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Evaluation of multi-loop multi-scale Feynman integrals for precision physics

Ievgen Dubovyk, Ayres Freitas, Janusz Gluza, Krzysztof Grzanka, Martijn Hidding, Johann Usovitsch

TL;DR

The paper introduces a semi-numerical framework to compute massive multi-loop, multi-scale Feynman integrals by solving a differential equation system with boundary terms specified at Euclidean kinematics. Boundary values are obtained numerically using sector decomposition and transported to physical Minkowski points via series expansions of the DE, achieving 8+ digits of precision with internal cross-checks. Demonstrations on 3-loop self-energy and vertex integrals and 2-loop box diagrams show the method’s ability to address corrections required for HL-LHC and future colliders, surpassing many existing approaches. The approach emphasizes Euclidean boundary conditions, automates the workflow with tools like Kira, Reduze, pySecDec, and DiffExp, and discusses bottlenecks and extensions to more loops and scales.

Abstract

Modern particle physics is increasingly becoming a precision science that relies on advanced theoretical predictions for the analysis and interpretation of experimental results. The planned physics program at the LHC and future colliders will require three-loop electroweak and mixed electroweak-QCD corrections to single-particle production and decay processes and two-loop electroweak corrections to pair production processes, all of which are beyond the reach of existing analytical and numerical techniques in their current form. This article presents a new semi-numerical approach based on differential equations with boundary terms specified at Euclidean kinematic points. These Euclidean boundary terms can be computed numerically with high accuracy using sector decomposition or other numerical methods. They are then mapped to the physical kinematic configuration with a series solution of the differential equation system. The method is able to deliver 8 or more digits precision, and it has a built-in mechanism for checking the accuracy of the obtained results. Its efficacy is illustrated with examples for three-loop self-energy and vertex integrals and two-loop box integrals.

Evaluation of multi-loop multi-scale Feynman integrals for precision physics

TL;DR

The paper introduces a semi-numerical framework to compute massive multi-loop, multi-scale Feynman integrals by solving a differential equation system with boundary terms specified at Euclidean kinematics. Boundary values are obtained numerically using sector decomposition and transported to physical Minkowski points via series expansions of the DE, achieving 8+ digits of precision with internal cross-checks. Demonstrations on 3-loop self-energy and vertex integrals and 2-loop box diagrams show the method’s ability to address corrections required for HL-LHC and future colliders, surpassing many existing approaches. The approach emphasizes Euclidean boundary conditions, automates the workflow with tools like Kira, Reduze, pySecDec, and DiffExp, and discusses bottlenecks and extensions to more loops and scales.

Abstract

Modern particle physics is increasingly becoming a precision science that relies on advanced theoretical predictions for the analysis and interpretation of experimental results. The planned physics program at the LHC and future colliders will require three-loop electroweak and mixed electroweak-QCD corrections to single-particle production and decay processes and two-loop electroweak corrections to pair production processes, all of which are beyond the reach of existing analytical and numerical techniques in their current form. This article presents a new semi-numerical approach based on differential equations with boundary terms specified at Euclidean kinematic points. These Euclidean boundary terms can be computed numerically with high accuracy using sector decomposition or other numerical methods. They are then mapped to the physical kinematic configuration with a series solution of the differential equation system. The method is able to deliver 8 or more digits precision, and it has a built-in mechanism for checking the accuracy of the obtained results. Its efficacy is illustrated with examples for three-loop self-energy and vertex integrals and two-loop box integrals.
Paper Structure (13 sections, 12 equations, 4 figures, 1 table)

This paper contains 13 sections, 12 equations, 4 figures, 1 table.

Table of Contents

  1. I. Introduction.
  2. II. Description of the method.
  3. III. Results and discussion.
  4. IV. Summary and Outlook.
  5. Acknowledgements
  6. Supplemental Material
  7. Numerical results for relevant Master Integrals
  8. Input parameters
  9. 3-loop self-energy integrals with top quarks
  10. Two-loop box diagram
  11. Tracing the run time of the calculations
  12. Tracing the error of the calculations
  13. Iterative approach for multi-scale kinematics

Figures (4)

  • Figure 1: Illustration of the DE transport method. The boundary conditions for the integral $f_i$ are evaluated at one or several Euclidean points $\rm A_k$, where the integral is purely real and one can obtain robustly converging numerical results with the SD method (using the package pySecDec in our case). The boundary value(s) are transported to the physical kinematic point, using solutions of the DE system eq. \ref{['eq:general-de']} derived with DiffExp, yielding the final result indicated by the red dot in the figure. The numerical uncertainty of a boundary value translates to an error estimate of the final result, as illustrated by the error bars in a zoomed-in area in the dotted circle, which permits a non-trivial cross-check if several boundary values $\rm A_k$ are employed.
  • Figure 2: Three loop self-energy non-planar and planar vertex diagrams which correspond to integrals in \ref{['eq:lhnp']} and \ref{['eq:vtwPl']}, respectively. W, Z and t stand for the W-boson, Z-boson and top quark, respectively.
  • Figure 3: Three loop self-energy non-planar integral defined in (\ref{['eq:taNp1']}). $\text{Z}$ and $\text{t}$ stands for the massive SM Z gauge boson and the top quark, respectively.
  • Figure 4: Two-loop box diagram with four scales: $s,t,m_1,m_2$.