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On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes

Pierpaolo Mastrolia, Giovanni Ossola

TL;DR

The paper tackles the challenge of reducing two-loop scattering amplitudes by extending the integrand-reduction framework from one loop. It introduces a four-dimensional, semi-analytic reduction where the integrand is decomposed into polynomial residues on multi-particle cuts, with coefficients determined via sampling and projection in irreducible scalar product bases. The authors demonstrate the method on four- and five-point ${\cal N}=4$ SYM amplitudes, extracting residues for ladder, crossed, pentabox, and pentacross topologies and showing how the resulting polynomials constrain the master-integral basis, including cases where spurious ISPs do not contribute to the final MI content. The approach provides a pathway toward automated, semi-analytic two-loop reductions in gauge theories and lays groundwork for extending to dimensional regularization, with potential broad impact on multi-loop computations in high-energy physics.

Abstract

We propose a first implementation of the integrand-reduction method for two-loop scattering amplitudes. We show that the residues of the amplitudes on multi-particle cuts are polynomials in the irreducible scalar products involving the loop momenta, and that the reduction of the amplitudes in terms of master integrals can be realized through polynomial fitting of the integrand, without any apriori knowledge of the integral basis. We discuss how the polynomial shapes of the residues determine the basis of master integrals appearing in the final result. We present a four-dimensional constructive algorithm that we apply to planar and non-planar contributions to the 4- and 5-point MHV amplitudes in N=4 SYM. The technique hereby discussed extends the well-established analogous method holding for one-loop amplitudes, and can be considered a preliminary study towards the systematic reduction at the integrand-level of two-loop amplitudes in any gauge theory, suitable for their automated semianalytic evaluation.

On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes

TL;DR

The paper tackles the challenge of reducing two-loop scattering amplitudes by extending the integrand-reduction framework from one loop. It introduces a four-dimensional, semi-analytic reduction where the integrand is decomposed into polynomial residues on multi-particle cuts, with coefficients determined via sampling and projection in irreducible scalar product bases. The authors demonstrate the method on four- and five-point SYM amplitudes, extracting residues for ladder, crossed, pentabox, and pentacross topologies and showing how the resulting polynomials constrain the master-integral basis, including cases where spurious ISPs do not contribute to the final MI content. The approach provides a pathway toward automated, semi-analytic two-loop reductions in gauge theories and lays groundwork for extending to dimensional regularization, with potential broad impact on multi-loop computations in high-energy physics.

Abstract

We propose a first implementation of the integrand-reduction method for two-loop scattering amplitudes. We show that the residues of the amplitudes on multi-particle cuts are polynomials in the irreducible scalar products involving the loop momenta, and that the reduction of the amplitudes in terms of master integrals can be realized through polynomial fitting of the integrand, without any apriori knowledge of the integral basis. We discuss how the polynomial shapes of the residues determine the basis of master integrals appearing in the final result. We present a four-dimensional constructive algorithm that we apply to planar and non-planar contributions to the 4- and 5-point MHV amplitudes in N=4 SYM. The technique hereby discussed extends the well-established analogous method holding for one-loop amplitudes, and can be considered a preliminary study towards the systematic reduction at the integrand-level of two-loop amplitudes in any gauge theory, suitable for their automated semianalytic evaluation.

Paper Structure

This paper contains 36 sections, 49 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Four-dimensional Reduction Algorithm
  3. Integrand Decomposition
  4. Residues
  5. Polynomial Structures and Master Integrals
  6. Numerator Sampling
  7. Momentum Basis
  8. Testing the Integrand-Decomposition
  9. Four-gluon MHV Amplitude in ${\cal N}=4$ SYM
  10. A contribution to the Ladder Amplitude
  11. On-shell Solutions.
  12. Residue.
  13. Coefficients.
  14. A contribution to the Crossed Amplitude
  15. On-Shell Solutions.
  16. ...and 21 more sections

Figures (11)

  • Figure 1: Two Master Integrals of the 4-gluon MHV amplitude in ${\cal N}=4$ SYM: the ladder (left) and crossed (right) $s$-channel double-box.
  • Figure 2: 7fold-cut of the 4-point ladder diagram ($s$-channel)
  • Figure 3: 7fold-cut of the 4-point crossed diagram ($s$-channel)
  • Figure 4: 5-point pentabox diagram.
  • Figure 5: 5-point 8fold-cut $\Delta_{12345678}$.
  • ...and 6 more figures