arXiv subjects

Hjalte Frellesvig

Publications and source records attributed to Hjalte Frellesvig.

At least 19 recordsLinked to original sources

arXiv2026

Magic Relations and Critical Varieties of Feynman Integrals

Magic relations are a class of integration-by-parts identities where all integrals in the generating sector drop out. Since their presence causes several otherwise successful methods in the Feynman-integral computational pipeline to break down, they are important to detect and understand. In this paper, we take a first step toward a systematic characterization of such identities. Specifically, we observe and argue that the occurrence of magic relations always coincides with the presence of higher-dimensional critical varieties in the generating sector. This provides a practical computational test to check if a family of Feynman integrals can contain magic relations and to find them, which we implement in the ancillary Mathematica file Magic-Test.m. Additionally, we discuss how to count the number of master integrals in the presence of higher-dimensional critical varieties, classify the behavior of magic relations under symmetries, and we discuss their interplay with cuts.

hep-th↗

arXiv2025

The spectrum of Feynman-integral geometries at two loops

We provide a complete classification of the Feynman-integral geometries at two-loop order in four-dimensional Quantum Field Theory with standard quadratic propagators. Concretely, we consider a finite basis of integrals in the 't Hooft--Veltman scheme, i.e. with $D$-dimensional loop momenta and four-dimensional external momenta, which belong to 79 independent topologies, or sectors. Then, we analyze the leading singularities of the integrals in those sectors for generic values of the masses and momenta, using the loop-by-loop Baikov representation. Aside from the Riemann sphere, we find that elliptic curves, hyperelliptic curves of genus 2 and 3 as well as K3 surfaces occur. Moreover, we find a smooth and non-degenerate Del Pezzo surface of degree 2, a particular Fano variety known to be rationalizable, resulting in a curve of geometric genus 3. These geometries determine the space of functions relevant for Quantum Field Theories at two-loop order, including in the Standard Model.

hep-th↗

arXiv2025

Classification of Feynman integral geometries for black-hole scattering at 5PM order

We provide a complete classification of the Feynman integral geometries relevant to the scattering of two black holes at fifth order in the post-Minkowskian (PM) expansion, i.e. at four loops. The analysis includes integrals relevant to both the conservative and dissipative dynamics, as well as to all orders in the self-force (SF) expansion, i.e. the 0SF, 1SF and 2SF orders. By relating the geometries of integrals across different loop orders and integral families, we find that out of the 16,596 potentially contributing integral topologies, only 70 need to be analyzed in detail. By further computing their leading singularities using the loop-by-loop Baikov representation, we show that there only appear two different three-dimensional Calabi-Yau geometries and two different K3 surfaces at this loop order, which together characterize the space of functions beyond polylogarithms to which the 5PM integrals evaluate.

hep-th↗

arXiv2025

Calabi-Yau Feynman integrals in gravity: $\varepsilon$-factorized form for apparent singularities

We study a recently identified four-loop Feynman integral that contains a three-dimensional Calabi-Yau geometry and contributes to the scattering of black holes in classical gravity at fifth post-Minkowskian and second self-force order (5PM 2SF) in the conservative sector. In contrast to previously studied Calabi-Yau Feynman integrals, the higher-order differential equation that this integral satisfies in dimensional regularization exhibits $\varepsilon$-dependent apparent singularities. We introduce an appropriate ansatz which allows us to bring such cases into an $\varepsilon$-factorized form. As a proof of principle, we apply it to the integral at hand.

hep-th↗

arXiv2025

The Loop-by-Loop Baikov Representation -- Strategies and Implementation

In this paper, we discuss the Baikov representation of Feynman integrals in its standard and loop-by-loop variants. The Baikov representation is a parametric representation, which has as its defining feature the fact that the integration variables are the propagators of the Feynman integral. For the loop-by-loop Baikov representation, we discuss in detail a strategy for how to make an optimal parametrization which is one that minimizes the number of extra integration variables that have to be introduced for a given Feynman integral. Furthermore, we present a Mathematica implementation, named BaikovPackage, that is able to generate the Baikov representation in its standard and loop-by-loop varieties. We also discuss some subtleties and open problems regarding Baikov representations.

