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The two-loop scalar and tensor pentabox graph with light-like legs

C. Anastasiou, E. W. N. Glover, C. Oleari

TL;DR

This work tackles the evaluation of the two-loop planar pentabox with light-like legs, a crucial component in NNLO 2→2 scattering computations. By exploiting integration-by-parts identities, the authors reduce the scalar pentabox to two master topologies and derive an explicit ε-expansion in terms of generalized polylogarithms, with careful treatment of region-dependent analytic structure. They then extend the reduction to tensor integrals using a Schwinger-parameter approach, mapping tensors to higher-dimensional scalar integrals tied to the same master topologies and providing explicit tensor decompositions. The paper also presents detailed analytic forms for the master topologies A^D and C^D (including A’s dimensional-shift relations) and shows how these master integrals underpin complete tensor reductions, enabling practical computations for two-loop processes.

Abstract

We study the scalar and tensor integrals associated with the pentabox topology: the class of two-loop box integrals with seven propagators - five in one loop and three in the other. We focus on the case where the external legs are light-like and use integration-by-parts identities to express the scalar integral in terms of two master-topology integrals and present an explicit analytic expression for the pentabox scalar integral as a series expansion in ep = (4-D)/2. We also give an algorithm based on integration by parts for relating the generic tensor integrals to the same two master integrals and provide general formulae describing the master integrals in arbitrary dimension and with general powers of propagators.

The two-loop scalar and tensor pentabox graph with light-like legs

TL;DR

This work tackles the evaluation of the two-loop planar pentabox with light-like legs, a crucial component in NNLO 2→2 scattering computations. By exploiting integration-by-parts identities, the authors reduce the scalar pentabox to two master topologies and derive an explicit ε-expansion in terms of generalized polylogarithms, with careful treatment of region-dependent analytic structure. They then extend the reduction to tensor integrals using a Schwinger-parameter approach, mapping tensors to higher-dimensional scalar integrals tied to the same master topologies and providing explicit tensor decompositions. The paper also presents detailed analytic forms for the master topologies A^D and C^D (including A’s dimensional-shift relations) and shows how these master integrals underpin complete tensor reductions, enabling practical computations for two-loop processes.

Abstract

We study the scalar and tensor integrals associated with the pentabox topology: the class of two-loop box integrals with seven propagators - five in one loop and three in the other. We focus on the case where the external legs are light-like and use integration-by-parts identities to express the scalar integral in terms of two master-topology integrals and present an explicit analytic expression for the pentabox scalar integral as a series expansion in ep = (4-D)/2. We also give an algorithm based on integration by parts for relating the generic tensor integrals to the same two master integrals and provide general formulae describing the master integrals in arbitrary dimension and with general powers of propagators.

Paper Structure

This paper contains 7 sections, 40 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. The scalar pentabox integral
  4. Application:
  5. The generic tensor integrals
  6. The master topologies
  7. Topology $\boldsymbol{A}$

Figures (3)

  • Figure 1: The pentabox topology. The propagators are labelled according to Eq. (\ref{['eq:props']}) and are each raised to the $\nu_i$ power.
  • Figure 2: One-loop self-energy insertion into a one-loop triangle. The propagators are labelled according to Eq. (\ref{['eq:props']}) and are each raised to the $\nu_i$ power.
  • Figure 3: The master topology $A$. The propagators are labelled according to Eq. (\ref{['eq:props']}) and are each raised to the $\nu_i$ power.