Recursive reduction of two-loop tensor integrals

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.

💡 Research Summary

The paper addresses a pressing need in high‑energy physics: the automated computation of next‑to‑next‑to‑leading order (NNLO) corrections for LHC and future collider processes. Building on the successful OpenLoops framework for one‑loop amplitudes, the authors focus on the second pillar of the OpenLoops strategy – the evaluation of tensor integrals – and present a novel recursive reduction algorithm that transforms arbitrary two‑loop tensor integrals into scalar master integrals suitable for numerical evaluation.

The authors begin by reviewing the standard decomposition of amplitudes in dimensional regularisation, separating four‑dimensional external quantities from D‑dimensional loop momenta (with D = 4 − 2ε). For a generic one‑loop diagram the numerator is expanded into a sum of tensors of rank r, each multiplied by a product of propagator denominators. The central technical tool is the exact identity (eq. 16) that rewrites a rank‑2 loop‑momentum product q_μ q_ν as a linear combination of propagator denominators and a term proportional to q_λ ∂_λ D_a. This identity reduces the tensor rank by one while simultaneously shifting the powers of the propagators via a configuration‑shift vector Δn_a. Crucially, the method permits negative indices (i.e. cancelled denominators), allowing the same set of propagators to be used throughout the recursion and avoiding the proliferation of distinct topologies.

Applying the identity recursively yields a systematic reduction of any rank‑r one‑loop tensor integral to a sum of rank‑1 and rank‑0 integrals (eq. 19). The coefficients A and B that appear at each step are themselves generated recursively from lower‑rank coefficients (eqs. 26‑27). Because the recursion never requires solving large systems of linear equations, it is computationally cheap and easily integrated into the on‑the‑fly construction of tensor coefficients used by OpenLoops 2. Once the integrals have been reduced to rank‑1, a standard Passarino‑Veltman (PV) decomposition converts them into scalar integrals and rational terms of type R₁, which arise from the (D − 4)‑dimensional part of the loop momentum (˜q²).

The core contribution of the paper is the extension of this one‑loop recursion to the two‑loop case, where two independent loop momenta q₁ and q₂ appear. The authors apply the same rank‑reduction identity separately to each momentum chain, introducing two independent sets of exponent vectors Ω and Ω′. The product of the two chains is then expressed as a sum of four structures: A·A, A·B, B·A, and B·B (eq. 33). Each structure contains tensor integrals with reduced rank in one or both loop momenta, ultimately yielding integrals of rank 0, rank 1, or mixed rank 1 × 1. At this stage the problem has been reduced to a collection of one‑loop‑like tensor integrals, which are again treated with the PV method. The remaining scalar integrals may contain powers of ˜q₁² and ˜q₂²; these contribute only through their interplay with 1/ε poles and are identified as higher‑order rational terms.

Implementation details are discussed: the recursion is performed on the fly, interleaved with the construction of process‑dependent tensor coefficients, thereby preserving the modularity of OpenLoops. The algorithm avoids the need for external reduction tools for the tensor part; only the final scalar master integrals are handed to established IBP‑based programs (e.g. FIRE, Reduze). Benchmark tests on irreducible two‑loop topologies (with N₁ ≥ N₂ ≥ N₃ ≥ 1) show a substantial reduction in both memory consumption and CPU time compared with traditional on‑the‑fly IBP reduction.

The paper also addresses the treatment of (D − 4)‑dimensional contributions. Since ˜q² is of order ε, its appearance in the numerator only yields finite pieces when multiplied by UV or IR poles. The authors reconstruct these contributions via rational counterterms, extending the known one‑loop R₁ machinery to two loops. While UV counterterms are fully incorporated, the treatment of IR rational terms is identified as ongoing work.

In conclusion, the authors deliver a practical, recursive algorithm that brings two‑loop tensor integral reduction to the same level of automation and efficiency as the one‑loop OpenLoops framework. By leveraging a simple algebraic identity, allowing negative propagator powers, and reusing one‑loop reduction coefficients, the method sidesteps the combinatorial explosion typical of IBP‑based approaches. This development paves the way for fully automated NNLO calculations of realistic scattering processes, a crucial step toward matching the experimental precision of current and future high‑energy colliders.