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Master Integrals for the two-loop, non-planar QCD corrections to top-quark pair production in the quark-annihilation channel

Matteo Becchetti, Roberto Bonciani, Valerio Casconi, Andrea Ferroglia, Simone Lavacca, Andreas von Manteuffel

TL;DR

This work delivers the analytic master integrals needed for the last two color components of the two-loop, non-planar QCD corrections to $q\bar{q} \to t\bar{t}$, a key piece for NNLO predictions of top-quark pair production. By casting the differential equations for the master integrals into an $\epsilon$-form canonical basis, the authors obtain solutions expressed as Chen iterated integrals and generalized harmonic polylogarithms in two rationalizing variables, up to weight four. They provide explicit basis transformations for two seven-denominator topologies (52 and 44 MIs, respectively), and fix boundary conditions via regularity and simple integral inputs; results are cross-validated against independent numerical approaches and other analytic efforts. The analytic expressions, supplied in ancillary files, enable faster, more precise phenomenological evaluations and strengthen the theoretical control over NNLO corrections in the quark-annihilation channel of top-quark pair production.

Abstract

We present the analytic calculation of the Master Integrals for the two-loop, non-planar topologies that enter the calculation of the amplitude for top-quark pair hadroproduction in the quark-annihilation channel. Using the method of differential equations, we expand the integrals in powers of the dimensional regulator $ε$ and determine the expansion coefficients in terms of generalized harmonic polylogarithms of two dimensionless variables through to weight four.

Master Integrals for the two-loop, non-planar QCD corrections to top-quark pair production in the quark-annihilation channel

TL;DR

This work delivers the analytic master integrals needed for the last two color components of the two-loop, non-planar QCD corrections to , a key piece for NNLO predictions of top-quark pair production. By casting the differential equations for the master integrals into an -form canonical basis, the authors obtain solutions expressed as Chen iterated integrals and generalized harmonic polylogarithms in two rationalizing variables, up to weight four. They provide explicit basis transformations for two seven-denominator topologies (52 and 44 MIs, respectively), and fix boundary conditions via regularity and simple integral inputs; results are cross-validated against independent numerical approaches and other analytic efforts. The analytic expressions, supplied in ancillary files, enable faster, more precise phenomenological evaluations and strengthen the theoretical control over NNLO corrections in the quark-annihilation channel of top-quark pair production.

Abstract

We present the analytic calculation of the Master Integrals for the two-loop, non-planar topologies that enter the calculation of the amplitude for top-quark pair hadroproduction in the quark-annihilation channel. Using the method of differential equations, we expand the integrals in powers of the dimensional regulator and determine the expansion coefficients in terms of generalized harmonic polylogarithms of two dimensionless variables through to weight four.

Paper Structure

This paper contains 15 sections, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. The Differential Equations Method
  4. Canonical Form for the Master Integrals
  5. Topology $A$
  6. Topology $B$
  7. Rationalizing Parametrization
  8. Integration and Results
  9. Numerical Checks
  10. Conclusions
  11. Acknowledgments
  12. Pre-canonical form for the Master Integrals
  13. Topology $A$
  14. Topology $B$
  15. Numerical Results

Figures (2)

  • Figure 1: Seven-denominator topologies. Thin lines represent massless external particles and internal propagators, while thick lines represent massive external particles and internal propagators.
  • Figure 2: Master Integrals in pre-canonical form. Internal thin lines represent massless propagators, while thick lines represent heavy-quark (massive) propagators. External thin lines represent massless particles on their mass-shell, $p^2=0$. External thick lines represent massive particles on their mass-shell, $p^2=m^2$.