Canonical differential equations beyond polylogs

Feynman integrals whose associated geometries extend beyond the Riemann sphere, such as elliptic curves and Calabi-Yau varieties, are increasingly relevant in modern precision calculations. They arise not only in collider cross-section calculations, but also in the post-Minkowskian expansion of gravitational-wave scattering. A powerful approach to compute integrals of this type is via differential equations, particularly when cast in a canonical form, which simplifies their $\varepsilon$-expansion and makes analytic properties manifest. In these proceedings, we will present a method to systematically construct canonical differential equations even for integrals that evaluate beyond multiple polylogarithms. The discussion is kept as light as possible, focusing on the two-loop sunrise integral, deferring the technical details to the original publications.

💡 Research Summary

The paper addresses the growing need to handle Feynman integrals whose underlying geometry is more complex than the Riemann sphere, focusing on elliptic curves and Calabi–Yau varieties. While traditional differential‑equation methods for Feynman integrals rely on the existence of a canonical ε‑factorised form d J = ε A(s) J, where A(s) is a matrix of simple‑pole d log‑forms, this structure breaks down when the integrals evaluate to functions beyond multiple polylogarithms (MPLs). The authors illustrate the problem using the two‑loop sunrise integral, which reduces to MPLs when one internal mass vanishes (M²=0) but requires elliptic functions when all three masses are non‑zero (M²≠0).

First, they review the MPL case, emphasizing that the only independent differential forms are of the type d x/(x‑aᵢ), leading to pure UT (uniform transcendental weight) integrals and a straightforward canonical system. They then explain why elliptic geometries demand additional forms, namely d x/y and d x y/(x²‑s₁), because a torus possesses two independent cycles beyond the punctures.

Using the Baikov representation, the authors compute maximal cuts of the sunrise family in d=2 dimensions. For M²=0 the quartic polynomial under the square root degenerates, yielding a Riemann sphere with two branch points. Two independent contours C₁ (around a simple pole) and C₂ (around the branch cut) give rise to leading singularities that are pure logarithmic. By normalising the master integrals with the inverse of these singularities they obtain a pure basis {J₁,J₂} that satisfies an ε‑factorised differential equation with d log entries.

When M²≠0 the quartic polynomial defines a genuine elliptic curve. The maximal cut now produces a holomorphic differential on the curve, and there are two independent cycles C₁ and C₂. Their integrals ψ₀ and ψ₁ correspond respectively to the first‑kind elliptic integral (holomorphic) and a logarithmically singular elliptic integral of the third kind. A third master integral, associated with a double pole at infinity, brings in an elliptic integral of the second kind. The presence of higher poles prevents a naïve ε‑expansion from preserving the pure‑logarithmic structure.

To overcome this, the authors introduce the period matrix W, whose columns are the independent solutions (ψ₀, ψ₁) and their s‑derivatives. They decompose W into a semisimple part W_ss (containing the algebraic leading singularities) and a unipotent part W_u (encoding the transcendental, weight‑one function τ=ψ₁/ψ₀). W_ss has weight zero and can be used to normalise the master integrals, restoring uniform transcendental weight. The unipotent part yields a pure function that satisfies a canonical differential equation.

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