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Recola2: REcursive Computation of One-Loop Amplitudes 2
Ansgar Denner, Jean-Nicolas Lang, Sandro Uccirati
TL;DR
Recola2 targets the difficult automation of one-loop electroweak corrections for general theories beyond the Standard Model. It provides recursive tree-level and one-loop amplitudes through model files and automated renormalization, including extended Higgs sectors. The release supports automated polarized and squared amplitudes, correlated leading-order quantities for dipole subtraction, and a Background-Field implementation that is 10−15% faster in practical calculations but takes ∼50% longer to generate processes.
Problem
Automating one-loop electroweak corrections for Beyond Standard Model theories requires several intermediate steps beyond established Standard Model automation.
Method
Recola2 uses recursive amplitude computation with model files, automated renormalization, and conventional and Background-Field formulations for extended Higgs sectors.
Results
The release includes 2HDM and HSESM model files and automates polarized amplitudes, squared amplitudes, and colour- and spin-correlated leading-order squared amplitudes for dipole subtraction.
Takeaways & Limitations
Recola2 provides automated next-to-leading-order amplitude calculations for supported Standard Model and extended-Higgs-sector processes, with multiple renormalization schemes.
Takeaways & Limitations
The present version is restricted to theories with scalars, Dirac fermions, and vector bosons; Majorana-fermion support and further model files are in preparation.
Abstract
from arXiv · showhide
We present the Fortran95 program Recola2 for the perturbative computation of next-to-leading-order transition amplitudes in the Standard Model of particle physics and extended Higgs sectors. New theories are implemented via model files in the 't Hooft-Feynman gauge in the conventional formulation of quantum field theory and in the Background-Field method. The present version includes model files for the Two-Higgs-Doublet Model and the Higgs-Singlet Extension of the Standard Model. We support standard renormalization schemes for the Standard Model as well as many commonly used renormalization schemes in extended Higgs sectors. Within these models the computation of next-to-leading-order polarized amplitudes and squared amplitudes, optionally summed over spin and colour, is fully automated for any process. Recola2 allows the computation of colour- and spin-correlated leading-order squared amplitudes that are needed in the dipole subtraction formalism. Recola2 is publicly available for download at http://recola.hepforge.org.
1. Introduction
Recola2 addresses the need for automated one-loop QCD and electroweak corrections in general weakly interacting theories beyond the Standard Model. The article then introduces the program's new features, installation, usage, and validation organization.
- Precision Standard Model predictions motivate automated one-loop corrections for general weakly interacting Beyond Standard Model theories.
- The introduction identifies automation of one-loop electroweak corrections for BSM theories as more involved because several intermediate steps are required.
- The article presents Recola2's new features, installation instructions, usage, and extended-Higgs-sector subroutines before concluding with validation efforts.
2. New features in Recola2
Recola2 extends recursive tree-level and one-loop amplitude computation to general quantum field theories and adds model files, renormalization schemes, and Background-Field Method support for extended Higgs sectors.
- Recola2 computes tree-level and one-loop amplitudes through recursion relations, using Dyson–Schwinger equations at tree level and tensor-integral decomposition at one loop.Tensor integrals are computed with Collier, while Recola2 computes process- and theory-dependent tensor coefficients, including new BSM structures.
- Scalar potentials: The implementation includes CP-conserving Z2-symmetric renormalizable 2HDM and HSESM scalar sectors, with physical-basis parameters expressed through masses, mixing angles, and electromagnetic coupling.The 2HDM additionally permits soft Z2 breaking, while the HSESM uses a Higgs doublet and singlet.
- Yukawa sector: The 2HDM supports natural flavour-conserving Yukawa structures selected directly by the user through Recola2 subroutines.These structures impose a diagonal CKM matrix and Z2 symmetry to suppress flavour-changing neutral currents at tree level.
- Renormalization schemes: Extended Higgs sectors require renormalization of mixing angles and additional parameters, with supported schemes including FJTS, MDTS, MS, p*, and on-shell variants.The FJ Tadpole Scheme is the default for non-SM model files, while alternative tadpole-counterterm schemes are also available.
3. Installation
Recola2 can be installed either through a combined Recola2-Collier package that resolves dependencies automatically or through a stand-alone package requiring manual dependency management. The packages provide model files, build configurations, and CMake-based compilation workflows.
- Installation packages: The Recola2-Collier package configures and compiles Recola2, Collier, and a model file together, resolving dependencies automatically.It includes the library, model files, CMake configuration, and build directory.
- Installation packages: The stand-alone package requires users to resolve dependencies on Collier and model files manually.This mode separates compilation of the Recola2 library, model files, and external Collier dependency.
- Requirements: Recola2 and its model files are publicly available, and compilation requires the CMake build system.The package is distributed from the Recola2 website.
- Package contents: Model files share the Recola2 library version and are included in the combined package alongside the main Recola2 source directory and CMake configuration.The package structure also includes a dedicated build directory for generated compilation files.
