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Challenges for analytic calculations of the massive three-loop form factors

J Blümlein, A. De Freitas, P. Marquard, C. Schneider

TL;DR

This work extends the large moments method to massive three-loop QCD form factors, focusing on the gluonic contributions beyond prior quarkonic results. It outlines a workflow that combines reduction to master integrals, differential equations, and a guessing approach from high-order expansion coefficients to reconstruct analytic structures across kinematic regimes. The authors achieve high-precision threshold and high-energy expansions by leveraging heavy computational techniques (including recurrence relations, PSLQ, and careful rational-function management) and report both progress and remaining challenges due to additional singularities in the gluonic sector. The results advance the analytic understanding of three-loop form factors with potential impact on precision predictions for heavy-quark production and decays at colliders, while demonstrating scalable strategies for handling very large integral systems.

Abstract

The calculation of massive three-loop QCD form factors using in particular the large moments method has been successfully applied to quarkonic contributions in [1]. We give a brief review of the different steps of the calculation and report on improvements of our methods that enabled us to push forward the calculations of the gluonic contributions to the form factors.

Challenges for analytic calculations of the massive three-loop form factors

TL;DR

This work extends the large moments method to massive three-loop QCD form factors, focusing on the gluonic contributions beyond prior quarkonic results. It outlines a workflow that combines reduction to master integrals, differential equations, and a guessing approach from high-order expansion coefficients to reconstruct analytic structures across kinematic regimes. The authors achieve high-precision threshold and high-energy expansions by leveraging heavy computational techniques (including recurrence relations, PSLQ, and careful rational-function management) and report both progress and remaining challenges due to additional singularities in the gluonic sector. The results advance the analytic understanding of three-loop form factors with potential impact on precision predictions for heavy-quark production and decays at colliders, while demonstrating scalable strategies for handling very large integral systems.

Abstract

The calculation of massive three-loop QCD form factors using in particular the large moments method has been successfully applied to quarkonic contributions in [1]. We give a brief review of the different steps of the calculation and report on improvements of our methods that enabled us to push forward the calculations of the gluonic contributions to the form factors.
Paper Structure (7 sections, 45 equations, 3 figures)

This paper contains 7 sections, 45 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review of the method
  3. Challenges arising in the large moments method
  4. The gluonic case
  5. High energy expansion
  6. The $\hat{s}=4$ threshold
  7. Conclusions

Figures (3)

  • Figure 1: MyTogether
  • Figure 2: Radii of convergence of the successive points around which expansions are matched in the case of the last four constants in (\ref{['eq:constants3']}). The points $\hat{s}=-1$ and $\hat{s}=-4$, shown in red, are singularities of the differential equations.
  • Figure 3: Radii of convergence of the successive points around which expansions are matched in order to obtain the threshold expansion at $\hat{s}=4$ (shown in red). The point $\hat{s}=3$ (shown in blue) is singular according to the differential equations, but the singularity disappears due to initial conditions. Left panel: radii of convergence of the points used in the case of the last four constants in (\ref{['eq:constants3']}). Right panel: radii of convergence of the points used in the case of $\ln(2) \zeta_2$.