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A Numerical Routine for the Crossed Vertex Diagram with a Massive-Particle Loop

Roberto Bonciani, Giuseppe Degrassi, Pier Paolo Giardino, Ramona Gröber

TL;DR

This work presents a semi-analytical, differential-equation–driven method to evaluate the two master integrals ${\mathcal T}_9$ and ${\mathcal T}_{10}$ for the crossed vertex diagram with a massive-loop. By expanding solutions as power series around the singular points $x=0$, $x=16$, and $x=\infty$ and then matching these series, the authors obtain highly accurate numerical representations across the entire real axis, including analytic continuation to the time-like region. The results are implemented in a Fortran routine that computes both master integrals for any real $x$ with double-precision accuracy, validated against PySecDec and exact results. This approach extends the toolbox for NNLO calculations in processes with massive loops and elliptic integral structures, providing a practical, fast, and precise numerical solution.

Abstract

We present an evaluation of the two master integrals for the crossed vertex diagram with a closed loop of top quarks that allows for an easy numerical implementation. The differential equations obeyed by the master integrals are used to generate power series expansions centered around all the singular points. The different series are then matched numerically with high accuracy in intermediate points. The expansions allow a fast and precise numerical calculation of the two master integrals in all the regions of the phase space. A numerical routine that implements these expansions is presented.

A Numerical Routine for the Crossed Vertex Diagram with a Massive-Particle Loop

TL;DR

This work presents a semi-analytical, differential-equation–driven method to evaluate the two master integrals and for the crossed vertex diagram with a massive-loop. By expanding solutions as power series around the singular points , , and and then matching these series, the authors obtain highly accurate numerical representations across the entire real axis, including analytic continuation to the time-like region. The results are implemented in a Fortran routine that computes both master integrals for any real with double-precision accuracy, validated against PySecDec and exact results. This approach extends the toolbox for NNLO calculations in processes with massive loops and elliptic integral structures, providing a practical, fast, and precise numerical solution.

Abstract

We present an evaluation of the two master integrals for the crossed vertex diagram with a closed loop of top quarks that allows for an easy numerical implementation. The differential equations obeyed by the master integrals are used to generate power series expansions centered around all the singular points. The different series are then matched numerically with high accuracy in intermediate points. The expansions allow a fast and precise numerical calculation of the two master integrals in all the regions of the phase space. A numerical routine that implements these expansions is presented.

Paper Structure

This paper contains 12 sections, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Differential Equations for the two crossed Master Integrals
  3. ${\mathcal{T}}_{9}$ evaluation for $x \geq 0$
  4. The solution around $x=0$
  5. The solution around $x \, = \, 16$
  6. Matching the series in $x=0$ and $x=16$
  7. The solution around $x \, = \, \infty$
  8. The solution around $s = 16$
  9. Expansions for the Master Integral ${\mathcal{T}}_{10}$
  10. The Fortran Routine
  11. Numerical Checks
  12. Conclusions

Figures (2)

  • Figure 1: The 6-denominator topology. Internal plain thin lines represent massless propagators, while thick lines represent the massive propagator. External plain thin lines represent massless particles on their mass-shell.
  • Figure 2: Master Integrals. The convention for the lines is as in Fig. \ref{['fig1']}. The dot represents a propagator raised to the second power.