hep-th↗

arXiv2024

General Relativity from Intersection Theory

This paper combines the post-Minkowskian expansion of general relativity with the language of intersection theory. Because of the nature of the soft limit inherent to the post-Minkowskian expansion, the intersection-based approach is of enhanced utility in that theory compared to a generic quantum field theory. In the language of intersection theory, Feynman integrals are rephrased in terms of twisted cocycles. The intersection number is a pairing between two such cocycles and its existence allows for the direct projection onto a basis of master integrals. In this paper we use this approach to compute the second post-Minkowskian contribution to the scattering of two compact astronomical objects, getting results in agreement with previous findings.

gr-qc↗

arXiv2024

Classifying post-Minkowskian geometries for gravitational waves via loop-by-loop Baikov

We use the loop-by-loop Baikov representation to investigate the geometries in Feynman integrals contributing to the classical dynamics of a black-hole two-body system in the post-Minkowskian expansion of general relativity. These geometries determine the spaces of functions to which the corresponding Feynman diagrams evaluate. As a proof of principle, we provide a full classification of the geometries appearing up to three loops, i.e. fourth post-Minkowskian order, for all diagrams relevant to the conservative as well as the dissipative dynamics, finding full agreement with the literature. Moreover, we show that the non-planar top topology at four loops, which is the most complicated sector with respect to integration-by-parts identities, has an algebraic leading singularity and thus can only depend on non-trivial geometries through its subsectors.

hep-th↗

arXiv2024

Calabi-Yau meets Gravity: A Calabi-Yau three-fold at fifth post-Minkowskian order

We study geometries occurring in Feynman integrals that contribute to the scattering of black holes in the post-Minkowskian expansion. These geometries become relevant to gravitational-wave production during the inspiralling phase of binary black hole mergers through the classical conservative potential. At fourth post-Minkowskian order, a K3 surface is known to occur in a three-loop integral, leading to elliptic integrals in the result. In this letter, we identify a Calabi-Yau three-fold in a four-loop integral, contributing at fifth post-Minkowskian order. The presence of this Calabi-Yau geometry indicates that completely new functions occur in the full analytical results at this order.

hep-th↗

arXiv2024

On $\varepsilon$-factorised bases and pure Feynman integrals

We investigate $\varepsilon$-factorised differential equations, uniform transcendental weight and purity for Feynman integrals. We are in particular interested in Feynman integrals beyond the ones which evaluate to multiple polylogarithms. We show that a $\varepsilon$-factorised differential equation does not necessarily lead to Feynman integrals of uniform transcendental weight. We also point out that a proposed definition of purity works locally, but not globally.

hep-th↗

arXiv2023

Intersection Numbers, Polynomial Division and Relative Cohomology

We present a simplification of the recursive algorithm for the evaluation of intersection numbers for differential $n$-forms, by combining the advantages emerging from the choice of delta-forms as generators of relative twisted cohomology groups and the polynomial division technique, recently proposed in the literature. We show that delta-forms capture the leading behaviour of the intersection numbers in presence of evanescent analytic regulators, whose use is, therefore, bypassed. This simplified algorithm is applied to derive the complete decomposition of two-loop planar and non-planar Feynman integrals in terms of a master integral basis. More generally, it can be applied to derive relations among twisted period integrals, relevant for physics and mathematical studies.

hep-th↗

arXiv2023

Intersection Numbers from Higher-order Partial Differential Equations

We propose a new method for the evaluation of intersection numbers for twisted meromorphic $n$-forms, through Stokes' theorem in $n$ dimensions. It is based on the solution of an $n$-th order partial differential equation and on the evaluation of multivariate residues. We also present an algebraic expression for the contribution from each multivariate residue. We illustrate our approach with a number of simple examples from mathematics and physics.

hep-th↗

arXiv2022

Functions Beyond Multiple Polylogarithms for Precision Collider Physics

Feynman diagrams constitute one of the essential ingredients for making precision predictions for collider experiments. Yet, while the simplest Feynman diagrams can be evaluated in terms of multiple polylogarithms -- whose properties as special functions are well understood -- more complex diagrams often involve integrals over complicated algebraic manifolds. Such diagrams already contribute at NNLO to the self-energy of the electron, $t \bar{t}$ production, $γγ$ production, and Higgs decay, and appear at two loops in the planar limit of maximally supersymmetric Yang-Mills theory. This makes the study of these more complicated types of integrals of phenomenological as well as conceptual importance. In this white paper contribution to the Snowmass community planning exercise, we provide an overview of the state of research on Feynman diagrams that involve special functions beyond multiple polylogarithms, and highlight a number of research directions that constitute essential avenues for future investigation.

