Rational Tracer: a Tool for Faster Rational Function Reconstruction

Rational Tracer (Ratracer) is a tool to simplify complicated arithmetic expressions using modular arithmetics and rational function reconstruction, with the main idea of separating the construction of expressions (via tracing, i.e. recording the list of operations) and their subsequent evaluation during rational reconstruction. Ratracer can simplify arithmetic expressions (provided as text files), solutions of linear equation systems (specifically targeting Integration-by-Parts (IBP) relations between Feynman integrals), and even more generally: arbitrary sequences of rational operations, defined in C++ using the provided library ratracer.h. Any of these can also be automatically expanded into series prior to reconstruction. This paper describes the usage of Ratracer specifically focusing on IBP reduction, and demonstrates its performance benefits by comparing with Kira+FireFly and Fire6. Specifically, Ratracer achieves a typical ~10x probe time and ~5x overall time speedup over Kira+FireFly, and even higher if only a few terms in $\varepsilon$ need to be reconstructed

Master integrals for all unitarity cuts of massless four-loop propagators

Among the unitarity cuts of massless 4-loop propagators two classes have remained unknown until recently: 2-loop 3-particle cuts, and 1-loop 4-particle cuts. In this article we shall discuss the calculation that completes the master integrals for these cuts: both the motivation and the methods (including dimensional recurrence relations and direct integration at higher space-time dimensions).Comment: 11 pages; contribution to the proceedings of RADCOR 2019; based on arXiv:1910.0752

Cutting massless four-loop propagators

Among the unitarity cuts of 4-loop massless propagators two kinds are currently fully known: the 2-particle cuts with 3 loops corresponding to form-factors, and the 5-particle phase-space integrals. In this paper we calculate master integrals for the remaining ones: 3-particle cuts with 2 loops, and 4-particle cuts with 1 loop. The 4-particle cuts are calculated by solving dimensional recurrence relations. The 3-particle cuts are integrated directly using 1->3 amplitudes with 2 loops, which we also re-derive here up to transcendentality weight 7. The results are verified both numerically, and by showing consistency with previously known integrals using Cutkosky rules. We provide the analytic results in the space-time dimension 4-2{\epsilon} as series in {\epsilon} with coefficients being multiple zeta values up to weight 12. In the ancillary files we also provide dimensional recurrence matrices and SummerTime files suitable for numerical evaluation of the series in arbitrary dimensions with any precision.Comment: 57 pages, 15 tables, 1 figure, auxiliary file

Two-loop amplitudes for t t ¯ H production: the quark-initiated N f -part

We present numerical results for the two-loop virtual amplitude entering the NNLO corrections to Higgs boson production in association with a top quark pair at the LHC, focusing, as a proof of concept of our method, on the part of the quark-initiated channel containing loops of massless or massive quarks. Results for the UV renormalised and IR subtracted two-loop amplitude for each colour structure are given at selected phase-space points and visualised in terms of surfaces as a function of two-dimensional slices of the full phase space

Targeting multi-loop integrals with neural networks

Numerical evaluations of Feynman integrals often proceed via a deformation of the integration contour into the complex plane. While valid contours are easy to construct, the numerical precision for a multi-loop integral can depend critically on the chosen contour. We present methods to optimize this contour using a combination of optimized, global complex shifts and a normalizing flow. They can lead to a significant gain in precision

Cutting massless four-loop propagators

Among the unitarity cuts of 4-loop massless propagators two kinds are currently fully known: the 2-particle cuts with 3 loops corresponding to form-factors, and the 5-particle phase-space integrals. In this paper we calculate master integrals for the remaining ones: 3-particle cuts with 2 loops, and 4-particle cuts with 1 loop. The 4-particle cuts are calculated by solving dimensional recurrence relations. The 3-particle cuts are integrated directly using 1→3 amplitudes with 2 loops, which we also re-derive here up to transcendentality weight 7. The results are verified both numerically, and by showing consistency with previously known integrals using Cutkosky rules. We provide the analytic results in the space-time dimension 4 − 2ε as series in ε with coefficients being multiple zeta values up to weight 12. In the supplementary material we also provide dimensional recurrence matrices and SummerTime files suitable for numerical evaluation of the series in arbitrary dimensions with any precision

Fuchsia : A tool for reducing differential equations for Feynman master integrals to epsilon form

We present Fuchsia — an implementation of the Lee algorithm, which for a given system of ordinary differential equations with rational coefficients $∂_xJ(x,ε)=\mathbb{A}(x,ε)J(x,ε)$ finds a basis transformation $\mathbb{T}(x,ε),i.e., J(x,ε)= \mathbb{T}(x,ε)J′(x,ε)$, such that the system turns into the epsilon form: $∂_x J′(x,ε)=ε \mathbb{S}(x)J′(x,ε)$, where $\mathbb{S}(x)$, where is a Fuchsian matrix. A system of this form can be trivially solved in terms of polylogarithms as a Laurent series in the dimensional regulator. That makes the construction of the transformation $\mathbb{T}(x,ε)$ crucial for obtaining solutions of the initial system.In principle, Fuchsia can deal with any regular systems, however its primary task is to reduce differential equations for Feynman master integrals. It ensures that solutions contain only regular singularities due to the properties of Feynman integrals

Master integrals for all unitarity cuts of massless four-loop propagators

Among the unitarity cuts of massless 4-loop propagators two classes have remained unknown until recently: 2-loop 3-particle cuts, and 1-loop 4-particle cuts. In this article we shall discuss the calculation that completes the master integrals for these cuts: both the motivation and the methods (including dimensional recurrence relations and direct integration at higher space-time dimensions)

Cutting massless four-loop propagators

Among the unitarity cuts of 4-loop massless propagators two kinds are currently fully known: the 2-particle cuts with 3 loops corresponding to form-factors, and the 5-particle phase-space integrals. In this paper we calculate master integrals for the remaining ones: 3-particle cuts with 2 loops, and 4-particle cuts with 1 loop. The 4-particle cuts are calculated by solving dimensional recurrence relations. The 3-particle cuts are integrated directly using 1→3 amplitudes with 2 loops, which we also re-derive here up to transcendentality weight 7. The results are verified both numerically, and by showing consistency with previously known integrals using Cutkosky rules. We provide the analytic results in the space-time dimension 4 − 2ε as series in ε with coefficients being multiple zeta values up to weight 12. In the supplementary material we also provide dimensional recurrence matrices and SummerTime files suitable for numerical evaluation of the series in arbitrary dimensions with any precision