OpenLoops 2

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. 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...

NNLO QCD + NLO EW with Matrix+OpenLoops: precise predictions for vector-boson pair production

“…The core of the Matrix framework [27] is the Monte Carlo program Munich 1 which includes a fully automated implementation of the dipole-subtraction method at NLO QCD [46,47] and NLO EW [43,[48][49][50][51], combined with an efficient multi-channel integration algorithm. The required tree-level and oneloop amplitudes, including spin and colour correlations for the subtraction of infrared divergences, are calculated with OpenLoops 2 [52]. This new version of the OpenLoops program supports the automated generation of matrix elements for any SM process at NLO QCD+NLO EW.…”

“…[54]. To avoid numerical instabilities in critical phase space regions, this new algorithm is equipped with a sophisticated system that automatically detects the occurrence of spurious singularities in the reduction identities and circumvents them by using algebraic tricks in combination with analytic expansions up to any order in rank-two Gram determinants [52,54]. Residual instabilities are avoided in a systematic way by means of an efficient hybrid-precision system, which carries out certain critical operations in quadruple precision while using double precision for the bulk of the calculation.…”

“…[43]. Unstable particles are treated in the complex-mass scheme [52,68], where width effects are absorbed into the complex-valued renormalised squared masses…”

W+W− production at the LHC: NLO QCD corrections to the loop-induced gluon fusion channel

“…We perform our calculation in the computational framework Matrix [75]. All tree-level and one-loop amplitudes are obtained with OpenLoops 2 [77,78]. 3 For validation of the loop-induced contribution, we have employed the independent matrix-element generator Recola [82,83], finding complete agreement.…”

Next-to-leading-order QCD and electroweak corrections to triple-W production with leptonic decays at the LHC

“…We have implemented two fully independent calculations for all ingredients, in particular for the virtual one-loop contributions, for the real emission parts, and for the multidimensional phase-space integration. The off-shell calculation and the TPA are carried out as follows: One calculation uses amplitudes provided by OpenLoops 2 [16][17][18] and, in the case of the pole (11) approximation, amplitudes generated by in-house software based on Feynman diagrams generated with FeynArts 1 [29]. The second implementation uses amplitudes generated with MadGraph [30] and Recola [14,15], and for the pole-approximated virtual corrections amplitudes created by FeynArts [31] and FormCalc [32], which are further processed and modified.…”