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-byparts identities, has an algebraic leading
singularity and thus can only depend on non-trivial geometries through its
subsectors.
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.
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.
We present the analytic computation of all the planar master integrals which contribute to the two-loop scattering amplitudes for Higgs arrow right 3 partons, with full heavy-quark mass dependence. ...These are relevant for the NNLO corrections to fully inclusive Higgs production and to the NLO corrections to Higgs production in association with a jet, in the full theory. The computation is performed using the differential equations method. Whenever possible, a basis of master integrals that are pure functions of uniform weight is used. The result is expressed in terms of one-fold integrals of polylogarithms and elementary functions up to transcendental weight four. Two integral sectors are expressed in terms of elliptic integrals. We show that by introducing a one-dimensional parametrization of the integrals the relevant second order differential equation can be readily solved, and the solution can be expressed to all orders of the dimensional regularization parameter in terms of iterated integrals over elliptic kernels. We express the result for the elliptic sectors in terms of two and three-fold iterated integrals, which we find suitable for numerical evaluations. This is the first time that four-point multiscale Feynman integrals have been computed in a fully analytic way in terms of elliptic 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.
Abstract We present the analytic calculation of the two-loop QCD corrections to the decay width of a Higgs boson into a photon and a Z boson. The calculation is carried out using integration-by-parts ...identities for the reduction to master integrals of the scalar integrals, in terms of which we express the amplitude. The calculation of the master integrals is performed using differential equations applied to a set of functions suitably chosen to be of uniform weight. The final result is expressed in terms of logarithms and polylogarithmic functions Li^sub 2^, Li^sub 3^, Li^sub 4^ and Li^sub 2,2^.
We present the analytic calculation of the two-loop QCD corrections to the decay width of a Higgs boson into a photon and a Z boson. The calculation is carried out using integration-by-parts ...identities for the reduction to master integrals of the scalar integrals, in terms of which we express the amplitude. The calculation of the master integrals is performed using differential equations applied to a set of functions suitably chosen to be of uniform weight. The final result is expressed in terms of logarithms and polylogarithmic functions Li sub(2), Li sub(3), Li sub(4) and Li sub(2,2).
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.
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 \(m_{Z}/p_\perp\).
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.
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