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. only little user intervention. Moreover, OpenLoops is used as a building block of Matrix [50] for the calculation of NNLO QCD observables. In this context, the automation of EW corrections in OpenLoops 2 opens the door to ubiquitous NLO QCD+NLO EW simulations in Sherpa [51, 52] and NNLO QCD+NLO EW calculations in Matrix [53].The OpenLoops 2 code is publicly available on the Hepforge webpage https://openloops.hepforge.org and via the Git repository https://gitlab.com/openloops/OpenLoops. It consists of a processindependent base code and a process library that covers several hundred partonic processes, including essentially all relevant processes at the LHC. The desired processes can be easily accessed through an automated download mechanism. The set of available processes is continuously extended, and possible missing processes can be promptly generated by the authors upon request.The paper is organised as follows. Section 2 presents the structure of the original open-loop recursion and the new on-the-fly reduction algorithm. Numerical instabilities and the new hybrid-precision system are discussed in detail. Section 3 deals with general aspects of NLO calculations and their automation in OpenLoops. This includes the bookkeeping of towers of terms of variable order α p s α q , the treatment of input parameters, optimal couplings for external photons, the renormalisation of the SM at O(α s ) and O(α), the on-shell and complex-mass schemes, and the I-operator. Section 4 provides instructions on how to use the program, starting from installation and process selection, and including the various interfaces for the calculation of matrix elements, colour/spin correlators, and tree amplitudes in colour space. Technical benchmarks concerning the speed and numerical sta...

Building on the open-loop algorithm we introduce a new method for the automated construction of oneloop amplitudes and their reduction to scalar integrals. The key idea is that the factorisation of one-loop integrands in a product of loop segments makes it possible to perform various operations on-the-fly while constructing the integrand. Reducing the integrand on-the-fly, after each segment multiplication, the construction of loop diagrams and their reduction are unified in a single numerical recursion. In this way we entirely avoid objects with high tensor rank, thereby reducing the complexity of the calculations in a drastic way. Thanks to the on-the-fly approach, which is applied also to helicity summation and for the merging of different diagrams, the speed of the original open-loop algorithm can be further augmented in a very significant way. Moreover, addressing spurious singularities of the employed reduction identities by means of simple expansions in rank-two Gram determinants, we achieve a remarkably high level of numerical stability. These features of the new algorithm, which will be made publicly available in a forthcoming release of the OpenLoops program, are particularly attractive for NLO multi-leg and NNLO real-virtual calculations.

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We present the leading colour and light fermionic planar two-loop corrections for the production of two photons and a jet in the quark-antiquark and quark-gluon channels. In particular, we compute the interference of the two-loop amplitudes with the corresponding tree level ones, summed over colours and polarisations. Our calculation uses the latest advancements in the algorithms for integration-by-parts reduction and multivariate partial fraction decomposition to produce compact and easy-to-use results. We have implemented our results in an efficient C++ numerical code. We also provide their analytic expressions in Mathematica format.

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