arXiv subjects

Fabian J. Wagner

Publications and source records attributed to Fabian J. Wagner.

7 recordsLinked to original sources

arXiv2026

Integrand Analysis, Leading Singularities and Canonical Bases beyond Polylogarithms

In this paper, we elaborate on the connection between leading singularities and canonical bases of Feynman integrals beyond polylogarithms. We start by discussing a notion of leading singularities in dimensional regularization, which can be generalized from the Riemann sphere to more complex geometries, and use it to demonstrate how selecting Feynman integrals with unit leading singularities necessitates introducing new transcendental functions related to the periods of the underlying geometries. Integrals with unit leading singularities in this generalized sense, satisfy $ε$-factorized differential equations, and the new transcendental functions are in direct correspondence to the new differential forms appearing in their Gauss-Manin connection. We argue that this construction is mathematically equivalent to the splitting of the period matrix into semi-simple and unipotent parts plus a clean-up step, and demonstrate its use with examples of increasing complexity that require the interplay of multiple geometries.

hep-th↗

arXiv2026

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.

hep-th↗

arXiv2025

Analytic two-loop amplitudes for di-jet and $γ+$jet production mediated by a heavy-quark loop

In this paper, we present analytical results for the two-loop QCD corrections to the production of two partons or a photon and a parton in hadronic collisions, mediated by loops of massive quarks. These amplitudes involve Feynman integrals defined on an elliptic curve. We compute them by generalizing our recent results for the production of two photons to include additional crossings of the corresponding master integrals, which we compute in terms of the same basis of independent iterated integrals. We discuss the analytical properties of the amplitudes, highlighting the cancellations of a large number of elliptic differential forms in their finite remainders. Finally, we elaborate on a strategy for their numerical evaluation based on generalized series expansions at singular points of the physical amplitude, through the introduction of suitable sets of variables that allow us to resolve all singularities.

hep-ph↗

arXiv2025

Aspects of canonical differential equations for Calabi-Yau geometries and beyond

We show how a method to construct canonical differential equations for multi-loop Feynman integrals recently introduced by some of the authors can be extended to cases where the associated geometry is of Calabi-Yau type and even beyond. This can be achieved by supplementing the method with information from the mixed Hodge structure of the underlying geometry. We apply these ideas to specific classes of integrals whose associated geometry is a one-parameter family of Calabi-Yau varieties, and we argue that the method can always be successfully applied to those cases. Moreover, we perform an in-depth study of the properties of the resulting canonical differential equations. In particular, we show that the resulting canonical basis is equivalent to the one obtained by an alternative method recently introduced in the literature. We apply our method to non-trivial and cutting-edge examples of Feynman integrals necessary for gravitational wave scattering, further showcasing its power and flexibility.

hep-th↗

arXiv2025

Analytic two-loop amplitudes for $q\bar{q}\to γγ$ and $gg \to γγ$ mediated by a heavy-quark loop

We address the analytic computation of the two-loop scattering amplitudes for the production of two photons in parton-parton scattering, mediated by loops of heavy quarks. Due to the presence of integrals of elliptic type, both partonic channels have been previously computed using semi-numerical methods. In this paper, leveraging new advances in the theory of differential equations for elliptic Feynman integrals, we derive a canonical basis for all integrals involved and compute them in terms of independent iterated integrals over elliptic and polylogarithmic differential forms. We use this representation to showcase interesting cancellations in the physical expressions for the scattering amplitudes. Furthermore, we address their numerical evaluation by producing series expansion representations for the whole amplitudes, which we demonstrate to be fast and numerically reliable across a large region of the phase space.

hep-ph↗

arXiv2023

An algorithmic approach to finding canonical differential equations for elliptic Feynman integrals

In recent years, differential equations have become the method of choice to compute multi-loop Feynman integrals. Whenever they can be cast into canonical form, their solution in terms of special functions is straightforward. Recently, progress has been made in understanding the precise canonical form for Feynman integrals involving elliptic polylogarithms. In this article, we make use of an algorithmic approach that proves powerful to find canonical forms for these cases. To illustrate the method, we reproduce several known canonical forms from the literature and present examples where a canonical form is deduced for the first time. Together with this article, we also release an update for INITIAL, a publicly available Mathematica implementation of the algorithm.

hep-ph↗

arXiv2023

On a procedure to derive $ε$-factorised differential equations beyond polylogarithms

In this manuscript, we elaborate on a procedure to derive $ε$-factorised differential equations for multi-scale, multi-loop classes of Feynman integrals that evaluate to special functions beyond multiple polylogarithms. We demonstrate the applicability of our approach to diverse classes of problems, by working out $ε$-factorised differential equations for single- and multi-scale problems of increasing complexity. To start we are reconsidering the well-studied equal-mass two-loop sunrise case, and move then to study other elliptic two-, three- and four-point problems depending on multiple different scales. Finally, we showcase how the same approach allows us to obtain $ε$-factorised differential equations also for Feynman integrals that involve geometries beyond a single elliptic curve.

hep-th↗