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...

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We present a consistent renormalization scheme for the CP-conserving TwoHiggs-Doublet Model based on MS renormalization of the mixing angles and the soft-Z 2 -symmetry-breaking scale M sb in the Higgs sector. This scheme requires to treat tadpoles fully consistently in all steps of the calculation in order to provide gauge-independent Smatrix elements. We show how bare physical parameters have to be defined and verify the gauge independence of physical quantities by explicit calculations in a general R ξ -gauge. The procedure is straightforward and applicable to other models with extended Higgs sectors. In contrast to the proposed scheme, the MS renormalization of the mixing angles combined with popular on-shell renormalization schemes gives rise to gauge-dependent results already at the one-loop level. We present explicit results for electroweak NLO corrections to selected processes in the appropriately renormalized Two-Higgs-Doublet Model and in particular discuss their scale dependence.

We discuss the renormalization of mixing angles for theories with extended scalar sectors. Motivated by shortcomings of existing schemes for mixing angles, we review existing renormalization schemes and introduce new ones based on on-shell conditions or symmetry requirements such as rigid or background-field gauge invariance. Considering in particular the renormalization of the mixing angles in the Two-Higgs-Doublet Model and the Higgs-Singlet Extension of the Standard Model, we compare electroweak corrections within these models for a selection of renormalization schemes. As specific examples, we present next-to-leading-order results on the four-fermion decays of heavy and light CPeven Higgs bosons, H 1 /H 2 → WW/ZZ → 4f , and on electroweak Higgs-boson production processes, i.e. Higgs-strahlung and vector-boson fusion. We find that our new proposals for on-shell and symmetry-based renormalization conditions are well-behaved for the considered benchmark scenarios in both models. arXiv:1808.03466v2 [hep-ph] 19 Nov 2018 1 We put "on-shell" in quotation marks here, since we want to reserve this word to conditions that are based on S-matrix elements rather than simply taking momenta on their mass shell in more general quantities such as Green functions, self-energies, etc..2 Following the arguments of Refs. [33,34] we consider the "pinch technique" just as one of many physically equivalent choices to fix the gauge arbitrariness in off-shell quantities (related to the 't Hooft-Feynman gauge of the quantum fields in the background-field method) rather than singling out "its gauge-invariant part" in any sense.

We present the computer code RECOLA2 along with the first NLO electroweak corrections to Higgs production in vector-boson fusion and updated results for Higgs strahlung in the Two-Higgs-Doublet Model and Higgs-Singlet extension of the Standard Model. A fully automated procedure for the generation of tree-level and one-loop matrix elements in general models, including renormalization, is presented. We discuss the application of the Background-Field Method to the extended models. Numerical results for NLO electroweak cross sections are presented for different renormalization schemes in the Two-Higgs-Doublet Model and the Higgs-Singlet extension of the Standard Model. Finally, we present distributions for the production of a heavy Higgs boson.