Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Subleading Poles in the Numerical Unitarity Method at Two Loops

S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier, B. Page

TL;DR

This work addresses the challenge of computing subleading-pole contributions in two-loop scattering amplitudes within the numerical unitarity framework. It introduces a universal algorithm that extracts subleading terms by relaxing certain on-shell constraints and constructs a hierarchical set of cut equations to determine all integrand coefficients, including those not directly expressible as products of tree amplitudes. The method is demonstrated on planar two-loop four-gluon amplitudes, where it reproduces known analytic results and passes multiple consistency checks against independent calculations. The approach is process-independent, generalizable to higher loops, and provides a practical path to fully numerical predictions for complex multi-loop processes with subleading-pole structure.

Abstract

We describe the unitarity approach for the numerical computation of two-loop integral coefficients of scattering amplitudes. It is well known that the leading propagator singularities of an amplitude's integrand are related to products of tree amplitudes. At two loops, Feynman diagrams with doubled propagators appear naturally, which lead to subleading pole contributions. In general, it is not known how these contributions can be directly expressed in terms of a product of on-shell tree amplitudes. We present a universal algorithm to extract these subleading pole terms by releasing some of the on-shell conditions. We demonstrate the new approach by numerically computing two-loop four-gluon integral coefficients.

Subleading Poles in the Numerical Unitarity Method at Two Loops

TL;DR

This work addresses the challenge of computing subleading-pole contributions in two-loop scattering amplitudes within the numerical unitarity framework. It introduces a universal algorithm that extracts subleading terms by relaxing certain on-shell constraints and constructs a hierarchical set of cut equations to determine all integrand coefficients, including those not directly expressible as products of tree amplitudes. The method is demonstrated on planar two-loop four-gluon amplitudes, where it reproduces known analytic results and passes multiple consistency checks against independent calculations. The approach is process-independent, generalizable to higher loops, and provides a practical path to fully numerical predictions for complex multi-loop processes with subleading-pole structure.

Abstract

We describe the unitarity approach for the numerical computation of two-loop integral coefficients of scattering amplitudes. It is well known that the leading propagator singularities of an amplitude's integrand are related to products of tree amplitudes. At two loops, Feynman diagrams with doubled propagators appear naturally, which lead to subleading pole contributions. In general, it is not known how these contributions can be directly expressed in terms of a product of on-shell tree amplitudes. We present a universal algorithm to extract these subleading pole terms by releasing some of the on-shell conditions. We demonstrate the new approach by numerically computing two-loop four-gluon integral coefficients.

Paper Structure

This paper contains 13 sections, 18 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setup for two-loop numerical unitarity
  3. Diagrammatic decomposition
  4. Master integrals and integrands
  5. Integrand coefficients and factorization
  6. Leading and subleading poles in generalized unitarity
  7. Extracting leading poles
  8. Extracting subleading poles
  9. Applications to four-point gluon amplitudes
  10. Box coefficients from the triangle phase-space
  11. The bubble-box hierarchy at two loops
  12. Numerical implementation and checks
  13. Conclusions

Figures (4)

  • Figure 1: A generic diagram depicting the propagator structure that appears in a two-loop planar amplitude. The momenta $q_i$ and $\tilde{q}_i$ are determined by momentum conservation.
  • Figure 2: Two diagrams with the same set of propagators. Propagator $1/\rho$ appears twice in diagram (a) but only once in diagram (b).
  • Figure 3: The planar $\Delta$ hierarchy in a 2 $\to$ 2 amplitude. Only topologically inequivalent diagrams are shown. The boxed diagrams do not belong to the cut hierarchy. The diagrams to the left of the dashed line are the members of the sunrise (cut) hierarchy.
  • Figure 4: The planar bubble-box hierarchy. The maximal diagrams are (a)-(d), next-to maximal are the (e)-(g) and at the bottom we find the bubble-box diagram (h).