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Tensorial Reconstruction at the Integrand Level

G. Heinrich, G. Ossola, T. Reiter, F. Tramontano

TL;DR

The paper introduces a novel tensorial reconstruction of the one-loop integrand at the integrand level by sampling real loop-momentum components, reformulating the numerator as a basis of tensor structures. This approach provides a robust rescue mechanism near Gram determinant instabilities, enables real-momentum sampling compatible with tree-level generators, and can preprocess the numerator to streamline reductions. It details a Level-0 to Level-4 extraction scheme, extends to μ^2-dependence for d-dimensional regularization, and validates the method through instability tests and performance benchmarks. The authors implement the technique within samurai and Golem 95, demonstrating improved stability and potential for efficiency gains in complex multi-leg processes.

Abstract

We present a new approach to the reduction of one-loop amplitudes obtained by reconstructing the tensorial expression of the scattering amplitudes. The reconstruction is performed at the integrand level by means of a sampling in the integration momentum. There are several interesting applications of this novel method within existing techniques for the reduction of one-loop multi-leg amplitudes: to deal with numerically unstable points, such as in the vicinity of a vanishing Gram determinant; to allow for a sampling of the numerator function based on real values of the integration momentum; to optimize the numerical reduction in the case of long expressions for the numerator functions.

Tensorial Reconstruction at the Integrand Level

TL;DR

The paper introduces a novel tensorial reconstruction of the one-loop integrand at the integrand level by sampling real loop-momentum components, reformulating the numerator as a basis of tensor structures. This approach provides a robust rescue mechanism near Gram determinant instabilities, enables real-momentum sampling compatible with tree-level generators, and can preprocess the numerator to streamline reductions. It details a Level-0 to Level-4 extraction scheme, extends to μ^2-dependence for d-dimensional regularization, and validates the method through instability tests and performance benchmarks. The authors implement the technique within samurai and Golem 95, demonstrating improved stability and potential for efficiency gains in complex multi-leg processes.

Abstract

We present a new approach to the reduction of one-loop amplitudes obtained by reconstructing the tensorial expression of the scattering amplitudes. The reconstruction is performed at the integrand level by means of a sampling in the integration momentum. There are several interesting applications of this novel method within existing techniques for the reduction of one-loop multi-leg amplitudes: to deal with numerically unstable points, such as in the vicinity of a vanishing Gram determinant; to allow for a sampling of the numerator function based on real values of the integration momentum; to optimize the numerical reduction in the case of long expressions for the numerator functions.

Paper Structure

This paper contains 20 sections, 23 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Tensorial Reconstruction Algorithm
  3. Algorithm
  4. Level-0
  5. Level-1
  6. Level-2
  7. Level-3
  8. Level-4
  9. Example of the reconstruction for rank two
  10. Reconstruction of the $\mu^2$-dependence
  11. Examples of Applications
  12. Dealing with unstable points
  13. Cancellation of the poles.
  14. The "$N=N$" local test.
  15. The "$N=N$" global test.
  16. ...and 5 more sections

Figures (2)

  • Figure 1: For the example in this section we use the above diagram in QED with two massless and two off-shell photons attached to a massless fermion loop.
  • Figure 2: Top panel: Comparison between standard reduction at the integrand level (Standard) and tensorial reconstruction combined with an evaluation of the tensor integrals (Tensorial). The standard method starts deviating from the correct result at $\det G/\det S\approx10^{-7}$, the standard method with quadruple precision (see the text) starts deviating at $\det G/\det S\approx10^{-8}$, while the tensorial method remains stable over the whole range. Middle panels: the behaviour of tests that trigger the detection of instabilities. Bottom panel: timing of tensorial reduction in double precision versus standard reduction in quadruple precision, normalized by the timing of standard reduction in double precision.