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On the Computation of Form Factors in Massless QCD with Finite Master Integrals

Andreas von Manteuffel, Erik Panzer, Robert M. Schabinger

TL;DR

This work rederives massless QCD quark and gluon form factors at one, two, and three loops using a basis of finite master integrals, enabling explicit ε-pole structures and exact ε-expansions. By combining dimension shifting with a 'dots' propagator framework and HyperInt, the authors obtain analytic results through weight eight and validate them against the literature, while revealing that only a subset of integrals contribute to cusp anomalous dimensions. They document four-loop indications, including a finite twelve-line example that does not affect the cusp poles, and discuss the practical potential of their approach for complete four-loop cusp computations. An automated, publicly available computational setup and ancillary data support reproducibility and future extensions to higher loops.

Abstract

We present the bare one-, two-, and three-loop form factors in massless Quantum Chromodynamics as linear combinations of finite master integrals. Using symbolic integration, we compute their $ε$ expansions and thereby reproduce all known results with an independent method. Remarkably, in our finite basis, only integrals with a less-than-maximal number of propagators contribute to the cusp anomalous dimensions. We report on indications of this phenomenon at four loops, including the result for a finite, irreducible, twelve-propagator form factor integral. Together with this article, we provide our automated software setup for the computation of finite master integrals.

On the Computation of Form Factors in Massless QCD with Finite Master Integrals

TL;DR

This work rederives massless QCD quark and gluon form factors at one, two, and three loops using a basis of finite master integrals, enabling explicit ε-pole structures and exact ε-expansions. By combining dimension shifting with a 'dots' propagator framework and HyperInt, the authors obtain analytic results through weight eight and validate them against the literature, while revealing that only a subset of integrals contribute to cusp anomalous dimensions. They document four-loop indications, including a finite twelve-line example that does not affect the cusp poles, and discuss the practical potential of their approach for complete four-loop cusp computations. An automated, publicly available computational setup and ancillary data support reproducibility and future extensions to higher loops.

Abstract

We present the bare one-, two-, and three-loop form factors in massless Quantum Chromodynamics as linear combinations of finite master integrals. Using symbolic integration, we compute their expansions and thereby reproduce all known results with an independent method. Remarkably, in our finite basis, only integrals with a less-than-maximal number of propagators contribute to the cusp anomalous dimensions. We report on indications of this phenomenon at four loops, including the result for a finite, irreducible, twelve-propagator form factor integral. Together with this article, we provide our automated software setup for the computation of finite master integrals.

Paper Structure

This paper contains 12 sections, 17 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notation and Conventions
  3. Computational Method
  4. One-Loop Form Factors
  5. Two-Loop Form Factors
  6. Three-Loop Form Factors
  7. Towards the Four-Loop Cusp Anomalous Dimensions
  8. Conclusions
  9. From Quasi-Finite Integrals to Finite Integrals
  10. Automated Feynman Integral Evaluations with HyperInt
  11. Linear Reducibility
  12. Practical Remarks

Figures (3)

  • Figure 1: The one-loop Feynman diagrams for the quark (upper panel) and gluon (lower panel) form factors in massless QCD. At each order in the bare strong coupling constant, the effective coupling of the Higgs boson to gluons can be obtained by matching full QCD with a massive top quark onto an effective field theory in which the massive top quark is integrated out.
  • Figure 2: The one-, two-, and three-loop finite form factor master integrals used in Sections \ref{['sec:1Lffs']}, \ref{['sec:2Lffs']}, and \ref{['sec:3Lffs']} for $\mathcal{F}_1^q(\epsilon)$, $\mathcal{F}_1^g(\epsilon)$, $\mathcal{F}_2^q(\epsilon)$, $\mathcal{F}_2^g(\epsilon)$, $\mathcal{F}_3^q(\epsilon)$, and $\mathcal{F}_3^g(\epsilon)$.
  • Figure 3: The finite three-loop form factor master integrals in $\mathcal{F}_3^q\left(\epsilon\right)$ and $\mathcal{F}_3^g\left(\epsilon\right)$ (Eqs. \ref{['eq:3Lffq']} and \ref{['eq:3Lffg']}) which do not contribute to the three-loop cusp anomalous dimensions.