A compact analytic formula for the one-loop triangle cosmological correlator

Abstract: We derive a compact analytic formula for the one-loop triangle correlator of conformally coupled scalars in de Sitter space. The result is organised as six leading-singularity prefactors multiplying pure weight-two functions of the six energy variables. It contains forty-two dilogarithms, compared with approximately one hundred and twenty in the previously known closed-form representation, and requires no auxiliary regulator. The dilogarithms occur in Galois-conjugate pairs, making each contribution separately real throughout the physical region. We validate the result numerically. Moreover, we show that its symbol can be derived directly from the dressed integral representation or, independently, bootstrapped from Landau singularities and general consistency conditions. Finally, we show that the correlator (in a suitable normalization) is a Stieltjes function of each squared energy separately and is jointly completely monotone in all six squared energies. These structures suggest a route towards higher-point one-loop cosmological correlators.

• The paper develops a compact, regulator-free formula containing 42 dilogarithms organized into six Galois-invariant leading-singularity sectors, substantially reducing the roughly 120 dilogarithms in earlier work.
• The authors derive the symbol independently from dressed-integral contour discontinuities and Landau-singularity bootstrap data, then validate the result numerically, through conformal Ward identities, and in key factorization limits.
• The correlator is shown to have positive analytic structure: after dividing by external energies, it is a multivariate Stieltjes function and jointly completely monotone in the six squared energy variables, providing constraints for future loop calculations.

Context and motivation

Cosmological correlators of primordial fluctuations are the boundary observables from which CMB and large-scale-structure data are inferred, and their non-Gaussian components encode the particle content and interactions during inflation. For conformally coupled scalars in de Sitter space with $\phi^4$ interactions, the mode functions are plane waves up to powers of conformal time, so time integrations reduce to flat-space-type energy integrals while retaining genuinely cosmological analytic structure. The paper studies the one-loop three-site ("triangle") correlator with two external legs per quartic vertex — the first loop case that is both nontrivial and finite. Its kinematics is captured by six energies: $x_i = |\vec p_{i1}|+|\vec p_{i2}|$ at each vertex and $y_i$, the magnitudes of the momenta injected at the vertices.

A closed-form evaluation was previously obtained by Pimentel et al., establishing that the correlator is polylogarithmic even though the corresponding wavefunction coefficient involves elliptic functions. That representation, however, contained approximately 120 dilogarithms with arguments involving two square roots per permutation, required an auxiliary regulator removed jointly with the $i\varepsilon$ prescription, and exhibited spurious singularities cancelling only after full assembly. The present work replaces this with a compact formula and provides two independent derivations of its symbol.

The main result

The central result organises the correlator as six leading-singularity prefactors multiplying pure weight-two functions:

$$C(x_1,x_2,x_3,y_1,y_2,y_3)=\sum_{a=1}^{6}R_a\,\Phi_a .$$

The prefactors take the form $R_a \sim 1/\sqrt{\Sigma_a}$ times rational combinations of $1/E_T$ and $1/u_i$, where $E_T=x_1+x_2+x_3$, $u_i$ are partial-energy combinations, and the radicands $x_i = |\vec p_{i1}|+|\vec p_{i2}|$0 are quadratic in the external energies built on the Heron combination $x_i = |\vec p_{i1}|+|\vec p_{i2}|$1 of the internal energies. Each pure function $x_i = |\vec p_{i1}|+|\vec p_{i2}|$2 is a sum of differences of Rogers dilogarithms evaluated on Galois-conjugate argument pairs (related by $x_i = |\vec p_{i1}|+|\vec p_{i2}|$3), plus logarithm products. Both $x_i = |\vec p_{i1}|+|\vec p_{i2}|$4 and $x_i = |\vec p_{i1}|+|\vec p_{i2}|$5 are odd under the Galois involution, so each product is separately real throughout the physical region.

