OpenLoops 2

TL;DR

OpenLoops 2 delivers automated tree and one-loop amplitudes across the full SM by unifying open-loop recursion with on-the-fly reduction, enabling simultaneous QCD and EW corrections and robust numerical stability. It introduces a stability-centric hybrid-precision framework and analytic Gram-determinant expansions to reliably handle challenging multi-leg processes and NNLO applications. The workflow includes comprehensive renormalisation, complex-mass schemes, infrared subtraction, and flexible external-photon treatment, with extensive interfaces and process libraries for seamless integration into MC frameworks. The results demonstrate significant speed and stability gains over OpenLoops 1, supporting broad NLO QCD+EW and NNLO-like computations with improved reliability.

Abstract

We present the new version of OpenLoops, an automated generator of tree and one-loop scattering amplitudes based on the open-loop recursion. One main novelty of OpenLoops 2 is the extension of the original algorithm from NLO QCD to the full Standard Model, including electroweak (EW) corrections from gauge, Higgs and Yukawa interactions. In this context, among several new features, we discuss the systematic bookkeeping of QCD-EW interferences, a flexible implementation of the complex-mass scheme for processes with on-shell and off-shell unstable particles, a special treatment of on-shell and off-shell external photons, and efficient scale variations. The other main novelty is the implementation of the recently proposed on-the-fly reduction algorithm, which supersedes the usage of external reduction libraries for the calculation of tree-loop interferences. This new algorithm is equipped with an automated system that avoids Gram-determinant instabilities through analytic methods in combination with a new hybrid-precision approach based on a highly targeted usage of quadruple precision with minimal CPU overhead. The resulting significant speed and stability improvements are especially relevant for challenging NLO multi-leg calculations and for NNLO applications.

Figures (5)

• Figure 1: Evolution of the tensor rank and the number of open-loop tensor coefficients (right vertical axis) as a function of the number $k$ of dressed segments during the open-loop recursion. The red diagonal lines illustrate the dressing steps, and the blue vertical lines the reduction steps.
• Figure 2: Example of parent-child relation between open loops. The parent $N$-point diagram $\mathcal{I}_{N}$ and the child $(N-1)$-point diagram $\tilde{\mathcal{I}}_{N-1}$ share the first $k$ segments (blue blobs). Thus $\mathcal{N}_k(\mathcal{I}_{N},q)$ and $\mathcal{N}_k(\tilde{\mathcal{I}}_{N-1},q)$ are identical and need to be constructed only once.
• Figure 4: Schematic representation of the towers of mixed QCD--EW terms at LO and NLO. The first row represents the LO tower \ref{['eq:SMborn2']}--\ref{['eq:SMborn2splitting']}, which consists of an alternating series of dominant squared Born terms (dark grey blobs) and sub-leading pure interference terms (light grey blobs). The second row corresponds to the NLO tower \ref{['eq:SMloopint']}--\ref{['eq:SMloopintsplittingB']}. Each LO term is connected to two NLO terms via QCD (red) and EW (blue) corrections, while each NLO term is connected to a unique squared Born term either via QCD or EW corrections. Apart from the outer most NLO terms of pure QCD and pure EW kind, QCD (EW) corrections to squared Born terms mix with EW (QCD) corrections to adjacent interference terms.
• Figure 5: Probability of finding an instability $\mathcal{A}>\mathcal{A}_{\mathrm{min}}$ as a function of $\mathcal{A}_{\mathrm{min}}$ in a sample of $10^6$ events for $gg\to t\bar{t} gg$ at NLO QCD (upper plot) and $\bar{u} u \to e^+e^-\mu^+\mu^-$ at NLO EW (lower plot). The stability of quad-precision benchmarks (blue) is compared to different variants of the OpenLoops 2 on-the-fly reduction (green, black, red) and to the OpenLoops 1 algorithm interfaced with Collier (yellow) or CutTools (turquoise). For OpenLoops 2, besides default stability settings (black) we show the effect of increasing the hybrid-precision target from 8 to 11 digits ($\texttt{hp\_loopacc=11}$, red), or disabling the hybrid precision system ($\texttt{hp\_mode=0}$, green). The OpenLoops 1 curves correspond to the level of stability that is obtained in dp without full re-evaluations of unstable points in qp.
• Figure 6: Relative numerical accuracy $\mathcal{A}$ for $gg\to t\bar{t} g$ (upper plot) and $u \bar{u} \to W^+W^-g$ (lower plot) at NLO QCD versus the degree of collinear ($\xi_{\mathrm{coll}}$) or soft singularity ($\xi_{\mathrm{soft}}$) as defined in \ref{['eq:softcolldeg']}. For each value of $\xi_{\mathrm{coll/soft}}$ the numerical accuracy is estimated with a sample of $10^4$ randomly distributed underlying $2\to 2$ hard events. The plotted central points and variation bands correspond, respectively, to the average and $99.9\%$ confidence interval of $\mathcal{A}$. Quad-precision benchmarks (blue) are compared to OpenLoops 2 with additional hybrid-precision improvements for IR regions (hp_mode=2, red) and also to OpenLoops 1 with Collier (yellow) or CutTools (turquoise) in dp.