arXiv subjects

Sara Maggio

Publications and source records attributed to Sara Maggio.

12 recordsLinked to original sources

arXiv2026

Discrete symmetries of Feynman integrals

We perform a comprehensive study of a certain class of discrete symmetries of families of Feynman integrals, defined as affine changes of variables that map different sectors of the family into each other. We show that these transformations are always encoded into permutations of the Feynman parameters that relate the Lee-Pomeransky polynomials of the two sectors, irrespective of the integral representation used to define the Feynman integrals. We then construct an affine map in loop-momentum space that encodes such a permutation. We also show that these symmetries can be naturally embedded into the framework of twisted cohomology theories, and the period and intersection parings are invariant under the symmetry transformations. If we focus on symmetries within a fixed sector, we obtain a group acting on the twisted cohomology group, and we study the decomposition of this action into irreducible representations. One of our main mathematical results is that the character of this representation is proportional to the Euler characteristic of the corresponding fixed-point set. We then study the implications for Feynman integrals, in particular for the intersection matrix in a canonical basis. We also present a formula for the number of master integrals in a given sector in the presence of a non-trivial symmetry group in terms of the Euler characteristics of fixed-point sets. As an application, we obtain the numbers of master integrals for banana integrals with up to four loops for arbitrary configurations of non-zero masses. In order to achieve our results, we had to combine tools from various different areas of mathematics, including graph theory, group theory and algebraic topology.

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↗

arXiv2026

Intersection theory and canonical differential equations

In these proceedings we will review recent progress in applying ideas from the mathematical framework of twisted cohomology to the study of canonical differential equations for Feynman integrals. Firstly, we will show how the intersection matrix can shed some light on the nature of the canonical basis of a Feynman integral family, a concept still not fully understood in the general case. In particular we will show how the intersection matrix can detect hidden linear dependencies of the iterated integrals resulting from an $\eps$-factorized differential equation, which are difficult to find otherwise. Furthermore, we will explain how the intersection matrix can help in deriving (polynomial) relations between the transcendental functions occurring in the rotation to the canonical basis. This allows us to simplify the rotation, and furthermore leads to simplifications in the final result. The discussion we be kept as light as possible, focusing on a simple running example and deferring the technical details to the original publications.

hep-th↗

arXiv2025

Three-loop banana integrals with three equal masses

We obtain and solve the canonical differential equations for the three-loop banana integrals in dimensional regularisation when three of the four masses are equal. The K3 surface associated with the maximal cuts factorises into a product of two elliptic curves. This allows us to express the differential forms in the canonical differential equations in terms of meromorphic modular forms. We present a rigorous proof that these differential forms only have simple poles and that they define independent cohomology classes. We also present explicit results for all master integrals in terms of iterated integrals of meromorphic modular forms (and integrals thereof). This is the first time that it was possible to express the results of a Feynman integral associated with a K3 geometry and depending on two dimensionless ratios in terms of functions that have previously been studied in the literature.

hep-th↗

arXiv2025

Canonical differential equations and intersection matrices

Differential equations are one of the main approaches to evaluate multi-loop Feynman integrals. The construction of a canonical or $\varepsilon$-factorised basis for multi-loop integrals remains a key bottleneck for this approach, especially when the differential equation involves non dlog-forms. Recently, several methods have been proposed to find $\varepsilon$-factorised differential equations. Many of them introduce new functions that are themselves defined as iterated integrals. If and when these iterated integrals can be explicitly evaluated in terms of other classes of functions remains an open problem. In this paper we elaborate on the recent proposal that one can use the fact that the intersection matrix computed in a canonical basis can be used to derive polynomial relations between these iterated integrals. On the one hand, we discuss properties of the canonical intersection matrix, in particular methods to determine the intersection matrix in a canonical basis. On the other hand we show how one can reduce the non-linear constraints on the iterated integrals to linear ones. We illustrate these ideas on examples involving Calabi-Yau varieties and higher-genus Riemann surfaces.