The quantitative improvement is substantial: the final expression contains forty-two dilogarithms, compared with roughly one hundred and twenty in the previous representation, uses a single quadratic extension per term rather than two square roots per permutation, requires no auxiliary regulator or $x_i = |\vec p_{i1}|+|\vec p_{i2}|$6 prescription, and is valid as written throughout the physical region. The method follows four steps familiar from amplitude computations: identify algebraically independent leading singularities; compute and simplify symbols; integrate weight-two symbols back to functions via symmetric log products and antisymmetric Rogers-dilogarithm pairs whose arguments factorise multiplicatively over the symbol letters; and fix beyond-the-symbol ambiguities ($x_i = |\vec p_{i1}|+|\vec p_{i2}|$7 terms and $x_i = |\vec p_{i1}|+|\vec p_{i2}|$8 constants) without numerical input, using reality in the physical region and Galois-even parity of the full correlator.

An important structural point concerns degenerate boundaries: the radicands $x_i = |\vec p_{i1}|+|\vec p_{i2}|$9 vanish only on collinear boundaries of the physical region, but since each $y_i$0 behaves locally as $y_i$1, the explicit pole cancels and no physical singularity arises there. Continuity of the building blocks across the apparent discontinuity loci of the piecewise-defined function $y_i$2 is proven analytically; the delicate case where two arguments cross $y_i$3 simultaneously in opposite directions produces cancelling $y_i$4 jumps.

Checks

Three complementary validations are performed. Against the prior analytic result evaluated with finite regulators $y_i$5, $y_i$6, the compact formula agrees at six physical reference points to relative deviations between $y_i$7 and $y_i$8, with deviations shrinking as regulators are reduced — indicating they stem from the finite-regulator prescription rather than a discrepancy. Independent numerical evaluations of the cut representation agree to $y_i$9 or better, and of the dressed representation (via double-exponential quadrature with Richardson extrapolation) to $i\varepsilon$0 or better, limited only by quadrature precision.

Analytically, the dilatation Ward identity (homogeneity of degree $i\varepsilon$1) is manifest, and the special conformal Ward identities — reduced, after stripping external two-point factors, to the condition $i\varepsilon$2 — are verified algebraically. Notably, each leading singularity satisfies these identities separately. Singular limits behave correctly: the total-energy residue reproduces exactly the symbol of the off-shell massless flat-space triangle with leading singularity $i\varepsilon$3; the partial-energy limit $i\varepsilon$4 exhibits the expected factorisation into a shifted tree-level factor times the flat-space triangle discontinuity with one null invariant; and the $i\varepsilon$5 poles of individual prefactors cancel in the full sum, confirming they are spurious.

Deriving the symbol from the dressed representation

The second contribution derives the complete symbol directly from the dressed integral representation — obtained by relaxing energy conservation, attaching auxiliary propagators to each vertex, and integrating over auxiliary energies — without performing the two dressing integrations. A two-fold dlog decomposition of the dressed integrand separates it into precisely the six sectors $i\varepsilon$6 of the master formula, with the $i\varepsilon$7 pulled outside the integrals.

First entries are determined by contour pinches: a first-entry singularity occurs when the auxiliary-energy contour can no longer be deformed away from singular divisors. For example, the pole $i\varepsilon$8 and branch point $i\varepsilon$9 approach the real axis from opposite sides as $$C(x_1,x_2,x_3,y_1,y_2,y_3)=\sum_{a=1}^{6}R_a\,\Phi_a .$$0, producing the divisor $$C(x_1,x_2,x_3,y_1,y_2,y_3)=\sum_{a=1}^{6}R_a\,\Phi_a .$$1; the conjugate locus $$C(x_1,x_2,x_3,y_1,y_2,y_3)=\sum_{a=1}^{6}R_a\,\Phi_a .$$2 does not pinch on the physical sheet. Six physical pinch circuits for $$C(x_1,x_2,x_3,y_1,y_2,y_3)=\sum_{a=1}^{6}R_a\,\Phi_a .$$3 yield exactly the factors appearing in its first entries — notably only sums of energies occur, selected by the positive-energy $$C(x_1,x_2,x_3,y_1,y_2,y_3)=\sum_{a=1}^{6}R_a\,\Phi_a .$$4 prescription.