hep-ph↗

arXiv2022

On Epsilon Factorized Differential Equations for Elliptic Feynman Integrals

In this paper we develop and demonstrate a method to obtain epsilon factorized differential equations for elliptic Feynman integrals. This method works by choosing an integral basis with the property that the period matrix obtained by integrating the basis over a complete set of integration cycles is diagonal. The method is a generalization of a similar method known to work for polylogarithmic Feynman integrals. We demonstrate the method explicitly for a number of Feynman integral families with an elliptic highest sector.

hep-th↗

arXiv2021

Decomposition of Feynman Integrals by Multivariate Intersection Numbers

We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers and provide three different approaches for a direct decomposition of Feynman integrals, which we dub the straight decomposition, the bottom-up decomposition, and the top-down decomposition. These algorithms exploit the unitarity structure of Feynman integrals by computing intersection numbers supported on cuts, in various orders, thus showing the synthesis of the intersection-theory concepts with unitarity-based methods and integrand decomposition. We perform explicit computations to exemplify all of these approaches applied to Feynman integrals, paving a way towards potential applications to generic multi-loop integrals.

hep-th↗

arXiv2021

Cuts and Isogenies

We consider the genus-one curves which arise in the cuts of the sunrise and in the elliptic double-box Feynman integrals. We compute and compare invariants of these curves in a number of ways, including Feynman parametrization, lightcone and Baikov (in full and loop-by-loop variants). We find that the same geometry for the genus-one curves arises in all cases, which lends support to the idea that there exists an invariant notion of genus-one geometry, independent on the way it is computed. We further indicate how to interpret some previous results which found that these curves are related by isogenies instead.

hep-th↗

arXiv2021

On the Application of Intersection Theory to Feynman Integrals: The Univariate Case

This document is a contribution to the proceedings of the MathemAmplitudes 2019 conference held in December 2019 in Padova, Italy. A key step in modern high energy physics scattering amplitudes computation is to express the latter in terms of a minimal set of Feynman integrals using linear relations. In this work we present an innovative approach based on intersection theory, in order to achieve this decomposition. This allows for the direct computation of the reduction, projecting integrals appearing in the scattering amplitudes onto an integral basis in the same fashion as vectors may be projected onto a vector basis. Specifically, we will derive and discuss few identities between maximally cut Feynman integrals, showing their direct decomposition. This contribution will focus on the univariate part of the story, with the multivariate generalisation being discussed in a different contribution by Gasparotto and Mandal.

hep-ph↗

arXiv2020

The complete set of two-loop master integrals for Higgs + jet production in QCD

In this paper we complete the computation of the two-loop master integrals relevant for Higgs plus one jet production initiated in arXiv:1609.06685, arXiv:1907.13156, arXiv:1907.13234. We compute the integrals by defining differential equations along contours in the kinematic space, and by solving them in terms of one-dimensional generalized power series. This method allows for the efficient evaluation of the integrals in all kinematic regions, with high numerical precision. We show the generality of our approach by considering both the top- and the bottom-quark contributions. This work along with arXiv:1609.06685, arXiv:1907.13156, arXiv:1907.13234 provides the full set of master integrals relevant for the NLO corrections to Higgs plus one jet production, and for the real-virtual contributions to the NNLO corrections to inclusive Higgs production in QCD in the full theory.

hep-ph↗

arXiv2020

Two-Loop QCD-EW Master Integrals for Z Plus Jet Production at Large Transverse Momentum

The production of electroweak $Z$ bosons that decay to neutrinos and recoil against jets with large transverse momentum $p_\perp$ is an important background process to searches for dark matter at the Large Hadron Collider (LHC). To fully benefit from opportunities offered by the future high-luminosity LHC, the theoretical description of the $pp \to Z+j$ process should be extended to include mixed QCD-electroweak corrections. The goal of this paper is to initiate the computation of such corrections starting with the calculation of the Feynman integrals needed to describe two-loop QCD-electroweak contributions to $q \bar q \to Z+g$ scattering amplitudes. Making use of the hierarchy between the large transverse momenta of the recoiling jet, relevant for heavy dark matter searches, and the $Z$ boson mass $m_{Z}$, we present the relevant master integrals as a series expansion in $m_{Z}/p_\perp$.

hep-ph↗