hep-th↗

arXiv2025

Three-loop banana integrals with four unequal masses

We present a system of canonical differential equations satisfied by the three-loop banana integrals with four distinct non-zero masses in $D = 2-2\eps$ dimensions. Together with the initial condition in the small-mass limit, this provides all the ingredients to find analytic results for three-loop banana integrals in terms of iterated integrals to any desired order in the dimensional regulator. To obtain this result, we rely on recent advances in understanding the K3 geometry underlying these integrals and in how to construct rotations to an $\eps$-factorized basis. This rotation typically involves the introduction of objects defined as integrals of (derivatives of) K3 periods and rational functions. We apply and extend a method based on results from twisted cohomology to identify relations among these functions, which allows us to reduce their number considerably. We expect that the methods that we have applied here will prove useful to compute further multiloop multiscale Feynman integrals attached to non-trivial geometries.

hep-th↗

arXiv2025

On canonical differential equations for Calabi-Yau multi-scale Feynman integrals

We generalise a method recently introduced in the literature, that derives canonical differential equations, to multi-scale Feynman integrals with an underlying Calabi-Yau geometry. We start by recomputing a canonical form for the sunrise integral with all unequal masses. Additionally, we compute for the first time a canonical form for the three-loop banana integral with two unequal masses and for a four-loop banana integral with two unequal masses. For the integrals we compute, we find an $\epsilon$-form whose connection has at most simple poles. We motivate our construction by studying the Picard-Fuchs operators acting on the integrals considered. In the appendices, we give a constructive explanation for why our generalisation works.

hep-th↗

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

Feynman integrals, elliptic integrals and two-parameter K3 surfaces

The three-loop banana integral with three equal masses and the conformal two-loop five-point traintrack integral in two dimensions are related to a two-parameter family of K3 surfaces. We compute the corresponding periods and the mirror map, and we show that they can be expressed in terms of ordinary modular forms and functions. In particular, we find that the maximal cuts of the three-loop banana integral with three equal masses can be written as a product of two copies of the maximal cuts of the two-loop equal-mass sunrise integral. Our computation reveals a hidden symmetry of the banana integral not manifest from the Feynman integral representation, which corresponds to exchanging the two copies of the sunrise elliptic curve.

hep-th↗

arXiv2024

Some conjectures around magnetic modular forms

We study a class of meromorphic modular forms characterised by Fourier coefficients that satisfy certain divisibility properties. We present new candidates for these so-called magnetic modular forms, and we conjecture properties that these functions should obey. In particular, we conjecture that magnetic modular forms are closed under the standard operators acting on spaces of modular forms (SL$_2(\mathbb{Z})$ action, Hecke and Atkin-Lehner operators), and that they are characterised by algebraic residues and vanishing period polynomials. We use our conjectures to construct examples of real-analytic modular forms with poles.

math.NT↗

arXiv2024

Two-loop non-planar four-point topology with massive internal loop

We study a set of two-loop non-planar master integrals needed for the NNLO QCD corrections to diphoton and dijet production at hadron colliders. The top-sector topology contains an internal massive fermion loop and is known to contain elliptic curves. Leveraging the method of differential equations, we provide a comprehensive discussion for deriving an $\epsilon$-factorized differential equation related to the most intricate sector within the Feynman integral family. Despite the dependence on multiple scales and the presence of two elliptic sectors, we demonstrate how to leverage the properties of their maximal cuts and the factorization of the Picard-Fuchs operator to deal with the complexity of the analytic computation. In particular, we construct a transformation matrix that brings the differential equations into a format enabling the convenient expression of analytic results in terms of Chen's iterated integrals.

hep-th↗

arXiv2023

Two-Loop Infrared Renormalization with On-shell Methods

Within the framework proposed by Caron-Huot and Wilhelm, we give a recipe for computing infrared anomalous dimensions purely on-shell, efficiently up to two loops in any massless theory. After introducing the general formalism and reviewing the one-loop recipe, we extract a practical formula that relates two-loop infrared anomalous dimensions to certain two- and three-particle phase space integrals with tree-level form factors of conserved operators. We then provide several examples of the use of the two-loop formula and comment on some of its formal aspects, especially the cancellation of 'one-loop squared' spurious terms. The present version of the paper is augmented with a detailed treatment of the structure of infrared divergences in massless theories of scalars and fermions up to two loops. In the calculation we encounter divergent phase space integrals and show in detail how these cancel among each other as required by the finiteness of the anomalous dimension. As a non-trivial check of the method, we also perform the computation with a standard diagrammatic approach, finding perfect agreement.

hep-th↗