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Two-Loop Five-Parton Leading-Colour Finite Remainders in the Spinor-Helicity Formalism

Giuseppe De Laurentis, Daniel Maître

TL;DR

The paper addresses the computational bottleneck of two-loop, five-parton amplitude predictions by reconstructing all leading-color finite remainders in the spinor-helicity framework. By reformulating the known Mandelstam-space expressions into spinor variables and applying a partial-fractioned, LCD-based reconstruction, the authors obtain compact, fast, and pole-structure–transparent expressions for the 740 coefficient functions $r_i$ multiplying transcendental bases $h_i$ in $\,\mathcal{R}^{(2)}$. This approach yields about an order-of-magnitude reduction in required numerical samples relative to previous finite-field reconstructions, while enhancing numerical stability, especially in soft and collinear regions. The improvements enable more practical phenomenological usage and point to future extensions to processes with external masses or additional legs (e.g., six-point amplitudes).

Abstract

We present all two-loop five-parton leading-colour finite remainders in the spinor-helicity formalism by analysing numerical evaluations of their known expressions in terms of Mandelstam invariants. Recasting them in terms of spinor-helicity variables allows us to obtain expressions which are more compact, faster to evaluate, numerically more stable and manifestly free from poles of higher order than necessary. At the same time, due to the better scaling of our reconstruction strategy with the complexity of the input, we required one order of magnitude fewer numerical samples to complete the analytical reconstruction than were needed by the authors of Ref. \cite{Abreu:2019odu}, albeit using higher numerical working precision. This places our reconstruction technique as an alternative to the finite-field single-numerator reconstruction for future applications.

Two-Loop Five-Parton Leading-Colour Finite Remainders in the Spinor-Helicity Formalism

TL;DR

The paper addresses the computational bottleneck of two-loop, five-parton amplitude predictions by reconstructing all leading-color finite remainders in the spinor-helicity framework. By reformulating the known Mandelstam-space expressions into spinor variables and applying a partial-fractioned, LCD-based reconstruction, the authors obtain compact, fast, and pole-structure–transparent expressions for the 740 coefficient functions multiplying transcendental bases in . This approach yields about an order-of-magnitude reduction in required numerical samples relative to previous finite-field reconstructions, while enhancing numerical stability, especially in soft and collinear regions. The improvements enable more practical phenomenological usage and point to future extensions to processes with external masses or additional legs (e.g., six-point amplitudes).

Abstract

We present all two-loop five-parton leading-colour finite remainders in the spinor-helicity formalism by analysing numerical evaluations of their known expressions in terms of Mandelstam invariants. Recasting them in terms of spinor-helicity variables allows us to obtain expressions which are more compact, faster to evaluate, numerically more stable and manifestly free from poles of higher order than necessary. At the same time, due to the better scaling of our reconstruction strategy with the complexity of the input, we required one order of magnitude fewer numerical samples to complete the analytical reconstruction than were needed by the authors of Ref. \cite{Abreu:2019odu}, albeit using higher numerical working precision. This places our reconstruction technique as an alternative to the finite-field single-numerator reconstruction for future applications.

Paper Structure

This paper contains 11 sections, 24 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Summary of the original calculation
  3. Spinor-Helicity Remainders
  4. Reconstruction strategy
  5. Results
  6. Algebraic complexity of the result
  7. Numerical stability
  8. Conclusion
  9. Appendix: concrete examples
  10. Expression for $r_{3}$ in $\mathcal{R}^{(2)[N_f^1]}_{u^{+}\overline{u}^{-}g^{+}g^{-}g^{+}}$
  11. Expression for $r_{10}$ in $\mathcal{R}^{(2)[N_f^1]}_{g^{-}g^{+}g^{-}g^{+}g^{+}}$

Figures (4)

  • Figure 1: Ratios of leaf counts of Mandelstam expressions to spinor-helicity ones.
  • Figure 2: Maximum mass dimension of numerator antäze to be reconstructed for each helicity configuration when using Mandelstam invariants or spinor variables.
  • Figure 3: Maximum mass dimension of the numerator ansatze. This can be equivalently thought of as the maximum degree of the polynomials in the angle and square spinor brackets. The 'Nf' labels refer to the respective terms in the expansion of Eq. \ref{['eq:nf_expansion_2l']}.
  • Figure 4: Relative errors over a thousand phase-space points in a collider configuration of double-precision evaluations of spinor and Mandelstam expressions.