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Numerical evaluation of the general massive 2-loop sunrise self-mass master integrals from differential equations

M. Caffo, H. Czyz, E. Remiddi

TL;DR

The paper develops a robust numerical framework to evaluate the four master integrals of the general massive two-loop sunrise self-mass by solving a system of differential equations in $p^2$ with a fourth-order Runge-Kutta method in the complex plane. It addresses the challenges posed by thresholds, pseudothresholds, and large-$|p^2|$ asymptotics, introducing specialized constructs and variable changes to maintain accuracy. The method is validated through extensive cross-checks, analytic expansions, and comparisons with established results, achieving high precision and demonstrating compatibility with existing literature. This approach offers a general, extensible tool for precise multi-loop integral calculations, scalable to more complex diagrams and additional external legs.

Abstract

The system of 4 differential equations in the external invariant satisfied by the 4 master integrals of the general massive 2-loop sunrise self-mass diagram is solved by the Runge-Kutta method in the complex plane. The method, whose features are discussed in details, offers a reliable and robust approach to the direct and precise numerical evaluation of Feynman graph integrals.

Numerical evaluation of the general massive 2-loop sunrise self-mass master integrals from differential equations

TL;DR

The paper develops a robust numerical framework to evaluate the four master integrals of the general massive two-loop sunrise self-mass by solving a system of differential equations in with a fourth-order Runge-Kutta method in the complex plane. It addresses the challenges posed by thresholds, pseudothresholds, and large- asymptotics, introducing specialized constructs and variable changes to maintain accuracy. The method is validated through extensive cross-checks, analytic expansions, and comparisons with established results, achieving high precision and demonstrating compatibility with existing literature. This approach offers a general, extensible tool for precise multi-loop integral calculations, scalable to more complex diagrams and additional external legs.

Abstract

The system of 4 differential equations in the external invariant satisfied by the 4 master integrals of the general massive 2-loop sunrise self-mass diagram is solved by the Runge-Kutta method in the complex plane. The method, whose features are discussed in details, offers a reliable and robust approach to the direct and precise numerical evaluation of Feynman graph integrals.

Paper Structure

This paper contains 6 sections, 18 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction.
  2. Analytical properties and behaviours of the MI.
  3. The numerical method.
  4. Tests and comparisons.
  5. Conclusions.
  6. Corrections to some analytic formulae of CCR1CCR2

Figures (5)

  • Figure 1: The general massive 2-loop sunrise self-mass diagram.
  • Figure 2: Plots of $\hbox{Re} F^{(0)}_{0,r}$ (labeled as (0)) and $\hbox{Re} F^{(0)}_{i}$ (labeled as ($i$)) as a function of $p_r^2$ for $m_1 \ = \ 2, \ m_2\ = \ 1, \ m_3\ =\ 4$ and $\mu = m_1+m_2+m_3$.
  • Figure 3: Plots of $\hbox{Re} F^{(0)}_{0,r}$ (labeled as (0)) and $\hbox{Re} F^{(0)}_{i}$ (labeled as ($i$)) as a function of $p_r^2$ for $m_1 \ = \ 1, m_2\ = \ 9, \ m_3\ =\ 200$ and $\mu = m_1+m_2+m_3$.
  • Figure 4: Plots of $\hbox{Im} F^{(0)}_{0,r}$ (labeled as (0)) and $\hbox{Im} F^{(0)}_{i}$ (labeled as ($i$)) as a function of $p_r^2$ for $m_1 \ = \ 2, \ m_2\ = \ 1, \ m_3\ =\ 4$ and $\mu = m_1+m_2+m_3$.
  • Figure 5: Plots of $\hbox{Im} F^{(0)}_{0,r}$ (labeled as (0)) and $\hbox{Im} F^{(0)}_{i}$ (labeled as ($i$)) as a function of $p_r^2$ for $m_1 \ = \ 1, \ m_2\ = \ 9, \ m_3\ =\ 200$ and $\mu = m_1+m_2+m_3$.