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The tensor reduction and master integrals of the two-loop massless crossed box with light-like legs

C. Anastasiou, T. Gehrmann, C. Oleari, E. Remiddi, J. B. Tausk

TL;DR

The paper resolves the long-standing problem of tensor reduction for two-loop massless crossed boxes with light-like legs by reducing any tensor crossed-box integral to two master integrals plus simpler diagrams. It develops a dual approach: (i) dimensional-shift relations connecting master integrals across dimensions and (ii) a coupled differential-equation system that determines the second master from the known first master. The authors obtain an explicit ε-expansion for the second master integral and validate it through two independent methods (raising/lowering operators and off-shell differential equations). The results provide essential building blocks for NNLO 2→2 massless amplitude calculations and establish a framework applicable to related topologies and future high-precision predictions.

Abstract

The class of the two-loop massless crossed boxes, with light-like external legs, is the final unresolved issue in the program of computing the scattering amplitudes of 2 --> 2 massless particles at next-to-next-to-leading order. In this paper, we describe an algorithm for the tensor reduction of such diagrams. After connecting tensor integrals to scalar ones with arbitrary powers of propagators in higher dimensions, we derive recurrence relations from integration-by-parts and Lorentz-invariance identities, that allow us to write the scalar integrals as a combination of two master crossed boxes plus simpler-topology diagrams. We derive the system of differential equations that the two master integrals satisfy using two different methods, and we use one of these equations to express the second master integral as a function of the first one, already known in the literature. We then give the analytic expansion of the second master integral as a function of epsilon=(4-D)/2, where D is the space-time dimension, up to order O(epsilon^0).

The tensor reduction and master integrals of the two-loop massless crossed box with light-like legs

TL;DR

The paper resolves the long-standing problem of tensor reduction for two-loop massless crossed boxes with light-like legs by reducing any tensor crossed-box integral to two master integrals plus simpler diagrams. It develops a dual approach: (i) dimensional-shift relations connecting master integrals across dimensions and (ii) a coupled differential-equation system that determines the second master from the known first master. The authors obtain an explicit ε-expansion for the second master integral and validate it through two independent methods (raising/lowering operators and off-shell differential equations). The results provide essential building blocks for NNLO 2→2 massless amplitude calculations and establish a framework applicable to related topologies and future high-precision predictions.

Abstract

The class of the two-loop massless crossed boxes, with light-like external legs, is the final unresolved issue in the program of computing the scattering amplitudes of 2 --> 2 massless particles at next-to-next-to-leading order. In this paper, we describe an algorithm for the tensor reduction of such diagrams. After connecting tensor integrals to scalar ones with arbitrary powers of propagators in higher dimensions, we derive recurrence relations from integration-by-parts and Lorentz-invariance identities, that allow us to write the scalar integrals as a combination of two master crossed boxes plus simpler-topology diagrams. We derive the system of differential equations that the two master integrals satisfy using two different methods, and we use one of these equations to express the second master integral as a function of the first one, already known in the literature. We then give the analytic expansion of the second master integral as a function of epsilon=(4-D)/2, where D is the space-time dimension, up to order O(epsilon^0).

Paper Structure

This paper contains 13 sections, 70 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Tensor reduction using raising and lowering operators
  4. The scalar crossed-box reduction
  5. The dimensional shift for the two master integrals
  6. Differential equations for the two master integrals
  7. The off-shell method
  8. Analytic expansion of the second master integral
  9. Conclusions
  10. Tensor reduction and dimensional shift of the crossed two-loop triangle
  11. Coefficients $\boldsymbol{c_i}$
  12. Coefficients of the dimensional-shift system
  13. Coefficients of the differential equations for the two master integrals

Figures (5)

  • Figure 1: The generic two-loop crossed box.
  • Figure 2: The master integral topologies of the pinchings coming from the reduction of the scalar crossed two-loop box.
  • Figure 3: The master integral ${\rm Rtri}$p_1^2,s$$, which becomes reducible when $p_1$ is on-shell.
  • Figure 4: The physical regions (i), (ii) and (iii) in the $(s,t,u)$-plane.
  • Figure 5: The generic two-loop crossed triangle.