_\triangle$, demonstrating the analytic structure and the location of various physical singularities.

The final result is analytic across the full kinematic range; all leading singularities are manifest, and the function is explicitly piecewise polylogarithmic.

Factorisation Theorem at Partial-Energy Singularities

The paper establishes and proves a rigorous factorisation theorem for the leading behaviour near partial-energy singularities of one-loop in-in correlators. Whenever such a singularity is approached, the correlator admits a universal factorisation: $$\mathcal{I}_{\text{triangle}} \sim \left[ \psi(X_1, \pm k_1) \right] \times \mathcal{A}_{\text{triangle}}(P_1, P_2, P_3)$$ where $\psi$ denotes a lower-point (contact) correlator, and $\mathcal{A}_{\text{triangle}}$ is the corresponding triangle amplitude with off-shell kinematics specified by the energies and three-momenta of the “far in past” and “near boundary” segments. This property is analogous to—yet distinct from—the familiar S-matrix factorisation at physical thresholds due to the lack of global energy conservation.

The authors generalise this theorem, proving that for all one-loop $V$-gon diagrams, in any partial-energy singularity channel associated to a set of $V-1$ vertices, the discontinuity of the cosmological correlator factorises into a product of a lower-point correlator and an on-shell amplitude, with explicit combinatorial rules.

Figure 7: Schematic time-ordered triangle graph in the partial energy factorisation limit; temporal separation induces effective classical propagation (“tube”) connecting subgraphs.

Numerical Validation and Analytic Structure

The analytic results are validated against direct numeric integration, and the mapping between analytic and discontinuity structure is checked through explicit multi-sheet analysis of the dilogarithmic solution. All leading and subleading singularities identified via Landau and algebraic methods are manifest in explicit plots of the real and imaginary parts of the correlator as a function of external energy variables.

Figure 9: Real and imaginary parts of the triangle solution plotted as a function of $X_3$; singular points and discontinuities are clearly visible at predicted kinematic loci.

Implications and Outlook

This work demonstrates that one-loop cosmological in-in correlators are substantially simpler, in terms of analytic structure and transcendental weight, than corresponding wavefunction coefficients. The simplified denominator structure, enabled by the time-ordered decomposition and the cosmological Baikov representation, is likely a generic feature and suggests the possibility of a “minimal basis” of loop functions (analogous to the box-triangle-bubble basis in amplitude theory) for cosmological correlators. Furthermore, the geometric structure of the integration domain and the explicit analytic control over the analytic continuation and branch cut structure provide a solid foundation for developing bootstrap approaches, as well as for efficiently producing phenomenological templates for observed non-Gaussianity signals.

These technical advancements open avenues for: (i) systematic characterisation and bootstrap of singularity loci for higher-site and higher-loop diagrams, (ii) importation of advanced algebraic geometry tools (such as PLD polynomials and homology), (iii) analytic continuation to curved backgrounds (e.g., de Sitter space), and (iv) integration with kinematic algebras uncovered in tree-level cosmological bootstrap frameworks. The rigorous connection between the singularities of in-in correlators and flat-space amplitudes, as established, is expected to be particularly powerful for future explorations of the analytic S-matrix/Cosmological Bootstrap correspondence.

Conclusion

This paper provides the most complete analytic treatment to date of one-loop cosmological correlators for massless scalars in flat space, culminating in explicit dilogarithmic forms for the triangle diagram. Through an overview of advanced integration variable change, algebraic and geometric singularity analysis, and integral-analytic methods, the authors clarify both the analytic structure and physical interpretation of loop corrections to cosmological observables. The general factorisation theorems and the demonstration of the relative “simplicity” of correlator (relative to wavefunction) loops represent key conceptual advances and set the stage for further progress in analytic cosmological QFT, with implications for both formal theory and precision phenomenology.