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Recursive reduction of two-loop tensor integrals

Fabian Lange, Max F. Zoller

Abstract

In order to meet the precision requirements for the LHC and future colliders, next-to-next-to-leading order corrections to a wide range of processes are essential, making general automated tools highly desirable. Extending the strategy of the widespread one-loop program OpenLoops to two loops, there are three major ingredients: process-dependent tensor coefficients, tensor integrals, and process-independent counterterms. In these proceedings, we focus on the second part and present a new recursive algorithm to reduce arbitrary two-loop tensor integrals to scalar integrals numerically.

Recursive reduction of two-loop tensor integrals

Abstract

In order to meet the precision requirements for the LHC and future colliders, next-to-next-to-leading order corrections to a wide range of processes are essential, making general automated tools highly desirable. Extending the strategy of the widespread one-loop program OpenLoops to two loops, there are three major ingredients: process-dependent tensor coefficients, tensor integrals, and process-independent counterterms. In these proceedings, we focus on the second part and present a new recursive algorithm to reduce arbitrary two-loop tensor integrals to scalar integrals numerically.
Paper Structure (11 sections, 40 equations, 4 figures, 1 table)

This paper contains 11 sections, 40 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Automated calculations with tensor integral decomposition
  3. One loop
  4. Two loops
  5. Recursive reduction of one-loop tensor integrals
  6. Integrand-level reduction by one rank
  7. Reduction of one-loop tensor integrals to rank one
  8. Reduction to rank zero
  9. Recursive reduction of two-loop tensor integrals
  10. Validation and performance tests of the two-loop reduction
  11. Conclusions

Figures (4)

  • Figure 1: General structure of a one-loop diagram. The blue blobs denote external subtrees.
  • Figure 2: General structure of an irreducible two-loop diagram. The blue blobs denote external subtrees. In general $\mathcal{V}_{0}, \mathcal{V}_{1}$ can be quartic vertices, in which case an external subtree is also attached there.
  • Figure 3: Two-loop pentagon-triangle topology used to test our implementation.
  • Figure 4: Stability of the recursive reduction of the rank-$(5, 1)$ pentagon-triangle integral to ranks $(0, 1)$ and $(1, 1)$.