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All Two-Loop Feynman Integrals for Five-Point One-Mass Scattering

Samuel Abreu, Dmitry Chicherin, Harald Ita, Ben Page, Vasily Sotnikov, Wladimir Tschernow, Simone Zoia

TL;DR

The paper delivers a complete set of two-loop master integrals for five-point scattering with one massive external leg, enabling NNLO QCD predictions for processes such as H/Z/W plus two jets. It introduces a canonical epsilon-form differential-equation framework with pure integral bases for the DPmz and DPzz families and constructs a minimal one-mass pentagon-function basis expressed via Chen iterated integrals. The authors develop a polynomial-in-pentagon-functions plus zeta-values ansatz to represent all master integrals, avoiding heavy MPL representations and PSLQ procedures, and provide a public C++ library for efficient numerical evaluation. The results illuminate the analytic structure of these integrals, support theoretical studies of Steinmann relations and amplitude cancellations, and remove a major bottleneck for precise NNLO and beyond collider predictions.

Abstract

We compute the complete set of two-loop master integrals for the scattering of four massless particles and a massive one. Our results are ready for phenomenological applications, removing a major obstacle to the computation of complete next-to-next-to-leading order (NNLO) QCD corrections to processes such as the production of a $H/Z/W$ boson in association with two jets at the LHC. Furthermore, they open the door to new investigations into the structure of quantum-field theories and provide precious analytic data for studying the mathematical properties of Feynman integrals.

All Two-Loop Feynman Integrals for Five-Point One-Mass Scattering

TL;DR

The paper delivers a complete set of two-loop master integrals for five-point scattering with one massive external leg, enabling NNLO QCD predictions for processes such as H/Z/W plus two jets. It introduces a canonical epsilon-form differential-equation framework with pure integral bases for the DPmz and DPzz families and constructs a minimal one-mass pentagon-function basis expressed via Chen iterated integrals. The authors develop a polynomial-in-pentagon-functions plus zeta-values ansatz to represent all master integrals, avoiding heavy MPL representations and PSLQ procedures, and provide a public C++ library for efficient numerical evaluation. The results illuminate the analytic structure of these integrals, support theoretical studies of Steinmann relations and amplitude cancellations, and remove a major bottleneck for precise NNLO and beyond collider predictions.

Abstract

We compute the complete set of two-loop master integrals for the scattering of four massless particles and a massive one. Our results are ready for phenomenological applications, removing a major obstacle to the computation of complete next-to-next-to-leading order (NNLO) QCD corrections to processes such as the production of a boson in association with two jets at the LHC. Furthermore, they open the door to new investigations into the structure of quantum-field theories and provide precious analytic data for studying the mathematical properties of Feynman integrals.
Paper Structure (6 sections, 12 equations, 3 figures, 1 table)

This paper contains 6 sections, 12 equations, 3 figures, 1 table.

Table of Contents

  1. Notation and Conventions
  2. Analytic Differential Equations
  3. Construction of One-Mass Pentagon Functions
  4. Numerical Evaluation and Analytic Structure
  5. Discussion and Outlook
  6. Analytic Continuation Across $\Sigma_5^{(3)}=0$ Surface

Figures (3)

  • Figure 1: Two-loop five-point one-mass families. The thick external line denotes the massive external leg.
  • Figure 2: Distribution of correct digits compared to quadruple-precision evaluations over 100k points.
  • Figure 3: A path that connects phase-space points with opposite $\Sigma_5^{(3)}$ signs without encountering the singularity at the $\Sigma_5^{(3)}=0$ surface.