- Build workflow: The combined package is built from a build directory using CMake and make, with the selected model linked to Recola2 and Collier.The model can be selected by identifier or by a path to model-file sources.
The Recola2 demo files
The Recola2 demos illustrate amplitude calculations, correlations, decay widths, coupling selections, and integration with external Fortran, C++, and Python programs. They also provide build examples and configuration options for compiling Recola2, model files, and dependencies.
- Demo coverage: The demo programs cover basic Recola2 usage, resonant-contribution selection, colour and spin correlations, decay widths, and coupling-power selection.Examples include neutral Higgs decays and a vector-boson-fusion partonic channel.
- Language interfaces: Recola2 demos support Fortran95, C++, and Python interfaces, with C++ examples compiled and Python examples run directly.The interfaces follow the documented Recola2 usage conventions.
- CMake integration: The CMake example defines a project, builds an executable from program.f90, finds Recola2, and links the executable against the Recola2 library.The template exposes library, header, and library-path variables through CMake package discovery.
- CMake integration: The CMake linking script works with shared or static Recola2 libraries, Fortran, C, or C++ programs, and different operating systems or compilers.The project language can be selected explicitly or combined across multiple languages.
- Model-file compilation: Model-file compilation requires compiling the desired model first, using CMake and make from its build directory.Collier can be located automatically, specified by path, or configured through environment variables.
- Model-file compilation: The model file is built as a shared library by default, with an option to create a static library instead.The generated module files are placed in the model-file include directory.
4. Usage of Recola2 in extended Higgs sectors
Recola2 usage follows an ordered five-step workflow for setting inputs, defining and generating processes, computing amplitudes, and resetting the library. Extended Higgs-sector routines add model-specific parameters, renormalization schemes, and general coupling-power selection.
- General usage: Recola2 applications proceed through five ordered steps: optional input setup, process definition, process generation, computation, and resetting.The sequence must be followed because new processes cannot be defined after generation unless Recola2 is reset.
- General usage: Process generation initializes the defined processes and creates the building blocks for recursively computing off-shell currents.The generated process-dependent information is then used during amplitude computation.
- General usage: The compute routine evaluates amplitudes and squared amplitudes for user-supplied momenta and is repeatedly called at different phase-space points in Monte Carlo integration.It uses the process-dependent recursive information created during generation.
- Extended Higgs sectors: Extended Higgs-sector routines support parameter input and renormalization schemes for the 2HDM and HSESM, including mixing angles, Higgs masses, widths, and soft-breaking parameters.Several routines are available only when supported by the selected model file.
- Extended Higgs sectors: The mixing-angle schemes include MS variants with FJ, MD, or minimal tadpole prescriptions, while only one scheme can be active at a time.The MS scheme 'l345' is restricted to the 2HDM.
- Coupling selection: Recola2 introduces general methods for selecting contributions by powers of coupling types, required for theories where selecting only strong-coupling powers is insufficient.The coupling-power information is defined by model files in UFO format.
5. Conclusions
Recola2 automates tree-level and one-loop amplitude computations for the Standard Model and general quantum field theories with available model files. It also supports correlated squared amplitudes, resonant-contribution selection, and model files for two extended Higgs sectors, subject to a stated particle-content scope.
- Capabilities: Recola2 computes tree-level and one-loop amplitudes in the Standard Model and general quantum field theories without a priori particle-multiplicity restrictions when model files are available.Amplitudes can be obtained for specific colour structures and helicities.
- Capabilities: The library provides squared amplitudes with optional helicity summation or averaging, plus colour- and spin-correlated leading-order squared amplitudes for dipole subtraction.It also supports selecting resonant contributions for factorizable corrections in pole approximations.
- Model files: The first Recola2 release includes model files for the Two-Higgs-Doublet Model and the Higgs-singlet extension of the Standard Model.These model files are generated in the complex-mass scheme and support multiple renormalization schemes.
- Scope: The present version is restricted to scalars, Dirac fermions, and vector bosons; Majorana-fermion support and additional model files were in preparation.The planned additions include anomalous couplings.
Appendix A. Checks
The implementation was checked against prior, independent, and Background-Field-method calculations across the Standard Model, 2HDM scalar two-point functions, and Higgs decays to four fermions.
- Validation: Recola2 was thoroughly tested against Recola, ensuring consistency with the Standard Model checks reported in Appendix B of the earlier work.This validates continuity with the prior implementation for the Standard Model.
- Validation: All renormalized 2HDM scalar two-point functions were verified off shell against an independent QGRAF and QGS calculation.QGS is described as an extension of GraphShot.
- Validation: Partonic channels for Higgs decays into four fermions were compared with an independent FeynArts/FormCalc calculation, and all models were tested against the BFM implementation.The checks cover both physical processes and the alternative field-theory formulation.