Second entries follow from localised discontinuities. Decomposing the contour into a regular part (analytic in the letter) plus a small loop around the pinched pole reduces the discontinuity to a residue; the remaining one-dimensional integral is performed by closing the contour onto the logarithmic branch cut of $$C(x_1,x_2,x_3,y_1,y_2,y_3)=\sum_{a=1}^{6}R_a\,\Phi_a .$$5. Since the jump across that cut is $$C(x_1,x_2,x_3,y_1,y_2,y_3)=\sum_{a=1}^{6}R_a\,\Phi_a .$$6, no new dilogarithm is generated, and the integral collapses to endpoint values of an explicitly constructed dlog quantity $$C(x_1,x_2,x_3,y_1,y_2,y_3)=\sum_{a=1}^{6}R_a\,\Phi_a .$$7. Careful branch fixing of $$C(x_1,x_2,x_3,y_1,y_2,y_3)=\sum_{a=1}^{6}R_a\,\Phi_a .$$8 at the endpoints — fixed by requiring the cut to connect a zero and a pole of $$C(x_1,x_2,x_3,y_1,y_2,y_3)=\sum_{a=1}^{6}R_a\,\Phi_a .$$9 — then reproduces exactly the odd letters $R_a \sim 1/\sqrt{\Sigma_a}$0, $R_a \sim 1/\sqrt{\Sigma_a}$1 of the master formula. This constitutes an independent derivation of the closed form.

Bootstrap from Landau singularities

The third result shows that still less information suffices. Given only the leading singularities, the Landau loci, and candidate algebraic letters — constructed by searching for polynomials $R_a \sim 1/\sqrt{\Sigma_a}$2 such that $R_a \sim 1/\sqrt{\Sigma_a}$3 factorises into rational products of signed energy combinations — the symbol is fixed uniquely by consistency conditions alone. Integrability of the weight-two ansatz leaves a one-dimensional solution space for each sector (e.g. a $R_a \sim 1/\sqrt{\Sigma_a}$4 ansatz with thirty coefficients for $R_a \sim 1/\sqrt{\Sigma_a}$5). The full correlator is then assembled by imposing $R_a \sim 1/\sqrt{\Sigma_a}$6 permutation symmetry, conformal Ward identities (which act letter by letter and hold identically, providing overdetermined consistency checks), and one boundary datum, the total-energy residue. Since none of these ingredients is triangle-specific, this constitutes a viable bootstrap strategy for higher-point one-loop correlators, whose dressing integrals are higher-dimensional and difficult to evaluate directly.

Positivity properties

As a further structural result, after division by $R_a \sim 1/\sqrt{\Sigma_a}$7 the correlator $R_a \sim 1/\sqrt{\Sigma_a}$8 is shown to be a Stieltjes function of each squared energy separately and jointly completely monotone in all six squared energies $R_a \sim 1/\sqrt{\Sigma_a}$9. Two representations establish this. A Feynman-parametric form displays the dependence on each $1/E_T$0 or $1/E_T$1 through a single non-negative Stieltjes kernel superposed with positive weight. More strongly, a fully parametric Schwinger representation gives a homogeneous generalised multivariate Stieltjes representation of order three,

$1/E_T$2

with manifestly non-negative measure, from which joint complete monotonicity follows immediately. These statements refer to the positive analytic extension in which the squared energies vary independently over the positive orthant; they hold in particular throughout the Euclidean region.

Limitations and open questions

The paper is candid about scope. The conformally coupled $1/E_T$3 seed theory is not itself realistic; whether tree-level weight-shifting constructions extend systematically to loop-level in–in correlators remains open. The Ward identity analysis assumes generic momenta to avoid contact terms. The Stieltjes and monotonicity results apply to the analytic extension beyond the strict physical region, not only to on-shell kinematics. Most significantly, the extension to the one-loop four-site (box) correlator is left open, hinging on basic unresolved questions: whether the box is expressible in multiple polylogarithms or requires more general functions, what its transcendental weight is, and whether it has uniform weight. The authors expect the dressed representation to bear on these questions but do not settle them.

Conclusion

This paper delivers a substantially more economical closed form for the one-loop triangle cosmological correlator — forty-two real dilogarithms organised into six Galois-invariant leading-singularity sectors, free of regulators and spurious singularities — validated numerically against three independent representations and checked against Ward identities and singular limits. Methodologically, it demonstrates that the symbol can be reconstructed either directly from dressed-integral contour discontinuities or bootstrapped from Landau data plus integrability, symmetry, and Ward identities, and it establishes exact positivity structure via multivariate Stieltjes representations. These techniques, being largely graph-independent, provide a concrete template for attacking higher-point one-loop cosmological correlators.